We present a simple model for multiphoton transitions between the quasi-bound states of a current-driven Josephson junction. The transitions are induced by applying an ac voltage with controllable frequency and amplitude across the junction. The voltage induces transitions when the ac frequency equals n times the splitting between the ground and first excited quasi-bound state of the junction. We calculate the transition matrix elements as functions of the dc bias current I, and the frequency and amplitude of the ac voltage, for representative junction parameters. We also calculate the frequency-dependent absorption coefficient by solving the relevant Bloch equations when the ac amplitude is sufficiently small. In this regime, the absorption coefficient is a sum of Lorentzian lines centered at the n-photon absorption frequency, of strength proportional to the squared matrix elements. For fixed ac voltage amplitude, the n-photon transition rate usually decreases with increasing n. We also find a characteristic even-odd effect: The absorption coefficient typically increases with I for n even but decreases for n odd. Our results agree qualitatively with recent experiments.
We have observed multiphoton transitions between two macroscopic quantum-mechanical superposition states formed by two opposite circulating currents in a superconducting loop with three Josephson junctions. Resonant peaks and dips of up to three-photon transitions were observed in spectroscopic measurements when the system was irradiated with a strong RF-photon field. The widths of the multiphoton absorption dips are shown to scale with the Bessel functions in agreement with theoretical predictions derived from the Bloch equation or from a spin-boson model.
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 investigated both theoretically and experimentally dynamic features of a phase-biased charge qubit consisting of a single-Cooper-pair transistor closed by a superconducting loop. The effective inductance of the qubit was probed by a high-quality tank circuit. In the presence of a microwave power, with a frequency of the order of the qubit energy level separation, an alteration of the qubit inductance was observed. We demonstrate that this effect is caused by the redistribution of the qubit level population. The excitation of the qubit by one-, two-, and three-photon processes was detected. …
We study the dynamics of a quantum superconducting circuit which consists of a Josephson charge qubit, coupled capacitively to a current biased Josephson junction. Under certain conditions, the eigenstates of the qubit and the junction become entangled. We obtain the time evolution of these states in the limit of weak coupling. Rabi oscillations occur, as a result of the spontaneous emission and re-absorption of a single oscillation quantum in the junction. We discuss a possible way to experimentally determine the quantum state of the junction and hence observe the Rabi oscillations.
We investigate electromagnetic radiation emitted by a small voltage-biased Josephson junction connected to a superconducting transmission line. At frequencies below the well known emission peak at the Josephson frequency (2eV/h), extra radiation is triggered by quantum fluctuations in the electromagnetic environment. For weak tunneling couplings and typical ohmic transmission lines, the corresponding photon flux spectrum is symmetric around half the Josephson frequency, indicating that the photons are predominately created in pairs. By establishing an input-output formalism for the microwave f…
This work reports a novel time-domain expression for the current-response kernels of a Josephson tunnel junction between BCS superconductors with in general different energy gaps, and its use to simulate the classical dynamics of such junctions. The simulations show a dynamic regime characterized by the resonance between the Josephson oscillations and the gap oscillations in the asymptotics of the current kernels that has not been studied previously. The resonance manifests itself as a hysteresis in the dc current-voltage characteristics of the current-biased junctions in the voltage range abo…
We investigate the coherent energy and thermal transport in a temperature-biased long Josephson tunnel junction, when a Josephson vortex, i.e., a soliton, steadily drifts driven by an electric bias current. We demonstrate that thermal transport through the junction can be controlled by the bias current, since it determines the steady-state velocity of the drifting soliton. We study the effects on thermal transport of the damping affecting the soliton dynamics. In fact, a soliton locally influences the power flowing through the junction and can cause the variation of the temperature of the devi…
We investigate the behavior of a Josephson junction consisting of a ferromagnetic insulator-superconductor (FI-S) bilayer tunnel-coupled to a superconducting electrode. We show that the Josephson coupling in the structure is strenghtened by the presence of the spin-splitting field induced in the FI-S bilayer. Such strenghtening manifests itself as an increase of the critical current $I_c$ with the amplitude of the exchange field. Furthermore, the effect can be strongly enhanced if the junction is taken out of equilibrium by a temperature bias. We propose a realistic setup to assess experimenta…
Periodic in time spin-Hall current and spin polarization induced by a dc electric bias has been calculated in a superconductor-normal 2DEG-superconductor (SNS) Josephson junction. We assumed that the band energies of electrons in the normal system are splitted due to Rashba spin-orbit coupling. The transport parameters have been calculated within the diffusion approximation and using perturbation expansion over a small SN contact transparency. We found out that in contrast to the stationary Josephson effect, the spin-Hall current does not turn to zero. Besides a direct proximity effect caused …
We calculate the current-voltage (I-V) characteristic of a Josephson junction containing a resonant level in the weakly coupled regime (resonance width small compared to the superconducting gap). The phase $ϕ$ across the junction becomes time dependent in response to a DC current bias. Rabi oscillations in the Andreev levels produce a staircase I-V characteristic. The number of voltage steps counts the number of Rabi oscillations per $2π$ increment of $ϕ$, providing a way to probe the coherence of the qubit in the absence of any external AC driving. The phenomenology is the same as the "Majora…
The dc transport properties of long arrays of small Al Josephson junctions, biased through on-chip Cr resistors, are studied. The IV-characteristics show a large Coulomb threshold for current as well as negative-slope regions indicating the regime of autonomous Bloch oscillations up to rather high frequencies of f = I/2e ~1 GHz, comparable to those reported by other groups for single junctions. On the other hand, a small depth of the back-bending implies a low duty cycle and a broad spectrum of the oscillations, which we attribute to the insufficiently high impedance of the bias resistors. A s…
Motivated by recent experiments, where a voltage biased Josephson junction is placed in series with a resonator, the classical dynamics of the circuit is studied in various domains of parameter space. This problem can be mapped onto the dissipative motion of a single degree of freedom in a nonlinear time-dependent potential, where in contrast to conventional settings the nonlinearity appears in the driving while the static potential is purely harmonic. For long times the system approaches steady states which are analyzed in the underdamped regime over the full range of driving parameters inclu…
Quantum fluctuations in a current biased Josephson junction, described in terms of the RCSJ-model, are considered. The fluctuations of the voltage and phase across the junction are assumed to be initiated by equilibrium current fluctuations in the shunting resistor. This corresponds to low enough temperatures, when fluctuations of the normal current in the junction itself can be neglected. We used the quantum Langevin equation in terms of random variables related to the limit cycle of the nonlinear Josephson oscillator. This allows to go beyond the perturbation theory and calculate the widths …
We study a planar Josephson junction under an applied DC voltage bias in the presence of an in-plane magnetic field. Upon tuning the bias voltage across the junction, $V_{\rm J}$, the two ends of the junction are shown to simultaneously host both $0$- and $π$-Majorana modes. These modes can be probed using either a Scanning-Tunneling-Microscopy measurement or through resonant Andreev tunneling from a lead coupled to the junction. While these modes are mostly bound to the junction's ends, they can hybridize with the bulk by absorbing or emitting photons. We analyze this process both numerically…
We theoretically study the behavior of the critical current of a thermally-biased tunnel Josephson junction with a particular design, in which the electrodes of the junction are enclosed in two different superconducting loops pierced by independent magnetic fluxes. In this setup, the superconducting gaps can be modified independently through the magnetic fluxes threading the loops. We investigate the response of the device as a function of the magnetic fluxes, by changing the asymmetry parameter, i.e., the ratio between the zero-temperature superconducting gaps $δ=Δ_{10}/Δ_{20}$, and the tempe…
By coupling a superconducting weak link to a microwave resonator, recent experiments probed the spectrum and achieved the quantum manipulation of Andreev states in various systems. However, the quantitative understanding of the response of the resonator to changes in the occupancy of the Andreev levels, which are of fermionic nature, is missing. Here, using Bogoliubov-de Gennes formalism to describe the weak link and a general formulation of the coupling to the resonator, we calculate the shift of the resonator frequency as a function of the levels occupancy and describe how transitions are in…
In a Josephson junction (JJ), Cooper pairs are transported via Andreev bound states (ABSs) between superconductors. The ABSs in the weak link of multi-terminal (MT) JJs can coherently hybridize two Cooper pairs among different superconducting electrodes, resulting in the Cooper quartet (CQ) involving four fermions entanglement. The energy spectrum of these CQ-ABS can be controlled by biasing MT-JJs due to the AC Josephson effect. Here, using gate tunable four-terminal graphene JJs complemented with a flux loop, we construct CQs with a tunable spectrum. The critical quartet supercurrent exhibit…
We study the dynamic behaviour of a quantum two-level system with periodically varying parameters by solving the master equation for the density matrix. Two limiting cases are considered: multiphoton Rabi oscillations and Landau-Zener transitions. The approach is applied to the description of the dynamics of superconducting qubits. In particular, the case of the interferometer-type charge qubit with periodically varying parameters (gate voltage or magnetic flux) is investigated. The time-averaged energy level populations are calculated as funtions of the qubit's control parameters.
We propose a method to excite and detect a rotor localized mode (rotobreather) in a Josephson-junction array biased by dc currents. In our numerical studies of the dynamics we have used experimentally realizable parameters and included self-inductances. We have uncovered two families of rotobreathers. Both types are stable under thermal fluctuations and exist for a broad range of array parameters and sizes including arrays as small as a single plaquette. We suggest a single Josephson-junction plaquette as an ideal system to experimentally investigate these solutions.
Theory of non-stationary coherent effects is developed for superconductor-normal-superconductor (SNS) structures with relatively strong normal scattering on S/N interfaces (interface resistance is large compared to intrinsic resistance of N metal). Analitical expressions are found for the time-dependent anomalous Green functions induced in the N region under the fixed-voltage-bias. The amplitude of the current oscillations is determined in non-equilibrium conditions. Non-stationary correction to the distribution function is calculated in high-temperature limit and found to be slowly decreasing…
We theoretically investigate an opportunity to implement a source of highly spin-polarized ac current on the basis of superconductor/weak ferromagnet/superconductor (SFS) voltage-biased junction in the regime of essential proximity effect and calculate the current flowing through the probe electrode tunnel coupled to the ferromagnetic interlayer region. It is shown that while the polarization of the dc current component is generally small in case of weak exchange field of the ferromagnet, there is an ac component of the current in the system. This ac current is highly spin-polarized and entire…
We have observed a dissipative phase diffusion branch in arrays of hysteretic high-Tc intrinsic Josephson junctions. By comparing the data with a thermal activation model we extract the impedance seen by the junction in which phase diffusion is occurring. At the plasma frequency this junction is isolated from its environment and it sees its own large (~ kilo Ohm) impedance. Our results suggest that stacks of Josephson junctions may be used for isolation purposes in the development of a solid state quantum computer.
An all-superconducting bijunction consists of a central superconductor contacted to two lateral superconductors, such that non-local crossed Andreev reflection is operating. Then new correlated transport channels for the Cooper pairs appear in addition to those of separated conventional Joseph- son junctions. We study this system in a configuration where the superconductors are connected through gate-controllable quantum dots. Multipair phase-coherent resonances and phase-dependent multiple Andreev reflections are both obtained when the voltages of the lateral superconductors are commensurate,…
The Josephson quasiparticle (JQP) cycle in a voltage-biased superconducting single-electron transistor (SSET) combines coherent Cooper pair tunneling with incoherent quasiparticle decay. We have measured the influence of current flow through an independently-biased SSET on the JQP cycle when the two SSET's have a strong mutual capacitive coupling. We find, among other effects, that the JQP current in one SSET is enhanced by the presence of a quasiparticle current in the other SSET. A simplified model of the coupled-SSET system is presented which reproduces the enhancement effect.
A Lagrangian formalism is used to derive the Hamiltonian for a $λ$/4 resonator shunted by a current-biased Josephson junction. The eigenstates and the quantum dynamics of the system are analyzed numerically, and we show that the system can function as an efficient detector of weak incident microwave fields.
Josephson qubits without direct interaction can be effectively coupled by sequentially connecting them to an information bus: a current-biased large Josephson junction treated as an oscillator with adjustable frequency. The coupling between any qubit and the bus can be controlled by modulating the magnetic flux applied to that qubit. This tunable and selective coupling provides two-qubit entangled states for implementing elementary quantum logic operations, and for experimentally testing Bell's inequality.
We examine the resonant activation of a current-biased Josephson junction near the crossover temperature in order to show the way in which the device transitions from the classical regime to the quantum regime. We use microwaves to probe the quantum energy states that exist in the potential well. The quantum features are visible until the junction is heated up to the crossover temperature, at which point the line widths of the energy levels overlap and become indistinguishable from one another. When well above this temperature, the junction behaves classically when resonantly activated with mi…
We investigate spin transport in voltage-biased spin-active Josephson junctions. The interplay of spin filtering, spin mixing, and multiple Andreev reflection leads to nonlinear voltage dependence of the dc and ac spin current. We compute the voltage characteristics of the spin current (I_S) for superconductor-ferromagnet-superconductor (SFS) Josephson junctions. The sub-harmonic gap structure of I_S(V) is shown to be sensitive to the degree of spin mixing generated by the ferromagnetic interface, and exhibits a pronounced even-odd effect associated with spin transport during multiple Andreev …
We show experimentally that a dc biased Josephson junction in series with a high-enough impedance microwave resonator emits antibunched photons. Our resonator is made of a simple micro-fabricated spiral coil that resonates at 4.4 GHz and reaches a 1.97 k$Ω$ characteristic impedance. The second order correlation function of the power leaking out of the resonator drops down to 0.3 at zero delay, which demonstrates the antibunching of the photons emitted by the circuit at a rate of 6 $10^7$ photons per second. Results are found in quantitative agreement with our theoretical predictions. This simp…
The behavior of a long Josephson tunnel junction drastically depends on the distribution of the dc bias current. We investigate the case in which the bias current is fed in the central point of a one-dimensional junction. Such junction configuration has been recently used to detect the persistent currents circulating in a superconducting loop. Analytical and numerical results indicate that the presence of fractional vortices leads to remarkable differences from the conventional case of uniformly distributed dc bias current. The theoretical findings are supported by detailed measurements on a n…
A three-terminal Josephson junction consists of three superconductors coupled coherently to a small nonsuperconducting island, such as a diffusive metal, a single or double quantum dot. A specific resonant single quantum dot three-terminal Josephson junction $(S_a,S_b,S_c)$ biased with voltages $(V,-V,0)$ is considered, but the conclusions hold more generally for resonant semi-conducting quantum wire set-ups. A simple physical picture of the steady state is developed, using Floquet theory. It is shown that the equilibrium Andreev bound states (for $V=0$) evolve into nonequilibrium Floquet-Wann…
We investigate the proximity effect in junctions between $N=3$ superconductors under commensurate voltage bias. The bias is chosen to highlight the role of transport processes that exchange multiple Cooper pairs coherently between more than two superconductors. Such non-local processes can be studied in the dc response, where local transport processes do not contribute. We focus on the proximity-induced normal density of states that we investigate in a wide parameter space. We reveal the presence of deep and highly tunable pseudogaps and other rich structures. These are due to a static proximi…
The escape of a small current-biased Josephson tunnel junction from the zero voltage state in the presence of weak microwave radiation is investigated experimentally at low temperatures. The measurements of the junction switching current distribution indicate the macroscopic quantum tunneling of the phase below a cross-over temperature of $T^{\star} \approx 280 \rm{mK}$. At temperatures below $T^{\star}$ we observe both single-photon and \emph{multi-photon} transitions between the junction energy levels by applying microwave radiation in the frequency range between $10 \rm{GHz}$ and $38 \rm{GH…
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 resona…
Exact Duality at Low Energy in a Josephson Tunnel Junction Coupled to a Transmission Line
Université Paris Cité, CNRS, Matériaux et Phénomènes Quantiques（巴黎西岱大学，法国国家科学研究中心，物质与量子现象）；Physics Department, University of California Santa Barbara（加州大学圣塔芭芭拉分校物理系）；Google Quantum AI（谷歌量子AI）
By use of the multi-Josephson junction model, we investigate voltage-biased I-V characteristics. Differently from the case of the single junction, I-V characteristics show a complicated behavior due to inter-layer couplings among superconducting phase differences mediated by the charging effect. We show that there exist three characteristic regions, which are identified by jumps and cusps in the I-V curve. In the low voltage region, the total current is periodic with trigonometric functional increases and rapid drops. Then a kind of chaotic region is followed. Above certain voltage, the total …
We study a current-biased 0-π-0 Josephson junction made by high-T_c superconductors, theoretically. When a length of the πjunction is large enough, this junction contains a vortex-antivortex pair at both ends of the πjunction. Magnetic flux carried by the vortices is calculated using the sine-Gordon equation. The result shows that the magnetic flux of the vortices is suppressed to zero as the distance between the vortices is reduced. By applying an external current, the orientation of the vortices is reversed, and a voltage pulse is generated. The current needed for this transition and generat…
Deep theoretical understanding of the electrical response of Josephson junctions is indispensable regarding both recent discoveries of new kinds of superconductivity and technological advances such as superconducting quantum computers. Here, we study the microscopic theory of the DC current-biased $I$-$V$ characteristics of Josephson tunnel junctions. We derive an analytical formula of the $I$-$V$ characteristics of generic junctions. We identify subharmonics of the $I$-$V$ characteristics and their underlying mechanism as the feedback effect of intrinsic AC currents generated by voltage pulse…
We investigate classical anomalous electrical transport in a driven, resistively and capacitively shunted Josephson junction device. Novel transport phenomena are identified in chaotic regimes when the junction is subjected to both, a time periodic (ac) and a constant, biasing (dc) current. The dependence of the voltage across the junction on the dc-current exhibits a rich diversity of anomalous transport characteristics: In particular, depending on the chosen parameter regime we can identify so termed absolute negative conductance around zero dc-bias, the occurrence of negative differential c…
We study analytically an underdamped current-biased topological Josephson junction. First, we consider a simplified model at zero temperature, where the parity of the non-local fermionic state formed by Majorana bound states (MBSs) localized on the junction is fixed, and show that a transition from insulating to conducting state in this case is governed by single-quasiparticle tunneling rather than by Cooper pair tunneling in contrast to a non-topological Josephson junction. This results in a significantly lower critical current for the transition from insulating to conducting state. We propos…
The results of chaos simulations in Josephson Fluxonic Diode (JFD) subject to the magnetic field, rf driving and dc biasing currents in the RCSJ model are reported. The dynamic behavior of this nonlinear device is mapped as a function of loss, bias, frequency and amplitude of rf excitation which leads to find sustainable fluxon oscillation regions in microwave application. As a result, chaos does not happen if either the dc bias or the rf excitation is below specific value, but the loss term can change this state without affected by rf frequency. An attempt is made to explain the boundaries of…
A three-terminal Josephson junction biased at opposite voltages can sustain a phase-sensitive dc-current carrying three-body static phase coherence, known as the "quartet current". We calculate the zero-frequency current noise cross-correlations and answer the question of whether this current is noisy (like a normal current in response to a voltage drop) or noiseless (like an equilibrium supercurrent in response to a phase drop). A quantum dot with a level at energy $ε_0$ is connected to three superconductors $S_a$, $S_b$ and $S_c$ with gap $Δ$, biased at $V_a=V$, $V_b=-V$ and $V_c=0$, and wit…
We propose an effective scheme for manipulating quantum information stored in a superconducting nanocircuit. The Josephson qubits are coupled via their separate interactions with an information bus, a large current-biased Josephson junction treated as an oscillator with adjustable frequency. The bus is sequentially coupled to only one qubit at a time. Distant Josephson qubits without any direct interaction can be indirectly coupled with each other by independently interacting with the bus sequentially, via exciting/de-exciting vibrational quanta in the bus. This is a superconducting analog of …
The dependence of microwave impedance of a dc-biased Josephson Fluxonic Diode (JFD) under application of both dc magnetic field and rf excitation is calculated with a variety of conditions. For finite length of a JFD excited by a very low microwave excitation below its plasma frequency, applied dc magnetic field increases the rate of Vortex and Anti-Vortex (VAV) pair generation which fine-tunes the microwave resistance up to several factors more than its zero field microwave resistance (R0). Under this circumstance, adding a dc bias for moving VAVs causes oscillation-like features in microwave…
We report the experimental evidence of the ac Josephson effect in a transition edge sensor (TES) operating in a frequency domain multiplexer and biased by ac voltage at MHz frequencies. The effect is observed by measuring the non-linear impedance of the sensor. The TES is treated as a weakly linked superconducting system and within the resistively shunted junction model framework. We provide a full theoretical explanation of the results by finding the analytic solution of the non-inertial Langevian equation of the system and calculating the non-linear response of the detector to a large ac bia…
We study analytically a current-biased topological Josephson junction supporting $\mathbb{Z}_n$ parafermions. First, we show that in an infinite-size system a pair of parafermions on the junction can be in $n$ different states; the $2π{n}$ periodicity of the phase potential of the junction results in a significant suppression of the maximal current $I_m$ for an insulating regime of the underdamped junction. Second, we study the behaviour of a realistic finite-size system with avoided level crossings characterized by splitting $δ$. We consider two limiting cases: when the phase evolution may be…
The Josephson current in a diffusive superconductor/ferromagnet/superconductor junction with precessing magnetization is calculated within the quasiclassical theory of superconductivity. When the junction is phase-biased, a stationary current (without a.c. component) can flow through it despite the non-equilibrium condition. A large critical current is predicted due to a dynamically induced long range triplet proximity effect. Such effect could be observed in a conventional hybrid device close to the ferromagnetic resonance.
Superconductivity is a result of quantum coherence at macroscopic scales. Two superconductors separated by a metallic or insulating weak link exhibit the AC Josephson effect - the conversion of a DC voltage bias into an AC supercurrent. This current may be used to activate mechanical oscillations in a suspended weak link. As the DC voltage bias condition is remarkably difficult to achieve in experiments, here we analyse theoretically how the Josephson effect can be exploited to activate and detect mechanical oscillations in the experimentally relevant condition with purely DC current bias. We …
In diffusive Josephson junctions the phase-difference $ϕ$ between the superconductors strongly influences the spectroscopic features of the layer separating them. The observation of a uniform minigap and its phase modulation were only recently experimentally reported, demonstrating agreement with theoretical predictions up to now - a vanishing minigap at $ϕ=π$. Remarkably, we find that in the presence of intrinsic spin-orbit coupling a giant proximity effect due to spin-triplet Cooper pairs can develop at $ϕ=π$, in complete contrast to the suppressed proximity effect without spin-orbit couplin…
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 have developed a scheme for a high resolution measurement of the switching current distribution of a current biased Josephson tunnel junction using a timing technique. The measurement setup is implemented such that the digital control and read-out electronics are optically decoupled from the analog bias electronics attached to the sample. We have successfully used this technique to measure the thermal activation and the macroscopic quantum tunneling of the phase in a small Josephson tunnel junction with a high experimental resolution. This technique may be employed to characterize current-b…
