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arXiv:0803.0059

Probing Mott lobes via the AC Josephson effect

Mingxia Huo, Ying Li, Zhida Song, C. P. Sun

Published Mar 1, 2008• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.0803.0059

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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.
In this paper, we propose to directly detect Mott lobes, i.e. the order parameter 〈a〉, describing the Mott insulator (MI) to superfluid (SF) quantum phase transition of the Bose–Hubbard (BH) model. By weakly coupling the system to an environment in the SF phase, the U(1) symmetry breaking of the system is simulated, and the order parameter can be read from the AC Josephson current between the system and the environment. The relation between the order parameter and the Josephson current is obtained from both the mean-field theory approach and an exact numerical simulation of a finite-size example. Our numerical simulations show that the profile of the order parameter read from the Josephson current is different from it predicted by the mean-field theory, but similar to it in a system whose U(1) symmetry is broken by a weak field proportional to a + a†. This proposal is feasible in optical lattices.
We propose an experiment that would demonstrate the dc and ac Josephson effects in two weakly linked Bose-Einstein condensates. We consider a time-dependent barrier, moving adiabatically across the trapping potential. The phase dynamics are governed by a "driven-pendulum" equation, as in current-driven superconducting Josephson junctions. At a critical velocity of the barrier (proportional to the critical tunneling current), there is a sharp transition between the dc and ac regimes. The signature is a sudden jump of a large fraction of the relative condensate population. Analytical results are compared with a numerical integration of the Gross-Pitaevskii equation, in an experimentally realistic situation.
In this study, we perform a strong-coupling expansion up to third order of the hopping parameter ttt for the spin-1 Bose-Hubbard model with antiferromagnetic interaction. As expected from previous studies, the Mott-insulator phase is considerably more stable against the superfluid phase when filling with an even number of bosons than when filling with an odd number of bosons. The phase-boundary curves are consistent with the perturbative mean-field theory in the limit of infinite dimensions. The critical value of the hopping parameter tC{t}_{\mathrm{C}}tC​ at the peak of the Mott lobe depends on the antiferromagnetic interaction. This result indicates the reliability of the strong-coupling expansion when U2{U}_{2}U2​ possesses large (intermediate) values for Mott lobe with an even (odd) number of bosons. Moreover, in order to improve our results, we apply a few extrapolation methods up to infinite order in ttt. The fitting results of the phase-boundary curves agree better with those of the perturbative mean-field approximation. In addition, the linear fit error of tC{t}_{\mathrm{C}}tC​ is very small for the strong antiferromagnetic interaction.
We calculate both the dc and the ac Josephson current through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. We treat the (repulsive) Coulomb interaction in the framework of the one-channel, spin-1/2 Luttinger model. The Josephson current is obtained for two geometries of experimental relevance: a quantum wire and a ring. At zero temperature, the critical current is found to decay algebraically with increasing distance d between the junctions. The decay is characterized by an exponent which depends on the strength of the interaction. At finite temperatures T, lower than the superconducting transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​, there is a crossover from algebraic to exponential decay of the critical current as a function of d, at a distance of the order of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/kBT{\mathit{k}}_{\mathit{BT}}kBT​. Moreover, the dependence of critical current on temperature shows nonmonotonic behavior. If the Luttinger liquid is confined to a ring of circumference L, coupled capacitively to a gate voltage and threaded by a magnetic flux, the Josephson current shows remarkable parity effects under the variation of these parameters. For some values of the gate voltage and applied flux, the ring acts as a \ensuremath{\pi} junction. These features are robust against thermal fluctuations up to temperatures on the order of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/kBL{\mathit{k}}_{\mathit{BL}}kBL​. For the wire geometry, we have also studied the ac-Josephson effect. The amplitude and the phase of the time-dependent Josephson current are affected by electron-electron interactions. Specifically, the amplitude shows pronounced oscillations as a function of the bias voltage due to the difference between the velocities of spin and charge excitations in the Luttinger liquid. Therefore, the ac-Josephson effect can be used as a tool for the observation of spin-charge separation. \textcopyright{} 1996 The American Physical Society.
Using the Aharonov-Casher (A-C) phase, we present a microscopic theory of the Josephson and persistent spin currents in quasiequilibrium Bose-Einstein condensates (BECs) of magnons in ferromagnetic insulators. Starting from a microscopic spin model that we map onto a Gross-Pitaevskii Hamiltonian, we derive a two-state model for the Josephson junction between the weakly coupled magnon-BECs. We then show how to obtain the alternating-current (ac) Josephson effect with magnons as well as macroscopic quantum self-trapping in a magnon-BEC. We next propose how to control the direct-current (dc) Josephson effect electrically using the A-C phase, which is the geometric phase acquired by magnons moving in an electric field. Finally, we introduce a magnon-BEC ring and show that persistent magnon-BEC currents flow due to the A-C phase. Focusing on the feature that the persistent magnon-BEC current is a steady flow of magnetic dipoles that produces an electric field, we propose a method to directly measure it experimentally.
Starting from the two-mode Bose-Hubbard model, we derive an exact version of the standard Mathieu equation governing the wave function of a Josephson junction. For a finite number of particles N, we find an additional $\mathrm{cos}2\ensuremath{\varphi}$ term in the potential. We also find that the inner product in this representation is nonlocal in $\ensuremath{\varphi}.$ Our model exhibits phenomena, such as $\ensuremath{\pi}$ oscillations, which are not found in the standard phase model, but have been predicted from Gross-Pitaevskii mean-field theory.