ABSTRACT
Title of dissertation: BACK-ACTION EVADING
MEASUREMENTS OF
NANOMECHANICAL MOTION
APPROACHING QUANTUM LIMITS
Jared B. Hertzberg
Doctor of Philosophy, 2009
Dissertation committee chair: Professor Christopher Monroe
The application of quantum mechanics to macroscopic motion suggests many
counterintuitive phenomena. While the quantum nature of the motion of individual
atoms and molecules has long been successfully studied, an equivalent demonstration
of the motion of a near-macroscopic structure remains a challenge in experimental
physics. A nanomechanical resonator is an excellent system for such a study. It
typically contains > 1010 atoms, and it may be modeled in terms of macroscopic
parameters such as bulk density and elasticity. Yet it behaves like a simple harmonic
oscillator, withmass lowenough andresonantfrequencyhigh enough foritsquantum
zero-point motion and single energy quanta to be experimentally accessible.
In pursuit of quantum phenomena in a mechanical oscillator, two important
goals are to prepare the oscillator in its quantum ground state, and to measure its
position with a precision limited by the Heisenberg uncertainty principle ?x?p ? ~2.
In this work we have demonstrated techniques that advance towards both of these
goals. Our system comprises a 30 micron ? 170 nm, 2.2 pg, 5.57 MHz nanomechani-
cal resonator capacitively coupled to a 5 GHz superconducting microwave resonator.
The microwave resonator and nanomechanical resonator are fabricated together onto
a single silicon chip and measured in a dilution refrigerator at temperatures below
150 mK. At these temperatures the coupling of the motion to the thermal environ-
ment is very small, resulting in a very high mechanical Q, approaching ? 106.
By driving with a microwave pump signal, we observed sidebands generated
by the mechanical motion and used these to measure the thermal motion of the
resonator. Applying a pump tone red-detuned from the microwave resonance, we
used the microwave fleld to damp the mechanical resonator, extracting energy and
\cooling" the motion in a manner similar to optical cooling of trapped atoms. Start-
ing from a mode temperature of ?150 mK, we reached ?40 mK by this \backaction
cooling" technique, corresponding to an occupation factor only ? 150 times above
the ground state of motion.
We also determined the precision of our device in measurement of position.
Quantum mechanics dictates that, in a continuous position measurement, the pre-
cision may be no better than the zero-point motion of the resonator. Increasing the
coupling of the resonator to detector will eventually result in back-action driving of
the motion, adding imprecision and enforcing this limit. We demonstrated that our
system is capable of precisions approaching this limit, and identifled the primary
experimental factors preventing us from reaching it: noise added to the measure-
ment by our amplifler, and excess dissipation appearing in our microwave resonator
at high pump powers.
Furthermore, by applying both red- and blue-detuned phase-coherent mi-
crowave pump signals, we demonstrated back-action evading (BAE) measurement
sensitive to only a single quadrature of the motion. By avoiding the back-action
driving in the measured quadrature, such a technique has the potential for preci-
sions surpassing the limit of the zero-point motion. With this method, we achieved
a measurement precision of ?100 fm, or 4 times the quantum zero-point motion of
the mechanical resonator. We found that the measured quadrature is insensitive to
back-action driving by at least a factor of 82 relative to the unmeasured quadrature.
We also identifled a mechanical parametric ampliflcation efiect which arises during
the BAE measurement. This efiect sets limits on the BAE performance but also
mechanically preamplifles the motion, resulting in a position resolution 1.3 times
the zero-point motion.
We discuss how to overcome the experimental limits set by amplifler noise,
pump power and parametric ampliflcation. These results serve to deflne the path
forward for demonstrating truly quantum-limited measurement and non-classical
states of motion in a nearly-macroscopic object.
BACK-ACTION EVADING MEASUREMENTS OF
NANOMECHANICAL MOTION APPROACHING QUANTUM
LIMITS
by
Jared Barney Hertzberg
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulflllment
of the requirements for the degree of
Doctor of Philosophy
2009
Advisory Committee:
Professor Christopher Monroe, Chair
Professor Keith C. Schwab (California Institute of Technology), Advisor
Professor Richard L. Greene
Professor Luis A. Orozco
Professor John Melngailis, Dean?s Representative
c Copyright by
Jared B. Hertzberg
2009
Dedication
To my parents, Fran and George Hertzberg, who have always encouraged me in
my education, and who did everything they could to give me the best opportunities.
To my late great-uncle Sam Ballen, friend and admirer of physicists.
ii
Acknowledgments
I have received funding during my Ph.D. from the Laboratory for Physical
Sciences (2003-07), and from FQXi (2007-09). The work appearing in this disserta-
tion was accomplished mostly in the the last two years, but was the culmination of
many more years of efiort, and re ects the aid and encouragement of many people.
I must thank the members of my examining committee: Prof. Christopher
Monroe, Prof. Keith Schwab, Prof. Richard Greene, Prof. Luis Orozco and Prof.
John Melngailis. I?m especially grateful to Prof. Monroe as chair of the committee
for helping me to resolve some administrative issues at the last minute.
The administration and stafi of the UMD Physics department have been very
helpful to me over the years, especially during the past two years when I left College
Park to continue my research work at Cornell. The department very graciously
arranged to cover my tuition and waive my student fees during this time. Prof.
Greg Sullivan and Prof. Nick Chant deserve my thanks in this, and Jane Hessing
has been a tireless help. Without her I could never have navigated the un-ending
paperwork.
I must also thank the UMD physics department for their support and advo-
cacy in 2005-07 when the nanomechanics group faced di?culties imposed by LPS
management. Prof. Drew Baden and Prof. Jordan Goodman, in addition to Prof.
Sullivan and Prof. Chant, were a great help.
In my time at Cornell I have had the great fortune to work closely with two
talented graduate students, Tristan Rocheleau and Tchefor Ndukum. By the time
iii
I arrived here in summer 2007, they had already put in nearly a year of hard efiort
making the lab ready for measurements. I have learned a great deal from both of
them, and enjoyed it, too. I feel like this work is as much theirs as it is mine. Tristan
did much of the lab setup and wiring, all of the LabView programming, and many
of the measurements. His creativity, curiousity and positive spirit were invaluable
to this work. Tchefor did much of the lab setup, most of the clean room fabrication,
and many measurements and calculations. His thoroughness, determination and
careful reading of the literature make him a formidable researcher. I feel that both
Tristan and Tchefor are destined for great careers in science.
Many other members of the Schwab group at Cornell have helped with the ex-
periment in practical ways, and just by their presence made my time here enjoyable:
Chris Macklin, Peter Hauck, Madeleine Corbert, Sara Brin Rothenthal, Adil Gan-
gat, Peter Swift, Darren Southworth and Dr. Matt Shaw. Manolis Savva?s design
and assembly of the microwave fllter cavities was essential to our results.
I should thank many others at Cornell. I shared many enjoyable conversations
with my fellow denizens of \H corridor," Dr. Andrew Fefierman, Dr. Kiran Thadani,
Sufei Shi and Dr. Ethan Bernard. Our group did all of our micro- and nanofab-
rication at the NSF-sponsored Cornell Nanoscale Facility, where I am grateful to
Dr. Rob Ilic and Meredith Metzler for invaluable fabrication advice. Our group
borrowed key pieces of equipment from the CCMR facilities. Stan Carpenter, Jefi
Koski, Chris Cowulich, Nate Ellis and others at the Cornell physics shop made crit-
ical components of high precision and quality. Dr. Eric Smith has dispensed helium
along with his encyclopedic knowledge of low-temperature systems and techniques.
iv
If you ask his advice, he?ll never simply tell you to \read my book."
I am indebted to several people beyond Cornell for enlarging my understand-
ing of quantum phenomena in mechanical motion. Prof. Miles Blencowe and Prof.
Roberto Onofrio looked over our draft paper and ofiered valuable suggestions. Prof.
Blencowe also generously read and ofiered comments on a partial draft of this dis-
sertation. In writing our paper, Prof. Aashish Clerk was patient with my questions
and tireless in his careful reading and analysis of our many manuscripts. Several
of his unpublished theoretical calculations are incorporated into this dissertation. I
also had many valuable discussions with Dr. John Lawall. Prof. Britton Plourde
generously contributed his resources to our early fabrication efiorts. I must also
thank Dr. Markus Aspelmeyer and his group at the IQOQI in Vienna, Dr. Syl-
vain Gigan, Dr. Hannes B?ohm, and Simon Gr?oblacher, for welcoming me into their
group as a collaborator. I learned much from their quantum optics perspective on
their work, which is so similar in concept to ours, and enjoyed immensely my time
working with them fabricating optomechanical devices.
From our four years together at LPS, and many meetings and conversations
afterward, I am indebted to the \old guard" of the Schwab group, now scattered
to the four corners of the globe but still together in spirit: Emrah Altunkaya, Dr.
Harish Bhaskaran, Dr. Benedetta Camarota, Dr. Akshay Naik, Dr. Matt LaHaye,
Dr. Alex Hutchinson, Dr. Olivier Buu and Dr. Patrick Truitt. Benedetta taught
me nanofabrication, Akshay taught me how to run a dilution fridge, and Patrick
and Emrah shared many long nights in the lab with me. Benedetta also read parts
of this dissertation in draft form and suggested helpful corrections.
v
Many LPS and UMD stafi made my time at LPS more enjoyable and produc-
tive: Les Lorenz, J.B. Dotellis, Dr. Ben Palmer, Dr. Barry Barker, Mike Khbeis,
Dr. Bruce Kane, Russell Frizzell, Toby Olver, Butch Bilger and Prof. Bob Ander-
son. The student and postdoc cohort, Dr. Kevin Eng, Dr. Dan Sullivan, Dr. Luyan
Sun, Dr. Hui Wang, Dr. Jonghee Lee, Dr. Kenton Brown and Robert McFarland
were all sources of valuable conversation and advice whether over lunch or in the
clean room.
Through six years of grad school, I have relied on many old friends to keep up
good cheer and diversion from my studies: Cristy Maldonado, Dr. Rudy Magyar,
Mariana Osorio, Juan Burwell, Lisa Winter, Angi Arya, Dr. Kani Ilangovan, Clifi
Lardin and Heather Miller Lardin, Dr. Jeremy Perlman, Umaa Rebbapragada,
Jessica Schattschneider and Ryan Senser, some of whom are still slogging through
their own graduate work. I should also thank the friends I made at UMD, including
Bijan Afsari, Dr. Doug Kelley and Andrew Tunnell. Thanks to Eran Goudes and
Gwen?s friends on U Street for letting me stay at their home when I did my defense.
I must thank Prof. Larry Hunter and the Amherst College physics department
for the inspiration, encouragement, and early training in physics that led me to this
point.
To my family, especially my parents, George and Fran Hertzberg, I owe a
debt I can never fully repay, not only for my earlier education that paved the way
for my graduate work, but also for their support, patience, advice, encouragement,
and a ready ear to listen to me during the past six years. My grandmother, Judy
Levine, has given her quiet strong support. My late grandfathers, Irving Harelick
vi
and Milton Levine, were always keenly interested in how I was coming along with
my Ph.D., and insistent that I do my very best academically. I wish they could be
here to see it completed.
These acknowledgements would be incomplete without a word about my ad-
visor, Prof. Keith Schwab. It has been a privilege to work for and with Keith
these past six years. I have learned that in experimental physics the most precious
commodity is not money or supplies or equipment but good ideas, good not just
in their novelty but in their ability to be practically realized, and for the level of
interest they will generate in the scientiflc community. The work in this dissertation
is the fruit of Keith?s good ideas. I am indebted to Keith for his leadership, for his
advice and guidance, for his willingness to introduce me to other key researchers,
for providing ample resources, and for his friendship. In the past half-dozen years
I have watched quantum nanomechanics emerge as a growing sub-fleld of physics,
largely due to his vision and efiorts.
Lastly, I must thank my dear partner Gwen Glazer, for her patience and love
as this Ph.D. grew inevitably in its flnal stages into my singular obsession. She has
been un appable, ofiering me limitless support, good cheer, patience, keen instincts,
a willingness to listen to me at any time, and on many occasions even feeding me or
bringing me cofiee in the morning. If this document is relatively free of bad wording,
misspellings and bad grammar, thank Gwen and her professional editor?s eye. I am
glad to share my life with her.
vii
Table of Contents
List of Tables x
List of Figures xi
List of Abbreviations and Symbols xiv
1 Introduction 1
2 Superconducting Microwave Resonator (SMR) Coupled to Nanomechanical
Resonator (NR) 5
2.1 Transmission line resonator . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Transmission line . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Circuit model of transmission line resonator . . . . . . . . . . 8
2.2 Model of driven SMR . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Loading of resonator . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Internal and output voltage of the SMR . . . . . . . . . . . . 15
2.2.3 Energy, power and current in the SMR . . . . . . . . . . . . . 16
2.3 Design of coplanar waveguide . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Nanomechanical resonator coupled to SMR . . . . . . . . . . . . . . . 21
2.4.1 Difierential equation for sideband voltage . . . . . . . . . . . . 24
2.4.2 Solution for sideband voltage . . . . . . . . . . . . . . . . . . 26
2.4.3 Solutions if sideband frequency equals !SMR . . . . . . . . . . 29
3 Theory and Literature Review: Backaction and Related Efiects 32
3.1 Backaction damping and cooling . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Classical analysis . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Recent work in the fleld . . . . . . . . . . . . . . . . . . . . . 37
3.1.3 Summary of quantum analysis . . . . . . . . . . . . . . . . . . 40
3.1.4 Experimental tradeofis . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Shift in NR frequency by optical spring efiect . . . . . . . . . . . . . 49
3.3 Backaction cooling while the SMR is thermally excited . . . . . . . . 51
3.4 Backaction-evading (BAE) single quadrature detector . . . . . . . . . 55
3.4.1 Quantum non-demolition (QND) measurements: formalism . . 56
3.4.2 Harmonic oscillator quadratures as QND observables . . . . . 58
3.4.3 Classical backaction evasion and limitations of real devices . . 64
3.4.4 Quantum squeezed states of mechanical motion . . . . . . . . 70
3.5 NR motion parametrically amplifled in BAE pump conflguration . . . 72
3.5.1 Electrostatic frequency shift due to @2C=@x2 . . . . . . . . . . 72
3.5.2 Parametric ampliflcation of NR motion . . . . . . . . . . . . . 75
3.6 Position sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
viii
4 Fabrication, Setup and Apparatus 84
4.1 Fridge setup and internal wiring . . . . . . . . . . . . . . . . . . . . . 84
4.2 Wiring external to fridge . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Device fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4 Sample boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.5 Microwave fllter cavities . . . . . . . . . . . . . . . . . . . . . . . . . 110
5 Measurement Methods 115
5.1 Characterization of SMR . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Measured and calculated NR characteristics . . . . . . . . . . . . . . 120
5.3 Sideband measurement using spectrum analyzer . . . . . . . . . . . . 124
5.4 Estimation of line loss, gain and ?ext . . . . . . . . . . . . . . . . . . 127
5.5 Thermal calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Results and Discussion 137
6.1 Summary of device parameters . . . . . . . . . . . . . . . . . . . . . . 137
6.2 Backaction cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3 Backaction cooling when SMR is excited by noise . . . . . . . . . . . 146
6.4 Optical-spring frequency shift . . . . . . . . . . . . . . . . . . . . . . 149
6.5 Backaction-evading single quadrature detection . . . . . . . . . . . . 153
6.5.1 Demonstration of single quadrature detection . . . . . . . . . 153
6.5.2 Demonstration of backaction evasion . . . . . . . . . . . . . . 157
6.5.3 Backaction evasion degraded by parametric ampliflcation . . . 163
6.6 Position sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.7 Force sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.8 Approach to quantum limits on Sx ?SF . . . . . . . . . . . . . . . . . 177
7 Conclusions 182
Bibliography 185
ix
List of Tables
6.1 Geometric parameters of device used for backaction cooling and eva-
sion measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 RF parameters of device used for backaction cooling and evasion mea-
surements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
x
List of Figures
2.1 Microphotograph of 3.5mm ? 10mm chip. . . . . . . . . . . . . . . . 10
2.2 Lumped-element model of transmission-line resonator. . . . . . . . . . 11
2.3 Circuit model of our SMR-NR device. . . . . . . . . . . . . . . . . . . 12
2.4 Norton equivalent model of SMR . . . . . . . . . . . . . . . . . . . . 14
2.5 Microphotograph of coplanar waveguide showing capacitor C?. . . . . 22
2.6 Diagram of NR, pump and sideband frequencies if ?!p = ?!NR . . . 30
3.1 Schematic energy level diagram for backaction cooling. . . . . . . . . 41
3.2 Conceptual illustration of cooling process. . . . . . . . . . . . . . . . 44
3.3 Calculated sideband spectrum of thermal and backaction-driven NR. 54
3.4 Schematic illustration of single-quadrature detection. . . . . . . . . . 68
4.1 Fridge wiring schematic. . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Annotated photograph of internal fridge wiring. . . . . . . . . . . . . 87
4.3 Photographs in shielded room and control room. . . . . . . . . . . . . 93
4.4 Wiring for backaction evasion or cooling. . . . . . . . . . . . . . . . . 94
4.5 Wiring for single quadrature detection. . . . . . . . . . . . . . . . . . 95
4.6 Photograph of wiring at top of fridge. . . . . . . . . . . . . . . . . . . 96
4.7 Schematic cross-section view of device fabrication. . . . . . . . . . . . 98
4.8 False-colored SEM microphotograph of NR coupled to CPW of SMR. 100
4.9 Top-view SEM microphotograph of NR and gate electrode. . . . . . . 101
4.10 SEM microphotograph indicating joints between SMR and NR. . . . 102
4.11 SEM microphotograph showing damage to Al fllm forming SMR. . . . 103
4.12 Chip in sample box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
xi
4.13 Close-up photo of chip in sample box. . . . . . . . . . . . . . . . . . . 107
4.14 Comparison photo of three sample box designs. . . . . . . . . . . . . 107
4.15 Comparison of microwave transmission through difierent sample box
designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.16 Filter cavity assembly and testing. . . . . . . . . . . . . . . . . . . . . 111
4.17 Spectra showing excitation of SMR by source phase noise suppressed
by fllter cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.18 Noise power density transmitted by SMR, with and without fllter
cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.1 Transmission spectrum of SMR. . . . . . . . . . . . . . . . . . . . . . 116
5.2 SMR linewidth vs circulating power in SMR. Four difierent devices. . 118
5.3 SMR frequency and linewidth vs temperature. . . . . . . . . . . . . . 120
5.4 NR frequency and linewidth vs temperature. . . . . . . . . . . . . . . 123
5.5 Typical motional sideband power spectral density. . . . . . . . . . . . 126
5.6 Thermal calibration: sideband power vs fridge T . . . . . . . . . . . . 134
6.1 Mechanical occupation and linewidth vs pump power in four pump
conflgurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2 Position noise spectra during backaction cooling measurements. . . . 139
6.3 Mechanical occupation and linewidth vs pump power during cooling
measurement while SMR is excited by microwave source phase noise. 147
6.4 Position noise spectra during backaction cooling measurements while
SMR is excited by microwave source phase noise. . . . . . . . . . . . 148
6.5 Fits of NR frequency and linewidth vs. pump frequency. . . . . . . . 151
6.6 BAE pump conflguration: sensitivity to single quadrature of NR mo-
tion, and sensitivity to thermal noise. . . . . . . . . . . . . . . . . . . 156
6.7 Demonstration of backaction evasion using noise injected to SMR . . 161
6.8 Position uncertainty ?x vs pump power in four pump conflgurations. 168
xii
6.9 Highest-achieved mechanically preamplifled thermal noise of NR. . . . 172
6.10 Force sensitivity vs. pump power, at T = 142 mK. . . . . . . . . . . 175
6.11 Force sensitivity vs. fridge T . . . . . . . . . . . . . . . . . . . . . . . 176
6.12 Estimate of excess backaction. Non-BAE double-pump conflguration . 179
xiii
List of Abbreviations and Symbols
SMR Superconducting Microwave Resonator
NR Nanomechanical Resonator
CPW Coplanar waveguide
SiN Silicon nitride
C Efiective capacitance of SMR
C? Capacitance coupling each end of SMR to external circuitry
Cg \Gate" capacitance coupling SMR to NR
?NR Linewidth of NR due to coupling to thermal bath
?opt Additional dissipation of NR due to coupling to microwave fleld
?tot Total measured NR linewidth
(Note, all symbols !, ? and ? are in units of angular frequency.)
gain Total gain of apparatus from SMR to measurement point
~ Reduced Planck constant
?int Linewidth of SMR due to internal dissipation
?ext Linewidth of SMR due to coupling to external circuitry
? Total linewidth of SMR (? = ?ext +?int)
k Spring constant of NR
kB Boltzmann?s constant
xiv
loss Loss in signal lines from apparatus input to SMR
m Efiective mass of NR
?nSMR Occupation of pump photons in SMR
?nTSMR Thermal occupation of SMR
?nTNR Thermal occupation of NR when coupled only to thermal bath
?noptNR Efiective occupation of energy bath formed by microwave fleld
?neffNR Apparent occupation of NR as directly measured
?nNR Equilibrium occupation of NR
!SMR Resonant frequency of SMR
!NR Resonant frequency of SMR
?! Difierence of frequency from !SMR
Pin Input pump power to apparatus
Pout Output pump power transmitted by apparatus
Psideb Power of sideband signal generated by NR motion
Sx Position noise spectral density
RL Load resistance of external circuitry (typically 50 ohms)
TNR Mode temperature of NR motion
T Fridge or \thermal bath" temperature
xv
X1 Quadrature of NR motion detected by BAE measurement
X2 Quadrature of NR motion perturbed by BAE measurement
x Instantaneous amplitude of mechanical motion
hx2i Mean-squared amplitude of mechanical motion
?x Position uncertainty
?xZP Zero-point motion
Z00 Characteristic impedance of transmission line
Z0 Characteristic impedance lumped-element SMR model
xvi
Chapter 1
Introduction
Quantum mechanics is a remarkably successful set of physical theories. In
principle there should be no limits on the size or scale of systems or their parameters
to which it may apply. There is no fundamental reason why the behavior of everyday
objects, for instance the motion of a baseball or the vibrations of a bridge, should not
be describable in terms of quantum mechanics. Yet until very recently, experimental
demonstration of physical systems that are described entirely in terms of quantum
mechanics was conflned to the atomic scale.
The development of such systems as Bose-Einstein condensates and supercon-
ducting qubits has brought the flrst opportunities to truly engineer devices in which
quantum-mechanical behavior of some variable is coherent across the entirety of a
macroscopic or near-macroscopic structure. In solid-state systems, it is appealing to
move beyond electrical degrees of freedom such as voltage or charge and try to apply
quantum mechanics to the center-of-mass position of a large mechanical structure.
While the quantum nature of the motion of individual atoms and molecules has
been successfully shown over macroscopic distances, the equivalent for the motion
of an actual macroscopic structure has not. In pursuit of such a demonstration, an
excellent system to work with is a harmonic oscillator. The quantum harmonic os-
cillator is a textbook case in every introductory quantum-mechanics class, and real
1
mechanical harmonic oscillators are easy to make in the laboratory at a wide range
of sizes and mass scales. The goal then is to take a real mechanical harmonic os-
cillator and make it behave quantum-mechanically. Most promising is to work with
mechanical resonators on the micron scale, which typically contain > 1010 atoms
and which are usually described entirely in terms of macroscopic parameters such
as bulk density and elasticity, but which are yet small and light enough to seem
promising for quantum-mechanical experiments.
Quantum mechanics describes a wide number of non-classical behaviors that
are inherently counterintuitive when applied to macroscopic scales. The possibilities
of manifesting these in the motion of a large structure are intriguing: for instance,
a superposition state of position states would place our mechanical oscillator simul-
taneously in two difierent locations. Micro- and nanomechanical resonators are the
focus of a number of proposals to create macroscopic Schr?odinger?s-cat states, and
use them to perform fundamental tests of quantum mechanics and the nature of
decoherence. [1]
There is increasing interest by many researchers in demonstrating quantum-
limited measurement of mechanical motion [2]. Heisenberg?s uncertainty principle
dictates limits for the precision of measurement of mechanical motion, which have
yet to be reached. In practice, the limits on measurement are readily conceived of in
terms of noise, either noise added to the measurement or noise that drives the me-
chanical resonator. The latter is how the uncertainty principle is \enforced": when
the measurement is made more strongly, i.e. with closer and closer precision, the
measurement will begin to perturb the motion, degrading the precision of subsequent
2
measurements. Here we come upon the concept of \measurement backaction". In
general this term has a very broad meaning, and can refer to any kind of efiect that
the measurement has on the device being measured. By making the measurement
in difierent ways, the backaction can be made to damp the motion or to drive it to
higher amplitudes. The most interesting and relevant type of backaction, however,
is quantum backaction, in which the shot noise of the electromagnetic fleld that is
used to make the measurement has the efiect of perturbing the motion. It is shot
noise that ultimately enforces the uncertainty principle. Shot-noise backaction on
mechanical motion has been demonstrated in a few instances, but full exploration
of this behavior remains elusive. [3]
The experimental study of these issues is nowadays a very rapidly-expanding
fleld, as manifested by the rising number of publications and active research groups.
All researchers in this fleld share a common set of goals: to put a mechanical res-
onator into its quantum ground state, to demonstrate quantum limited position de-
tection, andtogeneratenon-classicalstatesofmotionsuchas\squeezedstates." The
results presented in this dissertation represent a contribution towards these goals.
The work is as much the product of a whole research group as it is my own. Building
on earlier investigations by our group [4] [3], it represents a new type of experiment
and promising new type of measurement device - the measurement of the motion
of a nanomechanical resonator by exploiting its coupling to a superconducting mi-
crowave resonator. While I did not in this work attain any of the goals of quantum
measurement, I thoroughly investigate this device, explore a wide range measure-
ment techniques, identify the experimental challenges and present a path forward to
3
reaching all of the key goals in the fleld. In particular, I address the prospect of using
our new type of device to perform backaction-evading and quantum-nondemolition
measurements of the motion, which in certain circumstances can surpass quantum
limits on position measurement. While all my results are in the classical realm, I will
describe them interchangeably in quantum-mechanical terms as much as possible.
I expect the work presented in this dissertation to be merely the start of a series
of experiments carried on by the members of the research group. This dissertation
was designed to serve as baseline and reference for those further investigations. The
material is presented as follows: In chapter 2 I describe a model of our device based
on simple electrical circuit theory, and derive analytical expressions for the signal
amplitude representing the motion of the nanomechanical resonator. In chapter 3 I
survey the ways in which measurement backaction can be made to perturb the mo-
tion of the mechanical resonator, and review some of the theory and experimental
work performed by others in this fleld. In chapter 4 I describe our measurement
apparatus and device fabrication. Chapter 5 describes the ways in which we char-
acterize our device and our apparatus to enable us to properly understand the more
critical measurements. Chapter 6 details those measurements, in which we explored
the various efiects described in chapter 3. Chapter 7 ofiers some brief concluding
remarks.
4
Chapter 2
Superconducting Microwave Resonator (SMR) Coupled to
Nanomechanical Resonator (NR)
In this chapter we intoduce the system that we will use to study mechanical
motion near the quantum limit, consisting of a superconducting microwave resonator
(SMR) coupled capacitively to a nanomechanical resonator (NR). We present a
simple lumped-element LRC circuit model for the SMR and use this to describe the
voltages, currents, power and stored energy in the SMR for a given input signal. By
modeling the NR as an additional capacitance whose value changes with position,
we derive analytical expressions for signal levels measured in this work.
2.1 Transmission line resonator
Microwave resonators made of microfabricated superconducting transmission
line have been adopted widely in recent years for sensitive detection and measure-
ment applications. [5] By exploiting their similarity in principle to optical cavities,
researchers have adapted techniques developed in atomic physics and quantum op-
tics to the study of nano-fabricated electrical devices. [6] [7]
5
2.1.1 Transmission line
We base our analysis of the SMR on the well-known solutions for voltages and
currents  owing in a transmission line, often referred to as \transmission line theory"
and appearing in many textbooks [8] [9] [10]. Here we summarize the relevant points:
Consider a transmission line consisting of a centerline and ground line or
groundplane, having capacitance C0 per unit length between centerline and ground,
and inductance L0 per unit length along the centerline. The phase velocity of waves
traveling on the transmission line is vph = 1pL0C0 and the characteristic impedance
of the transmission line is deflned as
Z00 =
r
L0
C0 (2.1)
The transmission line may also include some ohmic loss represented by a re-
sistance R0 per unit length. R0 may be thought of as a combination of a resistance
R0series in series with the inductance and a shunt conductance G0shunt in parallel with
the capacitance: R0 = R0series + G0shunt(Z00)2. Using Kirchofi?s equations, with the
impedances per unit length in the transmission line, we can solve for the voltages
and currents on the line. We flnd that at frequency !, the transmission line supports
waves traveling in the +x direction (V +(x) and I+(x)) along with those traveling
in the ?x direction (V ?(x) and I?(x)):
V ?(x) = V ?0 ei!tcurrency1ix(!=vph)currency1(R0=2Z00)x (2.2)
I?(x) = ?V
?
0
Z00 e
i!tcurrency1ix(!=vph)currency1(R0=2Z00)x (2.3)
Here we assume that the wavelength is short enough that reactive voltage
6
drops dominate over ohmic ones: R0 ? !Z00=vph (\low loss" cable approximation).
The time-averaged power traveling down the cable may be calculated using P(x) =
Re(V(x)I?(x)), where by convention power  owing in the + direction has positive
sign and power  owing in the ? direction has negative sign.
If the transmission line is terminated with a (possibly complex) impedance ZL
connecting centerline to ground at position x = 0, this sets a boundary condition
ZL = V +(0)+V?(0)I+(0)+I?(0) , which we may use to solve for the amplitudes V ?0 of the waves
propagating in each direction. For a transmission line that extends in the?direction
from x = 0, the ? traveling wave V ?(x) will be a partial or total re ection of
the incoming + traveling wave, V +(x): V ?0 = ?V +0 (where both V ?0 and V +0 are
referenced to x = 0, the re ection point). The re ection coe?cient is ? = ZL?Z00Z
L+Z00
.
The voltage and current at point x (where in this geometry x < 0) are then
V(x) = V +0 ei!t(e?ix(!=vph)?(R0=2Z00)x +?eix(!=vph)+(R0=2Z00)x) (2.4)
I(x) = V
+
0
Z00 e
i!t(e?ix(!=vph)?(R0=2Z00)x ??eix(!=vph)+(R0=2Z00)x) (2.5)
Due to ohmic loss in the line, the forward-traveling power P+(x) diminishes
as it approaches the load, while the negative-traveling power P?(x) diminishes as
it recedes from the re ection point. These are found to be
P+(x) = jV
+
0 j
2
2Z00 e
?(R0=2Z00)x (2.6)
P?(x) = ?j?j2jV
+
0 j
2
2Z00 e
(R0=2Z00)x (2.7)
We can also deflne the input impedance Zin(x) at point x, i.e. the terminated
7
length of transmission line, of length jxj, treated as a single load impedance:
Zin(x) = Z00
ZL +Z00 tanh
?
i !v
ph
jxj+ R02Z0
0
jxj
?
Z00 +ZL tanh
?
i !v
ph
jxj+ R02Z0
0
jxj
? (2.8)
2.1.2 Circuit model of transmission line resonator
From equations (2.4) and (2.5) it is evident that the flelds on the terminated
transmission line are a combination of traveling waves and standing waves of wave-
length ? = 2?vph=!, and that for particular terminations (e.g. ? = ?1) only stand-
ing waves will result. Then if terminated properly at two ends, a transmission line of
length l becomes a resonant cavity with resonant frequency deflned by the boundary
conditions: A short (ZL = 0) at one end and open (ZL ? 1) at the other becomes
resonant for l = ?4 + n?2, i.e. ! = ?vphl (n + 12), with n = 0;1;2;::: (\quarter wave"
resonator and harmonics). An open termination at both ends becomes resonant for
l = n?2, i.e. ! = n?vphl , with n = 1;2;::: (\half wave" resonator and harmonics).
Of course, the open ends are not completely isolated; some energy leaks in and out,
which is how the cavity is energized and probed. Consider a resonator with open
terminations at both ends, with microwave power applied at one end. We can think
about the resonance this way: The fleld inside is in phase with the fleld entering at
the \input end". Inside the cavity, the microwaves circulate, i.e. bounce back and
forth between the two ends, forming a standing wave. In a single round-trip, the
microwaves lose an amount of power due to ohmic losses and the power emitted at
the opposite end of the cavity. From equations (2.6) and (2.7), we can see that this
should be a fraction 1?j?j2e?(R0=Z00)l of the power circulating within the cavity. In
8
steady state, the power injected at the \input end" must balance the amount lost
in the round-trip.
In our experiments, we will refer to the half-wave transmission-line resonator
of length l as the \superconducting microwave resonator" (SMR). For frequencies
around a resonant frequency !SMR = n?vphl it is convenient to model the SMR as
an equivalent LRC circuit. We start by setting ZL = 1 in equation (2.8), leaving
Zin = Z00tanh(i!l=v
ph+R0l=2Z00)
. We then substitute ! = !SMR +?! and perform a taylor
expansion on the denominator:
Zin = Z
0
0
tanh(in? +i?!l=vph +R0l=2Z00) (2.9)
? Z
0
0
i?!l=vph +R0l=2Z00 (2.10)
For a parallel LRC circuit (flgure 2.2 b), the total impedance Z is found from
1
Z =
1
R+
1
i!L+i!C. Considering frequencies ! = !0+?! around resonant frequency
!0 = 1pLC (assuming ?!!0 ? 1 and employing Z0 = pL=C) we have
Z = iRZ0(1+?!=!0)R +iZ
0(1+?!=!0)?R(1+?!=!0)2
Dividing numerator and denominator by iRZ0(1+?!=!0) and approximating
one term in the denominator using (1+?!=!0)?1 ? (1??!=!0) yields
Z = R1+2i R
Z0
?!
!0
(2.11)
Comparing to equation (2.10) and setting !SMR = !0 we can see that the
lumped-element LRC circuit models the SMR well. For the SMR of length l,
with resistance, capacitance and inductance per unit length R0, C0 and L0 , and
9
C?C?
NR
contact
point for
signal line
Figure 2.1: Microphotograph of 3.5mm ? 10mm chip. The meandered
line is coplanar waveguide. Barely distinguishable at this resolution are
the centerline and gaps separating it from the groundplane. At the
two ends of the chip the waveguide geometry gradually changes to an
approximate microstrip ending in a bond-pad wide enough to accom-
modate wirebonds. The coupling capacitors C? deflne a length of CPW
forming a 5.00684 GHz SMR having a measured ? = 2? ? 494kHz and
coupled to 5.5717 MHz nanomechanical resonator (NR). Labels indicate
positions of NR and coupling capacitors.
transmission-line characteristic impedance Z00 = pL0=C0, for resonance of order n
we can assign lumped-element equivalent circuit values
R =2Z
0
0
lR0 Z
0
0 (2.12)
C =lC
0
2 (2.13)
L = 2(n?)2lL0 (2.14)
Z0 =
r
L
C =
2
n?Z
0
0 (2.15)
!SMR =n?vphl = 1pLC (2.16)
10
Z0'R' L' C'
R CL
a)
b)
l
Figure 2.2: a) Half-wave resonator: transmission line of length l, char-
acteristic impedance Z00 and inductance, capacitance and resistance L0,
C0 and R0 per unit length. b) Lumped-element model of the resonator.
2.2 Model of driven SMR
A schematic of a typical measurement circuit appears in flgure 2.3 a. The
applied microwave drive signal is carried by signal lines having real characteris-
tic impedance RL, attenuated in power by an amount loss before entering the
SMR through coupling capacitor C?. The signal emitted by the SMR at the other
end through another coupling capacitor is amplifled by an amplifler having input
impedance RL matched to the signal lines. Typically, RL = 50?. The amplifler has
power gain gain.
We model this in a lumped-element fashion in flgure 2.3 b. The voltage V0
equals the voltage Vin applied by the microwave source, attenuated by an amount
ploss in its travel down the input line. In the lumped-element model the voltage is
11
SMR Half-wave
resonator, length l
NR (at voltage
antinode of SMR) Amplifier (power
gain and input
impedance RL)
Source
(power Pin)
Signal line
(impedance RL,
attenuation loss)
C?a)
b)
R CL
C?
RL
C?
RL C
g
C?
2V0
Figure 2.3: a) Physical circuit: half-wavelength of CPW transmission-
line forming superconducting microwave resonator (SMR), driven by mi-
crowave source, and detected with microwave amplifler. The SMR is
coupled capacitively to nanomechanical resonator (NR) by capacitance
Cg. b) Lumped-element model.
12
2V0 in series with a source resistance RL. In this way the load (in this case the SMR
plus input capacitance) sees a voltage matching that of a load on a transmission
line of characteristic impedance RL, i.e. Vload = V0 2ZLZ
L+RL
= V0(1+ ?) (cf. equation
(2.4) at x = 0).
Using this model we would like to flnd the voltage VSMR within the SMR
and the voltage Vamp emerging from it. For the time being we neglect the NR and
the capacitance Cg coupling it to the SMR. We treat the LRC circuit and the C?-
and-RL network at the amplifler end as a combined load impedance, and apply a
Norton-equivalent-circuit model to it. [11] Norton?s theorem holds that if a load is
driven by a source voltage in series with a source impedance, we may replace the
source voltage and series impedance by a source current equal to the source voltage
divided by source impedance, in parallel with the same source impedance. In our
model the source voltage is 2V0, and source impedance is RL + 1i!C?. Hence the
equivalent source current is I0;eq = 2V0R
L+ 1i!C?
. Norton?s theorem assumes that all
circuit elements are linear, which our circuit model satisfles.
The model is further clarifled by replacing the impedance at each end of
the SMR by an equivalent resistance RL;eq = 1+(!C?RL)2R
L(!C?)2
and capacitance C?;eq =
C?
1+(!C?RL)2 which when combined in parallel have a total impedance equaling RL +
1
i!C?. The Norton equivalent circuit with this further modiflcation appears in flg-
ure 2.4. We can make a simplifying approximation by noting that in our system,
13
R CLCgI0,eq RL,eq C?, eq RL,eqC?, eq
motion (x)
Figure 2.4: SMR model with Norton equivalent drive current I0;eq. The
Norton equivalent parallel impedance and the load impedance at the
amplifler end (C? and RL in series, see flgure 2.3 b) have each been
replaced with equivalent parallel RL;eq and C?;eq network.
! ? 2? ?5 GHz, RL ? 50?, and C? . 10 fF. Therefore !C?RL . 0:1 and
I0;eq ? 2V0 ?i!C? (2.17)
RL;eq ? 1R
L(!C?)2
(2.18)
C?;eq ? C? (2.19)
2.2.1 Loading of resonator
These equivalent parallel impedances allow us to clarify the loading of the SMR
by the external circuitry. The total capacitance becomes Ctot = C +2C? +Cg, and
the total resistance is likewise the parallel combination of R with RL;eq=2. Because
in our system, Cg ? C? ? C, we will neglect the shift in Z0 and !SMR due to
C?, and everywhere approximate Ctot = C. For the time being we will also neglect
the behavior of the NR, and the efiect of Cg on Z0 and !SMR. However, we must
consider the efiect of the loading resistance RL;eq. Here we can use the deflnition of
quality factor for a parallel LRC circuit, Q = R=Z0. Seeing that 1Q = Z0(1R + 2R
L;eq
),
14
we can restate this as 1Q = 1Qint + 1Qext, thereby deflning
Qint = RZ
0
(2.20)
Qext = RL;eq2Z
0
(2.21)
We can further introduce the angular-frequency linewidth ? = !SMR=Q.
? = ?int +?ext (2.22)
?int = !SMRZ0R (2.23)
= 1RC
?ext = !SMR 2Z0R
L;eq
= 2RL(!C?)2 1C (2.24)
= 4RL(!C?)2Z00!SMRn? (2.25)
2.2.2 Internal and output voltage of the SMR
Using the equivalent current and equivalent impedances, we can readily deter-
mine the voltage VSMR.
VSMR =I0;eq
 
i!C + 1i!L + 1R + 2R
L;eq
??1
=I0;eq
 
i !!
SMRZ0
?i!SMR!Z
0
+ ?int!
SMRZ0
+ ?ext!
SMRZ0
??1
=2V0 ?i!C? ? !SMRZ0
?+i(! ? !2SMR! )
=2iV0
r ?
ext
2RLC ?
1
?+i(! ?!SMR)(1+ !SMR! )
If ? ? !SMR and we are working with frequencies within only a few linewidths
15
of !SMR, we can further approximate !SMR=! ? 1. Then we have
VSMR = 2iV0
r ?
ext
2RLC ?
1
?+2i(! ?!SMR) (2.26)
We will see below in section 2.2.3 that this voltage is identical to the voltage
amplitude of the standing wave within the transmission line in the SMR.
The voltage Vamp at the amplifler is the SMR voltage VSMR, reduced by C?
and RL acting as a voltage divider:
Vamp = RLR
L +1=i!C?
VSMR
=?V0 ?ext1+i!R
LC?
? 1?+2i(! ?!
SMR)
??V0 ?ext?+2i?! (2.27)
Here we deflne ?! = ! ? !SMR. In the last step above we assume that
!C?RL ? 1, which is reasonable for our devices. Note that on resonance (! =
!SMR) we have
Vamp = ??ext? V0 (2.28)
Qint = 11?jVamp
V0 j
Q (2.29)
This is a useful expression. Since Qint is often not known a priori but Q is easy
to measure, we can determine the internal losses in the SMR if we can measure VampV0 .
In practice, however, this usually means knowing loss and gain to high precision.
2.2.3 Energy, power and current in the SMR
We can calculate the energy stored in the driven SMR using the expressions for
energy in reactive components: E = 12CV 2 + 12LI2, where V and I are RMS time-
16
averaged quantities. In our model we will treat VSMR as a voltage amplitude, for
consistency with the transmisison-line voltage amplitudes in equation (2.4). There-
fore VSMR;rms = 1p2VSMR. Assuming ?!!
SMR
? 1 we can approximate ISMR = VSMRZ0
and therefore
ESMR = 12CjVSMRj2 (2.30)
Using equation (2.26) we can state
ESMR = V 20 ?extR
L
? 1?2 +4?!2
Using equation (2.27) we can state
ESMR = 1R
L
1
?extjVampj
2
The characteristic impedance of the lines leading to the SMR is RL. Then by
comparison with equation (2.6) we have Pin ? loss = V 202R
L
and Pout = V 2amp2R
L
? gain.
Therefore
ESMR =Pin ?loss? 2?
ext
? ?
2
ext
?2 +4?!2 (2.31)
ESMR = 2?
ext
Pout
gain (2.32)
Equation (2.32) agrees with what we expect for a Lorentzian resonance. A
resonator with (angular-frequency) linewidth ? dissipates its stored energy at a rate
proportional to the energy times ?. Equation (2.22) shows the difierent dissipation
rates that make up ?. The SMR emits a power ??ext2 ?ESMR? out of each end, and
dissipates a power (?int ?ESMR) internally.
From equations (2.31) and (2.32), we can also express the energy stored in the
SMR as an average number ?nSMR of photons of energy ~! at the drive (i.e. pump)
17
frequency !:
?nSMR = 1~!Pin ?loss? 2?
ext
? ?
2
ext
?2 +4?!2 (2.33)
?nSMR = 1~! 2?
ext
Pout
gain (2.34)
In practice, in most measurements !  ?! and so when it is convenient we
may replace ~! with ~!SMR in equations (2.33) and (2.34).
It is also useful to square the magnitude of both sides of equation (2.27) in
order to express the power transmitted by the SMR. Considering also loss in signal
lines and gain of amplifler we have
Pout = Pin ?loss? ?
2
ext
?2 +4?!2 ?gain (2.35)
The standing wave within the SMR consists of a wave traveling back and forth
in phase, losing a fraction of power at each end and in the internal ohmic losses,
balanced by the microwaves admitted at the drive end. To determine this circulating
power, consider microwaves within the SMR, of power PSMR impinging on the \load
impedance" formed of C? and RL at the amplifler end. From equations (2.6) and
(2.7) it is clear that the power deposited into the load will be PSMR?(1?j?2j): This
power should be equal to the measured power Pamp = Pout=gain. For ZL = 1i!C +RL
and ? = ZL?Z00Z
L+Z00
we have
? = 1+i!C(RL ?Z
0
0)
1+i!C?(RL +Z00)
Taking j?j2, Taylor-expanding the denominator, multiplying through and re-
taining the lowest order terms yields
j?j2 = 1?4!2C2?RLZ00
18
Comparing to equations (2.25) and (2.20) we flnd
Pamp = n?Q
ext
PSMR
Therefore the circulating power in the SMR is
PSMR =Qextn? ? Poutgain (2.36)
=Qextn? Pin ?loss? ?
2
ext
?2 +4?!2 (2.37)
where n is the order of the resonance; typically we will work with the lowest-
order (half-wave) resonance, so that n = 1.
From equation (2.37) we can also determine the amplitude jV +SMRj of the trav-
eling wave within the SMR. Using jV +SMRj = p2Z00PSMR and P0 = Pin ?loss = V 202R
L
,
we have
jV +SMRj =
s
Qext
n?
Z00
RL ?
?extp
?2 +4?!2 ?V0 (2.38)
Because ? ? 1, at the voltage antinode at the ends of the SMR (where we place
our NR) the local voltage will be 2jV +SMRj. This is the amplitude of the standing
wave voltage in the SMR. By using equation (2.15) to substitute for Z00 and using the
deflnitions of Qext and Z0, we can see that 2jV +SMRj is identical to VSMR as deflned
in equation (2.26).
The current of the traveling wave in the SMR can be seen from equation (2.5)
to be
jI+SMRj = jV
+
SMRj
Z00 (2.39)
As with the standing wave voltage, the amplitude of the standing wave current
in the SMR is equal to 2jI+SMRj.
19
2.3 Design of coplanar waveguide
Coplanar waveguide (CPW) is a very convenient type of transmission line to
form the SMR because it is two-dimensional, can be microfabricated easily in a
single lithographic layer, concentrates most of the RF electric fleld in a very small
region between the centerline and groundplane, and isolates the RF region extremely
well via the groundplane. This isolation permits, for instance, forming tight curves
and meanders of the CPW with only minimal efiect on the behavior. [12] CPW is
thus preferable in comparison to other structures such as microstrip and stripline.
CPW supports quasi-TEM electromagnetic waves. Approximate analytical
solutions for Z00, C0 and L0 in CPW may be found in books such as references [13]
and [14]. As these involve elliptical integrals their presentation here will not shed
much useful light on CPW design, and we instead ofier a few rules of thumb:
A photograph of microfabricated CPW appears in flgure 2.5. In designing
CPW, the critical dimensions are the width of the centerline wCPW and the gap
between centerline and groundplane dCPW. In particular the waveguide parameters
Z00, C0 and L0 scale with the ratio wCPWw
CPW+2dCPW
, so the CPW may be shrunk or
enlarged yet have the same electrical behavior. The thickness of the conductor is
typically much smaller than the other dimensions and has negligible efiect. If both
wCPW and dCPW are much smaller than the substrate thickness then the latter
can be considered inflnite and neglected. As a general rule of thumb for design,
decreasing wCPWw
CPW+2dCPW
lowers C0 and increases Z00. In practice Z00 < 100? is easily
achievable but to achieve larger Z00 approaching 200?, dCPW must be made so wide
20
or wCPW so narrow that the CPW loses its favorable characteristics in comparison
to microstrip. The phase velocity is well approximated by
vph = cp(?
r +1)=2
where c is the speed of light and ?r is the relative dielectric constant of the substrate.
In silicon ?r is about 12.
As can be seen in flgure 2.1, we design the chip to have at each end a bond pad
for wire-bonding the CPW to the microstrip transmission line on the sample box.
To avoid step-changes in Z00 we progressively transition the geometry from CPW to
microstrip over a 1 mm distance.
The ohmic dissipation R0 of the waveguide is minimized by using a supercon-
ducting metal on a low-loss substrate. Dissipation in superconducting CPW should
be limited by dielectric losses, the fraction of normal-state conductors in the super-
conductor, and other loss mechanisms such as the motion of trapped magnetic  ux
vortices. These mechanisms have been the subject of many recent studies such as
[11], [15], [12] and [16].
2.4 Nanomechanical resonator coupled to SMR
In this dissertation, we employ the SMR as a detector of mechanical motion
and explore the backaction of the measurement on the motion. We would like to
identify a measured signal that is directly related to the amplitude of the nanome-
chanical resonator?s oscillation. Here we present a classical derivation based on the
circuit model of section 2.2. Although this derivation does not explicitly incorpo-
21
Figure 2.5: Microphotograph of coplanar waveguide. Centerline (width
wCPW) and centerline-to-groundplane gap (dCPW) are clearly distin-
guishable. Also visible is the interdigitated capacitor comprising C?.
22
rate backaction efiects, it is useful in being completely comprehensible in terms of
simple circuit theory, and it gives a precise value for measurable signal amplitude in
terms of the NR oscillation amplitude. If backaction is present, the expressions for
signal amplitude remain accurate as long as the backaction efiects are independently
incorporated into the NR behavior.
We return to the derivation presented in section 2.2.2 for the voltage in the
SMR, and introduce the motion of the NR. The capacitance Cg is a function Cg(x)
of the nanoresonator position x. We take the NR to be oscillating at its resonant
frequency !NR with an amplitude x0 that is much smaller than the gap d between
the NR and the opposing gate electrode. Thus we can approximate Cg as
Cg = Cg(0)+ @Cg@x x0 cos(!NRt+`NR)
Given Ctot = C + 2C? + Cg and !SMR = 1pLC
tot
, we can expect the resonant
frequency of the SMR to oscillate at frequency !NR. For small x0 amplitude we can
use a Taylor-expansion to approximate the oscillating !SMR(t). We take the partial
derivative
@!SMR
@x = ?!SMR
1
2C
@Cg
@x (2.40)
where here !SMR represents the value of the SMR frequency for x0 = 0, and
for simplicity the equilibrium value of Ctot is approximated as C. Then to flrst order,
the oscillating !SMR(t) is:
!SMR(t) = !SMR
 
1? 12C @Cg@x x0 cos(!NRt+`NR)
?
(2.41)
This suggests one way to measure the motion of the NR, by measuring the
oscillation frequency of the SMR. If the SMR is driven on resonance, and if the
23
oscillation amplitude of !SMR(t) is ? ?, the instantaneous phase of the response
should follow the instantaneous value of !SMR(t) and therefore the instantaneous
amplitude of the NR. This method was pursued by Regal et al. [17] in a system very
similar to ours and using a homodyne detection scheme. They fed the response of
their SMR+NR system into a phase-discriminating (\I-Q") microwave mixer, with
a the pump signal used as a reference. The portion of the response that was out-of-
phase with the reference was a direct measure of the phase of the system response
and therefore of the time-varying SMR frequency. This time-varying phase output
could also be examined on a spectrum analyzer to distinguish the noise spectrum of
thermally-driven mechanical motion. These researchers reported that this detection
scheme was susceptible to material-dependent noise in the frequency of the SMR
(\phase noise") due to small  uctuations in the dielectric constant of the substrate.
However, the !NR ? ? of their device did make it particularly suited for a time-
domain measurement. For devices having !NR  ? such as the one we have used,
the situation is somewhat difierent. The time constant of the SMR is 1?, whereas the
NR oscillates with period 2?!
NR
which is < 1?. The microwave resonator is too \slow"
to respond instantaneously to the NR motion, and the phase of the driven SMR
response does not accurately represent the instantaneous amplitude of the NR.
2.4.1 Difierential equation for sideband voltage
Instead of observing the NR motion in the time domain, we may observe it in
the frequency domain. From equation (2.41) and equation (2.27) it is evident that
24
the system response oscillates at !NR while the drive oscillates at pump frequency
!p and thus the emitted voltage will contain terms ? cos(!NRt)cos(!pt), which
decompose into terms oscillating at !p ?!NR. To calculate the expected signal we
must solve for the voltage within the SMR. As discussed in section 2.2 and depicted
in flgure 2.4, the microwave pump signal can be expressed as an equivalent current,
which we will take here to be oscillating at frequency !p with amplitude I0;eq. Thus
the pump signal is I0;eq cos(!pt). This current must equal the sum of the currents
through all of the parallel components. Thus
I0;eq cos(!pt) = @@t(CtotV)+ 1R
tot
V + 1L
Z
Vdt
where 1Rtot = 1R + 2R
L;eq
. We wish to solve for the voltage V(t) within the SMR.
Difierentiating once and plugging in our expression for Cg we have the difierential
equation
?I0;eq!p sin(!pt) = V
 1
L ?!
2
NR
@Cg
@x x0 cos(!NRt+`NR)
?
+ _V
 1
Rtot ?2!NR
@Cg
@x x0 sin(!NRt+`NR)
?
+ ?V
 
Ctot + @Cg@x x0 cos(!NRt+`NR)
?
Rearranging and substituting deflnitions of Z0, C, ? and !SMR yields
?I0;eqZ0!p sin(!pt) = !SMRV
?
1?
 !
NR
!SMR
?2 1
C
@Cg
@x x0 cos(!NRt+`NR)
!
+ ?!
SMR
_V
 
1?2!NR? 1C @Cg@x x0 sin(!NRt+`NR)
?
+ 1!
SMR
?V
 
1+ 1C @Cg@x x0 cos(!NRt+`NR)
?
(2.42)
25
The second term in each parentheses is necessarily small, because Cg ? C
and therefore 1C @Cg@x x0 ? 1. Note that if we ignore all of these small-valued terms,
i.e. we set x0 = 0, we can substitute the deflnitions of I0;eq, Z0, ?ext, and !SMR,
to readily flnd the voltage oscillating at !p. The result is identical in magnitude
and phase to the previous solution, equation (2.26). Thus the NR motion should
make only a negligible change in the voltage amplitude at !p, but we would like to
flnd the voltages oscillating at the sum frequency !s = !p +!NR and the difierence
frequency !d = !p ?!NR. We thus expect the solution to have the form
V(t) = Vpcos(!pt+`p)+Vscos(!st+`s)+Vdcos(!dt+`d) (2.43)
2.4.2 Solution for sideband voltage
By solving the difierential equation, we can flnd the amplitude and phase of
the sidebands. Conceptually we can think of the SMR-NR system as amplitude-
modulating the transmitted pump signal. The SMR resonance further afiects the
amplitude of the sidebands generated this way.
We take _V and ?V of trial solution (2.43) and plug into (2.42). We collect terms
and consider only the terms oscillating at !d or !s. Neglecting terms oscillating at
26
all other frequencies and neglecting as well any terms of order
?
!NR
!SMR
?2
, we have
0 = !SMR (Vs cos(!st+`s)+Vd cos(!dt+`d))
? ?!
SMR
(!sVs sin(!st+`s)+!dVd sin(!dt+`d))
? 1!
SMR
?!2
sVs cos(!st+`s)+!
2
dVd cos(!dt+`d)
?
? !p!
SMR
!pVp 1C @Cg@x x0 cos(!pt+`p)cos(!mt+`m)
+2 !NR!
SMR
!pVp 1C @Cg@x x0 sin(!pt+`p)sin(!mt+`m)
Applyingthetrigonometricidentitiescos 1 cos 2 = 12 (cos( 1 ? 2)+cos( 1 + 2))
and sin 1 sin 2 = 12 (cos( 1 ? 2)?cos( 1 + 2)), we flnd that we can further ne-
glect terms of order !NR!
SMR
. We approximate (!SMR ? !s!
SMR
!s) as 2(!SMR ?!s) and
(!SMR ? !d!
SMR
!d) as 2(!SMR ?!d), and separate into two equations oscillating at
!s and !d:
Vs
 
2(!SMR ?!s)cos(!st+`s)?? !s!
SMR
sin(!st+`s)
?
= 12 !p!
SMR
!pVp 1C @Cg@x x0 cos(!st+`p +`m)
Vd
 
2(!SMR ?!d)cos(!dt+`d)?? !d!
SMR
sin(!dt+`d)
?
= 12 !p!
SMR
!pVp 1C @Cg@x x0 cos(!dt+`p ?`m)
To proceed, we approximate !p!
SMR
= 1, !s!
SMR
= 1 and !d!
SMR
= 1. We denote
?!d = (!d ?!SMR), ?!s = (!s ?!SMR) and ?!p = (!p ?!SMR). From equation
(2.26), Vp and `p are known: Vp = I0;eqZ0!p 1p?2+4?!2
p
and tan`p = ?2?!p=?. We
then have solutions for the amplitude and phase of the upper and lower sideband
voltages produced by the mechanical motion. Approximating !p ? !SMR and using
27
the deflnitions of I0;eq and Z0 we flnd the sideband voltages within the SMR:
Vs = ?1C @Cg@x x0 ?
r ?
ext
2RLC ?
!SMRp
?2 +4?!2s ?
1p
?2 +4?!2p ?Vp;0 (2.44)
`s = arctan
 ?
2?!s
?
?arctan
 2?!
p
?
?
+`m (2.45)
Vd = ?1C @Cg@x x0 ?
r ?
ext
2RLC ?
!SMRp
?2 +4?!2d ?
1p
?2 +4?!2p ?Vp;0 (2.46)
`d = arctan
 ?
2?!d
?
?arctan
 2?!
p
?
?
?`m (2.47)
where Vp;0 is the pump voltage at the input of the SMR (equivalent to V0 in
flgure 2.3 b and equation (2.26)). Note that the phase of mechanical motion appears
with opposite sign in the upper and lower sideband signals.
As in section 2.2.2 we can further determine the voltages of the sidebands
emitted by the SMR, and appearing at the input of the amplifler. Here we also use
(2.40) to replace 12C @Cg@x with ?1!
SMR
@!SMR
@x .
Vs;amp = ?1!
SMR
@!SMR
@x x0 ?
!SMRp
?2 +4?!2s ?
?extp
?2 +4?!2p ?Vp;0 (2.48)
Vd;amp = ?1!
SMR
@!SMR
@x x0 ?
!SMRp
?2 +4?!2d ?
?extp
?2 +4?!2p ?Vp;0 (2.49)
The output power of each sideband can be determined as in equation 2.35.
Note that the mechanical amplitude x0 in equations (2.48) and (2.49) is the peak
amplitude of the NR oscillation. When describing power, it is more convenient to
work in terms of the RMS mechanical oscillation xRMS = phx2i = 1p2x0. In this
way the measured sideband power can be directly related to the energy khx2i in the
mechanical oscillation, where k = mNR!2NR is the spring constant of the mechanical
28
oscillator.
Ps;out = Pp;in ?loss(!p)?
 1
!SMR
@!SMR
@x
?2
?2hx2i? !
2
SMR
?2 +4?!2s ?
?2ext
?2 +4?!2p ?gain(!s)
(2.50)
Pd;out = Pp;in ?loss(!p)?
 1
!SMR
@!SMR
@x
?2
?2hx2i? !
2
SMR
?2 +4?!2d ?
?2ext
?2 +4?!2p ?gain(!d)
(2.51)
Since in general gain and loss may be frequency-dependent, we have explicitly
indicated the frequencies. This can be relevant in analyzing measured data.
2.4.3 Solutions if sideband frequency equals !SMR
In most of the measurements in this work, we use !p = !SMR ? !NR, i.e.
?!p = ?!NR. This places one sideband at !SMR and the other at !SMR ?2!NR.
The sideband at !SMR will be enhanced by the resonance of the SMR and in the
\good cavity" or \sideband resolved" limit of !NR > ?, the other sideband will be
suppressed. The suppression of the second sideband is crucial to both backaction
cooling and backaction-evading measurement. In most cases we will be interested in
the sideband appearing at !SMR and will neglect the suppressed sideband. As we
will see in chapter 3, the optimum frequency for backaction cooling is ?!p = ?!NR.
For the speciflc case of ?!p = ?!NR, then the upper (\sum") sideband falls at
the SMR resonant frequency, i.e. !s = !SMR. This conflguration appears in flgure
29
s48s46s48s48s48 s48s46s48s48s53 s52s46s57s57s53 s53s46s48s48s48 s53s46s48s48s53
s78s82
s78s82
s83s77s82 s78s82
s80
s111
s119
s101
s114
s70s114s101s113s117s101s110s99s121s32 s32s40s71s72s122s32s120s32s50s41
s83s77s82
s112
Figure 2.6: Diagram of NR, pump and sideband frequencies if ?!p =
?!NR. Narrow black line is measured S21, i.e. Pp;out=Pp;in of device
depicted in flgure 2.1, showing Lorentzian lineshape of SMR response as
in equation (2.35). Horizontal axis is to scale; note the break in the axis.
Vertical heights of pump and sidebands are not to scale; the size of the
suppressed sideband at !SMR ?2!NR is greatly exaggerated.
30
2.6. We can state the sideband voltage and power in this case:
Vs;amp = ?1? @!SMR@x x0 ? ?extp?2 +4?!2
p
?Vp;0 (2.52)
Ps;out = Pp;in ?loss(!p)?
 1
?
@!SMR
@x
?2
?2hx2i? ?
2
ext
?2 +4?!2p ?gain(!SMR) (2.53)
If instead ?!p = +!NR, then the \difierence" frequency !d equals !SMR. The
upper sideband is suppressed, and the voltage and power in the lower sideband will
be identical to equations (2.52) and (2.53).
Equations (2.48) through (2.53) ofier us a set of expressions to precisely relate
measured signal levels directly to mechanical motion of the NR. We derived these
expressions classically, so they do not include quantum efiects nor do they explicitly
include the efiects of measurement backaction on the mechanical motion. All of the
measurements in this work are in the classical limit, and in many of our measure-
ments, the backaction is negligible. In the case where backaction is strong, these
expressions remain valid if we incorporate the efiects of backaction independently
into the values of !NR and hx2i.
31
Chapter 3
Theory and Literature Review: Backaction and Related Efiects
Although a classical framework serves for the majority of the analyses and
derivations in this dissertation, it is worthwhile to present the hamiltonian of our
parametrically coupled SMR-NR system: [18], [19], [20]
^H =~
 
!SMR +g^x? ?2^x2
? 
^by^b+ 1
2
?
+~!NR
 
^ay^a+ 12
?
+ ^Hpump+ ^H?+ ^H? (3.1)
Where
g = @!SMR@x is the 1st-order coupling of SMR to NR.
? = !SMR2C @2Cg@x2 is the 2nd-order coupling.
^x = ?xZP(^ay + ^a) is the amplitude of NR motion.
^b (^by) are the lowering (raising) operators of the SMR.
^a (^ay) are the lowering (raising) operators of the NR.
^Hpump represents the microwave pump.
^H? represents SMR damping.
^H? represents NR damping.
The term ~g^x ?^by^b shows the parametric coupling (also commonly described
as a \ponderomotive" or Kerr-type coupling) of the SMR frequency to the mechan-
ical motion, discussed classically in section 2.4. This interaction becomes critical to
backaction damping and cooling of the NR motion, discussed below in section 3.1.3.
The term~?^x2?^by^b on the other hand results from the shifting of the NR frequency
32
due to energy in the SMR. Under the right conditions, this will lead to paramet-
ric ampliflcation of the NR, as described below in section 3.5. This behavior will
ultimately place a limitation on the potential of our system for backaction evading
measurement.
3.1 Backaction damping and cooling
Preparing a mechanical oscillator of frequency !m in its quantum ground state
remains an experimental challenge. The average mechanical energy khx2i must be
suppressed below a single quantum~!m. However, one must overcome the harmonic
oscillator?s coupling to its thermal environment, or \thermal bath", of temperature
T. As a single mode with equal-spaced energy levels, we expect the oscillator to
follow Bose-Einstein statistics, having average thermal occupation ?nth = (e~!m=kBT?
1)?1. At temperatures T  ~!mk
B
, we can approximate the thermal energy as khx2i =
kBT = ?nth~!m, but to reach the regime of the ground state, the temperature must
be made < ~!mk
B
, and this deflnes the experimental di?culty. For instance, for
!m = 2? ? 1MHz, this temperature is equivalent to ~!mk
B
= 0:05 mK. The \brute
force" approach is to cool the environment itself by techniques such as dilution
refrigeration. To date the best achieved by this method was to cool a 21.8 MHz
nanomechanical resonator to ? 26mK, or ?nth ? 25 [3]. Alternatively, if the mode?s
mechanical quality factor is large, then its coupling to the thermal bath is weak.
The mode can then be cooled below the thermal bath temperature by bringing it
out of equilibrium with the environment. By coupling it strongly to another system
33
which extracts energy from it, the average energy of the mode is suppressed while
the mechanical structure itself remains at the bath temperature.
Historically, backaction cooling of the motion of a single mechanical mode to
its ground state was flrst demonstrated for the motion of a single ion trapped in a
harmonic potential. [21] [22]. The harmonic motion of trapped charges in an elec-
tromagnetic trap, and the coupling of the motion to the thermal environment, was
a well-studied phenomenon [23] [24]. To reduce the energy of the ions to the ground
state, researchers led by D. Wineland adopted the doppler-cooling technique previ-
ously developed for neutral atoms [25]. In an electromagnetic trap, the harmonic
motion of the ion at frequency !v doppler-shifts the frequencies of its atomic transi-
tions. The spectral line of an atomic transition at frequency !0 acquires sidebands
at frequencies !0 ?!v, very similarly to our SMR-NR system discussed in section
2.4. By exciting the transition at !0 ? !v with an applied laser, while permitting
the ion to emit at !0, the ion is made to lose an energy ~!v with each transition.
For successful cooling, it was found to be critical that the system be in the \resolved
sideband" limit, where the linewidth ? of the atomic transition is ? !v. The quan-
tum number of the ion?s motion can then reach a theoretical limit ?n ? (?=2!v)2.
Using either stimulated Raman transitions or single-photon transitions, researchers
in references [21] and [22] were able to cool a !v = 3 to 30 MHz mechanical mode to
a quantum occupation ?n < 0:1. Such well-controlled state-preparation of trapped
ions has become instrumental in quantum measurements of trapped ions and in
designs for using trapped ions as qubits. [26]
34
3.1.1 Classical analysis
Schemes for using the backaction of a radiation fleld to damp and cool the mo-
tion of a micromechanical or nanomechanical oscillator adapt this concept to much
larger mass and size scales. [27] Rather than an atomic transition, the resonance of
an optical or RF cavity is employed to extract energy from the mechanical motion.
The concept is seen most readily if we consider an optical cavity one end of which
is a mirror flxed to a mechanical resonator. [28] Light circulating within the cavity
exerts a \ponderomotive" force on the mirror due to radiation pressure. As each
photon of frequency ! bounces ofi of the mirror, its momentum changes by 2~!=c,
and therefore for a power P circulating in the cavity, the mirror experiences a force
Frad = 2P=c. A harmonic oscillator subject to this force and to thermal forces Ftherm
has an equation of motion m?x + m?_x + kx = Ftherm + Frad. If the mirror is oscil-
lating at frequency !m, the resonant frequency of the cavity !c oscillates along with
it. A flxed-frequency ofi-resonance optical drive (\pump signal") !pump is therefore
brought slightly closer and slightly farther away from resonance each cycle. This
means that the amount of power admitted to the cavity and therefore a portion
of Frad will oscillate at !m. Yet the ring-up time ? of the cavity ensures that the
oscillating radiation-pressure force lags the motion slightly. If the oscillating motion
of the mirror is x = x0 cos(!mt), then the oscillating radiation-pressure force will
have components both in-phase and out-of-phase with x. The in-phase component
will appear as an extra contribution to the restoring-force (kx) term in the equation
of motion, leading to a shift in the mechanical resonance frequency (\optical spring"
35
efiect). The out-of-phase component on the other hand will appear as an extra con-
tribution to the m?_x damping term (\optical damping" efiect). The optical spring
and optical damping can be either positive or negative: for a red-detuned pump
(!pump < !c), the phase lag leads to total work per cycle H Fraddx < 0, meaning
the radiation-pressure force does negative work on the mechanical resonator, i.e.
positive damping. For a blue-detuned pump (!pump > !c), the opposite is true: the
radiation-pressure force does positive work, amplifying the mechanical motion, i.e.
negative damping. (A good illustration of this may be seen in flgure 1 of Ref. [29].)
The lag in the response of the force is the key to the damping efiect. While in
a later discussion we describe quantum analyses ([19] [30] [31] [32]), here we summa-
rize some results of classical theory. In general any delay mechanism may produce
similar damping behavior. One demonstration of optical damping has employed
the photothermal force due to difierential thermal expansion of a gold fllm on a
silicon micromechanical resonator. [33] The energy extraction, i.e. cooling, due to
such damping mechanisms will be limited by the power absorbed as heat into the
mechanical resonator. Only damping derived from non-dissipative interactions with
the electromagnetic fleld, i.e. radiation pressure or electromagnetic forces, can ofier
the prospect of cooling the mechanical mode into its quantum ground state. Xue et
al. [34] have analyzed a coupled NR-SMR system nearly identical to ours. By intro-
ducing a delay 1?e??t=2 into the electrostatic force on the NR due to the voltage on
the SMR, they flnd behavior identical to that of radiation pressure acting on a mov-
ing mirror in an optical cavity. Damping increases the mechanical linewidth ?NR to
?effNR, and shifts the frequency !NR to !effNR. If the un-damped mechanical resonator
36
has a temperature T0, and deflning the mechanical mode temperature TNR = khx2ik
B
,
they flnd the spectral density Sx of position noise (note that we use throughout this
work a convention of single-sided spectral densities)
Sx = 4kBTk !
2
NR?NR
(!2 ?(!effNR)2)2 +(?effNR!)2
Integrating to flnd hx2i = R10 Sxd!2? and therefore to flnd TNR, they conclude
that
TNR ? T0?NR?eff
NR
(3.2)
In this analysis, the authors assume that !NR ? ?, and in fact conclude that
the optimum cooling occurs for detuning ?! = !NR=2, and ?! = ?=4. These
conditions hold for the so-called \bad cavity" regime, in which ? & !m. A full
quantum analysis of this regime has also been done by Paternostro et al., for the
case of the moving mirror at one end of an optical cavity. [35] Cooling in the
\bad cavity" regime, however, will start to break down as the mechanical energy
approaches one quantum. As will be discussed below, the \resolved sideband" or
\good cavity" regime, !NR > ?, ofiers the prospect of cooling the mechanical motion
well below its quantum ground state.
3.1.2 Recent work in the fleld
Early implementations of backaction cooling were developed more than a
decade ago to improve the sensitivity of resonant-bar gravitational-wave antennas by
suppressing their Brownian motion. Blair et al. [36] studied ?700 kHz mechanical
modes of a 1300 kg niobium bar with a 10 GHz superconducting RF cavity attached
37
to one end so that vibrations of the bar modulated the resonant frequency of the
cavity. By driving the cavity ofi-resonance with RF power up to -12 dBm, they were
able to suppress the mechanical amplitude from an ambient temperature of 5 K to a
mode temperature of 2 mK. They found that further cooling was limited by an un-
desired backaction driving efiect due to amplitude noise in their microwave source.
(Similar efiects are discussed below in section 3.3.) A more recent demonstration of
backaction cooling of a macroscopic mechanical resonator was done by Brown et al.
[37], who demonstrated RF backaction cooling in the non-sideband-resolved regime,
using a 1 mm-long mechanical resonator of frequency 7 kHz, coupled to a 100 MHz
resonant RF cavity having ? ? 2? ?430 kHz.
For micro or nanomechanical resonators, active interest in backaction cooling
began earlier this decade, and initial demonstrations used non-sideband-resolved
devices. Gigan et al. [38] [39] cooled a 280 kHz mechanical resonator from room
temperature to 8K, while Arcizet et al. [40] cooled a 814 kHz microresonator to
about 10K. A related technique was reported by Naik et al. [41] [3] who used the
backaction of shot-noise in charge motion through a superconducting single-electron
transistor to cool the 21.9 MHz fundamental mode of a nanomechanical resonator
from a starting temperature of 550 mK to a flnal mode temperature of 300 mK.
Inmorerecentwork, mostresearchershaveimprovedtheirdevicestooperatein
the sideband-resolved regime. Gr?oblacher et al. have used the most \conventional"
geometry for their cooling experiments, i.e. a free-space optical cavity of linear
geometry, with a mechanically-resonant mirror at one end. With this system they
cooled the 945 kHz fundamental mechanical mode of a 100-micron-long resonator
38
having a high-re ectance mirror attached to it. Starting from a cryostat temperature
of 5.3 K, they achieved a backaction-cooled mode temperature of 1.5 mK, or ? 32
mechanical quanta. [42]
Schliesser et al. [43] have reported optical backaction cooling on the motion of
a radial breathing mode of a silica microtoroid that also functions as a whispering-
gallery-mode optical cavity. Laser power is coupled into the optical resonator by
the evanescent mode of an optical flber, and radiation pressure acts radially on the
structure as the light circulates. Initial measurements cooled the 58 MHz mechanical
mode from room temperature to 11 K, or ? 400 quanta. Later measurements
extended this system into the sideband-resolved regime, and achieve higher cooling
powers, broadening the mechanical linewidth of a 74.5 MHz mechanical mode up to
1.5 MHz. [44] This work was later improved by performing it in a 1.6 K cryostat.
This lower starting temperature enabled a 65 MHz mechanical mode to be cooled
to an occupation of ? 63 quanta. [45] Park and Wang [46] recently demonstrated a
similar interaction between mechanical modes and whispering-gallery optical modes
of a silica microsphere. By making the sphere slightly prolate, they were able to
excite its optical modes by evanescent coupling to a free-space laser beam. The
optical resonance of a 26.5 micron sphere had ? = 2? ? 29:7 kHz and permitted
cooling of its 118.6 MHz mechanical mode by a factor of 6.6 starting from 1.4K,
achieving an occupation of ? 37 mechanical quanta.
Thompson et al. [47] have demonstrated yet another technique in which a
partially-re ecting membrane resonator is placed within a high-flnesse optical cavity.
Radiation pressure then acts on the mechanical resonator from both sides. They
39
were able to backaction-cool the 134 kHz mechanical mode to a temperature of 6.8
mK, or ? 1080 quanta. This technique also has the advantage that the membrane
may be positioned in the cavity standing wave so that the emitted power is directly
sensitive to the mean squared amplitude hx2i of the motion. This may ultimately
permit a \quantum non-demolition" measurement of the total energy (see sections
3.4.1 and 3.4.2).
Teufel et al. [48] [49] used a coupled SMR-NR system very similar to our own,
except that the SMR comprised a quarter-wave resonant length of CPW transmis-
sion line. This system operated in the sideband-resolved regime and used backaction
of the microwave fleld to damp and cool a 1.52 MHz nanomechanical resonator from
the refrigerator temperature of 50 mK to ? 10 mK, or an occupation of 140 quanta.
3.1.3 Summary of quantum analysis
Quantum analyses of backaction cooling of a mechanical mode coupled to
an optical cavity have been presented in references [19], [30] and [31], focusing
speciflcally on the sideband-resolved or \good cavity" regime !m  ?. Another
valuable quantum analysis appears in Ref. [32], dealing with an SMR coupled to
NR motion via the current in a SQUID embedded in the SMR. Such a system obeys
the same form of hamiltonian as ours does in equation (3.1), with the g coupling
in the ~g^x?^by^b term deriving from the magnetic  ux and current in the SQUID. It
therefore can be analyzed in a similar way to the optical systems, with the same
results regarding backaction cooling. We will describe here the general concepts of
40
Figure 3.1: Schematic energy level diagram for optical backaction cooling
or heating processes. Adapted from Ref. [44].
such analyses and summarize their results.
Figure 3.1 shows a schematic energy level diagram for NR-SMR interactions.
Backaction cooling or heating of the NR may be understood in terms of the Stokes
and anti-Stokes scattering processes familiar from atomic and molecular physics.
Pumping the SMR mode with a photon of energy ~(!SMR ?!NR) (\red detuning",
anti-Stokes process) or energy ~(!SMR + !NR) (\blue detuning", Stokes process)
excites the coupled SMR-NR system to raise the SMR occupation one quantum
while lowering (raising) the NR occupation one quantum. The excited SMR mode
then decays, with the large density of states at the SMR resonance favoring decay
by an amount ~!SMR. The SMR returns to its initial state but the NR is left with
one less (more) quantum than before. Repetition of the anti-Stokes (Stokes) process
cools (heats) the NR. Intuitively, we expect the repetition rate, and thus the cooling
or heating rate, to scale with the number of incident scattering photons, i.e. the
41
total number of photons in the cavity.
Anti-Stokes
flfl
flnSMR;nNR
E
?!
flfl
fl(nSMR +1);(nNR ?1)
E
(excitation)
flfl
fl(nSMR +1);(nNR ?1)
E
?!
flfl
flnSMR;(nNR ?1)
E
(decay)
Stokes
flfl
flnSMR;nNR
E
?!
flfl
fl(nSMR +1);(nNR +1)
E
(excitation)
flfl
fl(nSMR +1);(nNR +1)
E
?!
flfl
flnSMR;(nNR +1)
E
(decay)
Figure 3.1 also illustrates the advantage of working in the sideband-resolved
regime !NR  ?. SMR linewidth ? represents broadening of the energy levels; if
!NR ? ? then the
flfl
fl(nSMR + 1);(nNR ? 1)
E
and
flfl
fl(nSMR + 1);(nNR + 1)
E
levels
overlap; there becomes a sizable probability that an excitation to the former state
will nonetheless decay to
flfl
flnSMR;(nNR + 1)
E
. This also makes sense in terms of
the energy-time uncertainty relationship ?t?E ?~, familiar from time-dependent
perturbation theory, considering the decay time of the SMR to be 1?. [50] The
decay to
flfl
flnSMR;(nNR +1)
E
will compete with the anti-Stokes-type cooling process
and diminish the cooling rate. The importance of suppressing this heating behavior
becomes more evident as the mechanical mode approaches its quantum ground state
nNR = 12. As there is no lower energy state of the NR, the anti-Stokes process cannot
extract further energy from the NR, but the competing heating process is unchanged.
We can think of the optical-cooling of the harmonic-oscillator mode of the NR
in terms of a rate equation for energy transfer to and from the mode. A similar
analysis was presented by Naik et al. in regard to backaction cooling of a NR
42
mode using the shot noise of charge moving through a single-electron transistor.
[3] The thermal environment, or thermal \bath", at temperature T, emits energy
to the NR at rate ?NR ? kBT, where ?NR is the natural, or thermal, linewidth
of the NR. On the other hand the NR mode having total energy ~!NR?nNR emits
energy to its environment at a rate equal to ?tot ?~!NR?nNR, where ?tot is the total
decay rate due to thermal and all other causes. If the NR mode is in equilibrium
with the thermal bath, then ?tot = ?NR and ?nNR = ?nTNR = kBT~!
NR
. Now introduce
another, \optical" temperature bath kBTopt = ~!NR?noptNR and damping rate ?opt.
Thus ?tot = ?NR + ?opt. Then in steady state the emissions to and from the NR
mode balance and we have ?tot ? ?nNR = ?NR ? ?nTNR +?opt ? ?noptNR, or
?nNR = ?opt?n
opt
NR +?NR?n
T
NR
?opt +?NR (3.3)
This detailed balance equation balances the emission and absorption processes
in the NR mode. This balance is illustrated in flgure 3.2. For the interaction of the
NR with the microwave mode in the SMR, we must determine ?noptNR and ?opt. The
NR coupled in this way to the microwave mode will then achieve a mode temperature
TNR = ~!NRk
B
?nNR (assuming that ?nNR >> 1 so that we may neglect Bose-Einstein
statistics for the NR mode occupation).
Marquardt et al. [19] have used a quantum noise analysis to flnd ?noptNR and
?opt in the case of a mechanically-resonant mirror coupled to an optical cavity mode
by radiation pressure. Their results are readily generalized to the SMR-NR system.
Wilson-Rae et al. [30] and Genes et al. [31] have found similar results using a variety
of approaches.
43
Figure 3.2: Conceptual illustration of cooling process, showing balance
of energy emission and absorption rates that result in NR mode tem-
perature TNR. The NR is coupled to thermal bath of average energy
kBT = ~!NR ? ?nTNR at rate ?NR. In addition, the mechanical mode is
coupled to \optical" bath kBTopt =~!NR ? ?noptNR, at rate ?opt.
The \optical" bath Topt =~!NR?noptNR may be thought of as an efiective temper-
ature for backaction heating of the NR by the microwave energy in the SMR [19].
This quantifles the efiect of the lower-probability decay processes described above
in reference to flgure 3.1. ?noptNR is the ratio of the rate of transitions that add energy
to the NR mode, to the total optical damping rate. [29] Marquardt et al. flnd
?noptNR = ?(!NR +?!)
2 +(?=2)2
4!NR?! (3.4)
where ?! = !SMR ? !pump. For the optimal heating and cooling pump fre-
quencies !RED = !SMR?!NR and !BLUE = !SMR +!NR, i.e. for ?! = currency1!NR, the
optical damping is given by
?opt = ?4
 
?xZP @!SMR@x
?2 1
??nSMR ?
1
1+(?=4!NR)2 (3.5)
where ?nSMR is the number of photons in the SMR due to the coherent pump
signal. This is consistent with our discussion of transition rates above, which sug-
gested that the cooling rate should scale with the total number of microwave photons
44
in the cavity. In equation (3.5), blue pumping yields negative optical damping, and
red pumping yields positive damping. In the sideband resolved limit, equation (3.5)
reduces to approximately
?opt = ?4
 
?xZP @!SMR@x
?2 1
??nSMR (3.6)
In this regime we can also consider ?noptNR. From equation 3.4 we see that for
?! = +!NR we have ?noptNR ??1 and for ?! = ?!NR we have
?noptNR =
 ?
4!NR
?2
(3.7)
This dependence on the square of the ratio ?!
NR
appears also in the limiting
occupation number determined for laser backaction cooling of trapped ions [25]. (In
fact, an exact calculation [29] flnds this value of ?noptNR in equation (3.7) only for
frequency ?! = ?!NR
q
1+( ?2!
NR
)2, but for !NR >> ? the difierence is small
enough to neglect. Also, such a small difierence in pump frequency has negligible
efiect on the damping rate ?opt and therefore would not be experimentally relevant
until an experiment is in a position to achieve ?nNR < 1.)
Considering the detailed balance equation, Eq. (3.3), we see that in the limit
?opt ! 1, ?noptNR sets a lower limit on the temperature to which the NR mode may
be cooled. This conflrms our expectation that sideband-resolved cooling enables
attainment of the lowest occupations. For blue pumping, on the other hand, ?nNR
diverges rapidly as ?opt approaches ??NR, and leads to self-oscillation of the NR
when ?tot = ?opt+?NR falls below zero. In nearly all cases experimentally, then, we
will have ?NR?nTNR  ?opt?noptNR, so we may neglect the flrst term in the numerator of
45
equation (3.3). The detailed balance equation reduces to
?nNR = ?NR?n
T
NR
?tot (3.8)
This closely resembles the classical result, Eq. (3.2). The more exact quantum
solution, however, predicts maximum cooling for the sideband-resolved regime and
for ?! = ?!NR, in contrast to the classical analysis which flnds maximum cooling
for ? ? !NR. We do note that Eq. (3.7) predicts a minimum possible cooled ?nNR
of 14 even for the non-sideband-resolved condition of ? = 2!NR. [19]
We can also gain further insights by plugging the expression for ?nSMR (Eq.
(2.33)) into Eq. (3.6):
?opt = 4
 
?xZP @!SMR@x
?2 1
?
1
~!SMRPin ?loss?
2
?ext ?
?2ext
?2 +4?!2 (3.9)
By equating khx2i = ?nNR~!NR we can express
hx2i = 2?nNR(?xZP)2 (3.10)
Comparing to Eq. (2.53) we see that
Psideb = ?opt?nNR ?~!SMR ? ?ext=2? ?gain(!SMR) (3.11)
This is exactly what we would expect. The power seen in the sideband equals
the rate ?opt??nNR at which photons are upconverted by the scattering process, times
the energy ~!SMR per upconverted photon, times the fraction ?ext=2? which emerges
from the SMR into the amplifler.
46
3.1.4 Experimental tradeofis
In the device used for backaction cooling in this dissertation, we had !NR ?
11?, meaning that the theoretical lower limit to which we could cool would be
?nNR ? 5?10?4. Of course we didn?t get anywhere near this due to other limitations.
In this light it?s worth examining equations (3.6) and (3.8) to see what experimental
adjustments will maximize the cooling power and minimize the NR occupation. We
can further approximate equation (3.8) in the limit ?opt  ?NR as
?nNR ? ?NR?
opt
?nTNR (3.12)
Plugging in the deflnition of ?xZP and of @!SMR@x (equation (2.40)) to Eq. (3.6)
we have
?opt = ~2 1m!
NR
 @C
g
@x
?2?!
SMR
C
?2 1
??nSMR (3.13)
We can further plug in Eq. (2.33) for ?nSMR. Assuming ?! = ?!NR, and that
!NR  ?, we have
?opt ? ~m!
NR
 @C
g
@x
?2?!
SMR
C
?2 ?ext
? ?
1
4!2NR
Pin ?loss
~!SMR (3.14)
The flrst thing to note is that there are distinct beneflts in increasing the SMR
resonance frequency !SMR, lowering its capacitance C and diminishing its internal
losses, so that ?ext ? ?. In practice, lowering the SMR capacitance C can be done by
lowering the line impedance Z00 of the transmission line resonator, or by switching to
a difierent design in which the SMR is a lumped-element oscillator whose C may be
adjusted more freely. Concerning loss, while losses in the signal lines will generally
be greater at greater frequencies, this probably won?t ofiset the beneflt of increasing
47
!SMR. Decreasing the mass of the NR is beneflcial if this can be done without
adversely afiecting @Cg@x or !NR. Increasing @Cg@x is a challenge in engineering the NR,
as will be seen in the discussion of fabrication, section 4.3. For our NR design, it can
be increased by reducing the gap between the NR and gate electrode across from
it, which both increases Cg and its derivative, or by lengthening the NR to increase
Cg. Lengthening the NR however decreases !NR proportionally. It would appear
from Eq. (3.14) that reducing !NR would be very beneflcial, as long as we maintain
!NR > ?, but this is not necessarily so. Comparing with Eq. (3.12) it is evident
that in terms of occupation number, because the thermal bath temperature rather
than ?nTNR is flxed by the environment, one factor of !NR will be canceled by the
factor in ?nTNR = kBT~!
NR
. The other two factors of !NR appearing in Eq. (3.14) serve
only to increase ?nSMR by placing the pump frequency closer to the SMR resonance.
However, in practice, unfavorable side efiects such as absorption heating of the NR
by microwaves will tend to increase with ?nSMR, so it is often wise to minimize the
?nSMR that will give a desired level of cooling.
Here Eq. (3.13) is a better guide. Diminishing ? improves the cooling e?ciency
for a given ?nSMR, but increases the pump power necessary to achieve that ?nSMR. A
very important consideration in backaction cooling is the ofi-resonance noise emitted
by commercial microwave sources. In contrast to lasers, microwave sources are
not quantum-limited photon sources, and exhibit a large amount of phase noise
and amplitude noise which will appear as white noise that falls ofi gradually at
frequencies away from the carrier. For !pump = !SMR ?!NR, the noise at !pump +
!NR will excite the SMR resonance, leading to backaction driving of the NR that
48
competes with the cooling process. (See section 3.3.) The noise level emitted by
the source is lower at a larger amount !NR away from the carrier, but it also scales
directly with pump power. Section 4.5 describes techniques we have developed to
suppress this noise, up to certain levels of pump power.
It is also important to keep in mind that the expression for the cooling rate,
Eq. (3.6), is based on an assumption of linear coupling between SMR and NR,
which becomes invalid if pump power is increased too far. For instance, attempting
to increase ?opt beyond ? has little beneflt, because the system enters a nonlinear
coupling regime, which limits the cooling rate. [19] At high powers, the resulting
large shifts in SMR frequency due to SMR-NR coupling can also lead to nonlinear
efiects. While the measurements described here do not enter such a regime, future
experiments probably will. It is possible to quantify these efiects by flnding expres-
sions for ?opt that include higher-order coupling terms. [32] These considerations
therefore emphasize the value of starting at a low thermal occupation ?nTNR of the
resonator. Thus because many of these various parameters also present fabrication
and testing challenges (for instance, whether or not a microwave source is available
to operate at !SMR), the correct trade-ofis are not always clear when designing a
device.
3.2 Shift in NR frequency by optical spring efiect
In addition to adding or subtracting damping from the NR mode, the electro-
magnetic fleld within the SMR also shifts the equilibrium position of the NR, and
49
shifts its resonance frequency. As described classically in section 3.1.1 for the case of
a mechanically-resonant mirror in an optical cavity, the force of the electromagnetic
radiation on the NR will oscillate with the NR motion because it is varying the
resonance frequency of the SMR. The force oscillating in phase with the NR adds
efiectively to the kx term in the equation of motion of the NR; this efiective change
in the spring constant k modifles the resonance frequency !NR =
q
k
m.
To estimate this \optical-spring" efiect, Marquardt et al. use a quantum noise
analysis as for calculating the backaction cooling. [19] (Identical results have been
found elsewhere, [31]) They flnd a shift ?!NR in the NR frequency !NR as a function
of the detuning of the pump ?! = !pump ?!SMR from the cavity resonance:
?!NR = 8
 
?xZP @!SMR@x
?2
??nSMR??! ?
2 +4(?!2 ?!2
NR)
(?2 +4(?! ?!NR)2)?(?2 +4(?! +!NR)2)
(3.15)
We note that in the special case where ?! = ?!NRp1+(?=2!NR)2, the
optical-spring frequency shift ?!NR will be zero. For a sideband-resolved device,
this condition occurs when ?! ??!NR.
The optical damping of the NR ?opt may also be expressed as a function of
?!. Marquardt et al. [19] flnd
?opt = ?8
 
?xZP @!SMR@x
?2
? ?nSMR ? 8?! ?!NR ??(?2 +4(?! ?!
NR)2)?(?2 +4(?! +!NR)2)
(3.16)
In the case ?! = currency1!NR, this reduces to equation (3.5). Marquardt et al.
also express the optical damping and optical-spring frequency shift compactly in a
convenient notation based on the response function of the optical cavity.
50
3.3 Backaction cooling while the SMR is thermally excited
Equation (3.3) allows us to consider two distinct cooling regimes. As described
in section 3.1.3, when we consider that the SMR is excited only by a pump tone
(and its associated shot noise), then for a sideband-resolved system ?noptNR ? 1. In
practice then we may neglect ?noptNR altogether.
However, if the SMR is excited by a second source of broadband classical noise
the situation is difierent. An SMR driven by photons of both energy~(!SMR?!NR)
and~!SMR enables transitions that emit quanta at the difierence energy, i.e. ~!NR,
thereby adding energy to the NR mode. This backaction heating process is distinct
from the undesired transitions discussed in section 3.1.3, which set lower limits
on ?noptNR as indicated in Eq. (3.4). Here the rate of the heating process will be
proportional to the photon  ux at !SMR.
One process that introduces such photons is thermal excitation of the SMR
above its ground state, i.e. Johnson noise within the SMR. In practice we may
treat any process that drives the cavity with broadband noise at frequencies near
!SMR as if the SMR were thermally excited. We let ?nTSMR represent the average
thermal occupation of the SMR. The actual source may be Johnson noise originating
in other components of the system, or phase noise of the microwave source, or RF
noise deliberately introduced to the SMR as in the demonstration of backaction
evasion: all will excite the SMR above its ground state to a level we will denote
?nTSMR. Phase noise in the microwave pump source is of particular interest here.
Recent theoretical work looking at the efiects of pump phase noise [51] shows that
51
as long as ?opt << ? and !NR >> ?, the efiects of phase noise are indistinguishable
from amplitude noise.
We can apply equations (2.31) and (2.32) to noise power densities SN;in and
SN;out rather than single-frequency powers Pin and Pout, and consider~!SMR??nTSMR
to be the total energy in the cavity integrated over all frequencies. Then we can
consider the thermal occupation of the SMR to be driven by a white noise (in units
of W/angular frequency)
SN;in = 1loss ?~!SMR?nTSMR ? ???
ext
(3.17)
The SMR excited to ?nTSMR will emit a noise spectrum (in units of W/angular
frequency)
SN;out = gain?~!SMR?nTSMR ? ???
ext
? ?
2
ext
?2 +4?!2 (3.18)
The thermal occupation of the SMR may be found from the measured noise
spectral density SN;out(!SMR) (in units of W/angular frequency) at the SMR peak
frequency:
?nTSMR = 1~!
SMR
? ???
ext
? 1gainSN;out(!SMR) (3.19)
If SN;out(!SMR) instead has units of W/Hz (which are the typical measurement
units on a spectrum analyzer) then equation (3.19) should be divided by 2?.
A. Clerk has extended the theoretical calculation of Ref [19] to include the
case where the SMR is excited to thermal occupation ?nTSMR. [52]. He flnds that
the optical damping (equation (3.6)) and the detailed balance expression (equation
(3.3)) remain valid to describe the cooling process, but that the efiective backaction
52
temperature is given by
?noptNR =
 ?
4!NR
?2
+ ?nTSMR
?
1+2
 ?
4!NR
?2!
(3.20)
In the sideband resolved limit, we then have approximately ?noptNR = ?nTSMR.
Considering this in equation (3.3) it is important to note that the NR occupation
?nNR can never be lower than ?nTSMR.
When ?nTSMR > 0, the resulting backaction driving of the NR will produce a
sideband signal at !SMR that is coherent with but 180 degrees out of phase with the
SMR noise at !SMR. (This will be discussed further in section 3.4.) The sideband
signal due to backaction thus subtracts from the SMR noise at !SMR while the signal
due to NR thermal noise adds to it incoherently, as illustrated in flgure 3.3. This
behavior resembles the \noise squashing" that has been seen in feedback cooling of
some optomechanical systems. [53] The calculation in Ref. [52] flnds the spectral
solution SN;out(?) (in units of W/angular frequency) at measured frequencies !SMR+
?, where ? ? ?
SN;out(?) = gain?~!SMR ? ?ext??
 ?
NR?opt
4?2 +?2tot?n
T
NR +
 
1? ?opt(?NR +?tot)4?2 +?2
tot
?
?nTSMR
?
(3.21)
To flnd the total measured sideband power, we neglect the term in equation
(3.21) equaling the background level at the SMR resonance frequency (see (3.19))
and integrate: Psideb = R (SN;out(?)?Sbgd)d?. Comparing the result to equation
(3.11), we see that when ?nTSMR > 0, what we actually measure is an efiective me-
chanical occupation, ?neffNR, given by
?neffNR = ?NR?
opt +?NR
?nTNR ?
 
1+ ?NR?
opt +?NR
?
?nTSMR (3.22)
53
a)
?400?2000 2004002
468
? / ?NR
246
8
Normalized 
S N,out 24
68
b)
?10?5 0 5 102
468
? / ?NR
246
8
Normalized 
S N,out 24
68
Figure 3.3: Noise spectrum of thermally-excited SMR and NR during
backaction cooling, calculated from Eq. (3.21). Parameters are not
experimental, only for illustration: ?opt = ?NR, ? = 160?NR. Green
curve: backaction only, ?nTSMR = 5, ?nTNR = 0. Red curve: thermal noise
only, ?nTSMR = 0, ?nTNR = 20. Black curve: ?nTSMR = 5, ?nTNR = 20. Noise
spectrum scaled by gain?~!SMR ? ?ext?? . a) Wide span showing full span
of emitted SNR thermal noise. Here equation (3.18) has been combined
with (3.21) to calculate the full SMR spectrum. b) Same calculation
over a narrower span showing NR sideband only.
54
Here the term ?NR?opt+?
NR
?nTNR represents the NR thermal noise while the term
?
?
1+ ?NR?opt+?
NR
?
?nTSMR represents the backaction signal subtracting coherently from
the SMR noise. Note that for ?nTSMR = ?nTNR ? ?NR?
NR+?tot
, no sideband at all will
appear, and for ?nTSMR > ?nTNR ? ?NR?
NR+?tot
the sideband will appear as a \dip" in the
SMR noise rather than a peak. We may use Eq. (3.22) along with equation (3.3)
with ?noptNR = ?nTSMR to calculate the actual NR occupation ?nNR during the combined
backaction cooling and backaction excitation due to SMR excitation ?nTSMR. In terms
of the measured quantities ?neffNR and ?nTSMR we flnd
?nNR = ?neffNR +2?nTSMR (3.23)
3.4 Backaction-evading (BAE) single quadrature detector
In trying to measure any physical quantity with high precision, one eventu-
ally runs up against limitations imposed by quantum mechanics. Any observable
quantity, represented by a quantum mechanical operator ^A, will have a conjugate
observable ^B, for which the uncertainties in the two observables h?A2i and h?B2i
must obey the uncertainty relation [50]
h?A2ih?B2i? 14jh[ ^A; ^B]ij2 (3.24)
In principle there is no problem in measuring A to arbitrary precision, if you
measure it only once. The complementary observable B absorbs the penalty in
imprecision. But for any practical measurement, in particular in measuring me-
chanical motion, one wants to measure some quantity of the system continuously, or
at least at regular intervals. We often think of measurements in quantum mechanics
55
in terms of a process wherein you prepare an ensemble of identical systems in the
same state, and then measure each one to establish the various outcomes and their
probabilities. Here the situation is completely difierent, because repeated measure-
ments on a single system are not in general independent. Each measurement of A
adds uncertainty to B, and if B is coupled to A in any way then the imprecision of
B will \contaminate" that of A. This is quantum measurement backaction.
3.4.1 Quantum non-demolition (QND) measurements: formalism
We would like to measure mechanical motion in a way that is immune from
such measurement backaction. The trick, then, is to identify observables which are
decoupled from their complementary observables, and flgure out how to measure
them. We want to do the measurement in such a way that one measurement does
not add to the uncertainty of later measurements; the measured value is entirely
predictable based on the result of earlier measurements. This is known as a quan-
tum non-demolition (QND) or back-action evading (BAE) measurement. A good
review of the quantum theory involved appears in the paper by Bocko and Onofrio.
[54] Braginsky and Khalili [55] provide a similar discussion, and a further analy-
sis of back-action-evading and quantum nondemolition measurements of mechanical
oscillators appears in the review by Caves et al. [56].
Here it is valuable to employ the \Heisenberg picture" of quantum analysis.
[50] As opposed to the Schr?odinger picture in which the state kets evolve in time
while an operator observable ^A does not, in the Heisenberg picture the operators
56
evolve according to a unitary transformation ^A(t) = ^Uy ^A^U, where ^U is the time-
evolution operator ^U = e?i ^Ht=~ and ^H is the system hamiltonian. The state kets
jfii representing the state of the system, meanwhile, are time-independent while the
base kets jai (here the eigenstates of operator ^A(t)) evolve in time via the conjugate
of the time evolution operator: ja(t)i = ^Uyja(0)i. The time evolution of the operator
will follow the Heisenberg equation of motion
d
dt
^A(t) = @
@t
^A(t)+ 1
i~[
^A(t); ^H] (3.25)
To deflne the requirements for a QND measurement, consider a system de-
scribed by hamiltonian ^H0. To measure observable ^A of this system we use a
measurement apparatus described by hamiltonian ^HM. The interaction between
the measurement apparatus and the system is deflned by interaction hamiltonian
^HI, dependent on both ^A and on some observable ^Q of the measurement appara-
tus. Our flrst measurement at time t0 using operator ^A(t0) casts the system into
an eigenstate ja(t0)i of ^A(t0). But because in the Heisenberg picture the state kets
do not evolve, the system remains in this same state until our next measurement
at time t1. In order for the second measurement using operator ^A(t1) (or subse-
quent measurements using ^A(tn)) to give predictable results, the system ought to
remain in the same state after the measurement. That means ja(t0)i should also be
an eigenstate of ^A(t1). This is satisfled if ^A(t1) is just a function of ^A(t0). That
is, the difierent-time operators must commute. Thus for a QND observable, the
\difierent-time commutator" must vanish:
[ ^A(tm); ^A(tn)] = 0 (3.26)
57
In general, Eq. (3.26) may be true only at speciflc discrete instances. In such
case, ^A(t) is a \stroboscopic" QND observable. [54] A more rigorous condition is
to require Eq. (3.26) to be true at all times, making ^A(t) a \continuous" QND
observable. If ^A(t) does not change at all, i.e. if it is a constant of the motion,
d
dt ^A(t) = 0, then this condition is satisfled. To identify such an observable, we use
the Heisenberg equation of motion (Eq. (3.25)), with ^H0 as the hamiltonian. If ^A(t)
satisfles the condition
@
@t
^A(t)+ 1
i~[
^A(t); ^H0] = 0
then it is a constant of the free evolution of the system. To measure it, we must
flnd an interaction hamiltonian ^HI that makes ddt ^A(t) = 0 still be true even if ^HI is
added to ^H0. This is satisfled if ^A(t) commutes with the interaction hamiltonian:
[ ^A(t); ^HI] = 0 (3.27)
A convenient form for ^HI is to be linearly dependent on both ^A and on mea-
surement observable ^Q, with some coupling constant K(t): [54]
^HI = K(t)? ^A(t)? ^Q
3.4.2 Harmonic oscillator quadratures as QND observables
To apply the formalism of QND observables to the harmonic oscillator, con-
sider the hamiltonian of our SMR-NR system, Eq. (3.1). In terms of the notation
used in section 3.4.1, the ~!SMR
?^
by^b+ 12
?
, ^Hpump and ^H? terms together comprise
^HM. The ~!NR?^ay^a+ 12? term meanwhile is ^H0. We will assume that we may ne-
glect the NR dissipation term ^H?. In terms of the QND formalism, it is legitimate
58
to do so as long as the measurement noise dominates over thermal noise, i.e. if the
measurement exchanges a quantum of energy with the measured system in a time
shorter than ?nTNR?NR. In practice, we will see that the backaction-evading nature
of our measurement scheme is valid even if the NR is strongly driven by thermal
noise.
We can go into some further detail on the operators
H0 = 12m^p2 + m!
2
NR
2 ^x
2
^x = ?xZP(^ay + ^a)
^p = im!NR ??xZP(^ay ?^a)
^a = 12?x
ZP
 
^x+i 1m!
NR
^p
?
^ay = 12?x
ZP
 
^x?i 1m!
NR
^p
?
The question is, what can we select as a QND observable, and what ^HI to use
to detect it? One readily appealing QND observable is the average energy in the NR,
ENR = ~!NR?nNR. Indeed this is the observable corresponding to the hamiltonian
^H0 itself, so it is thereby a constant of the motion. However, practical measurement
of ENR is very di?cult. Classically, the average mechanical energy is proportional
to the squared average position amplitude, ENR = khx2i, as described in section 2.4.
However, as we will see later, the ~?^x2
?^
by^b+ 12
?
term in our system is too weak
for easy measurement of hx2i. Other measurement schemes strongly sensitive to ^x2
have been demonstrated by e.g. Thompson et al. [47], but remain to be perfected.
We could consider position x as a QND observable. Taking the difierent-time
59
commutator we flnd [54]
[^x(tm); ^x(tn)] = i~m!
NR
sin(!NR(tn ?tm))
Evidently, the NR position serves as a QND observable only instantaneously
at time intervals (tn?tm) = ?=!NR and integer multiples thereof. The momentum p
has similar behavior. [54] Thus the position or momentum of the NR can conceivably
be measured in a \stroboscopic" QND fashion, making very brief measurements each
half-cycle of NR motion. This is technically very challenging, and requires a very
large coupling to achieve reasonable sensitivities in this very short duty cycle of
measurement.
Instead, we can deflne the quadrature amplitudes ^X1 and ^X1 with explicit
time-dependence [54] [55]
 
^x+i 1m!
NR
^p
?
=
? ^
X1 +i ^X2
?
e?i!NRt
Restating, we have
^X1(t) = ^xcos(!NRt)? 1
m!NR ^psin(!NRt)
^X2(t) = ^xsin(!NRt)? 1
m!NR ^pcos(!NRt)
^x = ^X1 cos(!NRt)+ ^X2 sin(!NRt) (3.28)
^p = m!NR
? ^
X2 cos(!NRt)? ^X1 sin(!NRt)
?
The raising and lowering operators may be restated in terms of ^X1 and ^X2.
^a = 12?x
ZP
? ^
X1 +i ^X2
?
e?i!NRt
^ay = 12?x
ZP
? ^
X1 ?i ^X2
?
ei!NRt
60
We can also flnd the commutator of ^X1 and ^X2.
[ ^X1; ^X2] = i ~m!
NR
From the commutator, and equation (3.24), we flnd the uncertainty relation
of the two quadratures to be
?X1?X2 ? ~2m!
NR
(3.29)
We can restate the hamiltonian ^H0 in terms of ^X1 and ^X2.
^H0 = m!2NR
2
? ^
X21 + ^X22
?
By determining the commutator of ^H0 with ^X1 and ^X2, and the partial time-
derivatives of ^X1 and ^X2
[ ^H0; ^X1] = ?i~!NR ^X2
[ ^H0; ^X2] = i~!NR ^X1
@
@t
^X1 = ?!NR ^X2
@
@t
^X2 = !NR ^X1
we see by the Heisenberg equation of motion (Eq. (3.25)) that the two quadra-
ture amplitudes are constants of the motion with respect to ^H0. They are thus
continuous QND observables.
d
dt
^X1 = 0
d
dt
^X2 = 0
61
It remains for us to choose either ^X1 or ^X2 and identify an interaction hamil-
tonian ^HI that commutes with it and also represents a feasible measurement. Our
best bet is to use the ~g^xbyb term in the hamiltonian (Eq. (3.1)), adjusting the
parameter byb experimentally by varying the SMR pump to make the interaction
commute with ^X1. We see however that the ~?^x2byb term is likely to cause trouble,
because it may not commute with ^X1 even if the ~g^xbyb term is made to do so. In
the following, we will neglect the weaker ^x2-dependent term. In practice, as we will
see in sections 3.5 and 6.5, this means a restriction on the pump strengths at which
we can work.
A good choice of interaction hamiltonian is to modulate the coupling at fre-
quency !NR, by modulating the electric fleld in the SMR. [54]
^HI = E0 cos(!NRt)^x^Q (3.30)
where E0 is the electric fleld amplitude and ^Q represents the charge on the
coupling capacitance Cg. Restating this in terms of quadrature amplitudes, we have
^HI = E0 cos(!NRt)^x^Q
= E0 cos(!NRt)
? ^
X1 cos(!NRt)+ ^X2 sin(!NRt)
? ^
Q
= E02
?
(1+cos(2!NRt)) ^X1 +sin(2!NRt)) ^X2
? ^
Q
If the oscillating components of this measurement are then flltered out, only
the DC portion remains:
^HI = E0
2
^X1 ^Q (3.31)
This ^HI commutes with ^X1, thus enabling a continuous QND measurement of
62
that quadrature of the NR oscillation. In practice, the way that we can make the
coupling have the form of equation (3.30) is to apply a pump tone at !SMR, fully
modulated at frequency !NR. This is equivalent to pumping simultaneously with
equal strength at two tones, !SMR ?!NR and !SMR +!NR.
^HI = E0 cos(!SMRt)cos(!NRt)^x^Q
= E02 (cos((!SMR ?!NR)t)+cos((!SMR +!NR)t)) ^x^Q (3.32)
The flltering needed to arrive at equation (3.31) will also fllter out the oscil-
lations at frequency !SMR. This pumping scheme thus ofiers a practical means for
QND measurement of quadrature amplitude ^X1.
The flltering is provided automatically by the SMR resonance. If our device
works in the sideband-resolved limit, ? ? !NR, then the SMR decay time is much
longer than the NR oscillation period, and the signal emerging from the SMR faith-
fully represents the measurement indicated by the desired interaction hamiltonian
(Eq. (3.31)). [54]
The selection of the X1 quadrature is illustrated in flgure 3.4. We can regard
this measurement scheme as acting sort of like a lock-in amplifler! Of the two
quadratures comprising the full motion ^x (Eq. (3.28)), the measurement selects
only the one in phase with the modulation of the pump. However, there is a subtle
and important difierence here from measuring ^x and feeding the measurement into
a lock-in amp. The difierence is that no information about ^X2 emerges from the
SMR, and that the flltering takes place before the ampliflcation of the signal, which
unavoidably adds noise. [54] Thus by avoiding measurement backaction the double-
63
pump measurement scheme should preserve the ability to continuously measure the
amplitude X1 with a precision beyond that which is physically possible for x.
We should keep in mind that this discussion of a QND or backaction-evading
measurement deals only with efiects on the NR by the measurement interaction
itself. The backaction evasion eliminates these efiects on the measured observable.
In practice, however, we expect the NR to be subject to forces arising from sources
other than the act of measurement. These will in general afiect both quadratures of
the NR motion, and will be fully visible during the QND measurement. In fact, in
many proposed applications the whole point of the QND measurement is to observe
such efiects - for instance, the impulse of gravitational waves on the mass of the
mechanical resonator. When we described a QND observable in section 3.4.1 as
having to be \predictable", this is in respect only to the efiects of measurement.
The point is to suppress the measurement backaction as a source of noise with
respect to more interesting signals.
3.4.3 Classical backaction evasion and limitations of real devices
If we model the backaction classically, the behavior of the double-pump QND-
type measurement is readily apparent. In section 3.3 we discussed classical backac-
tion due to noise exciting the SMR while the SMR is pumped ofi-resonance. Evasion
of classical noise in the SMR (due to amplifler noise, thermal excitation or noise in
the pump) is an important application of BAE techniques. The classical behavior
of backaction evasion serves as a useful analogue for the quantum regime in which
64
backaction driving of the NR would arise due to microwave shot noise. A measure-
ment that demonstrates sensitivity to only a single quadrature of NR motion, and
that demonstrates insensitivity to classical noise in the SMR, can be expected to
behave as a QND measurement when the measurement enters the quantum regime,
i.e. when pump power is su?ciently high to drive the NR with microwave shot
noise.
To describe the backaction force in classical terms, we refer to the circuit model
of the coupled SMR and NR, described in section 2.4. Consider the voltage within
the SMR to consist of a pump tone at frequency !SMR + !NR or !SMR ?!NR, as
well as a noise signal at frequency ! ? !SMR:
VSMR(t) = Vp cos((!SMR ?!NR)t)+Vn cos(!t+`n) (3.33)
where Vn is a noise amplitude that is random in time and corresponds to
a mean-squared average voltage noise density SV (in units of volts2 per angular
frequency) at frequencies around !SMR, while `n is a random phase. The voltage
on the SMR falls also across Cg, generating a backaction force on the NR equal to
FBA = 12 @Cg@x V 2SMR. (For simplicity we?ll denote @Cg@x as @C@x.) We can see that of the
various terms in V 2SMR, only the cross-term 2Vp cos((!SMR?!NR)t)?Vn cos(!t+`n)
will apply a force at the right frequencies to drive the NR. We decompose this term
into two terms, one of which goes as cos((!SMR?!NR)t?!t?`n), and discard the
other. If we denote ?! = !SMR ?!, we have, for !pump = !SMR ?!NR,
FBA = 12@C@xVpVn cos((!NR ??!)tcurrency1`n)
Considering only the components of noise that lead to driving at the NR
65
resonance, i.e. ?! ? 0, gives insight into the BAE efiect. For !pump = !SMR ?!NR,
we then have FBA = 12 @C@xVpVn cos(!NRtcurrency1`n). Yet if we apply both pumps in the
BAE conflguration, each of amplitude Vp, and make them phase-coherent with one
another, the force appearing at !NR due to each pump derives from the same voltage
noise at !SMR, making the two components of backaction force phase-coherent with
one another. The total force is then
FBA = 12@C@xVpVn(cos(!NRt?`n)+cos(!NRt+`n))
= @C@xVpVn cos(!NRt)cos(`n) (3.34)
Thus in the BAE pump conflguration it is evident that despite the random
phase of the noise at !SMR, the NR is driven only at a single phase, dictated by the
phase of the pump tones. The force in Eq. (3.34) is proportional to cos(!NRt) and
not sin(!NRt). In this condition, the backaction drives only one quadrature of NR
motion. The driven quadrature is designated as X2.
The backaction-driving being conflned to X2 is half of the story. The other half
is that the measurement measures exclusively X1. We can examine this behavior
using the classical analysis of sideband amplitudes described in section 2.4, where
the pump tone within the SMR was described as Vp cos(!pt+`p), and `p identifled
with respect to the phase of the pump signal at the input of the SMR as `p =
arctan
?
?2?!p
?
?
. Here we have !p = !SMR ? !NR and assume that the phase of
each input pump tone is adjusted to zero the phase of the voltage within the SMR
as in Eq. (3.33). Now we note that a harmonic oscillator driven on resonance
acquires a ?2 phase shift relative to the driving force. Thus the phase `m of the
66
mechanical motion driven by the backaction force will be ?2 relative to FBA. The
solution for sidebands appearing within the pumped SMR due to mechanical motion
is given in equations (2.44) to (2.47), where (Vs, `s) are amplitude and phase of an
upper (sum) sideband and (Vd, `d) apply to a lower (difierence) sideband. It can be
seen from these expressions that the phase of an upper sideband follows `m while
that of a lower sideband follows ?`m. These have been termed respectively \phase
preserving" and \phase conjugating" detection of the motion. [54] [32] For two
pumps in the BAE conflguration, one upper sideband will be generated by the red
pump and one lower sideband by the blue pump, overlapping at !SMR having a
difierence in phase `s ?`d = 2`m. For the backaction-driven motion, the phase is
`m = ?2, making the two sidebands cancel. Thus no backaction-driven motion will
appear as a sideband at !SMR. The X2 quadrature of mechanical motion is invisible
to the measurement. Mechanical motion of truly random phase, such as thermal
motion, will have amplitude in both X1 and X2 quadratures, and will thus generate
a sideband.
Calculation of the backaction force (Eq. (3.34)) enables calculation of the X2
amplitude as a function of SMR noise and pump amplitudes. This has been done
classically by Bocko and Onofrio [54] [57] as well as by Clerk et al. [20] [58] using a
quantum-noise analysis. It is important to note that the backaction force described
in Eq. (3.34) is an idealized case because it considers only backaction forces arising
from the noise at !SMR. In fact, the pump at !SMR ? !NR will also mix with
noise at !SMR ? 2!NR, while the pump at !SMR + !NR will mix with noise at
!SMR +2!NR, to generate forces at !NR. These forces are incoherent and will thus
67
Time
E field
Position
 
 X
1X
2
Figure 3.4: Schematic illustration of electric fleld in SMR for single-
quadraturedetection, asinequation(3.32). Beatingofpumpflelddeflnes
quadratures X1 and X2 of mechanical motion.
excite both X1 and X2. The driving of X1 imposes a limit on the efiectiveness of the
backaction evasion. However because of the Lorentzian lineshape, the voltage noise
at !SMR?2!NR will be much smaller, approximately a factor of ?4!
NR
times the level
at !SMR. Taking the X1 position noise density (the square of the amplitude, having
units of m2/Hz) due to these two forces and summing them to flnd the total SX1,
and comparing this to the X2 position noise density, yields a quantitative measure
of the BAE efiectiveness. The result strongly illustrates the value of being in the
sideband-resolved regime in order to make these measurements e?ciently: [57]
SX2
SX1 =
32!2NR
?2 +3 (3.35)
Past experimental investigations of backaction-evading single quadrature de-
tectionhavebeenundertakenprimarilyforimprovementofmeasurementofvibrating-
68
bar gravitational-wave antennas. In these devices, backaction evasion was sought
as a means of counteracting the efiects of amplifler noise on the measured device.
Noise in pump sources or thermal noise in electrically resonant transducers could
also be a concern. Two representative efiorts appear in the work by Marchese et
al. [57] and by Cinquegrana et al. [59]. The former group employed a 50 kHz
resonant bridge circuit consisting of two LC oscillators sharing a common inductor
and common ground connection. A 1.87 kHz, 0.2 kg torsional mechanical oscillator
simultaneously modulated the capacitances in the two arms of the bridge, in inverse
fashion, so that @C@x in one arm was equal and opposite to that in the other arm.
The resonant bridge was excited with coherent double pump signals and white noise,
while a separate actuator and transducer allowed the mechanical motion to be di-
rectly excited and monitored independently of the bridge. This work was able to
clearly demonstrate sensitivity to a single X1 quadrature of motion, and to demon-
strate insensitivity to the motion excited by the backaction of the white noise. A
particular feature of this work was the use of the independent transducer to observe
the backaction-driven mechanical motion in both the X1 and X2 quadratures, en-
abling measurement of the complete \noise elipse" for the backaction driving. For
their circuit having !mech = 0:5?, they achieved a value of SX2S
X1
= 14 ? 4:6, close
to that predicted by equation (3.35). In the result by Cinquegrana et al., a similar
bridge circuit, incorporating a 380 g, 928 Hz mechanical resonator and 129.4 kHz
electrical resonance, was measured at a temperature 4.2 K. The researchers were
able to demonstrate that their system was sensitive to only a single quadrature of
mechanical motion, and were able to distinguish the brownian motion of the me-
69
chanical resonance superimposed on the electrical resonance noise and not obscured
by backaction-driven motion. A more recent attempt at backaction-evading mea-
surement has been made by Caniard et al. [60] using two mechanically resonant
mirrors mounted at each end of an optical cavity. Their scheme difiers from the one
discussed here in exploiting a difierence in the two mirrors? resonance frequencies,
meaning that at intermediate frequencies, the phase response of the two mirrors
will be opposite. Because optical power circulating within the cavity exerts force in
opposite directions on the two mirrors, radiation-pressure noise at the intermediate
frequency will actually cause the two mirrors to move in unison, thus maintaining
the cavity length. The researchers were able to demonstrate the backaction-evading
behavior by injecting a noisy optical signal. This system, however, does not func-
tion as a single-quadrature detector of the mechanical motion, and because the BAE
does not appear at either mechanical resonance frequency, the sensitivity to external
forces is greatly reduced.
3.4.4 Quantum squeezed states of mechanical motion
Quantum squeezed states of an electromagnetic fleld have been extensively
studied, both at optical [61] and at microwave [62] wavelengths. In both regimes,
parametric ampliflcation processes were used to generate flelds having one quadra-
ture with variance below the vacuum  uctuation level 12~!. Optical squeezed states
have also been produced using parametric downconversion as well as four-wave mix-
ing in optical flbers and atomic vapors. [63] The electromagnetic fleld of a resonant
70
cavity behaves as a simple harmonic oscillator and serves as a model for applying
the same concept to a mechanical system. The mechanical case has been the subject
of theoretical study for more than twenty years. [64]
Because in the ideal case the X1 quadrature is not subject to any backaction
at all, the BAE technique holds the possibility of generating a squeezed state of
mechanical motion, i.e. one in which the uncertainty in one quadrature is reduced
below the zero-point motion. [20] [54] The uncertainty relation of the two quadra-
tures given in equation (3.29) shows that at the minimum uncertainty condition, if
each quadrature has equal uncertainty, we have ?X1 = ?X2 = ?xZP =
q
~
2m!NR.
To suppress ?X1 below ?xZP, all that need be done is to increase the measurement
strength by increasing the pump power (as described in section 3.6) until the addi-
tive uncertainty due to amplifler noise and noise in the SMR is reduced below ?xZP.
Because the measurement is QND, backaction will add nothing to the uncertainty
in X1.
As Clerk et al. point out however (Ref. [20]), a measurement that is truly de-
coupled from X1 is by nature unable to afiect the intrinsic  uctuations in X1 arising
from its zero point and thermal motion. In averages over many measurements, these
 uctuations would appear as extra uncertainty in the measured X1 value. While the
added uncertainty due to amplifler noise would be < ?xZP, the mechanical motion
itself would be only \conditionally" squeezed. To enforce the squeezing of the mo-
tion, the authors propose to use the precision of the X1 measurement in a feedback
scheme that would actively damp the X1 amplitude. An alternate proposal [65]
would employ parametric ampliflcation of the NR beyond the self-oscillation limit
71
for short durations, in order to squeeze one quadrature below the zero point motion.
This technique could be employed even if the average occupation of the mechani-
cal mode is initially well above ?n = 1, but a BAE type scheme would nonetheless
be required to measure the resulting squeezed state. A number of other schemes
have been proposed for generating quantum squeezed states of mechanical motion.
[66] The demonstration of such states would be a dramatic demonstration of truly
non-classical behavior, and backaction evading single-quadrature measurement is an
important step towards this goal.
3.5 NR motion parametrically amplifled in BAE pump conflguration
3.5.1 Electrostatic frequency shift due to @2C=@x2
Apart from the NR frequency shift imposed by optical-spring efiects, we can
identify a frequency shift due to electrostatic energy in the capacitance Cg coupling
NR to SMR. [67], [68] In flgure 2.4 we can see how the NR motion changes the gap
in Cg. We assume that the equilibrium position of the NR is x = 0. The total
energy stored in the capacitor is "NR = "mech +"electrostat, or
"NR = 12kx2 + 12CgV 2
Expanding Cg to 2nd order in x we have
"NR = 12kx2 + 12V 2
 
Cg;0 + @Cg@x x+ 12@
2Cg
@x2 x
2
?
(3.36)
where Cg;0, @Cg@x and @2Cg@x2 are evaluated at x = 0. Since Ctot = C + 2C? + Cg
and Cg is the only varying term, and in practice C  C?  Cg, we can simplify the
72
notation by replacing @Cg@x with @C@x and @2Cg@x2 with @2C@x2 . Rearranging equation (3.36)
we have
"NR = 12Cg;0V 2 ??"EM + 12(k +kEM)(x+?xEM)2 (3.37)
In practical application we can neglect the energy shift ?"EM =
1
2(
1
2
@C
@x V
2)2
k+12 @2C@x2 V 2 ,
and the shift in the equilibrium mechanical position ?xEM = 12 @C@x V 2k+1
2
@2C
@x2 V
2. However,
the \electrostatic spring constant" kEM becomes important experimentally
kEM = 12@
2C
@x2 V
2
The additional spring constant introduces a shift in the mechanical resonance
frequency !NR = pktot=m = p(k +kEM)=m. Taylor-expanding yields
!NR = !NR;0
 
1+ 12kEMk
?
(3.38)
Note that Cg scales roughly as 1d, where d is the gap between the NR and
the nearby gate electrode. Therefore, Cg ? 1d+x, meaning @2C@x2 will be negative.
Increasing voltage will tend to reduce the resonance frequency of the NR.
For V = VSMR cos(!pumpt), and !pump  !NR, the electrostatic spring con-
stant will have a time-averaged value 12 @2C@x2 ? 12V 2SMR. The NR is located at a voltage
antinode within the SMR, having a standing-wave amplitude twice the wave ampli-
tude given by equation (2.38); this is the same as the lumped-element model voltage
of Eq. (2.26). We can use equation (2.30) to express kEM in terms of the magnitude
?nSMR of pump energy within the SMR.
kEM = 12C @
2C
@x2 ? ?nSMR~!pump (3.39)
73
In most practical cases in Eq. (3.39) we may approximate !pump = !SMR. We
can deflne the parameter ?
? = !SMR2C @
2C
@x2 (3.40)
Then the electrostatic spring constant may be expressed
kEM = ??~?nSMR (3.41)
The NR frequency shift in Eq. (3.41) may be seen when a single pump tone is
applied. Another important condition is to apply two pump tones in the SMR for
backaction evasion. Then the SMR voltage is:
V = VSMR2 (cos((!SMR ?!NR)t)+cos((!SMR +!NR)t))
= VSMR cos(!SMRt)cos(!NRt)
Taking V 2 and time-averaging over a microwave cycle 1!
SMR
, we have
hV 2i = 14V 2SMR(1+cos(2!NRt))
Considering the SMR internal energy ?nSMR~!SMR to be the sum of the ener-
gies due to the two pump tones (and approximating ~(!SMR ?!NR) = ~(!SMR +
!NR) =~!SMR) we have ?nSMR~!SMR = 2? 12C?VSMR2 ?2 and we can express
hV 2i = 1C?nSMR~!SMR(1+cos(2!NRt))
Then we can express kEM = 12 @2C@x2 hV 2i and extend equation (3.39) to this
double-pump case
kEM = 12C @
2C
@x2 ?nSMR~!SMR(1+cos(2!NRt))
= ??~?nSMR(1+cos(2!NRt)) (3.42)
74
Note that in the limiting case !NR = 0, i.e. if we reduce to a single pump tone,
equation (3.42) reduces to equation (3.41) (keeping in mind that ?nSMR is deflned
slightly difierently in equation (3.42) as the sum of two pump energies, whereas in
(3.41) a single pump tone is assumed).
3.5.2 Parametric ampliflcation of NR motion
Oscillation of the mechanical spring constant and therefore of the mechanical
frequency is a condition for parametric ampliflcation of the mechanical motion. As
described by Rugar [69], this ampliflcation is phase-dependent: it amplifles one
quadrature by an amount 11?Q
NR
kEM
2k
while the other is amplifled by an amount
1
1+QNRkEM2k , i.e. it amplifles one quadrature and deamplifles the other. Thus we can
see from equation (3.38) that when the oscillating frequency shift ?!NR;0 ? 12 kEMk ? is
made ? ?NR, the NR will self-oscillate uncontrollably.
Parametric ampliflcation will amplify any motion of the NR and has the dis-
tinct beneflt of doing so without adding noise. It has been shown to readily amplify
the thermal motion of a mechanical resonator. [69] [70] The phase-dependent ampli-
flcation means that while one quadrature?s motion is increased, the other?s is reduced
in energy, that is, cooled. Such a behavior is referred to as \thermomechanical noise
squeezing." [69]
However, further analysis comparing equations (3.38) and (3.42) with the
single-quadrature detection described in section 3.4.3 indicates that the parametri-
cally amplifled and deamplifled quadratures are not the X1 and X2 deflned by the
75
double pump tone, but are instead deflned at a phase of ?2 relative to these. This
poses a problem for backaction-evading measurement, because X1 and X2 are no
longer independent. At su?ciently high levels of parametric ampliflcation, X1 will
re ect both quadratures of motion equally, destroying the BAE completely. When
not considering backaction, however, the parametrically-amplifled quadrature will
dominate over the deamplifled one in the measurement, and thus we can treat the
parametric ampliflcation as an efiective decrease ?!NR;0 ? 12 kEMk ? in the linewidth
?tot, and associated ampliflcation of the motion.
3.6 Position sensitivity
The Heisenberg uncertainty principle requires that the product of the uncer-
tainties in position ?x and momentum ?p, be ?x?p ? ~2. The minimum of this
condition is satisfled when ?x = ?xZP =
q
~
2m!m and ?p = m!m?xZP. Thus,
to reach the SQL, the total uncertainty in position must be ?xZP. When reach-
ing this limit, we expect the components of the noise spectral density Simpx due to
additive noise and SBAx due to measurement backaction to be equal. For a (angular-
frequency) linewidth ?m and position noise spectral density Sx the absolute position
uncertainty due to this noise is ?x =
q
?m
4 Sx. (Note that we use throughout this
work a convention of single-sided spectral densities.) Thus we have, at the standard
76
quantum limit
?xZP =
r
?m
4 S
totx
=
r
?m
4 (S
impx +SBA
x )
=
r
?m
4 ?2S
impx
Thus at the SQL we expect
Simpx (SQL) = ~m!
m?m
(3.43)
From a position noise Sx we can flnd an energy spectral density SN = m!2mSx
and from this the total energy in the Lorentzian line E = ?m4 SN. Thus at the SQL
we can deflne an equivalent energy of the measurement imprecision
Eimp(SQL) = 14~!m
Thus at the SQL, the imprecision noise raises the level of the measured am-
plitude by an amount equivalent to 14 quantum of mechanical energy. Because we
expect at the SQL for the backaction noise to equal the imprecision noise, the back-
action will also add 14 quantum of mechanical energy. Whereas the energy associated
with imprecision noise adds only to the measured value, the backaction energy, how-
ever, is in fact added to the mechanical resonator itself: the mechanical amplitude
is increased by backaction. Stronger coupling which further diminishes Simpx doesn?t
gain you anything, as it will only increase SBAx . For continuous position measure-
ment, detection of this backaction represents the true signature of quantum-limited
measurement: You can?t measure the motion any more precisely, because the mo-
77
tion you?re trying to measure is being perturbed by the measurement in a manner
dictated by quantum mechanics.
There are two important things to note here. First, there is in principle no
lower limit on Simpx . While the achievement of Simpx = Simpx (SQL) is an important
technological challenge and was an experimental goal in the work reported in this
dissertation, it does not represent a fundamental lower limit on anything. As long
as you continue to increase the coupling between the mechanical resonator and the
detector, and don?t add any extra noise to the system, this source of uncertainty
continues to diminish. The second thing to note is that experimentally, Simpx is
usually pretty easy to determine. It results from \additive noise", i.e. noise which
is added to the measurement by the amplifler or other components of the system,
but which does not drive the mechanical motion. Such noise is generally frequency-
independent over a relatively broad band and can therefore be resolved accurately
by averaging over a wide spectral span. Indeed, the fundamental lower limit possible
for this additive noise is one quantum: one-half quantum of white noise at the mea-
surement frequency, contributed each by vacuum  uctuations and by the amplifler,
if the amplifler is quantum-limited. This lower limit of 12 quantum of additive noise
from the amplifler is derived formally by Caves [71].
The backaction noise SBAx , on the other hand, is typically very di?cult to
measure because it may be seen only in its efiects on the mechanical motion. In
most practical experiments, evidence of such backaction driving will appear as a very
small change in the mechanical motion relative to the thermally-driven amplitude.
An ideal measurement would employ a mechanical resonator in its ground state,
78
but such a system remains yet out of reach. In practice in most experiments the
thermal motion is orders of magnitude greater than the expected magnitude of
quantum backaction. Moreover, in real systems the efiects of thermal heating and
other classical noise sources may mask and mimic the quantum backaction, making
it even more di?cult to distinguish. The magnitude of the backaction is a function of
the coupling strength of the measurement (which we vary in our system by varying
the pump power). Although in quantum mechanical terms Simpx and SBAx are related
by the Heisenberg uncertainty principle (see section 6.8), the classical noise added
after the measurement may raise or lower Simpx without afiecting SBAx . Thus the
measured Simpx may not be very useful in predicting quantum limits on SBAx . If one
wants to use the Heisenberg relations to predict the level of SBAx due to quantum
backaction, one must flrst estimate what the quantum-limited Simpx would be.
An interesting result reporting Simpx < Simpx (SQL) is given by Teufel at al,
in the research group led by K. Lehnert. [72] They reported the use of a pro-
totype ultra-low-noise microwave amplifler to read out the position of a 1.04 MHz
nanomechanical resonator coupled to a 7.49 GHz SMR. The amplifler is a Josephson
parametric amplifler, which operates nonlinearly and may thus in principle surpass
some of the quantum limits on linear ampliflers. Where a quantum-limited lin-
ear amplifler would add 12 quantum of energy at the measurement frequency, and
an ideal nonlinear amplifler is capable of adding zero quanta, this amplifler is re-
ported to add only 1.3 quanta. Considering the unavoidable 12 quantum of noise
contributed by vacuum  uctuations of the microwave fleld, this makes their total
added noise a factor of 3.6 above the shot noise limit. The readout scheme is very
79
similar to that of the Lehnert group?s earlier work (Ref. [17], discussed in section
2.4 of this dissertation), in which the SMR is driven on-resonance and its phase
tracks the motion of the NR. For this very small level of added noise, the coupling
strength that these researchers achieved between their SMR and NR was su?cient
to suppress the position uncertainty Simpx below the level Simpx (SQL), deflned in Eq.
(3.43). By applying a microwave power of > 103 pW to their SMR, they achieved
Sx = (0:80?0:03)SSQLx . In other words, they achieved a position uncertainty due
to additive noise that equaled p0:8? 1p2?xZP. Furthermore, by driving with a blue-
detuned microwave pump, they suppressed the linewidth of their NR su?ciently to
achieve Sx = 0:07SSQLx . (This latter technique is analogous to the parametric ampli-
flcation efiect that we employed to improve our position sensitivity, as discussed in
section 6.6, and similarly has the disadvantage of dramatically increasing the ther-
mal amplitude of the mechanical motion.) With such a low position uncertainty,
there should necessarily be quantum noise backaction driving the NR. For instance,
Naik et al. observed quantum noise backaction (reference [3]) using the shot noise
of a single-electron transistor, even though they did not achieve Sx < SSQLx . In-
deed, for Sx = 0:8SSQLx and a total additive noise 3.6 times the shot noise limit,
the backaction-driven position noise density should be 4:5SSQLx , corresponding to
1.1 added quanta of mechanical energy.
However, the device used by Teufel et al. would have great di?culty distin-
guishing this amount of additional mechanical amplitude, as the mechanical mode
temperature is 130 mK, or 2600 quanta. Lehnert?s research group has also reported
elsewhere observing in similar SMR-NR devices evidence of strong classical back-
80
action (most likely thermal heating) driving the NR for increasing pump powers
([17]). Here they report that the mechanical mode temperature increases with drive
power and that the linewidth also increases from its initial value of ?m = 2??1:7 Hz,
strongly suggestive of thermal heating. Thus it appears very challenging to use their
device to achieve true quantum-limited position measurement by detecting quantum
backaction. The recent results with this device do not include any measurements of
backaction driving the nanoresonator, or estimates of how much classical backaction
is contributed by their ultra-low-noise amplifler and how much may be obscuring
the quantum backaction. Furthermore, as their device had ? = 2? ? 2:88 MHz it
operates in the bad-cavity limit and would be poorly adaptable to backaction evad-
ing measurements. (See section 3.4.3) Nonetheless, the results reported by Teufel
et al. powerfully demonstrate the advantage of low-noise ampliflcation in seeking
quantum limited position measurement.
Calculating the position sensitivity due to additive noise is a matter of consid-
ering how the noise in the measurement mimics the way the actual motion appears
in the measurement. Indeed the actual thermal motion of a mechanical resonator or
the motion of a quantum state will after all appear in the measurement apparatus as
just another noise signal; we can call the added noise a \position noise". In spectral
terms this means that the noise contribution is just the additive white noise within
the noise bandwith of the mechanical resonator. Alternatively it can be thought of
as the ratio of noise background level to measured peak amplitude. [73] Thus at
the standard quantum limit the position noise, averaged over one decay time of the
NR, equals the zero-point motion of the resonator. [27]
81
Clearly the mechanical linewidth ?tot is critical to the measured position sen-
sitivity and for a given level of noise, a longer mechanical decay time, i.e. narrower
linewidth, allows us to more closely approach the SQL. While in many measure-
ments the linewidth is flxed, ?tot = ?NR, in some cases it is dependent on the
pump power, which is also the parameter which is increased in order to increase the
position sensitivity. The position sensitivity of an NR measured while undergoing
backaction-cooling (section 3.1.3) is an interesting case. For motion measured using
a single red-detuned pump, the increase in optical damping with increasing pump
power leads to a rollofi and saturation in the position sensitivity. This situation
may be analyzed following conventions presented by Braginsky and others [55], and
assuming the sideband-resolved limit and that the SMR is pumped at !SMR?!NR.
[74] For the ideal case of a dissipationless SMR in the ideal good-cavity limit, with
no losses between SMR and amplifler, the limiting-value of the position uncertainty
is given by
?x = p?namp + ?nvac ??xZP
where ?namp is the additive amplifler noise, expressed as a number of SMR
quanta, and ?nvac is the noise due to vacuum  uctuations. Because the lowest possible
namp equals 12 for a quantum-limited linear amplifler, and nvac is at least 12, the
position sensitivity attained using a single red pump reaches the standard quantum
limit only in the ideal case in the limit of high pump power, i.e. strong coupling. This
conclusion accords with our other discussions of backaction cooling (section 3.1.3) in
regard to its backaction temperature proportional to ?noptNR. In the good-cavity limit,
82
?noptNR = 0. In other words, when applying a single red pump in a sideband-resolved
device, the quantum back-action corresponds to a near-zero-temperature bath and
will not drive the NR no matter how high the pump power. [73] We will demonstrate
some of this kind of behavior in section 6.2.
83
Chapter 4
Fabrication, Setup and Apparatus
4.1 Fridge setup and internal wiring
In order to approach the goal of placing the mechanical resonator into its quan-
tum ground state, it is necessary to work at very low temperatures. Furthermore,
the need for a very high-Q microwave resonator requires operating well below the
superconducting transition temperature of aluminum, Tc = 1:14 K. Moreover, to
prevent the SMR from being thermally excited, we should operate at a temperature
well below the energy of one microwave quantum at 5 GHz, ~!SMR=kB = 240 mK.
We meet all of these requirements by using an Oxford Kelvinox 400 dilution refrig-
erator, capable of a base temperature below 10 mK. The refrigerator is mounted
to an optical table supported on sand-fllled concrete pillars. For operation, a coun-
terweighted helium dewar is raised from below by a winch and bolted to the table.
Low-temperature thermometry is provided by calibrated RuO resistance thermome-
ters supplied by Oxford and reliable down to 20 mK. An AVS-47B resistance bridge
and TS-530A temperature controller (Picowatt Inc) read the thermometers and pro-
vide temperature control to a heater at the mixing chamber stage.
Measurement wiring for microwave signals within the fridge employs coaxial
cables and additional inline components to meet several requirements: 1) Present a
continuous 50 ohm line impedance to the signals, attenuating the signal only where
84
needed for control of thermal noise. 2) Ofier minimal thermal conductance between
fridge stages so as not to add extra heat loads to the fridge. 3) Be well thermalized
to the fridge (particularly the inner conductor of the coaxial cable) so as not to heat
the sample. 4) Block or attenuate Johnson noise emitted by dissipative components
at room temperature or lower temperatures. Any lossy component or section of
cable at temperature T will radiate as blackbody radiation a white noise density
equal to kBT down the signal lines, which is liable to excite the SMR or thermally
heat it. 5) Shield against interference by signals that could drive the SMR or NR
resonances. Additional wiring for low-frequency signals must meet all these needs
except for a less strict need for impedance matching. The low frequency signals
reach the sample by coupling into the RF lines via a bias tee thermalized at the
mixing chamber stage.
Detailed solutions to these issues have been discussed elsewhere in e.g. the
Ph.D. dissertation by LaHaye [75]. In developing our experiments we went through
several iterations of fridge wiring. For instance, when initially prototyping SMRs
we added several coax lines to enable measuring multiple devices at once, but we
later removed these because they added too much heat load to the fridge. Here we
describe the wiring conflguration used to perform the backaction cooling and evasion
measurements reported in chapters 5 and 6. A wiring diagram appears in flg 4.1
and an annotated photograph appears in flg 4.2. Wiring variations used in earlier
prototyping will be noted in other chapters where needed.
Within the fridge are three signal lines: a microwave drive line, a microwave
return line and a low-frequency drive line. On the drive line, microwave frequency
85
+35
Driveline ReturnLineLowfrequncy
Drive
Microstripthermalizer
Microstripthermalizer
Microstripthermalizer
Circulators
HEMT
PowderFilter
PowderFilter
Microstripthermalizer
Microstripthermalizer
Microstripthermalizer
10dB
6dB
3dB
10dB
6dB
3dB
4K
1K
50mK
15mK
Sample
 
Figure 4.1: Schematic of interior wiring within dilution fridge. (Diagram
prepared by T. Rocheleau.)
86
Figure 4.2: Photograph of Kelvinox 400 dilution fridge with vacuum can
removed, showing internal wiring. Some components indicated in flgure
4.1 are hidden in this photo.
87
blackbody radiation is suppressed using cold NiCr attenuators (Midwest Microwave)
mounted at successive temperature stages, each of a magnitude to attenuate the
Johnson noise emitted from the preceding stage. Microwave signal lines between
300K and 4K are Au-plated, CuNi-inner, CuNi-outer, 50 ? semirigid 0.085" diam-
eter coax cables (Coax Co, Ltd, Japan), which provide similar low thermal con-
ductivity but better microwave transmission as compared to stainless-steel coaxial
cable used elsewhere. [75] On the drive line at stages below 100 mK, and on the
return line between the mixing stage and 1.5 K, microwave signals are carried by
Nb-inner, Nb-outer 50 ? semi-rigid 0.085" coax cables. On both drive and return
lines, segments of 50 ? superconducting Al microstrip on silicon serve to thermalize
the inner conductor of the microwave lines at successive temperature stages. A de-
tailed discussion of this kind of thermalizer appears in the PhD dissertation by M.
LaHaye. [75] T. Rocheleau designed and installed the microstrip thermalizers used
here.
On the return line, measured signals are amplifled by a cryogenic HEMT
amplifler (CITCRYO1-12A, S. Weinreb, Caltech). The builders of this amplifler
tested it before delivery and found a noise temperature of 3.56 ? 1 K and gain of
37 dB at 5 GHz. As discussed in section 5.4, we will use this noise temperature to
calibrate our signal lines in-situ. This amplifler is anchored to 4K and is isolated
from the sample by two cryogenic circulators (Quinstar QCY-050100CM0). Two
circulators in series were found to be necessary to prevent the 3.56 K noise radiating
from the input of the amplifler from exciting the SMR above its ground state. We
verifled this in situ when the fridge was cold, by applying no pump power and
88
observing the emitted spectrum in a range !SMR ??. After averaging the output
signal for a duration su?cient to resolve an emitted power corresponding to less
than 1 quantum stored in the SMR, we observed no resonance, and from this we
conclude that the amplifler noise did not excite the SMR above its ground state.
Considering equations (2.35), (2.33) and (2.53), it is clear that a precise knowl-
edge of the parameters loss and gain are necessary to know the signal levels within
the SMR and to compare the results accurately to theory. While the gains and losses
in wiring external to the fridge could be readily measured, the losses internal to the
fridge are di?cult to ascertain because they vary between room temperature and
low temperatures. Most signiflcantly, the microstrip thermalizers sufier from excess
dissipation at room temperature due to free charge in the Si substrates, and are fully
functional only at temperatures below 1 K where the Al is superconducting. (Our
estimates of their behavior rely on measurements made at 4 K, where the substrate
dissipation is eliminated by freeze-out of the substrate charge.) Nonetheless, based
on measurements of the various components at room temperature and at 4 K, we
estimated loss at 5 GHz to be 48.5 dB from the top of the fridge to the sample and
gain at 5 GHz to be 51 dB from the sample to the measurement point in the control
room. The uncertainties in each of these values are at least 3 dB, which motivated
us to calibrate our signal levels in situ, based on equipartition and on the HEMT
noise temperature. (See sections 5.5 and 5.4.)
It is important that the HEMT gain (37 dB) is great enough that the noise
emitted by the HEMT (3.56 K, amplifled 37 dB) will dominate the input noise of
the room temperature amplifler (Miteq AFS3-04700530-07-8P-4-GW, noise flgure
89
0.63 at 5 GHz, see section 4.2) even after approximately 5 dB attenuation in the
coax cable between the HEMT amp and the top of the fridge. The additional 27.7
dB gain added by the room temperature amplifler enabled our measurement to
remain limited by the HEMT amplifler noise despite further attenuation between
this amplifler and the control room. We verifled this by comparing the input noise of
our spectrum analyzer to the noise from the sample; when monitoring the sample,
the observed noise  oor was ? ?142dBm/Hz while the spectrum analyzer noise
 oor was ? ?154dBm/Hz. It is also important to estimate the loss between the
sample and the HEMT amp, because this cannot be determined in situ from RF
measurement. From measurements of the microstrip thermalizers and circulators at
4 K, we estimate this loss to be 1.5 ? 1 dB.
The low-frequency signal line is intended to carry < 10 MHz signals for direct
driving of the NR. This wiring is CuNi-inner, CuNi-outer 0.012" diameter cable
(Coax Co, Ltd, Japan) and attenuates the signal by a total of about 18 dB at 5.5
MHz. At the mixing stage, a bias-tee (Anritsu K252) couples this signal into the
RF line. On this line, Cu powder fllters at the 1.5 K and mixing stages suppress all
higher frequencies. Design and performance of this type of powder fllter is described
in the PhD dissertation by M. LaHaye. [75]
Studies in the literature have demonstrated that SMRs will exhibit excess
internal dissipation due to magnetic  ux trapped in the superconducting fllm. Early
measurements of our prototype SMRs exhibited excess dissipation. We surmised
that the ambient magnetic fleld of several gauss in the room where the fridge was
located (due to magnetic materials in the concrete pillars and supporting structure)
90
may have contributed to this. To eliminate this potential problem, we installed
a Cryoperm magnetic shield can which bolted to the 50 mK stage of the fridge
in place of the radiation shield can supplied by Oxford. This shield, which has
a magnetic permeability optimized for low temperatures, enclosed all components
beneath that stage. In addition, we wrapped two sheets of 0.010" thick Mu-Metal
shielding around the fridge vacuum can. While we did not measure the resulting
fleld in situ (which would require a magnetometer on the sample stage that could
be read during fridge operation), we estimate based on vendor speciflcations that
the shield reduced the fleld at the sample stage to tens of milligauss.
4.2 Wiring external to fridge
In all of these experiments, great care was taken to limit spurious signals and
interference. The dilution fridge and instrumentation were installed in a shielded
room. RF signal lines into and out of the shielded room were routed through DC-
block fllters. Due to limitations of cable attenuations, we operated the two Agilent
microwave sources and one SR844 lock-in inside the shielded room; all others were
placed outside. All other electronics was located in a separate control room. Pho-
tographs illustrating the inside of the shielded room and control room appear in
flgure 4.3. Every microwave pump signal passed through DC-block fllters before
entering the fridge. All instrumentation ampliflers within the shielded room were
powered from batteries. Circuitry for voltage regulation of the battery power sup-
plies was designed and installed by T. Rocheleau.
91
While wiring within the fridge is flxed during the experiment, the wiring out-
side the fridge was reconflgured for the several difierent kinds of measurements.
Figures 4.4 and 4.5 illustrate the wiring conflgurations needed for the two primary
types of measurement described in this work. Several other variants were needed
for other measurements such as driven response of the NR. A photograph of a por-
tion of the wiring appears in flgure 4.6. Further variations were also employed in
earlier prototyping of SMR-NR devices. For instance, when studying SMRs that
operated in the \bad cavity" regime !NR . ?, we found that the SMR transmit-
ted enough of the pump signal (see e.g. equation (2.35)) to overload the low-noise
room-temperature amplifler. To prevent this, a portion of the pump power was split
from the source, phase-shifted to cancel the undesired signal and injected at the
amp input. This kind of wiring change sometimes had to be done on-the- y during
measurements. Details of particular variants are described as necessary in other
chapters.
Referring to flgures 4.4 and 4.5, the microwave sources used for pump signals
were of type Agilent E8257D, which has a maxiumum output of 25 dBm; when a
third source was needed, we used a Rohde and Schwarz SMA100A, capable of 18
dBm maximum power. For 5 to 10 MHz driving signals on the low-frequency line,
we used a Tektronix AFG3252. Lock-in ampliflers were of type Stanford SR844,
and the spectrum analyzer was an Agilent N9020A. All sources and measurement
devices were synchronized to the 10MHz signal of a SRS FS725 rubidium frequency
standard.
The power-splitters are Pasternack model PE2028; directional couplers are
92
Figure 4.3: (Left) Photograph inside shielded room showing power sup-
ply rack on left and rack containing RF sources and lock-in on right.
(Right) Photograph in control room. Fridge valve panel is on left.
Spectrum analyzer is in lower middle of rack on right.
93
LowNoise
Amp
Isolator
Power
splitter/
combiner
Directional
coupler
50ohm
Phase-noise
fiter
cavities
Drive
line ReturnLine
Bandpass
fiter
microwave
source1
microwave
source2
+25
+28
Noiseinjectionamps
SA
+25+25
Spectrum
analyzer
 
Figure 4.4: Schematic for BAE demonstration or backaction cooling.
During cooling, either noise-injecting ampliflers were shut ofi, or the
entire subcircuit containing directional coupler and noise-injecting am-
pliflers was removed. Drive and return lines join to corresponding points
on fridge. (Diagram prepared by T. Rocheleau.)
94
26dB
detectordiode IQmixer
microwavesource1 microwavesource2 microwavesource3RFsource
20dB
Bandpassfiter
Directionalcoupler
PowerSplitter
Driveline ReturnLineLowfrequncy
Drive
I
Q
RF LO
sig Ref
LockinSRS844
LockinSRS844
sig Ref
 
Figure 4.5: Schematic for demonstrating single-quadrature detection of
driven NR. Signal lines join to corresponding points on fridge. (Diagram
prepared by T. Rocheleau.)
Pasternack model PE-2204-10 and PE-2210-20. The room temperature low noise
amplifler is a Miteq AFS3-04700530-07-8P-4-GW. The bandpass fllter is a Minicir-
cuits VBFZ-5500. The IQ mixer is a Marki Microwave IQ0307LXP; in some wiring
conflgurations this was replaced by a non-IQ-type mixer, Marki TL30007LA. To
inject broadband microwave noise for backaction evasion measurements (flgure 4.4,
and section 6.5) we used two Quinstar OLJ-04122025-J0 ampliflers in series with a
Miteq LCA-0408. The room temperature circulator is a Ditom D3C-4080 and the
diode detector used for tracking the phase of the double microwave pump signal
in BAE measurements (section 6.5.1) was an Eclipse DT4752A3. We used SMA-
terminated MiniCircuits type CBL-6FT-SMSM+ signal cables (in various lengths
from 3 to 6 ft), which proved far superior to other types in suppressing crosstalk of
95
Figure 4.6: Representative photograph of microwave components con-
nected at top of fridge. For scale, the splitter is about 1.5 in wide. (The
wiring conflguration is slightly difierent than those shown in flgures 4.4
and 4.5 and was used in measurements not discussed in this dissertation.)
signals. One practice we tried to maintain was to not disconnect and reconnect the
circuit too often, as the transmission through the connection points could change
very slightly each time. This could be a particular problem with some of the gold-
plated connectors. On repeated use the plating could  ake ofi and become lodged
between the pin and shield of the connector, partially shorting the connector and
adding up to a dB of loss! Stainless-steel connectors proved to be more reliable in
this regard.
96
4.3 Device fabrication
Our fabrication technique builds on procedures our group had developed in
past years. Devices are made on silicon-nitride (SiN)-coated silicon chips. The NR
is formed of a doubly-clamped SiN beam covered by an aluminum fllm; the nitride
acts as the structural element while the aluminum serves as both an electrode and
an etch mask. The mechanical structure is deflned by electron-beam lithography
and freed by undercutting with a dry etch rather than a wet etch, thus avoiding
the stresses of a liquid dip on these delicate structures. The development of the
fabrication procedure is documented extensively in Ph.D. dissertations by Naik [41]
and Truitt [76].
In this recent work, we adapted the procedure to couple the NR to an alu-
minum superconducting microwave resonator rather than a single-electron transistor
as in earlier work. This made for simpler and more versatile fabrication since the
SMR can be deflned entirely by photolithography. We also adapted the procedure
to use high-stress silicon nitride rather than low-stress nitride as in earlier devices,
to exploit the very low dissipations demonstrated by Verbridge et al. in high-stress
nitride resonators [77]. However, early prototype trials indicated that chips coated
with high-stress nitride exhibited unusually high dissipation at microwave frequen-
cies. The amount of dissipation exceeded by at least a factor of 100 the published
values of dissipation in nitride [11], and remains unexplained. To counter this, we
adapted the procedure to remove nearly all of the nitride from the chip except in
the NR itself. These developments required several iterations. The chip size was
97
Figure 4.7: Schematic cross-section view of device fabrication.
also revised during the prototyping process, which accompanied a revision in sam-
ple box design and will be discussed below in section 4.4. I will present here the
flnal procedure (shown schematically in flgure 4.7) that we used to form the device
employed for the measurements in chapters 5 and 6.
All fabrication was performed at the Cornell Nanoscale Fabrication facility,
except for some of the aluminum deposition, which was done in a dedicated Al
deposition chamber maintained by the Schwab group. T. Ndukum did the majority
of the device design and fabrication, and in particular all of the e-beam lithography.
1. Substrates are 100 mm dia, ultra-high resistivity Si wafers: <100> orientation,
500 micrometers thick, with resistivity > 10 k??cm.
2. Low pressure chemical vapor deposition (LPCVD) of 70nm of high-stress SiN,
98
which will form the structural material of the NR.
3. Usingphotolithographyand plasmaetch, place alignmentmarkersonthewafer
to deflne the 3.5mm ? 10mm die pattern.
4. Photolithographically deflne two 2 micron ? 37 micron \patches" where the
nano-resonators are to be located on each chip. Photoresist covers the patches;
the rest of the wafer is exposed.
5. Strip the nitride in all exposed regions, using plasma etching (150W, 60mTorr,
50sccm CHF3/5 sccm O2) followed by a smoothing bufiered oxide (BOE, 6:1)
wet etch. The latter etches the nitride at ?5-10?A/min but does not attack
the underlying Si.
6. Photolithographically deflne the SMR followed by thermal evaporation depo-
sition and lift-ofi of 260 nm of thermally evaporated (99.999% pure) Al.
7. Deflne the NR on top of the SiN patches using double-layer PMMA resist,
electron beam lithography, electron-beam evaporation of 105 nm of Al and
lift-ofi.
8. Dice the wafer into individual chips. Chips should be diced precisely with a
narrow (< 50 micron wide) dicing blade to permit a tight flt into the sample
box, minimizing the gaps that must be bridged for RF conduction.
9. Using e-beam lithography, deflne an \etch window" for plasma etch to free the
NR. This window deflnes the length of the doubly-clamped beam.
99
Figure 4.8: False-colored SEM microphotograph. NR coupled to center-
line and groundplane of coplanar waveguide in SMR.
10. Plasma etch (150W, 60mTorr, 50sccm CHF3/5 sccm O2) to vertically etch the
SiN.
11. Withoutremovingthechipfromtheetchchamber, plasmaetch(100W,125mTorr,
50 sccm SF6) to isotropically remove the silicon underneath the NR
12. Oxygen plasma clean to remove residual e-beam resist.
Figure 2.1 shows a full completed chip and flgure 4.8 shows a false-color angle-
view SEM image of the NR. The flnal NR consists of a 60 nm-thick SiN layer coated
with 105 nm of Al. The 85 micron gap between NR and gate electrode may be
more clearly seen in flgure 4.9. Dimensions of the device were verifled by high-
magniflcation SEM inspection.
To ofier redundancy and the possibility of more measurements per cooldown,
100
Figure 4.9: Top-view SEM microphotograph of NR and gate more clearly
showing ?85 nm gap between them.
two nanoresonators were fabricated onto each chip, both coupled to the SMR. The
lengths of the two NRs were made difierent to produce difierent resonance frequen-
cies. In practice, on cooling down the sample, we usually found that one NR would
be more easily-measurable than the other because its coupling was better or the
other one had broken in handling, and only the intact well-coupled NR was studied.
As discussed in section 3.1.4, making the gate capacitance Cg large is critical
to high-precision detection of the motion or to backaction cooling. In practice,
this means reducing the gap d between NR and gate. The lower practical limit
on d for these devices is about 60 nm. Smaller gaps are di?cult to deflne reliably
by our e-beam lithography and lift-ofi process, and even when they can be made
the NR has a tendency to snap to the gate. Shorter beams, having higher spring
101
Figure 4.10: SEM microphotograph indicating joints between SMR and NR.
constant, will be less likely to snap. Another challenge concerning Cg is that the
aluminum layers forming the SMR and NR are deposited in two separate steps,
and the flrst aluminum surface oxidizes in air in the meantime, thus the contact
points between the two layers can be a problem, indicated in flgure 4.10. The NR
aluminum is only half as thick as the SMR aluminum, but is deposited afterward,
meaning that the contact area may be limited to a narrow insulating joint where
the SMR centerline meets the gate material and another where the NR material
meets the groundplane. If these joints act as capacitors of order ? Cg, they would
capacitively divide the gate capacitance, thus reducing the efiective coupling of
SMR to NR. The capacitance of these structures is di?cult to calculate precisely,
but our evidence from RF measurements of the device behavior suggests that this
does indeed happen. (See section 6.2.) Fabrication of future devices should avoid
102
Figure 4.11: SEM microphotograph showing damage to Al fllm.
this issue by depositing the NR material prior to the SMR material, or some other
way.
Another possible problem concerns the quality of the Al fllm in the SMR.
We would like the SMR to carry as high a current as possible without dissipation;
impurities in the Al could potentially reduce the superconducting critical current.
Magnetic impurities are a particular concern, which motivated us to deposit the
SMR material in a thermal evaporation chamber dedicated only to Al and having
a base pressure of ? 5 ? 10?8 torr. However, other problems may arise in the fab-
rication. Figure 4.11 shows damage that occurred to the fllm probably owing to
organic contaminants left on the wafer surface before SMR Al deposition, which
later outgassed when the device was heated in the plasma etch. This device was
used for the measurements reported in this dissertation; the impact of the damage
103
is not known, but it may possibly account for the poor microwave power handling
seen in this device.
We design the characteristic impedance of the CPW that forms our SMR to
be Z00 = 50 ohms. Because we know the substrate dielectric constant and our
photolithography can reproduce features to better than a micron, by following the
designs described in section 2.3 we expect our device?s waveguide to closely match
this design. More di?cult to design precisely are the end-coupling capacitors of
the SMR, C?, which are made as interdigitated capacitors shown in flgure 2.5. The
capacitance may be estimated from the geometry, and governs the coupling ?ext of
the SMR to the input and output transmission lines as per Eq. (2.25). We would like
for ?int ? ?ext, but the internal loss ?int depends on dissipation in the aluminum
and the substrate (see equation (2.23)) which is not always easy to control. In
this device we designed C? to be 3.0 fF, which would make ?ext = 2? ?141kHz, or
Qext = 3:5?104. In section 5.4 we compare these estimates to values derived from
RF measurements of the device.
4.4 Sample boxes
Proper mounting of the sample is critical to making a precise measurement.
The sample package should transfer all the microwave input power and output sig-
nal to and from the on-chip waveguide without losses or impedance mismatches,
and without exciting microwave modes that bypass the SMR. Figures 4.12 and 4.13
show the 3.5mm ? 10mm ? 500 micron sample chip mounted in our sample box.
104
To minimize impedance mismatches, the coplanar waveguide on the circuit board is
designed to have a .016" wide centerline, closely matching the .010" pin size of the
SMA connectors to which it is soldered. The circuit board is designed to match as
closely as possible to the thickness and dielectric constant of the silicon chip, so that
the microwaves travel as nearly as possible along an unbroken waveguide with no
step changes. The board is made of Arlon AR1000 material, designed to have low
dielectric losses at microwave frequencies, 0.015" thick, ? = 9:7, with 1/2 oz/in Cu
cladding on each side, subsequently gold-plated by the board fabrication vendor as
protection against tarnishing. The waveguide centerline also matches the centerline
of the lead-in portion of waveguide at each end of the chip. At these dimensions, the
waveguide is a hybrid between pure CPW and microstrip, and the dimensions were
calculated using \TXline" software to produce 50 ohm characteristic impedance (as-
suming the manufacturer?s values of the substrate dielectric constant). Throughout
the groundplane, 0.014" dia \via" holes, spaced 0.04" apart, short the top surface
of the circuit board to the grounded sample box. The box itself (designed by T.
Rocheleau) is made of OFHC copper, gold-plated. The chip flts snugly into a hole
in the circuit board and is held flrmly against the gold-plated sample box surface
by two clips. (To remove the chip, it must be forced out from underneath by a
screw.) To ensure proper grounding of the SMR groundplane, a large number of Al
wirebonds are used to connect the Al groundplane of the chip to the groundplane
of the circuit board as well as to connect the two halves of the chip groundplane. It
is important to bond each end of the chip centerline and groundplane to the circuit
board centerline and groundplane with several short bonds, so that the microwaves
105
Figure 4.12: Chip in gold-plated copper sample box. SMA coaxial con-
nector at each end is soldered to coplanar waveguide on circuit board to
transmit signals to/from the chip. Gloved flngers holding box provide
scale.
are launched properly from board to chip and vice versa.
Both box and circuit board are plated with ?100 nm of gold as a protection
against tarnishing, to enable reliable thermal contacts and wire bonding. While
it is a standard practice to underlay gold plating with several microns of nickel
plating, in our case we omit the nickel, to avoid the presence of magnetic material
near the superconductor. Slow difiusion of the copper into the gold is expected to
gradually degrade the gold fllm. However, studies in the literature [78] indicate that
at room temperature a 100 nm Au layer should hold up for at least two years against
grain-boundary difiusion of Cu.
We revised the sample box design twice before arriving at a satisfactory de-
106
 
Figure 4.13: Close-up photo of chip in sample box showing wirebonds.
Figure 4.14: Photograph comparing three sample box designs. Note
smaller chip size and \mode fllling" lid in flnal design.
107
sign. Figure 4.14 shows the three designs. The flrst two types permitted too much
microwave power to bypass the SMR, as shown in flgure 4.15. The flrst type was
designed with interior dimensions as small as possible to suppress free-space mi-
crowave modes, and designed without a circuit board to avoid modes propagating
through the board. In this design the chip was wirebonded directly to the gold-
plated copper sample box and to the center pin of the SMA connector. However,
this was not successful because it was di?cult to make enough bonds for a good RF
connection. The large size (5?14 mm) of the chip designed for this box may also en-
able transmission through modes in its groundplane. A second design incorporated
a circuit board which could be more readily bonded to the chip, but its groundplane
was not su?ciently well-grounded with vias and the waveguide on the board was
not well matched to the chip. The interior of this box was also large enough to
permit propagating waveguide modes as low as 8 GHz. The flnal design (described
earlier in this section) remedied these problems. It was also designed with a \mode
fllling" lid, that presses flrmly against the circuit board everywhere except very near
to the chip. The remaining open cross-section will not support propagating modes
below 30 GHz. Although placing a copper surface close to the chip surface might
potentially increase the dissipation of the SMR, our measurements did not indicate
that the lid caused any such problems.
Verygooddiscussionsofmicrowavesamplepackagingdesignforlow-temperature
experiments may be found in the dissertations by Schuster [79] and Mazin [15].
108
a)
0 1 2 3 4 5 6 7 8 9 10111213
x 109
?120
?100
?80
?60
?40
?20
Frequency (Hz)
Transmission S21 (dB)
 
 
Final Sample Box Design, 3.5 by 10 mm chip2nd Sample Box Design, 5 by 14 mm chip
b)
4.87 4.92 4.97 5.02 5.07
x 109
?70
?60
?50
?40
?30
?20
Frequency (Hz)
Transmission S21 (dB)
 
 
Final Sample Box Design, 3.5 by 10 mm chip2nd Sample Box Design, 5 by 14 mm chip
Figure 4.15: Results of sample box design development. Early designs
held larger-area chip. Here the SMR in the early-type box has a fun-
damental mode at ? 2:5 GHz, the one in the later-type box ? 5 GHz.
Both are prototype designs of SMR only, no NR. Devices were tested in
fridge at T < 50 mK. a)Transmission spectra through device plus signal
lines. Early design transmitted excess \bypass" power through spurious
modes of box and chip. Final design achieved > 40 dB suppression of
all modes except through SMR itself, at frequencies up to about 9 GHz.
b) Same measurement on narrower span showing clearly how microwave
bypass power obscures SMR Lorentzian resonance. Signal line loss is
appx 20 dB at 5 GHz.
109
4.5 Microwave fllter cavities
As described in section 3.3, thermal noise in the signal lines at frequen-
cies around !SMR can excite the SMR resonance, thus leading to backaction driv-
ing of the NR. Our microwave sources (Agilent E8257D and Rohde and Schwarz
SMA100A) produce phase and amplitude noise totaling -145dBc/Hz at a frequency
ofiset ?5.5 MHz from the carrier. In these experiments we used pump strengths
at a frequency of !SMR ? !NR up to 14 dBm at the fridge input, meaning that
the noise power density at !SMR will be up to ?-131dBm/Hz. By comparison, the
Johnson noise emitted into the 50 ohm lines by dissipation at room temperature is
-174 dBm/Hz. The cold attenuators on the fridge (flg 4.1) are intended to suppress
the Johnson noise only.
To suppress the excess noise, we use TE011 cavity fllters, following a design
described in Ref. [80] and Ref. [81]. Figure 4.16 shows design and photographs of
our microwave fllter cavities. These were machined from OFHC copper, then heat
treated at > 400C, then electro-plated with Au (Alfa Aesar, stock number 42307).
To tune the frequency of each cavity, we insert or remove a 4mm diameter quartz
rod mounted on a threaded rod at one end of the cavity. Because the axis of the
cavity is a node of the resonance, this varies the efiective dielectric constant without
appreciably afiecting the quality factor or insertion loss. To increase the conductivity
of the copper and thereby improve the quality factor, we cool the cavities to 77K
by mounting in a sealed probe fllled with He gas and immersed in liquid nitrogen.
At an ofiset of 5.57 MHz from the carrier, we obtain a flltering factor greater
110
a)
 
b)
4.99 5.00 5.01 5.02
-70
-60
-50
-40
-30
-20
-10
S2
1/d
B
f/GHz  
c) d)
Figure 4.16: Filter cavity for suppressing phase noise. a) Design. b)
Transmission of fllter cavity tuned to maximum and minimum frequen-
cies, measured at 77 K. Q ranges from 35373 to 41826. Sharp \dip"
to the right of each peak is due to fortuitous interference with adjacent
mode of cavity. c) Cavity and mounting brackets with quartz tuning
rod. d) Assembled probe ready for cooling to 77 K.
111
than 50 dB. (see flgure 4.16(b)), suppressing the phase noise to?-195 dBc/Hz. Even
when a microwave pump power of +25dBm is applied, the phase noise at the input
of the device should be far below the level of quantum vacuum  uctuations, i.e.
the noise level of one quantum in the SMR (? -205 dBm/Hz). On resonance, the
insertion loss of these fllters was about 7 to 9 dB, thereby limiting the maximum
pump powers that could be applied to the fridge.
To test the efiectiveness of the fllter cavities, we applied a microwave pump
signal at several powers at frequency !SMR?!NR, with and without the use of fllter
cavities inline with the microwave source. These measurements were made through
the SMR used for cooling and backaction evasion in the fridge. Figure 4.17 shows
a wide span plot of the measured spectrum at two pump powers, and flgure 4.18
shows the noise power density at !SMR as a function of pump power. This data
demonstrates that even at the highest pump powers used in those measurements,
when the fllter cavities were employed, the phase noise of the microwave source did
not excite the SMR out of its ground state, and the measurement noise remained
dominated by amplifler noise.
Design, assembly and initial testing of the fllter cavities was carried out by T.
Rocheleau and M. Savva.
112
5.002 5.004 5.006 5.008
-140
-120
-100
-80
-60
No
ise
 P
ow
er 
(dB
m 
/ H
z)
Frequency (GHz)
 20 dBm pump, no filter
 20 dBm pump, with filter
 10 dBm pump, no filter
 10 dBm pump, with filter
 
Figure 4.17: Power measured on spectrum analyzer (10 kHz bandwidth)
while applying a flxed pump tone to the device at two difierent powers,
with and without copper fllter cavity. Pump at !SMR ?!NR is evident
in plots. Excitation of SMR by microwave source phase noise is clearly
seen when fllter is not used.
113
1E-6 1E-5
5.0x10-18
1.0x10-17
1.5x10-17
2.0x10-17
 filtered pump
 unfiltered pump
No
ise
 po
we
r a
t S
MR
 re
so
na
nc
e
fre
qu
en
cy
, 5
.00
68
4 G
Hz
 (W
/H
z)
Pump power at 5.00126826 GHz (W)
 
Figure 4.18: Noise power density Sbgd measured at SMR resonant fre-
quencywhilepumpingat!SMR?!NR. Filtercavitysuppressesexcitation
of SMR by source phase noise.
114
Chapter 5
Measurement Methods
5.1 Characterization of SMR
To characterize the SMR, we use a network analyzer at the drive and return
lines measure to its S21 parameter, i.e. transmission spectrum. A typical measure-
mentappearsinflgure5.1. WeflndtheS21tofollowaLorentzianformasinequation
(2.35), and flt the spectrum to flnd ? and !SMR. Because of the recent experimental
interest in SMRs, there has been much study of the various dissipation mechanisms
in microfabricated superconducting waveguide at millikelvin temperatures. [82] [11]
In the SMR that we used for backaction cooling and evasion experiments, at the
power levels employed in those experiments the quality factor of was approximately
1:014?104, corresponding to a linewidth of 494 kHz. Section 5.4 describes how we
estimate the portions attributable to internal dissipation and external loading, ?int
and ?ext.
High pump powers are essential to achieving high coupling between SMR and
NR for backaction cooling and sensitive position measurement. At su?ciently high
powers, however, the current in the SMR will approach the critical current of the
material; in this regime we expect an increasing population of quasiparticles in the
superconductor to manifest as excess internal RF dissipation in the SMR. Measure-
ments of the total dissipation ? of four SMR devices having two difierent designs of
115
0 2 4 6 8
-120
-100
-80
-60
-40
S2
1 (
dB)
Frequency (GHz)
 Fridge at 1.5K
 Fridge at 20mK
 
Figure 5.1: Wide span transmission spectrum of the sample. When the
SMR is superconducting, the 5 GHz resonance is clearly visible, but
vanishes when the SMR is normal (1.5K).
116
?ext appear in flgure 5.2 as a function of internal circulating power within the SMR.
Above 100 microwatts, we do see a distinct increase in internal dissipation. How-
ever, we flnd that in the sample we chose to use for backaction cooling and evasion
studies, the ? degrades at a power about 20 times lower. The sharp step-change
in ? at PSMR ? 5?W (equivalent to ?nSMR ? 3 ? 108) was accompanied by strong
internal dissipation and thermal heating in the SMR, setting an upper bound on
the power we can employ with this sample. The poorer performance of this device
has not been adequately explained and may be related to the damage to the alu-
minum fllm during fabrication, as described in section 4.3. Unfortunately, the other
samples either were not fabricated with nanomechanical resonators, or the NRs had
unsatisfactory coupling or were damaged on cooldown.
Using a simple model for the superconducting critical current density, Jc =
Hc=?0?L, where ?L is the London penetration depth of ? 20 nm, we flnd for alu-
minum Jc ? 4 ? 1011 A/m2, from which we flnd in our SMR geometry (assuming
the RF current in our SMR penetrates only to the London depth) critical current
Ic ? 100 mA. Using equation (2.39) this is equivalent to a power of ? 100 mW
in our 50-ohm waveguide, much higher than the onset of excess dissipation in our
devices. The higher current densities that may in practice appear at edges or defects
in the superconducting fllm may account for this discrepancy.
We found that the best way to measure dependence of ? on power was to add
a splitter to the input line, and apply a flxed power microwave tone ofi-resonance
from the SMR, while simultaneously using a network analyzer at a much weaker
power to trace out the S21 response. In this way the internal power in the SMR was
117
s49s48
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s32s83s97s109s112s108s101s32s117s115s101s100s32s102s111s114s32s67s111s111s108s105s110s103s32s38s32s66s65s69
s32s80s114s111s116s111s116s121s112s101s32s51s58s32s77s97s121s32s50s48s48s56s32s39s81s50s48s39s32s100s101s115s105s103s110s44s32s119s105s116s104s32s78s82
s32s80s114s111s116s111s116s121s112s101s32s50s58s32s65s112s114s105s108s32s50s48s48s56s32s39s81s53s48s39s32s100s101s115s105s103s110s44s32s110s111s32s78s82
s32s80s114s111s116s111s116s121s112s101s32s49s58s32s65s112s114s105s108s32s50s48s48s56s32s39s81s53s48s39s32s100s101s115s105s103s110s32s111s110s32s83s105s44s32s110s111s32s78s82
s32
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s41
s67s105s114s99s117s108s97s116s105s110s103s32s80s111s119s101s114s32s80
s83s77s82
s32s40s87s41
Figure 5.2: SMR linewidth vs circulating power in SMR. Four difierent
devices, all measured at T < 100 mK. Measurements made by varying
power of network analyzer. Circulating power PSMR was calculated from
equation (2.37) for ?! = 0. Black points are device used in measure-
ments of chapter 6; red points are a difierent device having identical
SMR design. For these two devices, calculation of PSMR employed ?ext
and loss found in section 5.4. Green and blue points are another SMR
design, for which the calculation of PSMR employed values of ?ext esti-
mated from geometry, and estimated linelosses of the particular cables
installed for those measurements.
118
maintained approximately flxed while the network analyzer swept frequencies. If we
instead used the network analyzer to apply high power, then at high pump powers
the ?int dependence on PSMR meant that difierent frequencies sampled difierent ?,
distorting the lineshape. (The measurements appearing in flgure 5.2 were not made
in this way, so at the highest pump powers the ? may be only approximate.)
In developing our device, we measured more than a dozen difierent SMRs,
varying materials and design seeking a design with low internal dissipation. One
remarkable observation we made is that SMRs fabricated on top of high-stress SiN
fllms on Si substrates exhibited severe excess internal dissipation, so much in fact
that no resonance could be observed. We also fabricated chips with identical waveg-
uide that had no breaks, whose transmission could be measured directly, and found
that the dissipation in the ?=2 length of CPW on the chip was? 1:5 dB. This behav-
ior was orders of magnitude greater than the published levels of dissipation in silicon
nitride [11], and remains unexplained. Identical designs made on silicon showed no
such dissipation. At least three SMRs were also fabricated on low-stress SiN fllms
on Si substrates and did not exhibit the severe dissipation. Among other factors
that we investigated were the use of high-resistivity Si substrates, the presence of
small amounts of platinum on the chips (used as lithography alignment marks in
locations far from the CPW gap), and the presence or absence of a superconducting
groundplane on the backside of the chip. None of these were found to dramatically
afiect the quality factor of the SMR.
We expect internal dissipation in the SMR and variation in the SMR fre-
quency to vary with temperature due to RF dissipation and kinetic-inductance of
119
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s72
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s41
s70s114s105s100s103s101s32s84s101s109s112s101s114s97s116s117s114s101s32s40s109s75s41
Figure 5.3: (a) SMR frequency vs. temperature, with linewidth bound-
aries overlaid. (b) SMR linewidth vs. temperature. Fit line is of the
form ? = ?0 +fi?e?fl=T predicted by BCS.
thermally-excited quasiparticles, as predicted by BCS theory. [10] Figure 5.3 shows
temperature dependence of !SMR and ?. The critical temperature of the Al fllm
was measured directly in the separate \transmission" waveguide device described
above, and found to be 1.15 K. However, as this Al fllm was fabricated separately
it may not accurately indicate the Tc of the Al used in the device for bacakction
cooling and measurement.
5.2 Measured and calculated NR characteristics
Fabrication of our nanomechanical beam resonator is described in section 4.3.
This resonator vibrates in tensile mode like a stretched string, with a restoring
120
force contributed by the tension in the material, and a sinusoidal mode shape. The
tension is intrinsic to the nitride fllm due to its growth at temperatures > 800 C
and subsequent difierential thermal contraction of the fllm and substrate, producing
tensile stress in the fllm. To grow our nitride, we used process conditions demon-
strated previously to produce a stress of 1200 MPa and density of 2.7 g/cc. [77] Our
earlier designs [68] employed low-stress silicon nitride which relies on a restoring
force due to the elastic modulus of the nitride. Such bending-mode resonators have
a non-sinusoidal mode shape and higher mechanical dissipation than tensile-mode
resonators. [83] The tensile mode is expected to have a frequency !NR = 2?? n2l
q
S
?A,
where n is the mode number (n = 1 for the fundamental mode), S is the tensile
stress, ? is the density and A is the cross-sectional area of the nitride. We assume
that the aluminum on top of the beam has density 2.7 g/cc and contributes only
added mass per unit length to the beam, not to the restoring force. For l = 30?,
beam width 170 ? 10 nm, SiN thickness 60 ? 5 nm and Al thickness 105 ? 10 nm,
we expect !NR = 2??6:7 MHz. The measured value of 5.57 MHz is somewhat lower
than is likely given our uncertainties in dimensions, but may indicate a lower than
expected stress in the nitride fllm. It is also possible that the undercut of the beam
ends during fabrication adds slightly to the efiective length, or that compressive
stress in the aluminum fllm partially counteracts the tensile stress in the nitride.
We tested several prototype coupled SMR-NR devices before selecting one to
make the backaction cooling and evasion measurements reported in chapter 6. It was
common to observe the NR resonating at two frequencies, typically separated by a
few tens of kHz. The measured amplitude at one frequency was smaller than at the
121
other, suggesting that the two frequencies represent orthogonal vibrational modes x
andy ofthedoubly-clampedbeamcomprisingtheNR.Oscillationy perpendicularto
the gate would have a coe?cient @Cg@y much smaller than for the in-plane x motion. In
subsequent measurements we focused on the motion exhibiting the higher coupling,
and ignored the presumably perpendicular mode.
Figure 5.4 shows measured values of the NR linewidth and frequency as a
function of fridge temperature. The linewidth ?NR exhibits a linear temperature
dependence which enables mechanical quality factors exceeding 106 at temperatures
below 50 mK. However, at temperatures below about 150 mK the linewidth also
appears to be perturbed by a time-varying dissipation which is also apparent in
the measurements of thermal excitation of the NR mode. (See section 5.5.) Since
these linewidths were found from Lorentzian flts of long averages of the NR response
excited by thermal noise, as described in section 5.3, it is also likely that the apparent
value is greater than the actual value of ?NR due to drifts in the frequency over time
scales of minutes. Very small values of ?NR at the lowest temperatures indicate weak
coupling to the thermal bath, and probably contribute to the poor thermalization
of the NR to the fridge temperature in this regime.
The oscillation of the sinusoidally-shaped de ection of the 30 micron long NR
is detected as a change of capacitance Cg deflned by a flxed gate electrode covering
the middle 26 microns of that length. It is convenient to reduce this two-dimensional
oscillatory motion to an equivalent one-dimensional motion that may be described as
a simple harmonic oscillator. Details of this kind of calculation have been described
elsewhere for the case of a bending-mode resonator. [75] If the beam length is
122
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s48 s49s48s48 s50s48s48 s51s48s48 s52s48s48
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s40
s72
s122
s41
s70s114s105s100s103s101s32s84s32s40s109s75s41
Figure 5.4: NR frequency and linewidth plotted against fridge temper-
ature. Three difierent pump conflgurations. Same dataset as flgure 5.6.
Note that the linewidth appears linear with temperature down to ? 150
mK, below which is is believed to be afiected by the same unexplained
force-noise bath that prevents the NR motion from thermalizing to the
fridge at these temperatures. (See section 5.5.) Considering only the dat-
apoints recorded using a single red pump tone, a linear flt of all points
above 150 mK yields a slope of 95 ? 9 Hz / K, passing through the
origin with standard error of 2 Hz. The adjusted R-squared coe?cient
of this flt is 0.89.
123
in dimension z and we denote the de ection in the x direction as u(z), then the
kinetic energy per unit length dz is 12?A
?
du(z)
dt
?2
dz. If u(z;t) = umax sin??zl ?ei!NRt,
where umax is the de ection amplitude at the midpoint of the resonator, then we
may integrate from 0 to l to flnd that the total energy amplitude of the NR is
1
4mtotu
2
max!
2
NR, where mtot = ?Al. However, instead of umax we?d like to re-express
this in terms of the average amplitude hugi over the length of the gate. This is
actually the quantity that we measure as a change in the capacitance Cg. In our
equivalent one-dimensional model we call this amplitude x0. (See e.g. equation
(2.41).) For beam length l = 30 microns, with a 26 micron gate, and mode shape
u(z) = umax sin??zl ?, we flnd hugi = 0:72?umax. We equate the total kinetic energy
calculated above to the potential energy of our equivalent 1-dimensional device,
1
2kx
2
0, where the spring constant k = m!
2
NR is deflned in terms of an \efiective
mass" m. Thus we have 12kx20 = 12m!2NRhugi2 equal to 14mtotu2max!2NR, from which
we flnd efiective mass m = 0:97?mtot. Using our values of the NR dimensions, we
flnd mtot = 2:3?0:3 pg, m = 2:2?0:3 pg and k = 2:70?0:37 N/m.
5.3 Sideband measurement using spectrum analyzer
To detect the motion of the mechanical resonator when pumping at !SMR ?
!NR, we observe the sideband falling at frequency !SMR = 2? ? 5:00684 GHz. As
found from circuit analysis (equations (2.53) and (2.52)), the sideband voltage is
proportional to mechanical amplitude x and total sideband power is proportional
to RMS mechanical amplitude hx2i. To detect the motion we merely measure the
124
sideband voltage or power directly at frequency !SMR. When the motion is driven
by a force noise such as thermal noise, we observe the noise power spectrum on a
spectrum analyzer. A wiring diagram for such a measurement appears in flgure 4.4.
Figure 5.5 shows a typical measured sideband when the NR is driven by thermal
noise. Using Matlab software, we flt the sideband to a Lorentzian function Sbgd +
Psideb? 4?tot?2
tot+4(!?!NR)2
, to determine the linewidth ?tot, background level Sbgd (in units
of W/Hz), mechanical frequency !NR and peak area equaling Psideb (in units of W).
Sbgd is a measure of amplifler noise. In some cases it will also have a contribution
due to excitation of the SMR, which may be driven by phase noise of the microwave
source, thermal excitation of the SMR, or noise deliberately injected to study back-
action evasion.
In addition to measurements of NR motion driven by thermal noise, we can
also drive the NR directly at a single frequency with an electrostatic force through
the \low frequency signal line" installed on the fridge and coupled to the SMR drive
line via a bias tee. (See flgure 4.1.) DC or low-frequency oscillating voltages on the
order of a volt applied at the top of the fridge will be attenuated in the signal line
and be capacitively divided by C?C within the SMR, but still produce voltages on
the order of a mV across Cg, enough to drive the NR at detectable levels. Sweeping
drive frequencies reveals the same Lorentzian lineshape as seen in flgure 5.5. Direct
drive of the motion with a coherent signal at the resonance frequency !NR becomes
important in our demonstration of single-quadrature detection (sections 6.5.1 and
6.5).
The NR may be driven by an electrostatic force 12 @Cg@x V 2 across the gate.
125
5.0068399 5.006840 5.00684010.4
0.6
0.8
1
1.2
1.4
1.6
1.8x 10
-17
Frequency (GHz)
S s
ide
b(
?)
 (W
 / 
Hz
)
 
 
data
fit
1-sigma conf bound
 
Figure 5.5: Typical motional sideband observed at frequency !SMR while
pumping at frequency !SMR ?!NR. Fridge T = 63 mK. Lorentzian flt
to function Sbgd+Psideb? 4?tot?2
tot+4(!?!NR)2
yields linewidth of 10.4 ? 0.2 Hz,
Psideb = 149 ? 3 aW and Sbgd = 6.56 ? 0.01 aW/Hz. This measurement
appears as a datapoint at T = 63 mK in flgure 5.6. Measurements
included in flgure 5.6 resulted from Lorentzian flts with adjusted R-
squared coe?cients ranging from 0.7 to 1.0. R-squared coe?cient in this
flt is 0.87.
126
Applying a voltage V = VDC + Vdrive cos(!NRt) should result in a driving force
@Cg
@x ?VDC ?Vdrive cos(!NRt). However, we found omitting VDC we were equally suc-
cessful in driving the NR. This suggests that at low temperatures the SMR centerline
is completely electrically isolated from ground and has trapped on it enough charge
to carry a DC voltage on the order of a mV. A charge of 10?15 coulombs, or only
about 6000 electrons, on the 1 pF SMR capacitance would produce a voltage of 1
mV. Slight variations or  uctuations in this charge due to leakage currents or other
efiects account for the variation in the drive amplitudes we were able to attain, and
may also contribute to the unexplained  uctuating force-noise that we observed to
drive the NR at low temperatures. (See section 5.5.)
5.4 Estimation of line loss, gain and ?ext
Our ability to compare measurements precisely with theory depends on how
well we know the energy~!SMR?nSMR stored in the SMR at any given applied pump
power. Wemaydetermine ?nSMR fromeithertheappliedortransmittedpumppowers
Pin or Pout using equations (2.33) or (2.34). Considering equations (2.35), (2.33) and
(2.53), we can see that this relies on precision of parameters loss, gain, ? and ?ext.
While ? is a directly measurable quantity, the others are not. In section 5.5 below we
show how we use the equipartition theorem to directly calibrate our measurement
of the mechanical mode temperature. It would be very appealing to replicate this
with the SMR in order to determine gain and ?ext. The thermal occupation ?nTSMR of
the SMR should follow the fridge temperature according to Bose-Einstein statistics,
127
and should emit a frequency spectrum following equation (3.18). This noise should
be detectable as a small temperature-dependent noise peak on top of the 3.56 K
amplifler noise. However, an attempt to detect this was unsuccessful because other
dissipative elements in the fridge wiring also emit Johnson noise at frequencies near
!SMR. Noise emitted by the microstrip thermalizers or circulators between the
SMR and HEMT amplifler partially re ect from the SMR, making it di?cult to
distinguish precisely the microwave noise emitted by the SMR.
To determine gain, we employ the noise emitted by our HEMT amplifler as
a calibrated signal level. The amplifler noise temperature (Tn = 3:56 ? 1 K at 5
GHz) was measured by the vendor at a temperature of 20 K on calibrated equip-
ment before delivery to us. The uncertainty of ?1 K was communicated to us by
the people who designed and built the amp, as an estimate of both systematic vari-
ations in the performance of the amplifler and their uncertainty in measuring the
device before delivery. For the gain calibration, we treat the noise temperature as a
broadband noise power density kBTn at the input of the amplifler, and neglect any
noise generated by dissipative elements between the device and amplifler, because
these are all at temperatures < 1 K. Because the HEMT amplifler noise dominates
the noise of all components beyond it in the circuit, the white noise power density
at the spectrum analyzer should simply equal kBTn ?gainamp, where gainamp is the
total gain from the HEMT amp input to our spectrum analyzer. The total gain,
however, must also include the losses between the sample and amplifler, which we
estimated (see section 4.1) to be 1.5 ? 1 dB. Thus gain = gainamp ?1:5 dB. The
circuits depicted in flgures 4.1 and 4.4 apply here, with microwave pumps shut ofi.
128
We expect gain to vary with frequency due to weak resonances in the lengths of
cable separated by many joints lying between the HEMT amplifler and our measure-
ment point. We also expect slight variations in gain over time, as the temperature
of the cabling in the fridge varied and as we had to disconnect and reconnect com-
ponents in the room-temperature wiring for difierent measurements. We observed
both kinds of efiects causing variations up to about 1 dB. In one early set of mea-
surements we recorded the noise spectral density over a range of frequencies from
!SMR?!NR to !SMR +!NR. From this data we calculated that gain(!SMR?!NR)
is 0.77 dB less than gain(!SMR), and gain(!SMR + !NR) is 0.11 dB greater than
gain(!SMR). To account for time-variation in gain, in each dataset we used the
Sbgd values determined from flts of the motional sideband (section 5.3) while ap-
plying low pump powers, to recalculate gain(!SMR). From this we could estimate
gain(!SMR?!NR) in each dataset as needed. (A better technique might be to shut
ofi the microwave pump periodically during each dataset and measure the noise
levels at gain(!SMR ?!NR) directly.) For the datasets appearing in chapter 6, the
gain values were,
Blue pump, 142 mK (flg 6.1):
gain(!SMR) = (1:1?0:4)?105, gain(!SMR +!NR) = (1:0?0:4)?105
Red pump, 142 mK (flg 6.1):
gain(!SMR) = (8:9?3:0)?104, gain(!SMR ?!NR) = (7:5?2:5)?104
Double pump, 142 mK (flg 6.1):
gain(!SMR) = (9:1?3:1)?104, gain(!SMR ?!NR) = (7:6?2:6)?104
Red pump, 17 mK (flg 6.1):
129
gain(!SMR) = (9:4?3:2)?104, gain(!SMR ?!NR) = (7:9?2:7)?104
Red pump, 20 mK (flg 6.3):
gain(!SMR) = (6:6?2:2)?104, gain(!SMR ?!NR) = (5:6?1:9)?104
These values range from 48 to 50 dB, similar the estimate of 51 dB gain
found from the HEMT amplifler gain speciflcation along with room-temperature
and 4 K measurements of various components (sections 4.1 and 4.2). Once we
know gain(!SMR), we can flnd the coupling ?ext of the SMR to the signal lines,
exploiting our good precision ability to measure the mechanical mode temperature
based on the thermal calibration (section 5.5), as well as our ability to directly
measure the backaction damping ?opt. The mechanical sideband during backaction
cooling measurements can then act as a calibrated power source. As discussed in
section 3.1.3 in reference to equation (3.11), we expect the total rate of photons
?nNR?opt upconverted from the NR into the SMR, multiplied by the SMR energy
per photon, to be the power in the sideband. The total measured sideband power
should then be the portion of this that is emitted by the end-coupling of SMR and
amplifled: Thus Psideb = ~!SMR?nNR?opt (?ext/2?) ? gain. For each set of cooling
data (section 6.2) we determine ?opt at each datapoint by a Lorentzian flt of the
thermal-noise sideband, to flnd ?tot = ?NR +?opt, then subtract the values found at
the lowest pump powers which should be equal to just ?NR. TNR was found from
measured sideband areas scaled by the thermal calibration, and ?nNR found from
TNRkB = ~!NR?nNR. Values of gain are known as described above, and Psideb is
directly measured. We calculated an average value of ?ext for each of the single-
pump datasets appearing in chapter 6 (i.e. the ones listed above), and averaged the
130
four values together to flnd ?ext/? = 0:61?0:24.
Using equation (2.25) we flnd the corresponding value of C? is 4:38?0:84 fF.
This is in somewhat larger than the design value of 3.0 fF, discussed in section 4.3,
but not unreasonable given the di?culty of designing such electrical microstructures
with good precision. We may also readily calculate the internal quality factor of the
SMR (section 2.2.1) to be Qint = (2:6?1:6)?104. Clearly the quality of our SMR
is limited by large internal losses. The uncertainties in these calculated values are
traceable ultimately to the uncertainty in the HEMT amplifler noise temperature
used as a calibration for gain, as well as the ?1 dB uncertainty in the losses between
the device and the amplifler.
If we know in a given measurement both power Pin applied at the input of
the fridge and Pout transmitted, and we know both gain and ?ext, then we can also
calculate loss using equation (2.35). Using the 142 mK red-pump and double-pump
datasets listed above, we flnd an average loss(!SMR?!NR) = (4:3?3:9)?10?5 and
loss(!SMR +!NR) = (4:2?3:8)?10?5. The loss is about 44 dB, somewhat less than
the 48.5 dB estimated from room-temperature measurements of cables and other
components (section 4.1). For the blue-pump dataset at 142 mK (flgure 6.1) and
for the red-pump dataset at 20 mK (flgure 6.3) these values were used to calculate
?nNR, because Pout was not recorded during those datasets.
131
5.5 Thermal calibration
Equations (2.53) and (2.35) enable us to flnd the NR mean-squared amplitude
from either Psideb/Pout or Psideb/Pin. The values of loss, gain, ?ext, @!SMR=@x,
@Cg=@x and C are known a priori to a poor precision, but we may instead exploit
the equipartition theorem to obtain a precise calibration of the mode temperature
of the mechanical motion, TNR. This technique has been discussed extensively
elsewhere in references [75], [41], [4] and [3].
We expect TNR = ~!NRk
B
? ?nTNR to follow the Bose-Einstein distribution for the
average thermal occupation ?nTNR of a single mode:
?nTNR = (e~!NR=kBT ?1)?1
At temperatures T  ~!NRk
B
, this reduces to TNR = T, so the measured NR
mean-squared amplitude should be proportional to the temperature of its thermal
environment, i.e. the fridge. Assuming a single-sided power spectral density, we
expect the power spectrum of thermal motion of the NR to be
SNRx (!) = 4kBTNR!NRmQ
NR
1
(!2 ?!2NR)2 +(!!NR/QNR)2 (5.1)
the integral of which gives R10 SNRx (!)d!/2? = hx2i = kBTNR/k, where k
is the spring constant of the NR, found from the estimated efiective mass m as
k = m!2NR. Thus comparing equations (2.35) and (2.53) we expect
Psideb
Pout =
" 
@!SMR
@x
1
?
?2
?2kBk
#
? gain(!SMR)gain(!
pump)
?TNR (5.2)
Psideb
Pin =
" 
@!SMR
@x
1
?
?2
?2kBk ?loss(!pump)? ?
2
ext
?2 +4(! ?!SMR)2 ?gain(!SMR)
#
TNR
(5.3)
132
The values of Psideb are found from Lorentzian flts of the measured sideband,
as described in section 5.3. Figure 5.6 shows Psideb/Pout, plotted against fridge
temperature, using all of the pump conflgurations employed with the device used
in cooling and BAE measurements (except for single-pump, blue detuned.) While
backaction driving or cooling efiects would make the measurements deviate from
equipartition, in these measurements backaction may be neglected because the SMR
was not excited above its ground state, and the measurements made with a single
red-detuned pump used a weak enough power so that ?opt ? ?NR. At temperatures
above 250 mK, the linewidth ? of the SMR was observed to increase due to the RF
dissipation of thermally excited quasiparticles in the aluminum. The data in flgure
5.6 was therefore corrected at each temperature by multiplying by
?
?(T)
2??494 kHz
?2
.
At temperatures above 60 mK, the response to all pump conflgurations closely
follows the fridge temperature and the measured power agrees closely with what
we expect from the device. In this regime a linear regression flt of all Psideb/Pout
measurements vs fridge temperature yields
Psideb/Pout = (4:20?8:79)?10?12 +(2:78?0:03)?10?9 ?TNR (5.4)
From this we see that the intercept is consistent with zero, and we deflne the
empirical calibration factor cal(Pout) = (2:78?0:03)?10?9 kelvin?1. (A similar anal-
ysis was also done for Psideb/Pin, yielding cal(Pin) = (8:33?0:05)?10?12 kelvin?1.
In the flt of Psideb/Pin vs fridge temperature, due to excess scatter and variations in
line loss or gain it was necessary to set the intercept to zero.)
As can be seen from expressions (5.2) and (5.3), PsidebPout is a much more reliable
133
0 100 200 300 400
0.0
5.0x10-10
1.0x10-9
 Pump ?RED only
 Double Pump ?BLUE = ?RED + 2?NR + 600 Hz 
 Double Pump BAE ?BLUE = ?RED + 2?NR
P s
ide
b 
/ P
ou
t /
 (n
orm
ali
ze
d Q
SM
R2
)
Fridge T (mK)  
Figure 5.6: Thermal calibration data for device used in cooling and BAE
measurements. Three pump conflgurations. Sideband power divided by
transmitted pump power, plotted vs. fridge temperature. The linear
flt excludes measurements below 63 mK where the device appears to be
poorly thermalized. Adjusted R-squared coe?cient of the flt is 0.98.
134
measure of the motion of the NR than PsidebPin , because gain(!SMR)gain(!pump) is unlikely to change
much over time, whereas absolute values of gain(!SMR) and loss(!pump) can vary
up to about a dB between datasets, as described in section 5.4.
Using equation (5.2) and the value of cal(Pout), and accounting for the difier-
ences in gain at !pump and !SMR (see section 5.4), we can calculate @!SMR@x . We flnd
that @!SMR@x = 2? ?(7:5?1:6 kHz=nm). This agrees reasonably well with what we
expect from the calculated value of efiective SMR capacitance C and estimates of
@Cg
@x based on the geometry of the device. Further discussion appears in section 6.2.
It is interesting to note that below 60 mK, the NR appears to decouple from
the thermal bath of the fridge temperature and be strongly coupled instead to
an unidentifled dissipative bath whose properties vary on a time-scale of seconds.
The dissipation in this bath causes the NR quality factor to  uctuate from ? 5 ?
105 to > 106, while the force noise in this bath drives the NR mode temperature
to varying levels up to 7 times the fridge temperature. In past measurements of
nanomechanical resonators it is not unusual to see the motion decouple from the
thermal bath at the lowest fridge temperatures. [4] However, it is novel to observe
the NR actually appear to be \heated" to greater mode temperatures as the fridge
temperature drops. From the observed mode temperature of ?50 to ?150 mK and
linewidths of 5 to 10 Hz (flgures 5.6 and 5.4 at fridge T below 60 mK) we can
estimate the NR to be driven by a white force noise at frequencies around !NR of
3?10?19 to 10?18N
.p
Hz. The nature of this  uctuating force noise is unclear. The
careful flltering we employed in all of the signal lines (section 4.1) suggests that it
is not due to noise or thermal heating entering via the signal lines. As discussed
135
in section 6.2, there is some evidence that microwave pump power can excite it.
This behavior presents an impediment to the goal of cooling the NR motion to its
quantum ground state, and is worth investigating further in future experiments.
136
Chapter 6
Results and Discussion
6.1 Summary of device parameters
Table 6.1: Geometric parameters of device used for backaction cooling and evasion
measurements.
Fabrication geometry, verifled by inspection in SEM:
wCPW 16 ?m Width of Al centerline
dCPW 10 ?m Gap between centerline and ground plane
tCPW 260 nm Thickness of Al fllm on SMR
l 30 ?m length of mechanical resonator
lg 26 ?m length of capacitive gate opposite NR
wNR 170 nm width of mechanical resonator
tSiN 60 nm thickness of SiN
tAl 105 nm thickness of Al fllm on NR
d 85 nm Distance between the NR and gate
Parameters calculated using the geometry:
m (2:2?0:3)?10?15 kg efiective mass of NR
k 2.70 ? 0.37 N/m spring constant of NR
Z00 50 ? Characteristic impedance of CPW waveg-
uide forming SMR
?xZP 26 fm zero-point motion of the NR
6.2 Backaction cooling
We studied backaction damping and cooling of the NR motion by applying
a microwave pump tone at !SMR ? !NR, and varying the pump power up to the
maximum that the SMR could withstand, equivalent to about 3?108 pump quanta
circulating within the SMR. At each pump power, we recorded the noise spectrum of
137
Figure 6.1: NR linewidth, ?tot=2?, and occupation factor, ?nNR, vs. aver-
age number of microwave photons ?nSMR circulating within the SMR. (a)
and (b) show NR behavior for a single pump tone: !red = !SMR ?!NR
(red and orange points) or !blue = !SMR + !NR (blue points). Solid
lines show flts of the data to equations (3.3) and (3.6). Note that in
(b) a flt of the base-temperature points (orange) to equation (3.3) was
not possible due to  uctuating ?nTNR at this temperature. (c) and (d)
show behavior for two simultaneous pump tones: The BAE condition
(purple points) !red = !SMR?!NR and !blue = !SMR+!NR, or the bal-
anced pump, non-BAE condition (black points) !red = !SMR?!NR and
!blue = !SMR +!NR +2??600Hz. The latter condition balances rates of
phonon upconversion and downconversion (note that 600 Hz ? ?=2?),
but ofiers no backaction evasion because the sidebands of the two pro-
cesses do not overlap. In the BAE conflguration (purple points), at high
pump power we observe narrowing of ?tot and mechanical ampliflcation
due to the parametric ampliflcation efiect of the double pump tones hav-
ing difierence frequency 2!NR. Signiflcant scatter in the parametrically-
amplifled datapoints results from unexplained drifts of up to ?5 Hz in
?NR and !NR. Shaded region is inaccessible to BAE due to parametric
instability. Red, blue, black and purple points taken at a fridge temper-
ature of 142mK, orange points taken at a fridge temperature of ?20mK.
138
s53s46s48s48s54s56s51s57s53 s53s46s48s48s54s56s52s48s48 s53s46s48s48s54s56s52s48s53
s49s69s45s50s56
s49s69s45s50s55
s49s69s45s50s54
s83
s120
s32
s40
s109
s50
s32
s47
s32
s72
s122
s41
s102s32s40s71s72s122s41
s32s110
s83s77s82
s32s61s32s49s46s52s51s32s215s32s49s48
s55
s44s32s110
s78s82
s32s61s32s53s55s54s44s32
s116s111s116
s47s50s32s61s32s50s55s46s56s32s72s122
s32s110
s83s77s82
s32s61s32s51s46s50s52s32s215s32s49s48
s55
s44s32s110
s78s82
s32s61s32s52s53s51s44s32
s116s111s116
s47s50s32s61s32s51s54s46s49s32s72s122
s32s110
s83s77s82
s32s61s32s56s46s50s56s32s215s32s49s48
s55
s44s32s110
s78s82
s32s61s32s50s55s48s44s32
s116s111s116
s47s50s32s61s32s52s56s46s51s32s72s122
s32s110
s83s77s82
s32s61s32s49s46s53s50s32s215s32s49s48
s56
s44s32s110
s78s82
s32s61s32s49s57s52s44s32
s116s111s116
s47s50s32s61s32s55s50s32s72s122
s32s110
s83s77s82
s32s61s32s50s46s52s57s32s215s32s49s48
s56
s44s32s110
s78s82
s32s61s32s49s52s57s44s32
s116s111s116
s47s50s32s61s32s49s48s49s46s50s32s72s122
Figure 6.2: NR position noise spectra. Five representative measurements
selected from 142 mK cooling dataset (flg 6.1, red points). For increasing
pump amplitude, sideband peak exhibits broader linewidth and smaller
total area (indicating cooling) and reduced background level (indicating
improved position sensitivity, see section 6.6). Position noise calculated
from measured power spectra via Eq. (6.5). Area of each Lorentzian
equals hx2i for that measurement.
139
Table 6.2: RF parameters of device used for backaction cooling and evasion mea-
surements.
Determined from direct RF measurement:
!SMR 2??5.00684 GHz SMR resonant frequency
? 2??494 kHz SMR damping rate
!NR 2?? 5.5717 MHz NR resonant frequency (varies by ?50 Hz
with temperature, ?10 Hz with random
drift)
Parameters derived from RF measurement:
C 1.0 pF Efiective SMR capacitance
L 1.0 nH Efiective SMR inductance
C? 4.38 ? 0.84 fF Coupling capacitance into and out of SMR
@C
@x 3:0?0:6 aF/nm Change in SMR capacitance for NR dis-placement
@!SMR
@x 2? ?(7:5?1:6)
kHz
nm Coupling constantC
g 253 ? 54 aF Capacitance between the NR and SMR
@2C
@x2 0.06 F/m
2 2nd derivative of capacitance with respect
to NR position
? 2? ?0:15 kHz=nm2 Nonlinear coupling constant
the thermally-driven sideband at !SMR representing NR motion. Each sideband was
then flt to a Lorentzian to flnd the NR frequency, linewidth and mode temperature,
as described in sections 5.3 and 5.5. To prevent phase noise of the microwave source
from exciting the SMR above its ground state, we used a microwave fllter cavity
inline with the microwave source. These measurements were repeated at several
fridge temperatures. As shown in the thermal calibration (section 5.5) at fridge
temperatures below 100 mK the NR was coupled to an additional force-noise bath
of unknown origin that had a time-varying efiective temperature greater than the
fridge temperature. Measurements taken at 142 mK showed much greater stability
and ofier the clearest demonstration of backaction cooling, appearing in panels a)
and b) of flgure 6.1. This dataset was also used to determine position sensitivity
140
of the device, which will be discussed below in section 6.6. Figure 6.1(a) and (b)
also shows backaction cooling data taken at the fridge base temperature. Figure 6.2
shows several of the spectral measurements used to compile the data appearing in
flgure 6.1(a) and (b).
To compare the results with the theory described in section 3.1.3, the NR
linewidth ?tot = ?NR +?opt and occupation number ?nNR are plotted in flgure 6.1(a)
and (b) against the pump energy stored in the SMR, expressed as an average num-
ber of pump photons ?nSMR. For the datasets represented by red, blue, black and
purple points, ?nSMR was calculated using equation (2.34) from the pump power Pout
transmitted through the SMR, ampliflers and signal lines (along with the parame-
ters ?ext, loss and gain determined as described in section 5.4). For the blue points,
Pout was not recorded, so Eq. (2.33) was used to flnd ?nSMR from Pin.
The data in flgure 6.1(a) and (b) may then be flt to the theory represented
by equations (3.6) and (3.3). The measured NR linewidth ?tot is flt to the equation
?tot = (?NR +?opt) = (?NR +fl ? ?nSMR) while the NR occupation ?nNR is flt to
?nNR = ?NR?nTNR?(?NR +fl ? ?nSMR) in a simultaneous least-squares flt, using free
flt parameters ?NR, ?nTNR and fl, using \Origin" software. The fltting routine uses
a Levenberg-Marquardt algorithm. From equation (3.6) we note that parameter
fl represents the coupling between SMR and NR: fl = 4? ??xZP @!SMR@x ?2. For the
dataset taken at 20 mK (orange points in flgure 6.1), ?NR could not be flt, so only
?tot was flt.
Values of ?NR and ?nTNR are set by the environment and may vary from one
dataset to another, but the coupling should be a flxed parameter of the device.
141
Indeed, independent flts of all the datasets shown in flgure 6.1 (a) and (b) as well
as the flt of ?tot measured while the SMR was excited (flgure 6.3, see below) found
values of fl agreeing within uncertainty. We used the average of these, fl = 2? ?
(3:49?10?7 Hz), in flgure 6.1 to produce all of the flt lines. The other flt parameters
were
Red pump, fridge T = 142 mK:
?nTNR = (752?41 ), ?NR = 2? ?(19:1? 3:7 Hz)
(42 points, reduced ?2 = 7:5?103, adjusted R2 = 0:89)
Blue pump, fridge T = 142 mK:
?nTNR = (777?39 ), ?NR = 2? ?(18:5?0:3 Hz)
(34 points, reduced ?2 = 3:5?104, adjusted R2 = 0:96)
Red pump, fridge T = 20 mK:
?NR = 2? ?(15:3? 1:2 Hz)
(linear flt, 17 points, adjusted R2 = 0:97)
Here the uncertainties are the standard errors of the flt values. Given that
the datasets were taken on difierent occasions separated by weeks, these results
represent reasonable flts of the data to equations (3.6) and (3.3) using parameters
that are well within expectation.
We can quantify the maximum achieved cooling rate of ?opt ? 2? ?100 Hz in
terms of a cooling power _Q. As ?opt is the rate at which NR quanta are extracted
by the upconversion process, we have _Q = ?opt ?~!NR = 2:3?10?24 W. The lowest
occupation achieved by the backaction cooling was ?nNR = 58?0:2, achieved at base
temperature. However, the  uctuating ?nTNR at this fridge temperature made this
142
result di?cult to reproduce. This behavior is discussed in section 5.5 as appearing to
indicate that the NR is coupled to an additional time-varying force-noise in addition
to the thermal bath, and that at the lowest fridge temperatures the coupling ?NR
to the thermal bath is weak enough for the additional unidentifled force-noise to
dominate.
At these low fridge temperatures, another efiect that may counteract the cool-
ing is thermal absorption of applied RF power by the SMR and/or the NR. Such
efiects are hard to quantify, but one observation may be made by looking at the
plot of NR frequency vs fridge temperature in flgure 5.4. In this data, the double
pump measurements employed a higher power than the single-pump measurements.
At temperatures above 100 mK, where the NR is well coupled to the thermal bath,
the NR frequency drops slightly with increasing pump power, as expected due to
electrostatic frequency shift (section 3.5). However, at the lowest fridge temper-
atures, the NR frequency increases with pump power, suggesting that the NR is
being thermally heated a few tens of mK by the applied RF power. There is also
some evidence that RF power excites the excess force noise, leading to a hysteresis
in the mode temperature with respect to microwave power levels. In one trial with
the fridge held at 17 mK, we observed the NR mode temperature initially to be
? 40 mK. After the microwave power was cycled to high levels and then reduced,
the mode temperature was > 100 mK.
At a fridge temperature of 142 mK, these poor thermalization efiects and
thermalheatingefiectswerelesssevere. (Seealsoforexamplethediscussionofexcess
backaction in section 6.8.) However the higher starting temperature meant that the
143
lowest occupation achieved through backaction cooling was ?nNR = 149:3?0:2. With
this behavior in mind, future experiments intending to cool the mechanical motion
to its quantum ground state should not anticipate that the mode will thermalize to
temperatures below 100 mK. Instead, other improvements are needed to the device
to enhance the cooling power. Our clearest limitation in this experiment is the
degradation in SMR dissipation for ?nSMR & 3?108. This is most likely due to the
critical current of the SMR being suppressed due to contamination of the aluminum
during the fabrication process. (See section 4.3.) Greater care in fabrication could
distinctly improve the power-handling of the SMR. Further improvements in the
power handling could be made by using a material such as Nb having a higher
critical current. Considering equations (3.6) and (2.40), other enhancements in
cooling power could also be made by reducing the linewidth ? of the SMR (using
smaller capacitances C? at the ends of the SMR), and improving the coupling @!SMR@x .
The latter could be readily improved by reducing the capacitance C of the SMR
by raising the characteristic impedance Z00 of the waveguide. (See section 2.3.)
Improvements to the capacitance derivative @Cg@x could be made by reducing the gap
d between NR and gate, but as this is already only 85 nm, reducing it much further
will be technologically challenging. These tradeofis are further discussed in section
3.1.4.
From the values of the flt parameter fl = 4? ??xZP @!SMR@x ?2, and the estimated
values of ?xZP and measured value of ?, we can derive values of the coupling @!SMR@x .
We flnd @!SMR@x = 2? ?(8:0?2:1 kHz=nm). This agrees well within uncertainty to
the value @!SMR@x = 2??(7:5?1:6 kHz=nm) derived from the temperature calibration
144
(Seesection5.5). Wecanalsousethesevaluestomakeanestimateofthecapacitance
Cg between NR and SMR. Using a value of 7:5?1:6 kHz=nm in the expressions in
equations (2.53), we can use equation (2.40) to flnd @Cg@x = 3:0 ? 0:6 aF/nm, and
from this we use the approximation @Cg@x ? Cgd to estimate Cg = 253?54 aF.
Capacitance of these complicated structures is di?cult to estimate accurately
from geometry. A very crude model is to treat Cg as a parallel-plate capacitor, i.e.
Cg = ?0 ?lg ?tAl=d, which yields a capacitance of 280 aF. This is an underestimate,
because fringing flelds contribute to the capacitance more in our device than in a
parallel-plate geometry. We can also compare to the gate capacitance seen in past
experiments. Naik (Ref. [41]) used an NR of similar geometry whose gate electrode
was a 1 micron long single-electron transistor, across a 100 nm gap. This device had
Cg = 33:6aF. Extrapolating to our much longer gate and narrower gap yields 1030
aF, which should be an overestimate because fringing flelds at the ends of the gate
are less important in a longer device. Nonetheless, these estimates suggest that our
measured value of gate capacitance is about a factor of two smaller than expected.
A probable cause of this discrepancy arises in the fabrication. Since the SMR and
NR are fabricated in two separate steps, they are separated by a joint containing a
thin insulating layer of native Al oxide. (See section 4.3 for fabrication details.) If
the capacitance thereby added in series with Cg is of the same order as Cg, it would
produce the observed discrepancy. Solving this problem would distinctly improve
the coupling and the cooling power. Possible solutions might be to greatly increase
the joint area, to ion-mill before the second deposition to remove the Al oxide, or
to deposit the material for NR and SMR in the same fabrication step.
145
6.3 Backaction cooling when SMR is excited by noise
In section 3.3 we discussed the behavior of backaction cooling when the SMR
is thermally excited. When ?nTSMR > 0, the backaction will no longer appear to the
NR as a zero-temperature bath, but will drive the NR with backaction, setting a
limit on the cooling. In this case we cannot neglect the ?opt?noptNR term in the detailed
balance equation (3.3), but approximately set the thermal occupation of the SMR
?nTSMR equal to ?noptNR. The sideband due to backaction-driven motion of the NR will
add coherently with a 180 degree phase shift to the thermal noise in the SMR, so
that the measured sideband does not accurately indicate the actual NR occupation.
The proper value of ?nNR may be found from the data using equation (3.23).
We studied this behavior by omitting the fllter cavity at the microwave source
during backaction cooling, thus allowing the phase noise of the microwave source to
excite the SMR. These measurements were done at fridge base temperature. Phase
noise power scales with pump power, and at the highest pump powers, the SMR
was excited up to ?nTSMR ? 52. Several measured spectra appear in flgure 6.4, and a
negative-going sideband appears at the highest pump power due to the backaction.
We determine ?neffNR from the sideband area by using the thermal calibration
described in section 5.5 (no difierently than in the case where ?nTSMR = 0). The
occupation ?nTSMR may be calculated from the measured white noise spectrum at
!SMR by flrst subtracting the portion attributable to amplifler noise, and scaling
the result by the system gain to flnd the noise spectral density Sout(!SMR) emitted
by the SMR at resonance. We then have (modifying Eq. (3.19) to account for the
146
s49s48
s49s48s48
s32
s32
s116
s111
s116
s47
s50
s32
s40
s72
s122
s41
s49s48
s54
s49s48
s55
s49s48
s56
s48
s49s48s48
s50s48s48
s51s48s48
s32s110
s101s102s102
s78s82
s32s110
s84
s83s77s82
s32s110
s78s82
s32
s110
s110
s83s77s82
s97s41
s98s41
Figure 6.3: Cooling measurement while SMR is pumped at !SMR?!NR
and also driven by microwave source phase noise. Fridge temperature 20
mK. (a) NR linewidth, ?tot=2?, not afiected by excitation of SMR. Solid
line is a linear flt (8 points, adjusted R2 = 0:99) to Eq. (3.6), as described
in section 6.2. (b) SMR excitation expressed as an equivalent thermal
occupation number ?nTSMR, and apparent NR occupation ?neffNR calculated
directly from measured sideband area using thermal calibration factor.
Correct NR occupation ?nNR is calculated from these by equation (3.23).
Note that at highest pump power ?neffNR is negative. See also flg 6.4.
147
s53s46s48s48s54s56s51s57s54 s53s46s48s48s54s56s51s57s56 s53s46s48s48s54s56s52s48s48 s53s46s48s48s54s56s52s48s50 s53s46s48s48s54s56s52s48s52
s49s48
s45s50s55
s49s48
s45s50s54
s83
s120
s32
s40
s109
s50
s32
s47
s32
s72
s122
s41
s102s32s40s71s72s122s41
s32s110
s83s77s82
s32s61s32s55s46s48s49s32s215s32s49s48
s54
s44s32s110
s101s102s102
s78s82
s32s61s32s51s51s48s44s32
s116s111s116
s47s50s32s61s32s57s46s50s32s72s122s44s32s110
s84
s83s77s82
s32s61s32s48s46s55
s32s110
s83s77s82
s32s61s32s55s46s48s49s32s215s32s49s48
s55
s44s32s110
s101s102s102
s78s82
s32s61s32s53s51s44s32
s116s111s116
s47s50s32s61s32s50s56s32s72s122s44s32s110
s84
s83s77s82
s32s61s32s49s55
s32s110
s83s77s82
s32s61s32s49s46s50s53s32s215s32s49s48
s56
s44s32s110
s101s102s102
s78s82
s32s61s32s49s53s44s32
s116s111s116
s47s50s32s61s32s52s57s32s72s122s44s32s110
s84
s83s77s82
s32s61s32s51s48
s32s110
s83s77s82
s32s61s32s50s46s50s50s32s215s32s49s48
s56
s44s32s110
s101s102s102
s78s82
s32s61s32s45s55s44s32
s116s111s116
s47s50s32s61s32s56s49s32s72s122s44s32s110
s84
s83s77s82
s32s61s32s53s50
Figure 6.4: NR position noise spectra during backaction cooling mea-
surements while microwave source phase noise excites SMR. Selected
from dataset shown in flg 6.3. Position noise calculated from measured
power spectra via Eq. (6.5). At high pump powers, background level
rises due to excitation of SMR. (Compare to measurements at similar
pump magnitude shown in flg 6.2). At highest pump power, backaction-
driven motion of NR produces sideband that subtracts from noise level
in SMR, yielding negative peak.
148
subtraction of amplifler noise)
?nTSMR = 1~!
SMR
? ?2?
ext
? 1gain(!
SMR)
?(SN;out(!SMR)?Samp) (6.1)
The flt values of linewidth ?tot and mode temperature ?neffNR appear in flgure
6.3, along with ?nNR calculated using equation (3.23). The optical damping is not
afiected by the SMR thermal occupation. The flt line in flgure 6.3(a) uses the
same slope as in flgure 6.1 (fl = 2? ?(3:49?10?7 Hz)) and has an intercept ?NR =
2? ?(5:4?1:0)Hz, indicating the very low natural NR linewidths that are possible
at low fridge temperatures.
The excitation of the SMR places limits on the cooling of the NR motion.
The lowest occupation achieved in this dataset was ?nNR = 74 ? 39, much poorer
than trials at the same fridge temperature with phase noise suppressed. This clearly
demonstrates the utility of suppressing phase noise using the microwave fllter cavi-
ties. The correction using Eq. (3.23) also imposes large uncertainties on the values
of ?nNR because of the uncertainties in ?nTSMR traceable to the uncertainty in ampli-
fler noise temperature via the determination of system gain and ?ext. (See section
5.4.) While in this device the excitation of the SMR was deliberately introduced,
in other cases it could conceivably arise from thermal heating within the SMR or
other system components, making it more di?cult to control.
6.4 Optical-spring frequency shift
To examine the behavior of the backaction damping and optical-spring fre-
quency shift, we applied a red-detuned pump tone at a flxed power Pin = ?7:4 dBm
149
into the fridge, varying the frequency ?! = !pump ? !SMR from 2? ??8:19 MHz
to 2? ??3:19 MHz, i.e. from ?1:47!NR to ?0:57!NR. The NR frequency !NR and
linewidth ?tot were then determined from Lorentzian flts of the NR thermal noise
sideband. Measurements were made at a fridge temperature of 145 mK. We were
careful to use a low enough pump power that even at the pump frequency closest
to !SMR, the power circulating inside the SMR was low enough not to degrade ?.
We expect the behavior to be described by equations (3.16) and (3.15).
In these measurements, a fllter cavity was not used to suppress phase noise
from the pump source, because re-tuning the fllter for each new frequency measure-
ment would have been very tedious. In principle, it is possible for backaction driving
of the NR as described in section 3.3 to afiect the measurement. While we would
not expect backaction driving to afiect !NR and ?tot, at detunings ?! > ?!NR and
?! < ?!NR, we would expect the backaction-driven NR signal to exhibit a phase
shift other than 180 degrees, which when added to the noise amplitude emitted by
the SMR would produce a distorted Lorentzian lineshape, which would be di?cult
to analyze. However, in this case the pump power and resulting phase noise was low
enough that ?nNR  ?nTSMR, so we may neglect backaction driving efiects.
The resulting data appears in flgure 6.5. When trying to flt the data to
equations (3.16) and (3.15), it is important to note that for a flxed applied pump
power, ?nSMR varies with ?! according to equation (2.33). The calculated value of
?nSMR appearsinflgure6.5(a)usingparameters?ext andlossdeterminedasdescribed
in section 5.4.
Furthermore, the measurements of !NR will re ect not only the optical-spring
150
a)
?9 ?8 ?7 ?6 ?5 ?4 ?3x 1060
5
10
15x 107
? ? / 2 pi   (Hz)
Pump energy  n
SMR
b)
?9 ?8 ?7 ?6 ?5 ?4 ?3x 1065.5717
5.5717
5.5717
5.5717
5.5717
5.5717x 106
? ? / 2 pi   (Hz)
? NR / 2 
pi   (Hz)
 
 datafit1?? conf bound
c)
?9 ?8 ?7 ?6 ?5 ?4 ?3x 106?20
?15?10
?50
510
? ? / 2 pi   (Hz)
? ? NR
 / 2 pi   (Hz)
 
 dataElectrostatic shift onlyOptical?spring shift only
d)
?9 ?8 ?7 ?6 ?5 ?4 ?3x 10620
30
40
50
60
? ? / 2 pi     (Hz)
? tot / 2 
pi     (Hz)
 
 datafit
1?? conf bound
Figure 6.5: NR frequency and linewidth vs. detuning ?! of pump fre-
quency from SMR frequency, at flxed pump power. (a) Calculation of
pump magnitude within SMR, ?nSMR. (b) Measured NR frequency with
flt to combined optical-spring and electrostatic frequency shifts (equa-
tions (3.15) and (3.41)). (c) Shift from !NR, plotted alongside separate
optical-spring and electrostatic frequency shifts, calculated using flt val-
ues from (b). (d) Measured NR linewidth with flt line (equation (3.16)).
151
frequency shift but also the electrostatic shift due to kEM as given by equation
(3.41). As shown in flgure 6.5(b), using \Matlab" software, we flt the measured
NR frequency against the frequency shift deflned by equation (3.15), plus the shift
deflned byequations (3.41) and (3.38). Nonlinear least-squares fltting was performed
using a Levenberg-Marquardt algorithm, yielding an R-squared value of 0.84. The
data is plotted again in flgure 6.5(c) as a shift from !NR, together with the flts of
optical-spring and electrostatic frequency shifts shown separately. Fit parameters
and their resulting values were !NR = 2? ?(5571741:1?0:7 Hz), ??xZP @!SMR@x ?2 =
4?2 ? (0:074 ? 0:004 Hz2), and ~?=2k = (?1:85 ? 0:20) ? 10?14. From these results
we can calculate
@!SMR
@x = 2? ?(10:3?0:8 kHz/nm)
? = 2? ?(?0:151?0:026 kHz/nm2)
@2C
@x2 = (?0:06?0:01) aF/nm
2
The flt of ?tot to equation (3.15) appears in flgure 6.5(d). In this flt, we use
the flt value of !NR found from the flrst flt. The flt parameter ??xZP @!SMR@x ?2 in
this result is 4?2 ?(0:074?0:004 Hz2), yielding @!SMR@x = 2??(8:85?0:68 kHz/nm).
The R-squared value in this flt was 0.95.
We can compare these values of @!SMR@x to the value 2? ? (7:5 ? 1:6 kHz/nm)
found from thermal calibration (section 5.5). Considering that the data in flgure
6.5 was taken over a 24 hour period in which drifts in !NR and ?NR may have been
several Hz, and that in calculating ?nSMR we could not account for dependence of
loss on frequency, and considering the large uncertainties in loss and ?ext that go
152
into determining ?nSMR, these results seem to be in reasonable agreement with other
measurements for this device (section 6.2).
These results also provide insight into the parametric-ampliflcation efiect that
willarisewhenpumpingthedevicesimultaneouslyatbothfrequencies?! = !SMR?
!NR forBAEmeasurement. GiventhatthenaturalNRlinewidthis?NR ? 15 to 25 Hz
at a temperature of about 140 mK, we see from flgure 6.5(a) and (c) that we can
readily generate electrostatic frequency shifts approaching ?NR at reasonable pump
powers. As described in section 3.5, in the BAE pump conflguration an oscillating
component of the electrostatic frequency shift will lead to ampliflcation of the ther-
mal motion, to degradation of the backaction evasion, and ultimately to uncontrolled
self-oscillation of the NR when the frequency shift exceeds ?NR.
6.5 Backaction-evading single quadrature detection
6.5.1 Demonstration of single quadrature detection
To demonstrate sensitivity to a single quadrature X1 of the NR motion, we
drive the SMR in the BAE conflguration of equal microwave pump tones at frequen-
cies !SMR?!NR and !SMR+!NR. The quadrature X1 is deflned by the phase `beat
of the beat frequency of the RF fleld in the SMR, as described in sections 3.4.2 and
3.4.3,
VSMR(t) = Vp cos(!SMRt)cos(!NRt+`beat)
which is the sum of the two pump tones, each of amplitude Vp2 . The phases
of the two pumps may be known at the input of the signal lines at the top of the
153
fridge; however they will acquire an unknown but flxed phase shift in the signal
lines and the coupling into the SMR. Thus the only way to identify the X1 phase
is to measure it. A circuit for this measurement was shown in flgure 4.5. For this
measurement, !NR was flrst determined precisely by applying pumps at !SMR?!NR
and !SMR+!NR+2??600 Hz and recording the thermal-noise sidebands. (We needed
to do this re-measurement of !NR before each dataset because !NR=2? could drift
5 to 10 Hz from one day to another. The 600 Hz frequency separation enables the
two sidebands to be individually resolved, but being ? ?=2? still enables us to treat
the measured sideband amplitude as if it were at !SMR.) Meanwhile, we drive the
NR with a flxed-phase RF signal at frequency !NR as described in section 5.3. The
sideband voltage appearing at !SMR due to this driven motion is downmixed to
!NR and monitored on a lockin amplifler, using the same RF source as a reference.
Alternatively, the sideband power could be monitored directly on the spectrum
analyzer. The phase of the RF drive was stepped progressively to identify the phase
of maximum response, i.e. X1.
The microwave tone synthesized by the Agilent E8257D exhibited phase drifts
of up to a few tens of degrees per hour, resulting in an equivalent drift of `beat. Since
varying the RF drive phase over 2? typically required half an hour to an hour of
stepping and signal averaging, this could be a problem. To compensate, we extract
a portion of the combined pump signal via a microwave directional coupler and
apply it to an RF diode, as shown in flgure 4.5. The diode bandwidth is  !NR
but ? !SMR so that it registers the envelope of pump power oscillating at twice
the pump beat frequency. An attenuator is used to prevent the pump power from
154
driving the diode nonlinear. The diode output is measured on a lock-in amplifler,
with the same RF source used to drive the NR used here as a reference, making
use of the 2f setting on the reference input of the SR844 lock-in. The measured
phase was divided by 2 and subtracted from the phase of the RF drive. While the
resulting phase value still contains an arbitrary flxed phase shift, it should negate
the efiects of pump phase slippage.
Figure 6.6(a) shows the results of this measurement, clearly showing sensitivity
to a single quadrature of motion, with X1 falling at about +20 degrees. The vertical
scale is the magnitude of the sideband voltage, normalized to its maximum value.
Several repetitions were averaged to make this trace, and the error bars are the stan-
dard error of the averaged values. It is interesting to note that the measurement
does not appear to be entirely insensitive to X2 motion at about -70 degrees. Im-
balance in the two pump amplitudes within the SMR could produce such an efiect,
which we tried to counteract with careful balancing as described in section 6.5.3.
Most likely the small apparent sensitivity to X2 is due to drifts of a few Hz in !NR
during the measurement, meaning that the NR was not driven exactly on resonance,
introducing an additional time-varying phase shift in its response. Nonetheless the
results indicate that the selectivity to X1 is at least a factor of 10. Figure 6.6(b)
shows thermally-driven motion of the NR using the same pump conflguration (the
RF drive is shut ofi to enable detecting the thermal motion), conflrming that intro-
ducing two pump tones to produce the BAE pumping conflguration has no efiect on
the device?s ability to act as a sensitive position detector.
155
a)
s45s49s56s48 s45s57s48 s48 s57s48
s48s46s48
s48s46s50
s48s46s52
s48s46s54
s48s46s56
s49s46s48
s82
s101
s108s97
s116s105
s118
s101
s32s83
s101
s110
s115
s105s116
s105s118
s105s116
s121
s112s104s97s115s101s32s40s100s101s103s41
b)
s53s46s48s48s54s56s51s57s53 s53s46s48s48s54s56s52s48s48 s53s46s48s48s54s56s52s48s53
s54s48
s56s48
s49s48s48
s70s114s101s113s117s101s110s99s121s32s40s71s72s122s41s32
s78
s111
s105s115
s101
s32s80
s111
s119
s101
s114
s32s40
s102s109
s32s47
s32s72
s122
s49
s47s50
s41
Figure 6.6: BAE pump conflguration. Equal pump powers at !red =
!SMR ? !NR and !blue = !SMR + !NR. (a) Demonstration of phase-
sensitive nature of this scheme. While applying BAE pumps, NR is
driven separately with a coherent drive signal at !NR. The phase of
this drive is varied over 2?, while the sideband amplitude at !SMR is
monitored. The response achieves a maximum when the drive excites
the X1 quadrature of NR motion deflned by the coherent pump tones,
and a minimum when X2 is excited. Response amplitude is normalized to
the X1 response. Phase ofiset from 0 degrees is due to flxed phase shifts
in signal lines. (b) Thermal motion of the X1 quadrature of NR mode,
measured at 142mK. Single-quadrature measurement scheme maintains
the same sensitivity to small-amplitude motion. Pump conditions are
same as in flgure 6.7. Each pump power was ? ?2 dBm at the top of
the fridge, for a total occupation of ?nSMR ? 1:2?107 pump photons in
the SMR.
156
6.5.2 Demonstration of backaction evasion
To ensure that backaction driving of the NR is restricted to the X2 quadrature
while only the X1 quadrature is detected, it is necessary for the pump amplitudes
within the SMR at the two frequencies !SMR ? !NR and !SMR + !NR to be of
equal amplitude. While we could attempt to balance the pump amplitudes using
the estimates of loss and gain at the two pump frequencies (see section 5.4), given
the uncertainties in these values we considered it more accurate to balance the
amplitudes using the sidebands themselves, which should be directly proportional to
the pump powers and should both undergo the same gain after being emitted by the
SMR. We applied pump tones at frequencies !SMR?!NR and !SMR+!NR+2??600
Hz. At a fridge temperature of 142 mK, we adjusted the balance of powers in
increments of 0.2 dB, recording the area of the thermally-driven sidebands of the
two pumps, then taking a linear regression to determine the exact pump power
ratio that equalized the sideband areas. These measurements were made with fllter
cavities inline with the microwave sources, so that the precise amplitudes of the
thermal noise sidebands would not be obscured by backaction. We repeated this
procedure at four difierent pump powers over a range of 10 dB, and found that
in all cases the sideband areas balanced when the transmitted pump powers had
the ratio Pout(!SMR + !NR) = Pout(!SMR ? !NR) + 1:1 dB. This is in reasonable
agreement with our separate measurement of 0.88 dB as the difierence in gain at
these frequencies (section 5.4).
Using this balance of applied pump powers, we measured the NR thermal noise
157
over a range of pump powers. In section 5.5 we showed that the thermal calibration
factor is equally applicable to the double-pump conflguration, and this was employed
to calculate the NR occupation ?nNR from Psideb=Pout;redpump, assuming that the blue
pump was balanced properly. Filter cavities were employed at each microwave source
to ensure that no source phase noise excited the SMR, so we should not expect any
backaction driving of the NR. The results are plotted in flgure 6.1(c) and (d). For
the horizontal axis, the occupation of pump photons in the SMR, ?nSMR, is calculated
in the non-BAE case as the power in one pump, in the BAE case as the total power
in both pumps, since in the latter case both pumps contribute to a single measured
sideband. In contrast to single-pump measurements made at the same pump powers
(panels (a) and (b) of the same flgure), no backaction damping or cooling of the NR
mode is observed. The absence of backaction damping in both cases is attributable
to the balancing of upconversion and downconversion processes described in section
3.1.3. It is notable in this data that the mode temperature of the NR is somewhat
higher than the fridge temperature, which we attribute to the unidentifled excess
force noise discussed in section 5.5. At low fridge temperatures that efiect can
dominate the NR amplitude but at these fridge temperatures it adds at most a few
tens of mK to the NR mode temperature. In the BAE conflguration, at the highest
pump powers, ?nNR > 108, we observe linewidth narrowing and ampliflcation of the
thermal noise by the parametric ampliflcation efiect described in section 3.5.
To demonstrate backaction evasion, we must drive the NR with backaction,
and demonstrate that it does not excite the measured X1 quadrature. The wiring
conflguration used for this measurement appears in flgure 4.4. The backaction is
158
conveniently provided by injecting white noise into the SMR at frequencies around
!SMR while applying pump tones in the BAE conflguration after careful measure-
ment of the NR resonant frequency !NR at the same total pump power. The white
noise is generated using a string of three noisy microwave ampliflers, producing a
measured noise power density of -96 dBm/Hz, injected into the drive line at the top
of the fridge through a 10 dB coupler. Noise of this level excites the SMR into a
thermal state with occupation factor ?nTSMR > 104. At the measurement temperature
of 142 mK, the thermally-driven motion of the NR (?nNR ? 600) is then much less
than backaction-driven motion. In the BAE case, thermal motion in the X2 quadra-
ture may be neglected, and in a case where backaction drives both quadratures
we may neglect the thermal motion in both quadratures. In this measurement we
used pump power weak enough that we may neglect parametric ampliflcation. Each
pump applied ??2 dBm at the top of the fridge, resulting in a total occupation of
?nSMR ? 1:2?107 pump photons in the SMR. Injection of noise at the SMR frequency
mimics the action of microwave shot noise which would provide the backaction in
the quantum regime, and follows a similar procedure used by other researchers to
demonstrate BAE [57]. While injecting noise, we recorded the spectrum around
!SMR. The result appears as the purple trace in flgure 6.7. Drifts of a few Hz in
!NR required readjustment of the blue pump frequency, limiting our averaging time.
No motional sideband is distinguishable in the noise spectrum at !SMR. As
described in section 3.4.3 the weak noise amplitude at !SMR ?2!NR can mix with
the pumps to drive the NR via backaction in both X1 and X2 quadratures. This
deviation from ideal behavior will appear as motion in the X1 quadrature and the
159
ratio of the two quadrature amplitudes is given as a ratio of peak noise spectral
densities SX2S
X1
= 32!2NR?2 + 3 (equation (3.35)). In our case we have !NR? = 11:3,
meaning we expect SX2S
X1
= 4 ? 103. Our averaging time is too short to resolve the
nonideal backaction motion or the thermal motion in the X1 quadrature. Instead,
we estimate the maximum possible sideband amplitude consistent with the random
noise observed at !SMR. We take this to be the standard error of the measured noise
power density within the noise bandwidth ?NR4 of the NR. Multiplying this standard
error by the noise bandwidth ?NR4 yields an estimate of the minimum resolvable
sideband power in the BAE measurement, denoted as PBAE. A Lorentzian line
having this area and linewidth ?NR is overlaid in black on flgure 6.7.
To determine SX2S
X1
we would ideally like to directly measure the backaction-
driven motion in the X2 quadrature, recording the full \noise ellipse" of backaction-
driven motion as demonstrated for instance by Marchese et al. [57]. However, the
X2 quadrature is by deflnition invisible to the BAE measurement. In the past,
researchers developing BAE measurements with gravitational-wave antennas were
able to add a separate transducer to their devices for independent measurement of
X2. [54] In our system, one option might be to apply a third microwave tone as a
\probe" signal whose sidebands will be sensitive to both X1 and X2 while the two
pump tones and noise injection are applied simultaneously.
Here we use a simpler technique of measuring the backaction due to a single
pump tone that excites and detects both quadratures, and using this result to calcu-
late the unseen X2 backaction in the double pump BAE case. (For this calculation
I rely on notes generously provided to my by A. Clerk. [84] [58]) After averaging
160
5.0068399 5.0068400 5.0068401
24
26
28
30
No
ise
 P
ow
er 
(fW
/H
z)
Frequency (GHz)  
Figure 6.7: Demonstration of backaction evasion. Red line is motional
sideband observed while pumping at !RED = !SMR?!NR and injecting
noise to the SMR. Lorentzian flt (599 points, reduced ?2 = 5:6?10?32,
adjusted R2 = 0:96, gray shaded area) yields linewidth ?tot = 2? ? (29
? 0.5) Hz and PBA;red = -234 ? 3 fW. Purple line is measured BAE noise
spectrum measured while injecting same amplitude of white noise and
pumping equally at !RED = !SMR ? !NR and !BLUE = !SMR + !NR.
Black shaded area is a Lorentzian with amplitude equal to standard error
of measured noise within span ?NR4 = ?2 ?(24:2?0:8) Hz. Area of this
region is PBAE = 2:46 fW, representing the maximum possible sideband
power consistent with the measured noise. Pump conditions are identical
to those in flgure 6.6.
161
the spectrum in the BAE conflguration for ?1 hour, we shut ofi the blue pump
and recorded the spectrum of the backaction given by a single red pump at the
same power as in the BAE case and with the same level of noise injection. The
backaction-driven single-pump spectrum appears as a red line in flgure 6.7. We
flt the peak area to flnd sideband power PBA;red and linewidth ?tot = ?NR + ?opt.
The peak area is shaded in gray in the flgure. A separate measurement using the
double-pump non-BAE conflguration permits us to measure the natural linewidth
?NR under identical conditions.
The expected noise spectra in the single-pump case with backaction-driven
motion was discussed in section 3.3. Because the SMR \thermal" occupation ?nTSMR
in this case is driven to such high levels by the injected noise, we may ignore the
mechanical thermal-noise term proportional to ?nTNR in equation (3.21), leaving an
expression that should correspond to the red curve in flgure 6.7.
SN;out(?) = gain?~!SMR ? ?ext??
 
1? ?opt(?NR +?tot)4?2 +?2
tot
?
?nTSMR
Integrating to flnd the apparent occupation due to backaction-driven motion
in the single-pump case, corresponding to PBA;red:
?neff;BANR = ?
 
1+ ?NR?
opt +?NR
?
?nTSMR (6.2)
From this expression and the measured value of PBA;red, we would like to esti-
mate the unseen backaction-driven occupation in the X2 quadrature in the double-
pump BAE case, ?nBA. Then this may be compared to the occupation of the X1
quadrature, ?nX1, as determined from the measured value PBAE. A calculation of
162
?nBA has been done in reference [20], assuming that ?nTSMR = 0, i.e. that the exci-
tation of the SMR is limited to vacuum noise equaling 12 microwave quantum. For
large ?nTSMR, that solution for ?nBA may be adapted by multiplying it by 2?nTSMR.
[58] Taking into account that the power in the BAE case is double that in the
single-pump case, ?nBA may then be expressed in terms of the single-pump result
?neff;BANR (equation (6.2)). The ratio SX2S
X1
should equal ?nBA?n
X1
. Expressing in terms of
the measured sideband powers we have [58] [84]
SX2
SX1 = 8
?opt
?NR
?tot
?tot +?NR
jPBA;redj
PBAE (6.3)
For measured values PBA;red = ?234 fW, PBAE = 2:46 fW, ?NR = 2??24 Hz
and ?tot = 2? ? 29 Hz , equation (6.3) yields SX2S
X1
= 82. This represents a lower
bound on the efiectiveness of our backaction evading measurement, limited by aver-
aging duration. The result compares favorably to other published demonstrations of
backaction-evading single quadrature detection, such as in Ref. [57], which achieved
SX2
SX1 = 26. Further limitations on our BAE scheme due to the accompanying para-
metric ampliflcation efiect are discused in section 6.5.3.
6.5.3 Backaction evasion degraded by parametric ampliflcation
In the BAE demonstration in section 6.5.3, the total pump power corresponded
to ?nSMR ? 1:2?107, which we can see from flgure 6.5(a) and (c) should lead to an
oscillating NR frequency shift of only ? 2 Hz. The resulting level of parametric
ampliflcation should degrade the BAE only slightly. To quantify the degradation
in SX2S
X1
to be expected from a given level of parametric ampliflcation, I rely on
163
calculations helpfully provided by A. Clerk. [58] For an electrostatic frequency shift
?!NR = 12 kEMk ?!NR oscillating at frequency 2!NR, he flnds that SX2S
X1
is limited to
SX2
SX1 =
1?(?!NR=?NR)2
(?!NR=?NR)2
For the data in flgure 6.7, given the measured ?NR = 2??24:2 Hz and assuming
?!NR = 2??2 Hz we flnd SX2S
X1
limited by parametric ampliflcation to a factor of 145.
This contrasts with the discussion in section in section 3.4.3 of limits to BAE due to
mixing of noise from frequencies !SMR?2!NR, which identifled a limit SX2S
X1
= 4?103
for our device. This indicates that over nearly the whole regime of pump powers
used in our measurements, our BAE e?ciency is limited by parametric ampliflcation
rather than the more widely understood limits on BAE. This is likely to be the case
foranyattemptatBAEmeasurementsusingcoupledSMR-NRdevices. Nonetheless,
in the measurement shown in section this limit still exceeded our ability to resolve
SX2
SX1 .
It is worth noting that when we applied higher pump powers where the para-
metric ampliflcation should signiflcantly degrade the BAE, and injected noise at
!SMR, the resulting sideband spectrum appeared distinctly non-Lorentzian. Instead
of a \dip" as in the red trace in flgure 6.7, it appeared to be a \dip" for one portion
of the linewidth, and a \peak" for another portion. From this I can surmise that
the mixing of the backaction-driven X1 and X2 by parametric ampliflcation results
in a varying phase shift relative to the noise injected at !SMR. When we shifted
the pump powers apart slightly in frequency, so that we pumped with microwave
tones at !SMR ?!NR and !SMR + !NR + ?, where ? was of the order of ?NR, this
164
slight detuning appeared to suppress the efiect of the parametric ampliflcation on
the BAE, recovering more of a  at noise spectrum. The exact behavior in these
conditions merits further study.
6.6 Position sensitivity
Following the discussion in section 3.6 and conventions presented elsewhere
[55], we would like to calculate the measured position uncertainty and compare
it to quantum limits. For a measurement of thermal motion of the mechanical
resonator, we expect the measured spectral density of motion to be a sum Stotx (!) =
Simpx +SBAx +Sthermx (!). The flrst term re ects measurement uncertainty added by
amplifler noise and vacuum  uctuations in the SMR. In our experiments the noise
contributed by the HEMT amplifler dominates by about a factor of 30 over the the
vacuum, or shot-noise, contribution. The second term re ects quantum backaction
driving of the NR. As discussed in section 3.6, for shot-noise limited detection, at
the standard quantum limit (SQL) the quantum backaction will contribute a mere
1
4 quantum of mechanical energy to the NR motion. However, as we will show
below, our position uncertainty never surpasses about 4 times ?xZP (neglecting the
parametrically-amplifled measurements in which the thermal noise is also amplifled).
Thus our coupling is barely adequate to reach the SQL even if our detection were
shot-noise limited. On the other hand, in all of our measurements, thermal noise
drives our NR to ?nNR > 100. Thus we will ignore the SBAx . We also assume
that no other backaction force drives the NR. In these measurements we ensured
165
that the SMR was not excited above its ground state, so there is no backaction
driving as described in sections 3.3 and 6.3. Here we also neglect other efiects of the
measurement on the NR amplitude, such as thermal heating. Although we saw some
evidence of pump-power-dependent thermal heating at the fridge base temperature
(see flgure 5.4 and discussions in section 6.2), at 142 mK we see no such efiects,
as discussed in section 6.8. In any case themal heating would appear merely as an
increase in Sthermx (!). Thus in calculating the measurement uncertainty we assume
that within the regime in which we are operating, the additive contribution of the
amplifler noise tells the whole story.
The total measured noise spectral density at frequencies near the SMR reso-
nanceshouldbethesumoftwocontributionscorrespondingrespectivelytoSthermx (!)
andSimpx : thenoise due tothemechanicalsideband, Ssideb(!), and aflxed-amplitude,
frequency-independent background noise due to the amplifler, Sbgd.
S(!) = Ssideb(!)+Sbgd (6.4)
The thermal calibrations and expressions for Psideb, equations (2.53) and (5.4),
allow us to readily calculate the position noise spectral density (in units of m2/Hz)
of the measured NR amplitude [4]. We use the constant cal(Pout) determined from
our thermal calibration data (see section 5.5) to relate mechanical amplitude to
the measured power spectrum at frequencies near !NR, divided by the transmitted
pump power Pout. (Note that we use throughout this work a convention of single-
sided spectral densities.) To flnd the thermal excitation of the NR in terms of a
166
noise spectral density we take
Sthermx (!) = (kB/k)(Ssideb(!)/Pout)/cal(Pout) (6.5)
The additive amplifler noise Sbgd contributes an uncertainty Simpx to our mea-
surement of the NR position Sx. Expressed as a position noise spectral density (in
units of m2/Hz):
Simpx = (kB/k)(Sbgd/Pout)/cal(Pout) (6.6)
In flgure 6.2, I show Sx(!) = Simpx +Sthermx (!) found from measured sideband
values Sbgd+Ssideb(!). Figure 6.4 shows the case where excitation of the SMR ?nTSMR
contributes additionally to Sbgd and therefor to Simpx , demonstrating how excitation
of the SMR not only causes the measurement to drive the NR with backaction but
also degrades the position sensitivity.
From Simpx , we can calculate the position uncertainty ?x in our measurement
of the motion. This method is similar to the methods described in past work [75] and
is equivalent to comparing additive noise with the mechanical response amplitude.
[73] For a NR having linewidth ?tot=2?, this is given by the total position noise
attributable to Simpx within the efiective noise bandwidth of the NR:
?x =
r
Simpx ? ?tot4 (6.7)
Measured values of ?x calculated in this way from Sbgd appear for all pump
conflgurations in flgure 6.8. (The dataset is the same as the one appearing in
flgure 6.1.) The values of ?x are expressed in real units as well as in multiples
of the zero point motion ?xZP. From equation (6.6) we expect the precision to
167
s49s48
s54
s49s48
s55
s49s48
s56
s50s48
s52s48
s54s48
s56s48
s49s48s48
s50s48s48
s52s48s48
s54s48s48
s56s48s48
s49s48s48s48
s32
s120
s32
s40
s102
s109
s41
s110
s83s77s82
s48s46s56
s49
s50
s52
s54
s56
s49s48
s50s48
s52s48
s32
s120s32
s47
s32
s120
s90
s80
s112
s97
s114
s97
s109
s101
s116
s114
s105
s99s97
s108
s108
s121s32
s117
s110
s115s116
s97
s98
s108
s101
s32
s105
s110
s32
s66
s65
s69
s32
s99s111
s110
s102
s105
s103
s117
s114
s97
s116
s105
s111
s110
Figure 6.8: Measured position uncertainty ?x vs. SMR occupation.
Same dataset as used in Fig. 6.1. Red-detuned pumping (red points)
!red = !SMR?!NR. Blue-detuned pumping (bluepoints)!blue = !SMR+
!NR. BAE pump condition (purple points) !red = !SMR ? !NR and
!blue = !SMR+!NR. Balanced pump, non-BAE condition (black points)
!red = !SMR?!NR and !blue = !SMR+!NR+2??600Hz. The horizontal
red line shows the limiting sensitivity for a single pump tone. The slanted
black dotted line shows the expected sensitivity proportional to 1p?n
SMR
if
linewidth ?tot is insensitive to pump power and equal to only the natural
linewidth ?NR of the nanoresonator. Shaded region is inaccessible to
BAE due to parametric instability.
168
improve inversely with pump power, i.e. inversely with the coupling of NR to the
readout. Thus we expect ?x to scale inversely with the square root of pump power.
At low pump powers, this behavior is followed by the measurements in all pump
conflgurations. Yet from equation (6.7) we see that the contribution of additive
noise to our position sensitivity is critically dependent on the NR linewidth ?tot.
We flnd that at high pump powers the narrow linewidths possible in the high-stress
SiN nanoresonator enable ?x to approach ?xZP.
The four pump conflgurations therefore present four distinct behaviors of po-
sition sensitivity as power is increased, because of the efiects on linewidth demon-
strated in flgure 6.1. The double-pump, non-BAE conflguration is the most straight-
forward. Because the pumps are balanced, there is no backaction damping, thus the
natural linewidth of the NR is maintained independent of pump power, ?tot = ?NR,
yet the measurement is sensitive to both quadratures of mechanical motion. In
this regard it is similar to the technique of detecting motion by pumping at !SMR,
employed by other researchers. [17] [72] [45] The best sensitivity achieved in this
measurement is roughly 5 ? ?xZP, or ? 7 times the SQL level, at a cavity pump
occupation of 8?107. We would expect that if the product cal(Pout)?Pout could be
raised a factor of 50, the measurement imprecision due to additive noise would reach
the SQL. Given that the additive noise of our amplifler is about 30 times above shot
noise, for such a measurement we would expect shot-noise backaction to add on the
order of 10 mechanical quanta to the NR motion.
In comparison, the BAE pump conflguration similarly preserves ?tot = ?NR,
but this measurement, as described in sections 3.4 and 6.5, will introduce backaction
169
only to quadrature X2 while measuring only X1. Thus the uncertainty is really an
uncertainty in only the measured quadrature, ?X1. The best sensitivity achieved
with backaction-evading measurement is seen in flgure 6.8 to be roughly 4 ? ?xZP
at ?nSMR = 108. At higher pump powers, where the sensitivity of the double-
pump, non-BAE measurement would degrade due to shot-noise backaction, the BAE
measurement should ideally exhibit no such limits on its sensitivity. This level of
?X1 = 4 ? ?xZP is as far as I?m aware the best sensitivity achieved to date in a
BAE measurement. Other published results report uncertainties that are orders
of magnitude greater. [54] It is also very interesting to note that this uncertainty
derives entirely from the noise added by the amplifler, with a noise temperature
of TN = 3:56 K, or 14.8 quanta at the measurement frequency of !SMR = 2? ?
5:00684 GHz. If the amplifler noise were eliminated, we must consider only the 12
quantum, i.e. 12~!SMR = 12kB?240 mK, of noise contributed by vacuum  uctuations
of the microwave fleld. This would reduce ?X1 a factor of 5.4, leading to ?X1 below
?xZP. This demonstrates that our coupling and measurement strength in the BAE
conflguration is capable of generating a conditionally squeezed state, as discussed in
sections 3.4.2 and 3.4.4. To my knowledge this is the flrst time such a measurement
has been shown.
In the BAE conflguration at slightly higher pump powers, however, the para-
metric ampliflcation efiect described in section 3.5 becomes signiflcant. This has
both beneflts and drawbacks. Because the parametric ampliflcation favors a phase
?
4 away from X1 and X2, it will efiectively combine the two quadratures, destroy-
ing the BAE efiect. However, the mechanical preampliflcation adds no noise to
170
the measurement, and also dramatically narrows the NR linewidth, making our
measurement less sensitive to the additive amplifler noise. Because small  uctu-
ations in !NR cause large variations in the parametric ampliflcation, and a slight
increase in the parametric ampliflcation could cause the NR to self-oscillate, this
efiect was challenging to control. However, linewidths below 3 Hz were readily
achievable and as low as 2.1 Hz were possible. In a measurement at SMR occupation
?nSMR = 1:1?108, we found ?tot=2? = 2:1 Hz, reaching our lowest value of position
uncertainty ?x = 1:3??xZP. The parametrically-amplifled thermal noise spectrum
appears in flgure 6.9, along with a Lorentzian flt having linewidth ?tot=2? = 2:1 Hz.
It is important to note that, even though the parametric ampliflcation narrows the
linewidth, reducing the efiect of Simpx , it also amplifles the thermal motion by the
same amount. Thus even though it helps approach the SQL, it makes it more dif-
flcult to observe shot-noise backaction. The parametric ampliflcation thus does not
improve our ability to achieve true quantum-limited measurement. This technique
could, however, be very useful in detecting the thermal noise or the response to
small forces in circumstances where the additive noise dominated the measurement.
The blue pump measurements appearing in flgure 6.8 exhibit a similar behavior
to the parametric ampliflcation efiect, narrowing the linewidth while amplifying the
thermal noise. In this case however, the NR motion is driven to self-oscillating levels
at pump powers too low to reliably achieve ?x as close to ?xZP. The red pump
conflguration presents another interesting case. This type of measurement has been
commonly used in prototype gravitational-wave antennas. [57] [54] The limiting
case for a single-pump measurement is discussed in section 3.6 as reaching ?xZP
171
5.5714 5.5715 5.5716 5.5717 5.5718 5.5719 5.5720 5.5721
1E-17
1E-16
1E-15
1E-14
1E-13
 f (MHz)
S s
ide
b 
(W
/H
z)
 
Figure 6.9: Highest-achieved mechanically-preamplifled thermal noise
of NR. Double-pump conflguration, pumps at !SMR +/- !NR. Fridge
temperature 142 mK, mechanically-amplifled mode temperature 2.26 K,
linewidth 2.1 Hz. (Here pump frequency is subtracted from horizontal
axis so that the response is centered around fNR). The measurement
in this flgure also appears in flg 6.1 (c) as the point with lowest ?tot, in
flg 6.1 (d) as the point with largest ?nNR, and in flg 6.8 as the point of
lowest ?x.
172
only in the limit of inflnitely strong coupling. It is instructive to calculate how
close we expect the measurement to reach and compare it to the measured limits
appearing in flgure 6.8. Combining equations 6.6 and 6.7, and using the deflnition
?xZP =
q
~
2m!NR =
q
~!NR
2k , we have
 ?x
?xZP
?2
= kB~!
NR
? SbgdP
out ?cal(Pout)
? ?tot2
As discussed in section 5.4, Sbgd equals kBTn ?gainamp, where gainamp is the
total gain from the HEMT amp input to our spectrum analyzer, as opposed to the
total gain which also includes the losses between the sample and amplifler. We
can also incorporate the deflnition from equations (5.2) and (5.4) of cal(Pout) =
?@!
SMR
@x
1
?
?2 ?2k
B
k ?
gain(!SMR)
gain(!pump), to flnd
 ?x
?xZP
?2
= k ?kBTn~!
NR
? gainamp(!SMR)gain(!
SMR)
? gain(!pump)P
out
?
 @!
SMR
@x
1
?
??2
? ?tot4
We can reduce this further by expressing the amplifler noise in terms of Tn =
?namp ? ~!SMRk
B
, and by using equation (2.34) to substitute for gain(!pump)Pout :
 ?x
?xZP
?2
= ?namp(?x
ZP)2
? gainamp(!SMR)gain(!
SMR)
? ??
ext
?
 @!
SMR
@x
??2
? ??n
SMR
? ?tot4 (6.8)
In the limiting case of high pump powers, ?opt  ?NR, thus we may take
?tot = ?opt. Using expression (3.6) we have the limiting value
 ?x
?xZP
?2
= ?namp ? gainamp(!SMR)gain(!
SMR)
? ??
ext
(6.9)
Note that the limiting precision is explicitly dependent on the losses between
the sample and amplifler, but for the ideal case of zero losses and a quantum-limited
amplifler ?namp = 12, this expression reduces to the theoretical expression discussed
173
in section 3.6. For our HEMT amplifler having Tn = 3:56 ? 1 K at 5 GHz, we
have ?namp = 14:8 ? 4:2. The loss between sample and amplifler is also estimated
(see section 4.1) to be gainamp(!SMR)gain(!
SMR)
= 1:5?1 dB. From these values, and our best
estimates of ?ext, and expression (6.9) we flnd a limiting value of ?x?x
ZP
= 5:8?1:5.
For comparison, we take the red-pump data in flgure 6.8 and flt it to 1 over the
pump power plus a constant:
?
?x
?xZP
?2
= A?n
SMR
+ B. We flnd an excellent flt to
this expression, with the flt parameter B = 44:4?2:5 yielding a limiting value of
?x
?xZP =
pB = 6:7 ? 0:2. This value is shown in flgure 6.8 as a horizontal red
line. The theoretical and flt values are thus in good agreement within uncertainty.
Or to look at it another way, equating expression (6.9) to the flt value of B yields
?namp = 19:3?8:4, or Tn = 4:6?2:0 K.
6.7 Force sensitivity
We can also determine the force sensitivity of the device, as described in refer-
ence [85], which reports the best published sensitivity of Fmin = 0:8 aN=pHz, with
a closely matching sensitivity also reported in Ref. [3]. For a resonator of mechani-
cal linewidth ?tot, the force sensitivity Fmin represents the driving force that would
produce an RMS mechanical amplitude equal to the apparent measured amplitude
of the NR due to both additive noise and thermally-driven motion of the NR. Or in
other words, what level of force driving the NR would produce a signal-to-noise ra-
tio of one with respect to measurement noise and thermal noise? The measurement
bandwidth is taken to be the noise bandwidth ?NR4 of the NR. Fmin is thus given (in
174
0.01 0.1 1 10
0.00E+000
1.00E-018
2.00E-018
3.00E-018
4.00E-018
5.00E-018
 Double Pump ?BLUE = ?RED + 2?NR (BAE, parametric amp)
 Double Pump ?BLUE = ?RED + 2?NR + 600 Hz
 Pump ?RED only
 Pump ?BLUE only
Se
ns
itiv
ity
 F m
in 
(N
/H
z1/
2 )
Pin (mW)
5.7x10-4 5.7x10-3 5.7x10-2 5.7x10-1
 faction of max power
 
Figure 6.10: Force sensitivity vs. pump power, at fridge temperature of
142 mK. Same dataset as in flgures 6.1 and 6.8.
units of N=pHz) by
Fmin = k!
NR
s
?tot ?
 
Simpx ?tot + 4kBTNRk
?
(6.10)
It is evident from expression (6.10) that if 4kBTNRk  Simpx ?tot then the thermal
noise dominates, and the additive noise may be neglected. It is also evident that
in this limit such processes as either positive or negative optical damping, or para-
metric ampliflcation have negligible efiect on the detectable force, because in the
limiting cases of these processes TNR scales inversely with ?tot. Figure 6.10 shows
the force sensitivity Fmin calculated for measurements at a temperature of 142 mK
for all pump conflgurations. At high pump powers, we have su?cient sensitivity
that in equation (6.10) the thermal noise of the NR dominates over Simpx . At this
175
0 100 200 300 400
0
1x10-18
2x10-18
3x10-18
4x10-18
Pump ?RED only
Double Pump ?BLUE = ?RED + 2?NR + 600 Hz 
Double Pump BAE ?BLUE = ?RED + 2?NR
Se
ns
itiv
ity
 F mi
n 
(N
/H
z1/
2 )
Fridge T (mK)
 
Figure 6.11: Force sensitivity vs. fridge temperature. Same dataset as
thermal calibration (flgure 5.6).
temperature we reliably achieve a force sensitivity of 1:7 ? 10?18N
.p
Hz. Figure
6.11 shows force sensitivity vs fridge temperature, calculated using equation (6.10)
from the same dataset used in our thermal calibration (flgure 5.6). At the lowest
temperature at which the sample thermalized to the fridge (63 mK), we reliably
achieved sensitivity of 8?10?19N
.p
Hz at high pump powers.
At our lowest fridge temperature of ?17mK, we occasionally observed a NR
mode temperature of 46mK and linewidth of 8 Hz, yielding a force sensitivity of
6?10?19N
.p
Hz. Each of our measurements of TNR is made simultaneously with
a measurement of ?tot by measuring and fltting the thermal noise of the NR, which
may tend to overestimate ?tot because of drifts in the NR frequency during the
176
averaging. Another way to consider it is to measure ?tot separately using a more
rapid measurement under the same conditions. This was the technique used by
Mamin et al. [85]. We have made separate measurements of ?tot using a time-domain
ring-down technique, and found ?tot = 2??1:9 Hz at 50 mK and ?tot = 2??3:2 Hz
at 75 mK. Taking TNR = 60 mK and ?tot = 2??3 Hz and assuming su?cient pump
power to overcome Simpx we flnd Fmin = 0:4 aN=pHz, an improvement on the best
published results.
6.8 Approach to quantum limits on Sx ?SF
The Heisenberg uncertainty principle ?x?p ? ~2 may be applied to the act of
position measurement as ?xmeas?pperturb ? ~2, where ?xmeas is the uncertainty in
the measurement and ?pperturb is the change in the momentum due to measurement
backaction. [55] This relation in turn may be re-expressed as a limit on the product
of the measurement precision Sx, and the resulting back-action force noise SF. For
single-sided noise spectral densities, the limit is SxSF ?~2 [55]. At su?ciently high
coupling strengths of the measuring system to the measured mechanical motion, the
shot noise of the microwave pump fleld will generate the back-action force noise SF to
enforce this limit. In our system, the coupling is most easily increased by increasing
the pump power. The measurement imprecision decreases with increasing pump
power as shown in equation (6.6). In our system, in the range of accessible pump
powers, we expect the shot-noise SF to drive the NR only very weakly, only to
amplitudes barely approaching one quantum of NR energy. In practice, however,
177
classical efiects may produce force noise stronger than the shot-noise SF. Any
such efiects that also increase with increasing pump power will mimic and obscure
the quantum back-action. In our measurements we avoid one important efiect by
making sure that the SMR is not excited above its ground state ?nTSMR = 0; thus
the classical backaction noise driving (described in sections 3.3 and 3.4.3) may be
neglected. However, other classical efiects such as thermal heating of the NR can
also drive the NR to increasing amplitudes as pump power is increased. We treat
this \classical backaction" as adding to the backaction force noise SF, and want to
set limits on the magnitude of this force. We may then estimate how closely such
classical backaction would permit us to observe quantum-mechanical contributions
to SF.
To distinguish the classical contribution to SF, we look for evidence of a white
force noise which drives the mechanics to a mode temperature TNR(BA) additional
to the average thermal bath temperature. To be a signature of backaction, TNR(BA)
should also increase with increasing pump power. As a force F at the resonant
frequency will drive a mechanical resonator to an amplitude FQ=k, a force noise SF
drives a resonator of linewidth ? to mean-squared motion hx2i = SF ??Qk?2 ? ?4, and
therefore we express
TNR = TNR(therm) +TNR(BA)
= TNR(therm) + SF(Ppump)4mk
B?NR
(6.11)
where SF has units of N2/Hz. Note that if we express TNR(BA) =~!NR ? ?nBA
and use equation (3.10) to express Sx in terms of an efiective imprecision occupation
178
1E-27
1E-26
 
 
 S
x  
ca
lcu
lat
ed
 fro
m 
S b
gd
 
(m
2  
/ H
z)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
150
200
250
300
350
400
 
 Fraction of Maximum Pump Power
Mo
de
 T
(m
K)
 
Figure 6.12: Position noise due to additive amplifler noise, and mode
temperature, plotted against pump power, when pumping with two tones
at !SMR ? !NR and !SMR + !NR + 2? ? 600Hz. Data in lower panel
also appears as black points in flgure 6.1 (d). Red line is a linear flt
to NR mode T as a function of pump power, consistent with no excess
backaction in the measurement.
179
Sx = 4?
NR
?2?nimp(?xZP)2, we have SxSF = 16~2??nimp??nBA, thus by the uncertainty
relation we must have p?nimp ? ?nBA ? 14. The increase in mode temperature in
equation (6.11) is most easily distinguished when TNR is otherwise expected to be
independent of pump power. Naik et al. [3] identifled TNR(BA) and therefore SF for
a superconducting SET coupled to a NR, by reducing TNR(therm) to negligible levels
by lowering their fridge temperature. Regal et al. [17] did something similar with
a coupled SMR-NR system, pumping their SMR on-resonance to avoid backaction
damping efiects while operating at the lowest fridge temperature and examining
the dependence of their NR mode temperature on microwave pump power. Here we
wish to avoid the behavior at our lowest fridge temperatures in which an unidentifled
non-thermal force-noise drives the NR to amplitudes far in excess of thermal noise
(section 5.5). Instead we use Eq. (6.11) to estimate SF as a function of pump power
based on measurements at 142 mK.
To avoid any backaction cooling or parametric ampliflcation efiects, we apply
equal microwave pumps at !SMR ?!NR and !SMR +!NR +2??600Hz. This data
is included in flgure 6.1 (d) and also appears in flgure 6.12, with a linear regression
flt line attempting to determine a trend in the mode temperature as a function
of pump power. The flt yields intercept 187 ? 12 mK, slope 40:4 ? 58:7 mK per
normalized power of 1, with adjusted R2 = ?0:02. Thus at the highest pump
power in this dataset, the flt flnds TNR(BA) = 14?21mK. The large uncertainty in
the flt means that the data is consistent with no backaction driving of the NR at
all. Nonetheless, we can employ the flt value to conservatively estimate an upper
bound on classical backaction contributions to SF. From TNR(BA) = 14mK we flnd
180
SF = 5:3 ? 10?37N2/Hz. At the same pump power, the additive amplifler noise
yields position uncertainty Sx = Simpx = 6:7 ? 10?28m2/Hz. From this we flnd
pS
xSF = 9:2?10?33J?s, or ? 90~. This represents an upper bound on our approach
to the quantum limit SxSF ? ~2. This calculation of SxSF pertains only to this
particular pump conflguration, i.e. when the SMR is driven with both red and blue
microwave pumps so as to balance up- and down-conversion but without BAE.
However, this estimate of the classical contribution to SF should be relevant
to any pump conflguration, because it is presumed to result from thermal heating
or other parasitic processes that depend directly on pump power. Further evidence
for this conclusion is seen in the backaction cooling data (red and blue points in
flgure 6.1 (a) and (b)) which flts well to the theory based on optical damping and
detailed balance equation (equations (3.3) and (3.6)). Excess thermal heating of the
NR would likely cause the flt of ?nNR to deviate from theory.
181
Chapter 7
Conclusions
The work presented in this dissertation represents a flrst demonstration in our
research group of high-precision position measurement and backaction cooling in
a coupled SMR-NR system. The close agreement of the backaction cooling with
theory (flgure 6.1 (a) and (b)) shows that the system is well understood. More-
over, we have for the flrst time demonstrated backaction-evading single-quadrature
detection in a nanomechanical system. With this technique we demonstrated po-
sition uncertainty only 4 times above the zero-point motion of the NR. Because
this uncertainty is contributed largely by the additive noise of our amplifler, our
device actually achieved coupling strengths that could permit the generation of a
conditionally squeezed state. Ultimately our ability to achieve lower position un-
certainties in continuous position measurement, and our ability to perform better
backaction cooling, is limited by the poor power-handling ability of the SMR sample
that we used (flgure 5.2). This problem is readily amenable to engineering solutions
involving improved device fabrication. Our ability to achieve better position sensi-
tivity in the BAE measurement is limited by the parametric ampliflcation behavior
arising in this conflguration. This limitation has not previously been explored in
the experimental literature.
Since the measurements in this dissertation were completed, the research group
182
has advanced rapidly to even more exciting results. Building on the techniques
proven in the work presented here, we made four major improvements to the device:
1) fabrication of the SMR from Nb rather than Al, for improved power handling,
2) preventing the capacitive-division between gate and SMR (section 4.3 and flgure
4.10), for improved coupling, 3) raising the SMR frequency to 7.5 GHz, and 4)
increasing the impedance Z00 of the CPW forming the SMR, in order to reduce its
capacitance, for improved coupling (equation (2.40)).
The results were just as we expected, achieving a dramatically improved cou-
pling of @!SMR@x = 2? ? 84 kHz / nm, and backaction cooling to ?nNR < 4 [86]. Our
understanding of backaction-driving of the NR during red pump measurements, de-
scribed in this dissertation (sections 3.3 and 6.3), was essential to the later work
because the later device exhibited excitation of the SMR during the cooling mea-
surements. Work is ongoing to improve the devices yet further. With modest
improvements in coupling and the elimination of the unwanted excitation of the
SMR, it appears that cooling to the ground state, demonstrating additive noise
uncertainties below the SQL, single-quadrature backaction-evading measurements
with uncertainty below zero-point motion, and generation of squeezed states, are all
within reach.
I should also mention here a related project that I participated in actively
for several years, but which is not discussed in this dissertation. The group led by
Markus Asplemeyer at the Institute for Quantum Optics and Quantum Information
in Vienna, Austria, has been studying backaction cooling and related phenomena
in micromechanical resonators coupled to optical cavities - the identical concept to
183
the system appearing in this dissertation, but in a difierent regime, with difierent
techniques [38] [87] [42]. The optical system ofiers advantages and disadvantages
over a microwave system. Lasers and photodiodes have far lower noise than sources
and ampliflers in the microwave regime. The optical cavity is not susceptible to
thermal occupation as the SMR is, but it is a free space cavity many thousands
of wavelengths long, which requires complicated alignment and locking techniques.
Although the optical system allows easy prototyping at room temperature, these
experiments are just beginning to be done at low temperatures and not yet at
millikelvin temperatures. I assisted with design and fabrication of the flrst and
second generations of device used in these measurements. The challenge was to make
> 1 MHz mechanical resonators having both high Q and high optical re ectivity.
This work recently succeeded in using radiation-pressure optical damping to cool
the motion of a 100?m ? 50?m ? 1?m, 945 kHz mechanical resonator to a factor
of 32 above the quantum ground state, to date the best published result of such a
technique [42]. In terms of the possibilities for demonstrating quantum behavior,
this optical system is as promising as the microwave and nanomechanics system,
and I am pleased to have contributed to its development.
184
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