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Physical review. B, Condensed matter

Time-domain violation of the optical Bloch equations for solids

R. N. Shakhmuratov, A. Szabó

Published Sep 1, 1993• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.6903

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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.
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.
The free-induction decay (FID) after saturation by a laser radiation pulse of finite duration is studied for systems with spectral diffusion. The exact solution of the FID signal shape has been obtained in the framework of the telegraph noise model. This solution takes into account the finite duration of the saturating field and is valid at arbitrary values of the spectral exchange rate and the amplitude of the coherent field. The exact expression of the FID signal is derived in the weak-external-field limit. It is shown that the FID signal suddenly disappears after a time equal to the pulse width. This confirms the validity of the theorem on coherent transients, which has been proved for the Bloch equations [A. Schenzle, N. C. Wong, and R. G. Brewer, Phys. Rev. A 22, 635 (1980)].
Measurements of optical rotary-echo decay of the R1{R}_{1}R1​(-1/2),\ensuremath{\sigma} transition of Cr3+{\mathrm{Cr}}^{3+}Cr3+ in dilute ruby at 2 K are described and compared with predictions of two recent modifications (Gauss-Markov and random-telegraph models) of the optical Bloch equations (OBE's). While the modified OBE approximately describes the free-induction decay behavior both in Pr3+{\mathrm{Pr}}^{3+}Pr3+:LaF3{\mathrm{LaF}}_{3}LaF3​ and ruby, we show that rotary-echo decay in ruby is instead described by the standard OBE. Various aspects of the spin flip-flop model used in the theories are discussed in regard to ruby, as well as other possible dephasing contributions.
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.