Quantum stability of Mott-insulator states of ultracold atoms in optical resonators

Abstract

We investigate a paradigm example of cavity quantum electrodynamics with many body systems: an ultracold atomic gas inside a pumped optical resonator, confined by the mechanical potential emerging from the cavity-field spatial mode structure. When the optical potential is sufficiently deep, the atomic gas is in the Mott-insulator state as in open space. Inside the cavity, however, the potential depends on the atomic distribution, which determines the refractive index of the medium, thus altering the intracavity-field amplitude. We derive the effective Bose-Hubbard model describing the physics of the system in one dimension and study the crossover between the superfluid -Mottinsulator quantum states. We predict the existence of overlapping stability regions corresponding to competing insulator-like states. Bistable behavior, controlled by the pump intensity, is encountered in the vicinity of the shifted cavity resonance.

Cold Fermi atomic gases in a pumped optical resonator

“…We again pointed out that by calculating the Heisenberg equations of motion we use the exact relation [â, g(n)] = (g(n) − g(n − 1))â, and do not impose any approximations involving derivatives as in (16). Still, the asymptotic solutions of the numerical integration agree well with the ones obtained from (22), which verifies the trustworthiness of the steady state results 22). (14) for different initial values n(0) and for a fixed number of fermionic atoms N .…”

“…From these expressions we find |J 1 | < 18J within the validity of the tight-binding approximation, which according to Ref. [22] is given by y < 0.5 − 1. Moreover, the contribution from ℓ-neighbours can be estimated to be…”

“…where we have taken care of the proper ordering between atomic and field operators, which is here normal (see also [22]). Let us now consider that the number of photons is fixed to the value n. Then, after tracing out the photonic degrees of freedom from Hamiltonian (4), the resulting Hamiltonian describes the motion of N Fermions in the optical lattice of the cavity field, whose depth is proportional to the number of photons n. Hence, the degree of localization of the atoms at the minima of the potential wells depends on n, and so will the Wannier functions in the tight-binding limit.…”

“…In the regime in which we can assume that the cavity field variables relax to a stationary value on a faster time scale than the time scale of the evolution of the atomic variables, we can eliminate the photon variables from the atomic Heisenberg equations. In doing this we follow the procedure discussed in [22], now applied to fermionic atoms. Hence finding that the atomic dynamics are determined by an effective Hamiltonian of the form…”

“…This Hamiltonian is derived at leading order in an expansion in 1/N , assuming a scaling such that the number of lattice sites K is proportional to the number of atoms N where we assume a fixed number of atoms N . The coefficients of this Hamiltonian depend on the atomic variables through the nonlinear coupling with the cavity field [21,22]. When they are independent of the atomic variables, for instance in the case of Fermi gases in deep optical lattices in free space, the ground state of this Hamiltonian is the usual Fermi sea, where the value N i=1 |k i | is minimized.…”