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Physical Review A

Coherent irradiation of multilevel atoms in branched and cyclic configurations

S. P. Krinitzky, David Thomas Pegg

Published Jan 1, 1986• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.33.403

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 3, 1994•1 citation
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 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.
Laser isotope separation schemes which require resonant photoexcitation of atoms or molecules with complex spectra will in general involve processes more complicated than a simple chain of transitions between successive pairs of energy levels. In the rotating-wave approximation these situations can be described by excitation graphs with side branches and cycles. For many cases a simple change of basis removes the time dependence of the interaction Hamiltonian and reduces the calculation of level populations and interlevel flows to an eigenvalue problem. By this method it is shown that a side branch with a large matrix element strongly inhibits the flow of population along the path to which it is attached. When several phase-locked lasers are employed, the populations and flows depend on certain linear combinations of the phases, one for each independent cycle in the excitation graph. In the simplest one-cycle example this phase dependence allows a continuous change from no population inversion to complete inversion.
By constructing a vector model we study the coherent dynamics of a special three-level quantum system--one with equal detunings and equal Rabi frequencies. This three-level problem can be fully related to a two-level problem. We find that complete population depletion can be achieved even in the case of constant nonzero detunings when the three-level system is prepared initially in the common level of the two transitions. Complete population depletion with constant detunings can also occur in other odd-number multilevel systems. An explanation in terms of an equivalent resonant excitation is presented.
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 develop the group-theoretical approach for analytically solving the time-dependent Schrodinger equations for multilevel atoms driven by amplitude- and/or frequency-modulated concurrent laser fields. The time evolution of externally driven N-level atoms is calculated in an explicit matrix form. The exact analytical conditions for complete population transfer and its return are simply connected with the parameters of the symmetry. We consider in detail the case of a four-level system driven by concurrent laser pulses of different forms.