Josephson Equation Research Articles

The number of limit cycles of Josephson equation

The number of limit cycles of Josephson equation

Global dynamics of the Josephson equation in TS1

The Josephson equation ϕ˙=y,y˙=−sin⁡ϕ+ϵ(a−(1+γcos⁡ϕ)y) was researched by Sanders and Cushman (1986) [12] for its phase portraits when ϵ>0 is small by applying the averaging method. The parameter ϵ can actually be large or even any real number in the practical application of this model. When |ϵ| is not small, we cannot apply the averaging method because the system is not near-Hamiltonian. For general ϵ∈R, we present complete dynamics and more complex bifurcations of the Josephson equation in TS1, including saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic loop bifurcation, two-saddle heteroclinic loop bifurcation, upper saddle connection bifurcation and lower saddle connection bifurcation. Moreover, we prove the monotonicity of bifurcation functions with respect to parameters and the nonexistence of a two-saddle heteroclinic loop for all a≠0.

Spin Mixing Dynamics in a Spin–Orbit Coupled Bose–Einstein Condensate

We investigate the spin mixing dynamics of a spin–orbit coupled Bose–Einstein condensate by using a variational approach. It is shown that the spin mixing dynamics can be well described by an internal Josephson equation. In the nonzero momentum phase, the spin mixing dynamics leads to a spin polarization, i.e., most atoms are suppressed in the one of the two spin states. In the zero momentum phase, the internal Josephson oscillation between the two spin states takes place. Thus, the coupling effects of spin–orbit coupling, Raman coupling and atomic interaction can result in the transition between the spin polarization and the internal Josephson oscillation of the two spin states, which provide a theoretical evidence for elaborating the spin mixing dynamics. Moreover, at the phase transition point, the oscillation period of the spin mixing dynamics tends to infinity. This can be used to explore the phase transition experimentally. The accurate manipulation of the spin mixing dynamics can provide a theoretical evidence for designing the atomic device.

Spin mixing dynamics in a spin-orbit coupled spin-1 Bose–Einstein condensate

We study the spin mixing dynamics of a spin-orbit coupled spin-1 Bose–Einstein condensate. Using mean-field theory and adopting plane wave solutions, the spin mixing dynamics can be well described by an internal Josephson equation. In order to establish the role of the different terms in spin mixing dynamics clearly, we consider the equilibrium state firstly, which shows that the spin-orbit coupling can significantly modify the dispersion relation and change the property of ground state of the condensate. For the non-equilibrium state, the spin-orbit coupling plays an important role in the spin mixing dynamics. Moreover, the oscillation period of spin mixing dynamics is determined by the coupled effects of momentum, magnetization, spin-dependent atomic interaction and spin-orbit coupling. When spin-orbit coupling is less than a critical value, it can prolong the oscillation period of spin mixing dynamics. Conversely, when the spin-orbit coupling is larger than the critical value, it can shorten the oscillation period of spin mixing dynamics. Particularly, when spin-orbit coupling is equal to the critical value, the period tends to infinity. Our study provides a complete spin mixing dynamics of spin-orbit coupled spin-1 BEC, which can be used to explore the spin mixing dynamics experimentally.

Theory of non-Abelian superfluid dynamics

We write down a theory for non-Abelian superfluids with a partially broken (semisimple) Lie group. We adapt the offshell formalism of hydrodynamics to superfluids and use it to comment on the superfluid transport compatible with the second law of thermodynamics. We find that the second law can be also used to derive the Josephson equation, which governs dynamics of the Goldstone modes. In the course of our analysis, we derive an alternate and mutually distinct parametrization of the recently proposed classification of hydrodynamic transport and generalize it to superfluids.

Comparison of phase slippage processes in Josephson junctions and charge-density wave stacked junctions

We discuss a time-space symmetry in phase slippage processes in Josephson junctions and charge density wave (CDW) stacked junctions. In Josephson junctions, above a critical current a phase slip of 2π occurs periodically in time with a period T corresponding to the rate given by the Josephson relation ν = 1/ T= 2eV/h, where V is the voltage on the junction. In CDW stacks, a CDW dislocation appears in the weakest junction of the stack when the voltage on the stack exceeds a threshold value. This phase dislocation corresponds to a local phase slippage of 2π. As the voltage is increased, new dislocations show up and form a periodic array of dislocations with period L. The inverse spacing 1/L obeys an analog of Josephson equation, vF/L = 2eV/h.

External electric field control in the charge qubits including three Josephson junctions

Based on the standard canonical quantization principle, this paper gives the quantization scheme for the charge qubits mesoscopic circuit including three Josephson junctions coupled capacitively. By virtue of the Heisenberg equation, the time evolution of the phase difference operators across the polar plates and the number operators of the Cooper-pairs on the island are investigated and the modification of the Josephson equation is discussed. The time evolution of the phase difference operators is analysed when the Josephson junctions are irradiated by the external electrical field, which is referred to as also the obtainable controlling parameter.

MODIFIED JOSEPHSON EQUATION AND MEASUREMENT DYNAMICS FOR THE JOSEPHSON TRANSMISSION LINE DETECTOR

By introducing the entangled state representation and following Feynman's assumption that "electron pairs are bosons … A bound pair acts as a Bose particle", we construct a Hamiltonian operator for the Josephson transmission line (JTL) detector, then we use the Heisenberg equation of motion to derive the modified Josephson equation. Based on the modified Josephson equation, the measurement dynamics of JTL detector is investigated, which will be helpful for us to analyze the quantum measurement of JTL detector.

Electromagnetic Phenomena in Superconductors

Electromagnetic phenomena in superconductors are reviewed. In superconductor in a transverse magnetic field, the electromagnetic phenomena are described by the critical state model assuming a balance between the Lorentz force and pinning force. In this case the Josephson equation for the induced electric field, E = B × v, holds for the motion of flux lines with velocity v. On the other hand, the electromagnetic phenomena in a longitudinal magnetic field of current-carrying superconductor are quite different from those in the transverse magnetic field. For example, the Josephson relation does not hold and even a negative potential drop is locally observed in the resistive state above the critical current. In this review it is shown that these peculiar phenomena are explainable using the flux motion driven by a force-free torque, a restoring torque against rotationally shearing deformation of flux lines due to the force-free current parallel to flux lines.

Modified Josephson Equation for Mesoscopic Parallel LC Circuit Including a Josephson Junction

By introducing the entangled state representation, the Cooper-pair number–phase quantization of the mesoscopic parallel LC circuit including a Josephson junction is realized. In the Heisenberg picture, the modified Josephson equation associated with the modification of the Faraday equation about the inductance is deduced from the motion equation.

OPERATOR JOSEPHSON EQUATION FOR A JOSEPHSON JUNCTION CONNECTED INTO A MESOSCOPIC LC CIRCUIT

We find that when a single Josephson junction is inserted into a mesoscopic LC circuit, the operator Josephson equation is modified accompanying with the modification of Farady equation describing the inductance. By virtue of the entangled state representation and using the appropriate phase operator in bosonic form we derive the modified equations.

Superconducting charge qubits from a microscopic many-body perspective

The quantized Josephson junction equation that underpins the behaviour of charge qubitsand other tunnel devices is usually derived through canonical quantization of theclassical macroscopic Josephson relations. However, this approach may neglecteffects due to the fact that the charge qubit consists of a superconducting island offinite size connected to a large superconductor. We show that the well-knownquantized Josephson equation can be derived directly and simply from a microscopicmany-body Hamiltonian. By choosing the appropriate strong-coupling limit we producea highly simplified Hamiltonian that nevertheless allows us to go beyond themean-field limit and predict further finite-size terms in addition to the basic equation.

On Bosonic Magnetic Flux Operator and Bosonic Faraday OperatorFormula

In the literature about mesoscopic Josephson devices the magneticflux is considered as an operator, the fundamental commutative relationbetween the magnetic flux operator and the Cooper-pair charge operator isusually preengaged. In this paper we show that such a relation can be deducedfrom the basic Bose operators' commutative relation through the entangled staterepresentation. The Faraday formula in bosonic form is then equivalent tothe second Josephson equation. The current operator equation for LCmesoscopic circuit is also derived.

Periodic Liénard-type delay equations with state-dependent impulses

Conditions are obtained for Liénard-type equations with delay and state-dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation (and consequently with Lebesgue integrable second derivative). The results are applied to Josephson's equation and the nonconservative forced pendulum equation.

Almost periodic solutions to Josephson's equation

Under certain a priori bounds on the mean of an almost periodic forcing f, Josephson's equation x′′+(2c+dcosx)x′+asinx = f, with 0<(a−cd)2+c2d2<c4, is shown to admit a twice continuously differentiable solution of the same type as f.

BIFURCATIONS OF PERIODIC ORBITS IN A JOSEPHSON EQUATION WITH A PHASE SHIFT

The Josephson equation with constant current and sinusoidal forcings and a phase shift is investigated in detail: the existence and the bifurcations of harmonics and subharmonics under small perturbations are given, by using the second-order averaging method and Melnikov function; the influence on bifurcations of periodic or subharmonics as the phase shift varies is considered; some numerical simulation results are reported in order to prove our theoretical predictions.

Chaotic dynamics of Josephson equation driven by constant dc and ac forcings

The chaotic dynamics of a Josephson equation driven by constant dc and ac current forcing with a shifted phase is investigated. We at first study whether there exist multiple saddle points with nontransverse homoclinic or heteroclinic orbits for unperturbed equation. Secondly, we obtain threshold for the existence of chaos by using Melnikov's method, and the threshold is influenced by the shifted phase. Finally some numerical results are reported in order to prove theoretical predictions.

Existence of periodic solutions to the driven Josephson equation

It is shown that the Josephson equation θ" + (c + d cos θ)θ' + a sin θ = ƒ with a,c,d ∈ ℝ and ƒ a T-periodic real-valued Lebesgue integrable function on [0,T] with [formula] dt suitably bounded, admits a continuously differentiable T-periodic solution with weak second derivative which is Lebesgue integrable on [0, T] whenever a2 + ധ2 &lt; ധ4 + ധ 2 C2 where ധ = 2π/T.

Bifurcations of periodic solutions and chaos in Josephson system

The Josephson equation is investigated in detail: the existence and bifurcationsfor harmonic and subharmonic solutions under small perturbations areobtained by using second-order averaging method and subharmonic Melnikov function,and the criterion of existence for chaos is proved by Melnikov analysis; thebifurcation curves about n-subharmonic and heteroclinic orbits and the driving frequency$\omega$ effects to the forms of chaotic behaviors are given by numerical simulations.

Quantization by Parts, Maximal Symmetric Operators, and Quantum Circuits

In the context of a generalized quantum theorywhich admits maximal symmetric operators as observables,we discuss a quantization scheme which cansystematically deal with what may be called quantumcircuits. The scheme, known as the method of quantizationby parts, has recently been applied to obtain a newderivation of the Josephson equation for thesupercurrent through a Josephson junction in asuperconducting circuit. This paper presents an application ofthis scheme to several circuit configurations, namely,from one branch to many-branch circuits. We also proposean experimental test on whether the condensate is always in a pure state, using a three-branchY-shape circuit.