Quantum dynamics of periodically driven superconducting nanocircuits

Testo completo

(1)Quantum dynamics of periodically driven superconducting nanocircuits Alessandro Romito. SCUOLA NORMALE SUPERIORE.

(2) Scuola Normale Superiore di Pisa. Ph. D. Thesis. Quantum dynamics of periodically driven superconducting nanocircuits. Alessandro Romito. Supervisor Prof. Rosario Fazio. Pisa, 2005.

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(4) Contents Contents. I. Introduction. 1. 1 Charging effects in mesoscopic superconductors. 6. I. 1.1. The Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 1.2. Charging effect in small grains: The Coulomb blockade . . . . . . . .. 10. 1.3. Josephson effect in presence of Coulomb blockade . . . . . . . . . . .. 15. 1.4. Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 1.5. Quantum communication using one-dimensional systems . . . . . . .. 22. THE COOPER PAIR SHUTTLE. 2 Josephson current. 26 27. 2.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. Implementation with SQUIDs . . . . . . . . . . . . . . . . . . . . . .. 2.3. Shuttle dynamics: EJ , EJ. 2.4. (L). (R). 28 31. ≪ EC . . . . . . . . . . . . . . . . . .. 32. DC Josephson current . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 2.4.1. Effect of driving fluctuations . . . . . . . . . . . . . . . . . .. 42. -I-.

(5) 2.5. AC Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 Josephson current fluctuations 3.1. 3.2. 51. Full counting statistics for the Cooper pair shuttle . . . . . . . . . .. 52. 3.1.1. General formalism . . . . . . . . . . . . . . . . . . . . . . . .. 53. 3.1.2. Application to the Cooper pair shuttle . . . . . . . . . . . . .. 56. Current Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62. 4 Chaotic dynamics in the Cooper pair shuttle. II. 45. 66. 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67. 4.2. From classical to quantum dynamics in the chaotic Cooper pair shuttle 69 4.2.1. Diagrammatic perturbation theory . . . . . . . . . . . . . . .. 71. 4.2.2. Time reversal symmetry breaking and COE to CUE crossover 78. 4.3. Fidelity measurement in the Cooper pair shuttle . . . . . . . . . . .. 81. 4.4. The effect of gate voltage fluctuations . . . . . . . . . . . . . . . . .. 86. 4.5. Transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90. JOSEPHSON JUNCTION CHAINS. 5 Quantum communication through Josephson junction arrays. 93 94. 5.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95. 5.2. Quantum state transfer . . . . . . . . . . . . . . . . . . . . . . . . .. 97. 5.3. Measurement protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 104. Conclusions. 110. A Full Counting Statistics and charge statistics. 112. A.1 Construction of the full counting statistics . . . . . . . . . . . . . . . 113 A.2 Interpretation of the full counting statistics . . . . . . . . . . . . . . 116. - II -.

(6) B Quantum effects of classically chaotic systems. 120. B.1 Chirikov Standard Map . . . . . . . . . . . . . . . . . . . . . . . . . 125 B.1.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 B.1.2 Quantum Chaos . . . . . . . . . . . . . . . . . . . . . . . . . 127 B.2 Quantum Chaos Characterization . . . . . . . . . . . . . . . . . . . . 128 B.2.1 Level Repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . 129 B.2.2 Level Spacing Statistics . . . . . . . . . . . . . . . . . . . . . 130 B.2.3 Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 B.3 Random Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . 133 B.3.1 Gaussian Ensembles . . . . . . . . . . . . . . . . . . . . . . . 134 B.3.2 Superposition of two independent spectra . . . . . . . . . . . 138 C Microscopic derivation of the Josephson Hamiltonian. 140. D The Cooper pair shuttle at constant gate voltage. 145. E Phase dependent corrections to the classical diffusion. 148. Bibliography. 152. List of publications. 163. Acknowledgments. 166. - III -.

(7) Introduction The Josephson effect [1] consists in a dissipation-less current between two superconducting electrodes connected through a weak link. [2, 3] The origin of the effect stems from the macroscopic coherence of the superconducting condensate. Since its discovery in 1962, the research on devices based on the Josephson effect has achieved a number of important breakthroughs both in pure [2] and applied physics. [3] One of the most recent and exciting developments is probably in the field of solid state quantum computation. Indeed both theoretical and experimental efforts have been made in last years focused toward the comprehension of the effects and phenomena arising in superconducting nanocircuits. Such an investigation has an immediate1 applicative counterpart in quantum information processing. [4] The underlying idea of quantum processing is based on exploiting the coherent quantum-mechanical evolution of the (entangled) states of some elementary logical objects, the qubits, which consists in a quantum two level system. The implementation of such processing requires the ability to coherently manipulate the quantum states realized in these devices. By now, this has been shown in several experiments in systems of small Josephson junctions. [5, 6, 7, 8, 9, 10] Important questions can also be addressed in several other areas of fundamental physics; superconducting nanocircuits appear as a natural systems to observe fea1 in. the logical, and not chronological, sense!. 1.

(8) Introduction. Pag. 2. tures unique of quantum mechanics, like Berry phases in mesoscopic systems. [11] The observation of the laws of quantum mechanics in a mesoscopic system is ultimately limited by decoherence effects. Decoherence refers, in a very broad sense, to all those effects leading a pure state of any quantum system toward a statistical mixture. Decoherence effects can be understood in terms of the interaction of the quantum system with a many-degrees of freedom body, [12]. A detailed description requires however a knowledge of the microscopic interaction mechanisms between the system and the macroscopic environment. [13] The resulting entangled state of the system plus the environment leads to an effective description of the state of the system alone in terms of a mixed state. Decoherence effects are therefore unavoidable when acting on a quantum system to control, measure, or manipulate it, and the dynamic of mesoscopic physics has always to do with it. Clear signatures of the decoherence mechanisms can be observed in the transport properties of superconducting nanocircuits. Physics of nanocircuits, again due to their small dimensions, is crucially governed by the Coulomb repulsion energy associated to single electron charge. In fact, at T = 0, in a very small capacitive junction of capacitance C, even the energy EC = e2 /(2C) associated to having a single charge at the opposite sides of the junction itself, can overcome the characteristic energy, eV , associated with electron transfer. This leads to an effects known as “Coulomb blockade”. [2, 14, 15] Within present day technology it is possible to obtain contacts with capacitance ∼ 10−16 F. resulting in a charging energy ∼ 10−3 eV , which can be larger than bias voltage. and temperature, typically of the order of 1K, of an experimental setup. The. Coulomb blockade is the fundamental building block to realize several different single electron devices, [16] whose transport properties are strongly sensitive to the quantum character of charge carriers. New features can be observed in systems where Coulomb blockade is combined with Nano-Electro-Mechanical devices (NEMs). Central to this area of investiga-.

(9) Introduction. Pag. 3. tion is a device named “Cooper pair shuttle”. It involves the coherent tunneling of Cooper pairs through a mechanically moving nanoscopic grain. [17] The investigation of shuttle systems in general is a consequence of the technological achievement of fabricating a wide class of NEMs. [18] In its essential realization, a shuttle system consists of a small conducting grain, in Coulomb blockade regime, periodically driven between two electrodes (source and drain). The essential condition to characterize the shuttling mechanism is that the grain must be in contact with a single electrode at any time. The interest in such systems is mainly due to their peculiar transport properties because of the interplay of mechanical and electromagnetic degrees of freedom. In last years single electron shuttling has been widely analyzed (for a review see [19]) both from experimental [20, 21, 22, 23, 24] and theoretical [25, 26, 27, 28, 29, 30, 31, 32, 33] point of view. In the case of Cooper pair shuttling, additional coherent interactions of the central island with superconducting electrodes, due to the Josephson effect, play a crucial role. The main question about Cooper pair shuttling is whether vibrations of the grain can effect the coherent transfer of Cooper pairs between the bulk electrodes. The results obtained by Gorelik et al. in Ref. [17, 34] show that, despite the non-equilibrium steady state of the system, and the fact that the grain is in contact with only one lead at a time, the shuttle does not only carry charge, as in the normal metal case, but it also establishes phase coherence between the superconductors. This is witnessed by the presence of a steady state Josephson current through the system. Although decoherence effects, for instance due to gate voltage fluctuations, do exist, they provide in driving the system toward a steady state, modify the current phase relation, but do not (in general) destroy the effect. [17, 35] Present day technology allows to have shuttles and, in the case of superconducting circuits, quantum oscillations have been already observed. These facts give a certain confidence the Cooper pair shuttle can be realized in the next future. Transport properties of the Cooper pair shuttle, as all nano-devices, are not.

(10) Introduction. Pag. 4. completely characterized by the averaged current. In many relevant cases noise measurements provide informations not accessible by a current measurement. [36] Noise, for example, allows to directly access the charge of the elementary carriers and it gives us informations about the statistics of the quasiparticles relevant to the transport process. [37] In the Cooper pair shuttle the coexistence of coherent charge transfer and dissipation (consequence of the out of the equilibrium steady state) makes the analysis of current fluctuations particularly enlightening.. In this thesis we investigate the effect of a periodic external driving on the transport properties of superconducting nanocircuits. We mainly focus on the properties of the Cooper pair shuttle. We will however consider also quantum dynamics of several superconducting grains connected in series by means of Josephson junctions, i.e. a Josephson junction chain. We will pay a particular attention to the role of a chain in transferring quantum information. The thesis is organized as follows. In Chapter 1 we introduce the relevant physics of superconducting nanocircuits. The main effect we discuss is the coherent transfer of Cooper pairs due to the Josephson effect. We consider the Josephson effect in bulk superconductors and then in mesoscopic systems where charging effects become important. In the rest of the thesis we discuss the effect of the external periodic driving on superconducting nanocircuit. It is divided into two parts. The first part is dedicated to the transport properties of the Cooper pair shuttle. We explore both charge (Chapters 2 and 3) and Josephson (Chapter 4) dominated regimes. In the first regime we are mostly interested in the effect of decoherence on the Josephson current. They are presented in Chapter 2 where we also propose a possible nonmechanical realization of the Cooper pair shuttle’s physics. Charge transfer in the Cooper pair shuttle is achieved as a result of an out of equilibrium process due to the mechanical motion usually associated with dissipation. The role of.

(11) Introduction. Pag. 5. coherent Cooper pair transfer in presence of dissipation is the key issue in Chapter 3 where we calculate the current noise. We determine the whole current distribution function by means of the formalism of the Full Counting Statistic which is reviewed in Appendix A. The opposite limit, where the Josephson energy dominates over the charging energy, is investigated in Chapter 4. In this case the Cooper pair shuttle behaves like a quantum kicked rotator, which is a chaotic system in the classical limit. We discuss how the Cooper pair shuttle can be used to observe characteristic features of classically chaotic quantum system and we investigate the effect of the chaotic dynamics on the transport properties of the shuttle. In the second part we analyze a Josephson junctions chain as a channel for quantum information transfer. We investigate the transfer of a quantum state between the two ends of the chain and we discuss a protocol to measure the transferred state. We include in the Appendices C, D, E all the technical discussions and calculations. Appendices A and B have a different role: They are extensive introductions to the subject of full counting statistics and quantum dynamics of classically chaotic system respectively. They are not the main topic of this PhD thesis, but turn out to be of some importance for what is discussed in Chapters 3 and 4. Throughout the whole thesis, the Boltzmann constant is the dimensionless unity, kB = 1..

(12) Chapter 1. Charging effects in mesoscopic superconductors The microscopic theory of superconductivity has been formulated by Bardeen, Cooper and Scrieffer.1 [40] It accounts for the ground state and low energy excitations of the superconductive phase of many simple metals (aluminum, niobium, etc.). The BCS ground state of a bulk superconductor is described by a collective degree of freedom, a wave function ∆(x) introduced as a complex order parameter in the Ginzburg-Landau theory, [38] with |∆(x)|2 representing the local density of Cooper pairs. The condensation of a macroscopic number of Cooper pairs in a quantum coherent ground state is the origin of quantum coherent effects in superconducting devices. A clear manifestation of the quantum coherent behavior of the superconducting condensate is provided by the coherent transfer of Cooper pairs through a tunnel junctions between superconducting leads. This is the Josephson 1 Hereafter. we will assume the knowledge of the BCS theory of superconductivity. The subject. is widely treated in the literature: For an exhaustive introduction see Ref. [2], or Ref. [39] for a detailed theoretical description.. 6.

(13) 1.1. Charging effects in mesoscopic superconductors. Pag. 7. effect resulting in a dissipation-less ground state current through the junction. In the following we will discuss the Josephson effect, then we will consider the effects of Coulomb energy in mesoscopic devices (both normal and superconductive) and finally we discuss the interplay between the charging and Josephson effects. This constitutes the ground on which the understanding of the transport properties of the Cooper pair shuttle and the quantum information transfer through a Josephson junction chain will be developed in the next Chapters.. 1.1. The Josephson effect. The coherence of Cooper pair transfer through a weak link is reflected in the existence of a ground state dissipation-less current flow. The current depends on the phase difference between the two superconductors and it is given by I = IC sin(ϕR − ϕL ) .. (1.1). Here ϕb is the phase of the Ginzburg-Landau wave function of the superconducting condensate at left (b = L) and right (b = R) side of the junction; IC is the critical current, i.e. the maximum current that the junction can sustain. This is the DC Josephson effect, which was originally predicted for a tunnel junction between two bulk BCS superconductors. [1] As previously mentioned, the junction does not need to be a tunnel junction, rather a more general “weak link” between the two superconductor is required: It can consists of an insulating barrier or a narrow constriction between the two bulk leads. Independently on the specific realization of the weak link, what is really crucial is that the junction allows the two condensate to interact through the exchange of Cooper pairs while keeping their phase difference fixed. In its original work, Josephson also analyzed the effect of a voltage bias, V = VR − VL , across the junction. As long as it is lower than the excitation gap ∆, the.

(14) 1.1. Charging effects in mesoscopic superconductors. Pag. 8. supercurrent in Eq. (1.1) is an alternating current due to the time dependence of the phase difference across the junction, 2e d (∆ϕ) = V , ∆ϕ = ϕR − ϕL . dt ~. (1.2). This is called the AC Josephson effect. Eq. (1.2) is simply a consequence of the electromagnetic gauge invariance: By means of a gauge transformations we can eliminate the electric potential and reabsorb it in the phase factor of the wave function according to Eq. (1.2). By integrating the electrical work made by the current source in changing the phase across the junction, we can determine the free R R energy of the junction, F = IV dt = (~/2e) I d∆ϕ = const. − (~IC /2e) cos ∆ϕ. We can then write. F = −EJ cos(ϕR − ϕL ) , EJ =. ~IC . 2e. (1.3). The Josephson energy, EJ , (or equivalently IC ) is a measure of how strong the coupling between the phases of the two superconductors is and it depends on the details of the junction. Typical values of the critical current go from a few milliamperes down to tenths of picoampere corresponding to EJ ∼ 10−5 eV . [3] The main ex-. perimental constraint in lowering the Josephson energy comes from requiring a low temperature T ≪ EJ to avoid thermal fluctuations to wash out the effect. Eq. (1.3). is one of the building blocks to understand the physics we are going to describe. We. present in Appendix C a quantitative microscopic derivation of the free energy of the Josephson coupling. Here we explore in more details the physical consequences of Eq. (1.3) The BCS ground state [2] for a bulk superconductors is characterized by a phase factor, eıϕ between terms differing by unity in the number of Cooper pair, Y |ψϕ i = |uk | + |vk | eıϕ c†k↑ c†−k↓ |Ψi ,. (1.4). k. |Ψi being the vacuum state of the theory. All the states differing by the value of ϕ. are degenerate. Each of them correspond to have an indefinite number of Cooper.

(15) 1.2. Charging effects in mesoscopic superconductors. Pag. 9. pairs. By Fourier transforming the previous state, we get the eigenstates of the number operator, |ni =. Z. 2π. 0. dφ e−ınϕ/2 |ψϕ i ,. (1.5). with a completely undefined phase. In any state, the superconducting phase difference and the charge cannot have simultaneously well defined values. There is an uncertainty relation ∆ϕ∆n & 1 ,. (1.6). between phase and charge in superconductors associated to this. The ground state of an isolated superconductor has to be an eigenstates of the number operator, |ni, chosen to exactly neutralize the positive charge of ion cores. Then, a second superconductor, separated by the first one would again be in a state with a well. definite charge, |mi. In case of a tunnel junction between the two, at T = 0, the system has to be characterized by a well defined phase difference ϕ across the. junction, so that the system can gain energy from a non vanishing value of the Josephson coupling, EJ cos ϕ. It means that the state of the whole system must be a superposition of products state of fixed number of Cooper pairs on the left and right superconductors of the form X aj |n + jiL |m − jiR . |ψ ′ i =. (1.7). j. Each terms in Eq. (1.7) corresponds to a charge 2ej on the capacitance of the junction and therefore to the transfer of j Cooper pairs through the junction. If the energy to transfer charges through the junction, i.e. the charging energy, is comparable to the Josephson energy, it is possible to create only states with a small charge difference across the junction. Therefore the state |ψ ′ i in Eq. (1.7). will involve only few charge states and cannot have a well defined phase. The competition between the Josephson and charging energies gives origin to interesting physical effects. As an introduction, in the next Section we discuss in detail the role of charging energy in absence of Josephson couplings..

(16) 1.2. Charging effects in mesoscopic superconductors. Pag. 10. 1.2. Charging effect in small grains: The Coulomb blockade. In sub-micron devices, the capacitance of a junction is so small that single electron charging energy, EC = e2 /2C, may be comparable with the typical energy scales of the system (the external voltage bias, eV , the temperature, T and the Josephson energy, EJ in case of superconducting systems). This can lead to a modification of the I −V characteristics of the device. In particular there can be a vanishing current. through the junction despite a finite voltage bias (in the case of normal junctions) across the junction itself. This phenomenon is known as Coulomb blockade.. The simplest device esibiting Coulomb blockade can be realized in an isolated capacitor of capacitance C = Q/V , Q > 0 and −Q being the charge on the two. opposite electrodes. The energy of the system is Q2 /2C. Consider the process of. transferring one electron from the negative to the positive lead. The energy of the system after the process is (Q − |e|)2 /2C. The final energy is larger than the initial. one as long as Q 6 |e|/2. It shows that the process is energetically forbidden at. Q 6 |e|/2, or equivalently at V 6 |e|/2C. In spite of a finite voltage bias, we have. a zero tunnel current between the electrodes, only due to charging energy. This is the Coulomb blockade regime.. The previous argument, based on energy considerations, is valid at T = 0. Physically we have to require that T ≪ EC to avoid thermal fluctuation to destroy charging effects. A more subtle requirement is that the total resistance “seen” by the. junction capacitance (the parallel combination of the internal tunneling resistance and any external resistance) have to be larger than the quantum resistance RQ = h/(4e2 ) ∼ 6KΩ. This requirement follows by the fact that charging energy EC =. e2 /(2C), to be observable, must exceed the energy uncertainty ∆E = ~/(RC) associated with the relaxation time of a charge on the capacitor. This requirement can be easily achieved for the zero frequency resistance of a tunnel junction, which.

(17) 1.2. Charging effects in mesoscopic superconductors. Pag. 11. can be made as big as ∼ 109 Ω. However the real problem rise when leads are taken. into account. In fact the characteristic frequency involved in the process of charge escape is ∼ 2π/(.

(18) 1.2. Pag. 12. Charging effects in mesoscopic superconductors. V2. R1. Ci. Vi. V1. C2. R2. C1. φ Ri. Figure 1.1: Schematic representation of a metallic grain connected to a given number of voltage sources through tunnel junctions. Each junction is characterized by given capacitance Ci and resistance Ri .. The previous equation includes also the energy automatically provided by the generators in the tunneling of ne charges onto the island. The work performed by the jth generator is Wj =. X Ci Cj 1− eVj − e eVi . CΣ CΣ. (1.11). i6=j. The total internal energy of the system is therefore given by X nj Wj , E=U−. (1.12). j. which depends on the number of electrons that reach the island through the jth junction. The energy can be rewritten as E=. XX 1 XX (ne)2 Ci ′ Ci Cj (Vi − Vj )2 + −e nj (Vj − Vi ) . 2CΣ i j>i 2CΣ C Σ i j. The symbol. P. ′. (1.13). means that the sum has to be extended over the junctions through. which charge has passed. It is worth notice that the total energy for having n charges.

(19) 1.2. Pag. 13. Charging effects in mesoscopic superconductors. n=0. Cg. CJ. n=-2. n=-1. EC. n. n=1. Vg. EJ. -2. -1. 0 ng. 1. 2. Figure 1.2: Left panel. Schematic representation of a single electron box. The colored junction represents a tunnel junction. In case of superconductors the system is a Cooper pair box and the tunnel junction is characterized by the Josephson energy EJ . Right panel. Charging energy of the single electron box plotted vs. the dimensionless gate charge ng = Cg Vg /e for different values of n. on the grain is not, in general, uniquely defined by knowing n, but it depends on the number of electrons passing through the junctions nj .. Single electron box The simplest case involving a single island instead of a single junction is that of a metallic grain put to ground through a tunnel junction (of capacitance CJ ) and connected to a gate voltage Vg through a capacitive junction (of capacitance Cg ). Such system is referred to as “single electron box” and is sketched in Fig. (1.2). It follows from Eq. (1.13) that the internal energy of such system is E=. e2 (n − ng )2 , 2CΣ. (1.14). up to an irrelevant constant term. The total capacitance is CΣ = CJ + CL eng = Vg Cg is referred as the “gate charge”. The energy dependence on the gate voltage.

(20) 1.2. Pag. 14. Charging effects in mesoscopic superconductors. CL. CR. n Cg. VL. Vg. VR. Figure 1.3: Schematic representation of the Single Electron Transistor (SET).. in Eq. (1.14) is shown in Fig. (1.2). It clearly shows that the average number of electrons on the island has a step-like dependence on the gate voltage. In general one expects the step-like dependence is washed out by increasing the temperature above the charging energy e2 /(2CΣ ). This has been observed in several experiments (see, for instance [42]).. Single electron transistor In the single electron box no electron transport can take place. The simplest mesoscopic device which allows for electron transport consists of a mesoscopic grain contacted to a drain and a source through two tunnel junction and to a gate voltage by means of a capacitive junction. It is called “Single Electron Transistor” (SET), and is represented in Fig. 1.2. The I − V characteristics of the device has. been widely investigated. We do not go further in the discussion of the rich physics present in the SET (for a detailed analysis, see chapter 3 in [16]). Here we are interested in writing its charging energy, which is VL − VR 2 E = EC (n − ng ) + [(CL − CR ) n + CΣ (nR − nL )] + const. , (1.15) 2e.

(21) 1.3. Charging effects in mesoscopic superconductors. Pag. 15. where EC = e2 /(2CΣ ) , ng = (Cg Vg )/e ,. (1.16). As for the single electron box, EC is the charging energy, and eng is the gate charge. The constant term is independent on the charge degree of freedom, and therefore unimportant. We note that the energy in Eq. (1.15) depends not only on the total excess charge of the grain, but also on the charge entering it from the left, nL , and right, nR , contacts. Such a dependence disappears in the case of a SET with a zero voltage bias, VL − VR = 0. In this special case, the total energy of the. system depends only on the total number of electrons on the island, and Eq. (1.15) becomes identical to Eq. (1.14), with the only difference in the definition of the total capacitance which now is CΣ = CL + CR + Cg . The case of a SET with zero voltage bias is of relevance if the grain and the leads are superconductors, in which case the transport is due to the Josephson effect. We will analyze this case in the next section.. 1.3. Josephson effect in presence of Coulomb blockade. The observation of charging effects in superconductors is possible in mesoscopic samples and the formalism of the previous Section can be applied as well. The only constraint is that the BCS pair condensate is well defined in the grain.2 In superconducting grains the excess of charge is due either to Cooper pairs or to quasiparticle. Quasiparticles are gapped excitations, with an energy gap ∆. That is why a T = 0 (or better at T ≪ ∆) quasiparticles do not play any. role in the physics of the system and we can neglect decoherence effects due to 2 The. condition for that is that δ ∼ ~/L ≪ ∆, where δ is the mean level spacing fixed by the. typical size of the grain, L. A detailed discussion on the subject including the fate of superconductivity at δ ≫ ∆ can be found in Ref. [43]..

(22) 1.3. Charging effects in mesoscopic superconductors. Pag. 16. quasiparticle tunneling (see Appendix C and references therein). Moreover, if in addiction EC ≪ ∆, quasiparticle are not present in the grain. In fact, the effect. of a gap ∆ for the quasiparticle excitations in the charging energy in Eq. (1.14) can be taken into account by inserting in the energy an additional term, equal to ∆, whenever n is odd. [44] As can be seen in Fig. 1.4, it results in shifting the parabolas labeled by an odd value of n by an amount of ∆. Indeed for EC ≪ ∆,. the ground state charging energy of the superconducting grain, as a function of the gate charge, is 2e periodic. This 2e periodicity is reflected in the low voltage current of a N −S −N tunnel junction with a mesoscopic superconducting grain. Of course, at high temperature (T ≫ ∆), one expects the effect is washed out and the. non superconducting e periodicity characteristic of the normal state is recovered, as experimentally verified. [45] In the regime EC ≪ ∆ we are interested in, due to. the shifting of ∆ for odd-n parabolas, we neglect quasiparticle degrees of freedom,. and we restrict to Cooper pairs coherent dynamics. The Josephson effect in small superconducting grains is modified by the existence of a comparable charging energy scale. The reason is due to the fact that because of charging energy fluctuations of Cooper pairs’ number are important and only a coherent superposition of states differing only by a small number of Cooper pairs is allowed. In this case the state |ψ ′ i of Eq. (1.7) describes a coherent superposition of. few charge eigenstates of the grain and cannot be an exact eigenstate of the number. operator as a consequence of the number-phase uncertainty ∆ϕ∆n & 1 introduced in Section 1.1. We now want to describe quantitatively such effects. The simplest system we take into consideration is the superconducting equivalent of the single electron box, named ”Cooper pair box” (see Fig. (1.2)). The total internal energy for the Cooper pair box is obtained by combining the charging energy in Eq. (1.14), 3 and the Josephson energy in Eq. (1.3). Moreover we remind that the charge on the grain and the superconducting phase difference across the 3 Note. that the replacement 2 → 2e has to be performed, the elementary charge carriers now. being the Cooper pairs..

(23) 1.3. Pag. 17. Charging effects in mesoscopic superconductors. E/EC. n = −2. 3.5. n = −1. 3. n=0. 2.5 2 1.5. n=1. 1 0.5 0. −1. 1. 2. n=2 ng. Coulomb blockade. (a) Normal state Degeneracy point E/EC 3.5. n=1. E/EC. n = −1. 3.5. 3. 3. 2.5. 2.5. 2. 2. 1.5. 1.5. 1. 1. 0.5. 0.5. n=0. n = −2. −1. 0. 1. n=2 2 ng. (b) Supeconducting state, ∆ < EC. n = −1. n=1. n = −2 n=2. n=0 −1. 0. 1. 2. ng. (C) Superconducting state, ∆ > EC. Figure 1.4: Charging energy of a single electron box and a Cooper pair box, in units of EC , vs. the dimensionless gate voltage ng for various charge eigenstates labeled by n. In the various panels we plotted the non superconducting case (panel (a)), and the superconducting one in the two cases of ∆ < EC (panel (b)) and ∆ > EC (panel (c)). In the last case the regions of charge degeneracy and Coulomb blockade are pointed out..

(24) 1.3. Charging effects in mesoscopic superconductors. Pag. 18. junctions are canonically conjugate variables. These considerations lead to the Hamiltonian 2 ˆ = EC (ˆ H n − ng ) − EJ cos(ϕ) ˆ .. (1.17). Here EC and ng are defined analogously to Eq. (1.16) after taking into account the doubling of the charge of Cooper pairs with respect to the electrons. ϕˆ is the phase difference across the Josephson junction. The classically conjugate variables n ˆ and ϕˆ are now non-commuting operators, [ˆ n, ϕ] ˆ = −i, acting on the Hilbert space. spanned by excess charge states on the grain. It can be useful to rewrite the previous Hamiltonian in terms of rising and lowering operators for the charge on the grain. The only non-trivial term is the Josephson coupling one. By rewriting cos ϕˆ = 1/2(eıϕˆ + e−ıϕˆ ), and noting that eıϕˆ is the operator describing the translation in n of an amount ∆n = 1, the Hamiltonian can be written as, X ˆ = − EJ H (|n + 1i hn| + |ni hn + 1|) + EC (ˆ n − ng )2 . 2. (1.18). N. The Hamiltonian for a ”Superconducting Single Electron Transistor” (SSET), for zero applied voltage bias, similarly is given by (R) (L) 2 ˆ = EC (ˆ H n − ng ) − EJ cos(ϕˆ − ϕL ) − EJ cos(ϕˆ − ϕR ) .. (1.19). The parameters are defined as in Eq. (1.13), the only change being in the definition of CΣ = CL + CR + Cg . In the following Chapters (with the only exception of Chapter 4) we will consider the case of small Josephson energy, EJ ≪ EC . In this regime, by tuning ng in the. range 0 < ng < 1, the energies of all states except those with n = 0, 1 are too high to be exited (see Fig. (1.4)). As a consequence of Coulomb blockade, we can therefore restrict the Hamiltonian to the two level Hilbert space spanned by |n = 0i, and |n = 1i. 4 The. 4. We also note that at ng = 1/2, the charging energy levels. choice of the states labeled by n = 0, 1 is completely arbitrary, any other pair of states. labeled by consecutive integer numbers, n, n + 1 is equally valid. Of course physics is independent of such choice..

(25) 1.4. Charging effects in mesoscopic superconductors. Pag. 19. of n = 1 and n = 0 states are degenerate. In the vicinity of such degenerate point, the superposition of Cooper pair states due to the Josephson coupling is favored. Instead, far away from such a point, the charge state of the grain is locked due to Coulomb energy EC ≫ EJ . Recently Nakamura et al. [5] have firstly demonstrated. the possibility to coherently manipulate the quantum state of a two level system realized in a solid state device consisting of a Cooper pair box.. 1.4. Superconducting qubits. The Cooper pair box, as described by the Hamiltonian in Eq. (1.17) provides an interesting possible implementation of a qubit, the elementary constitutive object of quantum information. In a classical system, the elementary logical object is a bit, which can assume the logical values 0 or 1. In any physical implementation the two logical values correspond to two different wide separated values of a classical variable, like a voltage bias. The information, in a classical system, is therefore encoded as a sequence of 0 and 1. The qubit is the quantum equivalent of the classical bit in which the classical states 0 and 1 are replaced by the quantum state |0i, |1i, of a. quantum two level system. In a quantum system the information is encoded in a state of the Hilbert space of the N qubits, H=. N O i=1. Hi ,. (1.20). where Hi = span{|0ii , |1ii } is the two dimensional Hilbert space of a single qubit.. The laws of quantum mechanics make possible to implement algorithms to solve. some problems in a time exponentially small compared to the classical ones: There are cases in which quantum algorithms can solve classical intractable problems (like prime factorization) in a polynomial time. This is the reason of the great interest recently addressed to the quantum computation. For a detailed discussion on the.

(26) 1.4. Charging effects in mesoscopic superconductors. Pag. 20. development of quantum information theory see Ref. [46], and Ref. [47]. Here we simply point out that the reason of such an exponential gain in the computational time relays on two key aspect of quantum mechanics having not correspondence in classical mechanics: (i) the possibility to create a quantum coherent superposition of the two qubit’s state |0i, and |1i; (ii) the possibility to create states involving. quantum correlations between distinct qubits, so called ”entangled” states. In fact. it has been realized that all the known quantum algorithms which are exponential faster than the classical one involve the creation of entangled states. The state of a qubit can be manipulated differently with respect to its classical counterpart. The procedure to manipulate the quantum state depends on the specific system used to implement the qubits. Quantum information theory analyze the state manipulation in an implementation-independent way. At this formal level any unitary operator of the the N -qubits Hilbert space is an allowed operation, also called quantum gate. A universal set of quantum gates, i .e. a set of elementary operations from which it is possible to generate all the others, consists of any two qubits gate combined with all single qubit gates.. 5. For instance, let us present a. simple example of a single qubit quantum gate with no classical analog. It is the so √ called NOT, defined as ! √ 1 1 + i −1 + i NOT = . (1.21) 2 −1 + i 1 + i in the quantum computational basis |0i, |1i. The name is clearly deduced by the √ √ fact that NOT ◦ NOT = NOT, where the NOT gate acts on the vector of the basis by permuting them, like the classical NOT does.. The implementation of quantum algorithms requires a suitable physical system as a qubit. Any physical system which is considered a candidate for quantum state 5 The. reduction of a given unitary manipulation into a sequence of the elementary gates is. not unique. This decomposition can, therefore, be optimized considering that in a given physical implementation some gates are realized easier than others. [4, 46].

(27) 1.4. Charging effects in mesoscopic superconductors. Pag. 21. engineering should satisfy the following criteria, as stressed out in Ref. [48]: (i) The system has to be a two level system, which means that the higher energy states (if any) are “never” excited during the external manipulations; (ii) The initial stats has to be prepared with high accuracy; (iii) The decoherence time, both for the phase and the populations should allow to perform an high number (∼ 104 ) of coherent manipulations; (iv) A high control on the value and time dependence of the external field is required to perform the unitary operations; (v) Finally one needs to read out the state of the system. Many system have been proposed as implementation of qubits, going from nuclear spins to trapped ions to ultra-small quantum dots to electron spins.6 Among the various proposals, solid state implementations present the great advantage of scalability, which is quite relevant in the eventual realization of systems involving a large number of qubits. Within such framework it has been realized that the Cooper pair box, for EJ ≪ EC may behaves as qubit in quantum information devices. [49, 50, 51, 52, 5] The experimental cornerstone in this context has been. the observation of the coherent superposition of two macroscopic quantum states and of coherent oscillations between the two states, as firstly presented in Ref. [5] and then confirmed in several experiments demonstrating the ability of coherently manipulate such states. [6, 7, 8, 9, 10] The Cooper pair box is also called “charge qubit” because the two computational states are charge eigenstates, in contrast to the “flux qubit” where the computational states are characterized by different values of the magnetic flux piercing a superconducting ring. [4] The Hamiltonian of the charge qubit is that in Eq. (1.17); in the standard notation of Pauli matrices 6 References. about some proposal for qubits can be found in Ref. [47]..

(28) 1.5. Charging effects in mesoscopic superconductors. Pag. 22. for a two level system it reads (b). ˆ = EC (1 − 2ng (t))σz − EJ (t) σx . H 2 2. (1.22). The time dependence corresponding to an external control on the Hamiltonian’s parameters is explicit in Eq. (1.22). The control on the parameter EJ can be obtained by replacing the simple Josephson junction with a squid and the Josephson energy is then controlled by an applied magnetic field. [4] By switching on a constant EC or EJ for a constant time ∆t in the Hamiltonian in Eq. (1.22) one can construct respectively Uz (α = EC ∆t/~) = exp(iEC ∆tˆ σz /(2~)) , Ux (α = EJ ∆t) = exp(iEJ ∆tˆ σx /(2~)) .. (1.23). Any single qubit gate can be realized by combining the two previous operations. √ In particular NOT = Ux (α = π/2). Again one can prepare any initial state. |ψi = cos θ |0i + eiφ sin θ |1i. The further step in view of the implementation of. quantum algorithms is the realization of two qubits gates. It can be achieved by a. coupling two charge qubits through a Josephson junction. This leads to an effective ˆ int = EJ /2(ˆ XY coupling between the two qubit of the form H σ+,1 ⊗ σ ˆ−,2 + h.c.).. The index 1, 2 label the two qubits. Different couplings which may turn to be more suitable to realize specific gates can be designed. [4]. 1.5. Quantum communication using one-dimensional systems. So far we have discussed how quantum algorithms can give, in some important case, an exponential gain in the computational time with respect to the classical ones. The outstanding performance of quantum protocols as compared to the classical ones is not confined to quantum computation. Quantum information processing [46].

(29) 1.5. Charging effects in mesoscopic superconductors. Pag. 23. has been shown to be extremely efficient in many other areas, as compared with its classical counterpart, for instance in communication or cryptography. As superconducting qubits are among the most promising candidate as the building blocks of quantum information processor, it would be desirable to have protocols for quantum information transfer in solid-state environments. To this purpose protocols for quantum state transfer through qubit networks can be suitable to be implemented in Josephson qubit based devices. The analysis of quantum state transfer by means of a careful tailored qubit network with an appropriate choice of the couplings between them and of the time over which the system is let evolve without any manipulation has been the subject of a number of recent papers. In this spirit, a simple protocol to transfer quantum information through a spin chain with ferromagnetic Heisenberg interactions has been proposed. [53] We will consider such protocol in this thesis, therefore here we present it in some detail. The Hamiltonian of the spin chain is HG = −. X. hi,ji. Ji,j σ i · σ j −. L X. Bi σzi ,. (1.24). i=1. where σ i = (σxi , σyi , σzi ) and L is the total number of qubits. Ji,j > 0 are the interaction strength and hi, ji represent pairs of connected qubits, the Bi > 0 are. static magnetic fields if we identify qubits with spins.7 The computational basis for the qubits is given by the eigenstates of σzi , σzi |0i = − |0i and σzi |1i = |1i.. The aim of the protocol is to transfer a quantum state between two sites of the. chain, i.e. from sth to the rth site. It works as follows. The chain is in its ground state |0i = |00 . . . 0i as determined by the external magnetic field. At t = 0 the. state to be transfered |ψi = cos(θ/2) |0i + eiφ sin(θ/2) |1i is prepared in the sth site and it is let evolve according to the Hamiltonian in Eq. (1.24). At t = t⋆ the rth. qubit is measured to read its state, which, generally speaking, will be a mixed state described by a density matrix ρr (t⋆ ) due to the entanglement of the rth qubit with 7 In. geneal, the physical meaning of Bi will depend on the specific implementation of the qubit..

(30) 1.5. Pag. 24. Charging effects in mesoscopic superconductors. the other qubits of the chain. One has to choose the time t⋆ so that the state ρr (t⋆ ) of the qubit to be measured is “as similar as possible” to the transmitted one. The quality of state transfer is quantified by the fidelity hψ|ρr (t⋆ )|ψi, of the (mixed). state ρr (t⋆ ) of the right-most island to the initial state. As we are interested in the. property of the transmission line independently on the prepared state we consider the fidelity averaged over all possible initial state |ψi on the Bloch sphere, Z 1 hψ|ρr (t⋆ )|ψi dΩ . F (t⋆ ) = 4π. (1.25). The fidelity equals 1, F (t⋆ ) = 1 if, and only if, the state of the rth qubit coincides with the transmitted one. The symmetries of the Hamiltonian in Eq. (1.24) allows us to write the averaged P fidelity in a simple way. In fact i σzi , i.e. the total magnetization in the z direction,. is a constant of motion. Therefore the initial state |ψi evolves only in the Hilbert. space spanned by |0i and |ji = |00 . . . 1 . . . 0i corresponding to an excited qubit at the jth site of the chain. The state of the whole chain at t = t⋆ is then |ψ(t⋆ )i = cos. θ X L ⋆ θ |0i + eiφ sin fj,s (t ) |ji , 2 2. (1.26). j=1,L. L with fj,s (t) = hj| exp(−iHG t/~)|si. The mixed state of the rth qubit, determined. by tracing out the state of the other qubits of the chain, is ρr (t⋆ ) =. L |fr,s (t⋆ )|2 sin2. L e−iφ (fr,s (t⋆ ))∗ sin. θ 2 θ θ 2 cos 2. L eiφ fr,s (t⋆ ) sin θ2 cos 2θ L 1 − |fr,s (t⋆ )|2 sin2. θ 2. !. .. (1.27). From Eq. (1.25), the averaged fidelity is easy determined, F =. L L (t⋆ )| cos α |fr,s (t⋆ )|2 1 |fr,s + + , 2 3 6. (1.28). L where α = Arg{fr,s }. The fidelity is maximized when α is a multiple of 2π. This. can be achieved by a suitable choice of various Bi ..

(31) 1.5. Charging effects in mesoscopic superconductors. Pag. 25. The protocol discussed here and its various modifications allows to transfer unknown quantum states over appreciable distances (∼ 102 lattice sites) with high fidelity [53, 54, 55, 56]. By modifying the protocol and the structure of the network, even perfect transfer could be achieved over arbitrary distances in spin chains [57, 58, 59]. The possibility of transferring a quantum coherent state through a chain of charge qubits coupled by means of Josephson junctions will be the subject of the second part of this thesis. One last point require some consideration. The operations described so far have to be coherent. However coupling to the external environment is unavoidable and leads to decoherence effects. The main sources of decoherence for a charge qubit are those affecting the charge level separation. They are due either to fluctuations of the external circuit parameters, or to the random switching of charges trapped in the substarte. [60] Decoherence due to background charges is a source of low frequency 1/f noise which recent experiments [61] have shown to be the most effective source of dephasing. An exhaustive discussion on the subject can be found in Ref. [62]..

(32) Part I. THE COOPER PAIR SHUTTLE. 26.

(33) Chapter 2. Josephson current The Cooper pair shuttle consists of a small superconducting island coupled to two macroscopic leads and forced to change its position periodically in time, with period T , from the Right (R) to the Left (L) electrode and back (see Fig.2.1). It has been shown that such a movable Cooper pair box can serve as conveyor of Josephson coupling, leading to coherent transfer of Cooper pairs between the electrodes. [17] In fact the Cooper pair shuttle exhibits a steady state Josephson current which can be controlled, both in magnitude and direction by modifying the motion of the grain. The coherence of Cooper pair transfer crucially relies on the possibility to create a quantum coherent superposition of charge state differing by a Cooper pair on the grain, and therefore on the decoherence mechanisms of the system. In this Chapter we consider the effect of a realistic decoherence mechanism due to gate voltage fluctuations. [35, 63] We first discuss the model of the system and present a possible realization based on Superconducting Quantum Interference Devices (SQIDs). We analyze the Josephson current emphasizing the effect of decoherence. Finally we consider the case in which a voltage bias is applied between the two electrodes of the Coper pair shuttle. 27.

(34) 2.1. Josephson current. 2.1. The model. Pag. 28. The model of the Cooper pair shuttle is schematically presented in Fig. (2.1). We assume that the superconducting grain is small enough so that charging effects are important, while the two leads are macroscopic and have definite phases φL,R . We assume that the position of the grain is a classical variable. Physically this means that the characteristic length, lJ , associated with the contact region is larger √ compared to the quantum zero-point vibration amplitude, x0 ∝ 1/ ω, associated to the shuttle oscillations with characteristic frequency ω. Therefore at sufficiently. high frequency oscillations of the grain, the motion of the shuttle is classical. The moving island is then described by the Hamiltonian X (b) H0 = EC (t)[ˆ n − ng (t)]2 − EJ (t) cos(ϕˆ − φb ). (2.1). b=L,R. where n ˆ is the number of excess Cooper pairs in the grain and ϕˆ is its conjugate phase, [ˆ n, ϕ] ˆ = −i. EC (t) = (2e)2 /2CΣ (t), is the charging energy, CΣ (t) = Cg (t) + (L,R). CL (t) + CR (t) being the total capacitance of the Cooper pair box. EJ. (t) are the. Josephson couplings to the left or right lead respectively, and ng (t) = Cg (t)Vg (t)/2e is the dimensionless gate charge which can be arbitrary tuned. We assume that the time dependence of the parameters in the Hamiltonian in Eq. (2.1) is fixed. This has to be contrasted with the case of single electron shuttle where the interplay of electrical and mechanical degree of freedom is considered. [22, 23] The Hamiltonian in Eq. (2.1) describes an externally driven system: a superconducting single electron transistor (SSET) in which Josephson energies and gate voltage vary in time, independently on the fact that this is a consequence of mechanical motion of the grain or not. As an interesting example we mention the recently realized Cooper pair sluice where the Josephson couplings are varied adiabatically. [64, 65] Here however we are interested in the different limit of nonadiabatic switching of the Josephson couplings. In fact, an adiabatic variation of the Josephson couplings is not realistic for the case of a moving grain. The time.

(35) 2.1. Pag. 29. Josephson current. (. C. ). (L). 2e. (. (R). EJ. L. (R). EJ = EJ. ). 2e. =0 (. (. (L). C. EJ. ). (R). EJ = EJ. x. J. t. o. e. s. r. e. n. a. p. l. h. l. s. y. c. o. n. o. n. j. u. t. n. r. o. c. l. t. l. i. a. o. b. n. l. e. n ˆ ϕ ˆ. CL. Cg. =0. =0. E. s. ER. φL. ). (L). =0. E. R. φR. (R). EJ (t). (L). EJ (t). EL. CR. Vg. 0. tL. tL + t →. T − t←. T. t. Figure 2.1: Upper panel. Time dependence of the position of the Cooper pair shuttle. The three intervals L, C and R, within the period T = tL + t→ + tR + t← , correspond to the situations: (L) EJ (L) (t) = EL , EJ (R) (t) = 0 (interaction time at left lead); (C) EJ (L) (t) = 0, EJ (R) (t) = 0 (free evolution time in forward and backward directions); (R) EJ (L) (t) = 0, EJ (R) (t) = ER (interaction time at right lead). Lower left panel. Schematic representation of the general system described by the Hamiltonian 2.1. It consists of a Cooper pair box coupled through externally switched Josephson junctions to phase biased superconductors. Lower right panel. Time dependence of the left and right Josephson energies within a single period.. dependence of various parameters is described in Fig. 2.1. When the grain is close to one of the leads, the corresponding Josephson coupling is non-zero (with value EL , ER ) (positions L and R in Fig.2.1). In the intermediate region (position C),.

(36) 2.1. Pag. 30. Josephson current. (L). (R). EJ (t) = EJ (t) = 0. As in Ref. [34] we employ a sudden approximation (which (L,R). requires a switching time ∆t ≪ 1/EL(R) ) and suppose EJ. (t) to be step functions. in each region (see Fig. 2.1). For later convenience we define the functions ΘL (t). = θ(t)θ(tL − t) ,. (2.2). ΘR (t). = θ(t − (tL + t→ ))θ(tL + t→ + tR − t) ,. (2.3). so that we can write (b). EJ (t) = Eb. X. n∈N. Θb (t − nT ) .. (2.4). The total capacitance CΣ (t) is weakly dependent on time in the contact region. We assume it to be constant during the intervals L and R (therefore the same hold for EC (t) = EC ). In the intermediate region (C) it is not necessary to specify the R exact time dependence of EC (t), only the time integrals [tL ,tL +t→ ] dt EC (t)/~ and R [T −t← ,T ] dt EC (t)/~ will enter the results. In both contact regions, instead, the simultaneous presence of the Josephson couplings and the charging term leads to. quite a different behavior according to the relative strength of the corresponding (L). (R). (L). (R). energy scales, EC and EJ , EJ . In the charge dominating regime EJ , EJ. ≪. EC , Coulomb blockade occurs. The Josephson coupling is not able to populate widely separate charge eigenstates, so that the state of the system is localized in the. charge space; only two charge states {|n = 0i , |n = 1i} are important. The opposite (L). (R). Josephson dominated regime EC ≪ EJ , EJ. allows the Josephson coupling to mix. several states and the Cooper pair shuttle behaves like a quantum kicked rotator, i.e. a quantum system displaying a chaotic dynamics in the classical limit. In this. case an initial charge eigenstate diffuse in the charge coordinate, until quantum interference effect lead to a dynamically localized state. In the rest of the thesis we will discuss both cases. We will be interested in the transport properties of the Cooper pair shuttle both because of the appearance of interesting effects and as a tool to gain information about the physics of the system. The transfer of charge is expressed by the presence.

(37) 2.2. Pag. 31. Josephson current. of a current at left and right contacts. The corresponding current operators are, in the Schrödinger picture, IˆL (t) IˆR (t). EL sin (ϕˆ − ϕL ) ΘL (t) , ~ ER = 2e sin (ϕˆ − ϕR ) ΘR (t) , ~ = 2e. (2.5) (2.6). corresponding to the coherent exchange of Cooper pairs between the grain and the left or right lead respectively. Due to the periodical external driving, any interaction with the external environment leads to a periodical steady state, it is any observable at the steady state is periodic. Then, at the steady state, the time averaged current, IL(R) = hIˆL(R) i ≡. 1 T. Z. T. dthIˆL(R) (t)i ,. (2.7). 0. can be measured.. 2.2. Implementation with SQUIDs. Before analyzing in detail the transport properties, we discuss a way to realize a Cooper pair shuttle which does not require any mechanically moving part. In the proposed device the time dependence of the Josephson couplings and ng is regulated by a time dependent magnetic field and gate voltage, respectively. The setup consists of a superconducting nanocircuit in a uniform magnetic field as sketched in Fig.2.2. By substituting the Josephson junction by SQUID loops, it is possible to control the EJ by tuning the applied magnetic field piercing the loop. The presence of three type of loops with different area, AL , AR , AC allows to achieve independently the three cases, where one of the two EJ ’s is zero (regions L,R) or both of them are zero (region C), by means of a uniform magnetic field. If the applied field is such that a half-flux quantum pierces the areas AL ,AR or AC , the Josephson couplings will be those of regions R,L and C, respectively and the.

(38) 2.3. Pag. 32. Josephson current. Hamiltonian of the system can be exactly mapped onto that of Eq.(2.1). Moreover, by choosing the ratios AC /AR = 0.146, and AC /AL = 0.292 the two Josephson (L). coupling are equal, EJ. (R). = EJ. = EJ . This implementation has several advan-. tages. It allows to control the coupling with the environment by simply varying the time dependence of the applied magnetic field. The time scale for the variation of the magnetic field should be controlled at the same level as it is done in the implementation of Josephson nanocircuits for quantum computation (see Ref. [4] for an extensive discussion). For a quantitative comparison with the results described here, the magnetic field should vary on a time scale shorter than ~/EJ , typically a few nanoseconds with the parameters of Ref. [5]. This is possible with present day technology. [66] At a qualitative level the results presented in this thesis (πjunction behavior, non-monotonous behavior in the damping) do not rely on the step-change approximation of the Josephson couplings (which leads to Eq.(2.17)). Those effects are observable even if the magnetic field changes on time-scales comparable or slower than EJ . The only strict requirement is that only one Josephson coupling at the time is switched on.. 2.3. (R). (L). Shuttle dynamics: EJ , EJ (L). ≪ EC. (R). We firstly consider the system operating in EJ , EJ. ≪ EC , i.e. in the Coulomb. blockade regime. In addition, gate voltage is chosen so that 0 < ng (t) < 1 and. the Hilbert space of the system is two dimensional spanned by the two charge states {|n = 0i , |n = 1i}. We mean that ng (t) = 1/2 as long as the system is in. contact with one of the lead and ng (t) = const. ∈ (0, 1) during the remaining time of the period. Our choice (the same of Ref. [17]) results in having exact. charge degeneracy during the Josephson contacts, then enhancing charge transfer. A different condition, of easier experimental realization, in which ng (t) = const. is discussed in Appendix D. The Hamiltonian of the system restricted to the two.

(39) 2.3. Pag. 33. Josephson current. X. X. X. X. AL. B/B 0. AC. AC. 1. B(t). AC /AL. AR X. AC X. AC /AR. X. t. AC. tJ. X. tC. tJ. tC. Figure 2.2: Left panel. Sketch of the setup for the implementation of the shuttle process by means of a time-dependent magnetic field. Crosses represent Josephson junctions. Right panel. Plot of the time variation of the applied field (in unity of B0 = Φ0 /(2AC ),Φ0 is the flux quantum) in order to realize Cooper pair shuttling. The different loop areas (L). can be chosen in order to obtain EJ. (R). = EJ. .. dimensional vector space, reads X E (b) (t) J ˆ 0 = EC (t) (1 − 2ng (t))σz − e−iφb σ+ + σ− eiφb , H 2 2. (2.8). b=L,R. written in terms of the 2 × 2 Pauli matrices σi (i = x, y, z) with the usual notation σ± = (σx ± σy )/2.. In order to evaluate the current, Eqs.(2.7), or the average value of any observable,. we need to compute the reduced density matrix of the grain ρ(t). The steady state density matrix will depend on the specific decoherence mechanism. We identify the source of decoherence in the gate voltage fluctuations, which is the most effective decoherence mechanism in the charge regime. The role played by such fluctuations is taken into account, at a classical level, by adding a classical stochastic term to ng (t). The Hamiltonian in Eq (2.8) si modified by the presence of a further term, ˆ =H ˆ 0 + ξ(t)σz , H. (2.9). where ξ(t) has white noise statistics hξ(t)istoc = 0 and hξ(t)ξ(t′ )istoc = ~2 γδ(t − t′ )..

(40) 2.3. Pag. 34. Josephson current. Thus defined, γ is just the inverse decoherence time of the two charge states. If we neglected the fluctuations, the time evolution of the system would be fully coherent. By including fluctuations, the shuttle will be described by 2 × 2 density matrix that obeys the following Bloch equation: ∂ ρˆ ı ˆ ˆ 0 (t) − 2γ (ˆ H0 (t)ˆ ρ − ρˆH ρ − σz ρˆσz ) . =− ∂t ~. (2.10). The only stationary solution of this equation is trivial: ρˆ ∝ ˆ1, this corresponds to the absence of any average superconducting current. This is a combined effect. of the decoherence term and Josephson coupling. In the absence of Josephson coupling, voltage fluctuations can not cause transitions between the charge states so no relaxation takes place. With Josephson coupling switched on, the voltage fluctuations cause transitions between the stationary states separated by energy (L,R). EJ. . Classical voltage fluctuations result in equal transition rates with increasing. and decreasing energy. In fact, the ratio of these rates is given by Boltzmann (L,R). factor exp(EJ. /Tb ), Tb being the temperature of the environment producing the (L,R). the interactions with the fluctuations. Therefore at low temperature, Tb . EJ ˆ bath can lead to an anisotropic ρˆ 6= 1 and then to a non-vanishing supercurrent. In. order to analyze the low temperature regime, we need to take into proper account the quantum dynamics of the bath. This is achieved by modeling the quantum dynamics of the external circuit coupled to the shuttle through the gate capacitance as a bath of bosonic oscillators. In this scheme the shuttle is coupled via the charge operator n ˆ to an environment described by the Caldeira–Leggett model, [12] X ˆ + Hbath = n λi (ai + a†i ) + Hbath . Hint = n Ô ˆ. (2.11). i. In Eq.(2.11), Hbath is the bath Hamiltonian, with boson operators ai , a†i for its ith mode. The form of the coupling in Eq.(2.11) describes gate voltage fluctuations [4] and, in some limits, random switching of background charges in the substrate. [60].

(41) 2.3. Pag. 35. Josephson current. Due to the periodicity of the parameters in the Hamiltonian, the time evolution of the system at long time, t ≫ T , is determined by iterating the evolution of the density matrix ρ(t) over one single period.. The evolution of ρ(t) after one period can be computed through a linear map Mt defined by. ρ(t + T ) = Mt→t+T [ρ(t)] .. (2.12). In the 2 × 2 matrix formalism of Hamiltonian Eq. (2.8), the reduced density matrix can be parameterized as ρ(t) = 1/2 [1I + σ · r(t)], where i = x, y, z and ri (t) = hσi i.. By using such parameterization, the map in Eq. (2.12) assumes the form of of a general affine map for the vector r(t): r(t + T ) = Mt r(t) + vt , r ∈ B1 (0) ⊂ R3. (2.13). where B1 is the Ball of unitary radius in R3 . The matrix Mt fulfill the property |Mt v| ≤ |v| ∀v ∈ B1 (0) ,. (2.14). as we will see from its explicit form determined below. This is enough to show that, in the long time limit, the system reaches a periodic steady state, r∞ (t) = (1I − Mt ) vt ,. (2.15). if, and only if, det(1I − Mt ) 6= 0. This condition ensures that the long time limit. for the map M exist: When it is not satisfied, in fact, our decoherence mechanism. introduced in Eq. (2.11) is not effective and the system never loses memory of the initial conditions. The stationary limit is the fixed point of Mt→t+T .. [67] The. knowledge of r∞ (t), and therefore of ρ∞ (t) = 1/2 [1I + σ · r∞ (t)], uniquely determines the steady state of the system. In particular the periodic time dependence of any observable A is given by hAi (t) = T r{ρ∞ (t)A} ,. (2.16).

(42) 2.3. Pag. 36. Josephson current. where the operator A is in the Shrödinger representation. The assumption of a stepwise varying Hamiltonian considerably simplifies the form of the map Mt→t+T , obtained as a composition of the time evolutions of ρ in. the intervals L, C, R (see Fig. 2.1). In each time interval it is straightforward to solve the corresponding master equation for the reduced density matrix. [68] In the time interval L corresponding to a Josephson interaction time, such master equation. reads ˙ r(t) = GL (t)r(t) + 2γL wL † with wL = tanh(EL /Tb ) cos φL , sin φL , 0 and   −2γL 0 − E~L sin φL   GL =  0 −2γL − E~L cos φL   .  EL ~. sin φL. EL ~. cos φL. (2.17). (2.18). 0. Here, γL is the dephasing rates in the regions L, depending on the temperature of the bath, which is taken in thermal equilibrium at temperature Tb . The treatment in the time interval R is exactly the same, ones the substitution L → R has been per-. formed, thus introducing a dephasing rate γR . Both dephasing rate can be obtained in the Born-Markov approximation, [68] which requires that the bath autocorrelation time is the smallest time scale in the problem. This treatment is then valid provided that γL(R) ≪ Tb /~, EL(R) /~, and that the time interval tL(R) is much longer −1 than both ~Tb−1 and ~EL(R) . As an example, for an Ohmic bath with coupling to. the environment α ≪ 1, one has γL(R) = (π/2)αEL(R) coth(EL(R) /2Tb )/~. [12] GL. is time independent as a consequence of the Born-Markov approximation, thus the solution of the equation can be obtained: r(tL ) = exp (GL tL ) r(0) − 2γL G−1 L [1I − exp (GL tL )] wL ,. (2.19). thus the parameters of the Hamiltonian enter the final results only through the combinations θL(R) = EJ tL(R) /~ and γL(R) tL(R) . Due to the condition γL(R) ≪ EJ /~ the parameter ~γL(R) /EJ does not enter the results at lowest order..

(43) 2.4. Josephson current. Pag. 37. In the free evolution time, the situations is simpler. Since n ˆ is conserved, the evolution can be determined exactly. It is     0 0 e−γ→ t→ cos(χ→ ) − sin(χ→ ) 0       r(t + tL ) =  e−γ→ t→ 0  0  ·  sin(χ→ ) cos(χ→ ) 0 r(tL ). (2.20) 0 0 1 0 0 1 where χ→ =. R tL +t→ tL. EC (t)(1 − 2ng )/~. The rate γ→ depends only on bath param-. eters and not on the system’s ones. Its explicit expression is different at different time scales compared with the inverse ultraviolet bath mode cut-off, 1/ωc , and the. inverse bath temperature, ~/Tb . [69, 70] An explicit expression of γ→ in terms of bath parameters can be obtained within the same Born-Markov approximation discussed above in the case of a weakly coupling between the bath and the system. It gives γ→ = 2παTb /~. in which case γ→ is independent on time and the decay is purely exponential. This is the case we will refer to from now on. The same equation holds in the backward free evolution time by replacing tL with T − t← , γ→. with γ← and χ→ with χ← . A part from the dynamical phases χ→(←) and θL(R) , it is also the phase difference φ = φL − φR that enters the physical results. The. effect of damping is characterized by the dimensionless quantities γL(R) tL(R) , and γ→(←) t→(←) . From Eqs. (2.17-2.20) it is easy to check that Mt fulfills the property in Eq. 2.14. For the following values of the parameter, and for those only, (γL , γR , γ→ , γ← ) = (0, 0, 0, 0) or (γL , γR , θL ), θR = (0, 0, kπ/2, hπ/2)), k, h integers, det(1I − Mt ) =. 0. In these cases, the system keeps memory of its initial conditions and it never approaches the steady state. This is however an artificial situation, because other sources of dissipation are present which will let the system reach a steady state..

(44) 2.4. Josephson current. 2.4. DC Josephson current. Pag. 38. The existence of a nontrivial fixed point (Eq. (2.15)) for the map M, gives a nontrivial current through the system. In the case considered by Gorelik et al. in. Ref. [17], the Josephson current does not depend on the dephasing rates. One expects, however, that this cannot be always the case. If, for example, the period T is much larger than the inverse dephasing rates, the shuttle mechanism is expected to be inefficient and the critical current is strongly suppressed. Indeed, we find a quite rich scenario, depending on the relative value of the various time scales and phase shifts. ˆ int ] = 0, it can As charge is conserved by the coupling to the environment, [ˆ n, H flow only trough the electrodes. Therefore, in the Heisenberg picture, IˆL (t)+ IˆR (t)+ n ˆ˙ = 0. By integrating over a period and due to the Θ functions in the definition of the current operators (Eq. (2.5, 2.6)), IL = −IR ≡ I = T r{ˆ n(ρ(tL ) − ρ(0))} =. T r{ˆ n(ρ(T /2) − ρ(0))}, setting the origin of times within a period at the beginning of contact with the left lead.. Using the steady state density matrix (Eq. 2.15) we have a formal expression for the DC Josephson current in the system: I=. e z · (1I − M0 )−1 v0 − z · (1I − MT /2 )−1 vT /2 T. (2.21). where z is the unitary vector (0, 0, 1)T , · stands for the usual scalar product in R3 ,. and MT /2 , vT /2 , M0 , v0 are defined in Eq. (2.13). We note that MT /2 , vT /2 are. obtained from M0 , v0 by the exchange of right and left Josephson contacts and of forward and backward free evolution time. It is MT /2 = PM0 , vT /2 = Pv0 with P. acting on the parameters as:. P : (L, →, R, ←) → (R, ←, L, ←) .. (2.22). The expression of the current I(φ, θL(R) , χ→(←) , γL(R) tL(R) , γ→(←) t→(←) ) can be obtained analytically from Eq.(2.21) by explicitly writing M0 and v0 in terms of.

(45) 2.4. Josephson current. Pag. 39. the various parameters. The current depends only on the phase difference between the two superconductors φR − φL . It is an odd function with respect to the action of P defined by Eq. (2.22). From this observation follows that, even for φ = 0, there. can be a supercurrent between the leads as long as the evolution over a cycle is not P-invariant. In this sense the system behaves like a (non-adiabatic) Cooper pair pump.. A typical plot of I in the P-invariant case is shown in Fig.2.3 as a function of. θ and φ. Depending on the value of θ (a similar behavior is observed as a function. of χ), the critical current can be negative, i.e. the system can behave as a πjunction. The phase shifts accumulated in the time intervals L,C and R, leading to the current-phase relation shown in Fig.2.3, are affected by the dephasing rates in a complicated way. By changing γJ tJ and γC tC , certain interference paths are suppressed, resulting in a shift of the interference pattern and ultimately in a change of the sign of the current, as shown in Fig.2.3. An analysis of the critical current as a function of the dephasing rates reveals another interesting aspect: the Josephson current is a non-monotonous function of γJ tJ , i.e. by increasing the damping, the Josephson current can increase. The behavior as a function of the dephasing rates is presented in Fig.2.3. The presence of a maximum Josephson current at a finite value of γJ tJ can be understood by analyzing the asymptotic behaviors in the strong and weak damping limits, where simple analytic expressions are available (in the following we do not explicitely write the temperature dependence in γJ , γC ). i)If the dephasing is strong, I can be expanded in powers of e−γL(R) tL(R) and e−γ→(←) t→(←) and, to leading order, EL 2e tanh e−(γL tL +γ← t← ) sin(2θL ) sin(φ − χ← )+ I ∼ T Tb ER + tanh e−(γR tR +γ→ t→ ) sin(2θR ) sin(φ + χ→ ) . Tb. (2.23). For simplicity we assume that the Josephson energies at left and right contact are.

(46) 2.4. Pag. 40. Josephson current. 2π 3π 2 π. 0. 0. π. 2π. 3π.

(47) 2.4. Josephson current. Pag. 41. Strong dephasing is reflected in the simple (i.e. ∝ sin φ) current-phase relationship and in the exponential suppression of the current itself. Strong dephasing, in fact,. suppresses coherent transport over multiple cycles, thus giving a corresponding suppression of higher harmonics in the current-phase relationship, i.e. a suppression of terms ∝ sin2m+1 (φ), m ∈ N .. For the sake of simplicity, from now on, we present all the result in the case of. perfect P-invariance of the time evolution of the density matrix in a period. This is not a serious limitation for the experimental setups.. ii)In the opposite limit of weak damping defined by γJ tJ ≪ γC tC ≪ 1 EJ γJ tJ (cos φ + cos 2χ) tan θ sin φ 2e tanh . Iweak ∼ T Tb γC tC 1 + cos φ cos 2χ. (2.26). The current tends to zero if the coupling with the bath is negligible during the interaction time. In this case, indeed, the time evolution in the intervals L, R is almost unitary, while, in the region C, pure dephasing leads to a suppression of the off-diagonal terms of the reduced density matrix ρ(t). As a result, in the stationary limit the system is described by a complete mixture with equal weights. At the point (γJ tJ , γC tC ) = (0, 0) our model is not defined as discussed at the end of Section 2.3. The limit value of the current in approaching such point is not unique. It depends on the relative strength γJ tJ ≶ γC tC between this two parameters. The current tends to zero in both limiting cases of large and small γJ tJ . Therefore one should expect an optimal coupling to the environment where the Josephson current is maximum. A regime where the crossover between the strong and weak damping cases can be described in simple terms is the limit γC → 0, for a fixed value of θ. For example, at θ = π/4 the current reads 2e EJ 2e−γJ tJ [2e−2γJ tJ cos φ + (1 + e−4γJ tJ ) cos 2χ] sin φ tanh I= T Tb (1 + e−2γJ tJ )(1 + e−2γJ tJ cos φ cos 2χ + e−4γJ tJ ). (2.27). In the limit of vanishing γJ tJ , Eq.(2.27) corresponds to the situation discussed in Ref. [17]. Indeed, both expressions are independent of the dephasing rates. The.