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Physical Review A

Effect of dissipation on phase periodicity and the quantum dynamics of Josephson junctions

Daniel Loss, Kieran J. Mullen

Published Mar 1, 1991• 17 citations
Publisher

Abstract

We consider systems described by compact, periodic variables, such as Josephson junctions and small normal metal rings. We examine how they can be coupled to a heat bath, via either momentum or position coupling, and how the two descriptions can be related by a unitary transformation. We show how it is critical to transform not only the Hamiltonian, but also the initial conditions. We then demonstrate that for certain types of initial conditions, paths of different winding number can interfere. Still other, ostensibly reasonable initial conditions, have no such interference. We conclude with a discussion of appropriate models to describe periodic systems.

DOI
10.1103/physreva.43.2129

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We consider systems described by compact, periodic variables, such as Josephson junctions and small normal metal rings. We examine how they can be coupled to a heat bath, via either momentum or position coupling, and how the two descriptions can be related by a unitary transformation. We show how it is critical to transform not only the Hamiltonian, but also the initial conditions. We then demonstrate that for certain types of initial conditions, paths of different winding number can interfere. Still other, ostensibly reasonable initial conditions, have no such interference. We conclude with a discussion of appropriate models to describe periodic systems.
We investigate the dynamics of a quantum particle coupled to a boson heat bath. Using a real-time path-integral formalism, we obtain the Wigner distribution of the particle in the form of a power series in the strength of the anharmonicity V0{V}_{0}V0​. This series is shown to converge for all V0{V}_{0}V0​ when t is fixed and for small V0{V}_{0}V0​ uniformly in t. The latter proves the convergence to equilibrium for small anharmonicities. The effects of initial conditions on the evolution are studied by explicitly considering two types of initial states: product states and mixed Gibbs states. We show that in certain cases the evolution of the mixed states which avoid many pathologies of the product states, arising from the removal of the cutoff on the frequency distribution of the bath, can be related to that of product states when the latter are started at t=-\ensuremath{\infty}. We also solve exactly a simple stationary nonequilibrium model, a harmonic system in contact with two thermal baths, and derive an alternative criterion for a practically useful quasiclassical approximation. Finally, some connections to the Josephson junction are discussed. These include (a) the Green-Kubo-Einstein relation between the mobility and the diffusion constant of the washboard potential and (b) the time evolution of a rf superconducting quantum interference device.
For the realistic theoretical descriptions of quantum devices, we have to take the unavoidable coupling to their surrounding environment into account. A lot of effort in theoretical physics has been made to describe decoherence effects and the corresponding quantum to classical transition in the past. In these studies, baths are coupled to certain observables making their behavior more classical. In the most general case, a quantum device is affected by environments coupled to different observables. In this thesis, we study the differences emerging from coupling the bath to different observables. In particular, we investigate the situation, where two baths are coupled simultaneously to two conjugate variables (e.g. xˆ and pˆ, denoted dissipative frustration). We study how the interplay of these two baths can lead to interesting effects. After an introduction of the topic, we discuss the results of this thesis in Part II and Part III. In Part II, we study the effect of quantum tunneling in presence of a dissipative interaction with two Ohmic environments to the position xˆ and/or to the momentum pˆ of a particle. In chapter 2, we consider a symmetric double well potential which is important in the context of coherent state preparation, as its ground state is a superposition of being in the left and in the right well. Using the semiclassical approximation to calculate instanton paths within the path integral method, we find that momentum dissipation enhances the coherent tunnel splitting. In presence of dissipative couplings to xˆ and pˆ, the momentum dissipation acts in favor of the delocalization of the particle shifting the critical coupling strength of the dissipative phase transition induced by the position dissipation. Chapter 3 contains a study of the escape rate out of a metastable well in presence of dissipation. Here we find that the incoherent tunneling rate is exponentially enhanced in presence of momentum dissipation. We further show that the influence of momentum dissipation depends on the slope of the part of barrier the particle escaping through. In presence of both dissipative couplings, we find a non-monotonic behavior of the escape rate as a function of the dissipative couplings strengths. In Part III, we show that it is possible to realize the situation of "dissipative frustration" in Josephson junction chains. In particular, we introduce charge dissipation as a physical realization of the afore introduced momentum coupling. After presenting the circuit elements needed for charge dissipation, we study a chain of Josephson junctions affected by Ohmic baths coupled to the charge and/or the phase. Using a self-consistent harmonic approximation, we determine the phase diagram at zero temperature which exhibits a quantum phase transition between an ordered phase, corresponding to the superconducting state, and a disordered phase, corresponding to the insulating state with localized superconducting charge. Interestingly, we find that the critical line separating the two phases has a non-monotonic behavior as a function of the dissipative coupling strength. Moreover, within the self-consistent harmonic approximation, we analyze the dissipation induced crossover between a first and second order phase transition, showing that dissipative frustration increases the range in which the phase transition is second order. The non-monotonic behavior is also reflected in the purity of the system that quantifies the degree of correlation between the system and the environment, and in the logarithmic negativity as an entanglement measure that encodes the internal quantum correlations in the chain.
We investigate the effect of dissipation in a Bose-Josephson junction (BJJ) coupled to baths of bosons at its two sites. Apart from the dynamical transition due to repulsive interactions, the BJJ undergoes a quantum phase transition by increasing the coupling strength with the bath modes. We analyze this system by mapping to an equivalent spin model coupled to the bosonic modes. The excitation energies and fluctuation of number imbalance are obtained within a Holstein-Primakoff approximation, which exhibits vanishing of the energy gap and enhanced quantum fluctuations at the critical point. We study the dynamics of BJJ using a time-dependent variational method and analyze stability of different types of steady states. As a special case we study in detail the phase space dynamics of BJJ coupled to a single mode, which reveals diffusive and incoherent behavior with increasing coupling to the bath mode. The dynamical steady states corresponding to the π oscillation and self-trapped state become unstable when their oscillation frequencies are in resonance with the bath modes. We study the Josephson dynamics in the presence of an Ohmic bath with Gaussian noise to incorporate the thermal fluctuations and obtain the Josephson oscillation frequency and damping analytically. We also observe the transition to the symmetry-broken state for strong coupling as well as decay of π oscillation and a self-trapped state to the ground state due to dissipation. Variation of the phase fluctuation with temperature of the bath shows similar behavior as observed in experiment. Finally we discuss the experimental setup to study the observable effects of dissipation in BJJ.
This chapter provides an analysis of macroscopic quantum dynamics of small-size Josephson junctions. It demonstrates that nontrivial interplay between quantum coherence, dissipation, and Coulomb effects yields a rich variety of properties, which can be accessed not only by means of the phase variable but also through the canonically conjugate quantum variable of the junction charge. The chapter argues that useful and sometimes complementary information can be obtained if one considers quantum dynamics of another collective variable—the junction charge. It discusses some (quasi-) equilibrium properties of ultrasmall Josephson junctions, which can be adequately accounted for by evaluating the grand partition function Z within the imaginary-time path integral formalism. In general, however, the imaginary-time technique may be insufficient and/or inconvenient to properly describe real-time evolution of Josephson junctions in the presence of macroscopic quantum coherence and dissipation.
We investigate the effect of dissipation in a Bose Josephson junction (BJJ) coupled to bath of bosons at two sites as an application of the generalized spin-boson model. Apart from the dynamical transition due to repulsive interactions, the BJJ undergoes a quantum phase transition by increasing the coupling strength with the bath modes. We analyze this system by mapping to an equivalent spin model coupled to the bosonic modes. The excitation energies and fluctuation of number imbalance are obtained within Holstein-Primakoff approximation, which exhibits vanishing of energy gap and enhanced quantum fluctuations at the critical coupling. We study the dynamics of BJJ using time-dependent variational method and analyze stability of different types of steady states. As a special case, we study in details the phase space dynamics of BJJ coupled to a single-mode, which reveals diffusive and incoherent behaviour with increasing coupling to the bath mode. The dynamical steady states corresponding to the Pi-oscillation and self-trapped state become unstable in the region where their oscillation frequencies are in resonance with the bath modes. We investigate the time evolution of number imbalance and relative phase in presence of Ohmic bath with Gaussian noise to incorporate thermal fluctuations. Apart from the damping of Josephson oscillations and transition to symmetry broken state for strong coupling, we observe the decay of Pi-oscillation and self-trapped state to the ground state as a result of dissipation. Variation of phase fluctuation with temperature of the bath shows similar behaviour as observed in the experiments.
We have studied quantum mechanically a system of several small identical Josephson junctions in a lossless single-mode cavity for different initial states, under conditions such that the system is at resonance. This system is analogous to a collection of identical atoms in a cavity, which is described under appropriate conditions by the Dicke model. We find that our system can be well approximated by a reduced Hamiltonian consisting of two levels per junction. The reduced Hamiltonian is similar to the Dicke Hamiltonian, but contains an additional term resembling a dipole-dipole interaction between the junctions. This extra term can be understood as a natural consequence of degenerate second-order (L\"owdin) perturbation theory. For typical, physically reasonable values of the junction-cavity coupling, we find that this perturbation treatment is an adequate way to include the junction energy levels beyond the lowest two. As in the Dicke model, we find that, when N junctions are present in the cavity, the junction-cavity interaction is enhanced by N,\sqrt{N},N​, with a corresponding decrease in the Rabi oscillation period. We find that this enhancement survives even if the junctions differ slightly from one another, as expected in a realistic system. Since coherence effects thus reduce the Rabi period, it may become smaller than the decoherence time due to dissipation, making these oscillations observable.