Steady state diagram of interacting fermionic atoms coupled to dissipative cavities

We investigate fermionic atoms subjected to an optical lattice and coupled to a high finesse optical cavity with photon losses. A transverse pump beam introduces a coupling between the atoms and the cavity field. We explore the steady state phase diagram taking fluctuations around the mean-field of the atoms-cavity coupling into account. Our approach allows us to investigate both one- and higher-dimensional atomic systems. The fluctuations beyond mean-field lead to an effective temperature which changes the nature of the self-organization transition. We find a strong dependence of the results on the atomic filling, in particular when contrasting the behavior at low filling and at half filling. At low filling the transition to a self-organized phase takes place at a critical value of the pump strength. In the self-organized phase the cavity field takes a finite expectation value and the atoms show a modulation in the density. Surprisingly, at even larger pump strengths a strongly non-monotonous behavior of the temperature is found and hints towards effects of cavity cooling at many-body resonances. Additionally multiple self-organized stable solutions of the cavity field and the atoms occur, signaling the presence of a fluctuation-induced bistability, with the two solutions having different effective temperatures previously discussed in [Tolle et al., Phys. Rev. Lett. 134, 133602 (2025)]. In contrast, at half filling a bistable region arises at the self-organization transition already neglecting the fluctuations. The presence of the fluctuations induce an effective temperature as at lower filling and change the behavior of the transition and the steady states drastically. We analyze the properties of the occurring steady states of the coupled atoms-cavity system.

💡 Research Summary

The paper investigates a many‑body system consisting of interacting fermionic atoms confined in an optical lattice and globally coupled to a single‑mode dissipative optical cavity. The atomic sector is modeled by the Hubbard Hamiltonian with tunnelling amplitude J and on‑site repulsion U, while the cavity is described by a detuned mode (detuning δ) subject to photon loss at rate Γ. A transverse pump laser mediates a density‑wave coupling g between the cavity field and the staggered atomic density imbalance Δ. The full dynamics is governed by a Lindblad master equation.

First, a zero‑temperature mean‑field (MF) treatment is applied: the cavity field is adiabatically eliminated, yielding a coherent amplitude λ that depends on g, δ, Γ and the atomic expectation ⟨Δ⟩. Substituting λ into the atomic Hamiltonian produces an effective ionic Hubbard model with a staggered potential ℏgλ. This MF approach predicts a self‑organization transition at a critical pump strength η_c, where the cavity field acquires a finite expectation value and the atoms develop a density wave. However, any eigenstate of the effective Hamiltonian is a steady state, so MF alone cannot select the physically realized state.

To go beyond MF, the authors employ a many‑body adiabatic elimination technique that treats the cavity fluctuations δĤ_ac perturbatively. They identify the decoherence‑free subspace Λ₀ (photon coherent state × atomic eigenstates) and the slowest decaying subspace Λ₁ (single‑photon excitations). By eliminating Λ₁ they derive an effective master equation for the reduced atomic density matrix. Assuming rapid thermalization of the atoms, they approximate the atomic state by a Gibbs ensemble ρ_at ∝ e^{‑β H_eff(λ)}. This introduces a self‑consistent effective temperature T_eff = 1/(k_Bβ) generated by photon‑induced fluctuations.

The analysis reveals that T_eff is a non‑monotonic function of the pump strength. At low filling, increasing the pump first drives the system into a self‑organized phase with a finite cavity field; beyond a certain pump value the effective temperature drops sharply, indicating efficient cavity‑mediated cooling at many‑body resonances. Moreover, two distinct stable solutions appear in the self‑organized regime: one with positive cavity field and low T_eff, another with negative field and higher T_eff. This fluctuation‑induced bistability was previously reported in Ref.