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Generation of Spin Squeezed States Via Quantum Non-Demolition Measurements

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Generation of Spin Squeezed States Via Quantum Non-Demolition Measurements Breaking Quantum Limits with Collective Cavity-QED: Generation of Spin Squeezed States via Quantum Non-Demolition Measurements by Zilong Chen B.S., Massachusetts Institute of Technology, 2004 M.S., University of Colorado at Boulder, 2009 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics 2013 This thesis entitled: Breaking Quantum Limits with Collective Cavity-QED: Generation of Spin Squeezed States via Quantum Non-Demolition Measurements written by Zilong Chen has been approved for the Department of Physics James K. Thompson Jun Ye Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Chen, Zilong (Ph.D., Physics) Breaking Quantum Limits with Collective Cavity-QED: Generation of Spin Squeezed States via Quantum Non-Demolition Measurements Thesis directed by Prof. James K. Thompson Large ensembles of uncorrelated atoms are extensively used as precise sensors of time, rota- tion, and gravity, and for tests of fundamental physics. The quantum nature of the sensors imposes a limit on their ultimate precision. Larger ensembles of N atoms can be used to average the quan- p tum noise as 1= N, a scaling known as the standard quantum limit. However, the ensemble size may be limited by technical constraints and/or atom-atom collisions { a fundamental distinction from photon-based sensors. Learning to prepare entangled states of large ensembles with noise properties below the standard quantum limit will be key to extending both the precision and/or bandwidth of atomic sensors. More broadly, the generation and application of entanglement to solve problems is a core goal of quantum information science being pursued in both atomic and solid state systems. In this thesis, we utilize the tools of cavity-QED to prepare entangled spin-squeezed states with 3.4(6) dB improvement in spectroscopic sensitivity over the standard quantum limit. The collective atomic spin is composed of the two-level clock states of 87Rb confined in a medium finesse F = 710 optical cavity. We employ cavity-aided quantum non-demolition measurements of the vacuum Rabi splitting to measure and subtract out the quantum projection noise of the collective spin state, preparing states with collective atomic spin projection noise 4.9(6) dB below the projection noise level. The conditionally reduced spin noise combined with the measured 1.5(3) dB reduction in the mean spin length implies a net 3.4(6) dB spectroscopic enhancement or conditional squeezing as defined by the Wineland criterion. Our method does not require single particle addressability and is applied to a spectroscopically large ensemble of N = 7 × 105 atoms using two collective population measurements, with the whole squeezing operation taking ∼ 150 µs. iv The gain in sensitivity is spectroscopically equivalent to the enhancement obtained had we created > 105 pairs of maximally entangled qubits, demonstrating the power of a top-down approach for entangling large ensembles. The nondemolition probing of atomic populations via the vacuum Rabi splitting is also of broad interest for non-destructively reading out a wide variety of both atomic and solid state qubits. Contents Chapter 1 Introduction and Motivation 1 1.1 Precision Metrology . .4 1.2 Contributions to Superradiant Lasers . .6 1.3 Organization of Thesis . .8 2 Spin Squeezing: Overview and Context 10 2.1 Collective Spins, Coherent Spin States, and the SQL . 10 2.1.1 Collective Spins . 11 2.1.2 Coherent Spin States and the SQL . 12 2.2 Ramsey Spectroscopy with Coherent Spin States . 14 2.3 Surpass SQL with Spin Squeezed States . 16 2.4 Generation of Spin Squeezed States . 19 2.4.1 Collisional Interactions . 19 2.4.2 Light-mediated One-Axis Twisting Hamiltonians . 20 2.4.3 Quantum Non-demolition Measurements . 23 2.5 Relative advantages and disadvantages to different approaches . 26 2.6 Other work on entanglement-enhanced metrology . 27 2.7 Outline of our squeezing experiment . 28 vi 3 Theory: Probing atoms-cavity system with projection-noise-limited sensitivity 32 3.1 Introduction . 32 3.2 Coupled atoms-cavity modes . 33 3.2.1 Linearized response . 34 3.2.2 Driven and damped dynamics . 38 3.2.3 Full complex field response to probing . 41 3.2.4 Probe vacuum noise and measurement imprecision . 43 3.3 Quantum-limited signal-to-noise and free space scattering . 44 3.3.1 Projection-noise-driven fluctuations of mode frequencies . 44 3.3.2 Fundamental measurement imprecision and free space scattering at arbitrary detuning δc ..................................... 45 proj 3.3.3 Minimizing ms at fixed maximum detuning . 52 3.4 Summary . 52 4 Theory: Probe-induced back-action 54 4.1 Back-action on Jy ..................................... 55 4.2 Back-action on jhJ^ij .................................... 56 4.3 Back-action on Jz ..................................... 57 4.3.1 Simple three-level model . 57 4.3.2 Diffusion of Jz ................................... 58 4.4 Measurement limits on Jz ................................. 58 4.4.1 Single spin measurement resolution . 60 4.5 Spectroscopic enhancement . 61 4.5.1 Spin-Flip Diffusion-Noise-Limited Squeezing/Raman Limit . 62 4.5.2 Decoherence-Limited Squeezing/Cycling Limit . 62 4.6 Conclusions . 64 vii 5 Experiment: Spin squeezing via cavity-aided QND measurements 65 5.1 Energy levels and probing scheme . 65 5.2 Atom-cavity coupling . 66 5.3 Key experimental details for Science Cavity and atomic ensemble . 68 5.3.1 Science Cavity . 68 5.3.2 Atomic ensemble properties . 69 5.4 Inhomogeneous coupling: effective coupling 2g and effective atom number N .... 71 5.5 Sample preparation . 72 5.6 Measuring Jz ........................................ 73 5.6.1 Single Rabi splitting measurement . 73 5.6.2 Measurement uncertainty ∆Jz .......................... 78 5.6.3 Probe-induced Raman scattering . 79 5.7 Observation of quantum projection noise in Jz ..................... 83 5.8 Conditional spin variance . 86 5.9 Loss of coherence . 87 5.10 Net spectroscopic enhancement . 90 5.11 Back-action . 91 5.12 Conclusions . 92 6 Advantages of operating in resonant-coupling regime 93 6.1 Fundamental scalings related to projection noise . 93 6.2 Noise rejection in the resonant limit . 94 6.3 ac-Stark shift cancellation . 96 6.4 Radial oscillations . 99 6.5 Spatial mode matching of atoms-cavity modes . 100 6.6 Summary . 102 viii 7 Experiment: Realization of precise rotations 104 7.1 Aligning rotation axis to collective spin vector . 104 7.2 Constraining noise from microwave rotations . 107 7.3 Low phase noise microwave source . 108 7.3.1 Context . 111 7.3.2 Low noise, fixed frequency source . 112 7.3.3 Agile control of Phase, Frequency and Amplitude . 117 7.3.4 Impact on QND measurements . 120 8 Theory: Impact Phase Noise on Bloch Vector Rotations 122 8.1 Introduction . 123 8.2 Deflection of Bloch Vector due to coherently phase modulated rotation . 125 8.2.1 System Hamiltonian . 125 8.2.2 LO Phase Noise . 125 8.2.3 Bloch Sphere Picture . 126 8.2.4 Small Rotation describing Phase Modulated Rotation . 127 8.2.5 Solution for Deflection Vector . 128 8.3 Experimental Verification . 129 8.3.1 Physical Implementation . 130 8.3.2 Response to Coherently Phase Modulated Rotation . 131 8.4 Noise in Bloch Vector due to Phase Noise in a Single Rotation . 132 8.4.1 Covariance Transfer Matrix . 132 8.4.2 Covariance Noise Matrix . 134 8.5 Noise in Bloch Vector due to White Phase Noise From Multiple Rotations . 136 8.5.1 Noise Propagation . 136 8.5.2 Average Infidelity . 138 8.5.3 Composite π-pulse Comparisons . 138 ix 8.5.4 Spin Echo Pulse Sequences . 142 8.6 Summary . 142 9 Broader Context, Future Prospects, and Conclusions 144 9.1 Framing our results in broader context . 144 9.2 Spectroscopic gain in the presence of decoherence . 147 9.3 Fundamental Limits on Squeezing via Probing the 87Rb cycling transition . 149 9.4 Future Experimental Upgrades . 157 9.4.1 Improving Cavity Finesse . 157 9.4.2 Single-ended Cavity to Improve Detection Efficiency . 158 9.4.3 Uniform Coupling of Cavity Mode to Atoms . 159 9.4.4 Removing 17 Hz Vibration Noise Source . 159 9.5 Summary and Conclusions . 160 Bibliography 162 Appendix A Coherent Reinitialization for Reduced Recoil Heating 174 B Heterodyne PDH Detection Scheme 177 C AD8015 Photodiode Characteristics 182 D Phase Noise Definitions 185 E Relation between Mean Squared Modulation Amplitude and Single Sideband Phase Noise 187 F Composite and Plain π-pulse Covariance Noise Matrix and Infidelity 188 F.1 CORPSE π-pulse . 189 F.2 SCROFULOUS π-pulse . 189 x F.3 BB1 π-pulse . ..
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