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Annalen der Physik

Saturated Multi‐Photon‐Absorption of a Nonmonochromatic Quantized Field

Ulrike Mohr

Published Jan 1, 1981• 4 citations
Publisher

Abstract

Abstract By extending a previously developed method [9] to the case of saturation, the interaction of a nonmonochromatic light beam with a saturable m‐photon‐absorber (m = 1, 2,…) is theoretically investigated for frequency widths of the incident light being small compared to the bandwidth of the absorber. It turns out that by averaging the exact Heisenberg equation of motion of the operator for the slowly varying field amplitude with respect to the atomic relaxation time and by taking the expectation value over the absorbing atomic system a reduced single‐mode equation can be derived, which refers to a mode volume precisely defined in physical terms. The resulting reduced Heisenberg equation of motion for the radiation field is compared to the single‐mode density matrix equation following from the method of SCULLY and LAMB. For ideal laser light and for chaotic light it is used to study the change of photon statistics in dependence on the degree of saturation and on the ratio of inhomogeneous and homogeneous linewidth.

DOI
10.1002/andp.19814930207

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Attenuation of a multimode field due to two-photon absorption

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Starting from an effective Hamiltonian describing the interaction of a multimode field with a two-photon absorber, an equation of motion for the quantised slowly varying field amplitude is derived which exhibits a memory effect due to the finite atomic linewidth. Averaging this equation, as a whole, over the memory time leads to a single-mode formalism, thus providing a basis for the physical interpretation of the conventional one-mode treatment of two-photon absorption, especially with respect to the photon antibunching phenomenon. In particular, a precise physical meaning can be attributed to the mode volume.

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The authors present a method of deriving a reduced density operator equation for a quantized field interacting with an arbitrary number of atoms that does not neglect atom-atom correlations. As a particular example they analyse a three-level laser operating well below saturation and solve the density matrix equations in steady state. For weak pumping the system is equivalent to the Scully-Lamb model (1966) and the corresponding equations are recovered. Using the saturation photon number as a (large) scaling parameter one can derive a Fokker-Planck equation for the Glauber-P-distribution. The stationary solution of the reduced density matrix equation is compared with the solution of the Fokker-Planck equation and-for the case of a single-atom laser-with a numerical solution of the total density matrix equation.

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Physical Review A•Dec 1, 1979•31 citations
Abstract By extending a previously developed method [9] to the case of saturation, the interaction of a nonmonochromatic light beam with a saturable m‐photon‐absorber (m = 1, 2,…) is theoretically investigated for frequency widths of the incident light being small compared to the bandwidth of the absorber. It turns out that by averaging the exact Heisenberg equation of motion of the operator for the slowly varying field amplitude with respect to the atomic relaxation time and by taking the expectation value over the absorbing atomic system a reduced single‐mode equation can be derived, which refers to a mode volume precisely defined in physical terms. The resulting reduced Heisenberg equation of motion for the radiation field is compared to the single‐mode density matrix equation following from the method of SCULLY and LAMB. For ideal laser light and for chaotic light it is used to study the change of photon statistics in dependence on the degree of saturation and on the ratio of inhomogeneous and homogeneous linewidth.
Starting from an effective Hamiltonian describing the interaction of a multimode field with a two-photon absorber, an equation of motion for the quantised slowly varying field amplitude is derived which exhibits a memory effect due to the finite atomic linewidth. Averaging this equation, as a whole, over the memory time leads to a single-mode formalism, thus providing a basis for the physical interpretation of the conventional one-mode treatment of two-photon absorption, especially with respect to the photon antibunching phenomenon. In particular, a precise physical meaning can be attributed to the mode volume.
Under usual laboratory conditions the absorption of a beam of light by a dilute two-level medium depends on the intensity and spectrum of the incident light. If the incident photon flux is small compared to the decay rate of the absorber’s excited state, saturation effects can be neglected and only the spectrum of the light is important; thus, weak-field absorption depends only on the firstorder correlation function of the incident light. This result relies, however, on a requirement that the incident beam cross-section be much larger than the absorption cross-section of an atom. Then individual atoms act as low efficiency scatterers and the statistics of the scattering process is governed by the law of large numbers applied to many scattering sites. In contrast, if it is arranged that just one atom significantly absorbs (scatters) a beam of photons, correlation functions of the incident light beyond the first-order must be considered. For a beam that is focused within an absorption cross-section, we might naively predict that every photon will be scattered and none transmitted. In fact, even for a very weak beam there is a finite probability for two photons to arrive at the atom within the excited state lifetime, and in this event at least one of them will be transmitted (statistical saturation). In this paper we calculate the transmitted photon flux for a weak beam of photons focussed strongly onto a single, resonant two-state atom. We study the dependence of the transmitted flux on the statistics of the incident photons. Photon beams derived from sources of coherent, chaotic, squeezed, coherent-antibunched, and broadband-antibunched light are considered. our calculations are based on a recently developed theory of cascaded open systems,1,2 and serve to illustrate the usefulness of this theory.
The authors present a method of deriving a reduced density operator equation for a quantized field interacting with an arbitrary number of atoms that does not neglect atom-atom correlations. As a particular example they analyse a three-level laser operating well below saturation and solve the density matrix equations in steady state. For weak pumping the system is equivalent to the Scully-Lamb model (1966) and the corresponding equations are recovered. Using the saturation photon number as a (large) scaling parameter one can derive a Fokker-Planck equation for the Glauber-P-distribution. The stationary solution of the reduced density matrix equation is compared with the solution of the Fokker-Planck equation and-for the case of a single-atom laser-with a numerical solution of the total density matrix equation.
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.