Antibunching in an optomechanical oscillator

We theoretically analyze antibunching of the phonon field in an optomechanical oscillator employing the membrane-in-the-middle geometry. More specifically, a single-mode mechanical oscillator is quadratically coupled to a single-mode cavity field in the regime in which the cavity dissipation is a dominant source of damping, and adiabatic elimination of the cavity field leads to an effective cubic nonlinearity for the mechanics. We show analytically in the weak-coupling regime that the mechanics displays a chaotic phonon field for small optomechanical cooperativity, whereas an antibunched single-phonon field appears for large optomechanical cooperativity. This opens the door to control of the second-order correlation function of a mechanical oscillator in the weak-coupling regime.

Figure 1

Steady-state mean phonon number of the mechanics as a function of the multiphoton cooperativity $C$ for different thermal occupation numbers ${\overline{n}}_{\mathrm{th}};{\overline{n}}_{\mathrm{th}}={10}^6$ (dotted red line), ${\overline{n}}_{\mathrm{th}}={10}^5$ (dot-dashed orange line), ${\overline{n}}_{\mathrm{th}}={10}^4$ (dashed green line), and ${\overline{n}}_{\mathrm{th}}={10}^3$ (solid blue line).

Figure 2

Steady-state second-order correlation function $g^{\left(2\right)}\left(0\right)$ in the high-temperature regime as a function of the multiphoton cooperativity $C$ for different thermal occupation numbers ${\overline{n}}_{\mathrm{th}};{\overline{n}}_{\mathrm{th}}={10}^6$ (dotted red line), ${\overline{n}}_{\mathrm{th}}={10}^5$ (dot-dashed orange line), ${\overline{n}}_{\mathrm{th}}={10}^4$ (dashed green line), and ${\overline{n}}_{\mathrm{th}}={10}^3$ (solid blue line). One-phonon absorption and emission processes are dominant so that the second-order correlation function is larger than unity.

Figure 3

Steady-state phonon number distribution $P\left(n\right)$ of the mechanical oscillator (green circles) for ${\overline{n}}_{\mathrm{th}}={10}^4$ and $C={10}^2$. For comparison, the phonon number distributions of the mechanics in a thermal state (red triangles) and a coherent state (blue squares) with the same mean phonon number are shown.

Figure 4

Steady-state mean phonon number $n_{\mathrm{ss}}$ of the mechanical oscillator as a function of the multiphoton cooperativity $C$ in the low-temperature regime: ${\overline{n}}_{\mathrm{th}}=1$ (solid blue line), ${\overline{n}}_{\mathrm{th}}=10$ (dashed green line), ${\overline{n}}_{\mathrm{th}}=20$ (dot-dashed orange line), and ${\overline{n}}_{\mathrm{th}}=40$ (dotted red line).

Figure 5

Steady-state second-order correlation function $g^{\left(2\right)}\left(0\right)$ of the mechanical oscillator as a function of both the multiphoton cooperativity and the thermal occupation number. The various lines show contours of constant $g^{\left(2\right)}\left(0\right)$: $g^{\left(2\right)}\left(0\right)=1.6$ (dotted line), $g^{\left(2\right)}\left(0\right)=1.3$ (dot-dashed line), $g^{\left(2\right)}\left(0\right)=1.0$ (thick solid line), $g^{\left(2\right)}\left(0\right)=0.7$ (thick dot-dashed line), and $g^{\left(2\right)}\left(0\right)=0.4$ (thick dashed line).

Figure 6

Steady-state phonon distributions $P\left(n\right)$ of the mechanical oscillator with different multiphoton cooperativities at the same temperature, ${\overline{n}}_{\mathrm{th}}=20$: $C=1$ (red circles), $C=41$ (green squares), and $C=1000$ (blue rhombi).