Optimal estimation with quantum optomechanical systems in the nonlinear regime

We study the fundamental bounds on precision measurements of parameters contained in a time-dependent nonlinear optomechanical Hamiltonian, which includes the nonlinear light-matter coupling, a mechanical displacement term, and a single-mode mechanical squeezing term. By using a recently developed method to solve the dynamics of this system, we derive a general expression for the quantum Fisher information and demonstrate its applicability through three concrete examples: estimation of the strength of a nonlinear light-matter coupling, the strength of a time-modulated mechanical displacement, and a single-mode mechanical squeezing parameter, all of which are modulated at resonance. Our results can be used to compute the sensitivity of a nonlinear optomechanical system to a number of external and internal effects, such as forces acting on the system or modulations of the light-matter coupling.

Figure 1

Cavity optomechanics is one realization of the Hamiltonian (1). A semitransparent mirror allows the electromagnetic field to enter the cavity and interact with a moving-end mirror, which therefore affects the frequency of the fundamental modes that can be trapped in the cavity [33]. The degree of freedom of the mirror (i.e., its position) can be modeled as a harmonic oscillator coherently interacting with the field.

Figure 2

QFI for estimation of (a) ${\tilde{g}}_0$ and (b) ${\tilde{d}}_1$ as a function of dimensionless time $\tau$ for different values of ${\mathrm{\Omega }}_g$ (respectively, ${\mathrm{\Omega }}_{{\tilde{d}}_1}$). While overall the QFI tends to increase with time in both cases, modulations with the period of the harmonic oscillators are clearly visible. For ${\mathrm{\Omega }}_{{\tilde{d}}_1}\ge 1$, ${\mathcal{I}}_{{\tilde{d}}_1}$ is bounded from above, and a doubling of the period is observed. For a discussion of the relation of the QFI plotted here to the Heisenberg limit, see the Discussion section (Sec. 6).

Figure 3

QFI for estimation of (a) ${\tilde{g}}_0$ and (b) ${\tilde{d}}_1$ for different frequencies. Parameters are ${\tilde{g}}_0=1$, $\epsilon =0.5$, and ${\mu }_{\mathrm{c}}=1$ for (a), and ${\tilde{g}}_0=1$ and ${\mu }_{\mathrm{c}}=1$ for (b). We find that the constant case and the resonances perform best. While the resonance at about ${\mathrm{\Omega }}_g=1$ gives the best QFI for the estimation of ${\tilde{g}}_0$, the QFI for the estimation of ${\tilde{d}}_1$ at the resonance ${\mathrm{\Omega }}_{d_1}=1$ is smaller compared with when ${\tilde{\mathcal{D}}}_1\left(\tau \right)$ is constant.