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Physical Review A

Second invariant in an excited three-level system

Hans P. W. Gottlieb

Published Jul 1, 1985• 19 citations
Publisher

Abstract

For a three-level atom excited by two laser fields whose amplitudes are sinusoidally modulated with a phase difference of \ensuremath{\pi}/2, we show that, rather surprisingly, a second (independent) linear constant of motion may be obtained even more directly by the coherence vector method than by the recent density-matrix formulation of Hioe. As a consequence it is found that the two corresponding eigenvectors of the time-independent time-evolution matrix in the degenerate subspace with two zero eigenvalues are orthogonal if a certain relation holds between the laser modulation and Rabi frequencies.

DOI
10.1103/physreva.32.653

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Optica Acta International Journal of Optics•Apr 1, 1986•8 citations
For a three-level atom excited by two laser fields whose amplitudes are sinusoidally modulated with a phase difference of \ensuremath{\pi}/2, we show that, rather surprisingly, a second (independent) linear constant of motion may be obtained even more directly by the coherence vector method than by the recent density-matrix formulation of Hioe. As a consequence it is found that the two corresponding eigenvectors of the time-independent time-evolution matrix in the degenerate subspace with two zero eigenvalues are orthogonal if a certain relation holds between the laser modulation and Rabi frequencies.
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.
When two-level atoms interact with laser light which is fully amplitude modulated at frequency omega , it is found that if omega is a particular fraction of the Rabi frequency then the coherence associated with the density matrix element rho 12 , or dipole, can be suppressed, along with the modulation in the inversion, provided omega also greatly exceeds the longitudinal decay rate Gamma . Increasing Gamma , by random collisions for example, induces a non-zero oscillating dipole and a modulation of the inversion, the enhancements of which are directly proportional to Gamma , giving an example of a coherent effect produced by an incoherent perturbation of a random nature.
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.