Interplay between dissipation and driving in nonlinear quantum systems [Elektronische Ressource] / Carmen Vierheilig
165 Pages
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Interplay between dissipation and driving in nonlinear quantum systems [Elektronische Ressource] / Carmen Vierheilig

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In this thesis we investigate the interplay between dis-sipation and driving in nonlinear quantum systems for a special setup: a flux qubit read out by a DC-SQUID - a nonlinear quantum oscillator. The latter is embedded in a harmonic bath, thereby mediating dissipation to the qubit. Two different approaches are elaborated: First we con-sider a composite qubit-SQUID system and add the bath afterwards. We derive analytical expressions for its ei-genstates beyond rotating wave approximation (RWA), by applying Van Vleck perturbation theory (VVPT) in the qubit-oscillator coupling. The second approach is an ef-fective bath approach based on a mapping procedure, where SQUID and bath form an effective bath seen by the qubit. Here the qubit dynamics is obtained by ap-plying standard procedures established for the spin-boson problem. This approach requires the knowledge of the steady-state response of the dissipative Duffing oscillator, which is studied within a resonant and an off- Carmen Vierheiligresonant approach: The first is applicable near and at an N-photon resonance using VVPT beyond a RWA. The Interplay between second is based on the exact Floquet states of the non-linear driven oscillator. dissipation and The dissipative qubit dynamics is described analytically for weak system-bath coupling and agrees well for both driving in nonlinear approaches.

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Published 01 January 2011
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In this thesis we investigate the interplay between dis-
sipation and driving in nonlinear quantum systems for
a special setup: a flux qubit read out by a DC-SQUID - a
nonlinear quantum oscillator. The latter is embedded in
a harmonic bath, thereby mediating dissipation to the
qubit.
Two different approaches are elaborated: First we con-
sider a composite qubit-SQUID system and add the bath
afterwards. We derive analytical expressions for its ei-
genstates beyond rotating wave approximation (RWA),
by applying Van Vleck perturbation theory (VVPT) in the
qubit-oscillator coupling. The second approach is an ef-
fective bath approach based on a mapping procedure,
where SQUID and bath form an effective bath seen by
the qubit. Here the qubit dynamics is obtained by ap-
plying standard procedures established for the spin-
boson problem. This approach requires the knowledge
of the steady-state response of the dissipative Duffing
oscillator, which is studied within a resonant and an off- Carmen Vierheilig
resonant approach: The first is applicable near and at
an N-photon resonance using VVPT beyond a RWA. The Interplay between second is based on the exact Floquet states of the non-
linear driven oscillator. dissipation and The dissipative qubit dynamics is described analytically
for weak system-bath coupling and agrees well for both driving in nonlinear
approaches. We derive the effect of the nonlinearity on
the qubit dynamics, on the Bloch-Siegert shift and on quantum systems
the vacuum Rabi splitting.
a
1 9
ISBN 978-3-86845-072-9 ISBN 978-3-86845-072-9
9 7 8 3 8 6 8 4 5 0 7 2 9
Carmen Vierheilig Dissertationsreihe Physik - Band 19Carmen Vierheilig
Interplay between
dissipation and
driving in nonlinear
quantum systemsPromotionskolloquium fand am 23.11.2010 statt.
Interplay between dissipation and driving
in nonlinear quantum systems
Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.)
der naturwissenschaftlichen Fakultät II - Physik der Universität Regensburg
vorgelegt von
Carmen Vierheilig
aus Tegernheim
2010
Die Arbeit wurde von Prof. Dr. Milena Grifoni angeleitet.
Das Promotionsgesuch wurde am 12.01.2010 eingereicht.
Das
Prüfungsausschuss: Vorsitzender: Prof. Dr. Jascha Repp
1. Gutachter: Prof. Dr. Milena Grifoni
2. Gutachter: Dr. Tobias Kramer
weiterer Prüfer: Prof. Dr. Vladimir Braun
Dissertationsreihe der Fakultät für Physik der Universität Regensburg,
Band 19
Herausgegeben vom Präsidium des Alumnivereins der Physikalischen Fakultät:
Klaus Richter, Andreas Schäfer, Werner Wegscheider, Dieter WeissCarmen Vierheilig
Interplay between
dissipation and
driving in nonlinear
quantum systemsBibliografische Informationen der Deutschen Bibliothek.
Die Deutsche Bibliothek verzeichnet diese Publikation
in der Deutschen Nationalbibliografie. Detailierte bibliografische Daten
sind im Internet über http://dnb.ddb.de abrufbar.
1. Auflage 2011
© 2011 Universitätsverlag, Regensburg
Leibnizstraße 13, 93055 Regensburg
Konzeption: Thomas Geiger
Umschlagentwurf: Franz Stadler, Designcooperative Nittenau eG
Layout: Carmen Vierh eilig
Druck: Docupoint, Magdeburg
ISBN: 978-3-86845-072-9
Alle Rechte vorbehalten. Ohne ausdrückliche Genehmigung des Verlags ist es
nicht gestattet, dieses Buch oder Teile daraus auf fototechnischem oder
elektronischem Weg zu vervielfältigen.
Weitere Informationen zum Verlagsprogramm erhalten Sie unter:
www.univerlag-regensburg.deInterplay between dissipation and driving
in nonlinear quantum systems
DISSERTATION ZUR ERLANGUNG DES DOKTORGRADES DER NATURWISSENSCHAFTEN (DR. RER. NAT.)
DER NATURWISSENSCHAFTLICHEN FAKULTÄT II - PHYSIK
DER UNIVERSITÄT REGENSBURG
vorgelegt von
Carmen Vierheilig

aus
Tegernheim
im Jahr 2010Die Arbeit wurde angeleitet von: Prof. Dr. Milena Grifoni
Promotionsgesuch eingereicht am: 12.01.2010
Das Promotionskolloqium fand am 23.11.2010 statt.
Prüfungsausschuss: Vorsitzender: Prof. Dr. Jascha Repp
1. Gutachter: Prof. Dr. Milena Grifoni
2. Gutachter: Dr. Tobias Kramer
weiterer Prüfer: Prof. Dr. Vladimir BraunContents
1 Introduction 7
1.1 Classicalnonlinearsystems......................... 10
1.2 Nonlinearquantumsystems........................ 12
2 Dissipative quantum systems 25
2.1 Systemplusbathmodel.......................... 26
2.2 Spin-boson-model.............................. 35
2.3 Populationdifferenceofaqubit...................... 36
3 The dissipative quantum Duffing oscillator 39
3.1 Quantum Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Complementary approaches for the quantum Duffing oscillator . . . . . 42
3.3 Perturbation theory for a time-periodic Hamiltonian with time-independent
perturbation................................. 45
3.4 Perturbative approach for the one-photon resonance . . . . . . . . . . . 49
3.5 Comparison of the outcomes of the two approaches . . . . . . . . . . . 53
3.6 Disipativedynamics............................ 58
3.7 Observableforthenonlinearresponse................... 59
3.8 Off-resonant approach based on the dissipative driven harmonic oscillator 64
3.9 Conclusions................................. 73
4 Qubit-nonlinear oscillator system coupled to an Ohmic bath 75
4.1 Q oscillator-bath system . . . . . . . . . . . . . . . . . . 76
4.2 Energy spectrum and dynamics of the non-dissipative TLS-NLO system 79
4.3 Influence of the environment . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Numerical versus analytical predictions for dissipative qubit dynamics . 93
4.5 Conclusions 95
5 Effective bath approach 99
5.1 Qubit-nonlinear oscillator-bath Hamiltonian . . . . . . . . . . . . . . . 100
5.2 Mapping to an effective bath . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Steady-state dynamics of a Duffing oscillator . . . . . . . . . . . . . . . 106
5.4 Effective spectral density for a nonlinear system . . . . . . . . . . . . . 107
5.5 Qubitdynamics...............................108
5.6 Analytical solution for the nonlinear peaked spectral density . . . . . . 1106 | CONTENTS
5.7 Qubit dynamics within different approaches . . . . . . . . . . . . . . . 113
5.8 Conclusions.................................15
6 Conclusions and perspectives 119
A Fourier components 123
B Rotating wave approximation for a driven linear oscillator 125
C Van Vleck perturbation theory 129
C.1 Van Vleck pe theory within the undriven qubit-NO system . 130
D Comparison for the states 137
D.1FloquetstatesinAppI...........................137
D.2FloquetstatesinAppI..........................138
E Oscillator matrix elements 141
F Rate coefficients for the off-diagonal density matrix elements 147
G Diagonal reduced density matrix elements 149
References 150
Acknowledgements 159
Publications 161Chapter 1
Introduction
Investigating the interplay of dissipation and driving for nonlinear systems, in par-
ticular for nonlinear oscillators, is essential to understand a large variety of physi-
cal systems, including electrical circuits, nanoresonators or SQUIDs (superconducting
quantum interference device) acting as read-out devices for qubits. Dissipation affects
the system dynamics in two ways: it leads to energy loss and hence to damping of
the motion as well as to decoherence as a consequence of dephasing. While dissipative
effects perturb or even strongly affect the motion of the system under consideration,
an additional tunable driving can act as a source of energy and stabilize the dyna-
mics. Therefore an efficient amplification that counteracts the dissipation is possible.
Moreover, taking into account an external driving is more appropriate to the experi-
mental situation, where external voltages or currents are applied. Within a quantum
mechanical interpretation dissipation enters a system which is coupled to an environ-
ment. The latter is treated as a bath containing infinitely many degrees of freedom.
The bath measures continuously the system thereby destructing phase correlations
and causing decoherence [1]. A famous model for including environmental effects on a
quantum level is the Ullersma-Zwanzig-Caldeira-Leggett model [2, 3, 4, 5]. The bath
is a reservoir composed of independent harmonic oscillators, where each one is coupled
bilinearily to the system of interest. This kind of coupling allows an exact elimination
of the bath degrees of freedom, when the reduced dynamics of the system is consi-
dered. Understanding dissipative effects and decoherence including driving allows to
construct efficient procedures in quantum computation to achieve longer coherence
times. When on top of this nonlinearity comes into play, we observe new behavior of
the system, like bistability, frequency doubling, higher harmonics generation and non-
linear response [6, 7]. Thus quantum computation schemes can benefit from the use
of nonlinear devices, which allow for e.g. efficient amplification of signals or improved
read-out schemes for qubits [8, 9, 10, 11]. To observe coherent effects a quantum de-
scription of nonlinear systems is essential. This will be elaborated in this thesis for
the case of a SQUID modeled as a nonlinear oscillator in the deep quantum regime.
Imposing a nonlinearity allows insight into the classical to quantum transition of a
system [12], which exists both in the deep quantum and in the classical regime. The
reason for testing the transition using a nonlinear system relies on the fact that Ehren-