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arXiv:1009.1389 · Physical Review A

Bogoliubov theory of interacting bosons on a lattice in a synthetic magnetic field

Stephen Powell, Ryan Barnett, Rajdeep Sensarma, S. Das Sarma

Published Jan 19, 2011• 36 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.83.013612

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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.
Motivated by experimental advances in the synthesis of gauge potentials for ultracold atoms, we consider the superfluid phase of interacting bosons on a square lattice in the presence of a magnetic field. We show that superfluid order implies spatial symmetry breaking, and predict clear signatures of many-body effects in time-of-flight measurements. By developing a Bogoliubov expansion based on the exact Hofstadter spectrum, we find the dispersion of the quasiparticle modes within the superfluid phase, and describe the consequences for Bragg spectroscopy measurements. The theory also provides an estimate of the critical interaction strength at the transition to the Mott insulator phase.
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, 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 consider effects of artificial magnetic fields on the ground state of the two-dimensional Bose-Hubbard model. Using an asymmetric Bose-Hubbard model, we demonstrate that the frustrating hopping energy localizes bosons and enlarges insulators and supersolid borders. We show these increments in the presence of artificial gauge fields up to a symmetric point of the field. Moreover, our calculations exhibit the real space modulations of the superfluid and supersolid phases. The bosonic current exhibits vortices in these phases, which their configurations depend on the commensuration between the magnetic field and the lattice. Although the magnetic field breaks the translational symmetry of the square lattice explicitly, this symmetry is restored in the Mott insulator phase. The filling factor exhibits a checkerboard pattern in the density-wave and supersolid phases, regardless of the magnetic field strength. We explore thermal fluctuations and demonstrate the robustness of insulators and SS phases up to temperatures comparable to the interaction energy, which support the feasibility of observing such phases in experiments.
In recent years the quantum simulation of condensed-matter physics problems has resulted from ex- citing experimental progress in the realm of ultracold atoms and molecules in optical lattices. In this thesis we analyze theoretically a spinor Bose gas loaded into a three-dimensional cubic optical lattice. In order to account for different superfluid phases of spin-1 bosons with a linear Zeeman effect, we work out a Ginzburg-Landau theory for the underlying spin-1 Bose-Hubbard model. To this end we add artificial symmetry- breaking currents to the spin-1 Bose-Hubbard Hamiltonian in order to break the global U (1) symmetry. With this we determine a diagrammatic expansion of the grand-canonical free energy up to fourth order in the symmetry-breaking currents and up to the leading non-trivial order in the hopping strength which is of first order. As a cross-check we demonstrate that the resulting grand- canonical free energy allows to recover the mean-field theory. Applying a Legendre transfor- mation to the grand-canonical free energy, where the symmetry-breaking currents are transformed to order parameters, we obtain the effective Ginzburg-Landau action. With this we calculate in detail at zero temperature the Mott insulator-superfluid quantum phase boundary as well as condensate and particle number density in the superfluid phase. We find that both mean-field and Ginzburg-Landau theory yield the same quantum phase transition between the Mott insulator and superfluid phases, but the range of validity of the mean-field theory turns out to be smaller than that of the Ginzburg-Landau theory. Due to this finding we expect that the Ginzburg-Landau theory gives better results for the superfluid phase and, thus, we restrict ourselves to extremize only the effective Ginzburg-Landau action with respect to the order parameters. With- out external magnetic field the superfluid phase is a polar (ferromagnetic) state for anti-ferromagnetic (ferromagnetic) interactions, i.e. only the hyperfine spin 0 (1) is macroscopically occupied, in accor- dance with previous mean-field results. On the other hand, in the presence of the external magnetic field for ferromagnetic interaction, the superfluid phase does not change as the minimization of the energy implies the maximal spin value. However, when an anti-ferromagnetic interaction competes with the linear Zeeman effect, we can distinguish various ferromagnetic and anti-ferromagnetic su- perfluid phases within the range of validity of the Ginzburg-Landau theory. Increasing the external magnetic field yields a breaking of spin singlet pairs and a subsequent alignment of spins, thus anti- ferromagnetic phases decrease until only a ferromagnetic superfluid phase prevails. In addition, we find that the superfluid-Mott insulator phase transition is always of second order for both ferromag- netic and anti- ferromagnetic interactions. However, the transitions between different superfluid phases for an anti-ferromagnetic interaction can be both of first and second order depending on whether the respective macroscopic occupation of hyperfine spin states changes discontinuously or continuously. The established Ginzburg-Landau theory for spin-1 bosons in optical lattices will certainly be the basis for many further applications as, for instance, time-of-flight absorption pictures or collective excitations, which are of experimental importance.
We investigate the effects of artificial magnetic fields on the ground-state of the two-dimensional Bose-Hubbard model. Using an asymmetric Bose-Hubbard model, we demonstrate that anisotropic hopping amplitude localizes bosons and enlarges the insulating and supersolid regions. We show these increases in the presence of artificial gauge fields up to the symmetric point of the field. Moreover, our calculations exhibit real-space modulations of the superfluid and supersolid phases. The bosonic current exhibits vortices in these phases, whose configurations depend on the commensuration between the magnetic field and the lattice. Although the magnetic field explicitly breaks the translational symmetry of the square lattice, this symmetry is restored in the Mott insulator phase. The local densities shows a checkerboard pattern in the density-wave and supersolid phases, regardless of the magnetic field strength. We investigate thermal fluctuations and demonstrate the robustness of insulating and supersolid phases up to temperatures comparable to the interaction energy, which supports the feasibility of observing such phases in experiments.
We analyze theoretically the emergence of different superfluid phases of spin-1 bosons in a three-dimensional cubic optical lattice by generalizing the recently developed Ginzburg–Landau theory for the Bose–Hubbard model to a spinor Bose gas. In particular at zero temperature, our theory distinguishes within its validity range between various superfluid phases for an anti-ferromagnetic interaction with an external magnetic field. In addition, we determine that the superfluid–Mott insulator phase transition is of second order and that the transitions between the respective superfluid phases with anti-ferromagnetic interaction can be both of first and second order.