An analytical variational method for the biased quantum Rabi model in the ultra-strong coupling regime

Abstract

An analytical variational method for the ground state of the biased quantum Rabi model in the ultra-strong coupling regime is presented. This analytical variational method can be obtained by a unitary transformation or alternatively by assuming the form of ground state wave function. The key point of the method is to introduce a variational parameter λ, which can be determined by minimizing the energy functional. Using this method, we calculate physical observables with high accuracy in comparison with the numerical exact one. Our method evidently improves the results from the widely used general rotating-wave approximation (GRWA) in both qualitative and quantitative ways.

Nonorthogonal-qubit-state expansion for the asymmetric quantum Rabi model

“…There are four free parameters: α, θ, γ and p, in the NOQ Ansatz (9). For practical reasons, we would 15) of the AQRM using the GRWA [34], GVM [35], GSRWA [36], NOQ Ansatz and exact diagonalization. The parameter values are displayed in the figures, with ω = 1.…”

“…2 we compare the ground state energies obtained using the NOQ Ansatz with the known approximation schemes, namely the GRWA, GVM and GSRWA [34][35][36]. Even though the authors of the GVM [35] and the GSRWA [36] obtained approximate analytic expressions for some specific parameter regimes, we still calculate the eigenvalues and other physical observables variationally, in order to make the best comparison with their trial functions. The exact results computed from numerical diagonalization in the truncated Hilbert space are also shown as a benchmark.…”

“…The GRWA, as well as its further generalizations, have been applied to other light-matter interaction models. The example of interest here, and most relevant to the current cQED experiments, is the application to the asymmetric quantum Rabi model (AQRM) [34][35][36]. The AQRM is a generalization of the QRM with a bias term [10,11,13,14,[37][38][39][40], which naturally appears in the cQED systems to describe qubits with an asymmetric potential such as for flux qubits [7,9,[22][23][24].…”

“…The AQRM is a generalization of the QRM with a bias term [10,11,13,14,[37][38][39][40], which naturally appears in the cQED systems to describe qubits with an asymmetric potential such as for flux qubits [7,9,[22][23][24]. The approximation methods for the AQRM may be classified into the GRWA [34], the GVM [35] and the GSRWA [36], following the treatment of the QRM. The work by arXiv:2006.08913v1 [quant-ph] 16 Jun 2020…”

“…Zhang et al [34] is a straightforward application of the GRWA, while Refs. [35,36] further introduce the variational displacement and squeezing parameters, respectively. These three approximations are seen to describe the ground state of the AQRM reasonably well in specific parameter regimes.…”