Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Resonance Transitions Induced by Perturbations at Two or More Different Frequencies

Norman F. Ramsey

Published Nov 15, 1955• 168 citations
Publisher

Abstract

Resonance transition probabilities for a system in the presence of oscillatory perturbations at two or more different frequencies are discussed. It is shown that if resonance transitions are induced by a perturbation at one frequency, then the presence of the other perturbations at nonresonant frequencies alters the resonance frequency for the first perturbation. Theoretical expressions for the alteration of the resonance frequency are derived. Various applications of the derived formulas are discussed, including resonance experiments with more than one oscillatory field, molecular beam experiments in a nonuniform constant field, evaluation of "collision-narrowing" effects in nuclear paramagnetic resonance, etc. The effects of the extraneous oscillatory perturbations are shown to be especially important if their frequencies are close to Bohr frequencies for the perturbation.

DOI
10.1103/physrev.100.1191

Related papers

Transition Probability due to Random Perturbations

Amnon Yariv, W. H. Louisell

The subject of transition probabilities induced by random perturbations is treated. The expression for average transition rates between two energy levels is derived for the general case when the frequency spectrum of the perturbation and the distribution of the energy levels have comparable widths. The general expression takes an especially simple form for the special cases where the frequency bandwidth of the perturbation Hamiltonian is either much larger or much smaller than that of the energy level distribution. The formulas are applied, by way of illustration, to a number of physical problems.

Physical Review•Jan 15, 1962•0 citations

Resonant four-wave frequency mixing in Doppler-broadened transitions

A. K. Popov, S. A. Myslivets

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.

Quantum Electronics•Nov 30, 1997•6 citations

Effect of a Non-Resonant Irradiation on Atomic Energy Levels–Application to Light-Shifts in Two-Photon Spectroscopy and to Perturbation of Rydberg States

Claude N. Cohen-Tannoudji

Atomic or molecular energy levels are shifted by a non-resonant electromagnetic irradiation. These so-called "light-shifts" must be carefully taken into account in high resolution spectroscopy experiments since they perturb the atomic Bohr frequencies which one wants to measure or to use as frequency standards. In this paper, we present a brief survey of various types of light-shifts corresponding to different physical situations. We use a dressed atom approach allowing simple physical discussions and simple calculations. Some new suggestions are made.

Metrologia•Jul 1, 1977•44 citations

Effects of coupled atomic states on the resonance scattering of radiation

Rajat Srivastava, Peter R. Fontana

The excitation and decay probabilities for resonance scattering of radiation from an atom with two coupled excited states in an external static field are calculated as a function of time and frequency. Various oscillatory terms are found in the probabilities. These oscillations depend on the frequency of radiation, the external coupling and the energy difference between the excited states. Two special cases are investigated in detail. In the first case where one of the excited states does not decay a 'hole' appears in the emission line at a frequency equal to the frequency difference between the ground state and the unperturbed non-decaying excited state. In the second case where the two excited states decay with the same rate, one of the two lines in the emission spectrum is suppressed and the other is enhanced.

Journal of Physics B Atomic and Molecular Physics•Jun 1, 1974•3 citations

Time-Dependent Perturbation Theory (TDPT)

Steven M. Pascal

If the perturbation is turned on for a brief period of time and then turned off, the particle ends up in the original potential. But it may have been induced to transition from one eigenstate to another during the perturbation. The calculation of the probability of that transition is the topic of time-dependent perturbation theory (TDPT). Again, formula are presented rather than derived, and two examples are provided: a linear perturbation, and an oscillating perturbation. These examples, respectively, show how selection rules arise, and how light of a given frequency excites energy transitions of the same frequency. This latter phenomenon is called resonance .

CRC Press eBooks•Jul 12, 2023•0 citations

Nonlinear resonance effects in multilevel quantum systems weakly coupled to a thermal reservoir. II. Multiple-quantum transitions

Clayton W. Bates

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 ${M}_{+}(t)={M}_{x}(t)+i{M}_{y}(t)$ of a system of $N$ noninteracting spins under the influence of a static field in the $z$ direction and an arbitrary number of rf fields in the $x$ 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.

Physical Review A•Dec 1, 1974•0 citations
Resonance transition probabilities for a system in the presence of oscillatory perturbations at two or more different frequencies are discussed. It is shown that if resonance transitions are induced by a perturbation at one frequency, then the presence of the other perturbations at nonresonant frequencies alters the resonance frequency for the first perturbation. Theoretical expressions for the alteration of the resonance frequency are derived. Various applications of the derived formulas are discussed, including resonance experiments with more than one oscillatory field, molecular beam experiments in a nonuniform constant field, evaluation of "collision-narrowing" effects in nuclear paramagnetic resonance, etc. The effects of the extraneous oscillatory perturbations are shown to be especially important if their frequencies are close to Bohr frequencies for the perturbation.
The subject of transition probabilities induced by random perturbations is treated. The expression for average transition rates between two energy levels is derived for the general case when the frequency spectrum of the perturbation and the distribution of the energy levels have comparable widths. The general expression takes an especially simple form for the special cases where the frequency bandwidth of the perturbation Hamiltonian is either much larger or much smaller than that of the energy level distribution. The formulas are applied, by way of illustration, to a number of physical problems.
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.
Atomic or molecular energy levels are shifted by a non-resonant electromagnetic irradiation. These so-called "light-shifts" must be carefully taken into account in high resolution spectroscopy experiments since they perturb the atomic Bohr frequencies which one wants to measure or to use as frequency standards. In this paper, we present a brief survey of various types of light-shifts corresponding to different physical situations. We use a dressed atom approach allowing simple physical discussions and simple calculations. Some new suggestions are made.
The excitation and decay probabilities for resonance scattering of radiation from an atom with two coupled excited states in an external static field are calculated as a function of time and frequency. Various oscillatory terms are found in the probabilities. These oscillations depend on the frequency of radiation, the external coupling and the energy difference between the excited states. Two special cases are investigated in detail. In the first case where one of the excited states does not decay a 'hole' appears in the emission line at a frequency equal to the frequency difference between the ground state and the unperturbed non-decaying excited state. In the second case where the two excited states decay with the same rate, one of the two lines in the emission spectrum is suppressed and the other is enhanced.
If the perturbation is turned on for a brief period of time and then turned off, the particle ends up in the original potential. But it may have been induced to transition from one eigenstate to another during the perturbation. The calculation of the probability of that transition is the topic of time-dependent perturbation theory (TDPT). Again, formula are presented rather than derived, and two examples are provided: a linear perturbation, and an oscillating perturbation. These examples, respectively, show how selection rules arise, and how light of a given frequency excites energy transitions of the same frequency. This latter phenomenon is called resonance .
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.