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Optica Acta International Journal of Optics

Interaction of Three-level Atoms with Modulated Lasers

David Thomas Pegg

Published Apr 1, 1986• 8 citations
Publisher

Abstract

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.

DOI
10.1080/713821963

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Journal of the Optical Society of America B•Aug 1, 1991•26 citations
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.
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 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.
An analytic solution is presented for the time development of an n-level atom in resonant interaction with n-1 lasers in a branch configuration, which is an alternative generalization of the three-level atom to the commonly studied cascade configuration. The same analytic approach is also applied to a four-laser resonant interaction with a four-level atom in a cyclic configuration, where it is found that a change of phase of any one of the lasers has a critical effect on the population dynamics and, with certain Rabi frequency and laser phase relationships, double-quantum transitions can be eliminated completely.
The trapping states of a quantum system occupy a very special place in quantum optics. Here, we propose a method to realize the population trapping (PT) in a two-level atom driven by a carrier envelope phase (CEP) locked laser pulse. We present detailed numerical results on the influence of CEP in different detuned atoms beyond the rotating wave approximation and find the population is strongly dependent on CEP, the detuning and the Rabi frequency. This allows us to choose proper parameters to realize three specific trapping states: upper trapping state, half trapping state and lower trapping state. To explain the fantastic influence of CEP, we present an analytical solution under weak-field approximation, which is consistent with the numerical solution. We further define the action quantity of the pulse acting on the atom as the integral for the interaction Hamiltonian and find the population varies approximatively linearly with the action quantity. This investigation provides directive significance for achieving PT at arbitrary quantum states and insightful schemes to regulate and control the quantum dynamics of atoms via accurately controlled optical field, and the results could be beneficial for applications to laser cooling, isotope separation and so on.
A quantum theory of two-photon correlated-spontaneous-emission lasers (CEL's) is developed, starting from the exact atom-field interaction Hamiltonian for cascade three-level atoms interacting with a single-mode radiation field. We consider the situation where the active atoms are prepared initially in a coherent superposition of three atomic levels and derive a master equation for the field-density operator by using a quantum theory for coherently pumped lasers. The master equation is transformed into a Fokker-Planck equation for the antinormal-ordering Q function. The drift coefficients of the Fokker-Planck equation enable us to study the steady-state operation of the two-photon CEL's analytically. We have studied both resonant two-photon CEL for which there is no threshold, and off-resonant two-photon CEL for which there exists a threshold. In both cases the initial atomic coherences provide phase locking, and squeezing in the phase quadrature of the field is found. The off-resonant two-photon CEL can build up from a vacuum when its linear gain is larger than the cavity loss (even without population inversion). Maximum squeezing is found in the no-population-inversion region with the laser intensities far below saturation in both cases, which are more than 90% for the resonant two-photon CEL and nearly 50% for the off-resonant one. Approximate steady-state Q functions are obtained for the resonant two-photon CEL and, in certain circumstances, for the off-resonant one.
We present the dynamics of nondegenerate as well as degenerate two-photon processes in a three-level Rydberg atom interacting with the eigenmodes of an ideal cavity. For large one-photon detuning, the results are compared with those obtained from an effective Hamiltonian and are found to be mostly different. However, for atom–field interaction times that are comparable with that in the two-photon micromaser experiment, the two models give similar results. We also discuss the time evolution of squeezing in the individual fields in the non-degenerate case. If the two modes are of frequencies ωf + ɛ and ωf − ɛ with ɛ ≪ ωf, the radiation field can be represented by a wave of frequency ωf with amplitudes modulated at ɛ. We find squeezing in the dynamical evolution of these amplitudes.