On quantum entanglement, measurement and decoherence in nanosystems

O

N

Q

UANTUM

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NTANGLEMENT

, M

EASUREMENT

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Q

UANTUM

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NTANGLEMENT

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EASUREMENT

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ECOHERENCE IN

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ANOSYSTEMS

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 15 oktober 2013 om 15:00 uur

door

Marcin Szymon D

UKALSKI

Master of Science in Theoretical Physics van Universiteit Utrecht, Nederland,

Dit concept-proefschrift is goedgekeurd door de promotor: Prof. dr. Ya. M. Blanter

Samenstelling promotiecommissie: Rector Magnificus, voorzitter

Prof. dr. Ya. M. Blanter, Technische Universiteit Delft, promotor Prof. dr. A. Borrás López Universitat de les Illes Balears

Prof. dr. Y. Gefen Weizmann Institute of Science Prof. dr. ir. R. Hanson Technische Universiteit Delft Prof. dr. A. Shnirman Karlsruhe Institute of Technology Prof. dr. ir. T. Klapwijk Technische Universiteit Delft Dr. L. DiCarlo Technische Universiteit Delft

Prof. dr. ir. L. Vandersypen Technische Universiteit Delft, reservelid

Keywords: entanglement, decoherence, (continuous) measurement, Cavity and Circuit QED, transmon qubit,

Printed by: Ipskamp Drukkers B.V.

Cover Design: R. Dukalski, K. Dukalska, and M. Dukalski,

using and adapting Moth by bramblejungle, http://www.flickr.com/photos/bramblejungle/7032993667. Licence at http://creativecommons.org/licenses/by-nc/2.0.

Copyright © 2013 by M. Dukalski

Casimir PhD Series, Delft-Leiden 2013-26 ISBN 978-90-8593-167-6

An electronic version of this dissertation is available at http://repository.tudelft.nl/.

C

ONTENTS

1 Introduction 1

References . . . 5

2 Theoretical Background 7 2.1 From classical to quantum physics. . . 7

2.2 Harmonic oscillator - the semi-universal quantum system . . . 9

2.3 Quantum Binary Digit (qubit). . . 13

2.4 Qubit-resonator systems. . . 19

2.5 Entanglement . . . 20

2.6 Dissipation and dephasing . . . 24

2.7 Measurement . . . 27

2.7.1 Basic concepts . . . 27

2.7.2 Theoretical description . . . 29

2.7.3 Positive Operator Value Measure(POVM) . . . 30

2.7.4 Continuous measurement. . . 31

References . . . 34

3 Periodic revival of entanglement 41 3.1 Introduction . . . 42

3.2 The model . . . 43

3.3 Entanglement Evolution . . . 46

3.4 Conclusions . . . 49

3.5 Appendix . . . 50

3.5.1 Single qubit master equations . . . 50

3.5.2 Some convenient identities . . . 54

References . . . 54

4 Entanglement revival in tripartite perspective 57 4.1 Introduction . . . 58

4.2 The Model and its Dynamics . . . 58

4.3 Entanglement Measures . . . 61

4.4 Dissipationless cavities. . . 65 vii

viii CONTENTS

4.5 Dissipative cavity . . . 69

4.6 Conclusions . . . 70

References . . . 71

5 High Jaynes-Cummings pseudospins eigenstates in the homogeneous Tavis-Cummings model 73 5.1 Introduction . . . 74 5.2 The system. . . 75 5.3 General case . . . 79 5.4 Switching pseudospins. . . 81 5.5 Effects of decoherence . . . 83

5.6 Solutions to the system with dephasing . . . 85

5.7 Strong coupling . . . 89

5.8 Conclusion. . . 89

5.9 Appendix . . . 90

5.9.1 I. Four qubits multiplets . . . 90

5.9.2 II. Details of the N -qubit propagator calculations . . . 91

References . . . 98

6 Entanglement by measurement in two strongly driven qubits 101 6.1 Introduction . . . 102

6.2 The Model . . . 103

6.3 Polaron Transformation and the Effective Stochastic Master Equation . . . 106

6.4 Solutions to the Effective Stochastic Master Equation. . . 108

6.5 Deterministic Protocol at the Cost of Entanglement Production Rate 114 6.6 Conclusion. . . 116

References . . . 116

7 Deteministic entanglement of superconducting qubits by parity measure-ment and feedback 119 7.1 Introduction . . . 120 7.2 Model. . . 121 7.3 Experimental Implementation . . . 125 7.4 Theory vs. Experiment . . . 130 7.5 Conclusion. . . 131 References . . . 132

CONTENTS ix

8 Dynamics of coupled vibration modes in a quantum non-linear

mechan-ical resonators 137 8.1 Introduction . . . 138 8.2 Model . . . 139 8.3 Classical results . . . 142 8.4 Quantum dynamics. . . 142 8.5 Conclusions . . . 146 References . . . 146

9 Quantum parametric oscillator in a Wei-Norman method setting 149 9.1 Introduction . . . 150

9.2 Effective squeezing.. . . 151

9.3 System initialised in a coherent state . . . 153

9.4 System initialised in a thermal state . . . 155

9.5 Steady state character of the resultant states. . . 155

9.6 Conclusions . . . 157

9.7 Appendix . . . 157

9.7.1 Wei-Norman Method. . . 157

9.7.2 Theso(3,2) Lie Algebra valued problem . . . 160

9.7.3 Resonant Driving . . . 162

9.7.4 Non-resonant driving . . . 164

9.7.5 Resonant Driving with Dissipation. . . 165

9.7.6 Algebra decomposition . . . 167

9.7.7 The complete problem: detuned squeezing subject to dissipa-tion . . . 169

9.7.8 Action of Lie group operators eαHi _{. . . 171}

9.7.9 Acting on a coherent state . . . 172

9.7.10 Acting on a thermal state . . . 175

References . . . 177

10 Quantifiable entanglement from two-mode squeezing subject to dissipa-tion 179 10.1 Introduction . . . 180

10.2 System . . . 181

10.3 Entanglement measures . . . 185

10.4 Entanglement generation on a thermalised states . . . 188

10.5 Other Lie algebra reductions . . . 190

10.6 Conclusion . . . 192

10.7 Appendix . . . 192

x CONTENTS 10.7.2 Equations of Motion . . . 193 References . . . 197 Summary 199 Samenvatting 203 Curriculum Vitæ 207 List of Publications 209 Acknowledgements 211

1

I

NTRODUCTION

It might be difficult to imagine our society in the early 21stcentury without the de-vices that changed the way we learn, work, exchange information, travel or even talk. A great number of these can trace its roots to the breakthroughs of what we today refer to as modern physics, a field most frequently associated to special and general relativity, and quantum physics. It is, for example, thanks to the former that the modern taxi drivers are able to do their job at all, since commercialisa-tion of the Global Posicommercialisa-tioning System, the development of which was only possible due to our revised understanding of space and time, first laid out by Einstein in 1905 and 1915. However, as far as the number and the scope of applications are concerned relativity is no match to the later, quantum physics.

Quantum physics not only shed new light and explained the behaviour of el-ementary particles, matter, or light, but also directly and indirectly contributed to a vast range of technological applications. The understanding of control of electronic behaviour in semiconductor structures lead to the digital revolution -the advent of integrated circuits and computers. Fur-ther manipulation of -these structures gave us the ability to controllably produce light with desired properties paving the way to such inventions as light emitting diode (LED), energy saving light bulbs and lasers which are now at the heart of fields ranging from heavy industry, IT, telecommunication,military or medicine. Furthermore, employing photoelec-tric and photovoltaic effect gave us for instance digital cameras, night vision or solar power, and handling elementary particles with their dual particle-wave-like nature lead to, among many others, the developments of electron microscopy or fine-structure etching of integrated circuit boards. Quantum physics is also

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2 1. INTRODUCTION

hind our well-being in magnetic resonance imaging (MRI) and positron emission tomography (PET) scans in hospital diagnostics, or modern radiotherapy in the oncology ward. All these devices and the ongoing progress resulting in greater ef-ficiency, lower power consumption, better resolution or better computational ca-pabilities, could be secured by further incremental improvements in the existing capabilities based on semi-classical many-body physics. One could ask, whether a complete paradigm shift introducing devices and tools employing a wider range of uniquely quantum mechanical features, would lead to another technological leap? The potential strength and the innovation of such devices would rely on three unique qualities of quantum mechanics: wave-like properties of matter (superpo-sition), measurement induced random state collapse and entanglement. Super-position allows the physical microscopic object to be present in multiple distinct states at once. Then, subject to a measurement capable of distinguishing these states, such a state probabilistically collapses to one of these, and upon repeat-ing the prepare-and-measure process the latter might give different outcomes and produce a different state every time. Entanglement, on the other hand, is property of multiple systems present in a superposition, such that the information some of their global property is known, but they cannot be described individually.

Already in the early 1980s – the time which one could call the infancy of mod-ern digital technology – implementations of these three qualities in future devices were being considered. In 1982 Richard Feynman argued that simulating quan-tum systems can be done much more effectively with another quanquan-tum system, rather than a "normal computer" or a "universal automaton" [1]. The reasoning was that a simulation of a quantum system composed of, say 300 two-level sys-tems (quantum bits, or qubits) would require a storage of 2300

≈ 1090 degrees of freedom, which already exceeds the number of all the atoms in the visible universe by a factor of 10 trillion. Another 300 atoms quantum system, on the other hand, could in theory contain this information. The question remains: can we design devices and protocols that could harness these special capabilities?

The emergence of such Quantum Information Processing protocols took an-other decade, however, soon after Feynman’s proposal, in 1984, Charles Bennett and Gilles Brassard proposed a scheme (BB84) of distributing a one-time pad (a form of a cypher) based on the superposition and state collapse principles [2]. Later Ekert [3] expanded on that using an entangled photons based protocol, in both cases allowing two parties to securely communicate by means of polarized photons, without the risk of an eavesdropper intercepting the pad. The power of quantum computers was first elucidated with the Deutsch-Jozsa algorithm [4] showing a substantial speed up in distinguishing between a balanced and a con-stant function. This early success was followed by the proposal of a quantum Fourier transform, which is an integral part of a much celebrated Shor algorithm

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grow-ing poly-logarithmically O£log N¤3rather than exponentially O£eN¤, which is what the best known classical algorithm could achieve. Following that Grover [6] pro-posed his quantum algorithm capable of searching a database of N objects and re-turning the result in computational time O£pN¤, as compared to classical [N ]. For details see reference [7]. The challenge then remains to find physical systems ca-pable of carrying out these algorithms, which requires the combination of a great deal of control, the ability to manipulate and access the information inside the quantum state encoded in a (typically) very fragile quantum system, at the same time preserving these qualities in a process of scaling up the device [8].

There are many theoretical proposals for a physical realisation of a working qubit, however the scope of experimental realisation is much narrower [9]. Some of these realisations made use of the more matured and better understood systems such as photon polarizations (optics), nuclear magnetic resonance (spectroscopy and imaging) or trapped atoms (high precision atomic clocks), while others were only made possible due to advancements in nanofabrication techniques. The lat-ter class includes single electrons trapped on an island on a semiconductor, elec-tronic circuits made out of superconducting materials, or crystallographic defects with unique luminescent properties. As the field was developing some of these devices were found to have a potential wider range of applications with theoret-ical proposals and some experimental implementations spanning single photon sources, single molecule scale magnetic imaging, and in combinations with nano-electro-mechanical systems (NEMS) they could act as a read-out of very sensitive motion sensors.

These technologies, however, are no longer confined to the laboratory work bench. For example, the first quantum secured bank transfer took place in 2004 [10], election results were sent using an unbreakable channel in 2007 [11], and three years later when Germany was securing a 4-0 win against Australia [12], the same small company from Geneva, IdQuantique, employed the BB84 protocol to secure all the communication between the stadium and the 2010 World Cup head-quarters [13]. The success and gradual wide commercial availability of such prod-ucts delivered by a growing number of companies, even inspired an academic pur-suit of quantum hacking, where it is device imperfections rather than quantum mechanical principles that are being put to the test. There are very few commer-cially available quantum computing solutions. Two devices worth mentioning are D-Wave One [14,15] and its more recent follow-up D-Wave Two [16], which use adiabatic quantum computing, rather than a circuit model, to solve discrete com-binatorial optimization problems. The large scale many-qubit quantum circuit model type device remains an outstanding challenge.

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4 1. INTRODUCTION

The main problem in meeting this challenge is in that there appears to be a non-fundamental physical trade-off in isolation and control. A greater degree of manipulability over these small and delicate quantum systems implies that the system in question has to be strongly coupled to the external controls; this how-ever potentially exposes other uncontrollable external degrees of freedom which randomise the qubit evolution during computation – a process known as decoher-ence. Very much like the Mark II computer needed to be debugged from a moth accidentally trapped in the relays, here too we need to learn to control the imme-diate environment of the quantum system in order to engineer a working quan-tum computer. It is the difficulty of doing that, which gradually made some of the realisations pull out of the race. Simultaneously, it is unreasonable to expect that the qubit coherence times will continue to grow indefinitely with experimen-tal progress, which is a reason why quantum error correction (QEC) [17] becomes an element with growing importance in the fault tolerant quantum computing technologies. QEC is based on the majority vote and requires entanglement of at least five physical qubits, and repeatedly (or continuously) detect (measure) an er-ror syndrome and perform an appropriate corrective measurement. In this way a number of physical qubits are turned into a single logical qubit protected against single-qubit errors. In order to accomplish that however one needs to be able to generate highly entangled states and perform joint-qubit measurements.

This thesis discusses the interplay between entanglement, measurement and decoherence in quantum nanosystems. Chapter 2 briefly introduces the neces-sary theoretical background. Chapters 3-7 describe qubit-resontator type systems, chapters 3, 4 and 6 specifically deal with a system of strongly driven qubits cou-pled to a dissipative resonator; chapter 3 describes a entanglement revival, chapter 4 shows how these entanglement fluctuations can be understood in terms of sec-ondary coupling to the environment, and chapter 6 shows how such system can be used to generate entanglement by means of measurement of the resonator state. Chapter 7 treats a specific realisation of this idea in a system of two-transmon qubits dispersively coupled to a resonator subject to a continuous homodyne mea-surement. Chapter 5 discusses a system of multiple qubits coupled to a single res-onator. Chapter 8 studies a nanomechanical oscillator entering a regime of para-metric driving and shows the subtle differences between the classical and quan-tum understanding of this system. Chapters 9 and 10 deal with this matter in a strong coupling regime, where in the Wei-Norman formalism we present a geo-metrical picture of the systems evolution subject to dissipation and show the ef-fects on the degree of squeezing and entanglement formation.

REFERENCES 5

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R

EFERENCES

[1] R. P. Feynman,Simulating Physics with Computers, International Journal of Theoretical Physics 21, 467+ (1982).

[2] C. H. Bennett and G. Brassard, in Proceedings of IEEE International Conference

on Computer Systems and Signal Processing (IEEE, 1984), pp. 175–179.

[3] A. K. Ekert,Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67, 661 (1991).

[4] D. Deutsch and R. Jozsa, Rapid solution of problems by quantum

computa-tion, Proc. R. Soc. Lond. A 439, 553 (1992).

[5] P. W. Shor,Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM Journal on Computing 26, 1484 (1997).

[6] L. K. Grover, Quantum Mechanics Helps in Searching for a Needle in a

Haystack, Phys. Rev. Lett. 79, 325 (1997).

[7] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum

Infor-mation (Cambridge Series on InforInfor-mation and the Natural Sciences)

(Cam-bridge University Press, 2004), 1st ed.

[8] D. P. DiVincenzo, The physical implementation of quantum computation, Fortschritte der Physik 48, 771 (2000).

[9] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien,

Quantum computers, Nature 464, 45 (2010).

[10] World premiere: Bank transfer via quantum cryptography based on

en-tangled photons, http://www.secoqc.net/downloads/pressrelease/ Banktransfer_english.pdf(2004).

[11] F. Jordans, Swiss call new vote encryption system ’unbreakable’,http://www. technewsworld.com/story/59793.html(2007).

[12] German lessons for stunned socceroos, http://www.fifa.com/ worldcup/archive/southafrica2010/matches/round=249722/match= 300111116/summary.html(2010).

[13] Quantum encryption to secure world cup link,http://www.idquantique. com/news-and-events/press-releases.html?id=107(2010).

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6 REFERENCES

[14] Learning to program the d-wave one, http://dwave.wordpress.com/ 2011/05/11/learning-to-program-the-d-wave-one/(2011).

[15] D-wave quantum computer solves protein folding

problem, http://blogs.nature.com/news/2012/08/

d-wave-quantum-computer-solves-protein-folding-problem.html (2012).

[16] D-wave two system, http://www.dwavesys.com/en/

products-services.html(2013).

[17] P. W. Shor,Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995).

2

T

HEORETICAL

B

ACKGROUND

In this chapter we will revisit a number of core theoretical concepts that the lat-ter chaplat-ters of this thesis are built upon. Following a brief introduction to quan-tum mechanics exemplified by the quanquan-tum harmonic oscillator, we will look at physical realisations of qubits, and the description of a qubit-resonator interac-tion. Towards the end of this chapter, we will look at entanglement, measurement and decoherence, and how these three are related to each other.

2.1

F

ROM CLASSICAL TO QUANTUM PHYSICS

Classical physics is a field of science dealing with quantities (observables) which

describe the state of a set of objects (a system). These observables are

continu-ous functions of space-time coordinates, and their dynamics can be understood in terms of a Hamiltonian containing information about their motion (kinetic en-ergy) and the system’s internal and external interactions (potential)

H¡_{p, x}¢_{= T +V =} ~

p2

2m+V (~x) ,

where ~p and ~x denote the momentum and position of a particle. The dynamics of

a time dependent physical quantity ~Q are given by the equations of motion

d~Q

dt = { ~

Q,H }P where {A,B}P=∂A

∂~x ∂B ∂~p− ∂B ∂~x ∂A ∂~p.

The early twentieth century experimental observations have shown a break-down of these classical laws at the smallest scales, and a new theory needed to

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8 2. THEORETICALBACKGROUND

emerge in order to explain them. Max Planck’s proposal that light comes in dis-crete packages (quanta) explained the black body radiation in 1900. Following years brought about the development of Bohr’s planetary model of the atom, later expanded up on by Peter Debye and Arnold Sommerfeld. Together with Einstein’s theory of the photoelectric effect, which later granted him a Nobel prize, these dis-coveries marked the so-called Old Quantum Theory (1900-1925). By the second part of the 1920s Werner Heisenberg, Max Born and Pascual Jordan made their contributions to what was then called matrix mechanics, coining the idea of non-commutativity leading directly to the famous uncertainty relation and abolition of the notion of the clockwork universe. Around the same time Louis de Broglie associated wavelengths with momentum and Erwin Schrödinger introduced the concept of the wave function, in what they called wave mechanics, and Wolfgang Pauli contributed to the theory of non-relativistic spin and the exclusion princi-ple now carrying his name. This eruption of new ideas was then formalised by David Hilbert, Paul Dirac and John von Neumann by 1930, leaving behind among others the Hilbert space, Dirac’s creation and annihilation operators, the von Neu-mann equation, and contributions to the theory of quantum measurement. These developments not only established quantum mechanics as a new discipline, but more importantly, they fuelled scientific developments throughout the rest of the century and allow today us to enter the stage of quantum control in quantum nanoscience.

Quantum mechanics uses states |i 〉 which describe the statistical information about the physical quantities of a system. Consider a pure superposition state in Dirac representation, describing the state of some physical system (e.g. an atom) with three discrete levels, such that the system is present to a different extent in all of the states simultaneously

¯

¯ψ®= c1¯¯j = 1®+ c2¯j = 2¯ ®+ c3¯¯j = 3®, ci∈ C.

This states reflects two important features. For one, a measurement of quantity j of this state will give an outcome j = x only with probability |cx|2, with a condition

P

x|cx|2= 1. Secondly, a subject to experimental manipulation this system could

undergo destructive interference leading to local or temporal disappearance (or suppression) of one of the states j in the superposition.

In this work, we will focus on the disappearance as well as the (re-)emergence of these quantum features (interference and the measurement outcome indeter-minacy). In order to mathematically describe the difference between quantum states and classical ones, one makes use of the density operator ρ. In this language, the state above takes the form ρpure=¯¯ψ®­ψ¯¯, as opposed to the mixed state

2.2. HARMONIC OSCILLATOR-THE SEMI-UNIVERSAL

QUANTUM SYSTEM 9

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dif-ference is that the latter, ρmixed, does not contain elements ¯¯j = x® ­j = y¯¯, with x 6= y, and hence does not describe a state present in all three states

simultane-ously. Instead ρmixed describes an ensemble of states, where given state¯¯j = x®

can be found with a probability |cx|2. Individually, none of these states would be

capable of interfering and it is already in a well-defined state in the j -basis. The evolution of quantum states is found by means of canonical quantization of a classical theory, a process involving a) promoting classical physical quanti-ties to operators acting on a Hilbert space (infinitely dimensional, complex valued space with an L2norm), and b) translating the outcome of a Poisson bracket to an outcome of a commutator£ ˆA, ˆB¤ = ˆA ˆB − ˆB ˆA multiplied by iħ. For example, the vector components of position xiand momentum pj are translated into

{xi, pj}P= δi j→£ ˆXi, ˆPj¤= i ħδi j. (2.1)

In the x-basis, the operators take the form ˆXi = xi, ˆPj = i ħdxdj, in accordance to

the commutation relations. It is the non-commutative nature of the observables which gives rise to such features as the Heisenberg (uncertainty) principle or Lie al-gebra applications in quantum mechanics. Moreover, upon quantization the time evolution of an operator ˆQ takes the form of the von Neumann equation

dQ

dt = {Q,H }P → i ħ d ˆQ

dt =£ ˆQ, ˆH¤ , (2.2)

which for ˆQ = ρ =¯¯ψ® ­ψ¯¯ (a pure state) can be factorised, and then becomes a

time-dependent Schrödinger equation i ħ∂t¯¯ψ(t)®= ˆH¯¯ψ(t)®. Both of these

equa-tions describe unitary (reversible) state evolution; the equaequa-tions describing ad-ditional interaction with a classical environment will be discussed in section2.6. One particular state¯¯ψn(t)®= e−iEnt/ħ¯¯ψn(0)®, the eigenstate of the Hamiltonian

ˆ

H¯¯ψn®= En¯¯ψn®, is known as a wave function, which in the presence of a

non-zero potential V¡ ˆX¢ subjected to boundary conditions, results in a quantized set of energies En. Other physical observables, given by self adjoint operators ˆQ†= ˆQ,

can give rise to (discrete) spectra of other physical quantities.

2.2

H

ARMONIC OSCILLATOR

-

THE SEMI

-

UNIVERSAL

QUANTUM SYSTEM

One of the simplest, well understood and yet most versatile systems is the one-dimensional harmonic oscillator with a parabolic form potential energy V (x) =

1

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10 2. THEORETICALBACKGROUND

Upon a variable redefinition α =¡mω2

¢1/2

x + i¡ 1 2mω

¢1/2

p the Hamiltonian of this

system becomes H = ω|α|2.

In order to see how important that system to this work, let us consider an elec-tromagnetic wave – a vacuum solution to Maxwell’s equations governing electro-magnetism. The electric field associated with the wave reads

~

E (~x, t) =~εsin³~

k ·~x´¡A (t) + A∗_(t)¢_,

where ~ε is the polarization vector, ~k is the k-vector satisfying¯¯_¯~_k¯¯

¯ = 2πωc , A (t) =

Aeiωt _{is the vector potential, and E = 2|A| is the magnitude of the electric field.}

Then, using Faraday’s law ~∇ ×~E = −µo∂ ~∂tH one arrives at the associated magnetic

field ~ H (t) = − Z dtˆεy µo ∂Ex ∂z = ˆεy µoωkzcoskzz ¡ A (t) − A∗_(t)¢_.

Substituting these solutions into the Hamiltonian of electromagnetic energy in a charge neutral volume

H ₌ Z d3~_r³ εo 2 ~ E2_{(~x, t) +}µo 2 ~ H2(~x, t)´ = ε₂o Z

d3~_r¡_{4Re(A (t))}2_sin2_{(kzz) + 4Im(A (t))}2_cos2_(kzz)¢

= 2εoVe f f|A (t)|2, (2.3)

we obtain that this Hamiltonian is identical to that of the harmonic oscillator with the rescaling α(t) =

q_2ε

oVe f f

ω A (t). This means that the mathematics of the two

will be identical, and it is easy to see from the construction of the Hamiltonian 2.3, that in this description the electric field plays the role of displacement and the magnetic field that of momentum. Later on we will see how the simplicity of har-monic oscillator allows one to study superconducting qubits, optical or microwave cavities, or nanomechanical resonators.

In quantum mechanics the Hamiltonian together with position, momentum, and variable α are promoted to operators,

ˆ H = Pˆ 2 2m+ 1 2mω 2_Xˆ2 = ħω µ ˆ a†a +ˆ 1₂ ¶ , (2.4)

where ˆ_{a =}¡mω_2ħ¢1/2_{X + i}ˆ ¡_2mωħ1 ¢1/2P, and where the factor ofˆ 1₂emerges due to the commutation relations£a, ˆˆ a†¤

= 1, based on the position-momentum commutator in equation2.1.

2.2. HARMONIC OSCILLATOR-THE SEMI-UNIVERSAL

QUANTUM SYSTEM 11

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2

wave functions of the harmonic oscillator. Based on this description together with the commutation relations

[ ˆn, ˆ_{a] = − ˆa ,} hn, ˆˆ a†i

= ˆa†,

we interpret ˆa†_{and ˆa as the rising and lowering operators respectively, i.e.}

ˆ

a†_{|n〉 =}p_{n + 1|n〉 , a |n〉 =}ˆ p_{n |n − 1〉 ,}

which in leads to a conclusion that the number operator ˆn is non-negative integer

valued, i.e. the spectrum of the system is not only quantized but also equidistant. This is of great consequence to the mathematically equivalent quantum electro-magnetic field Hamiltonian, as it allows one to refer to individual quanta of elec-tromagnetic energy, or as we call them today, photons.

The ground state of the harmonic oscillator Hamiltonian is defined by a state |0〉 (vacuum, a state that cannot be lowered), such that ˆa |0〉 = 0 or ˆH |0〉 =ħω

2 |0〉.

It is interesting to note that in the presence of an external displacement (electric) field ǫ¡aˆ†

+ ˆa¢, the Hamiltonian can take the "square-completed" form

ˆ Hǫ = ħω µ ˆ a†_{a +}ˆ 1 2 ¶ + ħǫ³aˆ†+ ˆa´= ħω³aˆ†+ ǫ ω ´³ ˆ a +_ωǫ´+ħω₂ −ǫ 2 ω2, = D¡β¢ ˆH D¡−β¢+ ħω 2 − β 2_, where β = ǫ

ω and where we define a displacement operator D

¡

β¢= e¡β ˆa†−β∗aˆ¢=

e−|β_|2_eβ ˆa†

eβ∗aˆ_{. It is easy to check that the operator is unitary i.e. D (α)}†

= D (−α) =

D (α)−1_.

By displacing the Hamiltonian ˆH along with its eigenstates, one can find the

ground state¯¯β®of ˆHǫ, such that

ˆ Hǫ ¯ ¯β® = µ ħω 2 − β 2¶¯_¯β®_,

where the state ¯ ¯β®= D¡β¢|0〉 = e−|β|2eβ ˆa† |0〉 = e−|β|2 ∞ X n=0 βn p n!|n〉 ,

is commonly referred to as the canonical coherent state, i.e. one which oscillates around a minimum of a parabolic potential displaced by the complex parameter

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12 2. THEORETICALBACKGROUND

β. Moreover, it is easy to see that¯¯β®is an eigenstate of the annihilation operator ˆ

a¯¯β®= β¯¯β®, and that the non-discrete valuedness of coherent state amplitudes makes the states |x〉 and¯¯y®(x, y ∈ C) form an over-complete set, meaning that ¯

¯〈 x|y 〉¯¯2

= e−|x−y|2, which is only orthogonal in the limit¯¯x − y¯¯ → ∞. Very im-portantly, unlike the number operator eigenstates |n〉, the coherent states are no longer quantized; their amplitudes β are arbitrary complex number which obey equations whose form is equivalent to the equations of motion of classical har-monic oscillator variable α. It is for this reason that the photon states |n〉 are re-ferred to as quantum, while the coherent states are often termed pseudo-classical. Lastly, due to the annihilation operator eigenstate property, coherent states have a number of elegant mathematical properties, where the mean and the variance of the number of photons in a coherent state |α〉 are equal,

〈 ˆn 〉α= |α|2, and Var ˆn = 〈 ˆn2〉α− 〈 ˆn 〉2α= |α|2,

which means that the number of photons present in the coherent state obeys Pois-sonian statistics, i.e. the outcomes of a photon number measurement coming from a coherent state source will be uncorrelated in time.

Another interesting property of the coherent state is its Heisenberg uncertainty minimising property, steaming from the non-commutative nature of the quantum observables

Std ˆA Std ˆ_{B ≥}1

2 ¯

¯〈£ ˆA, ˆB¤〉¯¯ ,

where¡Std ˆA¢2_{= Var ˆ}A denotes the standard deviation of observable A.

Consid-ering the quadrature and the in-phase components (the real and the imaginary parts, or dimensionless position and momentum) of the coherent state amplitude we see that the coherent state satisfies its lower bound

¡ Std ˆX Std ˆP¢_|α〉=ħ 2 4 , and ¡ Std ˆX Std ˆP¢_{|n〉= ħ}2 µ n +1₂ ¶2 .

Due to this and the fact that Std_{|α〉X = Std}ˆ _{|α〉P =}ˆ ħ

2, the coherent state is typically

represented as a circle of radius1₂centred at¡〈 ˆX 〉,〈 ˆP 〉¢on the ˆ_{X − ˆ}P plane. One

could decrease the quantum mechanical bound on uncertainty of one of the com-ponents at the cost of increasing the other through a process called squeezing. This is used for example in the Josephson Parametric Amplifiers (JPAs) used for coherent state read-out. The process of squeezing is mathematically achieved by forming a state |α,ξ〉 = S (ξ)D (α)|0〉 where S (ξ) = exph12

³

ξ¡aˆ†¢2

− ξ∗aˆ2´i_{is the}

2.3. QUANTUMBINARYDIGIT(QUBIT) 13

④ ④

2

direction (given by Arg(ξ) = θ) and forming an ellipse. This can be seen by defining a rotated set of quadratures

Y1=1 2 ³ ˆ ae−iθ/2_{+ ˆa}†_eiθ/2´_{, Y} 2= 1 2i ³ ˆ ae−iθ/2_{− ˆa}†_eiθ/2´_,

whose the uncertainties then read StdY1= e−|ξ| 2 , StdY2= e|ξ| 2 , StdY1StdY2= 1 4,

marking a clear trade-off between the two without changing the lower bound of the Heisenberg uncertainty principle. Additionally, an important quantity that we need to consider is

〈α,ξ| ˆa†a |α,ξ〉 = |α|ˆ ¡cosh 2|ξ| − cos¡θ − 2φ¢sinh 2|ξ|¢+ sinh2|ξ| ,

where θ and φ are arguments of ξ and α respectively. This shows that the squeezing operator, in the absence of the coherent state displacement operator (thus squeez-ing vacuum) also generates an average number of bosons, which are a rapidly growing functions of the absolute value of the squeezing parameter. This non-classical process distinguishes between the non-classical and quantum parametrically driven oscillator. Lastly, the process of squeezing could be done across several modes simultaneously - a process that results in (multi-modal) entanglement which we will return to in section2.5and Chapter 10.

2.3

Q

UANTUM

B

INARY

D

IGIT

(

QUBIT

)

The basic resource empowering the set of algorithms which form the field of quan-tum computing is a quanquan-tum bit (qubit). A (classical) bit (binary digit), is an ab-stract entity which can take two values, assigned typically with 0 or 1, whereas a qubit can take any quantum superposition of the two

¯

¯ψ®= α|0〉 + β|1〉 ,

where α and β are arbitrary complex numbers satisfying |α|2+¯¯β¯¯2= 1. If we parametrise the amplitudes with α = cos(θ/2) and β = eiφsin (θ/2), with 0 ≤ θ ≤ π and 0 ≤ φ < 2π, we can associate the state¯¯ψ®with a position on a so-called Bloch sphere, and any operation on the qubit can be understood as a rotation in three di-mensions ˆ_{R = exp}h_{i ξ ˆr ·~ˆσ}i, where ˆr is the unit vector normal to the rotation plane,

ξ is the rotation angle and ~ˆσ =©σˆx, ˆσy, ˆσzªis a vector of Pauli matrices

ˆ σx= µ _{0 1} 1 0 ¶ , σˆy= µ 0 −i i 0 ¶ , and ˆσz= µ _{1 0} 0 −1 ¶ . (2.5)

④ ④

2

14 2. THEORETICALBACKGROUND

The continuity of θ and φ makes a qubit seem to possess analogue information fea-tures, making it capable of storing an infinite amount of information. This however is not true, as the analogue nature of the Bloch sphere, is broken by a measurement with a binary outcome. In spite of this fact, one can build algorithms [1–3] capable of harnessing the power of the state superposition, which are described in great detail in the standard reference [4].

Information, classical or quantum, must always be physical, meaning that a (qu)bit is not only an abstract mathematical concept, but more importantly it needs to have a physical embodiment. A classical bit could take a form of: a position of an electrical switch; two distinct levels of light intensity, voltage or current; or two directions of local magnetization or polarization of a light wave. Such bit reali-sation has to be capable of reliably storing and manipulating a string of bits in the course of computation. In quantum information processing, in order to out-compete classical computer realisations, a qubit, typically realised by the two low-est energy levels in some system, has to obey similar conditions, known as DiVin-cenzo criteria [5], given by these five points:

1 Scalability: the qubit realisation (two well defined levels in a subspace of a Hilbert space) needs to be physically scalable, with the Hilbert space in-creasing exponentially with a linear increase in the number of qubits. Addi-tionally, the growth in the number of qubits must not come at the cost of the reduced lifetime of the quantum states.

2 Initialisation: the qubit can be reliably and sufficiently quickly initialised (or reset) to an arbitrary state prior to computation.

3 Logic gates: there must exist a physical realisation of a universal gate set (see [4,6] and references therein), where a minimum such set could involve arbi-trary rotations and a two- or more qubit conditional gate (like CNOT). These gates must be executed faster than the quantum coherence of the system, which requires strong coupling between the qubits and the qubit and con-trol apparatus.

4 Read-out: the state of the qubit must be determined easily and quickly by a measurement (limited only by the fundamental laws of quantum mechan-ics), and there needs to be a one-to-one correspondence between the state of the qubit and the physical signal.

5 Qubit coherence (relaxation and pure dephasing) times have to be sufficiently long.

2.3. QUANTUMBINARYDIGIT(QUBIT) 15

④ ④

2

the system could sometimes overlap, when for example the quantum system col-lapses and remains in a known pure state after a quantum non-demolition (QND) measurement. There is however an intrinsic contradiction among these condi-tions, as longer qubit lifetimes require a stronger degree of isolation from the sys-tem and the environment, while faster qubit manipulation necessitates more pos-sibility for external influence. In order to achieve a balance between the two, we need to be able to control and promptly switch between strong and weak system-external world interaction.

Seen from the smallest scale the whole world is quantum, however not ev-ery small scale system presenting truly quantum features satisfies the above cri-teria. In trying to find the most successful realisation one could consider the truly quantum system composed of such elements as (many-)electron or nuclear spins, charges, supercurrents, (atomic) energy levels, or a single photon polarizations. Their thorough and detailed description as well as the progress in their respective fields is beyond the scope of this chapter, so here we describe some of the most successful realisations to date along with their working principles and implemen-tation challenges. An interested reader is advised to consult reference [7].

Photons. The polarisation degree of freedom of a single quantum of light (a pho-ton) offers long coherence time and can be manipulated easily (single qubit gates) using the standard optics tool kit. The unavailability of sufficiently strong non-linearities in optical media inhibits photon interactions, however the so-called KLM scheme [8] offers a solution in the field of cluster state quantum computing and it has marked significant improvements in the recent years. The outstanding challenges include single photon detection, single photon source and reduction of photon losses due to waveguide attenuation. Most likely, the advancements in this field would lead to a spillover effect in other qubit realisations and nanotechnology as a whole. Meanwhile, photons are perfect candidates in hybrid architectures, as a flying qubit transferring quantum information in space, as well as in quantum key distribution, now successfully implemented commercial setting [7].

Liquid state Nuclear Magnetic Resonance. Due to expertise built by decades of research in the field of nuclear spins on organic molecules in solution, NMR quan-tum computing was ready with first implementations in 1996 [9,10], only years after the first algorithms were proposed. The qubit is encoded in two energy levels of a1_{H or}13_{C nuclear spin in a strong magnetic field, however since the}

shield-ing effects and magnetic field inhomogeneity give rise to resonant frequency dif-ferences among the ensemble of molecules, only the few with the right frequency splitting are selectively addressed and manipulated by radio frequency pulses. Two

④ ④

2

16 2. THEORETICALBACKGROUND

qubit gates were realised using indirect spin-spin coupling mediated by molecular electrons and the read-out is accomplished by means of the induced current mea-surement in the coil surrounding the sample. Despite devices reaching as many as twelve qubits [11] and successful implementation of some algorithms [12], ini-tialisation is an outstanding challenge as the quantum computer was always in a thermal state, and even pseudo-pure state techniques (singling out individual pure state spins treating the rest as a highly entropic background) have not proven to be scalable. Moreover, since the loss of quantum information takes place on the timescales comparable to gate times, this setup has been deemed to have a very poorly scaling signal-to-noise ratio with a growing number of qubits [13], and some went as far as claiming that NMR is capable of only performing classical sim-ulations of a quantum computer [14]. There are some that hope that solid state NMR will deliver better results, as for example gate times are expected to be much shorter, however initialisation and measurement still remain lengthy and any ad-ditional improvements will likely prove to be more relevant to other realisations [7].

Trapped atoms. Precision timing technology, driven by control of hyperfine tran-sitions in individual atoms set the stage for qubit realisations with trapped ions. Appropriately chosen atomic energy levels offer high coherence times, good level of control, and CNOT gates (entanglement formation) realised by a carefully cho-sen interaction. Initialisation is done through state pumping, the atom position can be maintained with great accuracy, and the read-out is realised through state dependent fluorescence. Scaling of these systems presents a challenge due to laser cooling inefficiencies, and inability to address individual qubits due to mode cross talk (or other emerging nonlinearities), as the number of qubits increases. Good coupling to photons and further control of the latter are expected to be a good can-didate for distributed probabilistic quantum computing.

Quantum dots. One of the first proposals of an engineered scalable quantum com-puter was proposed by Loss and DiVincenzo [15,16], where they imagined a large array of quantum dots (zero-dimensional systems) where a single electron is con-fined in all three dimensions in a semiconductor structure and its spin is manip-ulated. This is realised by means of a set of electrodes placed on top of a two-dimensional electron gas (2DEG), used for both confinement and manipulation, in gallium arsenide GaAS. Measurement is done by means of the spin to charge con-version, where the spin state is found through the spin-sensitive tunnelling rate out of the dot, and a charge measurement can disclose the spin state on the dot. The added advantage due to control of the motional degrees of freedom (unavail-able for photons, NMR or cold atoms), came at the price of increased decoherence

2.3. QUANTUMBINARYDIGIT(QUBIT) 17

④ ④

2

in-terfering with the dot spin state, however solutions involve defects in silicon lattice. A different type of device, a so-called self assembled quantum dot, is formed spon-taneously due to close proximity of lattice constants of two crystalline materials, one deposited on top of the other. A release of the resultant strain at the interface gives rise to a local island where electrons can be confined. The advantages of this solution include ∼ 4K (instead of millikelvin) operation temperatures, as well as control by optical means, with very fast gates in the order of picoseconds. The cur-rent research is focused on greater control of randomised dot formation.

Defect centres. Another promising solid-state qubit realisation is the Nitrogen Va-cancy (NV) center in diamond, where the exceptional physical properties of the host lattice ensure very high qubit lifetimes. The presence of a nitrogen impurity in diamond breaks the local tetrahedral symmetry reducing the point symmetry group to C3v, which combined with the spin symmetry upon absorption of an

ad-ditional electron gives rise to a complicated multilevel structure of two electrons spin states [17]. The qubit is realised by the two lowest laying states separated by energies corresponding to a microwave (MW) frequency range, where initialisa-tion is done by optical pumping in the order of 250 nanoseconds, manipulainitialisa-tion is achieved by means of MW pulses, and the read-out is accomplished by means of optical excitation and a spin-state dependent decay rates (at room temperatures), or a similar mechanism by means of carefully tuned laser transition in the ≈ 8K regime which circumvents the thermal broadening problems from before [18]. Di-amond (even when ultra-pure) has some degree of magnetic impurities (mainly

13_{C), which couple to the electronic spin state and give rise to decoherence with}

qubit lifetimes reaching milliseconds. It is the long spin state lifetime and the mag-netic nuclear spin state of nitrogen impurity that makes NV centres very good can-didates for quantum memories. The weak interaction between the NV centres and the weak coupling strength to light is the main weakness of this realisation, how-ever ensemble of states proposals, hybrid systems or elements of quantum net-works are some of the quantum information processing uses of this spin qubit. Moreover, it is postulated that the NV centers could act as excellent nanometer precision magnetic imaging probes, electric field or strain sensors; they are also expected to be widely applicable in the imaging of cellular processes in biologi-cal systems due to their luminescent properties and high compatibility with living systems.

Superconducting qubits. A qubit can be also realised by means of single circuits, which when made out of superconducting materials operating at very low temper-atures can do away with losses otherwise originating from finite resistance. A

com-④ ④

2

18 2. THEORETICALBACKGROUND

bination of an inductor and a capacitor in said circuit would give rise to a Hamil-tonian resembling a harmonic oscillator

H ₌Q

2

2C+ Φ2 2L,

where Φ and Q are the magnetic flux through the inductor and the charge on the capacitor respectively, which can be thought of as the position and momentum, respectively so thatQ_2C2³Φ2

2L

´

plays the role of kinetic (potential) energy. Upon quan-tising, these two become conjugate variables satisfying the Heisenberg algebra

£ ˆΦ_{, ˆQ}¤_{= i ħ,}

and it is easy to see that the spectrum of this setup is linear in the number of quanta

n, En= ħω¡n +1₂¢, with ω = (LC)−1/2. In order to form a qubit one needs to add a

nonlinearity, which will make the level separation different. This is accomplished by the insertion of a thin insulating layer in the circuit (a Josephson junction) in-troducing an additional potential energy U = −EJcos φ, where EJ is the

Joseph-son energy, proportional to the critical current. The mixture of the cosine and the parabolic potential give rise to a nonlinearity which defines the qubit. The rel-ative size of the charging energy e2_{/2C to the Josephson’s energy E}

J determines

whether we are dealing with a charge, phase, or a flux qubit, as different ratios put the cosine-plus-parabola potential in a different regime, and these define the qubits susceptability to charge or flux fluctuations.

One class of such qubits, without the inductance term i.e. relying only on the Josephson effect, uses charge quantisation on the superconducting island. In the regime where EJ≪ EC, this is called the charge qubit (or a Cooper pair box), and

for EJ≫ EC different specific designs are called transmon [19,20], quantronium

[21], or Xmon [22]. Flux qubit uses a combination of two potentials to create a double well system, where different wells correspond to currents flowing in oppo-site directions along the loop (hence the name persistent current qubit). To keep the transition between the two, the charging (kinetic) energy is much smaller than the Josephson energy, EJ≫ EC. Lastly in the same regime, the potential landscape

can be modified to form the phase qubit, where due to a different biasing point one well is much higher than the other and the two lowest levels in the upper lay-ing well form the logical states |0〉 and |1〉. In this architecture, the millikelvin oper-ating temperatures in combination with qubit frequencies in the microwave (MW) regime suppress the thermal excitations, and the resonant MW pulses are used for single qubit gates carried out on nanosecond scale, whereas the qubit coherence times are of the order of up to tens of micro-seconds. Superconducting qubits are frequently coupled to resonators (CircuitQED), which can mediate the qubit-qubit

2.4. QUBIT-RESONATOR SYSTEMS 19

④ ④

2

improving control, and understanding the sources of decoherence are the frontline of the research in this field [23]. More on superconducting qubits can be found in reference [24].

2.4

Q

UBIT

-

RESONATOR SYSTEMS

In order to carry out a read-out, or to control or manipulate the state of the qubit, one could couple it to a bosonic degree of freedom. The resultant mathematical description of such a system interaction is known as the cavity or circuit quantum electrodynamics (cQED/CircuitQED) [25,26], where the former refers to the light field trapped inside an optical cavity-resonator, and the latter refers to supercon-ducting circuits coupled to a planar or 3D microwave resonator. This section will briefly outline the theory of such systems.

A typical implementation of such interaction is brought about by introducing U(1) gauge invariance in a Hamiltonian. Then, in the experimental regime, typical to spin or superconducting qubits, one can use the dipole approximation, assum-ing that the electromagnetic field does not appreciably vary across the length of a system it is interacting with. Upon fixing the gauge, narrowing the scope to a single mode of radiation, quantising the electromagnetic field and reducing the matter system to a two level system, one arrives at what is known as a Rabi Hamiltonian

ˆ H = ˆHo+ ˆHi, with ˆ Ho= Ω 2σz+ ω ˆa †_{a ,ˆ Hˆ} i= g (σ++ σ−) ³ ˆ a† + ˆa´,

where the operators σz,σ±= 1/2¡σx± i σy¢, are the Pauli matrices Eq. 2.5and

ˆ

a†_{, ˆa are the bosonic creation and annihilation operators in Eg.2.4, Ω is the energy}

level difference of the qubit, ω stands for the frequency of the resonator, and g is the coupling strength given by the scalar product of the local electric field strength and the qubit’s electric dipole moment.

The interesting quantum features of this Hamiltonian, marked by the energy violating terms σ+_aˆ† _{and σ}−_{a, are only accessible in the ultra-strong couplingˆ}

regime g ∼ Ω,ω, which has only been realised in a handful experiments [27], and then has been a subject of theoretical research [28]. Numerous experimental re-alisations work in the strong coupling regime Ω,ω ≫ g , justifying the application of the rotating wave approximation (RWA), which in the interaction picture V =

ei ˆHot_Hˆ

ie−i ˆHotdeems the energy violating terms negligible as their time-dependence

ei(ω+Ω)t oscillates much faster than the other, energy conserving, ones ei(ω−Ω)t. Upon the RWA one obtains the Jaynes-Cummings model

ˆ HJC= Ω 2σz+ ω ˆa †_ˆ a + g³σ+a + σˆ −aˆ† ´ ,

④ ④

2

20 2. THEORETICALBACKGROUND

which predicts a temporal qubit-resonator entanglement, such that the popula-tions of the qubit and the resonator oscillate with a so-called Rabi frequency ΩR=

p

g2_{n + ∆}2_{set by the total quanta present in the system n and the qubit-resonator}

detuning ∆ = Ω − ω. Additionally, the generalisation of this model to multiple qubits coupled to a single mode is referred to as the Tavis-Cummings Hamiltonian. Transmon qubits, with typical parameters ∆/2π ≈ 1 GHz, g /2π ≈ 60 MHz, can be made to operate in the so-called dispersive regime ∆ ≫ gpn i.e. where a qubit

is sufficiently detuned from the cavity eigenmode and the average number of res-onator excitations are far below the critical number ncrit=∆

2

g2, which is typically of

the order of ten photons. In this regime the dynamics of a Tavis-Cummings system is accurately described by the effective Hamiltonian obtained using the transfor-mation ˆ H′_{= exp}¡ ˆA¢ ˆH TCexp ³ ˆ A†´, = ˆHTC+£ ˆA, ˆHTC¤+1 2£ ˆA,£ ˆA, ˆHTC ¤¤ + ··· , where A =X i gi ∆_i ³ σi_{+a − σ}ˆ i₋aˆ†´.

Keeping the terms up to first order in g∆gives

ˆ H =1₂X i ħΩ i_σi z+ Ã ħω +1₂X i g2 i ∆_iσz ! ˆ a†a +ˆ 1₂X i j gigj ∆_i ³ σ₊jσi₋+ σi+σ−j ´ ,

resulting in a number of effects: the cavity (qubit) state dependent frequency shift of the qubit (cavity), and a direct effective qubit-qubit flip-flop type interaction facilitated by an exchange of virtual photons. Appending an external drive of the resonator, turns the cavity field into a probe allowing for the qubit state determi-nation.

2.5

E

NTANGLEMENT

Erwin Schrödinger in his 1935 seminal paper [29] stated that Verschränkung (eng. entanglement) is "not one but rather the characteristic trait of quantum mechan-ics, the one that enforces its entire departure from classical lines of thought." Two subsystems are said to be entangled (or inseparable), when they are bound by a shared wave function in a superposition of two or more states, such that no in-formation is available about the individual subsystem entities. Upon the measure-ment of the first subsystem the state collapses and based on the measuremeasure-ment out-come the state of the second subsystem is immediately determined, but remains unknown prior to the measurement. When we consider either of two-qubit states

2.5. ENTANGLEMENT 21 ④ ④

2

we see that the measurement of the system in the state |Ψ〉 (|Φ〉) in either of the subspaces will give rise to (anti-)correlated outcomes, but the individual outcomes are not a priori known. Ignoring the outcome of the measurement, the state of the second qubit will not be in a superposition state, but rather it would be in state |0〉 or |1〉, each with probability P|0〉= P|1〉=12. Mathematically, this amounts to taking

a partial trace [30] of a density operator of state |Ψ〉 or |Φ〉

Tr1¡¯¯ψ®­ψ¯¯¢= 1

2(|0〉 〈0| + |1〉〈1|) ,

which is no longer a pure single-qubit state, but rather a statistical mixture of states, with a maximum level of entropy (uncertainty).

Maxwell in 1871, postulated the existence of a device which could reduce the total entropy, thus violating the second law of thermodynamics. Maxwell’s demon, as it was called, was a two-chamber box with a valve separating the two, and the demon’s goal was to separate a two-component gas present in the box into the two chambers by a well-timed closing and re-opening of the valve. This apparent vio-lation was resolved by Rolf Landauer [31], who argued that the apparent decrease in entropy of the box’s subsystem has to be compensated by the increase in the en-tropy associated with any erasure of information (which needs to take place due to demons finite memory). Moreover, Landauer’s limit states that there is a minimum amount of energy associated with information change equal to kbT ln 2 ∼ 10−21J

at room temperature, which recently has been successfully measured [32]. In the meantime Claude Shannon postulated the minimum unpredictability in a random variable, which can be attributed to its information content [33], and is known as Shannon entropy

E_{= −}X

i

λilogkλi,

where k is the size of the alphabet that the message is encoded in, and which is very closely related to the von Neumann entropy

S = −Tr¡ρ ln ρ¢= −X

i

νiln νi,

used to provide an extension of the entropy from classical thermodynamics to quantum theory. Here νi are the eigenvalues of ρ. It is easy to see that both the

④ ④

2

22 2. THEORETICALBACKGROUND

von Neumann as well as the Shannon entropies are maximised by states with max-imum uncertainty, which in quantum mechanics would be given by a uniformly distributed mixture of states. This inspired the introduction of the entropy of en-tanglement [34], where a pure state is maximally entangled, provided that the von Neumann entropy of a bipartite state is maximized upon partial tracing over one of the subspaces. Moreover, entropy of entanglement is equal to the entanglement cost and distillable entanglement for pure states. Other entanglement measures exist ranging from relative entropy of entanglement or entanglement of formation, but only very few of them are easily computable. Among those belonging to the lat-ter class we distinguish concurrence [35], a three-tangle [36], or the (logarithmic) negativity [37–39], where the last two can be generalised to a tri- and multi-partite cases. For an exploration of these and many more entanglement measures the reader is advised to consider references [40] and [41].

Long before we built the mathematical formalism, the fact that two qubits could share a single wave function was the reason that Einstein called quantum mechan-ics incomplete. Together with Boris Podolsky and Nathan Rosen he argued that it allowed for superluminal communication, which he called a spooky action at a

dis-tance and they used entangled states in the attempt to prove that the measurement

outcome of physical quantities must be predetermined prior to the measurement, as otherwise the quantum theory would stand in complete contradiction to special and general relativity. Nowadays it is understood that such a superluminal com-munication is impossible by virtue of the no-comcom-munication theorem [42].

The EPR paradox was finally resolved once John Stewart Bell in 1964 rephrased the EPR concept of a deterministic universe in terms of a local hidden variable theory [43,44], which assumes:

realism (or determinism) that particles posses prior properties, which are inde-pendent of the measurement and determine its outcome,

locality that measurement outcomes cannot be influenced by activities at space-like separations.

He set out to show that such a local hidden variable theory would be unfunded based on an inequality formed by a set of correlations of measurements of two systems left to interact and then separated a space-like distance. Violation of such an inequality would lead to a rejection of one of the two assumptions, and favour quantum mechanics. His form of the inequality is not used in practice, because of restrictions involving applicability to a rather narrow range of hidden variables theories, or truly two-outcome systems. A convenient generalisation of this the-orem was proposed by John Clauser, Michael Horne, Abner Shimony and Richard Holt [45], where they showed that two fully entangled spin−12particles violate what

2.5. ENTANGLEMENT 23

④ ④

2

〈QS 〉 + 〈 RS 〉 + 〈 RT 〉 − 〈QT 〉 ≤ 2.

Here operators Q and R (S and T ) act on the first (second) qubit. For any classical state this inequality is obeyed, however with the right choice of measurement op-erators on the left hand side of this inequality for bipartite entangled states could reach 2p2.

Since the initial proposal, there has been a number of experimental verifica-tions [46–57] of this inequality, however to date none of them succeeded in closing of all of the loopholes of the CHSH violation, which in the effort of the ongoing field of research. With every consecutive result our confidence in the soundness of quantum mechanics is enforced and despite the fact that the topic remains moot, local hidden variable theories receive decreasing support.

Already in the early days quantum entanglement was considered to be not only a quantum phenomenon, but also a valuable resource, which could be used in a number of tasks. For example quantum entanglement is used in superdense cod-ing, where two bits of classical information are sent using a single qubit [58], beat-ing the standard quantum limit [59] and reaching the Heisenberg limit, and quan-tum teleportation, which attempts to move the unknown state of the matter (qubit) in space without actually displacing the qubit itself [60], where experiments are capable to move photonic states over distances exceeding 100 km [61], or mat-ter states over tens of memat-ters [62]. Additionally, quantum key distribution also harnesses the power of polarisation-entangled photons where an eavesdropper’s presence can be detected by randomly verifying whether the transmitted entan-gled pair still does (no eavesdropper) or does not (eavesdropper detected) violate the Bell/CHSH inequalities [63].

Entangled states are also believed to be extremely important in the quantum computing, especially if in a course of performing an algorithm one of the qubits could be subject to a spin flip or a phase kick. It is thus crucial to be able to correct for such errors, without destroying the quantum superposition. Classically this is realised by means of redundancy, where every bit is copied twice. Subject to noise, one bit could flip, which can be detected and fixed by means of looking at the state of all three bits and applying the majority rule to determine if an error took place, and if so, which of the three copies was subjected to it. Unfortunately, this approach is impossible in quantum mechanics for two reasons: a) creating a copy of a yet unknown qubit state is forbidden by the no-cloning theorem, b) if one was able to produce three such copies, then the indeterminate measurement outcome of a superposition state would give us no grounds to conclude which qubit had been subjected to noise.

④ ④

2

24 2. THEORETICALBACKGROUND

Despite these restrictions, quantum mechanics offers a way to perform quan-tum error correction [64,65]. The way to detect, diagnose, and fix the spin-flip operation of a single physical qubit is by entangling it to two others

α|0〉 + β|1〉 → α|000〉 + β|111〉 ,

forming a single logical qubit. Let us suppose that the first physical qubit under-went a spin flip leading to a state α|100〉+β|011〉. Performing a joint measurement of three qubits using the so-called error syndrome operators

P0= |000〉 〈000| + |111〉 〈111| , P1= |100〉 〈100| + |011〉 〈011| , P2= |010〉 〈010| + |101〉 〈101| , P3= |001〉 〈001| + |110〉 〈110| ,

would return zero upon measurements P0,2,3and one upon P1, indicating that the

first qubit was flipped, without collapsing the state or disclosing what the ampli-tudes α and β could be. Similarly, we can define the so-called phase-flip errors, i.e. |±〉 → |∓〉, with states |±〉 = (|0〉 ± |1〉)/p2, could be detected using the same scheme upon replacement 0 → + and 1 → −, and arbitrary errors could be de-tected by means of a nine entangled qubits code [64], though more complicated seven [66] or even five [67] qubits codes exist.

A successful implementation of the ideas above relies on the ability to generate entanglement. Nonlinear crystals are currently used to create entangled photon pairs through the process of spontaneous down-conversion. Such pairs have been shown to be capable of establishing quantum communication protocols already over distances exceeding 100 km [61]. In solid state systems, generation of en-tanglement till recently has been a probabilistic process or has required heralding [68]. Recently upon using carefully engineered interaction [69] first conditional gates have been implemented. Entanglement could also be generated by means of a skilfully set up measurement scheme, which has been proposed in a number of physical systems, and recently realised probabilistically in diamond NV center qubits [70], and for the first time deterministically (presented in chapter 7).

The great problem behind a successful generation of entanglement and later sustaining it, is similar to that of single qubit coherence protection – the almost inevitable coupling between the system and its environment. This could lead to gradual or abrupt loss of entanglement, which sometimes could be followed by entanglement revival. The next section together with chapters 3 and 4, will address this issue.

2.6

D

ISSIPATION AND DEPHASING

In building quantum microscopic devices one of the problems is noise – a random, uncontrollable influence of the environment on a imperfectly isolated quantum