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Journal of Physics A Mathematical and General

Off-diagonal density matrix for single-beam two-photon absorbed light

H D Simaan, R. Loudon

Published Feb 1, 1978• 59 citations
Publisher

Abstract

The authors derive general expressions for the time dependence of off-diagonal matrix elements of the density operator for single-mode light subjected to two-photon absorption. In conjugation with previous results for the diagonal matrix elements, they enable all measurable properties of the light to be calculated. The case of electric-field measurements is briefly considered.

DOI
10.1088/0305-4470/11/2/018

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The authors derive general expressions for the time dependence of off-diagonal matrix elements of the density operator for single-mode light subjected to two-photon absorption. In conjugation with previous results for the diagonal matrix elements, they enable all measurable properties of the light to be calculated. The case of electric-field measurements is briefly considered.
Abstract In order to continue and generalize the studies of the density matrix of a light field undergoing k‐photon absorption, in this paper we put the emphasis on the off‐diagonal elements. The solution obtained earlier for the diagonal elements describing the photon statistics can be found as a special case but will not be discussed again. The general solution calculated by recursion shows an asymptotic behaviour if the initial photon number is sufficiently high. Only the initial phase information survives. Illustrating the solution we start with coherent light and a generalized coherent state.
The authors derive the equation of motion for the field density matrix of the non-degenerate two-photon laser under conditions of two-photon resonance starting from the full microscopic hamiltonian. They calculate the mean photon numbers and establish which solutions are stable. The results are compared with the corresponding quantities obtained from the effective hamiltonian. In general, there are significant differences. It is shown how the full diagonal density matrix equations tend to the effective hamiltonian density matrix equations in an appropriate limit. The same limit applied to the microscopic off-diagonal elements does not however give the off-diagonal equations of motion obtained in the effective hamiltonian approach. The off-diagonal elements determine, for example, the laser linewidths and cross-correlation coefficient so that the effective hamiltonian approach leads to correct results for these quantities. The results show that for a full understanding of the two-photon laser, the microscopic approach is essential.
The effects of two-photon absorption from a single beam of light on the statistical properties of the beam are considered. Simple expressions are obtained for the initial rates of change and limiting steady-state values of the mean photon number and degree of second-order coherence of the beam. The general time dependences of these properties of the light and of the complete photon probability distribution are evaluated for several initial types of beam. The physical significance of the results is described, and their relationship to calculations which use a classical representation of the light is discussed.
The nonlinear interaction of light with matter is described from a quantum-statistical point of view. The phenomena of two-photon emission and two-photon absorption including both the single- and two-mode cases and the Raman effect are discussed in detail. A master equation for the density operator of the light fields alone is derived. This operator equation is converted to a c number equation and analytic solutions are obtained for the diagonal matrix elements of the density operator in the Fock representation. No linearizing approximation is introduced. These solutions allow one to compute the moments of the photon distribution for the above nonlinear processes.
Two-photon absorption and emission for a collection of two level atoms from a single mode beam of light are considered and the statistical proper- ties of the field after the interaction are investigated without placing any restriction of the population of the atomic levels . The dynamical behaviour of the field is described by assuming the atomic levels maintained with a nearly fixed popula- tion distribution . By applying iteration methods the time development for the density operator of the field is written and an expression for the coherent component is obtained . This component appears to be present only when there is a coherent component in the field before the interaction . The statistical properties of the field after the interaction, when the mean photon number is large initially, are then discussed and explicit calculations are carried out for some special cases of initial fields . Finally, it is shown that two-photon spontaneous emission field can be expressed as a superposition of chaotic fields .
Solutions of the time-dependent Schrödinger equation, in the semi-classical electric dipole approximation, are developed for the interaction of a two-level system, possessing both transition and diagonal dipole matrix elements, with applied static and sinusoidal electric fields. Exact solutions can only be obtained subject to conditions involving the orientations and strengths of the applied fields. These, and perturbative extensions, are used to discuss some of the features of multi-photon processes that depend on the existence of non-zero diagonal dipole matrix elements. Some of the results apply, in the limit of vanishing permanent moments, to the atomic two-level problem.