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Optica Acta International Journal of Optics

Quantum Theory of Second-harmonic Generation

Miguel Orszag, Patrick Carrazana, H. Chuaqui

Published Mar 1, 1983• 9 citations
Publisher

Abstract

A quantum theory of second-harmonic generation is presented in which use is made of the formal equivalence that exists between, on the one hand, the emission of a system of N two-level atoms that interact via the electromagnetic field and, on the other, the generation of second harmonics. The N -atom system is described in terms of the Dicke model, but in the so-called ‘restricted rotating-wave approximation’. An analytical, closed-form expression is found for the number of second-harmonic photons as a function of time. Also, a comparison with the semiclassical theory and with experiments is given.

DOI
10.1080/713821173

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A quantum theory of second-harmonic generation is presented in which use is made of the formal equivalence that exists between, on the one hand, the emission of a system of N two-level atoms that interact via the electromagnetic field and, on the other, the generation of second harmonics. The N -atom system is described in terms of the Dicke model, but in the so-called ‘restricted rotating-wave approximation’. An analytical, closed-form expression is found for the number of second-harmonic photons as a function of time. Also, a comparison with the semiclassical theory and with experiments is given.
Motivated by recent experiments on second-harmonic generation in three-level systems [K. S. Yngvesson and E. L. Kollberg, Appl. Phys. Lett. 36, 104 (1980)], a semiclassical description of resonant three-wave parametric interactions in three nondegenerate levels is considered. Within the limitations of the rotating-wave approximation, analytical nonperturbative solutions for the nonlinear polarization are derived, which are then coupled to Maxwell's equations and integrated numerically. The results of sample calculations for second-harmonic generation show (a) a maximum harmonic conversion occurs on full resonance, (b) the conversion efficiency can be > 50% but requires a strongly saturating pump, (c) the conversion distance increases with pumping intensity, and (d) the inclusion of ac Stark splittings is crucial for the achievement of high conversion efficiency. The role of level splittings is further explored for the case of a resonant parametric oscillator.
A quantum model of N two-level atoms coupled by a single-mode radiation field under a restricted rotating-wave approximation is studied, assuming a dipole-interaction approximation and that the total number of excitations of the system is conserved. A solution is found for the case of spontaneous emission when the atoms are initially prepared in the state of complete inversion. This solution is valid for all times. A general solution of the model is also found for any initial configuration of the system, in the cases of both spontaneous and stimulated emission. This solution is in a closed form and valid for times tau < tau max .
A non-relativistic quantum theory of matter-photon interaction is formulated, within the framework of the dipole approximation, by employing a canonical transformation to diago­ nalize the A' term plus the free photon Hamiltonian. The matter-photon interaction is there­ by expressed rigorously in a renormalized A·p form. Using this form for the Dicke model of two-level atoms interacting with a single-mode radiation field, we examine a possibility of the occurrence of stationary coherent states which could arise for a photon and atomic polarization as the ground state of the system, deducing a condition for such coherent states. The condition, expressed as an inequality which imposes a stable nontrivial Bloch angle of the uniform atomic polarization, is identical with that for the occurrence of a second order phase transition, and indicates that the non-existence of a second order phase transition pointed out by Rza2ewski et al. io; in fact a consequence of the present treatment of the A'­ term renormalization. However, a fulfillment of the condition is recovered, if exchange­ type atom-atom interaction is taken into account in the matter system. We also present a method to construct an effective Hamiltonian of atom-atom interaction which is equivalent to the above Dicke model in the thermodynamic limit.
The interaction of electrons with strong laser fields is usually treated with semiclassical theory, where the laser is represented by an external field. There are analytic solutions for the free electron wave functions, which incorporate the interaction with the laser field exactly, but the joint effect of the atomic binding potential presents an obstacle for the analysis. Moreover, the radiation is a dynamical system, the number of photons changes during the interactions. Thus, it is legitimate to ask how can one treat the high order processes nonperturbatively, in such a way that the electron-atom interaction and the quantized nature of radiation be simultaneously taken into account? An analytic method is proposed to answer this question in the framework of nonrelativistic quantum electrodynamics. As an application, a quantum optical generalization of the strong-field Kramers-Heisenberg formula is derived for describing high-harmonic generation. Our formalism is suitable to analyse, among various quantal effects, the possible role of arbitrary photon statistics of the incoming field. The present paper is dedicated to the memory of Prof. Dr. Fritz Ehlotzky, who had significantly contributed to the theory of strong-field phenomena over many decades.
We present a theoretical treatment for third-harmonic generation in gaseous systems by describing the atomic system as well as the radiation field quantum mechanically. The atomic system is represented by two energy levels. The effects of detuning in the two-photon interactions and the atomic motion on the third-harmonic generation are taken into account. The photon statistics of the third-harmonic field are obtained. The average photon number, second-order coherence, and output variance of the third-harmonic field are discussed. The above results show a nonlinear dependence on the detuning in the two-photon interactions and the Doppler parameter.