Superconducting circuits, photonic cavities, and tiny mechanical oscillators are versatile systems in which quantum dynamics can be observed and controlled. These various technologies can be combined, and the resulting “circuits” are often referred to as nano-electromechanical or optomechanical systems. In the context of quantum physics, such devices are referred to as “mesoscopic” because they contain a large number of atoms. This terminology distinguishes them from the “microscopic” systems of atom-optics, such as single ions [364] and atoms [545, 233, 26], and the spins of individual nuclei [412]. Nano-engineered systems provide an especially powerful arena for exploring quantum measurement and control because of their potential to realize complex circuits. In this chapter we explain how to treat these systems and describe in detail a common method used to measure them. Superconducting circuits The electrical circuits that we describe in this chapter are quite different from those that use normal conductors, such as copper wire at room temperature. Normal conductors are subject to strong dissipation and decoherence from the environment. As a result the dynamical degrees of freedom of the circuits, the currents and voltages, behave as classical variables. In superconducting circuits, the resistance, and thus the dissipation, is so small that the currents and voltages behave as quantum mechanical degrees of freedom. Of course, the current in a wire is actually composed of many elementary particles that each obey quantum mechanics. Why then should the current act like a single quantum degree of freedom? A mechanical analogy is useful here. Recall that even though a macroscopic pendulum is composed of many particles, when one talks about the motion of the pendulum one means the motion of the center-of-mass of its constituent particles. Since the center of mass is merely a linear combination of the coordinates of each particle, it is itself a valid quantum mechanical coordinate with a conjugate momentum. In a similar way the total current in a superconducting wire is a collective degree of freedom that obeys quantum mechanics.
Superconducting circuits, photonic cavities, and tiny mechanical oscillators are versatile systems in which quantum dynamics can be observed and controlled. These various technologies can be combined, and the resulting “circuits” are often referred to as nano-electromechanical or optomechanical systems. In the context of quantum physics, such devices are referred to as “mesoscopic” because they contain a large number of atoms. This terminology distinguishes them from the “microscopic” systems of atom-optics, such as single ions [364] and atoms [545, 233, 26], and the spins of individual nuclei [412]. Nano-engineered systems provide an especially powerful arena for exploring quantum measurement and control because of their potential to realize complex circuits. In this chapter we explain how to treat these systems and describe in detail a common method used to measure them. Superconducting circuits The electrical circuits that we describe in this chapter are quite different from those that use normal conductors, such as copper wire at room temperature. Normal conductors are subject to strong dissipation and decoherence from the environment. As a result the dynamical degrees of freedom of the circuits, the currents and voltages, behave as classical variables. In superconducting circuits, the resistance, and thus the dissipation, is so small that the currents and voltages behave as quantum mechanical degrees of freedom. Of course, the current in a wire is actually composed of many elementary particles that each obey quantum mechanics. Why then should the current act like a single quantum degree of freedom? A mechanical analogy is useful here. Recall that even though a macroscopic pendulum is composed of many particles, when one talks about the motion of the pendulum one means the motion of the center-of-mass of its constituent particles. Since the center of mass is merely a linear combination of the coordinates of each particle, it is itself a valid quantum mechanical coordinate with a conjugate momentum. In a similar way the total current in a superconducting wire is a collective degree of freedom that obeys quantum mechanics.
Abstract This book gathers the lecture notes of courses given at the 2011 Les Houches Summer School in Theoretical Physics, Session XCVI. What is a quantum machine? Can we say that lasers and transistors are quantum machines? After all, physicists advertise these devices as the two main spin-offs of the understanding of quantum physics. However, while quantum mechanics must be used to predict the wavelength of a laser and the operation voltage of a transistor, it does not intervene at the level of the signals processed by these systems. Signals involve macroscopic collective variables such as voltages and currents in a circuit or the amplitude of the oscillating electric field in an electromagnetic cavity resonator. In a true quantum machine, the signal collective variables, which both inform the outside on the state of the machine and receive controlling instructions, must themselves be treated as quantum operators, just like the position of the electron in a hydrogen atom.Quantum superconducting circuits, quantum dots, and quantum nanomechanical resonators satisfy the definition of quantum machines. These mesoscopic systems exhibit a few collective dynamical variables, whose fluctuations are well within the quantum regime and whose measurement is essentially limited in precision by Heisenberg’s uncertainty principle. Other engineered quantum systems based on natural, rather than artificial, degrees of freedom can also qualify as quantum machines: trapped ions, single Rydberg atoms in superconducting cavities, and lattices of ultracold atoms. This book provides the basic knowledge needed to understand and investigate the physics of these novel systems.
Mesoscopic physics and nanoelectronics concentrate on systems with dimensions somewhere between atomic and everyday macroscopic scale. Modern technology enables construction of submicron nanostructures where modeling based on classical physics has proven inadequate. It is possible to design electric circuits where dynamics of single electrons and photons are controlled using state-of-the-art experimental methods. For quantitative understanding of these systems it is necessary to resort to a quantum-mechanical description. Quantum phenomena, such as tunneling and a wave-like interference of particles, are essential ingredients of physics in mesoscopic systems. Field of mesoscopic physics contains a rich variety of topics ranging from fundamental condensed matter physics to quantum information processing and possible future technological applications. \n \nThis thesis presents theoretical studies of mesoscopic quantum phenomena in nanostructures and small electronic devices. We have focused on effects of environment fluctuations and investigated connections between fluctuations and transport phenomena. Decoherence in quantum bits (qubits) and quantum state engineering in superconducting circuits are also studied. The theoretical analysis in each case requires an open-system treatment. \n \nEffects of current fluctuations on quantum probe systems have been studied in detail. We have calculated transitions induced by current noise and discussed how these could be used for characterization of fluctuations. We have also shown that electric fluctuations play a key role in radiation and photon heat transport in nanostructures. Motivated by recent advances in mesoscopic electron-photon systems, we have studied a response of a coupled resonator-qubit system, squeezing of quantum fluctuations in small superconducting circuits and investigated decoherence in Josephson flux qubits.
Superconducting qubits, realized by incorporating Josephson junctions into superconducting circuits, behave as artificial atoms with anharmonic energy spectra and can be precisely controlled and measured using microwave cavities within the framework of circuit quantum electrodynamics (cQED). Since its emergence in the early 2000s, cQED has established superconducting qubits as leading candidates for scalable quantum devices and has enabled the exploration of hybrid quantum systems that integrate disparate physical platformsThis review surveys superconducting hybrid quantum electromechanical systems in which mechanical resonators are coupled to superconducting qubits, with a focus on two widely used qubit platforms: the transmon and the fluxonium. We provide an overview of the underlying coupling mechanisms arising from interactions through the phase and charge degrees of freedom of the qubit, and discuss how these mechanisms give rise to both longitudinal and transverse qubit-mechanical interactions. We further review extensions of electromechanical platforms to electro-optomechanical architectures, in which optical cavities are integrated to enable coherent interfacing between superconducting circuits and optical photons. This review aims to present a unified framework and perspective on qubit-mechanical and qubit-mechanical-optical hybrid systems in superconducting quantum technologies and applications related to sensors.
Cavity optomechanics enables controlling mechanical motion via radiation pressure interaction, and has contributed to the quantum control of engineered mechanical systems ranging from kg scale LIGO mirrors to nano-mechanical systems, enabling entanglement, squeezing of mechanical objects, to position measurements at the standard quantum limit, and quantum transduction. Yet, nearly all prior schemes have employed single- or few-mode optomechanical systems. In contrast, novel dynamics and applications are expected when utilizing optomechanical arrays and lattices, which enable to synthesize non-trivial band structures, and have been actively studied in the field of circuit QED. Superconducting microwave optomechanical circuits are a promising platform to implement such lattices, but have been compounded by strict scaling limitations. Here we overcome this challenge and realize superconducting circuit optomechanical lattices. We demonstrate non-trivial topological microwave modes in 1-D optomechanical chains as well as 2-D honeycomb lattices, realizing the canonical Su-Schrieffer-Heeger (SSH) model. Exploiting the embedded optomechanical interaction, we show that it is possible to directly measure the mode shapes, without using any local probe or inducing perturbation. This enables us to reconstruct the full underlying lattice Hamiltonian beyond tight-binding approximations, and directly measure the existing residual disorder. The latter is found to be sufficiently small to observe fully hybridized topological edge modes. Such optomechanical lattices, accompanied by the measurement techniques introduced, offers an avenue to explore out of equilibrium physics in optomechanical lattices such as quantum and quench dynamics, topological properties and more broadly, emergent nonlinear dynamics in complex optomechanical systems with a large number of degrees of freedoms.
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.
Abstract Cavity optomechanics enables controlling mechanical motion via radiation pressure interaction [1–3], and has contributed to the quantum control of engineered mechanical systems ranging from kg scale LIGO mirrors to nano-mechanical systems, enabling entanglement [4, 5], squeezing of mechanical objects [6], to position measurements at the standard quantum limit [7], non-reciprocal [8] and quantum transduction [9]. Yet, nearly all prior schemes have employed single- or few-mode optomechanical systems. In contrast, novel dynamics and applications are expected when utilizing optomechanical arrays and lattices [10], which enable to synthesize non-trivial band structures, and have been actively studied in the field of circuit QED [11–14]. Superconducting microwave optomechanical circuits are a promising platform to implement such lattices [15], but have been compounded by strict scaling limitations. Here we overcome this challenge and realize superconducting circuit optomechanical lattices. We demonstrate non-trivial topological microwave modes in 1-D optomechanical chains as well as 2-D honeycomb lattices, realizing the canonical SuSchrieffer-Heeger (SSH) model [16–18]. Exploiting the embedded optomechanical interaction, we show that it is possible to directly measure the mode functions of the bulk band modes, as well as the topologically protected edge states, without using any local probe [19–21] or inducing perturbation [22, 23]. This enables us to reconstruct the full underlying lattice Hamiltonian beyond tight-binding approximations, and directly measure the existing residual disorder. The latter is found to be sufficiently small to observe fully hybridized topological edge modes. Such optomechanical lattices, accompanied by the measurement techniques introduced, of-fers an avenue to explore out of equilibrium physics in optomechanical lattices such as quan-tum [24] and quench [25] dynamics, topological properties [10, 26, 27] and more broadly, emergent nonlinear dynamics in complex optomechanical systems with a large number of degrees of freedoms [28–31].