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Acta Physica Sinica

QUANTUM THEORY OF TWO-PHOTON LASER

Wang Zhicheng, 兰州大学物理系,兰州,730001

Published Jan 1, 1991• 0 citations
Publisher

Abstract

Starting from the microscopic Hamiltonian of the three-level model and making use of the slowly varying amplitude approximation and the two-photon rotating wave approximation, we derive an effective Hamiltonian for the two-level two photon model under conditions where the two-photon process dominates the other radiartive processes, so that an exact relation between the three-level model and the two-level two-photon model is established. The effective Hamiltonian includes a term come from the optical Stark effect which was neglected in the conventional treatment. The coefficient of this term may have the same order of magnitude as the effective coupling constant. Based on this effective Hamiltonian, the standard method of Haken's laser theory is applied to study the threshold, the mean photon number, the self-induced frequency shift, the photon statistics and the fluctuation in steady states. Results are compared with previous ones which neglect the Strk effect.

DOI
10.7498/aps.40.1259-2

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Starting from the microscopic Hamiltonian of the three-level model and making use of the slowly varying amplitude approximation and the two-photon rotating wave approximation, we derive an effective Hamiltonian for the two-level two photon model under conditions where the two-photon process dominates the other radiartive processes, so that an exact relation between the three-level model and the two-level two-photon model is established. The effective Hamiltonian includes a term come from the optical Stark effect which was neglected in the conventional treatment. The coefficient of this term may have the same order of magnitude as the effective coupling constant. Based on this effective Hamiltonian, the standard method of Haken's laser theory is applied to study the threshold, the mean photon number, the self-induced frequency shift, the photon statistics and the fluctuation in steady states. Results are compared with previous ones which neglect the Strk effect.
We derive the equation of motion for the field density matrix of the degenerate two-photon laser under conditions of two-photon resonance starting from the full microscopic Hamiltonian. Our results are compared with the corresponding quantities obtained from the standard effective Hamiltonian. As we have shown for the nondegenerate case, the full diagonal density-matrix equations tend to the effective Hamiltonian density-matrix equations in an appropriate limit, but the equations of motion for the off-diagonal elements do not coincide. The equations obtained using a more accurate form of the effective Hamiltonian, in which Stark shifts are included, do agree. In this paper we concentrate on the photon-number distribution and the nature of the phase transition that takes place in the neighborhood of the two-photon lasing transition threshold. We determine the steady-state mean photon numbers and consider the stability of the solutions. A solution is found where the mean photon number is zero, but this is found to be a stable solution only for sufficiently weak pumping, whereas in the standard effective Hamiltonian approach it is always stable. As the pumping strength is increased, a range of detunings is reached in which there are two stable solutions. The amplitude of the zero-photon peak diminishes very rapidly as the pumping strength increases. Finally, for sufficiently large detunings, a single, stable, steady-state solution is obtained. The nature of the phase transition is illustrated by presenting plots of the photon-number distribution against detuning and pumping rate. The photon-number fluctuations about the mean are also discussed.
A quantum theory of a two-photon laser in a three-level atomic system of the cascade type is presented starting from an original microscopic Hamiltonian. The master equation of the laser operation is derived by using the method of Scully and Lamb [Phys. Rev. A 159, 208 (1967)] for a single-mode laser containing a two-level medium. The photon-number distribution of the laser is studied in detail. The curve of the distribution may have a two-peak structure. One of the peaks located at the origin comes from the single-photon process and reflects the thermal property of the laser, while another comes from the two-photon process and reflects the coherent property of the laser. We cannot find an exact criterion for the threshold of the two-photon laser. The two-photon laser can approximately be considered as a coherent state for certain values of A/C, which can even be much smaller than the threshold for the nonresonant single-mode laser in a two-level system, but it is much larger than the threshold for the resonant one.
A quantum theory of a two-photon laser is developed by making use of the antinormal-ordering Q function. Starting from the microscopic atom-field interaction Hamiltonian for cascade three-level atoms and including saturable absorbers, we first present a master equation for the field-density operator, and then transform the master equation into a Fokker-Planck equation for the Q function. The Q-function approach enables us to study the two-photon laser analytically, obtaining simple expressions for nonlinear gain, mean photon number, frequency pulling, natural linewidth, and photon-number variance in a unified method. We find that the field in a two-photon laser will build up from a vacuum without triggering if its linear gain is larger than the cavity loss. Also hysteresis can occur in the two-photon laser even without triggering. With an overall two-photon resonance, the normalized photon-number variance approaches a common value 11/8 well above ``threshold,'' independent of the one-photon detuning. We compare these results with previous results obtained from an effective interaction Hamiltonian. We find that the domain of validity of the effective Hamiltonian for the photon-number variance is much smaller than that for the mean photon number. Well above ``threshold'' the effective Hamiltonian overestimates both the natural linewidth and the photon-number variance.
An effective Hamiltonian for the two-photon two-level model is derived form the microscopic Hamiltonian of a three-level atom in a \ensuremath{\Lambda} configuration, interacting through two-photon transitions with a light field. The effective Hamiltonian includes a term describing the optical Stark shift. With this effective Hamiltonian, we study the steady-state equation and the squeezing spectrum for a cavity-field mode interacting with an ensemble of three-level ``\ensuremath{\Lambda}-configuration'' atoms by a two-photon transition. It is shown that significant changes in the transmission characteristics and the squeezing spectrum for the output field result from the inclusion of the Stark shift.
Multiphoton transitions are of great interest in laser spectroscopy because they permit the study of new physical phenomena and because of their potential applications. Malcuit et al.1 studied two-photon transitions in atomic sodium and showed that for appropriate phase-matching four-wave mixing can overcome the usual spontaneous decay processes. We have recently developed a quantum theory2 to treat such multiwave mixing processes in two-photon media. Because of the greater complexity of two-photon transitions, many effects arise that are absent in the one-photon case, some of which have important consequences in the generation of squeezed states. For the two-photon two-level model, the field modes have frequencies approximately one-half the frequency difference between the levels. We find that our coefficients differ from the corresponding one-photon theory in that dynamic Stark shifts have a major effect and that the pump-induced coherence between the two levels has a new term in the conjugate coupling terms, which leads to improved squeezing.
Steady-state interactions of one, two, and three waves with two-photon two-level media are treated semiclassically. Dynamic Stark shifts as well as dynamic Stark splittings are included in the theory. The intermediate levels are assumed to be nonresonant, and one wave is allowed to have arbitrarily large intensity. The theory parallels the one-photon theory of multiwave mixing couched in terms of population pulsations. We find the one-, two-, and three-wave two-photon absorption coefficients and the three-wave coupling coefficients. We find similarities and significant differences between the one- and two-photon polarizations of the medium. The work treats two-photon saturation spectroscopy, laser- and optical-bistability instabilities, and phase conjugation with a unified formalism.