Dressed-State Optomechanics in the Few-Photon Regime

Abstract

Efficient optomechanical cooling typically requires high photon occupancy to maximize cooling power, a constraint that generally limits the degree of coherent quantum control available in the few-photon regime. Here, we investigate this trade-off by considering a strongly nonlinear cavity operated as a discrete quantum system. In the weak-coupling limit, we derive a general connection between the optomechanical damping rate and the cavity's dressed-state manifold. This framework reveals that the damping rate (determined by the population imbalance across dressed states) is directly tunable via the coherent manipulation tools which are standard in circuit quantum electrodynamics. We illustrate this framework using a Josephson photonics architecture, where a dc-biased junction induces a photon blockade that truncates the cavity to an $N$-level system. By sacrificing raw cooling (or heating) power, this platform enables full quantum mechanical control over optomechanical properties, offering a versatile avenue for the quantum manipulation of mechanical modes.

Figures (4)

• Figure 1: (Color online.) Circuit schematic of a dc-biased Josephson junction providing a nonlinear drive to an optomechanical $LC$ resonator integrating a mechanical element as a drumhead capacitor. At the resonance condition $\omega_{\text{dc}} \simeq \omega_c$, energy conservation dictates that each inelastic Cooper pair tunneling event results in the emission of a photon into the cavity mode.
• Figure 2: (Color online.) Josephson optomechanics in the photon blockade regime. (a) Evidence of photon blockade in the dimensionless transition matrix elements $|t_{n,n+1}|^2 = 4|T_{n,n+1}|^2 / (E_J^*\phi_0)^2$. For the specific values of $\phi_0^2$ marked by green and yellow dots, Fock state transitions $|1\rangle \to |2\rangle$ (orange) and $|2\rangle \to |3\rangle$ (black) are respectively suppressed. (b) Dressed-state eigenenergies for a $3$-level system versus detuning $\Delta$, at $E_J = 300\hbar\gamma$. (c) Photon number spectral density $S_\text{nn}(\omega)$. Arrows identify the transition frequencies between the dressed states of the 3-level manifold ($E_J = 300\hbar\gamma$). The unresolved sideband regime is marked by pink shading.
• Figure 3: (Color online) Optomechanics using an effective $2$-level cavity. (a) Optomechanical damping rate $\Gamma_{\text{opt}}(\omega)$ in the sideband resolved regime $\omega\gg\gamma$. Curves correspond to detunings $\Delta/\gamma = \{-8, -26, -38, -50\}$ (left to right, colored from light blue to purple) at $E_J = 80\hbar\gamma$, with peaks centered at the dressed-state transition $\omega_{10}$. (b) Steady-state populations of the dressed states $\ket{\tilde{0}}$ ($P_0$, dashed) and $\ket{\tilde{1}}$ ($P_1$, solid) versus detuning $\Delta$. Colored curves correspond to different driving amplitudes $E_J/\hbar\gamma=80$ (blue), $150$ (orange). (c) Single-transition rate $\Gamma_{01}(\omega_{10})$ versus detuning $\Delta$. (d) Maximum optomechanical damping rate $\Gamma_{\text{opt}}(\omega_{10})$ plotted versus detuning $\Delta$. Colored round markers correspond to the specific detuning values illustrated in (a). (e) Residual heating phonon number $\bar{n}_m^{\text{r}}$ (logarithmic scale) plotted versus detuning $\Delta$ (linear scale), showing a strong suppression of this unwanted backaction at large detuning $|\Delta|\gg\gamma$.
• Figure 4: (Color online) Optomechanics using an effective $3$-level cavity. (a) Optomechanical damping rate $\Gamma_{\text{opt}}(\omega)$ for detunings $\Delta/\gamma =-30$ (light blue), and $-63$ (orange). (b) Steady-state populations of dressed states $\ket{\tilde{0}}$ ($P_0$, purple), $\ket{\tilde{1}}$ ($P_1$, light-brown) , and $\ket{\tilde{2}}$ ($P_2$, green) versus detuning $\Delta$. The vertical dashed line (brown) at $\Delta_c$ marks the population inversion threshold (where $P_2 = P_1$). (c) Single-transition rates $\Gamma_{01}$ (black), $\Gamma_{12}$ (red), and $\Gamma_{02}$ (light blue) vs. detuning. (d) Maximum optomechanical damping rate $\Gamma_{\text{opt}}$ extracted at each resonance: $\omega_{10}$ (black), $\omega_{21}$ (red), and $\omega_{20}$ (light blue). Coloured dashed lines denote the specific detunings illustrated in (a). The dashed region in the curve corresponding to $\omega_{21}$ (red) highlights heating at the red-detuned drive. (e) Transition frequencies $\omega_{\alpha\beta}$ as a function of detuning. (f) Residual heating $\bar{n}_m^{\text{r}}$ extracted for each transition at its respective optimal frequency. All results are calculated for $E_J = 300\hbar\gamma$.