Connecting nth order generalised quantum Rabi models: Emergence of nonlinear spin-boson coupling via spin rotations

Abstract

We establish an approximate equivalence between a generalised quantum Rabi model and its nth order counterparts where spin-boson interactions are nonlinear as they comprise a simultaneous exchange of n bosonic excitations. Although there exists no unitary transformation between these models, we demonstrate their equivalence to a good approximation in a wide range of parameters. This shows that nonlinear spin-boson couplings, i.e. nth order quantum Rabi models, are accessible to quantum systems with only linear coupling between boson and spin modes by simply adding spin rotations and after an appropriate transformation. Furthermore, our result prompts novel approximate analytical solutions to the dynamics of the quantum Rabi model in the ultrastrong coupling regime improving previous approaches.

Spin-Boson Model as A Simulator of Non-Markovian Multiphoton Jaynes-Cummings Models

“…In particular, for an underdamped spectral density, the reaction coordinate takes a simple form as it interacts with the residual environment in a Markovian fashion. The resulting Hamiltonian is then used to generate multiphoton interaction terms, following the theory explained in [43,44], while the dissipation effects must be transformed accordingly. Furthermore, we extend the scheme to spin-boson models with structured environments.…”

“…In addition, we comment that one could consider the application of n d drivings onto the spin. As discussed in [43,44], under certain conditions that we will explain in the following section, applying spin drivings enables the simultaneous realization of different multiphoton Jaynes-Cummings interaction terms. In this manner, while a multiphoton Jaynes-Cummings model can be attained without the need for any driving, n d = 0, the realization of a multiphoton quantum Rabi model requires the application of at least one, i.e., n d = 1.…”

“…The task now consists of bringing the spin-boson Hamiltonian H SB into the form of a n-photon model, i.e, into a model containing interaction terms of the form σ ± a n and σ ± (a † ) n . For that, one could perform the approximate mapping used in [43,44] directly onto H SB . This would require the selection of a particular bosonic mode out of the environment with frequency ω q to now play the role of a in the interaction with the spin (c q → a), while treating the rest of c k =q as a residual environment.…”

“…These latter schemes realize effective multiphoton exchange terms by exploiting the nonlinear fashion in which the spin and bosonic degrees of freedom couple. It is however still possible to realize such multiphoton models even when the setup comprises solely a linear, i.e., standard, interaction mechanism, and thus, it is not suited for a direct simulation of these models, as shown in [43].…”

“…In this article, we follow the theoretical framework developed in [43,44], combining the ideas of the reactioncoordinate mapping [22][23][24][25][26][27][28] to show that the paradigmatic spin-boson model, featuring a continuum of bosonic modes, can serve as an analogue quantum simulator for the realization of different dissipative multiphoton Jaynes-Cummings models by tuning the frequency and bias parameter of the spin. In this manner, we demonstrate the emergence of a connection between the dynamics of these paradigmatic and fundamental quantum models, which was not previously unveiled.…”