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Physical Review A

Theory of cooperative fluorescence from products of reactions or collisions: Identical neutral atomic fragments

Gershon Kurizki, A. Ben‐Reuven

Published Jul 1, 1987• 30 citations
Publisher

Abstract

The time-resolved cooperative emission from a system of correlated neutral dissociation fragments, or molecular collision products in beams, is investigated. The investigation is focused on emission at large fragment separations (between 1 nm and a few emission wavelengths), exceeding the domain of short-range interactions within the reactive or collisional molecular complex. A master-equation approach is used to obtain a general expression for the cooperative emission rate, which consists of nonexponential decay factors multiplied by temporal ringing patterns. These features result from the time-dependent radiative coupling between the receding fragments; they depend in an essential manner on the initial electronic state of the parent molecular complex and its symmetry which determine the correlations between the fragments. In the model system of a pair of identical two-level fragments two cases are considered separately. (a) A single photon shared by the fragments, where the emission is initially superradiant or subradiant (radiation trapping), depending on the spin and inversion symmetry of the parent molecular system and of the nascent fragments. The ringing pattern depends on the electronic angular momentum state of the parent molecule and on the polarization of the emitted light. (Such a ringing has been observed recently by Grangier, Aspect, and Vigu\'e [Phys. Rev. Lett. 54, 418 (1985)] in the emission of photodissociated ${\mathrm{Ca}}_{2}$.) (b) Two initially excited fragments, where the ringing pattern is of smaller amplitude, and is weakly dependent on the electronic angular momentum of the parent molecule. All the aforementioned cooperative features generally last until the fragments recede several radiation wavelengths away from each other. The application of this time-resolved analysis to various diagnostic problems is discussed, especially with regard to the identification of excited electronic states of the parent molecular complex, and the stereospecificity of the fragmental orbitals.

DOI
10.1103/physreva.36.90

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Gershon Kurizki, Gabriel Hose, A. Ben‐Reuven

Time-resolved cooperative fluorescence (TRCF) from electronically excited dissociating homonuclear diatoms is suggested as a new technique for studying orbital and spin-polarization effects in fragmented diatoms. The method is based on a theoretical prediction of a novel feature named here ‘superbeats’. This feature results from a combination of two oscillatory effects, namely, the ringing oscillations caused by the time variation of the interference phase between the receding atoms during emission (P. Grangier, A. Aspect and J. Vigué, Phys. Rev. Lett., 1985, 54, 418) and the quantum beats associated with the fine-structure splitting. The time-resolved superbeat pattern is shown to be sensitive to the amplitudes and phases of the superposed states, with different orbital polarization and spin, mixed by adiabatic and non-adiabatic couplings in the outcome of the dissociation process. Quantitative estimates are made, based on a detailed analysis, for the photodissociation of Li2. Under conditions in which rotational effects can be neglected, the two states 1Σu(g) and 3Πu(g) of Li2 are superposed in the final outcome of the reaction. This should lead to a superbeat pattern dependent on the amplitude and phase of the superposition. Conditions for the experimental observability of these features have been studied, showing that a time resolution of ≲0.1 ns is required.

Journal of the Chemical Society Faraday Transactions 2 Molecular and Chemical Physics•Jan 1, 1989•1 citation
The time-resolved cooperative emission from a system of correlated neutral dissociation fragments, or molecular collision products in beams, is investigated. The investigation is focused on emission at large fragment separations (between 1 nm and a few emission wavelengths), exceeding the domain of short-range interactions within the reactive or collisional molecular complex. A master-equation approach is used to obtain a general expression for the cooperative emission rate, which consists of nonexponential decay factors multiplied by temporal ringing patterns. These features result from the time-dependent radiative coupling between the receding fragments; they depend in an essential manner on the initial electronic state of the parent molecular complex and its symmetry which determine the correlations between the fragments. In the model system of a pair of identical two-level fragments two cases are considered separately. (a) A single photon shared by the fragments, where the emission is initially superradiant or subradiant (radiation trapping), depending on the spin and inversion symmetry of the parent molecular system and of the nascent fragments. The ringing pattern depends on the electronic angular momentum state of the parent molecule and on the polarization of the emitted light. (Such a ringing has been observed recently by Grangier, Aspect, and Vigu'e [Phys. Rev. Lett. 54, 418 (1985)] in the emission of photodissociated Ca2{\mathrm{Ca}}_{2}Ca2​.) (b) Two initially excited fragments, where the ringing pattern is of smaller amplitude, and is weakly dependent on the electronic angular momentum of the parent molecule. All the aforementioned cooperative features generally last until the fragments recede several radiation wavelengths away from each other. The application of this time-resolved analysis to various diagnostic problems is discussed, especially with regard to the identification of excited electronic states of the parent molecular complex, and the stereospecificity of the fragmental orbitals.
The recent experiment by Grangier, Aspect, and Vigu\'e on interference in the emission from fragments of electronically photodissociated molecules is treated as a special case of cooperative fluorescence (CF) from products of various molecular processes. This general analysis relates time-resolved features of the CF to characteristics (such as orbital symmetry) of the dissociating parent molecule (PM) suggests various PM state preparations (including formation of subradiant states), and discusses the persistence of CF in systems of nonidentical fragments. The diagnostic potentialities of such studies are emphasized.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms A1{A}_{1}A1​, A2{A}_{2}A2​, whose separation r21{r}_{21}r21​ can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only A1{A}_{1}A1​ inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators O1{O}_{1}O1​, O2{O}_{2}O2​. The atoms exhibit superradiant behavior, whereas O1{O}_{1}O1​, O2{O}_{2}O2​ tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that A2{A}_{2}A2​ and A2{A}_{2}A2​ radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
In photodissociation of a homonuclear diatomic molecule, conservation of linear and angular momenta requires that each atomic fragment follow identical but oppositely directed trajectories with respect to the molecular center of mass. For certain values of the incident photon energy, one of the atoms will be electronically excited while the other will be in its ground state. When a detector records the undispersed fluorescence from the decay of the excited atom it cannot distinguish which trajectory the excited atom followed and hence whether the photon possesses a blue or red Doppler shift corresponding to recoil towards or away from the detector. According to the principle of superposition in quantum mechanics, the wave function of the photon must be a sum over two paths corresponding to emission from an excited atom in either of the two recoil directions. Detection of such photons gives rise to an interference between the probability amplitudes for the two paths that is manifested in the output from the photodetector as a beat frequency. Since photodissociation of an ensemble of molecules gives rise to a distribution of recoil directions relative to the electric vector of the incident radiation, a range of beat frequencies must be present in the detected signal. This paper gives a calculation of expected output voltage waveforms from a photodetector viewing atomic fluorescence excited by molecular photodissociation. Expressions are given for partial Fourier transforms of the photodetector signals. The results show that the shape of the voltage waveforms depends strongly on the asymmetry parameter for recoil of the photofragments and on whether or not the magnetic sublevels of the excited atomic state are created in a coherent superposition.
We discuss from first principles the cooperative decay of a system of two-level atoms, initially prepared in an uncorrelated excited state with population inversion NNN, and we give the conditions under which the superfluorescence effect occurs. Describing the atomic system in terms of collective variables, we derive a master equation for the reduced atomic density operator, which gives rise both to a damping and to a time-dependent frequency shift in the dynamics of collective modes. The coupled equations of motion are solved with a self-consistent approach. It is found that the system goes through a nonexponential decay if the maximum length of the active volume is smaller than a "cooperation range" and larger than a "threshold length," in agreement with the one-mode theory. The radiation burst has a time width proportional to ${N}^{\ensuremath{-}1}$, and its intensity is proportional to N2{N}^{2}N2. Specializing to a pencil-shaped volume, we find that only two atomic modes need to be considered; in this case, the average emitted radiation is all condensed in the two diffraction patterns of the opposite axial modes.
We predict a new type of time-resolved cooperative fluorescence (CF) obtainable when a homonuclear alkali-metal diatom dissociating via a $^{1}\mathrm{\ensuremath{\Sigma}}^{\mathrm{*}}$ state evolves (by nonadiabatic radial coupling) into a superposition of two excited adiabatic states correlated to different doublet levels. CF intensity then combines ``ringing'' oscillations, caused by time variation of the interference phase between the receding emitting atoms, with quantum beats associated with fine-structure splitting. The calculated CF patterns for Li2{\mathrm{Li}}_{2}Li2​ are sensitive to the amplitudes and relative phase of the two superposed adiabatic states.
Time-resolved cooperative fluorescence (TRCF) from electronically excited dissociating homonuclear diatoms is suggested as a new technique for studying orbital and spin-polarization effects in fragmented diatoms. The method is based on a theoretical prediction of a novel feature named here ‘superbeats’. This feature results from a combination of two oscillatory effects, namely, the ringing oscillations caused by the time variation of the interference phase between the receding atoms during emission (P. Grangier, A. Aspect and J. Vigué, Phys. Rev. Lett., 1985, 54, 418) and the quantum beats associated with the fine-structure splitting. The time-resolved superbeat pattern is shown to be sensitive to the amplitudes and phases of the superposed states, with different orbital polarization and spin, mixed by adiabatic and non-adiabatic couplings in the outcome of the dissociation process. Quantitative estimates are made, based on a detailed analysis, for the photodissociation of Li2. Under conditions in which rotational effects can be neglected, the two states 1Σu(g) and 3Πu(g) of Li2 are superposed in the final outcome of the reaction. This should lead to a superbeat pattern dependent on the amplitude and phase of the superposition. Conditions for the experimental observability of these features have been studied, showing that a time resolution of ≲0.1 ns is required.