Nondegenerate multimode optomechanics

We theoretically investigate interactions between nondegenerate mechanical oscillators mediated by a time-dependent cavity field. We obtain a reduced master equation valid for all optomechanical systems operating in the weak-coupling regime. This master equation includes all forms of decoherence and backaction due to the dissipation of the field mediating the interaction. We apply the master equation to study two resonant coupling schemes within a rotating-wave approximation: the beam-splitter Hamiltonian and the two-mode parametric amplifier. In both cases, the effective unitary interaction can be made arbitrarily strong compared to the decoherence due to dissipation of the mediating field by choosing appropriate detunings.

Figure 1

(a) Sketch of a possible experimental realization involving separate optomechanical membranes. The mechanical modes could also be on the same resonator. (b) A scattering process contributing to the coherent exchange of phonons in the resonant beam-splitter interaction. The three other contributing processes result from time reversal and pump photons with frequency ${\omega }_{L1}$ scattering with oscillator 2. (c) A scattering process contributing to the parametric amplification of the mechanical modes. The three other processes are obtained in the same way as in (b).

Figure 2

Sketch of the intracavity photon power spectral density neglecting the cavity profile for clarity. (a),(b) The contributions of the two pumps in red (solid lines) and blue (dashed lines), respectively. The difference between the pump frequencies is arranged in the beam-splitter configuration specified by Eq. (27) with the pumps indicated by arrows representing $\delta$ peaks at ${\omega }_{L1}$ and ${\omega }_{L2}$. Each pump creates a pair of sidebands for each mechanical oscillator, as indicated in the sketch. The overlapping sidebands are emphasized by the dashed lines and formally captured by Eqs. (33). The exact behavior of this quantity including the cavity profile can be obtained from Eq. (18).

Figure 3

Normalized coupling strength as a function of central detuning and sideband resolution. The line gives parameters resulting in zero coupling, $J_{\mathrm{BS}}=0$. Other parameters are $G_1=G_2=0.01\overline{\mathrm{\Omega }}$.

Figure 4

Scaled total decoherence rate $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\max }$ as a function of detuning and optical dissipation rate. Other parameters are the same as in Fig. 3.

Figure 5

Sketch of the photon power spectral density neglecting the cavity profile. (a),(b) The power spectral density due to the two pumps in red (solid lines) and blue (dashed lines), respectively. The two pumps, indicated as arrows at ${\omega }_{L1}$ and ${\omega }_{L2}$, are arranged in the parametric amplifier configuration specified by Eq. (28). We emphasized the overlapping sidebands captured by Eqs. (44) with dashed lines.

Figure 6

Normalized parametric driving strength as a function of central detuning and sideband resolution. The line gives parameters with $J_{\mathrm{PA}}=0$. Other parameters are $G_1=G_2=0.01\overline{\mathrm{\Omega }}$. Note the formal equivalence up to a different normalization compared to Fig. 3.

Figure 7

Phase-space probability distributions (Wigner functions) for (a) oscillator 1, (b) oscillator 2, (c) collective mode under evolution given by Eq. (54). The dashed black line is the contour of half maximum, while the thick white one marks the same contour for a vacuum state.

Figure 8

Effective coherences ${\xi }_{\mathrm{PA}}$ [as defined in Eq. (55)] as a function of central detuning $\overline{\mathrm{\Delta }}$ and cavity decay rate $\kappa$. (a),(c) Widely separated oscillators, $\delta \omega /\overline{\mathrm{\Omega }}=1.9$; (b),(d) $\delta \omega /\overline{\mathrm{\Omega }}=0.1$. The red line denotes the value ${\xi }_{\mathrm{PA}}=1$. Other parameters are $G_1=G_2=0.05\overline{\mathrm{\Omega }}$.

Figure 9

Corrections of the resonant pump detunings from their bare value. Top row: beam-splitter setting $\frac{\delta {\mathrm{\Omega }}_1-\delta {\mathrm{\Omega }}_2}{\delta \omega }$; bottom row: parametric amplification $\frac{\delta {\mathrm{\Omega }}_1+\delta {\mathrm{\Omega }}_2}{2\overline{\mathrm{\Omega }}}$. All plots have $g_j{\alpha }_k=0.01\overline{\mathrm{\Omega }}$ and the optical dissipation rate is (a),(d) $\kappa =10\overline{\mathrm{\Omega }}$, (b),(e) $\kappa =\overline{\mathrm{\Omega }}$, and (c),(f) $\kappa =0.1\overline{\mathrm{\Omega }}$.