Quantum phases of incommensurate optical lattices due to cavity backaction

Ultracold bosonic atoms are confined by an optical lattice inside an optical resonator and interact with a cavity mode whose wavelength is incommensurate with the spatial periodicity of the confining potential. We predict that the intracavity photon number can be significantly different from zero when the atoms are driven by a transverse laser whose intensity exceeds a threshold value and whose frequency is suitably detuned from the cavity and the atomic transition frequency. In this parameter regime the atoms form clusters in which they emit in phase into the cavity. The clusters are phase locked, thereby maximizing the intracavity photon number. These predictions are based on a Bose-Hubbard model, whose derivation is reported here in detail. The Bose-Hubbard Hamiltonian has coefficients which are due to the cavity field and depend on the atomic density at all lattice sites. The corresponding phase diagram is evaluated using quantum Monte Carlo simulations in one dimension and mean-field calculations in two dimensions. Where the intracavity photon number is large, the ground state of the atomic gas lacks superfluidity and possesses finite compressibility, typical of a Bose glass.

Figure 1

Ultracold atoms are tightly confined by an optical lattice of periodicity ${\lambda }_0/2$. They are driven by a weak transverse laser at Rabi frequency $\Omega$ and strongly couple to the mode of a standing-wave cavity, both at wavelength $\lambda$. Since $\lambda$ and ${\lambda }_0$ are incommensurate, one would expect no coherent scattering into the cavity mode. The mechanical effects due to multiphoton scattering, however, give rise to an incommensurate quantum potential, which mediates an effective long-range interaction between the atoms and modifies the properties of the quantum ground state. As a result, the intracavity photon number can be large. The corresponding ground state can show features typical of a Bose glass.

Figure 2

Mean density $\overline{n}$ as a function of the chemical $\mu$ (in units of $U$) at $t=0$ in a 1D lattice. Curves were obtained by an exact diagonalization of Hamiltonian (30) for ${\delta }_c=5\kappa$ (dashed line) and ${\delta }_c=-5\kappa$ (solid line), both for $K=100$. Other parameters are $s_0=0.006\kappa$ (with $\kappa =2\pi \times 1.3$ MHz), $u_0=0.8\kappa$, and $U/\hslash =50$ Hz. Here, the chemical potential is reported without the constant shift, i.e., $\mu \to \mu -{\epsilon }^{\left(0\right)}-V_1{J}_0$. The dash-dotted line was evaluated for the same parameters as the solid line, except with $K=200$. It shows that the results remain invariant as the system size is scaled up.

Figure 3

Boson occupation number, $n_i$, as a function of the site $i$ for the two distributions corresponding to the two ground states at ${\delta }_c=-5\kappa$ and $\mu =0$. Other parameters are the same as in the caption of Fig. 2. Filled bars correspond to $\langle {\widehat{n}}_i\rangle =1$; white bars, to $\langle {\widehat{n}}_i\rangle =0$.

Figure 4

Mean density $\overline{n}$ as a function of $\mu$ (in units of $U$) for a 1D lattice of $K=100$ sites for $t=0$, ${\delta }_c=-5\kappa$, $u_0=0.8\kappa$, $U/\hslash =50$ Hz (with $\kappa /2\pi =1.3$ MHz), while the values of $s_0$ are reported in the legend. (a) Curves were evaluated by diagonalizing Eq. (30) after setting $\langle \widehat{\Phi }\rangle =1/4$ (i.e., by artificially removing cavity backaction). (b) Curves were found for the corresponding parameters by diagonalizing the full quantum model of Eq. (30). Other parameters are as described in the caption to Fig. 2.

Figure 5

(a) Mean density $\overline{n}$ and (b) expectation value $\langle \widehat{\Phi }\rangle$ versus the chemical potential $\mu$ (in units of $U$) for a 1D lattice at different tunneling values, $t/U=0.014$, 0.053, 0.096, 0.216. Other parameters are the same as for the black line at $s_0=0.008\kappa$ in Fig. 4b. Curves were evaluated by means of a QMC program described in Appendix . Dotted vertical lines indicate the values of $\mu$ in Fig. 6.

Figure 6

Upper row: On-site density distribution $\langle {\widehat{n}}_i\rangle$ and local density fluctuations $\langle {\widehat{n}}_i^2\rangle -{\langle {\widehat{n}}_i\rangle }^2$ as a function of the site $i$. The line connecting the points serves as a guide for the eyes. Lower row: On-site energy due to the cavity field, $\delta {\epsilon }_i$ (in units of $U$). Curves were evaluated for the parameters of the curve in Fig. 5 with $t=0.096U$ and $s_0=0.008\kappa$. Plots correspond to $\mu /U=0.2$, 0.7, and 1.2 (from left to right).

Figure 7

Results of QMC simulations for a 1D lattice with 74 sites and $t=0.053$ with periodic boundary conditions. (a) Mean density $\overline{n}$ and (b) modulus of the mean intracavity-field amplitude $\langle \widehat{\Phi }\rangle$ versus $\mu$ (in units of $U$) for $s_0=0.003\kappa$ (triangles), for $s_0=0.004\kappa$ (circles), and in the absence of the pump laser $s_0=0$ (squares). (c) Zoom of mean density in the region of parameters with $-0.18\le \mu \le 0.18$. (d) Fourier transform of the pseudocurrent-current correlation function $\tilde{J}\left(\omega \right)$ for the values of $\mu /U$ indicated by the arrows in (c) for $s_0=0.003\kappa$. Simulation parameters used are $L=128$ and $\Delta \tau =1$. Lines show cubic interpolations of the data. The extrapolated value at zero frequency is the estimation of the SF density. Other parameters are the same as in Fig. 5.

Figure 8

Upper row: On-site density distribution $\langle {\widehat{n}}_i\rangle$ and local density fluctuation $\langle {\widehat{n}}_i^2\rangle -{\langle {\widehat{n}}_i\rangle }^2$ as a function of the site $i$. The line connecting the points serves as a guide for the eyes. Lower row: Mean on-site energy $\langle \delta \widehat{{\epsilon }_i}\rangle$ (in units of $U$). Curves were evaluated for the parameters of the (red) curve with triangles in Figs. 7a, 7b, 7c with $t=0.053U$ and $s_0=0.003\kappa$. Plots correspond to $\mu /U=0$, 0.04, and 0.08 (from left to right).

Figure 9

The phase diagram obtained by QMC calculation for a 1D lattice with 74 sites for $s_0=0.004\kappa$. Other parameters are as in Fig. 7. Results are compared with the pure case (dotted curves).

Figure 10

Mean density $\overline{n}$ versus $\mu$ (in units of $U$) for a 2D lattice. The curve was evaluated by exact diagonalization of Hamiltonian (16) for $t=0$, $K=70\times 70$, and ${\Delta }_a<0$ (dashed line) and for $t=0$, $K=70\times 70$, and ${\Delta }_a>0$ (solid line). Parameters are $|{\Delta }_a|=2\pi \times 58$ GHz, $s_0=0.15\kappa$, ${\delta }_c=-5\kappa$, $u_0=237\kappa$ ($\kappa =2\pi \times 1.3$ MHz). Inset: Zoom of the curve at $K=70\times 70$ and ${\Delta }_a>0$ in the compressible phase. The dashed-dotted line for $K=100\times 100$ and ${\Delta }_a>0$ shows, compared with the solid line, that the qualitative behavior of the curves remains invariant as the system size is scaled up.

Figure 11

Mean density $\overline{n}$ [upper (blue) line] and $\langle \widehat{\Phi }\rangle$ [lower (green) line with crosses] versus $\mu$ (in units of $U$) for a 2D lattice. Curves were evaluated using the local mean field for $t=0.01U$ and $K=70\times 70$. Parameters are ${\Delta }_a=2\pi \times 58$ GHz, $s_0=0.15\kappa$, ${\delta }_c=-5\kappa$, $u_0=237\kappa$ ($\kappa =2\pi \times 1.3$ MHz). Inset: Same curves, but for ${\delta }_c=-300\kappa$. Note that the maximum value of $\langle \widehat{\Phi }\rangle$ is almost five orders of magnitude smaller than for ${\delta }_c=-5\kappa$.

Figure 12

(a,b) Order parameters in the $\mu$-$t$ plane (in units of $U$) obtained by the mean-field calculation for a 70 lattice with periodic boundary conditions. Dotted lines separate the region with vanishing order parameters, while the solid line identifies the border for the incompressible MI state at density $\overline{n}$. Regions with finite compressibility and vanishing order parameters correspond to BG phases. The dashed line separates the region where the photon number is two orders of magnitude larger than outside. Parameters are $s_0=0.15\kappa$, $u_0=237\kappa$, ${\Delta }_a=2\pi \times 58$ GHz, whereas (a) ${\delta }_c=-5\kappa$ and (b) ${\delta }_c=-300\kappa$. In the latter case the effect of the cavity potential is expected to be small. Local densities $\langle {\widehat{n}}_{i,j}\rangle$ of the phase diagram at $\mu =0.1U$ and $t=0.01\kappa$ are shown (c) for ${\delta }_c=-5\kappa$ and $\overline{n}=0.57$ [white square in (a)] and (d) for ${\delta }_c=-300\kappa$ and $\overline{n}=0.47$ [white square in (b)].

Figure 13

(a) Intensity at the cavity output, $n_{\mathrm{out}}=\langle a_{\mathrm{out}}^{†}{a}_{\mathrm{out}}\rangle$, as a function of $t$ (in units of $U$) for the parameters in Fig. 12a and by fixing $\overline{n}=0.5$. Here, $n_{\mathrm{out}}$ is in units of $n_{\mathrm{out}}^{\left(0\right)}=\kappa n_{\mathrm{cav}}^{\left(\max \right)}$, where $n_{\mathrm{cav}}^{\left(\max \right)}=s_0^{2}K/({\widehat{\delta }}_{\mathrm{eff}}^2+{\kappa }^2)$ is the maximum number of intracavity photons, obtained when all atoms scatter in phase into the cavity mode. The curve with circles (right $y$ axis) gives the corresponding order parameter. (b) Contour plot of the local density distributions at point (I) in (a), where $t=0.034U$, $\mu =0.106U$, $\langle \widehat{\Phi }\rangle =0.136$. (c) Local density distributions at point (II) in (a), where $t=0.039U$, $\mu =0.122U$, and $\langle \widehat{\Phi }\rangle \simeq 0$. (d) Local density $\langle {\widehat{n}}_{i,j}\rangle$ as a function of the site numbers along $z$ for lattice site 20 along $x$. The (blue) squares correspond to parameters in (b); (red) circles, to parameters in (c).

Figure 14

Ratio of the energy gap between the two Bloch bands, $\Delta E=\sqrt{4E_R\left|V_0\right|}$, and the interaction energy, $V_{\mathrm{int}}=U+\max |\delta {\epsilon }_{i,j}|$, as a function of the chemical potential $\mu$ (in units of $U$)and at zero tunneling. The single-band approximation is valid, $\Delta E/V_{\mathrm{int}}\gg 1$. The (green) curve with crosses shows the maximum values of $\langle \delta {\widehat{\epsilon }}_{i,j}\rangle$ at the corresponding values of $\mu /U$. Parameters are the same as in Fig. 12a.

Figure 15

Path integral representation of the system. The horizontal expansion displays the spatial dimension whereas the vertical slices represents the evolution in imaginary time. One valid configuration is described by a set of particle paths (world-lines) that obey periodic boundary conditions in spatial and temporal direction and particle conservation in each slice. World-lines can either exhibit a closed connection from bottom to top (dash line) with so-called winding number $W=0$ or winded around the system [here with $W=+1$ (solid line)]. The shaded squares of the checkerboard pattern indicates the plaquettes, where ${\mathcal{H}}_{i,i+1}$ acts and hence hopping can occur.