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Physical Review A

Nonclassical photon states generated by stimulated Compton scattering

Ching Tsung Lee

Published Aug 1, 1988• 7 citations
Publisher

Abstract

The stimulated Compton scattering of photons between two oppositely propagating coherent lights by a relativistic electron moving along the same line is studied analytically. The system is described in a moving frame so that the electron becomes nonrelativistic and the frequencies of the two lights coincide. Except for the replacement of the static wiggler by a propagating light, our system is essentially the same as a free-electron laser working in the Compton regime and described in the usual Bambini-Renieri frame. The Q representation of two-mode coherent states is adopted to describe the radiation fields, and the electron states are expressed in terms of plane waves. A perturbative solution, with the quantum recoil of the electron as the perturbation parameter, of the time evolution of the system is carried out far beyond the first order. It demonstrates that two nonclassical effects, i.e., squeezing and photon antibunching, occur as results of the scattering. It also confirms once more the fact that quantum recoils play the exclusive role in generating such nonclassical photon states. A very important improvement in the present treatment is that the electron part of the density matrix is traced out before the calculation of the squeezing effect. As a result, it is found that there is no squeezing if the initial state is a vacuum. This is, perhaps, a significant new discovery. Many-electron effects are ignored in the present study.

DOI
10.1103/physreva.38.1230

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The stimulated Compton scattering of photons between two oppositely propagating coherent lights by a relativistic electron moving along the same line is studied analytically. The system is described in a moving frame so that the electron becomes nonrelativistic and the frequencies of the two lights coincide. Except for the replacement of the static wiggler by a propagating light, our system is essentially the same as a free-electron laser working in the Compton regime and described in the usual Bambini-Renieri frame. The Q representation of two-mode coherent states is adopted to describe the radiation fields, and the electron states are expressed in terms of plane waves. A perturbative solution, with the quantum recoil of the electron as the perturbation parameter, of the time evolution of the system is carried out far beyond the first order. It demonstrates that two nonclassical effects, i.e., squeezing and photon antibunching, occur as results of the scattering. It also confirms once more the fact that quantum recoils play the exclusive role in generating such nonclassical photon states. A very important improvement in the present treatment is that the electron part of the density matrix is traced out before the calculation of the squeezing effect. As a result, it is found that there is no squeezing if the initial state is a vacuum. This is, perhaps, a significant new discovery. Many-electron effects are ignored in the present study.
When an electron is accelerated, it emits radiation. In the relativistic quantum realm the elementary radiation process is the emission of a single photon, a process known as nonlinear Compton scattering in the case of an electron moving in the presence of a strong electromagnetic wave. This process is typically described within the Furry picture, where the electromagnetic wave is described as a classical background field and the electron-positron field is quantized in the presence of that background field. Equivalently one can quantize the electron-positron field in the vacuum but then the photon emission process is described as a transition from an initial state to a final state both featuring, apart from the electron (the initial state) and the electron and the photon (the final state), the same coherent state of photons appropriately related to the electromagnetic wave. Here, we consider a more general situation where the initial and the final state feature the same squeezed coherent state. Then, we specialize to the case where the coherent state corresponds to a plane-wave field and the mostly populated modes of the coherent state are also squeezed. We show that, when quantum fluctuations induced by the squeezing in the coherent field are negligible, a condition well satisfied at available squeezing levels, the squeezing effects effectively reduce to a frequency modulation of the plane-wave field corresponding to the coherent state. By means of numerical examples we show that at already available squeezing levels the emission spectrum of nonlinear Compton scattering and the total photon yield can be altered significantly. Analytical explanations of the main numerical results are also provided.
The concept of photon antibunching as a manifestation of nonclassical character in single-mode radiation is extended to two-mode radiation. A simple criterion for the existence of nonclassical effects in two-mode radiation is established. It implies that the intermode photon bunching, rather than antibunching, can play a key role in rendering two-mode radiation nonclassical. The two-mode squeezed vacuum states are used as simple examples to illustrate the essential point of the criterion. The criterion is then used to study the nonclassical properties of two types of two-mode squeezed states. A state generated by applying the squeeze operator to the two-mode vacuum first, followed by the displacement operator, is called a two-mode coherent squeezed state. If the order of the two operators is reversed, a two-mode squeezed coherent state is obtained. It is found that the latter has a much stronger tendency to become a nonclassical state than the former, assuming that the parameters involved are all the same. A measure of nonclassical depth of radiation is adopted. It is found that, according to this measure, the effect of squeezing in making two-mode coherent squeezed states nonclassical is not monotonic. Such phenomena do not occur in squeezed coherent states.
Nonclassical light is now available in a number oflaboratories, being generated by Frank—Hertz experiment, optical parametric processes, four-wave mixing including optical fibres, by optical system with feedback, etc. Therefore it is senseful to reconsider various nonlinear optical interactions involving initially nonclassical light ,originally solved with coherent (laser) light. Adopting tools of quasidistributions related to antinormal ordering, the photocount distributions, its factorial moments and characteristics of squeezing of vacuum fluctuations can be investigated assuming initially nonclassical (squeezed) light in the framework of the Heisenberg—Langevin of Schrödinger approaches. Photon statistics of stimulated Raman scattering of squeezed light can be derived assuming phonon system in chaotic, coherent or squeezed states and including additive noise. The influence of nonlinear dynamics can be established on initially inclassical light and it can be shown that the initially nonclassical effects are encresed by the nonlinear dynamics provided that the Stokes mode is less amplified than the anti-Stokes mode is damped, whereas in the opposite case the nonclassical features are smoothed out. Also frequency detuning can be optimized to have maximum nonclassical effects, which are missing with increasing phonon number, even if the phonon system is squeezed. Small initial squeezing can by increased by the nonlinear dynamics, whereas strong squeezing can only be diminished.
An initially prepared coherent state coupled to a second-order nonlinear medium is responsible for stimulated and spontaneous hyper Raman processes. By using an intuitive approach based on perturbation theory, the Hamiltonian corresponding to the hyper Raman processes is analytically solved to obtain the temporal development of the field operators. It is true that these analytical solutions are valid for small coupling constants. However, the interesting part is that these solutions are valid for reasonably large time. Hence, the present analytical solutions are quite general and are fresh compared to those solutions under short-time approximations. By exploiting the analytical solutions of field operators for various modes, we investigate the squeezing, photon antibunching and nonclassical photon statistics for pure modes of the input coherent light responsible for hyper Raman processes. At least in one instance (stimulated hyper Raman processes for vibration phonon mode), we report the simultaneous appearance of classical (photon bunching) and nonclassical (squeezing) effects of the radiation field responsible for hyper Raman processes.
Quantum statistical properties are derived for stimulated Raman scattering by squeezed phonons including squeezing of vacuum fluctuations in radiation modes and sub-Poisson photon behaviour.Initial radiation can be in squeezed thermalized and/or sub-Poisson states, phonon system can also be squeezed.In particular, effects of initial phase conditions, losses, initial nonclassical behaviour and phonon squeezing are demonstrated in terms of reduced factorial moments up to the fifth order in combined photon-photon and photon-phonon modes.Also photon number distribution, quadrature variances and principał squeeze parameters are shown.
Nonclassical states of light may be generated by processes involving the creation or annihilation of photons in pairs. A quadratic coupling, characteristic of a parametric amplifier, generates a squeezed vacuum from a normal vacuum, and a two-photon absorber can also generate a squeezed state (though not a minimum-uncertainty state) even though it is a purely dissipative process. We consider here the simultaneous action of a quadratic pump on a two-photon absorber and demonstrate how superpositions of distinct coherent states may be generated by their combined effects. We use standard master equations to describe the time development, employing split operators and direct numerical integrations to determine the field density-matrix elements and quasiprobabilities. The purities of the nonclassical states are determined by evaluating the field entropy. Provided one-photon dissipative processes may be ignored, a pure superposition state is formed in the steady state. This superposition is destroyed if one-photon loss processes are important.