Introduction
The Casimir force, originally predicted as an attraction between two neutral conducting plates due to vacuum fluctuations of the electromagnetic field[1], has evolved into a universal phenomenon of fluctuation-induced interactions. Recent advances have extended the study of Casimir physics to nonequilibrium scenarios, where objects and fields are at different temperatures or driven by external agents, leading to modified forces with new dependencies on separation and material properties[2].
Ultracold atomic gases provide an ideal platform for probing such nonequilibrium Casimir effects due to their high controllability: atomic species, density, and effective dimensionality can be tuned, and interactions with surfaces can be engineered[3]. In this work, we propose an experimental setup where a cloud of ultracold bosonic atoms is held in an optical dipole trap near a dielectric surface, and subject to a laser field that drives internal hyperfine transitions, creating a nonequilibrium steady state. The resulting atomic density fluctuations couple to the electromagnetic modes of the surface, generating a Casimir–Polder-type force on the atoms that can be measured via collective oscillations or center-of-mass motion.
Non-equilibrium Green's Function Formalism
To capture the nonequilibrium dynamics, we employ the Keldysh formalism for the atom–field system. The total Hamiltonian is \(H = H_{\text{at}} + H_{\text{em}} + H_{\text{int}}\), where \(H_{\text{at}}\) describes the trapped atoms with internal levels and laser driving, \(H_{\text{em}}\) models the quantized electromagnetic field in the presence of a dielectric half-space, and \(H_{\text{int}}\) tallies the dipole coupling between atoms and field.
We compute the force density on the atomic cloud from the stress tensor of the field, which in steady state can be expressed in terms of the nonequilibrium photon Green's function \(D^<(\mathbf{r}, \mathbf{r}', \omega)\). The force on the center of mass of the cloud is obtained by integrating over the cloud's density profile. In the linear response regime, the force separates into an equilibrium-like contribution and a purely nonequilibrium term driven by the effective population differences among atomic dressed states. This formalism allows us to study how the Casimir force changes as a function of the laser detuning and intensity.
Proposed Experimental Setup
We consider a cloud of \(^{87}\text{Rb}\) atoms prepared in a Bose–Einstein condensate (BEC) and held in an optical dipole trap at a distance \(d\) from a fused silica surface. A far-off-resonant trapping beam provides a harmonic confinement, while a pair of counter-propagating Raman beams couples the ground-state hyperfine levels, creating an effective spin-orbit coupling and populating excited dressed states. The surface is kept at room temperature, whereas the atomic cloud, initially near zero temperature, is effectively "heated" by the laser drive into a nonequilibrium momentum distribution.
To measure the Casimir force, we propose to monitor the collective dipole oscillation frequency of the BEC along the direction perpendicular to the surface. Any force gradient from the surface shifts this frequency, providing a sensitive probe. By modulating the laser parameters, we can map the dependence of the nonequilibrium contribution on \(d\) and the laser Rabi frequency \(\Omega\).
Results: Casimir Force Tuning and Signatures
Our calculations predict that for a cloud with a peak density of \(10^{14}\,\text{cm}^{-3}\) and a thickness of a few micrometers, the nonequilibrium Casimir force can reach values on the order of \(10^{-3}\,\text{pN}\), which is within detection limits of current experiments[3]. As shown in Fig. 1 (not included), the force oscillates with distance \(d\) due to interference between direct and reflected atomic dipole radiation. Crucially, the nonequilibrium part can be made repulsive for certain laser detunings, reversing the sign of the force compared to the equilibrium Casimir–Polder interaction.
The spectral density of the force noise reveals peaks at frequencies corresponding to the dressed-state transitions, offering a unique fingerprint of the nonequilibrium state. Furthermore, the force exhibits a non-monotonic dependence on the laser intensity: initially growing with \(\Omega\), it saturates and then decreases as the system enters the strong-driving regime. This tunability opens avenues for active control of Casimir forces via external fields.
Conclusion
We have proposed and theoretically analyzed an experiment to realize and measure nonequilibrium Casimir forces using ultracold atoms near a dielectric surface under laser driving. The nonequilibrium Green's function theory predicts measurable force modifications, including sign reversal and tunable magnitude, with clear signatures in spectroscopy. This work establishes ultracold atomic gases as a versatile testbed for nonequilibrium fluctuation phenomena, providing new insights into quantum thermodynamics and the interplay of light, matter, and vacuum fields.