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Moscow University Physics Bulletin

Quantum nonlinear resonance in a strong electromagnetic field

P. V. Elyutin

Published Feb 1, 2009• 0 citations
Publisher

Abstract

The motion of a one-dimensional quantum weakly anharmonic system under the influence of harmonic external field is treated in the range where the transition frequency between the neighboring levels is close to the frequency of the harmonic field (quantum nonlinear resonance). The case of the strong external field, in which the Rabi frequency for the transitions is large in comparison with the field frequency, is studied. The expression for compact wavepackets of quasienergy states that change their content and width periodically (and, possibly, strongly), is obtained.

DOI
10.3103/s0027134909010111

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The motion of a one-dimensional quantum weakly anharmonic system under the influence of harmonic external field is treated in the range where the transition frequency between the neighboring levels is close to the frequency of the harmonic field (quantum nonlinear resonance). The case of the strong external field, in which the Rabi frequency for the transitions is large in comparison with the field frequency, is studied. The expression for compact wavepackets of quasienergy states that change their content and width periodically (and, possibly, strongly), is obtained.
We consider a two-level quantum system interacting with two classical time-periodic electromagnetic fields. The frequency of one of the fields far exceeds that of the other. The effect of the high-frequency field can be averaged out of the dynamics to realize an effective transition frequency of the field-dressed two-level system. We examine the linear response, second harmonic response and Stokes and anti-Stokes Raman response of the dressed two-level system, to the weak frequency field. The vibrational resonance enhancement in each case is demonstrated for optimal strength of the high-frequency field. Our theoretical scheme is corroborated by full numerical simulation of the two-level, two-field dynamics governed by loss-free Bloch equations. We suggest that quantum optics can offer an interesting arena for the study of the vibrational resonance. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.
We study theoretically the Rabi oscillations of a resonantly driven two-level system linearly coupled to a harmonic oscillator (vibrational mode) with frequency ${\ensuremath{\omega}}_{0}$. We show that for weak coupling, ${\ensuremath{\omega}}_{p}\ensuremath{\ll}{\ensuremath{\omega}}_{0}$, where ${\ensuremath{\omega}}_{p}$ is the polaronic shift, Rabi oscillations are strongly modified in the vicinity of the Rabi-vibronic resonance ${\ensuremath{\Omega}}_{R}={\ensuremath{\omega}}_{0}$, where ${\ensuremath{\Omega}}_{R}$ is the Rabi frequency. The width of the resonance is $({\ensuremath{\Omega}}_{R}\ensuremath{-}{\ensuremath{\omega}}_{0})\ensuremath{\sim}{\ensuremath{\omega}}_{p}^{2/3}{\ensuremath{\omega}}_{0}^{1/3}\ensuremath{\gg}{\ensuremath{\omega}}_{p}$. Within this domain of ${\ensuremath{\Omega}}_{R}$ the actual frequency of the Rabi oscillations exhibits a bistable behavior as a function of ${\ensuremath{\Omega}}_{R}$. Importantly, within the resonant domain, the oscillator is highly excited, which allows us to treat it classically.
The influence of nonlinear interference processes and of changes in populations in strong electromagnetic fields, and also of the Doppler broadening on resonant four-wave frequency mixing is investigated for a system of Raman-type transitions. The case in which a quantum system interacts with one weak field and two strong fields is considered. Each of the energy levels is assumed to be perturbed by just one strong field. It is shown that the Doppler broadening alters drastically the contributions of the populations of the various levels to the investigated process and that this can alter by several orders of magnitude and nonlinear susceptibilities averaged over the velocities and influence significantly the selection of optimal configurations of the transitions and of the experimental conditions for effective resonant four-wave conversion of the radiation frequencies. A numerical analysis of the derived expressions is used to account for the experimentally observed considerable differences between the spectral dependences of the output power and the conclusions drawn from perturbation theory.
The object of the research are the quantum transitions of a many-level system between its stationary states under the influence of an alternating external electric field. The transition amplitudes and probabilities are calculated by the method of perturbation theory in the approximation of a classical external field. It is illustrated that in the case when the external field frequency is equal to the frequency of any quantum transition, then the probability of this transition fluctuate nearly the average value which is oscillating with Rabi frequency. Probabilities of other quantum transitions are very small and change chaotically with time.
The effect of a double structure of saturated absorption resonance in the field of counterpropagating light waves interacting with an atomic gas is studied. The experimental observation of this effect was first reported in 2011 in a work by our colleagues at the P.N. Lebedev Physical Institute of the Russian Academy of Sciences (Laboratory of Frequency Standards). The essence of the effect lies in the fact that, on exciting an open dipole transition, another, narrower, resonance of an opposite sign can be observed at the center of the ordinary saturated absorption resonance. A theoretical analysis of this effect has also been performed in this work in terms of a simple spectroscopic model of an atom with two nondegenerate energy levels without taking into account higher spatial harmonics of atomic polarization and polarizations of light waves (scalar model). The present work is devoted to the development of a theory of the formation of a central narrow resonance for the example of a real F g = 1 → F e = 1 atomic transition and to the study of its main characteristics (amplitude, width, contrast, and amplitude-to-width ratio). In addition, the theoretical results obtained without taking into account the influence of higher spatial harmonics and with inclusion of the influence of first higher harmonics are compared. This comparison shows that their influence on the parameters of the new nonlinear resonance is strong even in moderately intense light fields (R ~ γ, where R is the Rabi frequency). The results of this study can be of interest for quantum metrology, as well as for many experiments in which the laser-radiation frequency is stabilized by the saturated absorption resonance on open dipole transitions in atoms and molecules.
This is the second in a series of three papers on nonlinear resonance effects in multilevel quantum systems in which quantum-mechanical systems are considered that are under the influence of any number of classically described driving fields (well defined in amplitude and phase) and are in weak contact with a thermal reservoir, such that relaxation may be described by means of phenomenological decay constants. In the first paper, referred to as I, a projection-operator formalism was developed for separating the equation of motion for the density operator into a part which may be treated by conventional time-dependent perturbation techniques and another part which, because it would give rise to secular terms using a time-dependent perturbation theory, must be treated by more exact means. In this paper three applications of the formalism developed in I are considered. The first application is to a system in which a multiple-quantum transition is taking place between two levels of the system which contains many levels between the two levels under consideration. In particular, the transition probability is calculated for these two levels, which are not directly connected by matrix elements of the interaction but are connected indirectly through intermediate states. The next application considered is the Raman effect in a four-level system. As a final application of the formalism developed in I an expression is derived for the magnetization M+(t)=Mx(t)+iMy(t){M}_{+}(t)={M}_{x}(t)+i{M}_{y}(t)M+​(t)=Mx​(t)+iMy​(t) of a system of NNN noninteracting spins under the influence of a static field in the zzz direction and an arbitrary number of rf fields in the xxx direction. As in I, a phenomenological form is assumed for the relaxation in all of these applications. It is shown in this problem that fields which are not involved in directly causing transitions in this system give rise to frequency shifts in the resonance conditions, which are accounted for in this treatment in a very natural and straightforward manner. For these fields, the rotating as well as the counter-rotating fields give shifts in the resonance condition.