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Physical Review A

Linear and nonlinear constants of motion for two-photon processes in three-level systems

F. T. Hioe

Published Jun 1, 1984• 40 citations
Publisher

Abstract

Explicit expressions for the constants of motion and the complete solution for the time evolution of the density matrix of a three-level system excited by two laser fields whose amplitudes are sinusoidally modulated and have a phase difference of $\frac{\ensuremath{\pi}}{2}$ are presented.

DOI
10.1103/physreva.29.3434

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Explicit expressions for the constants of motion and the complete solution for the time evolution of the density matrix of a three-level system excited by two laser fields whose amplitudes are sinusoidally modulated and have a phase difference of \ensuremathπ2\frac{\ensuremath{\pi}}{2}2\ensuremathπ​ are presented.
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.
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 nonlinear coherence in three-level systems exposed to two monochromatic radiation fields is treated using the unitary symmetries of SU(3). The Hamiltonian and the initial density operator are brought by a series of unitary transformations into a form in which the problem is reduced to a nutation between the two levels connected by the two-photon transition, i.e., to a time-dependent unitary rotation within the proper SU(2) subgroup. The net effect of the two modes in which the photons can combine in the nonlinear transition is shown to vanish gradually as the three levels approach an equidistant pattern. This certifies that two-photon processes can occur in systems which obey SU(3) symmetry, but not in those which obey the symmetry of SU(2). Four independent constants of motion are identified for the process of two-photon nutation in three-level systems.
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.
We study the evolution in time and in one space coordinate of picosecond laser pulses interacting with a nonlinear medium modeled by three quantum states. This is done by solving numerically the equation of evolution of the density matrix elements together with the Maxwell equation. The nonperturbative treatment of the density matrix allows us to understand the contribution of the different optical transitions and populations to the pulse deformation during its propagation. We apply the model to CuCl where the quantum levels are the fundamental, the excitonic, and the biexcitonic states. Near the two-photon resonance, the pulse is strongly altered and the excitonic and biexcitonic levels show a population inversion. A modulation of the amplitude and a superbroadening of the phase occur during the propagation.