Abstract
A quantum system weakly coupled to a zero-temperature environment will relax, via spontaneous emission, to its ground-state. However, when the coupling to the environment is ultra-strong the ground-state is expected to become dressed with virtual excitations. 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. 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. 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. 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.
Highlights
A quantum system weakly coupled to a zero-temperature environment will relax, via spontaneous emission, to its ground-state
In the limit of a discrete environment consisting of a single bosonic mode, as arises in cavity QED21, the nonperturbative limit, in which the coupling is a significant fraction of the cavity frequency, is sometimes referred to as the ultrastrong coupling (USC) regime[16,17]
In the ultra-strong coupling regime, we find that, via comparison with the reaction coordinate method[10,52,53,54] neglecting the Matsubara terms leads to an unphysical emission of photons from the ground state of the coupled light-matter system
Summary
A quantum system weakly coupled to a zero-temperature environment will relax, via spontaneous emission, to its ground-state. The advantage of considering a non-Hermitian Hamiltonian together with a pseudo-Schrödinger equation in this derivation is that, by doing so, the Dyson equation for the reduced dynamics of the system is formally equivalent to one where the system is physically interacting with a single environment via a Hermitian coupling operator characterized by the same correlation function
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