Bose-Glass Phases of Ultracold Atoms due to Cavity Backaction

We determine the quantum ground-state properties of ultracold bosonic atoms interacting with the mode of a high-finesse resonator. The atoms are confined by an external optical lattice, whose period is incommensurate with the cavity mode wavelength, and are driven by a transverse laser, which is resonant with the cavity mode. While for pointlike atoms photon scattering into the cavity is suppressed, for sufficiently strong lasers quantum fluctuations can support the buildup of an intracavity field, which in turn amplifies quantum fluctuations. The dynamics is described by a Bose-Hubbard model where the coefficients due to the cavity field depend on the atomic density at all lattice sites. Quantum Monte Carlo simulations and mean-field calculations show that, for large parameter regions, cavity backaction forces the atoms into clusters with a checkerboard density distribution. Here, the ground state lacks superfluidity and possesses finite compressibility, typical of a Bose glass. This system constitutes a novel setting where quantum fluctuations give rise to effects usually associated with disorder.

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 coupled to the mode of a standing-wave cavity both at wavelength $\lambda$. Since $\lambda$ and ${\lambda }_0$ are incommensurate, light is scattered into the cavity mode because of quantum fluctuations. It then gives rise to an incommensurate long-range interaction between the atoms which modifies the density distribution.

Figure 2

Results of quantum Monte Carlo simulations for a 1D lattice ($\beta \gg 1$) with 100 sites and periodic boundary conditions. (a) Phase diagram in the $\mu \mathrm{\text{−}}t$ plane for $s_0=0.004\kappa$. The gray regions indicate incompressible phases at density $\overline{n}=1$, 2, the blue region the gapless phases with vanishing SF density. The dotted lines indicate the incompressible phases in the absence of the pump laser ($s_0=0$) [27]. (b) Linear density $\overline{n}$ and (c) $⟨\widehat{\Phi }⟩$ versus $\mu$ for $t=0.053U$ and $s_0/\kappa =0.003$ (triangles), $s_0/\kappa =0.004$ (circles), and $s_0=0$ (squares): The number of photons is different from zero for the parameters of the BG regions in (a). Inset: Fourier transform of the pseudo current-current correlation function $J(\omega )$ [27] for the parameters indicated by the arrows in (b). (d) Local density distribution $⟨{\widehat{n}}_i⟩$ (empty points joined by the blue curve) and local density fluctuations $⟨{\widehat{n}}_i^2⟩-⟨{\widehat{n}}_i{⟩}^2$ (filled points joined by the red curve) as a function of the site for $\mu =0$ and $s_0=0.004\kappa$. The values of $s_0=\sqrt{K}\Omega g_0/{\Delta }_a$ are found from $g_0/2\pi =14.1\mathrm{MHz}$, $\kappa /2\pi =1.3\mathrm{MHz}$, and detunings about ${\Delta }_a/2\pi =58\mathrm{GHz}$ [13]. The other parameters are ${\delta }_c=-5\kappa$, $k/k_0=785/830$ as in Ref. 13, and ${\mathcal{G}}_s$ was taken from Ref. 37.

Figure 3

Results of a mean-field calculation for a 2D lattice with $70\times 70$ sites and periodic boundary conditions. (a) Order parameter $\Theta$ in the $\mu \mathrm{\text{−}}t$ plane when $s_0=0.15\kappa$ (${\delta }_c=-5\kappa$). Inset: Phase diagram when cavity backaction is negligible (${\delta }_c=-300\kappa$). The dotted and solid curves delineate the regions with a small, arbitrarily chosen threshold (here 0.02) for $\Theta$ and the density fluctuations, respectively, while inside the dashed curve the number of photons is at least 100 times larger than outside. The localized regions at smaller $\Theta$ deep in the SF phase are due to the transverse incommensurate pump. (b) Local density distribution $⟨{\widehat{n}}_i⟩$ at $\mu =0.156U$ and $t=0.01U$ [point in (a) at $\overline{n}=0.64$]. Inset: Corresponding density distribution in the $\mu \mathrm{\text{−}}t$ plane for ${\delta }_c=-300\kappa$ (here $\overline{n}=0.625$). (c) Photon emission rate at the cavity output, $n_{\mathrm{out}}=⟨a_{\mathrm{out}}^{†}{a}_{\mathrm{out}}⟩$, as a function of $t$ for $\overline{n}=0.64$ and in units of $n_{\mathrm{out}}^{(0)}=\kappa n_{\mathrm{cav}}^{(\max )}$. Inset: Corresponding order parameter. Here, $n_{\mathrm{cav}}^{(\max )}=s_0^{2}K/({\widehat{\delta }}_{\mathrm{eff}}^2+{\kappa }^2)$ is the intracavity photon number when all atoms scatter in phase into the cavity mode, $s_0=0.15\kappa$ is consistent with the parameters of Ref. 13 for $300\times 300$ sites, and ${\mathcal{G}}_s$ was taken from Ref. 38.