Quantum computation with nuclear spins in quantum dots [Elektronische Ressource] / Henning Christ

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• Physik

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• Publié par: technische_universitat_munchen
• Publié le: 01 janvier 2008
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 2 Mo

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Technische Universit¨at Munc¨ hen
Max-Planck-Institut fur¨ Quantenoptik
QUANTUM COMPUTATION
WITH NUCLEAR SPINS IN
QUANTUM DOTS
Henning Christ
Vollst¨andiger Abdruck der von der Fakult¨at fur¨ Physik
der Technischen Universit¨at Munc¨ hen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzender : Univ.-Prof. Dr. R. Gross
Prufer¨ der Dissertation : 1. Hon.-Prof. I. Cirac, Ph. D.
2. Univ.-Prof. J. J. Finley, Ph. D.
Die Dissertation wurde am 17.12.2007 bei der
Technischen Universit¨at Munc¨ hen eingereicht und
durch die Fakult¨at fur¨ Physik am 24.01.2008 angenommen.Abstract
The role of nuclear spins for quantum information processing in quantum
dots is theoretically investigated in this thesis. Building on the established
fact that the most strongly coupled environment for the potential electron
spinquantumbitarethesurroundinglatticenuclearspinsinteractingviathe
hyperfineinteraction, weturnthisviceintoavirtuebydesigningschemesfor
harnessing this strong coupling. In this perspective, the ensemble of nuclear
spins can be considered an asset, suitable for an active role in quantum
information processing due to its intrinsic long coherence times.
We present experimentally feasible protocols for the polarization, i.e. ini-
tialization, of the nuclear spins and a quantitative solution to our derived
master equation. The polarization limiting destructive interference effects,
caused by the collective nature of the nuclear coupling to the electron spin,
are studied in detail. Efficient ways of mitigating these constraints are pre-
sented, demonstrating that highly polarized nuclear ensembles in quantum
dots are feasible.
At high, but not perfect, polarization of the nuclei the evolution of an
electronspinincontactwiththespinbathcanbeefficientlystudiedbymeans
of a truncation of the Hilbert space. It is shown that the electron spin can
function as a mediator of universal quantum gates for collective nuclear spin
qubits,yieldingapromisingarchitectureforquantuminformationprocessing.
Furthermore, we show that at high polarization the hyperfine interaction of
electron and nuclear spins resembles the celebrated Jaynes-Cummings model
of quantum optics. This result opens the door for transfer of knowledge from
the mature field of quantum computation with atoms and photons. Addi-
tionally, tailored specifically for the quantum dot environment, we propose a
novel scheme for the generation of highly squeezed collective nuclear states.
Finally we demonstrate that even an unprepared completely mixed nu-
clear spin ensemble can be utilized for the important task of sequentially
generating entanglement between electrons. This is true despite the fact
that electrons and nuclei become only very weakly entangled through the
hyperfine interaction. Straightforward experimentally feasible protocols for
the generation of multipartite entangled (GHZ- and W-)states are presented.Contents
1 Introduction 7
1.1 Quantum Computation in Quantum Dots. . . . . . . . . . . . 9
1.2 Hyperfine Interaction in Quantum Dots . . . . . . . . . . . . . 10
1.3 Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . . 13
2 Nuclear Spin Cooling in a Quantum Dot 17
2.1 The Cooling Scheme . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Polarization Dynamics . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Approximation Schemes . . . . . . . . . . . . . . . . . 24
2.2.2 The Bosonic Description . . . . . . . . . . . . . . . . . 28
2.2.3 Polarization Time . . . . . . . . . . . . . . . . . . . . . 32
2.2.4 Enhanced Protocols . . . . . . . . . . . . . . . . . . . . 33
2.2.5 Imperfections . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Adapting the Model to Concrete Physical Settings . . . . . . . 40
2.4 Quantitative Treatment of Dipolar Interactions . . . . . . . . 47
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Nuclear Spin Cooling – The Homogeneous Limit 51
3.1 Achievable Polarization . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Time Evolution - Analytic Expressions . . . . . . . . . . . . . 54
3.3 Microscopic Description of Dark States . . . . . . . . . . . . . 57
3.4 Mode Changes . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Trapping States . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Effective Dynamics of Inhomogeneously Coupled Systems 65
4.1 Inhomogeneous Tavis-Cummings model . . . . . . . . . . . . . 67
4.2 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
56 CONTENTS
5 Quantum Computation with Nuclear Spin Ensembles 83
5.1 Qubits and Gates . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.1 Electron Spin Manipulation . . . . . . . . . . . . . . . 84
5.1.2 Nuclear Qubit Gates . . . . . . . . . . . . . . . . . . . 85
5.1.3 Long-range Entanglement . . . . . . . . . . . . . . . . 87
5.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.1 Quantitative Error Estimation for GaAs, InAs and CdSe 93
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Quantum Optical Description of the Hyperfine Interaction 95
6.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Effective Bosonic Hamiltonian . . . . . . . . . . . . . . . . . . 102
6.3 Study of Electron Spin Decay with the Bosonic Formalism . . 105
6.4 Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7 Entanglement Creation 119
7.1 Entanglement Generation . . . . . . . . . . . . . . . . . . . . 121
7.1.1 Two Qubit Entanglement . . . . . . . . . . . . . . . . 122
7.1.2 Multipartite Entanglement . . . . . . . . . . . . . . . . 124
7.2 Realization with Quantum Dots . . . . . . . . . . . . . . . . . 126
7.3 Iterative Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.4 Electron-Nuclear Entanglement . . . . . . . . . . . . . . . . . 134
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A Polarization with Static External Magnetic Field 139
B Bosonic Mode Picture of the Cooling Process 141
C Role of Dimensionality in Nuclear Spin Cooling 145
D Matrix-Element Approach to the Bosonic Approximation 149
E Derivation of the Effective Bosonic Hamiltonian 153
Bibliography 157Chapter 1
Introduction
A computing device properly harnessing the laws of quantum mechanics can
solve certain problems that are intractable, i.e. computationally hard, for
machines based on classical logic [1, 2, 3, 4]. Even more, quantum mechanics
guaranteesprovablysecurecommunicationbetweentwoparties[5]. Basedon
these deep results, quantum information processing and quantum cryptogra-
phyhavequicklygrownintoavibrant,veryactive,largeandinterdisciplinary
field of physics [4]. Ranging from fundamental insights into the structure of
quantum mechanics and reality [6] over ground breaking results in materials
science [7, 8] to first commercially available products [9], the achievements
are deeply impressive.
The challenges, however, are quite as remarkable: The quantum com-
puter (QC) has to be very well shielded from its environment in order to
avoid unwanted interruption of its coherent evolution, a phenomenon called
decoherence. At the same time the constituents (quantum bits, or qubits) of
the very same system typically need to be actively manipulated and finally
read out. Thus one arrives at the contradictory requirements, that on the
one hand isolation from the surrounding and on the other hand strong cou-
pling to some classical interface is needed. The quest for suitable physical
systems is still on, and has lead to a plethora of possible candidates for the
realization of quantum information processing (QIP), spanning a fascinating
range of systems [10, 11] from elementary (quasi-)two level systems, such as
hyperfine levels in ions [12] and electron spins [13], to complex macroscopic
structures like superconducting devices [14].
Since the early days of the field spins have been in the focus of both
experimental and theoretical research, as they are natural qubits and very
generally speaking possess long coherence times. The latter is particularly
true for nuclear spins [15]. Some of our fundamental understanding of QIP
has been triggered by liquid state nuclear magnetic resonance (NMR) exper-
78 Introduction
iments [16, 17, 18] (which are listed in the Guiness book of world records
as the largest running quantum computer). However, the two most promi-
nent approaches to nuclear spin based QIP suffer from serious disadvantages.
Liquid state NMR, relying on the nuclear spins of molecules in solution, is
intrinsically unscalable, which has caused this line of research to fade. The
Kane proposal [19, 20], with the qubits defined as (phosphorous) donor spins
(in a silicon matrix), is based on electronically mediated gates, but due to
the large extent of the electronic wave function, the clock cycle is rather low.
In addition, the required sub-lattice-site precision placement is a daunti