In the latter, the excitations are called virtual because they are energetically trapped in the hybridized light-matter ground state. This regime harbors a range of new physics, including higher-order coupling effects, the possibility to excite two atoms with one photon 22, the ability to prepare Bell and GHZ states in cQED 23, and the potential to generate a ground state which contains virtual excitations 12, 24, 25, 26. In the limit of a discrete environment consisting of a single bosonic mode, as arises in cavity QED (cQED) 21, the non-perturbative limit, in which the coupling is a significant fraction of the cavity frequency, is sometimes referred to as the ultra-strong coupling (USC) regime 16, 17. However, research areas, such as energy transport in photosynthetic systems 4, 5, 6, 7, 8, 9, quantum thermodynamics 10, 11, and the ultra-strong coupling regime in artificial light-matter systems 12, 13, 14, 15, 16, 17, have demanded the development of numerically exact methods to explore non-perturbative and non-Markovian parameter regimes 18, 19, 20, which are out of reach of traditional approaches. Practically speaking, a number of perturbative approaches and assumptions such as the Born–Markov and rotating-wave approximation (RWA) are usually employed to obtain tractable solutions 2. It not only allows us to understand the relationship between quantum dissipation and classical friction, but is a powerful model to study topics ranging from physical chemistry to quantum information. The spin-boson model is a cornerstone of the theory of open-quantum systems, and its elegance often belies its power to describe a wide range of phenomena 1, 2, 3. For the pseudomode method, we present a general proof of validity for the use of superficially unphysical Matsubara-modes, which mirror the mathematical essence of the Matsubara frequencies. We compare these methods to the reaction coordinate mapping, which helps show how these sometimes neglected Matsubara terms are important to regulate detailed balance and prevent the unphysical emission of virtual excitations. To access this regime we generalize both the hierarchical equations of motion and pseudomode methods, taking into account this explosion using only a biexponential fitting function. This regime is difficult to capture with some traditional methods because of the explosion in the number of Matsubara frequencies, i.e., exponential terms in the free-bath correlation function. However, when the coupling to the environment is ultra-strong the ground-state is expected to become dressed with virtual excitations. A quantum system weakly coupled to a zero-temperature environment will relax, via spontaneous emission, to its ground-state.
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