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Optica Acta International Journal of Optics

On the Photon Statistics of Two-photon Stimulated Emission

Jan Peřina, Vlasta Peřinová

Published Jul 1, 1983• 4 citations
Publisher

Abstract

Using the generalized Fokker-Planck equation approach it is shown that two-photon stimulated emission cannot be a source of light exhibiting antibunching of photons and therefore the two-photon coherent states, also providing antibunching, cannot serve as a fully appropriate description of two-photon stimulated emission, if losses are not considered.

DOI
10.1080/713821303

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•Sep 28, 2017•1 citation
Using the generalized Fokker-Planck equation approach it is shown that two-photon stimulated emission cannot be a source of light exhibiting antibunching of photons and therefore the two-photon coherent states, also providing antibunching, cannot serve as a fully appropriate description of two-photon stimulated emission, if losses are not considered.
Photon statistics are obtained from exact closed-form solutions for both spontaneous and stimulated coherent emissions by a large number of two-level atoms in a Dicke state. Depending on the initial excitation level of the atoms, three types of photon statistics are obtained for the emitted radiation field. Moments of the photon number of the emitted field are calculated in the case of initial atomic coherent states as well as Dicke states, and in the case of n-photon stimulating pulses as well as coherent stimulating pulses. In the particular case of an initial "superradiant" Dicke state stimulated by a coherent-radiation field, the emitted radiation pulse is also fully coherent. In other cases the photon distributions undergo large fluctuations.
Exact solutions of the master equations which describe single-mode mmm-photon absorption and mmm-photon emission processes in a two-level atomic system are obtained using a density-matrix approach. Some aspects of the photon statistics, e.g., antibunching, are discussed.
Stimulated emission can be defined as the process when an incoming photon stimulates an additional quantum of energy from an atom into the same electromagnetic mode as the impinging photon. Hence, the two outgoing photons are identical. In a waveguide or free space, this intuition is typically found through Fermi's Golden rule, however, this does not properly account for the wave-like nature of the photons. Here, I present an exact solution to stimulated emission from a quantum two-level atom that properly accounts for the incoming and outgoing electromagnetic modes. This result is valid over a huge range of incident photon numbers. For a single incident photon, it shows how the photon must properly mode match the two-level system to cause stimulated emission. For a Fock state drive with large photon number, the exact solution shows how a two-level system Rabi oscillates with the traveling Fock mode as it passes by. I additionally use the same formalism to show that stimulated emission by a coherent pulse cannot be understood as an additional photon being emitted into the incident coherent state because the two-level system's state factorizes with the electromagnetic field's coherent state. Recent advances in superconducting circuits make them an ideal platform to test these predictions.
Crucial experiments of physics, such as the quantum mechanical version of the double slit experiment, have so far not been performed with true single (antibunching) photons. Instead, attenuated classical light sources such as bulbs or lasers have been suggested to provide the photons in these experiments. Since attenuators do not change photon statistics, i.e. do not convert classical into non - classical light, reliable single photon beams cannot be generated on this way. Antibunching sources, for example single molecules, are required. Due to the lack of such sources in the past, some extrapolations from the many - photon case to the one photon case have remained speculative, for example the long wavetrain model for a single photon. Alternative hypotheses, elaborated by considering details of the emission of single photons by single molecules are equally probable. Arguments are presented which disfavour the wavetrain model and favour the short pulse as a model for a photon.
Interference of two beams produced at separate biphoton sources was first observed more than two decades ago. The phenomenon, often called coherence without induced emission, has recently gained attention after its applications to imaging, spectroscopy, and measuring biphoton correlations have been discovered. The sources used in the corresponding experiments are nonlinear crystals pumped by laser light. The use of a laser pump makes the occurrence of induced (stimulated) emission unavoidable and the effect of stimulated emission can be observed in the joint detection rate of the two beams. This fact raises the question whether the stimulated emission also lays a role in inducing the coherence. Here we investigate a case in which the crystals are pumped with a single-photon Fock state. We find that coherence is induced even though the possibility of stimulated emission is now fully ruled out. Furthermore, the joint detection rate of the two beams becomes ideally zero and does no longer change with the pump power. We illustrate our results by numerical simulations and by comparisons with experimental findings. Our results rule out any classical or semi-classical explanation of the phenomenon and also imply that similar experiments can be performed with fermions, for which stimulated emission is strictly forbidden.