Real photons from vacuum fluctuations in optomechanics: The role of polariton interactions

We study nonlinear interactions in a strongly driven optomechanical cavity, in regimes where the interactions give rise to resonant scattering between optomechanical polaritons and are thus strongly enhanced. We use a Keldysh formulation and self-consistent perturbation theory, allowing us to include self-energy diagrams at all orders in the interaction. Our main focus is understanding how nonequilibrium effects are modified by the polariton interactions, in particular the generation of nonzero effective polariton temperatures from vacuum fluctuations (both in the incident cavity drive and in the mechanical dissipation). We discuss how these effects could be observed in the output spectrum of the cavity. Our work also provides a technical toolkit that will be useful for studies of more complex optomechanical systems.

Figure 1

(a) A generic optomechanical cavity: a cavity mode of frequency ${\omega }_C$ and damping rate $\kappa$ is driven by a laser at frequency ${\omega }_L$, and is coupled via radiation pressure to a damped mechanical mode (resonance frequency ${\omega }_M$, damping rate $\gamma$). The temperature of the mechanical bath is $T_M$. (b) The single-photon optomechanical interaction $g$ can be made resonant in the basis of optomechanical polaritons; the resulting enhanced interaction shown schematically and described in Eq. (8).

Figure 2

(a) $G_{\mathrm{res}}$, the value of the many-photon coupling which leads a resonant nonlinear interaction, as a function of laser detuning $\Delta$ [cf. Eq. (7)]; the corresponding effective nonlinear coupling $\tilde{g}$ [cf. Eq. (8)] is also plotted. (b), (c) Coefficients relating the optomechanical polaritons (eigenstates of the linearized Hamiltonian) to original photon and phonon operators [cf. Eqs. (5)] as a function of laser detuning $\Delta$ with $G=G_{\mathrm{res}}\left[\Delta \right]$. In (b), we plot the “normal” coefficients which relate polariton destruction operators to photon and phonon destruction operators, i.e., ${\alpha }_{b,\sigma }=\langle 0,0|{\widehat{c}}_{\sigma }{\widehat{b}}^{†}|0,0\rangle$, with $|0,0\rangle$ being the polariton vacuum. In (c), we plot the “anomalous” coefficients, i.e., ${\overline{\alpha }}_{b,\sigma }=\langle 0,0|{\widehat{c}}_{\sigma }\widehat{b}|0,0\rangle$; these coefficients are directly related to the existence of “quantum heating” effects in the linearized theory [cf. Eq. (15)].

Figure 3

(a) Damping rates ${\kappa }_{\sigma }$ of the $\sigma =\pm$ polaritons [cf. Eq. (23)] in the linearized theory (i.e., $\tilde{g}=0$), as a function of the detuning $\Delta$. For each $\Delta$, we tune the cavity control drive to maintain $G=G_{\mathrm{res}}\left(\Delta \right)$. Near $\Delta =-2{\omega }_M$, the $- (+)$ polariton damping tends to $\gamma \left(\kappa \right)$ as it is mostly phononlike (photonlike); the converse is true near $\Delta =-{\omega }_M/2$. (b) Thermal occupation numbers ${\overline{n}}_{\sigma }^0$ of the effective bath coupled to the $\sigma$ polaritons [cf. Eq. (24)] also when $\tilde{g}=0$ and $G=G_{\mathrm{res}}$. Here, ${\overline{n}}_{\mathrm{th}}^{\mathrm{M}}$ characterizes the (physical) temperature of the mechanical bath [cf. Eq. (18)]. (c) The temperatures corresponding to the thermal occupation ${\overline{n}}_{\sigma }^0 \left(k_B=1\right)$ in the same regime as panels (a) and (b). For each curve, $\gamma /\kappa ={10}^{-4}$ and ${\omega }_M/\kappa =50$.

Figure 4

Diagrams representing the self-energies up to second order in $\tilde{g}$ in the Keldysh formalism. The classical vertices are composed of only one quantum component, while the quantum vertex is composed of three.

Figure 5

Solid lines: leading-order-in-$g$ effective cooperativities ${\mathcal{C}}_{\sigma }^{\mathrm{eff}}$ associated with the nonlinear interaction [defined in Eq. (41)], as a function of the detuning in the resonant regime $\left(G=G_{\mathrm{res}}\left[\Delta \right]\right)$. One sees that the ${\mathcal{C}}_{\sigma }^{\mathrm{eff}}$ are enhanced near $\Delta =-2{\omega }_M$. The inset zooms on the region where ${\mathcal{C}}_{-}^{\mathrm{eff}}$ goes below zero, which signals the possibility of a new kind of instability, as discussed explicitly in Sec. 5d. Circles: results of the self-consistent perturbation theory described in Sec. 4b, which includes diagrams at all orders in $g$. For all values of ${\mathcal{C}}_{\sigma }^{\mathrm{eff}}$ shown here, the approach converges and is in good agreement with numerical simulations of the Lindblad master equation [cf. Eq. (21)]. The parameters used are $\gamma /\kappa ={10}^{-4},g=\kappa ,{\omega }_M=50\kappa$, and ${\overline{n}}_{\mathrm{th}}^{\mathrm{M}}=0$.

Figure 6

Effective thermal occupancies for the various baths coupled to the two polariton species, as a function of $\Delta$ with $G=G_{\mathrm{res}}\left[\Delta \right]$. Solid lines: occupancies associated with the “interaction-induced” baths to leading order in $g$ [cf. Eqs. (44)]. Circles: same, but calculated using the all-orders self-consistent approach of Sec. 4b. Dashed lines: occupancies of the intrinsic polariton baths [cf. Eqs. (24)]. Note the leading-order occupancy for the $-$ polariton interaction bath $\left({\overline{n}}_{-}^{\mathrm{int}}\right)$ diverges when ${\overline{n}}_{-}^0={\overline{n}}_{+}^0 \left(\Delta /{\omega }_M\approx -0.8\right)$ and becomes negative when ${\overline{n}}_{+}^0>{\overline{n}}_{-}^0 \left(\Delta /{\omega }_M\gtrsim -0.8\right)$, as shown in the inset. This divergence persists in the self-consistent theory, but occurs at smaller-magnitude detunings. This negative occupancy signals the possibility of a different kind of instability, as discussed explicitly in Sec. 5d. All curves are plotted for ${\overline{n}}_{\mathrm{th}}^{\mathrm{M}}=0$ [cf. Eq. (18)], $\gamma /\kappa ={10}^{-4},{\omega }_M/\kappa =50$, and $g=\kappa$.

Figure 7

(a) Solid lines: net polariton occupancies in the presence of nonlinear interaction ${\overline{n}}_{\sigma }^{\mathrm{eff}}\left[E_{\sigma }\right]$, calculated using leading-order self-energies, as a function of $\Delta$ with $G=G_{\mathrm{res}}\left[\Delta \right]$. Circles: same, but calculated using the self-consistent approach (cf. Sec. 4b). Dashed curves: occupancies for $g=0$, i.e., calculated in the linearized theory. In each case, the mechanical temperature is zero $\left({\overline{n}}_{\mathrm{th}}^{\mathrm{M}}=0\right)$ and $g=\kappa$. (b) Same as (a), but now with ${\overline{n}}_{\mathrm{th}}^{\mathrm{M}}=100$ and $g=0.1\kappa$; main plot and inset show different ranges of $\Delta$. Near $-2{\omega }_M$, one sees an important contribution to the mean number of $+$ polaritons due to nonlinear interaction; this contribution is greatly enhanced by temperatures, as discussed in Sec. 6d. All the curves are plotted for $\gamma /\kappa ={10}^{-4}$ and ${\omega }_M/\kappa =50$.

Figure 8

(a), (b) Cavity DOS near the $-$ polariton resonance and the $+$ polariton resonance, respectively, for a cavity driven on the red sideband and when the nonlinear interaction is resonant (i.e., $G=G_{\mathrm{res}}$). We work in the frame rotating at the drive frequency so that $\omega =0$ refers to the drive frequency while $E_{+}=2E_{-}=63.24\kappa$ in this frame. The light dashed curves represent the linearized theory, the green dotted ones represent the results to leading orders in $\tilde{g}$ [Eq. (42) for the DOS], the full blue curves are the results of the self-consistent approach as described in Sec. 4b, and the black curves are for the numerical simulation of the Lindblad master equation shown in Eq. (21). (c), (d) Cavity spectrum in the same conditions; the results to leading orders in $\tilde{g}$ are given in Eqs. (52) and (59). (e), (f) Cavity energy distribution function [Eq. (61)] and (g), (h) the corresponding (frequency-dependent) effective temperatures [Eq. (62)]. The parameters used for all the curves are $\gamma /\kappa ={10}^{-4}$ and ${\omega }_M/\kappa =50$, which leads to the leading-order effective cooperativities [i.e., Eq. (41)] ${\mathcal{C}}_{-}^{\mathrm{eff}}=0.18$ and ${\mathcal{C}}_{+}^{\mathrm{eff}}=2.46$.

Figure 9

(a) Output photon flux in a bandwidth of $5\kappa$ near the $-$ polariton resonance $E_{-}$ [cf. Eq. (63)] as a function of the many-photon coupling $G$, and for a control laser detuning $\Delta =-{\omega }_M$. (b) Same as (a) but near the $+$ polariton resonance $E_{+}$. Note that $G$ can be varied by simply tuning the amplitude of the control laser. On resonance, i.e., $G=G_{\mathrm{res}}$, the nonlinear effects are enhanced by a factor of ${\omega }_M/\kappa =50$ compared to the off-resonance case, i.e., $(G-G_{\mathrm{res}})\gtrsim \kappa$. The light dashed curves represent the linearized theory $\left(g=0\right)$, the solid blue curves are for the self-consistent approach as described in Sec. 4b, and the black dashed curves are for the numerical simulation of the Lindblad master equation shown (21), but this time, using the full nonlinear part of the Hamiltonian in Eq. (6) that includes all the nonresonant nonlinear processes. For all the curves, we used $\gamma /\kappa ={10}^{-4}$ and ${\overline{n}}_{\mathrm{th}}^{\mathrm{M}}=0$.

Figure 10

Same as Fig. 8 but for laser detuning $\Delta =-1.8{\omega }_M$. In that case, $E_{+}=2E_{-}=92.08\kappa$ in the rotating frame and the corresponding leading order [i.e., Eq. (41)] effective cooperativities ${\mathcal{C}}_{-}^{\mathrm{eff}}=1.92$ and ${\mathcal{C}}_{+}^{\mathrm{eff}}=5.40$.

Figure 11

Same as Fig. 8 but for laser detuning $\Delta =-0.65{\omega }_M$ and finite temperature ${\overline{n}}_{\mathrm{th}}^{\mathrm{M}}=650$. In these circumstances, leading-order perturbation theory predicts that the system is close to a parametric instability (see Sec. 5d). Note the mechanical damping rate here is larger than in previous plots $\left(\gamma ={10}^{-3}\kappa \right)$, as this enhances the pumping effects of mechanical temperature. For ${\omega }_M=50\kappa$ and $g=\kappa$, the leading-order effective cooperativities [cf. Eq. (41)] are ${\mathcal{C}}_{-}^{\mathrm{eff}}=-0.97$ and ${\mathcal{C}}_{+}^{\mathrm{eff}}=0.75$. In the frame that rotates at the drive frequency, $E_{+}=2E_{-}=53.34\kappa$. One sees from these plots the crucial importance of higher-order-in-$g$ terms.

Figure 12

(a) Signatures of the effective two-phonon absorption in the cavity DOS near the $+$ resonance; the inset shows a larger range of frequency than the main plot. The light-blue dashed curve represents the linearized theory $\left(g=0\right)$, the green dotted one represents the results to leading order in $g$ [Eqs. (42), (52), and (59)], and the full dark-blue and black curves are the results of the self-consistent approach (S.C.) of Sec. 4b for ${\overline{n}}_{\mathrm{th}}^{\mathrm{M}}=100$ and 0, respectively. (b) Signatures of the effective two-phonon absorption in the cavity spectrum in the same circumstances than (a). It shows the striking temperature-enhanced effect of nonlinear interaction in the $+$ polariton population. The parameters used are $\gamma /\kappa ={10}^{-4}$ and ${\omega }_M/\kappa =50$.