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We review recent advances in the research on quantum parametric phenomena in superconducting circuits with Josephson junctions. We discuss physical processes in parametrically driven tunable cavity and outline theoretical foundations for their description. Amplification and frequency conversion are discussed in detail for degenerate and non-degenerate parametric resonance, including quantum noise squeezing and photon entanglement. Experimental advances in this area played decisive role in successful development of quantum limited parametric amplifiers for superconducting quantum information technology. We also discuss nonlinear down-conversion processes and experiments on self-sustained parametric and subharmonic oscillations., 22 pages, 25 figures

We develop a theory for the current-voltage characteristics of diffusive superconductor-normal metal-superconductor Josephson junctions with resistive interfaces and the distance between the electrodes smaller than the superconducting coherence length. The theory allows for a quantitative analytical and numerical analysis in the whole range of the interface transparencies and asymmetry. We focus on the regime of large interface resistance compared to the resistance of the normal region, when the electron-hole dephasing in the normal region is significant and the finite length of the junction plays a role. In the limit of strong asymmetry we find pronounced current structures at the combination subharmonics of $\Delta+\Delta_g$, where $\Delta_g$ is the proximity minigap in the normal region, in addition to the subharmonics of the energy gap $2\Delta$ in the electrodes. In the limit of rather transparent interfaces, our theory recovers a known formula for the current in a short mesoscopic connector - a convolution of the current through a single-channel point contact with the transparency distribution for an asymmetric double-barrier potential., Comment: 10 pages, 3 figures

Recent demonstrations of macroscopic quantum coherence in Josephson junction based electronic circuits have opened an entirely new dimension for research and applications in the established field of Josephson electronics. In this article we discuss basic Josephson circuits for qubit applications, methods of quantum description of these circuits, and circuit solutions for qubit couplings. Principles of manipulation and readout of superconducting qubits are reviewed and illustrated with recent experiments using various qubit types.

The formation of quasiparticle zero-energy bound states at surfaces and interfaces of high-Tc superconductors is a direct consequence of the d-wave symmetry of the order parameter. In many cases it is possible to understand the novel physics appearing at surfaces/interfaces of d-wave superconductors by a simple consideration of the specific properties of the zero-energy states. Here we discuss how the large zero-energy density of states makes breaking of time-reversal symmetry energetically favorable, both at surfaces and at Josephson junctions. 2001 Elsevier Science B.V. All rights reserved.

We present an extensive experimental and theoretical study of the proximity effect in InAs nanowires connected to superconducting electrodes. We fabricate and investigate devices with suspended gate-controlled nanowires and nonsuspended nanowires, with a broad range of lengths and normal-state resistances. We analyze the main features of the current-voltage characteristics: the Josephson current, excess current, and subgap current as functions of length, temperature, magnetic field, and gate voltage, and compare them with theory. The Josephson critical current for a short-length device, L = 30 nm, exhibits a record high magnitude of 800 nA at low temperature that comes close to the theoretically expected value. The critical current in all other devices is typically reduced compared to the theoretical values. The excess current is consistent with the normal resistance data and agrees well with the theory. The subgap current shows a large number of structures; some of them are identified as subharmonic gap structures generated by multiple Andreev reflection. The other structures, detected in both suspended and nonsuspended devices, have the form of voltage steps at voltages that are independent of either the superconducting gap or length of the wire. By varying the gate voltage in suspended devices, we are able to observe a crossover from typical tunneling transport at large negative gate voltage, with suppressed subgap current and negative excess current, to pronounced proximity junction behavior at large positive gate voltage, with enhanced Josephson current and subgap conductance as well as a large positive excess current., 12 pages, 15 figures

We calculate the current-voltage characteristics (IVC) of a biased superconducting junction with a tunnel resonance. Within a simple one-dimensional model we discuss how the IVCs depend on the position and width of the resonance. We consider both subgap region and excess current. The results provide a possible explanation of the anomaly in the onset of c-axis current in recent experiments on intrinsic Josephson effect in BSCCO.

It is found that the dc Josephson current in a ballistic superconductor{endash}normal-metal{endash}superconductor junction connected to an electron reservoir does not decay with the length of the junction if voltage is applied between the reservoir and the junction. At finite temperature, this nonequilibrium Josephson current is proportional to the applied voltage and saturates at eV{gt}{Delta} at a level typical for the critical current of short junctions. In asymmetric junctions the current-phase dependence is {pi} periodic. {copyright} {ital 1997} {ital The American Physical Society}

Multiple Andreev scattering in single-mode superconducting junctions with arbitrary normal electron transparency 01 is studied in the limit of low applied voltage eV\ensuremath{\ll}\ensuremath{\Delta}. A quasiclassical approach is developed for investigation of the dense lattice of inelastic sidebands associated with multiple Andreev scattering, which gives a global description of inelastic-scattering amplitudes and spectral distribution of the current. The crossover from the contact to the tunnel regime is investigated for the dc current and ac Josephson current as function of junction transparency and applied voltage. A mesoscopic interference effect in junctions with intermediate transparency is discussed. This effect shows up in oscillating features of the current of thermal excitations.

We consider the supercurrent flow through a gated two-dimensional normal constriction confined between superconducting electrodes. Photon-assisted Landau-Zener transitions between Andreev bound states in the constriction are shown to give rise to a series of reversals in the direction of supercurrent flow.

We study the microscopic structure of the Josephson current in a single-mode tunnel junction with a wide quasiclassical tunnel barrier. In such a junction each Andreev bound state carries a current of magnitude proportional to the {\em amplitude} of the normal electron transmission through the junction. Tremendous enhancement of the bound state current is caused by the resonance coupling of superconducting bound states at both superconductor-insulator interfaces of the junction. The possibility of experimental observation of the single bound state current is discussed., 11 pages, [aps,preprint]{revtex}

We solve the coherent multiple Andreev reflection (MAR) problem and calculate current-voltage characteristics (IVCs) for Josephson SINIS junctions, where S are local-equilibrium superconducting reservoirs, I denotes tunnel barriers, and N is a short diffusive normal wire, the length of which is much smaller than the coherence length, and the resistance is much smaller than the resistance of the tunnel barriers. The charge transport regime in such junctions qualitatively depends on a characteristic value \gamma = \Delta \tau_d of relative phase shifts between the electrons and retro-reflected holes accumulated during the dwell time \tau_d. In the limit of small electron-hole dephasing \gamma << 1, our solution recovers a known formula for a short mesoscopic connector extended to the MAR regime. At large dephasing, the subharmonic gap structure in the IVC scales with 1/ \gamma, which thus plays the role of an effective tunneling parameter. In this limit, the even gap subharmonics are resonantly enhanced, and the IVC exhibits portions with negative differential resistance., Comment: 8 pages, 3 figures, typos corrected, to be published in Phys. Rev. B

We develop a theoretical description of non-adiabatic Josephson dynamics in superconducting junctions containing low energy quasiparticles. Within this approach we investigate the effects of midgap states in junctions of unconventional d-wave superconductors. We identify a reentrance effect in the transition between thermal activation and macroscopic quantum tunneling, and connect this phenomenon to the experimental observations in Phys. Rev. Lett. 94, 087003 (2005). It is also shown that nonlinear Josephson dynamics can be defined by resonant interaction with midgap states reminiscent to nonlinear optical phenomena in media of two-level atoms., 16 pages, 2 figures

Conventional models of Josephson junction dynamics rely on the absence of low energy quasiparticle states due to a large superconducting gap. With this assumption the quasiparticle degrees of freedom become "frozen out" and the phase difference becomes the only free variable, acting as a fictitious particle in a local in time Josephson potential related to the adiabatic and non-dissipative supercurrent across the junction. In this article we develop a general framework to incorporate the effects of low energy quasiparticles interacting non-adiabatically with the phase degree of freedom. Such quasiparticle states exist generically in constriction type junctions with high transparency channels or resonant states, as well as in junctions of unconventional superconductors. Furthermore, recent experiments have revealed the existence of spurious low energy in-gap states in tunnel junctions of conventional superconductors - a system for which the adiabatic assumption typically is assumed to hold. We show that the resonant interaction with such low energy states rather than the Josephson potential defines nonlinear Josephson dynamics at small amplitudes., Comment: 9 pages, 1 figure

We investigate manipulation with Andreev bound states in Josephson quantum point contacts with magnetic scattering. The Rabi oscillations in the two-level Andreev subsystems are excited by resonant driving the direction of magnetic moment of the scatterer and by modulating the superconducting phase difference across the contact. The Andreev level dynamics is manifested by the temporal oscillation of the Josephson current, which is accompanied, in the case of magnetic manipulation, also by the oscillation of the Andreev states spin polarization. The interlevel transitions obey a selection rule that forbids manipulations in a certain region of external parameters and results from specific properties of Andreev bound states in magnetic contacts: 4pi periodicity with respect to the superconducting phase and strong spontaneous spin polarization.

Theoretical investigation of a chain of loop-shaped charge qubits, interacting via joint Josephson junctions is performed. Neighbouring qubits are coupled to each other only when the bias current through the joint Josephson junction is switched on. The maximum coupling strength is estimated and how to perform a universal set of quantum gates is shown.

We discuss the current situation concerning measurement and readout of Josephson-junction based qubits. In particular we focus attention of dispersive low-dissipation techniques involving reflection of radiation from an oscillator circuit coupled to a qubit, allowing single-shot determination of the state of the qubit. In particular we develop a formalism describing a charge qubit read out by measuring its effective (quantum) capacitance. To exemplify, we also give explicit formulas for the readout time., 20 pages, 7 figures. To be published in J. Phys.: Condensed Matter, 18 (2006) Special issue: Quantum computing

We study nonequilibrium effects in current transport through voltage biased tunnel junction with long diffusive superconducting leads at low applied voltage, $eV \ll 2\Delta$, and finite temperatures. Due to a small value of the Josephson frequency, the quasiparticle spectrum adiabatically follows the time evolution of the superconducting phase difference, which results in the formation of oscillating bound states in the vicinity of the tunnel junction (Andreev band). The quasiparticles trapped by the Andreev band generate higher even harmonics of the Josephson ac current, and also, in the presence of inelastic scattering, a non-equilibrium dc current, which may considerably exceed the dc quasiparticle current given by the tunnel model. The distribution of travelling quasiparticles also deviates from the equilibrium due to the spectrum oscillations, which results in an additional contribution to the dc current, proportional to $\sqrt{V}$., Comment: 11 pages, 7 figures, to be published in Phys. Rev. B

We investigate the design and functionality of a network of loop-shaped charge qubits with switchable nearest-neighbor coupling. The qubit coupling is achieved by placing large Josephson junctions at the intersections of the qubit loops and selectively applying bias currents. The network is scalable and makes it possible to perform a universal set of quantum gates. The coupling scheme allows gate operation at the charge degeneracy point of each qubit, and also applies to charge-phase qubits. Additional Josephson junctions included in the qubit loops for qubit readout can also be employed for qubit coupling., 23 pages, 4 figures

We present an experimental and theoretical investigation of coherent current transport in wide ballistic superconductor-two dimensional electron gas-superconductor junctions. It is found experimentally that upon increasing the junction length, the subharmonic gap structure in the current-voltage characteristics is shifted to lower voltages, and the excess current at voltages much larger than the superconducting gap decreases. Applying a theory of coherent multiple Andreev reflection, we show that these observations can be explained in terms of transport through Andreev resonances., 4 pages, 4 figures

We design and evaluate a scalable charge qubit chain network with controllable current-current coupling of neighbouring qubit loops via local dc-current gates. The network allows construction of general N-qubit gates. The proposed design is in line with current main stream experiments., 4 pages, 4 figures

We study non-equilibrium Josephson effect and phase-dependent conductance in three-terminal diffusive interferometers with short arms. We consider strong proximity effect and investigate an interplay of dissipative and Josephson currents co-existing within the same proximity region. In junctions with transparent interfaces, the suppression of the Josephson current appears at rather large voltage, $eV\sim \Delta$, and the current vanishes at $eV\geq\Delta$. Josephson current inversion becomes possible in junctions with resistive interfaces, where the inversion occurs within a finite interval of the applied voltage. Due to the presence of considerably large and phase-dependent injection current, the critical current measured in a current biased junction does not coincide with the maximum Josephson current, and remains finite when the true Josephson current is suppressed. The voltage dependence of the conductance shows two pronounced peaks, at the bulk gap energy, and at the proximity gap energy; the phase oscillation of the conductance exhibits qualitatively different form at small voltage $eV\Delta$., Comment: 11 pages, 9 figures, revised version, to be published in Phys. Rev. B

We study the dc-current transport in a voltage biased superconductor-chaotic dot-superconductor junction with an induced proximity effect(PE) in the dot. It is found that for a Thouless energy $E_{Th}$ of the dot smaller than the superconducting energy gap $\Delta$, the PE is manifested as peaks in the differential conductance at voltages of order $E_{Th}$ away from the even subharmonic gap structures $eV \approx 2(\Delta\pm E_{Th})/2n$. These peaks are insensitive to temperatures $kT \ll \Delta$ but are suppressed by a weak magnetic field. The current for suppressed PE is independent of $E_{Th}$ and magnetic field and is shown to be given by the Octavio-Tinkham-Blonder-Klapwijk theory., Comment: 4 pages, 3 figures

We present a detailed study of nonequilibrium Josephson currents and conductance in ballistic multiterminal SNS-devices. Nonequilibrium is created by means of quasiparticle injection from a normal reservoir connected to the normal part of the junction. By applying a voltage at the normal reservoir the Josephson current can be suppressed or the direction of the current can be reversed. For a junction longer than the thermal length, $L\gg\xi_T$, the nonequilibrium current increases linearly with applied voltage, saturating at a value equal to the equilibrium current of a short junction. The conductance exhibits a finite bias anomaly around $eV \sim \hbar v_F/L$. For symmetric injection, the conductance oscillates $2\pi$-periodically with the phase difference $\phi$ between the superconductors, with position of the minimum ($\phi=0$ or $\pi$) dependent on applied voltage and temperature. For asymmetric injection, both the nonequilibrium Josephson current and the conductance becomes $\pi$-periodic in phase difference. Inclusion of barriers at the NS-interfaces gives rise to a resonant behavior of the total Josephson current with respect to junction length with a period $\sim \lambda_F$. Both three and four terminal junctions are studied., Comment: 21 pages, 19 figures, submitted to Phys. Rev. B

We study theoretically the ac Josephson effect in voltage biased planar junctions of d-wave superconductors. For some orientations of the superconductors a current peak is found at finite voltage in the current-voltage characteristics. We pick out the relevant physical processes and write down an analytical formula for the current which clearly shows how the midgap state acts as a resonance and produces the peak. We present a possible explanation for the zero-bias conductance peak, recently found in experiments on grain boundary junctions of high-temperature superconductors, in terms of resonant transmission through midgap state of quasiparticles undergoing multiple Andreev reflections. We note that within our framework the zero-bias conductance peak appears in rather transparent Josephson junctions of d-wave superconductors., 10 pages, 5 figures, Submitted to a special volume of "Superlattices and Microstructures"

We discuss how to control dc Josephson current by influencing the structure and nonequilibrium population of Andreev levels via external electrostatic gates, external current injection and electromagnetic radiation. In particular we will consider the "giant" Josephson current in "long" SIS tunnel junctions and the regular and anomalous nonequilibrium Josephson currents in three terminal SNS junctions. We will briefly discuss applications to the Josephson field effect transistor (JOFET) and to the newly invented Josephson interference transistor (JOINT)., Comment: 10 pages, 3 figures; contribution to a special volume of Superlattices and Microstructures journal (ed. P.F. Bagwell)

We present a consistent theory of superconductive tunneling in single-mode junctions within a scattering formulation of Bogoliubov-de Gennes quantum mechanics. Both dc Josephson effect and dc quasiparticle transport in voltage biased junctions are considered. Elastic quasiparticle scattering by the junction determines equilibrium Josephson current. We discuss the origin of Andreev bound states in tunnel junctions and their role in equilibrium Josephson transport. In contrast, quasiparticle tunneling in voltage biased junctions is determined by inelastic scattering. We derive a general expression for inelastic scattering amplitudes and calculate the quasiparticle current at all voltages with emphasis on a discussion of the properties of subgap tunnel current and the nature of subharmonic gap structure., 47 pages, 9 figures, [preprint,eqsecnum,aps]{revtex}

We study the manifestations of superconducting current-induced bound states in the ac Josephson effect in tunnel contacts. Due to the normal elelctron backscattering by the contact, the bound states have a two-level structure, which allows direct interlevel transitions driven by an electromagnetic field introduced in the dielectric layer. We find substantial nonequilibrium suppression of the Josephson current, and nonlinear current oscillations induced by a resonant applied electromagnetic field.

We study quantum phase fluctuations in planar junctions between high-Tc superconductors for a range of relative orientations of the a-b crystal axes. Specifically, attention is directed towards the fermionic subspace of low energy Andreev bound states, midgap states (MGS). This subspace, being responsible for most of the Josephson current, strongly couples to the phase degree of freedom. Low frequency phase fluctuations may introduce intrinsic dynamics in this subspace, which would lead to a strong dynamical modulation of the Josephson current, and thus affect the phase quantum dynamics. In this article it is found that transitions between pairs of MGS induced by the phase fluctuations are forbidden due to specific symmetry properties of the MGS, and the macroscopic quantum Hamiltonian remains essentially adiabatic within the assumption of a specular interface.

We have fabricated and characterized tunable superconducting transmission line resonators. To change the resonance frequency, we modify the boundary condition at one end of the resonator through the tunable Josephson inductance of a superconducting quantum interference device. We demonstrate a large tuning range (several hundred megahertz), high quality factors (104), and that we can change the frequency of a few-photon field on a time scale orders of magnitude faster than the photon lifetime of the resonator. This demonstration has implications in a variety of applications.

We analyze the variation of the critical Josephson current with gate voltage in a ballistic SNS heterostructure. With increasing gate voltage the number of bound Andrev levels increases, the junction becomes effectively long, and the critical current is suppressed.