Gauge invariance is the cornerstone of modern quantum field theory [1][2][3][4]. Recently, it has been shown that the quantum Rabi model, describing the dipolar coupling between a two-level atom and a quantized electromagnetic field, violates this principle [5][6][7]. This widely used model describes a plethora of quantum systems and physical processes under different interaction regimes [8,9]. In the ultrastrong coupling regime, it provides predictions which drastically depend on the chosen gauge. This failure is attributed to the finite-level truncation of the matter system. We show that a careful application of the gauge principle is able to restore gauge invariance even for extreme lightmatter interaction regimes. The resulting quantum Rabi Hamiltonian in the Coulomb gauge differs significantly from the standard model and provides the same physical results obtained by using the dipole gauge. It contains field operators to all orders that cannot be neglected when the coupling strength is high. These results shed light on subtleties of gauge invariance in the nonperturbative and extreme interaction regimes, which are now experimentally accessible, and solve all the long-lasting controversies arising from gauge ambiguities in the quantum Rabi and Dicke models [5,[10][11][12][13][14][15][16][17][18].arXiv:1809.08749v3 [quant-ph]

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, quasi-harmonic, 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 understimate excited-state populations. Photonic ReservoirMatter Reservoir Photonic Reservoir Matter Reservoir LightMatter Light-Matter system

Two close parallel mirrors attract due to a small force (Casimir effect) originating from the quantum vacuum fluctuations of the electromagnetic field. These vacuum fluctuations can also induce motional forces exerted upon one mirror when the other one moves. Here we consider an optomechanical system consisting of two vibrating mirrors constituting an optical resonator. We find that motional forces can determine noticeable coupling rates between the two spatially separated vibrating mirrors. We show that, by tuning the two mechanical oscillators into resonance, energy is exchanged between them at the quantum level. This coherent motional coupling is enabled by the exchange of virtual photon pairs, originating from the dynamical Casimir effect. The process proposed here shows that the electromagnetic quantum vacuum is able to transfer mechanical energy somewhat like an ordinary fluid. We show that this system can also operate as a mechanical parametric down-converter even at very weak excitations. These results demonstrate that vacuuminduced motional forces open up new possibilities for the development of optomechanical quantum technologies.Effective interactions able to coherently couple spatially separated qubits [1] are highly desirable for any quantum computer architecture. Efficient cavity-QED schemes, where the effective long-range interaction is mediated by the vacuum field, have been proposed [2-4] and realized [1,5,6]. In these schemes, the cavity is only virtually excited and thus the requirement on its quality factor is greatly loosened. Based on these interactions mediated by vacuum fluctuations, a two-qubit gate has been realized [7] and two-qubit entanglement has been demonstrated [1]. Creation of multi-qubit entanglement [8] has also been demonstrated in circuit-QED systems. Very recently, it has been shown that the exchange of virtual photons between artificial atoms can give rise to effective interactions of multiple spatially-separated atoms [9, 10], opening the way to vacuum nonlinear optics. Moreover, it has been shown that systems where virtual photons can be created and annihilated can be used to realize many nonlinear optical processes with qubits [11,12]. Multiparticle entanglement and quantum logic gates, via virtual vibrational excitations in an ion trap, have also been implemented [13,14]. A recent proposal [15] suggests that classical driving fields can transfer quantum fluctuations between two suspended membranes in an optomechanical cavity system.Given these results, one may wonder whether it is possible for spatially separated mesoscopic or macroscopic bodies to interact at a quantum level by means of the vacuum fluctuations of the electromagnetic field. It is known that, owing to quantum fluctuations, the electromagnetic vacuum is able, in principle, to affect the * corresponding author: ssavasta@unime.it motion of objects through it, like a complex fluid [16]. For example, it can induce dissipation and decoherence effects on the motion of moving objects [17,18]. By using linear dis...

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The interaction between quantized electromagnetic fields in cavities and natural or artificial atoms has played a crucial role in developing our understanding of light-matter interactions and quantum technologies. Recently, new regimes beyond the weak and strong light-matter coupling of cavity-QED have been explored in several settings, wherein the light-matter coupling rate becomes comparable to (ultrastrong coupling) or even exceeds (deep-strong coupling) the photon frequency. These ultrastrong coupling regimes can give rise to new physical effects and applications, and they challenge our understanding of cavity QED; fundamental issues like the proper definition of subsystems, their quantum measurements, the structure of light-matter ground states, and the analysis of time-dependent interactions are subject to gauge ambiguities that lead to even qualitatively distinct predictions. The resolution of these ambiguities is important for understanding and designing next-generation quantum devices that can operate in extreme coupling regimes. Here we discuss and provide solutions to these ambiguities by adopting an approach based on operational procedures involving measurements on the individual light and matter components of the interacting system.