Quantum Computing in Solid State Systems

Quantum Computing in Solid State Systems With 170 Figures

Edited by:

B. Ruggiero Istituto di Cibernetica "E. Caianiello " CNR Italy

P. Delsing Chalmers University of Technology Sweden

C. Granata Istituto di Cibernetica "E. Caianiello " CNR Italy

Y. Pashkin RIKEN/NEC Tsukuba, Japan

P. Silvestrini Seconda Universita di Napoli Italy Under auspices of Regione Campania

Springer

Contents

Preface List of Corresponding Authors 1 Decoherence of a Josephson Quantum Bit during its Free Evolution: The Quantronium 1.1 Introduction 1.2 The Quantronium Circuit and its Decoherence Sources 1.2.1 Principle 1.2.2 Experimental implementation 1.2.3 Decoherence sources 1.2.4 Theoretical relaxation and decoherence rates 1.3 Experimental Characterization of Decoherence During - Free Evolution 1.3.1 Relaxation Time (T1) measurement 1.3.2 Coherence time (T2 and TE) measurements 1.3.3 Discussion 1.4 Conclusion

v xv

1 1 2 2 3 3 5 6 6 6 7 8

2 Conditional Gate Operation in Superconducting Charge Qubits 2.1 Introduction 2.2 Experimental Details 2.2.1 Device Structure 2.2.2 Operation Scheme 2.2.3 Experimental Setup 2.3 Results and Discussion 2.3.1 Operation Point 2.3.2 Conditional Gate Operation 2.4 Conclusions

10 10 11 11 12 13 15 15 16 16

3 Coupling and Dephasing in Josephson Charge-Phase Qubit with Radio Frequency Readout 3.1 Introduction 3.2 Qubit Parameters and the Model

19 19 20

Contents 3.3 3.4 3.5

Effect of Finite Inductance of the Ring Qubit Dephasing Conclusion

22 24 25

The Josephson Bifurcation Amplifier - for Quantum Measurements 4.1 Introduction 4.2 Theory 4.3 Devices and Setup 4.4 Results 4.5 Conclusion

28 28 29 31 32 36

Current-Controlled coupling - of superconducting charge qubits 5.1 Introduction 5.2 Current-Controlled Coupling of Two Qubits 5.3 Maximum Coupling Strength 5.4 Operating the System 5.5 Coupling via Measurement Junctions 5.6 Extension to Arbitrary Number of Qubits

38 38 38 40 41 41 43

Direct Measurements of Tunable Josephson Plasma Resonance in the L-Set 6.1 Introduction 6.2 The L-Set Circuit 6.3 Plasma Oscillations in L-Set 6.4 Simulation Scheme 6.5 Experiment 6.6 Results and Discussion 6.6.1 Charge detection 6.6.2 Harmonic Oscillations 6.6.3 Switching and nonlinear oscillations

45 45 46 47 48 49 49 49 50 52

Time Domain Analysis of Dynamical Switching in a Josephson Junction 7.1 Introduction 7.2 The Experiment 7.3 The Model 7.4 Results and Discussion 7.5 Summary

54 54 56 57 59 62

Cooper Pair Transistor in a Tunable Environment 8.1 Introduction 8.2 Sample Fabrication and Measurement Techniques 8.3 Squid Arrays Characterization 8.4 Measurement of the CPT

63 63 64 66 67

Phase Slip Phenomena in Ultra-Thin Superconducting Wires 9.1 Introduction 9.2 Theoretical 9.3 Experimental

70 70 70 73

Contents 9.4

Experimental Evidence of Quantum Phase Slips?

ix 74

10 Dynamics of a Qubit Coupled to a Harmonic Oscillator 10.1 Introduction 10.2 The System 10.3 Sample Fabrication, Setup and Characterization 10.4 Coupled Dynamics 10.4.1 Spectroscopy 10.4.2 Dynamics 10.5 Using the Coupled Dynamics to Probe the System 10.5.1 Measuring the Oscillator Temperature 10.5.2 Probing the Qubit State 10.6 Conclusion

76 76 77 78 79 79 81 82 82 82 84

11 Josephson junction Materials Research Using Phase Qubits 11.1 Introduction 11.2 Josephson Phase Qubits 11.3 Junction Fabrication Processes 11.3.1 The "Ion Mill Junction Process" 11.3.2 The "Standard Trilayer Junction Process" 11.3.3 The "Evaporated Trilayer Junction Process" 11.4 Josephson Junction and Qubit Characterization 11.4.2 Measurements of Qubit Spectroscopy 11.5 Concluding remarks

86 86 87 88 89 89 89 91 92 93

12 Energy level spectroscopy of a bound vortex-antivortex pair

95

13 Adiabatic Quantum Computation with Flux Qbits 13.1 Introduction 13.2 Adiabatic Quantum Computation 13.3 AQC with an Array of Flux Qbits 13.4 Vertical Two Josephson Junctions Interferometer 13.5 Conclusions

103 103 104 105 106 109

14 Anomalous Thermal Escape in Josephson Systems Perturbed by Microwaves 14.1 Introduction 14.2 Theory 14.3 Experiments and Simulations 14.4 Conclusions

111 111 113 114 117

15 Realization and Characterization of a Squid Flux Qubit with a Direct Readout Scheme 120 15.1 Introduction 120 15.2 The RF Squid Qubit 120 15.3 The Double Squid Qubit 122 15.4 Gradiometric Configuration 122 15.5 Direct Readout Scheme 123

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Contents

15.6 Experimental Characterization 15.7 Conclusions

124 126

16 A Critique of the Two Level Approximation 16.1 Introduction 16.2 System and Environment 16.3 Density Matrix Solutions 16.3.1 The effect of spectral distribution on Decoherence 16.3.2 The effect of number of levels on Decoherence 16.4 The Photon Number Calculations 16.5 Conclusions and Discussions

127 127 128 129 129 132 132 135

17 Josephson Junction Qubits with Symmetrized Couplings to a Resonant LC Bus 17.1 Introduction 17.2 Josephson Junction Triangular Prism Qubits 17.3 Coupling the Resonant LC Bus to the Qubits 17.3.1 Logical Qubits for a Decoherence Free Subspace 17.3.2 M0lmer-S0rensen Gate 17.4 Circulating Current Patterns for the Qubit States 17.4.1 Initializing the Qubits and the Effect of Critical Current and Geometric Defects 17.4.2 Connecting Buses into a Network 17.5 Conclusions

137 137 138 139 140 140 142

18 Spatial Bose-Einstein Condensation in Josephson Junction Arrays 18.1 Introduction 18.2 Preliminary Studies for Engineering Graph-Shaped Networks 18.3 Fabrication of Comb-Shaped JJNs 18.4 Measurements 18.5 Experimental Implications

147 147 149 150 152 152

19 Cooper Pair Shuttle: A Josephson Quantum Kicked Rotator 19.1 Introduction and System Description 19.2 Classical and Quantum Dynamics 19.3 Fidelity 19.4 The Effect of Noise 19.5 Fidelity Measurement

154 154 155 158 158 161

20 Size Dependence of the Superconductor-Insulator Transition in Josephson Junction Arrays 20.1 Introduction 20.2 Finite-size Effect in the 2DXY Model 20.3 The Effective Hamiltonian 20.4 DMRG 20.5 Spontaneous Symmetry Breaking 20.6 BKT as a SSB 20.7 Conclusions

163 163 164 164 166 166 167 170

143 144 145

Contents

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21 Monte Carlo Method for a Superconducting Cooper-Pair-Box Charge Qubit Measured by a Single-Electron Transfer 21.1 Introduction 21.2 Model Hamiltonian 21.3 Measurement records and conditional density matrix 21.4 Stochastic master equation 21.5 Connection to "Partially" Reduced Density Matrix 21.6 Conclusion

171 171 172 173 174 176 178

22 On the Conversion of Ultracold Fermionic Atoms to Bosonic Molecules via Feshbach Resonances 22.1 Introduction 22.2 Initial State Preparation 22.3 General symmetry arguments 22.4 Explaining the Experimental Limited Transfer Efficiency 22.5 Transfer efficiencies above 0.5 22.6 Summary

180 180 181 181 183 185 186

23 Revealing Anisotropy in a Paul Trap Through Berry Phase 23.1 Introduction 23.2 Physical System and Hamiltonian Model 23.3 Calculation of Berry Phase 23.4 Anisotropy vs. Berry Phase Effects 23.5 Conclusions

188 188 189 191 192 194

24 Distilling Angular Momentum Schrodinger Cats in Trapped Ions 24.1 Introduction 24.2 Distilling Angular Momentum 'cats' Through Vibronic Couplings 24.2.1 Vibronic Couplings in trapped ions 24.2.2 Distilling angular momentum eigenstates 24.3 Controllable Cat-Like Superposition 24.4 Conclusive Remarks 24.5 Acknowledgments

195 195 196 196 197 199 200 200

25 Linear-response conductance of the normal conducting single-electron pump 25.1 Introduction 25.2 Motivation for the Measurements 25.3 Sequential Model 25.4 Experimental Details and Results 25.5 Discussions 25.6 Summary

202 202 203 205 206 208 210

26 Transmission Eigenvalues' Statistics for a Quantum Point Contact 26.1 Introduction 26.2 Model of QPC with Impurities 26.3 Scattering Matrix Elements and Correction to the Conductance 26.4 Distribution function

212 212 213 215 216

Contents 26.4.1 One open channel 26.4.2 Circuit theory 26.5 Conclusions

216 217 218

27 Creating Entangled States between SQUID Rings and Electromagnetic Fields 27.1 Introduction 27.2 Background 27.3 The Hamiltonian's Spectrum 27.4 Rapid Passage 27.5 Towards the Adiabatic Limit 27.6 The Effect of Dissipation 27.7 Conclusions

219 219 220 221 223 223 223 225

28 Frequency Down Conversion and Entanglement 28.1 Introduction 28.2 Entanglement of em Fields via a SQUID Ring 28.3 Entanglement of field modes at similar frequencies 28.3.1 Entanglement of dissimilar input/output frequency field modes 28.4 Energy down Conversion at Higher Frequency Ratios 28.4.1 Factor 5 down conversion 28.5 Larger frequency ratio (factor of 10) down conversion 28.6 Conclusions

228 228 230 231 231 234 234 236 236

29 Entanglement of distant SQUID rings 29.1 Introduction 29.2 Interaction of a Single SQUID Ring with Nonclassical Microwaves 29.3 Interaction of two Distant SQUID Rings with Entangled Microwaves 29.3.1 Microwaves in number states 29.3.2 Microwaves in coherent states 29.3.3 Numerical results 29.4 Discussions

239 239 240 241 242 243 244 245

30 Time Evolution of two distant SQUID rings irradiated with entangled electromagnetic field 30.1 Introduction 30.2 Interaction of Two SQUID Rings with Nonclassical Microwaves 30.3 Discussions

247 247 248 252

31 Phase diagram of dissipative two-dimensional Josephson junction arrays 31.1 Introduction 31.2 Path Integral Monte Carlo for Dissipative Systems 31.3 Josephson Junction Arrays 31.4 Numerical Simulations and Phase Diagram 31.5 Discussion of the Results

254 254 254 256 257 260

32 Persistent currents in a superconductor/normal loop 32.1 Introduction

263 263

Contents 32.2 SNS Junction 32.3 NS Loop

xiii 264 266

33 Josephson junction ladders: a realization of topological order 33.1 Introduction 33.2 Josephson Junction Ladders 33.3 m-reduction Technique 33.4 Topological Order and "Protected" Qubits

271 271 272 274 276

34 Single-electron charge qubit in a double quantum dot 34.1 Introduction 34.2 Charge Qubit in a Double Quantum Dot 34.2.1 Rotation gate and phase-shift gate operation 34.2.2 Decoherence of the system 34.3 Charge Detection of a Double Quantum Dot 34.4 Summary

279 279 280 281 282 284 286

35 Quantum dots for single photon and photon pair technology

288

36 Semiconductor few-electron quantum dots as spin qubits 36.1 Qubit 36.2 Read-out 36.3 Initialization 36.4 Coherence Times 36.5 Coherent Single-spin Manipulation: ESR 36.6 Coherent Spin Interactions: ySWAP 36.7 Conclusion

298 298 299 300 301 301 303 304

37 Spin amplifier for single spin measurement 37.1 Introduction 37.2 Measurement Schemes 37.3 Experimental Demonstration 37.4 Conclusions

306 306 307 308 310

38 Entanglement in quantum-critical spin systems 38.1 Introduction 38.2 The Model 38.3 Linear Chain: Critical Point and Factorized State 38.4 Entanglement Estimators 38.5 Results 38.6 Two-leg Ladder 38.7 Conclusions

313 313 314 315 316 316 319 320

39 Control of nuclear spins by quantum Hall edge channels 39.1 Introduction 39.2 Samples 39.3 Dynamical Nuclear Polarization 39.4 Local and Coherent Control

322 322 322 324 324

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Contents

39.5 Discussion 39.6 Summary 40 Cloning of single photon by high gain amplifier

327 328 330