We study the quantum dynamics of optical fields in weakly confining resonators with overlapping modes. Employing a recently developed quantization scheme involving a discrete set of resonator modes and continua of external modes we derive Langevin equations and a master equation for the resonator modes. Langevin dynamics and the master equation are proved to be equivalent in the Markovian limit. Our open-resonator dynamics may be used as a starting point for a quantum theory of random lasers.
Feshbach's projector technique is employed to quantize the electromagnetic field in optical resonators with an arbitray number of escape channels. We find spectrally overlapping resonator modes coupled due to the damping and noise inflicted by the external radiation field. For wave chaotic resonators the mode dynamics is determined by a non--Hermitean random matrix. Upon including an amplifying medium, our dynamics of open-resonator modes may serve as a starting point for a quantum theory of random lasing.
A recently developed technique for the system--and--bath quantization of open optical cavities is applied to three resonator geometries: A one dimensional dielectric, a Fabry--Perot resonator, and a dielectric disk. The system--and--bath Hamiltonian for these geometries is derived starting from Maxwell's equations and employed to compute the electromagnetic fields, the resonances, and the cavity gain factors. Exact agreement is found with standard quantization methods based on a modes--of--the--universe description. Our analysis provides a microscopic justification for the system--and--bath qu…
Quantum thermal transport and two-photon statistics serve as two representative nonequilibrium features in circuit quantum electrodynamics systems. Here, we investigate quantum heat flow and two-photon correlation function at steady-state in a composite qubit-resonator model, where one qubit shows both transverse and longitudinal couplings to a single-mode optical resonator. With weak qubit-resonator interaction, we unravel two microscopic transport pictures, i.e., cotunneling and cyclic heat exchange processes, corresponding to transverse and longitudinal couplings respectively. At strong qub…
We study the quantum properties of the electromagnetic field in optical cavities coupled to an arbitrary number of escape channels. We consider both inhomogeneous dielectric resonators with a scalar dielectric constant $ε({\bf r})$ and cavities defined by mirrors of arbitrary shape. Using the Feshbach projector technique we quantize the field in terms of a set of resonator and bath modes. We rigorously show that the field Hamiltonian reduces to the system--and--bath Hamiltonian of quantum optics. The field dynamics is investigated using the input--output theory of Gardiner and Collet. In the c…
The concept of an acoustical analog of the optical laser has been developed recently both in theoretical as well as experimental works. We discuss here a model of a coherent phonon generator with a direct signature of quantum properties of the sound vibrations. The considered setup is made of a laser driven quantum dot (QD) embedded in an acoustical nanocavity. The system's dynamics is solved for a single phonon mode in the steady-state and in the strong QD-phonon coupling regime beyond the secular approximation. We demonstrate that the phonon statistics exhibits quantum features, i.e. sub-Poi…
We study the low temperature physics of an ultracold atomic gas in the potential formed inside a pumped optical resonator. Here, the height of the cavity potential, and hence the quantum state of the gas, depends not only on the pump parameters, but also on the atomic density through a dynamical a.c.-Stark shift of the cavity resonance. We derive the Bose-Hubbard model in one dimension, and use the strong coupling expansion to determine the parameter regime in which the system is in the Mott-insulator state. We predict the existence of overlapping, competing Mott states, and bistable behavior …
Electrical quantum conductors coupled to microwave resonators have in the last decade emerged as a versatile testbed for controllable light-matter interaction on the nanometer scale. Recent experimental progress with high impedance resonators has resulted in conductor-resonator systems with a large, dimensionless coupling parameter $λ\gtrsim 0.1$, well beyond the small coupling regime $λ\ll 1$. Motivated by this progress, we here analyse theoretically the joint statistics of transported electrons and emitted photons in a single level quantum dot coupled to a microwave resonator, for arbitraril…
We investigate steady-state thermal transport and photon statistics in a nonequilibrium hybrid quantum system, in which a qubit is longitudinally and quadratically coupled to an optical resonator. Our calculations are conducted with the method of the quantum dressed master equation combined with full counting statistics. The effect of negative differential thermal conductance is unravelled at finite temperature bias, which stems from the suppression of cyclic heat transitions and large mismatch of two squeezed photon modes at weak and strong qubit-resonator hybridizations, respectively. The gi…
We consider the statistics of time-integrated energy fluctuations of a driven bosonic resonator (as measured by a QND detector), using the standard Keldysh prescription to define higher moments. We find that due to an effective cascading of fluctuations, these statistics are surprisingly non-classical: the low-temperature, quantum probability distribution is not equivalent to the high-temperature classical distribution evaluated at some effective temperature. Moreover, for a sufficiently large drive detuning and low temperatures, the Keldysh-ordered quasi-probability distribution characterizin…
We report on the experimental investigation of the fluctuation properties in the resonance frequency spectra of a flat resonator simulating a dissipative quantum billiard subject to partial time-reversal invariance violation (TIV) which is induced by two magnetized ferrites. The cavity has the shape of a quarter bowtie billiard of which the corresponding classical dynamics is chaotic. Due to dissipation it is impossible to identify a complete list of resonance frequencies. Based on a random-matrix theory approach we derive analytical expressions for statistical measures of short- and long-rang…
Detection of single, itinerant microwave photons is an important functionality for emerging quantum technology applications as well as of fundamental interest in quantum thermodynamics experiments on heat transport. In a recent experiment [W. Khan et al., Nat. Commun. 12, 5130 (2021)], it was demonstrated that a double quantum dot (DQD) coupled to a microwave resonator can act as an efficient and continuous photodetector by converting an incoming stream of photons to an electrical photocurrent. In the experiment, average photon and electron flows were analyzed. Here we theoretically investigat…
We give a concrete experimental scheme for engineering the insulator-superfluid transition of light in a one-dimensional (1-D) array of coupled superconducting stripline resonators. In our proposed architecture, the on-site interaction and the photon hopping rate can be tuned independently by adjusting the transition frequencies of the charge qubits inside the resonators and at the resonator junctions, respectively, which permits us to systematically study the quantum phase transition of light in a complete parameter space. By combining the techniques of photon-number-dependent qubit transitio…
We investigate a paradigm example of cavity quantum electrodynamics with many body systems: an ultracold atomic gas inside a pumped optical resonator. In particular, we study the stability of atomic insulator-like states, confined by the mechanical potential emerging from the cavity field spatial mode structure. As in open space, when the optical potential is sufficiently deep, the atomic gas is in the Mott-like state. Inside the cavity, however, the potential depends on the atomic distribution, which determines the refractive index of the medium, thus altering the intracavity field amplitude.…
We propose a quantum heat engine composed of two superconducting transmission line resonators interacting with each other via an optomechanical-like coupling. One resonator is periodically excited by a thermal pump. The incoherently driven resonator induces coherent oscillations in the other one due to the coupling. A limit cycle, indicating finite power output, emerges in the thermodynamical phase space. The system implements an all-electrical analog of a photonic piston. Instead of mechanical motion, the power output is obtained as a coherent electrical charging in our case. We explore the d…
We theoretically propose a scheme for the generation of a non-classical single-mode motional state of a mechanical resonator (MR) in the three-mode optomechanical systems, in which two optical modes of the cavities are linearly coupled to each other and one mechanical mode of the MR is optomechanically coupled to the two optical modes with the same coupling strength simultaneously. One cavity is driven by a coherent laser light. By properly tuning the frequency of the weak driving field, we obtain engineered Liouvillian superoperator via engineering the selective interaction Hamiltonian confin…
Measuring thermodynamic quantities can be easy or not, depending on the system that is being studied. For a macroscopic object, measuring temperatures can be as simple as measuring how much a column of mercury rises when in contact with the object. At the small scale of quantum electromechanical systems, such simple methods are not available and invariably detection processes disturb the system state. Here we propose a method for measuring the temperature on a suspended semiconductor membrane clamped at both ends. In this method, the membrane is mediating a capacitive coupling between two tran…
We derive the photocount statistics of the radiation emitted from a chaotic laser resonator in the regime of single-mode lasing. Random spatial variations of the resonator eigenfunctions lead to strong mode-to-mode fluctuations of the laser emission. The distribution of the mean photocount over an ensemble of modes changes qualitatively at the lasing transition, and displays up to three peaks above the lasing threshold.
The parametrically pumped Kerr model describes a driven-dissipative nonlinear cavity, whose nonequilibrium phase diagram features both continuous and discontinuous quantum phase transitions. We consider the consequences of these critical phenomena for the fluctuations of the photocurrent obtained via continuous weak measurements on the cavity. Considering both direct photodetection and homodyne detection schemes, we find that the current fluctuations diverge exponentially at the discontinuous phase transition. However, we find strikingly different current fluctuations for these two detection s…
The concept of wavefunction reduction should be introduced to standard quantum mechanics in any physical processes where effective reduction of wavefunction occurs, as well as in the measurement processes. When the overlap is negligible, each particle obey Maxwell-Boltzmann statistics even if the particles are in principle described by totally symmetrized wavefunction [P.R.Holland, The Quantum Theory of Motion, Cambridge Unversity Press, 1993, p293]. We generalize the conjecture. That is, particles obey some generalized statistics that contains the quantum and classical statistics as special c…
We analyze the full counting statistics of photons emitted by a double quantum dot (DQD) coupled to a high-quality microwave resonator by electric dipole interaction. We show that at the resonant condition between the energy splitting of the DQD and the photon energy in the resonator, photon statistics exhibits both a sub-Poissonian distribution and antibunching. In the ideal case, when the system decoherence stems only from photodetection, the photon noise is reduced below one-half of the noise for the Poisson distribution and is consistent with current noise. The photon distribution remains …
We investigate temperature reservoir effects in a lossy Kerr nonlinear resonator considering selective excitation of ooscillatory mode driven by a sequence of Gaussian pulses. In this way, we analyze time-dependent populations of photon-number states and quantum statistics on the base of second-order photon correlation function in one-photon and two-photon transitions. The effects coming from thermal reservoirs are interesting for performing more realistic approach to generate Fock states and for study phenomena connecting quantum engineering and temperature. We also study the role of pulse-sh…
We study the far-from-equilibrium statistical mechanics of periodically driven fermionic atoms in a lossy optical resonator. We show that the interplay of the Fermi surface with cavity losses leads to sub-natural cavity linewidth narrowing, squeezed light, and out-of-equilibrium quantum statistics of the atoms. Adapting the Keldysh approach, we set-up and solve a quantum kinetic Boltzmann equation in a systematic $1/N$ expansion with $N$ the number of atoms. In the strict thermodynamic limit $N,V\rightarrow \infty$, $N/V=\text{const.}$ we find the atoms (fermions or bosons) remain immune again…
In experiments with superconducting quantum circuits, characterizing the photon statistics of propagating microwave fields is a fundamental task. We quantify the $n^{2}\,{+}\,n$ photon number variance of thermal microwave photons emitted from a black-body radiator for mean photon numbers $0.05\,{\lesssim}\,n\,{\lesssim}\,1.5$. We probe the fields using either correlation measurements or a transmon qubit coupled to a microwave resonator. Our experiments provide a precise quantitative characterization of weak microwave states and information on the noise emitted by a Josephson parametric amplifi…
We investigate two different kinds of quantum trajectories for a nonlinear photon resonator subject to two-photon pumping, a configuration recently studied for the generation of photonic Schroedinger cat states. In the absence of feedback control and in the strong-driving limit, the steady-state density matrix is a statistical mixture of two states with equal weight. While along a single photon-counting trajectory the systems intermittently switches between an odd and an even cat state, we show that upon homodyne detection the situation is different. Indeed, homodyne quantum trajectories revea…
In superconducting circuit architectures for quantum computing, microwave resonators are often used both to isolate qubits from the electromagnetic environment and to facilitate qubit state readout. We analyze the full counting statistics of photons emitted from such driven readout resonators both in and beyond the dispersive approximation. We calculate the overlap between emitted-photon distributions for the two qubit states and explore strategies for its minimization with the purpose of increasing fidelity of intensity-sensitive readout techniques. In the dispersive approximation and at negl…
We study the role of quantum statistics in the self-ordering of ultracold bosons and fermions moving inside an optical resonator with transverse coherent pumping. For few particles we numerically compute the nonequilibrium dynamics of the density matrix towards the self-ordered stationary state of the coupled atom-cavity system. We include quantum fluctuations of the particles and the cavity field. These fluctuations in conjunction with cavity cooling determine the stationary distribution of the particles, which exhibits a transition from a homogeneous to a spatially ordered phase with the app…
We formulate gaussian and circular random-matrix models representing a coupled system consisting of an absorbing and an amplifying resonator, which are mutually related by a generalized time-reversal symmetry. Motivated by optical realizations of such systems we consider a PT or a PTT' time-reversal symmetry, which impose different constraints on magneto-optical effects, and then focus on five common settings. For each of these, we determine the eigenvalue distribution in the complex plane in the short-wavelength limit, which reveals that the fraction of real eigenvalues among all eigenvalues …
We explore theoretically the physics of dynamic hysteresis for driven-dissipative nonlinear photonic resonators. In the regime where the semiclassical mean-field theory predicts bistability, the exact steady-state density matrix is known to be unique, being a statistical mixture of two states: in particular, no static hysteresis cycle of the excited population occurs as a function of the driving intensity. Here, we predict that in the quantum regime a dynamic hysteresis with a rich phenomenology does appear when sweeping the driving amplitude in a finite time. The hysteresis area as a function…
We study systems of fully polarized ultracold atomic gases obeying Fermi statistics. The atomic transition interacts dispersively with a mode of a standing-wave cavity, which is coherently pumped by a laser. In this setup, the intensity of the intracavity field is determined by the refractive index of the atomic medium, and thus by the atomic density distribution. Vice versa, the density distribution of the atom is determined by the cavity field potential, whose depth is proportional to the intracavity field amplitude. In this work we show that this nonlinearity leads to an instability in the …
We study the full field and frequency filtered output photon statistics of a resonator in thermal equilibrium with a bath and containing an arbitrarily large quartic nonlinearity. According to the general theory of photodetection, we derive general input-output relations valid for the ultra-anharmonic regime, where the nonlinearity becomes comparable to the energy of the resonator, and show how the emission properties are modified as compared to the generally assumed simple anharmonic regime. We analyse the impact of the nonlinearity on the full statistics of the emission and its spectral prop…
We investigate the dynamics of single- and multi-photon emission from detuned strongly coupled systems based on the quantum-dot-photonic-crystal resonator platform. Transmitting light through such systems can generate a range of non-classical states of light with tunable photon counting statistics due to the nonlinear ladder of hybridized light-matter states. By controlling the detuning between emitter and resonator, the transmission can be tuned to strongly enhance either single- or two-photon emission processes. Despite the strongly-dissipative nature of these systems, we find that by utiliz…
From the perspective of many body physics, the transmon qubit architectures currently developed for quantum computing are systems of coupled nonlinear quantum resonators. A significant amount of intentional frequency detuning (disorder) is required to protect individual qubit states against the destabilizing effects of nonlinear resonator coupling. Here we investigate the stability of this variant of a many-body localized (MBL) phase for system parameters relevant to current quantum processors of two different types, those using untunable qubits (IBM type) and those using tunable qubits (Delft…
We propose a model for scattering in a flat resonator with a thin antenna. The results are applied to rectangular microwave cavities. We compute the resonance spacing distribution and show that it agrees well with experimental data provided the antenna radius is much smaller than wavelengths of the resonance wavefunctions.
