Ultracold dipolar gas in an optical lattice: The fate of metastable states

We study the physics of ultracold dipolar bosons in optical lattices. We show that dipole-dipole interactions lead to the appearance of many insulating metastable states. We study the stability and lifetime of these states using a generalization of the instanton theory. We also investigate possibilities to prepare, control, and manipulate these states using time-dependent superlattice modifications and modulations. We show that the transfer from one metastable configuration to another necessarily occurs via superfluid states, but can be controlled fully at the quantum level. We show how the metastable states can be created in the presence of a trapping potential. Our findings open the way toward applications of the metastable states as quantum memories.

Figure 1

Gray sites are occupied by one atom and white sites are empty. A “classical” distribution $\mid {\Phi }_{\mathrm{I}}⟩$ of atoms in the lattice (I). The same distribution with one additional atom (Ia) and one removed atom (Ib).

Figure 2

(Color online) (a), (b), (c) Phase diagram with a range of the dipole-dipole interaction cut at the first, second, and fourth nearest neighbor, respectively. The thick line is the ground state and the other lobes correspond to the metastable states, the same color corresponding to the same filling factor. In (c) filling factors range from $\nu =1∕8$ to $\nu =1$. In the right column we present metastable configurations appearing at the first nearest neighbor (I), and second (IIa, IIb), and the corresponding ground state (g.s.); those metastable states remain stable for all larger ranges of the dipole-dipole interaction.

Figure 3

(Color online) Boundaries of the g.s. Mott lobes at zero tunneling, calculated for even filling factors from $2∕16$ to $1∕2$, as a function of the range (RNN) of the dipole-dipole interaction. The colors are the same as for the previous figures of the lobes. Notice the discontinuity in the g.s. after $\mathrm{RNN}=4$ that will be filled by other fractional filling factors.

Figure 4

Lobes of the g.s. at the minimal filling factor for $\mathrm{RNN}=1,\dots ,8$. As the range of dipole-dipole interaction increases the tip of the corresponding lobe gets smaller.

Figure 5

Insulating lobes of the checkerboard state with a negative local chemical potential that follow a checkerboard pattern (a) and a stripe pattern (b), see text for details. The continuous line shows the lobe without any superlattice applied.

Figure 6

(Color online) (a) Lowest excitation spectrum of metastable state (g.s.) (thick), (I) (dashed), (IIa) (dash-dotted), and (IIb) (dotted) of Fig. 2 calculated for $\mu =3.3U_{\mathrm{NN}}$, and $J=0.1U_{\mathrm{NN}}$. Population (b) of the metastable state (I) during real time evolution, the thick and dash-dotted lines are for a small perturbation of the metastable state for $J=0.04U_{\mathrm{NN}}$ and $J=0.12U_{\mathrm{NN}}$, while the dashed line corresponds to an initial big perturbation and $J=0.08U_{\mathrm{NN}}$; (c) the Mott insulating lobe of the state (I).

Figure 7

(Color online) Particle in a double well (a) and the instanton (b).

Figure 8

(I) Exchanging particles with holes, and (II) process where only in a region of the lattice (first and third row from top) the exchange of particles with holes takes place.

Figure 9

(Color online) Stationary paths (a), action per site (b), and barrier (c). The initial state (I) at $(q,P)=(0,0)$ and final state (III) at $(q,P)=(\sqrt{2},0)$. The state in (II) is at an intermediate point.

Figure 10

(Color online) Stationary paths (a), action per site (b), and barrier (c). The initial state (I) at $(q,P)=(0,0)$ and final state (III) at $(q,P)=(\sqrt{2},0)$. The state in (II) is at an intermediate point.

Figure 11

(Color online) Stationary paths (a), action per site (b), and barrier (c). The initial state (I) at $(q,P)=(0,0)$ and final state (III) at $(q,P)=(\sqrt{2},0)$. The state in (II) is the bouncing point at $(q,P)\simeq (0.89,0)$.

Figure 12

(Color online) Stationary paths (a), action per site (b), and barrier (c). The initial state (I) at $(q,P)=(0,0)$ and final state (III) at $(q,P)=(\sqrt{2},0)$. The state in (II) is the bouncing point at $(q,P)\simeq (1.36,0)$.

Figure 13

Disposition of two atoms in the lattice. In MF, CB and ${\mathrm{CB}}^{*}$ is the ground state, while $S1,\dots ,S4$ are degenerate metastable states.

Figure 14

(Color online) (a) Eigenenergies of the Bose-Hubbard Hamiltonian, thick and dashed line are the ground and first excited state, as a function of $\Delta \mu$. (b) The ground state of the system is initially prepared in a CB state (filled bar), while the first excited state consists of equally populated stripe states (empty bars), then an $S1$-type local chemical potential is applied. (c) The final populations is given by an $S1$ ground state (filled bar) and a CB first excited state (empty bar). (d) The variation of the tunneling coefficient.

Figure 15

(Color online) Spectrum of the Bose-Hubbard Hamiltonian for $J=0$ (a), and $J=0.18U_{\mathrm{NN}}$ (b). At $J=0$ the ground state coincides with the MF ground state. In (b) the shaded areas represent the superfluid region calculated in MF, while the round spots are the phase boundaries of filling factor $\nu =1∕2$ calculated with the Bose-Hubbard Hamiltonian.

Figure 16

(Color online) (a) The pulse of local chemical potential as a function of time. (b) The smoothed steplike function is the tunneling coefficient as a function of $\Delta \mu$, while the thick (dashed) line is the tip of $S1$ (CB) insulating lobe. (c) Population inversion, from CB to $S1$ at the end of the process. Notice the oscillation of populations when passing through the SF region of the phase diagram.

Figure 17

(Color online) Percentage of realizations (a) terminating with an $S1$ population bigger than threshold, versus threshold itself; as the threshold increases less realizations satisfy the required precision. (b)–(d) Slices of the discretized space of control parameters; the spots are for processes ending in $S1$ with at least 0.98 population. In (b) we fix $I_r=10\times {10}^{-3}$ and $\Delta {\mu }_o=-0.45U_{\mathrm{NN}}$, in (c) and (d) we fix $\alpha =40\times {10}^{-3}{U}_{\mathrm{NN}}^2$ and $\Delta {\mu }_o=-0.45U_{\mathrm{NN}}$, respectively, for $J_m=0.66U_{\mathrm{NN}}$.

Figure 18

(Color online) Density $\rho (x,y)$ and superfluid parameter ${\mid \phi (x,y)\mid }^2$ in the harmonic trap.

Figure 19

(Color online) (a) The thick line is the lobe of the ground state (I), while the thin line represents the lobe of metastable state (II). Thick and dashed vertical lines are the extension of the harmonic potential of Figs. 18c, 18d and 18a, 18b, respectively.