Superconducting order in a sufficiently narrow and infinitely long wire is destroyed at zero temperature by quantum fluctuations, which induce $2π$ slips of the phase of the order parameter. However, in a finite-length wire coherent quantum phase-slips would manifest themselves simply as shifts of energy levels in the excitations spectrum of an electrical circuit incorporating this wire. The higher the phase-slips probability amplitude, the larger are the shifts. Phase-slips occurring at different locations along the wire interfere with each other. Due to the Aharonov-Casher effect, the resulting full amplitude of a phase-slip depends on the offset charges surrounding the wire. Slow temporal fluctuations of the offset charges make the phase-slips amplitudes random functions of time, and therefore turn energy levels shifts into linewidths. We experimentally observed this effect on a long Josephson junction array acting as a "slippery" wire. The slip-induced linewidths, despite being only of order 100 kHz, were resolved from the flux-dependent dephasing of the fluxonium qubit.
We investigate experimentally the physics of quantum phase slips in one-dimensional Josephson Junction chains. These quantum phase-slips are induced by quantum phase fluctuations occurring on single junctions of the chain. In our experiment we can tune the strength of these fluctuations as each chain junction is realized in form of a SQUID leading to tunable Josephson coupling. We determine the ground state of the chain via switching current measurements of the chain shunted by a large Josephson junction. Our results can be well fitted with a tight binding Hamiltonian taking into account quant…
Fractional Josephson effect is a unique character of Majorana Fermions in topological superconductor system. This effect is very difficult to detect experimentally because of the disturbance of quasiparticle poisoning and unwanted couplings in the superconductor. Here, we propose a scheme to probe fractional DC Josephson effect of semiconductor nanowire-based topological Josephson junction through 4π phase slip. By exploiting a topological RF SQUID system we find that the dominant contribution for Josephson coupling comes from the interaction of Majorana Fermions, resulting the resonant tunnel…
We study coherent quantum phase-slips which lift the ground state degeneracy in a Josephson junction ring, pierced by a magnetic flux of the magnitude equal to half of a flux quantum. The quantum phase-slip amplitude is sensitive to the normal mode structure of superconducting phase oscillations in the ring (Mooij-Schön modes). These, in turn, are affected by spatial inhomogeneities in the ring. We analyze the case of weak periodic modulations of the system parameters and calculate the corresponding modification of the quantum phase-slip amplitude.
We theoretically study the macroscopic quantum tunneling (MQT) in a hetero Josephson junction formed by a conventional single-gap superconductor and a multi-gap one such as ${MgB}_{2}$ and iron-based superconductors. In such a Josephson junction more than one phase differences are defined. We clarify the quantum dynamics of the phase differences and construct a theory for the MQT in the multi-gap Josephson junctions. The dynamics of the phase differences are strongly affected by the Josephson-Leggett mode, which is the out-of-phase oscillation mode of the phase differences. The escape rate is …
Superconducting computing promises enhanced computational power in both classical and quantum approaches. Yet, scalable and fast superconducting memories are not implemented. Here, we propose a fully superconducting memory cell based on the hysteretic phase-slip transition existing in long aluminum nanowire Josephson junctions. Embraced by a superconducting ring, the memory cell codifies the logic state in the direction of the circulating persistent current, as commonly defined in flux-based superconducting memories. But, unlike the latter, the hysteresis here is a consequence of the phase-sli…
The investigation of the switching current probability distribution of a Josephson junction is a conventional tool to gain information on the phase slips dynamics as a function of the temperature. Here we adopt this well-established technique to probe the impact of an external static electric field on the occurrence of phase slips in gated all-metallic titanium (Ti) Josephson weak links. We show, in a temperature range between 20 mK and 420 mK, that the evolution of the phase slips dynamics as a function of the electrostatic field starkly differs from that observed as a function of the tempera…
Current state of the art devices for detecting and manipulating Majorana fermions commonly consist of networks of Majorana wires and tunnel junctions. We study a key ingredient of these networks - a topological Josephson junction with charging energy - and pinpoint crucial features for device implementation. The phase dependent tunneling term contains both the usual 2pi-periodic Josephson term and a 4pi-periodic Majorana tunneling term representing the coupling between Majoranas on both sides of the junction. In non-topological junctions when the charging energy is small compared to the Joseph…
We study the voltage noise properties including the switching rates and statistics of phase-slips in moderately damped Josephson junctions using a novel efficient numerical approach combining the matrix continued-fraction method with the full counting statistics. By analyzing the noise results obtained for the RCSJ model we identify different dominating components, namely the thermal noise close to equilibrium (small current-bias regime), the shot noise of (multiple) phase-slips in the intermediate range of biases and the switching noise for yet higher bias currents. We extract thus far inacce…
We investigate the Josephson transport through a ferromagnetic oxide film, e.g., La$_2$BaCuO$_5$, theoretically. Using the recursive Green's function technique, we found the formation of a pi-junction in such systems. Moreover the 0-pi phase transition is induced by increasing the temperature. Such ferromagnetic-oxide based Josephson junctions may become an element in the architecture of future quantum computers.
Current steps in the current-voltage characteristics of wide superconducting Sn films exposed to a microwave irradiation were observed in the resistive state with phase slip lines. The behaviour of the magnitude of the steps on the applied irradiation power was found to be similar to that for the current steps in narrow superconducting channels with phase slip centers and, to some extent, for the Shapiro steps in Josephson junctions. This provides evidence for the Josephson properties of the phase slip lines in wide superconducting films and supports the assumption about similarity between the…
We show that the time reversal symmetry inevitably breaks in a superconducting Josephson junction formed by two superconductors with different pairing symmetries dubbed as i-Josephson junction. While the leading conventional Josephson coupling vanishes in such an i-Josephson junction, the second order coupling from tunneling always generates chiral superconductivity orders with broken time reversal symmetry. Josephson frequency in the i-junction is doubled, namely $ω= 4eV /h$. The result provides a way to engineer topological superconductivity such as the d + id' -wave superconducting state ch…
Phase slips in a one-dimensional closed array of Josephson junctions hybridize the persistent current states (PCS) and plasmon branches of excitations. The interference between phase slips passing through different junctions of the array makes the hybridization sensitive to the charges of the superconducting islands comprising the array. This in turn results in the Aharonov-Casher effect for plasmons, which in absence of phase slips are insensitive to island charges.
The resistivity of a one-dimensional Josephson junction array at zero temperature is calculated by estimating the nucleation rate of quantum phase slips. We choose a certain collective coordinate which describes the nucleation process and estimate the corresponding effective mass. In the strong coupling regime, where Josephson coupling energy exceeds the charging energy, we calculate the nucleation rate by means of WKB method. Our estimation is in good agreement with recent experimental data. The superconductor-insulator transition point is also discussed.
A method was devised to construct a generalized dual field theory in the quantum field theory. As a simple example using this method, we examined the duality between coherent quantum phase slip and the Josephson effect in single junction systems and nanowires. The this method was proved to be reliable within the Villain approximation.
Charge formations on superconducting layers and creation of the longitudinal plasma wave in the stack of intrinsic Josephson junctions change crucially the superconducting current through the stack. Investigation of the correlations of superconducting currents in neighboring Josephson junctions and the charge correlations in neighboring superconducting layers allows us to predict the additional features in the current--voltage characteristics. The charge autocorrelation functions clearly demonstrate the difference between harmonic and chaotic behavior in the breakpoint region. Use of the corre…
We use a tight-binding Bogoliubov-de Gennes (BdG) formalism to self-consistently calculate the proximity effect, Josephson current, and local density of states in ballistic graphene SNS Josephson junctions. Both short and long junctions, with respect to the superconducting coherence length, are considered, as well as different doping levels of the graphene. We show that self-consistency does not notably change the current-phase relationship derived earlier for short junctions using the non-selfconsistent Dirac-BdG formalism but predict a significantly increased critical current with a stronger…
We derive a generic phenomenological model of a Majorana Josephson junction that accounts for avoided crossing of Andreev states, and investigate its dynamics at constant bias voltage to reveal an unexpected pattern of any-π Josephson effect in the limit of slow decoherence.
Coupling of Josephson-phase and spin-waves is theoretically studied in a superconductor/ferromagnetic insulator/superconductor (S/FI/S) junction. Electromagnetic (EM) field inside the junction and the Josephson current coupled with spin-waves in FI are calculated by combining Maxwell and Landau-Lifshitz-Gilbert equations. In the S/FI/S junction, it is found that the current-voltage (I-V) characteristic shows two resonant peaks. Voltages at the resonant peaks are obtained as a function of the normal modes of EM field, which indicates a composite excitation of the EM field and spin-waves in the …
This work deals with two pressing issues in the design and operation of Josephson qubits -- loss of coherence and measurement. (Longer abstract follows in the work.)
We perform extensive analysis of graphene Josephson junctions embedded in microwave circuits. By comparing a diffusive junction at 15 mK with a ballistic one at 15 mK and 1 K, we are able to reconstruct the current-phase relation.
Experiments on one-dimensional small capacitance Josephson Junction arrays are described. The arrays have a junction capacitance that is much larger than the stray capacitance of the electrodes, which we argue is important for observation of Coulomb blockade. The Josephson energy can be tuned in situ and an evolution from Josephson-like to Coulomb blockade behavior is observed. This evolution can be described as a superconducting to insulating, quantum phase transition. In the Coulomb blockade state, hysteretic current-voltage characteristics are described by a dynamic model which is dual to t…
Josephson junctions and SQUIDs with graphene or other 2D materials as the weak link between superconductors have become a hot topic of research in recent years, with respect to both fundamental physics and potential applications. We have previously reported ultra-wide Josephson junctions (up to 80 μm wide) based on CVD graphene where the critical current was found to be uniformly distributed in the direction perpendicular to the current. In this paper, we demonstrate that the unusually large Josephson penetration depth λ_J that this corresponds to is enabled by the unique geometric structure o…
The Josephson current between an s-wave and a spin-triplet superconductor Sr$_2$RuO$_4$ (SRO) is studied theoretically. In spin-singlet / spin-triplet superconductor junctions, there is no Josephson current proportional to $\sin ϕ$ in the absence of the spin-flip scattering near junction interfaces, where $ϕ$ is a phase-difference across junctions. Thus a dominant term of the Josephson current is proportional to $\sin 2ϕ$ . The spin-orbit scattering at the interfaces gives rise to the Josephson current proportional to $\cosϕ$, which is a direct consequence of the chiral paring symmetry in SRO.
We study coherent quantum phase-slips in a Josephson junction chain, including two types of quenched disorder: random spatial modulation of the junction areas and random induced background charges. Usually, the quantum phase-slip amplitude is sensitive to the normal mode structure of superconducting phase oscillations in the ring (Mooij-Schön modes, which are all localized by the area disorder). However, we show that the modes' contribution to the disorder-induced phase-slip action fluctuations is small, and the fluctuations of the action on different junctions are mainly determined by the loc…
We propose a realization of the superconducting diode effect in flux biased superconducting circuits of Josephson junctions. So far the observation of the superconducting diode effect has been limited to rather exotic material platforms. In theoretical proposals, it relied on a non-sinusoidal form of the current phase relation of Josephson junctions. Here, we show how the diode effect can be engineered in superconducting circuits without any such requirements. The only necessary ingredients are standard sinusoidal Josephson junctions and flux bias lines that are readily available in scalable i…
We theoretically investigate heat transport in temperature-biased Josephson tunnel junctions in the presence of an in-plane magnetic field. In full analogy with the Josephson critical current, the phase-dependent component of the heat flux through the junction displays coherent diffraction. Thermal transport is analyzed in three prototypical junction geometries highlighting their main differences. Notably, minimization of the Josephson coupling energy requires the quantum phase difference across the junction to undergo π-slips in suitable intervals of magnetic flux. An experimental setup suite…
We present a detailed study of photonic heat transport across a Josephson junction coupled to two arbitrary linear circuits having different temperatures. First, we consider the linear approximation, in which a nonlinear Josephson potential is replaced by a quadratic one and the junction acts as an inductor. Afterwards, we discuss the effects of junction anharmonicity. We separately consider the weak-coupling limit, in which the Bloch band structure of the junction energy spectrum plays an important role, and the opposite strong-coupling regime. We apply our general results to two specific mod…
We discuss dephasing by the dissipative electromagnetic environment and by measurement in circuits consisting of small Josephson junctions. We present quantitative estimates and determine in which case the circuit might qualify as a quantum bit. Specifically, we analyse a three junction Cooper pair pump and propose a measurement to determine the decoherence time $τ_ϕ$.
Topological Josephson junctions (JJs), which contain Majorana bound states, are expected to exhibit 4$π$-periodic current-phase relation, thereby resulting in doubled Shapiro steps under microwave irradiation. We performed numerical calculations of dynamical properties of topological JJs using a modified resistively and capacitively shunted junction model and extensively investigated the progressive evolution of Shapiro steps as a function of the junction parameters and microwave power and frequency. Our calculation results indicate that the suppression of odd-integer Shapiro steps, i.e., evid…
Control of the Josephson current by varying a gate current has recently been demonstrated in both 4-terminal and 3-terminal junctions. We show that, when the the gates are weakly coupled to the Josephson junction, the Josephson current versus gate current (or versus gate voltage) relation is the same for both the 4- and 3- terminal geometries. At low temperature, the supercurrent switches abruptly as a function of the gate voltage, but only slowly as a function of the gate current.
One of the consequences of Cooper pairs having a finite momentum in the interlayer of a Josephson junction, is $π$-junction behavior. The finite momentum can either be due to an exchange field in ferromagnetic Josephson junctions, or due to the Zeeman effect. Here, we report the observation of Zeeman effect induced 0-$π$ transitions in Bi$_{1-x}$Sb$_x$, 3D Dirac semimetal-based Josephson junctions. The large g-factor of the Zeeman effect from a magnetic field applied in the plane of the junction allows tuning of the Josephson junctions from 0- to $π$- regimes. This is revealed by sign changes …
We study the thermodynamic properties of topological Josephson junctions using a quantum spin Hall (QSH) insulator-based junction as an example. In particular, we propose that phase-dependent measurements of the heat capacity offer an alternative to Josephson-current measurements to demonstrate key topological features. Even in an equilibrium situation, where the fermion parity is not conserved, the heat capacity exhibits a pronounced double peak in its phase dependence as a signature of the protected zero-energy crossing in the Andreev spectrum. This double-peak feature is robust against chan…
We present measurements of the current-phase relation (CPR) of Superconductor-Ferromagnet-Superconductor (SFS) Josephson junctions as a function of temperature. The CPR is determined by incorporating the junction into a superconducting loop coupled to a dc SQUID, allowing measurement of the junction phase difference. Junctions fabricated with a thin (~ 22 nm) barrier of Cu0.47Ni0.53 sandwiched between Nb electrodes exhibit a re-entrant critical current with temperature, vanishing at T =T_pi ~ 2-4 K. We find that the critical current is negative for T < T_pi, indicating that the junction is a p…
We discuss heat transport in thermally-biased long Josephson tunnel junctions in the presence of an in-plane magnetic field. In full analogy with the Josephson critical current, the phase-dependent component of the heat current through the junction displays coherent diffraction. Thermal transport is analyzed as a function of both the length and the damping of the junction, highlighting deviations from the standard Fraunhofer pattern characteristic of short junctions. The heat current diffraction patterns show features strongly related to the formation and penetration of Josephson vortices, i.e…
The duality between coherent quantum phase slip and Josephson junction in nanosheets was investigated using the dual Hamiltonian method. This is equivalent to the duality between superconductivity and superinsulator in the 2 + 1 dimension at zero temperature. This method proved to be reliable within the Villain approximation. The possibility of the dual Ginzburg-Landau theory, which is the phenomenology of superinsulators and the dual BCS theory, which is a microscopic theory, is also shown.
We calculate the electrical currents through a superconductor - insulator - superconductor junction which is also weakly coupled to a normal metal side probe. The voltage $V$ applied to the normal metal terminal controls the occupation of Andreev energy levels $E_n$, and therefore controls the Josephson current flowing through these levels. Whenever the probe voltage crosses an Andreev level, the Josephson current changes abruptly by an amount equal to the current flowing through the Andreev level. The differential conductance along the normal metal terminal permits spectroscopy of the Andreev…
A non-band theory is developed to describe the superconductor-insulator (SI) transtition in resistively shunted, single Josephson junctions. The $I-V$ characteristic is formulated by a Landauer-like formula and evaluated by the path-integral transfer-matrix method. The result is consistent with the recent experiments at around 80 $mK$. However, the insulator phase shrinks with decreasing temperature indicating that the single Josephson junction becomes all superconducting at absolute zero temperature, as long as dissipation is present.
Wide mesoscopic superconducting - normal-metal - superconducting (S-N-S) junctions exhibit Andreev bound states which carry substantial supercurrents, even at temperatures for which the equilibrium Josephson effect is exponentially small --- the currents carried by different states can cancel each other. This cancellation is incomplete whenever the junctions are driven out of equilibrium, e.g., by a dc voltage. This leads to phenomena similar to the usual dc and ac Josephson effects, but dominated by the second harmonic of the Josephson frequency, which may explain some striking recent experim…
Superconducting circuits incorporating Josephson tunnel junctions are widely used for fundamental research as well as for applications in fields such as quantum information and magnetometry. The quantum coherent nature of Josephson junctions makes them especially suitable for metrology applications. Josephson junctions suffice to form two sides of the quantum metrology triangle, relating frequency to either voltage or current, but not its base, which directly links voltage to current. We propose a five Josephson tunnel junction circuit in which simultaneous pumping of flux and charge results i…
We demonstrate that by utilizing an over-screened Josephson junction as a noise detector it is possible to achieve the threshold regime, whereby the tails of the fluctuating current distribution are measured. This situation is realized by placing the Josephson junction and mesoscopic conductor in an external circuit with very low impedance. In the underdamped limit, over-screening the junction inhibits the energy diffusion in the junction, effectively creating a tunable activation barrier to the dissipative state. As a result, the activation rate is qualitatively different from the Arrhenius f…
We propose a theory for photon-assisted macroscopic quantum tunneling (MQT) in a stack of capacitively-coupled intrinsic Josephson junctions in which the longitudinal Josephson plasma, i.e., longitudinal collective phase oscillation modes, is excited. The scheme of energy-level quantization in the collective oscillatory states is clarified in the $N$-junction system. When the MQT occurs from the single-plasmon states excited by microwave irradiation in the multi-photon process to the uniform voltage state, our theory predicts that the escape rate is proportional to $N^2$. This result is consis…
We report on experiments with superconducting metamaterials containing Josephson junctions. In these structures, split-ring resonators used in conventional metamaterials are replaced by superconducting loops that are interrupted by Josephson junctions, so called rf-SQUIDs. Like the split-ring resonators, these elements can be seen as LC-resonators that couple to the magnetic field. The advantage of superconducting thin-film metamaterials is that, due to the tunable intrinsic inductance of the Josephson junction, the resonance frequency of the rf-SQUID can be changed by applying an external dc …
Josephson junctions with graphene as the weak link between superconductors have been intensely studied in recent years, with respect to both fundamental physics and potential applications. However, most of the previous work was based on mechanically exfoliated graphene, which is not compatible with mass production. Here we present our research using graphene grown by chemical vapour deposition (CVD) as the weak link of Josephson junctions. We demonstrate that CVD-graphene-based Josephson junctions with Nb electrodes can work effectively without any thermal hysteresis from 1.5 K down to a base …
Inspired by a recent experiment, we study the influence of thermal fluctuations on the $I$-$V$ characteristics of a Josephson junction, coupled to a strongly resistive environment. We obtain analytical results in the limit where the Josephson energy is larger than the charging energy and quasiparticles are absent.
Dislocations and disclinations in a lattice of Josephson junctions will affect the dynamics of vortex excitations within the array. These defects effectively distort the space in which the excitations move and interact. The interaction energy between such defects and excitations are determined and vortex trajectories in twisted lattices are calculated. Finally, possible experiments observing these effects are presented.
We consider a planar SIS-type Josephson junction between diffusive superconductors (S) through an insulating tunnel interface (I). We construct fully self-consistent perturbation theory with respect to the interface conductance. As a result, we find correction to the first Josephson harmonic and calculate the second Josephson harmonic. At arbitrary temperatures, we correct previous results for the nonsinusoidal current-phase relation in Josephson tunnel junctions, which were obtained with the help of conjectured form of solution. Our perturbation theory also describes the difference between th…
We discuss the Josephson current in superconductor / dirty normal conductor / superconductor junctions, where the superconductors have $d_{x^2-y^2}$ pairing symmetry. The low-temperature behavior of the Josephson current depends on the orientation angle between the crystalline axis and the normal of the junction interface. We show that the ensemble-averaged Josephson current vanishes when the orientation angle is $π/4$ and the normal conductor is in the diffusive transport regime. The $d_{x^2-y^2}$-wave pairing symmetry is responsible for this fact.
We study the Josephson effect in small one-dimensional (1D) Josephson junction arrays. For weak Josephson tunneling, topologically different regions in the charge-stability diagram generate distinct current-phase relationships (I$Φ$). We present results for a three-junction system in the vicinity of charge degeneracy lines and triple points. We explain the generalization to larger arrays, show that discontinuities of the I$Φ$ at phase $π$ persist and that, at maximum degeneracy, the problem can be mapped to a tight-binding model providing analytical results for arbitrary system size.
We study vortex structure in three different models of long Josephson junctions: exponentially shaped Josephson junction and Josephson junctions with resistor and shunt inhomogeneities in barrier layer. Numerical calculations of the possible magnetic flux distributions and corresponding bifurcation curves have done. For these three models the critical curves ``critical current-magnetic field'' are constructed. We develop an idea of the equivalence of exponentially shaped Josephson junction and rectangular junction with distributed inhomogeneity and demonstrate that at some parameters of shunt …
The ac Josephson effect in a ferromagnetic Josephson junction, which is composed of two superconductors separated by a ferromagnetic metal (FM), is studied by a tunneling Hamiltonian and Green's function method. We obtain two types of superconducting phase dependent current, i.e., Josephson current and quasiparticle-pair-interference current (QPIC). These currents change their signs with thickness of the FM layer due to the 0-$π$ transition characteristic to the ferromagnetic Josephson junction. As a function of applied voltage, the Josephson critical current shows a logarithmic divergence cal…
The current-phase relation (CPR) of a Josephson junction reveals valuable information about the microscopic processes and symmetries that influence the supercurrent. In this Letter, we present direct measurements of the CPR for Josephson junctions with a graphene barrier, obtained by a phase-sensitive SQUID interferometry technique. We find that the CPR is skewed with respect to the commonly observed sinusoidal behavior. The amount of skewness varies linearly with critical current (Ic) regardless of whether Ic is tuned by the carrier density or temperature.
We study Superconductor-Ferromagnet-Superconductor (Nb-Cu0.47 Ni0.53-Nb) Josephson junctions with spatial variations in the barrier thickness. Critical current vs. magnetic flux diffraction patterns indicate that the critical current density changes sign along the width of the junctions, creating interfaces between 0 and pi junction regions around which spontaneous currents can circulate. Shapiro steps observed at half-integer Josephson voltages can be explained by the phase-locking of the spontaneous circulating currents to the applied rf modulation.
We study the emergence of dissipation in an atomic Josephson junction between weakly-coupled superfluid Fermi gases. We find that vortex-induced phase slippage is the dominant microscopic source of dissipation across the BEC-BCS crossover. We explore different dynamical regimes by tuning the bias chemical potential between the two superfluid reservoirs. For small excitations, we observe dissipation and phase coherence to coexist, with a resistive current followed by well-defined Josephson oscillations. We link the junction transport properties to the phase-slippage mechanism, finding that vort…
In this paper we demonstrate that, if it exists, coherent quantum phase slip is the exact dual to Josephson tunneling. We use the duality to predict kinetic capacitance and a sharp resonance in narrow wires. Biased resistively and driven at high frequency, quantum phase slip junctions should exhibit current plateaus of interest for a fundamental standard.
Operation of a Superconducting Quantum Interference Device (SQUID) made of stacked Josephson junctions is analyzed numerically for a variety of junction parameters. Due to a magnetic coupling of junctions in the stack, such a SQUID has certain advantages as compared to an uncoupled multi-junction SQUID. Namely, metastability of current-flux modulation can be reduced and a voltage-flux modulation can be improved if junctions in the stack are phase-locked. Optimum operation of the SQUID is expected for moderately long, strongly coupled stacked Josephson junctions. A possibility of making a stack…
The transition from the model of a long Josephson junction of variable width to the model of a junction with a coordinate-dependent Josephson current amplitude is effected through a coordinate transformation. This establishes the correspondence between the classes of Josephson junctions of variable width and quasi-one-dimensional junctions with a variable thickness of the barrier layer. It is shown that for a junction of exponentially varying width the barrier layer of the equivalent quasi-one-dimensional junction has a distributed resistive inhomogeneity that acts as an attractor for magnetic…
