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Journal of Applied Physics

Multiple Quantum Transition and Raman Action in Three-Level Systems

Jack S. Margolis, GEORGE L. BIRNBAUM

Published Mar 1, 1965• 4 citations
Publisher

Abstract

A theory is presented of induced Raman emission in a three-level system whose parameters are appropriate for microwave to optical frequencies. Solutions of the time-dependent Schrödinger equation are obtained for arbitrary intensity of the pump and signal radiations, which are assumed to be not too many line-widths off resonance. It is shown how the quantum-mechanical aspects of the problem may be separated from the statistical-mechanical aspects, in contrast to the density matrix techniques. The frequency dependence of the Raman emission on the frequency and intensity of the pumping radiation obtained by this method is presented and compared with the results we have obtained by other methods. The intensity and line shape of the Raman emission when the pump frequency is far off resonance are also obtained. The case of three interacting fields is also considered.

DOI
10.1063/1.1714208

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The Journal of Chemical Physics•Feb 1, 1986•47 citations
A theory is presented of induced Raman emission in a three-level system whose parameters are appropriate for microwave to optical frequencies. Solutions of the time-dependent Schrödinger equation are obtained for arbitrary intensity of the pump and signal radiations, which are assumed to be not too many line-widths off resonance. It is shown how the quantum-mechanical aspects of the problem may be separated from the statistical-mechanical aspects, in contrast to the density matrix techniques. The frequency dependence of the Raman emission on the frequency and intensity of the pumping radiation obtained by this method is presented and compared with the results we have obtained by other methods. The intensity and line shape of the Raman emission when the pump frequency is far off resonance are also obtained. The case of three interacting fields is also considered.
The multiple quantum process in the off-resonant condition of the three-level gas maser is discussed. The intensity of the emission produced by this process is calculated by a perturbation theory and also by a more general treatment which is the application of Javan's theory to the off-resonant case. Some graphical representations of the signal line under off-resonant pumping are given for the purpose of comparison with experiment. The Doppler effect in three-level maser action is analyzed from both semiclassical and quantum mechanical points of view under the presence of collision relaxation. It was shown that in a saturation process between two levels, Doppler effect behaves as to contribute to both line-width and relaxation time for the transition, and that under strong pumping, Doppler effect in pumping transition has a marked influence on both intensity and shape of the signal line through the effect of multiple quantum transition.
We investigate the properties of a homogeneously broadened three-level system interacting with two optical fields, Eω1(x,t) and Eω2(x,t), with central frequencies ω1 and ω2 whose amplitudes are assumed to be slowly varying in time compared with dephasing times of the optical transitions. We calculate Raman susceptibilities allowing each transition to be induced by both fields and without employing the rotating wave approximation (RWA); this enables correct calculation of the optical response of the medium when the fields are on-resonance or arbitrarily far off-resonance with the Raman transition. In some frequency regions the RWA causes substantial errors. A distinction is drawn between two different types of RWA: (a) a neglect of off-resonant components of the polarization for direct optical transitions, and (b) neglect of off-resonant components of off-diagonal density matrix elements of order Eω1(x,t)Eω2(x,t)* in the field amplitudes. The first RWA is known to become inaccurate when the pump frequency is significantly off-resonance with the pump transition. We show that the second RWA becomes inaccurate when ‖ω1−ω2‖ is small or comparable to the active Raman mode frequency (even when the field are close to resonance). Calculations based upon our expressions show that when the pump frequency is close to resonance with the optical transition from the ground state, a nonlinear absorption at the Stokes shifted frequency occurs. This absorption can significantly decrease the Raman gain. Also, we find that Rabi splittings in a three-level system can occur even when the pulse durations of the incident fields are short compared with the T1 times of the system. We solve analytically for the optical response of the system to two fields of arbitrary strength and frequency. We analyze this solution and its perturbative expansion in the field amplitudes. We use the Maxwell–Bloch equations developed here to propagate time-dependent pulses.
The quantum mechanical behavior of the three-level maser was experimentally studied using the rotational spectrum of HCOOH molecule at microwave frequencies. By using the 2 11 ←1 10 line at the frequency of 46581 Mc as the pumping, the induced emission of the 2 11 →2 12 line at the frequency of 4916 Mc was obtained. For a strong pumping, the splitting of emission line due to resonant modulation effect was observed. The line-shape of emission signal under off-resonant pumping was studied in detail. The emission spectrum due to simultaneous emission and absorption of two photons corresponding to the signal and the pumping was observed for the condition of far off-resonant pumping. The results were compared with the theory described in the previous paper, and was found to be in good agreement in many respects. Some discrepancies concerning emission intensity are also discussed. An application of the multiple quantum transition to a four-level system of HC·OOH is described.
The rate of absorption and emission of radiation is calculated for two laser fields interacting with a homogeneously broadened, three-level molecular system. The time-dependent Schrödinger equation is used in the density matrix formalism and the laser fields are treated classically. The laser fields, at frequencies ωpand ωs, are off resonance by δpand δs, respectively, such that|\delta_{p}|, |\delta_{s}| \ll \omega_{p}, \omega_{s}. Analytic expressions are derived for the imaginary part of the electric susceptibility,\chi"(\omega_{p})and\chi"(\omega_{s}). The present work differs from previous calculations in that the laser fields may have arbitrary values of field strength and each field may be on or off resonance. A discussion is presented of the properties of the solutions for\chi"(\omega_{p})and\chi"(\omega_{s}), for both strong and weak laser fields, based on computer evaluation of the general analytic expressions. It is shown that the solutions may be divided into single-photon and two-photon, or Raman, contributions. Conditions for optimizingdI(\omega_{s})/dzand the photon conversion efficiency are derived. These results may be applied to analyzing the pulsed, optically pumped submillimeter laser and to studying two-photon absorption in a gas.
This is the second in a series of three papers on nonlinear resonance effects in multilevel quantum systems in which quantum-mechanical systems are considered that are under the influence of any number of classically described driving fields (well defined in amplitude and phase) and are in weak contact with a thermal reservoir, such that relaxation may be described by means of phenomenological decay constants. In the first paper, referred to as I, a projection-operator formalism was developed for separating the equation of motion for the density operator into a part which may be treated by conventional time-dependent perturbation techniques and another part which, because it would give rise to secular terms using a time-dependent perturbation theory, must be treated by more exact means. In this paper three applications of the formalism developed in I are considered. The first application is to a system in which a multiple-quantum transition is taking place between two levels of the system which contains many levels between the two levels under consideration. In particular, the transition probability is calculated for these two levels, which are not directly connected by matrix elements of the interaction but are connected indirectly through intermediate states. The next application considered is the Raman effect in a four-level system. As a final application of the formalism developed in I an expression is derived for the magnetization M+(t)=Mx(t)+iMy(t){M}_{+}(t)={M}_{x}(t)+i{M}_{y}(t)M+​(t)=Mx​(t)+iMy​(t) of a system of NNN noninteracting spins under the influence of a static field in the zzz direction and an arbitrary number of rf fields in the xxx direction. As in I, a phenomenological form is assumed for the relaxation in all of these applications. It is shown in this problem that fields which are not involved in directly causing transitions in this system give rise to frequency shifts in the resonance conditions, which are accounted for in this treatment in a very natural and straightforward manner. For these fields, the rotating as well as the counter-rotating fields give shifts in the resonance condition.
The perturbative density matrix formalism is used to study the time-dependent resonance Raman scattering (RRS) and resonance hot luminescence (RHL) components of the light-induced spontaneous emission process that has been called resonance secondary radiation (RSR). A three-level system coupled in a stochastic way to a bath is used to study the RSR emission driven by a coherent laser pulse with a variable Gaussian width in time. The spectral filter in a light detection system is explicitly included in the detection step of the problem. Observable time and frequency resolved RSR spectra are then calculated numerically. The spectral and temporal properties of the time-dependent RSR spectrum are very sensitive to the width of the driving pulse when the pulse width is of the same order of magnitude as the important system dephasing constants. In this domain the RRS contribution to the RSR spectrum weakens relative to the RHL as the light pulse is shortened. In the limit that the laser pulse becomes ‘‘delta function-like’’ the RSR spectrum consists essentially of RHL. When appropriate, the numerical results are compared to the stationary RSR problem as well as to the case where the RSR emission is driven by an exponential pulse. The analytical solutions to both of these cases are presented.