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

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.

Observation of Bose–Einstein condensation in a strong synthetic magnetic field

“…2b shows that the minima are most frequently populated with equal proportion, demonstrating the degeneracy of the minima and the robustness of the loading procedure to technical fluctuations. However, mean-field calculations predict that a superposition state, which would create spatial density modulation, is energetically unfavorable due to repulsive interactions [28]. Therefore, the simultaneous occupation of both minima most likely indicates the presence of domains, composed of atoms in only one of the dispersion minima.…”

“…In real space, there are two relevant unit cells: one is the unit cell of the cubic lattice, and the other is the unit cell of the Hamiltonian determined by the experimental gauge, which we call the magnetic unit cell. In highly symmetric gauges -such as our experimental gauge and the Landau gauge -a magnetic flux of α = p/q has a magnetic unit cell that is q times larger than the original unit cell and contains q indistinguishable sites [28]. In momentum space, these unit cells correspond to the Brillouin zone of the underlying lattice and to the magnetic Brillouin zone, respectively.…”

“…This reflects that the vector potential necessarily has a lower symmetry than a uniform magnetic field, since a transitionally invariant field can only be realized by a vector potential that breaks this translational invariance. In our experiment, we analyze the wavefunction directly and observe both its non-gauge invariance [28] and ground state degeneracy. Finally, we explore a many-body adiabatic state preparation protocol where the many-body gap of the Mott insulator is used to switch superfluid order parameters without creating excess entropy.…”

“…This modified translation symmetry connects all degenerate ground states in the magnetic Brillouin zone, and corresponds to a third relevant scale 2π/(qa) useful for understanding the diffraction pattern of a superfluid in the HH Hamiltonian. This length scale defines the doubly reduced Brillouin zone, which is the primitive cell of the band structure [28].…”

Optical lattice modulation spectroscopy for spin-orbit coupled bosons

“…a supersolid) with the incommensurate momentum k 0 providing the inverse lattice constant. However, in a cold atom system with the presence of a trapping potential which breaks translational symmetry, the more likely effect is the formation of domains 39 , each of which corresponds to choosing one of the four values of allowed k 0 . Assuming domains of size much larger than k −1 0 , the momentum distribution signal will be an incoherent weighted sum of the signal from a single domain with a fixed condensation wavevector.…”