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arXiv:1211.6469 · Physical Review A

Dynamical correlation functions and the quantum Rabi model

F. Alexander Wolf, Fabio Vallone, G. Romero, Marcus Kollar, E. Solano, Daniel Braak

Published Feb 22, 2013• 53 citations
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Abstract

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.

DOI
10.1103/physreva.87.023835

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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.
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.
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 present work focuses on the strong/weak interaction of many-body spin-systems with a cavity mode. It introduces the necessity of redefining the physical conditions determining the strong/weak coupling regime in those systems. In more complex systems, the effective coupling emerging from the collective dynamics may differ indeed from the actual coupling of each individual subsystem with the bosonic field. This is shown by highlighting some counter-intuitive dynamical effects properly related to the coupling regime: a Jaynes-Cummings dynamics emerging although a strong interaction is present. The universality of this result is demonstrated through the analysis of three distinct systems: a two-qubit, a two-qutrit, and an NNN-qubit chain quantum Rabi models.
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 ggg 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 ggg, but by a nontrivial dressing of the tunneling matrix element, which can be used to suppress specific frequencies through a variation of the coupling.
The Jaynes-Cummings and quantum Rabi models are fundamental to cavity and circuit quantum electrodynamics, as they describe the simplest form of light-matter interaction, where a single qubit is coupled to a single bosonic mode. A scenario that is commonly encountered in the experimental practice arises when the bosonic mode interacts with an external dissipative thermal bath, making the qubit-boson system open. In this work, we present a new analytical solution of the Lindblad master equations for the open dispersive Jaynes-Cummings model and a perturbative analytical solution of the open quantum Rabi model in the limit of weak qubit-boson coupling ggg, using the holomorphic formalism in Bargmann space. Specifically, we derive the most general solution of the local Lindblad master equation for the open dispersive Jaynes-Cummings model coupled to a thermal bath, with the only assumptions that the initial state of the qubit-boson system is separable. Additionally, we obtain a perturbative analytical solution for the open quantum Rabi model up to second order in ggg. Notably, our findings include a new formula for the qubit's steady state at zeroth order, showing that the stationary populations depend on both qubit and boson frequencies in the quantum Rabi model, but not in the Jaynes-Cummings model, regardless of the value of ggg. Our results are of general interest to the study of open quantum systems in the context of light-matter interaction.