ABSTRACT Title of dissertation: EXPERIMENTS WITH TRAPPED IONS AND ULTRAFAST LASER PULSES Kale Gifford Johnson Doctor of Philosophy 2016 Dissertation directed by: Professor Christopher R. Monroe Joint Quantum Institute University of Maryland Department of Physics and National Institute of Standards and Technology Since the dawn of quantum information science, laser-cooled trapped atomic ions have been one of the most compelling systems for the physical realization of a quantum computer. By applying qubit state dependent forces to the ions, their collective motional modes can be used as a bus to realize entangling quantum gates. Ultrafast state-dependent kicks [1] can provide a universal set of quantum logic op- erations, in conjunction with ultrafast single qubit rotations [2], which uses only ultrafast laser pulses. This may present a clearer route to scaling a trapped ion processor [3]. In addition to the role that spin-dependent kicks (SDKs) play in quantum computation, their utility in fundamental quantum mechanics research is also apparent. In this thesis, we present a set of experiments which demonstrate some of the principle properties of SDKs including ion motion independence (we demonstrate single ion thermometry from the ground state to near room tempera- ture and the largest Schro¨dinger cat state ever created in an oscillator), high speed operations (compared with conventional atom-laser interactions), and multi-qubit entanglement operations with speed that is not fundamentally limited by the trap oscillation frequency. We also present a method to provide higher stability in the radial mode ion oscillation frequencies of a linear radiofrequency (rf) Paul trap–a crucial factor when performing operations on the rf-sensitive modes. Finally, we present the highest atomic position sensitivity measurement of an isolated atom to date of ∼ 0.5 nm Hz−1/2 with a minimum uncertainty of 1.7 nm using a 0.6 numer- ical aperature (NA) lens system, along with a method to correct aberrations and a direct position measurement of ion micromotion (the inherent oscillations of an ion trapped in an oscillating rf field). This development could be used to directly image atom motion in the quantum regime, along with sensing forces at the yoctonewton (10−24 N) scale for gravity sensing, and 3D imaging of atoms from static to higher frequency motion. These ultrafast atomic qubit manipulation tools demonstrate inherent advantages over conventional techniques, offering a fundamentally distinct regime of control and speed not previously achievable. EXPERIMENTS WITH TRAPPED IONS AND ULTRAFAST LASER PULSES by Kale Gifford Johnson Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2016 Advisory Committee: Professor Christopher R. Monroe, Chair/Advisor Dr. William D. Phillips Professor James R. Williams Professor Alexey V. Gorshkov Professor Christopher C. Davis c© Copyright by Kale Gifford Johnson 2016 Dedication This thesis is dedicated to Chris, my advisor, and to the scientists in our group. For all the help you have given me, I hope this may be helpful in return. It is also dedicated to my other friends, who keep me going, and to my family, who love and support me. And finally, to Brittany, who would fall in love with a graduate student like me and be happy about it! ii Acknowledgments There are a lot of people that contributed (almost completely of their own volition) to the results of the research presented here, and also to my enjoyment as a graduate student. It all started when I was born...actually I won’t start that far back. Although I could with my family–thank you for giving me the freedom and support to find something I love to do. I love you, and I hope I give back as much as you give me. Thank you Chris, my academic father. It has been a pleasure learning the trade from you. Your busy schedule meant that I had to glean as much as I could from every one of our interactions. Thanks to your intensity to learn and compete, and your respect for skill and good comedy, music, sports, culture and family, I did enjoy our time together. You never wasted a moment of my time; even while I spent plenty of yours. With great delicacy, you smoothed out all the BS ideas I threw at you over these past years, and now I finally feel like a physicist. Thank you. Thanks David Wong-Campos, who came into the ”Ultrafast Lab” two years after me but had the audacity to direct some of our efforts towards optics research– his passion. You indeed taught me a great deal about optics, along with quite a few other things including reading science articles for pleasure, and that there is plenty out there which can still be greatly improved or invented. You have become a great friend, and I look forward to continuing the work we have started together. Thank you Jonathan Mizrahi for training me in the lab after I arrived, and for the technical and theoretical support you still provide. Your patience is something to aspire to. iii Thank you Brian Neyenhuis; in addition to the fierce nature with which you attack problems (I’m sure you’re laughing at that image), you know how to keep a team talking. While not all of the time spent is discussing physics, creative ideas spawned from many of our conversations while reloading the ion trap. Thank you Crystal Senko; our shared time on the ultrafast project was the shortest of the four, but I am very glad to have had it. Your methodical approach and long list of ion trap references taught me some important lessons early on. I also do not forget the help you gave when I moved, or all the advice you had for a new graduate student! Thanks also to all the other members of the ion trapping group that I’ve known over my 5 years in the group. Just listing your names isn’t thanks enough– but hopefully you remember what we’ve done, from useful ideas and discussions to doing smart and not-so-smart things at conferences and liquid lunches, long days in the lab, carrying furniture, wiring traps, discussing books, bicycling, trivia and friendly chalkboard arguments–David Hayes, Wes Campbell, Grahame Vittorini, Volkan Inlek, Clayton Crocker, Jake Smith, Ken Wright, Phil Richerme, Marty Lichtman, Nobert Linke, Kenny Lee, Emily Edwards, Rajibul Islam, Paul Hess, Ilka Geisel (visiting student), Simcha Korenblit, Susan Clark, Andrew Manning, Taeyoung Choi, Geofry Ji, Kate Collins, Caroline Figgatt, Aaron Lee, Kevin Lands- man, Shatanu Debnath, Guido Pagano, and Jiehang Zhang. Being around all of you was enjoyable, educational and rewarding. The exceptional quality of research that happens in our group and at the JQI is thanks to you. I also spent a few months in Australia working with Mike Biercuk. I would like also to thank him and his students MC Jarratt, Alex Soare, Harrison Ball and post iv doc David Hayes (again) for the refreshing time and the good research. Thanks also to our collaborator Alex Retzker and his students Itsik Cohen, and Amit Rotem. Your efforts will allow us to improve upon the work presented here. Finally, thanks to everyone else who was not mentioned but who helped me and this work. v Table of Contents List of Tables viii List of Figures ix List of Abbreviations xi 1 Introduction 1 2 The Apparatus 9 2.1 Linear Rf Ion Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The Blade Ion Trap and Vacuum Chamber . . . . . . . . . . . . . . . 12 2.3 Experimentation with One and Two Ions . . . . . . . . . . . . . . . . 20 2.3.1 The 171Yb+ Ion . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.2 Loading, Cooling and Detection . . . . . . . . . . . . . . . . . 22 2.3.3 Rotating Ion Trap Axes . . . . . . . . . . . . . . . . . . . . . 27 2.3.4 Doppler Cooling along multiple Axes . . . . . . . . . . . . . . 28 2.4 Second Ionization to Create 171Yb2+ with Laser Pulses . . . . . . . . 31 2.5 Imparting Ultrafast Laser Forces to Atoms . . . . . . . . . . . . . . . 34 2.5.1 Spin-Dependent Kick Generation . . . . . . . . . . . . . . . . 34 2.5.2 Frequency Domain Interpretation . . . . . . . . . . . . . . . . 42 2.5.3 The Mode-Locked Laser . . . . . . . . . . . . . . . . . . . . . 44 2.5.4 Picking and Rotating Pulses at 100 MHz . . . . . . . . . . . . 47 3 Stabilizing Ion Secular Motion 50 3.1 trap rf stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Ion Oscillation Frequency . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Limits and noise sources . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Sensing Atomic Motion 65 4.1 Experimental Description . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 Fock State Tomography . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Limits of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 75 vi 5 Ultrafast Schro¨dinger Cat States 77 5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 Three and Four-Component Cat Contrast . . . . . . . . . . . . . . . 89 5.3 Six and Eight-Component Cat Contrast . . . . . . . . . . . . . . . . 91 5.3.1 Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6 Highly Sensitive Atom Imaging 93 6.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.2 Aberration measurement and correction . . . . . . . . . . . . . . . . . 95 6.3 Position sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4 Sensing of Rf Induced Micromotion Position . . . . . . . . . . . . . . 103 6.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7 Realizing an Ultrafast Two Ion Gate 108 7.1 Single Mode Phase Accumulation . . . . . . . . . . . . . . . . . . . . 109 7.2 Two Ion Relative Phase Accumulation . . . . . . . . . . . . . . . . . 109 7.3 A Particular Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4 Measuring Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 118 8 Outlook and Future Directions 122 8.1 Imaging a Large Cat State and Other Motion . . . . . . . . . . . . . 122 8.2 Delta-Kicked Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . 123 8.3 Hamiltonian Engineering . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.4 Superfast Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.5 Two-dimensional Cat States . . . . . . . . . . . . . . . . . . . . . . . 124 8.6 Working in the Infrared . . . . . . . . . . . . . . . . . . . . . . . . . 124 Bibliography 126 vii List of Tables 2.1 Blade trap and chamber asembly in-vacuo parts . . . . . . . . . . . . 14 2.2 Blade trap and chamber assembly out-of-vacuum parts . . . . . . . . 15 3.1 Component Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 viii List of Figures 2.1 Linear Paul ”blade” trap . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Ion trap and vacuum chamber assembly . . . . . . . . . . . . . . . . . 13 2.3 Blade trap photographs and sketches . . . . . . . . . . . . . . . . . . 18 2.4 Circuit diagram of in-vacuo rf filter . . . . . . . . . . . . . . . . . . . 19 2.5 171Yb+ atomic energy level diagram . . . . . . . . . . . . . . . . . . . 21 2.6 Ion detection and imaging . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 Adjusting the trap axes . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.8 2D ion trajectory from scattering a single photon . . . . . . . . . . . 30 2.9 Observation of double ionization . . . . . . . . . . . . . . . . . . . . . 33 2.10 Conditions for a spin-dependent Kick . . . . . . . . . . . . . . . . . . 43 2.11 Experimental setup of Raman laser . . . . . . . . . . . . . . . . . . . 45 2.12 Triggering a pulse sequence . . . . . . . . . . . . . . . . . . . . . . . 49 3.1 Ion trap rf drives with active stabilization . . . . . . . . . . . . . . . 53 3.2 Helical resonator with a capacitive divider pickoff . . . . . . . . . . . 54 3.3 Rectifier circuit diagram and photograph . . . . . . . . . . . . . . . . 56 3.4 Suppression of injected noise . . . . . . . . . . . . . . . . . . . . . . . 57 3.5 Ramsey fringe contrast with and without feedback . . . . . . . . . . . 59 3.6 Secular frequency noise with and without feedback . . . . . . . . . . 59 3.7 Allan deviation of the secular frequency . . . . . . . . . . . . . . . . . 61 3.8 Simulation of rectifier response . . . . . . . . . . . . . . . . . . . . . . 64 4.1 Ultrafast atom interferometry . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Ultrafast sensing measurements of n¯ . . . . . . . . . . . . . . . . . . . 71 4.3 Ultrafast phase space tomography . . . . . . . . . . . . . . . . . . . . 74 5.1 Cat state experimental schedule and coherent state control . . . . . . 81 5.2 Cat state creation and verification . . . . . . . . . . . . . . . . . . . . 83 5.3 Three, four, six and eight-component cat states . . . . . . . . . . . . 86 6.1 Schematic of the imaging system . . . . . . . . . . . . . . . . . . . . 94 6.2 Aberration measurement results . . . . . . . . . . . . . . . . . . . . . 96 6.3 Position uncertainty of the trapped ion . . . . . . . . . . . . . . . . . 100 ix 6.4 Micromotion position measurement . . . . . . . . . . . . . . . . . . . 103 7.1 A two ion SDK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.2 Phase space and SDK timing for a gate . . . . . . . . . . . . . . . . . 114 7.3 Phase space trajectories for N1 = 1 . . . . . . . . . . . . . . . . . . . 115 7.4 Phase space trajectory for N1 = 2 . . . . . . . . . . . . . . . . . . . . 116 7.5 Phase space trajectory for N1 = 3, 4, 5, 6 . . . . . . . . . . . . . . . . 117 7.6 Ultrafast Entanglement Partiy Oscillations . . . . . . . . . . . . . . . 121 x List of Abbreviations AWG Arbitrary Waveform Generator BEC Bose-Einstein Condensate CCD Charge-Coupled Device CF ConFlat COM Center of Mass DC Direct Current EMCCD Electron Multiplying Charge-Coupled Device GVD Group Velocity Dispersion ICCD Intensified Charge-Coupled Device LO Local Oscillator MOT Magneto-Optical Trap NA Numerical Aperture PMT Photomultiplier Tube PSF Point Spread Function QIP Quantum Information Processing Rf Radiofrequency SDK Spin-Dependent Kick SQUID Superconducting Quantum Interference Device UHV Ultra-High Vacuum UV Ultraviolet xi Chapter 1: Introduction Since the proposal of an efficient quantum factoring algorithm [4], the past 22 years has seen a flurry of activity surrounding the development of quantum infor- mation processing (QIP). The number of experimental apparatuses that have been able to isolate individual quantum systems is now quite large–sufficiently describing each would require more than one thesis–and includes superconducting electronic circuits [5], microwave photon cavities [6], color centers in crystalline substrates [7], freely propagating photons [8], nanomechanical resonators [9–11], quantum ”dots” in Silicon [12], unbound room temperature molecules [13], cold, trapped, neutral [14] and charged atoms [15], atomic clouds [16] and molecules [17], plus more; this is not to mention the theoretical advances that have helped shape each of these fields. Each quantum technology has its advantages and disadvantages. In ion trapping, the story of isolating and controlling quantum systems started before the boom of QIP research. David Wineland’s 2012 Nobel Lecture [18] gives a clear account of the history, which is very closely related to this thesis and parallels part of, and I will try to summarize the relevant parts of his account. The first laser cooling of atoms occurred in 1978, in two different ion traps [19,20], following proposals that radiation pressure could be used to reduce the temperature of trapped 1 ions [21, 22]. Further progress in Doppler, then sub-Doppler cooling to near the ground state of the bound atom [23, 24], opened the door for quantum control of both the individual electronic states of an atom and of the harmonic motion of those trapped atoms. Through a relatively straightforward experimental process, by which a laser excites an optical transition in addition to changing the harmonic motion of the trapped atom by one quanta of energy–most commonly known as a Jaynes-Cummings interaction [25]–it became possible to prepare and control motion at the quantum level [18]. Following the theoretical groundwork [26], entangling gates between an internal and external degree of freedom of a single atom were carried out [27]. Spin-motion entanglement came shortly after, bringing about the creation of Schro¨dinger cat states [28] and eventually entanglement between two [29], and more [30] ions based on proposals using geometric phase as the entanglement generator [31]. Other methods for entangling quantum motion with internal states of an atom have emerged which hope to push the level of control further using near-field microwaves, or ultrafast laser pulses. These, and many other, advances of ion traps have built a foundation that is supporting a broad range of research in quantum science. The recent push to change the state-of-the-art in spin-motion entanglement has arisen because of several limitations associated with current methods. Performing high fidelity operations typically requires that the atom be within the ”Lamb-Dicke” regime where (kz0) 2(2n¯ + 1)  1 (k is the wavenumber of light driving the tran- sition, z0 is the ground state spread of the motion in 1-dimension, and n¯ is the average number of motional quanta, or phonons, stored in the oscillator). Remov- 2 ing this restriction allows the motion of trapped atoms to be controlled without the stringent requirement of near-ground-state cooling during experimental state preparation steps. Removing the restriction also allows larger quantum states of motion to be generated. Ultrafast laser pulses have proven to be one way of re- moving this restriction [1, 32], by manipulating states initially prepared to be far outside the Lamb-Dicke regime (n¯ up to 104) [33], and by generating entangled quantum states that extend outside of the Lamb-Dicke regime in just hundreds of nanoseconds–faster than a single period of harmonic oscillation. Using ultrafast laser pulses to manipulate quantum states also has the advan- tage of speed. In an experiment led by Wesley Campbell, before I joined the group, arbitrary, single-qubit pi-rotations were made on the 171Yb+ hyperfine ”clock” tran- sition (12.6 GHz splitting) in less than 50 ps [2]. This demonstration, where the qubit rotations occurred more than 104 times faster than a single period of har- monic oscillation of the trapped ion, with better than 0.99 fidelity took a small fraction, less than 10−8, of the qubit coherence time and demonstrated the first step towards a general quantum processor using only ultrafast pulses. As mentioned above, gains in computational speed are not limited to the internal degree of free- dom, and a technique has also been proposed (and implemented, as of this thesis) to generate two-qubit entanglement in a time shorter than a single motional oscillation period [3, 32, 34]. Demonstrating such a gate is an important benchmark because previous gates relied on spectral resolution of the phonons–quantized excitations in the eigenmodes of motion–which limits the gate time to be greater than the trap oscillation period. 3 Faster operations have additional advantages beyond reducing the effects of decoherence and increasing processing speeds: a fast enough entangling gate between two ions in a group can act through local modes of motion without exciting the collective motional modes of the larger group. This property could be a way to scale-up the number of operations available to a single group of trapped ions, which would complement other proposed methods such as shuttling ions into and out of different zones on a single apparatus [35], and using switchable interconnects to route photons between multiple quantum registers [36]. There are other reasons to use ultrafast operations, such as the ability to en- gineer Hamiltonians through trotterization (the decomposition of a desired unitary evolution operator into a stroboscopic sequence of shorter evolutions [37]) faster than the decoherence time of the ion for quantum simulation or cooling [38–40], studying quantum chaos in a delta-kicked harmonic oscillator [41], and others. Sim- ply having a nearly impulsive, qubit state-dependent force is generally useful, and demonstrations of some of the initial possibilities has been rewarding. This thesis follows closely behind the research laid out by Jonathan Mizrahi, Wes Campbell and others, who developed the first stages of this process and worked out how an ultrafast spin-dependent kick (SDK) can be made in the lab [1]. While creating and running the experiments in this thesis, focused on ultrafast control, several discov- eries and pieces of technology were made that turned out to be particularly useful. The second ionization of +Yb using ultrafast pulses was realized accidentally, while an improved method of stabilizing the harmonic oscillation frequency [42], and a generally useful microscopy technique used to localize a trapped ion position better 4 than 2 nm [43] were both realized intentionally. These early ultrafast, and other, experimental advances are the focus of this thesis. An outline is given below. • Chapter two is an overview of the experimental apparatus and techniques that were used while conducting this research. The chapter is focused mainly on the aspects to which I contributed directly, with supplemental details on theory and past work that is especially relevant. It covers the vacuum and trap assembly, the lasers, and a review of laser interactions detailed previously [44]. It also covers a short description of an effect we saw with one of our ultrafast lasers, where second ionization of 171Yb+ →171 Yb2+ was induced through a short exposure to high energy pulses. This was not pursued due to time constraints, but evidence of the process is clear. • Chapter three describes a method that we developed to stabilize the ion secular frequency by using active feedback control on the trap radiofrequency drive amplitude. This was important for ion trapping in general, because the secular frequency stability of most ion traps is about 1 part per thousand over an hour. We were able to improve this to 10 parts per million, in addition to reducing higher frequency noise. • Chapter four covers an experiment in which senses the initial motional state of a trapped ion. A single ion is prepared in a variety of different thermal and pure motional states. For any given state, the motion is divided into two copies, using SDKs, before being recombined some time later. The interference of the copies with each other is monitored. This process is referred to as atom 5 interferometry. By adjusting the time between SDKs, which open and close the atom interferometer, the motional state is mapped out. This was shown to work on thermal states ranging from ground-state-cooled to near room temperature, and also allows us to observe the quantum nature of a pure number state of the harmonic oscillator. • Chapter five describes an experiment where we explore the limits of creating Schro¨dinger cat states with spin-dependent kicks. In a harmonic oscillator, a cat state is often referred to as the quantum superposition of multiple classical- like oscillations which are localized to much less than the separation between them [18,45]. We create a cat state which grows in size by 4~k of momentum (k = 2pi/355 nm) by applying every pulse from a mode locked laser (repetition rate ≈ 80 MHz) to impart an SDK. This is accomplished using a switching technique that is discussed further in chapter 5. We also create the largest cat state ever made using a high fidelity technique that is resonant with half the trap frequency by kicking the ion every half trap period. Finally, we generate cat states with more than two localized components–up to eight–by applying additional microwave pulses during the kick sequences. • Chapter six gives a method by which we were able to correct the aberrations in a relatively high numerical aperture (NA) imaging system. The technique requires taking only a single shot intensity profile of light collected from a point-like emitter and fitting a truncated set of Zernike polynomials. Through this, we can determine the magnitude of standard aberrations the Zernike co- 6 efficients represent, such as defocus, coma, astigmatism and so on, and feed that information back to the lens position to create a diffraction limited spot. After using this technique with our high NA system, we integrated the im- age of a trapped 174Yb+ ion on an electron multiplying charge-coupled device (EMCCD) to achieve the best position localization ever measured from an isolated atom. In addition, we replaced the EMCCD with a razor blade and photomultiplier tube (PMT) to observe the motion of the ion by looking di- rectly at the position-dependent illumination of the mask and PMT. This is in contrast to all previous ion trap measurements at this small level of motion (10-20 nm), where the velocity-dependent fluorescence is correlated with the trap rf drive. • Chapter seven describes an experiment where two ions in the same trap are entangled with each other using a series of SDKs. This demonstrates the first temperature insensitive entangling gate, and also the first entangling gate which is not restricted to be temporally longer than a trap period. • Chapter eight is an outlook and covers current ideas on future directions for the experiments. These include pursuing higher fidelity entangling gate meth- ods by more effective power stabilization and possibly modifying the system to use a lower frequency optical or infrared laser pulse while pulse-shaping be- fore frequency multiplying to address the desired atom transition. This would allow us to use more sophisticated optical techniques and devices such as a spacial light modulator, which break down when used with even modest inten- 7 sities of ultraviolet laser light. Other experiments include looking at quantum chaos in a delta-kicked harmonic oscillator, faster ground state cooling using strong ultrafast pulses, directly imaging our large cat state and other mo- tional states, Hamiltonian engineering through Trotterization, or generating cat states in two dimensions. We believe there are many more possibilities with this experiment which have not been discovered. 8 Chapter 2: The Apparatus All of the experiments presented in this thesis were performed on 171 Yb+, 174 Yb+, and 171 Yb2+ trapped in a linear radiofrequency (rf) Paul trap. In my first summer with the group, I was fortunate to be given the responsibility of building our ion trap and vacuum chamber (with assistance from Jonathan Mizrahi and David Hucul among others). The account of that construction along with a brief introduction to linear rf ion traps is covered in this section. Additionally, the relevant atomic transitions and laser interactions required for trapping and controlling the motion and qubit state of the ions are presented. Finally, observation of a method to produce 171 Yb2+ ions from trapped 171 Yb+ is briefly discussed. There are many considerations which play into an experiment like this, but some are now standard and covered in previous theses and papers by our group and others. Other aspects of this experiment are less typical or even unique, and so more time is spent on them. 2.1 Linear Rf Ion Traps We trap individual ions in a linear rf trap, which consists of a two-dimensional rf quadrupole electric field superposed with a static quadrupole electric field to 9 provide confinement along the longitudinal direction [46]. Longitudinal confine- ment is typically much weaker than the transverse confinement, so that a crystal of laser-cooled ions resides along the x = y = 0 rf field null with minimal rf-induced micromotion [15, 47, 48]. The transverse confinement, dictated by the rf fields, is used as an information bus in many applications because motion along these direc- tions is at higher frequency, and the transverse normal mode spectrum for a chain of ions can be tuned [49]. When speed and independently tunable frequencies are not necessary, it is more common to use the stable, micromotion-free longitudinal modes. In the experiments presented here, we use a single transverse mode for all coherent operations. Linear ion traps exist in a variety of topologically equivalent electrode configurations, some even with electrodes all in a single plane for ease in lithographic fabrication [50]. The linear trap used in these experiments has four gold-plated ceramic “blade” electrodes with their edges running parallel to the longitudinal (z) axis of the trap, similar to the drawing in Fig. 2.1. Two opposite blades are driven with an rf potential with respect to the other two static blades, creating the transverse (x- y) quadrupole confinement potential. Appropriate static potentials applied to the longitudinally-segmented static blades serve to confine the ions along the z-axis. The rf electric quadrupole potential near the center of the trap V (x, y) = µV0 2R2 (x2 − y2) cos Ωt (2.1) is set by the rf amplitude on the trap electrode, V0, the distance from the trap center to the electrodes, R, the rf drive frequency Ω, and a dimensionless geometric 10 DETAIL A DETAIL B A B rf blade rf bladestatic blade static blade ion (radial confinement) (axial confinement) TOP FRONT z y x Figure 2.1: Linear Paul trap created with four gold-plated blade elec- trodes that are held in place by an insulating mount [42]. An ion is con- fined in between the electrodes through a combination of rf and static potentials applied to the electrodes. Each blade is split longitudinally into 5 segments that are electrically isolated on the static blades and electrically connected on the rf blades. The transverse distance from the ion axis to each electrode is R = 200 µm, and the length of the central longitudinal segments is 400 µm. 11 efficiency factor µ ∼ 0.3 for the geometry of Fig. 2.1. A particle with charge e and mass m inside the trap feels a resulting ponderomotive “pseudopotential” Upon = e2 4mΩ2 |∇V |2 = e 2µ2V 20 4mΩ2R4 (x2 + y2), (2.2) with harmonic oscillation frequency [51], ω = eµV0√ 2mΩR2 . (2.3) This expression is valid under the pseudopotential approximation where ω  Ω [47, 51], and we do not consider the residual transverse forces from the static potentials, because they are relatively small and stable. 2.2 The Blade Ion Trap and Vacuum Chamber The ion trap and chamber were assembled in-house, with a majority of the individual components being purchased or custom ordered. In order to minimize background gas collisions, which disturb experiments in a number of ways (formation of YbH+, population of the 2F7/2, and impulsive heating [44]), the ion trap is held at ultra-high vacuum, or UHV (< 10−9 torr), inside of a stainless steel chamber. (The chamber modules are connected using ConFlat (CF) technology.) Using clean assembly procedures and parts, we achieve vacuum pressures near 10−11 Torr, which allows a single trapped ion to be held without collision for ∼ 1 hour on average (collisions with H2, the dominant consideration for a clean system, are considered using the H2-ion Langevin cross-secion [52]). The trap and chamber shown in Fig. 2.2, drawn using the computer-aided design software Autodesk Inventor, was built using the parts in Tables 2.1 and 2.2. 12 End Chamber Support All Metal Valve 15-Pin Feedthrough NEG Getter Cartridge (inside nipple) Getter Cartridge Feedthrough Ion Pump Ion Gauge Blade Ion Trap Reentrant Viewport Octagon Ring Supports Main Chamber Support Viewports 2-Pin HV FeedthroughReducer Nipple Figure 2.2: Blade ion trap and vacuum chamber assembly. The ion trap is housed inside of a spherical octagon chamber for high optical access through fused silica viewports on the top, bottom, and sides of the chamber. Each section of the chamber is connected with stainless steel ”tubes” that are joined with ConFlat (CF) seals. Electrical connections to the ovens and blade segments are passed outside of the chamber with feedthroughs at either end of the chamber. 13 Table 2.1: Blade trap and chamber assembly in-vacuo parts. Viewports were anti- reflective coated by Spectra Thin Film for 355, 369 and/or 935 nm. Blades were gold coated by Sandia Nation Laboratories, two having static and two having AC style coating. Part Vendor Part Number Spherical Octagon Chamber Kimball Physics MCF450-SphOct-E2A8 Groove Grabbers (5) Kimball Physics MCF450-GrvGrb-C01 4-Way Standard Cross Kurt J. Lesker C-0275 Standard Tee (2) Kurt J. Lesker T-0275 Conical Reducer Nipple Kurt J. Lesker CRN275X133 Full Nipple Kurt J. Lesker FN-0337 All Metal Valve Kurt J. Lesker VZCR40R Zero-Length Reducer Flng. (2) Kurt J. Lesker RF337X275T Copper Gaskets Kurt J. Lesker GA-0133,... Macor Blade Holder Maryland Machine Blade (4) Laser Micromachining Ltd. Screws (to secure blades)(8) Small Parts Inc. 00-90, 0.125” SS 303 Washers (”)(8) Small Parts Inc. 0 outer diam-0.125” SS Lock Washers (”)(8) Small Parts Inc. 18-8 split lock washer SS Nuts (”)(8) J. I. Morris 00 SS 303 Washers (”)(16) J. I. Morris 00 SS 303 In Vacuo Capacitors (10) ATC 116UL821M100TT Gold Ribbon Semicond. Pkg. Materials 0.015”x0.0005” Constantan Foil Left-Handed Oven Holder (2) Mackenzie Machine Right-Handed Oven Holder Mackenzie Machine Oven Holder Mount (3) Mackenzie Machine 171Yb Enriched Oven Oak Ridge N.L. Yb Nat. Ab. Oven Alvasource AS-2-Yb-95-F Ba Nat. Ab. Oven Alvasource AS-2-Ba-55-F 15-Pin Vac. Side Cbl. Kit MDC 9921032 15-Pin Feedthrough MDC 9162002 2-Pin HV Feedthrough MDC 9422011 1.33” Viewport(6) MDC 9722013 4.5” Reenrant Viewport UKAEA 4.5” Viewport (Bottom) UKAEA Ion Gauge Agilent Technologies 9715007 Ion Pump Agilent Technologies 9191145 NEG Getter Cartridge SAES 4H04193 Getter Cartridge Feedthrough SAES 4H04023 Getter Ribbon SAES 4F0280D Getter Pellets SAES S5K0467 14 Table 2.2: Blade trap and chamber assembly out-of-vacuum parts. Part Vendor Part Number 15-Pin Air Side Cable Kit MDC 9921024 Ion Gauge Cable Duniway Stockroom ND-IGH-25-MultiGauge Ion Gauge Controller Agilent Technologies XGS600H1M0C0 Ion Pump Controller Agilent Technologies 9297000 Ion Pump Cable Agilent Technologies 9290705 Main Chamber Support (2) Mackenzie Machine End Chamber Support (1) Mackenzie Machine Octagon Ring Support (2) Mackenzie Machine Bolts (chamber assembly) Kurt J. Lesker SBK832050,... The chamber is designed to be compact, keeping the distances (spanned by ”tubes”) from the pumps to the trapping region as short as possible. The tubes are kept as wide as practicality allows. These designs are guided by standard vacuum principles, which can be formally described in an analogous manner to electrical circuits and Ohm’s law [53]. Consider first a tube under vacuum, well below the viscous flow pressure, in the molecular flow regime (.0.1 Torr). The conductance C of this tube is given by the equation Ctube ∝ d 3 l , (2.4) where d is the tube diameter and l is its length. Conductance C is in units of a volume flow rate (L s−1), and sets the throughput Q = C∆P (2.5) through a tube with pressure differential ∆P across the length l. Reducer nipples allow pipes of various dimensions to be connected, and in the limiting case of the zero-length reducer flange, the conductance is treated by considering an aperture 15 instead of a tube: Caperture ∝ d2. (2.6) The equivalent conductance of elements joined in parallel and in series is totaled in the same way as electrical conductors. It is clear, therefore, that short, wide tubes are best to increase pumping power at the ion trap. The higher pumping power reduces the amount of background gas which is produced by outgassing of all elements within the chamber. A getter cartridge in the form of St 172 (Zr-V-Fe) stacked disks pumps N2, CO, and H2 at a rate between 100 and 300 L s −1, while an ion pump at 20 L s−1 is able to pump virtually all gases and vapors. The speed at which these pump on the ion trap is governed by the conductance of the chamber, and the reducer nipple that lead between the elements. The total conductance, from air at room temperature Ctube,air ≈ d 3 l × 12.1 L s cm2 (2.7) Caperature,air ≈ d2 × 11.6 L s cm2 (2.8) Ctotal = ( 1 Ctube,air + 1 Caperature,air )−1 , (2.9) is about 70 L s−1. Between the conductance and pumping speed of both pumps, the total system speed is S = ( 1 Spump + 1 Ctotal )−1 = ( 1 100 + 1 70 )−1 ≈ 40 L s−1. (2.10) This tells us that the chamber is not throttling the pumps, which would be the case if Spump was much greater than Ctotal. Outgassing rates of 300 series stainless steel routinely reach 10−12 Torr L/s cm2 after having been baked at 250◦C (higher 16 than our usual 200◦C, but similar) for more than 15 hours [54]; if we estimate our chamber surface area at 200 cm2, the total outgassing throughput is Q = 2e−10 Torr L/s. Using the throughput relationship Q = PS, (2.11) the estimated final pressure would therefore be P = 5 × 10−12 Torr. This gives us some margin of error and indeed allows pressures deep into the UHV regime to be reached–we observe a pressure of ∼ 10−11 Torr in our system. As added assurance, NEG ribbon and getter pellets of the same material as the cartridge sit inside the spherical octagon. Although these are activated through the bakeout process alone, which takes them to a lower maximum temperature than achieved by the restive heating of the cartridge, they will still provide additional pumping. The ion trap itself was assembled by hand from a Macor holder and 4 gold- plated blade electrodes (Fig. 2.3). The blades were laser machined by Laser Micromachining Ltd. and subsequently coated by Sandia National labs. The blades are identical, except that the two rf blades are coated such that all 5 segments are shorted together, while the static blades have no connection between segments. The blades are wired using gold ribbon and wire bonding, with wire extensions consisting of Kapton coated wire. Each segment of the static blades is wired to the 15-pin feedthrough and filtered externally, before it is connected to a static voltage supply. The static blade segments are also capacitively coupled to a ground wire in order to filter rf pickup. Treating the ion trap as a simple capacitor, the system is 17 Figure 2.3: Blade ion trap photographs and sketches. The trap was assembled in house. Filters were assembled by a series of wire bonds and spot welds between Constantan foil (used for its high resistance, which is useful for spot welding), gold foil wire, capacitors and the blade segments. 18 Ctrap Clter Rlter Rfeed Vrf Vstatic blade Cfeed Z1 Z2 Vstatic Figure 2.4: Circuit diagram of in-vacuo rf filter. This filter is con- structed of the elements shown in Fig. 2.4, and is designed to reduce pickup on the static blades from the rf blades caused by a non-negligible resistance Rfeed between the static blades and ground (primarily the feedthrough). roughly equivalent to the voltage divider circuit diagram in Fig. 2.4, with Z2 ≈ ( 1 ZC,filter + ZR,filter + 1 ZR,feed + ZC,feed )−1 . (2.12) The impedance Z2 is the complex load on the trap and rf drive, and ZR,feed = Rfeed ≈ 30 Ω (due to the feedthrough and wire length), while |ZC,feed| = 1ωrfCfeed ≈ 0.01 Ω and can be ignored. The resistance to ground ZR,filter = Rfilter  1 Ω and can also be ignored when |ZC,filter| = 1ωrfCfilter & 1 Ω. It is clear that choosing the in- vacuo capacitor large enough allows us to ignore the resistance of the feedthrough. For this experiment, the in-vacuo capacitor has a value of 800 pF and so |ZC,filter| = 1 ωrfCfilter ≈ 10 Ω. Therefore, |Z2| ≈ 10 Ω. The trap capacitance Ctrap ≈ 10 pF, and so |Z1| ≈ 800 Ω. The rf pickup on the static blades is then Z2Z2+Z2Vrf ≈ 10−2Vrf . Further details of the bakeout procedure, drawings of custom parts, and exact trap and oven configurations inside of the spherical octagon are discussed elsewhere [44] and recored in the group laboratory files. 19 2.3 Experimentation with One and Two Ions In order to perform quantum mechanical experiments with trapped ions, a number of common laser techniques are implemented to ionize, prepare and detect the trapped ion qubit. Ionization consists of a two-color excitation of an electron to continuum. Preparation includes Doppler cooling and sometimes resolved side- band cooling [55], along with optical pumping to set the qubit in a known state. Detection consists of collecting state dependent fluorescence [56], which collapses the qubit wavefunction and allows for discrimination between the two possible out- comes. Ion fluorescence is typically collected with a lens and focused onto a PMT, while sometimes a camera is used to observe spatial information. 2.3.1 The 171Yb+ Ion The singly charged 171Yb+ ion is used in these experiments involving QIP for a number of reasons including its ground state hyperfine splitting, which provides a stable and accessible microwave qubit, and its fast cycling transition, which provides high efficiency qubit readout. An abridged energy level diagram in Fig. 2.5 shows the pertinent atomic structure. The 2S1/2 electronic ground state has a hyperfine splitting (interaction between single outer electron and spin-1/2 nuclei) of ωhf/2pi = 12.64281 GHz. The F=0 state is used as the logical |0〉 (denoted |↓〉 to avoid later confusion with motional state labels), and the F=1, mf=0 state is used as the logical |1〉 (|↑〉). The excited 2P1/2 level at 2pi/k ≈ 369 nm has a radiative lifetime of 8.12 ns 20 12.642812118466 + δ2z GHz δ2z = (310.8)B 2 Hz [B in gauss] 2S1/2 2.105 GHz 2P1/2 γ/2pi = 19.6 MHz δZeeman = 1.4 MHz/G 36 9. 52 61 n m (7 39 .0 52 1 / 2 ) 171Yb+ F=1 F=1 F=0 F=0 2D3/2 F=2 F=1 0.86 GHz 3[3/2]1/2 F=0 F=1 2.2095 GHz 93 5. 18 79 n m 2F7/2 F=4 F=3 F=3 F=2 1[5/2]5/2 63 8. 61 51 n m 63 8. 61 02 n m 0 1 2.438 μm 29 7. 1 nm τ = 8.12 ns γ/2pi = 4.2 MHz τ = 37.7 ns γ/2pi = 3.02 Hz τ = 52.7 ms 43 5.5 nm 467 n m τ = 5.4 yrs 2D5/2 F=2 F=3 41 1.0 nm γ/2pi = 22 Hz τ = 7.2 ms 2P3/2 100 THz 1.35 μm (9 9. 5% ) (0.5%) (0.2%) (1 .8 % ) (9 8. 2% ) 32 9 nm 1.65 μm(1.0%) (9 8. 8% ) (17 %) (83% ) 3.4 μm Zeeman Splittings (Δm=1): 2S1/2 F=1: +1.4 MHz/G 2P1/2 F=1: +0.47 MHz/G 2D3/2 F=1: +1.4 MHz/G 2D3/2 F=2: +0.84 MHz/G 2F7/2 F=3: +1.8 MHz/G 2F7/2 F=4: +1.4 MHz/G Percentages (XX.X%) show the branching ratio for that transition. Isotope Abundance 168: 0.13% 170: 3.05% 171: 14.3% 172: 21.9% 173: 16.1% 174: 31.8% 176: 12.7% (hfs/2 = 6.321406059233 GHz) (Shfs + Phfs = 14.748 GHz = 2 x 7.374 GHz) Figure 2.5: Atomic energy level diagram of 171Yb+. 21 dominated by the dipole allowed decay to the ground state. The F=0 and F=1 hyperfine levels are split by 2.105 GHz, and the F=1 states have a Zeeman splitting of 1.4 MHz/G. A second decay mechanism from the 2P1/2 manifold allows for popu- lation to fall to the 2D3/2 manifold, which has a radiative lifetime of 52.7 ms before decaying to the electronic ground state. The 2D3/2 state has a hyperfine splitting of 0.86 GHz. The 2P3/2 level is 100 THz higher in energy than the 2P1/2 level. This state is used for stimulated Raman transitions. A lower lying 2F7/2 manifold is not accessed through allowed transitions during our normal experimental operations, but sometimes becomes populated through collisions with background gas [57, 58]. This state has a radiative lifetime of over a year. The 2F7/2 and 2D3/2 can be excited to 1[5/2]5/2 and 3[3/2]1/2 bracket states, respectively. These bracket states involve excitations associated with lower lying electrons [59]. The 3[3/2]1/2 level, separated from the 2D3/2 manifold by 2pi/k ≈ 935 nm, has a radiative lifetime of 37.7 ns which will predominantly decay to the 2S1/2. It has a hyperfine splitting of 2.209 GHz. 2.3.2 Loading, Cooling and Detection During loading, a beam of neutral atoms is sent through the trap (ejected from a tungsten tube, stuffed with enriched 171Yb foil, that is resistively heated and aimed towards the trap), where ionization, cooling, and re-pump laser beams are focused to waists of ∼ 30 µm in the trapping region. As an atom passes through the trap, it is ionized by the beams and subsequently Doppler cooled. Ionization 22 of the neutral 171Yb into 171Yb+ is accomplished using 399 nm laser light resonant with the 6s2 1S0 ↔ 6s6p 1P1 transition in neutral Yb [60]. This isotope selective excitation allows a second photon from the cooling beam at 369 nm to drive the electron to the continuum, leaving an ion in the trapping region. Doppler cooling the ion is achieved by applying light that is red-detuned of the 2S1/2, F= 1↔2P1/2, F= 0 transition (Fig. 2.5), which scatters at a rate of [15] Γc(∆) = sγ 2 1 + s+ 4∆ 2 γ2 , (2.13) where ∆ is the laser detuning from resonance, γc is the decay rate, and s = I Isat = 2Ω2c/γ 2 c is the fractional saturation intensity of the transition(Ωc is the resonant Rabi frequency). When the Doppler width is much less than the natural linewdith of the transition, and the light is detuned by ∆min = γ 2 √ 1 + s, the minimum achievable energy (the Doppler limit) along a single dimension is given by [15] Emin = ~γ 4 √ 1 + s. (2.14) These laser interactions obey the dipole transition selection rules [61] ∆F = 0,±1 (2.15) F = 0 = F ′ = 0 (2.16) ∆MF = 0 for pi light (2.17) ∆MF = ±1 for σ± light. (2.18) While the ion cycles through this transition, it can off-resonantly excite the 2P1/2, F= 1 levels (with a probability that can be calculated from Eq. 2.13), which may 23 then decay to the 2S1/2, F= 0 or 2S1/2, F= 1 states. If it falls into the 2S1/2, F= 0 state, sidebands at 14.74 GHz (the hyperfine splitting of the 12.6 GHz qubit and 2.1 GHz 2P1/2 levels combined; generated as the second order sideband from a resonant EOM) will drive the 2S1/2, F= 0 ↔2P1/2, F= 1 transition, giving the atom more chances to decay back into the 2P1/2, F= 1 manifold and continue the cooling cycle. In addition to these 369 nm transitions, the atom may also decay from the 2P1/2, F= 0 state to the metastable 2D3/2, F= 1 manifold (lifetime of 52.7 ms) with a probability of 0.005. The Doppler cooling transition is cycled thousands of times during initialization, and the atom will fall to the 2D3/2, F= 1 states with a high probability. To solve this problem, a 935 nm ”re-pump” laser, with sidebands at 3.1 GHz to address all hyperfine levels, excites the 2D3/2 ↔3 [3/2]1/2 transition. The 3[3/2]1/2 manifold has a 37.7 ns lifetime and decays with probability 0.982 back to the cooling transition, thus requiring only a few cycles to ”re-pump”. Preparation of the |↓〉 state is achieved through optical pumping [56]. Light resonant with the 2S1/2, F= 1 ↔2P1/2, F= 1 will eventually trap the population in the 2S1/2, F= 0 (|↓〉) state with only a small probability of depopulation through 12.6 GHz-off-resonant excitation of the 2P1/2, F= 1 state. This process is performed while the cooling light is turned off. State detection is an incoherent process by which the population in each qubit level is determined. Light resonant with the 2S1/2, F= 1 ↔2P1/2, F= 0 transition is scattered, or not scattered, from the ion and collapses the state vector to either |↑〉 or |↓〉, respectively. This is a manifestation of the quantum measurement phe- nomenon known as the wave function collapse and, as mentioned, is the basis of our 24 measurement scheme. Continued application of this light produces a state depen- dent fluorescence signal [56], by which the number of photons scattered during a given period signals which state the ion has collapsed to. This must be performed in multiple repeated experiments (each experiment consists of cooling, qubit state preparation, coherent control of qubit and ion motion, and measurement) to pro- duce the probability distribution of the qubit state (i.e. a measurement in each experiment returns only a |↑〉 or a |↓〉). Additionally, experiments are run with a magnetic field applied to the ion, providing a clear quantization axis and splitting the F 6=0 hyperfine levels. This Zeeman splitting of 4 MHz (2.9 G x 1.4 MHz/G) prevents the ion from significant coherent population trapping during cooling and detection, but is modest enough to stay near resonance for efficient cooling from a single laser. Fluorescence from trapped ions is collected using a lens and PMT or camera. The lens is positioned to collect light from the reentrant window on top of the vacuum chamber, as seen in Fig. 2.6(a). The histogram, Fig. 2.6(b), shows the number of occurrences of a number of photons collected during an extended detection cycle after zero, one, and two trapped ions are prepared in the |↑〉 state. The ”zero” histogram is taken with no ions in the trap, and applying detection light to simulate scattering from trap electrodes during measurement. We measure no photon counts the majority of times (14932 occurrences of no counts in Fig. 2.6(b)), while the probability of counting a single photon is about the same when a single bright ion is trapped. The probability of counting two photons when there is no ion is essentially zero (much lower than counting two photons with a single bright ion). 25 tropweiV elohniP PMT/EMCCD snel AN 6.0 snel lacirdnilyC snel rotamilloC Ions 1 20 40 60 80 Number of collected photons in one detection cycle 0 1000 0 2000 (c) (b) Zero Bright Ions (max of 14932) One Bright Ion Two Bright Ions y po si tio n (μ m ) x position (μm) 0 2 64 8 0 1 2 3 (a) N um be r o f o cc ur an ce s Camera PMT 158 mm 6 mm 303 mm 11.6 mm 1.6 mm Figure 2.6: Ion detection and imaging. (a) Schematic diagram of the ion location, viewport, 0.6 NA lens stack, pinhole, collimator lens, cylindrical lens and the PMT or EMCCD. The lens stack is made by Photon Gear [43, 62], and is designed to focus onto the pinhole. The cylindrical lens is used to correct astigmatism from the 0.6 NA lens stack and viewport. The collimator lens is focusing light from a small solid angle, and so a plano-convex lens suffices. (b) Histograms are used to determine the qubit state of one or two ions. Light collected on the PMT during a single experimental detection cycle (scattering from the ion) produces a certain number of counts at the PMT. This number comes from one of three distributions localized around the average number of photons when there is zero, one or two fluorescing ions. By repeating the experiment many times, we build up a picture of this distribution and can determine the qubit state populations after an experiment (with this method, we cannot tell which ion is bright if there is only one fluorescing). (c) Light from two fluorescing ions collected on a camera gives spacial information about the ions. 26 So When working with a single ion, discrimination between the ”bright” (|↑〉) and ”dark” (|↓〉) state is achieved by calling any detection cycle in which 0 or 1 counts was detected as |↓〉. Any other number of counts is considered as measuring a |↑〉. By contrast, we see significant overlap between the one and two bright ion histograms in Fig. 2.6, and so a simple thresholding method does not work. Instead, we use these extended exposure (extended to reduce noise) detection histograms to fit a set of three Poisson distributions as a calibration. Then, after a set of a experiments, we fit the measurement histogram to a weighted sum of the three calibrated Poisson distributions and determine the probability of zero, one, and two bright ions from the weight of each coefficient. Figure 2.6(c) shows a high resolution image of two trapped ions taken by collecting the fluoresced photons on a camera. The details of this imaging are discussed further in Chapter 7. 2.3.3 Rotating Ion Trap Axes In most of our experiments, we use a laser beam to coherently excite the ex- ternal motion of the trapped ion through Raman transitions. Knowing the coupling strength of this laser to each axis of ion motion is critical for high fidelity oper- ations, such as atom interferometry, multi-qubit logic gates and others. One way to handle this issue, and probably the simplest, is to couple the laser to just one axis of motion. Because the laser beam is constrained to come through the trap at angles corresponding to available windows in the chamber, fine adjustments of the mode coupling are made by rotating the axis of the trapping potential. Figure 2.7 27 shows how the axes of the trap can be rotated and the relative strengths tuned for different trap geometries. The trap used in this experiment is not square, and so we are not able to make the potential degenerate, but we are able to rotate it so that the beam only couples to one axis of motion. 2.3.4 Doppler Cooling along multiple Axes So far, the discussion of Doppler cooling has been restricted to a single dimen- sion. At low laser beam intensities (compared to saturation), which is the regime of typical operation, the concepts are easily expanded to consider cooling along all three directions. Figure 2.8 gives an intuitive (albeit classical) understanding of how absorption (or emission) of a single photon will affect the motion of a trapped ion in two dimensions (similar to three dimensions). First, consider the two radial modes of motion, where the secular frequency along the x-direction is different than in the y-direction. This is called non-degenerate for an obvious reason. If a photon moves the ion (starting from rest at the center of the trap) along only the x- or y-direction, the resulting motion of free evolution is restricted to a single dimension (Fig. 2.8(a)). If the photon pushes the ion at an oblique angle to the axes (Fig. 2.8(b)), then motion is not only excited along both axes, but the trajectory is two- dimensional. This implies that a single laser beam at an oblique angle to the trap axes of a non-degenerate trap can Doppler cool all three degrees of freedom (or, a Raman beam can push the ion into three-dimensional trajectories). Finally, if the secular frequency is equal in all directions (degenerate), then a single photon can 28 Square Trap Rectangular Trap VR VR VR=0 VR VR VR Raman Laser ∆fmin (a) (b) Tr an si tio n Pr ob ab ili ty Raman Detuning (MHz) -1.0-1.1-1.2 Figure 2.7: Adjusting the trap axes. (a) Trap electrode geometry determines how the trapping potential will look. For a square system of four electrodes, modifying the relative static voltage (VR, a bias on the rf signal ranging from about -10 V to +10 V) while maintaining the ion at the trap center will change the ellipticity of the trapping potential, but will not rotate it. Doing the same thing on a trap that is not square will rotate the axis while modifying the ellipticity. The blue ellipse suggests the nature of the equipotential lines in the trap, and the light dashed lines along the major an minor axes of the ellipses indicate the direction of the trap modes. (b) The trap mode frequencies can be seen spectroscopically by observing the motional sidebands [55] of the ion with a Raman laser. Here we see the transition probability in a rectangular trap as a function of detuning from the carrier of a Raman transition. For VR 6= 0, the (red) sidebands of both modes are visible (the relative coupling is not equal and gives rise to the unequal heights of the peaks). For VR ≈ 0, the modes are separated by the minimum allowable distance ∆fmin, determined by the trap geometry (the left peak approaches zero because the Raman beam is along the direction drawn, and does not couple to the weaker mode when VR = 0). 29 1.5 1.0 0.5 0.5 1.0 1.5 1.5 1.0 0.5 0.5 1.0 1.5 1.5 1.0 0.5 0.5 1.0 1.5 1.5 1.0 0.5 0.5 1.0 1.5 x y x y (b) (c) Initial kick direction t=0 t=01.5 1.0 0.5 0.5 1.0 1.5 1.5 1.0 0.5 0.5 1.0 1.5 x y (a) Initial kick direction t=0 non-degenerate degeneratenon-degenerate Figure 2.8: 2D ion trajectory from scattering a single photon. (a) In a non-degenerate trap, an ion kicked along one principle axis will stay on that axis. This is important both for cooling and for driving SDKs. (b) In a non-degenerate trap, an ion kicked off-axis will undergo a trajectory through two (or three) dimensions. This means a single laser beam can cool in two (or three) dimensions, and a single SDK can excite motion in two or three dimensions. (c) A single kick in any direction of a degenerate trap will only excite one dimension of motion. only excite one-dimensional motion. Thus, a completely degenerate trap requires three beams to cool in all dimensions, or two beams if two of the modes are degen- erate. The Doppler limit for all configurations is discussed in detail elsewhere [63]. When multiple ions are trapped, we have found that higher intensity laser light is often required to maintain the chain of ions over the course of minutes and hours. We add this light in as an additional cooling step, before lower intensity Doppler cooling takes the ion to the Doppler limit. The high intensity light is also detuned to account for power broadening. This process is altered quite often, so the exact power and detuning change from week to week, but the intensity is typically at least twice that of saturation. We believe this additional light is required because background gas collisions can displace the ion chain enough to cause the trap to heat the chain [52]. 30 2.4 Second Ionization to Create 171Yb2+ with Laser Pulses Recent results have suggested the usefulness of highly charged trapped atoms for use in atomic clocks, quantum information and measuring the fine structure constant [64]. Although this was not an area of intended research during my PhD, we did encounter a method for reliable double ionization of 171Yb using a pulsed 355 nm laser. The details of this interaction were not deeply investigated because the laser causing this transition stopped functioning just days after seeing the process. The ionization was an undesirable effect of the laser in this case, but its significance is not lost on us. We investigate second ionization using a mode-locked 355 nm laser (High-Q Picotrain) with 6 W average power and transform limited pulse of duration τ ≈ 10 ps. We believe the ion undergoes a multi-step process by which it is excited and decays to the long-lived 2F7/2 state (Fig. 2.5) before a second excitation drives the outer electron to the continuum. Evidence for this comes from an experiment where we trap three ions in a linear chain. To the middle ion, we apply a series of SDK operations during a simple experiment–Doppler cooling, SDKs, detection–which is repeated. (Single pulses with no motional coupling instead of SDKs should have the same effect, but we did not get a chance to try.) In order to show that the ion enters the 2F7/2 state before second ionization, we apply a series of SDKs that is sufficiently large enough such that the 2F7/2 will be populated, but sufficiently small enough that we will detect the population change before ionization occurs. This is done as follows: when the middle ion goes dark 31 without pushing the outer ions farther away, we apply a pump beam at 638 nm to depopulate the 2F7/2 state [56]. If the ion returns to its ground state where we can cool and detect it, we presume it was in the 2F7/2 state. We indeed observe population of the 2F7/2 state (rate not measured). We then apply a short series of pulses which takes the ion to the 2F7/2 state, but instead of pumping out, we apply another series of SDKs and observe second ionization. It therefore appears that the atom only undergoes second ionization from the 2F7/2 state. To observe these processes we apply the short series of SDKs while looking at a continuous camera image feed to monitor the spatial configuration of the chain. When the center ion goes dark, but the chain spacing remains the same as when all three ions were bright (Fig. 2.9(a),(b)), the ion is in the 2F7/2 state. When the outer ions move farther apart, the image indicates that the middle ion has been doubly ionized (Fig. 2.9(c),(d)). This should give a good estimate of the ionization rate and the path it takes, but we were unable to complete this study. For added assurance that we are seeing double ionization, we fit the images along the axis to determine their positions (Fig. 2.9(b),(d)). Consider the forces acting on one of the two end ions at position x in equilibrium F = q2 (2x)2 + qqc (x)2 − Eqx = 0, (2.19) where q is the charge of the end ions, qc is the charge of the center ion, and E is the electric trapping field gradient. If we let xe be the ion position when the middle ion is singly charged (qc = e), and x2e be the position for qc = 2e, we can write down 32 100 200 300 400 500 600 700 0.85 0.90 0.95 1.00 100 200 300 400 500 600 700 0.70 0.80 0.85 0.90 0.95 1.00 586 .9(3 ) 173 .4(3 ) 213 .3(3 ) 383 .1(2 ) 553 .2(3 ) 7004000 x postion (pixels) 7004000 150 0 150 0 x postion (pixels) 7004000 7004000 y po st io n (p ix el s) y po st io n (p ix el s) co un ts co un ts (a) (c) (d) (b) Figure 2.9: Observation of double ionization. (a) ICCD image of three trapped, singly charged ions using the lens system described in Fig. (6.1). When the middle ion is in the 2F7/2 state, it appears dark in the image (not shown). (b) x-position dependent counts of the ion image in ”(a)” (blue points) with fit to Gaussian peaks (green line). The fit is used to determine the position of each ion, labeled above peaks. The dip in counts near pixel 500 is from a defect in the CCD chip. (c) ICCD image of three trapped ions, the middle is doubly charged and the outer two are singly charged. The middle ion appears dark because it does not fluoresce from the illuminating light. (d) x-position dependent counts of the ion image in ”(c)” (blue points) with fit to Gaussian peaks (green line). The fit is used to determine the position of each ion, labeled above peaks. 33 the simple relationship x2e xe = ( 9 5 )1/3 ≈ 1.216. (2.20) The ratio of the measured values of x2e and xe (from values shown in Fig. 2.9(b),(d)) yields 1.217(3), within error of Eq. 2.20, and strongly suggests that the center ion is double ionized. 2.5 Imparting Ultrafast Laser Forces to Atoms This section describes the method by which a pulse of light couples internal and motional degrees-of-freedom of a trapped 171Yb+ ion. These processes are the basic building blocks of much of the research described in this thesis. Much of our understanding was explicitly laid out by Wes Campbell and Jonathan Mizrahi [2, 32, 44]. Because the interactions are crucial to understanding how the results presented have come about, I will discuss the relevant details and consequences in the context of how I have worked on the whole project while leaving the first- principles derivation as it stands in other works. 2.5.1 Spin-Dependent Kick Generation Consider a single atomic 171Yb+ ion in a linear rf Paul trap, in a uniform magnetic field B oriented along the x-axis of the ion trap and under the presence of a laser beam with time-dependent electric field envelope X`(t). Single photon transitions between the qubit levels 0: F=0 and 1: F=1, mf=0, and the 2P1/2 levels 2: F=0; 3: F=1, mf=-1; 4:F=1, mf=0; 5:F=1, mf=1 (recall Fig. 2.5) have single 34 photon Rabi frequencies g`ij(t) = −X`(t)e−ıφ`〈i|µ · ˆ`|j〉, (2.21) where i and j are indices for the state (0-5), ` is the index of the laser beam (one beam for now, but two next), µ is the electric dipole moment, and φ` = k`x+ φ ` 0 is the position-dependent phase of the photon with wavenumber k` and offset φ ` 0. The photon polarization vector ˆ` = cos(β`)σˆ + + sin(β`)σˆ − is defined by the parameter β` (for instance, β` = pi/4 is pi-light). Now consider a single 355 nm laser pulse is divided and applied to the ion simultaneously with pulses counter-propagating along B. Choosing the quantization axis along the magnetic field direction, there is no pi-light, and only the transitions 0 ↔ 3, 1 ↔ 3, 0 ↔ 5, and 1 ↔ 5 are possible. The total Rabi frequency for two-photon Raman transitions is therefore Ω = 1 2∆ (g1∗03 + g 2∗ 03)(g 1 13 + g 2 13) + 1 2∆ (g1∗05 + g 2∗ 05)(g 1 15 + g 2 15), (2.22) where ∆/2pi = 33 THz is the detuning of the laser light from the 2P1/2 (which is much greater than the Zeeman and hyperfine splitting, and so is considered the same for each of the Raman Rabi frequencies). We see that transitions through level 3 are only made by σˆ− light, while transitions through level 5 are only made by σˆ+ light. The total Rabi frequency becomes Ω(t) = γ2 12∆ [ I1(t) Isat cos(2β1) + I2(t) Isat cos(2β2)+√ I1(t)I2(t) I2sat cos(β1 + β2) ( ei(2kxˆ+φ0) + e−i(2kxˆ+φ0) )] , (2.23) 35 where the single photon Rabi frequencies have been computed [61], I`(t) = (X`(t)c0/2) 2 are the beam intensities, Isat = ~ω30γ/(12pic2) is the saturation intensity, φ0 = φ10−φ20 is the relative phase offset of the two beams, γ is the excited state decay rate (∼ 20 MHz), and k1 − k2 = 2k. In these experiments, we apply the counter-propagating beams linearly polar- ized with mutually orthogonal configuration (lin-perp-lin) that arrive simultaneously at the ion 1. Equivalently stated, β1 = pi/4 and β2 = −pi/4, and so only the last term in Eq. 2.23 survives. The magnitude of the Rabi frequency in Eq. 2.23 should also include contributions from the 2P3/2 F=1 levels, which is not computed here but follows the same process [44]. This produces a Hamiltonian (including the qubit energy) of the form Hˆ(t) = √ Ω1(t)Ω2(t) cos[2kxˆ+ φ0]σˆx + ωhf 2 σˆz, (2.24) where Ω1(t) and Ω2(t) are the time-dependent effective Rabi frequencies of the counter-propagating pulses (not a physical process because a single beam does not drive transitions, but represents the beam strength), and σˆx,z are the Pauli spin operators of the qubit [44]. If we make the approximation that the effective Rabi frequencies are applied as pulses which are infinitely short in time compared to qubit and trap evolution (this has been proven to be a fine approximation [44]), and consider a string of these 1overlapping the pulses is not difficult–each pulse is about 3 mm long, which is managable for any micrometer. 36 pulses arriving at times tl, the Hamiltonian becomes Hˆ(t) = θδ(t− tl) cos[2kxˆ+ φ(tl)]σˆx (2.25) = θδ(t− tl) cos[η(aˆ+ aˆ†) + φ(tl)]σˆx, (2.26) where the Rabi frequencies have become Dirac delta functions with pulse area θ, the position operator xˆ has been substituted for the raising (aˆ†) and lowering (aˆ) operators and of the ion harmonic motion, and the phase of the light is φ(tl) = φ(0) + ωAtl. The Lamb-Dicke parameter η = 2kx0, (2.27) (≈ 0.2 in all our experiments) is something that comes up a lot in these atom-laser interactions, where x0 = √ ~ 2mω is the ground state spread of the harmonic oscillator. The time evolution of this interaction [44] Uˆtl = exp ( i ∫ Hˆ(t)dt ) (2.28) = exp(iθ cos[η(aˆ+ aˆ†) + ∆φ(tl)]σˆx) (2.29) = ∞∑ n=−∞ inJn(θσˆx)e in[η(aˆ+aˆ†)+∆φ(tl)] (2.30) = ∞∑ n=−∞ inJn(θ)e in∆φ(tl)Dˆ[inη]σˆnx , (2.31) produces a phenomenon known as Kapitsa-Dirac scattering where the ion is diffracted into an infinite number of momentum states by the polarization gradient of the light. The polarization gradient arises from the lin-perp-lin configuration of the laser beams [61]. Jn(θ) is the n th order Bessel function of the first kind, and the displacement operator Dˆ[α] = exp(αaˆ† − α∗aˆ), where the imaginary axis is mo- mentum pˆ and the real axis is position xˆ. Notice also that the diffracted orders 37 are associated with alternating spins. Although this appears to be a complicated result, there is a way to simplify it. We do so by stringing together multiple pulses arriving at different times tl and offsetting the frequency of the opposing light paths by an amount ωA provided by a set of acousto-optic modulators. This focuses the Kapitsa-Dirac diffractions into a nearly single, spin-dependent order, producing a spin-dependent kick (SDK). As far as we know, the only way to solve the problem of determining proper pulse timings is by numerical simulation, and this is an active area of research for our theory collaborators. For the experiments shown here, we have made two simplifying assumptions and found surprising good solutions. The assumptions are: 1) Every pulse has the same pulse area θ = Θ N . This simplifies the experimental apparatus. 2) The SDK occurs much faster than a trap period. The reason for these will become apparent as I run through a simple, but fairly effective, simulation as an example of what we do to solve for delay settings. There are several parameters to play with: pulse area θ, AOM frequency shift ωA, and pulse timings. First lets say we are going to have eight pulses. (Eight because it is manageable but effective using our beam splitter and delay setup.) The schematic for this is described in the next section (Fig. 2.11). If eight pulses are used, lets try the intuitive pulse area θ = pi/8 so that the total effect is a full spin flip for each diffracted order 2. Finally, lets say that population totaling . 0.001 is not worth 2Remember, this should all be played with for complete results. This is an area of research in its own right, and we did not explore all possible options but are working on expanding the search. 38 accounting for. This means we can probably drop orders J±3(pi/8) and above from Eq. 2.31 for a reasonable estimation: J0(pi/8) = 0.96; J±1(pi/8) = ±0.19; J±2(pi/8) = 0.019; J±3(pi/8) = ±0.0012. Now we can try a simulation, in which free evolution between each kick is governed by the evolution operator Uˆfe(t) = exp(−iωhf tσˆz/2). (2.32) From equation 2.31, the truncated pulse operator is Uˆtl ≈ J0(pi/8) + iJ1(pi/8) ( eiωAtlDˆ[iη] + e−iωAtlDˆ[−iη] ) σˆx − J2(pi/8) ( e2iωAtlDˆ[2iη] + e−2iωAtlDˆ[−2iη] ) , (2.33) in which the initial optical phase has been absorbed into a non-zero time-of-arrival for the first pulse, and subsequent optical phase evolution follows ωAtl (recall tl is the arrival time of the lth pulse). The final state after a pulse train is formulated as |Ψf〉 = Oˆ |Ψi〉 = Uˆt8Uˆfe(t8 − t7)Uˆt7Uˆfe(t7 − t6)...Uˆt1 |Ψi〉 . (2.34) I think the easiest way go from here is to use matrix multiplication on a computer. We write the operators in explicit matrix form in terms of the spin and a trun- cated momentum basis pertaining to absorbing+emitting pairs of photons. The eigenstates of the qubit manifold are: |↑〉 =  1 0  and |↓〉 =  0 1  . (2.35) 39 Relevant qubit state matrices are: σˆx = 0 1 1 0  and e−iωhf tσˆz/2 = e−iωhf t/2 0 0 eiωhf t/2  . (2.36) A vacuum state that has been displaced by 4~k, 2~k, 0, −2~k, and −4~k in mo- mentum can be written as: |2iη〉 =  1 0 0 0 0  ; |iη〉 =  0 1 0 0 0  ; |0〉 =  0 0 1 0 0  ; |−iη〉 =  0 0 0 1 0  |−2iη〉 =  0 0 0 0 1  ,(2.37) where higher order momentum states are ignored because only a tiny amount of population spreads into them. The displacement operators written in this basis are: Dˆ[2iη] =  0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0  ; ...; Dˆ[−2iη] =  0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0  . (2.38) Now all that is left is choosing ωA, and the timings. Let us imagine we are generating 8 pulses using delay lines in the shape of Mach-Zehnder interferometers as shown in Fig. 2.11. Additionally, let us try a solution where the pulses arrive at integer multiples of the hyperfine evolution plus the AOM shift. Intuitively, this is a good starting point considering it works for a standard Raman transition using an evenly spread train of low energy pulses [65] (although there are an infinite number of 40 possibilities). If we set the AOM shift to ωA/2pi = 466 MHz (arbitrary, besides being experimentally achievable, but most settings will give some good answers), and limit the maximum Mach-Zehnder arm to be less than 17 hyperfine evolutions of delay, we get a manageable set of solutions. The best solution in this example has delay arms (in units of hyperfine evolu- tion plus AOM evolution) of length tdelay(ωhf + ωA)/2pi =7, 8, and 16. For an ion initially in the state |Ψi〉 = |0〉 ⊗ (|↑〉 + |↓〉) (1/ √ 2 normalization factor ignored), the series of pulses gives Oˆ · |Ψi〉 = |Ψf〉 ≈  0.001 + 0.000i −0.315 + 0.948i 0.019 + 0.009i −0.009− 0.024i 0.013 + 0.011i  ⊗  1 0 +  0.013− 0.011i 0.009− 0.024i 0.019− 0.009i 0.315 + 0.948i −0.001 + 0.000i  ⊗  0 1  . (2.39) This means that an ion starting in state |Ψi〉 will have its spin and motion entangled with high fidelity. The fidelity of the transfer to the two desired diffracted momen- tum orders are |(〈iη| ⊗ 〈↑ |)|Ψf〉|2=0.997, and |(〈−iη| ⊗ 〈↓ |)|Ψf〉|2=0.997. That is 0.3% worse than the ideal SDK operator OˆSDK = ie iφ0D[iη]σˆ+ + ie−iφ0D[−iη]σˆ−, (2.40) where the phase φ0 is the initial optical phase. 41 2.5.2 Frequency Domain Interpretation Intuition for why certain pulse timings succeed in producing SDKs may be gained by looking at the frequency spectrum of the pulse train. In the well- understood case of a beatnote between two frequency combs coupling spin and motion [32], the condition for driving transitions is fhf = nfrep ± fA (2.41) where both solutions ± should not be satisfied at the same time lest the fidelity decreases. The repetition rate of the pulse train frep must be less than the bandwidth of the pulse, which must be greater than the hyperfine splitting (to drive qubit transitions) but much less than the splitting between the excited 2P3/2 and 2P1/2 states (lest Raman transitions will not work) [65]. A plot of these conditions is shown in Fig. 2.10(a). An uneven pulse train will produce an uneven frequency spectrum. However, it is still true that there should be preferential absorption depending on the spin state, which comes from a beatnote existing between spectral components at the hyperfine frequency and which always has the lower frequency component in one of the beams and the higher component in the other when shifted by an AOM (Fig. 2.10(b)). 42 (a) (b) ωA Figure 2.10: Conditions For a Spin-Dependent Kick. (a) The 355 nm mode locked laser pulse bandwidth must be greater than the hyperfine splitting. Drawn here, a single pulse bandwidth spans the hyperfine splitting of 12.6 GHz. This is not to scale with the excited states, which have a fine structure splitting of 100 THz. The pulse drives Raman transitions via off-resonantly coupling to the excited 2P states. (b) The frequency spectrum of a short train of unevenly spaced pulses and a beatnote resonant with the ground state hyperfine transition of the atom. The width of the spectrum envelop is greatly under-exaggerated to show how it drops off (proportional to the bandwidth of the laser ∼ 1/τ) . 43 2.5.3 The Mode-Locked Laser Each SDK is generated from a single 10 ps pulse of a frequency tripled, mode- locked Nd:YVO4 at wavelength λ = 355 nm [1, 32, 33]. I have used three different lasers during my PhD, the first being a High-Q Picotrain laser with 8 W average power and an 80.16 MHz repetition rate. This laser was poorly made, and rarely functioned long enough to produce reliable data. We used this laser the first three years of my time in the group before replacing it with a 4 W average power Paladin Compact from Coherent Inc. This laser worked well for the year that we used it, but we have since moved to using a laser capable of driving single-pulse Rabi flops on multiple ions–a 24 W Paladin Adavanced also made by Coherent Inc with a repetition rate of about 81 MHz. In the experiments described here, a pulse is divided into eight pulses using a series of beam splitters and delays in a Mach- Zehnder interferometer configuration (Fig. 2.11) and then applied to the ion in a linear-perpendicular-linear fashion to produce the spin-dependent displacement [32] OˆSDK = e iφλσˆ+Dˆ[iη] + e−iφλσˆ−Dˆ[−iη]. (2.42) The phase φλ is an optical phase that is assumed to be stable during the course of one experiment, but random over multiple experiments due to slow mechanical and other noise on the optics. The optical phase φλ cancels when an even number of applications of the operator OˆSDK are used during an experiment. Each of the Paladin lasers produce transform limited pulses with a temporal profile sech2(t/τp). At an average power of 24 W, a repetition rate of 80 MHz, and 44 AOM +230 MHz AOM -230 MHz Pockels Cell Pockels Cell beam dump beam cube beam cube beam cube trapped ions 355 nm pulsed laser pulsed light source kick timing SDK creation kick vector selection wave- plate t1 ti tN ... ... ti tN ... ... t1 t1 t1ti ti delay arms ... ... ... ... Figure 2.11: Experimental setup of Raman laser. The mode-locked Raman laser beam path is broken into four sections. The first section (outlined in red) is the laser itself, which produces ∼ 10 ps laser pulses at a repetition rate of ∼80 MHz. The second section (boxed in yellow) allows arbitrary pulses to be selected from the laser to be applied at the ion. The third section (green) takes each pulse and shapes it into two trains of 8 pulses each (2 ns duration) which are shifted in frequency by ∼500 MHz relative to each other. The last section (blue) recombines these trains in space and time while maintaining their orthogonal po- larization. A Pockels cell then selects which direction these beams are applied to the ion and determines the SDK direction. 45 a pulse duration of τp = 10 ps, the energy and peak power of a single pulse is Ep ≈ 24 W 80 MHz = 300 nJ (2.43) Pp ≈ 0.88Ep τp = 26.4 kW. (2.44) If all this power is focused to a 5 µm radius waist at the ion, it theoretically produces a total pulse area of [44] Θ ≈ 60× 2pi. (2.45) This is an upper limit, and only a fraction of the light is delivered to the ion after passing through the necessary optics. Although a common concern when working with short laser pulses, dispersion is not a a significant issue with 10 ps pulses. This can be understood by expanding the propagation constant β(ω) of the laser light to second order around the center frequency ω0 [66]: β(ω) = β(ω0) + β ′(ω − ω0) + 1 2 β′′(ω − ω0)2. (2.46) The second order term β′′(ω) = d dω ( 1 υg(ω) ) , in which υg(ω) is the group velocity, is typically referred to as the group-velocity dispersion (GVD) and is about 100 fs2/mm in fused silica at a laser wavelength of 400nm [67]. An initial pulse width of τp0 will broaden to τ 2p (z) = [ 1 + ( z(4 ln 2)β′′ τ 2p0 )2] τ 2p0, (2.47) and so even if the entire z ∼ 1 m beam path was fused silica (in actuality it is mostly air, which has lower GVD [68]), a 10 ps pulse would broaden by a factor of only about √ 1 + 10−5. 46 The laser was selected such that the repetition rate fell within specific ranges of frequencies to ensure high fidelity spin-dependent forces. As seen in Eq. 2.41, an integer multiple of the repetition rate plus the AOM shift must match the hyperfine splitting of the qubit. this is necessary if the laser is being used in the weak pulse regime for standard laser operations [32]. Additionally, the AOM shift must be larger than the width of a comb tooth for a given operation lest Raman transitions are driven with absorption and emission happening in both beams and reducing fidelity of the SDK. Finally, the AOM shift cannot be less than the width of a comb tooth away from half of the repetition rate (Eq. 2.41 again) or the force will not be spin dependent. In other words, there must be an asymmetry, and without this last condition, the ion will not preferentially absorb from one beam and emit into the other. 2.5.4 Picking and Rotating Pulses at 100 MHz The timing and direction of an SDK is determined by the repetition rate of the laser pulses along with two Pockels cells and beam cubes, which rotate and analyze the polarization of arbitrary pulses in the train (Fig. 2.11). Each of these pulses will become eight smaller pulses to form an SDK, and a train of these eight pulses, or an SDK, is always rotated together). The upstream Pockels cell is used to pick which pulses make it to the ion. If the voltage that drives the Pockels cell is at zero volts, the vertical input polarization remains unchanged, and the cube directs the light into a beam dump. If the drive voltage is high, the input light is rotated 47 to have horizontal output orientation, and passes through the cube and down the table. The extinction ratio of this process (the ratio of average power of intentional light to unintentional light sent to the ion) is about 300:1. The second Pockels cell is used to set the SDK direction. This method works by combining the linear-perpendicular-linear configured beams on a beam cube in- stead of at the ion–recall from Fig. 2.10 that two counter-propagating beams in a linear-perpendicular-linear fashion with a frequency difference ωA have a preferential direction of emission and absorption at the ion which determines the kick direction. After combination, the beams are sent through the Pockels cell and beam splitter (Fig. 2.11). If the cell drive voltage is at zero, then nothing happens and the exper- iment occurs as it would without the cell. If the drive voltage is high, then the two beams are rotated, maintaining their relative orientation, and applied from opposite directions of the ion. This reversal of kick direction allows us to concatenate every pulse that is emitted from the laser as an SDK3. In order to achieve proper timing between the pulse picker temporal window and arrival time of the pulses, we trigger the Pockels cell drive voltage sequence using a pulse from the laser. A pre-determined timing sequence is programmed into an arbitrary waveform-generator (AWG) which consists nominally of 0 V or 1 V (those voltages can be adjusted slightly to fine tune the driver) segments. This sequence is triggered when the experimental control program opens an rf switch and allows a pulse from the laser (converted to an electrical pulse on a photodiode) 3Without this switching, the ion would just be kicked back and forth without ever increase to a large momentum state because of the spin flip that occurs with each SDK. 48 Photodiode Power splitter Comparator Comparator AWG Clk in Trig in Pockels Cell Pockels Cell HV driverHV driver Ch 1 Ch 2 ttl Rf switch In Out 1 Out 2 ... Figure 2.12: Triggering a pulse sequence. In order to ensure that the experimental control program is able to temporally match the pulses from the laser, we clock and trigger the Pockels cell drive voltage using laser pulses converted to electrical pulses. to trigger the AWG sequence. The AWG, clocked by the same photodiode signal routed around the switch, runs the desired pulse sequence. This process is outlined in Fig. 2.12. It is important that the comparator (compares the input signal to an internal 50 mV and outputs 0 or 1 V depending on if the input is below or above the internal threshold) be after the rf switch, because the comparator fall time is much greater than the photodiode fall time; if the comparator were first, the rf switch would sometimes turn on during a comparator pulse making the timing random within that range and leading to varying optical pulse power when the pulse picker window does not align with the pulse. 49 Chapter 3: Stabilizing Ion Secular Motion Charged particles are often controlled with radiofrequency (rf) electrical poten- tials, whose field gradients provide time-averaged (ponderomotive) forces that form the basis for applications such as quadrupole mass filters, ion mass spectrometers, and rf (Paul) ion traps [47, 51]. These rf potentials, typically hundreds or thou- sands of volts at frequencies ranging from 1kHz to 100 MHz, drive high impedance loads in vacuum and are usually generated with rf amplifiers and resonant step-up transformers such as quarter-wave or helical resonators [69]. Such circuitry is sus- ceptible to fluctuations in amplifier gain, mechanical vibrations of the transformer, and temperature drifts in the system. Ion traps are particularly sensitive to these fluctuations, because the rf potential determines the harmonic oscillation frequency of the trapped ions. Stable trap frequencies are crucial in applications ranging from quantum information processing [36, 48] and quantum simulation [70, 71] to the preparation of quantum states of atomic motion [15], atom interferometry [33], and quantum-limited metrology [72]. Actively stabilizing rf ion trap potentials requires the faithful sampling of the rf potential. Probing the signal directly at the electrodes is difficult in a vacuum environment and can load the circuit or spoil the resonator quality factor. On the 50 other hand, sampling the potential too far upstream is not necessarily accurate, owing to changes in downstream inductance and capacitance. Here we actively stabilize the oscillation frequency of a trapped ion by noninvasively sampling and rectifying the high voltage rf potential between the step-up transformer and the vacuum feedthrough leading to the ion trap electrodes. We use this signal in a feedback loop to regulate the rf input amplitude to the circuit. We stabilize a 1 MHz trapped ion oscillation frequency to < 10 Hz for periods less than 200 s (slow drifts affect it on longer time scales), representing a maximum 34 dB reduction in the level of trap voltage (and therefore secular frequency) noise and drift at 200 s, with an adjustable locking bandwidth between 100 Hz and 30 kHz. One approach to stabilize the ion oscillation frequency is to control the ratio V0/Ωrf, which is important in cases where the rf drive frequency is itself dithered to maintain resonance with the step-up transformer. This is necessary when the trans- former resonance drifts, maybe due to mechanical or temperature fluctuations, by a significant amount of its linewidth. A feedback system of this style is shown in Fig. 3.1(a). One feedback loop (upper right section, blue) locks the rf drive frequency (tuned using frequency modulation–FM) to the step-up transformer resonance by deriving a zero crossing in the error signal from a phase shift across resonance of the reflected signal. This is done using a directional coupler to sample the drive and reflected signals and comparing the difference in phase using a frequency mixer. A second feedback loop (lower section, red) stabilizes the ratio V0/Ωrf using a digital divider. The main difficulty with this approach is the required performance of the digital divider circuit, which must have a precision as good as the desired stability, 51 and be fast enough to stabilize the system at the desired bandwidth. Moreover, higher order corrections to the trap frequency beyond the pseudopotential expres- sion of Eq. 2.3 depend on terms that do not scale simply as the ratio V0/Ωrf . Therefore, we instead stabilize the rf potential amplitude V0 alone, and use a fixed frequency rf oscillator and passively stable transformer circuit, as depicted in Fig. 3.1(b). 3.1 trap rf stabilization We stabilize the rf confinement potential by sampling the high voltage rf signal supplying the ion trap electrodes and feeding it back to a frequency mixer that controls the upstream rf oscillator amplitude. As shown in the schematic of Fig. 3.1(b), an rf signal at Ωrf/2pi = 17 MHz and −8 dBm is produced by an rf oscillator (SRS DS345) and sent through the local oscillator (LO) port of a level 3 frequency mixer (Mini-Circuits ZX05-1L-S), with a conversion loss of 5.6 dB. The RF port of the mixer is connected to a rf amplifier (Mini-Circuits TVA-R5-13) with a self- contained cooling system, providing a gain of 38 dB. The amplifier signal is fed into an antenna that inductively couples to a 17 MHz quarter-wave helical resonator and provides impedance matching between the rf source and the circuit formed by the resonator and ion trap electrode capacitance [69]. The antenna, resonator, and equivalent ion trap capacitance Ctrap are shown in Fig. 3.2, and exhibit an unloaded quality factor QU ∼ 600. A capacitive divider samples roughly 1% of the helical resonator output, using 52 rf oscillator (freq. = Ωrf/2pi) quarter- wave resonator amp LO RFIset point Servo Controller + - capacitive divider ion trap rf oscillator quarter- wave resonator amp LO RFIset point Proportional gain Integral gain Servo Controller + - capacitive divider ion trap directional coupler phase shifter CPL- in CPL- out LO I RF FM frequency counter digital divider Proportional gain Integral gain freq. mixer freq. mixer D AC A D C (a) (b) rectier rectier Figure 3.1: Ion trap rf drive with active stabilization [42]. (a) Stabiliza- tion of the ratio of rf potential amplitude to frequency V0/Ωrf (lower sec- tion, red) using a digital divider (ADC: analog-to-digital converter and DAC: digital-to-analog converter), with a separate feedback loop (upper right section, blue) that locks the rf drive frequency Ωrf to the resonant frequency of the quarter-wave resonator (a step-up transformer). (b) Stabilization of the rf potential amplitude V0 only, with fixed rf drive frequency (used in the experiment reported here). 53 from directional coupler to rectier to ion trap 0.2 pF 20 pF capacitive divider resonant coil antenna C trap Figure 3.2: Helical quarter-wave resonator (transformer) with a 1:100 capacitive divider (0.2 pF and 20 pF) mounted inside of the resonator near the high voltage side [42]. The divider samples V0 for feedback. A rigid wire is soldered from the output portion of the copper resonator coil to the copper-clad epoxy circuit board containing the dividing capacitors. The resonator drives the capacitance Ctrap of the vacuum feedthrough and ion trap electrodes. 54 C1 = 0.2 pF and C2 = 20 pF ceramic capacitors (Vishay’s QUAD HIFREQ Series) with temperature coefficients of 0±30 ppm/◦C. With C1  Ctrap and residual induc- tance between the divider and the trap electrodes much smaller than the resonator inductance itself, the divider faithfully samples the rf potential within a few cen- timeters of the trap electrodes and does not significantly load the trap/transformer circuit. The capacitors are surface-mounted to a milled copper-clad epoxy circuit board and installed inside the shielded resonator cavity, as diagrammed in Fig. 3.2. The sampled signal passes through a rectifier circuit (Fig. 3.3(a)) consisting of two Schottky diodes (Avago HMPS-2822 MiniPak) configured for passive tempera- ture compensation [73] and a low-pass filter giving a ripple amplitude 10 dB below the diode input signal amplitude. High quality foil resistors and ceramic capacitors are used to reduce the effect of temperature drifts. The entire rectifying circuit is mounted inside a brass housing (Crystek Corporation SMA-KIT-1.5MF) as shown in Fig. 3.3(b). The sampling circuit has a bandwidth of ∼ 500 kHz, limited by the 5 kΩ/68 pF RC filter. The ratio of dc output voltage to rf input voltage peak amplitude, including the capacitive divider, is 1 : 250 at a drive frequency of 17 MHz, 1 : 330 at 100 MHz, and 1 : 870 at a drive frequency of 1 MHz (see section IV for additional details about drive frequency response). The dc rectified signal is compared to a stable set-point voltage (Linear Tech- nology LTC6655 5V reference mounted on a DC2095A-C evaluation board) with variable control (Analog Devices EVAL-AD5791 and ADSP-BF527 interface board), giving 20-bit set-point precision and ±0.25ppm stability. The difference between these inputs – the error signal – is then amplified with proportional and integral 55 10 20(b) 2 K 5 K 68 pF HMPS2822 Output to servo controller 5 K HMPS2822 (a) Ctrap 20 pF 0.2 pF cpctv. divider Figure 3.3: Rectifier circuit diagram and photograph [42]. (a) Schematic circuit diagram depicting the components of the pick-off voltage divider and temperature-compensating rectifier. (b) Photograph of the connec- torized housing and mounted rectifier circuit. gain (New Focus LB1005 servo controller) and fed back to regulate the upstream rf oscillator amplitude via the frequency mixer described above. Figure 3.4 shows the response of the system for various servo controller bandwidth settings when signals over a range of frequencies with constant amplitude are injected into the system at the amplifier input. The overall frequency response of the feedback loop is limited to a bandwidth of 30 kHz, consistent with the linewidth Ωrf/(2piQU) of the helical resonator transformer. 3.2 Ion Oscillation Frequency We next characterize the rf amplitude stabilization system by directly mea- suring the transverse motional oscillation frequency of a single atomic 171Yb+ ion confined in the rf trap. We perform optical Raman sideband spectroscopy [15] be- tween the same |F = 0,mf = 0〉 ≡ |↓〉 and |F = 1,mf = 0〉 ≡ |↑〉 “clock” hyperfine levels of the 2S1/2 electronic ground state discussed in the last chapter. Recall, this atomic transition has a frequency splitting of ωhf/2pi = 12.642815 GHz and acquires 56 10 100 1000 104 105 -50 -40 -30 -20 -10 0 Frequency (Hz) No ise P ow er o n Er ro r S ig na l ( dB m ) no feedback 0.100 kHz 3 kHz 30 kHz 300 kHz servo bandwidth Figure 3.4: Suppression of injected noise in the stabilization circuit for various levels of feedback [42]. The rf drive is weakly amplitude- modulated at frequencies swept from 4 Hz to 100 kHz via a variable attenuator inserted before the rf amplifier (labeled ”amp” in Fig. 3.1). The amplitude of the resulting ripple on the error signal (set point minus feedback at servo controller input) is measured for several servo controller bandwidth settings. The observed overall loop bandwidth of ∼30 kHz is consistent with the linewidth of the helical resonator transformer. frequency-modulated sidebands at ωhf ±ω due to the harmonic motion of the ion in the trap, with ω/2pi ∼ 1 MHz. Before each measurement, the ion is Doppler cooled. The ion is next prepared in the |↓〉 state through optical pumping, and following the sideband spectroscopy described below, the state (|↓〉 or |↑〉) is measured with state-dependent fluorescence techniques [56]. The oscillation frequency is determined by performing Ramsey spectroscopy [74] on the upper (blue) vibrational sideband of the clock transition at frequency ωhf + ω. Because the atomic clock frequency ωhf is stable and accurate down to a level better than 1 Hz, drifts and noise on the sideband frequency are dominated by the oscillation frequency ω. The sideband is driven by a stimulated Raman process from two counter-propagating laser light fields with a beatnote ωL tuned 57 near the upper vibrational sideband frequency [32,65]. Following the usual Ramsey interferometric procedure [74], two pi/2 pulses separated by time τ = 0.4 ms drive the Raman transition. After the pulses are applied, the probability of finding the ion in the |↑〉 state P (δ) = (1 +C cos τδ)/2 is sampled, where δ = ωL− (ωhf +ω) is the detuning of the beatnote from the sideband and C is the contrast of the Ramsey fringes. The Ramsey experiment is repeated 150 times for each value of δ in order to observe the Ramsey fringe pattern P (δ) and track the value of ω. Because this Raman transition involves a change in the motional quantum state of the ion, the Ramsey fringe contrast depends on the purity and coherence of atomic motion. For short Ramsey times, the measured contrast of ∼ 0.8 is limited by the initial thermal distribution of motional quantum states, and for Ramsey times τ > 0.5 ms, the fringe contrast degrades further (Fig. 3.5), which is consistent with a decoherence timescale (2n¯0 ˙¯n) −1 for initial thermal state n¯0 = 15 quanta and motional heating rate ˙¯n = 100 quanta/s [75]. Through Ramsey spectroscopy, we sample the ion trap oscillation frequency ω at a rate of 2.1 Hz for 80 minutes with no feedback on the rf potential, and then for another 80 minutes while actively stabilizing the rf potential. A typical time record of the the measurements over these 160 minutes is shown in Fig. 3.6. Feedback control clearly improves the stability of the ion oscillation frequency. From these measurements, we plot the Allan deviation [76] of the oscillation frequency in Fig. 3.7 as a function of integration time T . When the system is stabilized, the Allan deviation in ω is nearly shot-noise limited (decreasing as 1/ √ T ) up to ∼ 200 s of integration time, with a minimum uncertainty of better than 10 58 0.01 0.05 0.10 0.50 1 0.04 0.10 0.20 0.40 1.00 R a m se y F ri n g e C o n tr a st τ (ms) δ (MHz) P (δ ) -0.006 0.003 0.013 0.0 0.2 0.4 0.6 0.8 -0.006 0.003 0.013 a b no feedback stabilized model Figure 3.5: Ramsey fringe contrast as a function of the Ramsey time τ between pi/2 pulses, with and without feedback [42]. The gray line is a model in which motional heating causes Ramsey fringe decoherence in ∼ 0.5 ms. Inlays a and b show full Ramsey fringe measurements and fits for two different values of τ . 5 6 7 8 9 0.9955 0.9960 0.9965 0.9970 0.9975 0.9980 85 86 87 88 89 0.9955 0.9960 0.9965 0.9970 0.9975 0.9980 Time (min) 0 20 40 60 80 100 120 140 0.995 0.996 0.997 0.998 0.999 ω /2 π ( M H z) 0.992 0.993 0.994 ba no feedback stabilized a b Figure 3.6: Time dependence of the ion harmonic oscillation frequency ω plotted over the course of 160 minutes with and without active stabi- lization [42]. With feedback there is a clear reduction in noise and drifts (apart from measurement shot noise, reflected by the fast fluctuations in the data). Inlays a and b show magnified sections of the plot covering 4 minutes of integration. 59 Hz, or 10 ppm, representing a 34 dB suppression of ambient noise and drifts in the drive voltage or secular frequency. Without feedback, the trap frequency deviation drifts upward with integration time. For integration times shorter than 7 s, there is not sufficient signal/noise in the measurements to see the effects of feedback stabilization. However, as shown in Fig. 3.4, the lock is able to respond to error signals up to a bandwidth of∼ 30 kHz, and we expect significant suppression of noise at these higher frequencies as well. Although the Allan deviation of the oscillation frequency in the stabilized system improves with longer averaging time as expected, it drifts upward for a period just after T = 50 s (likely caused by a temperature drift affecting the capacitive divider pick-off). We confirm this drift appears in the ion oscillation frequency ω and not the driving field ωL or the ion hyperfine splitting ωhf by performing the same experiment on the clock “carrier” transition near beatnote frequency ωL = ωhf instead of the upper sideband ωL = ωhf + ω. As shown in Fig. 3.7, the measured Allan deviation of the carrier continues downward beyond T = 50 s, meaning that the ion oscillation frequency is indeed the limiting factor at long times. 3.3 Limits and noise sources It should be possible to stabilize the rf trap frequency much better than the observed 10 ppm by improving passive drifts outside of feedback control. These include the capacitive divider that samples the rf, the rectifier, the stable voltage reference, rf source frequency, and certain cables in the rf circuitry. Most of these 60 1 5 10 A lla n D e v ia ti o n ( H z) 500 100 50 5 10 Averaging Time (s) no feedback stabilized carrier 50 100 Figure 3.7: Allan deviation data of the secular frequency ω while the system is with and without feedback, as well as the Allan deviation of the qubit carrier transition [42]. The Allan deviation curves are calculated from the time record shown in Fig. 3.6, along with a similar measurement performed on the carrier transition. components will have residual drifts with temperature, mechanical strains, or other uncontrolled noise. Below is a table of all crucial components outside of feedback control and their estimated contribution to the instability. Table 3.1: Estimated stability of components outside of feedback control. Component Stability Capacitive Divider 0 ≤ 6 ppm Rectifier 0.01 ppm Voltage Reference 0.25 ppm rf source freq. 0.1 ppb Cables Unknown The capacitive divider pick-off is comprised of two capacitors each with a temperature coefficient of ± 30 ppm/◦C. Given the voltage divider configuration, the net temperature coefficient can range from ∼ 0− 60 ppm/◦C depending on how well the capacitors are matched. Because temperature drifts on the order of ∼ 0.1◦C 61 are expected without active temperature stabilization, the capacitive divider is likely limiting the ultimate stability of the system. Instabilities in the rectifier can arise from variability in the junction resistance of the diodes. In series with a 5 kΩ resistor, the ∼ 0.01Ω/◦C junction resistance gives a net temperature coefficient of about 0.2 ppm/◦C in the rectifier response. This is roughly equal to the temperature coefficient of the resistors used in the rectifier circuit. By using the circuit configured for passive temperature compensation [73] shown in Fig. 3.3, we estimate the net temperature coefficient of the rectifier response is reduced to ∼ 0.1 ppm/◦C. Performance of the circuit is also helped by passively stabilizing components within the feedback loop as much as possible, such as temperature regulating the rf amplifier which feeds the resonator and using a passive mixer instead of a pow- ered voltage variable attenuator. The helical transformer is particularly sensitive to temperature fluctuations and mechanical vibrations, which alters the resonance frequency and quality factor. (Ensuring the helical coil is sealed against air cur- rents can be more important than correcting small drifts in ambient temperature.) If the resonant frequency of the transformer drifts too far, then a feedback circuit with a fixed frequency source (as used here and shown in Fig. 3.1(b)) will call for more input power, and the servo system could possibly run away and become un- stable. However, the resulting impedance mismatch from the off-resonant coupling will cause the servo to maintain the same amount of dissipated power in the res- onator [69] and not necessarily affect further drifts. In any case, we do not observe such servo runaway. This is true even when the set point is ramped down and back up to cycle the trap rf voltage during instances in which ions are too hot for effec- 62 tive laser cooling in the tighter trapping potential. The resulting transient thermal response has no apparent effect on the secular frequency stability. Based on simulations, this system is capable of stabilizing the rf amplitude in ion trapping apparatuses using a range of rf drive frequencies. Figure 3.8 shows the transient turn-on and steady state responses of the rectifier output for drive frequencies ranging from 1 MHz to 150 MHz. We see that while 17 MHz gives near maximum direct current (DC) voltage (a higher DC voltage is like a higher gain), increasing or decreasing the drive frequency by up to an order of magnitude still provides an appreciable DC voltage (the optimum frequency can also be shifted by modifying the rectifier circuit). So long as the rectifier output offset voltage (ripple is filtered by the servo controller) is appreciable, the feedback loop will maintain performance near the demonstrated fractional secular frequency stability of better than 10 ppm, independent of ion mass (see Eq. 2.3). If the temperature coefficients of the capacitors in the capacitive divider are properly matched and the divider is actively temperature-stabilized, we believe the technique presented in this article would provide a stability in radial secular fre- quency of ∼ 0.3 ppm. This stability could likely be made even better by further stabilization of the voltage reference in addition to improved design of the whole ap- paratus including mechanical and thermal stabilization, improved electrical shield- ing, and shortened distances between components. 63 R e ct if ie r O u tp u t (V ) 0.5 1.0 1.5 0.0 Time After Turn On (μs) 0 1 2 17 MHz 60 MHz 100 MHz Drive Freq. 150 MHz 10 MHz 3 MHz 1 MHz 0 1 2 Drive Freq. Figure 3.8: Simulations of the transient voltage output (into a 1MΩ load) of the rectifier circuit with a 700 Vpp rf trap drive turned on at t = 0 s [42]. The plots shows a response time, ripple, and dc offset at steady state for a range of drive frequencies. 64 Chapter 4: Sensing Atomic Motion Ultrafast sensing of atomic motion works over a wide range of energies, from the zero-point (phonon occupation number n = 0) to potentially above room- temperature (n¯ ∼ 106 for typical ion traps). Ultrafast partial state tomography (defined later in this chapter) on thermal states improves upon the dynamic range achieved with thermometry using dark resonances [77]. It also complements conven- tional methods of thermometry, including measurements of the motionally-induced upper and lower sideband asymmetries [78] and the thermal effects on induced tran- sitions (Debye-Waller factors) [79]. However, both of these other methods break down when the atomic motion is outside of the Lamb-Dicke regime, typically around n¯ > 10. Measuring the entire Doppler-broadened envelope of sidebands [80] pro- vides a more general measurement of thermal states, but can be difficult due to the bandwidth required to excite multiple sidebands. In this chapter, we use ultrafast techniques for accurate thermometry of ion motion ranging from n¯ ∼ 0.1 to n¯ ∼ 104 and show how this method extends to higher energies. We also measure particular quantum states through more complete motional tomography. 65 4.1 Experimental Description In this experiment, we again trap a 171Yb+ ion in a linear radio frequency Paul trap and probe the motion along a single radial mode of motion with secular trap frequency ω/2pi ≈ 1 MHz. The |F = 0,mf = 0〉 ≡ |↓〉 and |F = 1,mf = 0〉 ≡ |↑〉 hyperfine levels of the 2S1/2 electronic ground state are again used as the qubit, or effective spin. The ion is laser-cooled to near the Doppler limit (n¯ ≈ 10) and optically pumped during initial state preparation. Qubit state detection is performed by collecting state-dependent fluorescence [56]. The qubit state in these experiments is detected with a fidelity above 0.997 using an imaging objective with 0.6 numerical aperture and a photomultiplier tube [62]. We create a spin-dependent kick (SDK) using the method discussed in the previous section–by modifying individual pulses extracted from a mode-locked laser with center wavelength 2pi/k ≈ 355 nm, pulse duration τ ∼ 10 ps, and repetition rate of frep = 118 MHz (note that this is a higher repetition rate laser than discussed in Chapter 2). Each SDK has a spin population transfer from |↓〉 to |↑〉 with measured fidelity of 0.993(2) [32]. Because each SDK operation provides a momentum kick and flips the spin, immediately applying a second SDK would simply undo the first. However, by waiting one half of the trap period between SDKs, we can concatenate N individual kicks to create a larger effective SDK with ∆p = ±2N~k = ±Nηp0. (p0 = √ ~mω/2 is the momentum spread of the ground state, and m is the ion mass.) Techniques using Ramsey spectroscopy on states coherently displaced by spin 66 dependent forces have been demonstrated in creating Schro¨dinger cat states [28] and measuring spin dephasing in 2D ion crystals [81]. In this experiment, we create an interferometer to sense motion by applying two sets of N SDK operations within a Ramsey experiment on the qubit levels with time duration T (time separation of microwave pi/2 pulses). First the ion is prepared in a coherent superposition of |↓〉 and |↑〉 by applying a near-resonant microwave pi/2 pulse of duration τµ. A set of N SDKs is applied, and following this first set, the ion evolves for a time θ/ωt before a second set of N SDKs is applied. After a time T from the first microwave pi/2 pulse, another microwave pi/2 pulse with the same duration and tuning drives the qubit to close the Ramsey interferometer. This sequence is diagrammed in Fig. 4.1(a). By scanning the microwave detuning δ  1/τµ from resonance, we observe sets of Ramsey fringes with phase φ = δT that chronicle the ion motion (shown in Fig. 4.1(b) and 4.1(c)). For a pure initial state |Ψα〉i = |↓〉 |α〉, where α is a coherent state of the ion motion, the state following the Ramsey experiment is [44] |Ψα〉 = 1 2 [ eiγ ( |↓〉+ ie−iφ |↑〉 ) |(α + iNη)e−iθ − iNη〉 +ie−iγ ( |↑〉+ ieiφ |↓〉 ) |(α− iNη)e−iθ + iNη〉 ], (4.1) where γ = Nη[Re(α)(1− cos θ)− Im(α) sin θ]. Given an arbitrary initial state of motion in phase space described by the Glauber P-distribution [82,83], the final density matrix is ρˆ = ∫ P (α) |Ψα〉 〈Ψα| d2α. The probability of measuring the state spin-up after the Ramsey experiment is 67 - 4 - 2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 after rst SDK set p x θ/ωt SDK set SDK set microwave pi/2 pulse(a) (b) ... 1 2 N ... 1 2 N... ... initial state after second SDK setat various delays α θ = 0, 2pi θ = pi/2 (c) θ = 0 p x p x θ = pi t = 0 time t = T τμ microwave pi/2 pulse τμ + +α ¡Nη -α ¡Nη θ = pi/2 - 4 - 2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 δ (kHz) S( pi /2 ,N ;φ ) S( 0, N; φ ) ( ) α Figure 4.1: Ultrafast atom interferometry [33]. (a) Timeline of a single experiment, where a full SDK set is made of N single SDKs. (b) Phase space diagram of an initial state (|↓〉 + |↑〉) |α〉 evolving under two sets of SDKs separated by time delay θ/ωt, where |α〉 is a coherent state of motion. (c) Typical Ramsey fringes as a function of microwave frequency detuning δ. These two plots correspond to the points θ = 0 and θ = pi/2 of an initial thermal state (N = 1 for the data shown). The function S(θ,N ;φ) is described by Eq. 4.2. 68 therefore S(θ,N ;φ) = 〈↑| ρˆ |↑〉 = 1 2 + 1 2 ∫ P (α)e−4(Nη) 2(1−cos θ) cos(4γ − φ)d2α. (4.2) Two types of motional state that are readily accessible in the laboratory are thermal states and small Fock states. First we discuss ultrafast partial state tomog- raphy to determine the average phonon number in a thermal state. Then we extend this method to create a nearly complete map of the motion of an n = 1 Fock state in phase space, showing clear nonclassical signatures. 4.2 Thermometry For an ion prepared in a thermal state with mean phonon number n¯ and P- function Ptherm(α) = 1 pin¯ e−|α| 2/n¯, Eq. 4.2 yields an expected Ramsey fringe pattern Stherm(θ,N ;φ) = 1 2 + 1 2 e−4(Nη) 2(2n¯+1)(1−cosθ)cosφ. (4.3) The fringe contrast has periodic revival peaks at θ = 2pim, where m is a positive integer. For a hot ion where n¯  1/(Nη)2, these revivals in contrast become narrow and approximately Gaussian with full width at half maximum FWHM= 0.83/(Nη √ n¯). With N = 1, we measure the Ramsey fringe contrast as a function of θ for a variety of initial thermal states of motion, and fit the contrast revival peaks to Eq. 3 to determine the average phonon number n¯ of the thermal state [75,84,85]. In the fit, we allow the peak Ramsey contrast at θ = 2pim to be less than unity in order to parametrize imperfect fidelity of the SDK operations. This reduction in fidelity is mainly attributed to variations in the Raman beam intensity over the 69 spatial extent of the ion wave packet (beam waist is ≈ 2 µm), and becomes apparent at high n¯ (n¯ = 10000 has a spread of ≈ 1 µm). This does not affect the width (it does affect the height) of the contrast revival peak, and thus the accuracy of the thermometer, and can be mended by widening the beam waist. Ramsey contrast revival lineshapes are measured in experiments spanning over five orders of magnitude in n¯, as shown in Figs. 4.2(a)-(c). Figure 4.2(d) shows these measurements plotted versus the expected value of n¯ from theory and other measurements. The figure is broken into three regions according to the manner in which the motional state is prepared and calibrated before measurement of the contrast revival lineshapes. Low energy thermal states (n¯ < 10) are generated by first sideband-cooling the ion to its zero point motion and then allowing the ion to weakly heat (the trap has a natural heating rate of 310(10) quanta/s due to trap electrode noise and anomalous heating [86]) in the trap by known amounts. In this regime, we compare ultrafast interferometric measurements of n¯ (shown in Fig. 4.2(b)) to values extracted from measured sideband asymmetries [55]. The deviation of the two measurements are shown in the red section of Fig. 4.2(a). For thermal states 10 < n¯ < 150, the ion is prepared by Doppler cooling with various frequency detunings from resonance. Ultrafast measurements in this regime are shown in Fig. 4.2(d). Each of these measurements and the predicted value of n¯ from Doppler cooling theory [87] (Fig. 4.2(e)) are plotted against each other in the green section of Fig. 4.2(a). As a check on the expected values of n¯ in this range, we also measure the Debye-Waller suppression of Rabi flopping amplitude transitions between the ion qubit states [55] for several cases, resulting in expected 70 n = 13.06(62) n = 34.3(2.0) n = 90.5(5.8) Δ = 6 MHz Δ = 2.25 MHz Δ = 1 MHz 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.0 0.2 0.4 0.6 0.8 1.0 ω tT delay /2pi S[ θ ,0 ] 0.1 1 10 100 1000 104 0.1 1 10 100 1000 104 Predicted n M ea su re d n model of Doppler cooling limit 0 1 2 3 4 5 6 0 20 40 60 80 100 120 140 cooling detuning : Δ (MHz ) av g ph on on nu m be r: n model of heating from applied noise 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2000 4000 6000 8000 10 000 12 000 heating time (ms) av g Ph on on N um be r: nsideband asymmetry measurement of heating 0 5 10 15 20 - 1 0 1 2 3 4 5 6 wait time after RSB cooling (ms) A vg ph on on nu m be r: n (a) (b) (c) (d) (e) (f) (g) compared with estimates using sideband assymetry compared with model for Doppler cooling limit compared with model of heating from noise n = .210(37) n = 1.83(12) n = 5.49(40) t = 0.0 ms t = 5.0 ms t = 20.0 ms 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.0 0.2 0.4 0.6 0.8 1.0 ω tT delay /2pi S[ θ ,0 ] n = 452(39) n = 909(58) n = 3540(380) t = 0.1 ms t = 0.3 ms t = 1.0 ms 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.0 0.2 0.4 0.6 0.8 1.0 ω tT delay /2pi S[ θ ,0 ] Figure 4.2: Ultrafast sensing measurements of n¯ (with N = 1) [33]: (a) Measurements of n¯ versus predicted values. There are three regimes of thermal state preparation–red being sideband-cooling then heating (see (b)), green being Doppler cooling with different detunings (see (d)) and orange being heating with applied noise (see (f)). (b) Sampling of Ram- sey revival contrast lineshapes with initial states prepared by resolved sideband cooling to the ground state and subsequent heating. Data is fit to Stherm(θ, 1;φ). The amplitude of each fit is a free parameter to account for SDK infidelity (also done in (d) and (f)). This does not sig- nificantly affect the width of the peak, which is used to determine n¯. (c) Using identical state preparation to (b) but then a conventional sideband asymmetry measurement to determine ion temperature, a heating rate is determined and used to model the phonon number for wait times (solid line). The thermometry measurements from (b) are plotted to compare. (d) Sampling of Ramsey revival contrast lineshapes with initial states prepared by Doppler cooling only, with n¯ varied by changing the cool- ing beam detuning. (e) A model for average phonon occupation when preparing each state with various Doppler cooling beam detunings–the Doppler limit is a function of detuning (solid lines). Data from (d) is compared to the model. (f) Sampling of Ramsey revival contrast line- shapes with initial states prepared by inducing a high heating rate with white noise applied to a trap electrode. (g) A model for the phonon oc- cupation after applying electrode noise is shown (solid line). Data from fits like that of (f) are plotted for comparison. 71 values consistent with Doppler theory. Hot thermal states are prepared by inducing a high heating rate with a noisy electrical potential to a trap electrode for varied amounts of time after Doppler cooling. The ultrafast measurements of these states are shown in Fig. 4.2(f). Measurements in this regime are compared to a predicted n¯ given by the equa- tion ˙¯n = e 2SV (ωt) 4M~ωtd2 [78], where e is the ion charge, and SV (ωt) (V 2/Hz) is the applied power noise spectral density of the electric-potential, which is white over the mea- surement bandwidth (Fig. 4.2(g)). The effective distance d of the electrode to the ion is calibrated by applying a static potential offset to the same electrode and observing the resulting displacement of the ion in space [88]. The predicted and measured values for this regime are plotted against each other in the orange region of Fig. 4.2(a). 4.3 Fock State Tomography We next perform more complete tomography of a nearly pure quantum state of motion by extracting the characteristic function χW (α) = e −|α|2/2 ∫ P (β)e2iIm(αβ ∗)d2β, (4.4) where P (β) is again the Glauber P-distribution (integrated over the complex plane). This quasiprobability distribution contains all the information about the quantum state and is the Fourier transform of the better-known Wigner distribution [89,90]. 72 In terms of the observable S(θ,N ;φ), χW (α) is given by Re[χW (α)] = 2S(θ,N ; 0)− 1 (4.5) Im[χW (α)] = 2S(θ,N ; pi 2 )− 1, (4.6) where α = 2Nη[sin θ + i(1− cos θ)]. Scanning θ and N while measuring S(θ,N ;φ) maps the characteristic function over rings in phase space, shown in Fig. 4.3(a). In order to scan the negative imaginary part of α, we can change the direction of the initial momentum kick associated with the spin flip operators by shifting the relative optical phase of the counter-propagating beams by pi [1]. These reversed kicks can be thought of as effectively flipping the sign of η, and for simplicity, we represent them here by negative values of N . We measure the characteristic function χW (α) of the ion in the n=1 Fock state, prepared by sideband cooling to the ground state and transferring population to the n=1 state through application of a blue sideband operation [15]. To have a grid that spans the domain of the state, we scan around 16 rings in phase space set by ±N , where N = 1, 2, 3, 4, 5, 6, 8, 10. Two of the 16 rings along which we measure are highlighted in Fig. 4.3(a), and plots of S(θ,N ; 0) versus θ along those two rings are shown in Fig. 4.3(b). Notice in Fig. 4.3(b) that the larger SDK set (N = 5) separates the interferometer enough to see the oscillation of the motional distribution, while the smaller SDK set does not. Mapping along all 16 curves gives a nearly complete motional state map. The real part of the characteristic function is shown in Fig. 4.3(c) alongside the corresponding model of Re[χW (α)] for an n = 1 Fock state in Fig. 4.3(d). The negative values of the characteristic quasiprobability 73 (a) (b) - 4 - 2 0 2 4 - 4 - 2 0 2 4 Position (x 0= 2Mωt ) M om en tu m (p 0 = 2M ω t ) h h / θ = pi θ = pi N = 5 N= -2 θ = pi/2θ = 3pi/2 θ = 3pi/2θ = pi/2 N = 3 N = -6 N = -5 (c) model N = 5 data 0.0 0.2 0.4 0.6 0.8 1.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 S( θ, 5; 0) model N= - 2 data 0.0 0.2 0.4 0.6 0.8 1.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 θ/2pi S( θ, -2 ;0 ) Position Momentum data points meshed data points Re[χW(α)] n=1 model d 1.00 0.75 0.50 0.25 0.00 -0.25 Figure 4.3: Ultrafast phase space tomography [33] (a) Points in phase space accessible in our tomographic measurements. The radius of each circle (2Nη) is set by the number of kicks N , and the angular position on each circle is set by the SDK delay θ. The sign of N represents the direction of the initial momentum kicks associated with the spin flip operators. (b) Sample of measurements of the Ramsey fringe at φ = 0 for a nominal n = 1 Fock state, using two sets of kicks with N = 5 (red) and N = −2 (blue) and scanning the delay θ. [The coordinates of these particular scans in phase space are highlighed in (a)]. (c) Motional state tomography of an ion prepared in the n = 1 Fock state. In ascending order: the value-colored data points of Re[χW (α)] taken on 16 rings in phase space set by ±N where N = 1, 2, 3, 4, 5, 6, 8, 10, the interpolated data mapped in contour, and a 3D interpolation of the data. (d) Theory prediction for a Fock state with n = 1. 74 function highlight the nonclassical nature of the motional state of the ion. 4.4 Limits of Measurement These ultrafast tomographic techniques are capable of measuring motional energies far beyond the data presented here, which was limited to n¯ ∼ 104 because of re-cooling issues during state preparation (it becomes difficult to cool the ion in a reasonable amount of time to start a new experiment after it has been heated to a very high tempertaure). In the experiment, we scan the motional interferometric angular delay θ in steps set by the repetition rate of the laser, giving a resolution of ωt/frep ∼ 50 mrad. For revival lineshapes narrower than this laser repetition rate limit, we scan θ by changing the trap frequency ωt though accurate control of the trap rf drive voltage. With fine drive-voltage control, we can achieve a resolution in θ of 0.1 mrad, which would correspond to a contrast revival linewidth from a thermal state with n¯ ∼ 109 (that is also about where trap anharmonicity may start to play a role). Other factors also come into play when measuring such high-energy states: First, the spatial extent of motion swells beyond the laser beam waist. At n¯ = 106 for instance, or equivalent temperature T = ~ωtn¯/kB = 80K, the ion would experience a significant gradient in the Rabi frequency across a beam with a 3µm waist. A second factor is the decreased detection fluorescence due to larger Doppler shifts at these energies. The detection fluorescence at n¯ = 106 would be reduced by a factor of ∼ 103 from a cold ion [87]. Finally, when measuring these very narrow lineshapes, instabilities in the trap frequency ωt and laser repetition rate frep would 75 have to be sufficiently stable over the measurement time. At n¯ = 106, this would require a fractional stability from both the trap frequency and laser repetition rate of better than 0.1%. These factors put ultrafast interferometric measurements of n¯ ≥ 106 neither fundamentally nor technically beyond reach. 76 Chapter 5: Ultrafast Schro¨dinger Cat States Mesoscopic quantum superpositions, or “Schro¨dinger cat states,” are widely studied for fundamental investigations of quantum measurement and decoherence [91] as well as potential applications in sensing [92] and quantum information sci- ence [93]. The generation and maintenance of such states relies upon a balance between efficient external coherent control of the system and sufficient isolation from the environment. Here we create a variety of cat states of a single trapped atom’s motion in a harmonic oscillator using ultrafast laser pulses. These pulses produce high fidelity impulsive forces that separate the atom into widely-separated positions, without restrictions that typically limit the speed of the interaction or the size and complexity of the resulting motional superposition. This allows us to quickly generate and measure cat states larger than previously achieved in a har- monic oscillator, and create complex multi-component cat state superpositions in atoms. Quantum superposition is the primary conceptual departure of quantum me- chanics from classical physics, giving rise to fundamentally probabilistic measure- ments, nonlocal correlations in spacetime [94], and the ability to process informa- tion in ways that are impossible using classical means [93]. Quantum superposi- 77 tions of widely separated but localized states, sometimes called “Schro¨dinger cat states” [95], exacerbate the quantum/classical divide. These states can be created in systems such as cold atoms and ions [28, 96, 97], microwave cavity QED with Rydberg atoms [98] and superconducting circuits [99–101], nanomechanical oscilla- tors [9], and van der Waals clusters and biomolecules [102, 103]. All these systems gain sensitivity to outside influences with larger separations. The natural localized quantum state of a harmonic oscillator is the coherent state |α〉 [82], which is a Poissonian distribution of oscillator quanta with mean |α|2. For a mechanical oscillator with mass m and frequency ω, the complex num- ber α characterizes the position xˆ and momentum pˆ operators of the oscillator, with Re[α] = 〈xˆ〉/(2x0) and Im[α] = 〈pˆ〉x0/~, where x0 = √ ~/(2mω) is the zero- point width. Schro¨dinger Cat superpositions of coherent states |α1〉 + |α2〉 of size ∆α = |α1 − α2|  1 have been created in the harmonic motion of massive particles (phonons) [18] and in single mode electromagnetic fields (photons) [45]. In trapped ion systems, coherent states of motional oscillations are split using a qubit derived from internal electronic energy states [28, 104]. For photonic cat states, coherent states in a single mode microwave cavity are split using atoms or superconducting Josephson junctions. Recent experiments have created cat states with more than two components [105] for qubit storage and error protection [101]. In superconduct- ing cavities, the size of the cat state is restricted to a maximum photon number of ∆α2 ∼ 100, due to nonlinearity of the self-Kerr and dispersive shift [101]. For trapped ions, cat states have been restricted to a regime where the motion is smaller than the wavelength of the light providing the dispersive force, or the “Lamb-Dicke” 78 regime, which usually restricts phonon numbers also to ∆α2 ∼ 100 in the previous largest case (to our knowledge) [104] and ∆α2  100 for the heavier Yb atom. Mul- ticomponent cat states have not previously been created in the motion of atoms. Here we use ultrafast laser pulses to create cat states in the motion of a single 171Y b+ ion confined in a harmonic trap with frequency ω/2pi = 1 MHz [1]. We characterize the coherence of the cat state by interfering the components of the superposition and observing fringes in the atomic populations mapped to the qubit. We achieve the largest phase space separation in any quantum oscillator to date–a superposition with ∆α ≈ 20 (209 nm maximum separation compared to a x0 = 5.4 nm spread of each component) and involving up to 400 phonons, with 20% interference contrast. The ultrafast nature of the cat generation is less restrictive on nonlinearities in the forces on the atom, and allows for very fast state creation with ∆α = 0.4 per laser pulse period (12 ns). Finally, we demonstrate a method to create 3-, 4-, 6- and 8-component cat states by timing the laser pulses at particular phases of the harmonic motion in the trap. These tools allow us to create and measure fragile mesoscopic states before they lose coherence. In these experiments, the ion is subjected to 3-dimensional harmonic con- finement (resonant frequencies ωx 6= ωy 6= ωz) within the radiofrequency Paul trap detailed in past work by our group [42]. We prepare cat states in the x- direction oscillator mode (ω = ωx). The two hyperfine ground states of 171Y b+ (|↓〉 ≡ |F = 0,mf = 0〉 and |↑〉 ≡ |F = 1,mf = 0〉, with qubit splitting ωhf/2pi = 12.642815 GHz) are used to split coherent states of the atom motion through a strong state-dependent kick (SDK) [33]. The qubit can also be coherently manipu- 79 lated without motional coupling using resonant microwave pulses. Each experiment follows the same general procedure. We initialize the atom’s motion by Doppler laser cooling followed by resolved sideband cooling [33]. (Ground state population is 87%, and a thermal average is used when comparing data to theory [33], but the state will be represented from here as |n = 0〉 for simplicity.) Optical pumping initializes the qubit state to |↓〉 [56], and then a pair of microwave pi/2 pulses with variable relative phase are applied to the ion with a delay between them. During the time between microwave pulses, the ion motion is excited using two sets of SDKs (separated by time T ) to create a cat state and then reverse the process. The state is measured at the end of each experiment using qubit state- dependent fluorescence [56]. This sequence is detailed in the upper part of Fig. 5.1a. A single SDK is created with a series of eight laser pulses of duration τ ≈ 10 ps and center optical wavelength 2pi/k = 355 nm. Each of the pulses is divided and applied to the ion simultaneously in counter-propagating directions and orthogonal linear polarizations (Fig. 5.1a, lowest box). The counter-propagating pules, each with effective intensity envelope [32] sech(pit/τ) and bandwidth spanning the hyper- fine structure but much less than the fine structure, produce a polarization gradient at the ion [61] and couple the qubit and ion motion with a strength modulated along the x-direction. A pair of pulses arriving at time t = 0 drives such a modulated Raman process (Fig. 5.1b) with the approximate Hamiltonian [32] Hˆ(t) = Ω(t) sin[2kx0(aˆ † + aˆ) + φ]σˆx + ωhf 2 σˆz, (5.1) 80 π/2 π/2 measurementT SDK sets SDK SDK SDK ... ... V H ion laser pulses laser pulses SDK SDK set c p x d η iη 2S1/2 2P1/2 2P3/2 ωhf k 355 nm +2hk -2hk p 0 a b -iη + e Figure 5.1: Experimental schedule and coherent state control. (a) Spin-dependent kicks (SDKs) are concatenated in various ways to gen- erate large cat states. This process is performed between two microwave pi/2 pulses with variable relative phase before the final state is mea- sured. For simplicity, only kicks that affect the initial population in |↓〉 are drawn with arrowheads. (b) Stimulated Raman transitions through virtual excited states near 355 nm couples the qubit to the ion motion. The ωhf = 12.642815 GHz hyperfine splitting of the 2S1/2 serves as a qubit. (c) The Wigner phase space representation of a state-dependent kick (SDK). A single SDK displaces a coherent state by 2~k (iη in mo- mentum space) in a direction that depends on the initial qubit state (red:|↑〉 or blue:|↓〉), and splits a coherent state associated with a qubit superposition (purple:∝ |↑〉+ |↓〉). (d) By changing the direction of the laser beams between each SDK, every pulse from a mode-locked laser is used to create a large cat state. (e) Applying SDKs at intervals syn- chronous with half of the trap period oscillation cycle, the cat state grows without the need to change the laser direction. Free evolution appears in these plots as circular orbits. 81 where σˆx,z are Pauli spin operators, φ is the relative phase between the counter- propagating light fields and is considered constant during a pulse, aˆ† and aˆ are the raising and lowering operators of the ion motion, and Ω(t) = (1/2)(Ξ/τ)sech(pit/τ) is the Rabi frequency with pulse area Ξ. The interaction described in Eq. 5.1 yields the well known Kapitza-Dirac scattering process [1] by which the atomic motional wave diffracts from a light field grating into all momenta classes n2~k with population dictated by the Bessel func- tion Jn(Ξ), and n ∈ Z [32]. By applying a series of eight pulses (Ξ ≈ pi/8 for each) in which the phase φ is appropriately shifted between each pulse using an acousto- optic modulator, we achieve a qubit state-dependent coherent momentum transfer of ±2~k (+ for |↓〉 and − for |↑〉) along a single oscillator mode while the atom’s mo- tion is effectively frozen in time [1,32,33]. This SDK process approximately evolves the quantum state as UˆSDK = σˆ+Dˆ[iη] + σˆ−Dˆ[−iη], (5.2) without making the Lamb-Dicke approximation η √ 2n+ 1  1, where σˆ± are the qubit raising and lowering operators, and η = 2kx0 = 0.2 is the Lamb-Dicke pa- rameter, and Dˆ is the displacement operator. Figure 5.1c depicts the SDK process in which the coherent state is shown in its Wigner representation [106] as a disk in phase space and the color represents the associated qubit state (the superposition states in this letter are drawn for intuitive purposes and are not scale). Note that each momentum displacement is associated with a qubit flip. Each SDK has a fi- delity of 0.991, and this operation may be concatenated multiple times (Fig. 5.1, 82 22 1 θ θ Δα = 20p x a b c d x(t) p(t) p(0) x(0)θ 111 ns 62 ns 14 ns |α| = 0.4 |α| = 1.2 |α| = 2.0 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 C at st at e fid el ity C at st at e fid el ity Δα Δα θ/2� θ/2� |α| = 0.2 |α| = 1.0 |α| = 2.0 |α| = 10 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 C on tra st θ/2� 0.98 1.00 1.02 1.04 0.00 0.05 0.10 0.15 0.20 0.25 0.30 α -α 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 Figure 5.2: Cat state creation and verification. (a) The state |ψ1〉 (labeled “1”) is split using a set of SDKs to create the cat state |ψ2〉 (“2”). After evolution θ = ωT , a second set of SDKs drives the state to |Ψcat〉. (b) The cat state |ψ2〉 with α = 0.4 is generated in about 14 ns, α = 1.2 in 62ns, and α = 2.0 in 111 ns. The states are verified by observing contrast in the state |Ψcat〉 (lower plot). We find the fidelity of each cat state |ψ2〉 to be 0.88(2), 0.76(2), and 0.59(3), respectively (upper plot). (c) Using a higher fidelity technique which grows in α at an average rate ηω/pi, cat states are generated and verified by observing contrast revival (lower plot). Shown in the upper plot, cat state fidelity decays with the number of SDKs applied, and the effective single SDK fidelities are 0.9912(6) and 0.978(2) for Doppler (black, circles) and ground state cooled atoms(purple, triangles). (d) A cat state of ∆α = 20 is measured with a contrast revival peak of C0 = 0.19(3). Error bars are calculated with confidence interval of one sigma. middle box) to generate arbitrary cat states, so long as the atom’s motion stays confined within the harmonic trapping region |α| . 104. In the first of three experiment types, we demonstrate our fastest method for generating cat states by using every pulse that is emitted from a mode locked laser (repetition rate frep=81.4 MHz) to generate a set of N SDKs. This is achieved by separating each pulse from the laser into the eight pulses required for an SDK, in addition to adding optical elements capable of physically swapping the direction of the counter-propagating pulses (see later section on Experimental Setup). Swapping 83 the direction compensates for the spin flip that occurs after each SDK and allows for fast concatenation of constructive momentum transfers (Fig. 5.1d). Starting each experiment, the ion is initialized in the state |ψ1〉 = 1√2(|↓〉+ |↑〉) |0〉 using resonant microwaves. We apply a series of N SDKs which cause the superposition to grow in size at a rate d|α| dt ≈ ηfrep (this rate holds for small enough N, see Experimental Setup section), approximately generating the cat state |ψ2〉 = 1√ 2(1+e−|α|2 ) (|↑〉 |α〉+ |↓〉 |−α〉). After allowing the state to evolve for varying amounts of time T , then applying a second identical set of displacement operators, the state |Ψcat〉 ∝ |↑〉 |−αe−iθ + α〉+ |↓〉 |αe−iθ − α〉 (5.3) is ideally created, where θ = ωT (Fig. 5.2a). The phase of the second microwave pi/2 pulse is scanned to probe the qubit contrast [32] C(θ) = C0e −4|α|2(1−cosθ) (5.4) where C0 < 1 accounts for imperfect operations. At integer multiples of the trap period θ = 2pim;m ∈ Z, we observe revivals in contrasts, and when |α|  1√ 2 , the revival lineshape is approximately Gaussian with a FWHM of 1.18/|α|. In Fig. 5.2b, revival lineshapes at θ = 2pi are shown in which the state |ψ2〉 is generated for (up to) ∆α = 4.0 in 111 ns with fidelity of F = 0.59(3) estimated using the relation F = C 1/2 0 . This gives and effective single SDK fidelity of 0.951(4), which is lower than that of a true single SDK because of power fluctuations associated with swapping laser directions. Such a demonstration is an important benchmark for ultrafast quantum information processing. 84 In a second set of experiments, we create large cat states using a technique that does not require switching laser beam paths and instead works by delivering an SDK at every half trap period to excite large superpositions (Fig. 5.1e). This maintains high SDK fidelity by leaving the beam paths stationary, and the cat state grows at an average rate of d|α| dt = ηω/pi. Using this method we produce and verify states |Ψcat〉 up to ∆α = 20 (Fig. 5.2c,d). This largest state, with 100~k of momentum in each coherent state, has a 209 nm maximum separation and contrast C0 = 0.19(2). Generating the large superposition state requires a high level of trap stability, which is achieved using a rf stabilization procedure [42]. Additionally, the trap frequency ω is scanned for fine control in θ [33]. The total measured fidelity of each SDK is found to be 0.978(2) for displacing coherent states, and 0.9912(6) for Doppler cooled states. This discrepancy is most likely due to the slower rate of coherent cat state creation due to ground state cooling allowing slower drifts to have effects. The speed, fidelity, and high level of control in ultrafast operations allows us to make more complicated, multicomponent cat states. First, we create three and four component cat states with one additional microwave pulse and SDK set. Starting from the state |ψ2〉, a microwave pi/2 pulse rotates the state to |ψ3〉 ∝ (|↑〉 − |↓〉) |α〉 + (|↑〉 + |↓〉) |−α〉. A set of SDKs then produces three and four component cat states of the form |Ψ3,4cat〉 ∝ |↑〉 (eiφ1 |αe−iθ + α〉+ eiφ2 |αe−iθ − α〉) (5.5) + |↓〉 (eiφ3 |−αe−iθ + α〉+ eiφ4 |−αe−iθ − α〉), with configuration depending on θ (Fig. 5.3a). (phases φ1, φ2, φ3, and φ4 discussed 85 22 1 θ 3 3 a b c θ/2π 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 C on tra st θ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 C on tra st 4θ 4 44θ θ θ 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 θ/2π C on tra st d e π/2 T SDK sets 2 3 π/2 �/2ω SDK sets 2 3 π/2 T 4 Figure 5.3: Three, four, six and eight-component cat states. (a) Cre- ation of a multicomponent cat state begins by applying a set of SDKs to take the state |ψ1〉 (1) to the state |ψ2〉 (2). A microwave pi/2 pulse ro- tates the qubit to produce the state |ψ〉 ∝ (|↑〉+|↓〉) |α〉+(|↑〉−|↓〉) |−α〉) (3). Another set of SDKs generates the three or four-component cat state. The diagram within the dashed line replaces the one in Fig. 5.1a for these experiments. (b) If θ = 0, two of the components rejoin and the state has the form |α〉+ |0〉+ |−α〉. If θ = pi/4, for instance, then a four- component cat state of the form |α〉+ |−α〉+ |iα〉+ |−iα〉 is generated. The final microwave pulse analyzes the state contrast, and is plotted as a function of θ, which is compared with the predicted contrast curve with only the amplitude as a fitting parameter. Error bars are calculated with confidence interval of one sigma. (c) If the microwave pi/2 pulse in (a) is replaced by a mpi pulse, then the second SDK set behaves as it would in the 2-component experiment, with the exception that odd values of m are shifted by half of a trap period. We see this behavior fits the predicted model well. (d) The six and eight-component state is created by extending the technique for the three and four-component state with an additional microwave pulse and SDK set. (d) Contrast as a function of θ is used to verify the creation of the cat state when compared to the model (solid line). 86 in Three and Four-Component Cat Contrast section). It is evident from Eq. 5.5 that a three-component cat state is created when θ = mpi, and a four-component cat state is generated for other values of θ. Scanning θ and the phase of a final analysis microwave pi/2 pulse, we observe a contrast lineshape indicative of the desired state (Fig. 5.3b). To further verify that these multicomponent states are being created, we run the same sequence but apply either no microwave pulse, or a pi pulse, to the state |ψ2〉. An SDK set then generates the cat states |Ψcat,0〉 ∝ |↑〉 |−αe−iθ + α〉 + |↓〉 |αe−iθ − α〉 and |Ψcat,pi〉 ∝ |↓〉 |αe−iθ + α〉 + |↑〉 |−αe−iθ − α〉. These states revive at the same frequency, but out of phase by pi, which is verified in Fig. 5.3c. Continuing to unfold the state in phase space, another microwave pi/2 rotation and SDK set generates a six and eight-component cat state (Fig. 5.3d). In this case, the four component cat state is generated with a separation along one quadrature double that of the other to allow for a square lattice once the eight component state is created. Again, scanning θ and the phase of a final microwave pulse, Ramsey fringes are observed which compare well with the expected behavior (Fig. 5.3e). (See Six and Eight-Component Cat Contrast for more details.) Ultrafast laser pulses are capable of generating Schro¨dinger cat states larger than presented here, theoretically limited by the anharmonicity of the trap at large displacements. This technique can also be used to make even more complicated multicomponent states, as well as generate them in two and three dimensions by modifying the trapping potential and orientation. If a larger separation is desired for a measurement such as rotation sensing [107], lowering the trap frequency by 10 87 times would increase the separation by 10 times. 5.1 Experimental Setup Laser pulses are generated from a frequency tripled, mode-locked Nd:YVO4 laser. A pulse is divided into eight using a series of beam splitters and delays in a Mach-Zehnder interferometer configuration. The spin-dependent displacement in Eq. 5.2 has the more precise form OˆSDK = e iφλσˆ+Dˆ[iη] + e−iφλσˆ−Dˆ[−iη]. (5.6) The phase φλ is an optical phase that is assumed to be stable during the course of one experiment, but random over multiple experiments due to slow mechanical and other noise on the optics. Effects of the phase φλ cancel when an even number of applications of the operator OˆSDK are used during an experiment and so is dropped in Eq. 5.2. The first method discussed for generating cat states uses every pulse from the mode-locked laser to produce SDKs (Fig. 5.1d). This works by swapping the directions of the counter-propagating beams, countering the spin flip that occurs with each SDK. To make this swap, we combine the perpendicular linearly polarized beams on a polarizing beam splitter and pass them through a Pockels cell. The cell can rotate the polarizations by 0 or pi/2 radians arbitrarily for pulses arriving every 12 ns, here we alternate every pulse. A polarizing beam cube downstream of the Pockels cell separates the two beams after which they are directed, counter- propagating, onto the ion with simultaneous arrivals. As mentioned, the rate at 88 which the state grows is d|α| dt ≈ ηfrep for small enough N . This holds for N  2pifrep/ω, but for large enough N , the rate decrease as trap evolution counteracts the kicks. This rate cycles every half trap period because the photon momentum adds most constructively when applied during the high momentum periods of the oscillation. This is evident in Fig. 5.1d by the deviation of the coherent state from the p-axis. 5.2 Three and Four-Component Cat Contrast The contrast function which overlays the data in Fig. 5.3b is derived here. We write the time evolution operator for a coherent state as UˆT [θ] |α〉 = |αe−iθ〉. The microwave rotation operator in the z-basis is written as Rˆµ[φµ] = 1√ 2 1ˆ⊗  1 eiφµ −e−iφµ 1  , (5.7) where all rotations have pulse area pi/2. A full Ramsey experiment to create three and four-component cat states, including microwave rotations, SDKs, free evolution, and a final analysis microwave pulse produces the final state |Ψβf 〉 = Rˆµ[φ′′′µ ] · OˆSDK · UˆT [pi] · OˆSDK · UˆT [θ]· Rˆµ[φ ′′ µ] · OˆSDK · UˆT [pi] · OˆSDK · Rˆµ[φ′µ] · |↓〉 |β〉 . (5.8) 89 The spin-up portion of the final state is given as exp(−2iηβR + 2iηRe[e−iθ(2iη − β)] + iφ′′µ − iφ′µ − iφ′′′µ ) |−2iη − e−iθ(2iη − β)〉 − exp(−2iηβR − 2iηRe[e−iθ(2iη − β)]− iφ′µ) |2iη − e−iθ(2iη − β)〉 − exp(2iηβR − 2iηRe[e−iθ(−2iη − β)]− iφ′′µ) |2iη − e−iθ(−2iη − β)〉 − exp(2iηβR + 2iηRe[e−iθ(−2iη − β)]− iφ′′′µ ) |−2iη − e−iθ(−2iη − β)〉 , (5.9) where the normalization factor and spin-up ket is left out for simplicity. The bright- ness for any thermal state with average phonon occupation n¯ is given as B = 1 pin¯ ∫ ∞ −∞ e−|β| 2/n¯ 〈↑ |Ψβf 〉 〈ψβf | ↑〉 d2β. (5.10) For an ion initially in a thermal motional state the brightness is 1 4 [ 1 + e16(1+2n¯)η 2(cos θ−1) cos(φ′µ − φ′′′µ ) ] + 1 4 [ 1− e−32(1+2n¯)η2 cos2( θ2 ) cos(2φ′′µ − φ′µ − φ′′′µ ) ] + 1√ 8 e−8(1+2n¯)η 2 sin(16η2 sin θ) sin(φ′′µ − φ′′′µ ). (5.11) 90 5.3 Six and Eight-Component Cat Contrast This calculation is carried out in the same fashion, using the full set of opera- tions |Ψβf 〉 = Rˆµ[φ′′′′µ ] · OˆSDK · UˆT [pi] · OˆSDK · UˆT [θ] ·Rˆµ[φ′′′µ ] · OˆSDK · UˆT [pi] · OˆSDK · UˆT [pi] ·OˆSDK · UˆT [pi] · OˆSDK · UˆT [pi 2 ] · Rˆµ[φ′′µ] ·OˆSDK · UˆT [pi] · OˆSDK · Rˆµ[φ′µ] · |↑〉 |β〉 . (5.12) We do not show the full brightness calculation here because of its length. The solid line in Fig. 5.3e is a fit assuming that the initial motional state is β = 0. Our initial thermal occupation number is n¯ = 0.15, or about 87% in the ground state. We do not take the thermal average because our computer could not perform the intensive calculation in less than a couple of days per run. A simpler calculation including only the lowest phonon states would be simpler, but was not done because of the good agreement without averaging. 5.3.1 Sources of Error Several factors lead to less than perfect fidelity of the cat states we create. One restriction on the size of cat states we can generate and measure comes from Doppler cooling issues. Frequency instability leads to fluctuations in the initial thermal state, leading to slower data taking (cooling takes longer, and the noisier data requires more averaging) and sensitivity to slow noise. The ion is exposed 91 to off resonant light during the time that SDKs are being applied. This causes a Stark shift in the qubit splitting. SDK fidelity is discussed in other sources [1, 33]. The trap axes are rotated so that the Raman beam couples only to a single mode. Misalignment of this means some amount of motion is excited in other directions, and is not recovered. Detection fidelity is discussed in other work [62]. Finally, it is worth acknowledging that the trap is about 1 mm across in both directions (the motional wave packet for the larges state has a spread of about 200 nm), and non-harmonic contributions are negligible to the motion behavior. 92 Chapter 6: Highly Sensitive Atom Imaging The optical imaging of isolated emitters, such as individual molecules [108, 109], optically active defects in solids [110], fluorescent dyes in a solution [111], or trapped atoms [48,112], relies on efficient light collection and excellent image quality [113]. Such high resolution imaging underlies many methods in quantum control and quantum information science [48,112], such as quantum networks [114], fundamental atom-light interactions [115], and sensing small scale forces [116]. Individual atoms in particular have been resolved and imaged for many such applications [62,117–123], with performance that depends critically on minimizing misalignments and optical aberrations from intervening optical surfaces such as a vacuum window. In this chapter we develop a general method for suppressing aberrations by characterizing and adapting the imaging system, and report the highest performance optical imaging of an isolated atom to date. We image a single 174Yb+ atomic ion with a position sensitivity of ≈ 0.5 nm/√Hz for averaging times less than 0.1 s, observe a minimum uncertainty of 1.7(3) nm, and obtain direct measurements of the nanoscale dynamics of atomic motion. Complete knowledge on the wavefront distortions is obtained through the Zernike expansion of the point spread function and we adapt this information to correct aberrations and misalignments. The gen- 93 b a J=1/2 J=1/2 S2 1/2 P2 1/2 c 369.5 nm trop weiV el ohni P Ca m er a snel A N 6. 0 s nel l acir dnil yC snel rota milloC So ur ce y axis in micrometers x ax is in m ic ro m et er s 3 1 8 6 4 0 2 2 1 0 (a) ( ) (c) Figure 6.1: Schematic of the imaging system [43]. (a) Atomic energy diagram of 174Yb+. The atom is excited with laser radiation at 369.5 nm driving the 2S1/2 →2 P1/2 cycling transition and the resulting fluo- rescence is collected by the imaging system. (b) Transverse cut of the optical setup depicting the source, vacuum window, 0.6 NA objective lense, pinhole, short focal length lens, cylindrical lens and camera. (c) Image of two atomic ions separated by ∼ 5 µm. erality of the described work opens the door for improvement in adaptive optimal imaging in many other quantum optical systems as well as other contexts, such as biological microscopy or astronomy. 6.1 Experimental Apparatus The atomic imaging system is shown in Fig. 6.1. In this experiment, we confine a single 174Yb+ ion (the 174 isotope has no hyperfine splitting and so can fluoresce more brightly) in the linear Paul trap with 3D harmonic oscillation frequencies (ωx, ωy, ωz)/2pi = (1, 1.2, 0.8) MHz. Just as in the experiments with 171Yb+, laser light at a wavelength of λ = 369.5 nm is incident on the ion and resonantly excites the 2S1/2 →2 P1/2 cycling transition (radiative linewidth γ/2pi = 20 MHz) as shown in Fig. 1(a). The ion is laser-cooled and localized in each of the three dimensions 94 of position to ∆x = √ (2n¯+ 1)x0, where x0 = √ ~/2mωx ≈ 5 nm is the zero-point spread, n¯ is the mean thermal vibrational occupation number along each of the dimensions of motion, and m is the atomic mass. From Eq. 2.14, the cooling laser at an oblique angle to all directions of motion produces n¯ ≈ γ/2ωx ∼ 10. Thus, ∆x ∼ 20 nm λ and the trapped ion acts as an excellent approximation to a point source. The approximately isotropic fluorescence from the atom at λ = 369.5 nm is transmitted through the vacuum viewport and collected by an objective lens of numerical aperture NA = 0.6 with 10x magnification [62] (Fig. 6.1(b)). An in- termediate image from the objective is formed at a pinhole, which spatially filters light from background sources. A second lens, after the pinhole, re-images the ion at the face of an electron-multiplying-charge-coupled-device (EMCCD) array (camera) with about 50x magnification (Fig. 6.1(c)). This makes the total magnifi- cation about (10x)(50x)=500x. The objective lens is mounted on a precision 5-axis alignment stage to compensate for comatic aberrations, and cylindrical optics are inserted after the magnifier lens to compensate for astigmatic aberrations (chosen after looking at the aberrations on the camera). 6.2 Aberration measurement and correction The measured spatial distribution of the image is the point spread function (PSF) [124] which contains information about the ultimate resolution achievable in an imaging system and is the building block for more complex image formation 95 Figure 6.2: Aberration retrieval results [43]. (a), (b), (c) Single shot images of the misaligned system. (d), (e), (f) The optimally aligned sys- tem at various distances from the focal plane, with (f) at the best focus. For (d) and (e) a high contribution from the defocus term is evident with low contributions of astigmatism and coma. Large contributions of coma and astigmatism (a)-(c) are corrected with a 5-axis stage and cylindrical lens. Coefficients of determination are 0.989, 0.965, 0.958, 0.957, 0.983 and 0.994 for images (a), (b), (c), (d), (e) and (f) respectively. These images are integrated for ∼ 0.5 s. 96 through convolution techniques. The PSF can be decomposed into Zernike polyno- mials Zmn (ρ, θ) in space PSF(ρ, θ) = ∣∣∣∣∣F { exp ( −ik ∑ m,n cmn Z m n (ρ, θ) )}∣∣∣∣∣ 2 , (6.1) where F{} is the Fourier transform operator, k = 2pi/λ is the wavenumber and the cmn coefficients are contributions of each Zernike component defined in the polar coordinates ρ and θ. The cmn coefficients correspond to particular optical aberrations, so detailed characterization of the imaging system follows from the determination of the sign and magnitude of these coefficients. Although optical aberrations can be described in terms of a Taylor expansion of the object height and pupil coordinates [125], Zernike polynomials Zmn (ρ, θ) are better suited since they form an orthogonal basis set of functions. Zernike polyno- mials are expressed in polar coordinates ρ and θ as [126] Zmn (ρ, θ) =  Nmn R m n (ρ) cos(mθ) for m ≥ 0 Nmn R m n (ρ) sin(mθ) for m < 0, (6.2) Nmn = √ 2(n+ 1) 1 + δm0 , (6.3) R|m|n (ρ) = (n−|m|)/2∑ s=0 (−1)s s![(n+ |m|)/2− s]! × (n− s)! [(n− |m|)/2− s]!ρ n−2s, (6.4) where n is an integer number, m can only take values n, n − 2, n − 4, ...,−n for each n, and δm0 is the Kronecker delta. The radial coordinate is normalized to the exit pupil radius (the radius of the image of the input aperture at the camera). Importantly, each term of this polynomial expansion has a one-to-one relation with 97 a specific kind of aberration. Given the Zernike expansion of a wavefront, we can calculate its deviation from a perfect wavefront using the cmn coefficients of eq. (6.1). Decomposing an image into Zernike polynomials relies on numerical algorithms [127, 128] or semi-analytical calculations [129]. Here we obtain a full aberration characterization by using a least-squares fit to the measured data, using the cmn coefficients and the exit pupil radius as fitting parameters. Although this method omits consideration of vector (polarization) effects, it remains a generally applicable technique since these effects can be neglected at numerical apertures above 0.6 NA [130]. Fig. 6.2 shows six single-shot images of a single 174Yb+ ion. Figures 6.2(a)-(c) were taken during alignment and Figs. 6.2(d)-(f) were taken at different distances from the focal plane of the optimally aligned system. The images were integrated for ∼ 0.5 s, collecting ≈ 7× 105 photons and fitted according to Eq. 6.1 to a linear superposition of the first twelve Zernike polynomial basis functions. The overall fitting function is then smoothed by convolving with a Gaussian function that best fits the data and accounts for spatial drifts over long exposures. The Gaussian function parameters are added to the fitting algorithm and are only important for integration times longer than 0.2 s (See next section and supplemental materials). We find that the optimal image (Fig. 6.2(f)) has a characteristic radius of ρ0 = 363(18) nm, consistent with the diffraction-limited Airy radius of ρ0 = 0.61λ/NA = 375.1 nm given the system numerical aperture. Based on the one-to-one mapping of the Zernike polynomials to optical aber- rations, we plot an aberration budget which shows the leading order aberration 98 contributions to each of the images. For example, the contribution of the domi- nating negative (positive) defocus term of Fig. 6.2(d) (Fig. 6.2(e)) shows that we can map axial displacement on a transverse image distribution, with the position of best focus shown in Fig. 6.2(f). Moreover, a contribution of the comatic aberration indicates angular tilt errors and non-zero values of astigmatism indicate anisotropic foci in the system, seen in Figs. 6.2(a)-(c). Fitting results show parameter uncertainties on the order of 1 nm, providing a full quantitative basis for analyzing systems that rely on aberrations to extract information on particle dynamics. Examples of these experiments involve 3d off- focus tracking [131] and imaging of atoms arranged in 3d lattices [132]. Although we describe an atomic emitter, this method can also be applied to the imaging of microbiological test samples. 6.3 Position sensitivity The precision of measuring atomic position is dependent on the imaging system light collection and quality. As a result of the optical aberration characterization, even if it is not possible to directly correct the aberrations in the imaging system by alignment, it is feasible to post process and actively feedback the aberrated image and obtain a diffraction-limited performance through a digital filter with the information of the Zernike expansion. In this experiment we only correct the aberrations by direct alignment. We measure the sensitivity to the position by takingN images at 1 ms exposure 99 10−2 10−1 100 100 101 τ (s) Po si tio n un ce rt ai nt y δ x (n m ) 0.2 s 1.7(3) nm Figure 6.3: Measured position uncertainty δx of the trapped ion cen- troid position versus image integration time τ [43]. The blue line shows the expected uncertainty limited by photon counting shot noise in the imaging system. A sensitivity of ∼ 0.5nm/√Hz is measured for τ < 0.1 s, which is ∼ 3 times worse than shot noise, presumably because of cam- era noise. The ultimate position sensitivity is found to be 1.7(3) nm at τ = 0.2 s. These measurements include small corrections for dead time bias, as described in Methods. The error bars on each point are given by the root-mean-square error. 100 time, binning them over total time duration intervals τ and calculating the Allan variance of the central position [133]. σ2(τ) = 1 2(M − 1) M−1∑ n=1 (xn+1 − xn)2, (6.5) where M is the number of samples per bin and xn is the centroid of the ion image integrated over time τ . Each image was integrated along one direction and fit to a one dimensional Gaussian linear count density function. The same procedure taken at different times τ leads to a curve of position uncertainty δx vs integration time as shown in Fig. 6.3. The data is corrected for a dead time of 5 ms between each 1 ms frame, allowing for state preparation and laser cooling [133,134]. Dead times were corrected using the Allan B-functions [134] σ2(τ) = σ2(2,MT0,Mτ0) B3(µ)B2(µ) (6.6) where µ is the noise model coefficient that ranges between −1 < µ < 1, M is the binning parameter, T0 is the time between data acquisitions and τ0 is the sampling time. Dead times are then defined as tdead = T0 − τ0 for single acquisition times. The integration time for the Allan variance is τ =Mτ0. The noise model coefficient upon which the B-functions depend at each τ were found solving B1(µ) B3(µ) = σ2(N, T, τ) σ2(2,MT0,Mτ0) (6.7) for µ with σ2(N, T, τ) defined as the standard variance. The theoretical net position sensitivity for diffraction-limited imaging is a quadrature sum of three main (uncorrelated) sources of uncertainty: shot noise, 101 pixelation and background noise [135,136] δx = √ 2ρ20 R0τ + l2p 12R0τ + 16piρ40b R20τ 2 , (6.8) where b ≈ 0.07 is the mean background count rate per pixel, and lp ≈ 33 nm is the pixel size refered to the object (image pixel size divided by magnification). Recall ρ0 = 363(18) nm is the characteristic image radius. R0 = ηDFγ/2 is the maximum (saturated) measured fluorescence count rate from the atom, where F ≈ 10% is the solid angle fraction of fluorescence collected, γ/2pi = 20 MHz is the radiative linewidth, and ηD ≈ 25% is the quantum efficiency of the camera. Finite pixel size and background counts have negligible impact on the measured position sensitivity in this experiment. The observed sensitivity of ∼ 0.5nm/√Hz at small integration times is somewhat higher than the expected level of shot noise (shown as the blue line in Fig. 6.3), and is consistent with observed super-Poissonian noise on the camera. We measure a minimum uncertainty of δx ≈ 1.7(3) nm at an integration time of τ = 0.2 s. For longer integration times, drifts in the relative position between the optical objective and the trapped ion degrade the position uncertainty as shown in Fig. 6.3, and with simple mechanical improvements in the imaging setup the resolution can likely be pushed well below 1 nm. Given this uncertainty in the position of the harmonically-bound ion, the sensi- tivity to detecting external forces is δF = mω2xδx. Imagine trapping a single 174Yb+ ion with ωx/2pi = 10 kHz (which is experimentally possible); this would correspond to a force sensitivity in the yoctonewton (10−24 N) scale, or an electric field at the µV/cm scale. Unlike earlier work [116], this imaging force sensor applies to single 102 PMT x z v k 369.5 nm a=0 a b −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0 0.5 1 0 0.1 0.135 Position am plitude Velocity Position Mask position scan Mask position a (mm) Ve lo ci ty a m pl itu de ( ) ( ) Figure 6.4: Micromotion position measurement [43]. (a) The ion’s velocity ~v (solid black arrows) is colinear with the direction ~k of the detection light, taken to be the x-axis. The fluorescence is modulated from the micromotion of the ion along x by the first order Doppler effect as well as the obscuration by a mask with variable position a along the x-axis. (b) Contributions of the velocity (left y axis) and position (right y axis) of a single atom when a mask is scanned along one transversal direction x. The solid (dashed) line depict a fit to the data of the velocity (position) component of eq. 6.10 given by the cosine (sine) term alone. All the values are normalized with the signal amplitude at a = −∞. The horizontal error bars are given by the uncertainty of the scanning stage (0.01 mm) and the vertical errors are computed by the uncertainty propagation using Eq 6.10. ions and does not require resolution of optical sidebands. This could be used to mea- sure tiny forces between trapped ionic molecules or atoms in a chain or gravitational effects. 6.4 Sensing of Rf Induced Micromotion Position Confinement of atomic ions in a Paul trap is achieved through oscillating rf electric field gradients that create a harmonic ponderomotive potential [47]. In the 103 presence of a static uniform electric field E, which pushes the ion off of the rf null of the trap and into the oscillating field, the ion acquires a “micromotion” modulation in position x(t) = Xµ sin Ωt to first order in the pseudopotential approximation [15, 47], where Ω is the drive frequency of the rf trapping field (≈ 17 MHz in this experiment) and Xµ = √ 2eE/(mΩω) is the micromotion amplitude. The conventional approach for sensing micromotion is based on the first order Doppler modulation in the scattering of light from a laser beam of wavenumber k propagating along the micromotion velocity [137] (See Fig. 6.4(a)). This is based on a principle where a moving atom sees a velocity-dependent laser frequency from the Doppler effect, and when the laser frequency is tuned to be on the slope of the resonance lineshape of the atom’s cooling transition, the scattering rate depends on the ion velocity. The correlation between the photon arrival times (measured with a photomultiplier tube) and the micromotion velocity is obtained with a time-to- digital converter. With the excitation laser red-detuned from resonance of order γ and for small levels of micromotion kXµ  1, the measured fluorescence signal takes the form [138], R(t) = αR0 + βR0 ( kXµΩ γ ) cos Ωt, (6.9) where α, β ≤ 1 are dimensionless constants that depend on the precise detuning and intensity of the excitation laser [138], and R0 is the same as in Eq. 6.8. In order to also sense a direct position sensitivity to motion, we spatially mask the ion image with a sharp edge aperture, normal to the (x) direction of motion. The mask position can be adjusted from, effectively, a = −∞ (completely exposed) 104 to a = +∞ (completely masked) with a = 0 covering exactly half of the image. The total fluorescence behind the mask is then the integrated fluorescence behind the exposed area, R(a, t) = αF (a)R0 + βF (a)R0 ( kXµΩ γ ) cos Ωt+ αR0 Xµ σ √ pi e−a 2/2σ2 sin Ωt, (6.10) where we assume a Gaussian image distribution (approximating an Airy function) in space with root-mean-square radius σ = 0.36ρ0 (conversion between Gaussian width and Airy width) and the scale of the mask position a is referred to the object. The first term in Eq. 6.10 is the average scattering rate spread over a Gaussian distribution and integrated over space where the mask is not present. This integra- tion yields the error function F (x) = [1−erf( x σ √ 2 )]/2. The second term is found in the same way. The third term comes from a linear approximation of the scattering rate modulation due to small micromotion position fluctuations of the image at the mask. We extract the two quadratures of the modulated fluorescence from Eq. 6.10 by performing sine and cosine transforms of the data. The phases of the modulated signal are calibrated by opening the aperture (a = −∞) and taking the modulation as proportional to cos Ωt. Figure 6.4(b) shows the position (sin Ωt) and velocity (cos Ωt) quadrature am- plitudes (normalized to the amplitude at a = −∞) as the mask position is scanned. Based on the observed velocity-induced modulation in the count rate with full ex- posure (a = −∞), we infer a micromotion amplitude of Xµ ∼ 20 nm. As the mask is scanned along x, a position-dependent modulation in the fluorescence rate arises, 105 reaching a maximum level at a = 0. The absolute level of this position-dependent modulation is observed to be 15 times smaller than expected from Eq. 6.10. This may be due to fluctuations and drifts in the relative position of the ion with respect to the mask: fluctuations or drift of just 30 nm during the 300 s integration time required to obtain sufficient signal/noise ratio in the measurement would explain the observed reduction in the modulation. This could either be vibrations of the optical table, or the trapping system relaxing after loading (it takes up to one hour for the ion position to stop drifting after loading–an effect where it starts about 1 µm out of place after loading and returns to a repeatable spot over time). This type of measurement could allow us to minimize micromotion, although we did not because of the smaller than expected modulation. 6.5 Outlook In the single atom emitter presented here, ultimate average position determi- nation to the level of angstroms (10−10 m) should be possible for longer integration times, when drifts slower than 0.2 s are eliminated or actively corrected. Drifts may be reduced by further isolating the vacuum chamber from the environment and sta- bilizing the relative position between the sample and the objective. Since we obtain information of the centroid position, the zero-point ion motion does not affect these measurements. More generally, the isolation and correction of wavefront distortions by fitting intensity images can be extended to the adaptive imaging of a variety of source 106 objects. If the fluorescence from the emitter is incoherent (not the case for multiple trapped ions illuminated with the same detection laser beam and within the coher- ence length of each other [139]), then imaging errors should not accumulate, and images from multiple emitters localized in distinct regions of space could be cor- rected. This technique can also be used for the optimization of laser output cavity spatial modes [140]. Most misalignments provide an unambiguous signature in the image decomposition, but for certain symmetric misalignments such as axial dis- placements about the focus for single-atoms, introducing additional aberrations like astigmatism [132] or coma will yield an unique fit. Given sufficient emitter bright- ness, active shot-to-shot adjustments and feedback to appropriate optical elements should allow the continuous optimization of image quality. The bandwidth of this type of adaptive imaging would be limited by the speed of the correction elements and the computing time for numerically extracting the error signal (this is ∼ 1 s in our experiment). 107 Chapter 7: Realizing an Ultrafast Two Ion Gate Trapped ions have proven to offer excellent quantum memory [141], in addition to serving as useful tools for quantum gates [2, 27, 29] and modular entanglement (quantum processing modules arbitrarily connected by photons in fibers and fiber switches) for scalable quantum computation [142–144]. Although they have a small gate-to-coherence time ratio, gate speeds have been limited by the trap evolution time. In this chapter, we demonstrate an entangling gate which is not fundamentally limited by trap evolution but instead by the Coulomb interaction between ions. Entanglement between two ion qubits is generated using a series of SDKs with arbitrary kick direction, and periods of free evolution. As in the experiments with cat states, we restrict this entangling gate to a single axis of motion–a radial mode in this case. The radial mode is necessary because of the tunable ratio of the CoM (Center of Mass) and Relative modes of motion. Tuning the ratio of CoM and Relative mode of motion allows both paths to close in phase space, which disentangles the spin and motion and completes an entangling gate sequence. 108 7.1 Single Mode Phase Accumulation First consider a single oscillator (frequency ω) in an initial coherent state |α0〉, which is kicked with magnitudes zk at times tk a total of n times yielding a final state eiξn |α(tn)〉 with [44] α(tn) = e −iωtn ( α0 + i n∑ k=1 zke iωtk ) (7.1) ξn = Re [ α0 n∑ k=1 zke −iωtk ] − n∑ k=2 k−1∑ j=1 zkzj sin[ω(tj − tk)]. (7.2) When applying a sequence of kicks that is designed to open and close the motional phase space trajectory, satisfying the condition n∑ k=1 zke iωtk = 0 (7.3) (imagine a set of vectors in the complex plane placed tip to tail; then closing phase space means this chain ends up back where it started), the accumulated phase is independent of the initial motion, and the final state is given by α(tn) = e −iωtnα0 (7.4) ξn = n∑ k=2 k−1∑ j=1 zkzj sin[ω(tk − tj)]. (7.5) 7.2 Two Ion Relative Phase Accumulation Now consider again two trapped 171Yb+ ions which are being kicked simulta- neously by the same laser (Fig. 7.1) along one principle radial direction of the trap. In [34, 44], the SDK operator is recast in terms of the CoM and relative mode to 109 tl1 tl1 tl2 tl2 tl8 tl8 ... k k ... 21 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 truth table for t1 +2hk 2hk p 0 (a) (b) Figure 7.1: A two ion SDK. (a) An SDK on two ions is made using the same eight-pulse sequence discussed in previous chapters. The beam is applied to both ions simultaneously. The color of the ion specifies the spin state (black is |↓〉 and white is |↑〉), and the number is the physical position of ions. (b) Truth table for the two ion SDK. Depending on the initial qubit state of the ion, the SDK kicks the ion in the positive or negative direction. 110 be OˆSDK(t) = e 2iφ(t)σˆ+,1σˆ+,2DˆC [iηC ] + σˆ+,1σˆ−,2DˆR[iηR] +e−2iφ(t)σˆ−,1σˆ−,2DˆC [−iηC ] + σˆ−,1σˆ+,2DˆR[iηR], (7.6) where DˆC [] and DˆR[] are the displacement operators for the normal modes, and ηC and ηR are the modified Lambe-Dicke parameters ηC = √ 2η (7.7) ηR = √ 2 √ ω ωR η. (7.8) With kick size zk = bkηC,R, where bk = ±1 signifies the kick direction, and Eq. 7.3 being satisfied for both mode frequencies (which we guarantee by proper tuning of the trap voltages), the accumulated phase difference between the modes of motion is Φ = n∑ k=2 k−1∑ j=1 bkbj(η 2 R sin[ωR(tj − tk)]− η2C sin[ωC(tj − tk)]). (7.9) In terms of a truth table (where the two qubit state is represented in a single ket for simplicity), the gate has the effect |↓↓〉 ⇒ ei(Φ+γ)i |↓↓〉 (7.10) |↓↑〉 ⇒ |↓↑〉 (7.11) |↑↓〉 ⇒ |↑↓〉 (7.12) |↑↑〉 ⇒ ei(Φ−γ)i |↑↑〉 , (7.13) 111 where the additional phase γ = φ(t1)− φ(t2) + φ(t3)− ... = ωA(t1 − t2 + t3 − ...) +  0, if n even −φ0, if n odd (7.14) is imparted on the CoM mode and pertains to the phase of the light at the occurrence of each kick [44] (as previously mentioned, the relative frequency ωA between the two, counter-propagating laser beams is introduced with AOMs). It is crucial to notice that the optical phase φ0 only cancels for an even number of total kicks during a gate sequence. The phase that remains does not affect the amount of entanglement but does affect how the entanglement is witnessed through parity oscillations. This is discussed further in the next section. 7.3 A Particular Solution We realize a gate by following a technique which has previously been outlined [32, 34]. The gate utilizes a particular symmetry to reduce the number of available paths in phase space, which also allows us to calibrate the gate without adjusting the motional mode frequencies. We build this gate using a sequence outlined as: N1 kicks, followed by an absence of kicks with duration equal to the time it would take for N2 kicks, then applying another N1 kicks which can be in the same, or opposite direction, then waiting in the absence of N3 kicks before repeating the first three stages, possibly inverting the kick directions. The pulses and waits are separated in time by 1/frep and occur at times ti. This is outlined in Fig. 7.2. The values of N1, 112 N2, and N3 are given by solutions to Eq. 7.3, which is satisfied when either 1− eiN1ωC,R/frep = 0, or (7.15) 1± ei(N1+N2)ωC,R/frep = 0, or (7.16) 1± ei(2N1+N2+N3)ωC,R/frep = 0 (7.17) for both ωC and ωR (mixing and matching is alright, as long as at least one equation is satisfied for each). It is not immediately clear that Eq. 7.15 is too restrictive, but we will see that because ωC,R/frep have an additional constraint, we cannot use Eq. 7.15 to provide a solution. We may, however, use Eq. 7.16 (→ Eq. 7.18) and 7.17 (→ Eq. 7.19) to provide solutions of the form (N1 +N2) = n ωrep 2ωC,R ; n ∈ N (7.18) (2N1 +N2 +N3) = m ωrep 2ωR,C ; m ∈ N. (7.19) The integers n and m are free parameters which are proportional to the number of trap evolutions that occur during the wait times N2 and N3, meaning lower values yield faster gate times (but still tied to the trap period), while larger values often give more relative phase accumulation. The values n and m also set the kick directions (see Fig. 7.2). Also note that if ωrep/2ωC,R is an even integer, these equations are satisfied for all values of n and m. We now have all the tools to pick parameters for an entangling gate. One solution that works for our particular setup is fC = frep/66 = 1.234 MHz and fR = frep/72 = 1.1308 MHz. We set N1 to be 1, 2, 3, 4, 5, and 6, and solve N2 and N3 for each. In practice, this allows us to observe phase accumulation while 113 N1 N2 N3 N1 N1 N1 N2 N1N2 N3 N1 N1 N1 N2 N1 N2 ... ... N1 ... N3 ... N1 N2 ... ... N1 ... Sign of Kicks 1 (-1)(n+1) (-1)(2n+m+1) (-1)(n+m) CoM Mode Phase Space Relative Mode Phase Space x(t) p(t) p(0) x(0) x(t) p(t) p(0) x(0) bi= i=1 2 Figure 7.2: Phase space and SDK timing for a gate. This is displayed in frames that rotate at the respective mode frequencies, and the area enclosed is the accumulated phase of each mode. N1 kicks are separated by N2 and N3 periods of waiting. The period between each kick (kicks are shown as vertical lines connected by a node to the horizontal time axis) is given by the repetition rate of the mode locked laser–a fixed number. The direction (sign) of each kick is given by bi and is the same for each group of N1, but can be different between groups depending on the chosen solution. Definitions for n and m are given in Eqs. 7.18 and 7.19. 114 N1 = 1 CoM mode closes rst Rel. mode closes second fC fR {N2, N3, Φ}{n, m} {1, 1} {32, 2, 0.045} {6, 6} {2, 2} {3, 3} {65, 5, 0.087} {98, 8, 0.123} {197, 17, 0.175} {4, 4} {131, 11, 0.151} {5, 5} {164, 14, 0.169} Figure 7.3: Phase space trajectory for N1 = 1 following the scheme of Fig. 7.1. The trajectories for the CoM mode follow paths that perfectly backtrack (zero phase accumulation) because there is only one kick in each direction and so there is no trap evolution between kicks. Images are scaled to fit the table. 115 {31, 1, 0.18} {64, 4, 0.348} {97, 7, 0.493} {196, 16, 0.697} {130, 10, 0.603} {163, 13, 0.673} fC fR {N2, N3, Φ}{n, m} {1, 1} {6, 6} {2, 2} {3, 3} {4, 4} {5, 5} N1 = 2 CoM mode closes rst Rel. mode closes second Figure 7.4: Phase space trajectory for N1 = 2 following the scheme of Fig. 7.1. Images are scaled to fit the table. 116 {30, 0, 0.405} {195, 15, 1.563} {194, 14, 2.767} {94, 4, 3.034} {60, 0, 3.075} N1 = 3 fC fR {N2, N3, Φ}{n, m} {1, 1} {6, 6} {6, 6} {3, 3} {1, 1} N1 = 4 N1 = 5 N1 = 6 Figure 7.5: Phase space trajectory forN1 = 3, 4, 5, 6 following the scheme of Fig. 7.1. Images are scaled to fit the table. 117 keeping the main source of infidelity–the kick number N1–down before scaling up the phase accumulation to generate more entanglement between the ions. Tables of some solutions, along with a phase space trajectory for several values of n and m, are shown in Figs. 7.3-7.5. It is clear that only m ≥ n works for our trap frequencies. Furthermore, having m > n only increases time and not phase accumulation, so those cases are not considered. Thus, for these solutions m = n, but other solutions exist and would be useful in certain situations. The phase Φ is calculated using Eq. 7.9, with the values of bi chosen to close phase space, and shown in Fig. 7.2. 7.4 Measuring Entanglement Knowing the phase accumulation for each kick sequence, it is possible to de- scribe how an entanglement witness, in this case parity, should look. We describe a global microwave qubit rotation as Rˆµ(θµ, φµ) = [cos θµ 2 1ˆ1 + i sin θµ 2 ~ˆσ1 · ~u(φµ)]⊗ [cos θµ 2 1ˆ2 + i sin θµ 2 ~ˆσ2 · ~u(φµ)], (7.20) where the qubit is rotated about the Bloch sphere vector ~u(φµ) = cosφµ~ex + sinφµ~ey (7.21) using the Pauli matrices ~ˆσ1,2 = σˆ1,2x~ex + σˆ1,2y~ey + σˆ1,2z~ez, (7.22) and the quantities θµ and φµ are the microwave pulse area and phase. The ultrafast entangling gate operation may be written as Gˆ = ei(Φ+γ) |00〉 〈00|+ |01〉 〈01|+ |10〉 〈10|+ ei(Φ−γ) |11〉 〈11| , (7.23) 118 where Φ and γ are from Eqs. 7.9 and 7.14. A full entangling experiment is described with a set of operations taking an initial state |00〉 to the final state |ψf〉: |ψf〉 = Rˆµ(θµ,a, φµ,a)Rˆµ(θµ,2, φµ,2)GˆRˆµ(θµ,1, φµ,1) |00〉 . (7.24) Fluorescence collected during detection over many experiments yields the probability of 0, 1, and 2 bright ions (see Fig. 2.6(b)) and is used to find the parity [30] P = | 〈11|ψf〉 |2 + | 〈00|ψf〉 |2 − (| 〈10|ψf〉 |2 + | 〈01|ψf〉 |2) . (7.25) In practice, because we are using a single-channel PMT there is no discrimination between the quantities | 〈10|ψf〉 |2 and | 〈01|ψf〉 |2; both register as one bright ion (again, recall Fig. 2.6(b)). This is not a problem because the quantity is summed in Eq. 7.25, and so we use the total quantity in that bin. A critical feature of Eq. 7.24 is the second microwave rotation Rˆµ(θµ,2, φµ,2), which occurs before the final analysis pulse. This rotation allows us to compensate for the effect that the phase γ has on the parity oscillation curve which is important because γ can be  1. This experiment is currently underway, and only one set of preliminary data has been taken. We have set N1 = 1, while n = 3 and m = 3, giving a modest phase of Φ = pi/25 and a gate time of about 4 µs. Taking a parity curve for 3 different values of N3, with the phase φµ,2 set at the theoretical optimum to cancel γ (ideally the optimum would be measured experimentally), we see the curves in Fig. 7.6(a). When N3 = 8 (red data in Fig. 7.6(a)), which satisfies the condition to completely close phase space in this case, parity oscillations are consistent in amplitude with the model in Fig. 7.6(b). When N3 = 23, in which the spin and motion do not disentangle, we see the parity signature disappear (black data in Fig. 119 7.6). Additionally, the effects of the phase γ are only present when N3 is even. When N3 = 13 (blue data in Fig. 7.6(a)), the parity oscillation is decreased in amplitude but still present indicating it is a result of entanglement. The decrease in parity amplitude from N3 = 8 to N3 = 13 to N3 = 23 is consistent with the motional linewidth (see Chapter 4 on atom interferometry) at the measured thermal phonon occupation of n¯ = 13. The reason for the offset of the parity curve is likely calibration error of the histogram fitting, as described in Chapter 2.3.2. These measurements need to be improved, and there are a number of things that should increase fidelity such as pointing stability and reducing aberrations in the Raman laser beam, Pockels cell upgrades, higher experimental duty cycle to experimentally find γ, and more experience with histogram calibration for multiple ions. With this, there should be a clear path to increase this to Φ = pi/2 (a fully entangled state). 120 0 1 2 3 4 5 6 0.3 0.2 0.1 0.0 0.1 a P (a) (b) Figure 7.6: Ultrafast Entanglement Partiy Oscillations. (a) Parity data for N1 = 1 with N3 = 8 (red, proper solution for entangling gate), N3 = 13 (blue) and N3 = 23 (black). When, N3 = 8, the gate phase Φ = pi/25. Decreased amplitude in the blue and zero amplitude in the black plots is due to reduced and zero overlap of final motional states, showing that there is a motional dependence in the parity curves and entanglement for N3 = 8 and N3 = 13. The offset is likely due to histogram calibration error. (b) Theoretical parity oscillation overlying the N3 = 8 entangling gate data. The theory curve is shifted down to match the offset of the data, but the amplitude, phase, and frequency are not used as fitting parameters. 121 Chapter 8: Outlook and Future Directions The experiments presented here provide a sense of what ultrafast laser-atom interactions can be used for. In addition to improving the ultrafast entangling gate to its theoretical limit, there are still a number of experiments that only this apparatus (as of now) is capable of doing. Here is a short list of ones we find particularly interesting. 8.1 Imaging a Large Cat State and Other Motion Based on the research that is presented in Chapters 5 and 6, it should be possible to directly image the separation between two parts of a large cat state. With the current trap frequency of ω/2pi = 1 MHz, 100 SDKs produces separation of ∆x = 2 p mω ∼ 400 nm, (8.1) where p = 200~k is the momentum absorbed by each component of the cat state, and m = 171 amu. The zero-point spread of the wave function is given by x0 = √ ~ 2mω ∼ 1 nm, (8.2) 122 which is also the size of the components of a cat state made from displaced vacuum states. This zero point spread, however, is much smaller than the wavelength of the detection light, which governs the diffraction limited spot size, even when the trap frequency is decreased by an order of magnitude. At a trap frequency of ω/2pi = 100 kHz, only 10 SDKs are required to create the same 400 nm of separation. On the other hand, 100 SDKs would make a cat state with a 4 µm separation between the states. By averaging a large number of experiments where we stroboscopically image the ion at points when the cat state is largest, we should be able to collect enough light to see separation between the components. This same technique may be useful for other states of motion. 8.2 Delta-Kicked Harmonic Oscillator A proposal in 1997 for studying quantum chaos in an ion trap delta-kicked harmonic oscillator remains to be implemented [41]. This is because there is no system capable of providing a delta kick to a quantum harmonic oscillator. The relationship to a delta kicked rotor is discussed in theory [145]. With our system ideally suited for this, we should be able to study some of the properties of such a system. 8.3 Hamiltonian Engineering By creating SDK laser-atom interactions, which occurs on a time scale shorter than other relevant time scales in the system, it should be possible to engineer 123 Hamiltonians that would otherwise not be possible using the process of Trotteri- zation [38–40]. This method provides a way of building a time evolution operator that is approximately equal to the evolution operator of the desired Hamiltonian by using a series of different shorter interactions. 8.4 Superfast Cooling A proposal in 2010 showed another method to perform sub-Doppler cooling using ultrafast pulses [39]. Since then, we have worked with the authors of the proposal to implement this in our system. It is a work in progress. 8.5 Two-dimensional Cat States By providing a set of SDKs that is not along either axis of a non-degenerate trap (recall Fig. 2.8), it would be possible to create a cat state in two dimensions. We have not worked this out, but it may be useful for studying decoherence or for metrology [107]. 8.6 Working in the Infrared Because of the difficulties involved in working with high power Ultraviolet laser light, we have considered using a visible or infrared laser, which might be doubled or tripled just as it passes into the trap. 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