Enhancing quantum effects via periodic modulations in optomechanical systems

Parametrically modulated optomechanical systems have been recently proposed as a simple and efficient setting for the quantum control of a micromechanical oscillator: relevant possibilities include the generation of squeezing in the oscillator position (or momentum) and the enhancement of entanglement between mechanical and radiation modes. In this paper we further investigate this modulation regime, considering an optomechanical system with one or more parameters being modulated over time. We first apply a sinusoidal modulation of the mechanical frequency and characterize the optimal regime in which the visibility of purely quantum effects is maximal. We then introduce a second modulation on the input laser intensity and analyze the interplay between the two. We find that an interference pattern shows up, so that different choices of the relative phase between the two modulations can either enhance or cancel the desired quantum effects, opening new possibilities for optimal quantum control strategies.

Figure 1

Schematic description of system. A Fabry-Perot cavity is driven by an external laser and the radiation interacts with the movable mirror on the right, exchanging momentum.

Figure 2

Evolution of mirror position $Q\left(t\right)$ and momentum $P\left(t\right)$ mean values obtained by numerically integrating Eq. (8) from $t=0$ to $t=50\tau$, with $\tau =2\pi /\Omega$ being the period of the modulation (thin blue line). The plot was obtained by setting the system parameters as detailed in Sec. 3. In particular, here the modulation frequency $\Omega$ is twice the natural frequency ${\omega }_M$ of the mechanical oscillator which, in turn, is resonant with the detuning ${\Delta }_0$ that governs the free evolution of the optical field $A\left(t\right)$. The analytic solution for the asymptotic orbit (see Appendix 3) is also shown for comparison (thick red line).

Figure 3

Asymptotic quantum features as a function of $\Omega /{\omega }_M$ ($x$ axis) and $\epsilon$ ($y$ axis). (a) Maximum number of phonons in the mirror [Eq. (16)]. (b) Maximum of logarithmic negativity (21). (c) Maximum of quantum discord (20). (d) Minimum of the generalized quadratures of the mirror (18). In all the plots the system parameters are fixed as in Sec. 3.

Figure 4

Response of system in the presence of two different modulations as a function of their relative phase $\phi$: maximum number of phonons in the mirror (a), maximum of the logarithmic negativity (b), maximum of the quantum discord (c), and minimum of the generalized quadratures of the mirror (d). In all plots the two straight lines show the variation of the function in the case when only the mechanical frequency (blue horizontal line) or the laser amplitude (red dashed horizontal line) is modulated. Parameters are as detailed in the text.

Figure 5

First-order (red dashed curve) and second-order (blue solid curve) approximation to the classical position-momentum ($Q$-$P$) orbit of the mirror. The asymptotic numerical orbit (black dotted curve) is also plotted from $t=49\tau$ to $t=50\tau$, with $\tau =2\pi /\Omega$ being the period of the modulation. It is clear that a first-order approximation is not enough and fails to describe the dynamics of the system. On the other hand, the second-order approximation catches all relevant aspects and reproduces the correct behavior, at least on a qualitative level. Quantitative convergence to the numeric solution is found including higher-order terms. Units as in Fig. 2.

Figure 6

In blue (upper solid curve) is the maximum number of phonons, from Eq. (A21), plotted against relative phase $\phi$. Contributions due to different expansion orders are explicitly included: in black (lower solid curve) is the number of phonons in the unmodulated case (from ${\overline{C}}^{\left(0\right)}$); in red (upper dashed curve) is the second-order time-independent contribution (from ${\overline{C}}^{\left(2\right)}$); in green (lower dashed curve) is the maximum over one period of the time-dependent contributions (from first order $C^{\left(1\right)}$ and second order $C^{\left(2\right)}$). The total (blue curve) is equal to the sum of the other three curves.