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Journal of Physics A Mathematical and General

Competition between one- and two-photon absorption processes

V. V. Dodonov, Salomon Sylvain Mizrahi

Published May 7, 1997• 13 citations
Publisher

Abstract

We obtain an exact analytical solution to the master equation for the diagonal density matrix elements of the one-mode quantized field, when both one- and two-photon absorption processes are present. Explicit expressions for the time dependences of the factorial moments are found. The special cases of the initial Fock's, binomial, negative binomial, thermal, and coherent states, as well as of their even/odd counterparts are considered in detail. The existence of the universal time-dependent distribution of initially highly excited states is discovered, and simple explicit expressions are given for some specific values of parameters. This distribution holds for times exceeding the transition time of the order of , , being the two-photon absorption coefficient and the initial mean photon number, respectively. The transition time from any initial state to the ground state is shown to be finite even for highly excited states, provided that . Although the final stage of evolution is characterized by the sub-Poissonian statistics for any initial state, Mandel's parameter is shown to be very sensitive to small differences in high-order initial factorial moments at the intermediate stage.

DOI
10.1088/0305-4470/30/9/008

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We obtain an exact analytical solution to the master equation for the diagonal density matrix elements of the one-mode quantized field, when both one- and two-photon absorption processes are present. Explicit expressions for the time dependences of the factorial moments are found. The special cases of the initial Fock's, binomial, negative binomial, thermal, and coherent states, as well as of their even/odd counterparts are considered in detail. The existence of the universal time-dependent distribution of initially highly excited states is discovered, and simple explicit expressions are given for some specific values of parameters. This distribution holds for times exceeding the transition time of the order of , , being the two-photon absorption coefficient and the initial mean photon number, respectively. The transition time from any initial state to the ground state is shown to be finite even for highly excited states, provided that . Although the final stage of evolution is characterized by the sub-Poissonian statistics for any initial state, Mandel's parameter is shown to be very sensitive to small differences in high-order initial factorial moments at the intermediate stage.
We consider steady states of the one-mode quantized field interacting with two independent baths, each characterized by the one- and two-photon absorption and emission processes. In the absense of two-photon emission, using an exact analytical solution to the master equation for the diagonal elements of the density matrix in the Fock basis in terms of the confluent hypergeometric function, we obtain simple explicit expressions for the photon distribution function and for the factorial moments in the limiting cases of weak and strong two-photon absorption. If the two-photon absorption is strong enough, the steady state exhibits a sub-Poissonian photon statistics characterizing nonclassical behaviour, but Mandel's -parameter cannot be less . However, the distribution depends essentially on the temperature of the `one-photon bath'. For weak two-photon absorption, the stationary distribution is Gaussian, provided that the temperature of the `one-photon' bath is high enough. For an inversely populated `one-photon' bath, the -parameter is close to . In a generic case of nonzero two-photon emission probability, approximate asymptotic expressions for the factorial moments are found.
The dependence of the multiphoton absorption probability for an atomic system on the statistical properties of the incident light is investigated. It is shown, by means of nthn\mathrm{th}nth-order time-dependent perturbation theory, that the multiphoton absorption probability depends on the (n,n)th(n, n)\mathrm{th}(n,n)th normally ordered correlation functions of the electromagnetic field. For stationary fields, the transition probability per unit time is constant and is directly related to the reduced cross-spectral correlation function of the electromagnetic field. This expression for the transition probability is then analyzed under various conditions of coherence. Explicit calculations are carried out for thermal light and for laser light, and the transition rates are compared when each has a Lorentzian spectrum. In particular, when the width of the final atomic level is much larger than the bandwidth of the field, the transition rate for thermal light is nnn! times higher than with laser light of the same mean intensity. The other extreme case, i.e., when the width of the final atomic level is much smaller than the bandwidth of the field, is also considered in detail. An equation for the density operator of the field is derived; this equation governs the change in the statistical properties of the photon system in nnn-photon absorption processes. On the basis of this equation, the time-dependent properties of the photon system are then studied and some physical consequences of these equations are discussed. Some numerical solutions relating to the time dependence of the factorial moments in two-photon absorption processes are presented.
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.
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 process of two-photon absorption from two beams of light is treated theoretically. Expressions are derived for the time dependences of the joint photon probability distribution of the beams, their photon-number factorial moments and the photon-number correlation between the beams. These expressions are illustrated by numerical results for special cases and their significance is discussed in physical terms. Simpler approximate expressions are also derived for the case where one beam is much more intense than the other.