ABSTRACT
Title of dissertation: EXPERIMENTS IN
TWO-MODE CAVITY QED
David Glenn Norris, Doctor of Philosophy, 2011
Dissertation directed by: Professor Luis Orozco
Department of Physics
Cavity quantum electrodynamics (QED) allows the study of light-matter in-
teractions at the most basic level, through precise identi cation of the coherent and
incoherent (dissipative) parts of the system evolution. We present measurements of
light from a cavity QED system consisting of a high- nesse optical resonator coupled
to a beam of cold Rb atoms. The novelty of the design lies in the interplay of two
degenerate and orthogonal polarization modes. One mode (driven) behaves as the
canonical cavity mode of the Jaynes-Cummings Hamiltonian, coherently exciting
the atoms with a modest coupling strength; the other mode (undriven) collects a
small fraction of spontaneously emitted light and provides a probe of the dissipative
processes.
We  rst demonstrate the ability to detect individual atoms passing through
the cavity modes in real time by coincidence detection of photons from the un-
driven mode. Calculation of statistics and correlation functions from the complete
photon detection record allows the determination of detection probabilities and the
reconstruction of atomic trajectories. We next present evidence of quantum coher-
ence that is created, modi ed, and measured in the excitation-spontaneous emission
cycle. The coherence appears as a long-lived quantum beat at the ground-state
Larmor frequency, visible in the intensity autocorrelation function of the undriven
mode. Quantum jumps of the atomic state, occurring in between the detections of
photons from the cavity, result in substantial changes in the frequency and spectral
width of the beats. We present the results of a full quantum Monte Carlo calculation
in order to quantitatively explain the measurements.
EXPERIMENTS IN
TWO-MODE CAVITY QED
by
David Glenn Norris
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial ful llment
of the requirements for the degree of
Doctor of Philosophy
2011
Advisory Committee:
Professor Luis A. Orozco, Chair/Advisor
Professor Howard J. Carmichael
Professor Mario Dagenais
Professor William D. Phillips
Professor Steven L. Rolston
c Copyright by
David Glenn Norris
2011
Acknowledgments
The work presented in this thesis is the product of six years of collaborative
e ort. The largest debt of gratitude is owed to my advisor and mentor, Professor
Luis Orozco, who has most greatly shaped my development as a physicist. I know
of no one more committed to science and to inculcating a respect for science in
his students. Another great deal is owed to Professor Howard Carmichael, whose
year with us proved crucial to understanding the physics involved in this work. The
instruction happened chie y over co ee, and I am privileged to count such a brilliant
person as teacher and friend.
Many others have contributed to this project over the years, including current
and former members of the UMD cavity QED group (Michael Scholten, Basudev
Roy, Rebecca Olson Knell, Matt Terraciano, and Andres Cimmarusti, in whose ca-
pable hands I leave the apparatus); our former postdoctoral researcher, Jietai Jing;
our undergraduate researchers, Josh Crawford and Eric Cahoon; and a host of vis-
iting student researchers. In addition, we have bene ted from theory collaboration
with Pablo Barberis in Mexico, and Perry Rice and James Clemens in Ohio.
Finally, I expressly acknowledge the contributions of the National Science
Foundation, which has supported me monetarily for the past six years through
a Graduate Research Fellowship, grants to the cavity QED program, and establish-
ment of the Physics Frontier Center at the Joint Quantum Institute. Their generous
funding made this research possible.
ii
Table of Contents
List of Figures iv
List of Abbreviations v
1 Two-Mode Cavity Quantum Electrodynamics 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 The Jaynes-Cummings model . . . . . . . . . . . . . . . . . . 2
1.2.2 JC model versus reality . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Two modes and three levels . . . . . . . . . . . . . . . . . . . 7
1.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Physics cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Atomic source . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.4 Detection system . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Detecting Single Atoms 20
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Detection scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Preliminary data analysis . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Faraday rotation . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Photons per atom . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Analysis and optimization of detection . . . . . . . . . . . . . . . . . 33
2.6 Intensity autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Ground-State Quantum Beats 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Backward Light Shifts and Quantum Jumps 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Experimental system . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Conclusions and Future Directions 79
Bibliography 82
iii
List of Figures
1.1 The optical cavity QED system . . . . . . . . . . . . . . . . . . . . . 2
1.2 85Rb level diagram and simpli ed three-level model . . . . . . . . . . 5
1.3 Vacuum chamber containing optical cavity and LVIS apparatus . . . 10
1.4 E ects of optical pumping beam . . . . . . . . . . . . . . . . . . . . . 13
1.5 Schematic of laser system . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Schematic of coincidence measurement apparatus . . . . . . . . . . . 22
2.2 Diagram of atomic levels used for detection . . . . . . . . . . . . . . . 23
2.3 Measured photon statistics and linear  ts . . . . . . . . . . . . . . . . 31
2.4 Counts detected from a single atom . . . . . . . . . . . . . . . . . . . 32
2.5 Signal-to-background ratios . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 g(2)( ) vs. Faraday rotation . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 g(2)( ) and  t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.8 g(2)( ) vs. drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.9 Atom number and count rates . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Experimental system for measuring quantum beats . . . . . . . . . . 45
3.2 Simpli ed model of quantum eraser process . . . . . . . . . . . . . . . 46
3.3 g(2)( ) vs. atom number . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 g(2)( ) and power spectrum . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 g(2)( ) vs. magnetic  eld . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6 g(2)( ) vs. drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.7 g(2)( ) vs. waveplate mixing angle (composite) . . . . . . . . . . . . . 56
3.8 g(2)( ) vs. waveplate mixing angle (series) . . . . . . . . . . . . . . . 57
3.9 Calculated g(2)( ) and constituent terms . . . . . . . . . . . . . . . . 59
4.1 Four-level atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Schematic of apparatus for measuring g(2)( ) . . . . . . . . . . . . . . 63
4.3 Frequency shifts in calculated g(2)( ) . . . . . . . . . . . . . . . . . . 65
4.4 Frequency shifts in measured g(2)( ) . . . . . . . . . . . . . . . . . . . 65
4.5 Frequency of beat resonance vs. magnetic  eld . . . . . . . . . . . . . 70
4.6 Frequency of beat resonance vs. drive . . . . . . . . . . . . . . . . . . 71
4.7 FWHM of beat resonance vs. drive . . . . . . . . . . . . . . . . . . . 73
4.8 Sample peaks from FFT amplitude spectrum . . . . . . . . . . . . . . 74
4.9 Frequency and FWHM vs. drive detuning . . . . . . . . . . . . . . . 76
iv
List of Abbreviations
AM amplitude modulation
AOM acousto-optic modulator
APD avalanche photodiode
DAVLL dichroic absorption vapor laser lock
DC direct current
EOM electro-optic modulator
FM frequency modulation
FSR free spectral range
FWHM full width at half maximum
HWP half-wave plate
JC Jaynes-Cummings
LVIS low-velocity intense source
MOT magneto-optical trap
NIM nuclear instrumentation module
PBS polarizing beam splitter
PDH Pound-Drever-Hall
PID proportional-integral-di erential
PMT photomultiplier tube
PZT piezo-electric transducer
QED quantum electrodynamics
RF radio frequency
TEM transverse electromagnetic
TTL transistor-transistor logic
v
Chapter 1
Two-Mode Cavity Quantum Electrodynamics
1.1 Introduction
A cavity quantum electrodynamic (QED) system consists of an atom or other
quantum object coupled to a small number of modes of a cavity [1]. Such a system
provides a simple structure in which to investigate fundamental aspects of matter-
light interactions, and may also be used for applications in quantum information
science [2, 3, 4, 5]. Several cavity QED geometries operating over a wide range of
the electromagnetic spectrum have now been demonstrated experimentally (see for
example [1, 6, 7, 8].)
This thesis presents a collection of experiments performed on a particular cav-
ity QED system, a two-mode optical cavity and cold atomic source  rst described
in [9] and [10]. We begin with a theoretical overview of the system and the mea-
surement goals. As the basic setup is common to all of the experiments discussed in
this thesis, we give a thorough account of the technical details in this chapter, with
additional details for each speci c measurement in the subsequent chapters.
1
g
?
?
?
drive
Figure 1.1: The optical cavity QED system. g: atom-photon dipole
coupling frequency;  : free-space atomic decay rate;  : cavity  eld mode
decay rate; Edrive: coherent driving  eld amplitude.
1.2 Theoretical models
1.2.1 The Jaynes-Cummings model
The Jaynes-Cummings (JC) model provides the simplest realization of quan-
tized light-atom interactions [11]. The model considers a single two-level atom, a
saturable absorber, coupled to a mode of the electromagnetic  eld, a harmonic os-
cillator of equally-spaced energy levels; the mode is typically de ned by the mirror
boundaries in an optical cavity (see Fig. 1.1.) Coherent electric-dipole coupling
between the atom and a single photon (an excitation quantum of the mode) takes
place at the Rabi frequency
g =  E= h =   h
s  h!
c
2 0V; (1.1)
where  is the atomic transition dipole moment and E is the amplitude of the
electric  eld of a single photon, with mode volume V and frequency !c. The system
2
Hamiltonian without dissipation is
HJC =  h!a z +  h!caya i hg( +a ay  ): (1.2)
Here !a is the natural atomic transition frequency, a is the annihilation operator
for the cavity  eld, and   ;z are the Pauli spin operators for a two-level atom. The
evolution in this case is simply a continuous, reversible exchange of energy between
the atom and the cavity mode.
Dissipation may be added by considering separate reservoirs coupled to the
cavity mode and atomic radiation pattern. Once traced over, non-reversible decay
into these reservoirs brings lifetimes 1= and 1= for the cavity  eld and atomic ex-
cited state population, respectively [12]. The e ect of a single photon on the atomic
state is characterized by the saturation photon number, nsat   2=(8g2), which may
reach values much less than one for su ciently strong coupling. The e ect of an
atom on the cavity  eld is similarly characterized by the single-atom cooperativity
parameter, C1  g2=(  ), which expresses the ratio of coherent coupling to dissipa-
tion rates. For the case of N atoms coupled identically to the mode, constructive
interference enhances the emissions into the cavity mode, and the cooperativity
parameter increases to C = NC1 [13].
Driving the cavity on resonance with a laser  eld of amplitude Edrive (outside
the cavity) causes the build-up of a coherent  eld inside the cavity, the amplitude of
which depends on C. The steady-state behavior of the system is given by the state
equation,  rst introduced in the context of optical bistability [13]:
Y = X(1 + 2C=(1 +X))2; (1.3)
3
where Y is the intracavity photon number without atoms, and X is the number
with atoms (both are normalized to nsat.) This form is exact for a plane-wave mode
in a traveling-wave (ring) cavity, and it exhibits bistable behavior for large values of
C and X. In the limit of weak driving (X 1), the equation takes on the simpler
form:
Y = X(1 + 2C)2: (1.4)
This equation shows that X is smaller than Y by the factor 1=(1 + 2C)2, primarily
due to the drive light radiated into free space by the atoms.
1.2.2 JC model versus reality
Experimental realization of the JC Hamiltonian entails a number of approxi-
mations. True two-level atomic systems are rare. Even the alkalis, a staple of atomic
physics experiments because of their relatively simple spectra, exhibit complicated
magnetic hyper ne structure and are susceptible to optical pumping. An example is
the 85Rb D2-line F = 3!F0 = 4 manifold shown in  gure Fig. 1.2a. An e ective
two-level atom in such a level structure can be realized experimentally by optically
pumping to the F = 3;m = +3( 3) ground state and using a  + (  ) circularly
polarized cavity mode to drive the F = 3;m = +3( 3) !F0 = 4;m0 = +4( 4)
cycling transition, provided that light polarization and magnetic  elds are carefully
controlled.
Real cavity mode structure is likewise complicated. In a standing-wave cavity
of length l and in the transverse electromagnetic (TEM) basis, longitudinal modes
4
1 3
2
g G? ?
5S
1/2
,F = 3, m =
5P
3/2
,F' = 4, m' =
0
0 1-1
-1-4 -3 -2
-2-3 2 3
1 2 3 4
85
Rb
a)
b)
Figure 1.2: a) Diagram of 85Rb levels used in the experiments. b) Simpli-
 ed three-level atomic model with coupling constants for the two cavity
modes.
are separated by a free spectral range, FSR = c=2l, with each of these split into
multiple transverse modes. The requirement of a single spatial mode in the JC
model means that all other modes must be separated from the mode of interest by
several cavity linewidths. In addition, each transverse mode supports two orthogonal
polarizations. The frequency splitting between the modes is determined by the rate
of cross-coupling between polarizations due to cavity mirror birefringence. True
single-mode excitation is accomplished by driving one of the polarization eigenmodes
(generally elliptically polarized), but for su ciently small birefringence an arbitrary
polarization may be used.
A greater and still outstanding problem experimentally is keeping a single atom
localized in the cavity mode within a region of much less than one half-wavelength,
such that the value of g, which varies spatially with the mode structure of E, is
5
constant and known. Early cavity QED experiments operated with a thermal beam
of atoms, giving a random and changing distribution of coupling strengths and
atom number, and non-adiabatic interaction dynamics due to the high speeds [14].
Individual neutral atoms trapped inside a cavity mode have been demonstrated, but
still with time-varying values of g and a substantial warping of the level structure
due to light shifts [15, 16]. Single ions trapped in a cavity hold more promise, but
face their own technical challenge owing to the close presence of dielectric mirrors
[17].
Our goal in this work is to explore the dynamics of a real cavity QED system
in full quantitative detail beyond the limits of the single-mode JC Hamiltonian.
The atoms are multitudinous and freely moving, but can be reduced in number
to allow detection and manipulation of individual atoms. The magnetic structure
entails all sixteen levels shown in Fig. 1.2a, and supports the creation of ground-
state coherence that does not exist for two-level atoms. The cavity birefringence
is small enough to permit two degenerate polarization modes, but large enough to
give substantial contributions to the measured signals.
We  rst consider a simple extension to the JC model in order to motivate
the measurements that follow. In the subsequent chapters we will introduce greater
complexity into the model, requiring a quantum Monte Carlo simulation for the full
calculations.
6
1.2.3 Two modes and three levels
As a  rst complication, we consider a three-level  -type atom (Fig. 1.2b)
inside a cavity with two orthogonal and degenerate polarization eigenmodes, with
annihilation operators a and b ([10]). The j1i-to-j2i transition dipole couples to a
photon in mode a with Rabi frequency g, and thej2i-to-j3itransition dipole couples
to a photon in mode b with Rabi frequency G. The total spontaneous emission rate
is the sum of the emission in the two channels,  tot =  +  . The atom starts in
state j1i and the excitation is su ciently weak such that the probability of being
transferred to statej3iand subsequently re-excited by the driving light is negligibly
small. This avoids the problem that the atom in Fig. 1.2b gets pumped to a dark
state of the driving laser ( polarization), while the real level structure in Fig. 1.2a
has no dark states.
The cavity is driven with a coherent  eld (inside the cavity) of amplitude " in
mode a. The e ective Hamiltonian in between quantum jumps and in the rotating-
wave approximation is [12, 10]:
Hmode = " 
 
a ay
 
+i hg
 
a
NX
i=1
j2iih1j ay
NX
i=1
j1iih2j
!
+i hG
 
b
NX
i=1
j2iih3j by
NX
i=1
j3iih2j
!
 i h 
 
aya+byb
 
 i h +  2
NX
i=1
j2iih2j; (1.5)
where the sums are over the N atoms, each considered to be identically coupled to
the mode.
We now have two single-atom cooperativity parameters, C1 = g2=(  tot) for
7
the driven mode and eC1 = G2=(  tot) for the orthogonal mode. The e ect of eC1 on
the driven mode is to reduce the cooperativity by an amount commensurate with
emission into the orthogonal mode; instead of C = NC1, we de ne a new driven
mode cooperativity,
C0 = C1N1 + 2 eC
1
: (1.6)
De ning separate values of the intracavity photon number X for the driven (Xk)
and orthogonal (X?) modes, the state equations become [10]:
Xk = Y 1(1 + 2C0)2  Y (1 4C0); (1.7)
X? = Y
" 2 eC
1
1 + 2 eC1
#" C0
(1 + 2C0)2
#
 Y2 eC1C0; (1.8)
where the lowest-order approximations are valid for 2C 1 and 2 eC1  1, and we
assume weak drive (X 1). The population of the driven mode decreases with C0
as before, while the population of the orthogonal mode increases with C0; in other
words, X? contains a fraction of the light Y  Xk that would otherwise have been
radiated into free space.
This thesis studies light emission from the two modes for a variety of ex-
perimental parameters. In particular, we study time-resolved photon correlation
measurements of light from X?, which reveal information on atomic dynamics and
coherences that are invisible in the steady-state average.
8
1.3 Apparatus
1.3.1 Physics cavity
We perform measurements on light escaping from a high- nesse Fabry-Perot
interferometer (referred to below as the \physics cavity".) The mirrors that form
the cavity have re ective coatings from Research Electro-Optics of Boulder, CO,
with transmissions T1 = 15 ppm and T2 = 270 ppm at a wavelength of  = 780
nm, where the larger transmission T2 corresponds to the mirror on the output side
[9]. The spacing between the re ective surfaces is l = 2:2 mm, and the mirrors have
radii of curvature 25 cm and 10 cm, respectively. The cavity supports a TEM00
mode with 1=e  eld waist w0 = 56  m and Rayleigh range  w20= = 1:2 cm, i.e. the
mode diameter changes little over the distance between the mirrors. The cavity free
spectral range is FSR = c=(2l) = 68 GHz.
Barring additional losses, we expect cavity  nesse F = 2 =(T1 + T2) = 22
000 and resonance full width at half maximum (FWHM) 2 =(2 ) = FSR=F = 3:1
MHz. We measure instead a FWHM of 5.8 MHz and a  nesse of F = 12 000. We
infer from this an additional loss channel approximately equal in magnitude to the
larger mirror transmission. This loss is most likely due to residual glue deposited on
the mirror surface during construction. When inferring the photon number inside
the cavity, we increase our estimate by a factor of two due to this loss. This agrees
with the results of full Quantum Monte Carlo simulations of the system discussed
in Chapter 3. The cavity also has a small intrinsic birefringence, which results
in a slight (less than 200 kHz) splitting between the two (elliptical) polarization
9
cavity
mirror
MOT beams
7.5 cm
1.5-mm
hole
Figure 1.3: Schematic (not to scale) of vacuum chamber containing op-
tical cavity and LVIS apparatus.
eigenmodes. Because this splitting is much less than  =(2 ), we can e ectively
choose an arbitrary polarization basis for driving the cavity (typically vertical and
horizontal) and su er only a small background of coherent light in the undriven
mode due to birefringence coupling.
The cavity length is controlled by two shear piezo-electric transducers (PZT)
to which the mirrors are glued. The PZT?s are glued to a stainless steel base,
which rests on a stack of alternating layers of Sorbothane (a synthetic polymer from
Sorbothane, Inc.) and lead for acoustic isolation [18]. The unit resides in a small
stainless steel vacuum chamber at 10 8 Torr (Fig. 1.3), with electrical feedthroughs
10
for the PZT?s. One PZT is driven by the ampli ed output of a Stanford Research
Systems SIM960 Proportional-Integral-Di erential (PID) controller, with a feedback
bandwidth of about 1 kHz. The other PZT is connected to a high-voltage direct
current (DC) power supply to correct for slow thermal drift. A LabVIEW program
monitors the output of the PID controller and adjusts the DC voltage to keep the
error signal output near zero. The power supply voltage spans about three cavity
FSR?s.
1.3.2 Atomic source
We extract a continuous cold beam of 85Rb atoms from a modi ed magneto-
optical trap (MOT). The atoms are supplied by a Rb dispenser from SAES Get-
ters, Inc., operated at a current of 3.4 A. The trap uses a pair of coils in near
\anti-Helmholtz" con guration (8 cm diameter and 5 cm separation) to produce a
magnetic  eld gradient of 10 G/cm. We cool the atoms with three orthogonal retro-
re ected beams at 780 nm tuned below the F = 3 ! F0 = 4 cycling transition.
Each beam has a power of approximately 15 mW and a diameter of 2 cm. The
detuning from resonance is approximately -20 MHz, which we vary by changing the
frequency of a double-passed acousto-optic modulator (AOM). The exact parame-
ters are optimized to give the greatest  ux of atoms. A repumping beam resonant
with transitions from the F =2 ground state to the F0 = 3 or 4 excited states is
mixed with the MOT beams on a polarizing beam splitter (PBS).
The atoms exit via a 1.5 mm-diameter hole in the vertical beam retro-re ecting
11
optic, which consists of a quarter-wave plate ( =4) atop a gold mirror, placed at the
bottom of the MOT chamber. The radiation imbalance above the hole pushes out
a continuous beam of atoms in a Low-Velocity Intense Source (LVIS) con guration
[19]. The atoms have a mean longitudinal velocity of about 20 m/s, determined from
comparison of measured data with Monte Carlo simulation (see Chapter 3.) The
beam intersects the cavity mode at near normal incidence, 3-4 cm below the hole
in the mirror. We estimate a maximum  ux of about 1 107 atoms s 1 that pass
through the cavity mode, corresponding to a mean mode occupation of 50 atoms
with an average transit time of 5  s. For some measurements, we reduce the  ux
of atoms in order to have a mean mode occupation of less than one atom. This is
accomplished by changing the detuning or power of the MOT beams.
As part of the measurements, we require control of the magnetic  eld inside
the cavity, in particular the components parallel to the input laser polarization
(vertical) and parallel to the cavity axis (horizontal). To adjust the  eld in the
vertical direction, we implement a third coil below the \anti-Helmholtz" coil pair.
This allows us to vary the  eld in the cavity between  12 G (corresponding to a
splitting between ground-state Zeeman levels of approximately  =2) while a ecting
the  eld in the MOT region only slightly. We use an additional pair of coils oriented
with axis parallel to the cavity axis to control the magnetic  eld in that direction
as well.
Coming out of the MOT in LVIS con guration with a vertical magnetic  eld
applied, the atoms are left primarily in the m = 3 ground state, with the exact
distribution sensitive to laser alignment and detuning. Before they enter the cavity
12
0
2
4
6
8
10
12
14
16
-20 -15 -10 -5 0 5 10 15 20
U
n
d
r
i
v
e
n
 
m
o
d
e
 
c
o
u
n
ts (1000?s 
/
 
s
e
c
)
Detuning (MHz)
with optical pumping
no optical pumping
0
0.2
0.4
0.6
0.8
1
1.2
-20 -15 -10 -5 0 5 10 15 20
X
/
Y
Detuning (MHz)
with optical pumping
no optical pumping
a)
b)
Figure 1.4: a.) Example of absorption fraction X=Y in driven mode
as a function of common detuning of laser and cavity frequency from
central m =0 to m0 =0 transition. The optical pumping beam shifts the
population toward the center. The magnetic  eld value is approximately
5 G and drive strength is approximately one photon in the cavity, with
drive polarization parallel to the magnetic  eld. Solid lines are least-
square  ts of the data points to a Lorentzian line shape. b.) Count rates
from the undriven mode measured concurrently with data in (a), with
Lorentzian  ts.
13
mode, we optically pump the atoms to the m = 0 state using a beam resonant with
the F = 3!F0 = 3 transition and with polarization vector parallel to the vertical
magnetic  eld. The beam is combined with light from the repumper laser in a
50/50 polarization-maintaining single-mode  ber splitter, and collimated to a waist
diameter of 0.5 mm in order to pass between the top of the cavity mirrors and the
upper edge of the viewport. Due to strong scattering into the photon detectors from
multiple re ections, we are unable to use the beam in a retro-re ected con guration.
Therefore, the beam imparts a net momentum kick to the atoms, which must be
compensated for with atomic beam alignment. Alternatively, a con guration of
MOT beams may be found which leaves a broader distribution of population over
the ground states. In this case the optical pumping beam acts as a  lter, ejecting
those atoms which are not already in m = 0. The intensity of the optical pumping
beam and repumper are optimized as a compromise between moving most of the
atoms to m = 0 while not ejecting too many from the beam.
Figure 1.4 shows a typical measurement sequence used for optimization of the
optical pumping beam. The e ect of the beam is to shift and narrow the line shapes
for both absorption and emission, with the goal of having narrow Lorentzian peaks
centered around 0 MHz (the resonance frequency of the F = 3;m = 0 ! F0 =
4;m0 = 0 transition.) The distributions do not always shift completely to the center
(an e ect evident in Fig. 1.4); this is likely due to some combination of incomplete
optical pumping and a small ( 1 MHz) drift in the frequency setpoint of the
saturated absorption spectroscopy reference, arising from temperature-dependent
amplitude modulation (AM) in the electro-optic modulator (EOM) that generates
14
820 nm
transfer
cavity
Pump laser
532 nm
Ti:Sapph
780 nm
Lock system for
stabilizing Ti:Sapph
Rb vapor cell
LVIS
Repumping
laser system
physics cavity
Figure 1.5: Schematic of the laser system used for stabilizing and probing
the optical cavity, and for cooling and trapping the atoms.
frequency modulation (FM) sidebands for the error signal (see below).
1.3.3 Laser system
Figure 1.5 shows a schematic of the apparatus, focusing on the laser sys-
tem. The primary source of light at 780 nm is a Titanium:Sapphire MBR-110 laser
(Ti:Sapph), driven by a 10 W pump laser at 532 nm (Verdi V-10), both from Co-
herent, Inc. A reference cavity-based lock internal to the MBR housing narrows the
Ti:Sapph linewidth to less than 100 kHz. Long-term frequency stability and abso-
lute frequency referencing are accomplished by saturated-absorption spectroscopy
in a magnetically-shielded Rb vapor cell. The error signal is derived from the 85Rb
F = 3 ! (F0 = 3;F0 = 4) hyper ne crossover transition. We generate the signal
by imprinting FM sidebands at 12 MHz with an EOM, detecting the saturated ab-
15
sorption with a fast photodiode, and mixing the ampli ed photodiode output with
the radio frequency (RF) local oscillator. Locking is accomplished with a SIM960
PID controller, the output of which is fed to the external scan input on the MBR
control box. The light output of the MBR at this frequency is approximately 1 W.
We stabilize the length of the physics cavity with a Toptica DL-100 diode
laser at 820 nm, which interacts negligibly with the atomic beam but still re ects
strongly from the cavity mirrors. The diode laser frequency itself is stabilized to the
Ti:Sapph frequency with a transfer cavity technique. First, we adjust the current
to the Toptica laser diode such that both lasers are simultaneously resonant with a
TEM00 mode of the physics cavity, changing the cavity length if necessary to  nd
a suitable point. We then  nd a similar resonance overlap for a 25 cm confocal
(transfer) cavity into which light from both lasers has been injected, adjusting the
length of the transfer cavity as necessary. We use the Ti:Sapph light to lock the
transfer cavity length with the Pound-Drever-Hall (PDH) technique [20], a PID
controller, and PZT?s. Finally, we stabilize the diode laser frequency to the transfer
cavity resonance with another PDH signal, derived from FM sidebands at 9 MHz
generated by directly modulating the laser diode current.
The rest of the light from the Toptica diode laser is used to generate a PDH
signal for stabilizing the length of the physics cavity. Before reaching the cavity, the
light goes through an AOM in double-passed con guration and enters a single-mode
polarization-maintaining  ber. The AOM allows for scanning the cavity frequency
by 40 MHz. A monitor photodiode after the  ber output is read by an analog-to-
digital sampling card and monitored by a LabVIEW program, which controls the
16
RF drive power to the AOM in order to maintain the light output from the  ber at a
constant power. This is especially important when scanning the cavity frequency, as
residual AM sidebands from modulating the diode current give a power-dependent
DC o set to the PDH error signal for the physics cavity; this results in a nonlinear
scan due to the frequency-dependent AOM di raction e ciency.
We drive the physics cavity with Ti:Sapph light that passes through a 780 nm
single-mode  ber electro-optical modulator (EOSpace) in Mach-Zender con gura-
tion to imprint AM sidebands at 150 MHz from the carrier. One sideband is near
resonance with the atomic transition and the cavity, becoming the system drive,
while the carrier and all other sidebands are re ected from the front cavity mirror.
Changing the sideband frequency allows us to probe the frequency response of the
cavity-atom system without misalignment of the input laser coupling. Polarization
elements (better than 5 10 5 extinction) and mode-matching optics prepare the
driving laser before it enters the cavity.
Light at 780 nm used for repumping atoms in the MOT is generated from a
Sharp laser diode housed in a ThorLabs mount. The laser is slaved to a New Focus
Vortex diode laser, which is frequency stabilized to the repumping transition with
a Dichroic Absorption Vapor Laser Lock (DAVLL) and PID controller [21]. This
system generates about 20 mW of repumper light.
17
1.3.4 Detection system
A lens at the cavity output collimates the escaping light, and a zero-order half-
wave plate (HWP) aligns the polarization to a Wollaston prism, which separates
vertically and horizontally polarized light into separate beams. Additional optics
allow the choice of sending both beams to the detectors, or of blocking the vertically
polarized beam and splitting the horizontally polarized beam between two paths for
coincidence measurements. We split o a small fraction of the vertically polarized
beam  rst in order to monitor the intensity of transmitted light at 820 nm; the
rest of this light is removed from the signal with 5-nm wide interference  lters
centered at 780 nm (Semrock MaxLine), such that no detectable level of light at
820 nm reaches the detectors. Passing through spatial  ltering optics to remove
background light, the signal beams enter a dark box where they are detected by
two photomultiplier tube (PMT) units (Hamamatsu R636-10), which measure the
average light intensity; or, by remotely activating motorized  ip mirrors, to two
avalanche photodiode (APD) units (Perkin Elmer SPCM-AQR-12 and -13) for time-
resolved photon counting, each with quantum e ciency greater than 50%. The total
detection e ciency of light from the cavity is about 30%, as calibrated by an optical
power meter and neutral density  lters of known attenuation.
The electronic output pulses from the two APDs are  rst duplicated in a
transistor-transistor logic (TTL) voltage divider circuit. One copy of each goes to
a counter unit (SR400) for measuring average count rates, or to a nuclear instru-
mentation module (NIM) logical gate module for coincidence measurements. The
18
other copy of each goes to a computer time-stamp correlator card (Becker and Hickl
DPC-230), which records both channels of TTL events with 164 ps resolution. For
measuring photon correlation functions, we record the arrival times of all photons
and compute the detector cross-correlation in post-processing with a C++ program,
with typical user-de ned time resolution of 16.4 ns bins.
19
Chapter 2
Detecting Single Atoms
2.1 Introduction
Many applications in atomic physics and quantum information hinge on the
ability to detect the presence of individual neutral atoms or molecules in order
to perform subsequent state manipulation protocols [22, 23]. Optical detection of
atoms involves identifying a change in average light intensity due to the presence of
the atom. This typically happens by one of two methods:  uorescence detection,
in which the atom adds light on top of a background level; or absorption detection,
in which the atom removes light from an incident  eld. Dispersive detection can
e ectively operate in either of these limits, depending on whether the signal with no
atoms is bright or dark. All these techniques require the integration of photon  ux
until a targeted signal-to-noise ratio is reached; this sets the probability of obtaining
a false positive for atom detection.
Fluorescence detection requires high-numerical aperture optics and careful
suppression of background light, and generally operates on time scales of tens of
milliseconds. The integration time limits the bandwidth with which new atoms can
be identi ed and then manipulated, making this method much better suited for
identifying stationary trapped atoms [24, 25, 26]. When detection speed is more
important (as when detecting freely moving particles that spend a limited time in
20
an interaction region), a more useful technique is  uorescence burst detection [27],
which looks for above-average count rates over short time intervals [28, 29]. One
particularly impressive example [30] showed detection of falling atoms in free space
in 60  s using highly-e cient mirrors and lenses.
In contrast to free-space methods that collect light from as many spatial modes
as possible, cavity-based methods limit the optical interactions to one or a few
resonant modes, yet give signal-to-noise ratios that may be orders of magnitude
larger. Collecting  uorescence in an optical cavity with axis perpendicular to the
driving laser gains the bene t of Purcell-enhanced emission [31] into the cavity mode
[32, 33, 34]. Experiments based on changes in cavity transmission, which require
strong atom-cavity coupling [35], have achieved single atom detection times of 20  s
for moving atoms [7, 32, 36, 37], and as low as 10  s [34, 38, 39] for trapped atoms.
Both methods enable strong suppression of background light through spatial  ltering
of the output quasimode of the cavity.
We have developed a cavity technique for atom detection that combines ele-
ments of driven cavity QED detection and spontaneous emission collection [40]. The
technique allows rapid identi cation of atoms with near maximal coupling to the
cavity mode, and their subsequent emission dynamics. This chapter presents our
approach, in particular autocorrelation studies of the transmitted light. We identify
contributions to the autocorrelation signal from di erent physical processes, in the
short time (atomic decay) and in the long time (transit of atoms through the mode).
21
2.2 Detection scheme
The basic components of our system are shown in Fig. 2.1. We resonantly
drive a TEM00 mode of the optical cavity, previously discussed, with a laser sideband
tuned to the 85Rb F = 3 to F0 = 4 transition. Intersecting the cavity axis at near-
normal incidence is the beam of cold atoms, each of which interacts with the cavity
mode for some time before leaving. Because of the large di erence in time scales
(5  s transit time through the mode versus 26 ns excited state lifetime), the atoms
undergo dozens of spontaneous emission cycles while in the mode volume.
Atomic Beam
PBS
BSB
APD A
(start)
APD B
(stop)
Figure 2.1: Schematic of the apparatus with the basic optical elements
necessary for coincidence measurements. Vertically polarized laser light
(parallel) drives the high  nesse cavity traversed by a continuous beam
of cold 85Rb atoms from a magneto-optical trap. The output light passes
through a polarizing beam splitter (PBS) that separates the horizontal
(perpendicular) polarization, sending it to a nonpolarizing beam split-
ter (BS) for coincidence measurements using two avalanche photodiodes
(APD). The direction of the magnetic  eld (B) inside the cavity can be
parallel to the exciting polarization (yellow arrow,  -polarized drive) or
parallel to the direction of light propagation (blue arrow,  -polarized
drive).
22
B
Detection of
spontaneous
emission
B
Detection of forward
scattering
?
?
+
?
?
5P
3/2
, F?=4
5S
1/2
, F=3
m = n-1 n+1n
m? = n
5P
3/2
, F?=4
5S
1/2
, F=3
m = n-1 n+1n
m? = n
?
+
?
?
a) b)
Figure 2.2: a) A linearly polarized driving  eld excites the atom, but a
transition to a ground state level with di erent m results in emission in
the orthogonal polarization. b) Simpli ed diagram of 85Rb levels relevant
to Faraday rotation in an applied magnetic  eld.
The crux of our detection scheme is the separation of the cavity output  eld
into two distinct polarization components relative to the polarization of the exciting
laser: one parallel (driven mode) which is little a ected by the transiting atoms
(as the cooperativity parameter is relatively small), and the other perpendicular
(undriven mode) which in the ideal case is populated only by scattered light from
the atoms. This con guration in cavity QED was  rst used in [41]. As the coupling
of the two modes by mirror birefringence is small, the two eigenfrequencies are split
by less than 10% of the cavity linewidth, and we can use the two modes for excitation
and signal collection, respectively.
We consider two di erent con gurations for excitation. The  rst has a weak
(< 10 G) magnetic  eld parallel to the incident polarization, forming the quan-
tization axis. The light drives  ( m=0) transitions in the atoms in this basis
23
(see Fig. 2.2a). An excited atom can return to its original ground-state m-level
through spontaneous or stimulated emission into the parallel (driven) mode; how-
ever, there is also some probability to relax into a di erent m-level through a  +
or   ( m =  1) spontaneous emission transition, which produces light that is
circularly polarized with respect to the magnetic  eld axis and may be collected by
the perpendicular (undriven) cavity mode. Although the cavity coupling strength is
modest, the amount of collected light increases measurably from that in free space
[10].
The second con guration has a weak magnetic  eld along the cavity axis, while
keeping the same polarization arrangement (see Fig. 2.2b). The linearly polarized
drive can then be considered as the sum of two circular components with di erent
indices of refraction due to the Zeeman shift. This results in a Faraday rotation of
the drive into the perpendicular mode. Though the single-pass Faraday rotation is
small, the high cavity  nesse allows the light to compound thousands of rotations
before escaping, and the e ect becomes appreciable.
A polarizing beam splitter (PBS) sends the two modes to separate detectors,
which can resolve individual photon events. From the record of photon detections,
we analyze the statistics of emission to look for signs of single atom transits. One
advantage of collection into a cavity mode is that we can spatially  lter out most
sources of background light. However, there are remaining contributions to the
perpendicular mode from cavity mirror birefringence and background light that
cannot be completely eliminated. The optimal parameters for operation minimize
the background in uence on the atom detection con dence.
24
2.3 Theory
To include the e ects of Faraday rotation, we consider an atom with one ex-
cited state (m0 = 0) and three ground states (m =  1;0). (See Fig 2.2b). A
magnetic  eld B along the direction of light propagation lifts the ground state de-
generacy through the Zeeman e ect, resulting in a shift in the dispersion curves and
a di erence in the phase velocities of the two circular components of light, causing
Faraday rotation [42]. We excite the system with vertically polarized light, which
is an equal superposition of right- and left-circularly polarized light. The light in-
teracts with the atom and becomes elliptically polarized; the plane of polarization
rotates through an angle. The light then passes again through the atom after re-
 ection on the mirror and compounds the rotation. When the light exits the cavity,
the total rotation is that of a single pass multiplied by the number of times the
light passes through the atom, indicated here by the presence of the cooperativity
C (which is proportional to F):
 = 2gL BB= h tot1 + (2g
L BB= h tot)2
C; (2.1)
where gL is the Land e factor and  B is the Bohr magneton.
The addition of this e ect changes Eq. 1.8 (assuming j j 1) into:
X? = Y
 
2 eC1C +j j2
 
: (2.2)
Equation 2.2 shows that the signal in the perpendicular mode now consists of two
parts: a background level of spontaneous emission (which may be considered as
( +   ) polarization if the drive is ( + +   ), in full analogy to the basis with
25
 drive), and Faraday-rotated drive which increases from zero to a maximum value
of X?Faraday = YC2=4 when B =  h tot=(2gL B). For simplicity in what follows we
treat the atom-cavity coupling in the basis of  drive.
We analyze the two-time intensity autocorrelation function of the light from
the perpendicular mode to understand the dynamics of photon emissions that con-
tribute to the steady-state value of X?. We do not distinguish between the two
components in Eq. 2.2, treating both as arising from near-resonant scattering of
incident light (resonance  uorescence) in which the cavity serves only as a mode in
which to collect the light.
For a stream of photons scattered from a single two-level atom undergoing sta-
tionary, resonant excitation, the intensity autocorrelation function takes the familiar
form for resonance  uorescence antibunching [12]:
g(2)A ( ) = 1 e (3 =4) (cosh  + 3 4 sinh  ); (2.3)
where  = ( =4)p1 8Y is a function of the drive intensity Y. The antibunching
referred to here means an increasing slope for increasing times: dg(2)( )=d j >0 > 0.
The condition g(2)(0) < 1 indicates that the photons obey sub-Poisson statistics,
and is not a necessary condition for the photons to exhibit antibunching [43].
The modi cation of Eq. 2.3 for the case of an atomic beam passing through
a localized excitation region appears in Ref. [44], which adds a number of e ects:
i., the multiplication of Eq. 2.3 by a window function to include the shape of the
mode function encountered by each transiting atom; ii., the scaling of Eq. 2.3 by the
inverse of the steady-state mean number of interacting atoms, N; iii., the addition
26
of a term jg(1)A ( )j2 to include the beating of  elds emitted by di erent atoms (the
Hanbury Brown and Twiss e ect [45]); iv., the addition of a constant o set of one
arising from uncorrelated emissions from di erent atoms; and v., the reduction of the
correlation factor due to the presence of completely uncorrelated background light.
Further generalizations appear in Refs. [46, 47], where the authors also consider the
product of the  elds that come from atomic emission with a temporally uncorrelated
but spatially mode-matched  eld from background light.
We are able to describe the total correlation function as:
g(2)( ) = 1 + 1(1 +R
b=Rs)2
f( )g(2)A ( )
N +F( ;Rb=Rs;N) (2.4)
where Rb and Rs are the average count rates for background and signal, respectively,
and
f( ) = [cos(  )e   + 1]e ( =T)2 (2.5)
is an empirically-determined window function describing motion of the atom through
the cavity mode with Gaussian temporal width T, where a small tilt of the atomic
beam with respect to the cavity axis normal results in motion through the standing-
wave lobes at frequency  . These oscillations decay at a rate  much faster than the
transit time, due to the spread in velocities from the atomic source. The function F
corresponds to any additional terms related to the beating of signals from di erent
atoms or of signal with background light; since this term represents classical noise
sources, it can only contribute light which is bunched around  = 0 [48]. Any
observation of antibunching in our signal must come from the single-atom term
g(2)A ( ). An explicit observation of F would require a high atom density (N  1)
27
since the width is comparable to that of the antibunching signal. (A technique in
which the antibunching width was arti cially made wider was demonstrated in [49]
in order to see this term.)
2.4 Preliminary data analysis
2.4.1 Faraday rotation
The Stokes parameter formalism [50] allows a measurement of the rotation
angle  if the amplitudes of the electric  eld in the two polarizations "? and "k are
known:
j j= j"k"?jj"
kj2 j"?j2
 
  
  
 
"?
"k
  
  
 ; (2.6)
where the approximation is made forj"kj j"?j. We measure the count rates in the
perpendicular mode as a function of applied magnetic  eld along the cavity axis to
determine the location of maximum Faraday rotation. For the measurements in this
chapter we do not optically pump the atoms to m = 0 before they enter the cavity.
The peak rotation is di erent depending on the sign of the magnetic  eld, probably
due to an asymmetry in the population distribution. We obtain a maximum of
j j = 0:035 0:007 rad from the measured increase in the count rate with a  eld
magnitude of 3.3 Gauss (calibrated from Zeeman shifts in absorption.) To make
explicit the e ects of the cavity, we estimate the e ective number of maximally-
coupled atoms by using Eq. 1.7 and measuring the driven mode transmission. This
was greater than 0.95, giving a bound of C <0.01 or N <0.1 e ective maximally-
28
coupled atoms (for  transitions from m = 0). Calculating the free-space absorption
length for 0.1 atoms and our cavity waist, we would expect a maximum rotation of
the order of 3  rad [42]. The enhancement by a factor of approximately 10,000 is a
result of the cavity  nesse.
2.4.2 Photons per atom
The formula of Mandel relates the photon number distribution P(n) with the
atom number distribution Patom(m):
P(n) = X
m
P(njm)Patom(m); (2.7)
where P(njm) = ( m)n exp (  m)=n! is the conditional probability of detecting n
photons when there are m atoms in the cavity volume, each contributing a mean of  
photons to the signal with a Poisson distribution of number [51]. This shows that the
super-Poisson  uctuations of the light arise from the combined Poisson  uctuations
in atom number and photons scattered, with  an extractable parameter.
We follow the method of Ref. [32] to relate the mean and mean-squared photon
number (hni, hn2i) to  :
hn2i
hni  1 =hnigaa + ; (2.8)
where the atom-atom correlation function is gaa = (hm2i hmi)=hmi2, which for
Poisson atomic  uctuations reduces to gaa = 1.
Figure 2.3 shows a detail of the plot of the experimental values for the left side
of equation (2.8) with the two di erent detection methods: Faraday rotation on the
29
top and no Faraday rotation on the bottom. We calculate the statistics based on
300 s time series of photon detections that we divide into a series of time bins of
equal width. Longer bin sizes have a larger average number of counts (horizontal
axis). The intercept gives the value of  . A linear  t for bins from 50  s to 100  s
(hni= 1.81 to 3.63, o the scale of the plot in Fig. 2.3) in the Faraday case gives a
vertical axis intercept of  = 0:196 0:003 photons detected per atom, and a slope
of 1.033 0.001. The intercept in the no Faraday case (for a  t fromhni= 0.50 to
1.00) is  = 0:036 0:002 photons detected per atom and a slope of 1:026 0.003.
The drive in both cases is two photons. The values of the slopes indicate that the
assumption of Poisson statistics for the atom number is well justi ed.
Figure 2.4 shows the extracted values of  as a function of cavity photon
number (driving intensity) spanning more than two orders of magnitude. There are
data points (rhombs) that come from Faraday rotation and one with spontaneous
emission alone (circles). The latter is smaller by almost a factor of  ve.  increases
linearly with drive until atomic saturation intensity (nsat = 5:3 photons for the
m = 0 to m0 = 0 transition), where in addition to saturation of the scattering rate,
optical pumping e ects also start to enter. Each point on Fig. 2.4 comes from a
least-squares  t similar to those in Fig. 2.3. The count rates in the perpendicular
mode and the values of  give an atomic  ux through the mode of approximately
160,000 atoms s 1. (The  ux is set much smaller than the maximum available in
order to isolate primarily single-atom events.)
30
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.00 0.05 0.10 0.15 0.20 0.25 0.30
<n>
<n>
<n
2
>/<n>-1 (no Faraday)
<n
2
>/<n>-1 (Faraday)
0.0
Figure 2.3: Examples of data (circles) and least-squares  ts (continuous
line) to Eq. (2.8) in the large-hni region for Faraday rotation (top) and
for no Faraday rotation (bottom) in the cavity as a function of average
number of photons in a time bin (hni).
31
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.01 0.1 1 10
?
 
(photons/atom)
<n>/n
sat
 (driven mode photons)
Figure 2.4: Counts detected from a single atom (rhombs for Faraday,
circle for non Faraday) as a function of intensity of the exciting laser
normalized by the saturation photon number. The arrows indicate the
points extracted from the plots of Fig. 2.3. The statistical error bars are
smaller than the size of the data points.
32
2.5 Analysis and optimization of detection
Three identi able sources of background counts contribute to the measurement
of the perpendicular mode: intrinsic detector dark counts (approximately 300 s 1
each), scattered light from the MOT beams (approximately 2000 s 1), and the
birefringence of the cavity (less than 5 10 5 of the driven mode count rate). The
light from the MOT dominates the background count rate at low driving intensity;
however, at higher intensity, light from cavity birefringence is the main contributor
to the degradation of the signal-to-background ratio. A single photon escaping in
the perpendicular mode most likely comes from an atom in the cavity; however,
there is at best still a 4% probability that it comes from background counts. We
suppress the probability of a false atom detection by requiring photon coincidences
in a time window smaller than the transit time of an atom across the cavity mode
[7]. We have implemented this in real time for preliminary measurements using
a NIM electronic coincidence module. The  rst perpendicularly polarized photon
detection (\start") opens a gate of variable width which allows counting pulses from
the \stop" APD. Detection of one or more of these yields an atom detection event.
Figure 2.5 shows the signal-to-background ratio calculated from photon time
series for the rate of single photon detection (open rhombs), two-photon coincidence
(open squares), and three-or-more-photon coincidence events (open triangles) in a
1  s window as a function of driving intensity. The maximum occurs at the same
driving intensity for all three. Two photons within a 1  s window improves the
signal-to-background ratio by more than an order of magnitude compared to de-
33
<n>/n
sat
 (driven mode photons)
signal-to-background ratio
10
0
10
1
10
2
10
3
0.1 1 10
10
4
Figure 2.5: Signal-to-background ratio as a function of normalized driv-
ing laser intensity for single detection (open rhombs), double coincidence
detection (open squares), and triple coincidence detection (open trian-
gles) in a 1  s window with Faraday rotation of the drive.
tection of single photons. The three-photon coincidence gives a better ratio but
signi cantly decreases the rate of detection as evident in the error bars associated
with each point. The three sets achieve their maxima just before the atomic satu-
ration intensity, due to background counts from cavity birefringence that continue
to increase linearly with drive. The contribution to the coincidences from detector
afterpulsing is small (less than 1%) and does not a ect the results.
To evaluate the probability of making a false detection for various coincidence
gate times, we extract the  delity of detection F versus gate length for two-photon
coincidence based on the time record. We calculate the waiting time distribution
34
of photon arrivals on a single APD by making a histogram of the time separation
between consecutive detections. The integral of the waiting time distribution gives
the number of two-photon coincidences c in a given time window. We calculate c
with and without atomic  ux to obtain:
F = 1 cwithout=cwith: (2.9)
The  delity (one minus the probability of error) reaches more than 99% at 0.1
 s, and 99.7% at 1  s with Faraday rotation, but only 96.7% at 1  s with detection
of spontaneous emission alone. The  delity decreases at longer times. The  delity
is optimal for times between 1 and 5  s, determined by the distribution of atom
transit times.
2.6 Intensity autocorrelation
We ensure that our coincidence detection scheme is sensitive to single atoms
by measuring the intensity autocorrelation function (g(2)( )) of the perpendicular
mode under very weak driving intensity at the same atomic beam density used
during detection measurements. If we consider the light in the perpendicular mode
as coming from resonance  uorescence, the antibunching can only arise from photon
pairs from the same atom. The observation of antibunching in our signal is strong
indication of our ability to detect photon coincidences from a single atom in less
than 1  s.
Figure 2.6 shows two examples of an autocorrelation function around  = 0
with and without Faraday rotation for the same driving intensity (0.4 nsat). The
35
-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
g
(2)
(
?
)
? (?s)
Figure 2.6: Intensity autocorrelation function (g(2)( )) of the perpendic-
ular mode with (triangles) and without (diamonds) Faraday rotation, at
a driving intensity of 0.4 nsat.
antibunching is visible and lasts for a time of the order of the excited state lifetime
(26 ns). The area under the curve is larger with Faraday rotation, indicating a
substantial increase in photon  ux from individual atoms. The antibunching is less
pronounced without the rotation because of the lower signal-to-noise ratio.
Since we record the photon detections as continuous time series, we can also
look at the long-term behavior of the autocorrelation in the context of resonance
 uorescence as presented in Ref. [44, 46, 47]. We  t the data to Eq. 2.4 with free
parameters N, T,  , and  , excluding the term F( ) and the region j j <50 ns
36
where it is important. We  nd excellent agreement in this outer region (reduced
 2=1.03 for points out to  5  s). Fig. 2.7 shows the  t to the data taken at a
driving intensity of 0.24 nsat. The damped oscillation at  =1.5 MHz corresponds
to motion through the standing wave with a period of 0.67  s (a velocity of 0.58
m/s along the cavity axis) and a damping time of 1= =0.29  s. Monte Carlo
simulations of the transit show good agreement with this damping time based on
the geometric collimation of transverse velocities from the source [19]. The Gaussian
background gives a mean atom number of N=0.88, and a 1/e waist of T=2.7  s (a
mean velocity of 20.7 m/s across the Gaussian mode). The ratio of the velocities
gives a beam tilt of 1.6 with respect to the cavity axis normal. Although we take
care to align the cavity under the exit hole of the atomic source, the cavity mode is
not necessarily centered on the two mirrors, and the transverse beam width allows
for small inclinations of the beam.
The full structure of the correlation function depends not only on the above
parameters (properties of the atomic beam), but also on the intensity of the drive,
which can change the atomic response or increase the amount of background light
from cavity birefringence. Fig. 2.8 shows the evolution of the autocorrelation func-
tion as the driving intensity increases by two orders of magnitude. We observe a
transition from antibunching to bunching with higher drive. The behavior comes
from the term F( ) in Eq. 2.4 which includes beating of the birefringence light
with the atomic emissions. This is in contrast to the transition seen in Ref. [49],
where the atomic density is increased such that the constant jg(1)A ( )j2 term (part
of our F( )) becomes visible. In separate experiments at low intensity we have also
37
-5 -4 -3 -2 -1 0 1 2 3 4 5
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
? (?s)
g
(2)
(
?
)
Figure 2.7: Intensity autocorrelation function (g(2)( )) of the perpen-
dicular mode at 0.24 nsat. The continuous line is the theoretical  t to
Eq. 2.4.
38
-8
-6
-4
-2
0
2
4
6
8
10
-1
10
0
10
1
0.5
1.0
1.5
2.0
2.5
? (?s)
n/n
sat
g
(2)
(
?
)
Figure 2.8: Intensity autocorrelation function (g(2)( )) of the perpendic-
ular mode for di erent values of photon number (intensity) in the driven
mode.
followed the disappearance of the antibunching as we increase the number of atoms.
Even when the actual density of atoms is kept constant, the e ective excitation
volume and thus the value of N increases with power, as atoms near the nodes and
wings of the mode start to interact [52]. This process reaches a maximum above
saturation intensity, when the only unsaturated atoms are too weakly coupled to
emit into the mode. Fig. 2.9a. shows the extracted values of N versus driving
39
intensity for the autocorrelations in Fig. 2.8. The line shows the expected shape
of the saturation, assuming that the probability of collecting a photon from an
atom is proportional to the product of a constant g2(x;y;z) for collection and a
saturating strength of excitation ([(n=nsat)g2(x;y;z)]=[1+(n=nsat)g2(x;y;z)]), which
is consistent with the expression Y2 eC1C in Eq. 1.8, modi ed for strong driving. We
obtain the theoretical line by integrating this product over a volume much larger
than the cavity mode, and using a least-squares  t to scale the vertical axis and
nsat.
Using the extracted parameters of mean atom number and mean transit time
from the two-APD correlation measurements, together with the extracted values of
 from the one-APD measurements, we can predict the expected macroscopic count
rates as Rs = N =2T. Fig. 2.9b shows that we obtain excellent agreement for
low intensities, while the highest intensity drive (largest birefringence background)
disagrees signi cantly. This comes from the large contribution from the beating be-
tween background and signal, which generates the large central peak in the measured
autocorrelation.
We emphasize that although the mean number of interacting atoms is ap-
proximately one, this is not the same as the e ective number of maximally-coupled
atoms, since most are weakly coupled. To explore this numerically, we use the work
of Carmichael and Sanders [53]. We distribute atoms randomly and uniformly across
the cavity mode function such that the mean density is 0.88 atoms within two mode
waists (similar to the measured value for low driving intensity). Allowing the atom
number to  uctuate with a Poisson distribution, we sum the individual coupling
40
0 1 2 3 4 5 6 7 8
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
n/n
sat
atoms
0
1
2
3
4
5
6
7
8
9
10
0.01 0.1 1 10
n/n
sat
10
-4
 counts/s
a)
b)
Figure 2.9: a) Measured mean number of interacting atoms, and theory
curve showing the expected saturation. b) Measured ( lled rhombs)
count rates in the perpendicular mode together with those predicted
(open squares) from the extracted mean number of atoms, photons per
atom, and transit time.
41
strengths for each con guration of atoms and weight by the atomic number distri-
bution to obtain the mean steady-state e ective number of atoms, which in this case
is 0.04. Using the mean value of C1=0.09 for  transitions from the ground state,
this gives a driven mode transmission of approximately 0.99 from Eq. 1.7, consis-
tent with the experimental conditions for single atom detection. A rapid decrease in
probability above one e ective atom indicates that we operate in a density regime
of single-atom coupling, with a low probability of having two atoms simultaneously
coupled by more than half of the maximum value. This numerical result together
with the antibunching in the perpendicular mode counts con rms that the measured
signals indeed arise primarily from single-atom emission bursts.
42
Chapter 3
Ground-State Quantum Beats
3.1 Introduction
Quantum beats are oscillations in the radiation intensity of an ensemble of ex-
cited atoms due to interfering emission pathways. They must be counted amongst
the earliest predictions of quantum mechanics [54]. So-called \Type-I" atoms ex-
hibit beats at the separation frequency of two excited states which, prepared in a
superposition, decay to a common ground state. The preparation may be achieved
in a number of ways, e.g., through pulsed optical excitation [55, 56] or cascade
emission [57]. In all cases the coherence lasts for the excited state lifetime, unlike
ground-state coherence, which can last long enough to be interrogated later and, for
this reason, is favored by the  eld of quantum information.
We consider an unusual situation where ground-state coherence gives rise to
a long-lived quantum beat in spontaneous emission [58]. As repeatedly noted [54,
59, 60], QED predicts no beat in the decay of a \Type-II" atom to non-degenerate
ground states, since they are orthogonal. Nevertheless, while beats may be absent
from the mean intensity, they can still lie hidden in the  uctuations. In an elegant
experiment in the 1950?s, Forrester et al. [61] showed this for the (classical) beating
of light from a pair of incoherent sources. We proceed in similar spirit; we recover
a long-lived quantum beat from  uctuations.
43
In contrast to recent experiments [62, 63], which aim for deterministic quantum
control, the ground-state coherence in our experiment is both prepared and read out
by spontaneous emission. Moreover, our measured beat is di erent from that seen
by Schubert et al. [64], where an interference occurs in absorption rather than
emission.
Creation of coherence through spontaneous emission, so-called spontaneously
generated coherence, has been discussed in the theoretical literature [65, 66, 67]
and indirect experimental evidence reported for spontaneous creation of electron
spin coherence in charged GaAs quantum dots [68]. We detect only spontaneous
emission, and therefore make a direct and unambiguous observation of spontaneously
generated coherence. We realize, in a continuously driven variation, the scheme
of Zajonc [69] for generating ground-state quantum beats on the principle of the
quantum eraser [70].
3.2 Theoretical model
Beginning with an outline of the scheme, we consider  rst one idealized atom
then a realistic atomic ensemble. Consider an atom with Zeeman structure in its
ground and excited states (Fig. 3.1a) interacting with degenerate, orthogonally po-
larized cavity modes, H and V; a weak magnetic  eld sets the quantization axis
in the V direction, and mode V is weakly and continuously driven (Fig. 3.1b).
The atom is prepared in state jg0i from which it is excited to je0i by the V mode
(Fig. 3.2A). It may return to the ground state emitting a  ; + or   photon, or
44
Figure 3.1: Experimental system for observation of ground-state quan-
tum beats. (a)  -excitation of an F = 3 to F0 = 4 transition with
scattering of a  rst (red) and second (blue) photon into the H mode;
b) schematic of the apparatus, HWP: half-wave plate, PBS: polarizing
beam-splitter, BS: beam-splitter, APD: avalanche photodiode.
45
e
-1
?
?
?
?
e
0
e
+1
e
-1
e
0
e
+1
e
-1
e
0
e
+1
e
-1
e
0
e
+1
g
-1
g
0
g
+1
g
-1
g
0
g
+1
g
-1
g
0
g
+1
g
-1
g
0
g
+1
A B
C D
} ?
e
}?
g
Figure 3.2: Simpli ed model of quantum eraser process. A)  excitation
from g0 to e0; (B) Spontaneous decay by  transitions to superposition
of g 1 and g+1; (C)  excitation to superposition of e 1 and e+1; (D)
Spontaneous decay by  transitions back to g0.
46
any linear combination conserving angular momentum. In the given geometry, only
 + or   light couples to the H mode, with the helicity undetermined; thus, if the
emitted photon escapes the cavity before being reabsorbed, its detection places the
atom in the superposition j 0i=jg 1i+jg+1i (Fig. 3.2B). The atom is now in the
ground state with angular momentum perpendicular to the magnetic  eld, and thus
performs Larmor precession. When subsequently reexcited by the driven V mode,
state
j 0i= ei ( )je 1i+e i ( )je+1i (3.1)
is reached, with phase   ( ) gained through its precession in the ground state
(Fig. 3.2C). From here the atom can decay back tojg0i, emitting a second H-mode
photon (Fig. 3.2D). The probability to do so depends on  (t), giving rise to quantum
beats.
In summary, there are two paths for scattering a pair of photons into the H
mode: jg0i!je0i!jg+1i!je+1i!jg0i and jg0i!je0i!jg 1i!je 1i!
jg0i. The phase gained from the ground-state Zeeman shift (precession) di ers
along the two paths, which interfere to produce oscillations in the rate of delayed
coincidences|i.e., in the correlation function g(2)( ). Note that after the  rst pho-
ton is detected \which path" information is available, since jg+1i and jg 1i are
distinguishable in principle. This information is erased by the second detection.
What we have presented is a single-atom idealization. It neglects the pres-
ence of more than one atom in the cavity (not admissible for an atomic beam),
spontaneous emission to non-cavity modes, the  nite cavity decay rate, and the full
47
complement of magnetic sublevels for the employed F = 3 to F0 = 4 transition
(Fig. 3.1a). These features are included in a full quantum trajectory treatment,
including a Monte Carlo simulation of an atomic beam [71]. Two approximations
are adopted: (i) the driven mode is treated semiclassically (with absorption still
taken into account), and (ii) reabsorption of H-mode photons is neglected. The
approximations are justi ed by our moderate dipole coupling strength.
Consider  rst an atom prepared injg0ithat has not yet su ered a spontaneous
emission. Letjaii,ja0ii, andja00iidenote unnormalized states expanded, respectively,
over the mi = 0, mi =  1, and mi = 0; 2 sub-manifolds, as indicated by the
black, red, and black plus blue levels of Fig. 3.1a. These states correlate with the
scattering of zero, one, and two photons into the H mode. Should the atom undergo
a spontaneous emission (to non-cavity modes), the expansion manifolds, after the
quantum jump, are unchanged for a  -emission but move one step to the right or
left for a  -emission (mi!mi 1). Keeping track of these shifts, the system state
is expanded as
j i = j0ijAi+j1i
 NX
i=1
ja0iijAii
!
+p2j2i
NX
i=1
0
@1
2
NX
j6=i=1
ja0iija0jijAiij +ja00iijAii
1
A;
where j0i, j1i, and j2i denote zero, one, and two photons in the H mode, N is the
(time-varying) number of interacting atoms,jAi=ja1ija2i:::jaNi,jAii is the state
jAi with jaii omitted from the product, and jAiij is the state jAi with jaii and
jaji omitted from the product. This base state evolves under the coherent drive
and coupling to the H mode, the coming and going of atoms as they transit the
48
cavity, and spontaneous emission. From it, at regular sample times, we initiate
the collapsed state j 0i = ^bj i, where ^b annihilates an H-mode photon; thus, the
ground-state coherence is prepared as an entangled state of N atoms, which then
evolves in parallel with j i:
j 0i = j0i
N0X
i=1
jb0iijAii
+j1i
N0X
i=1
0
@
NX
j6=i=1
ja0jijb0iijAiij +jb00iijAii
1
A; (3.2)
where jb0ii = ja0ii and jb00ii = 2ja00ii at the start of a sample, after which jb0ii (jb00ii)
andja0ii(2ja00ii) di er due to their correlation with zero (one) rather than one (two)
H-mode photons; N0  N is the number of surviving entangled atoms, i.e., those
remaining in the cavity.
The source of the quantum beat is the atomic ground-state coherence preserved
in the vacuum of the cavity, i.e., the  rst term in Eq. (3.2). The system ground
state|both atoms and cavity|does not decay, and through  -excitation drives a
sustained excited-state oscillation in the atom mirroring Eq. (3.1):
gmi 1ei g 
  i(mi 1) jemi 1i+
gmi+1e i g 
 +i(mi + 1) jemi+1i; (3.3)
with  =  e  g, where  g ( e) are ground-state (excited-state) Zeeman detunings,
gmi 1 are Clebsch-Gordan coe cients, and  is the excited state linewidth; mi tracks
the state reached by atom i through possible spontaneous emissions. From Eq. (3.3),
oscillation at the ground-state frequency  g is passed to the probability amplitudes
for emitting a second H-mode photon through jb00ii and ja0jijb0ii in Eq. (3.2); note
thatjb0iiis a source term driving the equation of motion forjb00ii(the blue wavy lines
49
plus  -excitation in Fig. 3.1a). Only the amplitude and phase, but not the frequency
of the oscillation, will be a ected by a detuning of the drive from the atom.
The probability for emitting a second H-mode photon contains a term pro-
portional to hb00ijb00ii, summed over all surviving entangled atoms. It accounts for
the scattering of a  rst and second photon by the same atom and shows the quan-
tum beat introduced above. There is also a term computed from the norm of
ja0jijb0ii+ja0iijb0ji, i6= j, which adds probability amplitudes for \a  rst photon from
atom i and a second from atom j," and \a  rst photon from atom j and a second
from atom i." If the scattered  elds were classical,Ebi at time t andEai at time t+ ,
one would have the intensityjEajEbi +EaiEbjj2, where the interference 2Re(Ea j EbjEb i Eai )
disallows assignment of the  rst detection (superscript b) to the intensity of either
source (similarly the second). An assignment may be made in principle, however,
since one and only one atom changes its ground state when the photon is detected.
The change could be seen if one looked, thus providing \which-path" information.
As we do not look, the two quantum paths, \atom i then j" and \atom j then i",
interfere|after a second detection both atoms have changed state, so the \which-
path" information is erased. This interference also creates a quantum beat, as a
term analogous to the Hanbury Brown and Twiss thermal bunching peak [45], or to
the beating of light from independent atoms observed in Ref. [61].
50
-4 -3 -2 -1 0 1 2 3 4
1.0
1.2
1.4
1.6
1.8
2.0
b
 
 
 
1.0
1.2
1.4
1.6
1.8
2.0
a
 
 
 
?(?s)
g
(2)
(
?
)
g
(2)
(
?
)
Figure 3.3: Intensity correlation function of the H mode for 4{6 photons
in the V mode (when no atoms are present) and a 5G magnetic  eld: a)
 N = 0:2 and b)  N = 2:0.
51
3.3 Measurements
Figure 3.3a displays a measured correlation function at a magnetic  eld of 5G,
with a beat frequency of 4:9MHz. The V mode is populated with 4{6 photons, on
average, when no atoms are present, and the atomic  ux corresponds to  N  0:2
e ective maximally coupled atoms; most of the time there are no atoms well-coupled
to the mode [53]. The oscillation has low visibility and sits atop a raised Gaussian
background whose correlation time is given by the transit time of an atom ( 4 s).
The dominant beat for small  N is that arising from the emission of two photons
by the same atom (see Fig. 3.9c). Figure 3.3b displays the measured correlation
function with  N larger by a factor of 10 but otherwise similar conditions. The beat
visibility is improved. The background is also removed, evidence that there is now
an equal (at least) two-atom quantum beat (see Fig. 3.9d). The beat frequency
is reduced to 4:7MHz, an indirect e ect of increased absorption by the additional
atomic  ux, which reduces the V-mode photon number|by a factor of two|and
thus also the light shifts. A detailed study of light shifts is presented in Chapter 4.
Figure 3.4 shows a similar measured correlation function for a magnetic  eld of
4 G (A) and its calculated FFT power spectrum (B). The main peak near 3.5 MHz
corresponds to the quantum beat resonance. A smaller peak at half this frequency
is also visible, the result of homodyne interference with drive light mixed in by
cavity birefringence (see discussion below). The small sidebands on the main peak
correspond to a slight modulation of the beat envelope visible in (A). This is the
result of the small (1{2 ) deviation from normal incidence at which the atomic beam
52
g
(2)
(
?
)
1.5
1.4
1.3
1.2
1.1
1.0
0.9
? [?s]
0.0 2.0-2.0-4.0 4.0 6.0-6.0
0.0 2.0 4.0 6.01.0 3.0 5.0
0.0
1.0
0.2
0.4
0.6
0.8
Power [arb. units]
? [MHz]
A
B
Figure 3.4: Intensity correlation function and spectrum. A) g(2)( ) mea-
sured at 4 G magnetic  eld. B) FFT power spectrum of data in (A).
The peak at 3.5 MHz corresponds to approximately twice the Larmor
frequency for the ground states in a 4 G  eld.
53
-5
-4
-3
-2
-1
0
1
2
3
4
5 0
1
2
3
4
5
6
7
8
0.8
1.0
1.2
1.4
1.6
1.8
magnetic field (G)
g
(2)
(
?
)
?
(
?
s)
Figure 3.5: Evolution of g(2)( ) with increasing magnetic  eld.
traverses the cavity mode standing wave, resulting in a sinusoidally varying value
of g and amplitude modulation in the spontaneous emission rate. (This is the same
modulation envelope from Fig. 2.7.)
Figure 3.5 illustrates the change of the autocorrelation function with increas-
ing magnetic  eld. The beat frequency increases while the envelope from the atomic
transits stays constant. The beat visibility decreases for large magnetic  eld. Fig.
3.6 shows the e ect of stronger driving on the beats. The amplitude of oscillation
damps until at high drive only a thermal bunching peak remains. At even higher
drives this peak also disappears, the result of the larger fraction of coherent light in
the count rate, due to cavity birefringence and possibly the splitting of the sponta-
54
?3 ?2 ?1 0 1 2 3
2
1
1.5
?3 ?2 ?1 0 1 2 3
2
1
1.5
?3 ?2 ?1 0 1 2 3
2
1
1.5
?3 ?2 ?1 0 1 2 3
2
1
1.5
?3 ?2 ?1 0 1 2 3
2
1
1.5
n=0.4
n=1.2
n=4.2
n=13
n=46
g
(2)
(
?
)
?(?s)
Figure 3.6: Evolution of g(2)( ) with increasing photon number in V.
55
Figure 3.7: Evolution of the quantum beat as V-mode light is mixed with
the H mode. Upper and lower extremes correspond to approximately six
times more detector counts from the V mode than spontaneous emission.
The parameters are 1:2 photons in the V mode, a magnetic  eld of 4G,
and  N = 1.
neous emission spectrum with strong driving. The mechanism for the disappearance
of the beats in these two  gures will be explained in Chapter 4.
Figures 3.7 and 3.8 illustrate the change in the observed beat when the po-
larization presented to the detector is not taken orthogonal to the polarization of
the drive but is allowed to rotate by a few degrees. The rotation is controlled by
changing the angle of the HWP placed between the cavity and the PBS (Fig. 3.1b).
This mixes a small amount of drive light with the scattered light. With increas-
ing fraction of drive light, the beat is eventually dominated by a homodyne term
(Fig. 3.9e) arising from the correlation of a photon scattered into the H mode with
a photon from the drive; thus, as in the two-atom case, interfering time orders yield
a quantum beat. This beat oscillates at half the frequency and allows the correla-
tion function to dip below one. Generally, some drive light is coupled into the H
56
?3 ?2 ?1 0 1 2 3
0.5
1
1.5
2
?3 ?2 ?1 0 1 2 3
0.5
1
1.5
2
?3 ?2 ?1 0 1 2 3
0.5
1
1.5
2
?3 ?2 ?1 0 1 2 3
0.5
1
1.5
2
?3 ?2 ?1 0 1 2 3
0.5
1
1.5
2
0.0
?
0.5
?
1.0
?
1.5
?
2.0
?
?(?s)
g
(2)
(
?
)
Figure 3.8: Evolution of g(2)( ) with homodyne mixing. Parameters
as in Fig. 3.7, with angle of HWP labeled with respect to position of
maximum extinction.
57
mode through a small birefringence of the cavity mirrors. In Fig. 3.7, the evident
asymmetry with respect to angle is probably due to a small residual magnetic  eld
component in the direction of the cavity axis.
Quantum trajectory simulations agree well with the measurements. Figure 3.9a
displays a computed correlation function overlaying the data of Fig. 3.3b. A mean
velocity of 22 m/s  ts the decay of coherence well. Other parameters, such as the
 delity of the optical pumping, atomic beam tilt, and background from birefringence
or elsewhere are more di cult to accurately determine (no background correction
is made to the data). Note that with a background light amplitude  from birefrin-
gence, the post-detection state is j 0i+  j i, and  carries noise from the atomic
beam and spontaneous emission (absorption on the V mode). Plausible parameters
are used for the plots of Fig. 3.9, with the quality of the  t primarily determined by
the atomic beam density, which controls the relative size of the one- and two-atom
quantum beats, and the strength of the drive, which controls the level of sponta-
neous emission. For the parameters of Fig. 3.9a, an atom passing near the cavity
axis (within half a mode waist) typically undergoes  10 spontaneous emissions
to non-cavity modes during its transit, yet the coherence, merely passed between
di erent mi-multiplets, is preserved. In contrast, spontaneous emission decoheres
an excited-state beat.
Figure 3.9b displays an example of a correlation function with mixed drive light
(as in Fig. 3.7), together with its breakdown into three contributing quantum beats:
one-atom interference (frame c), two-atom interference (frame d), and homodyne
interference (frame e). The pieces lie in one-to-one correspondence with known
58
0.0
0.1
0.2
c
 
1
I
0.0
0.4
0.8
d
 
 
2
I
-3 -2 -1 0 1 2 3
-0.3
0.0
0.3
e
 
 
?
 
?
( s)
I
3
1.0
1.2
1.4
1.6
1.8
2.0
b
 
 
(
2
) (
 
 
)
?
 
g
0.8
1.0
1.2
1.4
1.6
1.8
2.0
 
(
2
)
a
(
 
 
)
?
 
g
Figure 3.9: Calculated intensity correlation function with the birefrin-
gence background set at 1% a) and 10% b) of the H-mode photon num-
ber; c), d), e) the three interference terms contributing in b); curve a) is
plotted against the data of Fig. 3.3 b). Parameters are  N = 4, 8 photons
in the V mode with no atoms present, and an atomic beam tilt of 1:3
degrees.
59
terms in the intensity correlation function for a source comprised of many scatterers
and a coherent background, e.g. Eq. (11) of [46], where the correlation function is
the sum of a single-atom term, g(2)A ( ), two-atom term, jg(1)A ( )j2, and a homodyne
term, Re[g(1)A ( )].
60
Chapter 4
Backward Light Shifts and Quantum Jumps
4.1 Introduction
Spontaneous emission is a fundamental damping mechanism for coherence
stored in optically driven atoms [72]. At random intervals, the atomic dipole be-
tween ground and excited energy levels disappears in a sudden projection onto a
single state, an event referred to as a \quantum jump" [12]. Under certain condi-
tions quantum jumps are directly observable in the measurement record, as when an
atom jumps intermittently between \bright" and \dark" states [73, 74]; experimen-
tal measurements of switching in atomic [75, 76, 77] and photonic [8] systems provide
clear visualization of the discrete nature of the dynamics. Although quantum jumps
often bring decoherence and spectral broadening{they give breadth to the natural
linewidth of resonance  uorescence{certain coherences may also be preserved or even
created in these processes under appropriate conditions [58, 65].
In contrast to the dynamical energy changes associated with quantum jumps
in a driven atom, light coupled to an atomic transition can also induce a structural
change in the energy levels through the AC Stark shift [78]. The energy eigenvalues
move in a direction determined by the sign of the light detuning from resonance,
but populations and coherences among the levels are preserved. For example, the
ground states of Fig. 4.1 exhibit equal but opposite AC Stark shifts when the laser
61
is tuned halfway between the two  transitions; the energy ofjg+idecreases and the
energy of jg i increases, reducing the ground-state energy separation (a negative
di erential light shift.) When preservation of coherence is paramount, quantum
state manipulation with lasers is typically performed far from resonance, where the
AC Stark shift is large but the spontaneous emission rate is small [79, 80]. Perhaps
for this reason, the evolution of coherence in the opposite regime, where AC Stark
shifts coexist with a large spontaneous emission rate, has not been exhaustively
investigated.
In this chapter we present measurements on a system near resonance where
high rates of spontaneous emission and quantum jumps not only preserve but dy-
namically modify a coherence, e ectively changing the sign of the ground-state di er-
ential light shift of the con guration illustrated in Fig. 4.1. Our analysis generalizes
a decoherence mechanism in Rayleigh scattering recently reported in the literature
[81]. We show that it encompasses a coherent as well as incoherent aspect, depending
on the size of the detunings.
4.2 Experimental system
We measure di erential light shifts in a beam of cold 85Rb atoms passing
through an optical cavity, using a ground-state interference technique [58]. A verti-
cally polarized cavity mode, V, is weakly driven with coherent light at 780 nm and
oriented with polarization vector parallel to a weak magnetic  eld (see Fig. 4.2).
62
e
+
e
-
g
-
g
+
?
e
?
g
?
g
?
e
g?
g?
Figure 4.1: Four-level atom with unequal Zeeman shifts in the ground
( g) and excited ( e) states. A coherent  eld  is resonant with the un-
shifted transition between ground and excited states (dashed lines) and
drives the two  transitions with equal coupling constants g.  transi-
tions (not shown) allow the spontaneous creation of an initial coherence
between the ground states, starting from an unshifted and driven m = 0
to m0 = 0 transition.
PBS
Magnetic field 
BS
APDs
V
H
Figure 4.2: Schematic of the apparatus for measuring g(2)( ), with a
single atom represented in green and the magnetic  eld inside the cavity
aligned to the exciting polarization; PBS: polarizing beam-splitter, BS:
beam-splitter, APD: avalanche photodiode.
63
Photons in V excite  transitions in the atoms between the F = 3 and F0 = 4
manifolds, and also cause light shifts when a detuning exists between a transition
frequency and the laser frequency. A degenerate but horizontally polarized mode,
H, collects photons emitted spontaneously on  transitions between the same two
manifolds; these photons have a bandwidth similar to that of the cavity, and can be
distinguished from those of the drive by polarization. Detection of a photon from H
spontaneously creates a coherent superposition among the magnetic ground states,
and conditional detection of a second photon from H reads out the coherence. This
setup enables a direct measurement of the Zeeman energy splitting of the ground
states as the frequency of a long-lived quantum beat in the intensity autocorrela-
tion function g(2)( ). The measurement is sensitive to a di erential light shift of
the ground states, which appears as an intensity-dependent change in the beat fre-
quency; and also to decoherence of the spontaneously-created superpositions, which
appears as a broadening of the resonance in the power spectrum of g(2)( ).
Examples of the beat signals taken from a full quantum Monte Carlo simulation
of our system appear in Fig.4.3, and from recorded data in Fig.4.4. Both show the
frequency of oscillation increasing as a function of average photon number in V,
evidently the result of a light shift. However, this behavior is contrary to that
expected from the AC Stark shift; our system roughly corresponds to the situation
depicted in Fig. 4.1, since most atoms are in a superposition of the m = 1 ground
states while the laser is resonant with the unshifted m = 0 to m0 = 0 transition.
Stronger driving rapidly damps the coherence, bringing faster decay times to the
oscillations.
64
?(?s)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
1.0
1.2
1.4
1.6
1.8
2.0
n=0.003
n=0.550
n=2.630
n=6.170
 
g
(2)
(
?
)
Figure 4.3: Quantum Monte Carlo simulation of intensity autocorrela-
tion signal from H for four di erent photon numbers n in the driven
mode V. (B) Composite experimental autocorrelation data as in (A)
with amplitude color scaled to show detail.
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
0
4
8
12
16
? (?s)
photons
1.0
1.1
1.2
1.3
1.4
1.5
2.0
g
(2)
(?)
Figure 4.4: Composite experimental autocorrelation data as in Fig. 4.3
with amplitude color scaled to show detail.
65
4.3 Theoretical model
We explain the origin of the light shifts and damping with the four-level atom
shown in Fig. 4.1, using the quantum trajectory formalism [12]. Our analysis applies
generally to any quantum system of this type and does not explicitly require a cavity,
which essentially provides a spatial mode for the optical excitation and detection.
The system is prepared at time t = 0 in the ground-state superposition j i=
1p
2(jg i+jg+i). (This occurs spontaneously in our apparatus after detection of a
single photon from H.) The weak coherent drive couples the ground and excited
states, with coe cients satisfying the equations:
_ce =   2ce  i ece +g cg ;
_cg =  i gcg  g  ce ; (4.1)
where g is the Rabi frequency, and  g and  e are ground and excited-state Zeeman
shifts. The ground-state superposition evolves in time as:
j g(t)i= 1p2(ei( g+ AC)tjg i+e i( g+ AC)tjg+i); (4.2)
where
 AC = g
2j j2 
( =2)2 +  2 (4.3)
is the ground-state AC Stark shift obtained from diagonalizing Eqs. 4.1 to lowest
order in g2j j2, and  =  e  g is the magnitude of the laser detuning from each
transition as depicted in Fig. 4.1. The di erential ground-state AC Stark shift is
2 AC, and it reduces the ground-state Zeeman splitting.
66
Under weak driving and with no quantum jumps, the amplitudes in Eq. 4.2
drive a steady-state superposition in the excited state:
j e(t)i=
 g 
p2
! ei( 
g+ AC)t
 =2 i je i+
e i( g+ AC)t
 =2 +i je+i
!
: (4.4)
The amplitudes of Eq. 4.4 oscillate at the ground-state splitting; the excited-state
splitting enters only through the factors  =2 i , which imply a phase shift. Inter-
ference of the amplitudes in Eq. 4.4 produces the quantum beats which we measure.
We show now that spontaneous emission in the form of quantum jumps adds
both a coherent and incoherent contribution to the time evolution, changing the
measured di erential light shift from 2 AC to  2 AC, while at the same time
decohering the ground-state superposition. (Here for simplicity we consider only
 jumps occurring in between the measured  emissions; additional  jumps serve
only to pump the coherence elsewhere in the manifold, changing Rabi frequencies
and adding additional detunings.) At jump rate  =  g2j j2=(( =2)2 +  2), the
driven dipole between ground and excited states turns o and the amplitudes of Eq.
4.4 are transferred to the ground states; in place of Eq. 4.2:
j g(t)i= 1p2
2
4  =2 +i q
( =2)2 +  2
ei( g+ AC)tjg i+  =2 i q
( =2)2 +  2
e i( g+ AC)tjg+i
3
5;
(4.5)
where the ground-state superposition has acquired a phase advance. The process
repeats for additional quantum jumps, so the phase advance accumulates over time.
For a jump rate  , we average thejg+ihg j-coherence against a Poisson distribution
of mean  t, to obtain the expectation value of the ground-state coherence:
 g+;g = e 2i( g+ AC)t12
1X
n=0
 ( =2 i )2
( =2)2 +  2
!n ( t)n
n! e
  t
67
= e 2i( g+ AC)t12e ( decoh+i2 jump)t: (4.6)
The imaginary part of the exponent contains a new term,  2 jumpt, where
 jump =    =2( =2)2 +  2 = 2g2j j2 
  =2
( =2)2 +  2
!2
: (4.7)
This term represents an additional frequency shift arising from the mean rate of
phase accumulation from quantum jumps. It is opposite in sign to the AC Stark
shift brought by the same driving laser (Eq. 4.3), and bigger than the AC Stark
shift by a factor of two when the detuning is small (i.e. when    =2). The net
di erential ground-state light shift in this case becomes
2 light = 2( AC +  jump)  2 AC; (4.8)
with the sign of the di erential AC Stark shift e ectively reversed. The e ect
requires at least the structure of a four-level atom, since jumps bring only an unob-
servable overall phase shift in a two-level atom.
The exponent in Eq. 4.6 also contains a damping term,   decoht, which deco-
heres the quantum beats at a rate
 decoh =  2 
2
( =2)2 +  2 = 2g
2j j2 
  
( =2)2 +  2
!2
: (4.9)
The origin of the decoherence is phase di usion of the ground-state superposition,
which accompanies the net phase drift that constitutes the frequency shift. Both
aspects are expected from the stochastic nature of the jump process. Noting that
 decoh=(2 jump) = 2 = , we may speak of a well-resolved frequency shift when
   =2. When this condition is not met, a single jump can result in a phase
68
change of  , smearing out the fringes over multiple measurements. In general, we
require the product of the jump number  uctuation with the phase shift per jump
to be much less than  in order to maintain coherence in the averaging process.
Increasing the magnetic  eld ( ) or the drive strength ( ) causes this product to
increase; this is the origin of the beat decoherence presented in Chapter 3.
The above analysis is valid for a four-level atom with constant coupling strength
to a weak coherent drive. Our experimental system brings many additional com-
plications. These include optical pumping within the full level structure of 85Rb, a
 nite interaction time with the laser and  uctuations in coupling strength due to
the atomic beam traversing the cavity mode, (weak) multi-atom and cavity e ects,
and saturation behavior with strong driving. For these reasons we implement the
theory as a full quantum Monte Carlo calculation in order to make quantitative
comparison with our measurements [58].
4.4 Data analysis
We analyze the data by calculating the Fast Fourier Transform (FFT) am-
plitude spectrum of each measured g(2)( ) and extracting the center frequency and
FWHM of each ground-state beat resonance peak. Figure 4.5 shows the weak drive
region for four di erent magnetic  eld values, with linear  ts to the data as dashed
lines; the slope increase is approximately linear with magnetic  eld (detuning). We
extract the natural (non-light-shifted) Larmor frequency for each magnetic  eld
value by extrapolating this portion of data to n=0. This point corresponds to the
69
0.0 
0.1 
0.2 
0.3 
0.0 1.0 2.0 3.0 
photons 
Shift (MHz) 
5 G
2 G
8 G
11 G
Figure 4.5: Center frequency of beat resonance at low photon number for
four magnetic  eld values. The dashed lines are linear  ts to the data
used to extrapolate the unshifted beat frequency. In all the following
 gures, error bars show  one standard deviation con dence intervals
of Voigt pro le  ts to the resonance peaks. Experimentally determined
photon numbers have an uncertainty of 20%.
70
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6 8 10 12 14 16 18 20
Freq
uen
cy 
shift
 (M
Hz
)
photons
4.9
Figure 4.6: Center frequency of beat resonance versus photon number
in V from FFTs of measured data (red  lled triangles) and simulations
(blue open squares). The solid line is a  t to a simple saturation func-
tional form.
unshifted beat frequency 2 g = 2 BBmag=3, where  B=1.4 MHz/G is the Bohr
magneton and Bmag is the magnitude of the magnetic  eld. From this frequency we
extract the value of Bmag and the detuning,  =  g=2.
Figure 4.6 tracks the frequency shift (red  lled triangles) for the data of Fig.
4.4, where a shift of 0 MHz corresponds to the natural Larmor frequency of 4.7 MHz
at 5 G magnetic  eld. The results taken from the quantum Monte Carlo calculations
are plotted as blue empty squares. The di erence between the two is primarily due
71
to the experimental uncertainty in intracavity photon number.
We  nd that despite the complications, the data points in the high intensity
region are described well by considering saturation of the jump rate as in a two-level
atom, i.e.  ! ( =2)(n=nsat)=(1 + n=nsat), where nsat   2=(8g2) is the saturation
photon number for a two-level atom in a cavity. In the absence of optical pumping
e ects, the e ective saturation photon number for an atom in the m = 1 ground
states of our system should be nsat =  2=(3g2max) = 5:7 photons, where gmax is the
peak value of g at the center of an antinode of the cavity Gaussian standing wave
mode function, and a factor of 3/8 has been included to account for the averaging
of g2 over the mode structure.
To match the frequency shift data in Fig. 4.6, we use the simple saturation
form for  together with Eq. 4.8 at lowest order in  :
2 light = Ashift (n=n
0
sat)
1 +n=n0sat :
We expect the e ective saturation photon number n0sat to be larger than nsat = 5:7 as
optical pumping puts the population in states with weaker coupling to the mode and
larger detunings. The scale factor of Ashift is to include any other optical pumping
and detuning e ects. The two-parameter least-squares  t shown by the solid line
in Fig. 4.6 gives Ashift = 0:81 0:01, n0sat = 9:2 0:2, with a reduced  2 = 0:32.
The statistical error bars in all the  gures are the 68.3% con dence intervals for
the given parameter (center frequency or linewidth) from a least-squares  t of each
FFT resonance peak to a Gaussian pro le.
Figure 4.7 shows the onset of decoherence with stronger driving, as extracted
72
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 2 4 6 8 10 12 14 16 18 20
F
W
H
M
 
(
M
H
z)
photons
i
ii
iii
Figure 4.7: FWHM of beat resonance versus photon number in V for
measured data (red  lled circles) and simulations (blue open squares).
The solid line is a  t to the data for a simple saturation functional form.
73
0.0 
0.5 
1.0 
4.0 4.5 5.0 5.5 
Frequency (MHz)
Amplitude (arb. units)
i
ii
iii
Figure 4.8: Three sample peaks from FFT amplitude spectra showing
shifts and broadening with increased drive; the labels refer to the three
points in Fig. 4.7 (i, ii, and iii).
74
from the widths of the resonance peaks from measured data (red  lled circles) and
from the quantum Monte Carlo simulation (blue empty squares). The zero-photon
linewidth (0.23 MHz) arises from the  nite transit time of the atoms traversing the
Gaussian cavity mode. To match the data, we apply a similar saturation form as
above to Eq. 4.9, at lowest order in  :
 decoh = Awidth (n=n
0
sat)(2 = )
2
1 +n=n0sat :
We convolve this Lorentzian width with the Gaussian FWHM arising from the
atom transits, 2 gauss = 2  0:23 MHz, in a Voigt pro le, using the numerical
approximation:
2 voigt 0:5346(2 decoh) +
q
0:2166(2 decoh)2 + (2 gauss)2:
A two-parameter least-squares  t to the composite width (solid line) gives Awidth =
1:31 0:03, n0sat = 14:2 0:5, with a reduced  2 = 1:6. Fig. 4.8 shows the broadening
and shift of three sample resonance peaks from the experimental data.
The results of the four-level model can be extended to include a detuning of
the drive laser from the unshifted (zero magnetic  eld) transition frequency (dashed
lines in Fig. 4.1), denoted below as  . The formulas for the di erential ground-state
light shift and decoherence rate become:
2 light = 2( AC +  jump) = 2g2j j2 ( =2)
2  2 + 2
[( =2)2 + ( +  )2][( =2)2 + (   )2]
(4.10)
and
 decoh = 2g2j j2  
2
[( =2)2 + ( +  )2][( =2)2 + (   )2]: (4.11)
75
-0.05 
0.00 
0.05 
0.10 
0.15 
0.20 
0.25 
0.30 
0.35 
-10 -8 -6 -4 -2 0 2 4 6 8 10 
Frequency (MHz)
Laser detuning (MHz) 
Full width at half maximum 
Shift 
Figure 4.9: Sensitivity of light shift and linewidth to drive detuning
 from the central (zero magnetic  eld) transition frequency, with 5
G magnetic  eld and n =2 photons in V. Blue circles show shift in
the center frequency from the zero-intensity value of 4.6 MHz, and red
squares show the resonance FWHM. The solid lines are two-parameter
 ts to Eqs. 4.10 and 4.11. Error bars show  one standard deviation
con dence intervals of Gaussian pro le  ts to the resonance peaks, and
for the case of the linewidths are smaller than the squares used to plot
the points.
76
Figure 4.9 presents the predicted and measured detuning dependence at a
magnetic  eld of 5 G. The photon number inside the cavity is kept constant at
n = 2 photons, compensating for the o -resonant response of the cavity and atoms.
The frequency shift (blue circles) decreases in both directions away from resonance.
We  t the data to Eq. 4.10 (blue line) with an amplitude scaling factor Ashift, while
keeping the same value obtained before for n0sat, which e ectively scales g2 for the
multi-level atomic structure. We  nd Ashift = 0:71 0:01 with a reduced  2 = 7:6,
due to a poor  t at large detuning and some possible  ne structure at small.
The resonance peak width (red squares) presents a similar structure, reducing
to the limiting value of 0.20 MHz (atom transit width) at a detuning of approx-
imately 10 MHz, where the di erential phase shift from each quantum jump is
approximately zero. The red line is a  t to Eq. 4.11 convolved with transit broad-
ening, with amplitude scaling as a free parameter. We  nd Awidth = 0:64 0:01 for
Eq. 4.11, with an order-of-magnitude discrepancy between the statistical error bars
from the resonance peak  ts and the  uctuations about the theoretical form.
4.5 Comparisons
Our results generalize the work of Uys et al. [81], who analyzed the e ect
of Rayleigh scattering on the decoherence of trapped ion qubits. Their result is
recovered from Eq. 4.11 in the limit    =2, i.e. under conditions where jumps
rarely occur but each carries a  phase shift, fully destroying the ground-state
coherence. More generally the character of the decoherence di ers depending on
77
the relative sizes of  ,  , and  =2.
We have presented in the above an analysis of ground-state superpositions
driven weakly by coherent light. A related situation was investigated by Cohen-
Tannoudji [82, 83], who studied atoms excited by broadband thermal light. In this
case, the ground-state superposition (induced by an external RF  eld) jumps at
random times completely to an excited-state superposition, with no coherent dipole
between ground and excited states, and the Larmor frequency changes from 2 g to
2 e. Remaining in the excited state for an average time of 1= , the relative phase
advance per jump is thus 2( e  g)= = 2 = . This should be compared to the
coherent driving case, where the relative phase advance per jump is 4 = (from Eq.
4.5 with    =2.) The extra factor of two arises because the damping rate for
the dipole is  =2, in contrast to the damping rate for the population, which is  ;
however, there is no AC Stark shift for broadband light near resonance, so the net
light shift in the two cases is exactly the same when  = 0. This is a remarkable
result, considering that the situations di er completely in detuning dependence and
require entirely separate physical descriptions. We also note that the e ect with
coherent drive is  ve orders of magnitude larger than that observed previously with
thermal light [82, 84, 85] and thus may need to be considered in some applications.
78
Chapter 5
Conclusions and Future Directions
The experimental results reported in this thesis all rely on conditional mea-
surements of light from one mode of an optical cavity that probe the atom-cavity
interactions occurring in the other mode. Although the basic concept of the appa-
ratus is simple, the complexity of the components{in particular the presence of an
atomic beam, multi-level atoms, and a mode-matched source of background light{
has allowed observation of several surprising e ects that may not appear in simpler
systems, e.g. in ones optimized for quantum information applications and precise
state control.
We demonstrated in Chapter 2 that a two-mode cavity QED system can func-
tion as a high- delity detector of single atoms. Our method can operate with a wide
range of photon  uxes and works for atoms traveling as fast as 20 m/s across the
mode. Because the cavity is frequency-selective and has a short lifetime (approxi-
mately equal to the atomic excited-state lifetime), the method guarantees that the
atom is in the F = 3 hyper ne ground state immediately after the detection of the
second photon. The scheme is sensitive enough to work without strong coupling of
the cavity and atoms, meaning that the detection volume can be kept large. We
also demonstrated how autocorrelation measurements of the emitted light show the
underlying dynamics of the atom as it traverses the Gaussian mode of the cavity
79
and encounters the standing wave.
In Chapter 3 we presented observations of ground-state quantum beats in the
spontaneous emission from a continuously driven atomic ensemble. Contrasting de-
terministic manipulations [62, 63], we demonstrated the spontaneous creation and
readout of ground-state coherence, where in the spirit of Forrester et al. [61], we re-
trieve a hidden beat from the  uctuations. Our theoretical treatment via a quantum
Monte Carlo calculation, which is in good agreement with the measurements, de-
composes the signal into a one-atom beat, a two-atom beat (interference of emission
time order), and a homodyne beat due to interference with a drive photon mixed
through birefringenece.
In Chapter 4 we reported evidence of repeated quantum jumps modifying the
spontaneously-generated coherence without destroying it, as a net drift guiding a
stochastic phase di usion. We presented a microscopic quantum trajectory model
that gives an intuitive explanation of the light shifts, and which also explains the
state decoherence at large detuning reported in Ref. [81]. Moreover, our results sug-
gest the tantalizing possibility of using dissipation as an additional tool for coherent
control.
Future upgrades to the experimental apparatus are planned. These include
adding additional viewports to the cavity chamber to allow for better control of
atomic state initialization as well as to give optical access to the cavity from the
side; building a new cavity with slightly stronger coupling strength and much smaller
absorptive photon losses; and implementing optical focusing of the atomic beam to
increase the density of atoms in the cavity. Details are given in Ref. [18].
80
We plan to study conditional quantum feedback in this system, similar to that
 rst demonstrated in Ref. [86], in which the conditional evolution of the atom-cavity
system was frozen by application of a control pulse. Compared to the time scale
of coherence in that experiment (set by the atom and cavity lifetimes), we have
access to ground-state coherences that can last as long as the atom remains in the
cavity, which relaxes some of the technical requirements for conditional control. We
discuss details of one possible implementation in Ref. [87]. Other possibilities may
arise with the additional optical access to the side of the cavity, such as measuring
the correlation functions conditioned on a click directly from the (non-collective)
atomic emission; this spontaneous emission arises from the same quantum jumps
that give the measured light shifts. Another possibility is direct optical manipulation
of the atomic states through the side of the cavity, using the output of one of the
modes or some auxiliary control  eld. The  exibility of the system presents many
opportunities, some of which are bound to yield even more surprising results.
81
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