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Physical Review

Multiphoton Absorption and Coherence

Salvatore Carusotto, G. Fornaca, ERSEO POLACCO

Published Jan 25, 1968• 21 citations
Publisher

Abstract

We study the probability for multiphoton absorption processes induced by both laser and thermal light acting on an atomic system. When the thermal radiation, suitably filtered, has the same frequency spectrum and the same total energy as the laser packet, we find that the ratio between the relative probabilities for $s$-photon absorption can be expressed by ${(4\ensuremath{\pi})}^{\frac{\ensuremath{-}1}{2(s\ensuremath{-}1)}}s!{s}^{\frac{1}{2}}{(\frac{{\ensuremath{\tau}}_{c}}{\ensuremath{\tau}})}^{s\ensuremath{-}1},$ where $\ensuremath{\tau}$ is the time during which the atomic system is illuminated by thermal radiation with coherence time ${\ensuremath{\tau}}_{c}$. This shows that multiphoton absorption processes depend on the statistical properties of the incident light, and that the probability induced by the laser packet is certainly greater than that induced by the thermal light if $s<\frac{\ensuremath{\tau}}{{\ensuremath{\tau}}_{c}}$.

DOI
10.1103/physrev.165.1391

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We study the probability for multiphoton absorption processes induced by both laser and thermal light acting on an atomic system. When the thermal radiation, suitably filtered, has the same frequency spectrum and the same total energy as the laser packet, we find that the ratio between the relative probabilities for sss-photon absorption can be expressed by (4\ensuremathπ)\ensuremath−12(s\ensuremath−1)s!s12(\ensuremathτc\ensuremathτ)s\ensuremath−1,{(4\ensuremath{\pi})}^{\frac{\ensuremath{-}1}{2(s\ensuremath{-}1)}}s!{s}^{\frac{1}{2}}{(\frac{{\ensuremath{\tau}}_{c}}{\ensuremath{\tau}})}^{s\ensuremath{-}1},(4\ensuremathπ)2(s\ensuremath−1)\ensuremath−1​s!s21​(\ensuremathτ\ensuremathτc​​)s\ensuremath−1, where \ensuremathτ\ensuremath{\tau}\ensuremathτ is the time during which the atomic system is illuminated by thermal radiation with coherence time \ensuremathτc{\ensuremath{\tau}}_{c}\ensuremathτc​. This shows that multiphoton absorption processes depend on the statistical properties of the incident light, and that the probability induced by the laser packet is certainly greater than that induced by the thermal light if s<\ensuremathτ\ensuremathτcs<\frac{\ensuremath{\tau}}{{\ensuremath{\tau}}_{c}}s<\ensuremathτc​\ensuremathτ​.
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 non linear interaction between an intense laser radiation and atoms leads to ionization through the absorption of N photons from the laser radiation via laser-induced virtual states. The multiphoton ionization rate varies as a function of the laser intensity I as IN. We discuss the two most important effects which govern multiphoton ionization processes : resonance effects and laser-coherence effects. In a moderate laser intensity range (107 - 109 W cm-2) corresponding to the two, three or four-photon ionization of atoms, resonance effects play a dominant role while the photon statistics of the laser radiation are relatively unimportant. On the contrary, in the very high intensity range (1012 - 1013 W cm-2) required to observe large Nth - order ionization processes, coherence effects play a dominant role through N! enhancement in the N-photon ionization rate, while resonance effects are completely damped due to the high laser intensity. We conclude by reviewing the possible applications of multiphoton absorption processes in other fields.
The probability of an atom undergoing a transition to a given final state by means of two-photon absorption is expressed for arbitrary field states in terms of the second-order normally-ordered field correlation function, evaluated at the position of the atom. In the case of stationary fields, an expression for the absorption rate is found which reduces to a particularly simple form for narrow-bandwidth fields near resonance. The effect of field statistics is illustrated by comparing the absorption rate w2(ch)w_{2}^{}{}_{}{}^{(\mathrm{ch})}w2​​(ch) for chaotic or Gaussian light to the rate w2(l)w_{2}^{}{}_{}{}^{(l)}w2​​(l) for laser light. The rate for laser light is calculated within the context of a particular model based on the assumption of fixed field amplitude and random frequency modulation, and the chaotic light to which it is compared is assumed to have the same (Lorentzian) power spectrum. When the width ${\ensuremath{\kappa}}_{f}$ of the final atomic level is much greater than the bandwidth bbb of the field, we obtain the previously derived result w2(ch)w2(l)=2\frac{w_{2}^{}{}_{}{}^{(\mathrm{ch})}}{w_{2}^{}{}_{}{}^{(l)}}=2w2​​(l)w2​​(ch)​=2. When ${\ensuremath{\kappa}}_{f}\ensuremath{\ll}b$, on the other hand, the ratio between the two rates depends on the (mean) frequency of the field, and assumes the (maximum) value 4 when the field is exactly on resonance.