This article presents a full operator analytical method for studying the quadratic nonlinear interactions in quantum optomechanics. The method is based on the application of higher-order operators, using a six-dimensional basis of second order operators which constitute an exactly closed commutators. We consider both types of standard position-field and the recently predicted non-standard momentum-field quadratic interactions, which is significant when the ratio of mechanical frequency to optical frequency is not negligible. This unexplored regime of large mechanical frequency can be investigated in few platforms including the superconducting electromechanics and simulating quantum cavity electrodynamic circuits. It has been shown that the existence of non-standard quadratic interaction could be observable under appropriate conditions.
This article presents a full operator analytical method for studying the quadratic nonlinear interactions in quantum optomechanics. The method is based on the application of higher-order operators, using a six-dimensional basis of second order operators which constitute an exactly closed commutators. We consider both types of standard position-field and the recently predicted non-standard momentum-field quadratic interactions, which is significant when the ratio of mechanical frequency to optical frequency is not negligible. This unexplored regime of large mechanical frequency can be investigated in few platforms including the superconducting electromechanics and simulating quantum cavity electrodynamic circuits. It has been shown that the existence of non-standard quadratic interaction could be observable under appropriate conditions.
A method is described to solve the nonlinear Langevin equations arising from quadratic interactions in quantum mechanics. While the zeroth order linearization approximation to the operators is normally used, here, first and second order truncation perturbation schemes are proposed. These schemes employ higher-order system operators, and then approximate number operators with their corresponding mean boson numbers only where needed. Spectral densities of higher-order operators are derived, and an expression for the second-order correlation function at zero time-delay has been found, which reveals that the cavity photon occupation of an ideal laser at threshold reaches 6 − 2 , in good agreement with extensive numerical calculations. As further applications, analysis of the quantum anharmonic oscillator, calculation of Q-functions, analysis of quantum limited amplifiers, and nondemoliton measurements are provided.
Mechanical systems prepared in quantum non-Gaussian states constitute a new advanced class of phenomena breaking the laws of classical physics. Specifically, such mechanical states cannot be described as any mixture of the Gaussian states produced by operations described by Hamiltonians at most quadratic in position and momentum, such as phase rotations, squeezing operations and linear driving. Therefore, they form a class of resourceful states for quantum technological tasks such as metrology, sensing, simulation and computation. Quantum opto- and electromechanics are advanced platforms for quantum mechanical experiments with broad applications offering various methods for preparing such mechanical quantum non-Gaussian states. The suitability of these platforms as transducers additionally allows the integration of such mechanical states into a variety of other related platforms. Here, we summarise the current techniques for creating these states, emphasizing those that have had experimental success and looking to methods that show promise for future experiments. By collating these results, we expect to stimulate new ideas for non-Gaussian state preparation in these fields, resulting in the realisation of further experiments. Moreover, we provide concise and clear explanations of the milestones of research in the quantum non-Gaussianity of mechanical states and its implementation and verification in a laboratory setting.
Cavity-mediated collective-spin interactions are commonly described by a quadratic one-axis twisting Hamiltonian. However, the underlying atom-light interaction naturally generates nonlinearities to arbitrary order. Here, we derive a closed-form analytical expression for the complete hierarchy of cavity-mediated collective-spin interactions. We show that the nonlinear coefficients χk are governed by Chebyshev polynomials, with k the order of nonlinearity. This yields a universal scaling χk∝ηk with the single-atom cooperativity η and a description of their dependence on cavity detuning. The result provides a systematic framework for determining when higher-order nonlinearities become relevant and when the quadratic approximation breaks down. We identify experimentally relevant regimes in which higher-order terms substantially modify collective-spin dynamics, accelerating the generation of quantum correlations and quantum Fisher information, and demonstrate that finite-order expansions can accurately reproduce the full cavity-mediated evolution. Our results establish a general framework for understanding higher-order nonlinearities in cavity quantum electrodynamics and their role in collective entanglement and quantum-enhanced sensing.
In a nutshell, a cavity optomechanical system consists of a mechanical oscillator inside an optical cavity. Cavity optomechanics studies the interplay between the light-matter radiation-pressure interaction and the change in the cavity frequency induced by mechanical motion. By leveraging the tools of quantum optics and quantum measurement, cavity quantum optomechanics enables the engineering of nonclassical quantum states of bulk matter. In this thesis, I present protocols for engineering such nonclassical quantum states in Part II. First, I introduce a scheme to ‘grow’ a quantum kitten state of a mechanical oscillator into a macroscopic Schrödinger cat state using a series of nonlinear pulsed optomechanical interactions. Next, I detail our proposed schemes to generate and verify entanglement between two mechanical oscillators. Moving on from pulsed optomechanics, I describe the theory behind an experiment that achieved multi-phonon subtraction from a mechanical oscillator in a continuously driven Brillouin optomechanical system. In the final chapter of Part II, I provide a theoretical preview of an experiment that investigates the behaviour of the mechanical oscillator when no light is detected at the output. In Part III, I introduce a new framework for nonlinear cavity quantum optomechanics that incorporates the intrinsic nonlinearity of the radiation-pressure interaction, the nonlinearity from the cavity enhancement, and higher-order mechanical position terms in the expansion of the cavity frequency. I then propose a scheme for measuring mechanical position using general-dyne detection, applicable to pulsed and continuous measurements. This predicts a new regime of operation, the “stochastic Gaussian” regime. Finally, I discuss a protocol for deterministic mechanical squeezing that utilizes the cavity response.
Quantum control of phonons has become a focus of attention for developing quantum technologies. Here, we present a proposal to realize phonon blockade in a quadratically coupled optomechanical system, where a strong nonlinear interaction between photons and phonons can be induced by an external field coherently driving the cavity and the effective coupling strength is tunable by adjusting the amplitude of the driving field. This optically induced nonlinearity is different from standard methods for the realization of phonon blockade, where the nonlinearity is achieved by coupling the mechanical system to superconducting qubits. We both analytically and numerically study the phonon statistical properties via the steady-state solution of the second-order correlation function and find phonon blockade can be efficiently realized for a large cooperativity of the system, which is achievable based on the optically enhanced nonlinear coupling and high quality mechanical system.
The success of quantum physics in description of various physical interaction phenomena relies primarily on the accuracy of analytical methods used. In quantum mechanics, many of such interactions such as those found in quantum optomechanics and quantum computing have a highly nonlinear nature, which makes their analysis extraordinarily difficult using classical schemes. Typically, modern quantum systems of interest nowadays come with four basic properties: (i) quantumness, (ii) openness, (iii) randomness, and (iv) nonlinearity. The newly introduced method of higher-order operators targets analytical solutions to such systems, and while providing at least mathematically approximate expressions with improved accuracy over the fully linearized schemes, some cases admit exact solutions. Many different applications of this method in quantum and classically nonlinear systems are demonstrated throughout. This review is purposed to provide the reader with ease of access to this recent and well-established operator algebra, while going over a moderate amount of literature review. The reader with basic knowledge of quantum mechanics and quantum noise theory should be able to start using this scheme to his or her own problem of interest.