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Physical Review Letters

Experimental Test of the Optical Bloch Equations for Solids

Ralph G. DeVoe, Richard G. Brewer

Published Apr 25, 1983• 197 citations
Publisher

Abstract

This paper provides the first experimental test of the Bloch equations for an optical transition in a low-temperature impurity-ion solid, ${\mathrm{Pr}}^{3+}$: La${\mathrm{F}}_{3}$, where the condition ${T}_{1}\ensuremath{\gg}{T}_{2}$ applies. Intensity-dependent optical free-induction-decay measurements are performed by use of an ultrastable cw dye laser (linewidth, 300 Hz) and reveal an $\mathrm{exp}(\ensuremath{-}\frac{2t}{{T}_{2}})$ decay law at low Rabi frequency ($\ensuremath{\chi}$), as predicted by the Bloch equations, and an $\mathrm{exp}(\ensuremath{-}\ensuremath{\chi}t)$ decay at high $\ensuremath{\chi}$, which violates the Bloch prediction but agrees with Redfield's characterization of NMR saturation in solids.

DOI
10.1103/physrevlett.50.1269

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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 ${D}_{1}$ line in the solid ${\mathrm{Pr}}^{3+}$:${\mathrm{LaF}}_{3}$. In this paper we extend these studies to another solid, ruby, using free-induction-decay observations obtained by pulse excitation of the ${R}_{1}$ 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 ${\ensuremath{\tau}}_{c}$=${T}_{2}$, the dephasing time. This result is remarkably similar to that obtained for ${\mathrm{Pr}}^{3+}$:${\mathrm{LaF}}_{3}$. 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.

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P. R. Berman, Richard G. Brewer

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
This paper provides the first experimental test of the Bloch equations for an optical transition in a low-temperature impurity-ion solid, Pr3+{\mathrm{Pr}}^{3+}Pr3+: LaF3{\mathrm{F}}_{3}F3​, where the condition T1\ensuremath≫T2{T}_{1}\ensuremath{\gg}{T}_{2}T1​\ensuremath≫T2​ applies. Intensity-dependent optical free-induction-decay measurements are performed by use of an ultrastable cw dye laser (linewidth, 300 Hz) and reveal an exp(\ensuremath−2tT2)\mathrm{exp}(\ensuremath{-}\frac{2t}{{T}_{2}})exp(\ensuremath−T2​2t​) decay law at low Rabi frequency (\ensuremathχ\ensuremath{\chi}\ensuremathχ), as predicted by the Bloch equations, and an exp(\ensuremath−\ensuremathχt)\mathrm{exp}(\ensuremath{-}\ensuremath{\chi}t)exp(\ensuremath−\ensuremathχt) decay at high \ensuremathχ\ensuremath{\chi}\ensuremathχ, which violates the Bloch prediction but agrees with Redfield's characterization of NMR saturation in solids.
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.
An optical coherence phenomenon–power- dependent optical line narrowing—is reported for the first known time. The effect is observed in a low-temperature impurity- ion solid, Pr3+:LaF3, using a coherent transient technique for which saturation conditions prevail. Current theories of saturation, based on the optical Bloch equations or on rate equations, fail to explain our experimental results. We have therefore developed a microscopic coherence theory that introduces dephasing into the density-matrix equations of motion in a manner equally valid at low and high power and that agrees closely with our experimental results. This work thus represents the first reported observation of the failure of the optical Bloch equations in a low-temperature impurity-ion solid.
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 optical Bloch equations which incorporate the phenomenological population (T1)({T}_{1})(T1​) and dipole dephasing (T2)({T}_{2})(T2​) times have been tested recently by optical free-induction-decay (FID) measurements on an impurity-ion crystal Pr3+{\mathrm{Pr}}^{3+}Pr3+:La3{\mathrm{La}}_{3}La3​ at 1.6 K. At low optical fields, the observed Pr3+{\mathrm{Pr}}^{3+}Pr3+ optical linewidth is dominated by magnetic fluctuations arising from pairs of fluorine nuclear flip-flops where the condition ${T}_{1}\ensuremath{\gg}{T}_{2}$ prevails. At elevated fields, this nuclear broadening mechanism is quenched and the Bloch equations are violated with ${T}_{2}\ensuremath{\rightarrow}{T}_{1}$. In this paper, a microscopic theory appropriate for a low-temperature impurity solid is presented which reveals the above features both for optical and radio frequencies, and a simple physical interpretation of this line narrowing phenomenon is given. Modified Bloch equations of a novel form are derived to second order and yield analytic FID solutions over the entire range of optical-field strength. A discussion of the earlier NMR theories is given, pointing out similarities and differences.
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.