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arXiv:1612.03090 · Physical Review A

Spectral classification of coupling regimes in the quantum Rabi model

Daniel Z. Rossatto, Celso J. Villas-Bôas, Mikel Sanz, E. Solano

Published Jul 24, 2017• 111 citations

Abstract

The quantum Rabi model is in the scientific spotlight due to the recent theoretical and experimental progress. Nevertheless, a full-fledged classification of its coupling regimes remains as a relevant open question. We propose a spectral classification dividing the coupling regimes into three regions based on the validity of perturbative criteria on the quantum Rabi model, which allows us the use of exactly solvable effective Hamiltonians. These coupling regimes are (i) the perturbative ultrastrong coupling regime which comprises the Jaynes-Cummings model, (ii) a region where nonperturbative ultrastrong and nonperturbative deep strong coupling regimes coexist, and (iii) the perturbative deep strong coupling regime. We show that this spectral classification depends not only on the ratio between the coupling strength and the natural frequencies of the unperturbed parts, but also on the energy to which the system can access. These regimes additionally discriminate the completely different behaviors of several static physical properties, namely the total number of excitations, the photon statistics of the field, and the cavity-qubit entanglement. Finally, we explain the dynamical properties which are traditionally associated with the deep strong coupling regime, such as the collapses and revivals of the state population, in the frame of the proposed spectral classification.

DOI: 10.1103/physreva.96.013849

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We study the quantum Rabi model within the framework of the analytical solution developed in Phys. Rev. Lett. 107, 100401 (2011). In particular, through time-dependent correlation functions, we give a quantitative criterion for classifying two regions of the quantum Rabi model, involving the Jaynes-Cummings, the ultrastrong-coupling, and the deep strong-coupling regimes. In addition, we find a stationary qubit-field entangled basis that governs the whole dynamics as the coupling strength overcomes the mode frequency.

We revisit the analytic solution of the two-qubit quantum Rabi model with inhomogeneous coupling and transition frequencies using a displaced oscillator basis. This approach enables us to apply the same truncation rules and techniques adopted in the Rabi model to the two qubits system. The derived analytical spectra match perfectly with the numerical solutions in the parameter regime where the qubits’ transition frequencies are far off-resonance with the field frequency and the interaction strengths reach the ultrastrong coupling regime. We further explore the dynamical behavior of the two qubits as well as the evolution of entanglement. The analytical methods provide unexpectedly accurate results in describing the dynamics of the two qubits in the present experimentally accessible coupling regime. The time evolutions of the probability for the qubits show that the collapse-revival phenomena emerge, survive and finally disappear when one coupling strength increases from weak to strong coupling regimes and the other coupling strength is well into the ultrastrong coupling regime. The inhomogeneous coupling system exhibits new dynamics, which are different from the homogeneous coupling case.

The prototypical system constituted by a two-level atom interacting with a quantized single-mode electromagnetic field is described by the quantum Rabi model (QRM). The QRM is potentially valid at any light-matter interaction regime, ranging from the weak (where the decay rates exceeds the coupling rate) to the deep strong coupling (where the interaction rate exceeds the bare transition frequencies of the subsystems). However, when reaching the ultrastrong coupling regime, several theoretical issues may prevent the correct description of the observable dynamics of such a system: (i) the standard quantum optics master equation fails to correctly describe the interaction of this system with the reservoirs; (ii) the correct output photon rate is no longer proportional to the intracavity photon number; and (iii) they appear to violate gauge invariance. Here, we study the photon flux emission rate of this system under the incoherent excitation of the two-level atom for any light-matter interaction strength and consider different effective temperatures. The dependence of the emission spectra on the coupling strength is the result of the interplay between energy levels, matrix elements of the observables, and the density of states of the reservoirs. Within this approach, we also study the occurrence of light-matter decoupling in the deep strong-coupling regime and show how all of the obtained results are gauge invariant.

We examine a two-level system coupled to a quantum oscillator, typically representing experiments in cavity and circuit quantum electrodynamics. We show how such a system can be treated analytically in the ultrastrong coupling limit, where the ratio $g/\ensuremath{\Omega}$ between coupling strength and oscillator frequency approaches unity and goes beyond. In this regime the Jaynes-Cummings model is known to fail because counter-rotating terms have to be taken into account. By using Van Vleck perturbation theory to higher orders in the qubit tunneling matrix element $\ensuremath{\Delta}$ we are able to enlarge the regime of applicability of existing analytical treatments, including, in particular, also the finite-bias case. We present a detailed discussion on the energy spectrum of the system and on the dynamics of the qubit for an oscillator at low temperature. We consider the coupling strength

g

to all orders, and the validity of our approach is even enhanced in the ultrastrong coupling regime. Looking at the Fourier spectrum of the population difference, we find that many frequencies contribute to the dynamics. They are gathered into groups whose spacing depends on the qubit-oscillator detuning. Furthermore, the dynamics is not governed anymore by a vacuum Rabi splitting which scales linearly with

g

, but by a nontrivial dressing of the tunneling matrix element, which can be used to suppress specific frequencies through a variation of the coupling.

Abstract We consider a generalization of the quantum Rabi model where the two-level system and the single-mode cavity oscillator are coupled by an additional Stark-like term. By adapting a method recently introduced by Braak (2011 Phys. Rev. Lett . 107 100401), we solve the model exactly. The low-lying spectrum in the experimentally relevant ultrastrong and deep strong regimes of the Rabi coupling is found to exhibit two striking features absent from the original quantum Rabi model: avoided level crossings for states of the same parity and an anomalously rapid onset of two-fold near-degenerate levels as the Rabi coupling increases.