Nonlinearities in the quantum measurement process of superconducting qubits [Elektronische Ressource] / vorgelegt von Ioana Şerban

Ioana Serban

Sujets

• Physik

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2008
• Nombre de lectures: 20
• Poids de l'ouvrage: 2 Mo

Extrait

Nonlinearities in the quantum
measurement process of
superconducting qubits
Ioana S¸erban
Mun¨ chen, 2008Nonlinearities in the quantum
measurement process of
superconducting qubits
Ioana S¸erban
Dissertation
an der Fakult¨at fur¨ Physik
der Ludwig–Maximilians–Universit¨at
Munc¨ hen
vorgelegt von
Ioana S¸erban
aus Bukarest, Rum¨anien
Munc¨ hen, Mai 2008Erstgutachter: Prof. Dr. Jan von Delft
Zweitgutachter: Prof. Dr. Frank K. Wilhelm
Tag der mundlic¨ hen Prufung:¨ 09.07.2008Contents
Abstract 1
I Introduction 3
1 Quantum measurement 5
1.1 Quantum decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Quantum optical methods for the solid state . . . . . . . . . . . . . . . . . 9
2 Quantum computing with superconducting devices 11
2.1 The Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Flux qubit readout . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
II Flux qubit readout and control in the presence of decoher-
ence 21
3 Dispersive readout protocols 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 From circuit to Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.2 Back-action on the qubit . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.3 Comparison of dephasing and measurement times . . . . . . . . . . 43
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 Additional information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6.1 Parameter conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6.2 Parameters for the generalized Fokker-Planck equation . . . . . . . 47vi CONTENTS
4 Qubit dephasing for dispersive readout 49
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Weak qubit-oscillator coupling . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Strong qubit-oscillator coupling . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.6 Additional information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 QND-like, fast qubit measurement 59
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Model and method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.1 The interaction phase. . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.2 The post-interaction phase . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.1 Qubit decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.2 Detector dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.4 Practical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.6 Additional information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.6.1 Solution for the Wigner characteristic functions . . . . . . . . . . . 74
5.6.2 Conversion to circuit parameters . . . . . . . . . . . . . . . . . . . 76
6 Macroscopic dynamical tunneling in a dissipative system 77
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2 The Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3 Transformation to the rotating frame . . . . . . . . . . . . . . . . . . . . . 80
6.4 Thermal activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.6 Additional information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.6.1 WKB quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.6.2 Analytic formulas for the action . . . . . . . . . . . . . . . . . . . . 89
7 Qubit relaxation from the Duffing oscillator 91
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.3 Relaxation from detector noise . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.3.1 The rotating frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.3.2 The initial state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3.3 Using the quantum regression theorem . . . . . . . . . . . . . . . . 97
7.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Table of contents vii
8 Optimal control of a qubit coupled to a two-level fluctuator 101
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.2 Decoherence model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.3 Optimal control and results . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Conclusion 106
A System-bath model 111
A.1 The quantum regression theorem . . . . . . . . . . . . . . . . . . . . . . . 115
B Wigner representation 117
B.1 Useful relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C Floquet states 121
Bibliography 123
List of publications 138
Acknowledgment 140
Zusammenfassung 143
Curriculum vitae 144viii Table of contentsList of Figures
2.1 Schematic representation of a Josephson junction . . . . . . . . . . . . . . 14
2.2 Tilted washboard potential of a Josephson junction . . . . . . . . . . . . . 16
2.3 Schematic representation of a superconducting flux qubit . . . . . . . . . . 18
2.4 Sctation of a sup quantum interference device 19
3.1 Simplified circuit for dispersive flux qubit readout using a SQUID . . . . . 26
3.2 Detector output: probability distribution of momentum . . . . . . . . . . . 37
3.3 Discrimination time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Dephasing rate dependence on driving . . . . . . . . . . . . . . . . . . . . 42
3.5 Comparison of dephasing and measurement rates . . . . . . . . . . . . . . 44
3.6rison ofsing and discrimination rates (short time measurement) 45
4.1 Illustration of the phase Purcell effect . . . . . . . . . . . . . . . . . . . . . 53
4.2 The qubit dephasing rate overview . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Qubit dephasing rate in the strong coupling regime . . . . . . . . . . . . . 55
4.4 Illustration of the dephasing in the weak coupling regime . . . . . . . . . . 57
5.1 Simplified circuit for fast QND-like flux qubit readout . . . . . . . . . . . . 62
5.2 Qubit dephasing and relaxation for the QND-like readout . . . . . . . . . . 67
5.3 Detector dynamics in the phase-space during and after the qubit interaction 70
5.4 output: voltage probability distribution . . . . . . . . . . . . . . 71
5.5 Circuit diagram for the practical implementation of the QND-like readout . 72
5.6 Voltage probability distribution with additional reference SQUID . . . . . 73
6.1 Behavior of the classical Duffing oscillator: a double well parallel . . . . . . 79
6.2 The Hamiltonian in the phase-space and the classical trajectories . . . . . 82
6.3 Quantized energies of the system in the rotating frame . . . . . . . . . . . 83
6.4 Estimation of the effective temperature . . . . . . . . . . . . . . . . . . . . 85
6.5 Tunneling and activation rates . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.6 Effective double well potential in the rotating frame . . . . . . . . . . . . . 88
7.1 Relaxation rate for the qubit coupled to the Duffing oscillator . . . . . . . 99
7.2 Effective mass and frequency of the Duffing oscillator . . . . . . . . . . . . 100x List of figures
8.1 Illustration of the coupled system qubit-fluctuator-bath . . . . . . . . . . . 103
8.2 Typical behavior of relaxation in the coupled qubit-TLF system . . . . . . 104
8.3 Gate error overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.4 Comparison of Rabi and optimized pulses . . . . . . . . . . . . . . . . . . 107
8.5 Gate error dependence on the bath parameters . . . . . . . . . . . . . . . . 108