Nonequilibrium Casimir-Polder Force: Motion-induced Thermal-like Effect

Abstract

The Casimir-Polder force is analyzed when an atom is moving at a constant velocity relative to a collection of translationally invariant macroscopic bodies with generic shapes and compositions. The interaction is described within an approach that accurately treats the atom-field coupling and accounts for the backaction from the environment onto the moving particle. Previously overlooked aspects are uncovered and linked to the nonequilibrium and nonconservative nature of the interaction. Specifically, we examine a behavior that can be understood by characterizing the underlying physical processes in terms of a motional-induced effective temperature. This phenomenon shares similarities with the Fulling-Davies-Unruh effect, opening new perspectives for the understanding of nonequilibrium physics at work in the system.

Figures (2)

• Figure 1: Sketch of the analogy discussed here. The Casimir-Polder force on a particle driven at constant velocity $v$ through in the electromagnetic vacuum parallel to an object (right) is similar to the force on a static particle maintained locally in thermal equilibrium at $T>0$ (left).
• Figure 2: The nonequilibrium Casimir-Polder force acting on a particle moving in vacuum within the near field of a planar metallic interface. The different components and the total force are considered for a Cesium atom with $\underline{\alpha}_{0}=\alpha_{0}\mathbf{z}\mathbf{z}$ ($\omega_{a}=1.45$ eV and $\alpha_{0}=4 \pi \epsilon_{0}\times 59.45$ Å$^{3}$Steck10a). The metal is described in terms of the local permittivity function given by the Drude model: $\epsilon(\omega)=1-\omega_{p}^{2}/[\omega(\omega+\mathrm{i} \gamma)]$ with $\omega_{p}=8.39$ eV and $\gamma/\omega_{p}=10^{-2}$ which corresponds to values for gold. The force and its components are normalized by $F_{0}$, i.e the total force at $v=0$. They are represented as a function of $\tilde{T}_{v}/T_{a}$ for $z_{a}\omega_{a}/c=10^{-1.5}$. With these parameters, we obtain $\tilde{T}_{v}/T_{a}\approx 1.6$ for $v/c=10^{-1}$. We marked $\tilde{T}_v/T_{a}=10^{-1}$ with a vertical dotted line in all plots for visual reference. [a]$F^{\rm Ds}$ (blue curve) tends to $F_{0}$ for $v\to 0$, deviating from this value as $\propto v^{2}/z_{a}^{6}$ in agreement with the corresponding low velocity asymptotic expression (black dashed curve) SuppMat. It is visible that $F^{\rm Ds}$ exhibits an extremum, becoming less intense than $F_{0}$ at larger velocities. Inset: $F^{\rm Ds}$ calculated using the bare polarizability (orange) and full dressed polarizability (blue), see Eqs. \ref{['FCPNEqST']} and \ref{['model-dressed-pol']}. [b]$F^{\rm Th}$ (green curve) scales as $v^{2}/z_{a}^{9}$ at low velocity, see the Supplemental Material SuppMat. The inset highlights that the force is smaller by about a factor two if the thermal-like effects are neglected. In agreement with the behavior in Eq. \ref{['FThHighV']} (black dashed line), $F^{\rm Th}$ suddenly increases in magnitude when $\tilde{T}_{v}/T_{a}\gtrsim 5\times 10^{-2}$ and tends to flatten. Right inset: The effective temperature $T_{v}$ in Eq. \ref{['effective-temp']} normalized by $T_{a}$ as a function of $\tilde{T}_{v}/T_{a}$. The curve quantifies the quality of the approximation $T_{v}\approx\tilde{T}_{v}$. [c] The total nonequilibrium Casimir-Polder force $F$ (red curve) is dominated by $F^{\rm Th}$ for $\tilde{T}_{v}/T_{a}> 10^{-1}$, where $F^{\rm Ds}$ (blue curve) becomes a subleading contribution. The black dashed curve describes the total force when $F^{\rm Th}$ is described by Eq. \ref{['FThHighV']}. The total interaction reaches about twice the equilibrium value for $\tilde{T}_{v}/T_{a}\sim 1$. The red dashed curve shows the force in the LTE approximation, i.e. ignoring the terms proportional to $n_{T_{v}}(\omega_a)$ in Eq. \ref{['FThHighV']}.