Two-dimensional Josephson Junction Arrays Research Articles

Quantum Simulation of the Tricritical Ising Model in Tunable Josephson Junction Ladders.

Modern hybrid superconductor-semiconductor Josephson junction arrays are a promising platform for analog quantum simulations. Their controllable and nonsinusoidal energy-phase relation opens the path to implement nontrivial interactions and study the emergence of exotic quantum phase transitions. Here, we propose the analysis of an array of hybrid Josephson junctions defining a two-leg ladder geometry for the quantum simulation of the tricritical Ising phase transition. This transition provides the paradigmatic example of minimal conformal models beyond Ising criticality and its excitations are intimately related to Fibonacci non-Abelian anyons and topological order in two dimensions. We study this superconducting system and its thermodynamic phases based on bosonization and matrix-product-state techniques. Its effective continuous description in terms of a three-frequency sine-Gordon quantum field theory suggests the presence of the targeted tricritical point and the numerical simulations confirm this picture. Our results indicate which experimental observables can be adopted in realistic devices to probe the physics and the phase transitions of the model. Additionally, our proposal provides a useful one-dimensional building block to design exotic topological order in two-dimensional scalable Josephson junction arrays.

Resistivity Tensor of Vortex-Lattice States in Josephson Junction Arrays.

Two-dimensional Josephson junction arrays frustrated by a perpendicular magnetic field are predicted to form a cascade of distinct vortex lattice states. Here, we show that the resistivity tensor provides both structural and dynamical information on the vortex-lattice states and intervening phase transitions, which allows for experimental identification of these symmetry-breaking ground states. We illustrate our general approach by a microscopic theory of the resistivity tensor for a range of magnetic fields exhibiting a rich set of vortex lattices as well as transitions to liquid-crystalline vortex states.

Dynamical vortex transitions in a gate-tunable two-dimensional Josephson junction array

Dynamical vortex transitions in a gate-tunable two-dimensional Josephson junction array

Insulating phase in two-dimensional Josephson junction arrays investigated by nonlinear transport

We present experimental investigations of transport properties in the insulating phase of two-dimensional Josephson-junction arrays (JJAs) by systematically changing the ratio of Josephson energy $E_\mathrm{J}$ and charging energy $E_\mathrm{C}$. The observed temperature dependence of resistance indicates that the JJAs do not show a sharp phase transition but exhibit a gradual crossover to the insulating phase. At low temperatures, the current-voltage ($I$-$V$) characteristics become nonlinear as described by $I=cV+bV^a$ ($a$, $b$, and $c$ are temperature dependent coefficients). This nonlinear behavior is understood in terms of the Berezinskii-Kosterlitz-Thouless (BKT) mechanism by taking into account the influence of a finite-range cutoff of the logarithmic interaction between Cooper pairs. From the analysis of the nonlinearity, we deduce the crossover temperature to the insulating phase and determine the phase diagram in the insulating side as a function of $E_\mathrm{J} /E_\mathrm{C}$. We also show that, at very low temperatures, the $I$-$V$ characteristics continuously develop into the negative differential conductance caused by coherent single-Cooper-pair tunneling.

Magnetic Monopoles and Superinsulation in Josephson Junction Arrays

Electric-magnetic duality or S-duality, extending the symmetry of Maxwell’s equations by including the symmetry between Noether electric charges and topological magnetic monopoles, is one of the most fundamental concepts of modern physics. In two-dimensional systems harboring Cooper pairs, S-duality manifests in the emergence of superinsulation, a state dual to superconductivity, which exhibits an infinite resistance at finite temperatures. The mechanism behind this infinite resistance is the linear charge confinement by a magnetic monopole plasma. This plasma constricts electric field lines connecting the charge–anti-charge pairs into electric strings, in analogy to quarks within hadrons. However, the origin of the monopole plasma remains an open question. Here, we consider a two-dimensional Josephson junction array (JJA) and reveal that the magnetic monopole plasma arises as quantum instantons, thus establishing the underlying mechanism of superinsulation as two-dimensional quantum tunneling events. We calculate the string tension and the dimension of an electric pion determining the minimal size of a system capable of hosting superinsulation. Our findings pave the way for study of fundamental S-duality in desktop experiments on JJA and superconducting films.

Multidimensional washboard ratchet potentials for frustrated two-dimensional Josephson junctions arrays on square lattices

In this work, we derive an analytical procedure that allows us to write the multidimensional washboard ratchet potential (MDWBP) Uf for a two-dimensional Josephson junction array. The array has an applied perpendicular magnetic field. The magnetic field is given in units of the quantum flux per plaquette or frustration of the form , where Φ0 is the flux quantum. The derivation is done under the assumption that the checkerboard pattern ground state or unit cell of a two-dimensional Josephson junction array is preserved under current biasing. The resistively and capacitively shunted Josephson junction model with a white noise term describes the dynamics for each junction in the array. The multidimensional potential is the unique expression of the collective effects that emerge from the array in contrast to the single junction. The first step in the procedure is to write the equation for the phases for the unit cell. In doing this, one takes into account the constraints imposed for the gauge invariant phases due to frustration. Second, and the key idea of the procedure, is to perform a variable transformation from the original systems of stochastic equations to a system of variables where the condition for the equality of mixed second partial happens. This is achieved via Poincaré's theorem for differential forms. In this way, we find to a nonlinear matrix equation (equation () in the text), that permits us to find the new coordinate variables xf where the potential exists. The transformation matrix also permits the correct transformation of the original white noise terms of each junction to the intensities in the xf variables. The commensurate symmetries of the ground state pinned vortex lattice leads to discrete symmetries to the part of the washboard potential that does not contain a tilt due to the external bias current (equation () in the text). In this work we apply the procedure for the important cases . For , we show that previously efforts for finding the potential are restricted, leading to a reduced dimension of the potential. The correct potential is given in equation (). We examine this issue in detail. New physics emerge when currents are applied in the x and y directions, in particular, we confirm analytically previous numerical work for , concerning the border of stationary states, a landmark of the potential. For , we give a generalization of previous work, in which we include both the currents in the x and y directions as well the noise terms. We find the MDWBP realizes tilted ratchets analogous to a combustion motor.

Multistability of phase-locking and topological winding numbers in locally coupled Kuramoto models on single-loop networks

Determining the number of stable phase-locked solutions for locally coupled Kuramoto models is a long-standing mathematical problem with important implications in biology, condensed matter physics, and electrical engineering among others. We investigate Kuramoto models on networks with various topologies and show that different phase-locked solutions are related to one another by loop currents. The latter take only discrete values, as they are characterized by topological winding numbers. This result is generically valid for any network and also applies beyond the Kuramoto model, as long as the coupling between oscillators is antisymmetric in the oscillators’ coordinates. Motivated by these results, we further investigate loop currents in Kuramoto-like models. We consider loop currents in nonoriented n-node cycle networks with nearest-neighbor coupling. Amplifying on earlier works, we give an algebraic upper bound N≤2 Int[n/4]+1 for the number N of different, linearly stable phase-locked solutions. We show that the number of different stable solutions monotonically decreases as the coupling strength is decreased. Furthermore stable solutions with a single angle difference exceeding π/2 emerge as the coupling constant K is reduced, as smooth continuations of solutions with all angle differences smaller than π/2 at higher K. In a cycle network with nearest-neighbor coupling, we further show that phase-locked solutions with two or more angle differences larger than π/2 are all linearly unstable. We point out similarities between loop currents and vortices in superfluids and superconductors as well as persistent currents in superconducting rings and two-dimensional Josephson junction arrays.

Berezinskii–Kosterlitz–Thouless transition in two-dimensional arrays of Josephson coupled Bose–Einstein condensates

Abstract We study the phase transition occurring in Bose–Einstein condensates placed in an optical lattice where the system is arranged in a form of a two-dimensional bosonic Josephson junction array. It is shown that the Josephson interaction between adjacent condensates (trapped in the valleys of the periodic lattice potential) can trigger Berezinskii–Kosterlitz–Thouless transition at a finite temperature T KT to the ordered state composed of bound vortex–antivortex phase-field configurations of individual condensates. Using a lattice model of the bosonic Josephson junction array, we derive the effective phase-only Hamiltonian and calculate the critical temperature T KT . Finally, we discuss the results in the context of system parameters and possible experiments.

Vortex dynamics in two-dimensional Josephson junction arrays with asymmetrically bimodulated potential

On the basis of resistively-shunted junction dynamics, we study vortex dynamics in two-dimensional Josephson junction arrays with asymmetrically single and bimodulated periodic pinning potential for the full range of vortex density f. The ratchet effect occurring at a certain range of temperature, current, and f, is observed in our simulation. We explain the microscopic behavior behind this effect by analyzing the vortex distribution and interaction. The reversal of the ratchet effect can be observed at several f values for a small driven current. This effect is stronger when the asymmetric potential is simultaneously introduced in two directions.

Simulated radiation effects in the superinsulating phase of titanium nitride films

This paper investigates possible effects of alpha particle and ion beam irradiation on the properties of the superinsulating phase, recently observed in titanium nitride films, by using numerical simulation of particle transport. Unique physical properties of the superinsulating state are considered by relying on a two-dimensional Josephson junction array as a model of material structure. It is suggested that radiation-induced change of the Josephson junction charging energy would not affect the current-voltage characteristics of the superinsulating film significantly. However, it is theorized that a relapse to an insulating state with thermally activated resistance is possible, due to radiation-induced disruption of the fine-tuned granular structure. The breaking of Cooper pairs caused by incident and displaced ions may also destroy the conditions for a superinsulating phase to exist. Finally, even the energy loss to phonons can influence the superinsulating state, by increasing the effective temperature of the phonon thermostat, thereby reestablishing means for an energy exchange that can support Cooper pair tunneling.

Transport and its enhancement caused by coupling

Using the Weiss mean-field approximation theory and the particles' transport theory for the spatially periodic stochastic systems, we derive an exact analytical expression for the stationary probability current of a coupled lattice system driven by dichotomous noise. It is shown that, for this coupled lattice system, the spatial asymmetry of the system, the asymmetry of the dichotomous noise, and the coupling among nearest neighbors are the ingredients for the stationary probability current. By applying our theory to two special models, we find that (1) the coupling can lead to the directional transport of the particles (even when the potential and the dichotomous noise are symmetric) and (2) the coupling among nearest neighbors can enhance the transport of the particles in some circumstances. Our results are applied to a device of two-dimensional Josephson-junction arrays and a large protein motors cluster.

Instanton Glass Generated by Noise in a Josephson-Junction Array

We compute the correlation function of a superconducting order parameter in a continuous model of a two-dimensional Josephson-junction array in the presence of a weak Gaussian noise. When the Josephson coupling is large compared to the charging energy, the correlations in the Euclidian space decay exponentially at low temperatures regardless of the strength of the noise. We interpret such a state as a collective Cooper-pair insulator and argue that it resembles properties of disordered superconducting films.

Phase transitions in Josephson junction arrays in a weak magnetic field

Based on the resistively shunted junction dynamics, we numerically investigate the superconducting phase transitions of two-dimensional Josephson junction arrays exposed to a dilute external field with the magnetic flux density f=1/25. Dynamic scaling analysis of current-voltage characteristics shows that a Kosterlitz–Thouless–Berezinskii (KTB) phase transition occurs at a finite temperature in the absence of disorder. When the disorder is introduced into the bond, the phase transition would be driven into a non-KTB type, where a possible glass transition is suggested. The effects of the bond disorder on the critical temperature and critical exponents are also discussed. The further experimental works to realize well known KTB and glass transitions in Josephson junction arrays are clearly called for.

Phase transition in site-diluted Josephson junction arrays: A numerical study

We numerically investigate the intriguing effects produced by random percolative disorder in two-dimensional Josephson junction arrays. By dynamic scaling analysis, we evaluate critical temperatures and critical exponents with high accuracy. It is observed that with the introduction of site-diluted disorder, the Kosterlitz-Thouless phase transition is eliminated and evolves into a continuous transition with power-law divergent correlation length. Moreover, genuine depinning transition and creep motion are studied; evidence for distinct creep motion types is provided. Our results not only are in good agreement with the recent experimental findings but also shed some light on the relevant phase transitions.

Uniformly frustrated bosonic Josephson-junction arrays

We derive a uniformly frustrated $XY$ model that describes two-dimensional Josephson-junction arrays consisting of rotating Bose-Einstein condensates trapped by both a harmonic trap and a corotating deep optical lattice. The harmonic trap makes the coupling constant of the model have a nonuniform parabolic dependence. We study the ground state through Monte Carlo simulations in a wide range of the frustration parameter $f$, revealing a rich variety of vortex patterns.

Uniformly frustrated Josephson junction array in trapped Bose-Einstein condensates

We investigate the ground state structure of two-dimensional Josephson junction arrays consisting of Bose-Einstein condensates trapped by both a rotating harmonic potential and a corotating deep optical lattice with square symmetry. Monte Carlo simulations of the uniformly frustrated XY model in a wide range of the frustration parameter f reveal that the stable configuration of vortices for a simple fractional number f is a staircase structure, consisting of diagonal chains of vortices with periodic lattice of unit cells.

Magnetic-field-induced phase transition of vortex glass states in disordered Josephson junction arrays

Using the resistively shunted junction model，we study magnetic-field-induced phase transitions of the vortex glass states in a disordered two-dimensional Josephson junction array. From calculated results of the vorticity of the vortex system and the voltage noise generated by current-driven vortices as a function of the magnetic field， the following conclusions can be drawn. 1） With decreasing the magnetic field， the vortex system exhibits a reentrant transition from a vortex glass phase to a pinned dilute vortex liquid. 2） There is a peak of voltage noise driven by the current in the vortex glass state. It follows that the system is in a sub-stable state as a result of the competition between ordered and disordered interactions， which results in peak effects of the critical current. The calculated results are consistent with recent experimental reports by Okuma and Kamada on disordered and superconducting MoxSi1-x films.

Dissipation-Driven Quantum Phase Transition in Superconductor-Graphene Systems

We show that a system of Josephson junctions coupled via low-resistance tunneling contacts to graphene substrate(s) may effectively operate as a current switching device. The effect is based on the dissipation-driven superconductor-to-insulator quantum phase transition, which happens due to the interplay of the Josephson effect and Coulomb blockade. Coupling to a graphene substrate with gapless excitations further enhances charge fluctuations favoring superconductivity. The effect is shown to scale exponentially with the Fermi energy in graphene, which can be controlled by the gate voltage. We develop a theory that quantitatively describes the quantum phase transition in a two-dimensional Josephson junction array, but it is expected to provide a reliable qualitative description for one-dimensional systems as well.

Depinning and creep in Josephson junction arrays in weak magnetic fields

Based on resistively shunted junction dynamics, we have numerically studied the depinning transition and thermally activated creep motion of vortices in two-dimensional Josephson junction arrays exposed to a weak external field with the filling factor $f=1/25$. Whether the bond disorder is introduced into the system or not, a continuous depinning transition is found at zero temperature. By means of scaling analysis of the current-voltage characteristics, a non-Arrhenius creep law is observed at finite temperatures, with the existence of two universality classes depending on the strength of the bond disorder. The effects of the disorder on the critical current and critical exponents are also discussed.

Collective Cooper-Pair Transport in the Insulating State of Josephson-Junction Arrays

We investigate collective Cooper-pair transport of one- and two-dimensional Josephson-junction arrays. We derive an analytical expression for the current-voltage characteristic revealing thermally activated conductivity at small voltages and threshold voltage depinning. The activation energy and the related depinning voltage represent a dynamic Coulomb barrier for collective charge transfer over the whole system and scale with the system size. We show that both quantities are nonmonotonic functions of the magnetic field. We propose that formation of the dynamic Coulomb barrier and its size scaling are consequences of the mutual Josephson phase synchronization across the system. We apply the results for interpretation of experimental data in disordered films near the superconductor-insulator transition.