Strong coupling and active cooling in a finite-temperature hybrid atom-cavity system

For a standard two-level atom coupled to the quantized field of a resonant cavity, finite-temperature effects lead to thermal occupation of the cavity modes that obfuscates measurement of the quantum nature of the atom-light interaction. In this paper we demonstrate that using a hybrid system of a superconducting cavity coupled to a multilevel Rydberg atom, it is possible to observe the quantum nature of strong coupling even at finite temperatures and to exploit this coupling to permit cooling of the thermal microwave mode towards the ground state, enabling observation of coherent vacuum Rabi oscillations even at 4 K for realistic experimental parameters. Cooling for multiple atoms is also explored, showing maximal cooling for small samples, making this a viable approach to cavity cooling with potential applications in long-range coupling of superconducting qubits via thermal waveguides.

Figure 1

Hybrid atom-cavity scheme: (a) atom strongly coupled to a superconducting microwave coplanar waveguide resonator, (b) $\mathrm{\Lambda }$ level scheme, (c) ladder level scheme $\mathrm{\Xi }$, and (d) cavity cooling scheme.

Figure 2

Ground-state transfer as a function of $\mathrm{\Delta }$ and $T$ for (a) $\mathrm{\Xi }$ and (b) $\mathrm{\Lambda }$ systems after a pulse duration $\tau =\pi /\mathrm{\Omega }$ for $g/2\pi =4\mathrm{MHz}, \mathrm{\Omega }=0.5g$, and $Q={10}^5$.

Figure 3

Ground-state transfer as a function of $Q$ for (a) $\mathrm{\Xi }$ and (b) $\mathrm{\Lambda }$ systems for $g/2\pi =4\mathrm{MHz}, \mathrm{\Omega }=0.5g$, and $T=4\mathrm{K}$.

Figure 4

Cavity cooling. (a) Temporal evolution for the cooling scheme in Fig. 1 using $\mathrm{\Omega }={\mathrm{\Omega }}^{\prime }=2\pi \times 10\mathrm{MHz}, g/2\pi =4\mathrm{MHz}, Q={10}^5$, and $T=4\mathrm{K}$. (b) Steady-state solution vs thermal field distribution. (c) Steady-state effective temperature $T_{\mathrm{eff}}$ as a function of $g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$. (d) Steady-state effective temperature $T_{\mathrm{eff}}$ as a function of $Q$ and $T$.

Figure 5

Cavity cooling optimization for $g/2\pi =4\mathrm{MHz}, Q={10}^5$, and $T=4\mathrm{K}$. (a) Optimal cooling gives $\mathrm{\Omega }/2\pi =14.5\mathrm{MHz}$ and ${\mathrm{\Omega }}^{\prime }/2\pi =17.6\mathrm{MHz}$ for $\mathrm{\Delta }={\mathrm{\Delta }}^{\prime }=0$. (b) Effective temperature as a function of cavity detuning ${\mathrm{\Delta }}_c$ and $\mathrm{\Delta }$ for ${\mathrm{\Delta }}^{\prime }=0$ showing maximum cooling efficiency occurs for $\mathrm{\Delta }={\mathrm{\Delta }}_c=0$.

Figure 6

(a) Vacuum Rabi oscillation of state $|s\rangle$ and (b) ground-state transfer for the $\mathrm{\Lambda }$ system for a cavity at 0, 4, and 4 K after active cooling to $T_{\mathrm{eff}}=0.87\mathrm{K}$.

Figure 7

Steady-state effective temperature from active cavity cooling vs atom number for initial cavity temperatures of (a) $T=1\mathrm{K}$ and (b) $T=4\mathrm{K}$.