Superconducting Charge Qubits Denzil Anthony Rodrigues

Superconducting Charge Qubits

Denzil Anthony Rodrigues

H. H. Wills Physics Laboratory University of Bristol

A thesis submitted to the University of Bristol in accordance with the requirements of the degree of Ph.D. in the Faculty of Science

November 2003

Word Count: 36, 000

Abstract

In this thesis we discuss a particular type of superconducting qubit, the charge qubit. We review how the Hamiltonian for a single qubit can be constructed by quantising the Josephson relations as if they were classical equations of motion. We examine the charging energy of various qubit circuits, in particular focussing on the effect of connecting a superconducting reservoir to small islands. We establish the correct form for the charging energies and show that the naive method of constructing a quantum Hamiltonian by adding the charging energy of the circuit and the tunnelling energy for each junction leads to the correct Josephson relations for each junction. We describe a coupled two-qubit system and describe the operations necessary to test a Bell inequality in this system. We show how passing a magnetic flux through the circuit leads to oscillations as a function of both time and flux and show that these oscillations can be considered as being caused by interference between virtual tunnelling paths between states. We describe the strong coupling limit of the BCS Hamiltonian, where the sums over electron creation and annihilation operators can be replaced by operators representing large quantum spins. We then solve this Hamiltonian in both the exact and mean-field limits. We show that the spin solutions are equal to the full solutions in the limit where the interaction energy is much larger than the cutoff energy. We show how this simple model can be used to easily investigate various phenomena. We describe an analogue of the quantum optical effects of superradiance where the current through an array of Cooper pair boxes is proportional to the square of the number of boxes. We also describe an analogue of quantum revival, where the coherent oscillations of a Cooper pair box coupled to a reservoir decay, only to revive again later. We describe a similar two-qubit effect, where the oscillations of two entangled Cooper pair boxes decay and revive. If the entanglement between the boxes is calculated, we find that the entanglement also decays and then revives. We describe how quantum spins can be represented using a wavefunction that is a function of phase. This representation is used to find a microscopic derivation of the charge qubit Hamiltonian in terms of phase. We find that in the limit of large size, the Hamiltonian found by quantising the Josephson equations is recovered. Corrections to this phase Hamiltonian are described.

To my family.

Acknowledgments

I have been lucky enough to have two supervisors to whom I could turn in times of distress; I would like to thank Balazs for instructing me in basic physics as if I was stupid, and Tim for reminding me that I’m not. As well as technical advice and encouragement, both have provided instruction by example in the appropriate attitude to research, and in how to ﬁt research in to a (relatively) normal life. My fellow PhD students have been of great importance to me over the last three years. I would like to thank Ben and Mark for proving that it was possible to survive a PhD and to thank Andy for giving me something to aim for. I thank my roommate Danny for asking me stupid questions, both ones that I knew the answer to and those I did not. I thank David and Maria for stopping me when I was talking rubbish and Jamie and Emma for not stopping me when I was talking rubbish. Finally I thank Liz giving me something to look forward to at the end of the day.

Authors Declaration

I declare that the work in this thesis was carried out in accordance with the regulations of the University of Bristol. The work is original except where indicated by special reference in the text and no part of the thesis has been submitted for any other degree. Any views expressed in the thesis are those of the author and in no way represent those of the University of Bristol. The thesis has not been presented to any other University for examination either in the United Kingdom or overseas.

“You can never take your badge oﬀ.”

Jamie Walker.

Contents

1 Superconductivity and The Josephson Effect 1 1.1 Introduction to the Thesis ...... 1 1.2 Introduction to Superconductivity ...... 4 1.3 Ginzburg-Landau Theory ...... 5 1.3.1 The Ginzburg-Landau Equations ...... 6 1.3.2 The Josephson Effect in Ginzburg-Landau Theory ...... 9 1.4 Creation and Annihilation Operators ...... 10 1.5 The Fermi Sea ...... 11 1.6 BCS and BdG ...... 13 1.7 The Josephson Effect ...... 17 1.8 The Pair Tunnelling Hamiltonian ...... 20 1.9 Small Superconductors ...... 22 1.9.1 The Parity Effect ...... 23 1.9.2 The Richardson Exact Solution ...... 25

2 The Superconducting Charge Qubit 31 2.1 Experimental Progress ...... 31 2.2 The Josephson Equations for Small Circuits ...... 33 2.2.1 The Classical Lagrangian ...... 35 2.2.2 The Classical Hamiltonian and the Conjugate Variables ...... 36 2.2.3 Quantising the Classical Josephson Equations ...... 37 2.3 Tuneable Josephson Energy ...... 38 2.4 The Two-Level System ...... 41 2.5 Qubit Dynamics and the NOT Gate ...... 46 2.6 Decoherence and Noise ...... 50

xiii xiv CONTENTS

2.6.1 Gaussian Noise ...... 51 2.6.2 Discrete Noise in the Josephson Energy ...... 52

3 The Josephson Circuit Hamiltonian 57 3.1 The Capacitance of Qubit Circuits ...... 57 3.1.1 Single Junction with Gate Voltage ...... 58 3.1.2 Single Junction with Gate Voltage and Self-Capacitance ...... 60 3.1.3 The 2 Qubit Circuit ...... 62 3.1.4 Multiple Islands ...... 64 3.2 Quantising the Josephson Relations ...... 65 3.2.1 Single Junction with Gate Voltage ...... 65 3.2.2 Single Junction with Gate Voltage and Self-Capacitance ...... 67 3.2.3 The 2 Qubit Circuit ...... 67

4 The Dynamics of a Two Qubit System 71 4.1 The Two Qubit Circuit ...... 71 4.1.1 Energy Levels ...... 72 4.1.2 Two-Qubit Gates ...... 73 4.2 A Bell Inequality ...... 74 4.3 Virtual Tunnelling ...... 80 4.4 Path Interference ...... 87

5 Finite Superconductors as Spins 93 5.1 The Strong Coupling Approximation ...... 93 5.2 Quantum Spins ...... 95 5.3 The Majorana Representation ...... 99 l 1 5.3.1 Spin 2 in terms of Spin 2 ...... 99 5.3.2 The Coherent State ...... 100 5.3.3 The Majorana Sphere ...... 101 5.4 Mean Field Spin Solution ...... 103 5.4.1 Rotation of Spin Operators ...... 104 5.4.2 The Ground State ...... 107 5.4.3 Self Consistency ...... 108 5.4.4 Comparison to the BCS Theory ...... 109 CONTENTS xv

5.5 Solving the Spin Hamiltonian Exactly ...... 110 5.5.1 Broken Symmetry: Finite size ...... 112 5.5.2 Broken Symmetry: Inﬁnite Size ...... 114 5.6 Expanding the Richardson Solution to First Order ...... 115

6 The Super Josephson Eﬀect 119 6.1 The Cooper Pair Box Array ...... 119 6.2 Superradiant Tunnelling Between Number Eigenstates ...... 122 6.3 Superradiant Tunnelling: Number State to Coherent State ...... 124 6.4 Superradiant Tunnelling Between Coherent States ...... 126

7 Revival 129 7.1 The Superconducting Analogue of the Jaynes-Cummings Model ...... 129 7.1.1 Quantum Revival of the Initial State ...... 132 7.2 Revival of Entanglement ...... 136

8 An Electronic Derivation of The Qubit Hamiltonian 141 8.1 Introduction ...... 141 8.2 The Phase Representation ...... 142 8.2.1 Phase Representation of Spins ...... 143 8.2.2 The Limit of Large Size ...... 146 8.2.3 Phase Representation of BCS States ...... 148 8.3 Island - Reservoir Tunnelling ...... 151 8.4 Inter - Island Tunnelling ...... 155

9 Conclusion 157

A Eigenstates of the Two Qubit Hamiltonian 161

A.1 Exact Eigenstates with Real T12 ...... 161

A.2 Perturbation in T12 ...... 163

B Limit of BCS Self Consistency Equations 165 xvi CONTENTS List of Figures

1.1 The Ginzburg-Landau free energy as a function of the order parameter ψ. . 7 1.2 The Temperature dependence of the superconducting gap ∆(T ) measured in units of the zero-temperature gap ∆(0)...... 16

2.1 A simple circuit diagram for a charge qubit consisting of two regions of

superconductor connected by a Josephson junction with critical current IC whose capacitance is represented by a capacitor of capacitance C ...... 34 2.2 A simple circuit diagram for a charge qubit with a gate voltage applied and the single Josephson junction replaced by a magnetic ﬂux dependent double-junction...... 39 2.3 A double junction consisting of a ring of superconductor broken by two junction and threaded by a ﬂux Φ. The two paths from point 1 to point 2

are labelled ca and cb...... 40

2.4 The charging energy as a periodic function of ng...... 43

2.5 The qubit energy levels as a periodic function of ng...... 44

2.6 The avoided crossing at half-integer values of ng...... 45 2.7 A not gate takes a state 0 and returns a state 1 , and vice versa. . . . . 47 | i | i 2.8 Oscillations in the probability of the qubit being in state 1 (solid line) or i state 0 (dotted line)...... 49 | i 2.9 The probability of a Cooper pair box being in the state 1 as a function of | i time, with Gaussian noise in the Josephson energy...... 52 2.10 The probability of a Cooper pair box being in the state 1 as a function of | i time with discrete noise in the Josephson energy...... 55

3.1 Single Voltage with Gate Voltage Circuit Diagram ...... 59

xvii xviii LIST OF FIGURES

3.2 Single junction with gate voltage and self-capacitance, which is represented by a capacitive coupling to a third region, labelled w (for ‘world’)...... 60 3.3 Two Qubit circuit diagram. The system consists of two islands (1, 2) connected to a reservoir (r), with gate voltages applied to each. We wish to solve the problem for a reservoir of general size, and so a further region, w is included to represent the rest of the world...... 63

4.1 The energy levels of a two qubit system as a function of n1g = n2g, when the tunnelling between the islands is zero. The parameters of the system

are EC1 = EC2 = 100, EC12 = 10, T1 = 10, T2 = 10 ...... 73

4.2 The energy levels of a two qubit system as a function of n1g = n2g, when the tunnelling between the islands is included. The parameters of the system

are EC1 = EC2 = 100, EC12 = 10, T1 = 10, T2 = 10, T12 = 30 ...... 74 4.3 Oscillations between state 01 (solid line), 10, (dashed line) and the states | i | i 00 and 11 (dotted lines). Halfway between oscillations, the system is in | i | i a maximally entangled state (vertical lines)...... 75 4.4 The bases for measurements that violate a Bell inequality...... 76 4.5 A two qubit circuit with tunnelling between the qubits and a magnetic flux passed through the circuit ...... 80 4.6 The Probability of the Two Qubit system being in the state 00 (upper | i plot) and 01 (lower plot). The horizontal axis is time and the vertical axis | i Φ0 is the flux through the loop in units 2π . White represents high probability of occupation and black represents low probability...... 82 4.7 The Probability of the Two Qubit system being in the state 10 (upper | i plot) and 11 (lower plot). The horizontal axis is time and the vertical axis | i Φ0 is the flux through the loop in units 2π . White represents high probability of occupation and Black represents low probability...... 83 4.8 The Probability of the Two Qubit system being in each of the four charging states...... 84 4.9 The change in frequency of the oscillations in the probability of being in the state 00 ...... 86 | i 4.10 The two second order (a, b) and third order (c, d) tunnelling processes involved in the oscillation 00 11 ...... 87 | i → | i LIST OF FIGURES xix

4.11 The three third order tunnelling processes involved in the oscillation 01 | i → 10 ...... 89 | i 4.12 The oscillations of the two qubit system when the system is not in the limit T E ...... 90 ¿ C12 4.13 The Probability of the Two Qubit system being in the state 01 or 11 i.e. | i | i the probability that the ﬁrst qubit is occupied regardless of the occupation of the second. The z axis represents increasing probability...... 91

5.1 The Majorana Sphere...... 102 5.2 Each point on the Majorana sphere can be represented by its projection onto the complex plane, z...... 103 5.3 Rotation of basis of spin operators...... 104

6.1 An array of lb superconducting islands, or Cooper Pair Boxes, that are individually coupled to a superconducting reservoir, r, through capacitive Josephson tunnel junctions (with no inter-box coupling)...... 120

7.1 Spin Coherent State Revival for Nr = 10, lr = 50...... 133 7.2 Decay and revival of a single Cooper pair island state, showing the degree of linear entropy...... 134 7.3 Analytic asymptotic form for decay oscillations in the probability against

time in units of 2π/EJ (solid line) compared with the numeric evolution

(dashed line) Nr = 10, lr = 100 ...... 135

7.4 Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50. . . 137

7.5 Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50, showing the degree of entanglement...... 138

8.1 Energy levels of a charge qubit as a function of ng including the ﬁnite size correction to the tunnelling energy. The dashed lines indicate the uncor-

rected energy levels. The parameters are EC = 10 EJ = 1 and l = 100 . . . 154 xx LIST OF FIGURES Chapter 1

Superconductivity and The Josephson Eﬀect

We discuss the basic principles of quantum computing and list the requirements of a physical system used to construct a quantum computer. We discuss the advantages of using superconductors to construct qubits. We describe both phenomenological and microscopic theories used to describe superconductivity and the Josephson effect. We also examine how bulk superconductivity is modified in a superconductor of finite size.

1.1 Introduction to the Thesis

At the beginning of the Twentieth Century, a new description of the world appeared: Quantum Mechanics. Since then it has developed into the most accurate physical theory ever constructed, and explained many phenomena that would otherwise remain complete mysteries. Despite the overwhelming success of quantum mechanics, an understanding of the theory is far from complete, and new avenues of exploration are continually opening. One such avenue that is currently attracting much interest among physicists, mathematicians and computer scientists is the field of Quantum Computing. This field involves finding a quantum mechanical description of not only physical objects, but also the information that they represent and manipulate [1, 2]. Just as quantum mechanics tells us more about the world than classical mechanics, a quantum computer could do calculations that would be impossible on a classical computer.

1 2 Superconductivity and The Josephson Eﬀect

The first theoretical demonstration that a quantum mechanical algorithm could out- perform anything that is possible classically was made by David Deutsch and Richard Jozsa [3]. Whilst this sparked interest in the field of quantum computing, it was after the discovery that quantum computing could factorise large numbers exponentially more quickly than any known classical algorithm [4], and could give a √N improvement in the time needed for a database search [5] that the field really took off. A great boost was given to the credibility of quantum computing when the discovery of quantum error correction codes [6, 7, 8] showed that despite the fact that a quantum state cannot be determined by a single measurement, errors occurring below a certain rate can be corrected algorithmi- cally. This means the number of gate operations performable is no longer limited to that which can be performed within the ‘bare’ decoherence time of the system (as long as the decoherence time is long enough to allow the error correction codes to operate). The basic idea that underlies all modern (classical) computation is binary logic. This is the idea that information is stored in switches, or ‘bits,’ that are either on or off (1 or 0) and never anywhere in between. The bits take many forms, from the voltages on a transistor, to the direction of magnetic moments on a hard drive, to the presence or absence of pits on a CD. The underlying idea of quantum computation is that these switches behave according to the rules of quantum mechanics - in particular, they can be placed in a superposition state. If a bit is placed in a superposition of its two states, then it can be thought of as being in both the ‘on’ and ‘off’ states at the same time. In terms of the binary number stored, the qubit (quantum bit) represents both the number ‘0’ and ‘1’ simultaneously, and a series of n such bits (a register), each placed in a superposition, can represent all the numbers from 0 to 2n 1 simultaneously. Any operation performed − on this register of qubits then performs a calculation on all of the numbers stored by the qubits in one operation. It is this ‘quantum parallelism’ that gives quantum computing its power [9, 10]. At present there is no single preferred way of constructing quantum bits, and in fact there is a vast range of different systems that have been suggested as possible qubits, including the energy levels of ions [11] or atoms [12] held in electromagnetic traps or cavities, impurities embedded in a substrate [13], or quantum dots formed from silicon nanostructures [14]. With such a large array of possibilities, a set of criteria by which these potential qubits could be assessed was needed, and the so-called ‘DiVincenzo checklist’ [15] has become 1.1 Introduction to the Thesis 3 the standard for determining the viability of different proposals. The five main criteria outlined in this checklist are:

Well characterised qubits, scalable in number. •

The ability to initialise these qubits in a simple ‘starting state.’ •

Long decoherence times, compared with the gate operation time. •

A universal set of quantum gates. •

The ability to measure qubits. •

As is to be expected, some realisations of quantum bits are more successful at meeting different criteria than others. Two of the most successful realisations to date are ion traps and NMR, in which multiple sequential gates have been applied [16, 17] and, in the case of NMR, even Shor’s factoring algorithm has been performed [18]. Whilst these realisations have well-defined qubits upon which quantum gates can be accurately performed, they suffer from the problem of scalability - it is difficult to see how the techniques can be applied to the hundreds, thousands or millions of qubits that would be required in a working quantum computer. By contrast, realisations of qubits in the solid state, such as implanted ions or quantum dots, are intrinsically scalable in that a micro-fabricated array of qubits could be produced, each of which is well defined and is distinguishable due to its position. The main problem that solid-state realisations face is the problem of decoherence. Whilst ions in a trap are isolated to a large degree from the external world, solid state qubits are manufactured devices subject to noise from, for example, the movement of charges in the substrate. Although quantum error-correction schemes have been developed [6, 7, 8], these require that a large number of gate operations ( 104) can be performed ∼ within the decoherence time of a qubit. If this limit can be obtained then, in principle, very long calculations can be performed in a fault tolerant manner, as long as the additional qubits required for error-correction can be provided. One possible way of constructing a solid-state qubit that is less vulnerable to decoherence is to make it from a superconductor, as superconductors are known to posses some remarkable quantum coherent features that may be of value in the construction of a qubit. 4 Superconductivity and The Josephson Effect

The ability to describe the phenomenon of superconductivity is one of quantum mechanics’ great successes. Superconductivity was discovered, experimentally, around the start of the last century, when it was discovered that when certain metals are cooled down to close to absolute zero, their electrical resistance vanished. One of the key features of superconductivity is the existence of an energy gap: a minimum amount of energy is required to produce excitations above the ground state. This means the superconducting state is protected against low-energy excitations, and to some extent the superconductor is isolated from the external world. A further problem for many qubits is the difficulty of measurement. Qubits constructed from ‘fundamental’ objects (such as ions) may produce only a small signal, which may be difficult to isolate from the other qubits in the system. Solid state qubits in general have the problem that a measurement device connected to them will cause decoherence during the operation of the gates. Qubits constructed from qubits take advantage of the macroscopic quantum coherence of the electrons to provide a large signal that may be easier to measure, and also benefit from decades of research into Josephson junctions and other superconducting circuits. These factors suggest that a superconductor might be the ideal material from which to construct a quantum computer. In this thesis we discuss one of the ways of making a qubit out of a superconducting circuit, known as the superconducting charge qubit, and various other interesting phenomena that could possibly be observed in such systems.

1.2 Introduction to Superconductivity

In 1901, Kammerlingh-Onnes developed a technique for liquifying Helium, and proceeded to investigate the properties of materials at the low temperatures that this allowed. In 1911, he noticed that the electrical resistance of Mercury became very low below 4.2K [19]. Further investigation revealed two things: that the resistance was not just low, but so low that it could not be measured, and that the resistance did not smoothly decrease as the temperature was reduced, but dropped off suddenly at 4.2K. Kammerlingh-Onnes named the state of the Mercury below this temperature the superconducting state. A century later, experimental and theoretical work on superconductivity continues apace, and it is a major research topic in condensed matter physics. Supercon- ductivity turned out to be a common phenomenon, and most metals will superconduct if their temperature is reduced enough. Recently, many other systems have been shown 1.3 Ginzburg-Landau Theory 5 to exhibit superconductivity, including organic superconductors [20], magnesium diboride [21], and carbon nanotubes [22] among others. The superconducting state was found to have several distinctive characteristics other than zero resistance. In particular, the Meiss- ner effect [23], in which magnetic fields are excluded from a superconductor, indicates that a superconductor is not merely a perfect conductor (which would trap any flux present before the sample was cooled to its superconducting state) but a perfect diamagnet as well. Another important characteristic of superconductors is an exponential specific heat capacity at low temperatures, which suggests a finite energy gap between the Several phenomenological theories of superconductivity were developed in the first half of the last century, notably the London - London equations [24] which describe both the infinite conductivity and the exclusion of the magnetic field. A more complete phenomenological description is given by Ginzburg-Landau theory [25] which treats the onset of superconductivity as a second-order phase transition with a complex order parameter. Despite the success of the various phenomenological theories, it was half a century after Kammeling-Onnes’ discovery before a microscopic theory of superconductivity was formulated. A key insight came when it was shown [26] that a small attractive interaction between electrons could lead to a bound pair of electrons (a Cooper pair), explaining the energy gap. This insight led to the formulation of a microscopic theory of superconductivity by Bardeen, Cooper and Schrieffer now known as BCS theory [27]. Excitations above the BCS ground state are described using Bogoliubov-de Gennes (BdG) theory [28]. In BCS theory, the ground state of a superconductor is a state where all the Cooper pairs have a common phase. This is reminiscent of Ginzburg-Landau theory, which uses a complex order parameter with both an amplitude and a phase to describe the collective state of the electrons as a whole. In fact, it can be shown that the Ginzburg-Landau order parameter is proportional to the BCS energy gap ∆ [29]. The truly macroscopic nature of this phase is illustrated in the Josephson effect [30], where the zero-voltage current between two superconductors is dependent on the difference between the phases of the superconductors.

1.3 Ginzburg-Landau Theory

Ginzburg-Landau theory [25] is a phenomenological description of superconductivity. An expression for the free energy is given in terms of an order parameter, which must be 6 Superconductivity and The Josephson Eﬀect minimised with respect to the order parameter and the electromagnetic vector potential. This leads to the Ginzburg-Landau equations, which give a clear, intuitive picture of the appearance of the superconducting order parameter with temperature, and allow the treatment of spatially inhomogeneous situations, such as the Meissner eﬀect.

1.3.1 The Ginzburg-Landau Equations

The Ginzburg-Landau equations can be derived from a spatially inhomogeneous microscopic theory [29], but in this section we shall treat them in the way they were originally derived, from the phenomenological theory of second-order phase transitions. A ﬁrst-order phase transition shows a discontinuity in the entropy of a system. A second order transition shows a discontinuity in the derivative of the entropy, and we need to construct a theory that replicates this behaviour. The Ginzburg-Landau term for free energy can be viewed as an expansion of the free energy density, f, in terms of a complex order parameter ψ. The free energy must of course be real, so the ﬁrst two allowed terms are the quadratic and quartic terms:

f = α ψ 2 + β ψ 4 (1.3.1) | | | | The order parameter is then chosen to minimise the free energy. From this we see that β must be positive, otherwise the minimum of free energy would be at ψ = . For there | | ∞ to be a non-zero minimum of free energy, α must be negative. Finally, we want ψ to be | | zero above some temperature TC , and non-zero below this temperature. Thus, α must be a function of temperature, going from negative to positive at T = TC . We expand α to ﬁrst order in (T T )/T , and obtain: C − C

(T T ) f = a C − ψ 2 + β ψ 4 (1.3.2) − TC | | | | Where a and β are positive real constants. Minimising this equation (which has only two parameters) gives the behaviour of ψ around the critical temperature. Above this temperature, the free energy is minimised by an order parameter of zero, i.e. no superconductivity. Below this temperature, the order parameter has a non-zero value

2 a (TC T ) ψ = − . The subscript denotes the fact that this is the value for a bulk, or | ∞| 2β TC inﬁnite, superconductor. 1.3 Ginzburg-Landau Theory 7

Figure 1.1: The Ginzburg-Landau free energy as a function of the order parameter ψ, for the cases when: a) α is positive (above TC ) and b) α is negative (below TC ). Above TC the minimum in free energy is at ψ = 0, whereas below TC , the free energy has a minimum at a non-zero value ψ = α/2β.

Note that as 1.3.2 depends only on the magnitude of ψ, the phase of the parameter can take any value, and the free energy is independent of this. The above expression is for a perfectly homogenous superconductor of infinite size with no currents or magnetic fields. In general, the free energy is an integral over space, with 1 2 a position dependent order parameter, ψ(r). We add a kinetic energy term, 2m∗ pˆ , where ∗ the canonical momentum operator pˆ = i~ e A(r) and A(r) is the vector potential. − ∇ − c We also add the contribution of the magnetic field, H(r) = A(r), to the energy. ∇ ×

2 3 2 4 1 ~ µrµ0 2 f = dr α ψ(r) + β ψ(r) + e∗A(r) ψ(r) + H(r) (1.3.3) | | | | 2m i ∇ − 2 Z ∗ ¯µ ¶ ¯ ¯ ¯ ¯ ¯ Note that eq. 1.3.3 contains the expressions¯ m∗ and e∗, the eﬀectiv¯ e mass and eﬀective charge. These replace the usual electron mass and charge because the charge carriers are Cooper pairs rather than individual electrons. The free energy must now be minimised with respect to ψ(r) and A(r). Discarding a surface term, the minimisation condition with respect to ψ(r) leads to:

2 2 ~ ie∗ 0 = αψ + 2β ψ 2ψ A ψ (1.3.4) | | − 2m ∇ − ∗ ¯µ ~ ¶¯ ¯ ¯ ¯ ¯ This is the ﬁrst Ginzburg-Landau equation,¯ and from this¯ we see a natural length 8 Superconductivity and The Josephson Eﬀect

2 ~2 scale for the variation of the order parameter, ζ (T ) = 2m∗ α . Minimising eq. 1.3.3 with | | respect to A(r) gives:

e∗ 0 = A (ψ∗( i~ e∗A)ψ + ψ(i~ e∗A)ψ∗) (1.3.5) ∇ × ∇ × − 2m∗ − ∇ − ∇ − Inserting Ampere’s law, A = µ µ J, we ﬁnd an expression for the current J: ∇ × ∇ × r 0

e∗ J = (ψ∗( i~ e∗A)ψ + ψ(i~ e∗A)ψ∗) 2m∗ − ∇ − ∇ − 2 e∗ e∗ 2 = i~ (ψ∗ ψ ψ ψ∗) ψ A (1.3.6) − 2m∗ ∇ − ∇ − m∗ | | This is the second Ginzburg-Landau equation. We can use eq. 1.3.6 to derive the Meissner eﬀect and screening currents at the surface of a superconductor. In the limit where (ψ ψ ψ ψ ) is negligible, taking the curl of ∗∇ − ∇ ∗ both sides of the equation, we ﬁnd:

1 H = ψ 2H (1.3.7) ∇ × ∇ × −λ2 | | 2 m∗ with λ = ∗2 2 . First we consider the component of the field (HZ ) perpendicular to µrµ0e ψ | | the surface of a superconductor. We note that no current can flow in the z direction which, using J = H implies that the left hand side of eq. 1.3.7 is zero, so that H = 0, i.e. ∇ × there can be no field perpendicular to the surface of a superconductor. Considering the field parallel to the surface, H = H(z)xˆ, we use the identity H = ( H) 2H ∇×∇× ∇ ∇· −∇ to obtain:

∂2H 1 x = H (1.3.8) ∂z2 λ2 x Which give us the expression for the ﬁeld parallel to the ﬁeld of a superconductor:

z/λ Hx(z) = Hx(0)e− (1.3.9)

We see that the magnetic ﬁeld at the surface decays exponentially inside the superconductor with a characteristic length λ. This also implies the existence of a surface current: 1.3 Ginzburg-Landau Theory 9

c z/λ J (z) = H (0)e− (1.3.10) y 4πλ x This current is known as the screening current, and it is this that excludes (or screens) the magnetic field from the interior of the superconductor. Despite their apparent simplicity and phenomenological basis, the Ginzburg-Landau equations can be used to describe many of the important phenomena in superconductivity, and the 2003 Nobel Prize in physics was awarded for such work [31, 25, 32]. Phenomena that can be described include the proximity effect, the flux lattice in the mixed state of type-II superconductors and, of particular interest to us, the Josephson effect.

1.3.2 The Josephson Eﬀect in Ginzburg-Landau Theory

In 1962, Brian Josephson made the remarkable discovery that a zero-voltage current could flow between two superconductors separated by a barrier of insulator or normal metal [30]. Before discussing the microscopic theory, following [33] we shall use Ginzburg-Landau theory to deduce the existence of the effect between two bulk superconductors separated by a ‘weak link’ consisting of a short, one-dimensional bridge of superconductor. If we introduce a normalised order parameter ψ0 = ψ/ψ , then in the absence of magnetic ∞ fields, the condition for the minimisation of the free energy in one dimension is:

2 2 1 2 d ψ0 0 = ψ0 ψ0 ψ0 + ζ (T ) 2 (1.3.11) − | | 2m∗ dx We choose the boundary conditions that in both bulk superconductors (where we consider the gradient of ψ to be zero), the magnitude of the order parameter is at the value, ψ = ψ , i.e. the normalised wavefunction ψ0 = 1 at either end of the bridge. The ∞ | | Ginzburg-Landau free energy is independent of phase in the absence of gradients, and so

iφ1 iφ2 we set ψ0 = e at one end of the bridge and ψ0 = e at the other. If the coherence length ζ is much greater than the length of the bridge, L, then the 2 0 kinetic term is much greater than the other terms and we have 0 = d ψ , i.e. ψ x. The dx2 0 ∝ form for the order parameter that satisﬁes eq. 1.3.11 and the boundary conditions at each end of the bridge is:

x iφ1 x iφ2 ψ0(x) = 1 e + e (1.3.12) − L L ³ ´ 10 Superconductivity and The Josephson Eﬀect

If we now calculate the current using the standard Ginzburg-Landau current expression eq. 1.3.6, we ﬁnd:

I = I sin (φ φ ) (1.3.13) C 2 − 1

∗ 2 2~e ψ∞ A Where IC = m| ∗ | L is the critical current, the maximum supercurrent that can be sustained by a junction of area A. Although this is a highly simplistic derivation done for the particular case of a constric- tion weak link, we see a central feature of the Josephson eﬀect - the sinusoidal dependence of the current on the phase diﬀerence between the two superconducting regions.

1.4 Creation and Annihilation Operators

One of the reasons that a microscopic theory was so elusive is that superconductivity is intrinsically a coherent collective phenomenon, and that any description must be a true many-body description that describes the behaviour of the many-body wavefunction of the electrons. Whilst the wavefunction of a system of identical Bosons must be symmetric, the wavefunction of a system of identical Fermions (such as electrons) must be anti-symmetric under particle exchange:

ψ(r , r , r r ) = ψ(r , r , r r ) (1.4.1) 1 2 3 · · · i − 2 1 3 · · · i

This fact alone has many deep consequences, the most obvious of which is the Pauli exclusion principle stating that two fermions cannot occupy the same quantum state. Any many-body calculation must ensure that the wavefunction is at all times anti-symmetric.

For two fermions, with single electron states k1 and k2, the two-particle state of the system is of the form:

ψ(r , r ) = ψ (r )ψ (r ) ψ (r )ψ (r ) (1.4.2) 1 2 k1 1 k2 2 − k1 2 k2 1

Performing calculations with wavefunctions that have been symmetrised ‘by hand’ such as eq. 1.4.2 will obviously be very laborious, especially considering the vast number of 1.5 The Fermi Sea 11 electrons present in a condensed matter system. To deal with this we use a notation known as ‘second quantisation’ in which we have creation and annihilation operators ck† , ck that ↑ ↑ add electrons to the wavefunction in such a way as to ensure that the symmetrisation properties are at all times taken care of (see eg. [34]). Thus, if 0 is the vacuum state, | i then c† 0 is a state with one electron in the state k with spin up, and c† c† 0 is k1 1 k2 k1 ↑| i ↓ ↓| i a state with one electron in the state k1 and one electron in the state k2 (with correct anti-symmetry as given in eq. 1.4.2). Rather than describing the action of the operators in terms of their commutators [A,ˆ Bˆ] = AˆBˆ BˆAˆ as we would for Bosons, we describe it − in terms of the anti-commutation relations [A,ˆ Bˆ]+ = AˆBˆ + BˆAˆ:

ck†σck† 0σ0 + ck† 0σ0 ck†σ = 0

ckσck0σ0 + ck0σ0 ckσ = 0

ck†σck0σ0 + ck0σ0 ck†σ = δk,k0 δσ,σ0 (1.4.3)

It is clear from eq. 1.4.3 that ck†σck†σ = 0, i.e. two electrons cannot be put into the same state. Thus the commutation relations of the creation and annihilation operators embody, in a very direct way, the Pauli exclusion principle.

1.5 The Fermi Sea

The operators deﬁned in section 1.4 can be used to formulate the solutions to problems in many-body physics in a clear way. Perhaps the simplest problem of condensed matter physics is that of a free-electron solid (see eg. [34]). In a free electron solid, the potential that the electrons see is considered constant - the atoms in the lattice are ‘smeared out’ to form a constant background of positive charge or ‘Jellium.’ The Coulomb interaction between the electrons is also neglected, although of course the Pauli exclusion principle in not, and is taken into account by the properties of the creation and annihilation operators (eq. 1.4.3). The Hamiltonian of the system is just the sum of the single-electron energy levels:

Hfe = ²kck†σckσ (1.5.1) Xk,σ b 12 Superconductivity and The Josephson Eﬀect

In the zero temperature state of a free-electron solid, the electrons in the system occupy the lowest energy single-electron states with energy ²k. The maximum occupied energy level is known as the Fermi energy, ²F . This state can be written as:

ψ = ck† 0σ 0 (1.5.2) | i 0 | i kY,σ where the product k0 runs over the single electron states with the lowest energy. It is by no means obvious that this free-electron description will be adequate; in fact it seems unlikely given that the interactions between electrons have been neglected. When interactions are accounted for, the elementary excitations of the system are not excitations of a single electron, but instead are collective excitations of all of the electrons. The reason that Fermi-liquid theory is successful is that (close to the Fermi energy) these elementary excitations are long-lived and behave like independent particles [35, 36]. Both momentum and spin are good quantum numbers, and so the system of interacting electrons can be described by a system of non-interacting excitations. An important part of a microscopic understanding of superconductivity is the formation of pairs. In 1956, Cooper considered the behaviour of two electrons above a filled Fermi sea [26]. The Fermi sea is assumed to have no effect on the two electrons other than excluding them from the occupied levels. Cooper found that if there is an attractive potential between the electrons, they will form a bound state with opposite momenta. The bound state is formed no matter how weak the interaction between the electrons is, in contrast to the situation without the presence of the Fermi sea where bound states are only formed when the interaction is greater than some minimum value. It is perhaps surprising that there should be an attractive interaction between two charged electrons. The attraction arises from interaction of the electrons with the positive ions in the solid [37]. Although there is always a repulsive Coulomb interaction between two electrons, this is screened by the other electrons, and so the phonons (which describe the quantised vibrational modes of the ions - see [34]) can provide an attraction that outweighs the screened Coulomb repulsion. A simple picture of how this occurs is that the first electron disturbs the lattice, which in turn affects the second electron, leading to a net interaction between electrons. In the BCS treatment of superconductivity, this interaction is assumed to be attractive for electrons within a small energy region around the Fermi energy, ²k ²F < ~ωc, and | − | 1.6 BCS and BdG 13 zero otherwise. The discovery that the Fermi sea was unstable to the formation of bound pairs paved the way for Bardeen, Cooper and Schrieffer to formulate a microscopic theory of superconductivity, which we discuss in the next section.

1.6 BCS and BdG

The reduced BCS Hamiltonian [27] describes a system with interactions between paired electrons:

HBCS = (²k µ)ck†σckσ Vk,k0 ck† c† k c k0 ck0 (1.6.1) − − 0 ↑ − ↓ − ↓ ↑ Xk,σ Xk,k b The diﬃculty in solving this Hamiltonian comes from the interaction term; this term is quartic rather than quadratic. Following [28], we eliminate this problem by making the mean-ﬁeld approximation. This means that each electron only feels an interaction due to the average of all the other electrons around it. Each electron only interact with the average of all the other electrons. This is a valid approximation when the operator ck† c† k ↑ − ↓ does not deviate much from its average value, i.e. when ck† c† k ck† c† k ck† c† k . ↑ − ↓ − h ↑ − ↓i ¿ h ↑ − ↓i We write ck† c† k = ck† c† k ( ck† c† k ck† c† k ) and discard terms in the Hamiltonian ↑ − ↓ h ↑ − ↓i− h ↑ − ↓i− ↑ − ↓ that are quadratic in ( ck† c† k ck† c† k ). h ↑ − ↓i − ↑ − ↓

HBCS = (²k µ)ck†σckσ Vk,k0 ( ck† c† k c k0 ck0 + ck† c† k c k0 ck0 − − 0 h ↑ − ↓i − ↓ ↑ ↑ − ↓h − ↓ ↑i Xk,σ Xk,k b ck† c† k c k0 ck0 ) (1.6.2) −h ↑ − ↓ih − ↓ ↑i

We now introduce the pairing parameter ∆k = Vk,k0 c k0 ck0 and write the mean- k0 h − ↓ ↑i ﬁeld Hamiltonian as: P

2 HBCS = (²k µ)ck†σckσ ∆k∗c k ck + ck† c† k ∆k ∆k∗ (1.6.3) − − − ↓ ↑ ↑ − ↓ − | | Xk,σ Xk b We wish to diagonalise this Hamiltonian, i.e. to ﬁnd operators γk,σ so that the Hamil- tonian can be written Ekγk†,σγk,σ k,σ P 14 Superconductivity and The Josephson Eﬀect

We assume that these operators can be written as a linear combination of creation and annihilation operators of time reversed single-electron states, i.e. γ k, = ukc k + − ↓ − ↓ vk ck† , where uk, vk are unknown parameters. We require the elementary excitations of the ↑ ↑ system to be Fermions, and so we demand that the operators γ obey the Fermionic anti- commutation relations (eq. 1.4.3). This requirement implies that u 2 + v 2 = 1 (from | k| | k| [γ k , γ† k ]+ = 1) and also implies that γk† = ukck vk∗c† k (from [γ k , γk† ]+ = 0). − ↓ − ↓ ↑ ↑ − − ↓ − ↓ ↑ We ﬁnd the value uk, vk by noting that if the γ operators diagonalise the Hamiltonian, then:

[ Ekγk†,σγk,σ, γ k, ] = [HBCS, ukc k + vk ck† ] (1.6.4) − ↓ − ↓ ↑ ↑ Xk Equating these commutators leads directlyb to the equations for uk and vk:

1 ² u 2 = 1 + k | k| 2 E µ k ¶ 1 ² v 2 = 1 k | k| 2 − E µ k ¶ E = (² µ)2 + ∆ 2 (1.6.5) k k − | k| p The operators γ create and destroy elementary excitations of the system. Each γk†,σ creates a combination of particle and hole, which we call a quasiparticle. The form of Ek means that all quasiparticles have an excitation energy of at least ∆ so that there is a gap in the energy spectrum. The energy of the whole system is given by the sum over the energies of all the quasiparticles. From this we see that the lowest energy state, the ground state, is the state with no quasiparticles present. We ﬁnd this state by noting that if there are no quasiparticles in the ground state, then it will be destroyed by any of the quasiparticle annihilation operators γk,σ. The state deﬁned by this property is the famous BCS state:

iφ BCS = (uk + vke ck† c† k ) 0 (1.6.6) | i ↑ − ↓ | i Yk where uk and vk are now considered real and we have explicitly included the phase of vk.

In the mean-field Hamiltonian 1.6.3, each electron sees an average pairing field ∆k. If we calculate the behaviour of each electron in this field and find that averaging over all electrons returns ∆k, the solution is self-consistent. 1.6 BCS and BdG 15

∆k = Vk,k0 BCS c k0 ck0 BCS 0 h | − ↓ ↑| i Xk iφ = Vk,k0 uk0 vk0 e 0 Xk ∆k0 = Vk,k0 (1.6.7) 0 0 2Ek Xk

If we make the approximation that the interaction potential Vk,k0 is constant for ² ² < ω , then ∆ is also constant with this region, and zero outside. The gap | k − F | c k equation becomes:

² +ω V F c 1 1 = (1.6.8) 2 (² µ)2 + ∆ 2 ²F ωc k X− − | | p The chemical potential must be chosen so that the average number in the ground state is correct N¯ = BCS Nˆ BCS . This leads to the equation: h | | i

² +ω 1 F c (² µ) N¯ = 1 k − (1.6.9) 2 − (² µ)2 + ∆ 2 ²F ωc k X− − | | p These two equations must be solved self-consistently. At zero temperature, this can be done analytically (see Appendix B). The zero temperature value for the gap is:

ω 1 ~ c N V ∆ = 2~ωce− (0) (1.6.10) sinh(1/ (0)V ) ≈ N This expression for the zero-temperature gap reveals one of the reasons that a theory of superconductivity proved so hard to formulate; this expression cannot be expanded about V = 0 i.e. superconductivity can not be treated as a perturbation about the normal state. We can introduce temperature dependence by noting that the quasiparticles are non- interacting fermions. This means that they obey the statistics of a free Fermi gas. The thermally averaged expectation values of the quasiparticle occupations are given by the E /K T Fermi occupation factor, γ† γ = f , where f = 1/(e k B + 1). We have h k, σ k, σi k, σ k, σ a temperature dependent pairing parameter which is the thermally averaged expectation value Vk,k0 c k0 ck0 . This leads to the two temperature self-consistency equations: k0 h − ↓ ↑i P 16 Superconductivity and The Josephson Eﬀect

1.4

1.2

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 T/Tc

Figure 1.2: The Temperature dependence of the superconducting gap ∆(T ) measured in units of the zero-temperature gap ∆(0).

² +ω E V F c tanh k 1 = KB T (1.6.11) 2 Ek ²F ωc X− and:

²F +ωc Ek 1 (²k µ) tanh N¯ = 1 − KB T (1.6.12) 2 − Ek ²F ωc X− Although these temperature-dependent self consistency equations cannot be solved exactly, we can approximate the sum by an integral over ² and solve the equations numerically (figure 1.2). We find that the gap drops to zero at a critical temperature, TC , which when calculated is given by ∆(0) = 1.76TC . Solving the self-consistency equations means that we have a value for ∆, and thus for the parameters uk, vk. With the parameters of the ground state (eq. 1.6.6) found, we examine this state in more detail. One of the most noticeable things about the BCS state is that it does not contain an exact number of electrons, but is a superposition over terms with different number eigenvalues. This is to be expected, as the Hamiltonian eq. 1.6.3 does not commute with the number operator. This is due to the fact that we have made a mean field approximation; the Hamiltonian 1.6.1 does commute with the number operator. Although the BCS state is not an exact number eigenstate, we have chosen µ so that the expectation value of the number operator is correct, and furthermore, the 1.7 The Josephson Effect 17

fluctuation in the number operator ( Nˆ 2 Nˆ 2)/ Nˆ 2 goes to zero as the size of the h i − h i h i superconductor goes to infinity, meaning that for bulk superconductors the mean field treatment is adequate. A number eigenstate BCS, N can be extracted from the BCS state (which we denote | i BCS, φ for clarity) by integration over the phase φ. We can reverse the process by | i summing over all N:

2π iφN BCS, N = BCS, φ e− dφ | i | i Z0

+ N = uk (SBCS) l, 0 Ã ! | i Yk BCS, φ = BCS, N eiφN (1.6.13) | i | i XN + v Where the operator S = k c† c† increases the number of Cooper pairs on the BCS uk k k k ↑ − ↓ superconductor by one. P By integrating over φ, i.e. making the phase completely uncertain, we obtain an exact number eigenstate. Conversely, by making the number completely uncertain, we regain the usual BCS state where the phase is well-deﬁned. While this is not important for a bulk superconductor, where the number of Cooper pairs is of order 109 or more, this uncertainty relation between the phase and number will come into play when we consider superconductors of ﬁnite size.

1.7 The Josephson Eﬀect

The Josephson eﬀect [30] can be derived from a microscopic theory, which we shall sum- marise here. We begin with a Hamiltonian that describes two regions of superconductor, and the tunnelling between them.

H = H1 + H2 + Ht (1.7.1)

H1 and H2 are the BCS Hamiltoniansb b (eq.b 1.6.1)b for each superconductor, and Ht is a Hamiltonian that describes the tunnelling of single electrons between two metals [38]: b b b 18 Superconductivity and The Josephson Eﬀect

Ht = tk,q(c† c q + c kcq†) (1.7.2) − k − − Xk,q b The single electron levels in the two separate superconductors are labelled by k and q, with a single electron tunnelling matrix element tk,q.

The tunnelling term is treated as a perturbation on H1 + H2. We assume H1 + H2 to be solved in the sense that we assume that the Hamiltonian for each superconductor b b b b can be written in the form Ekγk†,σγk,σ where the γ’s are the quasiparticle operators. k,σ We shall use the interaction Ppicture to represent the time dependence. In the interaction picture, the time dependence of the operators is given by the commutator with the unperturbed Hamiltonian (see eg. [39]). The time evolution of the annihilation operator ck ↑ is determined by:

dck i ↑ = [H1, ck ] (1.7.3) dt ~ ↑ We wish to incorporate a voltage diﬀerenceb V across the junction. We do this by replacing H1 with H1 eV ck†,σck,σ where ck†,σck,σ measures the number of electrons − k,σ k,σ in superconductor 1. This is equivalent to setting the chemical potentials µ , µ on each b b P P 1 2 side to the junction to diﬀer by eV . If we represent the annihilation operator at zero voltage by ck and the operator at voltage V by c¯k , the time evolution of c¯k is given by: ↑ ↑ ↑

dc¯k i i2eV ↑ = [H1, c¯k ] [N1, c¯k ] dt ~ ↑ − ~ ↑ i i2eV = [Hb1, c¯k ] + c¯kb ~ ↑ ~ ↑ b (1.7.4)

i φ Substituting the trial solution c¯k = ck e 2 (φ/2 is chosen so that the phase of a Cooper ↑ ↑ pair will turn out to be φ).

i φ dck i dφ i i φ i2eV e 2 ↑ + c¯k = e 2 [H1, ck ] + c¯k dt 2 dt ↑ ~ ↑ ~ ↑ b (1.7.5)

i φ Comparing eq. 1.7.3 with eq. 1.7.4, we see that c¯k = ck e 2 is a solution if: ↑ ↑ 1.7 The Josephson Eﬀect 19

dφ 2eV = (1.7.6) dt ~ Thus we find that the rate of change of the phase across the junction is directly proportional to the voltage difference across it. This equation for the time dependence of the phase across the junction is known as the AC Josephson effect [30]. We now move on to discuss the current across the junction. In the discussion of tunnelling current, we are considering the rate of change of the number of electrons on each superconductor. To allow us to deal with this, we modify the quasiparticle operators so that each operator creates or destroys exactly one electron on each side.

+ γk† = ukck† vkS c k ↑ ↑ − − ↓ + γ† k = ukc† k + vkS ck (1.7.7) − ↓ − ↓ ↑

The current is the rate of change of electrons on one side of the junction, the expectation value of which is given by the commutator of the number operator with the Hamiltonian

[(H1 + H2), N1] = tk,q(c kcq† ck† c q). Time dependent perturbation theory gives the k,q − − − state of the system at time τ: b b b P

t i ητ 0 ψ(t) 1 e H (t0)dt0 ψ(0) (1.7.8) | i ≈  − It  | i ~ Z  −∞ b  where HIt is the tunnelling Hamiltonian in the interaction picture. The expectation value of the current operator can then be taken with the state at time t, using the form for b the quasiparticles given in eq. 1.7.7. The expectation value contains terms relating to the tunnelling of quasiparticles. Rather than get bogged down in details, we shall merely quote the term relating to the Cooper pair tunnelling [40, 41]:

2 ∞ 2e t ∆1∆2 f(²1) f(²2) I = | | 1(0) 2(0) d²1d²2 − sin φ ~ N N ²1²2 ²1 ²2 eV12 Z − − −∞ = IC sin φ (1.7.9) 20 Superconductivity and The Josephson Eﬀect where t 2 is the square of the single-electron tunnelling matrix element, which we have | | approximated as being independent of k, q and φ is the phase diﬀerence across the junction.

IC is known as the critical Josephson current, and is the maximum supercurrent the junction can support. We see we have derived an expression proportional to eq. 1.3.13. The equations eq. 1.7.6 and eq. 1.7.9 are the AC and DC Josephson eﬀects respectively, and together determine the behaviour of a Josephson junction. The current does not vanish when the voltage is reduced to zero if the superconductors on either side of the junction have diﬀerent phases. Thus a current can be passed through the junction by a current source, with no voltage appearing across the junction, up to the critical current IC . Alternatively, if the voltage across the junction is held constant, eq. 1.7.6 gives the eV 2eV evolution of phase as φ(t) = φ(0)+2 ~ t, and the current I = IC sin φ(0) + ~ t oscillates 2eV as a function of time with frequency ~ . ¡ ¢ In what follows we shall be interested in the pair tunnelling current. Rather than continue to work with a Hamiltonian describing the single-electron tunnelling from which pair tunnelling can be derived, we shall follow [42] and instead show that a pair tunnelling Hamiltonian can be derived from a single electron tunnelling Hamiltonian.

1.8 The Pair Tunnelling Hamiltonian

The Hamiltonian 1.7.1 describes two superconductors and the tunnelling between them. We wish to extract the tunnelling of Cooper pairs from this Hamiltonian, so we make a canonical transformation to eliminate the ﬁrst order terms, which correspond to single- electron tunnelling.

can K K H = e− (H1 + H2)e = (H1 + H2) + [(H1 + H2), K] b b 1 + [[(H + H ), K], K] + (1.8.1) b b b b2 b1 2 b b ·b· · b b b b First order terms are eliminated by setting Ht + [(H1 + H2), K] = 0. If we take the expectation value of this condition with a complete set of states of H0, n , we ﬁnd: b b b b | i

n K m = n H m /(E E ) (1.8.2) h | | i h | t| i m − n b b 1 From this we ﬁnd the matrix elements of the second order term 2 [Ht, K], which are:

b b 1.8 The Pair Tunnelling Hamiltonian 21

1 1 1 n H p p H m (1.8.3) 2 h | t| ih | t| i E E − E E p m p p n X µ − − ¶ b b If a pair is transferred from one side of the junction to the other, the ﬁnal state has no quasiparticle excitations and therefore has the same energy as the initial state. In the intermediate state p , an electron has been removed from state q on one side and | i − placed in state k on the other, creating a broken pair on either side. The energy of the intermediate state is then given by E E = E + E , where E are the quasiparticle p − n k q k/q energies E = (² µ)2 + ∆ : k/q k/q − k/q q

n H p p H m h | t| ih | t| i (1.8.4) E + E p k q X b b As states n and m are both states with no broken pairs and state p is a state | i | i | i with an extra electron in state k and extra hole in state q, with p H m the matrix − h | t| i element for the creation of this state, then n H p is the matrix element for the transfer h | t| i of an electron from the state q to the state k, and eq. 1.8.4 is the matrix element for the − transfer of a pair from q, q on one side to k, k: − −

tk,qt k, q Tk,q = − − − Ek + Eq t 2 = | k,q| (1.8.5) −Ek + Eq This derivation justiﬁes the use of the pair tunnelling Hamiltonian used in the rest of the thesis. Note that this matrix element is proportional to the square of the single- electron tunnelling element, as is expected - the single-electron tunnelling element has to act twice to transfer a pair across the junction. We note that this tunnelling element is not independent of k, q. However we shall approximate by an independent tunnelling element in the same way as we have approximated the BCS interaction Vk,k0 by one that is k - independent, V . We now take the Hamiltonian for the junction to be:

H = H1 + H2 + HT (1.8.6)

b b b b 22 Superconductivity and The Josephson Eﬀect

Where H1,2 are the Hamiltonians describing each superconductor (eq. 1.6.1) and the pair tunnelling is described by H : b T b

HT = T ck† c† kc qcq + cq†c† qc kck (1.8.7) − − − − − Xk,q b 1.9 Small Superconductors

When superconductivity is generally discussed (eg. the BCS wavefunction), the standard condensed matter assumption of infinite size is made. However, at very small sizes of island, various effects come into play due to the departure from bulk behaviour. Recent fabrication techniques have allowed the fabrication of extremely small particles of aluminium, with radii of order 5nm. This allowed Ralph, Black and Tinkham [43] to study the nature of superconductivity in ultrasmall grains. The grains were studied by constructing a Single Electron Transistor (SET) by placing the grain between two leads, and applying a gate voltage to the grain. The current through the grain can then be studied as a function of the bias voltage between the two leads for different applied gate voltages. These particles were so small that the single-electron energy level spacing d = 1/ (² ) N F needed to be regarded as discrete; in particular, d becomes comparable to the bulk superconducting gap, ∆. Furthermore, as the number of electrons on the island was well-defined, a canonical rather than grand-canonical description was required. An extensive review of the theory of finite size effects can be found in [44] The spectroscopic gap of the grains in these experiments is particularly significant. For grains with radii & 5nm, a spectroscopic gap could be observed when an even number of electrons was present on an island, but could not be observed for islands with an odd number of electrons. This rather striking parity effect is a clear indication that superconducting pairing correlations are taking place in these grains. For smaller grains, there was no observed spectroscopic gap, even for particles with an even number of electrons. Thus, these experiments show both the existence of superconducting pairing correlations in small grains, and the disappearance of a spectroscopic gap as the size of the grains is reduced even further and so a description of these grains requires an understanding of how superconductivity changes from the bulk to grains that are so small that superconductivity is suppressed. 1.9 Small Superconductors 23

1.9.1 The Parity Eﬀect

One of the more startling effects in small metallic grains is the parity effect; despite the fact that the number of electrons is of order 109, there are measurable differences in the properties of grains with odd and even numbers of electrons. The main difference is that the condensation energy of the two is different. This can be simply understood in the bulk limit; in an odd grain there will always be one unpaired electron, and so the difference in the condensation energies will be ∆, the energy of a single quasiparticle. In fact, this energy difference is observed in Coulomb blockade phenomena [45, 46], where the periodicity of the Coulomb oscillations in the gate voltage changes from e to 2e.

As well as a change in condensation energy, the pairing parameter (∆P , where the P = 0, 1 indicates the parity of the grain) is also parity dependent. The bulk limit (ﬁrst order in d/∆) of the parity dependence has been studied by Janko et al. [47] and von Delft et al. [48, 49]. Starting with the BCS Hamiltonian for a ﬁnite system:

H = ²kc† ck,σ V c† 0 c† 0 c k ck (1.9.1) k,σ k k − ↓ ↑ − 0 ↑ − ↓ Xk,σ Xk,k b where the labels k, k refer to generic time-reversed single electron states. In ultrasmall − grains, the number of electrons is well defined, and so the canonical distribution should really be used. However, the grand-canonical distribution is used to approximate the canonical distribution, and can in fact be regarded as a saddle-point approximation to it. Von Delft et al. treated the problem using a parity-projected grand canonical distribution. In this, the exact number of electrons present can vary, but the parity cannot. This technique allows the use of mean-field techniques, but ensures that parity effects are included.

G G 1 NL β(HL µNL) Z (µ) T r [1 ( ) ]e− − (1.9.2) P ≡ 2 § − b b b The procedure followed by von Delft et al. was to use the parity-projected partition function (eq. 1.9.2) to calculate the ﬁnite-temperature properties of the superconducting grain. Essentially, the grains are described by two gap equations, one for even grains and one for odd grains. 24 Superconductivity and The Josephson Eﬀect

1 1 = d (1 2fqP ) (1.9.3) λ 2EP − q<ωXc/d where the Fermi distribution function f = γq†γ also depends on the parity of the qP h qiP grains. The chemical potential must be chosen to give either an odd or even number of electrons on the grain, which determines the Fermi function. At zero temperature, fqp = δj0δP 0 i.e. there are no quasiparticles in the even case and only one, at the chemical potential, in the odd case. Equation 1.9.3 can be solved numerically at ﬁnite temperatures, but at T = 0 it can be evaluated exactly:

ωc 1 dω πE = tanh( e,ω ) (1.9.4) λd Eeven,ω d Z0 for even, and:

ωc 1 dω 1 d = πE (1.9.5) λd Eodd,ω tanh( odd,ω ) − Eodd,ω Z0 d for odd. In the limit where the level spacing d is much smaller than the bulk gap, ∆, the form for the parity-projected pairing parameters is:

2π∆/d ∆ (d, T = 0) = ∆(1 ∆/de− ) (1.9.6) even − p ∆ (d, T = 0) = ∆ d/2 (1.9.7) odd −

We have two different values for the pairing parameter ∆P depending on whether there are an odd or even number of electrons on the island, which also leads to two different values for the condensation energy. We find that the forms for ∆P described in eqs. 1.9.6, 1.9.7 lead to the intuitive result that, in the limit d/∆ 0, the difference in the → condensation energies is equal to the energy caused by having one unpaired quasiparticle. We also see from these results that the pairing parameter for an odd grain is always smaller than for an even grain, implying that there is a size region in which an even grain is superconducting, but an odd grain is not. In fact, as the level spacing becomes larger, the mean-field approach is no longer adequate. Nevertheless, these results serve to show that there can be a significant difference in the pairing parameters and condensation energies of grains with odd or even number of electrons. 1.9 Small Superconductors 25

Whilst these results were originally derived to treat ultrasmall superconducting grains, they are present at larger grain sizes, just smaller in magnitude. In terms of superconducting qubits, the parity effect means that there may be fluctuations in the values of EC and EJ between the values for odd and even grains. Some possible effects of this kind of fluctuation are discussed in section 2.6.2. Although a range of techniques for treating ultrasmall superconducting grains have been developed, there is in fact an exact solution of the BCS Hamiltonian for finite islands, known as the Richardson solution.

1.9.2 The Richardson Exact Solution

Recently it has come to the attention of the condensed matter community that the reduced BCS Hamiltonian for a ﬁnite superconductor:

H = ²kc† ck,σ V c† 0 c† 0 c k ck (1.9.8) k,σ k k − ↓ ↑ − 0 ↑ − ↓ Xk,σ Xk,k has an exact solution, whicb h reproduces the BCS results in the bulk limit [50, 51]. This solution was discovered by Richardson as far back as 1963 in the context of nuclear physics [52]. The existence of an exact solution to this Hamiltonian came as a great surprise to the community and we will take a while to discuss it. We can write the above Hamiltonian in terms of the Nambu spins [53]. The creation, annihilation and number operators for each electron pair k are represented as spin half operators. These are deﬁned as:

Z 1 σk = (ck† ck + c† k c k 1) 2 ↑ ↑ − ↓ − ↓ − + σk = ck† c† k ↑ − ↓ σ− = c k ck (1.9.9) k − ↓ ↑ (1.9.10)

The labels k are generic electron spin labels, rather than wavevectors. We also note Z + that σ = σ σ− 1/2, and write the reduced BCS Hamiltonian: k k k −

+ (2²kδk,k0 V )σk σk0 (1.9.11) 0 − Xk,k 26 Superconductivity and The Josephson Eﬀect

Our task is to diagonalise this Hamiltonian. We write the diagonalised Hamiltonian + and eigenstates in terms of the operators BJν, and our job is now to ﬁnd these operators.

N + H = EJνBJνBJ−ν ν=1 X b N ψ = B+ 0 (1.9.12) | N i Jν| i ν=1 Y Note that these eigenstates of the Hamiltonian are number eigenstates, and there are many diﬀerent eigenstates that all have the same number of Cooper pair on the island, N. + We shall see that the form for the operators BJν that diagonalises the Hamiltonian is:

+ + σk BJν = (1.9.13) 2²k EJν Xk − Our task is now to ﬁnd the parameters EJν. To do this we note that, if the form for the N N N N eigenstates is correct, then H B+ 0 = E B+ 0 = ε˜ B+ 0 Jν| i Jν Jν| i Jν| i µν=1 ¶ ν=1 µν=1 ¶ µν=1 ¶ b Q P Q Q N N N H B+ 0 = E B+ 0 Jν| i Jν Jν| i ν=1 ν=1 ν=1 Y X Y N bN N N [H, B+ ] 0 + B+ H 0 = E B+ 0 Jν | i Jν | i Jν Jν| i ν=1 ν=1 ν=1 ν=1 Y Y X Y b N b N [H, B+ ] 0 = ε˜ B+ 0 (1.9.14) Jν | i Jν| i ν=1 ν=1 Y Y b We need to calculate the commutator of the Hamiltonian with the product over all the + BJ operators. Calculating the commutator with just a single operator:

+ + + 1 [H, BJν] = EJνBJν + S 1 V − 2²k EJν Ã k − ! + X b 2σ σ− +S+V k k (1.9.15) 2²k EJν Xk − Following [54] we notice that this is a generalisation of the boson case. Our Nambu spin operators (sometimes referred to as hard-core boson operators) diﬀer from true bosons by one of the three commutation relations. In true bosons, we ﬁnd [bk, bk† ] = 1, whereas for 1.9 Small Superconductors 27

+ + Z our hard-core bosons, we have [σ−, σ ] = 1 2σ σ− = 2σ . If we were discussing true k k − k k − k bosons, then the commutator above (1.9.15 would only consist of the ﬁrst two terms. We could eliminate the second term by setting the condition for the EJν parameters:

1 0 = 1 V Ã − 2²k EJν ! Xk − (1.9.16)

+ + This would mean that the commutator would become [H, BJν] = EJνBJν and therefore that b

N N H B+ 0 = [H, B+ ] 0 Jν| i Jν | i ν=1 ν=1 Y Y b = b [H, B+ ] 0 Jν | i µ ν=µ X Y6 b N = E B+ 0 (1.9.17) Jµ Jν | i Ã µ ! ν=1 X Y

That is, in the true boson case, choosing the parameters EJν so that the condition + 1.9.16 is satisﬁed means that the operators BJν diagonalise the Hamiltonian. We, however, wish to consider the full Nambu spin case. This means that the ﬁnal term in 1.9.15 is not zero, and must be taken into account. Returning to the commutator + of the Hamiltonian with the product over BJν operators:

N N ν 1 N − [H, B+ ] = B+ [H, B+ ] B+ (1.9.18) Jν  Jη Jν  Jµ ν=1 ν=1 η=1 µ=ν+1 Y X Y Y b   b   + Examining our expression (eq. 1.9.15) for [H, BJν], we see that the first two terms + + 1 + (EJνB and S 1 V ) commute with the products over the remaining B Jν − 2²k EJ b J µ k − ¶ operators, and so we canPsimply move these terms past the products in eq. 1.9.20. We need to commute the third term of eq. 1.9.15 past the products as well. To do this, we first need to know its commutator with the individual operators: 28 Superconductivity and The Josephson Effect

+ + + 2σk σk− + + 1 σk [S V , BJµ] = S 2V 2²k EJν 2²k EJν 2²k EJµ Xk − Xk − − 1 σ+ σ+ = k k E E 2² E − 2² E Jν − Jµ µ k − Jν k − Jµ ¶ B+ B+ = S+2V Jν − Jµ (1.9.19) E E Jν − Jµ We can now move the third term of eq. 1.9.15 past the products:

N ν 1 + N − 2σ σ− B+ S+V k k B+ = Jη 2² E Jµ ν=1 η=1 Ã k Jν ! µ=ν+1 X Y Xk − Y N ν 1 N µ 1 + + ν 1 − − B B − B+ B+ S+2V Jν − Jµ B+ (1.9.20) Jη  Jρ E E Jτ  ν=1 η=1 µ=ν+1 ρ=ν+1 Ã Jν Jµ ! τ=µ+1 X Y X  Y − Y 

+ +   All the BJν and S operators commute, so we can collect up all the products, re-order the sums in the second term and then interchange the dummy indices µ, ν in the sums in the ﬁrst term:

N N + + + + N N 2S V BJη N N 2S V BJη η=µ η=ν = 6 6 E QE − E QE ν=1 µ=ν+1 Jν Jµ ν=1 µ=ν+1 Jν Jµ X X − X X − N N + + + + N N 2S V BJη N µ 1 2S V BJη η=µ − η=ν = 6 6 E QE − E QE ν=1 µ=ν+1 Jν Jµ µ=1 ν=1 Jν Jµ X X − X X − N N + + + + N N 2S V BJη N µ 1 2S V BJη η=ν − η=ν = 6 6 E QE − E QE µ=1 ν=ν+1 Jµ Jν µ=1 ν=1 Jν Jµ X X − X X − N + + N 2S V BJη η=ν = Q6 (1.9.21) EJµ EJν µ,ν=µ X6 −

We now have the third term of eq. 1.9.20. Writing out the commutator of the Hamil- + tonian with the product over BJ operators in full, we get: 1.9 Small Superconductors 29

N N N + + [H, BJν] = EJν BJν ν=1 ν=1 ν=1 Y X Y b N N + 1 + + S 1 V BJη − 2²k EJν ν=1 Ã k ! η=ν X X − Y6 N N N + 2V + + S BJη (1.9.22)  EJµ EJν  ν=1 µ=ν η=ν X X6 − Y6   + From eq. 1.9.14, we know that the BJ operators diagonalise the Hamiltonian if the second two terms above vanish. That is, we have the correct diagonal form for the Hamil- tonian if, for all ν:

1 2V 1 V + = 0 (1.9.23) − 2²k EJν EJµ EJν k µ=ν X − X6 −

Chapter 2

The Superconducting Charge Qubit

The progress in constructing superconducting qubits to date is reviewed. A phenomenological theoretical description of superconducting charge qubits is formulated, and the dynamics of such systems examined. We look at the eﬀect of noise in the qubit parameters on the evolution of the system, and describe new work on how discrete noise in the qubit parameters aﬀects the system.

In section 1.1 we mentioned brieﬂy why small superconducting circuits might be well suited for use as quantum bits. As with all solid-state realisations, each qubit is well- deﬁned as each is physically separated, and techniques of circuit fabrication suggest that large numbers of interacting qubits could be manufactured. In addition, superconducting qubits have the advantage over other solid-state implementations in that the superconducting gap protects the condensate from the environment to some extent. Whilst these arguments are suggestive, we require a physical theory of superconducting circuits in order to describe how they can be used as qubits, and where any possible problems arise.

2.1 Experimental Progress

Although far behind such implementations as ion trap [11, 16, 17] and NMR-based [55, 56, 57, 18] quantum computing realisations, superconducting charge qubits have developed

31 32 The Superconducting Charge Qubit quickly in recent years, as have other types of superconducting qubit. As an indication of the rapid growth of this field, superconducting charge qubits were first suggested in 1997 by Shnirman et al. [58], and by 2001 coherent oscillations were being observed [59]. A superconducting charge qubit consists of a small island of superconducting material (usually aluminium) connected via a Josephson tunnel junction to a superconducting reservoir. The island is small enough (the capacitance low enough) that the charging energy of a single Cooper pair becomes significant - it costs an appreciable amount of energy to add extra Cooper pairs to the island. The two qubit states 0 and 1 are the | i | i states where there are 0 or 1 excess Cooper pairs on the island. The separation of the two charging energy states is controlled by a gate voltage, which gives a diagonal term in the Hamiltonian (σZ in the spin-half notation where 0 = ( 0 ), 1 = ( 1 ) and σX , σY , σZ | i 1 | i 0 are the Pauli matrices). The Josephson coupling to the reservoir allows Cooper pairs to tunnel on and off the island, and provides an off-diagonal term (σX ). Oscillations between the two charge states were observed in 1999 by Nakamura et al. [59]. These oscillations are an indication that the system is behaving quantum mechanically and can be used to infer the existence of a superposition of charge states at a midpoint of the oscillation. Similar oscillations have since been seen by several groups in various superconducting systems [60, 61, 62], and spectroscopic measurements made on a range of others including those based on flux [63, 64] and phase [65, 66] Coupling between two qubits can be by use of capacitive or Josephson coupling between the superconducting islands, or through a common (or ‘bus’) inductance [67] or LC- oscillator circuit [68]. Two-qubit oscillations have now been observed in charge qubits [69] and there is spectroscopic evidence for coupling between superconducting flux qubits separated by a dis- tance of 0.7mm [70]. Recently, the first implementation of a controlled not gate (CNOT) was performed between two coupled charge qubits [71] As well as demonstrations of basic quantum behaviour, experiments illustrating the possibility of more sophisticated manipulations such as spin-echo have been performed [72]. This experiment in particular gives a clear indication of phase coherent evolution, and suggests that NMR-style techniques to reduce the effects of decoherence and noise may be viable. If superconducting qubits are to be used in quantum computers, they must also satisfy the final DiVincenzo criteria, which requires that the qubits be individually measurable. 2.2 The Josephson Equations for Small Circuits 33

In particular, in most of the current experiments, an ensemble measurement is made of the state of the qubit. The Nakamura experiments measure the charge on the qubit by placing a readout lead close to the qubit island. If the qubit is in the state with an extra Cooper pair on the island, the Cooper pair will decay through quasiparticle tunnelling to the lead. Individual Cooper pair decays cannot be detected in this manner, so each oscillation is repeated many times during one measurement so that there are enough quasiparticle decays to be detected as a current. The current on the readout lead is then proportional to the probability of the state 1 being occupied. A further problem with this measurement | i scheme is that it is ‘always on’; the tunnelling to the leads is present during the evolution of the qubit, and relies on the fact that the quasiparticle decay time is much longer than the oscillation time to ensure that the evolution is unaffected. Whilst this is adequate for demonstrations of coherent oscillations, over a long period of evolution a coupling to a measurement device will cause decoherence. Ideally, we would like to be able to decouple (or ‘turn off’) the measurement whilst the quantum evolution of the qubit takes place. A quantum computer will require single-shot measurements, i.e. measurements returning 1 or 0, rather than a probability. In theory these sorts of measurements could be done using a single electron transistor (SET) [58, 73]. In such a scheme, the presence of a Cooper pair on the qubit modifies the voltage on the central island of the transistor, and allows current to flow. A further advantage of this readout scheme is that if no bias voltage is applied to the SET, no measurement takes place, and thus the qubit dynamics are not affected. Single shot measurements are being performed in ‘quantronium’ qubit systems [60], and experiments using SETs performed [61], but as yet SETs have not been used to perform single-shot experiments on charge qubits.

2.2 The Josephson Equations for Small Circuits

Figure 2.1 shows a simple schematic of a circuit that could be used as a charge qubit. A small island of superconductor is connected via a capacitive Josephson tunnel junction to a superconducting reservoir. If the island was a bulk region, then the evolution of the phase and voltage across the Junction would be adequately described by the Josephson equations ( 1.7.6, 1.7.9 ). Although the Josephson eﬀect is an intrinsically quantum eﬀect, derived from the quantum behaviour of electrons, the equations of motion 1.7.6, 1.7.9 34 The Superconducting Charge Qubit

C IC

-q

Figure 2.1: A simple circuit diagram for a charge qubit consisting of two regions of superconductor connected by a Josephson junction with critical current IC whose capacitance is represented by a capacitor of capacitance C

are classical - the superconductors on either side of the junction are considered infinitely large, and the capacitance of the junction can be ignored. The superconductors can be in a superposition of different number eigenstates and therefore the classical equations for phase are adequate. As the size of the island becomes small, the capacitance of the junction becomes small and the charging energy large. A large charging energy means that states with too many or too few Cooper pairs are unfavourable. We can no longer sum over all number states, and so the phase is not a well-defined variable, and we need to take into account its quantum mechanical nature. In particular, we need to account for the conjugate nature of the phase and number operators.

As has been stated, the Josephson equations describe a large junction that can be adequately described classically. To describe non-classical eﬀects, there are two possible approaches we could take: a microscopic, or ‘bottom up’ approach, and a semiclassical, ‘top down’ approach. The microscopic approach is to return to the full quantum description of the superconductors, and try to re-derive a Josephson equation without making the assumption of inﬁnite size made in 1.7.

In this chapter, and in chapters 3 and 4, we consider the semiclassical approach. The motivation for this is the fact that the classical Josephson equations describe very well the behaviour of large Josephson junctions, and that effects due to the finite size of the circuit will be a modification to these. In the semiclassical approach, we ignore the quantum 2.2 The Josephson Equations for Small Circuits 35 origin of the Josephson equations, and treat them as purely classical equations of motion. We then quantise these ‘classical’ equations in the same way as we would for any other classical equations. As this semiclassical treatment, and its differences from the microscopic theory will feature heavily in later chapters of this thesis, we shall spend some time on quantising the simplest quantum Josephson circuit, shown in figure 2.1. Later we shall show (chapter 8) how these semiclassical results can be derived as a limiting form of a microscopic theory.

2.2.1 The Classical Lagrangian

We would like a quantum description of the circuit shown in ﬁgure 2.1, which consists of two regions of superconductor connected by a Josephson junction of critical current

IC and capacitance C. The charge on the capacitor is equal to the excess charge on the island, q, and so the voltage across the capacitor is given by V = q/C. The ﬁrst stage in quantising any classical equations of motion is to write the equations in terms of Lagrangian mechanics (see eg. [74]). That is, we need to ﬁnd a classical Lagrangian that leads to the equations of motion we wish to quantise. The Josephson equations derived in 1.7 are:

I = IC sin φ (2.2.1) dφ 2eV = (2.2.2) dt ~ We note that the voltage across the junction is given by V = q/C, and that the current across the junction is the rate of change of charge on the island. This leads to an equation of motion for the phase diﬀerence:

d2φ 2e = IC sin φ (2.2.3) dt2 ~C We need to ﬁnd a Lagrangian that corresponds to this equation. The Euler - Lagrange equations of the desired Lagrangian give eq. 2.2.3. Using educated guesswork, we try the Lagrangian:

2 2 1 ~ C dφ ~IC = + cos φ (2.2.4) L 2 4e2 dt 2e µ ¶ 36 The Superconducting Charge Qubit

∂ ∂ Putting this form for the Lagrangian into the Euler-Lagrange equations for φ, L ∂t ∂φ˙ − ∂ ³ ´ ∂Lφ = 0 indeed returns eq. 2.2.3.

2.2.2 The Classical Hamiltonian and the Conjugate Variables

If we choose φ to be the canonical position variable, then we ﬁnd the canonical momentum, π to be:

∂ π = L ∂φ˙ ~2C = φ˙ (2.2.5) 4e2

From eqs. 2.2.4 and 2.2.5 we get the classical Hamiltonian:

H = πφ˙ − L 1 2 = EC π EJ cos φ (2.2.6) ~2 −

Let us examine this Hamiltonian and the conjugate variables further. We know the physical meaning of the canonical position, φ, is the phase diﬀerence between the two superconducting condensates on either side of the junction. From eq. 2.2.2, we ﬁnd that the canonical momentum π can be rewritten:

~2C ~ CV π = φ˙ = 4e2 2e ~q = = ~N (2.2.7) 2e

4e2 The phase and the number of Cooper pairs are conjugate variables. EC = 2C is the charging energy of a single Cooper pair, i.e. the energy required to increase the number

~IC of Cooper pairs on the island by one. EJ = 2e is the Josephson tunnelling energy. We can calculate the equations of motion for φ and π (or N) from the Hamiltonian directly, just as we did from the Lagrangian. However, as we intend to quantise this system, let us calculate the equations of motion in another way - in terms of Poisson brackets. 2.2 The Josephson Equations for Small Circuits 37

Poisson brackets in classical mechanics are analogous to commutators in quantum mechanics. The Poisson bracket, A, B , of two functions of the canonical positions and { } momentums of the system, A, B, is deﬁned as:

∂A ∂B ∂A ∂B A, B = (2.2.8) { } ∂qi ∂πi − ∂πi ∂qi Xi µ ¶ If we assume that A, B have no explicit time dependence, then the equations of motion can be derived simply using:

dA = A, H (2.2.9) dt { }

Substituting A = φ, A = π and the Hamiltonian in eq. 2.2.6 into eq. 2.2.9 gives back the Josephson relations (2.2.1,2.2.2) and conﬁrms that we have the correct Hamiltonian for the system.

2.2.3 Quantising the Classical Josephson Equations

Now we have the correct classical Hamiltonian that described the circuit, we obtain a quantum description by promoting the conjugate variables (φ, π) to be operators (φˆ, πˆ).

Hˆ = E Nˆ 2 E cos φˆ (2.2.10) C − J

As these operators are conjugate, they have the commutation relation [φ,ˆ πˆ] = i~. Using the number operator Nˆ = πˆ/~, we have the relation [φ,ˆ Nˆ] = i (see also [75, 76]). We describe the state of the system by a wavefunction ψ(φ) which is a complex function of the canonical position φ. The operators are represented by diﬀerential operators that give the correct commutation relations:

φˆ = φ d Nˆ = i (2.2.11) dφ

This leads to a diﬀerential form for the Hamiltonian: 38 The Superconducting Charge Qubit

d 2 Hˆ = E i E cos φ (2.2.12) C dφ − J µ ¶ Having found our quantum Hamiltonian, we note that we can recover the classical equations of motion for the circuit through the use of Ehrenfest’s theorem, which gives the expectation value of the rate of change of an operator Oˆ in terms of its commutation relation with the Hamiltonian:

dOˆ i~ = [Hˆ , Oˆ] (2.2.13) * dt + D E This leads to a set of equations of motion for the expectation values of current and phase diﬀerence across the junction:

dNˆ 2e Iˆ = 2e = EJ sin φˆ h i * dt + ~ h i dφˆ 2e 2e = Nˆ (2.2.14) * dt + ~ C h i where 2e Nˆ is identiﬁed as the voltage across the capacitor, V . C h i Having found the correct classical Hamiltonian, quantised it, and found a convenient diﬀerential form for the operators, we go on to investigate how a two-level system can be extracted.

2.3 Tuneable Josephson Energy

In the previous section we formulated a quantum mechanical description of our superconducting circuit. For this system to be used as a qubit, it must behave as a two level system, and we must be able to adjust the relative charging and coupling energies in order to perform controlled operations.

In order to be able adjust the charging and coupling energies EC and EJ , we replace the circuit shown in figure 2.1 with a slightly more complicated one (figure 2.2). This circuit can be described by a Hamiltonian with a gate voltage dependent charging energy, and a Josephson energy that depends on the flux. We shall leave the dependence on the gate voltage until chapter 3, where it will be discussed in some detail. For the present, we 2.3 Tuneable Josephson Energy 39

Island

C IC IC Vg

Reservoir

Figure 2.2: A simple circuit diagram for a charge qubit with a gate voltage applied and the single Josephson junction replaced by a magnetic ﬂux dependent double-junction.

shall examine how a double-junction through which a magnetic ﬂux is passed can act like a single junction with a controllable Josephson energy. The line integral of the vector potential A around a closed contour is equal to the ﬂux passing through that contour:

Φ = A.ds (2.3.1) Zc

We write our integral around the double junction 2.3 as a sum of the contribution of two paths ca and cb. Note that the contour cb is in the opposite sense to contour ca and thus its contribution is negative.

Φ = A.ds A.ds (2.3.2) − cZa cZb

The contour passes through the centre of the superconducting electrodes and crosses the junction. We can make the junctions inﬁnitely thin, so that their contribution to the integral is essentially zero. The path goes through the centre of the electrodes, where the supercurrent density given in eq. 1.3.6 is zero, i.e. , where ( φ 2πA/Φ ) = 0, with φ ∇ − 0 the phase of the order parameter ψ = ψ eiφ. We can write the integral as: | | 40 The Superconducting Charge Qubit

1 2

Figure 2.3: A double junction consisting of a ring of superconductor broken by two junction and threaded by a ﬂux Φ. The two paths from point 1 to point 2 are labelled ca and cb.

Φ Φ Φ = 0 φ.ds 0 φ.ds (2.3.3) 2π ∇ − 2π ∇ cZa cZb

Doing the integrals, we ﬁnd that the diﬀerence in phase acquired by going from 1 to

2 by following path ca instead of cb is determined by the magnetic ﬂux threading the two paths:

2πΦ = ∆φca ∆φcb (2.3.4) Φ0 − We can now calculate the current from point 1 to point 2, passing through both paths.

πΦ πΦ I = I sin φ + + I sin φ (2.3.5) ca ca Φ cb ca − Φ µ 0 ¶ µ 0 ¶

If we assume the junctions are equal, Ica = Icb, then:

πΦ πΦ I = 2I sin ∆φ + cos (2.3.6) ca ca Φ Φ µ 0 ¶ µ 0 ¶ If we consider the double junction as a whole, we see that we have a critical Josephson current that is dependent on the magnetic ﬂux through the loop I = 2I cos πΦ . c ca Φ0 ³ ´ 2.4 The Two-Level System 41

Substituting this into the expression for Josephson energy and assuming the inductance of the circuit is negligible, we have a Josephson energy that can be adjusted by an external control parameter, the magnetic ﬁeld.

~Ica πΦ E (Φ) = cos (2.3.7) J e Φ µ 0 ¶ Note that while we have derived the flux dependence of the current using Ginzburg - Landau theory for a particular junction configuration, this dependence is essentially nothing more than the Aharonov-Bohm effect [77], and as such is a much more general phenomena than may be suggested by the above derivation. The phase of an electron wavefunction is changed by:

ie exp A.ds (2.3.8) c µ~ Z ¶ The charge carrier in a superconductor is a Cooper pair, and so the phase changes by twice this amount. This dependence of the behaviour of the qubit on the applied magnetic ﬁeld has been impressively demonstrated in the experiments of Nakamura et al. [59, 69].

2.4 The Two-Level System

The Hamiltonian of the system as a function of gate voltage and magnetic ﬂux can be written:

d 2 Hˆ = E i n E (Φ) cos φ (2.4.1) C dφ − g − J µ ¶ Where Φ is the magnetic flux through the double-junction, and we have redefined the origin of phase compared to 2.3.7 so that the cosine term has no flux dependence. For now, we note that the interaction energy between a charge 2eN and an external − voltage V will be of the form 2eNV = E Nn and assume that a gate voltage can be g − g − c g incorporated into the Hamiltonian as above with 2en = C V defining a ‘gate charge’; − g g g this shall discussed in detail in chapter 3. 42 The Superconducting Charge Qubit

Adjusting the gate voltage allows us to control the ‘zero position’ of charge on the circuit; the energy depends on how far N is from ng. In effect, ng determines at which value of N the island will be electrically neutral. In a superconducting charge qubit, E E and (for most values of n ) the charge C À J g states are a good approximation to the energy eigenstates. If we consider an island with a fixed number of Cooper pairs N, then we see that the charging energy E (Nˆ n )2 varies quadratically with n ; a plot of E against n produces C − g g C g a parabola centred on N. This is only for a fixed number of Cooper pairs, however, and the Josephson junction allows Cooper pairs to tunnel on and off the island. If ng is set close to N, then the charging energy is small and the state N will be the ground state. | i As n increases, however, the charging energy goes up, until the state N + 1 has a lower g | i energy and becomes the ground state. At this point, if the system is allowed to relax, a Cooper pair will tunnel from the reservoir to the island. This tunnelling means that the charging energy of the ground state is a periodic function of n . If we plot the charging energy for all states N against n (figure 2.4), the g | i g charging energy for each is a parabola centred on ng = N, and the parabolas cross at half-integer values of ng, where the states N and N + 1 are degenerate. The effect of the Josephson energy on this picture is to break the degeneracy of the charge states N, N + 1 at the points ng = N + 1/2 (figure 2.5). Around these points, the two lowest energy eigenstates will be the symmetric and anti- symmetric combinations of the N and N + 1 states (figure 2.6), and the other states will be separated in energy by EC . If EC is large compared to EJ , then only the N, N + 1 states will be occupied and we can regard the Cooper pair box as a two level system. That is, if we write the Hamiltonian 2.2.12 in terms of the number eigenstates:

E Hˆ = E N (N n )2 N J ( N N + 1 + N + 1 N ) (2.4.2) C | i − g h | − 2 | ih | | ih | XN XN We keep only the states N, N + 1 as the other are all suppressed by the charging energy and obtain a two-level system with the Hamiltonian:

Hˆ = E (1/2 n ) ( 1 1 0 0 ) C − g | ih | − | ih | E + J ( 1 0 + 0 1 ) (2.4.3) 2 | ih | | ih | 2.4 The Two-Level System 43

0.8

0.6 Ec

0.4

0.2

0 1 2 ng

Figure 2.4: The charging energy as a periodic function of ng. The plot is a series of parabolas centred on ng = N + 0, ng = N + 1, etc. At half-integer values of ng, the parabolas cross and the ground state changes from N to N + 1. 44 The Superconducting Charge Qubit

0 3 4 5 6

Figure 2.5: The qubit energy levels as a periodic function of ng. The degeneracy of the charging energies at half-integer values of ng is broken by the tunnelling energy EJ . The solid line represents the ground state energy, whilst the ﬁrst and second excited states are represented by the dashed and dotted lines respectively. 2.4 The Two-Level System 45

0.5

Ec 0

–0.5

–1 0.2 0.4 0.6 0.8 1 ng

Figure 2.6: The avoided crossing at half-integer values of ng. The degeneracy of the charge eigenstates (dotted lines) is broken by the Josephson tunnelling, and the energy eigenstates become the symmetric and anti-symmetric combinations of the charge states (solid lines). 46 The Superconducting Charge Qubit

Where we have discarded a term proportional to the identity. We can conveniently write this in matrix form, or in terms of the Pauli matrices:

Ech/2 EJ /2 Hˆ = −  E /2 E /2 − J − ch E E  = ch σZ + J σX 2 2 Where E = E (1 2n ) is the eﬀective charging energy, as determined by the gate ch C − g voltage. The eigenvalues E and eigenvectors v of this Hamiltonian are: § | §i

1 E = E2 + E2 § §2 ch J q1 v = E 2E§ (2.4.4) | §i ch− µ EJ ¶ Our gate voltage allows us to manipulate the Hamiltonian and change the eigenstates and eigenvalues. Two cases of particular interest are the case where we set E E , ch À J and the case where E E (but E E , so the two-level approximation is still ch ¿ J C À J valid). If the charging energy E E , then as long as we set n away from 1/2, then C À J g 1 the Hamiltonian will be diagonal and the eigenstates will be the charge states 0 and 0 with eigenvalues Ech. If, however, we set ng = 1/2, then the two charge states¡ ¢ are 1 § degenerate.¡ ¢ This degeneracy is broken by the Josephson coupling, and the eigenstates are 1 1 and 1 1 with eigenvalues E /2. √2 1 √2 1 J − § ¡This¢ abilit¡y to¢ manipulate the Hamiltonian by adjusting the gate voltage (and also the ﬂux through the double junction, discussed later in section 2.3) allows us to perform controlled quantum operations, or quantum logic gates, by applying a series of pules to the gate capacitor. We shall discuss an example of this in section 2.5.

2.5 Qubit Dynamics and the NOT Gate

From a quantum algorithmic point of view, a not gate is trivial. This is a gate that takes a qubit that is initially in the state 0 = 0 and returns the state 1 = 1 (ﬁgure 2.7). | i 1 | i 0 This is perhaps the simplest quantum¡gate,¢ acting on single qubit alone,¡ ¢ and from a algorithmic perspective merits no more than a mention. However, the notion of a ‘gate’ is 2.5 Qubit Dynamics and the NOT Gate 47

0 1

1 0

Figure 2.7: A not gate takes a state 0 and returns a state 1 , and vice versa. | i | i useful precisely because it is an abstract concept independent of the physics of the process that implements it. In order to even consider discussing ‘gates,’ we must understand the dynamics of our system thoroughly. Only when we have a complete understanding of the physics can we ignore it! In terms of the dynamics of our two-level system, implementing a not gate is not as trivial as figure 2.7 would suggest, and is certainly challenging experimentally, as the coherent quantum oscillation of a qubit must be observed. Perhaps the most famous experimental paper on superconducting charge qubits is the 1999 paper by Nakamura et al. [59] in which the coherent evolution of a charge qubit was first demonstrated. The procedure described in this paper is essentially that required to perform a not gate. In order to illustrate the difficulties and subtleties involved in even the simplest quantum operation, we shall describe this experiment in some detail. The operation can be broken down into three separate parts: Initialisation, evolution and readout. It will turn out that the complete operation can be achieved by applying a single voltage pulse to the capacitor. In the first part of the operation, initialisation, we wish to set the state of the system to be a charge eigenstate. This is done by adjusting the parameters of the Hamiltonian 2.4.3 so that one of the charge eigenstates, 0 is the ground state. We do this by setting the | i gate charge far from the degeneracy point, n = 1/2. If E E then the Hamiltonian g C À J will be essentially diagonal in the charge basis and the state 0 will be the ground state. | i At zero temperature, if we wait long enough the system will decay into this state. At low but finite temperature, the system will decay to an equilibrium state that is close to the ground state. A true NOT gate must invert any general input state, but for demonstration purposes it is easier to use a charge eigenstate as an input state. We now wish to control the evolution of the system so that the state of the system oscillates between the states 0 and 1 . We do this by applying a voltage pulse that takes | i | i 48 The Superconducting Charge Qubit

the gate charge to ng = 1/2 for the period t0 < t < t0 + ∆t. If the gate voltage is ramped rapidly enough, we can use the sudden approximation and assume that the system remains in the state 0 at the start of the pulse. | i The time evolution of a quantum system is governed by a unitary operator that follows directly from the Schr¨odinger equation:

t i ψ(t) = exp Tˆ Hˆ (t0)dt0 ψ(t0) (2.5.1) | i −~  | i tZ0   If the Hamiltonian is constant, we can ignore the time-ordering operator Tˆ and do the integral:

i Hˆ (t t0) ψ(t) = e− ~ − ψ(t ) (2.5.2) | i | 0 i

With ng set to 1/2, the diagonal parts of the Hamiltonian vanish, and the eigenstates are the states 1 1 and 1 1 . We write the state of the system at t in terms of these √2 1 √2 1 0 − states. Each of these¡ ¢ eigenstates¡ ¢ then picks up a phase factor when acted on by the time evolution operator. This phase factor depends on the energy eigenvalues which in this case are E /2: § J

i Hˆ (t t0) 1 1 ψ(t) = e− ~ − | i 1 − 1 µµ ¶ µ− ¶¶ i EJ i EJ + (t t0) 1 (t t0) 1 = e ~ 2 − e− ~ 2 − 1 − 1 µ ¶ µ− ¶ i EJ i EJ + (t t0) (t t0) e ~ 2 − e− ~ 2 − E − E = + i J (t t ) i J (t t ) (2.5.3) e ~ 2 0 + e ~ 2 0 µ − − − ¶ At the end of a pulse, at time t = t0 + ∆t, the gate voltage is again moved away from ng = 1/2, and the Hamiltonian is again diagonal in the charge basis. The ﬁnal stage of the operation is the readout stage. This is implemented simply by keeping the Hamiltonian in the charge basis, so that the probability of being in each charge state does not change. If the box is in the state with an extra Cooper pair, then this Cooper pair will decay after a time tqp via quasiparticle tunnelling to the lead, which can be detected. This decay also has the advantage of re-initialising the system to the state 0 . | i 2.5 Qubit Dynamics and the NOT Gate 49

0.8

0.6

Prob

0.4

0.2

0 1 2 3 4 (Ej t) /(2 h)

Figure 2.8: Oscillations in the probability of the qubit being in state 1 (solid line) or state i 0 (dotted line). The times required to perform a not (π/2) and to prepare a superposition | i (π/4) are indicated by vertical dashed lines.

In theory the individual quasiparticle decays could be detected, for instance by means of a single-electron transistor and of course, a single-shot measurement like this is essential for a quantum computing implementation. If instead we are interested in demonstrating coherent quantum oscillations, we only wish to know the probability that a Cooper pair was present, and furthermore would like the measurement to be as simple (experimentally) as possible. With this in mind, the oscillation is repeated many times leading to many quasiparticle decays, to get a (macroscopic) current proportional to the probability of there being an extra Cooper pair on the box. The beauty of this experiment is that a whole sequence of measurements can be performed by applying a sequence of voltage pulses of length ∆t and separated by a time trest > tqp, and that the probability of a Cooper pair being on the box can be found by making a current measurement on the lead. At the end of each pulse, the probabilities of the system being in the states 0 and | i 1 are given by 0 ψ(t + ∆t) 2 and 1 ψ(t + ∆t) 2 which evaluate to cos2( EJ ∆t) and | i |h | 0 i| |h | 0 i| 2~ 2 EJ sin ( 2~ ∆t).

EJ These probabilities oscillate with an angular frequency ~ (ﬁgure 2.8.) In Nakamura’s experiment [59], the value of EJ was altered by manipulating the magnetic ﬂux through the double junction (as in section 2.3), and the change in the oscillation frequency was observed. 50 The Superconducting Charge Qubit

To perform a not gate, we just need to set the length of the pulse to ∆t = π~/EJ . From eq. 2.5.3, the state of a system starting in state 0 = 0 is then 1 = 1 and vice versa. | i 1 | i 0 It is also worth noting that the system can be put into an¡ ¢equal superposition¡ ¢ of the charge states by setting ∆t = π /2E , in which case the state of the system becomes 1 i if it ~ J √2 1 0 was in the state 1 at the start of the pulse. This can be thought of as the square¡ ¢root of a NOT gate (√¡N¢ OT ) in the sense that performing this gate twice implements a NOT gate. There is no analogue to this gate in classical computing [78]. The oscillations between the two charge states can be taken as evidence that the system at some point passes through a superposition of the two states [79].

2.6 Decoherence and Noise

We discussed the DiVincenzo checklist in section 1.1, and mentioned that one of the advantages of solid-state qubits is that they are easily identifiable by their location and should be scalable to large numbers by means of microfabrication. Their main disadvan- tage is that they are particularly prone to decoherence. Although the superconducting energy gap goes some way to reducing this, it does not eliminate decoherence entirely. One source of error that is intrinsic to many types of qubit is the truncation of the state space. Many realisations of qubits approximate a two level system by ensuring that there is a large separation between two low-lying energy levels (the qubit states or computational subspace) and other states of the system. In atomic or ionic implementations, two low-lying electronic energy levels are chosen from the entire spectrum. In the case of our superconducting charge qubits, the computational subspace is the two charge states closest to the gate charge, and the other states are those with greater (or smaller) numbers of Cooper pairs on the island. The effect of this approximation has been considered in detail in [80], in which the number of one-qubit gates feasible before decoherence is around 4000 when the ratio of the Josephson and charging energies is 0.02. For two-qubit gates, the number of operations possible is less, but measurement on the non-computational space may reduce this error. Some techniques to reduce this problem are considered in [81]. Other sources of decoherence in charge qubits include the fact the voltage pulses applied to perform the gates will not be perfectly square but will have a finite rise time [82], coupling to charge noise in the environment, and non-zero inductances [58, 83, 84, 85, 86]. A further effect that needs consideration is the effect of noise in the control parame- 2.6 Decoherence and Noise 51 ters. In subsection 2.6.1 we review the effect of Gaussian noise, which could arise due to imperfect control of the external control parameters, and in subsection 2.6.2 we present a calculation demonstrating the effect of discrete jumps in the tunnelling energy, which could occur due to physical processes such as the parity effect (1.9.1) leading to jumps in the tunnelling energy with quasiparticle tunnelling.

2.6.1 Gaussian Noise

A simple example of the effect that noise in the parameters has on the dynamics of a qubit is the effect of Gaussian noise in the Josephson energy EJ . When the gate voltage is set to the degeneracy point (which is the relevant voltage to demonstrate oscillations, or to perform a not gate), the evolution in the presence of noise can be calculated analytically. From eq. 2.5.3, the probability of a system initialised in the state 0 evolving to the | i state 1 after a time ∆t is given by P (1) sin2( EJ ∆t). For this calculation, we consider | i 2~ a Josephson energy that fluctuates on an longer timescale that of the voltage pulse, i.e. tfluc > ∆t, so that the Josephson energy can be considered constant for each evolution. We can then find the current measured by the lead by calculating the evolution for a particular EJ and then averaging over the ensemble of all possible values of EJ , which we take to have a Gaussian distribution.

The time evolution of a charge qubit set at the degeneracy point ng = 1/2 is given in eq. 2.5.3, and the probability 1 ψ(t) 2 that the system will be in state 1 is given by |h | i| | i 2 EJ P (1) = sin ( 2~ ∆t). To calculate the evolution in the presence of noise, we average the probability over all EJ :

¯ 2 ∞ (EJ −EJ ) 1 2(∆E )2 2 EJ Pav(1) = e− J sin ∆t dEJ (2.6.1) √2π∆E 2 J Z µ ~ ¶ −∞ where we have assumed that EJ has a Gaussian distribution of width ∆EJ centred on E¯J . We do the integral by completing the square and ﬁnd:

∆t∆E 2 1 J E¯J ∆t Pav = 1 e− ~ cos (2.6.2) 2 − ³ ´ µ ~ ¶ ¯ After averaging, we have oscillations with a frequency EJ /~, determined by the average

Josephson energy, that decay exponentially with a time constant ∆EJ /~, determined by the amount of noise. 52 The Superconducting Charge Qubit

0.8

0.6

Prob

0.4

0.2

0 2 4 6 8 10 12 14 (Ej t) /(2 h)

Figure 2.9: The probability of a Cooper pair box being in the state 1 as a function of | i time, with Gaussian noise in the Josephson energy. The exponential decay envelope is plotted as dotted lines.

Although these analytic results are only for the case ng = 1/2, an exponential decay over time is a generic feature of the time evolution when the parameters have a Gaussian distribution over their average value. Numeric analysis of the eﬀect of Gaussian noise in

EJ and EC can be found in [87].

2.6.2 Discrete Noise in the Josephson Energy

Whilst Gaussian noise is a good generic model of noise in the control parameters, there are other types of noise which may have different effects on the dynamics. in this section, we present new work on a simple model of discrete noise. In this model we consider the case where the Josephson energy can fluctuate, but only between two discrete values. Again, we consider the case when the gate charge is set to ng = 1/2.

We model the ﬂuctuation in EJ by assuming that the distribution in the number of jumps between the two Josephson energy values follows the Poissonian distribution, i.e. that over a period of time t, the probability of there being j jumps in the Josephson energy βt (βt)j is given by e− j! . The calculation of the evolution of the system is found by calculating the evolution for a particular number of jumps j, and then averaging over all j. For this calculation it will be helpful to use a density matrix formulation ρ(t). The 2.6 Decoherence and Noise 53

time evolution of the density matrix is given by ρ(t) = U(t)ρ(0)U †(t) where U(t) is the time evolution operator that operates on the wavefunction in eq. 2.5.1. If ρch is the density matrix in the charge basis, then we can make a transformation to the energy eigenbasis,

ρeg:

1 1 1 1 1 ρeg = ρch 2 1 1 1 1 − −     1 1 1 ρ00 ρ01 1 1 ρeg = (2.6.3) 2 1 1 ρ ρ  1 1 − 10 11 −       As before, we make the sudden approximation, so that for a given series of jumps at times τ1, τ2 etc. the time evolution of the density matrix in the energy eigenbasis is given by:

ρeg(t) = U(t)ρeg(0)U †(t) = U (t τ ) U (τ τ )U (τ ) j − j · · · 1 2 − 1 0 1 ρ (0) U †(τ )U †(τ τ ) U †(t τ ) (2.6.4) eg 0 1 1 2 − 1 · · · j − j The oﬀ-diagonal elements become:

E τ E (τ −τ ) E (t−τ ) i J0 1 i J1 2 1 i Jj j ρ+/ (t) = ρ+/ (0)e− ~ e− ~ e− ~ (2.6.5) − − · · · and its complex conjugate. We have speciﬁed the number of jumps, and need to average over all possible times τ of those jumps by integrating over all possible times for the jumps, and then dividing by the integration time:

t t t E τ E (τ −τ ) E (t−τ ) av1 dτ1 dτ2 dτj i J0 1 i J1 2 1 i Jj j ρ+/ (t) = ρ+/ (0) e− ~ e− ~ e− ~ − − t t τ1 · · · t τj · · · Z0 τZ1 − τZj − av ρ+/ (t) = χ(t)ρ+/ (0) (2.6.6) − − where χ denotes the series of integrals over t. This means that we can write the density matrix in the charge basis as: 54 The Superconducting Charge Qubit

1 1 1 ρ+/+(0) ρ+/ (0)χ(t) 1 1 ρ = − (2.6.7) ch 2       1 1 ρ /+(0)χ∗(t) ρ / (0) 1 1 − − − − −       The problem of the time averaged evolution is now reduced to the problem of calculating the integral χ(t). For instance, in the case of a single jump.

t E τ E (t−tau ) 1 i J0 1 i J1 1 χ(t) = dτ e− ~ e− ~ t 1 Z0 E t E t i J0 i J1 e− ~ e− ~ = − (2.6.8) i(EJ0 EJ1)t/~ Ã− − !

If we started the system in the state 0 = 0 , we ﬁnd that the average probability of | i 1 being in the state 1 after a time t given that¡a¢single jump has occurred is: | i

av1 1 ~ 2EJ0t 2EJ1t ρ11 (t) = sin sin (2.6.9) 2 − 4(EJ0 EJ1)t ~ − ~ − µ ¶ So the overall time averaged evolution, including the case where there is no jump, and the case with only one jump (we assume 2 or more jumps are unlikely, i.e. β 1) becomes ¿ (ﬁgure 2.10):

av0,1 βt 2 EJ0 ρ (t) = e− 1 sin 11 − µ ~ ¶ βt 1 ~ 2EJ0t 2EJ1t +(e− βt) sin sin (2.6.10) 2 − 4(EJ0 EJ1)t ~ − ~ µ − µ ¶¶ There are two timescales in this averaging; 1/β, which determines the time between jumps, and 1/(EJ0 EJ1)/~. At times longer than the latter timescale, the average − evolution reduces to the exponential decay seen in section 2.6.1. Further terms can be found by calculating the integrals χ(t) (eq. 2.6.6) over more jumps, and the evolution of the density matrix calculated. The evolution of the density matrix element ρ11, which represents the probability that the system will be found in the state 1 , is given by: | i 2.6 Decoherence and Noise 55

0.8

0.6

0.4

0.2

10 20 30 40

Figure 2.10: The probability of a Cooper pair box being in the state 1 as a function of | i time with discrete noise in the Josephson energy. One jump is included, with parameters

EJ0 = 1, EJ1 = 0.5, β = 0.1. The Gaussian decay is plotted as dashed lines with the ideal, non-decaying oscillations denoted by dotted lines.

j av βt (βt) 1 1 ρ11(t) = e− (ρ+/ (0)χ(t) + ρ /+(0)χ∗(t)) (2.6.11) j! 2 − 2 − − Xj µ ¶

Chapter 3

The Josephson Circuit Hamiltonian

In this chapter we examine in greater detail the construction of the qubit Hamiltonian for superconducting charge qubits. We first examine the circuit diagrams of individual and multiple qubits, and consider the capacitive charging energy of these circuits. We confirm that the capacitive term found is correct by deriving the Josephson equations of motion for each junction, and finally produce a quantum Hamiltonian for each circuit.

3.1 The Capacitance of Qubit Circuits

In section 2.2 we derived a Hamiltonian for a single Josephson junction (2.2.10). This Hamiltonian was written in terms of phase. In order to derive it, we took the Josephson junction relations, i.e. the equations of motion for the phase diﬀerence between the two superconductors, found a classical Hamiltonian and a set of conjugate variables from which these equations of motion could be derived using Poisson brackets, and then elevated the conjugate variables to operators. A realistic superconducting charge qubit will be much more complicated than a single Josephson junction. For instance, a single qubit consisting of an island coupled to a reservoir will not only contain a Josephson junction, but will also need a gate voltage applied to the island in order to adjust the relative energies of the two charge states. The island we are discussing is small, and so we will have to consider its self-capacitance as well as the capacitance between it and the reservoir.

57 58 The Josephson Circuit Hamiltonian

When quantising the Josephson junction relations, we write down equations for motion of phase and voltage difference. In order to derive the Josephson relations from single- electron operators, it will be useful to have a charging term which is written in terms of total charge (number of Cooper pairs) on an island, rather than the charge stored on a capacitor. In this, we are following the spirit of [68, 47] in calculating the effective capacitance of a quantum circuit. However, we go further in that we show that this leads to a quantum Hamiltonian that reproduces the correct Josephson equations for the circuit as its equations of motion. We also show a general method of performing such calculations. Finally, a single qubit consists of an island coupled to an infinite reservoir. We would like to be sure we are taking the correct limit as we allow the size of the reservoir to become large.

3.1.1 Single Junction with Gate Voltage

We wish to consider more complicated circuits including gate voltages and self-capacitance. As the circuits become more complex, it will become convenient to solve the circuit equations using a computer. As with any calculation done on computer, we need to be especially clear of the procedure used, and so we shall go through a simpler problem, in order to illustrate the details of the calculation. After a single Josephson junction, the most simple case to consider is that of two regions of superconductor coupled by a capacitive Josephson junction, with a gate voltage between the two regions. For the present, we ignore any self capacitance. The regions of superconductor are labelled 1 i for the small islands, and r for the · · · reservoir (in this case 1, r). The charges on these regions are labelled with a single index qi. The capacitors between the regions i, j have capacitances Cij. The charge on the capacitor Cij is qij (with a double index) and the voltage across it is Vij. The gate voltage and their capacitances are labelled with a g. The sign of the charges on the capacitors and voltage sources is arbitrary, but needs to be consistent within each calculation. We wish to write the capacitive energy of the circuit in terms of the charges on the superconductors. We start with Kirchoﬀ’s laws and have one equation (3.1.1) representing the fact that the sum of the voltages around the circuit must sum to zero, and another (3.1.2) that indicates that the charge on each island is given by the sum of the charges on the capacitors connected to it. 3.1 The Capacitance of Qubit Circuits 59

q1 -q1g q1g q1r V -q1g 1g -q1r q1g qr

Figure 3.1: Single Voltage with Gate Voltage Circuit Diagram

q1r q1g + + V1g = 0 (3.1.1) C1r C1g q q = q (3.1.2) 1r − 1g 1

Note that we have no equation for the charge on the reservoir. In this calculation we are assuming that the circuit is electrically neutral overall, and so q = q . We shall 1 − r relax this restriction in the next calculation.

We solve equations 3.1.1 and 3.1.2 for q1r, q1g:

C C q q = 1r 1g 1 V 1r C + C C − 1g µ 1r 1g ¶ µ 1g ¶ C C q q = 1r 1g 1 + V (3.1.3) 1g − C + C C 1g µ 1r 1g ¶ µ 1r ¶ We write the energy due to the charges in the circuit in terms of the potential energy of the charges on the voltage source and the work done in placing the charges on the capacitors:

2 2 q1r q1g EC = + + V1gq1g (3.1.4) 2C1r 2C1g We now substitute the expressions for the charges on the capacitor (eqns. 3.1.3) into the above expression for the charging energy: 60 The Josephson Circuit Hamiltonian

-q 1w q1w

q1 -q1g q1g q1r q V w -q1g 1g -q1r q1g qr

qrw -qrw

Figure 3.2: Single junction with gate voltage and self-capacitance, which is represented by a capacitive coupling to a third region, labelled w (for ‘world’).

C C 2 1 1 E = 1r 1g (q /C V )2 + (q /C V )2 + V q C C + C 2C 1 1g − 1g 2C 1 1r − 1g 1g 1g µ 1r 1g ¶ µ 1r 1g ¶ 2 1 2 V1gC1g = (q1 V1gC1g) (3.1.5) 2(C1r + C1g) − − 2 This result shows that the gate voltage has had the effect we expected; the junction can be described with an effective capacitance, given by 1/Ceff = 1/(C1r + C1g), and a term that effectively resets the zero of capacitance. This term we call the gate charge, and denote qg = V1gC1g.

3.1.2 Single Junction with Gate Voltage and Self-Capacitance

As mentioned earlier, in the previous section we have made the restriction that our circuit is electrically neutral, i.e. that q1 + qr = 0. As we want to consider the region r as a reservoir, this is an undesirable assumption. In order to deal with this, we introduce a third region that represents ‘the rest of the world.’ This means that we can use the restriction that the entire circuit (including the ‘world,’ w), is neutral, and our qubit circuit itself (1 + r) does not have to be. The self-capacitance of the regions is represented by their coupling to w. This time we have two voltage loops around the circuit, and two charges on the regions 1, r. Again, the charge on w gives us no new information as we take the entire circuit to 3.1 The Capacitance of Qubit Circuits 61 be charge neutral.

q1r q1g + + V1g = 0 C1r C1g q q q 1w + rw 1r = 0 C1w Crw − C1r q q q = q 1g − 1r − 1w 1 q + q q = q (3.1.6) 1r rw − 1g r

We have four equations, which is enough to give the four charges on the capacitors in terms of q1, qr. Solving four simultaneous equations by hand would be time consuming, but can easily be solved on a computer, using a package such as Maple or Mathmatica (as long as we are careful - this is why we have gone through the previous calculations in such detail). As an example, the expression found for q1g is:

C1g q1g = 2 (C1w(qr + V1gC1r) + Crw( q1 + C1rV1g) + V1gC1wCrw) (3.1.7) − CΣ −

2 2 where CΣ is deﬁned by 1/CΣ = 1/(C1wC1g +C1wC1r +CrwC1g +CrwC1r +C1wCrw). There are similar expressions for the other capacitor charges. We can substitute these into the equation for charging energy, and then rearrange to get a charging energy in terms of q1 and qr.

2 2 2 2 q1r q1w qrw q1g EC = + + + + V1gq1g 2C1r 2C1w 2Crw 2C1g

(C1r + C1g + Crw) 2 (C1r + C1g + C1w) 2 = 2 (q1 V1gC1g) + 2 (qr + V1gC1g) 2CΣ − 2CΣ (C1r + C1g) + 2 (q1 V1gC1g)(qr + V1gC1g) + K (3.1.8) CΣ − where K represents constant terms. We see that we now have three terms, a term pro- 2 2 portional to q1, a term proportional to qr , and a cross term. Again, the ‘zero position’ is determined by the gate charge. We have specifically included a region, w, representing the external world, allowing us to consider the case where the charge on the circuit is non-zero. Now this has been done with general capacitances, we can take the limit where C 0 and C , i.e. the 1w → rw → ∞ 62 The Josephson Circuit Hamiltonian limit where the island is totally unconnected to the outside world, and the reservoir, r is strongly connected. 2 2 In this limit, the coefficients of qr and q1qr go to zero. The limit of the q1 coefficient is:

(C1r + C1g + Crw) Crw 2 2CΣ → 2(CrwC1g + CrwC1r + CrwC1w) 1 (3.1.9) → 2(C1g + C1r)

By comparing this result with eq. 3.1.5 we see that we have regained the previous result. This confirms that eq. 3.1.5 is indeed a valid expression for the charging energy of an isolated island connected via a capacitive tunnel junction to a reservoir. We have one further task remaining; the energy is given in terms of the charge on the island, q1, and the Josephson equations are given in terms of phase and voltage differences. This was unimportant in the previous case, as we were only considering two isolated islands, and therefore q = q = 1/2(q q ). To express our charging energy for this 1 − r 1 − r system in terms of charge differences, we substitute q = (q + q )/2 and q = (q q )/2 s 1 r 1r 1 − r into the expression for charging energy (eq. 3.1.8) before we take the limit.

3.1.3 The 2 Qubit Circuit

Now we have confirmed the form for the charging energy of a single junction, we can move onto a larger system. We wish to find the charging energy of a two-qubit system. This consists of two islands coupled to a large superconductor. As before, we wish to find a result when the reservoir has a finite size and is coupled to the outside world, and only then take the limit of an infinitely large reservoir. We also include a gate voltage on each of the two islands. Examining the circuit diagram (fig. 3.3), we see that we can define five loop equations and three equations for the charges on the islands. This gives us enough information to calculate the eight unknown capacitor charges, qij. As before, inserting these values for the capacitances into the expression for charging energy (eq. 3.1.10) gives us the charging energy in terms of the charges on the islands q1, q2, qr. 3.1 The Capacitance of Qubit Circuits 63

q q1w 2w -q 1w -q2w -q12 q12 q q -q1g 1 2 q2g

q -q -q 1g q1r 2r 2g -q q 1g -q1r q2r 2g V1g V2g q -q q1g r 2g

qrw

-qrw

Figure 3.3: Two Qubit circuit diagram. The system consists of two islands (1, 2) connected to a reservoir (r), with gate voltages applied to each. We wish to solve the problem for a reservoir of general size, and so a further region, w is included to represent the rest of the world. 64 The Josephson Circuit Hamiltonian

2 qij EC = + Vigqig (3.1.10) 2Cij i,j=i i=g X6 X6 As before, we are considering region r to be a large reservoir, and so we now wish to take the limit where q , q 0 and q . We ﬁnd, as before, that the terms 1w 2w → rw → ∞ containing the charge on the reservoir qr vanish. We ﬁnd that the charging energy only depends on the charges on the two islands.

2 2 (q1 q1g) (q2 q2g) (q1 q1g)(q2 q2g) EC = − + − + − − (3.1.11) 2Ceff 1 2Ceff 2 2Ceff 12

We have a term containing only q1, one containing only q2, and a cross term which is the capacitive coupling between the qubits. Again, the gate charges qg modify the zero-position of charge. The eﬀective capacitances are given by:

1 (C2g + C2r + C12) = 2 Ceff 1 CΣ 1 (C1g + C1r + C12) = 2 Ceff 2 CΣ 1 2C12 = 2 Ceff 12 CΣ 2 CΣ = C2gC12 + C2gC1g + C2gC12 + C12C2r

+C2rC1g + C1rC2r + C1gC12 + C1rC12 (3.1.12)

3.1.4 Multiple Islands

The same process as above can be applied to three or more islands. We ﬁnd very similar results; there is a quadratic term dependent on the charge on each island alone, and a cross term between any two charges:

2 (qi qig) (qi qig)(qj qjg) EC = − + − − (3.1.13) Ceff i Ceff ij i i,j=i X X6 We examine some of the eﬀective capacitances of a three-island system, in particular the cross terms between islands: 3.2 Quantising the Josephson Relations 65

1 2C12(C3g + C23 + C3r) = 3 Ceff 12 CΣ 1 2C12C23 = 3 (3.1.14) Ceff 13 CΣ

3 where CΣ is a term with units of capacitance cubed. Every island in the system has an interaction with every other. However, if we consider the interaction capacitances to be much smaller than the capacitances between the islands and the reservoir, we ﬁnd that eﬀective capacitances between islands that are not adjacent are correspondingly reduced.

3.2 Quantising the Josephson Relations

In section 2.2 we quantised the equations of motion of a single capacitive Josephson junction between two islands. In this section we wish to ensure that this procedure is still valid when we consider more complicated circuits. We expect the Hamiltonian of a circuit to consist of a capacitive term and a Josephson tunnelling term. If we have the correct Hamiltonian and conjugate variables, we should ﬁnd that the equations of motion given by the Poisson brackets (classically) or commutators (quantum mechanically) are just the Josephson equations for that circuit. Again, we shall consider the simplest case in detail ﬁrst before we examine more complicated circuits.

3.2.1 Single Junction with Gate Voltage

In order to ﬁnd a classical Hamiltonian for the circuit in 3.1, we shall guess a Hamiltonian based on the charging energy found in 3.1.1 and a Josephson tunnelling energy, and a pair of conjugate variables. We shall then check if this Hamiltonian is indeed correct by comparing the equations of motion derived from it to the Josephson equations for the circuit. We consider the Hamiltonian:

2 (q1 V1gC1g) H = − 2T cos (φ1 φr) (3.2.1) 2(C1r + C1g) − −

The conjugate variables are the phase φ1, and the conjugate momentum to this, N1 =

~q1/2e. The Poisson brackets for these two variables give: 66 The Josephson Circuit Hamiltonian

dH φ , H = { 1 } dπ 2e (q C V ) = 1 − 1g 1g (3.2.2) ~ 2(C1r + C1g)

Next we note that the voltage across the junction is given by V1r = q1r/C1r:

C1rC1g q1/C1g V1g V1r = − C1r + C1g C1r q V C = 1 − 1g 1g (3.2.3) C1r + C1g

Inserting this value for the voltage into equation 3.2.2, we ﬁnd:

d (φ φ ) 1 − r = φ , H dt { 1 } 1 = (2e)V1r (3.2.4) ~ This is the expected Josephson equation for rate of change of phase diﬀerence, i.e. that it is proportional to the voltage across the junction. Similarly, the rate of change of charge is given by its Poisson bracket:

d q1 2e = π1, H dt ~ { } 2T (2e) dH = − ~ − dφ1 2T (2e) = − sin(φ1 φr) (3.2.5) ~ − That is, the rate of change of charge on the island is given by the current onto the island. We have found that the Hamiltonian (eq. 3.2.1) for a single junction with a gate charge leads to the equations of motion for the charge and phase diﬀerence given by the Josephson relations for that junction. This conﬁrms that this indeed the correct classical

Hamiltonian. We obtain the quantum Hamiltonian by elevating the variables N1 and φ1 to be operators with the commutation relation [φˆ1, Nˆ1] = i. We can now use the commutation relations to ﬁnd the equations of motion for the expectation values of the operators. 3.2 Quantising the Josephson Relations 67

d (φˆ φˆ ) 2e 2e Nˆ C V h 1 − r i = h 1i − 1g 1g dt ~ C1r + C1g

d 2e Nˆ1 2T (2e) h i = − sin(φˆ1 φˆr) (3.2.6) dt ~ h − i We now move on to more complicated circuits.

3.2.2 Single Junction with Gate Voltage and Self-Capacitance

In section 3.1.2 we considered two regions of superconductor coupled to each other and to a third region representing the rest of the world. We found the charging energy in terms of the charges on the islands, and then took the limit C 0, C . We can also 1w → rw → ∞ check that the Hamiltonian gives the correct equations of motion before the limit is taken. Using the Hamiltonian H = E 2T cos(φ φ ) (we drop the hats on operators Nˆ , φˆ C − 1 − r i i for convenience) with the EC found in section 3.1.2, we ﬁnd the commutator for the phase diﬀerence:

i (2e)(C1w2eNr Crw2eN1 CrwC1gV1g C1wC1gV1g) [H, (φ1 φr)] = − − − (3.2.7) − (C1rCrw + C1rC1w + C1wCrw + C1wC1g + CrwC1g)

Again, we can compare this with the voltage across the junction, V1r = q1r/C1r.

C1r(C1w2e Nr Crw2e N1 CrwC1gV1g C1wC1gV1g) 1 V1r = h i − h i − − (3.2.8) (C1rCrw + C1rC1w + C1wCrw + C1wC1g + CrwC1g) C1r This leads to the relation we had in the last section, i.e. :

d φ1 φr 1 h − i = [H, φ1] dt i~h i 2e = V1r (3.2.9) ~ The same calculation as done in section 3.2.1 conﬁrms the second Josephson equation for this system.

3.2.3 The 2 Qubit Circuit

We can now check the Hamiltonian for the 2 qubit circuit described in section 3.1.3. We have gate voltages on islands 1, 2 and include a region w to represent the self-capacitance 68 The Josephson Circuit Hamiltonian of the other regions. The Hamiltonian consists of the charging energy (eq. 3.1.11) and terms describing the tunnelling between the regions 1,2 and r.

H = E 2T cos (φ φ ) 2T cos (φ φ ) 2T cos (φ φ ) (3.2.10) C − 1r 1 − r − 2r 2 − r − 12 1 − 2

The three tunnelling junctions are the junctions between the reservoir (r) and the islands 1, 2, and the junction between the two islands. If we check (again, using a computer program as before) we find that the rate of change of phase difference across each of these junctions is indeed proportional to the voltage difference. This is true before we take the limits C , C 0, C . The full results are unwieldy, so the results after the 1w 2w → rw → ∞ limit has been taken shall be presented here. When the limit is taken, the expression for the charging energy (eq. 3.1.11) does not contain qr and so the phase φr commutes with the Hamiltonian. The commutation relations for the three phase differences are:

(C12 + C2r + C2g) C12 [H, φ1 φr] = 2e− 2 (2e N1 V1gC1g) 2 (2e N2 + V2gC2g) − CΣ h i − − CΣ h i (C12 + C2r + C2g) C12 [H, φ2 φr] = 2e− 2 (2e N1 V1gC1g) 2 (2e N2 + V2gC2g) − CΣ h i − − CΣ h i (C2r + C2g) [H, φ1 φ2] = 2e − 2 (2e N1 V1gC1g) − CΣ h i − (C1r + C1g) + 2 (2e N2 + V2gC2g) (3.2.11) CΣ h i

2 where CΣ is as deﬁned as in eq. 3.1.12. If these are compared to the voltages across each of the three junctions (V1r = q1r/C1r etc.) we see that the equations of motion for phase calculated using the commutators are those given by the Josephson relations. We also calculate the rate of change of charge on the islands. The rate of change of the charge on island one is given by:

d 2e N1 2e h i = [H, N1] dt i~ 2T1r2e 2T122e = [ cos(φ1 φr) , N1] [ cos(φ1 φ2) , N1] − i~ h − i − i~ h − i 2T1r(2e) 2T12(2e) = sin(φ1 φr) sin(φ1 φ2) (3.2.12) − ~ h − i − ~ h − i This is a sum of the currents across the two junctions onto island one, as expected. 3.2 Quantising the Josephson Relations 69

The rate of change of charge on island one is given by the sum of the currents onto the island. We ﬁnd a similar expression for island two:

d 2e N2 2T2r(2e) 2T12(2e) h i = sin(φ2 φr) + sin(φ1 φ2) (3.2.13) dt − ~ h − i ~ h − i The sign on the current across the junction 1, 2 is reversed as a current onto island one is also a current oﬀ island two. In this chapter we have shown in detail that there is a consistent procedure quantising Josephson circuits with arbitrarily many gate capacitors, junctions, etc. and that the simplistic technique of adding the charging energy of the circuit to the tunnelling energy for each junction produces a Hamiltonian that generates equations of motion for the phase diﬀerences across each junction and the charges on each island that are equivalent to the Josephson equations for each junction. Whilst we have been dealing in this chapter with the macroscopic quantisation of the circuits in terms of phase, in later chapters we will go beyond this to a microscopic derivation of these equations.

Chapter 4

The Dynamics of a Two Qubit System

Below we examine a system consisting of two interacting charge islands, each of which is used as a qubit. In this chapter we consider the qubits as purely two-level systems and investigate their energy level structure and dynamics. We show how oscillations in the behaviour of the system with the magnetic ﬂux through the circuit can be considered in terms of virtual tunnelling of the Cooper pairs.

4.1 The Two Qubit Circuit

In the previous chapter we found the Hamiltonian of a system of two superconducting islands coupled to a reservoir. We included gate voltages on each of the two islands, and allowed for ﬂuctuations in the number of Cooper pairs on the reservoir by considering a capacitive coupling to the rest of the world before taking the limit.

H = E (nˆ n )2 + E (nˆ n )2 + E (nˆ n )(nˆ n ) C1 1 − 1g C2 1 − 1g C12 1 − 1g 2 − 2g T cos (φˆ φˆ ) T cos (φˆ φˆ ) T cos (φˆ φˆ ) (4.1.1) b − 1 1 − r − 2 2 − r − 12 1 − 2

Here we have changed notation so that EC1, T1 are the eﬀective charging energy and

Josephson tunnelling energy respectively between island one and the reservoir, and EC12,

T12 are the charging and tunnelling energies between the two islands. As before, nˆ1 and

φˆ1 are conjugate variables that have been elevated to quantum mechanical operators.

71 72 The Dynamics of a Two Qubit System

For a system to behave as a true qubit, it should be well described by a two-level system. For systems that are not intrinsically two level systems (as opposed to, say, the spin of an electron in a quantum dot) this means that the two lowest energy levels of each qubit are well separated in energy from the others, making transitions to other levels unlikely. For a charge qubit, the energies are well within the charging regime, i.e. T , T E , E . In this case, the two lowest levels for each qubit are the charge states 1 2 ¿ C1 C2 closest to n1g,n2g. Any other states have an extra charge of at least 2e and are therefore unlikely to be occupied (note that the ﬁnite probability of occupation of other levels leads to decoherence - see [80]). If we discard all but the lowest lying levels, we can write the Hamiltonian as a four by four matrix acting on the charge basis. The single qubit (i.e. uncoupled) part of the Hamiltonian can be written as:

E E T T 0 − ch1 − ch2 − 1 − 2  T1 Ech1 Ech2 0 T2  HS = − − − (4.1.2)  T 0 E + E T   2 ch1 ch2 1   − − −   0 T T +E + E   − 2 − 1 ch1 ch2    where E = E (1/2 n ) is a charging energy term that depends on the gate charge ch1 C1 − 1,g n = C V . The four basis states are the charge states 00 , 01 , 10 , 11 respectively. 1,g 1g 1g | i | i | i | i Note that if both charging energies are set to 1/2 the four charge states are degenerate. The interaction between the qubits is:

0 0 0 0

 0 EC12n2g T12 0  HI = − − (4.1.3)  0 T E n 0   12 C12 1g   − −   0 0 0 E (1 n n )   C12 − 1g − 2g    4.1.1 Energy Levels

The energy levels and of the Hamiltonian HS + HI are plotted in for the case when the inter-qubit tunnelling is zero (ﬁgure 4.1) and when it is included (ﬁgure 4.2). The parameters are chosen to be experimentally reasonable, considering those quoted in [69]. When the gate voltages are set far from the degeneracy point, the eigenstates are close to 4.1 The Two Qubit Circuit 73

100

–50

–100

0 0.2 0.4 0.6 0.8 1

Figure 4.1: The energy levels of a two qubit system as a function of n1g = n2g, when the tunnelling between the islands is zero. The parameters of the system are EC1 = EC2 = 100,

EC12 = 10, T1 = 10, T2 = 10

the charge states. The energies of the states 00 and 11 , where either both islands have | i | i an excess Cooper pair or neither do, are well separated in energy. The states with only one island having an extra Cooper pair are degenerate if the boxes are identical.

When the gate charges are set to the degeneracy position, n1g = n2g = 1/2, all four charge states of the non-interacting system would be degenerate. The degeneracy between 00 , 11 and the two states 01 and 10 is broken by the tunnelling energies T , T . The | i | i | i | i 1 2 additional degeneracy between 01 and 10 is broken by the charging interaction E . | i | i C12 The degeneracy in ﬁgure 4.2 is due to the fact that we have treated the islands as identical. Any diﬀerence in the tunnelling energies T1, T2 will break the degeneracy.

4.1.2 Two-Qubit Gates

Universal quantum computation can be achieved if we have a complete set of one-qubit gates and a two-qubit gate that can produce a maximally entangled state. In this section we show how an entangling two-qubit gate can be performed. As in section 2.5, the parameters of the qubits are set in such a way that the time evolution of the system performs the gate. We are considering a system with no tunnelling between the islands, 74 The Dynamics of a Two Qubit System

100

–50

–100

0 0.2 0.4 0.6 0.8 1

Figure 4.2: The energy levels of a two qubit system as a function of n1g = n2g, when the tunnelling between the islands is included. The parameters of the system are EC1 =

EC2 = 100, EC12 = 10, T1 = 10, T2 = 10, T12 = 30

i.e. T12 = 0, and equal tunnellings from the two islands to the reservoir T1 = T2. The evolution is plotted in ﬁgure 4.3. If the system is initialised in the state 01 then it will oscillate to the state 10 after | i | i a time determined by the diﬀerences in the eigenenergies (denoted e2 and e4 in appendix B). Halfway through the oscillation, the state will be:

1 ψ = ((1 + i) 01 (1 i) 10 ) (4.1.4) | i 2 | i − − | i

Which is a maximally entangled state. This is an alternative to the square root of imaginary swap, (√iSW AP ) gate described in [88].

4.2 A Bell Inequality

For a small superconducting island to function as a true qubit, we must be able to perform many complicated operations on it. One set of operations of particular interest is a set 4.2 A Bell Inequality 75

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 t Figure 4.3: Oscillations between state 01 (solid line), 10, (dashed line) and the states | i | i 00 and 11 (dotted lines). Halfway between oscillations, the system is in a maximally | i | i entangled state (vertical lines).

of operations to test a Bell inequality [89]. As an example of the level of control that we must have over the charge qubit, we shall go through the operations in detail.

An example of an experiment to test a Bell inequality that is particularly easy to visualise is that of a pair of spin-half particles being emitted from a source to widely separated locations. The particles are in a spin singlet state s that has zero total angular | i momentum:

1 s = ( A B B A) (4.2.1) | i √2 | ↑i | ↓i − | ↑i | ↓i where represents the state when a particle is in the spin-up eigenstate of the σ | ↑i Z operator, and represents spin down. | ↓i At the two locations (A and B), there are devices to measure the spin along a given direction (SˆA(θA), SˆB(θB)). We measure the spin along two directions, θ, θ0 at each location and take the correllators, C(θA, θB), between them. 76 The Dynamics of a Two Qubit System

z-y z+y

Figure 4.4: The Bell inequality is violated if a pair of particles in the singlet state are measured along the bases shown. Note that in three of the combinations of measurements, the bases are at 45 deg to each other, and in one of the measurements (y, (z y)/√2) they − lie at 135 deg.

C(θ , θ ) = Sˆ (θ )Sˆ (θ ) A B h A A B B i = P ( ) + P ( ) P ( ) P ( ) (4.2.2) θA,θB ↑↑ θA,θB ↓↓ − θA,θB ↑↓ − θA,θB ↓↑ where P ( ) is the probability that when the spins of the two particle are measured θA,θB ↑↑ along the directions θA, θB, both will be found in the up state.

A Bell inequality is a statement about the values of correlations in measurements made by separated observers. Any local realistic theory predicts that the measurements made at two separated locations must obey the condition:

C(θ , θ ) + C(θ0 , θ ) + C(θ , θ0 ) C(θ0 , θ0 ) 2 (4.2.3) | A B A B A B − A B | ≤

The measurements at A are made in the perpendicular directions zˆ and yˆ (with the corresponding operators SZ and SY ) , and the measurements at B are made along 1 (zˆ+yˆ) A A √2 and 1 (zˆ yˆ), (ﬁgure 4.4). √2 − 4.2 A Bell Inequality 77

1 1 C(zˆ, (zˆ + yˆ)) = s SZ , (SZ + SY ) s √2 h | A √2 B B | i = 1/√2 − 1 C(zˆ, (zˆ yˆ)) = 1/√2 √2 − − 1 C(yˆ, (zˆ + yˆ)) = 1/√2 √2 − 1 C(yˆ, (zˆ yˆ)) = +1/√2 (4.2.4) √2 −

We ﬁnd that these correllators violate the Bell inequality:

1 1 1 1 C(zˆ, (zˆ + yˆ)) + C(zˆ, (zˆ yˆ)) + C(yˆ, (zˆ + yˆ)) C(yˆ, (zˆ yˆ)) √2 √2 − √2 − √2 − = 1/√2 1/√2 1/√2 (+1/√2) | − − − − | £ 2 (4.2.5)

The violation of the Bell inequality shows in a very clear and simple way that quantum mechanics produces results that are fundamentally different from any possible classical results. As such, a demonstration of a violation of a Bell inequality in a system is an indication of quantum behaviour. Furthermore, recent experiments have offered hints of entanglement in superconducting qubit systems, [69, 62], and the ultimate proof of the existence of entanglement is the violation of a Bell inequality. Although a coupled pair of charge qubits is a very different physical system from the two spin-half particles described above (in particular, they are not spatially separated in the relativistic sense and so cannot violate relativistic locality) we can use Bell inequality demonstration as an example of the kind of control necessary to implement quantum computing protocols. In the above discussion, we stated that the particles began in a spin singlet state. We need to describe how we can produce such a state. We initialise the system in the state 01 | i by setting the gate charges to n = 1, n = 0 so that E = E (1/2 n ) = E /2 1g 2g ch1 C1 − 1g − C1 and E = E (1/2 n ) = E /2. With the tunnelling energies much smaller than the ch2 C2 − 2g C2 charging energies, the Hamiltonian of the system becomes: 78 The Dynamics of a Two Qubit System

EC1 EC2 −2 0 0 0 EC1 EC2  0 − 2− 0 0  HS = (4.2.6)  0 0 EC1+EC2 0   2   EC +EC   0 0 0 − 1 2   2   

The ground state of the system is now 01 if we wait long enough, the system will decay | i to this state. If we now suddenly set n1g = n2g = 1/2 and set the individual tunnelling energies T1, T2 to zero, then only the tunnelling between the two islands remains. If we i Ht π~ allow the system to evolve (under the evolution operator U = e− ~ ) for time t = the 4T12 ( 01 i 10 ) system evolves to the state | i− | i . We need to perform a further pulse to change this √2 state to the singlet state. If we suddenly set n1g = 1/2 and n2g = 1, and evolve for a time t = π~ then the system evolves to be in the singlet state. Ech2 SZ +SY SZ SY Z Y B B B − B We now wish to make measurements of the four operators SA , SA , 2 and 2 . In a system of superconducting charge qubits however, we can only measure in the SZ (charge) basis. To get round this we must ‘rotate’ the state of each qubit so that measuring SZ is equivalent to measuring in some other basis. We achieve this by setting the tunnelling energies to the values given in eq. 4.2.7.

Z T1t SA : ~ = 0 π SY : T1t = A ~ − 4 SZ + SY π B B : T2t = √2 ~ − 8 SZ SY π B − B : T2t = + (4.2.7) √2 ~ 4

For example, if we wish to measure qubit A in the y basis, and qubit 2 in the zB +yB √2 basis, we would set the tunnelling energies to T1t = π and T2t = π and evolve for a ~ − 4 ~ − 8 time t.

After this rotation, the system will be in one of four states: 4.2 A Bell Inequality 79

i(1 √2) − − 1  1  1 U(zˆ, (zˆ + yˆ)) s = √2 | i  1  2 2 √2    −  −  i(1 √2)  p  −    i(1 √2) − 1  1  1 U(zˆ, (zˆ yˆ)) s = √2 − | i  1  2 2 √2    −  −  i(1 √2) p − −    i(1 √2) − 1  1  1 U(zˆ, (zˆ + yˆ)) s = √2 | i  1  2 2 √2    −  −  i(1 √2) p − −    i − 1  (1 √2) 1 U(zˆ, (zˆ yˆ)) s = − − √2 − | i  (1 √2)  2 2 √2    −  −  i  p     (4.2.8)

We now measure both qubits in the zˆ basis. The probabilities of each result are given by the squares of the elements in the vectors representing the states after rotation given in eq. 4.2.8, eg. P (00) = 00 U(zˆ, 1 (zˆ + yˆ)) s 2. These, when inserted into 4.2.2 lead |h | √2 | i| to the values of the correllators given in eq. 4.2.4. Again, we ﬁnd that the Bell inequality is violated: 2√2 £ 2. We have given a brief description of how a Bell inequality violation could be observed in superconducting charge qubits. It is apparent from even this simple analysis that the series of gates that need to be applied is complicated. Current experiments are now at the point where a single 2-qubit gate is possible, and so a simple signature of 2-qubit quantum behaviour would be useful experimentally. In particular, it would be useful if we could observe a phenomenon that is intrinsically quantum mechanical but does not require the complex control of a demonstration of the violation of a Bell inequality. 80 The Dynamics of a Two Qubit System

T12

1 EC12 2

E C1 T1 EC2 T2

Figure 4.5: A two qubit circuit with tunnelling between the qubits and a magnetic ﬂux passed through the circuit

4.3 Virtual Tunnelling

In section 4.1.2 we considered a two-qubit system with no qubit-qubit tunnelling, and looked at oscillations between states that occurred when the gate voltages were set to the degeneracy positions (figure 4.3) An simple extension to the investigation of these oscillations is to consider the oscillations when tunnelling between the qubits is allowed and a magnetic field is passed through the circuit (figure 4.5). When a Cooper pair completes a circuit it picks up a phase equal to the line integral of the vector potential A around that circuit (which is equal to the flux through the circuit - see section 2.3). In terms of our Hamiltonian describing the two two-level systems, we 2π i A.ds 2πΦ Φ0 i ensure that the correct phase is picked up by including a phase factor e c = e Φ0 R in one of the tunnelling matrix elements, T12. Note that it does not matter which of the tunnelling elements the phase is included in as long as the correct phase is picked up on 4.3 Virtual Tunnelling 81 completing the loop - the overall phase is unimportant. For simplicity, we consider the gate charges to be set to 1/2, where the charge states are close to degeneracy.

0 T T 0 − 1 − 2  T1 EC12/2 T ∗ T2  H = − − − 12 − (4.3.1)  T T E /2 T   2 12 C12 1   − − − −   0 T T 0   − 2 − 1    We wish to investigate the evolution of the system over time, and in particular, how this evolution is aﬀected by the magnetic ﬁeld. As usual, we calculate the eigenvectors of the Hamiltonian, write the initial state in terms of these eigenvectors, and multiply each eigenvector by a phase factor dependent on the eigenenergy. Unfortunately, the above

Hamiltonian cannot be solved exactly (unless we force T12 to be real) and so the time evolution must be calculated numerically. Starting in the initial state 00 , the evolution of the system is as shown in Figs. 4.6 - | i 4.7: We note that not only do the probabilities oscillate in time as seen in recent papers (eg. [69]), but also with the magnetic flux through the loop. Whilst it is easy to calculate the evolution of the system numerically, we would like to find a framework for understanding these oscillations. In order to gain an understanding of this system, we set the charging interaction between the qubits EC12 to be much larger than the tunnelling either between the islands and reservoir, or between the two islands themselves. In this case, the charge states form two pairs, each of which is degenerate. The states where only one of the island has a charge on ( 01 and 01 ) have lower energy, E /2. The other pair of degenerate states | i | i − C12 ( 00 and 11 ) is higher in energy. | i | i The tunnelling elements break the degeneracies, and so the eigenstates of the Hamil- tonian are then (to first order) given by the symmetric and anti-symmetric combinations of each set of two states. That is:

00 11 v = | i § | i | 1,4i 2 01 10 v = | i § | i (4.3.2) | 2,3i 2 82 The Dynamics of a Two Qubit System

Time

Flux 3

0 Time

Flux 3

Figure 4.6: The Probability of the Two Qubit system being in the state 00 (upper plot) | i and 01 (lower plot). The horizontal axis is time and the vertical axis is the ﬂux through | i Φ0 the loop in units 2π . White represents high probability of occupation and black represents low probability. 4.3 Virtual Tunnelling 83

Time

Flux 3

0 Time

Flux 3

Figure 4.7: The Probability of the Two Qubit system being in the state 10 (upper plot) | i and 11 (lower plot). The horizontal axis is time and the vertical axis is the ﬂux through | i Φ0 the loop in units 2π . White represents high probability of occupation and Black represents low probability. 84 The Dynamics of a Two Qubit System

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 t Figure 4.8: The Probability of the Two Qubit system being in each of the four charging states. The solid line is the probability of being in 00 , the dashed line state 11 , and | i | i the probability of occupation of the other two states are just visible above the time axis.

The parameters are: EC12 = 500, T1 = T2 = T12 = 10

It is easy to see that if we start the system in the charge eigenstate 01 it will oscillate | i between this state and the state 10 . Similarly, if the system starts in the state 00 , the | i | i oscillations will be between 00 and 11 . This oscillation is plotted in ﬁgure 4.8. | i | i It is interesting to note two facts about this oscillation: the system never has a significant probability of being in state 01 or 10 , and there is no tunnelling element in the | i | i Hamiltonian between the states 00 and 11 . We have an apparent contradiction - if we | i | i were to think classically, we would say that the system must move through either 01 or | i 10 , but at no time is there any probability of being in either state. | i This is an example of the quantum mechanical phenomenon of virtual tunnelling where a system ‘passes through’ a state without occupying that state. If we think of this process physically, the system starts with no charge on either island i.e. the two excess Cooper pairs are both in the reservoir. The system tunnels to the state where there is a Cooper pair on each island without passing through the intermediate step of having only one island occupied by a Cooper pair. The frequency of these oscillations can be calculated, and is given by the diﬀerence of 4.3 Virtual Tunnelling 85

the two energy eigenvalues e1, e4. Because the Hamiltonian (eq. 4.3.1) cannot be solved analytically, we have to calculate this numerically in general. We do ﬁnd an analytic solution, however, when all the tunnelling elements are real. This means the eigenvectors and eigenvalues can be found when the phase on T12 is 0 or π, i.e. when there is an integer or half-integer number of ﬂux quanta through the circuit. The eigenvalues are:

1 1 1 e = E T + (E 2T )2 + 16(T T )2 1,4 −4 C12 ∓ 2 12 4 C12 § 12 1 § 2 1 1 1p e = E T (E 2T )2 + 16(T T )2 (4.3.3) 2,3 −4 C12 ∓ 2 12 − 4 C12 § 12 1 § 2 p To find the frequency of the oscillations between 00 and 11 , we need to find e e . | i | i 1 − 4 To first order in T/EC12, this is zero. To third order, we find:

8T T 8T (T 2 + T 2) T 4 e e = 1 2 12 1 2 + (4.3.4) 1 − 4 − E − E2 O E C12 C12 µ C12 ¶ The oscillation between 01 and 10 has a frequency that is non-zero to ﬁrst order in | i | i T/EC12:

8T T T 3 e e = 2T + 1 2 + (4.3.5) 2 − 3 − 12 E O E C12 µ C12 ¶ These differences give a good estimate of the frequencies of the oscillations, as is shown in figure 4.9. Examining the terms in the oscillation frequencies, we can see that they give an indication of the physical processes at work. There is a direct tunnelling matrix element between 01 and 10 , and T appears to first order in the oscillation frequency. There | i | i 12 is no direct tunnelling between 00 and 11 , and so the first non-zero term is 8T1T2 . This | i | i EC12 can be thought of representing the virtual tunnelling processes of a Cooper pair tunnelling through one of the intermediate states 01 or 10 . | i | i

a) 11 = Tˆ 01 = Tˆ (Tˆ 00 ) | i 2| i 2 1| i b) 11 = Tˆ 10 = Tˆ (Tˆ 00 ) | i 1| i 1 2| i

where Tˆ1 represents that part of the Hamiltonian describing tunnelling from the reservoir to island 1. The labels a and b correspond to the diagrams ﬁg. 4.10. We can interpret 86 The Dynamics of a Two Qubit System

0.8

0.6

0.4

0.2

0 1 2 3 4 5 t Figure 4.9: The change in frequency of the oscillations in the probability of being in the state 00 . The solid line is the oscillation with T = +10 and the dashed line is | i 12 the oscillation with T = 10. The vertical lines are the periods of these oscillations 12 − calculated from the energy diﬀerence in eq. 4.3.4. 4.4 Path Interference 87

a) b)

1 2 2 1

c) d)

2 2

3 1 1 3

Figure 4.10: The two second order (a, b) and third order (c, d) tunnelling processes involved in the oscillation 00 11 . | i → | i

the other terms in the frequencies in a similar manner; the third order term in e1 e4 is 2 2 − 8T12(T1 +T2 ) 2 . This represents the two third-order tunnelling processes whereby the state EC12 00 can tunnel to 11 : | i | i

c) 11 = Tˆ 10 = Tˆ (Tˆ 01 ) = Tˆ (Tˆ (Tˆ 00 )) | i 1| i 1 12| i 1 12 1| i d) 11 = Tˆ 01 = Tˆ (Tˆ 10 ) = Tˆ (Tˆ (Tˆ 00 )) | i 2| i 2 12| i 2 12 2| i These processes can be illustrated graphically (ﬁg. 4.10): We see that the oscillation frequencies consist of terms representing all possible tunnelling paths. This observation can give us some insight into the behaviour of the oscillations as a function of the magnetic ﬂux through the circuit.

4.4 Path Interference

We can view the interference fringes in ﬁg. 4.6, 4.7 as being due to interference between all possible paths the Cooper pairs could take to go from the initial to the ﬁnal state. Unfortunately, the Hamiltonian can not be solved analytically for a general phase, and so we must go to perturbation theory. Here again we have a problem, as the charging Hamiltonian on its own has doubly-degenerate energy eigenvalues, and so considering the tunnelling as a perturbation on the charging-only Hamiltonian will not be simple! Instead, 88 The Dynamics of a Two Qubit System we can take our unperturbed Hamiltonian as including both the charging interaction and the tunnelling to the reservoir from the islands, and then considering the tunnelling between islands as a perturbation on this.

0 H = H + H0 0 T T 0 0 0 0 0 − 1 − 2  T1 EC12/2 0 T2   0 0 T ∗ 0  = − − − + − 12 (4.4.1)  T 0 E /2 T   0 T 0 0   2 C12 1   12   − − −   −   0 T T 0   0 0 0 0   − 2 − 1        The unperturbed energy eigenvalues are:

1 1 e0 = E + (E )2 + 16(T T )2 1,4 −4 C12 4 C12 1 § 2 1 1p e0 = E (E )2 + 16(T T )2 (4.4.2) 2,3 −4 C12 − 4 C12 1 § 2 p The eigenvectors can be simply written in terms of the eigenenergies, for example:

1 0 0  e1/(T1 + T2) 1 v1 = − (4.4.3) 0 (e0)2 1/2 | i  e /(T1 + T2) 1  1  2 + 2 2 −  (T1+T2)  1    ³ ´   We can now do perturbation theory in T12. This allows us to investigate the oscillations as a function of phase on T12, i.e. as a function of magnetic flux through the circuit. We can, for instance, find the oscillations by expanding the energy differences in all three tunnelling energies. Using perturbation theory to first order in T12 and expanding to second order in T1, T2, we find that the eigenvalues, including the secound order perturabtions e0, are:

2 2 0 0 8T1T2 8T12(T1 + T2 ) (e1 e4) + (e10 e40 ) = 2 − − − EC12 − EC12 0 0 8T1T2 T1T2 (e2 e3) + (e20 e30 ) = (T12 + T12∗ ) 8(T12 + T12∗ ) 2 (4.4.4) − − EC12 − − EC12 4.4 Path Interference 89

3 a) 2

b) c)

3 3

2 1 1 2

Figure 4.11: The three third order tunnelling processes involved in the oscillation 01 | i → 10 . | i

These expressions agree with those given above (eqns. 4.3.4, 4.3.5) for the case when

T12 is real. It should be noted we have not done a full expansion to third order in T/EC12, but have instead expanded to different orders in the different matrix elements. To get a full third order expansion, we would have to expand to third order in T1, T2 and do third order perturbation theory in T12. Nevertheless, these calculations indicate that we can indeed think of the oscillations as interference between different tunnelling paths, and also suggests that a diagrammatic approach may be used to ‘keep track’ of the terms in the expansion. For instance, we know that the energy difference e e = (e0 e0) + (e e ) above (eq. 4.4.4) is correct 1 − 4 1 − 4 10 − 40 to third order in all tunnelling matrices as there are no other possible diagrams going from the state 00 to 11 . We also know that the correction to e e = (e0 e0) + (e e ) | i | i 2 − 3 2 − 3 20 − 30 3 is incomplete, as there is a (T12) diagram missing (figure 4.11). It is also worth noting that this idea of using diagrams to keep track of terms does not only apply to the degeneracy point (n1g, n2g = 1/2), but would work for any perturbation theory about the charging-only Hamiltonian. Away from the degeneracy point, of course, we also have the advantage that the charge states are not degenerate, making the perturbations easier to calculate.

When the condition T E is relaxed, more and more paths contribute to the ¿ C12 frequencies, but the ﬂux dependence can still be observed. We also observe oscillations 90 The Dynamics of a Two Qubit System

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1 t Figure 4.12: The oscillations of the two qubit system when the system is not in the limit T E . The solid line is probability of state 00 , the dashed line is the probability ¿ C12 | i of 11 and the grey lines are the probability of being in state 01 or 10 . The vertical | i | i | i lines are the periods of these oscillations calculated from the energies in eq. 4.3.3. The parameters are: EC12 = 50, T1 = T2 = T12 = 10 4.4 Path Interference 91

1 0.8 0.6 0.4 0.2 0 0 0 0.2 1 0.4 2 0.6 3 0.8 4 1 1.2 5 1.4 6 Figure 4.13: The Probability of the Two Qubit system being in the state 01 or 11 i.e. | i | i the probability that the ﬁrst qubit is occupied regardless of the occupation of the second. The z axis represents increasing probability.

from 00 into the states 01 and 10 , with a frequency given by e e (figure 4.12). | i | i | i 1 − 2 These oscillations are in the probabilities of the system being in the two-qubit states. To calculate them, all four states must be distinguished from each other. This would require instantaneous measurements of the charge on each island at the same time, for a range of times over the evolution. These single shot measurements may be difficult to perform at present, so the tunnelling current off each island may be measured instead. This measures the probability of a Cooper pair being on each island independently. In effect these currents will be measuring P (01) + P (11) and P (10) + P (11), where P (00) is the probability of finding the system in state 00 , etc. | i The interference fringes can still be observed, as in fig. 4.13. The importance of this is that the interference fringes are a signature of coherent quantum tunnelling that can be detected with an ensemble (current) measurement on an individual qubit, and as such may be much a more experimentally accessible method of inferring quantum behaviour than, for example, measuring a Bell inequality.

Chapter 5

Finite Superconductors as Spins: The Strong Coupling Limit

In this chapter we discuss a microscopic model of a finite region of superconductor. This model is known as the strong coupling approximation where the interaction potential is much larger than the cutoff energy. This allows us to write the well-known BCS Hamilto- nian in terms of large but finite quantum spins. After recapping the quantum mechanics of spins, we look at the solution of this Hamiltonian, both with and without the mean field approximation, which we compare to the standard BCS solution and to the Richardson solution.

5.1 The Strong Coupling Approximation

We wish to consider the behaviour of small islands of superconductor, which are small enough for ﬁnite size eﬀects to come into play. Eventually we will wish to consider the interactions between such islands, and interactions of these islands with a superconducting reservoir, as has already been done in chapter 3, by quantising the Josephson equations for each circuit. We can describe a single isolated island using the BCS Hamiltonian [27]:

ˆ H = ²kck†, σck, σ Vk, k0 ck† c† k c k0 ck0 (5.1.1) − 0 ↑ − ↓ − ↓ ↑ Xk, σ kX, k As we are discussing a ﬁnite superconducting island, the label k does not refer to a free electron wavevector but to a generic single-electron eigenstate.

93 94 Finite Superconductors as Spins

A common approximation to this equation is made by assuming the pairing potential

Vk, k0 is equal for all k, k0 in a region around the Fermi energy determined by the cutoﬀ energy ~ωc and zero outside this region. That is, Vk, k0 = V for ²k ²F < ~ωc and Vk, k0 = 0 | − | otherwise. This greatly simpliﬁes the above equation whilst retaining the essential physics.

²F +~ωc ˆ H = ²kck†, σck, σ V ck† c† k c k0 ck0 (5.1.2) − 0 ↑ − ↓ − ↓ ↑ Xk, σ kX, k ²k=²F ~ωc − To render the problem tractable, we make a further approximation; we take all the single electron energy levels ²k within the cutoﬀ region around the Fermi energy to be equal to the Fermi energy ²F . This crude approximation turns out to be equivalent to taking the strong coupling limit [90] as will be discussed further in 5.4.4. We can write the kinetic energy term as:

²F +~ωc

Ho = ²kck†, σck, σ + ²kck†, σck, σ + ²kck†, σck, σ (5.1.3) k, σ k, σ k, σ ²k<²XF ~ωc ²k=X²F ~ωc ²k>²XF +~ωc b − − where k is no longer a wavevector, but is a generic label for the single electron energy levels, with k representing the time-reversed state. Outside the cutoﬀ region, there is no − scattering and the Hamiltonian is diagonal in k. Considering only the terms within the cutoﬀ region (the dashes on the sums indicate they are taken over the region ²F ~ωc < − ²k < ²F ~ωc), we have: −

0 H = (² + (² ² )) c† c (5.1.4) o F k − F k, σ k, σ Xk, σ b The term (²k ²F ) never has a magnitude greater than ~ωc. In the strong coupling − limit, ~ωc V , so (²k ²F ) can be discarded. We are left with a Hamiltonian acting on ¿ − terms only within the cutoﬀ region:

0 0 H = ²F ck†, σck, σ V ck† c† k c k0 ck0 (5.1.5) − 0 ↑ − ↓ − ↓ ↑ Xk kX, k b Having made this approximation, we now deﬁne operators that correspond to the sums in the Hamiltonian: 5.2 Quantum Spins 95

Z 1 0 S = ck† ck + c† k c k 1 2 ↑ ↑ − ↓ − ↓ − Xk + 0 S = ck† c† k ↑ − ↓ Xk 0 S− = c k ck (5.1.6) − ↓ ↑ Xk These sums obey the usual commutation relations for quantum spins (see section 5.2) and are essentially Cooper pair number, creation and annihilation operators. Therefore, we can express our approximated Hamiltonian solely in terms of quantum mechanical spin operators. Discarding constants, the Hamiltonian becomes:

Z + Hˆ = 2² S V S S− (5.1.7) sp F − We have now rewritten our Hamiltonian in a vastly simpliﬁed form. The strong coupling limit has been studied since early days of BCS theory [91, 92, 93], for materials such as lead. More recently it has been studied, in particular, as a treatment of high temperature superconductivity (see eg. [94]) and as a description of nanoscopic superconducting grains [95] where the symmetry of the grain can cause the degeneracy of the energy levels to be very large. Obviously, we need to check the validity of this approximation and its relation to well known existing results. Before doing this, however, we shall recap the theory of quantum spins.

5.2 Quantum Spins

We begin our discussion of quantum spins with the commutation relations of our three operators (5.1.6). All the properties of quantum spins can be derived from these relations without reference to their physical ‘spin’ nature - i.e. once we have the commutation relations, we do not need any further physical intuition to ﬁnd their properties. It is important to emphasise this as we are discussing a system which is not a real quantum spin, but merely shares the same commutation relations. From our superconducting perspective, these commutation relations follow directly the deﬁnition of the operators in terms of electron creation and annihilation operators. 96 Finite Superconductors as Spins

Z 0 1 [S , S−] = [ (c† ck + c† c k 1), c k0 ck0 ] k ↑ k − ↓ 0 2 ↑ − ↓ − − ↓ ↑ kX, k 0 1 1 = [c† ck , c k0 ck0 ] + [c† c k , c k0 ck0 ] k ↑ k − ↓ 0 2 ↑ − ↓ ↑ 2 − ↓ − ↓ ↑ kX, k 0 1 = δk, k0 ( c k0 ck0 c k0 ck0 ) 0 2 − − ↓ ↑ − − ↓ ↑ kX, k 0 = c k ck − − ↓ ↑ Xk = S− (5.2.1) −

Similar calculations give the other two commutation relations.

[SZ , S+] = S+

Z [S , S−] = S− − + Z [S , S−] = 2S (5.2.2)

Next we deﬁne another operator, S2:

S+ + S 2 S+ S 2 S2 = (SZ )2 + − + − − (5.2.3) 2 2i µ ¶ µ ¶ In the interpretation of these operators as quantum spin operators, this operator mea- l l 2 Z 2 X 2 Y 2 sures the total spin 2 ( 2 + 1) and can be written S = (S ) + (S ) + (S ) , with the Hermitian operators SX = (S+ + S )/2 and SY = (S+ S )/2i. We do not need to − − − make this physical interpretation yet however, and instead note that [S2, SZ ] = 0. This means that the operators share a common set of eigenvectors:

S2 l, N = α l, N | i | i SZ l, N = β l, N (5.2.4) | i | i

We have given the eigenstates these labels because we are anticipating their physical meaning. The label l will turn out to be the number of levels within the cutoﬀ region l around the Fermi energy ²F ~ωc. In terms of physical spins, gives the size of the spin, § 2 5.2 Quantum Spins 97 sometimes denoted J. N will turn out to be the number of Cooper pairs on the island and relates to the eigenvalue of SZ through N l = m. For now though, l and N are merely − 2 labels of the eigenstates with unknown eigenvalues α and β. We take the expectation value of S2 with one of these common eigenstates:

l, N S2 l, N = l, N (SZ )2 l, N + l, N (SX )2 l, N + l, N (SY )2 l, N (5.2.5) h | | i h | | i h | | i h | | i

We have deﬁned SX and SY to be Hermitian, and therefore the expectation values of these operators are the magnitudes of the vectors SX l, N and SY l, N . The magnitude | i | i of a vector is always 0, and so we get: ≥

l, N S2 l, N l, N (SZ )2 l, N h | | i ≥ h | | i α β ≥

This means that there is a state with a minimum SZ eigenvalue. We call this state l, 0 | i and deﬁne l as its eigenvalue. Again, the eigenvalue has been chosen in anticipation of − 2 its physical meaning. As well as the minimum SZ eigenstate, we wish to ﬁnd the other eigenstates of SZ . We use a proof by induction to show that the states (S+)N l, 0 are eigenstates of the | i operator SZ . If l, N 1 is an eigenstate with eigenvalue (N 1) l , then: | − i − − 2

As the state l, 0 is an eigenstate of SZ with eigenvalue l , we have proved: | i − 2

l SZ l, N = (N ) l, N (5.2.7) | i − 2 | i Similarly, we can use another proof by induction to ﬁnd the correct normalisation constants. Consider the action of the S operator on the state l, N + 1 . Assuming that − | i 98 Finite Superconductors as Spins

S (S+)N 1 0 = ((N 1)l (N 2)(N 1))(S+)N 2 0 , which is trivially true for N = 1, − − | i − − − − − | i we ﬁnd:

So the ﬁnal line is true for all N. We use this to ﬁnd the magnitude of (S+)N 0 . | i

N + N N 1 + 1 l, 0 (S−) (S ) l, 0 = (Nl N(N 1)) l, 0 (S−) − (S )− N l, 0 h | | i − − h | | i l!(N 1)! = (Nl N(N 1)) − − − (l (N 1))! − − (l!N!) = (l (N 1))) (5.2.9) − − (l (N 1))! − − l!(N 1)! Thus, the assumption l, N 1 l, N 1 = (l (N− 1))! coupled with the observation that h − | − i − − l, 0 l, 0 = 1 proves that the magnitudes of the un-normalised states is given by: h | i

N + N (N)!l! l, 0 (S−) (S ) l, 0 = (5.2.10) h | | i (l N)! − With this value for the normalisation constant, we can deﬁne the orthonormal eigenstates of the SZ operator:

(l N)! l, N = (S+)N l, 0 − (5.2.11) | i | ir l!N! and the previous calculations also allow us to calculate the action of the spin operators:

l SZ l, N = (N ) l, N | i − 2 | i S+ l, N = (N + 1)(l N) l, N + 1 | i − | i S− l, N = pN(l (N 1)) l, N 1 | i − − | − i l l S2 l, N = p( + 1) l, N (5.2.12) | i 2 2 | i 5.3 The Majorana Representation 99

In summary, we have a set of operators that act on the state space l, N . For each | i eigenvalue of S2, l , we have l states l, N where N runs from zero to l, with eigenvalues 2 | i N l . − 2 It is worth taking a moment to re-connect these results, which we have been thinking of in terms of quantum spins, with the original problem of superconductivity in terms of single-electron creation and annihilation operators. An intuitive way of doing this can be done be relating our problem to a representation of quantum spins due to Majorana. This is the topic of the next section.

5.3 The Majorana Representation

l 1 5.3.1 Spin 2 in terms of Spin 2

l In his 1932 paper [96], Majorana noted that the Hilbert space of a spin 2 system could be 1 represented by the product of l spin 2 systems. In the description of the system above, this has a very clear meaning. The spin operators deﬁned in 5.1.6 can be viewed as a sum of operators labelled k, where k runs from 1 to l:

Z 1 σk = (ck† ck + c† k c k 1) 2 ↑ ↑ − ↓ − ↓ − + σk = ck† c† k ↑ − ↓ σ− = c k ck (5.3.1) k − ↓ ↑ (5.3.2)

Each of the k sets of operators acts like a set of spin-half operators, e.g.:

+ σk 0 k 0 k = ck† c† k 0 k 0 k | i ↑| i− ↓ ↑ − ↓| i ↑| i− ↓ σ+ = k | ↓ik | ↑ik 1 σZk 0 k 0 k = (ck† ck + c† k c k 1) | i ↑| i− ↓ 2 ↑ ↑ − ↓ − ↓ − 1 = 0 k 0 k −2| i ↑| i− ↓ 1 σ = (5.3.3) Zk| ↓ik −2| ↓ik 100 Finite Superconductors as Spins

The vacuum state l, 0 refers to the product of the l vacuum states 0 k 0 k , i.e. | i | i ↑| i− ↓ the Hilbert space of our spin J operators can be represented as the product of l spin-half down spaces, where each spin-half space refers to the absence or presence of a pair in a particular energy level. The action of the S+ operator is to add a Cooper pair to the system, in such a way that there is an equal amplitude of the pair being in any of the l levels. In spin language, this gives:

l, 0 = | i | ↓↓↓ · · · ↓↓i S+ l, 0 = + + + | i | ↓↓↓ · · · ↓↑i | ↓↓↓ · · · ↑↓i · · · | ↑↓↓ · · · ↓↓i (S+)2 l, 0 = + + + | i | ↓↓↓ · · · ↑↑i | ↓↓↑ · · · ↑↓i · · · | ↑↑↓ · · · ↓↓i (S+)N l, 0 = N! (5.3.4) | i | ↓↓ · · · ↓ ↑ · · · ↑isym N (l N) − | {z } | {z } where the sym subscript refers to a symmetric sum of all possible states with N up-spins, divided by N!. This last equation gives a description of the l, N states in terms of spin- | i half particles. A general state will be a superposition of such states. It turns out that a general state can also be represented by l spin-half states, but with general orientation, instead of just up or down, i.e.:

The complex numbers zk∗ are the coeﬃcients of spin-up for each of the spin-half states. They will be of particular interest in a special set of states: coherent states.

5.3.2 The Coherent State

When considering the harmonic oscillator we deﬁne a state as the eigenstate of the destruction operator a†. This is the coherent state (see eg. [97]): 5.3 The Majorana Representation 101

∞ αN (a )N 0 α = † | i (5.3.6) | i N! NX=0 We can define a similar state for our system. This is usually referred to as a (spin) coherent state [97], but it should be noted that whilst the harmonic oscillator coherent state (eq. 5.3.6) is defined as an eigenstate of the destruction operator, the analogous spin state is not an eigenstate because the sum does not run to infinity.

l 1 (z S+)N l, 0 l, z = ∗ | i (5.3.7) | i √2l N! NX=0 The coherent state is a state where all the component spin-half particles are pointing in the same direction, i.e. all the points on the Majorana sphere (see section 5.3.3) converge θ iφ l to one location. The coherent state with z = cot 2 e is the state with maximal spin 2 along the axis θ, φ. That is, if we deﬁne an operator¡ ¢ that gives the component of spin in the direction (θ, φ), then:

l S l, z = l, z (5.3.8) θ,φ| i 2| i We can easily see that this is true for the axis z, i.e. θ = 0 or θ = π. These cases correspond to the limits z 0 and z respectively, and examining the deﬁnition | | → | | → ∞ above (eq. 5.3.7) show that in these cases the coherent state tends to l, 0 or l, l . | i | 2 i In a real spin, these states are important states; they are the maximal eigenstates of spin in each direction. The states l, 0 and l, l are only two such states, determined by | i | 2 i the deﬁnition of the z-axis, and they can be transformed into any other coherent state by a rotation of co-ordinate basis.

5.3.3 The Majorana Sphere

The values zk in eq. 5.3.5 have a geometric interpretation. Consider the Bloch Sphere, as a representation of a spin-half particle. Here every pure state l = 1, ψ (i.e. j = 1 , ψ ) | i | 2 i can be uniquely represented by a point on the sphere, with the north pole corresponding to spin up, the south pole to spin down, and all other states lying on the surface of the sphere. Each state can also be speciﬁed by a complex number z. If the complex plane passes through the equator of the sphere, and a line is passed from a point on the sphere to the 102 Finite Superconductors as Spins

a) b)

Figure 5.1: The Majorana Sphere. Each state is represented by a series of points on the surface of a sphere. In the coherent state, all the points coincide (a), whereas for Z Z eigenstates, they are distributed around the sphere at a constant ‘latitude.’

north pole, then it crosses the plane at a single point z. Thus, a complex number can represent a point on the sphere, and thus a spin-half state.

l = 1, ψ = a + a | i 1| ↑i 0| ↓i θ i φ θ i φ = cos e− 2 + sin e 2 (5.3.9) 2 | ↑i 2 | ↓i 1 iφ z = θ e tan 2 a = 0 (5.3.10) a1

l In the Majorana representation [96, 98], a spin 2 state is represented by l spin-half states, i.e. l points on the sphere (ﬁgure 5.1), or l projections, zk, onto the complex plane (ﬁgure 5.2).

Any state, then, can be represented by a series of points on a sphere. In particular, for the coherent state, all the zk are equal, and all the points on the Majorana Sphere coincide. This point corresponds to the direction in which the state is a maximal spin state.

As mentioned above, in the case of the maximal and minimal eigenstates of the SZ operator, the value of z is zero or inﬁnity. The point z = 0 on the complex sphere corresponds to the south pole, and z = corresponds to the north pole. Thinking about ∞ the states in this way makes it very clear that l, 0 and l, l are merely coherent states, | i | 2 i and occupy a special position only in reference to a chosen co-ordinate basis. 5.4 Mean Field Spin Solution 103

Figure 5.2: Each point on the Majorana sphere can be represented by its projection onto the complex plane, z.

5.4 Mean Field Spin Solution

We wish to ensure that our approximation to eq. 5.1.2 is valid. One way to increase our confidence in the model is to compare its solutions to solutions to the full model. In particular, we will find the mean-field solution of 5.1.7 and compare it to the mean- field solution of 5.1.2. When making the mean field approximation, we assume that the operators S are close to their expectation values. Writing S = S + (S S ), we § § h §i § − h §i can rewrite the Hamiltonian 5.1.7. The mean field Hamiltonian is derived by discarding terms to second order or higher in (S S ). § − h §i

Z + + + H = 2(² µ)Sˆ V S + S S S− + S− S− (5.4.1) sp F − − h i − h i h i − h i Z + + + 2(² µ)Sˆ V ¡ S S−¡+ S− S ¢¢ ¡S S− ¡ ¢¢ b ≈ F − − h i h i − h ih i 2 Z + ∆ = 2(² µ)Sˆ ∆S¡ ∆∗S− + | | ¢ (5.4.2) F − − − V The term S+ S is a constant, and will be unimportant for most purposes, and we h ih −i shall discard it below when we find the mean field solution. We will need to take note of it later, however, when we compare the mean field solution to the exact solution (section 5.5).

Z + H = 2(² µ)Sˆ ∆S ∆∗S− MF F − − − Z X Y = 2ξ Sˆ S (∆ + ∆∗) S i(∆ ∆∗) (5.4.3) b f − − − Now the use of this spin-model becomes evident. Observing the above Hamiltonian, 104 Finite Superconductors as Spins

Zo o n Zn

Yo Xo

Z + Figure 5.3: Rotation of basis of spin operators. The operators Soˆ , Soˆ , Soˆ− refer to a Z + particular basis oˆ. We can consider another basis, nˆ with three operators Snˆ , Snˆ , Snˆ−. The operators corresponding to any basis can be written as a linear combination of any other basis. we see it is a linear combination of SZ , SY and SX operators. This means that the whole Hamiltonian is proportional to a spin operator in a diﬀerent direction. We have reduced the problem of ﬁnding the eigenvalues of eq. 5.4.3 to the problem of rotating a quantum mechanical spin.

5.4.1 Rotation of Spin Operators

+ The operators SZ , S , S− respectively represent a measurement, raising and lowering of angular momentum with respect to a particular frame of reference, say oˆ. For clarity, we Z + call these three original operators Soˆ , Soˆ , Soˆ−. Other unit vectors, nˆ, define other frames of reference, and therefore each direction nˆ Z + has three operators, Snˆ , Snˆ , Snˆ−, associated with it. An operator in one frame of reference can be expressed as a combination of the three operators in any other frame of reference. We want to find a diagonal representation for our Hamiltonian; we want to find a Z direction nˆ, such that HMF = γSnˆ . If we can do this, the eigenstates of the Hamiltonian are the spin states l, N . | nî We need to find the raising and lowering operators for this new reference frame. Again 5.4 Mean Field Spin Solution 105

+ Z + we express the new operator Snˆ in terms of the old operators Soˆ , Soˆ , Soˆ−, with d, e, f the unknown parameters.

+ ˆZ + Snˆ = dSoˆ + eSoˆ + fSoˆ− (5.4.4)

Z + Inserting the expressions for Snˆ and Snˆ (5.4.4, 5.4.3) into the commutation relation for spins will allow us to ﬁnd d, e and f:

Z + + [Snˆ , Snˆ ] = Snˆ 1 Z + Z + Z + [ (2ξ S ∆S ∆∗S−), (dS + eS + fS−)] = dS + eS + fS− γ F oˆ − oˆ − oˆ oˆ oˆ oˆ oˆ oˆ oˆ Z + Z + (2∆∗e 2∆f)S + (2ξ e + ∆d)S + ( ∆∗d 2ξ f)S− = γ(dS + eS + fS−) − oˆ F oˆ − − F oˆ oˆ oˆ oˆ (5.4.5)

Equating the coeﬃcients in the commutation relation gives three simultaneous equations:

(2∆∗e 2∆f) = γd − (2ξF e + ∆d) = γe

( ∆∗d 2ξ f) = γf (5.4.6) − − F

Solving these gives:

+ Z ∆ + ∆∗ S = d(S + − S + − S−) nˆ 2ξ γ 2ξ + γ F − F γ = (2ξ )2 + 4 ∆ 2 (5.4.7) F | | p Where we note that γ is equal to twice the quasiparticle energy at the Fermi level, EF . + A similar process gives an expression for Snˆ− (the Hermitian conjugate of Snˆ ). Finally, we + Z can use the commutation relation [Snˆ , Snˆ−] = 2Snˆ to find the parameter d. Summarising, we found that we could express a mean-field Hamiltonian in terms of Z + spin operators Soˆ , Soˆ , Soˆ−. This Hamiltonian is then recognized as a spin operator in another direction, nˆ. The spin commutation relations allow us to find the raising and lowering operator as well. 106 Finite Superconductors as Spins

Z Z + γS = 2ξ S S ∆ S−∆∗ nˆ F oˆ − oˆ − oˆ + Z + Snˆ = dSoˆ + eSoˆ + fSoˆ− d = ∆ /E | | | | F d∆ e = − 2ξ 2E F − F d∆ f = − ∗ 2ξF + 2EF (5.4.8)

We can write the above transformation as a matrix. Using the deﬁnitions for d, e, and f given in eq. 5.4.8, we have:

Z Z 2ξF /γ ∆/γ ∆∗/γ Soˆ Snˆ + +  d e f   Soˆ  =  Snˆ  (5.4.9)  d f e   S   S   ∗ ∗ ∗   oˆ−   nˆ−        Note that this matrix is not unitary. This is due to the fact the matrix acts on the Z + Z X Y operators S , S , S− rather than on S , S , S . The inverse of the above matrix is given by:

Z Z 2ξF /γ d∗/2 d/2 Snˆ Soˆ + +  2∆∗/γ e∗ f   Snˆ  =  Soˆ  (5.4.10)  2∆/γ f e   S   S   ∗   nˆ−   oˆ−        The fact that the above matrix is not unitary is just a normalisation problem that arises because the raising and lowering operators are deﬁned as S = (SX iSY ) without § § a normalisation constant of 1/√2. For completeness, we can write the rotation operators in terms of the operators S Z , SX , SY .

Z Z 2ξF /γ (∆ + ∆∗)/γ i(∆ ∆∗)/γ Soˆ Snˆ ∗ ∗ ∗ − ∗ ∗ (d+d ) (e+f+e +f ) (e f e +f ) X = X (5.4.11)  2 2 i − −2   Soˆ   Snˆ  (d d∗) (e+f e∗ f ∗) (e f+e∗ f ∗) Y Y  − − − − −   S   S   2i 2i 2   oˆ   nˆ        This matrix is indeed unitary, as expected. 5.4 Mean Field Spin Solution 107

5.4.2 The Ground State

Our identiﬁcation of the S operators as spin operators allows us to recognise that the mean-ﬁeld Hamiltonian is proportional to an operator that measures the spin in some direction, nˆ. We can simply rewrite the Hamiltonian (eq. 5.4.3) as:

Z + H = 2(² µ)S ∆S ∆∗S− MF F − oˆ − oˆ − oˆ Z b = γSnˆ (5.4.12) From this, it is obvious that the ground state will be the minimum eigenstate of spin in the direction nˆ and that the eigenvalue will be γ l . This does not tell us how to write − 2 Z,+, the ground state in terms of our original operators Soˆ −, however. To do this we utilise the results of section 5.3.2, and note that the minimum spin state in the direction nˆ will be Z,+, a coherent state in terms of the operators Soˆ −. We still do not know what the direction nˆ is, and thus do not know the parameter z in equation 5.3.7. We ﬁnd the parameter z (and therefore ﬁnd nˆ) by noting that if a state is the minimal spin state along the direction nˆ, it will be destroyed by the lowering operator Snˆ− along that direction i.e. S− l, z = 0. nˆ | i

S− l, z = 0 nˆ | i l + N Z ∆ + ∆∗ 1 (z∗Soˆ ) l, 0 = Soˆ + Soˆ + Soˆ− | i 2ξF + γ 2ξF γ √2l N! µ − ¶ NX=0 l l (z )N ∆(z )N 1 = (N ) ∗ + ∗ − Ã − 2 N! (2ξF + γ)(N 1)! NX=0 − ∆ (N + 1)(l N)(z )N+1 + ∗ − ∗ (S+)N l, 0 (5.4.13) (2ξ γ)(N + 1)! oˆ | i F − ! If this state is to be equal to zero, the coeﬃcients must be zero for all N. For example, considering the coeﬃcient of the l = 0 term:

l ∆ z l + ∗ ∗ = 0 −2 2ξ γ F − l ∆ z l = ∗ ∗ 2 2ξ γ F − 2ξF γ z∗ = − (5.4.14) 2∆∗ 108 Finite Superconductors as Spins

Therefore we have found that the ground state of eq. 5.4.3 is a coherent state (eq.

2ξF γ Z 5.3.7) with z = 2∆− . Directly calculating the action of Snˆ on this state conﬁrms it is indeed the ground state.

H l, z = γSZ l, z MF | i nˆ | i l = γ l, z −2 | i l = 4(² µ)2 + 4∆2 l, z −2 F − | i = l p(² µ)2 + ∆2 l, z (5.4.15) − F − | i p Remembering that we wish to compare the solution to the mean-ﬁeld spin Hamiltonian, it may not be immediately obvious how the spin coherent state above relates to the BCS state (eq. 1.6.6). We can, however, rewrite the coherent state so that the connection is clear:

l 1 (z S+)N l, 0 l, z = ∗ | i | i √2l N! NX=0 1 = (1 + z∗ ck† c† k ) l, 0 (5.4.16) (1 + z 2)l ↑ − ↓ | i | | Yk Above we see that the coherenpt state is equal to the BCS state when all the parameters uk and vk are equal.

5.4.3 Self Consistency

With the ground state of the mean-ﬁeld Hamiltonian found, we need to choose the parameters ∆ and µ to give self-consistency and also the correct value for the average number of particles. We want to choose µ such that the ground state α gives l, z Nˆ l, z = N¯. The section h | | i above showed that the state l, z is the ground state of the Hamiltonian. We calculate | i ˆ Z l the expectation value of the (pair) number operator, N = Soˆ + 2 with this state:

N¯ = l, z Nˆ l, z h | | i l ² µ N¯ = 1 F − (5.4.17) 2 2 2 Ã − (²F µ) ∆ ! − − | | p 5.4 Mean Field Spin Solution 109

We also have to chose ∆/V to be equal to the expectation value of the S+ operator:

+ ∆∗ = V l, z S l, z h | | i l ∆ = V ∗ (5.4.18) 2 (² µ)2 ∆ 2 F − − | | In full BCS theory, these equations inpvolve a sum over levels k, and we have two integral equations. For our simpliﬁed spin model, the fact that the ²k are all equal means that we can trivially sum over k, giving two coupled algebraic relations instead of integral equations. Thus we can evaluate ∆ and µ exactly. We ﬁnd:

∆ 2 = V 2N¯(l N¯) (5.4.19) | | − ² µ = V (l/2 N¯) (5.4.20) F − − With these values we have the mean ﬁeld solution for a given average number N¯. Including the constant that we discarded from 5.4.1, we have:

2 Z + ∆ H = 2(² µ)S ∆S ∆∗S− + | | SMF F − oˆ − oˆ − oˆ V ∆ 2 = γSZ + | | nˆ V (5.4.21)

When we insert 5.4.19 and 5.4.20 into γ and ∆, we obtain:

5.4.4 Comparison to the BCS Theory

In order to check the viability of neglecting the diﬀerences between electron energy levels

²k, we solve the gap equations for a general BCS case, and then take the limit where 110 Finite Superconductors as Spins all the levels have the same energy. To take this limit, we consider a top-hat density of states centred around ²F with width w and height l/w. This allows the limit where the bandwidth w 0 to be taken whilst keeping a constant number of levels, l. As shown in → appendix B, at zero temperature the integrals can be calculated exactly. We ﬁnd:

(x + 1)2 w 2 (x 1)2 ∆2 = (ξ )2 1 + − 1 (5.4.23) F r (x 1)2 − 2 (x + 1)2 − µ − ¶ ³ ´ µ ¶ l w w w N¯ = r (ξ + )2 + ∆2 + (ξ )2 + ∆2 (5.4.24) r w 2 − F r 2 F r − 2 µ r r ¶

2w N¯ With x = e V lr and n = r . These two equations can be solved to give ∆ and ξ . lr F r When the limit V/w (the strong coupling limit) is taken, we find that we obtain the → ∞ same results as given by the spin model, (5.4.19, 5.4.20). Evidently, the fact that this spin representation reproduces the BCS results, albeit only in the strong coupling limit, lends credibility to our simplified model, and allows us to consider the situations we wish to describe, namely those of small regions of superconductor, or ‘Cooper Pair Boxes’, coupled to a superconducting reservoir. Before we discuss these systems, let us examine the results we obtain for our single isolated island if we do not make the mean field approximation.

5.5 Solving the Spin Hamiltonian Exactly

Examining the mean field solution of our Hamiltonian confirms that, at least in the mean field, our spin Hamiltonian behaves regularly, and the results are those that would be obtained by solving the full (k-dependent) problem and then taking the limit ~ω V . ¿ However, we do not need to take the mean field approximation to solve the Hamiltonian. In fact the solution is rather simple.

Z + Hˆ = 2² S V S S− (5.5.1) sp F − Equation 5.5.1 is the strong coupling limit of the BCS Hamiltonian. It is easy to see that this is diagonal in l, N , the eigenstates of SZ , as the S operator reduces by one | i − the number of Cooper Pairs (N) on the island, and the raising operator increases it by + one. Equations 5.2.12 allow us to ﬁnd the eﬀect of the operator S S− on an eigenstate of 5.5 Solving the Spin Hamiltonian Exactly 111

SZ . This is: S+S l, N = N(l N + 1) l, N . With this we can ﬁnd the action of the −| i − | i Hamiltonian on its eigenstates:

H l, N = E l, N (5.5.2) sp| i N | i l E = 2(² µ)(N ) V N(l N + 1) (5.5.3) b N F − − 2 − − We need to find which of these eigenstates is the ground state. We find the lowest energy solution by differentiating the eigenvalue with respect to N and setting equal to 0:

dE N = 2(² µ) V (l + 1) + 2V N = 0 dN F − − 2(² µ) = V (l + 1) 2V N¯ (5.5.4) F − − This determines the chemical potential for a given N¯. If we insert this value back into the Hamiltonian, we have a Hamiltonian which is restricted so that the expectation of the number operator in its ground state is always equal to a speciﬁed value, N¯.

Z + H = (V (l + 1) 2N¯)S V S S− (5.5.5) sp − − l H l, N¯ = (V (l + 1) 2N¯)(N¯ ) V N¯(l N¯ + 1) l, N¯ sp| b i − − 2 − − | i l b = V (l + 1) + V N¯(l N¯) l, N¯ − 2 − | i µ ¶ (5.5.6)

If we compare this to the mean-ﬁeld result:

l2 H l, z = V + V N¯(l N¯) l, z SMF | i − 2 − | i µ ¶ b (5.5.7)

l2 We see that the mean field result has a term 2 which in the exact result is replaced l by 2 (l + 1). This difference is due to quantum fluctuations, and becomes especially clear ¯ l 2 l 2 at half-filling (N = 2 ). The classical energy of a spin J is J (= 2 ), but quantum ˆ2 l l fluctuations mean the energy is given by S , i.e. J(J + 1) (= 2 2 +¡ 1¢ ). The quantum fluctuations are not captured by the mean field theory but are ¡describ¢ ed by the exact solution. 112 Finite Superconductors as Spins

The mean-field solution gives a non-zero pairing parameter l, z S l, z = ∆ = l . The h | −| i V 2 exact solution gives l, l S l, l = 0. Thus, the exact solution has zero pairing parameter, h 2 | −| 2 i and therefore cannot be said to be superconducting in the usual sense. This is only to be expected, as we are dealing with a finite system. None the less, the system does possess pairing fluctuations:

l + l l + l l l l l l, S S− l, l, S l, l, S− l, = + 1 (5.5.8) h 2| | 2i − h 2| | 2ih 2| | 2i 2 2 µ ¶ 5.5.1 Broken Symmetry: Finite size

We have established that the solution to the exact Hamiltonian has zero pairing parameter. It may be however, that we could find a solution with broken symmetry that has a lower energy. To find if this is true, we consider a Hamiltonian which includes a symmetry- breaking field, η (see eg. [99]). We find the ground state of this Hamiltonian, and the pairing parameter in this ground state. We then let the magnitude of the field go to zero, and see if we still have a non-zero pairing parameter. This would indicate a spontaneously broken symmetry. Take the broken symmetry Hamiltonian:

Z + + H = 2(² µ)Sˆ V S S− ηS η∗S− (5.5.9) BS F − − − −

To investigate thisbwe consider the symmetry-breaking terms η, η∗ as a perturbation on the exact spin Hamiltonian 5.1.7 (since η 0 is the limit of interest). → From eq. 5.5.2 we have:

l E(0) = 2(² µ)(N ) V N(l N + 1) (5.5.10) N F − − 2 − − Using time-independent perturbation theory, we ﬁnd the ﬁrst order perturbation in the energy:

(1) E = l, N H0 l, N N h | | i + = l, N ηS η∗S− l, N h | − − | i = 0 (5.5.11) 5.5 Solving the Spin Hamiltonian Exactly 113

So there is no ﬁrst-order correction to the energy. The ﬁrst-order correction to the eigenstates is:

ψ(0) = l, N | N i | i H 0 ψ(1) = 0N ,N ψ(0) (5.5.12) | N i (0) (0) | N i N 0=N EN EN 0 X6 − + where the matrix elements H 0 of the perturbing Hamiltonian H = ηS η S are: 0N ,N 0 − − ∗ −

+ H0 0 = l, N 0 ηS η∗S− l, N N ,N h | − − | i = η (N + 1)(l N)δ 0 − − N ,N+1 ηp∗ N(l N + 1) δN 0,N 1 (5.5.13) − − − p and the diﬀerence in the energy eigenstates EN of the unperturbed Hamiltonian is given by:

E(0) E(0) = 2(² µ) + V (l + 1) 2NV V N − N+1 − F − − − = 2(N¯ N) V − − − (0) (0) EN EN 1 = 2(²F µ) V (l + 1) + 2NV V − − − − − = 2(N N¯) V − − − (5.5.14)

Substituting 5.5.14 and 5.5.13 into 5.5.12 gives the ﬁrst order corrections to the eigenstates:

(N + 1)(l N) ψ(1) = η − ψ(0) | N i − 2(N¯ N) V | N+1i p− − − N(l N + 1) (0) η∗ − ψN 1 (5.5.15) − 2(N N¯) V | − i −p − − The second order correction to the energy is:

2 H 0 E(2) = 0N ,N (5.5.16) N | (0) |(0) N 0=N EN EN 0 X6 − 114 Finite Superconductors as Spins

Inserting N = N¯ into 5.5.13 and 5.5.14 allows us to calculate the correction to the ground state:

η2(N + 1)(l N) η2(N)(l N + 1) E(2) = − − (5.5.17) N − V − V By looking at 5.5.15 and 5.5.17 we see that the broken symmetry solution does have a lower ground state energy than the non-broken symmetry solution, but that when we take the η 0 limit, we ﬁnd that the correction to the states disappears; we do not → have spontaneous symmetry breaking. The symmetry will break, however, if we have an external term of the form ηS+ η S . This happens when we connect the system to a − − ∗ − larger superconductor, as we shall see later.

5.5.2 Broken Symmetry: Inﬁnite Size

Above we have considered a finite spin, and checked to see if there is a broken symmetry solution. We find that there is not, for any size of the spin. If, however, we allow the size of the spin to become infinite first, and only then check for a broken symmetry solution, we may find a different result. Starting from eq. 5.5.9 we allow the size to go to infinity. By examining the differential 2 forms of S+, S , we see that in the l limit, S+S becomes l . This can also be seen − → ∞ − 4 without recourse to the differential forms if we recall that S+S = Sˆ2 SZ (SZ + 1) and − − make the assumption that SZ l in the infinite limit. Thus, 5.5.9 becomes: ¿ 2

2 Z l + H = 2(² µ)Sˆ V ηS η∗S− (5.5.18) BS F − − 4 − − We see then that thisb Hamiltonian has the same form as the mean-ﬁeld Hamiltonian. We can just read oﬀ the result, inserting the relevant parameters. The ground state is then the coherent state l, z , with: | i l2 H l, z = γSZ V l, z BS| i nˆ − 4 | i µ ¶ l l2 = 4(² µ)2 + 4 η 2 V l, z −2 F − | | − 4 | i µ ¶ ³p ´ (5.5.19)

2 As η 0, the energy at half ﬁlling becomes simply V l , so we have conﬁrmed → − 4 that the broken symmetry solution is indeed degenerate with the non-broken symmetry solution. 5.6 Expanding the Richardson Solution to First Order 115

We need to examine ground states, to see if the broken symmetry ground state reverts to the non-broken symmetry ground state as η 0. If it does not, we have a spontaneous → broken symmetry.

2 2 2(²F µ) 4(²F µ) + 4 η z∗ = − − − | | p2η∗ (5.5.20)

If we are not at half filling, then (² µ) is either negative or positive. If it is negative, F − then as η 0 z goes to infinity. If (² µ) is positive, then we use l’Hopital’s rule and → ∗ F − differentiate both the denominator and the numerator. As η 0 we get: →

4η∗ z∗ = − 4(² µ)2 + 4 η 2 F − | | p (5.5.21) which goes to zero. So, in the limit, z∗ goes to either 0 or infinity. A coherent state with z = 0 is the state J, m = J , and a coherent state with z = is the state J, m = +J , | − i ∞ | i so the system is only interesting precisely at half-filling. At half filling, +4 η 2 z∗ = − | | p2η∗ η = | | = eiφ (5.5.22) η∗

l 1 (eiφS+)N l, z = oˆ l, 0 (5.5.23) | i √2l N! | i NX=0 1 iφ = 1 + e ck† c† k 0 (5.5.24) √2l ↑ − ↓ | i Yk ³ ´ i.e. the ground state is independent of the magnitude of η in the inﬁnite limit, and the symmetry remains broken for η 0. Note that the phase of η still enters, and it is this → that gives the phase of the coherent state.

5.6 Expanding the Richardson Solution to First Order

In section 5.4 we compared the mean ﬁeld solution of our spin model with the strong- coupling limit of the mean ﬁeld solution of the BCS Hamiltonian. In this section we shall 116 Finite Superconductors as Spins derive the exact solution found in section 5.5 as a large-coupling limit of the Richardson Solution described in section 1.9.2.

N l 1 2 1 + = (5.6.1) −V E E E 2² ν=1 Jµ Jν Jµ k X − Xk=1 − We shall be taking the strong-coupling limit, that is where λ , where λ = V l/ω → ∞ c is the dimensionless coupling constant and we write V = λd, with d = ωc/l the mean level spacing. In what follows, we reproduce the derivation of Yuzbashyan et al. [90], with a small but signiﬁcant diﬀerence. In [90], Yusbashyan et al. were concerned with only isolated dots, and so after taking the limit, where ² ² , could disregard the kinetic energy part k → F of the Hamiltonian (essentially setting ²F to zero). We will be discussing coupling between more than one island, and so will need to keep the kinetic energy term. As λ , a number of the Richardson parameters E diverge (proportional to V ). → ∞ J The number which diverge is related to the symmetry of the solution. For any given number of Cooper pairs on the island, the ground state is the symmetrised solution l, N , | i and the number of diverging parameters for this state is N, i.e. all EJ diverge. As stated above, we wish to retain the kinetic energy term. We can regard (² ² )/V k − F as zero, because ² ² < ω , but we cannot discard ² /V as negligible. We expand the | k − F | c F Richardson equations (eq. 5.6.1) in 2(² ² )/(E 2² ), where we have not made any k − F Jµ − F assumptions about the size of ²F /V .

N l 1 2 l 2(² ² ) + = + k − F (5.6.2) −V E E (E 2² ) (E 2² )2 ν=1 Jµ Jν Jµ F Jµ F X − − Xk=1 − We discard the second term on the right as negligible, multiply by E 2² , and sum Jµ − F over the N parameters EJ µ

N N E 2² 2E 4² Jµ − F + Jµ F = l − V E E − E E ν=1 Jµ Jν ν=1 Jµ Jν X − X − N E 2² Jµ − F + N(N 1) + 0 = Nl (5.6.3) − V − µ=1 X 5.6 Expanding the Richardson Solution to First Order 117 where the double sums over µ, ν have either vanished, or gone to N(N 1) due to symmetry. − Finally, we recall that the energy of the Richardson solution is given by a sum over EJ , and rewrite 5.6.3 to get the energy of an island containing N Cooper pairs:

E = N2² V N(l N + 1) (5.6.4) N F − −

This should be compared with the energy as calculated directly from the spin Hamil- tonian:

Z l + (S + )2² V S S− l, N = E l, N 2 F − | i N | i µ ¶ E = N2² V N(l N + 1) (5.6.5) N F − − µ ¶ This derivation shows that the spin Hamiltonian can be thought of as the ﬁrst term in an expansion in ωc/V . Including all the terms in the expansion allows an exact treatment of the BCS Hamiltonian in terms of the spin operators SZ , S+, S . −

Chapter 6

The Super Josephson Eﬀect

The spin model for a superconducting system described in chapter 5 allows the easy investigation of various phenomena. Here we apply the model to an analogue of a well-known quantum optical eﬀect known as superradiance, in which the intensity of the radiation emitted by the atoms in a cavity is proportional to the square of the number of atoms.

We shall investigate an analogue of a quantum optical effect known as superradiance. In superradiance, a group of atoms in a quantised electromagnetic field (i.e. a cavity) are placed in a particular state. These atoms then emit light with an intensity that is proportional to the square of the number of atoms [100]. This is an interesting effect, as its observation relies on the fact that the atoms are placed in a highly entangled state, but is observable simply by measurement of the intensity of light emitted. In this chapter, we will describe a similar effect that occurs in an array of superconducting islands coupled to a superconducting reservoir, in which the current out of the array is proportional to the square of the number of islands. This work is presented in [101].

6.1 The Cooper Pair Box Array

The system we wish to describe is an array of Cooper pair boxes, each of which is coupled individually to a superconducting reservoir. In this chapter, we assume that we can represent the boxes as two level systems, and we consider the reservoir to be a quantum spin, the size of which we allow to go to inﬁnity at the end of the calculation.

119 120 The Super Josephson Eﬀect

Figure 6.1: An array of lb superconducting islands, or Cooper Pair Boxes, that are individually coupled to a superconducting reservoir, r, through capacitive Josephson tunnel junctions (with no inter-box coupling).

The Hamiltonian of this system is:

H = Hb + Hr + HT (6.1.1) where the individual Hamiltonians refer to the array of Cooper pair boxes, the reservoir, and the tunnelling between the two respectively. The ﬁrst term is the Hamiltonian of the boxes: lb ch z Hb = Ei σi (6.1.2) Xi=0 There are lb Cooper pair boxes, each of which is described as a two-level system where ch the two states are zero or one excess Cooper pair on the box, with Ei the charging energy of each, as described in chapter 3.

The second term Hr is the well known BCS Hamiltonian [27] (see section 1.6), and the ﬁnal term:

+ HT = Ti,k (σi c k ck + σi−ck† c† k ) (6.1.3) − − ↓ ↑ ↑ − ↓ Xk,i 6.1 The Cooper Pair Box Array 121 describes the tunnelling of Cooper pairs from each of the boxes to the reservoir. The charging energy of each box can effectively be tuned by applying a gate voltage to that box, and a tuneable tunnelling energy can be achieved by adjusting the flux through a system of two tunnel junctions in parallel ([68], section 2.3). In chapter 5 we used a set of finite spin operators to represent a finite superconductor. ch We can also use this formalism to describe the array of boxes. If the charging energies Ei and tunnelling elements Ti,k of each box are all adjusted to be the same, then the charging energy for all the boxes acts like the z-component of a quantum spin with J = lb/2. Similarly the sum over all the raising (lowering) operators acts like a large-spin raising (lowering) operator.

lb σZ SZ i → b Xi=0 lb σ§ S§ (6.1.4) i → b Xi=0 If we consider transitions caused by only these operators, then the symmetrised number Z states form a complete set. The eigenstates of Sb represent states where a given number N of the boxes are occupied, with SZ N = (N l /2) N . We now have a description b b | bi b − b | bi of the Cooper pair boxes in terms of a single quantum spin:

ch Z Hb = Eb Sb + HT = Tk(Sb c k ck + Sb−ck† c† k ) (6.1.5) − − ↓ ↑ ↑ − ↓ Xk As mentioned above, superradiance is observed when the atoms are placed in a particular initial state. The initial state is one of the form:

ψ(0) = 000..111.. + 001..110.. + + 111..000 (6.1.6) | i {| i | i · · · | i} where 0 and 1 indicate that an atom (identified by the place of these numbers in the array which specifies the state n , n , n . . . ) is in its ground or excited state. Clearly, this | 1 2 3 i Z state is an eigenstate of the operator for the total number of excited states (Sb ), but not Z of the individual σi operators. Note that it is the symmetrised state with a total number of excited states Nb. Evidently, these are exactly the states described by our model. 122 The Super Josephson Effect

We use the model discussed in chapter 5 to describe the superconducting reservoir. This model allows us to take into account the discrete nature of the Cooper pairs, whilst being simple enough to allow ease of calculation. In the superconducting analogue we consider three cases. The first is when the system is in an eigenstate of both the number operator of the large superconductor, and the operator representing the total number of Cooper pairs on the boxes. This is an artificial case if we think of the boxes coupled to a truly bulk superconductor, but instead would represent many boxes coupled to an island with only a few levels. It is, however, the closest to superradiance, where the atoms are in a symmetrised number eigenstate, and the field contains a given number of photons. The second case is when the boxes are in a number eigenstate and the reservoir in a coherent state, and finally we consider when the boxes are in a coherent state as well as the reservoir.

6.2 Superradiant Tunnelling Between Number Eigenstates

The simplest case is when both the boxes and the reservoir are in eigenstates of the Z Z number operators Sb , Sr respectively. We consider the unperturbed Hamiltonian as the sum of the Hamiltonians that act individually on the array and on the reservoir. As we are discussing the case of superradiance between number eigenstates, we take the Hamiltonian Z of the reservoir to be diagonal in Sr , i.e. , we do not break the symmetry with a pairing parameter.

ch Z Z H = Eb Sb + 2ξF rSr (6.2.1)

Z Z As the Hamiltonian is diagonal in Sr and Sb , the eigenstates are product states of the number states of the reservoir, N and of the array, N . | ri | bi This Hamiltonian is perturbed by the tunnelling Hamiltonian:

+ + H = T (S S− + S−S ) (6.2.2) T − b r b r and the current, i.e. the rate of change of charge on the islands, is given by the commutator of the number operator with the Hamiltonian: 6.2 Superradiant Tunnelling Between Number Eigenstates 123

ˆ ˆZ dNb dSb 1 ˆZ = = [Sb , H] dt dt i~ + + Iˆ = T (S S− S−S ) (6.2.3) b r − b r

The expectation value of the current operator with the ground state is equal to zero, so we go to time dependent perturbation theory:

ψ(t) = U(t) ψ(0) | i | i t 1 iHt0/~ iHt0/~ ψ(t) = 1 + e− HT e− dt0 + ψ(0) (6.2.4) | i  i~ · · ·  | i Z0  

The expectation value of any operator at a given time is, to ﬁrst order in HT :

ψ(t) Iˆ ψ(t) = ψ(0) Iˆ ψ(0) h | | i h | | i t 1 iHt0/~ iHt0/~ +2 ψ(0) Iˆ e HT e− dt0 ψ(0) (6.2.5) <{h | i~ | i} Z0

The symbol indicates we are taking the real part of the expression. The tunnelling < Hamiltonian and current consist of a sum (or diﬀerence) of raising and lowering operators. When calculating the time-dependent current given a number eigenstate as the initial state, the only non-zero terms are those that raise the number eigenstate and then lower + + + + it, or lower it then raise it, i.e. only the terms Sb−Sr Sb Sr− and Sb Sr−Sb−Sr survive.

Thus the expectation value of the current when the system is initialised in a number eigenstate is a simple sum over two other number eigenstates, with a time dependent factor from the exponentials. 124 The Super Josephson Eﬀect

t ˆ + 1 iHt0/~ + iHt0/~ ψ(t) I ψ(t) = 2 Nr, Nb T S Sr− e− ( T )S−Sr e dt0 Nr, Nb h | | i <{h | b (i~)2 − b | i} Z0 t + 1 iHt0/~ + iHt0/~ 2 Nr, Nb T S−Sr e− ( T )S Sr−e dt0 Nr, Nb − <{h | b (i~)2 − b | i} Z0 2 + + = 2 T N S S− N N S−S N <{ − h b| b b | bih r| r r | ri t b b r r 0 1 i(EN − EN ) i(EN EN ) t /~ e− b 1− b − r+1− r dt0 (i~)2 } Z0 2 + + +2 T N S−S N N S S− N <{ h b| b b | bih r| r r | ri t b b r r 0 1 i(EN EN ) i(EN − EN ) t /~ e− b+1− b − r 1− r dt0 (i~)2 } Z0 (6.2.6)

Taking the real part and noting that we have even level spacing (i.e. Eb and Er are Nb Nr equal for all Nb, Nr), we ﬁnd the current is:

The time dependence is determined by the energy difference between adjacent number ch eigenstates, i.e. by Eb and 2ξF r. If we evaluate the expectation values, we find terms like N (l N )(l 2N ) which is quadratic in l and linear in l (and vice versa, i.e. terms b b − b r − r b r quadratic in lr and linear in lb). In particular, to have the largest effect, we set Nb = lb/2 i.e. half the boxes are occupied.

2 ch ˆ 2T sin (Eb 2ξF rt/~ lb lb ψ(t) I ψ(t) = − 2 ch − + 1 (lr 2Nr) h | | i (i~) (E 2ξF r/~ 2 2 − ¡ b − ¢ µ µ ¶ ¶ ¡ ¢ (6.2.7) 6.3 Superradiant Tunnelling: Number State to Coherent State 125

2 We see that the current is proportional to lb , the square of the number of Cooper Pair boxes. This is in contrast to the situation where the initial state of the boxes is a product state of lb independent wavefunctions, in which any current can be at most linear in the number of boxes. The current is also proportional to (n 1/2)l , where n = N /l , i.e. r − r r r r how far away the large superconductor is from half ﬁlling.

6.3 Superradiant Tunnelling Between a Number Eigenstate and a Coherent State

The second case we consider is when the Cooper Pair boxes are in an eigenstate of the number operator, and the large superconductor is in a coherent, that is to say BCS-like, state. We ensure the coherent state of the large superconductor by introducing a pairing parameter into the unperturbed Hamiltonian:

ch Z Z + H = E S + 2ξ S ∆ S ∆∗S− (6.3.1) b b F r r − r r − r r The ground state of the large superconductor is a coherent state and ∆ = S can r h r−i now be found self-consistently (section 5.4). We make a basis transform (section 5.4.1) and write the Hamiltonian in terms of this new basis, nˆ:

ch Z Z H = Eb Sb + 2EF rSr,nˆ (6.3.2)

This basis transform makes the unperturbed Hamiltonian simpler, but at the expense of making the perturbation more complicated in the new basis:

+ + H = T (S S− + S−S ) T − b r b r + 2∆r Z + = T Sb ( Sr,nˆ + fr∗Sr,nˆ + erSr−,nˆ) − EF r 2∆r∗ Z + T Sb−( Sr,nˆ + er∗Sr,nˆ + frSr−,nˆ) (6.3.3) − EF r

Again, the tunnelling current is zero to ﬁrst order in T, as the boxes are in a number eigenstate. With this basis transform, we can easily calculate the second-order current. Following the same procedure as before, we get: 126 The Super Josephson Eﬀect

t 2 2 ˆ 2T ∆r Z Z ch ψ(t) I ψ(t) = − 2 | 2 | 0r,nˆ Sr,nˆSr,nˆ 0r,nˆ dt0 cos Eb t0/~ h | | i (i~) EF r h | | i Z0 + + ( N S S− N N S−S N ) × h b| b b | bi − h b| b b | bi 2 2T + 0r,nˆ S− S 0r,nˆ −(i~)2 h | r,nˆ r,nˆ| i × t 2 ch + e dt0 cos (2EF r E ) t0/~ Nb S S− Nb {| | − b h | b b | i Z0 t 2 ch + f dt0 cos (2EF r + E ) t0/~ Nb S−S Nb (6.3.4) −| | b h | b b | i} Z0 The terms e and f are as deﬁned in 5.4.8. The time dependence is again given by the level spacing, but for the large superconductor this is now given by E = ξ2 ∆ 2 rather than (² µ). F r F r − | | F r − q

2 ch 2 2 2T sin Eb t/~ ∆ lr ψ(t) Iˆ ψ(t) = − | | (lb 2Nb) h | | i (i )2 Ech/ E2 2 − ~ b ~ F r µ ¶ 2 ch 2T sin (Eb + γ2ξF )t/~ 2 + + 2 ch lr f Nb Sb−Sb NB (i~) (Eb + γ2ξF )/~ | | h | | i 2 ch 2T sin (Eb γ2ξF )t/~ 2 + 2 ch − lr e Nb Sb Sb− NB (6.3.5) −(i~) (E γ2ξF )/~ | | h | | i b − If we assume the levels of the large superconductor are much more ﬁnely spaced than those of the boxes, i.e. E Ech, then we obtain the expression below. If we do not F r ¿ b 2 make this assumption, we ﬁnd a similar result (current proportional to lb ), but with a more complicated time dependence.

ψ(t) Iˆ ψ(t) (6.3.6) h | | i 2 ch 2 2 2T sin Eb t/~ ∆ lr = − | | (lb 2Nb) (i )2 Ech/ E2 2 − ~ b ~ F r µ ¶ 2 ch 2T sin Eb t/~ ξF r 2 2 2 ch lr Nb(lb Nb) + e Nb f (lb Nb) −(i~) E /~ EF r − | | − | | − b n o Remembering that l ξ /E = l /2 N¯ , (where N¯ is the expectation value of r F r F r r − r r the reservoir number operator, see section 5.4.17) and setting Nb = lb/2, we ﬁnd this is proportional to: 6.4 Superradiant Tunnelling Between Coherent States 127

l l l (1 2n¯ ) b b + 1 (6.3.7) r − r 2 2 µ ¶ This is very similar to the case described in (6.2.7) where both the boxes and the reservoir are in number eigenstates. In both cases, the effect is largest when the boxes are at half-filling, and the large superconductor is far from half-filling. The difference is that here it is the average number N¯r that is relevant, not the number eigenvalue.

6.4 Superradiant Tunnelling Between Coherent States

The third case is where both the boxes and the large superconductors are in the coherent state. Pairing parameters are introduced for each:

Z + H = 2ξ S ∆ S ∆∗S− F b b − b b − b b Z + +2ξ S ∆ S ∆∗S− (6.4.1) F r r − r r − r r

The ground state of this Hamiltonian is for both superconductors to be in a coherent state, which again is determined self consistently. As before we make a change of basis, this time for both the boxes and the large superconductor, which makes the unperturbed Hamiltonian:

Z Z H = 2EF bSb,nˆ + 2EF rSr,nˆ (6.4.2)

The tunnelling Hamiltonian becomes:

+ + H = T (S S− + S−S ) (6.4.3) T − b r b r ∆b∗ Z + ∆r Z + = T ( Sb,nˆ + eb∗Sb,nˆ + fbSb,−nˆ)( Sr,nˆ + fr∗Sr,nˆ + erSr−,nˆ) − EF b EF r ∆b Z + ∆r∗ Z + T ( Sb,nˆ + fb∗Sb,nˆ + ebSb,−nˆ)( Sr,nˆ + er∗Sr,nˆ + frSr−,nˆ) − EF b EF r

If we insert this into (6.2.5) we ﬁnd that we do have a non-zero term linear in T , i.e. the expectation value of Iˆ with ψ(0) . To second order in T we have terms proportional | i to lr and quadratic in lb and vice versa, and terms linear in both, i.e. 128 The Super Josephson Eﬀect

T ∆b ∆r ψ(0) Iˆ ψ(0) = | | | | lb lr 2 sin (φb φr) h | | i i~ 2EF b 2EF r − T ∆b ∆r = | | | | 2 sin (φb φr) (6.4.5) i~ Vb Vr − where φb, φr are the phases of ∆b, ∆r. This is the usual Josephson eﬀect.

The term quadratic in lb is:

ˆ ψ(t) I ψ(t) l2l h | | i b r 2 2 T 2 sin 2EF rt/~ ∆b ξF r = − 2 lb lr | 2| (i~) 2EF r/~ 2EF b EF r 2 2 2 T 2 cos 2EF rt/~ ∆b ∆r 2 lb lr | 2| | 2| 2 sin 2(φb φr) (6.4.6) −(i~) 2EF r/~ 4EF b 4EF r − The first term is proportional to ∆ 2/E2 and ξ /E , and so is largest when the | b| F b F r F r boxes are half occupied and the large superconductor is far from half occupied. This term is also phase-independent and is like the effects seen in sections 6.2 and 6.3 The second term is largest when both are half occupied, and also depends on the phase difference of the two pairing parameters with a frequency twice that of the usual Josephson effect. As such, it can be considered a higher harmonic in the Josephson current. The term quadratic in lr is identical apart from an anti-symmetric exchange of the labels, b, r. Finally, the term that is linear in both contains a phase - independent term that is largest when both the boxes and the reservoir are far from half-filling, and a term that is phase dependent with twice the Josephson frequency that is largest at half filling.

ψ(t) Iˆ ψ(t) h | | ilblr 2 2 2 T sin 2(EF b + EF r)t/~ ξ ξF r ξ ξF b = − l l F b F r (i )2 b r 2(E + E )/ 4E2 E − 4E2 E ~ F b F r ~ µ F b F r F r F b ¶ 2 2 2 T cos 2(EF b + EF r)t/~ ∆b ∆r + l l | | | | 2 sin 2(φ φ ) (6.4.7) (i )2 b r 2(E + E )/ 4E2 4E2 b − r ~ F b F r ~ µ F b F r ¶ 6.4 Superradiant Tunnelling Between Coherent States 129

In summary we conclude that tunnelling into or out of a Cooper pair reservoir from a coherent ensemble of Cooper Pair Boxes can lead to a tunnelling current proportional to the square of the number of ‘boxes.’ Interestingly, an observation of such scaling, in turn, could be taken as a demonstration of coherence and entanglement of the ‘boxes’ as such states are necessary to produce the phenomena.

Chapter 7

Revival

A further quantum optical effect that can be observed in an analogous superconducting system is the effect of revival, where the state of a superconducting island connected to a reservoir, initially in a pure state, decays to a mixed state but returns to a pure state at a later time. The fact that the island is in a pure state is indicated by oscillations in the probability of occupation of the charge states. The effect can be extended to the case of multiple islands connected to a reservoir

7.1 The Superconducting Analogue of the Jaynes-Cummings Model

In the previous chapter we investigated an analogue of superradiance. Another well-known quantum optical phenomena is that of quantum revival (e.g. [102]), which occurs when an atom is coupled to a single mode of the electromagnetic field. The initial pure state of the atom decays into a mixed state, but reappears at a later time. This apparent decay and subsequent revival occur because the environment of the atom is a discrete quantum field. Here we examine a Josephson system analogue. As in the last chapter, we consider an array of Cooper pair boxes individually connected to a superconducting reservoir (see fig. 6.1). Again, we treat each of the boxes as a simple two-level system and treat the reservoir as a superconductor represented by a large quantum spin (see chapters 5, 6). We consider a Hamiltonian (eq. 6.3.1) discussed in section 6.3.

131 132 Revival

ch Z Z + H = E S + 2ξ S ∆S ∆∗S− b b F r r − r − r + + H = T (S S− + S−S ) (7.1.1) T − b r b r

As in section 6.3, we make a basis transform so that the reservoir is written as a quantum spin in another direction.

ch Z Z H = Eb Sb + 2EF rSr,nˆ + 2∆r Z + HT = T Sb ( Sr,nˆ + fr∗Sr,nˆ + erSr−,nˆ) − EF r 2∆r∗ Z + T Sb−( Sr,nˆ + er∗Sr,nˆ + frSr−,nˆ) (7.1.2) − EF r

We can write a general state of the system in terms of the number eigenstates:

lr,lb ψ(t) = a (t) N N (7.1.3) | i Nr,Nb | ri| bi NrX,Nb=0

For a single two level system, lb = 1, but we can also do the calculations for a general number of symmetrised boxes. We can calculate the time evolution of the system using the Schr¨odinger equations:

d aNr,Nb (t) ch lb lr i~ Nr Nb = aNr,Nb (t) E (Nb ) + 2ξF r(Nr ) Nr Nb dt | i| i b − 2 − 2 | i| i NXr,Nb NXr,Nb µ ¶ + 2∆r Z + aNr,Nb (t) T Sb ( Sr,nˆ + fr∗Sr,nˆ + erSr−,nˆ) Nr Nb − EF r | i| i NXr,Nb 2∆r∗ Z + aNr,Nb (t) T Sb−( Sr,nˆ + er∗Sr,nˆ + frSr−,nˆ) Nr Nb − EF r | i| i NXr,Nb (7.1.4)

More explicitly, we have a set of coupled diﬀerential equations for the coeﬃcients aNr,Nb (t): 7.1 The Superconducting Analogue of the Jaynes-Cummings Model 133

d aNr,Nb (t) ch lb lr i~ = aNr,Nb (t) E (Nb ) + 2ξF r(Nr ) dt b − 2 − 2 µ ¶ 2∆r∗ aNr,Nb+1(t) T (Nb + 1)(lb Nb) (Nr lr/2) − − EF r − p aNr 1,Nb+1(t) T (Nb + 1)(lb Nb)er∗ Nr(lr Nr + 1) − − − − a (t) T p(N + 1)(l N )f p(N + 1)(l N ) − Nr+1,Nb+1 b b − b r r r − r p 2∆r p aNr,Nb 1(t) T Nb(lb Nb + 1) (Nr lr/2) − − − EF r − p aNr 1,Nb 1(t) T Nb(lb Nb + 1)fr∗ Nr(lr Nr + 1) − − − − − p p aNr+1,N 1(t) T Nb(lb Nb + 1)er (Nr + 1)(lr Nr) − b− − − p p (7.1.5)

We consider an approximation that allows a simple analytical model.

The ∆ terms in eq. 7.1.1 mean that the total particle number Nr +Nb is not conserved. We note that whilst the commutator of the total number operator [Nˆ + Nˆ , Hˆ ] = ∆S+ r b r − ∆∗Sr− is not zero, its expectation value in the coherent state is. With this in mind, we approximate and assume no fluctuations in the total number (although we of course retain fluctuations in N N ) and discard any terms in eq. 7.1.5 that change the total number. r − b With these terms discarded, we have a set of coupled differential equations in which each coefficient aNr,N (t) only couples to aNr 1,N +1(t) and aNr+1,N 1(t). b − b b−

d aNr,Nb (t) ch lb lr i~ = aNr,Nb (t) E (Nb ) + EF r(Nr ) dt b − 2 − 2 µ ¶

aNr+1,N 1(t) T Nb(lb Nb + 1) (Nr + 1)(lr Nr) − b− − − p p aNr 1,N +1(t) T (Nb + 1)(lb Nb) Nr(lr Nr + 1) − − b − − p p (7.1.6)

These equations can be restated in terms of an eigenvalue problem. Each level Nr has a set of eigenvectors for the boxes. If the boxes and the reservoir are initially in a product state, the probability to be in a given value of Nb (regardless of Nr) is:

lb 2 2 iEi,Nr t/~ PNb = aNr (0) Nb νi,Nr e− νi,Nr ψNb (0) (7.1.7) | | ¯ h | i h | i¯ Nb=0 ¯ i ¯ X ¯X ¯ ¯ ¯ ¯ ¯ 134 Revival where ν (E ) are the eigenvectors (values) of the N -level system associated with | i,Nr i i,Nr b the reservoir level N , a (0) are the initial amplitudes of N , and ψ (0) is the initial | ri Nr | ri | Nb i state of the boxes. This may not be exactly solvable in general, but can be solved for few-level systems, lb. For example, if we have a single two level system and specify the initial state to be a product state of the box state 0 with some state a (0) N of the reservoir, we have | bi Nr | ri the probabilities that the box and reservoir are in aPgiven state:

T 2 2 N0 r 2 P1,Nr 1(t) = aNr (0) 2 sin (ΩNr t/~) − | | ΩNr 2 P0,Nr (t) = aNr (0) (1 P1,Nr 1(t)) (7.1.8) | | − − with T = T N (l N + 1) and Ω = (Ech E )2/4 + T 2. This is the same N0 r r r − r Nr b − F r 0 result as in quanptum optics, except that the photonq creation/ annihilation operators give √ TN0 r = T Nr .

7.1.1 Quantum Revival of the Initial State

In quantum optics, the phenomenon of revival is seen when the electromagnetic ﬁeld is placed in a coherent state (i.e. a 2 = exp( N¯)N¯ N /N!, with the sum over N running to | N | − inﬁnity). We place the reservoir in the spin coherent state, i.e.

lr P (t) = a (0) 2P (t) (7.1.10) 0 | Nr | 0,Nr XNr ch For simplicity we consider the case when Eb = EF r, i.e. ΩNr = TN0 r . This is shown in Figure 7.1, for the values N¯r = 10, lr = 50. The initial state dies away, to be revived at a later time. We have chosen the above values for N¯r and lr in order to show the revival clearly, and also to show the diﬀerence between the spin and quantum optics cases. If the value of lr is too small (compared to N¯r) the oscillations never fully decay. On the other hand, if l N¯ , the system behaves identically to the quantum optics case. The eﬀects r À r 7.1 The Superconducting Analogue of the Jaynes-Cummings Model 135

0.8

0.6

P0(t)

0.4

0.2

0 1 2 3 4 5 6 time

Figure 7.1: Spin Coherent State Revival for Nr = 10, lr = 50 (solid line). The faint line indicates the usual quantum optical coherent state revival with Nr = 10, scaled along the t axis by √lr and the vertical dash-dot line indicates the calculated revival time, where the time is in units of 2π/EJ .

of revival are much more generic than this, and also can be observed in systems with, for example, non-identical parameters for the Cooper pair boxes. We can calculate the linear entropy of the system over time (ﬁg. 7.2). This is deﬁned as 2(1 T r ρ2) and is a measure of how mixed the state is, with 0 indicating a pure state, and − 1 indicating a maximally mixed qubit state. As expected, the entropy [9] increases (purity decreases) as the oscillations decay, and then the entropy is reduced (purity increased) as the oscillations revive. The revival occurs when the terms in the sum are in phase. We can make an estimate of this by requiring the terms close to N¯r to be in phase:

2π = 2T (N¯ + 1)(l N¯ ) t 2T N¯ (l N¯ + 1) t r r − r rev − r r − r rev q1/2 1/2 q 2π n¯r (1 n¯ ) t = − r (7.1.11) rev T (1 2n¯ ) − r where, as before n = N /l . Note that the revival time remains ﬁnite in the l r r r r → ∞ limit, as long as Nr/lr is ﬁnite. 136 Revival

0.8

0.6

0.4

0.2

1 2 3 4 5 6

Figure 7.2: Decay and revival of a single Cooper pair island state, showing the degree of linear entropy, which increases at ﬁrst, and is then reduced. The oscillations are the probability that the island is in state 0 (solid line) or state 1 (dashed line) against time | i | i in units of 2π/EJ . 7.1 The Superconducting Analogue of the Jaynes-Cummings Model 137

0.8

0.6

P0(t)

0.4

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 time

Figure 7.3: Analytic asymptotic form for decay oscillations in the probability against time in units of 2π/EJ (solid line) compared with the numeric evolution (dashed line) Nr = 10, lr = 100

We can ﬁnd an analytic form for the initial decay, and also demonstrate the equivalence of this ‘spin revival’ to the well known quantum optics revival, in the limit l , r → ∞ l /N . Note that this is an unrealistic limit for our case, where we would keep r r → ∞ the ﬁlling factor Nr/lr constant, but it allows us to make a connection with the quantum optics case.

The coeﬃcients, aNr (0) for the coherent and spin-coherent states are the Poisson and binomial distributions respectively, and both can be approximated by a Gaussian distribution.

Nr 1/2 2 exp( N¯ )N¯ /N ! (2πN¯ )− exp (N N¯ ) /2N¯ − r r r ' r − r − r r 2Nr lr! 1/2 2 α (2πN¯ )− exp £ (N N¯ ) /2N¯ ¤ (7.1.12) | | (l N )!N ! ' r − r − r r r − r r £ ¤ For the coherent state we have taken the limit N¯ 1. For the spin-coherent case we r À have also assumed l N¯ . The Gaussian factor suppresses any terms not around the r À r 138 Revival average N¯ , so we can expand N N¯ + (N N¯ ). r r ' r r − r In the limit, the spin coherent state gives:

2 1 1 (Nr N¯r) P0(t) = + dN exp − 2 2(2πN¯)1/2 − 2N¯ Z µ r ¶ (N N¯ )2 cos 2T (l N¯ )1/2t 1 + r − r (7.1.13) × r r 2N¯ · µ r ¶¸

1/2 This is the same as for the coherent state apart from the factor of lr in the frequency. Doing the integral gives:

1 1 (T l t)2 P (t) = + cos(2T (l N¯ )1/2t) exp( r ) (7.1.14) 0 2 2 r r − 2 This is plotted in Figure 7.3. Thus the phenomenon of quantum revival has a direct analogy in our ‘spin superconductor’ model. In both cases, the revival is due to constructive interference between the terms in the sum over the reservoir (ﬁeld) number states.

7.2 Revival of Entanglement

Revival is usually considered for single two-level systems, but the formula in (7.1.7) gives the state for a general system. In particular, the dynamics of a pair of two-level systems can be easily calculated, either numerically, or analytically, remembering that the singlet state is uncoupled due to radiation trapping. The Hamiltonian, 7.1.1, does not cause transitions between the triplet states and the singlet state, so if we start the system in any of the triplet states, we essentially have only a three level system to solve. Starting in the state 00 , we find a very similar effect to the single-island case. Figure | i 7.4 shows the oscillations of one Cooper pair box other traced over, compared to the oscillations of a single box. Note that whilst the decays in the single and double box systems are in phase, the first revival of the double box is out of phase with that of the single box, and the second revival is in phase. If we do not trace over one island, but consider the revival of the state 00 , we find | i oscillations in the probability of returning to this state earlier than in the case of a single island. 7.2 Revival of Entanglement 139

1 2 3 4 5

0.8

0.6

0.4

0.2

Figure 7.4: Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50. The solid line represents the probability that, when the system is initialised in the state 00 , | i one of the Cooper pair boxes is in the state 0 at time t (in units of 2π/E ), after tracing | i J over the other box. The dashed line indicates the revival of a single isolated Cooper Pair box connected to a reservoir. Note that the revivals occur at the same time, but out of phase.

To investigate the revival of entanglement, we start the system in the state ( 01 + | i 10 )/√2. As before we note that the oscillations in the probability of being in the state | i ( 01 + 10 )/√2 decay and then revive. It is interesting to note, however, that in this case | i | i the initial state is an entangled one. One measure of entanglement is the negativity [103], i.e. the sum of the negative eigenvalues of the partially transposed density matrix. If we plot the negativity over time, we see that it dies away at ﬁrst, as do the oscillations. As the oscillations revive, we also see an revival in the negativity of the two islands (Figure 7.5). This revival of entanglement, an initially counterintuitive phenomenon, is of course due to the fact that we have considered the entanglement between the two islands only, just as when we considered a single island, we calculated the purity of the island as an isolated system and not the entropy of the whole system. When the entanglement ‘disappears,’ it is only the entanglement between the islands that has disappeared. If we were to consider the whole system, including the reservoir, the evolution would be unitary and the entanglement would remain constant. It is worth noting that this ‘entanglement revival’ is not a unique feature of our spin model, and would be expected whenever two-level systems are coupled to a ﬁeld with (a 140 Revival

0.8

0.6

0.4

0.2

1 2 3 4

Figure 7.5: Two Cooper Pair Box Spin Coherent State Revival for Nr = 10, lr = 50. The dashed line represents the probability that the Cooper Pair boxes have returned to their initial state ( 01 + 10 )/√2 at time t,in units of 2π/E . The dashed line indicates the | i | i J negativity of the boxes as a function of time. 7.2 Revival of Entanglement 141 possibly inﬁnite number of) discrete energy levels.

Chapter 8

An Electronic Derivation of The Qubit Hamiltonian

In this chapter we examine in greater detail the derivation of the tunnelling term of the qubit Hamiltonian for superconducting charge qubits. We consider a rigorous derivation of the qubit Hamiltonian from single-electron creation and annihilation operators using the spin model described in chapter 5.

8.1 Introduction

In chapter 3 we derived a Hamiltonian for the qubit circuit from the Josephson relations. These equations give the rate of change of phase and the current across each junction. To attempt to capture the behaviour of junctions with small capacitances, we quantise these relations. A Lagrangian is found from which the Euler-Lagrange equations reproduce the Josephson equations. This leads to a Hamiltonian for the system, and tells us that the phase and charge are conjugate variables. This procedure is somewhat empirical, in the sense that the equations of motion for the phase and charge are treated as if they were the equations of motion for any classical system. These equations are, of course, derived from a microscopic theory of the electrons in the superconductor. In the derivation, it is assume that the superconductors are bulk so that BCS theory is the correct description. In particular, taking the inﬁnite limit means that the phase becomes a well deﬁned classical variable. In the course of this derivation, then, we have taken a microscopic quantum description

143 144 An Electronic Derivation of The Qubit Hamiltonian

(superconductivity theory), taken its classical limit (the Josephson relations) and then ‘re-quantised’ it (the qubit Hamiltonian) in order to attempt to capture the ﬁnite size behaviour. If nothing else, this procedure is inelegant, and it may be that important features have been neglected. Furthermore, quantising the Josephson relations in this way implies the existence of a ‘phase operator’ analogous to the position operator Xˆ, which is problematic. We would like to go directly from an electronic description of the system to the qubit Hamiltonian. The model described in chapter 5 will allow us to do this.

8.2 The Phase Representation

Quantum mechanics is usually first taught to undergraduates in terms of wavefunctions (the co-ordinate representation). The position wavefunction ψ(x) gives the amplitude that a particle will be found between x and x + dx. The whole of the function ψ(x) defines the state of the particle, and we may have a set of such states ψn(x) that are of interest to us, for example that are energy eigenstates. Operators OC are represented by differential operators that act on these states. An alternative representation is ket notation. In this, each state is denoted by an abstract ket, n , and the operators O are defined by their actions on these states. | i K We can easily translate between these two equivalent representations. The position wavefunction is given in terms of ket notation by ψ (x) = x n , where x is an eigenket n h | i | i of the position operator Xˆ with eigenvalue x (see eg. [39]). The operators are represented by the differential operators in a way that is consistent with this.

O ψ = λ ψ K | ni n| ni O x ψ = λ x ψ (8.2.2) C h | ni nh | ni In going from an electronic description of the qubit to one in terms of phase, we have to go from ket to coordinate representation. 8.2 The Phase Representation 145

8.2.1 Phase Representation of Spins

The model set up in chapter 5 describes a small region of superconductor in terms of a large but ﬁnite quantum spin. Before returning to our superconducting system, we shall examine how to describe quantum spins in terms of phase. A quantum spin of size l has a complete set of l states, l, N . Operations on this 2 | i system can be written in terms of the diagonal z-operator, SZ , and the raising and lowering operators, S . The action of these operators on the states l, N is: § i

l SZ l, N = (N ) l, N | i − 2 | i S+ l, N = (N + 1)(l N) l, N + 1 | i − | i S− l, N = pN(l (N 1)) l, N 1 (8.2.3) | i − − | − i p We wish to write both the states and the operator in some coordinate representation, speciﬁcally a phase coordinate representation. We deﬁne a state φ : | i

φ = eiφN l, N (8.2.4) | i | i X We now follow the procedure described above. An SZ eigenstate l, N is now described | i by a wavefunction ψ (φ) given by ψ (φ) = φ l, N = e iφN . The wavefunction of any N N h | i − general state a can be expressed by a linear combination of these eigenfunctions: | i

a = a l, N | i N | i XN ψa(φ) = aN ψN (φ) XN iφN = aN e− (8.2.5) XN We now need the coordinate representation of the operators. Our task is to find differential operators which are equivalent to our spin operators (eq. 8.2.3) in the sense defined by eq. 8.2.2. The action of SZ is reproduced by id/dφ l/2: − 146 An Electronic Derivation of The Qubit Hamiltonian

Z d l iφN S φ l, N = i e− h | i dφ − 2 µ ¶ l iφN = N e− − 2 µ ¶ l SZ φ l, N = N φ l, N (8.2.6) h | i − 2 h | i µ ¶ So our coordinate representation of SZ is id/dφ l/2. We now want to do the same − thing for the raising and lowering operators. Examining the action of the raising and lowering operators on the state l, N (eq. 8.2.3) tell that the that the appropriate diﬀerential | i operator form is:

+ iφ d d S = e− i + 1 l i dφ − dφ sµ ¶ µ ¶

iφ d d S− = e i l i + 1 (8.2.7) dφ − dφ s µ ¶ We write these in a form that will be more convenient. For example, it is not imme- + diately obvious from the above form that S and S− are Hermitian conjugates of each other.

+ iφ l d l l d l S = e− + i + 1 i 2 dφ − 2 2 − dφ − 2 sµ µ ¶ ¶sµ µ ¶¶ (8.2.8)

iφ We now want to commute the e− term past the ﬁrst square root. To do this we expand the square root in i d l . dφ − 2 ³ ´

p l d l l ∞ 1 id/dφ l/2 + 1 + i + 1 = − (8.2.9) 2 dφ − 2 2 p ! l/2 s r p=0 µ µ ¶ ¶ X µ ¶ Note that this is an inﬁnite expansion, and so it is valid regardless of the size of the iφ Z p term we are expanding in. We have to commute e− past terms like (S + 1) , where we have just used SZ as a shorthand for id/dφ l/2. Using the commutation relation − [e iφ, i d ] = e iφ, it is easy to show that e iφ(SZ + 1)p = (SZ )pe iφ. We ﬁnd that we − dφ − − − − can write the coordinate representation of the raising and lowering operators as: 8.2 The Phase Representation 147

+ l d l iφ l d l S = + i e− i 2 dφ − 2 2 − dφ − 2 sµ µ ¶¶ sµ µ ¶¶

l d l iφ l d l S− = i e + i 2 − dφ − 2 2 dφ − 2 sµ µ ¶¶ sµ µ ¶¶ (8.2.10)

This form is much more convenient; the two operators are obviously Hermitian conju- + gates of each other, and the action of S S− is much clearer. As an example of how these operators act on a phase wavefunction, we can see how the raising operator acts on the wavefunction ψl(φ) for the highest spin state, where N = l:

+ l d l iφ l d l il S ψ (φ) = + i e− i e− l 2 dφ − 2 2 − dφ − 2 sµ µ ¶¶ sµ µ ¶¶

l l l l i(l+1) = + l + 1 l e− 2 − 2 2 − − 2 sµ µ ¶¶sµ µ ¶¶ i(l+1) = (l + 1) (l l) e− − = 0p p (8.2.11)

The raising operator destroys the N = l wavefunction, as we would expect. A similar calculation conﬁrms that the lowering operator destroys the N = 0 wavefunction, and that these forms act on all the N wavefunctions in the manner required by eq. 8.2.3. We also require that the coordinate forms for the operators have the commutation relations given in eq. 5.2.2:

Z + Z l Z iφ l Z [S , S ] = S , S e− + S " r2 − r2 #

Z l Z iφ l Z l Z iφ l Z Z = S S e− + S S e− + S S Ãr2 − r2 ! − Ãr2 − r2 !

Z l Z iφ l Z l Z Z iφ iφ l Z = S S e− + S S S e− e− + S Ã 2 − 2 ! − 2 − − 2 r r r ³ ´ r l Z iφ l Z = S e− + S r2 − r2 [SZ , S+] = S+ (8.2.12) 148 An Electronic Derivation of The Qubit Hamiltonian

So the above commutation relation matches the form we desired. A similar calculation conﬁrms that the other two commutation relations match also.

8.2.2 The Limit of Large Size

We would like to find the form of these operators in the limit where the size of the spin (or the size of the superconductor) becomes infinite, l . We have written our operators 2 → ∞ in a form that will make this easy. The limit we will be taking in order to make the expansion is the limit SZ l . In ¿ 2 a spin, this refers to the z-value of the spin always being close to zero, i.e. the spin lies close to the x y plane. In a superconducting island, this corresponds to the number of − l Cooper pairs being close to 2 . Specifically, we assume:

1 l l 2 l, N ψ 0 for N & (8.2.13) |h | i| ∼ | − 2| 2 µ ¶

For the states ψ that the system occupies. As a particular example of a state that | i obeys this condition, we consider a coherent state:

1 l! l, N l, z 2 = z 2 (8.2.14) |h | i| (1 + z 2)l (l N)! N!| | | | −

2 1 (N lp)2/2lp(1 p) l, N l, z e− − − (8.2.15) |h | i| ' 2πlp(1 p) − p ¯ l l If the average number N is set to 2 , this is a Gaussian centred around 2 with width l 1/2 2 , and satisﬁes our condition for expansion. We shall take the full expansion later, but¡ ¢ for now we only take the leading term. In this case, Our three operators have the simple form: 8.2 The Phase Representation 149

d l SZ (i ) → dφ − 2

+ l iφ S e− → 2 µ ¶ l iφ S− e (8.2.16) → 2 µ ¶ The commutation relation between the raising and lowering operators is now zero: + [S , S−] = 0 We can use this representation to show how the qubit Hamiltonian described in chapter 3 can be derived from electronic theory, in the limit where the size of both the island and reservoir become large. If they are not both infinite, we shall need to calculate the corrections to this. We take an expansion of the differential forms of the raising and lowering operators. For the case of an infinitely large spin, we took the limit described in eq. 8.2.13. Here we SZ find the next order terms by expanding in l/2 .

1 1 2 2 + l Z iφ l Z S = + S e− S 2 2 − µ ¶ µ ¶ Z Z 2 Z Z 2 S (S ) iφ S (S ) = l/2 1 + + . . . e− 1 + . . . l − 2l2 − l − 4l2 µ ¶ µ ¶ Z Z Z 2 Z 2 S iφ iφ S (S ) iφ (S ) iφ = l/2 1 + e− e− e− e− l − l − 4l2 − 4l2 − µ (SZ e iφSZ ) − . . . − l2 ¶ (8.2.17)

To combine these terms we must commute the exponential past the SZ operators.

Z iφ Z iφ + iφ 1 (S e− S ) e− S = l/2 e− (1 + + + . . .) (8.2.18) l − l2/2 4l2 µ ¶ SZ Our original expansion was an expansion in l/2 . Commuting the exponential has given 1 terms like l/2 , so we need to consider the limit. The assumption in eq. 8.2.13 means that l the only states that contribute are those such that N l l 2 . We therefore consider | − 2 | ≤ 2 l 1 2 N to be of order O((l/2) 2 ) and discard terms of order O¡((l¢/2)− ). − 2 150 An Electronic Derivation of The Qubit Hamiltonian

Z iφ Z + l 1 iφ (S e− S ) 2 S ( + )e− + O( ) ' 2 2 − l l Z iφ Z l 1 iφ (S e S ) 2 S− ( + )e + O( ) (8.2.19) ' 2 2 − l l

8.2.3 Phase Representation of BCS States

Instead of treating the superconductors as spins, we consider the BCS state. As has been discussed in chapter 5, the ω V limit of the BCS state is simply the spin coherent state ¿ l, z (eq. 5.3.7). Instead of considering the superconductor to be in a state in the Hilbert | i space spanned by l, N (or equivalently, by l, z ), we consider the space spanned by the | i | i BCS states. The BCS states, like the spin-coherent states, are parameterised by a phase φ, and can be written:

BCS, φ = (u + v eiφ) l, 0 (8.2.20) | i k k | i Yk By integrating over φ we can extract a number eigenstate from the BCS states:

2π iφN BCS, N = BCS, φ e− dφ | i | i Z0

+ N = uk (SBCS) l, 0 (8.2.21) Ã ! | i Yk + where SBCS is an operator that raises the number of Cooper pairs by one but does not distribute the extra Cooper pair over all levels k equally.

+ vk SBCS = ck† c† k (8.2.22) uk ↑ − ↓ Xk The number states described above are eigenstates of the operator SZ , and the operator + SBCS raises the number of the state. As in the case of spin states, we have a phase and number state, and it might be desirable to describe the superconductor in terms of these states rather than the spin states, which do not take into account the distribution functions uk, vk. 8.2 The Phase Representation 151

In order to investigate whether this would be a sensible description, we ﬁrst need to calculate the commutation relations of these operators.

Z + 0 1 vk [S , SBCS] = [ (ck† 0 ck0 + c† k0 c k0 1), ck† c† k ] 0 2 ↑ ↑ − ↓ − ↓ − uk ↑ − ↓ kX, k vk = ck† c† k uk ↑ − ↓ k X+ = SBCS (8.2.23)

Z We ﬁnd that two of the commutation relations are as desired: [S , S§ ] = S§ . BCS § BCS However, when we calculate the third, we ﬁnd:

+ 0 vk vk∗ [SBCS, SB−CS] = [ c† c† , c k0 ck0 ] k k − ↓ ↑ 0 uk ↑ − ↓ uk∗ kX, k 2 vk † 0 † 0 = | |2 (ck0 ck + c k0 c k 1) uk ↑ ↑ − ↓ − ↓ − Xk | | = 2SZ (8.2.24) 6

The operators do not form a closed basis, and so we cannot form an exact representation as we have done for the spin operators. We can, however, attempt to ﬁnd an approximate representation. We wish to produce a phase coordinate representation for these states, as described in sections 8.2 and 8.2.1. Following the procedure we used before, we examine the matrix Z + elements of the operators S , S , S− with BCS states with diﬀerent φ.

Z d l BCS, φ S BCS, φ0 = i BCS, φ BCS, φ0 h | | i dφ − 2 h | i µ ¶ + iφ BCS, φ S BCS, φ0 = u v e− BCS, φ BCS, φ0 + D h | | i k k h | i Xk iφ BCS, φ S− BCS, φ0 = u v e BCS, φ BCS, φ0 + D h | | i k k h | i k X 4 vk k D = 2 (8.2.25) ( Pukvk) k P This shows that, as before, we can make the connection: 152 An Electronic Derivation of The Qubit Hamiltonian

d l SZ (i ) (8.2.26) ⇒ dφ − 2 In addition, we can make the approximate connection:

+ iφ S ukvk e− ∼ Ã ! Xk iφ S− ukvk e (8.2.27) ∼ Ã ! Xk where D is a measure of the error incurred by making this approximation. We can estimate the size of this error term. We use the deﬁnition for v 2, and make | k| the usual approximation of the sum by an integral:

2 4 1 2ξk ξk vk = 1 + 4 − Ek Ek Xk Xk µ ¶ ~ωc 1 2ξ ξ2 = (0)V ol dξ 1 + N 4 − (ξ2 + ∆2)1/2) (ξ2 + ∆2)1/2) ~ZωC µ ¶ − ∆ 1 ~ωc = (0)V ol ~ωc tan− (8.2.28) N − 2 ∆ µ ¶ with V ol being the volume of the system. From the gap equation, we know ukvk = ∆/V . In the weak coupling limit, we ﬁnd: P

~ωc D = V 2 (0)V ol N ∆2 d ~ωc = λ2 (8.2.29) ∆ ∆ where in the last line we have written D in terms of the dimensionless coupling constant λ = V ol (0)V and the level spacing d = 1/ (0)V ol. N N This term goes to zero as the superconductor becomes larger (as 1/V olume), so in the bulk limit the above forms for the operators are exact. 1 If we consider D in the strong coupling limit, we expand tan− in ~ωc/V , and ﬁnd:

2 V (0)V ol ~ωc D = N 2 ∆2 1 = (8.2.30) l 8.3 Island - Reservoir Tunnelling 153

where we have use ~ωc/d = l/2 to regain the spin result. So we see that the term indicating the size of the error in expressing the operators + S , S− as differential operators acting on a phase, D, goes to zero when we take the size of the system to infinity, even in the weak coupling limit (as it also does in the strong coupling limit). We are therefore justified in saying that we can approximate the operators + iφ iφ S , S− as proportional to e− , e , and that this approximation gets better as the size of the superconductor becomes larger. In particular, we note that this representation is not a vagary of the spin representation or strong coupling limit, but a general feature. The large size limit of the operators in the spin case (eq. 8.2.16) and the BCS case (eq. 8.2.26 8.2.27 are identical up to a constant.

8.3 Island - Reservoir Tunnelling

With the spin operators written in terms of differential operators, we are now ready to write the Hamiltonian for our superconducting charge qubit in a differential operator form. Initially we consider tunnelling between one finite and one bulk superconductor. We write the Hamiltonian in four parts:

H = H1 + Hr + HC + HT (8.3.1) where H1 is the standard BCS Hamiltonian for the island,

kF +ωc

H1 = (²k µ)c† ck,σ V1 c† 0 c† 0 c k ck (8.3.2) k,σ k k − ↓ ↑ − − 0 ↑ − ↓ k,σ k,k =kF ωc X X− The term Hr is the standard BCS Hamiltonian for the reservoir. The Charging Hamil- tonian measures the electrostatic energy due to charges on the island, as discussed in chapter 3.

4e2 H = (N n )2 (8.3.3) C 2C 1 − g where N1 represents the number of pairs on the island, and ng allows us to incorporate a gate voltage. The ﬁnal term is the pair - tunnelling term derived in section 1.8: 154 An Electronic Derivation of The Qubit Hamiltonian

HT = T ck† c† kc qcq + cq†c† qc kck − − − − − Xk,q (8.3.4) where the subscripts k and q refer to the island and reservoir superconductors respectively. If we represent both superconductors by spins, the Hamiltonian can be written in terms of two sets of spin operators.

Z + H = 2(² µ )S V S S− 1 F 1 − 1 1 − 1 1 1 Z + H = 2(² µ )S V S S− r F r − r r − r r r 4e2 H = (SZ n )2 C 2C 1 − g + + H = T S S− + S−S (8.3.5) T − 1 r 1 r ¡ ¢ We would like to consider the case where we have one small superconductor (the island, 1) and one bulk superconductor (the reservoir, r). We can treat the large superconductor in two ways; as the large - spin limit of our spin representation, or as a BCS superconductor. As described in section 8.2, we get the same result up to a multiplicative constant. When the reservoir is considered inﬁnitely large, we can replace the raising and lowering operators + Sr , Sr− by the diﬀerential forms (eq. 8.2.16). The tunnelling operator then becomes:

lr + iφr iφr H = T (S e + S−e− ) (8.3.6) T − 2 1 1

The eﬀect of considering the reservoir as a BCS superconductor is just to replace the

lr term 2 with ukvk. Combining the tunnelling Hamiltonian with the charging energy k found in sectionP 3.1.2, we ﬁnd the Hamiltonian for a single qubit is:

2 H = E0 (N n0 ) C 1 − g lr + iφr iφr T (S e + S−e− ) (8.3.7) − 2 1 1 where EC0 and ng0 are the renormalised charging energy and gate charge which take into account the terms in H linear and quadratic in SZ , and N = SZ l1 . 1 1 1 1 − 2 8.3 Island - Reservoir Tunnelling 155

We see that this has the same form as the broken-symmetry Hamiltonian discussed in 5.5.1; the presence of the bulk superconductor has broken the symmetry of the small one. The eigenstates are no longer the l , N states, and there will be a non-zero pairing | 1 1i parameter in the ground state. In the limit where the size of the island as well as the reservoir becomes large, the raising and lowering operators for the island also take the form 8.2.16, and eq. 8.3.7 becomes:

2 H = E0 (N n0 ) C 1 − g lrl1 i(φ1 φr) i(φ1 φr) T (e − + e− − ) − 4 2 lrl1 = E0 (N n0 ) T cos(φ φ ) (8.3.8) C 1 − g − 2 1 − r

We have regained the Hamiltonian (eq. 2.2.12) obtained by quantising the Josephson relations. Thus we see that eq. 2.2.12 is the semiclassical limit of 8.3.7, which becomes valid as the island becomes large as well as the reservoir. + The corrections to this can be found by using the form for S1 , S1− found in section 8.2.2:

2 lr H = E0 (N n0 ) T (l + 1) cos(φ φ ) C 1 − g − 2 1 1 − r ½ l1 2 l1 2(N1 ) 2(N1 ) − 2 cos(φ φ ) − 2 sin(φ φ ) (8.3.9) − l 1 − r − l i 1 − r 1 1 ¾ We have written the correction terms above as cosine and sine terms with velocity (SZ ) dependent pre-factors. Note that in this form it is not so obvious that the Hamiltonian is Hermitian (it is) but this form shows the sine and cosine dependence on phase of the correction terms explicitly.

These corrections destroy the periodicity with ng in the energy levels (ﬁgure 8.1). It then becomes relevant what the absolute value of ng is (rather than just its value modulo 1). We can consider the extra terms in the expansion to be corrections to the existing cosine term. In this case, using the analogy of a particle in one dimension, the extra terms give rise to a change in the cosine potential which depends on SZ = N l1 , i.e. , these 1 − 2 terms are a velocity dependent potential. The tunnelling rate is then dependent on the value of SZ , i.e. the rate of tunnelling depends on the number of Cooper pairs on the 156 An Electronic Derivation of The Qubit Hamiltonian

0 2 4 6 8 10 12 14

–2

Figure 8.1: Energy levels of a charge qubit as a function of ng including the finite size correction to the tunnelling energy. The dashed lines indicate the uncorrected energy levels. The parameters are EC = 10 EJ = 1 and l = 100 8.4 Inter - Island Tunnelling 157 island. This can be seen in figure 8.1, where the energy gap between the ground and first excited states depends on ng. Alternatively, we can consider the terms to be corrections to the charging energy - they represent a position dependent mass (a phase - dependent capacitance). Although this interpretation is perhaps less natural, as the correction terms are proportional to the tunnelling matrix element T , it might be useful when E E . J À C The variation of the level splitting with ng, as well as being of interest from a fundamental point of view, offers the possibility of control of the effective Josephson tunnelling with ng, and as such may be of interest in quatum computing applications.

8.4 Inter - Island Tunnelling

The next case we consider is that of two small superconductors with tunnelling between them. We treat both as ﬁnite quantum spins. The tunnelling Hamiltonian contains terms like:

+ l1 Z iφ1 l1 Z S1 S2− = + S1 e− S1 r 2 r 2 −

l2 Z +iφ2 l2 Z S2 e + S2 (8.4.1) r 2 − r 2 We re-arrange this:

+ l1l2 2 Z 2 Z 4 Z Z S S− = 1 + S S S S 1 2 4 l 1 − l 2 − l l 1 2 r 1 2 1 2 iφ1 +iφ2 e− e l l 2 2 4 1 2 1 SZ + SZ SZ SZ (8.4.2) 4 − l 1 l 2 − l l 1 2 r 1 2 1 2 As before, we expand and combine the terms, discarding those of order ( 1 ) and O l2 higher (assuming l1 and l2 are of the same order).

+ l1l2 iφ 1 1 iφ S S− = e− + ( + )e− 1 2 4 l l ½ 1 2 2 Z iφ Z 2 Z iφ Z (S e− S ) (S e− S ) (8.4.3) −l2 1 1 − l2 2 2 1 2 ¾ where we have written φ = φ φ . The tunnelling Hamiltonian is then given by: 1 − 2 158 An Electronic Derivation of The Qubit Hamiltonian

1 H = T (l l + l + l ) cos φ T − 2 1 2 1 2 ½ l l l l 2 (SZ )2 + 1 (SZ )2 cos φ 2 SZ + 1 SZ sin φ (8.4.4) − l 1 l 2 − l i 1 l i 2 1 2 1 2 ¾ ¡ ¢ ¡ ¢ Z Z As before, the tunnelling energy is no longer constant with respect to S1 and S2 , but varies as the number of Cooper pairs on each island changes. Chapter 9

Conclusion

In this thesis we have discussed some of the advantages of quantum computing and discussed some of the problems that the construction of a quantum computer might face. We have described how constructing qubits from superconductors might help with these problems. We have detailed theories of superconductivity, both phenomenological and microscopic and shown how these lead to the Josephson effect, where the current across a superconducting tunnel junction is proportional to the sine of the difference between the phases of the superconducting condensates on either side of the junction. We have also described how for small superconducting grains, finite size effects can lead to the parity effect, where the condensation energy and pairing parameter are dependent on whether there is an odd or even number of electrons on the grain. Experimental progress on superconducting qubits has been reviewed, and the basic circuit layout of a charge qubit described. A Hamiltonian for such a circuit was derived, by treating the Josephson equations as classical equations of motion, and then quantising them in the usual manner by constructing a classical Hamiltonian that leads to those equations, and then elevating the conjugate variables to operators. We explained how if the charging energy is much larger than the tunnelling energy, the circuit can function as a two- level system, and investigated the dynamics of this two level system. We briefly discussed the effect of noise on a charge qubit, and presented a general method for evaluating the effect of discrete noise in the Josephson energy. We examined the charging energy of various relevant circuits, in particular focussing on the effect of connecting a reservoir to small islands. We found a general computational

159 160 Conclusion method to establish the correct form for the charging energy by taking the limit where the capacitance between the islands and the outside world goes to zero and the capacitance between the reservoir and the outside world goes to zero. These detailed calculations allowed us to construct a quantum Hamiltonian from the charging energy and the tunnelling energy across each junction. We have shown that this näive method indeed produces the correct equations of motion for each junction, and confirmed that the gate voltages applied to each island act as independent tunings to the ‘zero position’ of the individual charging energies, as is generally assumed. We described the energy levels of a coupled two-qubit system and described the operations necessary to test a Bell inequality in this system. We showed how passing a magnetic flux through the circuit lead to oscillations in both time and flux. We showed that these oscillations can be considered as being caused by interference between virtual tunnelling paths between states and showed how diagrams could be a useful visual aid in ‘keeping track’ of perturbative expansions of the occupation probabilities of each state. We described the strong coupling limit of the BCS Hamiltonian, where the sums over electron creation and annihilation operators can be replaced by operators representing large quantum spins. We then solved this Hamiltonian in both the exact and mean- field limits, where we found the ground state was the spin coherent state. As part of this solution, we investigated the rotation of finite spin operators. We investigated the relationship between the solutions of the spin Hamiltonian and the solutions of the full Hamiltonian, exact and mean-field. We increased our confidence in the model by showing that the spin solutions are equal to the full solutions in the limit where the interaction energy is much larger than the cutoff energy. We showed how this simple model could be used to easily investigate various phenomena. We described an analogue of the quantum optical effects of superradiance where the current out of an array of Cooper pair boxes is proportional to the square of the number of boxes. We showed that this effect remains present when the reservoir is in the spin coherent state. This effect illustrates clearly the quantum mechanical nature of the Cooper pair boxes, and does so using only current measurements, rather than the experimentally more difficult single-shot measurements. We also described an analogue of quantum revival, where the coherent oscillations of a Cooper pair box coupled to a reservoir decay, only to revive again later. The revival is due to the fact that the Cooper pairs are discrete, and so the phase information is transferred Conclusion 161 to the levels of the reservoir. We described a similar two-qubit effect, where the oscillations of two entangled Cooper pair boxes decay and revive. If the entanglement between the boxes is calculated, we find that the entanglement also decays and then revives. This effect not only demonstrates the quantum nature of the Cooper pair boxes, but also of the reservoir, and it is possible that this behaviour could be used as a probe of the reservoir. We described how quantum spins can be represented using a wavefunction that is a function of phase. Rather than the standard derivation which treats the Josephson relations as (classical) equations of motion to be quantised, we derived these equations directly from a microscopic electronic Hamiltonian, using the spin-Hamiltonian described in chapter 5. The form for the spin operators in the phase representation were found. It was found that in the limit of large size, the Hamiltonian found by quantising the Josephson equations was regained. Corrections to this phase Hamiltonian were described. In this thesis we have developed a simple model of superconductors that can be easily applied to the study of various interesting aspects of superconducting systems. We have used this model to derive the standard macroscopic qubit Hamiltonian directly from a quantum description of the electrons. This effectively justifies the use of the ‘quantised Josephson junction Hamiltonian’ that is commonly used to study these systems. The microscopic derivation allowed us to calculate the corrections to this Hamiltonian, and show how these corrections affect the energy levels of the system. We have also investigated some interesting new phenomena (superradiance and revival) in superconducting systems, which in principle could be observed by measuring the occupation probabilities of the Cooper pair boxes rather than requiring single - shot measurements. Given that coherent oscillations between two qubits have been observed, it is hoped that the effects described above could be observed and illuminate the quantum behaviour of these systems. Superconductivity remains a fascinating and broad subject nearly a century after its first discovery. In particular the macroscopic quantum coherent nature of the electron condensate offers an intriguing glimpse of the connections between the microscopic world of quantum mechanics and the large-scale everyday macroscopic world. This is clearly seen in superconducting qubits, where superpositions of macroscopically differing states can be prepared. For these reasons it is clear that mesoscopic superconducting systems such as charge qubits will continue to be of great interest for quantum computing as well as condensed matter physics, and also for demonstrating and investigating basic aspects of quantum 162 Conclusion mechanics. Appendix A

Eigenstates of the Two Qubit Hamiltonian

In this Appendix we examine the eigenstates and eigenvalues of a coupled two-qubit system. In particular we ﬁnd these exactly for the case when all the tunnelling matrix elements are real and use perturbation theory to generate approximate expressions when the tunnelling matrix elements are complex, up to third order in T/EC12.

A.1 Exact Eigenstates with Real T12

We are interested in the Hamiltonian of the two - qubit system when the gate charges are set to the degeneracy point. We can solve this analytically when all the tunnelling matrix elements are real, i.e. when there is an integer or half integer number of ﬂux quanta through the circuit (see section 2.3).

0 T T 0 − 1 − 2  T1 EC12/2 T12 T2  H = − − − − (A.1.1)  T T E /2 T   2 12 C12 1   − − − −   0 T T 0   − 2 − 1   

The eigenvalues v are given below. | ni 163 164 Appendix A. Eigenstates of the Two Qubit Hamiltonian

1 1 1 e = E T + (E 2T )2 + 16(T T )2 1,4 −4 C12 ∓ 2 12 4 C12 § 12 1 § 2 1 1 1p e = E T (E 2T )2 + 16(T T )2 (A.1.2) 2,3 −4 C12 ∓ 2 12 − 4 C12 § 12 1 § 2 p In the limit of small tunnelling energies,the eigenvectors v and v are the symmetric | 1i | 4i and anti-symmetric combinations of 00 and 11 , and v and v are the symmetric | i | i | 2i | 3i and anti-symmetric combinations of 01 and 10 . When the tunnelling energies go to | i | i zero,e go to zero, and e go to E /2 as would be expected. 1,4 2,3 − C12

1 1

 e1/(T1 + T2) 1  e2/(T1 + T2) 1 v1 = − v2 = − 2 e 2 | i 2(e1) | i 2+ 2( 2)  e1/(T1 + T2) 2 + 2  e2/(T1 + T2) (T +T )2   (T1+T2)   1 2 −  −  r  1  q  1          1 1

 e3/(T1 T2) 1  e4/(T1 T2) 1 v3 = − − v4 = − − 2 e 2 | i 2(e3) | i 2+ 2( 4)  e3/(T1 T2)  2 + 2  e4/(T1 T2)  (T −T )2  −  (T1 T2)  −  1 2   −   r  1  q  1   −   −      (A.1.3)

These expressions for the eigenvectors contain terms like e2/(T1 + T2), which go to inﬁnity as the tunnelling energies go to zero. When doing calculations, it is helpful to note that each of the eigenvectors can be rewritten in a diﬀerent form:

e2/(T1 + T2) e1/(T1 + T2)

 1  1  1  1 v1 = v2 = 2 e 2 | i 2(e2) | i 2+ 2( 1)  1  2 + 2  1  (T +T )2   (T1+T2)   1 2     r e /(T + T ) q e /(T + T )  2 1 2   1 1 2      e /(T T ) e /(T T ) 4 1 − 2 3 1 − 2  1)  1  1  1 v3 = v4 = 2 e 2 | i 2(e4) | i 2+ 2( 3)  1  2 + 2  1  (T −T )2  −  (T1 T2)  −  1 2   −   r  e /(T T ) q  e /(T T ) − 4 1 − 2  − 3 1 − 2      (A.1.4) A.2 Perturbation in T12 165

These two forms for the eigenvalues are equal, and the most convenient expression can be chosen for each calculation.

A.2 Perturbation in T12

In section 4.4 we considered the Hamiltonian with no inter-qubit tunnelling as our unperturbed Hamiltonian, and then did perturbation theory in a complex T12.

0 H = H + H0 0 T T 0 0 0 0 0 − 1 − 2  T1 EC12/2 0 T2   0 0 T ∗ 0  = − − − + − 12 (A.2.1)  T 0 E /2 T   0 T 0 0   2 C12 1   12   − − −   −   0 T T 0   0 0 0 0   − 2 − 1        The unperturbed eigenvalues (e0 ) and eigenvectors ( v0 ) can be trivially found from n | ni the above expressions by setting T = 0. The ﬁrst order correction e1 = v0 H v0 12 n h n| 0| ni to the energies can then be calculated.

1 0 0 e1 = v1 H0 v1 (A.2.2) h | | i 0 2 (e1) 1 = (T + T ∗ ) (A.2.3) 12 12 2 0 2 (T1 + T2) 2(e1) 2 + 2 (T1+T2) q This expression is correct to all orders in T1 and T2, but if we want to do an expansion (we do!) we ﬁnd that:

1 1 e0 = E + (E )2 + 16(T T )2 1,4 −4 C12 4 C12 1 § 2 1 Ep (T T )2 1/2 = E + C12 1 + 16 1 § 2 −4 C12 4 (E )2 µ C12 ¶ 2(T T )2 1 § 2 (A.2.4) ≈ EC12

Similarly, 166 Appendix A. Eigenstates of the Two Qubit Hamiltonian

1 1 e0 = E (E )2 + 16(T T )2 2,3 −4 C12 − 4 C12 1 § 2 1 Ep (T T )2 1/2 = E C12 1 + 16 1 § 2 −4 C12 − 4 (E )2 µ C12 ¶ E 2(T T )2 C12 1 § 2 (A.2.5) ≈ − 2 − EC12 We can now calculate the correction to the unperturbed energies:

1 0 0 e = v H0 v 1 h 1| | 1i 2 2 1 2(T1 + T2) 4(T1 + T2) − (T + T ∗ ) 1 + ≈ 12 12 (E )2 (E )2 C12 µ C12 ¶ 2 2(T1 + T2) (T12 + T12∗ ) 2 (A.2.6) ≈ (EC12)

0 When calculating the correction to e2 it is more convenient to use the second form (eq.

A.1.4) for the unperturbed eigenvectors. To second order in T1, T2:

1 0 0 e = v H0 v 2 h 2| | 2i 1 = (T + T ) 12 12∗ 0 2 2(e2) 2 + 2 (T1+T2) q 1 (T + T ) 4(T + T )2 − 12 12∗ 1 + 1 2 ≈ 2 (E )2 µ C12 ¶ 2 (T12 + T12∗ ) 2(T1 + T2) (T12 + T12∗ ) 2 (A.2.7) ≈ 2 − (EC12) We calculate the corrections to the remaining two energies in the same way;

2 1 (T12 + T12∗ ) 2(T1 + T2) e3 + (T12 + T12∗ ) 2 ≈ − 2 (EC12) 2 1 2(T1 + T2) e4 (T12 + T12∗ ) 2 (A.2.8) ≈ − (EC12) The energy diﬀerences, and therefore the frequencies, can be easily calculated from this. Appendix B

Limit of BCS Self Consistency Equations

In this Appendix we solve the BCS gap equations exactly at zero temperature, using a top hat density of states. This then allows us to take the strong coupling limit where the pairing potential is much larger than the cutoff energy. We show that solving the BCS gap equations with the energy levels and then taking the strong coupling limit yields the same results as assuming all the energy levels to be equal initially, and then finding the mean-field solution

As confirmation of the validity of our spin model, we wish to compare it with the standard BCS theory. In particular, we wish to check that solving the gap equations from the spin model gives the same result (5.4.19, 5.4.20) as if we solve the usual BCS gap equations (1.6.8,1.6.9) first, and then take the limit. We wish to take the limit where the single-electron energy levels, ²k are equal. To do this we consider a density of states which is a top-hat function centred around ²F with width ω and height lr/ω. Taking the limit ω 0 ensures that the interaction occurs in a narrow region around ² whilst ensuring → F the integral is over a constant number of levels, lr. We have two integral equations that must be solved self-consistently. The first is an equation for the pairing parameter, ∆:

²F +ω ∆ 1 ∆ = d² 1 (B.0.1) V 2 ((² µ)2 + ∆ 2)) 2 ²FZ ω − − | | 167 168 Appendix B. Limit of BCS Self Consistency Equations and the second is an equation for the average number of Cooper Pairs, N¯:

²F +ω ¯ 1 ² µ N = d² 1 − 1 (B.0.2) 2 Ã − ((² µ)2 + ∆ 2)) 2 ! ²FZ ω − − | | As we are at zero temperature, both these integrals can be done exactly:

1 l (ξ + ω/2) + (ξ + ω/2)2 + ∆ 2 = ln F F | | (B.0.3) 2 2 V 2ω Ã(ξF ω/2) + (ξF ω/2) + ∆ ! − p − | | l ω N¯ = (ξ + ω/2)p2 + ∆ 2 + (ξ ω/2)2 + ∆ 2 (B.0.4) ω 2 − F | | F − | | ³ p p ´ Again, ξ is shorthand for ² µ. These can be rearranged to give two equations for F F − ∆ 2 (B.0.5, B.0.6), or two equations for ξ2 (B.0.8, B.0.9). The equations for ∆ 2 are: | | F | |

(x + 1)2 w 2 (x 1)2 ∆ 2 = (ξ )2 1 + − 1 (B.0.5) | | F (x 1)2 − 2 (x + 1)2 − µ − ¶ ³ ´ µ ¶ N¯(l N¯) ∆ 2 = − 4ξ 2l2 ω2(l 2N¯)2 (B.0.6) | | l2(l 2N¯)2 F − − − ¡ ¢ 2 Equating the two and rearranging gives us an expression for ξF :

ω 2 (x + 1)2 4n(1 n) (x 1)2 ξ 2 = 1 + 4(1 2n)2 − − + 1 (B.0.7) F 2 (x 1)2 − − (1 2n)2 − (x + 1)2 ³ ´ µ − ¶ Á µ − ¶ 2w ¯ V l N where x = e and n = l . If we rearrange eqns. (B.0.3, B.0.4) for ξF we get:

ω 2 (x + 1)2 (x + 1)2 ξ 2 = ∆ 2 (B.0.8) F 2 (x 1)2 − | | 4x ³ ´ − ω 2 (1 2n)2 ξ 2 = (1 2n)2 + ∆ 2 − (B.0.9) F 2 − | | 4n(1 n) ³ ´ − Which yield:

ω 2 (x + 1)2 (1 2n)2 (x + 1)2 ∆ 2 = (1 2n)2 − + (B.0.10) | | 2 (x 1)2 − − 4n(1 n) 4x ³ ´ µ − ¶ Á µ − ¶ From eqns. (B.0.7, B.0.10), it is clear that the correct limit to take is ω/V 0. Taking → this limit of eqns. (B.0.7, B.0.10), we get: Appendix B. Limit of BCS Self Consistency Equations 169

V 2l2 ξ 2 = (1 2n)2 (B.0.11) F 4 − ∆ 2 = V 2l2n(1 n) (B.0.12) | | −

That is, we have regained the forms for ∆ and ξ given by the spin model (5.4.19, 5.4.20).

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