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Journal of Physics A Mathematical and General

Attenuation of a multimode field due to two-photon absorption

Ulrike Mohr, Henryk Paul

Published Jan 1, 1979• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/12/1/011

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Physical Review A•Jun 8, 2016•24 citations
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.
The two-photon absorption process is studied by considering the interaction of a quantized single-mode field with an effective two-level atom through intermediate states. The equations of motion of the coupled atom–field probability amplitudes for the effective two-level atom are obtained after adiabatically eliminating the intermediate states. These equations are then solved in the rotating-wave approximation. The time constants associated with the Rabi oscillation, collapse, and revival of the atomic inversion are derived, and the photon statistics are discussed with special reference to photon antibunching.
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.
A theory of a quantum amplifier based on two-photon transitions is presented. The model consists of an assemblage of atoms interacting with a single mode of the light field via two-photon absorption and emission. The treatment is quantum mechanical, and the emphasis is on the photon-statistical aspects of the problem. Difference-differential equations for the density matrix of the field mode are derived. The equations are solved for a special case. Differential equations for the expectation vaues of the number of photons and of the field amplitudes, and for their higher statistical moments, are also obtained. Approximate solutions for the average number of photons are found, and the effect of the amplification process on the statistical fluctuations of the number of photons and the field amplitudes is discussed. The results of this study suggest that such an amplifier does not have the coherence properties of a laser amplifier because it tends to increase the statistical fluctuations of the field.
A review is given of recent theoretical studies devoted to the problem of generating radiation fields that exhibit the opposite of the well-known bunching of photons observed in light from thermal sources, the so-called antibunching effect. It is made clear that this phenomenon reflects the corpuscular nature of light and, hence, cannot be interpreted in terms of classical electrodynamics, needing, instead, the quantum-mechanical formalism for its description. It is shown in some detail that nonlinear interaction mechanisms like multiphoton absorption and parametric three-wave interaction are suited to change the photon statistical properties of incident (in most cases coherent) light such that the output field will be endowed with antibunching properties. Special emphasis is given to the problem of correctly specifying the dimensions of the mode volume occurring in the usual single-mode treatment of the field, which is, in fact, of great practical interest, since the magnitude of the antibunching effect is determined by the inverse average number of photons contained in that volume. In a later section it is pointed out that destructive interference with a coherent reference beam provides a means of (a) effectively enhancing photon antibunching that is already present in a high-intensity field, through reduction of the intensity, and (b) transforming phase fluctuations produced in a Kerr medium into antibunching-type intensity fluctuations. On the other hand, there exists a way of directly generating light with antibunching properties, the physical mechanism being resonance fluorescence from a single atom. The main features of this technique, both theoretical and experimental, are outlined, including a discussion of the first experimental results obtained a few years ago.
A quantum theory of a two-photon laser is developed by making use of the antinormal-ordering Q function. Starting from the microscopic atom-field interaction Hamiltonian for cascade three-level atoms and including saturable absorbers, we first present a master equation for the field-density operator, and then transform the master equation into a Fokker-Planck equation for the Q function. The Q-function approach enables us to study the two-photon laser analytically, obtaining simple expressions for nonlinear gain, mean photon number, frequency pulling, natural linewidth, and photon-number variance in a unified method. We find that the field in a two-photon laser will build up from a vacuum without triggering if its linear gain is larger than the cavity loss. Also hysteresis can occur in the two-photon laser even without triggering. With an overall two-photon resonance, the normalized photon-number variance approaches a common value 11/8 well above ``threshold,'' independent of the one-photon detuning. We compare these results with previous results obtained from an effective interaction Hamiltonian. We find that the domain of validity of the effective Hamiltonian for the photon-number variance is much smaller than that for the mean photon number. Well above ``threshold'' the effective Hamiltonian overestimates both the natural linewidth and the photon-number variance.
We present a full multimode treatment of the interaction of the quantized radiation field with a single two-level atom in a one-dimensional waveguide configuration. Starting with an incident pulse consisting of an arbitrary (finite) number of photons in a general initial state, we derive the equations of motion and a closed-form expression for the shape of the pulse after its interaction with the atom. We then specialize our results to the two-photon case where a number of analytical results can be derived, for both unidirectional and bidirectional systems. We study the effects of different pulse shapes, the manifestations of the entangled, so-called bound state of the two photons, single- and two-photon detection probabilities, and provide simple approximate results for the strong-coupling (or adiabatic, or long-pulse) regime. We also discuss the requirements for a true unidirectional setup, and the application of such a setup to photon sorting proposed by Witthaut, Lukin, and S\o{}rensen [Europhys. Lett. 97, 50007 (2012)].