The interaction among the components of a hybrid quantum system is often neglected when considering the coupling of these components to an environment. However, if the interaction strength is large, this approximation leads to unphysical predictions, as has been shown for cavity-QED and optomechanical systems in the ultrastrong-coupling regime. To deal with these cases, master equations with dissipators retaining the interaction between these components have been derived for the quantum Rabi model and for the standard optomechanical Hamiltonian. In this article, we go beyond these previous derivations and present a general master equation approach for arbitrary hybrid quantum systems interacting with thermal reservoirs. Specifically, our approach can be applied to describe the dynamics of open hybrid systems with harmonic, quasiharmonic, and anharmonic transitions. We apply our approach to study the influence of temperature on multiphoton vacuum Rabi oscillations in circuit QED. We also analyze the influence of temperature on the conversion of mechanical energy into photon pairs in an optomechanical system, which has been recently described at zero temperature. We compare our results with previous approaches, finding that these sometimes overestimate decoherence rates and underestimate excited-state populations.
The interaction among the components of a hybrid quantum system is often neglected when considering the coupling of these components to an environment. However, if the interaction strength is large, this approximation leads to unphysical predictions, as has been shown for cavity-QED and optomechanical systems in the ultrastrong-coupling regime. To deal with these cases, master equations with dissipators retaining the interaction between these components have been derived for the quantum Rabi model and for the standard optomechanical Hamiltonian. In this article, we go beyond these previous derivations and present a general master equation approach for arbitrary hybrid quantum systems interacting with thermal reservoirs. Specifically, our approach can be applied to describe the dynamics of open hybrid systems with harmonic, quasiharmonic, and anharmonic transitions. We apply our approach to study the influence of temperature on multiphoton vacuum Rabi oscillations in circuit QED. We also analyze the influence of temperature on the conversion of mechanical energy into photon pairs in an optomechanical system, which has been recently described at zero temperature. We compare our results with previous approaches, finding that these sometimes overestimate decoherence rates and underestimate excited-state populations.
Cavity and circuit QED study light-matter interaction at its most fundamental level. Yet, this interaction is most often neglected when considering the coupling of this system with an environment. In this paper, we show how this simplification, which leads to the standard quantum optics master equation, is at the root of unphysical effects. Including qubit relaxation and dephasing, and cavity relaxation, we derive a master equation that takes into account the qubit-resonator coupling. Special attention is given to the ultrastrong coupling regime, where the failure of the quantum optical master equation is manifest. In this situation, our model predicts an asymmetry in the vacuum Rabi splitting that could be used to probe dephasing noise at unexplored frequencies. We also show how fluctuations in the qubit frequency can cause sideband transitions, squeezing, and Casimir-like photon generation.
With the recent advancements made in the coupling of quantum systems, reaching the (ultra)strong coupling regime, where the coupling constant exceeds the dissipation rate and approaches the frequencies of the coupled systems, is getting closer. In this regime, the much used semiclassical model is no longer valid, and the solving of the full quantum master equation can be computationally too demanding. Thus, we must develop a new approach to be able to simulate the system. One solution is to use the semiquantum approximation. In this thesis, we present a systematic way of employing the semiquantum approximation and compare it to the full master equation. Our aim is to determine the range for the coupling strength where the semiquantum model is applicable. We chose to do the comparison of the semiquantum approximation and the master equation within the Rabi model, which can be used to describe e.g. the interaction of light and matter. The Rabi model describes a system with a coupled two-level system and a harmonic oscillator. It was chosen because its behaviour is reasonably well known. We compared the semiquantum approximation to the quantum master equation in two cases. First we studied the steady state results from the two models, and then moved on to the spectral properties. The results show that the steady state and the spectrum obtained from the semiquantum model agree with the ones obtained by using the master equation, until the coupling reaches a considerable fraction of the resonant frequency of the oscillator, while still exceeding the experimentally reasonable dissipation rate by an order of magnitude. In the future, one could use the semiquantum approximation in the eld of cavity optomechanics. There a mechanical oscillator is coupled to optical radiation con ned in a cavity, e.g. a Fabry-Perot cavity. Especially in optomechanics, the solving of the master equation can turn out to be a formidable task, and by using the semiquantum approximation one could reduce the computation time considerably.
Mechanical resonators ultrastrongly coupled to quantum two-level systems provide a promising route towards mechanical qubits by introducing significant anharmonicity to the mechanical modes, particularly in the slow-oscillator regime. Although the resulting hybrid system is well described by the quantum Rabi model, a consistent treatment of dissipation remains challenging across the broad parameter space routinely probed in current nanotube electromechanical devices. Here, we investigate dissipation in the open quantum Rabi model using a Born-Markov framework based on the slowly varying bath spectrum approximation, yielding a Lindblad master equation applicable far beyond conventional descriptions while recovering them in their respective limits. Using this framework, we analyze experimentally accessible observables across this parameter space. As the secular approximation breaks down, phonon blockade progressively washes out. Our approach remains valid in this regime, enabling a quantitative description of the continuous evolution of phonon blockade with coupling strength and dissipation. At finite temperature, we find a suppression of the temperature-induced increase of coherence decay rate for weak anharmonicity. Under driving, our approach remains applicable to substantially stronger perturbations than conventional dressed-state master equations and shows that an apparently classical observable can coexist with Wigner negativity. We further capture the weakly anharmonic regime arising from finite detuning in the double-quantum dot. These results establish a unified description of dissipation from weakly anharmonic operating regimes to the strongly anharmonic mechanical-qubit regime and provide experimentally relevant predictions for ultrastrong electromechanical systems.
The goal of this chapter is to investigate the effects of relaxation and dephasing on the quantum Rabi model in the ultrastrong coupling regime, and to provide explicit formulas that enable students and researchers to implement and numerically solve the resulting nonunitary dynamics from first principles. We numerically solve the standard Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation and the dressed-picture master equation (DME), derived by Beaudoin, Gambetta, and Blais, for various initial field states, including coherent states, odd Schrödinger cat states, squeezed vacuum states, squeezed coherent states, and thermal states. The coupling strength is varied in the range 0.05ω to 0.8ω. We also examine photon generation from the vacuum induced by external time-dependent modulation of the qubit parameters, as well as multiphoton Rabi oscillations for an initially excited qubit. Two different reservoir spectral densities are considered: white noise and Ohmic noise. The differences between the two master-equation approaches are illustrated through numerical results for several physical observables, including the qubit excited-state population, the mean photon number, the Mandel Q-factor, the negativity (used as a measure of entanglement), the subsystem purities, and the photon-number probability distribution at selected times. As expected, for many of these quantities the predictions of the standard GKSL master equation differ substantially from those of the dressed-picture master equation. However, in certain parameter regimes and for specific observables, the discrepancies remain comparatively small.
We present a full quantum theory for the dissipative dynamics of an optical cavity in the ultrastrong light-matter coupling regime, in which the vacuum Rabi frequency is a significant fraction of the active electronic transition frequency and the antiresonant terms of the light-matter coupling play an important role. In particular, our model can be applied to the case of intersubband transitions in doped semiconductor quantum wells embedded in a microcavity. The coupling of the intracavity photonic mode and of the electronic polarization to the external, frequency-dependent, dissipation baths is taken into account by means of quantum Langevin equations in the input-output formalism. In the case of a time-independent vacuum Rabi frequency, exact analytical expressions for the operators are obtained, which allows us to characterize the quantum dissipative response of the cavity to an arbitrary initial condition (vacuum, coherent field, thermal excitation). For a vacuum input in both the photonic and electronic polarization modes, the ground state of the cavity system is a two-mode squeezed vacuum state with a finite population in both photonic and electronic modes. These excitations are, however, virtual and cannot escape from the cavity: for a vacuum input, a vacuum output is found, without any trace of the intracavity squeezing. For a coherent photonic input the linear optical response spectra (reflectivity, absorption, transmission) have been studied, and signatures of the ultrastrong coupling have been identified in the asymmetric and peculiar anticrossing of the polaritonic eigenmodes. Finally, we have calculated the electroluminescence spectra in the case of an incoherent electronic input: the emission intensity in the ultrastrong coupling regime results in being significantly enhanced as compared to the case of an isolated quantum well without a surrounding cavity.
We discuss both the spectrum and the dynamics of cavity QED in the presence of dissipation beyond the standard perturbative treatment of losses. Using the dynamical polaron ansatz and matrix-product state simulations, we discuss the case where both light-matter g coupling and system-bath interaction are in the ultra-strong-coupling regime. We provide a critical g for the onset of Rabi oscillations. Besides, we demonstrate that the qubit is dressed by the cavity and dissipation. Such a dressing governs the dynamics and, thus, it can be measured. Finally, we sketch an implementation for our theoretical ideas within circuit QED technology.