Decoherence of spatially separated quantum bits [Elektronische Ressource] / vorgelegt von Roland Doll

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• Physik

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• Publié par: universitat_augsburg
• Publié le: 01 janvier 2008
• Nombre de lectures: 44
• Langue: English
• Poids de l'ouvrage: 7 Mo

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Decoherence of spatially separated quantum bits
Dissertation
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften,
vorgelegt der Mathematisch-Naturwissenschaftlichen Fakultät
der Universität Augsburg
von
Roland Doll
Augsburg, im Januar 2008Prüfungskommission
Priv. Doz. Dr. Sigmund Kohler (Erstgutachter)
Prof. Dr. Stefan Kehrein (Zweitgutachter)
Prof. Dr. Peter Hänggi
Prof. Dr. Achim Wixforth
Tag der mündlichen Prüfung: 22.02.2008Contents
1 Introduction 1
Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Quantum information processing . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Qubit realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Collective vs. independent noise . . . . . . . . . . . . . . . . . . . . . . . 7
2 Coupling qubits to bosonic fields 11
2.1 Heat-bath model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 System-bath coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 The one-dimensional case . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Generalization to higher dimensions . . . . . . . . . . . . . . . . 16
2.3 Microscopic coupling mechanism . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Interaction with photons . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Carrier-phonon interaction . . . . . . . . . . . . . . . . . . . . . . 17
2.3.3 Spin-phonon interaction . . . . . . . . . . . . . . . . . . . . . . . 21
3 Pure phase noise 25
3.1 Exact reduced dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 results in explicit form . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Exact solutions from approximate master equations 33
4.1 Time-local master equation approach . . . . . . . . . . . . . . . . . . . . 33
4.1.1 Weak system-bath coupling: Born master equation . . . . . . . . 34
4.1.2 The Markov approximation: Bloch-Redfield theory . . . . . . . . 35
4.2 When second order is exact . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 Comparison with the exact solution . . . . . . . . . . . . . . . . 36
4.2.2 Time ordered cumulants and Gaussian bath initial state . . . . . 37
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Fast initial decoherence 41
5.1 Single qubit dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1.1 Ohmic spectral density . . . . . . . . . . . . . . . . . . . . . . . . 42
5.1.2 Super-ohmic spectral densities . . . . . . . . . . . . . . . . . . . . 46
iiiContents
6 Pure dephasing of spatially separated qubits 49
6.1 Robust and fragile entangled qubit pairs . . . . . . . . . . . . . . . . . . 50
6.1.1 Robust Bell state . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.1.2 Fragile Bell state . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Incomplete pure dephasing of a qubit register . . . . . . . . . . . . . . . 56
6.2.1 Frequency shifts and damping factors . . . . . . . . . . . . . . . . 58
6.2.2 N-qubit fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7 Causal master equation 65
7.1 Spurious effects from Bloch-Redfield theory . . . . . . . . . . . . . . . . 66
7.2 Taking causality into account . . . . . . . . . . . . . . . . . . . . . . . . 66
7.3 Incomplete pure dephasing revisited . . . . . . . . . . . . . . . . . . . . 68
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8 Spatially separated qubits subject to bit-flip noise 75
8.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.2 Causal master equation in energy eigenbasis . . . . . . . . . . . . . . . . 77
8.3 Super- and subradiance at a distance . . . . . . . . . . . . . . . . . . . . 78
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9 Summary and Conclusion 87
A Exact reduced dynamics 91
A.1 Preconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.2 Derivation of the exact solution . . . . . . . . . . . . . . . . . . . . . . . 92
B Damping rates, Lamb-shifts, and correlation functions 97
B.1 Solution of the integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
B.2 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
C Quantum master equations 99
C.1 Nakajima-Zwanzig projection operator formalism . . . . . . . . . . . . . 100
C.2 Time-convolutionless projection operator method . . . . . . . . . . . . . 102
Bibliography 105
Index 123
Acknowledgments 125
iv1
Introduction
Substantial technical progress in the last two decades enabled the development and
fabrication of nanoscale devices that exhibit explicit quantum mechanical properties on
amacroscopiclevel. Theabilitytogainexternalcontrolofsuchdevicesallowsonetoput
the fundamental laws of quantum mechanics to a test. A lot of effort in this direction
certainlywasdrivenbytheemergingdisciplineofquantuminformationprocessingwhose
basic aim is not only to test quantum mechanics, but rather to understand how its
principles can actually be used for the manipulation, storage, and communication of
information.
Computers based on intrinsic quantum mechanical devices will not only process infor-
mation faster than today’s computers. Rather, they are able to run specially designed
quantum algorithms to perform tasks that go beyond the capability of any classical
approach. For the implementation of a quantum algorithm it is necessary to ensure
and control the unitary evolution of an array of quantum mechanical two-level systems,
i.e. a qubit register. Solid-state quantum systems using charge or spin degrees of free-
dom of confined electrons or holes, and also superconducting qubits based on Josephson
junctions are currently very promising candidates for its realization. However, one of
the major remaining challenges is decoherence: The interaction of the qubits with their
environment affects the indispensable quantum coherence and entanglement of the quan-
tum states. Thus, understanding of decoherence in quantum computer architectures is
crucial for the development of successful qubit operations in scalable solid state systems.
Several strategies are pursued to beat decoherence [1,2]. An active scheme is quan-
tum error correction [3–5], which requires a redundant encoding of a logical qubit by
several two-level systems, so-called physical qubits. Standard error correction protocols
are designed to work efficiently if the physical qubits are subject to independent errors.
This condition can be realized by putting the far apart so that it is reasonable to
assume that they couple to uncorrelated noise sources. Decoherence-free subspaces are
a passive variant of quantum error correction [6–9]. In this scheme, one logical qubit
is encoded by several physical qubits in such a way that the logical qubit states do
not couple to the environment at all. Consequently, the quantum code works perfectly
coherent and neither the detection nor the active correction of errors is needed. Ideal
decoherence-free subspaces occur when the qubit-environment coupling exhibits symme-
tries such that the physical qubits interact with perfectly correlated noise, an idealized
situation that can be achieved by co-located qubits. In several physical situations,
11 Introduction
however, spatial correlations in the fluctuations of the environment can be present and
neither of the ideal cases outlined above is perfectly realized. In the present thesis we
focus on these non-ideal situations. Our goal is to study the consequences of spatially
correlated quantum noise for the dissipative entanglement dynamics and the fidelity of a
qubit register. We investigate the interplay of decoherence and spatial qubit separation.
The following sections give an overview of the basics of quantum information pro-
cessing and briefly review the ideas of quantum error correction and decoherence-free
subspace encoding and their respective relations to independent and collective noise
models. We then present in Chap. 2 a system-bath model that takes spatial separations
of qubits explicitly into account. It is shown how various physical situations can be
mapped to our model. For the case in which the environment induces pure phase noise,
the reduced qubit dynamics possesses an exact solution which we present in an explicit
form in Chap. 3. Since it is not always feasible to achieve exact results for the dissipa-
tive system dynamics, we consider in Chap. 4 a time-convolutionless master equation
approach and derive a non-Markovian master-equation for weak qubit-bath coupling.
An application of this approximate equation to our pure phase noise model allows for
an unambiguous comparison with an exact solution and shows that the emerging results
can even be exact, despite the fact that they are based on second-order perturbation
theory. As a first application of the analytical results, we discuss in Chap. 5 the dephas-
ing of a single qubit for various spectral densities of the environment. We concentrate
on the short-time dynamics of the qubit c