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Physical Review A

Linear constants of motion for a three-level atom excited by two modulated electromagnetic waves

Hans P. W. Gottlieb

Published Dec 1, 1982• 36 citations
Publisher

Abstract

The excitation of a three-level atom by two laser fields is analyzed in the case when the fields are sinusoidally amplitude modulated and have a phase difference of $\frac{\ensuremath{\pi}}{2}$. Two independent linear combinations of components of the coherence vector may be constructed which are constants of the motion. The time-evolution matrix in this case is constant, so the equations of motion may be solved in terms of its eigenvalues.

DOI
10.1103/physreva.26.3713

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Physical Review A•Sep 1, 1998•62 citations
The excitation of a three-level atom by two laser fields is analyzed in the case when the fields are sinusoidally amplitude modulated and have a phase difference of \ensuremathπ2\frac{\ensuremath{\pi}}{2}2\ensuremathπ​. Two independent linear combinations of components of the coherence vector may be constructed which are constants of the motion. The time-evolution matrix in this case is constant, so the equations of motion may be solved in terms of its eigenvalues.
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.
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.
We present a detailed experimental study of the frequency-modulated excitation of a two-level atom, using a microwave field to drive transitions between two Rydberg Stark states of potassium. In the absence of a modulation the interaction is the standard model of the Rabi problem, producing sinusoidal oscillations of the population between the two states. In the presence of a frequency modulation of the interacting field, however, the time evolution of the system is significantly modified, producing square wave oscillations of the population, sinusoidal oscillations at a different frequency, or even sinusoidal oscillations built up in a series of stair steps. The three responses described above are each found in a different regime for the frequency of the modulation with respect to the unmodulated Rabi frequency: the low-, high-, and intermediate-frequency regimes, respectively.