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Physica Scripta

Non-Hermitian interaction of matter and light

Kh. Saaidi, Ebrahim Karimi, Khabat Heshami, P Seifpanahi

Published May 21, 2008• 6 citations
Publisher

Abstract

We investigate the non-Hermitian Hamiltonian which governs the system including two-level atom and electromagnetical field with a circular polarization vector. We find the Hamiltonian by using dipole approximation and rotating wave approximation (RWA), which lead to a non-Hermitian Hamiltonian. We solve the time-independent non-Hermitian Hamiltonian and obtain real eigenvalues of energy for this system. Finally, by solving the Schrodinger equation in the pseudo interaction picture, we show that our results and the results of Jaynes‐Cummings (JC) Hamiltonian are in excellent agreement.

DOI
10.1088/0031-8949/77/06/065002

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We investigate the non-Hermitian Hamiltonian which governs the system including two-level atom and electromagnetical field with a circular polarization vector. We find the Hamiltonian by using dipole approximation and rotating wave approximation (RWA), which lead to a non-Hermitian Hamiltonian. We solve the time-independent non-Hermitian Hamiltonian and obtain real eigenvalues of energy for this system. Finally, by solving the Schrodinger equation in the pseudo interaction picture, we show that our results and the results of Jaynes‐Cummings (JC) Hamiltonian are in excellent agreement.
The interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic field is usually analyzed in the dipole approximation by applying the rotating-wave approximation (RWA) to obtain the exactly solvable Jaynes-Cummings model (JCM). In this paper, it is shown that for a spin-1/2 particle, at rest in a constant magnetic field and for a system that obeys the electric dipole \ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1 selection rules, a choice of a perturbing field that consists of circularly polarized photons can make the RWA unnecessary for the solution of the quantum-electrodynamic (QED) problem. For these cases, the only approximation necessary to obtain an exact solution of the QED problem is to neglect the spontaneous emission of photons with a sense of polarization opposite that of the perturbing field. The matrix elements corresponding to these ``virtual'' processes are independent of the number of photons n in the applied field and become relatively small, compared with those that correspond to energy-conserving processes, as n becomes large.However, a two-level system that obeys the \ensuremath{\Delta}m=0 selection rule will require the RWA in order to reduce it to the JCM, regardless of the choice of polarization of the perturbing field. For this case, the emission of the photons that do not conserve energy is an induced process that corresponds to matrix elements that increase as \ensuremath{\surd}n+1 . It is thus concluded that, provided the right choice of selection rules and polarization of the applied field is made, the JCM provides a more accurate description of some systems than the accuracy of the RWA. Furthermore, it is concluded that an experimental test of the validity of the RWA can be made by performing the same experiments on an electric dipole \ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1 transition, first with circularly polarized light and then with linearly polarized light of the same intensity. Any difference in response of the atomic system for these two experiments can be identified with terms that are neglected in the RWA. This experimental technique may prove useful for observing the optical Bloch-Siegert shift and for investigating the failure of the RWA and the chaotic quantum behavior predicted by Milonni, Ackerhalt, and Galbraith [Phys. Rev. Lett. 50, 966 (1983)].
We consider a two-level system such as a two-level atom, interacting with a cavity field mode in the rotating wave approximation, when the atomic transition frequency or the field mode frequency is periodically driven in time. We show that in both cases, for an appropriate choice of the modulation parameters, the state amplitudes in a generic nnn-excitation subspace obey the same equations of motion that can be obtained from a static non-Hermitian Jaynes-Cummings Hamiltonian with PT\mathcal{PT}PT symmetry, that is with an imaginary coupling constant. This gives further support to recent results showing the possible physical interest of PT\mathcal{PT}PT symmetric non-Hermitian Hamiltonians. We also generalize the well-known diagonalization of the Jaynes-Cummings Hamiltonian to the non-Hermitian case in terms of pseudobosons and pseudofermions, and discuss relevant mathematical and physical aspects.
In this paper the dynamics of a trapped two-level atom interacting with a beam of electromagnetic wave of circular polarization is studied. In a frame rotating with the atomic transition frequency ω0, the motion of the atom is governed by a nonlinear coupled Jaynes-Cummings (JC) Hamiltonian. By means of this nonlinear Hamiltonian, we study the atomic inversion, squeezing effect of the vibration mode and that of the atomic dipole, and the phase property as well as its fluctuation of the vibration mode. In the Lamb-Dicke limit, we get the results similar to those of the JC model; beyond the Lamb-Dicke limit, we find the evolution of these quantities becomes very complicated. The nonlinearity weakens the collapse-revival phenomenon of the atomic inversion and the squeezing effect; the evolution of the cosine phase operator becomes less predetermined with its fluctuation not enlarged.
The scientific interest in the analytical solution of the quantum Rabi model is due to the widespread use of this simple model in quantum optics, quantum computing, cavity QED, and nanoelectromechanical systems. This interest is related to the need for the theoretical description of the interaction of a two-level system with a quantum oscillator in the case when the rotating wave approximation fails. In this Letter, we present an approach to the exact diagonalization of the quantum Rabi Hamiltonian. This approach is based on the properties of the Pauli operators and allows us to readily solve the stationary Schrodinger equation for a two-level system. First, we demonstrate the applicability of the approach to the Jaynes-Cummings Hamiltonian to get the well-known solution. Then, we obtain the eigenvalues and eigenstates for the quantum Rabi Hamiltonian using the proposed approach. It is shown that the obtained eigenstates can be represented in the basis of the eigenstates of the Jaynes-Cummings Hamiltonian.