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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Revisiting the Jaynes-Cummings-Paul model in the limit of ultrastrong coupling

Elaheh Ahmadi, Hamid Reza Chalabi, Abbas Arab, Sina Ataollah Khorasani

Published Feb 4, 2011• 6 citations
Publisher

Abstract

The interaction of quantum radiation and a two-level atom is described in the context through the Jaynes-Cummings-Paul Hamiltonian which is obtained through Heisenberg's interaction picture of the Atom-Radiation Hamiltonian. We argue that such a transformation is not mathematically exact in case of ultrastrong coupling, where the coupling rate is comparable to the transition frequency, and leads to erroneous results. In addition, we introduce an exact mathematical solution to calculate optical spectrum of this system.

DOI
10.1117/12.872174

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The interaction of quantum radiation and a two-level atom is described in the context through the Jaynes-Cummings-Paul Hamiltonian which is obtained through Heisenberg's interaction picture of the Atom-Radiation Hamiltonian. We argue that such a transformation is not mathematically exact in case of ultrastrong coupling, where the coupling rate is comparable to the transition frequency, and leads to erroneous results. In addition, we introduce an exact mathematical solution to calculate optical spectrum of this system.
We revisit the mathematical formulation of the famous Jaynes–Cummings–Paul Hamiltonian, which describes the interaction of a two-level atom with a single mode of an electromagnetic cavity reservoir. We rigorously show that under the condition of ultrastrong coupling between the atom and cavity in which the transition frequency is comparable to the coupling frequency, the bosonic field operators undergo non-sinusoidal time variations. As a result, the well-known solutions to the Jaynes–Cummings–Paul model are no longer valid, even when the rotating wave approximation is not used. We show how a correct mathematical solution could be found instead.
Ultrastrong-coupling between two-level systems and radiation is important for both fundamental and applied quantum electrodynamics (QED). Such regimes are identified by the breakdown of the rotating-wave approximation, which applied to the quantum Rabi model (QRM) yields the apparently less fundamental Jaynes-Cummings model (JCM). We show that when truncating the material system to two levels, each gauge gives a different description whose predictions vary significantly for ultrastrong-coupling. QRMs are obtained through specific gauge choices, but so too is a JCM without needing the rotating-wave approximation. Analysing a circuit QED setup, we find that this JCM provides more accurate predictions than the QRM for the ground state, and often for the first excited state as well. Thus, Jaynes-Cummings physics is not restricted to light-matter coupling below the ultrastrong limit. Among the many implications is that the system's ground state is not necessarily highly entangled, which is usually considered a hallmark of ultrastrong-coupling.
We propose a reliable scheme to realize the ultrastrong Jaynes-Cummings (JC) model by simultaneously modulating the resonance frequencies of the two-level system and the bosonic mode in the ultrastrong quantum Rabi model. We find that in both the high- and low-frequency modulation regimes, the counter-rotating terms can be completely suppressed without reducing the coupling strength of the rotating-wave terms, and hence the ultrastrong JC Hamiltonian is achieved. The ultrastrong JC interaction can not only be used to implement ultrafast quantum operations such as ultrafast quantum state transfer and quantum logic gates but also will open up a new route to the demonstration of quantum phase transition associated with the JC Hamiltonian across the deep-strong-coupling point. Some discussions on the experimental implementation of this scheme with circuit-QED systems are presented.
We study the dynamics of a system consisting of a two-level atom and a quantum oscillator coupled by the vacuum radiation field. The oscillator and the atom are assumed to have the same resonance frequency. In contrast to the two-atom or two-oscillator case, the atom-oscillator system leads to an unclosed hierarchy of equations for the operators that determine the radiation pattern emitted by the atom-oscillator system. Instead of using the operator equations, we formulate the problem in terms of density-matrix equations which can be solved once the initial conditions are specified. As an example, we looked at the radiation pattern, the decay dynamics, and the spectrum for an initial condition in which the oscillator is in its n=1n=1n=1 state and the atom is in its excited state. Dressed states (dressed by the static interaction between the atom and oscillator) prove to be especially useful for interpreting the results and comparing them with the two-atom and two-oscillator systems. All calculations are carried out using the resonance or rotating-wave approximation. The relationship of our work to damped Jaynes-Cummings models is noted.
The authors calculate scattered, transmitted and reflected spectra for a coupled two-level atom and quantized electromagnetic cavity mode driven by broad-band chaotic light. They obtain analytical results in the secular approximation and compare these with exact numerical results. Under cavity QED conditions (strong dipole coupling) the spectra show multiple resonances at the incommensurate transition frequencies of the Jaynes-Cummings Hamiltonian. They obtain expressions for linewidths and transition amplitudes that help them to assess the prospects for an experimental observation of these Jaynes-Cummings resonances.