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arXiv:1107.3635 · Physical Review A

Analytical ground state for the Jaynes-Cummings model with ultrastrong coupling

Yuanwei Zhang, Gang Chen, Lixian Yu, Qifeng Liang, J.-Q. Liang, Suotang Jia

Published Jun 17, 2011• 34 citations

Abstract

We present a generalized variational method to analytically obtain the ground-state properties of the Jaynes-Cummings model with the ultrastrong coupling. An explicit expression for the ground-state energy, which agrees well with the numerical simulation in a wide range of the experimental parameters, is given. In particular, the introduced method can successfully solve this Jaynes-Cummings model with the positive detuning (the atomic resonant level is larger than the photon frequency), which cannot be treated in the adiabatical approximation and the generalized rotating-wave approximation. Finally, we also demonstrate analytically how to control the mean photon number by means of the current experimental parameters including the photon frequency, the coupling strength, and especially the atomic resonant level.

DOI: 10.1103/physreva.83.065802

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We present a generalized variational method to analytically obtain the ground-state properties of the Jaynes-Cummings model with the ultrastrong coupling. An explicit expression for the ground-state energy, which agrees well with the numerical simulation in a wide range of the experimental parameters, is given. In particular, the introduced method can successfully solve this Jaynes-Cummings model with the positive detuning (the atomic resonant level is larger than the photon frequency), which cannot be treated in the adiabatical approximation and the generalized rotating-wave approximation. Finally, we also demonstrate analytically how to control the mean photon number by means of the current experimental parameters including the photon frequency, the coupling strength, and especially the atomic resonant level.

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In this Brief Report, we present a perturbation theory that omits the weak multiphoton process to obtain analytical solutions for the Rabi model. Under this approximation, the Rabi model can be mapped into the solvable Jaynes-Cummings-like model by choosing a proper variation parameter. As a result, both the ground- and excited-state-energy spectrums and wave functions can be derived easily. Moreover, these explicit results agree well with the direct numerical simulation in a wide range of the experimental parameters. In addition, based on our obtained energy spectrums, the recent experimental observation of the Bloch-Siegert shift in the circuit quantum electrodynamics with ultrastrong coupling can be explained perfectly. We believe that our results have potential applications in solid-state quantum-information processing.

The eigenenergy spectrum of the Jaynes–Cummings (JC) model with and without the rotating-wave approximation (RWA) is investigated. The numerical analysis indicates that the non-RWA spectrum can only be approximated by the RWA in the range of sufficiently small coupling constant and detuning. In other region, the counter-rotating terms remarkably change the nature of the RWA energy spectrum. A simple expression with high accuracy for ground eigenenergy and eigenvector for non-RWA is available. The ground eigenvector for non-RWA shows that the ground state is not a dark state and very different from that of RWA which is a dark state.

The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by an extended Swain ansatz with conserved parity. Analytical approximations are then performed at different levels. The well-known rotating-wave approximation (RWA) is naturally covered in the present zeroth- and first-order approximations. A first-order correction to the RWA can be obtained in a second-order approximation, by which the effect of the counter-rotating-wave term emerges clearly. Concise analytical expressions are given explicitly and can be applicable up to the ultrastrong-coupling regime. A preliminary application to vacuum Rabi splitting is shown to be very successful.

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.