We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combing Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equation. We give a closed expression for the relaxation and dephasing rates of the TLS and discuss their behavior under variation of the driving amplitude. Further, we examine the robustness of coherent destruction of tunneling (CDT) and driving-induced tunneling oscillations (DITO). We show that also for a moderate driving frequency an almost complete suppression of tunneling can be achieved for short times and demonstrate the sensitiveness of DITO to a change of the external parameters.

We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using Van-Vleck perturbation theory and going to second order in the coupling between TLS and HO, we show how the Hamiltonian of the TLS-HO system can be diagonalized analytically. Our model represents an improvement to the usually used Jaynes-Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are found: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in closed analytic form.

We examine a two-level system coupled to a quantum oscillator, typically representing experiments in cavity and circuit quantum electrodynamics. We show how such a system can be treated analytically in the ultrastrong coupling limit, where the ratio g/ between coupling strength and oscillator frequency approaches unity and goes beyond. In this regime the Jaynes-Cummings model is known to fail because counter-rotating terms have to be taken into account. By using Van Vleck perturbation theory to higher orders in the qubit tunneling matrix element we are able to enlarge the regime of applicability of existing analytical treatments, including, in particular, also the finite-bias case. We present a detailed discussion on the energy spectrum of the system and on the dynamics of the qubit for an oscillator at low temperature. We consider the coupling strength g to all orders, and the validity of our approach is even enhanced in the ultrastrong coupling regime. Looking at the Fourier spectrum of the population difference, we find that many frequencies contribute to the dynamics. They are gathered into groups whose spacing depends on the qubit-oscillator detuning. Furthermore, the dynamics is not governed anymore by a vacuum Rabi splitting which scales linearly with g, but by a nontrivial dressing of the tunneling matrix element, which can be used to suppress specific frequencies through a variation of the coupling.

We introduce an approach to studying a driven qubit-oscillator system in the ultrastrong coupling regime, where the ratio g/ between coupling strength and oscillator frequency approaches unity or goes beyond, and simultaneously for driving strengths much bigger than the qubit energy splitting (extreme driving). Both qubit-oscillator coupling and external driving lead to a dressing of the qubit tunneling matrix element of different nature: the former can be used to suppress selectively certain oscillator modes in the spectrum, while the latter can bring the qubit's dynamics to a standstill at short times (coherent destruction of tunneling) even in the case of ultrastrong coupling. The model of a two-level system coupled to a harmonic oscillator has been a standard applied in many different fields of physics. For instance, in quantum optics it is used to describe the interaction between light and matter, leading to the field of cavity quantum electrodynamics (QED), where an atom interacts with the electromagnetic field of a resonator [1,2]. In the regime of strong coupling, where coherent exchange of excitations between atom and cavity is possible, those setups have become interesting for the field of quantum information with the atom being used as qubit and the cavity as information carrier. Additionally, the enormous progress in the field of circuit QED, where atom and cavity are replaced by superconducting circuits [3][4][5], opens the door to the ultrastrong coupling regime [6][7][8] with coupling strengths g between qubit and oscillator which are of the order of the oscillator frequency (typical values for cavity QED experiments are g/ ≈ 10 −6 ). Experimental realizations beyond the strong coupling regime have recently been reported [9,10]. The physics behind the qubit-oscillator system is usually analyzed within the Jaynes-Cummings model (JCM) [11], which relies on a rotating-wave approximation (RWA) with respect to g and provides deep insight into various effects of cavity and circuit QED. However, for the ultrastrong coupling regime the RWA fails, and theories beyond the JCM are needed, see, e.g., [12]. An external probing of the qubit, e.g., by microwave radiation, can be modeled by the driven JCM [13], where a RWA is additionally invoked for the coupling between the qubit and the classical driving field, limiting the validity of the model to moderate driving amplitudes. It has been shown [14] that a strong external driving of the oscillator makes an inclusion of counter-rotating terms necessary even in the regime where the qubit's tunneling splitting equals the oscillator frequency ( = ) and for couplings g/ ≈ 0.1, parameters for which the JCM is commonly used in the undriven case. Similar effects are expected if instead of the cavity the atom is driven. Such extreme driving strengths have already been experimentally realized [15][16][17][18] leading to a dressed qubit state [19].In this work we examine analytically the spectrum and dynamics of a system exposed to both ultrastrong coupling and extreme dr...

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