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Physical Review A

Quantum theory of laser-field effects on relaxation in solids: Slow versus fast stochastic modulation and applications to optical transients

S. Ya. Kilin, Alexander P. Nizovtsev

Published Oct 1, 1990• 16 citations
Publisher

Abstract

A consistent explanation of recent studies by Szabo and Muramoto [Phys. Rev. A 39, 3992 (1989); 38, 5928 (1988)] on the experimental test of modified optical Bloch equations is presented. It is shown that the observed peculiarities of optical transients in solids are connected with the mechanism of relaxation in systems with slow stochastic modulation. A theory of laser-modified relaxation and analysis of free induction decay (FID), hole burning, and rotary echo (RE) in such systems are presented. The universal character of the FID time dependence and field-independent RE decay is predicted.

DOI
10.1103/physreva.42.4403

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Recently, DeVoe and Brewer [Phys. Rev. Lett. 50, 1269 (1983)] observed a striking deviation from a prediction of the optical Bloch equations by monitoring optical free-induction decay in the impurity-ion crystal ${\mathrm{Pr}}^{3+}$:${\mathrm{LaF}}_{3}$. At low optical fields, the ${\mathrm{Pr}}^{3+}$ optical dephasing time ${T}_{2}$ arises from magnetic fluctuations of the local environment, but at elevated optical fields, ${T}_{2}$ is no longer a constant, as assumed in the Bloch equations, because the magnetic line-broadening process is quenched. Several theories have been developed to explain the phenomenon. In this paper we present a simple theory of relaxation in solids which allows for comparison with earlier work. A ``strong-redistribution'' model is proposed where the optically excited impurity ions experience frequency shifts \ensuremath{\epsilon} induced by a thermal bath. Frequency jumps occur at an average rate \ensuremath{\Gamma} and with an rms value of ${\ensuremath{\epsilon}}_{0}$/ \ensuremath{\surd}2 , where ${\ensuremath{\epsilon}}_{0}$ is the thermal width associated with the frequency shifts. Modified Bloch equations (MBE) follow that are solved explicitly for two limiting cases, ${\ensuremath{\epsilon}}_{0}$\ensuremath{\ll}\ensuremath{\Gamma} and ${\ensuremath{\epsilon}}_{0}$\ensuremath{\gg}\ensuremath{\Gamma}, and qualitatively for the more general case where the ratio \ensuremath{\Gamma}/${\ensuremath{\epsilon}}_{0}$ is arbitrary. Since many of the earlier theories are equivalent to the strong-redistribution model in the limit ${\ensuremath{\epsilon}}_{0}$\ensuremath{\ll}\ensuremath{\Gamma}, we can assess the validity of the approximations made by these authors. Finally, we compare our MBE to the analogous transport equations that describe the effects of collisions in atomic vapors. We conclude that the problem addressed here is a general one, common to solids and gases alike, and is not restricted to impurity-ion crystals but will occur whenever frequency fluctuations are important.

Physical Review A•Nov 1, 1985•85 citations
A consistent explanation of recent studies by Szabo and Muramoto [Phys. Rev. A 39, 3992 (1989); 38, 5928 (1988)] on the experimental test of modified optical Bloch equations is presented. It is shown that the observed peculiarities of optical transients in solids are connected with the mechanism of relaxation in systems with slow stochastic modulation. A theory of laser-modified relaxation and analysis of free induction decay (FID), hole burning, and rotary echo (RE) in such systems are presented. The universal character of the FID time dependence and field-independent RE decay is predicted.
In a previous paper [ J. Opt. Soc. Am. B3, 564 ( 1986)], the validity conditions for the optical Bloch equations were reviewed. It was shown that, even within the limits of an impact or Markovian approximation, the optical Bloch equations fail to account properly for fluctuation-induced changes in atomic transition frequencies. Such changes are properly incorporated in a quantum-mechanical transport equation (QMTE) in which the fluctuation-induced frequency shifts are totally characterized by kernels W(∊′ → ∊) that give the probability density per unit time for a fluctuation to change the frequency shift from ∊′ to ∊. The QMTE describes the interaction of atoms with both an external radiation field and the perturber bath producing the fluctuations. A general method for solving the QMTE as a perturbation series in the external field is presented. Specific calculations are carried out for strong-redistribution, difference [W(∊′ → ∊) is a function of (∊–∊′) only], and Brownian motion kernels. It is shown that, although the kernels possess fundamental differences, they can yield similar results in certain limits. As an example, a perturbation calculation is performed for the free-induction decay (FID) of atoms prepared by a cw laser field and then allowed to radiate when the field is suddenly removed. Radical departures from the predictions of the conventional Bloch equations are found in certain limits, including a first-order contribution to FID in vapors and a nonexponential FID decay for atoms in vapors or solids. The implications of these results to a consistent interpretation of a recent experiment [ Phys. Rev. Lett.50, 1269 ( 1983)] on FID in the impurity ion crystal Pr3+:LaF3 are explored.
Photon echo and free induction decay, i.e., representative coherent optical transients, are described for two level system whose transition frequency is modulated by the Gaussian-Markoffian process. Photon echo signal is shown to decay biexponentially and to be effective in fixing the magnitude and the correlation time of the frequency modulation. Then we present the transition of the free induction decay rate against laser intensity from the Bloch type at the weak laser field into the Redfield one at the strong laser field.
Recent studies of the failure of the optical Bloch equations (OBE) in describing optical saturation in a solid are extended to test another aspect of the OBE, namely the time dependence of free-induction decay (FID) predicted by the SWB theorem on optical transients [Schenzle, Wong, and Brewer, Phys. Rev. A 22, 635 (1980)]. This theorem states that, following an arbitrarily shaped excitation pulse of length T, the FID will vanish at time T following the end of the excitation pulse. Experiments with ruby violate SWB in that FID decay at least twice as long as the excitation pulse is observed. Numerical and analytical calculations using a Gauss-Markov model for the modified OBE show an extended FID decay, however, the theoretical decay shape does not agree with experiment. Thus as with earlier studies on saturation broadening, we find that the modified OBE provides only a qualitative description of radiation-atom interaction in a solid.
Following a brief review of laser frequency and time domain techniques for overcoming inhomogeneous broadening of optical transitions in solids, recent studies on the failure of the optical Bloch equations (OBE) in describing saturation and coherent transient phenomena in ruby will be discussed. Experiments using free induction decay (FID), as in the original work on the system Pr3+:LaF3 by DeVoe and Brewer1, as well as extensions to rotary echos2 and hole-burning3 will be described. At low fields (~ 0.4 T), the time and frequency domain linewidth variation with Rabi frequency agree provided care is taken to avoid line broadening caused by spectral diffusion4. Recent work at high fields (~ 5 T), where Cr-Cr electron spin flipping is suppressed5, shows that spectral diffusion decreases and the dephasing time increases. Observed frequency domain hole widths as narrow as 18 kHz (FWHM) have been obtained for the 4 A2(−1/2)→E¯(−1/2) optical transition in dilute ruby using a pump-probe technique4. Work with such narrow holes emphasizes the fundamental limits imposed by the uncertainty principle and as an aside, a thought experiment where single photons are transmitted through "frequency slits" will be discussed.
Recently, DeVoe and Brewer [Phys. Rev. Lett. 50, 1269 (1983)] observed a striking deviation from a prediction of the optical Bloch equations by monitoring optical free-induction decay in the impurity-ion crystal Pr3+{\mathrm{Pr}}^{3+}Pr3+:LaF3{\mathrm{LaF}}_{3}LaF3​. At low optical fields, the Pr3+{\mathrm{Pr}}^{3+}Pr3+ optical dephasing time T2{T}_{2}T2​ arises from magnetic fluctuations of the local environment, but at elevated optical fields, T2{T}_{2}T2​ is no longer a constant, as assumed in the Bloch equations, because the magnetic line-broadening process is quenched. Several theories have been developed to explain the phenomenon. In this paper we present a simple theory of relaxation in solids which allows for comparison with earlier work. A ``strong-redistribution'' model is proposed where the optically excited impurity ions experience frequency shifts \ensuremath{\epsilon} induced by a thermal bath. Frequency jumps occur at an average rate \ensuremath{\Gamma} and with an rms value of ${\ensuremath{\epsilon}}_{0}$/ \ensuremath{\surd}2 , where ${\ensuremath{\epsilon}}_{0}$ is the thermal width associated with the frequency shifts. Modified Bloch equations (MBE) follow that are solved explicitly for two limiting cases, ${\ensuremath{\epsilon}}_{0}$\ensuremath{\ll}\ensuremath{\Gamma} and ${\ensuremath{\epsilon}}_{0}$\ensuremath{\gg}\ensuremath{\Gamma}, and qualitatively for the more general case where the ratio \ensuremath{\Gamma}/${\ensuremath{\epsilon}}_{0}$ is arbitrary. Since many of the earlier theories are equivalent to the strong-redistribution model in the limit ${\ensuremath{\epsilon}}_{0}$\ensuremath{\ll}\ensuremath{\Gamma}, we can assess the validity of the approximations made by these authors. Finally, we compare our MBE to the analogous transport equations that describe the effects of collisions in atomic vapors. We conclude that the problem addressed here is a general one, common to solids and gases alike, and is not restricted to impurity-ion crystals but will occur whenever frequency fluctuations are important.