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Physical Review B

Quantum tunneling of vortices in two-dimensional superfluids

Daniel P. Arovas, Assa Auerbach

Published Sep 12, 2008• 14 citations
Publisher

Abstract

We examine the problem of quantum vortex tunneling in the Gross-Pitaevskii model. The effect of gapless phonons is to produce a super-Ohmic quantum dissipation that renormalizes the effective tunneling parameters but does not destroy quantum coherence. The renormalization of the instanton frequency is computed within a specific model tunneling potential.

DOI
10.1103/physrevb.78.094508

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We examine the problem of quantum vortex tunneling in the Gross-Pitaevskii model. The effect of gapless phonons is to produce a super-Ohmic quantum dissipation that renormalizes the effective tunneling parameters but does not destroy quantum coherence. The renormalization of the instanton frequency is computed within a specific model tunneling potential.
The macroscopic quantum tunneling in a double-well potential with dissipation is formulated without referring to instantons. This formulation, in the domain of small dissipation, goes beyond the instanton approximation which corresponds to the two-state truncated system (or, the so-called spin-boson Hamiltonian). It is first confirmed that our formulation, when applied to the two-state truncated system, reproduces all the known results of the instanton approximation. It is then shown that the ground state is mixed with excited states by dissipative interactions and this effect tends to enhance the ground-state quantum tunneling. In the Ohmic dissipation case this enhancement factor, even for a shallow double-well potential, can be at most comparable to the suppression factor which has been known from the work by Caldeira and Leggett. In the super-Ohmic case, however, the enhancement factor generally exceeds the suppression factor and the two-state approximation fails. A salient feature in the choice of counter terms in macroscopic quantum tunneling, which is essential to the above conclusion, is emphasized.
In two dimensional superfluid, annihilation processes of vortices are investigated by numerical simulation within the dissipative Gross-Pitaevskii equation (GPE) model. First, quantum vortex solution is obtained and its fitting function is found. Second, the simulation show that positive and negative vortices accelerate in both x,y directions, until they annihilate into a soliton and then a crescent-shaped shock wave. The processes are found to be controlled by the dissipative parameter and the general Magnus force. For the behavior of separation distance between vortices d(t), an universal scaling exponent 1/2 is found which is same with the three dimensional cases. Third, system's energy is surprisingly found to be determined by system's configuration and their relation are obtained. Then we derive the general Magnus force which decreases with the increases of d(t) for large d(t) and increases with the increases of d(t) for small d(t).
We study reconnections of quantum vortices by numerically solving the governing Gross-Pitaevskii equation. We find that the minimum distance between vortices scales differently with time before and after the vortex reconnection. We also compute vortex reconnections using the Biot-Savart law for vortex filaments of infinitesimal thickness, and find that, in this model, reconnections are time symmetric. We argue that the likely cause of the difference between the Gross-Pitaevskii model and the Biot-Savart model is the intense rarefaction wave which is radiated away from a Gross-Pitaeveskii reconnection. Finally we compare our results to experimental observations in superfluid helium and discuss the different length scales probed by the two models and by experiments.
Superfluids are inviscid flows in which vorticity is supported on filaments with quantized circulation. Such objects, known as quantum vortices, exhibit a hydrodynamical behavior. Experimentally, the dynamics of superfluids has been studied by using particles, which nowadays have become the main tool for visualizing quantum vortices. In this Thesis, we study numerically and analytically the dynamics of active and finite-size particles in superfluids. The superfluid is modeled with the Gross–Pitaevskii equation, while the particles are implemented as moving repulsive potentials coupled with the macroscopic wave function describing the superfluid. Firstly, the model is used to investigate the interaction between particles and quantum vortices at very low temperatures. This part aims to give a theoretical background to the current experiments in which macroscopic particles are used to sample superfluid vortices and quantum turbulence. Specifically, we address the following problems: the capture of a particle by a quantum vortex, the reconnections of vortex filaments and the propagation of Kelvin waves in presence trapped particles and the dynamics of particles in decaying quantum turbulence. In the last part of the manuscript, finite temperature effects are studied in the Fourier-truncated Gross–Pitaevskii model. The goal is to characterize the dynamics of impurities immersed in a thermal bath and how their presence modifies the statistical properties of the fluid. In particular, the random motion of the impurities and the temperature dependence of the friction coefficient are studied. Finally, the clustering of impurities and its effect on the phase transitions of the condensate are investigated.
At low temperature T we expect vacuum tunneling processes to occur in superfluid 4 He films. We distinguish between extrinsic processes, in which single vortices nucleate by tunneling off boundaries in the system, and intrinsic processes, in which vortex/anti-vortex pairs nucleate far from boundaries. It is crucial to incorporate the varying effective mass of the vortex in tunneling calculations. The intrinsic processes are the superfluid analogue of the Schwinger mechanism in quantum field theory; here they appear as a quantum phase transition at T = 0, driven by an external supercurrent. We calculate the tunneling rate for these processes, and describe a means of testing the predictions using a specific “vortex counting” experiment.