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arXiv:1103.0669 · Physical Review B

Ring of BEC pools as a trap for persistent flow

Jacek Dziarmaga, Marek Tylutki, Wojciech Hubert Zurek

Published Sep 26, 2011• 7 citations

Abstract

Mott insulator-superfluid transition in a periodic lattice of Josephson junctions can be driven by tunneling rate increase. The resulting winding numbers $W$ of the condensate wave function decrease with increasing quench time in accord with the Kibble-Zurek mechanism (KZM). However, in very slow quenches, Bose-Hubbard dynamics rearranges the wave-function phase so that its random walk cools, $\overline{{W}^{2}}$ decreases and eventually the wave function becomes too cold to overcome potential barriers separating different $W$. Thus, in contrast with KZM, in very slow quenches $\overline{{W}^{2}}$ is set by random walk with ``critical'' step size, independently of ${\ensuremath{\tau}}_{Q}$. As our study requires use of the truncated Wigner approximation (TWA) over relatively long time intervals, we investigate the validity of TWA by comparing its predictions with exact calculations for suitably small quantum systems.

DOI: 10.1103/physrevb.84.094528

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Mott insulator-superfluid transition in a periodic lattice of Josephson junctions can be driven by tunneling rate increase. The resulting winding numbers

W

of the condensate wave function decrease with increasing quench time in accord with the Kibble-Zurek mechanism (KZM). However, in very slow quenches, Bose-Hubbard dynamics rearranges the wave-function phase so that its random walk cools,

\overline{{W}^{2}}

decreases and eventually the wave function becomes too cold to overcome potential barriers separating different

W

. Thus, in contrast with KZM, in very slow quenches

\overline{{W}^{2}}

is set by random walk with ``critical'' step size, independently of

{\ensuremath{\tau}}_{Q}

. As our study requires use of the truncated Wigner approximation (TWA) over relatively long time intervals, we investigate the validity of TWA by comparing its predictions with exact calculations for suitably small quantum systems.

We study a linear ramp of the nearest-neighbor tunneling rate in the Bose-Hubbard model driving the system from the Mott insulator state into the superfluid phase. We employ the truncated Wigner approximation to simulate linear quenches of a uniform system in one, two, and three dimensions, and in a harmonic trap, in three dimensions. In all these setups, the excitation energy decays like one over third root of the quench time. The $\ensuremath{-}1/3$ scaling arises from an impulse-adiabatic approximation---a variant of the Kibble-Zurek mechanism---describing a crossover from nonadiabatic to adiabatic evolution when the system begins to keep pace with the increasing tunneling rate.

Kibble-Zurek mechanism (KZM) uses critical scaling to predict density of topological defects and other excitations created in second order phase transitions. We point out that simply inserting asymptotic critical exponents deduced from the immediate vicinity of the critical point to obtain KZM predictions can lead to results that are inconsistent with a more careful KZM-like analysis based on causality -- on the comparison of the relaxation time of the order parameter with the time "distance" from the critical point. As a result, scaling of quench-generated excitations with quench rates can exhibit behavior that is locally (i.e., in the neighborhood of any given quench rate) well approximated by the power law, but with exponents that depend on that rate, and that are quite different from the naive prediction based on the critical exponents relevant for asymptotically long quench times. Kosterlitz-Thouless scaling (that governs e.g. Mott insulator to superfluid transition in the Bose-Hubbard model in one dimension) is investigated as an example of this phenomenon.

Kibble-Zurek mechanism (KZM) uses critical scaling to predict density of topological defects and other excitations created in second order phase transitions. We point out that simply inserting asymptotic critical exponents deduced from the immediate vicinity of the critical point to obtain predictions can lead to results that are inconsistent with a more careful KZM analysis based on causality - on the comparison of the relaxation time of the order parameter with the "time distance" from the critical point. As a result, scaling of quench-generated excitations with quench rates can exhibit behavior that is locally (i.e., in the neighborhood of any given quench rate) well approximated by the power law, but with exponents that depend on that rate, and that are quite different from the naive prediction based on the critical exponents relevant for asymptotically long quench times. Kosterlitz-Thouless scaling (that governs e.g. Mott insulator to superfluid transition in the Bose-Hubbard model in one dimension) is investigated as an example of this phenomenon.

The quantum phase transition from the Mott insulator state to the superfluid in the Bose-Hubbard model is investigated. We research one, two and three dimensional lattices in the truncated Wigner approximation. We compute both kinetic and potential energy and they turn out to have a power law behaviour as a function of the transition rate, with the power equal to 1/3. The same applies to the total energy in a system with a harmonic trap, which is usually present in the experimental set-up. These observations are in agreement with the experiment of [8], where such scalings were also observed and the power of the decay was numerically close to 1/3. The results confirm the Kibble-Zurek (adiabatic-impulse-adiabatic approximation) scenario for this transition.