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Journal of Physics B Atomic and Molecular Physics

Two-photon resonance with fields amplitude modulated in quadrature

David Thomas Pegg

Published Feb 14, 1985• 22 citations
Publisher

Abstract

Cyclic commutation relations of the set of spin-one-operators J x , (J x ,J y ) and (J z ,J x ) are used to simplify and obtain a formal solution to the time evolution of a three-level atom in two-photon resonance with two laser fields which are fully amplitude modulated in quadrature. The formalism also leads to a simple physical explanation of the two linear constants of motion found by Gottlieb (1982) and can also be used to examine population trapping when the modulations of the laser fields are random but identical.

DOI
10.1088/0022-3700/18/3/017

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Cyclic commutation relations of the set of spin-one-operators J x , (J x ,J y ) and (J z ,J x ) are used to simplify and obtain a formal solution to the time evolution of a three-level atom in two-photon resonance with two laser fields which are fully amplitude modulated in quadrature. The formalism also leads to a simple physical explanation of the two linear constants of motion found by Gottlieb (1982) and can also be used to examine population trapping when the modulations of the laser fields are random but identical.
The problem of finding the time development of a three-level atom in interaction with two modulated laser beams is discussed. Formulation of the Hamiltonian in terms of spin-one operators, their squares and their anti-commutators allows exact and approximate analytical solutions for the time evolution operator to be found for a wide variety of interactions involving fields which are amplitude or frequency modulated. A fortuitous set of cyclic commutation relations among the subset of operator combinations Jx, {Jx, Jy} and {Jz, Jx} which allows an exact solution to be found for double photon resonance when the fields are amplitude modulated in quadrature, also allows the possibility of finding approximate solutions for more general modulation phase relations. The use of the formalism for describing population trapping and for finding a method of transient atomic state squeezing is also briefly discussed. The optical-frequency rotating-wave approximation is used throughout.
The author examines the problem of an unequally-spaced three-level atom irradiated by two laser beams of arbitrary intensity which have passed through the same modulator, and identify some solvable limiting cases. They find in particular the time evolution of the atom under the action of intense arbitrary shaped pulses with different Rabi frequencies, and also of tuned sinusoidally modulated beams. In the former case the author finds it is possible to achieve direct population transfer between the outer states without affecting the population of the middle state, and in the latter they find a resonance effect when the modulation frequency is in the region of a combination of the two different Rabi frequencies.
We present a detailed theoretical treatment of the response of a two-level atom to a fully-amplitude-modulated, narrow-band optical field. Specifically, we calculate the resonance behavior of modulated fluorescence and susceptibility components from the scalar continued-fraction solution of the optical Bloch equations. In the limit of vanishing damping the resonances are generally doubly branched: One set of branches reproduces the resonances of the time-averaged fluorescence, while the second set is different for each harmonic component. However, each branch is uniquely associated with an odd multiple quantum resonance. In analogy with magnetic resonance work, we derive analytical expressions for the generalized Bloch-Siegert shifts of the additional resonances. We discuss also the possibility of experimental verification of the theory.