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Journal of the Physical Society of Japan

Stochastic Theory of Coherent Optical Transients. II. Free Induction Decay in Pr3+: LaF3

Eiichi Hanamura

Published Oct 1, 1983• 35 citations
Publisher

Abstract

We study how the phase relaxation of the relevant electronic excitation is modified when the laser intensity increases. Especially the change of the free induction decay rate against the Rabi frequency is analytically derived for the intermediate region between the Bloch and the Redfield regimes, which corresponds to the observations by DeVoe and Brewer. Satisfactory coincidence is shown to be obtained, and we propose new experiments that will provide further tests of the theory.

DOI
10.1143/jpsj.52.3678

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In a recent experiment of DEVOE and BREWER [1], it was concluded that the optical Bloch equations are incapable of describing the saturation phenomena observed. Optical free-induction decay (FID) measurements of the impurity ion crystal Pr3+:LaF3 were conducted,where the Pr3+ ions are cohertntly prepared by a laser field under steady-state conditions,and then freely precess when the driving field is removed. At low optical fields, the observed Pr3+ optical line width is dominated by magnetic fluctuations.arising from pairs of fluorine nuclear flip-flops. At high optical fields, this nuclear broadening mechanism is quenched.and the Bloch equations are seriously violated. On physical grounds, this failure is due to a time-averaging of the magnetic interaction as the optical nutation frequency increases [2]. The phenomenological dipole dephasing time T2 of the Bloch equations is therefore not a true constant,but lengthens with increasing field strength.

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Experimental Tests of the Optical Bloch Equations for Solids

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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.

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We study how the phase relaxation of the relevant electronic excitation is modified when the laser intensity increases. Especially the change of the free induction decay rate against the Rabi frequency is analytically derived for the intermediate region between the Bloch and the Redfield regimes, which corresponds to the observations by DeVoe and Brewer. Satisfactory coincidence is shown to be obtained, and we propose new experiments that will provide further tests of the theory.
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.
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.
The effect of frequency modulation on optical phenomena is reduced critically when the Rabi frequency increases beyond the inverse correlation time or the amplitude of the frequency modulation. This is observed in the time region as the change of the free induction decay rate from the Bloch type into the Redfield one. That is shown in this paper to be observable also in the frequency region as the narrowing of the central line and the side bands split-off by the Rabi frequency above and below it in the emission spectrum beyond the Rabi frequency equal to the inverse correlation time.
Recent studies by DeVoe and Brewer [Phys. Rev. Lett. 50, 1269 (1983)] have shown that the conventional optical Bloch equations markedly fail to describe the optical saturation behavior of the D1{D}_{1}D1​ line in the solid Pr3+{\mathrm{Pr}}^{3+}Pr3+:LaF3{\mathrm{LaF}}_{3}LaF3​. In this paper we extend these studies to another solid, ruby, using free-induction-decay observations obtained by pulse excitation of the R1{R}_{1}R1​ line at 693.4 nm with an ultranarrow-linewidth dye laser. Comparison of the results with Gauss-Markov and random-telegraph-dephasing theories shows approximate agreement for a fluctuation correlation time ${\ensuremath{\tau}}_{c}$=T2{T}_{2}T2​, the dephasing time. This result is remarkably similar to that obtained for Pr3+{\mathrm{Pr}}^{3+}Pr3+:LaF3{\mathrm{LaF}}_{3}LaF3​. However, for theoretical and experimental reasons, we conclude that the theories do not consistently explain the current as well as other data. A qualitative discussion of another dephasing model is given.
In a recent experiment of DEVOE and BREWER [1], it was concluded that the optical Bloch equations are incapable of describing the saturation phenomena observed. Optical free-induction decay (FID) measurements of the impurity ion crystal Pr3+:LaF3 were conducted,where the Pr3+ ions are cohertntly prepared by a laser field under steady-state conditions,and then freely precess when the driving field is removed. At low optical fields, the observed Pr3+ optical line width is dominated by magnetic fluctuations.arising from pairs of fluorine nuclear flip-flops. At high optical fields, this nuclear broadening mechanism is quenched.and the Bloch equations are seriously violated. On physical grounds, this failure is due to a time-averaging of the magnetic interaction as the optical nutation frequency increases [2]. The phenomenological dipole dephasing time T2 of the Bloch equations is therefore not a true constant,but lengthens with increasing field strength.
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.