Quantized vortex states of strongly interacting bosons in a rotating optical lattice

Bose gases in rotating optical lattices combine two important topics in quantum physics: superfluid rotation and strong correlations. In this paper, we examine square two-dimensional systems at zero temperature comprised of strongly repulsive bosons with filling factors of up to one atom per lattice site. The entry of vortices into the system is characterized by jumps of $2\pi$ in the phase winding of the condensate wave function. A lattice of size $L\times L$ can have at most $L-1$ quantized vortices in the lowest Bloch band. In contrast to homogeneous systems, angular momentum is not a good quantum number since the continuous rotational symmetry is broken by the lattice. Instead, a quasiangular momentum captures the discrete rotational symmetry of the system. Energy level crossings indicative of quantum phase transitions are observed when the quasiangular momentum of the ground state changes.

Figure 1

(Color online) Schematic for a $4\times 4$ lattice rotating in the counter-clockwise direction. Sites labeled $i$ and $j$ are nearest neighbors. The rotation-driven hopping between neighboring sites is governed by the parameter $K_{ij}=\beta (x_i{y}_j-x_j{y}_i)∕d$ where $\beta$ is given by the overlap integral Eq. (8) and the second factor is the perpendicular distance of the line joining the two neighboring sites from the center of rotation.

Figure 2

(Color online) Overlap integrals $t$ [Eq. (4)] and $\beta$ [Eq. (8)] for a standing wave optical lattice described by $V=V_0[{\sin }^2(\pi x∕d)+{\sin }^2(\pi y∕d)]$. The lattice depth is given in units of the recoil energy $E_R={\hslash }^2{\pi }^2∕2M{d}^2$. (a) The hopping parameter $t$ decreases exponentially as a function of $V_0$. (b) $\beta d{E}_R∕t$ as a function of lattice depth. In the tight-binding regime, $\beta d$ scales roughly linearly with $t∕E_R$. In this paper we use $\beta =4.93t∕E_{R}d$ and $t=0.02E_R$, corresponding to a standing wave optical lattice of depth $V_0∕E_R=10$.

Figure 3

(Color online) Projection of the single particle ground state obtained using imaginary time propagation (ITP) onto the Hilbert space spanned by the eigenvectors obtained using $\widehat{H}$ (solid line) [Eq. (3)] and ${\widehat{H}}_2$ (dashed line) [Eq. (11)] for a $2\times 2$ lattice with a lattice depth of $V_0=10E_R$. $\Omega$ is in units of the recoil energy. The overlap is good even up to $\Omega =E_R∕\hslash \sim 50t$ where $t$ is the hopping energy.

Figure 4

(Color online) Spatial ground state number density (left-hand side) and phase information (right-hand side) for one particle in a $2\times 2$ sinusoidal lattice with $\Omega =0.5E_R∕\hslash$ and $V_0=10E_R$ obtained using (a) imaginary time propagation, (b) Hamiltonian $\widehat{H}$ [Eq. (3)], and (c) Hamiltonian ${\widehat{H}}_2$ [Eq. (11)].

Figure 5

(Color online) Current $J_{12}$ between adjacent sites of a $2\times 2$ unit cell lattice as a function of total number of particles $n$ for different values of the ratio between repulsive interaction and hopping $U∕t$ at large rotation $(\hslash \Omega \sim 0.6E_R⪢t)$. For weak interaction, $U=0.5t$, the particles can freely cross each other and the current is proportional to the number of particles. As the interaction increases, the current per particle drops as a function of filling. For $U=1000t$, the current for three particles is the same as that for one particle, indicating a particle-hole symmetry. The current was evaluated using a four-state Fock basis on each site, which allowed three particles per site.

Figure 6

(a) Mott insulator zone boundaries for bosons in a $2\times 2$ lattice as a function of chemical potential $\mu$ and of the real and imaginary parts of the complex hopping parameter described in Eq. (16). The volume inside the shaded surfaces corresponds to Mott insulating states with average fillings of one and two particles per site in the lower and upper volume, respectively. (b) The $x\text{−}z$ cross section of the phase diagram is the same as that obtained using the Bose-Hubbard Hamiltonian for a nonrotating system. (c) The $y\text{−}z$ cross section of the phase diagram is similar to the $x\text{−}z$ cross section. Note that the shape of the Mott zones will change on going beyond the two-state approximation [34].

Figure 7

(Color online) Number density distribution for one particle in a $3\times 3$ lattice of depth $V_0=10E_R$ at a lattice rotation of $\hslash \Omega =E_R$. There is almost complete number depletion in the central site because it coincides with the center of rotation and hence the vortex core.

Figure 8

(Color online) Number density distribution for one particle in a $4\times 4$ lattice at $V_0=10E_R$. (a) $\hslash \Omega =0.1E_R$, rotation has yet to enter the system. The center-peaked distribution is due to the infinite potential walls at the perimeter of the lattice. (b) $\hslash \Omega =0.2E_R$, (c) $\hslash \Omega =0.4E_R$, (d) $\hslash \Omega =0.8E_R$. At large rotation, particles get pushed to the outermost sites creating a washing machine effect.

Figure 9

(Color online) (a) Energy-quasiangular momentum dispersion relationship for a 12-site ring. Positive and negative $m_{12}$ values indicate rotation in opposite directions. The velocity of the particles in any state is given by the slope of the dispersion curve at the point. Accordingly, the velocity is maximum at $m=3$. (b) Energy-quasiangular momentum dispersion relationship for a 12-site ring with a small fourfold periodic potential. Note the lines joining states are obtained by extrapolating for an infinite system.

Figure 10

(Color online) One particle in an $8\times 8$ lattice. (a) Ground state energy, $E_0$ vs $\Omega$. (b) Average angular momentum, $⟨L_z⟩$ vs $\Omega$. Note that the expectation value of angular momentum [see Eq. (13)] is not quantized. (c) Quasiangular momentum $m$ vs $\Omega$. In direct analogy to quasimomentum values for linear lattices, $m$ repeats itself. (d) The phase winding, $\Theta ∕2\pi$ vs $\Omega$. For the $8\times 8$ lattice, the maximum phase winding is $7\times 2\pi$.

Figure 11

(Color online) Quasiangular momenta for one through four strongly repulsive bosons in a $6\times 6$ lattice. For multiple particles in the lattice and increasing rotation, the quasiangular momenta cycles through values given by $m=nl$ mod 4, $l=0,1,2,3$, where $n$ is the number of particles. (a) $n=1$, $m=0,1,2,3,0,1$; (b) $n=2$, $m=0,2,0,2,0,2$; (c) $n=3$, $m=0,3,2,1,0,3$; (d) $n=4$, $m=0$.

Figure 12

(Color online) Four particles in a $4\times 4$ lattice. (a) Number of vortices, ${\Theta }_{cf}∕2\pi$ vs $\Omega$. Three vortices enter the $4\times 4$ lattice. (b) Quasiangular momenta, $m$ vs $\Omega$. The symmetry of the ground state as indicated by the quasimomentum $m=0$ does not change even with three vortices entering the system. (c) Zoom in of lowest two energy levels around the entry of the second vortex shows an avoided energy level crossing. The mixing of states is possible because the ground states on either side have the same discrete rotational symmetry.

Figure 13

(Color online) Five particles in a $4\times 4$ lattice. (a) Number of vortices ${\Theta }_{cf}∕2\pi$ vs $\Omega$. As before, a maximum of three vortices enter the system. (b) Quasiangular momenta $m$ vs $\Omega$. Since the filling is incommensurate with the symmetry, the quasiangular momenta take on values $m=0,1,2,3$ [Eq. (19)]. (c) Zoom-in of lowest two energy levels around the entry of the second vortex shows an energy level crossing as the ground state symmetry changes as a function of a parameter in the Hamiltonian, $\Omega$.