In the late 1960s and early 1970s, the development of mechanical resonators was pushed forward in efforts to record gravitational waves. In order to push down the limit of the smallest detectable force, the radiation circulating in a Fabry-Perot cavity was demonstrated to modify the damping of the cavity end-mirror suspended to act as a mechanical resonator. A few years later, motivated by the study of the internal properties of atoms and ions, the radiation pressure of light was realized as a means to cool atomic motion. In the following years, the ground-state cooling of ion motion was achieved in landmark studies. For mechanical resonators parametrically coupled to Fabry-Perot cavities, it took much longer to reach quantum ground-state cooling, both in the microwave domain, where the original experiments took place, and in the optical domain. Ground-state cooling in both regimes was reached by three groups during the preparation of this thesis and was largely due to advancements in the fabrication methods. \n \nGround-state cooling allows, in principle, the quantum nature of mechanical resonators to be probed and harnessed. Microwave domain devices, which are focused on in this thesis, have the advantage that nonlinear elements (superconducting quantum bits, qubits) are readily available. These elements enable nonclassical state preparation of the qubit and the subsequent transfer to the microwave cavity. The goal is to transfer these quantum states to mechanical resonators. This would allow mechanics to be used as a quantum mechanical memory in quantum information processing thanks to their long lifetimes. \n \nIn this thesis, various aspects of microwave domain circuit optomechanics are studied. The main goal of this thesis is to study how to enhance the coupling between a quantum regime microwave device and a mechanical resonator by using Josephson junction –based qubits. Coherent interaction between a superconducting qubit and a classical driven mechanical field is demonstrated. The analysis presented in this thesis predicts that ground-state cooling and coherent state transfer is possible in the single-phonon regime with transmon-regime qubits. Also, traditional linear optomechanical systems are considered in which the mechanical resonator is coupled to an electrical resonator. Such systems exhibit a rich spectrum of phenomena from which the quantum-limited amplification and multimode effects are considered in this thesis. It is shown that the system operates near quantum-limited amplification, which could be reached if the mechanical resonator was in the ground state.
In the late 1960s and early 1970s, the development of mechanical resonators was pushed forward in efforts to record gravitational waves. In order to push down the limit of the smallest detectable force, the radiation circulating in a Fabry-Perot cavity was demonstrated to modify the damping of the cavity end-mirror suspended to act as a mechanical resonator. A few years later, motivated by the study of the internal properties of atoms and ions, the radiation pressure of light was realized as a means to cool atomic motion. In the following years, the ground-state cooling of ion motion was achieved in landmark studies. For mechanical resonators parametrically coupled to Fabry-Perot cavities, it took much longer to reach quantum ground-state cooling, both in the microwave domain, where the original experiments took place, and in the optical domain. Ground-state cooling in both regimes was reached by three groups during the preparation of this thesis and was largely due to advancements in the fabrication methods. \n \nGround-state cooling allows, in principle, the quantum nature of mechanical resonators to be probed and harnessed. Microwave domain devices, which are focused on in this thesis, have the advantage that nonlinear elements (superconducting quantum bits, qubits) are readily available. These elements enable nonclassical state preparation of the qubit and the subsequent transfer to the microwave cavity. The goal is to transfer these quantum states to mechanical resonators. This would allow mechanics to be used as a quantum mechanical memory in quantum information processing thanks to their long lifetimes. \n \nIn this thesis, various aspects of microwave domain circuit optomechanics are studied. The main goal of this thesis is to study how to enhance the coupling between a quantum regime microwave device and a mechanical resonator by using Josephson junction –based qubits. Coherent interaction between a superconducting qubit and a classical driven mechanical field is demonstrated. The analysis presented in this thesis predicts that ground-state cooling and coherent state transfer is possible in the single-phonon regime with transmon-regime qubits. Also, traditional linear optomechanical systems are considered in which the mechanical resonator is coupled to an electrical resonator. Such systems exhibit a rich spectrum of phenomena from which the quantum-limited amplification and multimode effects are considered in this thesis. It is shown that the system operates near quantum-limited amplification, which could be reached if the mechanical resonator was in the ground state.
Due to its high coherence in transmission over a large distance in the ambient environment, the quantum optical system has been a prevailing platform for long-distance quantum communication, which was recently realized over a continental distance with a low earth orbit satellite and ground stations [102, 70]. However, the pure quantum optical system has so far shown weak interactions between photon and matter, which makes it inefficient in carrying out deterministic quantum gates for quantum repeater based scalable quantum network and quantum computing. On the other hand, superconducting quantum systems operating in the microwave domain with Josephson junction transmon qubits have proven to be capable of efficient deterministic quantum operations on quantum states [86, 87, 66]. Nevertheless, such architecture is prone to errors and decoherence due to cross-talk between microwave elements in a large-scale superconducting quantum circuit. Furthermore, superconducting systems, in general, also have large footprint (100s um) elements (resonators and superconducting quantum bits) [92, 60] that limit the ability to scale up a superconducting quantum system. Moreover, microwave quantum circuits require cooling to around 10 mK, making it unsuitable for communicating quantum information outside a dilution refrigerator (DF). Micro- and nano- acoustic elements have been extensively used in conventional integrated information processing systems due to their compactness and high coherence [97]. Acoustic systems in quantum engineering also have the advantage of being a platform for universal couplings between various quantum systems including spins, optical photons, and superconducting circuits. As it will be discussed in this thesis, elements critical to scalable optical quantum network and superconducting quantum circuit can be constructed relying on the cavity optomechanics and piezoelectric interactions. Optomechanical interaction is concerned with the light pressure coupling of cavity mechanical deformation to a strong optical field. This interaction has allowed the close to mechanical ground state cooling of mechanical resonators using laser and the ultra-sensitive displacement measurement that led to the detection of gravitational waves in the LIGO collaboration [125, 25]. Optomechanical crystals (OMCs) are lithographically patterned devices which contain a periodic structure that host bandgaps for both optical band electromagnetic waves and microwave band acoustic waves. A properly engineered defect in the crystal can confine and localize acoustic and electromagnetic modes of similar wavelengths into a small mode volume [17, 20, 21]. A strong optomechanical coupling, which can be achieved between such strongly confined co-localized optical and acoustic modes, can be used in engineering the quantum state of mechanical motion to realize useful quantum devices such as a high-coherence quantum memory [74] and an optomechanical high efficiency optical isolator for unidirectionally connecting distant optical cavities via an acoustic bus [37]. To strongly couple the mechanical degree of freedom with a superconducting quantum circuit, various methods can be used, ranging from electromechanic coupling (electric coupling to a mechanically compliant capacitor), magnetomechanical coupling (magnetic coupling to a vibrating SQUID loop), and piezoelectric coupling. The recent advent of quantum acoustics [23, 8, 9] was realized with the strong piezoelectric coupling between a superconducting transmon qubit and a high-coherence mechanical resonator. The engineered strong piezoacoustic coupling provides the possibility to carry out deterministic ultra-high fidelity two-qubit quantum gates on non-classical mechanical quantum states [52]. This ability together with the recent demonstration of ultra-long phonon lifetime mechanical resonators show the possibility of integrating the ultra-high quality mechanical resonator as a compact quantum memory element and even a new ultra-compact (10s um) quantum bit architecture for scalable superconducting quantum circuits. Furthermore, the strong piezoelectric coupling that can transduce quantum state in a superconducting circuit into mechanical wave also makes it possible to efficiently transduce a quantum state between a superconducting quantum circuit and a telecommunication band optical channel via a mechanical waveguide connected to an optomechanical crystal cavity.
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.
Microscale and nanoscale mechanical resonators have been used in advanced technological applications, from high precision time keeping and mass sensing, to processing high frequency signals in mobile communications. In the last few decades, they have been an important part of progress in the field of quantum information and metrology and have been proposed as quantum memories or transducers for measuring or connecting different types of quantum systems. The field of cavity optomechanics and electromechanics is concerned with coupling the electromagnetic field of a resonant optical cavity or electrical circuit to mechanical motion. These systems provide potential means to control and engineer the state of a mechanical object at the quantum level. This thesis contains the description of mechanical systems in megahertz to a few hundred megahertz frequency range formed by nano-fabricating photonic, phononic, and electrical circuits on a chip. These structures are designed to provide a large radiation pressure coupling between mechanical motion and electromagnetic fields to address and manipulate motional degrees of freedom. Qualitatively novel quantum effects are expected when one takes a step beyond linear coupling and exploits higher order interactions. To that end, we integrate electrical, mechanical and photonic structures in a multimode photonic crystal structure to observe "x2-coupling", where the optical cavity frequency is coupled to the square of the mechanical displacement. Moreover, we have developed two integrated on-chip platforms based on Si3N4 and Si nanomembranes capable of interfacing superconducting qubits and optical photons and realizing reversible microwave-to-optical conversion. We employ radiation pressure to cool these mechanical resonators to their quantum ground state. Finally, we demonstrate a form of electromechanical crystal for coupling microwave photons and hypersonic phonons of frequency ωm/2π = 0.425 GHz by capacitively coupling a phononic crystal acoustic cavity to a superconducting microwave resonator. Moving to higher frequency acoustic cavities not only facilitates the integration of electromechanical circuits and nanophotonic systems capable of operation in the resolved sideband limit of optomechanics for noise-free quantum signal conversion, but it opens up the possibility of using phonons as information carriers via phononic circuits. Utilizing a two-photon resonance condition for efficient microwave pumping and phononic bandgap shield to eliminate acoustic radiation, we achieve large cooperative electromechanical coupling (C ≈ 30) and intrinsic decay time of 2.3 ms. Moreover, electrical read-out of the phonon occupancy shows that the acoustic mode thermalizes close to its quantum ground-state of motion (phonon occupancy nm=1.5) at a fridge temperature of Tf = 10 mK. We conclude by considering several designs and fabrication improvements to the hypersonic electromechanical crystals that would enable them to perform quantum conversion between the electrical and acoustic domain.
The nonlinearity is essential for creation of non-classical states of the cavity or mechanical resonator such as squeezed or cat states. A microwave cavity can be made nonlinear by, for instance, adding Josephson junctions. The mechanical resonator is inherently nonlinear. The radiation pressure interaction between cavity and mechanical resonator is also inherently nonlinear but typically under strong drive of the cavity interaction can be linearized. However, if the optomechanical system is in the strong coupling regime nonlinear quantum effects become observable. These three cases provide the motivation for our studies. We start with backaction analysis of the classical regime of a dc SQUID (superconducting quantum interference device) with an embedded mechanical resonator. Then, we perform quantum analysis to understand how the asymmetry of two Josephson junctions influences the coupling strength of optomechanical interactions such as radiation pressure, cross-Kerr and single-photon beam splitter. Using variational method and self-consistent harmonic approximation we estimate the influence of the radiation pressure coupling and Kerr nonlinearity of the cavity on the effective frequency and dissipation of the cavity. The second part of this thesis focuses on the nonlinearity of the mechanical resonator coupled to the optical/microwave cavity. We study in details the nonlinear optomechanical response and draw the response map to summarize all results of the overcoupled and undercoupled cavity for the red and blue sidebands additionally to the weak and strong drive powers. In the end, w e confirm the developed theory by using it to numerically fit the experimental results.
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
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.