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arXiv:1302.3036 · Journal of Physics B Atomic Molecular and Optical Physics

Off-resonant transitions in the collective dynamics of multi-level atomic ensembles

Y. Miroshnychenko, Klaus Mølmer

Published Nov 7, 2013• 4 citations
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Abstract

We study the contributions of off-resonant transitions to the dynamics of a system of N multi-level atoms sharing one excitation and interacting with the quantized vector electromagnetic field. The rotating wave approximation significantly simplifies the derivation of the equations of motion describing the collective atomic dynamics, but it leads to an incorrect expression for the dispersive part of the atom–atom interaction terms. For the case of two-level atoms and a scalar electromagnetic field, it turns out that the atom–atom interaction can be recovered correctly if integrals over the photon mode frequencies are extended to incorporate negative values. We explicitly derive the atom–atom interaction for multi-level atoms, coupled to the full vector electromagnetic field, and we recover also in this general case the validity of the results obtained by the extension to negative frequencies of the formulas derived with the rotating wave approximation.

DOI
10.1088/0953-4075/46/22/224009

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We study the contributions of off-resonant transitions to the dynamics of a system of N multi-level atoms sharing one excitation and interacting with the quantized vector electromagnetic field. The rotating wave approximation significantly simplifies the derivation of the equations of motion describing the collective atomic dynamics, but it leads to an incorrect expression for the dispersive part of the atom–atom interaction terms. For the case of two-level atoms and a scalar electromagnetic field, it turns out that the atom–atom interaction can be recovered correctly if integrals over the photon mode frequencies are extended to incorporate negative values. We explicitly derive the atom–atom interaction for multi-level atoms, coupled to the full vector electromagnetic field, and we recover also in this general case the validity of the results obtained by the extension to negative frequencies of the formulas derived with the rotating wave approximation.
We propose a general perturbative approach to quantum-optical models without the rotating-wave approximation. We show that a generic Hamiltonian describing interaction between two subsystems can be represented as a series of operators corresponding to different transitions between bare energy levels of the whole system. Under certain relations between frequencies of interacting subsystems one of these transitions becomes resonant. The rotating-wave approximation leads to separation of the resonant transition and to appearance of the integral of motion, which makes the problem exactly solvable in this approximation. (Different resonance conditions lead to different integrals of motion.) All of the other terms in these expansion can be considered as a perturbation. They result in dynamic Stark shifts and small corrections to the integrals of motion. All possible resonances are classified, and approximate integrals of motion are found for each resonance. Examples of field-field, field-atom, and atom-atom interactions are considered.
One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.
We consider N a three-level atoms (or systems) interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations. The order of the quantum phase transitions is determined explicitly for each of the configurations Ξ, Λ and V , with and without detuning. The semi-classical and exact quantum calculations for both the expectation values of the total number of excitations and photon number have excellent correspondence as functions of the control parameters. We prove that the ground state of the collective regime obeys sub-Poissonian statistics for the and n distribution functions. Therefore, their corresponding fluctuations are not well described by the semi-classical approximation. We show that this can be corrected by projecting the variational state to a definite value of .
We present a general procedure to describe the dynamics of N degenerate quantized fields interacting resonantly with a two–level atom, all coupled with the same strength, within the rotating–wave approximation. Starting from the analysis of the two and three field cases, we generalize the method by identifying dynamical invariants that lead to a factorized form of the time–evolution operator. A unitary transformation reduces the problem to an effective Jaynes–Cummings Hamiltonian, where only one field interacts with the atom and the remaining modes contribute as free fields. Assuming initially coherent fields and an atomic superposition, we compute the atomic inversion and the mean photon number, revealing vacuum Rabi oscillations with a frequency determined by an effective coupling constant that exceeds the individual atom–field coupling, as well as the characteristic collapse-revival behavior.
Recalling that the rotating wave approximation (RWA) is only valid in the weak coupling regimes, the purpose of this paper is to study the Hamiltonian dynamics describing the full quantum mechanical approach of the interaction between various configurations of three-level atoms Lambda, V and ladder distinctly with a two-mode radiation field, while the RWA is not considered; the counter-rotating terms (CRTs) are taken into account. Generally, the presence of CRTs in the Hamiltonian prevents one to achieve an analytical solution. Moreover, as we will show in the present work, using the perturbation theory, analytical solvable Hamiltonians can be successfully obtained. According to our calculations, the contribution of CRTs within the ordinary Hamiltonian is equivalent to the replacement of the constant detuning with a specific intensity dependent detuning in the first order, and the constant atom field coupling with a particular intensity dependent (f deformed) coupling in the second order of the associated perturbation parameter. Moreover, noticing that according to the initial expression of the Hamiltonian, each mode of the field interacts only with a specific pair of the allowed transitions of each type of the three-level atom, it is surprisingly observed that via applying the mentioned approach, the obtained intensity dependent coupling functions depend on both modes of the field, f(na,nb). In this way, it is seen that the CRTs are removed with the price of arriving at some intermixed intensity dependent atom field coupling functions of the two modes of the field. In this way, the obtained final Hamiltonians are analytically solvable. At last, by determining the time evolution of the atom field wave function, we study the effects of CRTs on a few nonclassical properties of the state of the system, including the atomic population inversion and photon statistics.