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Physical review. B, Solid state

Behavior of Electron-Spin Echoes and Photon Echoes in High Fields

D. Grischkowsky, Sven R. Hartmann

Published Jul 1, 1970• 78 citations
Publisher

Abstract

The amplitude of the electron-spin or photon echo from an ensemble of isolated atoms (echo atoms) interacting with magnetic-nuclear neighbors is calculated as a function of the magnitude and the orientation of an applied magnetic field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ and the separation between the two excitation pulses, in the limit that the magnitude of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ is large compared to the effective local fields at the echo-atom sites due to the nuclear neighbors. The general problem is reduced to the equivalent problem of two-level echo atoms interacting with nuclear neighbors. The latter problem is readily solved and applied to both spin and photon echoes. Experimentally observed spin-echo behavior in ruby is compared to calculated spin-echo behavior using no adjustable parameters, and is found to be in good agreement even when the magnetic field is tilted away from the optic axis thereby causing the echoes to disappear. Calculated photon-echo behavior, in which it is assumed that the interaction constants in the excited state are simply related to those in the ground state, shows the same general features as those that have been observed in experiment. Several results are presented to show the wide range of photon-echo behavior for different values of the interaction constants. A simple theory is presented which explains most observed echo behavior in a straightforward way.

DOI
10.1103/physrevb.2.60

Related papers

Effects of Superhyperfine Interactions on Photon-Echo Behavior in Dilute Ruby

L. Q. Lambert

A detailed analysis of the photon-echo decay in dilute ruby is presented. The experimental results are in quantitative agreement with an echo-decay theory based on the magnetic interaction between the Cr electronic spin and neighboring Al nuclear magnetic moments. Specifically, the following photon-echo properties are explained by the Cr-Al interaction: (i) modulation of the echo as a function of pulse separation time, (ii) the rapid decay of the echo from the $^{4}A({m}_{s}=\frac{3}{2})$ to $^{2}E({m}_{s}=\frac{1}{2})$ transition, (iii) the dependence of the echo on the direction and magnitude of an external magnetic field, (iv) the echo behavior at the 2.06- and 4.12-kG level crossings, and (v) the photon-echo decay in low magnetic fields. In addition, the experiments presented here show that the Cr-Al magnetic interactions in the $^{2}E$ excited state are roughly the same as the corresponding interactions in the $^{4}A_{2}$ ground state. The relationship of the photon-echo results to recent fluorescence-linewidth measurements is discussed.

Physical review. B, Solid state•Mar 1, 1973•59 citations

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Experiments are described in which a dilute ruby crystal is found to emit spontaneously a short pulse of light, the photon echo, at a time $\ensuremath{\approx}{\ensuremath{\tau}}_{s}$ after irradiation by two successive ruby-laser pulses separated by ${\ensuremath{\tau}}_{s}$. The phenomenon is explained in terms of a macroscopic oscillating electric dipole moment, which is momentarily reformed at the time the photon echo is observed. The analysis predicts the echo polarization as well as the propagation direction relative to the input pulses. A necessary condition for obtaining echoes in ruby is the application of a moderate magnetic field close to the optic axis of the crystal, and a simple model based on ${\mathrm{Cr}}^{3+}$-Al interactions is offered to account for this magnetic-field behavior. The relaxation time of the echo is found to exceed 250 nsec at 4.2\ifmmode^\circ\else\textdegree\fi{}K but to be less than 70 nsec at 14\ifmmode^\circ\else\textdegree\fi{}K, and is thought to be due to phonon-induced transitions in the excited $^{2}E(\overline{E})$ level. Multiple echo formation is also described.

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Arthur G. Szabo

Theoretical and experimental studies of photon-echo modulation for the four σ transitions of the R1 line in ruby with an on-axis magnetic field are described. Echoes originating from the Cr3+ ground-state 4A2 (−3/2) spin level (−3/2 echo) are studied for the first time. Unlike ±½ echoes, the −3/2-echo modulation is found to be dominated by Cr–Al interactions occurring outside the 13 inner-shell ions. The +½-echo modulation is observed to be in major disagreement with the theory using previously determined superhyperfine parameters. Regression-analysis studies indicate that this cannot be due solely to parameter errors. Finite nuclear linewidths and/or optical pumping effects are shown to be a plausible source of the disagreement.

Journal of the Optical Society of America B•Apr 1, 1986•15 citations

Nonexponential photon-echo decays of paramagnetic ions in the superhyperfine limit

Joseph Ganem, Y. P. Wang, D.M. Boye +3 more

Nonexponential photon-echo decays are observed in ruby, ${\mathrm{Er}}^{3+}$:${\mathrm{YLiF}}_{4}$, and ${\mathrm{Er}}^{3+}$:${\mathrm{LaF}}_{3}$ with samples of low ion concentrations in high (10--50 kG) magnetic fields. At the highest fields, photon-echo decays are field independent and their temporal forms are identical for all three systems. This behavior is attributed to optical dephasing arising from electron-spin nuclear-spin superhyperfine interactions in the presence of a frozen core.

Physical Review Letters•Feb 11, 1991•66 citations
The amplitude of the electron-spin or photon echo from an ensemble of isolated atoms (echo atoms) interacting with magnetic-nuclear neighbors is calculated as a function of the magnitude and the orientation of an applied magnetic field H\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}H\ensuremath→​ and the separation between the two excitation pulses, in the limit that the magnitude of H\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}H\ensuremath→​ is large compared to the effective local fields at the echo-atom sites due to the nuclear neighbors. The general problem is reduced to the equivalent problem of two-level echo atoms interacting with nuclear neighbors. The latter problem is readily solved and applied to both spin and photon echoes. Experimentally observed spin-echo behavior in ruby is compared to calculated spin-echo behavior using no adjustable parameters, and is found to be in good agreement even when the magnetic field is tilted away from the optic axis thereby causing the echoes to disappear. Calculated photon-echo behavior, in which it is assumed that the interaction constants in the excited state are simply related to those in the ground state, shows the same general features as those that have been observed in experiment. Several results are presented to show the wide range of photon-echo behavior for different values of the interaction constants. A simple theory is presented which explains most observed echo behavior in a straightforward way.
A detailed analysis of the photon-echo decay in dilute ruby is presented. The experimental results are in quantitative agreement with an echo-decay theory based on the magnetic interaction between the Cr electronic spin and neighboring Al nuclear magnetic moments. Specifically, the following photon-echo properties are explained by the Cr-Al interaction: (i) modulation of the echo as a function of pulse separation time, (ii) the rapid decay of the echo from the 4A(ms=32)^{4}A({m}_{s}=\frac{3}{2})4A(ms​=23​) to 2E(ms=12)^{2}E({m}_{s}=\frac{1}{2})2E(ms​=21​) transition, (iii) the dependence of the echo on the direction and magnitude of an external magnetic field, (iv) the echo behavior at the 2.06- and 4.12-kG level crossings, and (v) the photon-echo decay in low magnetic fields. In addition, the experiments presented here show that the Cr-Al magnetic interactions in the 2E^{2}E2E excited state are roughly the same as the corresponding interactions in the 4A2^{4}A_{2}4A2​ ground state. The relationship of the photon-echo results to recent fluorescence-linewidth measurements is discussed.
The behavior of photon echoes near level anticrossings is described in terms of the magnetic-field-dependent mixing parameters. The general features of level-crossing behavior are obscured for most cases in dilute ruby and are dominated instead by strong spin-dependent decay resulting from Cr-Al interactions. However, new experiments at short time separation ($\ensuremath{\tau}=32$ nsec) between pulses have permitted observation of the predicted photon-echo anticrossing signals in ruby, and may be applied to other atomic systems.
Photon echoes, observed recently in ruby and in NH2{\mathrm{NH}}_{2}NH2​D gas by two-pulse laser excitation, exhibit a modulation behavior as the pulse delay time is varied. The effect, which arises in a three-level quantum system, is due to the interference of two optically coupled transitions that share a common level. In this article, the three-level echo problem is treated exactly. Wave-function solutions are incorporated that are handled conveniently by a Laplace-transform technique. We, therefore, extend the earlier theory of echo modulation by Lambert et al., by removing their restriction of on-resonance pulse excitation. This procedure allows us to examine the effect of inhomogeneous broadening on echo amplitude explicitly, and we treat the problem by numerical and also by analytical means for the case of a Doppler-broadened gas. Analytic expressions are derived either for a high or a low laser power regime corresponding to excitation of either a large or a small fraction of the Doppler linewidth. The former case is analogous to pulsed nuclear magnetic resonance and is well known, whereas the latter result is new and applies to current photon echo measurements.
Experiments are described in which a dilute ruby crystal is found to emit spontaneously a short pulse of light, the photon echo, at a time $\ensuremath{\approx}{\ensuremath{\tau}}_{s}$ after irradiation by two successive ruby-laser pulses separated by ${\ensuremath{\tau}}_{s}$. The phenomenon is explained in terms of a macroscopic oscillating electric dipole moment, which is momentarily reformed at the time the photon echo is observed. The analysis predicts the echo polarization as well as the propagation direction relative to the input pulses. A necessary condition for obtaining echoes in ruby is the application of a moderate magnetic field close to the optic axis of the crystal, and a simple model based on Cr3+{\mathrm{Cr}}^{3+}Cr3+-Al interactions is offered to account for this magnetic-field behavior. The relaxation time of the echo is found to exceed 250 nsec at 4.2\ifmmode^\circ\else\textdegree\fi{}K but to be less than 70 nsec at 14\ifmmode^\circ\else\textdegree\fi{}K, and is thought to be due to phonon-induced transitions in the excited 2E(E‾)^{2}E(\overline{E})2E(E) level. Multiple echo formation is also described.
Theoretical and experimental studies of photon-echo modulation for the four σ transitions of the R1 line in ruby with an on-axis magnetic field are described. Echoes originating from the Cr3+ ground-state 4A2 (−3/2) spin level (−3/2 echo) are studied for the first time. Unlike ±½ echoes, the −3/2-echo modulation is found to be dominated by Cr–Al interactions occurring outside the 13 inner-shell ions. The +½-echo modulation is observed to be in major disagreement with the theory using previously determined superhyperfine parameters. Regression-analysis studies indicate that this cannot be due solely to parameter errors. Finite nuclear linewidths and/or optical pumping effects are shown to be a plausible source of the disagreement.
Nonexponential photon-echo decays are observed in ruby, Er3+{\mathrm{Er}}^{3+}Er3+:YLiF4{\mathrm{YLiF}}_{4}YLiF4​, and Er3+{\mathrm{Er}}^{3+}Er3+:LaF3{\mathrm{LaF}}_{3}LaF3​ with samples of low ion concentrations in high (10--50 kG) magnetic fields. At the highest fields, photon-echo decays are field independent and their temporal forms are identical for all three systems. This behavior is attributed to optical dephasing arising from electron-spin nuclear-spin superhyperfine interactions in the presence of a frozen core.