Quantum ground state of self-organized atomic crystals in optical resonators

Cold atoms, driven by a laser and simultaneously coupled to the quantum field of an optical resonator, may self-organize in periodic structures. These structures are supported by the optical lattice, which emerges from the laser light they scatter into the cavity mode and form when the laser intensity exceeds a threshold value. We study theoretically the quantum ground state of these structures above the pump threshold of self-organization by mapping the atomic dynamics of the self-organized crystal to a Bose-Hubbard model. We find that the quantum ground state of the self-organized structure can be the one of a Mott insulator, depending on the pump strength of the driving laser. For very large pump strengths, where the intracavity-field intensity is maximum and one would expect a Mott-insulator state, we find intervals of parameters where the phase is compressible. These states could be realized in existing experimental setups.

Figure 1

Setup of the system, showing matter-wave density distribution (blue region) in the effective potential created inside an optical resonator by coherent scattering of laser photons (red line). Here, $\Omega$ is the Rabi frequency, giving the strength of the coupling between the atoms and the laser field, $\lambda$ is the wavelength of laser and cavity mode, and $\kappa$ is the rate at which photons leak from the cavity mode. The quantum state of the system could be inferred by measuring the light at the cavity output, or by Bragg spectroscopy using a weak probe (see Sec. 5).

Figure 2

Phase diagram, showing the Mott-insulator states at different densities $n_0=1,2,3$ in the plane reporting the chemical potential $\mathrm{\mu ̃}=\mu /U_1$, Eq. (31), and the inverse of the laser amplitude, $1/s_0$ (with $s_0$ in units of cavity loss rate $\kappa$). The parameters are ${\Delta }_c=-50\kappa$ and $u_0=-0.1\kappa$. The arrow indicates the threshold value of the laser amplitude, for which the system at $n_0=1$ self-organizes [13]. The threshold value increases with the atomic density (not shown). The chemical potential is here reported in units of the on-site energy $U_1$, corresponding to the value of Eq. (18) for the density of one atom per site, with $g_{1D}/(E_R\lambda )=3.74\times {10}^{-4}$ and $E_R$ the recoil energy, considering a gas of ${}^{87}\mathrm{Rb}$ atoms whose dipole-transition, at wavelength $\lambda =830$ nm, is coupled to the cavity mode. Note, that the Mott-insulator zones are evaluated under the assumption that self-organization took place, hence that $s_0$ is above the corresponding self-organization threshold.

Figure 3

On-site density $n_0$ as a function of the chemical potential $\mathrm{\mu ̃}$ for $s_0\to \infty$ (corresponding to $t_1,t_2\to 0$). Same parameters as in Fig. 2.

Figure 4

Mott-insulator states at $n_0=1,2$ in the plane $\mathrm{\mu ̃}$-$\kappa /s_0$ for ${\Delta }_c=-50\kappa$ and (i) $u_0=-2\kappa$ (red solid line), (ii) $u_0=0.1\kappa$ (green dashed line), and (iii) $u_0=10\kappa$ (blue dotted line). The other parameters are the same as in Fig. 2.