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Journal of the Physical Society of Japan

Theory of ac Josephson Effect in Superfluid Helium

T. Kebukawa, Shosuke Sasaki

Published Feb 15, 1990• 1 citation
Publisher

Abstract

A microscopic theory of the ac Josephson effect in superfluid helium is proposed on the basis of the Heisenberg equation of the dressed-boson operators. The equation is solved exactly. The interference of the two eigen-oscillations in the exact solution produces the Josephson oscillation. It is verified that the oscillation occurs even when the quantum fluctuation of the relative phase is infinitely large. On the basis of these results, we propose an experiment to detect the tunnelling effect of the dressed-bosons. The formulation of this paper can be applied to the Josephson effect in superconductor.

DOI
10.1143/jpsj.59.601

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A microscopic theory of the ac Josephson effect in superfluid helium is proposed on the basis of the Heisenberg equation of the dressed-boson operators. The equation is solved exactly. The interference of the two eigen-oscillations in the exact solution produces the Josephson oscillation. It is verified that the oscillation occurs even when the quantum fluctuation of the relative phase is infinitely large. On the basis of these results, we propose an experiment to detect the tunnelling effect of the dressed-bosons. The formulation of this paper can be applied to the Josephson effect in superconductor.
When a chemical potential difference exists between the superfluid helium on two sides of an orifice, superfluid flow takes place at a critical velocity accompanied by the creation of quantized vortices. The frequency with which vortices are created is expected to be equal to the chemical potential difference (the analog of the ac Josephson frequency in superconducting tunneling). An experiment is described in which this frequency condition is verified by a method analogous to that used in superconductors. Experimental conditions were selected such that the gravitational head difference mgZ\mathrm{mgZ}mgZ was the dominant contribution to the chemical potential difference. ZZZ was measured by a capacitance technique. An ultrasonic transducer was used to modulate the flow through the orifice at the frequency $\ensuremath{\omega}$ so that the rate of creation of vortices was synchronized with the modulation frequency. The system was observed to exhibit dynamic stability at values of head difference $Z=\frac{n\ensuremath{\hbar}\ensuremath{\omega}}{{n}^{\ensuremath{'}}\mathrm{mg}}$ corresponding to nnn quantized vortices created every ${n}^{\ensuremath{'}}$ cycles of the modulation. The correspondence between theory and experiment is most convincing under experimental conditions such that the strength of the stability decreases rapidly with increasing values of the integer ${n}^{\ensuremath{'}}$. This was found to occur with small orifices and with moderate modulation amplitude, in agreement with the results of the analogous experiments in superconductors.
The alternating-current (AC) Josephson effect is studied in a system consisting of two weakly coupled Bose Hubbard models. In the framework of the mean field theory, Gross-Pitaevskii equations show that the amplitude of the Josephson current is proportional to the product of superfluid order parameters. In addition, the chemical potential--current relation for a small size system is obtained via the exact numerical computation. This allows us to propose a feasible experimental scheme to measure the Mott lobes of the quantum phase transition.
We investigate the macroscopic quantum tunneling of fermionic superfluids in the two-dimensional BCS-BEC crossover by using an effective tunneling energy which explicitly depends on the condensate fraction and the chemical potential of the system. We compare the mean-field effective tunneling energy with the beyond-mean-field one, finding that the mean-field tunneling energy is not reliable in the BEC regime of the crossover. Then we solve the Josephson equations of the population imbalance and the relative phase, calculating the frequency of tunneling oscillation both in the linear regime and in the nonlinear one. Our results show that the Josephson frequency is higher in the intermediate regime of the BCS-BEC crossover due to the peculiar behavior of the effective tunneling energy in the crossover.