Refrigeration of a 1D gas of microwave photons

TL;DR

The paper presents a reservoir-engineered cooling scheme for a 1D gas of microwave photons in a superconducting transmission line, using a waste mode coupled via a nonlinear Josephson element to enable photon-number-conserving energy exchange between neighboring modes. By deriving a cooling plus bath-thermalization master equation and introducing an operatively defined effective temperature, it characterizes nonequilibrium steady states that, under energy conservation, exhibit strong ground-state condensation in 1D. The work shows that substantial cooling to sub-millikelvin temperatures is possible for realistic parameters while maintaining a finite photon number, and it reveals a sharp nonequilibrium condensation transition unique to this driven-dissipative setting. An implementation based on a SNAIL-based three-wave mixer demonstrates feasibility and highlights practical considerations (Kerr suppression, parameter regimes), pointing to applications in quantum simulation and the cooling of interacting photonic systems.

Abstract

We discuss a conceptually simple scheme for cooling a one dimensional gas of microwave photons in a superconducting transmission line. By shunting one end of the transmission line by a nonlinear Josephson element, we show how a cooling mechanism can be engineered that transfers photons from high- into low-frequency modes, while preserving their total number. We evaluate the resulting nonequilibrium steady state of the photon gas, which arises from a competition between this engineered cooling process and the natural, number non-conserving thermalization with the surrounding bath. Our analysis predicts that for realistic experimental parameters, this mechanism can be used to prepare photonic gases at sub-millikelvin temperatures, considerably below the typical base temperature of a dilution refrigerator. In addition, the system exhibits a new type of condensation transition that does not occur in the corresponding equilibrium scenario. As an outlook, we discuss potential applications of this cooling approach for quantum simulation schemes with interacting microwave photons.

Figures (7)

• Figure 1: (a) Sketch of a refrigerator for microwave photons. A 1D transmission line resonator (blue) of length $\ell$ and mode spacing $\Delta$ is coupled to its thermal environment and to an engineered bath consisting of a lossy $LC$-resonator (orange), the waste mode. The coupling to the waste mode is mediated by an externally driven, nonlinear Josephson coupler to induce a resonant three-wave-mixing process. (b) Through this process, the photons in the waveguide can be scattered between neighboring modes, with the excess energy $\hbar \Delta$ being absorbed by the waste resonator and successively dissipated into the environment. This process is incoherent, but conserves the total number of photons in the 1D gas. The steady-state occupations $n_k$ are governed by the competition between this engineered cooling mechanism and the direct interaction of the waveguide with the environment.
• Figure 2: Typical steady-state distributions of the average mode energies $E_k$ for a temperature of $T=T_{\rm w} =50\,$mK and a mode spacing of $\Delta/(2\pi)=60$ MHz, which corresponds to $k_BT/(\hbar \Delta)\approx 20$. (a) Energy per mode for different values of the dissipation parameter $G=\gamma \Gamma/g^2$ and $\Delta/\omega_\mathrm{w}=1/3$. The dotted and the dashed lines indicate the behavior the Bose-Einstein distributions given in Eq. \ref{['eq:BE_eq']} and Eq. \ref{['eq:efTh']} for temperatures $T$ and $T_\text{low}$, respectively. The inset shows the occupation of the lowest mode as a function of $G$. As this ratio is reduced, almost all photons end up in the ground mode. (b) Plot of the steady-state energy distribution for varying $\omega_{\rm w}/\Delta=T/T_{\rm low}$ and for a fixed value of $G=10^{-6}$.
• Figure 3: Performance of the engineered cooling scheme for a representative set of experimental parameters and quantified in terms of the effective temperature $T_{\rm eff}$ introduced in Eq. \ref{['eq:Teff']}. (a) Plot of the cooling factor $T/T_{\rm eff}$ as a function of $\omega_{\rm w}/\Delta$ and for different spectral ranges $K$. (b) Plot of the absolute value of the effective temperature as a function of the support temperature and for different values of $\omega_{\rm w}/\Delta$ and $K$.
• Figure 4: Nonequilibrium condensation of microwave photons. The solid lines show the exact numerical results for the condensation fraction $n_1/N_{\rm ph}$ and the effective chemical potential $\mu_{\rm eff}$ for an increasing length $\ell$ of the transmission line (decreasing mode spacing $\Delta$) and as a function of $T_\text{low}/T=T_{\rm w}\Delta/(T \omega_\text{w})$. The dotted lines show the corresponding quantities for a 1D gas of bosons in equilibrium at temperature $T_{\rm low}$ and with a fixed particle number $N_{\rm ph}=N^{\rm th}$, which does not exhibit a sharp transition. In the thermodynamic limit $\ell\propto(\beta\hbar\Delta)^{-1}\rightarrow\infty$ the condensation transition of the photon gas leads to a step function in the condensation fraction plotted in panel (a), and a sudden increase of the chemical potential from a large negative value to $\mu_{\rm eff}=\omega_1$, which is shown in panel (b) and in panel (c). For all plots, the idealized limit $G=0^+$ has been assumed, in which case the nonequilibrium occupation numbers are determined by Eq. \ref{['eq:efTh']}.
• Figure 5: Photon-number distributions. (a) The average steady-state occupation numbers $n_k$ obtained from Monte Carlo simulations (MC) are compared with the corresponding mean-field (MF) results and with the equilibrium distributions given in Eq. \ref{['eq:BE_eq']} and Eq. \ref{['eq:efTh']}. (b) Plots of the steady-state distributions $P(n)$ of photon number states $|n\rangle_k$ for a selection of modes, together with the corresponding value of the second-order correlation function $g^{(2)}(0)$. These distributions have been obtained from Monte Carlo simulations. For both plots, the same parameters $T=100\;$mK, $\Delta/(2\pi)=80\;$MHz, $G=10^{-4}$, $\omega_\mathrm{w}/\Delta=5$ and a cutoff of $k_{\rm max}=10$ have been assumed.