Dissipation and non-thermal states in cryogenic cavities

TL;DR

This work analyzes photons in a multimode cryogenic cavity coupled to two thermal reservoirs at temperatures $T_m$ and $T_e$ using a Lindblad master equation. It derives microscopic expressions for the dissipation rates into the mirrors and into the external environment, connecting them to measurable macroscopic quantities like conductivity and geometry, and shows that each cavity mode attains a mode-dependent effective temperature $T^*_ u$, leading to a non-thermal steady state. The paper also explores structured dissipation rates to engineer photon statistics and investigates weak non-linear mode mixing, finding only modest deviations from the non-thermal framework. These results provide practical tools for predicting and tailoring dissipation and photon statistics in cavity-based quantum devices and materials, with potential applications in cavity-engineered phase behavior and energy transport. The analysis is applicable to Fabry-Perot and nanoplasmonic cavities and emphasizes that total quality factors alone are insufficient to infer microscopic dissipation channels.

Abstract

We study the properties of photons in a cryogenic cavity, made by cryo-cooled mirrors surrounded by a room temperature environment. We model such a system as a multimode cavity coupled to two thermal reservoirs at different temperatures. Using a Lindblad master equation approach, we derive the photon distribution and the statistical properties of the cavity modes, finding an overall non-thermal state described by a mode-dependent effective temperature. We also calculate the dissipation rates arising from the interaction of the cavity field with the external environment and the mirrors, relating such rates to measurable macroscopic quantities. These results provide a simple theory to calculate the dissipative properties and the effective temperature of a cavity coupled to different thermal reservoirs, offering potential pathways for engineering dissipations and photon statistics in cavity settings.

Figures (5)

• Figure 1: Sketch of the prototypical system under consideration. (a) Fabry Perot cavity with the mirrors at temperature $T_m$ and the surrounding environment at temperature $T_e$. Photons are exchanged by the cavity with the two reservoirs at the rates given in Eq. \ref{['Eq:Rates']}. (b) Approximate sketch of a nanoplasmonic cavity where the field is localized near a metal-vacuum interface. Photons are again lost due to interaction with the metal -- at temperature $T_m$ and with dielectric function $\varepsilon(\omega)$ -- and with free space -- at temperature $T_e$. The plot on the left sketches the decay of the field in the metal and in the vacuum.
• Figure 2: Skectch of the electric field inside and outside of a Fabry-Perot cavity. The cavity is formed by two mirrors of thickness $d$ located at $z=\pm L_c/2$. The fields modes inside the cavity are represented by $\mathbf{e}_\nu(\mathbf{r})$ while the free space modes are $\mathbf{f}_{\mathbf{k},\lambda}(\mathbf{r})$. The zoomed panel shows the overlap between the cavity modes and the free space modes, both of which decay exponentially inside the mirrors.
• Figure 3: Mode dependent effective temperature $T^*_\nu$ (a) and occupation $n_{\mathrm{ph},\nu}$ (b) for different constant bath coupling rates $\gamma_{\nu,e/m}=\gamma_{e/m}$ as a function of the mode frequency $\omega_\nu$. In particular lines go from electromagnetic enviroment dominated $\gamma_{e}/\gamma_{m}=10$ (blue) to mirror dominated $\gamma_{\nu,e}/\gamma_{\nu,m}=0.1$ (yellow). The mirror temperature is taken as half the electromagnetic enviroment temperature $T_m=0.5 T_e$. Note that the bosonic occupation is shown relative to the occupation at a fixed temperature $T_e$ which becomes exponentially small when $\omega_\nu/T_e\gg 1$.
• Figure 4: (a) Cavity-mode dependent effective temperature for frequency-structured mirror dissipation rates as function of the frequency. The mirror dissipation is given by Eq. \ref{['Eq:gamma_mir_RES']} with a resonance at $\omega_0/T_e=1$, $1/\tau=0.1T_e$ and $f_{\rm{res}}=10$. The dashed lines indicate the behavior of $T_\nu^*$ if no resonance is present in $\gamma_{\nu,m}$ (i.e. if $f_{\textrm{res}}=0$). (b) Photon occupation number as function of the frequency $\omega_\nu$. A population inversion can be observed near the resonance frequency $\omega_\nu/T_e=1$. The inset shows the behavior of $\gamma_{e}/\gamma_{\nu,m}$.
• Figure 5: Effective temperature $T_\nu^*$ for each of the three modes considered, as function of the non linear coupling g. The shaded regions indicate the interval of the effective temperature, due to deviations from thermal statistic for each mode.