Physical Review A

Experimental test of the modified optical Bloch equations for solids using rotary echoes

T. Muramoto, Arthur G. Szabo

Published Dec 1, 1988• 19 citations

Abstract

Measurements of optical rotary-echo decay of the ${R}_{1}$(-1/2),\ensuremath{\sigma} transition of ${\mathrm{Cr}}^{3+}$ 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 ${\mathrm{Pr}}^{3+}$:${\mathrm{LaF}}_{3}$ 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.

DOI: 10.1103/physreva.38.5928

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Physical Review A

Experimental test of the modified optical Bloch equations for solids using rotary echoes

T. Muramoto, Arthur G. Szabo

Published Dec 1, 1988• 19 citations

Publisher

Abstract

Measurements of optical rotary-echo decay of the

{R}_{1}

(-1/2),\ensuremath{\sigma} transition of

{\mathrm{Cr}}^{3+}

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

{\mathrm{Pr}}^{3+}

{\mathrm{LaF}}_{3}

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.

DOI: 10.1103/physreva.38.5928

Related papers

Experimental test of the optical Bloch equations for solids using free-induction decay

Arthur G. Szabo, T. Muramoto

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.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.

Physical Review AApr 1, 198942 citations

Experimental Tests of the Optical Bloch Equations for Solids

Alex Szabo

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.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.

Spectral Hole-Burning and Related Spectroscopies: Science and ApplicationsJan 1, 19920 citations

Time-domain violation of the optical Bloch equations for solids

R. N. Shakhmuratov, A. Szabó

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.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.

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Field-fluctuation-independent Coherence Decay in the Transition- 1/2(^4A_2) ～ - 1/2(E^^-) of the ruby R_1 line

孝夫村本

By a modified two manifold model, we explain the following experimental results performed for the test of the optical Bloch equation on the transition -1/2(4A2)～-1/2(E) of the ruby R1 line. 1)The optical hole width, including that derived from the FID increases linearly with x(Rabi frequency) under the strong excitation condition. 2)The decay of two pulse and rotary photon echo is a single exponential and the decay constant is power independent as is predicted from the ordinal optical Bloch equation. In this theory, a fluctuating local field is not assumed.By a modified two manifold model, we explain the following experimental results performed for the test of the optical Bloch equation on the transition -1/2(4A2)～-1/2(E) of the ruby R1 line. 1)The optical hole width, including that derived from the FID increases linearly with x(Rabi frequency) under the strong excitation condition. 2)The decay of two pulse and rotary photon echo is a single exponential and the decay constant is power independent as is predicted from the ordinal optical Bloch equation. In this theory, a fluctuating local field is not assumed.

Jan 1, 19980 citations

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

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.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.

Physical Review AOct 1, 199016 citations

Modified optical Bloch equations for solids

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.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 ANov 1, 198585 citations

Measurements of optical rotary-echo decay of the

{R}_{1}

(-1/2),\ensuremath{\sigma} transition of

{\mathrm{Cr}}^{3+}

{\mathrm{Pr}}^{3+}

{\mathrm{LaF}}_{3}

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.

{\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.