nThis thesis comprises of investigations relating to the creation, manipulation, and detection of quantum states in specific configurations of optomechanical and circuit QED systems. Both the unconditional and conditional master equation formalisms are employed toward this end. In the optomechanical context, the lmembrane-in-the-middler system is investigated with regard to its various experimentally demonstrated parametric couplings between optical fields and membrane displacement. Via these couplings, the optical fields are functionalized as probes and controls of the quantum state of a mechanical membrane vibrational mode. In one case, continuous measurement of the optical field is shown to both induce and detect quantum jumps of mechanical energy; the conditions under which this occurs are established and verified in numerical simulations. In a separate case, a scheme is developed that employs simultaneous continuous measurement of two optical fields with distinct parametric couplings to the mem- brane. In this case, it is shown that in situ monitoring and control of the quantum noise profile of a mechanical mode becomes possible. Specifically, the variance of the mechanical energy dis- tribution is numerically shown to monotonically decay with increasing measurement strength of one of the optical fields. In the circuit QED setting, a many-body system is proposed compris- ing of microwave coplanar waveguide resonators arranged in a one-dimensional lattice. Each of the resonators contains a tuneable Kerr nonlinearity that provides an attractive interaction energy for the microwave quanta in the lattice, and each of the resonators is tuneably coupled to its nearest neighbours in the lattice. A scheme is developed under which the tuneability of the parameters is used to effect quantum phase transitions such that initially local microwave resonator quantum states become non-locally superposed across the whole system, thereby providing a feasible route toward deterministically entangling complex microwave resonator states across forty or more resonators. Numerical simulations of the scheme are performed to demonstrate its effectiveness under experimentally accessible parameters.
nThis thesis comprises of investigations relating to the creation, manipulation, and detection of quantum states in specific configurations of optomechanical and circuit QED systems. Both the unconditional and conditional master equation formalisms are employed toward this end. In the optomechanical context, the lmembrane-in-the-middler system is investigated with regard to its various experimentally demonstrated parametric couplings between optical fields and membrane displacement. Via these couplings, the optical fields are functionalized as probes and controls of the quantum state of a mechanical membrane vibrational mode. In one case, continuous measurement of the optical field is shown to both induce and detect quantum jumps of mechanical energy; the conditions under which this occurs are established and verified in numerical simulations. In a separate case, a scheme is developed that employs simultaneous continuous measurement of two optical fields with distinct parametric couplings to the mem- brane. In this case, it is shown that in situ monitoring and control of the quantum noise profile of a mechanical mode becomes possible. Specifically, the variance of the mechanical energy dis- tribution is numerically shown to monotonically decay with increasing measurement strength of one of the optical fields. In the circuit QED setting, a many-body system is proposed compris- ing of microwave coplanar waveguide resonators arranged in a one-dimensional lattice. Each of the resonators contains a tuneable Kerr nonlinearity that provides an attractive interaction energy for the microwave quanta in the lattice, and each of the resonators is tuneably coupled to its nearest neighbours in the lattice. A scheme is developed under which the tuneability of the parameters is used to effect quantum phase transitions such that initially local microwave resonator quantum states become non-locally superposed across the whole system, thereby providing a feasible route toward deterministically entangling complex microwave resonator states across forty or more resonators. Numerical simulations of the scheme are performed to demonstrate its effectiveness under experimentally accessible parameters.
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.
Harmonic oscillators might be one of the most fundamental entities described by physics. Yet they stay relevant in recent research. The topological properties associated with exceptional points that can occur when two modes interact have generated much interest in recent years. Harmonic oscillators are also at the heart of new quantum technological applications: the long lifetime of high-Q resonators make them advantageous as quantum memories, and they are employed for narrowband processing of quantum signals, as in Josephson parametric amplifiers. The goal of this thesis is to explore different fundamental regimes of two coupled harmonic oscillators using cavity optomechanics as the experimen- tal platform. With consistent progress in attaining ever increasing Q factors, mechanical and electromagnetic resonators realize near-ideal harmonic oscillators. By parametrically modulating the nonlinear optomechanical interaction between them, an effective linear coupling is achieved, which is tunable in strength and in the relative frequencies of the two modes. Thus cavity optomechanics provides a framework with excellent control over system parameters for the study of two coupled harmonic modes. The specific optomechanical implementation employed are superconducting circuits with the vibrating top plate of a capacitor as the mechanical element. Multimode optomechanical circuits are realized, with two microwave modes interacting with one or two mechanical oscillators. The supplementary modes serve either as intermediaries in the relation of the two modes of interest, or as auxiliaries used to control a parameter of the system. Three main experimental results are achieved. First, an auxiliary microwave mode allows the engineering of the effective dissipation rate of a mechanical oscillator. The latter then acts as a reservoir for the main microwave mode with which it interacts. The microwave mode susceptibility can be tuned, resulting in an instability akin to that of a maser and in resonant amplification of incoming microwave signals with an added noise close to the quantum minimum. Second, we study the conditions for a nonreciprocal interaction between two microwave modes, when the information flows in one direction but not in the other. The two modes interact through two mechanical oscillators, leading to frequency conversion between the two cavities. Dissipation in the mechanical modes is essential to the scheme in two ways: it provides a reciprocal phase necessary for the interference and eliminates the unwanted signals. Third, level attraction between a microwave and a mechanical mode is demonstrated, where the eigenfrequencies of the system are drawn closer as the result of interaction, rather distancing themselves as in the more usual case of level repulsion. The phenomenon is theoretically connected to exceptional points, and a general classification of the possible regimes of interaction between two harmonic modes is exposed, including level repulsion and attraction as special cases.
In the field of optomechanics, one typically studies the interaction of a single mechanical resonator and a single optical cavity mode. In this thesis, we investigate collective quantum effects in optomechanical systems with two or more optical and mechanical modes. First, we study the generation and destruction of entanglement between two mechanical resonators effected by optical cavity fields. We also describe how the optomechanical interaction can be employed to mediate strong coherent coupling and entanglement between a micromechanical membrane and a single atom. Secondly, we show that a configuration with two optical and a single mechanical mode can lead to enhanced nonlinear interactions, which can be exploited for quantum non-demolition measurements of the phonon and photon numbers. Finally, we explore the quantum many-body dynamics of optomechanical arrays. In particular, we research into the transition from a disordered state due to quantum noise to a state with phasecoherent mechanical oscillations. As an intermediate step towards this study, we also investigate the nonlinear classical dynamics of single- and multi-mode optomechanical setups and compare theoretical predictions to experimental data.
In this thesis, we study two different physical systems, namely superconducting circuits and optomechanical cavities. \nIn the first part of the thesis, we study superconducting qubits and resonators and their potential to implement quantum information processing tasks. We propose a circuit quantum electrodynamics realization of a protocol to generate a Greenberger-Horne-Zeilinger (GHZ) state for transmon qubits homogeneously coupled to a microwave cavity in the dispersive limit. We derive an effective Hamiltonian with pairwise qubit exchange interactions of the XY type that can be globally controlled. Starting from a separable initial state, these interactions allow to generate a multi-qubit GHZ state within a time that does not depend on the number of qubits. We discuss how to probe the non-local nature and the genuine multipartite entanglement of the generated state. Finally, we investigate the stability of the proposed scheme to inhomogeneities in the physical parameters and the weak anharmonicity of transmon qubits. \nIn the second part of the thesis, we study optomechanical systems in which the position of a mechanical resonator modulates the resonance frequency of an optical cavity. The resulting radiation-pressure interaction is intrinsically nonlinear and can be used to implement strong Kerr nonlinearities and an effective interaction between photons. We investigate the optical bistability of such a system. The steady-state mean-field equation of the optical mode is identical to the one for a Kerr medium, and thus we expect it to have the same characteristic behavior with a lower, a middle, and an upper branch. However, the presence of position fluctuations of the mechanical resonator leads to a new feature: the upper branch will become unstable at sufficiently strong driving in certain parameter regimes. We identify the appropriate parameter regime for the upper branch to be stable, and we confirm, by numerical investigation of the quantum steady state, that the mechanical mode indeed acts as a Kerr nonlinearity for the optical mode in the low-temperature limit. This equivalence of the optomechanical system and the Kerr medium will be important for future applications of cavity optomechanics in quantum nonlinear optics and quantum information science. \n
This thesis explores nonlinear couplings in superconducting circuits with the purpose of achieving quantum control over the elementary excitations of light and motion. It consists mainly of three research themes. The first theme concerns the experimental realisation of a tuneable coupling scheme, giving rise to different interactions with adjustable ratios, between two transmon qubits. The tuneable capabilities of the device together with its versatile geometry and high coherence make it an interesting building block for analog quantum simulators of certain classes of complex problems. The second theme concerns two theoretical proposals for controlling mechanical resonators using transmon qubits. In the first part we analyse an electromechanical circuit architecture giving rise to tuneable three-body interactions between two qubits and a mechanical beam. Several protocols are performed numerically demonstrating ground-state cooling and the creation of mechanical quantum states, such as single-phonon and multi-phonon superposition states as well as qubit-phonon entanglement. Additionally, schemes for generating arbitrary quantum states are explored. The second part relies on the same concept for coupling a mechanical resonator to a transmon qubit and explores the qubit-resonator system in the ultrastrong coupling regime, where the optomechanical coupling approaches or even exceeds the mechanical frequency. We find that for certain coupling strengths ground-state cooling is possible and devise a protocol for generating macroscopic quantum superposition states, known as “Shrödinger cats”, on the mechanical resonator. Finally, the third research theme presented in the thesis concerns an experiment where a cold superconducting qubit is employed to readout and control a thermally populated radio-frequency resonator coupled via a strong dispersive coupling. By means of reservoir engineering, ground-state cooling as well as stabilisation of Fock states in the resonator are demonstrated reaching a new operating regime for circuit quantum electrodynamics.
Detecting an itinerant microwave photon with high efficiency is an outstanding problem in quantum optics at microwave frequencies and in the fundamental quantum mechanics experimental toolbox. This subject has attracted a lot of attention and there are growing number of excellent experimental demonstrations and theoretical proposals towards this aim. However, due to the extremely low energy of microwave quanta, efficiently detecting a propagating microwave photon in a non-absorbing manner is still an unresolved challenge and it is the focus of the major part of this thesis. Motivated by a recent experiment that shows a superconducting artificial atom (a transmon) in a one-dimensional microwave transmission line exhibits gigantic cross-Kerr nonlinearities, 1000 times larger than in comparable optical systems, I investigate the possibility of detection single microwave photons using a transmon induced cross-Kerr-like interaction between two microwave fields. The photon number in one field is inferred through the measurement of the displacement of the other field. The first project on this subject is to investigate the feasibility of detecting a single microwave photon with signal-to-noise ratio (SNR) greater than unity using a three-level approximation for the transmon in an open transmission line, due to the saturation of transmon response to the probe intensity. After that, an improved scheme was proposed to overcome the limitations in the preceding work. By using a cavity probe field the signal photon induced probe displacement is strongly enhanced. Further additional transmon-cavity units can be cascaded to further improve the detection efficiency. It is shown that with only two transmons the distinguishability to resolve a single-photon state from the vacuum is up to 90%. One advantage of this photon detection scheme is that it does not rely on the absorption of single microwave photons, which enables repeated measurements and further applications. Moreover, I also show how the measurement diminishes coherence in the photon number basis thereby illustrating a fundamental principle of quantum measurement: the higher the measurement efficiency, the greater is the decoherence. Apart from the investigation on the single microwave photon detection, this thesis also studies nonlinear dynamics of a hybrid coupled-resonator and opto-mechanical system. It is shown that the saddle-node bifurcation and the Hopf bifurcation appear in the system, which leads to bistability and limit cycle dynamics of the mechanical position and the output light intensity. More importantly, the opto-mechanical nonlinearity dramatically changes the transparency window of the coupled resonator induced transparency (CRIT) and based on the narrow and bistable CRIT transparency window, the proposed system is shown to be a good platform for the weak impulsive force detection with a good sensitivity.