Bose-Einstein Condensates in Rotating Lattices

Strongly interacting bosons in a two-dimensional rotating square lattice are investigated via a modified Bose-Hubbard Hamiltonian. Such a system corresponds to a rotating lattice potential imprinted on a trapped Bose-Einstein condensate. Second-order quantum phase transitions between states of different symmetries are observed at discrete rotation rates. For the square lattice we study, there are four possible ground-state symmetries.

Figure 1

Sketch of a rotating lattice. A laser beam passes through a rotating mask. A lens focuses the resulting lattice beam profile onto a stationary, trapped BEC. The lattice potential is imprinted via off-resonant interactions between the laser and the atoms.

Figure 2

The total dimensionless current ${\Lambda }_C$ around a lattice unit cell is shown as a function of the average total number of atoms $n$ in the system for $\hslash \Omega K>t$. The curves from top to bottom are for increasing on-site interaction strength $U$ in units of the lattice recoil energy $E_r$. For weak interactions, the atoms behave independently; for strong interactions, there is a particle-hole symmetry, as evident in the bottommost curve, where $n=1$ and $n=3$ have the same ${\Lambda }_C$. The connecting lines here are a guide to the eye.

Figure 3

(a) Index $m$ of the eigenvalue $e^{i2\pi m/4}$ of the discrete rotational symmetry operator $\mathcal{R}$ as a function of rotation rate, showing the discontinuous changes in the ground-state symmetry. The rotation $\Omega$ is in units of $E_r/\hslash$, with total number of atoms $n=1$, hopping $t/E_r=1$, and $\beta =4.93$ (as appropriate for a sinusoidal square lattice). (b) The derivative of the ground-state energy with respect to $\Omega$ in units of $\hslash$, showing discrete jumps at the points of level crossings.

Figure 4

The red arrows indicate direction and strength of current flow in the rotating frame for $t/E_r=1$ and $n=1$ at increasing rotations on a $4\times 4$ lattice, corresponding to $m=0$, 1, 2, and 3. (a) $\hslash \Omega /E_r=0.1$; (b) $\hslash \Omega /E_r=0.2$; (c) $\hslash \Omega /E_r=0.4$; (d) $\hslash \Omega /E_r=0.8$.

Figure 5

Observables for a $4\times 4$ lattice with five particles and $t/E_r=1$. (a) Site-dependent number density $n_i$ for $\hslash \Omega /E_r=0.1$. There is noticeable depletion on the inner sites. (b) Site-dependent normalized variance ${\nu }_i$ for $\hslash \Omega /E_r=0.1$. Note that ${\nu }_i<1$, so this is a number-squeezed ground state. (c) The normalized, scaled current ${\Lambda }_C/n_C$ (solid curve) and $m$ (dashed curve), all on the boundary as a function of $\Omega$ (in units of $E_r/\hslash$). Between each quantum phase transition, where the symmetry of the ground state changes abruptly as indicated by the jump in the value of $m$, the normalized current depends linearly on the rotation $\Omega$. (d) Is the same as (c) but describes four particles.