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arXiv:cond-mat/0408500 · Physical Review A

Tunneling in a uniform one-dimensional superfluid: Emergence of a complex instanton

Sergei Khlebnikov

Published Jan 14, 2005• 6 citations
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Abstract

In a uniform ring-shaped one-dimensional superfluid, quantum fluctuations that unwind the order parameter need to transfer momentum to quasiparticles (phonons). We present a detailed calculation of the leading exponential factor governing the rate of such phonon-assisted tunneling in a weakly coupled Bose gas at a low temperature $T$. We also estimate the preexponent. We find that for small superfluid velocities the $T$ dependence of the rate is given mainly by $\mathrm{exp}(\ensuremath{-}{c}_{s}P∕2T)$, where $P$ is the momentum transfer and ${c}_{s}$ is the phonon speed. At low $T$, this represents a strong suppression of the rate compared to the nonuniform case. As a part of our calculation, we identify a complex instanton whose analytical continuation to suitable real-time segments is real and describes formation and decay of coherent quasiparticle states with nonzero total momenta.

DOI
10.1103/physreva.71.013602

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In a uniform ring-shaped one-dimensional superfluid, quantum fluctuations that unwind the order parameter need to transfer momentum to quasiparticles (phonons). We present a detailed calculation of the leading exponential factor governing the rate of such phonon-assisted tunneling in a weakly coupled Bose gas at a low temperature TTT. We also estimate the preexponent. We find that for small superfluid velocities the TTT dependence of the rate is given mainly by exp(\ensuremath−csP∕2T)\mathrm{exp}(\ensuremath{-}{c}_{s}P∕2T)exp(\ensuremath−cs​P∕2T), where PPP is the momentum transfer and cs{c}_{s}cs​ is the phonon speed. At low TTT, this represents a strong suppression of the rate compared to the nonuniform case. As a part of our calculation, we identify a complex instanton whose analytical continuation to suitable real-time segments is real and describes formation and decay of coherent quasiparticle states with nonzero total momenta.
Quantum phase slips (QPS) in narrow superfluid channels generate momentum by unwinding the supercurrent. In a uniform Bose gas, this momentum needs to be absorbed by quasiparticles (phonons). We show that this requirement results in an additional exponential suppression of the QPS rate (compared to the rate of QPS induced by a sharply localized perturbation). In BCS-paired fluids, momentum can be transferred to fermionic quasiparticles, and we find an interesting interplay between quasiparticle scattering on QPS and on disorder.
We report results of a classical simulation of thermal phase slips, and the associated relaxation of supercurrent, in a ring-shaped one-dimensional superfluid. We find that the classical relaxation rate vanishes in the uniform limit. This leaves the quantum relaxation, with momentum transfer to phonons, the only mechanism of supercurrent decay in the uniform system. In the presence of a smooth periodic potential, classical decay becomes possible, and we identify a family of moving critical droplets that can mediate it.
In a one-dimensional weakly interacting Bose-Fermi mixture, one branch of elementary excitations is well described by the Bogoliubov spectrum. Here we use the microscopic theory to study the decay of such quasiparticle excitations. The main scattering process which leads to their decay is the backscattering of a Bogoliubov quasiparticle off the Fermi sea, where a particle-hole pair is excited. For a low-momentum quasiparticle (phonon) of momentum qqq, we find that the decay rate scales as q3{q}^{3}q3 provided qqq is smaller than the Fermi momentum kF{k}_{F}kF​, while in the opposite case the decay behaves as q2{q}^{2}q2. If the ratio of the masses of fermions and bosons is equal to the ratio of the boson-fermion and the boson-boson interaction strengths, the decay rate changes dramatically. It scales as q7{q}^{7}q7 for q<kFq<{k}_{F}q<kF​, while we find q6{q}^{6}q6 scaling at q>kFq>{k}_{F}q>kF​. For a high-momentum Bogoliubov quasiparticle, we find a constant decay rate for q<kFq<{k}_{F}q<kF​, while it scales as 1/q1/q1/q for q>kFq>{k}_{F}q>kF​. We also find an analytic expression for the decay rate in the crossover region between low and high momenta. The decay rate is a continuous, but nonanalytic function of the momentum at q=kFq={k}_{F}q=kF​. In the special case when the parameters of our system correspond to the integrable model, we observe that the decay rate vanishes.
The flow of a uniform Bose gas at speeds greater than the Landau critical velocity vc{v}_{c}vc​ does not necessarily destroy superfluidity, but rather needs only to lead to a decrease of the superfluid mass density ${\ensuremath{\rho}}_{s}$. Analyzing a weakly interacting Bose gas with a finite-range interparticle interaction that leads to a Landau critical velocity at nonzero quasiparticle momentum, we explicitly construct the (nonuniform) condensate for fluid flow faster than vc{v}_{c}vc​ and calculate the accompanying decrease in ${\ensuremath{\rho}}_{s}$. We briefly comment on the relation of the physics to other problems in superfluids, e.g., solitons and vortices in Bose-Einstein condensates, and critical currents in superconductors.
We examine the superfluid properties of a one-dimensional (1D) Bose gas in a ring trap based on the model of Lieb and Liniger. While the 1D Bose gas has nonclassical rotational inertia and exhibits quantization of velocities, the metastability of currents depends sensitively on the strength of interactions in the gas: the stronger the interactions, the faster the current decays. It is shown that the Landau critical velocity is zero in the thermodynamic limit due to the first supercurrent state, which has zero energy and finite probability of excitation. We calculate the energy dissipation rate of ring currents in the presence of weak defects, which should be observable on experimental time scales.
We study the quantum dynamics of supercurrents of one-dimensional Bose gases in a ring optical lattice to verify instanton methods applied to coherent macroscopic quantum tunneling (MQT). We directly simulate the real-time quantum dynamics of supercurrents, where a coherent oscillation between two macroscopically distinct current states occurs due to MQT. The tunneling rate extracted from the coherent oscillation is compared with that given by the instanton method. We find that the instanton method is quantitatively accurate when the effective Planck's constant is sufficiently small. We also find phase slips associated with the oscillations.