Cold Atom Dynamics in a Quantum Optical Lattice Potential

We study a generalized cold atom Bose-Hubbard model, where the periodic optical potential is formed by a cavity field with quantum properties. On the one hand, the common coupling of all atoms to the same mode introduces cavity-mediated long-range atom-atom interactions, and, on the other hand, atomic backaction on the field introduces atom-field entanglement. This modifies the properties of the associated quantum phase transitions and allows for new correlated atom-field states, including superposition of different atomic quantum phases. After deriving an approximative Hamiltonian including the new long-range interaction terms, we exhibit central physical phenomena at generic configurations of few atoms in few wells. We find strong modifications of population fluctuations and next-nearest-neighbor correlations near the phase transition point.

Figure 1

Sketch of setup.

Figure 2

(a) Numerically found average potential depth $V_{\mathrm{eff}}=V_{\mathrm{cl}}+\hslash U_0⟨a^{†}a⟩$ of the ground state of (6) for $N=2$ atoms in two wells. (b) Relative error in this potential using expansion (9). The parameters are $\eta =2\kappa$, $V_{\mathrm{cl}}=-4E_R$, and $a_s=0.1E_R$.

Figure 3

(a) Energy difference $\Delta E$ as a function of ${\Delta }_c$ for a single atom in two wells for $U_0=-1.2\kappa$ (solid line) and $U_0=-0.4\kappa$ (dashed line). The associated lattice depth is shown in (b). The other parameters are $\eta =2\kappa$ and $V_{\mathrm{cl}}=-4E_R$.

Figure 4

(a) Occupation probabilities $p_i$ for $i$ atoms in one well as a function of scattering length for the ground state of 4 atoms in 4 wells. The parameters are $U_0=-\kappa$, ${\Delta }_c=-3.75\kappa$, and $V_{\mathrm{eff}}=\hslash U_0⟨a^{†}a⟩=-4E_R$. For comparison, the dashed lines correspond to an equivalent classical potential, i.e., $⟨a^{†}a⟩=0$ and $V_{\mathrm{cl}}=-4E_R$. (b) Fluctuations of atom number in one well as a function of $a_s$ for different cavity detunings ${\Delta }_c=-5\kappa$ (solid line), ${\Delta }_c=-3\kappa$ (dashed line), and a classical field (dashed-dotted line).

Figure 5

(a) Difference of the density-density correlation functions: $⟨n_1{n}_3⟩-⟨n_1{n}_2⟩$. Parameters are as in Fig. 4b. (b) Dynamical evolution of atom number fluctuations in one well starting from an interaction free ground state and a sudden turn on of on-site interaction. A linear fit is depicted by the dashed line. Here we chose $U_0=-\kappa$, ${\Delta }_c=-4.2\kappa$, $V_{\mathrm{cl}}=0$, $a_s=3E_R$, and $\eta$ such that $V_{\mathrm{eff}}=-4E_R$.