Casimir-Lifshitz Theory for Cavity Modification of Ground-State Energy

TL;DR

This work addresses the problem of how ground-state energy and related properties of matter inside Fabry-Perot cavities are modified by vacuum fluctuations, challenging the adequacy of single-mode Hopfield models. It develops a nonperturbative macroscopic QED framework based on a Lorentz permittivity and Barash–Ginzburg regularization, linking the cavity-induced ground-state energy shift to the Casimir–Lifshitz energy and enabling a consistent treatment of losses and temperature. The authors derive an exact Lifshitz-type expression for the cavity-mediated energy shift that accounts for the full continuum of cavity modes and demonstrate that the effect is predominantly nonresonant, governed by static screening of the low-frequency response, rather than resonant polariton features. They compare this to the traditional single-mode Hopfield picture, show its limitations, and propose feasible experimental tests using Casimir-force measurements in liquid-filled FP cavities to probe ground-state modifications, thereby bridging Casimir and polariton physics with potential implications for polaritonic chemistry.

Abstract

A theory for ground-state modifications of matter embedded in a Fabry-Perot cavity and whose excitations are described as harmonic oscillators is presented. Based on Lifshitz's theory for vacuum energy and employing a Lorentz model for the material permittivity, a nonperturbative macroscopic QED model that accounts for the infinite number of cavity modes with a continuum of their wavevectors was built. Differences from the commonly used single-mode Hopfield Hamiltonian are revealed. The nonresonant role of polaritons in the ground-state energy shift is also demonstrated, showing that the cavity effect is mainly caused by static screening occurring at very low frequencies. The theory allows for a straightforward incorporation of losses and temperature effects.

Figures (6)

• Figure 1: (a) Sketch of a resonant system consisting of identical harmonic oscillators strongly coupled to the vacuum EM modes in a PEC FP cavity. (b) Cavity polaritons dispersion at $\mathrm{g}=0.2\omega_0$ for $\omega_0=2\omega_L$; the empty cavity modes are shown in dashed lines. (c) Single-mode Hopfield Hamiltonian spectrum for the cavity polaritons at normal incidence, fixed polarization and zero detuning ($\omega_0=\omega_L$) as a function of $\mathrm{g}$. (d) The real (top) and imaginary (bottom) frequency dependence of the Lifshitz integrand at $T=0$ in the empty cavity (gray dashed) and polaritonic cavity (red solid). (e) Casimir-Lifshitz energy at $T=0$ and $\omega_0=\omega_L$, normalized to the empty cavity case as a function of the coupling energy, $\mathrm{g}$.
• Figure 2: (a) Relative change of the ground-state energy vs. coupling $\mathrm{g}$ in a PEC cavity at $T=0$K and $L=100$ nm according to Lifshitz (red) and single-mode Hopfield (gray) approaches. The static screening approximation (SSA) is shown in dashed lines. USC and deep strong coupling (DSC) denote ultrastrong and deep stong coupling regimes, respectively. (b) Absolute change of the ground-state energy vs $L$ at $T=0$K, with the left axis for Lifshitz and the right one for single-mode Hopfield solutions. (c) Casimir-Lifshitz energy of the cavity with 30-nm gold mirrors on a glass substrate at $T=300$K without (black) and with (red) a medium (with bulk coupling $\mathrm{g} =\omega_0$). SSA results are depicted by dashed lines. (d) Absolute change in ground-state energy at different temperatures. SSA works perfectly at $T=0$K, as expected. In all the plots $\omega_0$ is tuned to the main mode of $L=100$ nm cavity ($\omega_0=\omega_{L=100{\space \rm nm}}$).
• Figure 3: (a) Non-resonant behavior of the Casimir-Lifshitz energy for a gold cavity on a glass substrate filled with molecules floating in water at room temperature (see the sketch). Different concentrations of molecules leads to $\mathrm{g}/\omega_0$ varying from 0 (black dashed) to 0.1, 0.5, 1 (light to dark red), where $\omega_0$ equals to the main mode of 100-nm-cavity. The inset shows the corresponding transmission spectrum with the resonant splittings having USC and DSC features. (b) For the same system, the change of the Lifshitz energy per molecule, showing a similar scaling law as non-retarded Casimir-Polder energy at small $L$ and vanishing of the energy shift at large $L$.
• Figure 4: The Lorentzian permittivity function at real and imaginary frequencies, plotted on top of each other (without offset), demonstrating how rotation to the imaginary frequency axis eliminates the material resonance, making the dielectric function monotonic and positive everywhere, while its high and low frequency limits remain unchanged.
• Figure 5: Matsubara frequencies contributing to the normalized Lifshitz integrand with PEC mirrors at room temperature. As the cavity size $L$ increases, their number diminishes, so that at $L>4\mu$m only $\xi=0$ survives, corresponding to the classical regime in which thermal fluctuations completely dominate the quantum ones. The frequency axis is normalized to the main cavity mode $\omega_L=\pi c/L$.