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Physical Review A

Two-photon optical transients: Effects of spontaneous emission and phase-interrupting collisions

Maryna D'Souza, Arvind Kumar

Published Aug 1, 1981• 0 citations
Publisher

Abstract

We present solutions for two-photon optical transients from a three-level system which include the effects of (a) direct spontaneous emission to and from the nonresonant intermediate state and (b) phase-interrupting collisions. These effects are, in general, expected to be important for electronic transitions in the visible or ultraviolet region relevant for many two-photon transient experiments on free atoms. The steady-state absorption profile shows power narrowing and desaturation due to spontaneous relaxation. The two-photon delayed optical nutation signal is obtained explicitly. A perturbative calculation of optical nutation shows intensity-dependent damping. We find that for a particular case, namely, when the upper level spontaneous decay rate is twice that of the intermediate level, transient signals have especially simple forms and can be usefully employed for experimental determination of decay constants. The paper is a direct but nontrivial generalization of Schenzle and Brewer's corresponding investigation on one-photon transients.

DOI
10.1103/physreva.24.955

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We present solutions for two-photon optical transients from a three-level system which include the effects of (a) direct spontaneous emission to and from the nonresonant intermediate state and (b) phase-interrupting collisions. These effects are, in general, expected to be important for electronic transitions in the visible or ultraviolet region relevant for many two-photon transient experiments on free atoms. The steady-state absorption profile shows power narrowing and desaturation due to spontaneous relaxation. The two-photon delayed optical nutation signal is obtained explicitly. A perturbative calculation of optical nutation shows intensity-dependent damping. We find that for a particular case, namely, when the upper level spontaneous decay rate is twice that of the intermediate level, transient signals have especially simple forms and can be usefully employed for experimental determination of decay constants. The paper is a direct but nontrivial generalization of Schenzle and Brewer's corresponding investigation on one-photon transients.
We present a theory of optical two-photon rotary echo for the case when the direct couplings between the three levels due to spontaneous decay to and from the intermediate state dominate the indirect couplings (collisional, etc.) to the passive levels and the phase-interrupting (P.I.) collisions cannot be ignored. For this situation which generally obtains in the visible/U.V. region which is relevant for many two-photon transient experiments, the ‘two-photon Bloch equations’ with non-resonant intermediate state are non-trivially different from the familiar (two-level) Bloch equations with or without direct coupling. An explicit analytical calculation demonstrates two-photon rotary echo with the usual dependence on the total phase shift. The dependence of the echo peak on pulse delay time is governed by a damping constant in which ‘intensity modification’ involves only the P.I. collision constant and not the spontaneous decay constant in sharp contrast to the one-photon case. For the case of counter-propagating beams, the intensity modified echo damping constant turns out to be especially simple. We suggest that the features unique to the two-photon rotary echo can be usefully exploited for the empirical determination of the polarization decay constant.
Pulsed radio-frequency transient phenomena of resonant two-level spins are often described by the Bloch equations where the phenomenological dipole (T2{T}_{2}T2​) and population (T1{T}_{1}T1​) decay times appear. A similar description has been carried over recently into the microwave and infrared regions as well. At visible-ultraviolet wavelengths, however, a more general solution is required because the transition levels do not constitute an isolated system and also spontaneous emission can no longer be disregarded. As a consequence, the lower (1) and upper (2) transition levels are characterized by vastly different decay rates (${\ensuremath{\gamma}}_{1}\ensuremath{\ne}{\ensuremath{\gamma}}_{2}$) and the lifetime T1{T}_{1}T1​ is no longer a defined quantity. In view of laser-induced coherent transient phenomena now being observed, we present general solutions of the density-matrix equations of motion for an optically resonant two-level system in contact with a multilevel reservoir where ${\ensuremath{\gamma}}_{1}\ensuremath{\ne}{\ensuremath{\gamma}}_{2}$. Particular and perturbative solutions involving intensity-dependent damping and frequency shifts are included with applications to echo and nutation phenomena.
A new method of observing two-photon optical transients is proposed, namely, by intensity switching of incident cw laser beams. This method, which is an alternative to the standard dc Stark or frequency-switching techniques, employs the second-order optical (ac) Stark effect to throw the two-photon active levels in or out of resonance by appropriately switching the intensities of the two laser beams. Theoretical treatment for the three basic transients optical nutation, free induction decay, and two-pulse echo in thin samples using intensity switching, is given for both the cases: co- and counter-propagating beams. The feasibility of the method for the $5{S}_{\frac{1}{2}}\ensuremath{-}3{S}_{\frac{1}{2}}$ transition in Na vapor is demonstrated. An important feature is the apearance of an intensity-dependent beat frequency in the optical FID signal.
A class of molecular two-photon processes, occurring under transient or steady-state conditions, is analyzed exactly for certain cases using a density-matrix formalism. Our results are relevant to the recently observed phenomena of transient coherent Raman beats and steady-state two-photon absorption of oppositely directed laser beams. Lowest-order perturbation theories have shown that these processes are insensitive to the Doppler effect, and consequently the linewidth is unaffected by Doppler broadening and by elastic collisions that change molecular velocity. The solutions presented here are of the same form for the Raman and two-photon problems and reveal significant power-dependent frequency shifts and line broadening that correct the ideal Doppler-free solutions. For the transient case, we assume a molecular three-level quantum structure that can be switched in or out of optical resonance with cw laser radiation by means of a pulsed dc Stark field, but the solutions also apply when resonant optical pulses are introduced. During the resonant condition when the Stark pulse is on, the three levels are prepared in coherent superposition, and during the nonresonant condition following the Stark pulse, the laser field(s) stimulates the coherently prepared sample in a transient two-photon process. One of these processes, the Ramanbeat effect, has been observed in forward scattering, but the second one, transient two-photon emission, should be observable in backward scattering. Bloch-like equations are derived for this three-level problem that facilitate an exact pulse solution for state preparation. Following the pulse, transient decay is well-approximated by a perturbation calculation. For the steady-state case, an exact solution is also obtained and is of interest for continuous-wave spectroscopy or for transient experiments requiring an initial preparation of quantum states, prior to Stark switching. These solutions exhibit a power-dependent line broadening and a frequency shift of magnitude $\ensuremath{\sim}\frac{\ensuremath{\Delta}({\ensuremath{\alpha}}^{2}\ensuremath{-}{\ensuremath{\beta}}^{2})}{({\ensuremath{\Delta}}^{2}+\frac{1}{{T}_{2}^{2}})}$, in agreement with our earlier estimate, where $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ are the Rabi flopping frequencies for the two intermediate transitions and $\ensuremath{\Delta}$ is the off-resonance tuning behavior for one of them.
Detailed descriptions of one- and two-photon ionization are presented. We go beyond conventional perturbation theory to investigate the dynamics of both processes for a broad range of field intensities. The general theory of unstable decaying states is applied to the problem of one-photon ionization (the photoeffect). It is found that a non-trivial dependence of the transition dipole moment on the continuum state energy can lead to corrections to exponential decay, and even oscillations in the bound state population. As examples we consider the photoionization of hydrogen and atoms with autoionizing states. Only in the absence of autoionizing states corrections to exponential decay remain inconsequential for any experimentally relevant intensity. Our description of two-photon ionization is formulated in the Heisenberg picture. Besides the usual two-level atom projection operators we introduce explicitly the operators referring to the atomic continuum. By eliminating from the equations of motion the terms related to the electromagnetic self field as well as the terms referring to the continuum, transverse and longitudinal damping corresponding to spontaneous emission and photoionization is obtained. The use of field correlation functions allows us to account for the partial coherence of the applied field. We develop a technique to extract from the equations of motion a unique two-photon ionization rate. Such a characterization of the ionization process is not only instructive because of its simplicity, it is also in many cases completely satisfactory. The resulting general expression for the two-photon ionization rate is the central result of our treatment. From it we recover several familiar results in different limiting cases. In addition, our general rate expression provides the first description of the complete hierarchy of saturation effects that one should expect. We present a number of examples where we calculate the two-photon ionization rate as a function of the detuning, and where we emphasize the saturation and field correlation effects.