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Physical Review A

Coherence of interacting bosons in optical lattices in synthetic magnetic fields with a large number of subbands

B. Grygiel, K. Patucha, T. A. Zaleski

Published May 9, 2016• 6 citations
Publisher

Abstract

We study the behavior of interacting ultracold bosons in optical lattices in synthetic magnetic fields with wide range of in-cell fluxes $\ensuremath{\alpha}=p/q$. The problem is similar to the one of an electron moving in a tight-binding scheme in the magnetic field and becomes difficult to tackle for a growing number of magnetic subbands, $q$. To overcome this, we focus on the interplay of the width, shape, and number of the subbands on the formation of the coherent state of cold bosons. Using the quantum rotor approach, which goes beyond the mean-field approximation, we are able to pinpoint the elements of the band structure, which are the most significant in a proper theoretical description of the synthetic magnetic field in a bosonic lattice system. As a result, we propose a method of reconstruction of the Hofstadter butterfly spectrum by replacing the magnetic subbands with renormalized bands of a square lattice. This allows us to effectively investigate the properties of the studied system for a wide range of magnetic fluxes and their impact on the Mott-insulator--superfluid transition.

DOI
10.1103/physreva.93.053607

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We study the behavior of interacting ultracold bosons in optical lattices in synthetic magnetic fields with wide range of in-cell fluxes \ensuremathα=p/q\ensuremath{\alpha}=p/q\ensuremathα=p/q. The problem is similar to the one of an electron moving in a tight-binding scheme in the magnetic field and becomes difficult to tackle for a growing number of magnetic subbands, qqq. To overcome this, we focus on the interplay of the width, shape, and number of the subbands on the formation of the coherent state of cold bosons. Using the quantum rotor approach, which goes beyond the mean-field approximation, we are able to pinpoint the elements of the band structure, which are the most significant in a proper theoretical description of the synthetic magnetic field in a bosonic lattice system. As a result, we propose a method of reconstruction of the Hofstadter butterfly spectrum by replacing the magnetic subbands with renormalized bands of a square lattice. This allows us to effectively investigate the properties of the studied system for a wide range of magnetic fluxes and their impact on the Mott-insulator--superfluid transition.
We study the properties of ultra-cold bosons in optical lattice in arbitrary gauge potentials.Using quantum rotor approach we are able to go beyond mean-field approximation thus taking into account subtleties of the band structure of the artificial magnetic field.This allows us to elucidate the interplay of the subbands widths and energy gaps on the formation of the coherent state.As a result, we are able to pinpoint the elements of the band structure, which are crucial to proper theoretical description of the synthetic magnetic field in a lattice bosonic system.This leads us finally to a method of approximation of the Hofstadter butterfly spectrum with a simpler band structure and use it to investigate the ground state of the system for a wide range of magnetic fluxes.
We study the Mott-insulator–superfluid transition of ultracold bosonic atoms in a two-dimensional square optical lattice in the presence of a synthetic magnetic field with p / q ( p and q being co-prime integers) flux quanta passing through each lattice plaquette. We show that on approach to the transition from the Mott side, the momentum distribution of the bosons exhibits q precursor peaks within the first magnetic Brillouin zone. We also provide an effective theory for the transition and show that it involves q interacting boson fields. We construct, from a mean-field analysis of this effective theory, the superfluid ground states near the transition and compute, for q =2, 3, both the gapped and the gapless collective modes of these states. We suggest experiments to test our theory.
We consider theoretically the problem of an artificial gauge potential applied to a cold atomic system of interacting neutral bosons in a tight-binding optical lattice. Using the Bose-Hubbard model, we show that an effective magnetic field leads to superfluid phases with simultaneous spatial order, which we analyze using Bogliubov theory. This gives a consistent expansion in terms of quantum and thermal fluctuations, in which the lowest order gives a Gross-Pitaevskii equation determining the condensate configuration. We apply an analysis based on the magnetic symmetry group to show how the spatial structure of this configuration depends on commensuration between the magnetic field and the lattice. Higher orders describe the quasiparticle excitations, whose spectrum combines the intricacy of the Hofstadter butterfly with the characteristic features of the superfluid phase. We use the depletion of the condensate to determine the range of validity of our approximations and also to find an estimate for the onset of the Mott insulator phase. Our theory provides concrete experimental predictions for both time-of-flight imagery and Bragg spectroscopy.
We study Mott phases and superfluid-insulator (SI) transitions of ultracold bosonic atoms in a two-dimensional square optical lattice at commensurate filling and in the presence of a synthetic periodic vector potential characterized by a strength ppp and a period l=qal=qal=qa, where qqq is an integer and aaa is the lattice spacing. We show that the Schr\"odinger equation for the noninteracting bosons in the presence of such a periodic vector potential can be reduced to an one-dimensional Harper-like equation which yields qqq energy bands. The lowest of these bands have either single or double minima whose position within the magnetic Brillouin zone can be tuned by varying ppp for a given qqq. Using these energies and a strong-coupling expansion technique, we compute the phase diagram of these bosons in the presence of a deep optical lattice. We chart out the ppp and qqq dependence of the momentum distribution of the bosons in the Mott phases near the SI transitions and demonstrate that the bosons exhibit several reentrant field-induced SI transitions for any fixed period qqq. We also predict that the superfluid density of the resultant superfluid state near such a SI transition has a periodicity qqq (q/2)(q/2)(q/2) in real space for odd (even) qqq and suggest experiments to test our theory.
Ultracold atoms in optical lattices undergo a quantum phase transition from a superfluid to a Mott insulator as the lattice potential depth is increased. We describe an approximate theory of interacting bosons in optical lattices which provides a qualitative description of both superfluid and insulator states. The theory is based on a change of variables in which the boson coherent state amplitude is replaced by an effective potential which promotes phase coherence between different number states on each lattice site. It is illustrated here by applying it to uniform and fully frustrated lattice cases but is simple enough that it can be applied to spatially inhomogeneous lattice systems.
Topological phase engineering of neutral bosons loaded in an optical lattice opens a new window for manipulating of transport phenomena in such systems. Exploiting the Bose-Hubbard model and using the magnetic Kubo formula proposed in this paper we show that the optical conductivity abruptly changes for different flux densities in the Mott phase. Especially, when the frequency of the applied field corresponds to the on-site boson interaction energy, we observe insulator or metallic behavior for a given Hofstadter spectrum. We also prove that for different synthetic magnetic-field configurations the critical conductivity at the tip of the lobe is nonuniversal and depends on the energy minima of the spectrum. In the case of 1/21/21/2 and 1/31/31/3 flux per plaquette, our results are in good agreement with those of the previous Monte Carlo study. Moreover, we show that for half magnetic flux through the cell the critical conductivity suddenly changes in the presence of a superlattice potential with uniaxial periodicity.