Quantum statistics of light after two-photon absorption and emission processest

Salvatore Carusotto

Published Jan 1, 1980• 1 citation

Abstract

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 .

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Quantum statistics of light after two-photon absorption and emission processest

Salvatore Carusotto

Published Jan 1, 1980• 1 citation

OpenAlex

Abstract

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Two-Photon Absorption and Field Correlation Functions

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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 $w_{2}^{}{}_{}{}^{(\mathrm{ch})}$ for chaotic or Gaussian light to the rate $w_{2}^{}{}_{}{}^{(l)}$ 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 $b$ of the field, we obtain the previously derived result $\frac{w_{2}^{}{}_{}{}^{(\mathrm{ch})}}{w_{2}^{}{}_{}{}^{(l)}}=2$ . 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.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 $w_{2}^{}{}_{}{}^{(\mathrm{ch})}$ for chaotic or Gaussian light to the rate $w_{2}^{}{}_{}{}^{(l)}$ 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 $b$ of the field, we obtain the previously derived result $\frac{w_{2}^{}{}_{}{}^{(\mathrm{ch})}}{w_{2}^{}{}_{}{}^{(l)}}=2$. 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.

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Jan 1, 20070 citations

w_{2}^{}{}_{}{}^{(\mathrm{ch})}

for chaotic or Gaussian light to the rate

w_{2}^{}{}_{}{}^{(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

b

of the field, we obtain the previously derived result

\frac{w_{2}^{}{}_{}{}^{(\mathrm{ch})}}{w_{2}^{}{}_{}{}^{(l)}}=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.

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

n\mathrm{th}

-order time-dependent perturbation theory, that the multiphoton absorption probability depends on the

(n, n)\mathrm{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

n

! 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

n

-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.