Description and control of decoherence in quantum bit systems [Elektronische Ressource] / vorgelegt von Henryk Peter Gregor Gutmann

Henryk Peter Gregor Gutmann

Sujets

• Physik

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2005
• Nombre de lectures: 31
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Description and control of decoherence
in quantum bit systems
Henryk Peter Gregor Gutmann
Munchen, Juni 2005Description and control of decoherence
in quantum bit systems
Henryk Peter Gregor Gutmann
Dissertation
an der Fakult at fur Physik
der Ludwig{Maximilians{Universit at
Munc hen
vorgelegt von
Henryk Peter Gregor Gutmann
aus Bonn
Munc hen, Juni 2005Erstgutachter: PD Frank K. Wilhelm
Zweitgutachter: Prof. Dr. Axel Schenzle
Tag der mundlic hen Prufung: 03.08.2005Contents
List of Publications ix
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.1.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 Derivation of Lindblad type master equations at nite temperatures 7
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Lindblad equation in the Bloch sphere representation . . . . . . . . . . . . 10
1.2.1 Lindblad requirements and the concept of dynamical semigroups . . 10
1.2.2 GKS formulation of the Lindblad equation . . . . . . . . . . . . . . 12
1.2.3 Bloch-sphere formalism . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Born-Markov master equations . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Markov approximations . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Spin-Boson Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.1 Thermodynamic limit and Ohmic bath . . . . . . . . . . . . . . . . 25
1.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.1 Na v e Markov approximation . . . . . . . . . . . . . . . . . . . . . 27
1.5.2 Davies-Luczk a appro . . . . . . . . . . . . . . . . . . . . . 28
1.5.3 Bloch-Red eld approximation . . . . . . . . . . . . . . . . . . . . . 28
1.5.4 Celio-Loss approximation . . . . . . . . . . . . . . . . . . . . . . . 29
1.6 Quantitative comparison of the Born-Markovian approximations . . . . . . 30
1.7 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Qubit decoherence due to bistable uctuators 35
2.1 Qubit-b -bath model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Bloch-Red eld master equation . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.1 Bloch-Red eld tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.2 Qubit dephasing and relaxation rates . . . . . . . . . . . . . . . . . 40
2.2.3 decoherence spectra at xed
for variable T . . . . . . . . 41q
2.2.4 Qubit spectra at xed T for v
. . . . . . . . 43q
2.2.5 1=f noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Stochastic Schr odinger equation and random walk model . . . . . . . . . . 48
2.3.1 Stochastic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 49vi CONTENTS
2.3.2 Random walk model . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.3 Symmetrical noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.4 B -noise at nite temperatures . . . . . . . . . . . . . . . . . . . . 56
2.3.5 Criteria for an appropriate choice of b parameter . . . . . . . . . . 59
2.3.6 Numerical and analytical results for temperature dependent b noise 61
2.3.7 Derivation of dephasing and relaxation rates . . . . . . . . . . . . . 64
2.4 B -noise induced by an SET-measurement setup . . . . . . . . . . . . . . . 67
2.4.1 Flipping rates of the SET-electron . . . . . . . . . . . . . . . . . . . 68
2.5 Refocusing of b -noise by means of dynamical decoupling . . . . . . . . . . 73
2.5.1 Refocusing (bang-bang) scheme . . . . . . . . . . . . . . . . . . . . 74
2.5.2 Random walk model . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5.3 Pulse shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.5.4 Distributions of the random walks deviation . . . . . . . . . . . . . 79
2.5.5 Bang-bang control working as a high-pass lter . . . . . . . . . . . 80
2.5.6 refocusing of nite temperature b -noise . . . . . . . . . 82
2.5.7 Applicability of imperfect bang-bang pulses . . . . . . . . . . . . . 85
2.5.8 Numerical and analytical results . . . . . . . . . . . . . . . . . . . . 87
3 Quantum phase diagram of a coupled qubit system 93
3.1 The 2-spins-3-bosonic-baths model . . . . . . . . . . . . . . . . . . . . . . 93
3.2 The dressed double-spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . 96
3.2.1 The Emery-Kivelson-transformation . . . . . . . . . . . . . . . . . . 97
3.3 Scaling analysis and quantum phase diagram . . . . . . . . . . . . . . . . . 100
st3.3.1 Scaling equations 1 order . . . . . . . . . . . . . . . . . . . . . . . 100
3.3.2 Entanglement capability of the xed point Hamiltonian . . . . . . . 102
nd3.4 Scaling equations 2 order . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.4.1 Operator product expansion . . . . . . . . . . . . . . . . . . . . . . 105
A Born master equation of the spin-Boson model 109
A.1 Derivation of the Born approximation correlation functions . . . . . . . . . 109
B Born-Markovian approximations of the Spin-Boson model 113
B.1 Na v e Markovximation . . . . . . . . . . . . . . . . . . . . . . . . . 113
B.2 Bloch-Red eld appro . . . . . . . . . . . . . . . . . . . . . . . . . 120
B.3 Davies-Luczk a approximation . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.4 Lindblad approximation according Celio and Loss . . . . . . . . . . . . . . 129
C Random walk analysis 131
C.1 Symmetrical random walk (driftless) . . . . . . . . . . . . . . . . . . . . . 131
C.1.1 Pure b -noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
C.1.2 Bang-bang refocused random walk . . . . . . . . . . . . . . . . . . 133Table of Contents vii
D Bosonic elds scaling formalism 135
D.1 Derivation of the rst order scaling equations . . . . . . . . . . . . . . . . 135
D.2 Derivation of the second order scaling . . . . . . . . . . . . . . . 136
D.2.1 Calculation of the operator product expansions . . . . . . . . . . . . 137
Acknowledgements 154
Deutsche Zusammenfassung 160viii Table of ContentsList of Publications
Major parts of this thesis are discussed in the following publications
chapter 1
1. Derivation of Lindblad type master equations by means of
Born Markov approximations at nite temperature
Henryk Gutmann and Frank K. Wilhelm
in preparation.
chapter 2
2. Bang-bang refocusing of a qubit exposed to telegraph noise
Henryk Gutmann, F.K. Wilhelm, W.M. Kaminsky and S. Lloyd
Quantum Information Processing 3, 247 (2004).
3. Compensation of decoherence from telegraph noise by means of
an open loop quantum-control technique
Henryk Gutmann, W.M. Kaminsky, Seth Lloyd and Frank K. Wilhelm
Physical Review A 71, 020302 (2005).
4. Dynamical decoupling of bistable uctuator noise at nite temperature
Henryk Gutmann, A. Holzner, F.K. Wilhelm, W.M. Kaminsky and S. Lloyd
in preparation.
5. Random Walk description of backaction during quantum charge detection
Henryk Gutmann, A. Holzner and F.K. Wilhelm
in preparation.
chapter 3
6. Scaling analysis of a coupled two-spin system exposed to
collective and localized noise
Henryk Gutmann, Gergely Zarand and Frank K. Wilhelm
in preparation.x Introduction