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Physical Review Letters

Quantum Dynamics of Tunneling between Superconductors

Vinay Ambegaokar, Ulrich Eckern, Gerd Schön

Published Jun 21, 1982• 445 citations
Publisher

Abstract

A functional-integral formulation is used to treat the quantum dynamics of a microscopic model of a Josephson junction, including the dissipative effects of quasiparticle tunneling. The calculation is carried to a point where it makes contact with, and therefore substantiates, recent work by Caldeira and Leggett in which the system is treated by analogy with the quantum Brownian motion of a massive particle coupled to a phenomenological heat bath.

DOI
10.1103/physrevlett.48.1745

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A functional-integral formulation is used to treat the quantum dynamics of a microscopic model of a Josephson junction, including the dissipative effects of quasiparticle tunneling. The calculation is carried to a point where it makes contact with, and therefore substantiates, recent work by Caldeira and Leggett in which the system is treated by analogy with the quantum Brownian motion of a massive particle coupled to a phenomenological heat bath.
Abstract We derive analytic solutions for the tunneling dynamics of two weakly coupled finite BCS–condensates. Pairing interaction between the finite–size condensates is taken into account. Using particle–number dependent chemical potentials the time–dependent transfer of Cooper pairs is obtained from a phenomenological calculation. The results of this theory are compared to a microscopic calculation within the quasispin formulation in its semiclassical limit. In both cases the tunneling current can be mapped onto the motion of a simple pendulum: The results are analogous to the Josephson current between two superconductors and can be used as a starting point to include quantum fluctuations and Josephson radiation.
In view of recent interest in the problem of macroscopic quantum tunneling in systems involving the Josephson effect, we present an accurate numerical calculation of the tunneling rate of a system from a metastable well, at zero temperature, in the presence of dissipative coupling to the environment. Although we concentrate on a specific form of dissipation, as discussed by Caldeira and Leggett, we believe that such a numerical method can be extended to other forms of dissipation as well. Our method is based on the framework recently described by Caldeira and Leggett, and requires (a) a novel treatment of a nonlinear integro-differential equation and (b) an extension of the usual Fredholm scattering theory so as to be applicable to the present dissipative problem. We present explicit results for wide ranges of dissipation and estimate our error in the calculation of the exponent to be no larger than 0.1% and of the prefactor to be no larger than 2%.
Quantum fluctuations of the charge at a current-biased Josephson junction and of the Josephson current couple to the quasiparticle tunneling when the system is at equilibrium in the zero-voltage state. The effect of reducing the capacitance of the junction is studied by means of a model Hamiltonian. When this becomes smaller, charge fluctuations are reduced, but the effective dissipation is also lowered so that the junction displays an underdamped behavior. The Josephson branch is shifted at the voltage e/2C because of charging due to the quasiparticles. A path-integral formulation of the model leads to a high dynamical subgap resistance at I\ensuremath{\rightarrow}0, as recently found experimentally.
We elaborate a theoretical description of large Josephson junctions which is based on Werthamer's microscopic tunneling theory. The model naturally incorporates coupling of electromagnetic radiation to the tunnel currents and, therefore, is particularly suitable for description of the self-coupling effect in Josephson junction. In our numerical calculations we treat the arising integro-differential equation, which describes temporal evolution of the superconducting phase difference coupled to the electromagnetic field, by the Odintsov-Semenov-Zorin algorithm. This allows us to avoid evaluation of the time integrals at each time step while taking into account all the memory effects. To validate the obtained microscopic model of large Josephson junction we focus our attention on the Josephson flux-flow oscillator. The proposed microscopic model of flux-flow oscillator does not involve the phenomenological damping parameter, rather the damping is taken into account naturally in the tunnel current amplitudes calculated at a given temperature. The theoretically calculated current-voltage characteristics is compared to our experimental results obtained for a set of fabricated flux-flow oscillators of different lengths.
Macroscopic quantum tunneling (MQT) for a single fluxon moving along a long Josephson junction is studied theoretically. To introduce a fluxon-pinning force, we consider inhomogeneities made by modifying thickness of an insulating layer locally. Two different situations are studied: one is the quantum tunneling from a metastable state caused by a single inhomogeneity, and the other is the quantum tunneling in a two-state system made by two inhomogeneities. In the quantum tunneling from a metastable state, the decay rate is estimated within the WKB approximation. Dissipation effects on a fluxon dynamics are taken into account by the Caldeira-Leggett theory. We propose a device to observe quantum tunneling of a fluxon experimentally. Required experimental resolutions to observe MQT of a fluxon seem attainable within the presently available micro-fabrication technique. For the two-state system, we study quantum resonance between two stable states, i.e., macroscopic quantum coherence (MQC). From the estimate for dissipation coefficients due to quasiparticle tunneling, the observation of MQC appears to be possible within the Caldeira-Leggett theory.
The complex energy shifts of the energy levels of a macroscopic system subject to dissipation are calculated as a function of the phenomenological damping parameters describing the classical motion of the system. These results are applied to the energy levels of the zero-voltage state of a current-biased Josephson junction in parallel with an arbitrary dissipative circuit. Following the approach of Leggett, the influence of the same dissipative circuit on the tunneling rate out of the zero-voltage state is also calculated. The dependences of both phenomena, quantization of energy levels, and quantum tunneling, on the admittance of the circuit are compared.