This thesis explores the field of optomechanics and the theory of multi-parameter estimation for quantum metrology applications. An optomechanical state inside a microresonator-taper system is studied experimentally, where the optical and mechanical modes interact by means of Brillouin anti-Stokes scattering. The scattering leads to sideband cooling of the mechanical (acoustic) mode and the underlying optomechanical coupling takes the form of a beam-splitter interaction Hamiltonian. The mechanical mode is further cooled by continuously monitoring the optical anti-Stokes mode by a heterodyne detector. The resulting time-domain data is input offline to a stochastic master equation model for Gaussian states to optimally estimate the mechanical phase space trajectories and subsequently reduce the observer’s uncertainty associated with the phase space coordinates. It is shown that more information about the mechanical mode can be obtained by heterodyne monitoring than by the commonly used in optomechanics homodyne detection for the beam-splitter type interaction Hamiltonians. The second part of this thesis theoretically considers the problem of simultaneous estimation of phase and phase diffusion using fixed-particle number states in the framework of quantum Cramér-Rao bound. Quantum Fisher information matrices are derived analytically in the limits of large and small diffusive noise to quantify the maximum amount of information available in the individual estimation of the parameters. The former is for a general fixed-particle number state and the latter for Holland-Burnett states for which quantum-enhanced estimation of phase as well as diffusion can be obtained. The precision limits given by quantum Fisher information may not be achievable in the joint estimation of the parameters due to the possible non-commutativity of the optimal measurements and this can be quantified in terms of a trade-off relation. This trade-off relation can reach the maximum of 2 in the large diffusion regime, whereas in the small diffusion regime a numerical evidence is shown that the optimal trade-off relation approaches 1 for Holland-Burnett states. These numerical results are valid in the small particle number regime. Finally, numerical results showing behaviour of the trade-off for a general value of phase diffusion when using Holland-Burnett states are provided.
This thesis explores the field of optomechanics and the theory of multi-parameter estimation for quantum metrology applications. An optomechanical state inside a microresonator-taper system is studied experimentally, where the optical and mechanical modes interact by means of Brillouin anti-Stokes scattering. The scattering leads to sideband cooling of the mechanical (acoustic) mode and the underlying optomechanical coupling takes the form of a beam-splitter interaction Hamiltonian. The mechanical mode is further cooled by continuously monitoring the optical anti-Stokes mode by a heterodyne detector. The resulting time-domain data is input offline to a stochastic master equation model for Gaussian states to optimally estimate the mechanical phase space trajectories and subsequently reduce the observer’s uncertainty associated with the phase space coordinates. It is shown that more information about the mechanical mode can be obtained by heterodyne monitoring than by the commonly used in optomechanics homodyne detection for the beam-splitter type interaction Hamiltonians. The second part of this thesis theoretically considers the problem of simultaneous estimation of phase and phase diffusion using fixed-particle number states in the framework of quantum Cramér-Rao bound. Quantum Fisher information matrices are derived analytically in the limits of large and small diffusive noise to quantify the maximum amount of information available in the individual estimation of the parameters. The former is for a general fixed-particle number state and the latter for Holland-Burnett states for which quantum-enhanced estimation of phase as well as diffusion can be obtained. The precision limits given by quantum Fisher information may not be achievable in the joint estimation of the parameters due to the possible non-commutativity of the optimal measurements and this can be quantified in terms of a trade-off relation. This trade-off relation can reach the maximum of 2 in the large diffusion regime, whereas in the small diffusion regime a numerical evidence is shown that the optimal trade-off relation approaches 1 for Holland-Burnett states. These numerical results are valid in the small particle number regime. Finally, numerical results showing behaviour of the trade-off for a general value of phase diffusion when using Holland-Burnett states are provided.
Multiparameter quantum estimation has emerged as a central task in quantum metrology. In this work, we investigate multiparameter quantum estimation in a hybrid exciton--optomechanical (EOM) system. The system consists of a semiconductor quantum well embedded inside a driven optomechanical microcavity, where the excitonic, optical, and mechanical modes interact coherently through exciton--photon and radiation-pressure couplings. Using the Gaussian-state formalism, we derive the covariance matrix of the steady-state quantum fluctuations and employ both the symmetric logarithmic derivative (SLD) and right logarithmic derivative (RLD) approaches to evaluate the quantum Fisher information matrix associated with the simultaneous estimation of the exciton--photon coupling strength g and the excitonic decay rate kx. We analyze the corresponding quantum Cramér--Rao bounds and determine the most informative precision limit governing the attainable estimation accuracy. The influence of several experimentally relevant parameters, including temperature, driving power, optomechanical coupling strength, and dissipation rates, is investigated in detail. Our results show that strong hybrid interactions and low-temperature regimes significantly enhance the estimation precision, whereas thermal fluctuations and dissipation processes deteriorate the metrological performance. Furthermore, we compare the ultimate quantum limits with experimentally feasible Gaussian measurement strategies based on homodyne and heterodyne detection. We show that heterodyne detection provides better estimation performance than homodyne schemes and can approach the optimal quantum precision limit in suitable parameter regimes.
Accurate simultaneous estimation of temperature and effective detuning is crucial for understanding, controlling, and optimizing optomechanical systems. Temperature characterizes the thermal state and mechanical properties, while detuning governs the light-motion interaction, influencing stability and enabling diverse quantum applications. Precise estimates of these parameters are therefore essential to exploit the full potential of such systems in high-precision metrology and the exploration of macroscopic quantum phenomena. In this work we investigate the simultaneous estimation of the effective detuning Δ and the mirror temperature T in a Fabry-Pérot cavity system, with a particular focus on the performance of heterodyne detection. Using analytical expressions for the quantum Fisher information matrices (QFIM) derived from the symmetric logarithmic derivative (SLD) and the right logarithmic derivative (RLD), we demonstrate that the quantum Cramér-Rao bound associated with the RLD is consistently lower than that based on the SLD, highlighting superior estimation precision. This gap between the SLD- and RLD-based bounds becomes especially pronounced in certain physical regimes, emphasizing the relevance of choosing the appropriate formalism in noncommutative estimation scenarios. From a physical perspective, the improved estimation performance stems from the fact that temperature directly influences thermal fluctuations in the mechanical mode, while the effective detuning modulates the strength of the optomechanical interaction, shaping the information exchange between optical and mechanical subsystems. We showed that using higher optical power, greater cavity decay rate, increased photon and phonon numbers, and reduced coupling strength improves the precision of both parameter estimates. These physical mechanisms affect the available quantum resources in the system—such as squeezing, entanglement, and state purity—which in turn determine the ultimate sensitivity achievable. Furthermore, our analysis shows that heterodyne detection can, under certain conditions, approach the ultimate precision set by the QFIM. Thus, a measurement strategy based on heterodyne detection could enable the efficient estimation of temperature, achieving performance close to that of ideal quantum-limited strategies.
This thesis focuses on the mathematical description and application of nonlinear cavity optomechanical systems. The first part is concerned with solving the dynamics of the standard nonlinear optomechanical Hamiltonian with an additional time-dependent mechanical displacement and single-mode squeezing term. The solution is based on identifying a Lie algebra that generates the time-evolution of the system, which reduces the problem to considering a finite set of coupled ordinary differential equations of real functions. The second part applies the solutions of the extended optomechanical Hamiltonian to the study of non-Gaussianity. We compute the non-Gaussian character of an optomechanical state as a function of the parameters in the Hamiltonian, and investigate the interplay between the non-Gaussianity, the strength of the nonlinear coupling and the strength of the single-mode mechanical squeezing term. We find that the strength and form of the nonlinear coupling strongly impacts the non-Gaussianity, and that its relationship with the squeezing term is highly complex. The third part concerns the use of nonlinear optomechanical systems as quantum sensors. We derive a general expression of the quantum Fisher information given the extended optomechanical Hamiltonian and demonstrate its applicability through three concrete examples: estimating the strength of a nonlinear light--matter coupling, the strength of a time-modulated mechanical displacement, and the strength of a single-mode mechanical squeezing parameter, all of which are modulated at resonance. In the last Chapter of the thesis, we consider the estimation of a constant gravitational acceleration with an optomechanical system. Our results suggest that optomechanical systems could, in principle, be used as powerful quantum sensors.
Multiparameter quantum estimation in open optomechanical systems is fundamentally constrained by dissipation, thermal fluctuations, and measurement incompatibility. In this work, we investigate a coupled cavity optomechanical platform in which two mechanical modes interact with driven optical cavities, two mode squeezed vacuum, intracavity degenerate parametric amplification, and coherent optical feedback. Using the continuous variable Gaussian state formalism, we derive the linearized quantum Langevin dynamics and steady state covariance matrix and evaluate the quantum Fisher information matrices associated with simultaneous estimation of the optomechanical coupling strength and cavity dissipation rate. We characterize the precision bounds using the symmetric and right logarithmic derivative formalisms and employ BMI=max{BS,BR} as a comparative figure of merit within the SLD/RLD framework. We find that parametric amplification can substantially reduce the most informative bound, BMI, thereby enhancing multiparameter estimation precision over a broad range of operating conditions. Coherent feedback can further reduce BMI and improve the estimation precision, particularly in the strong feedback regime. Moreover, the simultaneous presence of coherent feedback, parametric amplification, and squeezed light injection provides a combined mechanism for controlling quantum fluctuations and parameter dependent correlations, leading to enhanced and more robust multiparameter estimation. We further analyze the dynamical and steady state regimes to identify the parameter regions in which these quantum resources provide the largest metrological gain. Our results establish coupled cavity optomechanics as a flexible platform for quantum-noise engineering in multiparameter sensing.
Optical and optomechanical systems are exploited for quantum technologies to perform highly precise measurements for fundamental physics as well as biological and engineering tasks. Theoretical studies using tools from statistics and quantum information can greatly aid studies of the sensing capabilities of quantum systems and experimental designs. These allow us to quantify the amount of information about a parameter encoded in a quantum state itself and examine how that can be extracted through measurements. In this thesis we show how mechanical squeezing and measurements beyond position can be utilised to improve the precision of wavepacket expansion measurements which can test collapse models of quantum mechanics. Particularly, that squeezing can compensate for the free-fall time which is often the most significant limiting factor in such experiments, and measurements of other quadratures can offer increased precision. We demonstrate that the use of additional optical fields to measure the displacement of a free mass in a radiation-pressure limited interferometer cannot surpass the ultimate precision of the single-mode interferometer. This work applies to the likes of laser-interferometric gravitational wave detectors. Finally, we venture into multi-parameter estimation: we calculate bounds for multi-mode Gaussian states used to estimate many phase shifts and optimise over the input states. In contrast to the—far less experimentally feasible—multi-mode generalisations of N00N states we see no significant improvement when using multi-mode Gaussian states for the task. Given their comparable performance in single-phase estimation this is a new limitation of Gaussian states appearing only at the multi-parameter level.
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.