Optomechanical systems with linear and quadratic position couplings: Dynamics and optimal estimation

TL;DR

This paper addresses parameter estimation of the quadratic optomechanical coupling in a single-mode cavity coupled to a mechanical oscillator with both linear and quadratic interactions. It derives an exact analytical solution using two-phonon coherent states under adiabatic, single-mode assumptions and analyzes the optical-field state in terms of quantum and classical Fisher information. The authors show that, for a two-level optical subspace, balanced homodyne detection can saturate the QFI by tuning the local-oscillator phase, and they explore both independent and interdependent coupling regimes, including a phenomenological model linking $g_1$ and $g_2$ with realistic experimental parameters. The results highlight how FI peaks depend on time, temperature, and measurement strategy, informing practical high-precision estimation of quadratic optomechanical couplings in experimental platforms.

Abstract

We study the dynamics of an optomechanical system consisting of a single-mode optical field coupled to a mechanical oscillator, where the nonlinear interaction includes both linear and quadratic terms in the oscillator's position. We present an analytical solution to this quantum-mechanical Hamiltonian problem by employing the formalism of two-phonon coherent states. Quantum estimation theory is applied to the resulting state of the optical field, with a focus on evaluating the quantum Fisher information with respect to the strength of the quadratic coupling. Our estimation scheme employs balanced homodyne photodetection and demonstrates that the corresponding classical Fisher information can reach the quantum Fisher information limit, with the phase of the local coherent oscillator playing a crucial role.

Figures (10)

• Figure 1: Wigner phase-space distribution of the optical field initially prepared in an coherent state, with $\alpha = 10$ and system parameters defined by Eq. \ref{['eq:unit_free']}. In the figures, $g_1$ takes the values from $0$, $1$ and $10$ left to right, while $g_2$ top to bottom. The plots are snapshots of the distribution taken at time equals $T_\omega$, defined in Eq. \ref{['eq:chtime']}. With the border outlined in red, the top left figure is both a snapshot of the system at $T_{\omega}$ for $g_1= g_2 = 0$ but also the initial state for all the other figures at $t=0$.
• Figure 2: Trajectories in the X–Y plane of the Bloch sphere showing the evolution of the quantum states given in Eqs. \ref{['eq:pure_state']} and \ref{['eq:mixed_state']}, parameterized by $s = 0.1$ (red) and $s = 0.2$ (blue). The system evolves from $t = 0$ to $t = 2T_\omega$, where $T_\omega$ is defined in Eq. \ref{['eq:chtime']}. The first cycle is shown with solid lines, and the second with dotted lines. The evolution depends solely on the parameter $s$ and is independent of the state’s purity. All other parameters are defined in Eq. \ref{['eq:unit_free']}.
• Figure 3: Time evolution of $r_x$, the $x$-component of the system on the Bloch sphere, plotted for the initial pure state $|\psi_s\rangle$ with $s = 0.2$. Time is expressed in units of the fast oscillation period, two full cycles of which are shown in Fig. \ref{['fig:bloch_sphere']}. The fast oscillation is modulated by a slower beating pattern at frequency $\Delta = \Omega_1 - \Omega_0$. In all subplots, $g_1 = 1.0$ and $g_2 = 0.01$, while the remaining parameters are defined in Eq. \ref{['eq:unit_free']}.
• Figure 4: Quantum Fisher information as a function of time for the mixed state with $s = 0.2$, computed analytically. Periodic peaks appear at regular intervals of $2\pi/\Delta$, with $\Delta = \Omega_1 - \Omega_0$. The parameters are set as $g_1 = 1$, $g_2 = 0.01$, with all other relevant parameters set to unity, as in Eq. \ref{['eq:unit_free']}.
• Figure 5: Comparison of quantum Fisher information (blue) and classical Fisher information (red) as a function of time for an initially mixed two-level optical field state given by Eq. \ref{['eq:mixed_state']} with $s = 0.1$ and for $\Delta=\Omega_1-\Omega_0$. The time interval spans from $0$ to $4T_\Omega$, where $T_\Omega$ is defined in Eq. \ref{['eq:chtime']}. All other system parameters are set to unity, as in Eq. \ref{['eq:unit_free']}.