Quantum and classical phases in optomechanics

The control of quantum systems requires the ability to change and read-out the phase of a system. The noncommutativity of canonical conjugate operators can induce phases on quantum systems, which can be employed for implementing phase gates and for precision measurements. Here we study the phase acquired by a radiation field after its radiation pressure interaction with a mechanical oscillator, and compare the classical and quantum contributions. The classical description can reproduce the nonlinearity induced by the mechanical oscillator and the loss of correlations between mechanics and optical field at certain interaction times. Such features alone are therefore insufficient for probing the quantum nature of the interaction. Our results thus isolate genuine quantum contributions of the optomechanical interaction that could be probed in current experiments.

Figure 1

(a) Optomechanical cavity with a harmonically oscillating mirror at one end. (b) Scheme of a four displacement operation in the phase space of the mechanical oscillator.

Figure 2

(a) Michelson interferometer: A coherent field ${|\alpha \rangle }_f$ is split by a beam splitter (BS) into the two arms of the interferometer. Arm 1 ends with an optomechanical cavity with a movable oscillator, while arm 2 is composed by a phase shifter and a stationary cavity. (b) Quantum visibilities ${\nu }_q\left(t\right)$ in Eq. (7) for $T={10}^{-5}$ K (blue dotted line), $T={10}^{-2}$ K (red continuous line), and $T=1$ K (green dashed line); optomechanical coupling $k={10}^{-2}$, number of photons $N_p={10}^5$, and period $\tau ={10}^{-5}$ s. For relatively high temperature the visibility is strongly suppressed within every single oscillating period. Instead, in the low temperature limit visibility is slightly lowered and the main effect is due to the Kerr nonlinearity experienced by the field. (c) Quantum (red continuous line) and classical (blue dotted line) visibilities in Eqs. (7) and (12) for temperature $T=5\times {10}^{-2}$ K. Other parameters are as in (b).

Figure 3

Phase space description of the dynamics of the light pulse–mechanical oscillator interaction in the classical picture. (a) Four-pulse interaction model: the oscillator is assumed at rest at the origin of the phase space. The oscillator gains a momentum $\mathcal{I}$ due to the interaction at $t=t_0$. Then it freely evolves to the maximum amplitude $x\left(t_1\right)$ when the second pulse happens, this causes another momentum gain of the oscillator at time $t_1$. At this time the oscillator evolves to $x\left(t_2\right)=x\left(t_1\right)$ where the third pulse interaction brings its momentum to zero. Now, it evolves to $x\left(t_3\right)=0$ where its momentum becomes $-\mathcal{I}$. Finally, the oscillator is brought back to the origin of the phase space by the last pulse-oscillator interaction. (b) A similar dynamics is plotted for the six pulse interaction.