Exactly solvable approximating models for Rabi Hamiltonian dynamics

Abstract

The interaction between an atom and a one mode external driving field is an ubiquitous problem in many branches of physics and is often modeled using the Rabi Hamiltonian. In this paper we present a series of analytically solvable Hamiltonians that approximate the Rabi Hamiltonian and compare our results to the Jaynes-Cummings model which neglects the so-called counter-rotating term in the Rabi Hamiltonian. Through a unitary transformation that diagonlizes the Jaynes-Cummings model, we transform the counter-rotating term into separate terms representing several different physical processes. By keeping only certain terms, we can achieve an excellent approximation to the exact dynamics within specified parameter ranges.

Dynamical correlation functions and the quantum Rabi model

“…It is possible to improve the adiabatic basis in a systematic fashion [34,35] or use the RWA on top of the adiabatic approximation [36]. There are other approaches which reduce the approximative diagonalization of H R in H ± to diagonalization of finite matrices [37,38]. All these techniques are equivalent to a break-up of H ± into a set of invariant subspaces with finite dimension.…”

“…Instead, the off-diagonal components of the true eigenstates in terms of shifted oscillator states are distributed rather smoothly on both sides of the central diagonal being at the same time smaller by at least two orders of magnitude. It can be concluded that no self-consistent reduction to finitedimensional invariant subspaces [36,37] is possible which improves the adiabatic basis without creating a strong (and unphysical) symmetry. Most of the physics in the DSC regime (in fact, already for ḡ 0.7) is captured by the simple adiabatic basis.…”

Rabi model beyond the rotating-wave approximation: Generation of photons from vacuum through decoherence

“…a (a † ) is the annihilation (creation) operator of the EM field, σ z = |e e| − |g g| and σ + = |e g| (σ − = σ † + ) are atomic operators, with |g and |e denoting the ground and excited atomic states, respectively. Although largely studied over the last decades, up to now its exact analytical solution is lacking and only numerical [2,3,4,5] and approximate analytical solutions are available [6,7,8], despite the conjecture by Reik and Doucha [9] that an exact solution of RH in terms of known functions is possible. The most used analytical approach to RH is to make the rotate wave approximation (RWA), where the antirotating term g a † σ + + aσ − is neglected , since in the weak coupling regime g/ω 1, small detuning |∆| ω (∆ = ω 0 − ω), and weak field amplitude its contribution to the evolution of the system is quite small [10,11].…”

“…Therefore, the total energy of the system H is conserved and no violation of physical laws occur. Besides, recent works questioned the validity of the RWA [35,36,37] and proposed alternative analytical approximate methods [6,8]. Moreover, it was shown that the antirotating term is responsible for several novel quantum mechanical phenomena, such as quantum irreversibility and chaos [38,39], quantum phase transitions [40], implementation of Landau-Zener transitions of a qubit in circuit QED architecture [41,42], generation of atom-cavity entanglement [43,44], and simulation of the dynamical Casimir effect (DCE [45]) in semiconducting microcavities [46,47,48] or circuit QED [44].…”

Parity chain and parity chain breaking in the two-level cavity quantum electrodynamics system

“…However, this parity symmetry can be broken, such as the local bias fields [16] or the Ising interaction [17] . Recent progresses draw the extensive attention to the ultrastrong-coupling (USC) regime in the superconducting circuit cavity QED with the normalized coupling strength λ∕ω c reaching 0.1 [18,19] , where the CRT of the interaction is no longer ignored and induces the BlochSiegert (B-S) shift, and the population dynamics no longer show strictly periodic Rabi oscillation, but complicated chaotic behavior instead [20][21][22] . Moreover, in the strongcoupling regime, we have demonstrated that the permanent dipole moment (PDM) term comparable with the CRT cannot be neglected in the molecule-cavity coupling system [23] .…”