Recently demonstrated techniques for measuring collision-induced phase in macroscopic optical coherence are described and analyzed in detail. The time-dependent phase measured by this method is used to study imaginary collision kernels for optical radiators that are related to the phase by Fourier transformation. The results indicate that classical velocity changes may make an important contribution to the imaginary kernel, in contrast to the real kernel, which is dominated by diffractive velocity changes.
Collisional processes occurring within an atomic vapor can be conveniently described in terms of collision kernels. The population kernel ${W}_{\mathrm{ii}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{'}}\ensuremath{\rightarrow}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ gives the probability density per unit time that an "active" atom in state
i undergoes a collision with a perturber that changes the active atom's velocity from ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{'}}$ to $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$. For active atoms in a linear superposition of states
i and
j, there is an analogous coherence kernel ${W}_{\mathrm{ij}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{'}}\ensuremath{\rightarrow}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}) (i\ensuremath{\ne}j)$ reflecting the effects of collisions on the off-diagonal density-matrix element ${\ensuremath{\rho}}_{\mathrm{ij}}$. In this work, we discuss the general properties of the collision kernels which characterize a two-level active atom which, owing to the action of a radiation field, is in a linear superposition of its two levels. Using arguments based on the uncertainty principle, we show that collisions can be divided roughly into the following two categories: (1) collisions having impact parameters less than some characteristic radius which may be described classically and (2) collisions having impact parameters larger than this characteristic radius which give rise to diffractive scattering and must be treated using a quantum-mechanical theory. For the population kernels, collisions of type (1) can lead to a large-angle scattering component, while those of type (2) lead to a small-angle (diffractive) scattering component. For the coherence kernel, however, assuming that the collisional interaction for states
i and
j differ appreciably, only collisions of type (2) contribute, and the coherence kernel contains a small-angle scattering component only. The absence of a large-angle scattering component in the coherence kernel is linked to a collision-induced spatial separation of the trajectories associated with states
i and
j. Interestingly enough, the width of the diffractive kernel, as measured in the laboratory frame, is found to be insensitive to the perturber to active-atom mass ratio. To illustrate these features, a specific calculation of the kernels is carried out using a hard-sphere model for the scattering. The relationship of the present description of collisions to that of traditional pressure-broadening theory in which trajectory separation effects are ignored is discussed. It is explained why traditional pressure-broadening theory correctly describes collision effects in linear spectroscopy, but fails to provide an adequate description of some saturation spectroscopy and photonecho experiments in which velocity-changing collisions associated with the coherence kernel play a significant role. An expression for the collisionally modified photon-echo amplitude is derived which clearly displays the role played by velocity-changing collisions associated with the coherence kernel.
It has been shown recently that molecular gas samples excited with coherent light can display a variety of transient phenomena, similar to those found in nuclear magnetic resonance. This article elucidates how these coherence effects can be used to isolate or unfold molecular collision mechanisms that normally remain hidden within the optical line shape. Elastic collisions, for example, are easily detected here in two-pulse photon echo experiments for a
13CH3F vibration-rotation transition. The echo-decay function which provides a signature for the velocity-changing collision diffusion mechanism, is not just a simple exponential in time but exhibits an $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ contribution for short times and an $\mathrm{exp}(\ensuremath{-}\ensuremath{\Gamma}t)$ decay for long times. This behavior, which is unknown heretofore, contrasts with spin echoes in molecular liquids where Brownian motion leads only to the cubic decay law. An $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ behavior can be understood in terms of a solution of the Fokker-Planck equation which describes the effect of Brownian motion on echo decay. Such a treatment is valid for small phase excursions; in the case of a gas, this implies the Doppler phase factor $k\ensuremath{\Delta}u\ensuremath{\tau}\ensuremath{\ll}1$, where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ is the propagation vector of light, $\ensuremath{\Delta}u$ is a characteristic velocity jump for a binary collision, and $\ensuremath{\tau}$ is the echo-pulse delay time. When $k\ensuremath{\Delta}u\ensuremath{\tau}\ensuremath{\gg}1$ as in the long-time regime, the Fokker-Planck solution fails. We, therefore, present a new solution to the Boltzmann transport equation using a weak collision model and find agreement with the entire echo time dependence observed. The echo measurements indicate very small changes in longitudinal velocity per
13CH3F-
13CH3F collision, i.e., $\ensuremath{\Delta}u=85$ cm/sec, thereby justifying the weak-collision model. The total elastic collision cross section is 430
A˚2. It follows that elastic collisions lead to velocity thermalization in a time of \ensuremath{\sim}5 sec when the
13CH3F pressure is 1 mTorr. A comparison is also made of the
13CH3F dephasing time ${\ensuremath{\tau}}_{2}$ in a coherent Raman beat decay, which is independent of velocity-changing collisions, with the longitudinal decay time
T1. Here,
T1 represents the molecule-optical interaction time, due largely to jumps in molecular rotation (
J) and orientation (
M) state, and is obtained from a delayed nutation measurement. The fact that the pressure dependent part of ${\ensuremath{\tau}}_{2}={T}_{1}$ shows that
T1 is also independent of velocity diffusion. Futhermore, when a hole is burned in the Doppler distribution, population recovery must be due to inelastic rather than elastic collisions. Optical Carr-Purcell echoes, multiple pulse echoes, provide a direct measure of the
13CH3F transverse dephasing time
T2 without the effect of elastic collisions while being sensitive to "phase interrupting collisions." Again, we find that
T2=T1 so that phase interruptions are negligible. Had such a process dominated the two-pulse echo, an $\mathrm{exp}(\ensuremath{-}\frac{t}{{T}_{2}})$ damping would have been noticed with no $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ contribution. Thus, the present study covers several new aspects of molecular collisions. It represents the first detailed examination of velocity-changing collisions by coherence methods and without the complication of Doppler broadening.
Collision-induced coherence transfer between infrared molecular transitions is studied using a technique that permits the complete distribution of accompanying velocity changes to be determined in a simple way. The experiments are carried out by measuring the intensity of two pulse photon echoes as a function of applied Stark field at fixed large echo time delay. The velocity change distribution for coherence-transfer collisions is obtained directly as the shape of the echo intensity versus Stark field curve. Previously unpublished data for dipole-quadrupole and van der Waals interactions are presented, and a theory of coherence-transfer velocity-changing kernels is developed that is in good agreement with the data.