The dispersive regime of circuit QED is the main workhorse for todays quantum computing prototypes based on superconducting qubits. Analytic descriptions of this model typically rely on the rotating wave approximation of the interaction between the qubits and resonators, using the Jaynes-Cummings model as starting point for the dispersive transformation. Here we present analytic results on the dispersive regime of the dissipative Rabi model, without taking the rotating wave approximation of the underlying Hamiltonian. Using a recently developed hybrid perturbation theory based on the expansion of the time evolution on the Keldysh contour [Phys. Rev. A 95, 013847 (2017)], we derive simple analytic expressions for all experimentally relevant dynamical parameters like dispersive shift and resonator induced Purcell decay rate, focussing our analysis on a generic multi-level qubit. The analytical equations are easily tractable and reduce to the known Jaynes-Cummings results in the relevant limit. They however show qualitative differences at intermediate and large detuning, allowing for more accurate modelling of the interaction between superconducting qubits and resonators. In the limit of strong resonator driving, our results additionally predict new types of drive induced qubit dissipation and dephasing, not present in previous theories.
We present a simple theoretical description of two recent experiments where damping of Rabi oscillations, which cannot be attributed to dissipative decoherence, has been observed. This is obtained considering the evolution time or the Hamiltonian as random variables and then averaging the usual unitary evolution on a properly defined, model-independent, probability distribution.
We present a protocol to generate a large degree of squeezing of a boson (light) field mode strongly coupled to a two-level system in the dispersive regime. Our protocol exploits the strong dispersive coupling to introduce a time dependent frequency change of the boson field. With an appropriately timed sequence of sudden frequency changes, the quantum noise fluctuations in one quadrature of the field can be reduced well below the standard quantum limit, with a correspondingly increased uncertainty in the orthogonal quadrature. Even in the presence of realistic noise and imperfections, the pro…
The Rabi Hamiltonian is studied in the dispersive regime and ultra-strong coupling. We employ a recent unitary transformation to obtain not only the approximate Hamiltonian and its energy levels but also its eigenfunctions. The relationship of the approximation with other regimes and their approximations are also discussed.
We investigate the quantum phase transition in the dissipative anisotropic quantum Rabi model in the framework of quantum dressed master equation. From perspectives of both numerical and analytical analysis, we unravel the implication of the giant photon-bunching feature on the first-order quantum phase transition. The observed two-photon statistics can be well described analytically within a few lowest eigenstates at the low temperature. Moreover, such significant photon-bunching peak is generally exhibited at the deep-strong qubit-photon coupling, which is however lacking in the dissipative …
Rabi oscillations may be viewed as an interference phenomenon due to a coherent superposition of different quantum paths, like in the Young's two-slit experiment. The inclusion of the atomic external variables causes a non dissipative damping of the Rabi oscillations. More generally, the atomic translational dynamics induces damping in the correlation functions which describe non classical behaviors of the field and internal atomic variables, leading to the separability of these two subsystems. We discuss on the possibility of interpreting this intrinsic decoherence as a "which-way" informatio…
We discuss the equilibrium and out of equilibrium dynamics of cavity QED in presence of dissipation beyond the standard perturbative treatment of losses. Using the dynamical polaron \emph{ansatz} and Matrix Product State simulations, we discuss the case where both light-matter $g$-coupling and system-bath interaction are in the ultrastrong coupling regime. We provide a critical $g$ for the onset of Rabi oscillations. Besides, we demonstrate that the qubit is \emph{dressed} by the cavity and dissipation. That such dressing governs the dynamics and, thus, it can be measured. Finally, we sketch a…
The dispersive interaction between a two-level atom and a quantized field is studied. We consider besides a time dependent linear amplification and dissipative processes. In order to solve the master equation for this system, we use superoperator techniques.
We present the dissipative dynamics of the field of two-photon Jaynes-Cummings model (JCM) with Stark shift in dispersive approximation and investigate the influence of dissipation on entanglement. We show the coherence properties of the field can be affected by the dissipative cavity when nonlinear two-photon process is involved.
In this work, we investigate the quantum phase transition of light in the dissipative Rabi-Hubbard lattice under the framework of the mean-field theory and quantum dressed master equation. The order parameter of photons in strong qubit-photon coupling regime is derived analytically both at zero and low temperatures. Interestingly, we can locate the localization and delocalization phase transition very well in a wide parameter region. {In particular for the zero-temperature limit, the critical tunneling strength approaches zero generally in the deep-strong qubit-photon coupling regime, regardle…
Generation of squeezed light and optomechanical instability for dissipative type of opto- mechanical coupling is theoretically addressed for a cavity with the input mirror, serving as a mechanical oscillator, or an equivalent system. The problem is treated analytically for the case of resonance excitation or small detunings, mainly focusing on the bad cavity limit. A qualitative difference between the dissipative and purely dispersive coupling is reported. In particular, it is shown that, for the purely dissipative coupling in the bad cavity regime, the backaction is strongly reduced and the s…
We present theoretical transmission spectra of a strongly driven, damped, flux qubit coupled to a dissipative resonator in the ultrastrong coupling regime. Such a qubit-oscillator system, described within a dissipative Rabi model, constitutes the building block of superconducting circuit QED platforms. The addition of a strong drive allows one to characterize the system properties and study novel phenomena, leading to a better understanding and control of the qubit-oscillator system. In this work, the calculated transmission of a weak probe field quantifies the response of the qubit, in freque…
Recent progress on qubit manipulation allows application of periodic driving signals on qubits. In this study, a harmonic driving field is added to a Rabi dimer to engineer photon and qubit dynamics in a circuit quantum electrodynamics device. To model environmental effects, qubits in the Rabi dimer are coupled to a phonon bath with a sub-Ohmic spectral density. A non-perturbative treatment, the Dirac-Frenkel time-dependent variational principle together with the multiple Davydov D$_2$ {\it Ansatz} is employed to explore the dynamical behavior of the tunable Rabi dimer. In the absence of the p…
We investigate the entanglement dynamics and coherence of two two-level atoms interacting with two coherent fields of two spatially separated and dissipative cavities. It is in particular shown that entanglement sudden death is obtained within a short interacting time. However, after a long interaction time a long-lived entanglement is shown, that is, the initial entanglement of two atoms could be partially preserved. In addition, the coherence of the two atoms will not be lost during the evolution.
Photon-based strongly-correlated lattice models like the Jaynes-Cummings and Rabi lattices differ from their more conventional relatives like the Bose-Hubbard model by the presence of an additional tunable parameter: the frequency detuning between the pseudo-spin degree of freedom and the harmonic mode frequency on each site. Whenever this detuning is large compared to relevant coupling strengths, the system is said to be in the dispersive regime. The physics of this regime is well-understood at the level of a single Jaynes-Cummings or Rabi site. Here, we extend the theoretical description of …
We explore the long-time dynamics of Rabi model in a driven-dissipative setting and show that, as the atom-cavity coupling strength becomes larger than the cavity frequency, a new time scale emerges. This time scale, much larger than the natural relaxation time of the atom and the cavity, leads to long-lived metastable states susceptible to being observed experimentally. By applying a Floquet-Liouville approach to the time-dependent master equation, we systematically investigate the set of possible metastable states. We find that the properties of the metastable states can differ drastically f…
A Rabi dimer is used to model a recently reported circuit quantum electrodynamics system composed of two coupled transmission-line resonators with each coupled to one qubit. In this study, a phonon bath is adopted to mimic the multimode micromechanical resonators and is coupled to the qubits in the Rabi dimer. The dynamical behavior of the composite system is studied by the Dirac-Frenkel time-dependent variational principle combined with the multiple Davydov D$_{2}$ ansätze. Initially all the photons are pumped into the left resonator, and the two qubits are in the down state coupled with the …
The mean-field steady states of a generalized model of $N$ two-state systems interacting with one mode of the radiation field in the presence of external driving and dissipation are surveyed as a function of three control parameters: one governs the interaction strength relative to the resonance frequency, thus accessing the Dicke quantum phase transition, a second the relative strength of counter-rotating to rotating-wave interactions, and a third the amplitude of an external field driving the cavity mode. We unify the dissipative extension of the Dicke quantum phase transition with the recen…
In this paper we give fully analytical description of the dynamics of a collection of $N$-Frenkel excitons in high density regime dispersively coupled to a single mode cavity field, in the presence of both exciton and cavity-field dissipations. By using excitonic operators as q-deformed bosonic operators for the system, we solve analytically the Liouville equation for the density operator at zero temperature and investigate the influence of the number of excitons and the effect of both dissipations on dynamical behavior of the system. We use the solution of master equation to explore the dissi…
We develop a dual description of quantum optics adapted to dielectric systems without magnetic property. Our formalism, which is shown to be equivalent to the standard one within some dipolar approximations discussed in the article, is applied to the description of polaritons in dielectric media. We show that the dual formalism leads to the Huttner-Barnett equations [B. Huttner, S. M. Barnett, Phys. Rev. A \textbf{46}, 4306 (1992)] for QED in dielectric systems. More generally, we discuss the role of electromagnetic duality in the quantization procedure for optical systems and derive the struc…
We demonstrate that the open quantum Rabi model (QRM) exhibits a second-order dissipative phase transition (DPT) and propose a method to observe this transition with trapped ions. The interplay between the ultrastrong qubit-oscillator coupling and the oscillator damping brings the system into a steady-state with a diverging number of excitations, in which a DPT is allowed to occur even with a finite number of system components. The universality class of the open QRM, modified from the closed QRM by a Markovian bath, is identified by finding critical exponents and scaling functions using the Ke…
A comparison of different master equations for driven-dissipative dynamics in composite quantum systems: Dispersive readout in structured electromagnetic environments
The Rabi model considers a two-level system (or spin-1/2) coupled to a quantized harmonic oscillator and describes the simplest interaction between matter and light. The recent experimental progress in solid-state circuit quantum electrodynamics has engendered theoretical efforts to quantitatively describe the mathematical and physical aspects of the light-matter interaction beyond the rotating wave approximation. We develop a stochastic Schrödinger equation approach which enables us to access the strong-coupling limit of the Rabi model and study the effects of dissipation, and AC drive in an …
We have studied the coherent evolution of ultracold atomic rubidium clouds subjected to a microwave field driving Rabi oscillations between the stretched states of the F=1 and F=2 hyperfine levels. A phase winding of the two-level system pseudo-spin vector is encountered for elongated samples of atoms exposed to an axial magnetic field gradient and can be observed directly in state-selective absorption imaging. When dispersively recording the sample-integrated spin population during the Rabi drive, we observe a damped oscillation directly related to the magnetic field gradient, which we quanti…
Hyperfine atomic states are among the most promising candidates for qubit encoding in quantum information processing. In atomic systems, hyperfine transitions are typically driven through a two-photon Raman process by a laser field which is amplitude modulated at the hyperfine qubit frequency. Here, we introduce a new method for generating amplitude modulation by phase modulating a laser and reflecting it from a highly dispersive optical element known as a chirped Bragg grating (CBG). This approach is passively stable, offers high efficiency, and is compatible with high-power laser sources, en…
We present a theoretical study of optomechanical systems in which the mechanical resonator modulates both the resonant frequency (dispersive coupling) and the decay rates (dissipative coupling) of the optical cavity. We extend the generic dispersive framework to a more general case in which the dissipative coupling is split between its external and intrinsic contribution. We report a complete analysis of the influence of each kind of optical losses (intrinsic and external) on the three coupling mechanisms and highlight the interest of each external decay rate regime. The basic tools to experim…
The quantum dynamics of two entangled two-level atoms initially in extended Werner-like (EWL) states is studied. Each of the two atoms is located within an isolated and dissipative cavity. If the interaction time of atoms and cavities is not very long, the amount of quantum discord and entanglement between two atoms decreases as the system evolves. The sudden death of quantum discord and entanglement of two atoms occurs within a short interaction time. However, after a long interaction time, quantum discord and entanglement of two atoms could be partially preserved due to the long-lived nature…
We follow the passage from complex amplitude bistability to phase bistability in the driven dissipative Jaynes-Cummings oscillator. Quasidistribution functions in the steady state are employed, for varying qubit-cavity detuning and drive parameters, in order to track a first-order dissipative quantum phase transition up to the critical point marking a second-order transition and spontaneous symmetry breaking. We demonstrate the photon blockade breakdown in the dispersive regime, and find that the coexistence of cavity states in the regime of quantum bistability is accompanied by pronounced qub…
We experimentally study an optomechanical cavity that is formed between a mechanical resonator, which serves as a movable mirror, and a stationary on-fiber dielectric mirror. A significant change in the behavior of the system is observed when the distance between the fiber's tip and the mechanical resonator is made smaller than about 1 micrometer. The observed effects are attributed to the combined influence of Casimir force, Coulomb interaction due to trapped charges, and optomechanical coupling. The comparison between experimental results and theory yields a partial agreement.
Optomechanical system with tunable dissipative and dispersive couplings
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University（山西大学光电研究所量子光学与量子光学器件国家重点实验室）；Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学极端光学协同创新中心）
Dissipative and dispersive optomechanical couplings are experimentally observed in a photonic crystal split-beam nanocavity optimized for detecting nanoscale sources of torque. Dissipative coupling of up to approximately $500$ MHz/nm and dispersive coupling of $2$ GHz/nm enable measurements of sub-pg torsional and cantilever-like mechanical resonances with a thermally-limited torque detection sensitivity of 1.2$\times 10^{-20} \text{N} \, \text{m}/\sqrt{\text{Hz}}$ in ambient conditions and 1.3$\times 10^{-21} \text{N} \, \text{m}/\sqrt{\text{Hz}}$ in low vacuum. Interference between optomecha…
Combination of dissipative and dispersive coupling in the cavity optomechanical systems
Faculty of Physics, M.V. Lomonosov Moscow State University（莫斯科国立大学物理系）；Quantum Technology Centre, M.V. Lomonosov Moscow State University（莫斯科国立大学量子技术中心）
We have analyzed the interaction of a dissipative two level quantum system with high and low frequency excitation. The system is continuously and simultaneously irradiated by these two waves. If the frequency of the first signal is close to the level separation and the second one is tuned to the Rabi frequency, it is shown that the response of the system exhibits an undamped low frequency oscillation. The method can be useful for low frequency Rabi spectroscopy in various physical systems which are described by a two level Hamiltonian, such as nuclei spins in NMR, double well quantum dots, sup…
We propose a superconducting circuit to implement a two-photon quantum Rabi model in a solid-state device, where a qubit and a resonator are coupled by a two-photon interaction. We analyze the input-output relations for this circuit in the strong coupling regime and find that fundamental quantum optical phenomena are qualitatively modified. For instance, two-photon interactions are shown to yield single- or two-photon blockade when a pumping field is either applied to the cavity mode or to the qubit, respectively. In addition, we derive an effective Hamiltonian for perturbative ultrastrong two…
The recently identified possibility of ground-state cooling of a mechanical oscillator in the unresolved sideband regime by combination of the dissipative and dispersive optomechanical coupling under the red sideband excitation [Phys. Rev. A 88, 023850 (2013)], is currently viewed as a remarkable finding. We present a comprehensive analysis of this protocol, which reveals its very high sensitivity to small imperfections such as an additional dissipation, the inaccuracy of the optimized experimental settings, and the inaccuracy of the theoretical framework adopted. The impact of these imperfect…
Dissipative generation of spin squeezing in the resolved vacuum Rabi splitting limit
JILA, University of Colorado and National Institute of Standards and Technology（科罗拉多大学与美国国家标准与技术研究院联合JILA研究所）；Department of Physics, University of Colorado（科罗拉多大学物理系）；Center for Theory of Quantum Matter, University of Colorado（科罗拉多大学量子物质理论中心）；T.C.M. Group, Cavendish Laboratory, University of Cambridge（剑桥大学卡文迪许实验室TCM小组）；Institute of Physics（物理研究所）
Transitions as the Native Objects of Dispersive Light-Matter Dynamics
Telecom Paris, Institut Polytechnique de Paris（巴黎电信学院，巴黎理工学院）；Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS - Université de Bourgogne Franche-Comté（勃艮第卡尔多跨学科实验室，CNRS联合研究单位6303，勃艮第弗朗什-孔泰大学）；Technical University of Munich, TUM School of Natural Sciences, Physics Department（慕尼黑工业大学，自然科学学院，物理系）；Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften（瓦尔特·迈斯纳研究所，巴伐利亚科学院）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Centre for Nanosciences and Nanotechnology, Université Paris-Saclay（纳米科学与工程中心，巴黎萨克雷大学）
The synchronization of coherent states of light has long been an important subject of basic research and technology. Recently, a new concept for analog computers has emerged where this synchronization process can be exploited to solve computationally hard problems - potentially faster and more energy-efficient than what can be achieved with conventional computer technology today. The unit cell of such systems consists of two coherent centers that are coupled to one another in a controlled manner. Here, we experimentally characterize and analyze the synchronization process of two photon Bose-Ei…
We give a fully description of the dynamics of an atom dispersively coupled to a field mode in a dissipative environment fed by an external source. The competition between the unitary atom-field (which leads to entanglement) and the dissipative field-environment couplings are investigated in detail. We find the time evolution of the global atom-field system for any initial state and we show that atom-field steady state is at most classically correlated. For an initial state chosen, we evaluate the purity loss of the global system and of atomic and field subsystems as a function of time. We fin…
The recent demonstration of cooling of a macroscopic silicon nitride membrane based on dissipative coupling makes dissipatively coupled optomechanical systems as promising candidates for squeezing. We theoretically show that such a system in a cavity on resonance can yield good squeezing which is comparable to that produced by dispersive coupling. We also report the squeezing resulting from the combined effects of dispersive and dissipative couplings and thus the device can be operated in one regime or the other. We derive the maximal frequency and quadrature angles to observe squeezing for gi…
We study a dissipative, mechanically coupled optomechanical system that accommodates gain and loss. The gain (loss) is engineered by driven a purely dispersive optomechanical cavity with a blue-detuned (red-detuned) electromagnetic field. By taking into account the dissipative coupling, the Exceptional Point (EP), which is the $\mathcal{PT}-$symmetry phase transition, occurs at low threshold driving strength compared to the purely dispersive system. In the linear regime, the $\mathcal{PT}-$symmetry is unbroken and the dissipative coupling induces strong coupling between the mechanical resonato…
Anisotropic quantum Rabi model is a generalization of quantum Rabi model, which allows its rotating and counter-rotating terms to have two different coupling constants. It provides us with a fundamental model to understand various physical features concerning quantum optics, solid-state physics, and mesoscopic physics. In this paper, we propose an experimental feasible scheme to implement anisotropic quantum Rabi model in a circuit quantum electrodynamics system via periodic frequency modulation. An effective Hamiltonian describing the tunable anisotropic quantum Rabi model can be derived from…
We investigate the behavior of a circuit QED device when the resonator is initially populated with a mesoscopic coherent field. The strong coupling between the cavity and the qubit produces an entangled state involving mesoscopic quasi-pointer states with respect to cavity dissipation. The overlap of the associated field components results in collapse and revivals for the Rabi oscillation. Although qubit relaxation and dephasing do not preserve these states, a simple analytical description of the dissipative dynamics of the circuit QED device including cavity relaxation as well as qubit dissip…
A microscopic derivation of the master equation for the Jaynes-Cummings model with cavity losses is given, taking into account the terms in the dissipator which vary with frequencies of the order of the vacuum Rabi frequency. Our approach allows to single out physical contexts wherein the usual phenomenological dissipator turns out to be fully justified and constitutes an extension of our previous analysis [Scala M. {\em et al.} 2007 Phys. Rev. A {\bf 75}, 013811], where a microscopic derivation was given in the framework of the Rotating Wave Approximation.
We explore the phase diagram of the dissipative Rabi-Hubbard model, as could be realized by a Raman-pumping scheme applied to a coupled cavity array. There exist various exotic attractors, including ferroelectric, antiferroelectric, and inccomensurate fixed points, as well as regions of persistent oscillations. Many of these features can be understood analytically by truncating to the two lowest lying states of the Rabi model on each site. We also show that these features survive beyond mean-field, using Matrix Product Operator simulations.
