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Open quantum systems and quantum information dynamics [Elektronische Ressource] / Ángel Rivas Vargas

217 pages
˜Universitat UlmInstitut fur˜ Theoretische PhysikOpen Quantum Systems andQuantum Information Dynamics¶Angel Rivas VargasDissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult˜at fur˜Naturwissenschaften der Universit˜at UlmJanuary 2011Amtierender Dekan Prof. Dr. Axel Gro…Erstgutachter Prof. Dr. Martin B. PlenioZweitgutachter Prof. Dr. Joachim AnkerholdTag der Promotion 16. M˜arz 2011DedicationA mis abuelos, F¶elix, Gertrudis, Carlos y Luisa.Et si habuero prophetiam,et noverim mysteria omnia, et omnem scientiam,[:::] caritatem autem non habuero, nihil sum.1 Cor 13, 2.AbstractJust like other theories of physics, quantum theory was developed to explain someexperimental facts that were incomprehensible within the previously existing theo-ries (e.g. the black-body radiation or the photoelectric efiect). After the Planck’sintroduction of the concept of quanta at the beginning of the XX century [1], manyphysical situations have been explained by means of quantum mechanics, rangingfrom the behavior of elementary particles to the operational mechanism of stars.Nowadays, about 100 years from its foundation, a lot of efiort is being made in try-ing to exploit the genuine features of quantum systems as a resource to solve a widevariety of problems, such as the secure transmission of information, its processingin a more e–cient way or to make extremely accurate measurements of physicalquantities.
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˜Universitat Ulm
Institut fur˜ Theoretische Physik
Open Quantum Systems and
Quantum Information Dynamics
¶Angel Rivas Vargas
Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult˜at fur˜
Naturwissenschaften der Universit˜at Ulm
January 2011Amtierender Dekan Prof. Dr. Axel Gro…
Erstgutachter Prof. Dr. Martin B. Plenio
Zweitgutachter Prof. Dr. Joachim Ankerhold
Tag der Promotion 16. M˜arz 2011Dedication
A mis abuelos, F¶elix, Gertrudis, Carlos y Luisa.
Et si habuero prophetiam,
et noverim mysteria omnia, et omnem scientiam,
[:::] caritatem autem non habuero, nihil sum.
1 Cor 13, 2.Abstract
Just like other theories of physics, quantum theory was developed to explain some
experimental facts that were incomprehensible within the previously existing theo-
ries (e.g. the black-body radiation or the photoelectric efiect). After the Planck’s
introduction of the concept of quanta at the beginning of the XX century [1], many
physical situations have been explained by means of quantum mechanics, ranging
from the behavior of elementary particles to the operational mechanism of stars.
Nowadays, about 100 years from its foundation, a lot of efiort is being made in try-
ing to exploit the genuine features of quantum systems as a resource to solve a wide
variety of problems, such as the secure transmission of information, its processing
in a more e–cient way or to make extremely accurate measurements of physical
quantities.
To carry out this task we require an unprecedented control in the fabrication
and manipulation of devices operating in the quantum regime. However, quantum
systems are much more fragile in the presence of noise than their classical coun-
terparts, and relevant quantum states allowing novel forms to store, transmit and
process information vanish very easily. Actually, this is the ultimate reason why we
cannot see systems exhibiting quantum behavior in our daily life. As a result, the
studyofthepropertiesofquantumsystemsinthepresenceofnoiseisafundamental
issue in order to develop any form of reliable quantum technology.
Noise in physical systems can be explained theoretically using the concept of
\open system". Noise arises due to a lack of insulation, in such a way that it is the
result of an efiective interaction between our system and its environment. In this
thesis, we present new results concerning the dynamics of open quantum systems.
We propose to deflne rigourously the concept of quantum Markov processes and
introduce quantitative, novel, ways to detect deviations from strict Markovianity.
We explore the validity of the usual techniques employed to describe simple open
quantum systems, and their extrapolation to more complicated scenarios involving
interacting and driven systems. In addition we present examples of noise assisted
processes,andtransformationstomaketheseusuallytrickyproblemsmoree–ciently
tractable.
In the flrst part of this thesis we extensively review the general theory of open
quantum systems. We introduce new approaches to describe, in a hopefully more
transparent way, some aspects which are usually misleading in a broad range of the
literature. In addition, we revise new concepts and methods developed in recent
years that complement the general theory mainly developed in the 1970s.
In the second part, we focus our attention on the so-called Markovian processes
which have been described in the previous part. We study the validity of typical
equations used in simple systems such as harmonic oscillators, and propose propose
and check difierent strategies to describe Markovian dynamics in more complicatedinteracting systems. Finally we propose a counterintuitive situation where some
level of noise can be actually useful for information tasks.
In the third and last part, we approach the problem of the non-Markovian de-
scription of open quantum systems. To quantify how non-Markovian an open quan-
tum system is we introduce a computable measure and apply it to some relevant
situations. We also propose a way of probing the complexity of an environment by
visualizing properties of a pair of qubits embedded in it. In addition, we develop
a technique based on orthogonal polynomial theory to transform a wide range of
environments into simple chains of particles which can be e–ciently simulated with
numerical techniques.Contents
Acknowledgments 13
1 Introduction 15
I Time Evolution Theory of Open Quantum Systems 19
2 Mathematical tools 21
2.1 Banach spaces, norms and linear operators . . . . . . . . . . . . . . . 21
2.2 Exponential of an operator . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Semigroups of operators . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Contraction semigroups . . . . . . . . . . . . . . . . . . . . . 27
2.4 Evolution families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Quantum time evolutions 35
3.1 Time evolution in closed quantum systems . . . . . . . . . . . . . . . 35
3.2 Time evolution in open quantum systems. . . . . . . . . . . . . . . . 37
3.2.1 Dynamical maps . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Universal dynamical maps . . . . . . . . . . . . . . . . . . . . 42
3.2.3 Universal maps as contractions . . . . . . . . . . . 44
3.2.4 The inverse of a universal dynamical map . . . . . . . . . . . 46
3.2.5 Temporal continuity. Markovian evolutions . . . . . . . . . . . 47
3.3 Quantum Markov process: mathematical structure. . . . . . . . . . . 50
3.3.1 Classical Markovian processes . . . . . . . . . . . . . . . . . . 50
3.3.2 Quantum Markov evolution as a difierential equation . . . . . 51
3.3.3 Kossakowski conditions . . . . . . . . . . . . . . . . . . . . . . 55
3.3.4 Steady states of homogeneous Markov processes . . . . . . . . 58
4 Microscopic derivations 65
4.1 Markovian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 Nakajima-Zwanzig equation . . . . . . . . . . . . . . . . . . . 66
74.1.2 Weak coupling limit . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.3 Singular limit . . . . . . . . . . . . . . . . . . . . . . 86
4.1.4 Extensions of the weak coupling limit . . . . . . . . . . . . . . 89
4.2 Non-Markovian case . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.1 Integro-difierential models . . . . . . . . . . . . . . . . . . . . 94
4.2.2 Time-convolutionless forms. . . . . . . . . . . . . . . . . . . . 95
4.2.3 Dynamical coarse graining method . . . . . . . . . . . . . . . 97
II Interacting Systems under Markovian Dynamics 101
5 Composite systems of harmonic oscillators 103
5.1 A closer look into the damped harmonic oscillator . . . . . . . . . . . 103
5.1.1 Gaussian states . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1.2 Damped harmonic oscillator . . . . . . . . . . . . . . . . . . . 106
5.2 Markovian master equations for interacting systems . . . . . . . . . . 116
5.2.1 Two coupled damped harmonic oscillators . . . . . . . . . . . 117
5.2.2 Driven damped harmonic oscillator . . . . . . . . . . . . . . . 125
5.2.3 Some conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Stochastic resonance phenomena in spin chains 135
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 Steady state entanglement in qubit chains subject to longitudinal
decoherence (pure decay) . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3 XXZ Heisenberg interaction . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Steady state entanglement under transverse decoherence (pure de-
phasing) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.5 SR phenomena under both longitudinal and transverse decoherence . 142
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
III Identifying Elements of non-Markovianity 147
7 Measures of non-Markovianity 149
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2 Previous attempts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2.1 Proposal of Wolf et. al. . . . . . . . . . . . . . . . . . . . . . . 150
7.2.2 Proposal of Breuer et. al. . . . . . . . . . . . . . . . . . . . . 150
7.2.3 Geometric measures . . . . . . . . . . . . . . . . . . . . . . . 150
7.2.4 Microscopic approach . . . . . . . . . . . . . . . . . . . . . . . 150
7.3 Witnessing non-Markovianity . . . . . . . . . . . . . . . . . . . . . . 151
7.4 Measuringovianity . . . . . . . . . . . . . . . . . . . . . . . 154
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578 Probing a composite spin-boson environment 159
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.2 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.2.1 Master equation . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.2.2 Observable quantities . . . . . . . . . . . . . . . . . . . . . . . 164
8.3 Results: detecting the presence of coupling between the TLFs . . . . 164
^8.3.1 Power spectrum ofhM i . . . . . . . . . . . . . . . . . . . . . 165x
8.3.2 Probe entanglement. . . . . . . . . . . . . . . . . . . . . . . . 167
8.3.3 Discussion of the results . . . . . . . . . . . . . . . . . . . . . 169
8.4 Decoherence of entangled states . . . . . . . . . . . . . . . . . . . . . 171
8.5 Efiect of spin-boson environment on entangling gate operations . . . . 173
8.6 Some conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9 Mapping reservoirs to semi-inflnite discrete chains 177
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.2 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.2.1 Basic concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.2.2 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . 180
9.2.3 Boundness properties of the recurrence coe–cients . . . . . . . 182
9.3 System-reservoir structures . . . . . . . . . . . . . . . . . . . . . . . . 185
9.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.4.1 Continuous spin-boson spectral densities . . . . . . . . . . . . 189
9.4.2 Discrete spectral densities. The logarithmically discretized
spin-boson model . . . . . . . . . . . . . . . . . . . . . . . . . 191
9.4.3 Linearly-discretised baths . . . . . . . . . . . . . . . . . . . . 196
9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Prospects 201
Bibliography 203Journal Publications
Parts of this thesis are based on material published in the following papers:
† Introduction to the time evolution theory of open quantum systems
A. Rivas and S. F. Huelga
In preparation
(Part I)
† Markovian Master Equations: A Critical Study
A. Rivas, A. D. K. Plato, S. F. Huelga and M. B. Plenio
New J. Phys. 12 113032 (2010)
(Chapter 5)
† Exactmappingbetweensystem-reservoirquantummodelsandsemi-
inflnite discrete chains using orthogonal polynomials
A. W. Chin, A. Rivas, S. F. Huelga and M. B. Plenio
J. Math. Phys. 51, 092109 (2010)
(Chapter 9).
† Entanglement and non-Markovianity of quantum evolutions
A. Rivas, S. F. Huelga and M. B. Plenio
Phys. Rev. Lett. 105, 050403 (2010)
(Chapter 7).
† Probing a composite spin-boson environment
N. P. Oxtoby, A. Rivas, S. F. Huelga and R. Fazio
New J. Phys. 11, 063028 (2009)
(Chapter 8).
† Stochastic resonance phenomena in spin chains
A. Rivas, N. P. Oxtoby and S. F. Huelga
Eur. Phys. J. B 69, 51 (2009)
(Chapter 6).
Other papers not covered in this thesis to which the author contributed:
† Precision quantum metrology and nonclassicality in linear and non-
linear detection schemes
A. Rivas and A. Luis
Phys. Rev. Lett. 105, 010403 (2010).
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