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Physical review. B, Condensed matter

Monte Carlo theory of optical dephasing in La F3 : Pr3+

Ralph G. DeVoe, Alexander J. Wokaun, Stephen Colby Rand, Richard G. Brewer

Published Apr 1, 1981• 79 citations
Publisher

Abstract

Recent optical free-induction-decay (FID) measurements of the impurity ion ${\mathrm{Pr}}^{3+}$ in La${\mathrm{F}}_{3}$ at 2 \ifmmode^\circ\else\textdegree\fi{}K reveal optical homogeneous linewidths of only a few kilohertz, considerably narrower than the inhomogeneous broadening due to crystalline strains (5 GHz) or the static local magnetic fields of the $^{19}\mathrm{F}$ nuclei (100 kHz). In this regime, the homogeneous broadening arises from local field fluctuations, as in NMR, and is due to the $^{19}\mathrm{F}$ nuclei which undergo resonant flip-flops and modulate the ${\mathrm{Pr}}^{3+}$ optical transition frequency. We treat the optical response of a two-level quantum system to an intense coherent field and to fluctuating perturbations using a Monte Carlo computer routine that assumes (1) the La${\mathrm{F}}_{3}$ crystal structure and (2) a sudden fluorine spin-flip model. This procedure avoids many of the approximations of previous analytic theories of spectral diffusion in magnetic resonance and extends the calculation specifically to optical FID. The decay behavior is obtained by sampling statistically the ${\mathrm{Pr}}^{3+}$ phase history as subgroups of $^{19}\mathrm{F}$ spins flip randomly in space and time. These fluctuations modify the Bloch equations where the solutions for the preparative and post-preparative periods are obtained by numerical integration. In spite of the large lattice size assumed (2250 fluorines), only a few $^{19}\mathrm{F}$ spins contribute substantially to the homogeneous width, a result which shows for the first time that spin-flip correlations are not significant. Furthermore, a ${\mathrm{Pr}}^{3+}$ ion polarizes and detunes the nearest fluorines forming a frozen core that is incapable of resonant spin flipping with the bulk fluorines. We demonstrate that the core grows radially as the $^{141}\mathrm{Pr}(I=\frac{5}{2})$ magnetic moment increases with ${I}_{z}$, but the Pr optical linewidth changes little, producing essentially one rather than three linewidths. Our calculations utilize no free parameters and predict a Lorentzian line shape of 8.4 kHz half-width at half maximum which compares to the optical FID observation of a 10.1-kHz Lorentzian. The Monte Carlo algorithm is verified further by the static local magnetic broadening of a Pr quadrupole transition which is found to be Gaussian, 82 kHz full width at half maximum, in agreement with a second-moment calculation and current observations.

DOI
10.1103/physrevb.23.3125

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Recent optical free-induction-decay (FID) measurements of the impurity ion Pr3+{\mathrm{Pr}}^{3+}Pr3+ in LaF3{\mathrm{F}}_{3}F3​ at 2 \ifmmode^\circ\else\textdegree\fi{}K reveal optical homogeneous linewidths of only a few kilohertz, considerably narrower than the inhomogeneous broadening due to crystalline strains (5 GHz) or the static local magnetic fields of the 19F^{19}\mathrm{F}19F nuclei (100 kHz). In this regime, the homogeneous broadening arises from local field fluctuations, as in NMR, and is due to the 19F^{19}\mathrm{F}19F nuclei which undergo resonant flip-flops and modulate the Pr3+{\mathrm{Pr}}^{3+}Pr3+ optical transition frequency. We treat the optical response of a two-level quantum system to an intense coherent field and to fluctuating perturbations using a Monte Carlo computer routine that assumes (1) the LaF3{\mathrm{F}}_{3}F3​ crystal structure and (2) a sudden fluorine spin-flip model. This procedure avoids many of the approximations of previous analytic theories of spectral diffusion in magnetic resonance and extends the calculation specifically to optical FID. The decay behavior is obtained by sampling statistically the Pr3+{\mathrm{Pr}}^{3+}Pr3+ phase history as subgroups of 19F^{19}\mathrm{F}19F spins flip randomly in space and time. These fluctuations modify the Bloch equations where the solutions for the preparative and post-preparative periods are obtained by numerical integration. In spite of the large lattice size assumed (2250 fluorines), only a few 19F^{19}\mathrm{F}19F spins contribute substantially to the homogeneous width, a result which shows for the first time that spin-flip correlations are not significant. Furthermore, a Pr3+{\mathrm{Pr}}^{3+}Pr3+ ion polarizes and detunes the nearest fluorines forming a frozen core that is incapable of resonant spin flipping with the bulk fluorines. We demonstrate that the core grows radially as the 141Pr(I=52)^{141}\mathrm{Pr}(I=\frac{5}{2})141Pr(I=25​) magnetic moment increases with Iz{I}_{z}Iz​, but the Pr optical linewidth changes little, producing essentially one rather than three linewidths. Our calculations utilize no free parameters and predict a Lorentzian line shape of 8.4 kHz half-width at half maximum which compares to the optical FID observation of a 10.1-kHz Lorentzian. The Monte Carlo algorithm is verified further by the static local magnetic broadening of a Pr quadrupole transition which is found to be Gaussian, 82 kHz full width at half maximum, in agreement with a second-moment calculation and current observations.
Photon echo measurements made at 2 °K on the 3H4 - 1D2 transition in Pr3+:LaF3 show that magnetic dipolar couplings between the Pr and F nuclei account for the 56 kHz homogeneous linewidth of this transition.1 The homogeneous broadening arises from the enhanced 141Pr nuclear moment (I=5/2) interacting with the local field fluctuations of the 19F nuclear moments undergoing mutual spin flip transitions. Such resonant fluctuations should, in the absence of a fluorine spin diffusion barrier, produce a homogeneous linewidth of about 200 kHz which is, in fact, roughly what is observed for the inhomogeneous broadening of the Pr3+ hyperfine levels and is considerably broader than that obtained by the photon echo measurements. Shelby et al2 proposed a simple model analogous to the spin diffusion barriers responsible for narrowing the homogeneous lines in certain electron paramagnetic resonance transitions3. In such systems, the field produced by the electron magnetic dipole moment (2-3 orders of magnitude larger than the enhanced nuclear moment associated with the ground state of Pr3+ in LaF3) de-tunes the nearest neighbors from each other, prohibiting mutual spin flips among them. Thus, the fields produced by the neighboring spins are static and their interaction with the paramagnetic ion contributes to the inhomogeneous linewidth and not to homogeneous broadening. The de-tuned neighbors are referred to as the "frozen core." Application of this picture to the case of Pr3+:LaF3 is supported by the calculation of Devoe et al.4 Their Monte Carlo calculation of the Pr3+ optical dephasing times shows that the observed homogeneous linewidth could be explained by deleting the nearest neighbor F spins from the lattice of rapidly fluctuating fluorine moments, in effect, treating them as part of a frozen core.
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.
Spin decoupling and line narrowing are observed for the first time in an opticaltransition, 3H4elDZ of Pr3~ in LaF 3 at zOK, using optical FID. The 19F nuclei, when irradiated by an appropriate rf field, undergo forced precession about an effective field at the magic angle in the rotating frame. The fluctuating 19F_19F dipolar interaction is thereby quenched and the optical linewidth drops from -10 to - Z kHz, as predicted in a theory of spin diffusion. In the field of nuclear magnetic resonance (NMR) , there exist several ways of reducing the time-dependent magnetic dipolar interaction between spins. Examples are motional narrowing (l,Z), macroscopic sampIe rotation (3), spontaneous spin flip-flop processes (4), and forced spin precession (5-10). In this article, we report the first observation of this kind in an optical transition of a low temperature solid, LaF 3:Pr 3+, where the dilute nuclear spin (I) is praseodymium and the abundant spin (S) is fluorine. The Pr 3+ ions are coherently prepared by a laser field and thereafter exhibit nonlinear optical free induction decay (FID). The Pr3+ dephasing time, as suggested in earlier work (11), is limited by spin diffusion among the 19F nuclei which undergo resonant flip-flops and impress weak fluctuating fields on the l4lpr nuclei. This action adiabatically modulates the Pr 3+ optical transition frequency through the dipolar I-S interaction and broadens the line. We now show that the half-width half-maximum (HWHM) optical linewidth is reduced from -10 to - Z kHz when the fluorine spin diffusion process is quenched by application of suitable magnetic static snd rf fields, causing the 19F nuclei to precess about an effective magnetic field at the magie angle in the rotating frame. This observation enables us to identify in an unambiguous way that the 19F_19F dipolar interaction is the dominant optical line broadening mechanism and provides the first test of spin diffusion theory' in an optical transition. As we shall see, the behavior at optical and rf frequencies is different. 3+
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.
Optical free-induction dephasing times as long as 16 \ensuremath{\mu}sec, corresponding to an optical homogenous linewidth of 10 kHz, have been observed for the $^{3}H_{4}\ensuremath{\leftrightarrow}^{1}D_{2}$ transition of Pr3+{\mathrm{Pr}}^{3+}Pr3+ ions in LaF3{\mathrm{F}}_{3}F3​ at 2\ifmmode^\circ\else\textdegree\fi{}K. Measurements are facilitated by a frequency-locked cw dye laser and a new form of laser frequency switching. Zeeman studies reveal a Pr-F dipole-dipole dephasing mechanism where the Pr nuclear moment is enhanced in both 1D2^{1}D_{2}1D2​ and 3H4^{3}H_{4}3H4​.
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.