Superconducting Charge Qubits. Denzil Anthony Rodrigues

Transcription

1 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 003 Word Count: 36, 000

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3 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.

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5 To my family.

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7 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.

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9 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.

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11 You can never take your badge off. Jamie Walker.

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13 Contents Superconductivity and The Josephson Effect. Introduction to the Thesis Introduction to Superconductivity Ginzburg-Landau Theory The Ginzburg-Landau Equations The Josephson Effect in Ginzburg-Landau Theory Creation and Annihilation Operators The Fermi Sea BCS and BdG The Josephson Effect The Pair Tunnelling Hamiltonian Small Superconductors The Parity Effect The Richardson Exact Solution The Superconducting Charge Qubit 3. Experimental Progress The Josephson Equations for Small Circuits The Classical Lagrangian The Classical Hamiltonian and the Conjugate Variables Quantising the Classical Josephson Equations Tuneable Josephson Energy The Two-Level System Qubit Dynamics and the NOT Gate Decoherence and Noise xiii

14 xiv CONTENTS.6. Gaussian Noise Discrete Noise in the Josephson Energy The Josephson Circuit Hamiltonian The Capacitance of Qubit Circuits Single Junction with Gate Voltage Single Junction with Gate Voltage and Self-Capacitance The Qubit Circuit Multiple Islands Quantising the Josephson Relations Single Junction with Gate Voltage Single Junction with Gate Voltage and Self-Capacitance The Qubit Circuit The Dynamics of a Two Qubit System 7 4. The Two Qubit Circuit Energy Levels Two-Qubit Gates A Bell Inequality Virtual Tunnelling Path Interference Finite Superconductors as Spins The Strong Coupling Approximation Quantum Spins The Majorana Representation Spin l in terms of Spin The Coherent State The Majorana Sphere Mean Field Spin Solution Rotation of Spin Operators The Ground State Self Consistency Comparison to the BCS Theory

15 CONTENTS xv 5.5 Solving the Spin Hamiltonian Exactly Broken Symmetry: Finite size Broken Symmetry: Infinite Size Expanding the Richardson Solution to First Order The Super Josephson Effect 9 6. The Cooper Pair Box Array Superradiant Tunnelling Between Number Eigenstates Superradiant Tunnelling: Number State to Coherent State Superradiant Tunnelling Between Coherent States Revival 9 7. The Superconducting Analogue of the Jaynes-Cummings Model Quantum Revival of the Initial State Revival of Entanglement An Electronic Derivation of The Qubit Hamiltonian 4 8. Introduction The Phase Representation Phase Representation of Spins The Limit of Large Size Phase Representation of BCS States Island - Reservoir Tunnelling Inter - Island Tunnelling Conclusion 57 A Eigenstates of the Two Qubit Hamiltonian 6 A. Exact Eigenstates with Real T A. Perturbation in T B Limit of BCS Self Consistency Equations 65

16 xvi CONTENTS

17 List of Figures. The Ginzburg-Landau free energy as a function of the order parameter ψ.. 7. The Temperature dependence of the superconducting gap (T ) measured in units of the zero-temperature gap (0) A simple circuit diagram for a charge qubit consisting of two regions of superconductor connected by a Josephson junction with critical current I C whose capacitance is represented by a capacitor of capacitance C 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 A double junction consisting of a ring of superconductor broken by two junction and threaded by a flux Φ. The two paths from point to point are labelled c a and c b The charging energy as a periodic function of n g The qubit energy levels as a periodic function of n g The avoided crossing at half-integer values of n g A not gate takes a state 0 and returns a state, and vice versa Oscillations in the probability of the qubit being in state (solid line) or state 0 (dotted line) The probability of a Cooper pair box being in the state as a function of time, with Gaussian noise in the Josephson energy The probability of a Cooper pair box being in the state as a function of time with discrete noise in the Josephson energy Single Voltage with Gate Voltage Circuit Diagram xvii

18 xviii LIST OF FIGURES 3. Single junction with gate voltage and self-capacitance, which is represented by a capacitive coupling to a third region, labelled w (for world ) Two Qubit circuit diagram. The system consists of two islands (, ) 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 The energy levels of a two qubit system as a function of n g = n g, when the tunnelling between the islands is zero. The parameters of the system are E C = E C = 00, E C = 0, T = 0, T = The energy levels of a two qubit system as a function of n g = n g, when the tunnelling between the islands is included. The parameters of the system are E C = E C = 00, E C = 0, T = 0, T = 0, T = Oscillations between state 0 (solid line), 0, (dashed line) and the states 00 and (dotted lines). Halfway between oscillations, the system is in a maximally entangled state (vertical lines) The bases for measurements that violate a Bell inequality A two qubit circuit with tunnelling between the qubits and a magnetic flux passed through the circuit The Probability of the Two Qubit system being in the state 00 (upper plot) and 0 (lower plot). The horizontal axis is time and the vertical axis is the flux through the loop in units Φ 0 π. White represents high probability of occupation and black represents low probability The Probability of the Two Qubit system being in the state 0 (upper plot) and (lower plot). The horizontal axis is time and the vertical axis is the flux through the loop in units Φ 0 π. White represents high probability of occupation and Black represents low probability The Probability of the Two Qubit system being in each of the four charging states The change in frequency of the oscillations in the probability of being in the state The two second order (a, b) and third order (c, d) tunnelling processes involved in the oscillation

19 LIST OF FIGURES xix 4. The three third order tunnelling processes involved in the oscillation The oscillations of the two qubit system when the system is not in the limit T E C The Probability of the Two Qubit system being in the state 0 or i.e. the probability that the first qubit is occupied regardless of the occupation of the second. The z axis represents increasing probability The Majorana Sphere Each point on the Majorana sphere can be represented by its projection onto the complex plane, z Rotation of basis of spin operators An array of l b 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) Spin Coherent State Revival for N r = 0, l r = Decay and revival of a single Cooper pair island state, showing the degree of linear entropy Analytic asymptotic form for decay oscillations in the probability against time in units of π/e J (solid line) compared with the numeric evolution (dashed line) N r = 0, l r = Two Cooper Pair Box Spin Coherent State Revival for N r = 0, l r = Two Cooper Pair Box Spin Coherent State Revival for N r = 0, l r = 50, showing the degree of entanglement Energy levels of a charge qubit as a function of n g including the finite size correction to the tunnelling energy. The dashed lines indicate the uncorrected energy levels. The parameters are E C = 0 E J = and l =

20 xx LIST OF FIGURES

21 Chapter 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.. 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 [, ]. 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.

22 Superconductivity and The Josephson Effect The first theoretical demonstration that a quantum mechanical algorithm could outperform 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 algorithmically. 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 ( 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 simultaneously, and a series of n such bits (a register), each placed in a superposition, can represent all the numbers from 0 to n 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, 0]. 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 [] or atoms [] held in electromagnetic traps or cavities, impurities embedded in a substrate [3], or quantum dots formed from silicon nanostructures [4]. 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 [5] has become

23 . 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 [6, 7] and, in the case of NMR, even Shor s factoring algorithm has been performed [8]. 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 ( 0 4 ) 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.

24 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.. Introduction to Superconductivity In 90, Kammerlingh-Onnes developed a technique for liquifying Helium, and proceeded to investigate the properties of materials at the low temperatures that this allowed. In 9, he noticed that the electrical resistance of Mercury became very low below 4.K [9]. 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.K. 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. Superconductivity 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

25 .3 Ginzburg-Landau Theory 5 to exhibit superconductivity, including organic superconductors [0], magnesium diboride [], and carbon nanotubes [] among others. The superconducting state was found to have several distinctive characteristics other than zero resistance. In particular, the Meissner effect [3], 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 [4] which describe both the infinite conductivity and the exclusion of the magnetic field. A more complete phenomenological description is given by Ginzburg-Landau theory [5] 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 [6] 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 [7]. Excitations above the BCS ground state are described using Bogoliubov-de Gennes (BdG) theory [8]. 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 [9]. 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..3 Ginzburg-Landau Theory Ginzburg-Landau theory [5] is a phenomenological description of superconductivity. An expression for the free energy is given in terms of an order parameter, which must be

26 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..3. The Ginzburg-Landau Equations The Ginzburg-Landau equations can be derived from a spatially inhomogeneous microscopic theory [9], 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 + β ψ 4 (.3.) 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 T C, and non-zero below this temperature. Thus, α must be a function of temperature, going from negative to positive at T = T C. We expand α to first order in (T C T )/T C, and obtain: Where a and β are positive real constants. f = a (T C T ) T C ψ + β ψ 4 (.3.) 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. ψ = a β Below this temperature, the order parameter has a non-zero value (T C T ) T C. The subscript denotes the fact that this is the value for a bulk, or infinite, superconductor.

27 .3 Ginzburg-Landau Theory 7 Figure.: The Ginzburg-Landau free energy as a function of the order parameter ψ, for the cases when: a) α is positive (above T C ) and b) α is negative (below T C ). Above T C the minimum in free energy is at ψ = 0, whereas below T C, the free energy has a minimum at a non-zero value ψ = α/β. Note that as.3. 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 a position dependent order parameter, ψ(r). We add a kinetic energy term, m ˆp, where the canonical momentum operator ˆp = i e c A(r) and A(r) is the vector potential. We also add the contribution of the magnetic field, H(r) = A(r), to the energy. f = dr 3 α ψ(r) + β ψ(r) 4 + m ( ) i e A(r) ψ(r) + µ rµ 0 H(r) (.3.3) Note that eq..3.3 contains the expressions m and e, the effective 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: 0 = αψ + β ψ ψ m ) ( ie A ψ (.3.4) This is the first Ginzburg-Landau equation, and from this we see a natural length

28 8 Superconductivity and The Josephson Effect scale for the variation of the order parameter, ζ (T ) = respect to A(r) gives: m α. Minimising eq..3.3 with 0 = A e m (ψ ( i e A)ψ + ψ(i e A)ψ ) (.3.5) Inserting Ampere s law, A = µ r µ 0 J, we find an expression for the current J: J = e m (ψ ( i e A)ψ + ψ(i e A)ψ ) = i e m (ψ ψ ψ ψ ) e m ψ A (.3.6) This is the second Ginzburg-Landau equation. We can use eq..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: H = λ ψ H (.3.7) with λ m = µ rµ. First we consider the component of the field (H 0 e ψ Z ) perpendicular to 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..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)ˆx, we use the identity H = ( H) H to obtain: H x z = λ H x (.3.8) Which give us the expression for the field parallel to the field of a superconductor: H x (z) = H x (0)e z/λ (.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:

29 .3 Ginzburg-Landau Theory 9 J y (z) = c 4πλ H x(0)e z/λ (.3.0) 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 003 Nobel Prize in physics was awarded for such work [3, 5, 3]. 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..3. The Josephson Effect in Ginzburg-Landau Theory In 96, 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 ψ = ψ/ψ, then in the absence of magnetic fields, the condition for the minimisation of the free energy in one dimension is: 0 = ψ ψ ψ + m ζ (T ) d ψ dx (.3.) 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 ψ = at either end of the bridge. The Ginzburg-Landau free energy is independent of phase in the absence of gradients, and so we set ψ = e iφ at one end of the bridge and ψ = e iφ at the other. If the coherence length ζ is much greater than the length of the bridge, L, then the kinetic term is much greater than the other terms and we have 0 = d ψ dx, i.e. ψ x. The form for the order parameter that satisfies eq..3. and the boundary conditions at each end of the bridge is: ψ (x) = ( x ) e iφ + x L L eiφ (.3.)

30 0 Superconductivity and The Josephson Effect If we now calculate the current using the standard Ginzburg-Landau current expression eq..3.6, we find: I = I C sin (φ φ ) (.3.3) Where I C = e ψ m A L sustained by a junction of area A. is the critical current, the maximum supercurrent that can be Although this is a highly simplistic derivation done for the particular case of a constriction 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..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 3 r i ) = ψ(r, r, r 3 r i ) (.4.) 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 k and k, the two-particle state of the system is of the form: ψ(r, r ) = ψ k (r )ψ k (r ) ψ k (r )ψ k (r ) (.4.) Performing calculations with wavefunctions that have been symmetrised by hand such as eq..4. will obviously be very laborious, especially considering the vast number of

31 .5 The Fermi Sea 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 c k, c k 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, then c k 0 is a state with one electron in the state k with spin up, and c k c k 0 is a state with one electron in the state k and one electron in the state k (with correct anti-symmetry as given in eq..4.). Rather than describing the action of the operators in terms of their commutators [Â, ˆB] = Â ˆB ˆBÂ as we would for Bosons, we describe it in terms of the anti-commutation relations [Â, ˆB] + = Â ˆB + ˆBÂ: c kσ c k σ + c k σ c kσ = 0 c kσ c k σ + c k σ c kσ = 0 c kσ c k σ + c k σ c kσ = δ k,k δ σ,σ (.4.3) It is clear from eq..4.3 that c kσ c kσ = 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..5 The Fermi Sea The operators defined in section.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..4.3). The Hamiltonian of the system is just the sum of the single-electron energy levels: Ĥ fe = k,σ ɛ k c kσ c kσ (.5.)

32 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 ɛ k. The maximum occupied energy level is known as the Fermi energy, ɛ F. This state can be written as: ψ = c k σ 0 (.5.) k,σ where the product k 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 956, Cooper considered the behaviour of two electrons above a filled Fermi sea [6]. 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, ɛ k ɛ F < ω c, and

33 .6 BCS and BdG 3 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..6 BCS and BdG The reduced BCS Hamiltonian [7] describes a system with interactions between paired electrons: Ĥ BCS = k,σ (ɛ k µ)c kσ c kσ k,k V k,k c k c k c k c k (.6.) The difficulty in solving this Hamiltonian comes from the interaction term; this term is quartic rather than quadratic. Following [8], 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 c k c k does not deviate much from its average value, i.e. when c k c k c k c k c k c k. We write c k c k = c k c k ( c k c k c k c k ) and discard terms in the Hamiltonian that are quadratic in ( c k c k c k c k ). Ĥ BCS = k,σ (ɛ k µ)c kσ c kσ k,k V k,k ( c k c k c k c k + c k c k c k c k c k c k c k c k ) (.6.) We now introduce the pairing parameter k = k V k,k c k c k and write the meanfield Hamiltonian as: Ĥ BCS = k,σ (ɛ k µ)c kσ c kσ k k c k c k + c k c k k k (.6.3) We wish to diagonalise this Hamiltonian, i.e. to find operators γ k,σ so that the Hamiltonian can be written k,σ E k γ k,σ γ k,σ

34 4 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, = u k c k + v k c k, where u k, v k 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 anticommutation relations (eq..4.3). This requirement implies that u k + v k = (from [γ k, γ k ] + = ) and also implies that γ k = u kc k vk c k (from [γ k, γ k ] + = 0). We find the value u k, v k by noting that if the γ operators diagonalise the Hamiltonian, then: [ k E k γ k,σ γ k,σ, γ k, ] = [ĤBCS, u k c k + v k c k ] (.6.4) Equating these commutators leads directly to the equations for u k and v k : u k = ( + ɛ ) k E k v k = ( ɛ ) k E k E k = (ɛ k µ) + k (.6.5) The operators γ create and destroy elementary excitations of the system. Each γ k,σ creates a combination of particle and hole, which we call a quasiparticle. The form of E k 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 γ k,σ. The state defined by this property is the famous BCS state: BCS = k (u k + v k e iφ c k c k ) 0 (.6.6) where u k and v k are now considered real and we have explicitly included the phase of v k. In the mean-field Hamiltonian.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.

35 .6 BCS and BdG 5 k = k V k,k BCS c k c k BCS = k V k,k u k v k e iφ k = V k,k E k k (.6.7) If we make the approximation that the interaction potential V k,k is constant for ɛ k ɛ F < ω c, then k is also constant with this region, and zero outside. The gap equation becomes: = V ɛ F +ω c (ɛk µ) + (.6.8) ɛ F ω c The chemical potential must be chosen so that the average number in the ground state is correct N = BCS ˆN BCS. This leads to the equation: ɛ F +ω c N = ɛ F ω c (ɛ k µ) (ɛk µ) + (.6.9) 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 c sinh(/n (0)V ) ω ce N (0)V (.6.0) 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 noninteracting 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 Fermi occupation factor, γ k, σ γ k, σ = f k, σ, where f k, σ = /(e E k/k B T + ). We have a temperature dependent pairing parameter which is the thermally averaged expectation value V k,k c k c k. This leads to the two temperature self-consistency equations: k

36 6 Superconductivity and The Josephson Effect T/Tc Figure.: The Temperature dependence of the superconducting gap (T ) measured in units of the zero-temperature gap (0). = V ɛ F +ω c ɛ F ω c tanh E k K B T E k (.6.) and: ɛ F +ω c N = ɛ F ω c (ɛ k µ) tanh E k K B T E k (.6.) Although these temperature-dependent self consistency equations cannot be solved exactly, we can approximate the sum by an integral over ɛ and solve the equations numerically (figure.). We find that the gap drops to zero at a critical temperature, T C, which when calculated is given by (0) =.76T C. Solving the self-consistency equations means that we have a value for, and thus for the parameters u k, v k. With the parameters of the ground state (eq..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..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.6. 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

37 .7 The Josephson Effect 7 fluctuation in the number operator ( ˆN ˆN )/ ˆN goes to zero as the size of the 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 BCS, φ for clarity) by integration over the phase φ. We can reverse the process by summing over all N: π BCS, N = BCS, φ e iφn dφ = 0 ( ) u k (S + BCS )N l, 0 k BCS, φ = BCS, N e iφn (.6.3) N Where the operator S + BCS = k superconductor by one. v k u k c k c k increases the number of Cooper pairs on the 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 0 9 or more, this uncertainty relation between the phase and number will come into play when we consider superconductors of finite size..7 The Josephson Effect The Josephson effect [30] can be derived from a microscopic theory, which we shall summarise here. We begin with a Hamiltonian that describes two regions of superconductor, and the tunnelling between them. Ĥ = Ĥ + Ĥ + Ĥt (.7.) Ĥ and Ĥ are the BCS Hamiltonians (eq..6.) for each superconductor, and Ĥt is a Hamiltonian that describes the tunnelling of single electrons between two metals [38]:

38 8 Superconductivity and The Josephson Effect Ĥ t = k,q t k,q (c k c q + c k c q) (.7.) The single electron levels in the two separate superconductors are labelled by k and q, with a single electron tunnelling matrix element t k,q. The tunnelling term is treated as a perturbation on Ĥ + Ĥ. We assume Ĥ + Ĥ to be solved in the sense that we assume that the Hamiltonian for each superconductor can be written in the form E k γ k,σ γ k,σ where the γ s are the quasiparticle operators. k,σ We shall use the interaction picture 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 c k is determined by: dc k dt = i [Ĥ, c k ] (.7.3) We wish to incorporate a voltage difference V across the junction. We do this by replacing Ĥ with Ĥ ev c k,σ c k,σ where c k,σ c k,σ measures the number of electrons k,σ k,σ in superconductor. This is equivalent to setting the chemical potentials µ, µ on each side to the junction to differ by ev. If we represent the annihilation operator at zero voltage by c k and the operator at voltage V by c k, the time evolution of c k is given by: d c k dt = i [Ĥ, c k ] iev [ N, c k ] = i [Ĥ, c k ] + iev c k (.7.4) Substituting the trial solution c k = c k e i φ pair will turn out to be φ). (φ/ is chosen so that the phase of a Cooper e i φ dc k dt + i dφ dt c k = i φ ei [ Ĥ, c k ] + iev c k (.7.5) Comparing eq..7.3 with eq..7.4, we see that c k = c k e i φ is a solution if:

39 .7 The Josephson Effect 9 dφ dt = ev (.7.6) 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. γ k = u kc k v ks + c k γ k = u kc k + v ks + c k (.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 [(Ĥ + Ĥ), N ] = t k,q (c k c q c k c q). Time dependent perturbation theory gives the k,q state of the system at time τ: ψ(t) i t e ητ Ĥ It (t )dt ψ(0) (.7.8) where ĤIt 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 the quasiparticles given in eq 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, 4]: I = e t N (0)N (0) dɛ dɛ ɛ ɛ f(ɛ ) f(ɛ ) ɛ ɛ ev sin φ = I C sin φ (.7.9)