Quantum sensing is a field of study concerned with understanding the fundamental limits of sensitivity imposed by quantum mechanics as well as using quantum effects to improve sensitivity beyond classical limits. Optomechanical systems; that is, systems where light or another probe field interacts with a mechanical degree of freedom, are promising platforms for quantum sensing. Currently, almost all optomechanical experiments operate in the so-called linearised regime, using a strong drive of the probe field to achieve sufficient radiation pressure. However, it has been suggested that the nonlinear regime could improve sensing, in particular by offering a better scaling of the sensitivity with the occupation number of the probe field, N_a. One of the main bottlenecks in developing better optomechanics platforms is to engineer a sufficiently large light-matter coupling g. Measuring g is therefore key when characterising new optomechanical setups. In this thesis, we investigate whether g can be measured more precisely in the nonlinear or the linearised regime. To this end, we solved the dynamics of both the nonlinear optomechanical Hamiltonian and of a linearised Hamiltonian that has been further simplified through the rotating-wave approximation (beam-splitter Hamiltonian). We then computed the Quantum Fisher Information (QFI) for both Hamiltonians as well as for two different classes of initial states - (non-Gaussian) Fock-state superpositions (FSSPs) and (Gaussian) coherent states. We found that in general, the scaling of the QFI with N_a is better for the nonlinear Hamiltonian than for the linearised Hamiltonian. As for the initial states, the scaling with N_a is better for FSSP states than for coherent states. However, when comparing the QFI between the two Hamiltonians, the outcome of the comparison also depends heavily on the power of the probe field drive and the strength of g, as well as on the detuning between the optical and the mechanical frequency. Thus, which regime - nonlinear or linearised - is more suitable for measuring the coupling depends on what parameter regimes are experimentally accessible. Thus, the advantage conferred by the nonlinear Hamiltonian - and to a lesser extend also that conferred by non-Gaussian initial states - might not be experimentally accessible at the moment. It is also worth noting that while this work could in principle inform experiments, the predictive power of this work is limited by the fact that open system dynamics where not taken into account; as well as by the fact that the drive of the probe field was neglected in the nonlinear regime.
Quantum sensing is a field of study concerned with understanding the fundamental limits of sensitivity imposed by quantum mechanics as well as using quantum effects to improve sensitivity beyond classical limits. Optomechanical systems; that is, systems where light or another probe field interacts with a mechanical degree of freedom, are promising platforms for quantum sensing. Currently, almost all optomechanical experiments operate in the so-called linearised regime, using a strong drive of the probe field to achieve sufficient radiation pressure. However, it has been suggested that the nonlinear regime could improve sensing, in particular by offering a better scaling of the sensitivity with the occupation number of the probe field, N_a. One of the main bottlenecks in developing better optomechanics platforms is to engineer a sufficiently large light-matter coupling g. Measuring g is therefore key when characterising new optomechanical setups. In this thesis, we investigate whether g can be measured more precisely in the nonlinear or the linearised regime. To this end, we solved the dynamics of both the nonlinear optomechanical Hamiltonian and of a linearised Hamiltonian that has been further simplified through the rotating-wave approximation (beam-splitter Hamiltonian). We then computed the Quantum Fisher Information (QFI) for both Hamiltonians as well as for two different classes of initial states - (non-Gaussian) Fock-state superpositions (FSSPs) and (Gaussian) coherent states. We found that in general, the scaling of the QFI with N_a is better for the nonlinear Hamiltonian than for the linearised Hamiltonian. As for the initial states, the scaling with N_a is better for FSSP states than for coherent states. However, when comparing the QFI between the two Hamiltonians, the outcome of the comparison also depends heavily on the power of the probe field drive and the strength of g, as well as on the detuning between the optical and the mechanical frequency. Thus, which regime - nonlinear or linearised - is more suitable for measuring the coupling depends on what parameter regimes are experimentally accessible. Thus, the advantage conferred by the nonlinear Hamiltonian - and to a lesser extend also that conferred by non-Gaussian initial states - might not be experimentally accessible at the moment. It is also worth noting that while this work could in principle inform experiments, the predictive power of this work is limited by the fact that open system dynamics where not taken into account; as well as by the fact that the drive of the probe field was neglected in the nonlinear regime.
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.
We study the fundamental sensitivity that can be achieved with an ideal optomechanical system in the nonlinear regime for measurements of time-dependent gravitational fields. The optomechanical system consists of a cavity mode and a mechanical oscillator coupled to it via (non-linear) radiation pressure coupling. Using recently developed methods to solve the dynamics of a nonlinear optomechanical system with a time-dependent Hamiltonian, we compute the quantum Fisher information for linear displacements of the mechanical element due to gravity. We demonstrate that the sensitivity can be further enhanced by injecting squeezed states of the cavity field or by modulating the light–matter coupling of the optomechanical system. We specifically apply our results to the measurement of gravitational fields from small oscillating masses, where we show that, in principle, the gravitational field of an oscillating nanogram mass can be detected based on experimental parameters that will likely be accessible in the near-term future. Finally, we identify the experimental parameter regime necessary for gravitational wave detection with a quantum optomechanical sensor.
We address the precise characterization of coupling strength of nonlinear media in continuous-variable (CV) quantum systems using coherent and squeezed vacuum states as Gaussian probes, together with their photon-added and photon-subtracted counterparts as non-Gaussian probes. We consider three main classes of nonlinear Hamiltonians, namely quadrature nonlinearities, generalized squeezing and Kerr-type interactions. By analytically evaluating the quantum Fisher information (QFI), we compare the performance of Gaussian and non-Gaussian probes and assess the optimal probe based on the probe parameters, energy resource and non-Gaussianity. Our results are twofold as follows: first, for coherent-state family, the improvement provided by photon addition at fixed coherent amplitude originates mainly from the extra energy carried by the probe and does not provide a genuine metrological resource, since the same precision can be achieved by a Gaussian coherent-state signal of a larger energy, which can be more easily produced. Second, in contrast, photon addition and subtraction become effective resources when applied to already nonclassical states such as squeezed vacuum states. In this case, they lead to a significant enhancement of the QFI, particularly for higher-order interactions. Although Gaussian squeezed states remain optimal at equal energy constraint, photon-added and photon-subtracted squeezed states achieve comparable sensitives with significantly lower squeezing requirements. Since large squeezing level remains experimentally challenging, these non-Gaussian probes offer a practical route towards enhanced estimation of the nonlinear coupling strength within currently accessible squeezing regimes.
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.
We study the fundamental sensitivity that can be achieved with an ideal optomechanical system in the nonlinear regime for measurements of time-dependent gravitational fields. Using recently developed methods to solve the dynamics of a nonlinear optomechanical system with a time-dependent Hamiltonian, we compute the quantum Fisher information for linear displacements of the mechanical element due to gravity. We demonstrate that the sensitivity can not only be further enhanced by injecting squeezed states of the cavity field, but also by modulating the light--matter coupling of the optomechanical system. We specifically apply our results to the measurement of gravitational fields from small oscillating masses, where we show that, in principle, the gravitational field of an oscillating nano-gram mass can be detected based on experimental parameters that will likely be accessible in the near-term future. Finally, we identify the experimental parameter regime necessary for gravitational wave detection with a quantum optomechanical sensor.
Quantum opto-mechanics is a fast moving, broad research field which aims to investigate and control the interaction of light and the quantized motion of mechanical objects and the coupling between them. This allows one to engineer the mechanical quantum state by controlling and measuring the electromagnetic field, and conversely, to control the dynamical evolution of the optical field via its interaction with a mechanical system. Mechanical resonators are accessible in many different shapes and sizes, from kilogram scales in experiments that test gravitational theories to micro- and nano-scales in resolved sideband regime, where the frequency of the mechanical resonator is larger than the bandwidth of the optical cavity. In this regime, it is possible to cool the mechanical object to the ground state which allows a high degree of control and the ability to prepare interesting non-classical states, states that could aid in investigating the quantum dynamics of macroscopic objects. The field of quantum opto-mechanics is a promising testbed for quantum control technologies, such as fundamental tests of quantum mechanics, ultra-sensitive quantum sensors for precise measurements, phonon lasing, mechanical memories with long lifetime and quantum information processing. Furthermore, experiments are progressing toward strong opto-mechanical coupling regimes with single photons. This creates a platform that can take advantage of the nonlinear optical interactions to generate highly non-classical states in such light-matter interfaces. Photonic systems working in the single photon regime will be promising for low power applications and also for implementations of quantum information and computation. Motivated by these applications, this thesis consists of theoretical studies in the context of single photonics. This includes investigating non-classical mechanical state generation and mechanically controlling the dynamical evolution of optical fields in the strong coupling regime single photon opto-mechanics and furthermore, proposing a quantum sensor using single photonics. In this regard, we employ three important formalisms to treat and measure open quantum systems with non-Gaussian input fields: the conditional and unconditional cascaded master equations, Fock-state master equations and input-output Langevin equations. In the context of opto-mechanics, the system to be investigated is composed of two optical cavities with a coupling modulated via a mechanical resonator. In one case, a sequence of single photons are irreversibly sent to one of the optical cavities. After photons go through the opto-mechanical interaction, photon counting measurements are performed on the cavity output, conditionally steering the mechanical state to a highly non-classical Fock state. Numerical calculations are performed to predict the experimental parameters needed to generate a mechanical Fock state before the mechanical object is thermalized. In another case, the same opto-mechanical scheme is employed to conversely modify the state of the optical mode by engineering the state of the mechanical degree of freedom. In this case, it is shown that the opto-mechanical system operates as an effective beam splitter for the input single photons, with a transmission that can be controlled by tuning the number of excitations in the bulk mechanical resonator. When the mechanical resonator has a small coherent amplitude it acts as a quantum control, entangling the optical and mechanical degrees of freedom. As the coherent amplitude of the resonator increases, it acts as a classical control for the input photons. Finally, we propose a novel multi-purpose sensor architecture that can be used for force, refractive index and possibly local temperature detection. In this scheme, two coupled cavities behave as an effective beam splitter. This sensor is based on fourth order interference (the Hong-Ou-Mandel effect) and requires a sequence of single photon pulses and consequently has low pulse power. Changes in the parameter to be measured induce variations in the effective beam splitter reflectivity and result in changes to the visibility of interference. We propose that this generic scheme could be implemented with coupled L3 photonic crystal cavities and find that this system, which only relies on photon coincidence detection and does not need any spectral resolution, can estimate forces as small as 10-7 Newtons and can measure one part per million change in refractive index using a very low input power of 10-10 W.