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Physica Scripta

Quantum dissipative behaviour of a superconducting weak link constriction ring

Antonio Luis Extremera

Published Aug 1, 1988• 2 citations
Publisher

Abstract

We analyze the flux mode regime in a superconducting ring closed with a Josephson junction by means of a Feynman path-integral treatment. We derive the propagator for the quantum dissipative system, when a damping term in the Anderson-Josephson lagrangian is included. The wavefunctions and the energy levels are obtained from this propagator. Agreement with the exact results for the undamped case is discussed. Once found, the classical limit appears to be a formal way to obtain a relation between the conjugate variables for the weak link. The requirements of the equivalence between Feynman's and Schrödinger's approach allow to speculate on the Feynman path differential measure in flux space for the system under consideration.

DOI
10.1088/0031-8949/38/2/019

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Journal of Physics A Mathematical and General•Aug 21, 1984•14 citations
We analyze the flux mode regime in a superconducting ring closed with a Josephson junction by means of a Feynman path-integral treatment. We derive the propagator for the quantum dissipative system, when a damping term in the Anderson-Josephson lagrangian is included. The wavefunctions and the energy levels are obtained from this propagator. Agreement with the exact results for the undamped case is discussed. Once found, the classical limit appears to be a formal way to obtain a relation between the conjugate variables for the weak link. The requirements of the equivalence between Feynman's and Schrödinger's approach allow to speculate on the Feynman path differential measure in flux space for the system under consideration.
In this talk I will address the problem of supercurrent instability in highly transparent weak links. I will demonstrate that at subgap energies quantum behavior of superconducting weak links can be exactly accounted for by an effective Hamiltonian for a Josephson particle in a quantum dissipative environment formed by Andreev levels. I will identify several macroscopic quantum tunneling (MQT) regimes and predict a non-trivial non-monotonous dependence of the switching current distribution on temperature which can be observed in MQT experiments with transparent superconducting weak links. I will also argue that intrinsic dissipation in such weak links can be studied by means of qubit spectroscopy. Varying either the bias current or magnetic flux through a superconducting ring in the rf-SQUID setup, one can tune the level splitting value close to the bottom of the Josephson potential well. Monitoring the qubit energy relaxation time one can probe the spectral density of the effective dissipative bath and unambiguously identify the contribution emerging from Andreev levels.
Starting with a simple Lagrangian for the electromagnetic field with broken gauge symmetry, we derive an effective circuit Hamiltonian for a superconducting weak-link ring. The energy eigenstates of this Hamiltonian exhibit sensitivity to both external magnetic flux and an applied Faraday law voltage. We show that the flux periodicity of the rf SQUID ring, and the voltage periodic behaviour found in ultra-small-capacitance weak-link rings, can be seen as limiting cases of a more general phenomenon.
We introduce a numerically exact and computationally feasible nonlinear-response theory developed for lossy superconducting quantum circuits based on a framework of quantum dissipation in a minimally extended state space. Starting from the Feynman-Vernon path-integral formalism for open quantum systems with the system degrees of freedom being the nonlinear elements of the circuit, we eliminate the temporally nonlocal influence functional of all linear elements by introducing auxiliary harmonic modes with complex-valued frequencies coupled to the nonlinear elements. In our work, we propose a concept of time-averaged observables, inspired by experiment, and provide an explicit formula for producing their quasiprobability distribution. We illustrate the consistency of our formalism with the well-established Markovian input-output theory by applying them the dispersive readout of a superconducting transmon qubit. For an important demonstration of our approach beyond weak coupling, we analyze the low-frequency linear response of a capacitively and resistively shunted Josephson junction and observe signatures of a much-debated quantum phase transition at a finite temperature. The developed framework enables a comprehensive fully quantum-mechanical treatment of nonlinear quantum circuits coupled to their environment, without the limitations of typical approaches to weak dissipation, high temperature, and weak drive. This versatile tool paves the way for accurate models of quantum devices and increased fundamental understanding of quanutm mechanics such as that of the quantum measurement.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.