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Coherent Phenomena

Quantum and semiclassical Cooper–pair tunneling in finite systems

M. Kleber

Published Jan 30, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1515/coph-2016-0005

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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.
Based on a tunneling Hamiltonian description, we calculate the Josephson, normal, and interference currents between two Bose-Einstein condensates described by the Bogoliubov theory. The dominant Josephson term is of first order in the tunneling with a critical current density proportional to the ground state pressure. In contrast to superconductors, the normal current remains finite at zero temperature. We discuss the dynamics of the relative phase in a semiclassical approximation derived from an exact functional integral approach, which includes the interaction effects at fixed total particle number. It is shown that the normal current leads to a damping of the Josephson oscillations and, at long times, eliminates the macroscopic quantum self-trapping predicted by Smerzi et al. Finally, we give estimates for an experimental realization of Josephson tunneling in cold atomic gases, which indicate that coherent transfer of atoms might be realized with a 23Na{}^{23}\mathrm{Na}23Na condensate.
Within a semiclassical approximation and at the weak-coupling limit, where the charging energy dominates the Josephson coupling energy, the dynamics of small-capacitance superconducting tunnel junctions can be described by a stochastic master equation for the probability distribution of junction charges. We develop a method of solving this equation and obtain an analytic solution when Cooper-pair, quasiparticle, and Zener tunneling are all present under an arbitrary Ohmic conductance at zero temperature. We use this solution to systematically investigate the static and dynamic properties of small Josephson junctions. The charge distributions, current-voltage characteristics, and voltage spectra are analytically calculated and numerically evaluated for wide ranges of parameters. The separate contributions from quasiparticle and Cooper-pair tunneling to voltage spectra are evaluated and compared based on their respective sum rules. The linewidths of the resonance peaks for single-electron tunneling and Bloch oscillations are also discussed.
We develop a microscopic model to describe the Josephson tunneling between two superfluid reservoirs of ultracold fermionic atoms which accounts for the dependence of the critical current on both the barrier height and the interaction strength along the crossover from the BCS to the BEC regime. Building on a previous study [F. Meier and W. Zwerger, Phys. Rev. A 64, 033610 (2001)] of weakly interacting bosons, we derive analytic results for the Josephson critical current at zero temperature for homogeneous and trapped systems at arbitrary coupling. The critical current exhibits a maximum near the unitarity limit which arises from the competition between the increasing condensate fraction and a decrease in the chemical potential along the evolution from the BCS to the BEC limit. Our results agree quantitatively with numerical simulations and recent experimental data.
There exists a large number of experimental and theoretical results supporting the picture of macroscopic qubits implemented by nanoscopic Josephson junctions of three different types -- charge qubit, flux qubit and phase qubit. The standard unified description of such systems is based on the formal quantization of the phenomenological Kirchhoff equations for the corresponding circuits. In this paper a simplified version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions. For these models the state-dependent individual tunneling of Cooper pairs couples ground pair states with excited pair states what leads to a more complicated structure of the lowest lying energy levels. In particular, the highly degenerate levels emerge, which act as probability sinks for the qubit. These models allow also for the coupling to phonons as an efficient mechanism of relaxation for all types of junctions. The alternative formulas concerning basic spectral parameters of superconducting qubits are presented and compared with the experimental data. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.
We study the macroscopic quantum tunneling of two weakly linked superfluids made of interacting fermionic atoms. We derive atomic Josephson junction equations and find that zero-mode and $\ensuremath{\pi}$-mode frequencies of coherent atomic oscillations depend on the tunneling coefficient and the sound velocity of the superfluid. By considering a superfluid of 40K^{40}\text{K}40K atoms, we calculate these oscillation frequencies in the crossover from the Bardeen-Cooper-Schrieffer state of weakly bound Cooper pairs to the Bose-Einstein condensate of strongly bound molecular dimers.