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Physical Review A

Resonance expansion versus the rotating-wave approximation

A. B. Klimov, Isabel Sainz, S. M. Chumakov

Published Dec 11, 2003• 48 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.68.063811

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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.
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.
The interaction between an atom and a one mode external driving field is an ubiquitous problem in many branches of physics and is often modeled using the Rabi Hamiltonian. In this paper we present a series of analytically solvable Hamiltonians that approximate the Rabi Hamiltonian and compare our results to the Jaynes-Cummings model which neglects the so-called counter-rotating term in the Rabi Hamiltonian. Through a unitary transformation that diagonalizes the Jaynes-Cummings model, we transform the counter-rotating term into separate terms representing several different physical processes. By keeping only certain terms, we can achieve an excellent approximation to the exact dynamics within specified parameter ranges.
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 use the path-integral technique to solve the quantum-mechanical problem of interaction of a single two-level atom with a single-mode quantized field without the rotating-wave approximation (RWA). We calculate the full spin-Bose propagator in the form of a matrix as an expansion around a path corresponding to the RWA. The effect of energy-nonconserving terms on the atomic population inversion is studied, and we show that even under the conditions in which the RWA is considered to be valid there is a significant effect on the atomic inversion due to these terms.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2{\mathrm{cm}}^{2}cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.
Perturbative corrections are derived for the rotating-wave approximation (RWA) for the single-photon and multiphoton resonance profiles due to the interaction between a two-level molecule, with nonzero permanent dipoles, and a sinusoidal time-dependent electric field. The derivation is carried out through the use of a Floquet secular equation that introduces the effects of the permanent dipole moments into the problem through the argument, d\ensuremath{\cdot}e^scrE/\ensuremath{\omega}, of Bessel functions contained in molecule-laser coupling parameters; here d=${\ensuremath{\mu}}_{22}$-${\ensuremath{\mu}}_{11}$, where ${\ensuremath{\mu}}_{\mathrm{ii}}$ is the permanent dipole of molecular state i, and scrE, e^, and \ensuremath{\omega} are the field strength, direction of polarization unit vector, and the circular frequency of the electromagnetic field (EMF). Expansions for the N-photon resonance profiles and the associated resonance frequencies and full widths at half maximum, in powers of the couplings between the transition and permanent dipole moments and the applied EMF, are obtained by expanding the Bessel functions in the perturbative results and are compared with those in the literature (available mostly for d=0 only). The perturbative results, and a series of exactly calculated two-level model absorption spectra, are used to discuss (1) the ability of the explicit perturbative corrections to the RWA to explain the differences between exact and RWA resonance profiles (for example, the positive or negative Bloch-Siegert shifts and dynamic backgrounds absent in the RWA), (2) the usefulness of the perturbative corrections in general, and (3) the effects of permanent dipoles on single-photon and multiphoton absorption spectra.