Optomechanical Cooling without Residual Heating

TL;DR

The paper tackles the residual quantum backaction heating that limits conventional optomechanical cooling by introducing an active nonlinear drive that generates intracavity squeezing, enabling zero residual heating. It develops a general semi-classical framework for arbitrary cavity Hamiltonians, yielding a universal photon-number spectrum $S_{nn}(omega)$ and an optomechanical damping rate $Γ_{opt}(omega)$ that depend on squeezing parameters (r1, r2) and an effective detuning tildeDelta. The Josephson optomechanics circuit is analyzed as a concrete realization, showing that by tuning driving strength $E_J^*$ and detuning $Δ$, one can achieve large intracavity occupation $n$, control tildeDelta and $r$, and enter the zero-heating regime, reducing the minimum phonon number $n_m$ by orders of magnitude compared to conventional schemes. In the strong-cooling limit, the minimum phonon number follows $n_m ≈ n_m^r + n_m^T (γ_m/Γ_{opt})$, with the residual term $n_m^r$ vanishing when the squeezing parameters satisfy $r_1 = -γ/2$ and $ω_m = r_2 - tildeDelta$, enabling ground-state-like cooling in low-frequency mechanical modes.

Abstract

Resolved-sideband cooling is a standard technique in cavity optomechanics enabling quantum control of mechanical motion, but its performance is ultimately limited by quantum backaction heating. This fundamental effect imposes a limit on the minimum achievable mechanical phonon number, establishing a finite-temperature floor regardless of the applied cooling strength. We generalize the semi-classical model for optomechanical cooling to describe universal cavity Hamiltonians incorporating both passive and active nonlinearities. As a concrete demonstration, we analyze the simplest circuit optomechanical system that implements a nonlinear drive via a Josephson junction. Our analysis reveals that this active nonlinear drive can eliminate the residual heating backaction, thereby comparing favorably with alternative optomechanical cooling schemes based on passive nonlinearities arXiv:2202.13228. By successfully overcoming the finite-temperature floor that limits conventional schemes, our method paves the way for unprecedented quantum control over mechanical systems and establishes the experimental viability of zero-heating optomechanical cooling.

Figures (4)

• Figure 1: (Color online.) Classical fixed points of the Josephson optomechanics circuit. (a) Sketch of the simplest circuit that implements a nonlinear drive. A dc-biased Josephson junction connects in series with a superconducting $LC$-circuit where the capacitance (or inductance) is modulated by a mechanical element. (b) Cavity photon number $n$ as function of driving strength $E_J^*$ for different detuning $\Delta$. The curves are symmetric for $\pm\Delta$. (c) Phase $\theta_0$ as function of driving strength $E_J^*$ for $\Delta=0,\pm0.4\gamma$. Monostable solutions (black) bifurcate into bistable solutions ($\theta_0^+$ orange and $\theta_0^-$ blue). In panels (c)-(e), blue and pink shading indicate the cooling and heating regimes. (d) Resonance curves $n(\Delta)$ for $E_j^*/\hbar\gamma=$$100$ (purple), $200$ (light-blue), $300$ (navy), $404.40$ is $E_{J,\Delta=0}^\text{bif}/\hbar\gamma$ (dark-red), $750$ (blue/orange). (e) Classical fixed points in phase space, with $E_j^*$ indicated by the color scale. Thick lines represent monostable solutions; thin lines show bistable solutions. Curves correspond to different detunings $\Delta/\gamma$ (blue values). The blue arrow connects points (black) at $E_j^*/\hbar\gamma=750$, corresponding to the blue curve in (d). In all figures $\phi_0=0.06$.
• Figure 2: (Color online) Zero residual heating conditions for Josephson optomechanics. (a) Amplitude squeezing parameter $r_1$ versus normalized driving strength $E_J^*/E_{J,\Delta=0}^\text{bif}$ for specified detunings $\Delta$. Color indicates the monostable regime (black) and the bistable regime at $\theta_0^+$ (orange). (b) Mechanical frequency $\omega_m$ satisfying the minimal residual heating condition ($\omega_m=r_2-\tilde{\Delta}$), plotted against the driving strength, using the same parameters as (a).
• Figure 3: (Color online) Optomechanical damping and residual phonon number. (a,b) $\Gamma_\text{opt}$ and $\overline{n}_m^{\text{r}}$ at the optimal cooling frequency versus driving strength. Note the bifurcation of the monostable (black) solution into bistable solutions (orange and blue). (c,d) Complex eigenvalues $\lambda_\pm$ corresponding to the normal modes of fluctuations, shown for the same parameters as (a) and (b). The pink shading indicates regions where $\lambda_\pm$ are real, bounded by exceptional points. Detunings for (a,c) and (b,d) are indicated.
• Figure 4: (Color online.) Photon number spectrum and minimum phonon number. Spectrum $S_\text{nn}$ at finite detuning, for the (a) monostable $E_J^*/E_{J,\Delta}^\text{bif}=0.92$, and (b) bistable $E_J^*/E_{J,\Delta}^\text{bif}=2.06$ cases, showing $S_\text{nn}(\theta_0^+)$ (orange) and $S_\text{nn}(\theta_0^-)$ (blue). The full quantum solution (green, dashed) fits to a weighted average $0.875 S_\text{nn}(\theta_0^+)+0.125 S_\text{nn}(\theta_0^-)$, biased towards $\theta_0^+$ for $\Delta<0$ [$\phi_0=0.2$ in (a) and (b)]. Minimum phonon number for various drive strengths and detunings with (c) negative detunings (cooling below and above bifurcation); (d) positive detunings (cooling only above bifurcation). The color scale indicates the detuning that maximizes $\Gamma_\text{opt}(\omega_m)$, while the distance to bifurcation, $(E_J^*-E_{J,\Delta}^\text{bif})/\hbar\gamma$, is kept fixed. Other parameters are given in \ref{['tab:proposed_param']}, Appendix D.