Optically mediated nonlinear quantum optomechanics

We consider theoretically the optomechanical interaction of several mechanical modes with a single quantized cavity-field mode for linear and quadratic coupling. We focus specifically on situations where the optical dissipation is the dominant source of damping, in which case the optical field can be adiabatically eliminated, resulting in effective multimode interactions between the mechanical modes. In the case of linear coupling, the coherent contribution to the interaction can be exploited (e.g., in quantum state swapping protocols), while the incoherent part leads to significant modifications of cold damping or amplification from the single-mode situation. Quadratic coupling can result in a wealth of possible effective interactions including the analogs of second-harmonic generation and four-wave mixing in nonlinear optics, with specific forms depending sensitively on the sign of the coupling. The cavity-mediated mechanical interaction of two modes is investigated in two limiting cases: the resolved sideband and the Doppler regime. As an illustrative application of the formal analysis we discuss in some detail a two-mode system where a Bose-Einstein condensate is optomechanically linearly coupled to the moving end mirror of a Fabry-Pérot cavity.

Figure 1

Nonabsorptive dielectric membranes inside a Fabry-Pérot resonator which interact with a single-mode cavity field.

Figure 2

Effective frequencies of the first (red, solid) and second (blue, dashed) mechanical modes in the resolved sideband regime ${\omega }_1,{\omega }_2\gg \kappa$ for $g_1>g_2$. The dotted lines denote the bare frequencies of the mechanical modes. The shifted frequencies are matched at the intersections. The inset gives a detailed view on the intersections occurring in the red-detuned side. The parameters used are ${\omega }_1=2\pi \times 20$ MHz, ${\omega }_2=2\pi \times 19.95$ MHz, $\kappa =2\pi \times 1$ MHz, $g_1=2\pi \times 0.3$ MHz, and $g_2=2\pi \times 0.12$ MHz.

Figure 3

Upper plot shows short-time (linear time scale) and lower plot shows long-time (log time scale) evolution of the motional quadratures of the mechanical modes in the single- and two-mode scenarios. The mechanical modes of frequencies ${\omega }_1$ (red, solid) and ${\omega }_2$ (blue, dashed) are both cooled down while interacting with each other. For comparison the uncoupled cold damping of the modes ${\omega }_1$ (orange, dotted) and ${\omega }_2$ (green, dot-dashed) are also shown. Two-mode coupling results in slowing down of the individual cold damping rates. Here ${\omega }_1=2\pi \times 20$ MHz, ${\omega }_2=2\pi \times 19.99$ MHz, $\kappa =2\pi \times 0.95$ MHz, $g_1=2\pi \times 50$ kHz, and $g_2=2\pi \times 10$ kHz.

Figure 4

Qualitative behavior of effective mechanical frequencies for the first (red, solid) and second (blue, dashed) oscillators in the Doppler regime. The shifted frequencies are matched at the intersections. Here we chose $g_1>g_2$. A set of parameters is ${\omega }_1=2\pi \times 100$ kHz, ${\omega }_2=2\pi \times 93$ kHz, $\kappa =2\pi \times 1$ MHz, $g_1=2\pi \times 90$ kHz, and $g_2=2\pi \times 50$ kHz.

Figure 5

Motional quadrature of moving mirror (red, solid) and BEC (blue, dashed) in the presence of vacuum noise for ${\omega }_1=2\pi \times 101$ kHz, ${\omega }_2=2\pi \times 100$ kHz, $\kappa =2\pi \times 1$ MHz, $g_1=2\pi \times 100$ kHz, and $g_2=2\pi \times 10$ kHz.

Figure 6

Motional quadrature of moving mirror (red, solid) and BEC (blue, dashed) for squeezed vacuum noise with $N=1$ and squeezing phase ${\theta }_s-{\theta }_c=\pi /2$. Same parameters as in Fig 5.

Figure 7

State transfer fidelity for squeezed vacuum noise with $N=1$ (red, solid), $N=10$ (blue, dashed), and vacuum noise (gray, dotted) as a function of the phase difference between squeezing input and cavity field. The initial states of the BEC and the moving mirror are the ground state and a thermal state with mean phonon number $\overline{n}=1$, respectively.

Figure 8

Same as Fig. 7, but for BEC initially in squeezed state with position quadrature variance of $0.025$ and momentum quadrature variance of 10 and moving mirror in a thermal state with mean phonon number $\overline{n}=1$, respectively.