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Physical review. B, Condensed matter

Symmetry considerations regarding light propagation and light polarization for coherent interactions with ions in crystals

Yiwen Sun, G. M. Wang, R. L. Cone, Randy W. Equall, M. J. M. Leask

Published Dec 15, 2000• 71 citations
Publisher

Abstract

Spectroscopic studies and the efficiency of optical coherent transient devices can be adversely affected by interference arising from several distinct Rabi frequencies associated with crystallographically equivalent but orientationally inequivalent crystal sites. General symmetry principles have been used to find light field directions that guarantee both optimum single-Rabi-frequency interaction and the most efficient coherent transient generation in arbitrary crystals. This theoretical analysis has been applied experimentally to a wide range of rare earth crystals, many of which are important for technological applications. Site interference in optical nutation provides a simple illustration of this effect. Optimum single-Rabi-frequency nutation signal was obtained for the ${}^{3}{H}_{6}(1)$ to ${}^{3}{H}_{4}(1)$ transition of 0.1% ${\mathrm{Tm}}^{3+}:{\mathrm{Y}}_{3}{\mathrm{Al}}_{5}{\mathrm{O}}_{12}$ when the light $\stackrel{\ensuremath{\rightarrow}}{E}$ vector is along special crystal directions $〈111〉$ or $〈001〉.$ Quantitative comparison of nutation frequencies for light polarized along different directions allowed the determination of the optical transition dipole moment to be along $〈110〉$ for this transition. The transition dipole moment derived from the nutation signal agrees within 15% with that obtained from absorption experiments. Other applications of nutation measurements are discussed, including a procedure for determining site occupancy in crystals with crystallographically inequivalent sites.

DOI
10.1103/physrevb.62.15443

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Spectroscopic studies and the efficiency of optical coherent transient devices can be adversely affected by interference arising from several distinct Rabi frequencies associated with crystallographically equivalent but orientationally inequivalent crystal sites. General symmetry principles have been used to find light field directions that guarantee both optimum single-Rabi-frequency interaction and the most efficient coherent transient generation in arbitrary crystals. This theoretical analysis has been applied experimentally to a wide range of rare earth crystals, many of which are important for technological applications. Site interference in optical nutation provides a simple illustration of this effect. Optimum single-Rabi-frequency nutation signal was obtained for the 3H6(1){}^{3}{H}_{6}(1)3H6​(1) to 3H4(1){}^{3}{H}_{4}(1)3H4​(1) transition of 0.1% Tm3+:Y3Al5O12{\mathrm{Tm}}^{3+}:{\mathrm{Y}}_{3}{\mathrm{Al}}_{5}{\mathrm{O}}_{12}Tm3+:Y3​Al5​O12​ when the light E\ensuremath→\stackrel{\ensuremath{\rightarrow}}{E}E\ensuremath→​ vector is along special crystal directions 〈111〉〈111〉〈111〉 or 〈001〉.〈001〉.〈001〉. Quantitative comparison of nutation frequencies for light polarized along different directions allowed the determination of the optical transition dipole moment to be along 〈110〉〈110〉〈110〉 for this transition. The transition dipole moment derived from the nutation signal agrees within 15% with that obtained from absorption experiments. Other applications of nutation measurements are discussed, including a procedure for determining site occupancy in crystals with crystallographically inequivalent sites.
Optical nutation has been used as an important spectroscopic tool to study the interaction between atoms and coherent light. It is based on the fact that the input pulse area needed to reach the first nutation peak is a constant value that only depends on the transverse mode of the optical beam. The interaction strength in terms of Rabi frequency can be measured directly from the time to reach the first nutation peak. However, the assumption of constant pulse area is only valid in the limit of the optically thin media, while optically thick media are of great interest for applications in classical and quantum information storage and processing. This paper studies the behavior of optical nutation in optically thick media through theoretical simulation and experiments in rare-earth atomic ensembles doped in a crystal lattice. A simple formula for the input pulse area as a function of the absorption length has been found, which covers optically thick media and is consistent with the value given by the low absorption limit.
When introduced as dopants into Y2O3{\mathrm{Y}}_{2}{\mathrm{O}}_{3}Y2​O3​, Er3+\mathrm{E}{\mathrm{r}}^{3+}Er3+ ions replace Y3+{\mathrm{Y}}^{3+}Y3+ ions at two crystallographic sites of C2{\mathrm{C}}_{2}C2​ and C3i{\mathrm{C}}_{3\mathrm{i}}C3i​ point symmetries. The C3i{\mathrm{C}}_{3\mathrm{i}}C3i​ site is particularly interesting, given that the inversion symmetry prohibits static electric-dipole moments and electric-dipole transitions, resulting in 1.5 \textmu{}m optical transitions characterized by purely magnetic-dipole properties. Employing $\mathrm{E}{\mathrm{r}}^{3+}\phantom{\rule{4pt}{0ex}}^{4}I_{15/2}\ensuremath{\leftrightarrow}^{4}I_{13/2}$ transitions, we examined the inherent C3i{\mathrm{C}}_{3\mathrm{i}}C3i​ site of Er3+\mathrm{E}{\mathrm{r}}^{3+}Er3+ in Y2O3{\mathrm{Y}}_{2}{\mathrm{O}}_{3}Y2​O3​ for its potential in quantum information science and spectral hole burning applications. Our approach involved absorption and site-selective fluorescence spectroscopy to discern Er3+\mathrm{E}{\mathrm{r}}^{3+}Er3+ energy levels and revealed an excited-state lifetime of 14.6 ms. High-resolution laser spectroscopy at 3.7 K indicated an inhomogeneous linewidth of 1.3 GHz in our sample. Orientation-dependent Zeeman laser absorption spectroscopy facilitated the determination of the full ggg tensors for both the ground and excited states, confirming the C3i{\mathrm{C}}_{3\mathrm{i}}C3i​ site assignment. Furthermore, it identified a suitable magnetic field direction for mitigating decoherence, a finding corroborated through two-pulse photon echo spectroscopy. Our study presents photon echo coherent transient measurements related to magnetic-dipole-allowed rare-earth ion transitions at the C3i{\mathrm{C}}_{3\mathrm{i}}C3i​ site. The observed decoherence, attributed to spectral diffusion arising from $\mathrm{E}{\mathrm{r}}^{3+}\text{\ensuremath{-}}\mathrm{E}{\mathrm{r}}^{3+}$ magnetic-dipole interactions, underscores the complex spin dynamics at play in systems with multiple inequivalent dopant sites. Coupled with the practicality of telecom laser wavelengths, our investigation highlights the promise of magnetic-dipole transitions in advancing quantum information science and spectral hole burning applications.
We present a scheme for quantum computing in rare-earth-ion doped inorganic crystals. We use two of the ground state hyperfine levels as the qubit states, with different qubits absorbing light at different frequencies within an inhomogeneously broadened optical transition. Interaction between different qubits is accomplished using the change in permanent dipole moment that is induced at the optical excitation of the ions. The two most important aspects of the scheme have been accomplished experimentally: The isolation of a single qubit in a pure state and interaction between ions absorbing at different frequencies.
Rare-earth ions in solid-state hosts are a compelling hardware platform for quantum networking due to their exceptional optical, and long-lived spin transitions. Erbium's transitions couple to photons in the Telecom-C band, making it compatible with fibre, and has demonstrated high coherence and low spectral diffusion in a range of materials. However, it has a low optical dipole moment, leading to low light-matter coupling. This limits the rate at which operations performed within a network. Despite decades of research, Er transitions remain poorly understood. Hosts are chosen based on convention, convenience, and (well-informed) heuristics, since calculations of the properties of lanthanide 4f shell are prohibitively difficult. This thesis addresses the challenge of this weak dipole through two complementary approaches; both of which constitute engineering the erbium-crystal system. Firstly, we engineer the emitter itself, from the bottom up. We seek an optimal crystal site for ions, which maintains coherence but increases transition strength. To this end, we have characterised four Er sites in CaF2 to understand how the crystal field affects ion properties. We identify sites with dipole moments as high as 1.63x10^-32 Cm despite lifetimes in excess of 10 ms, and coherence of up to 43.9 µs. This informs our next steps, in selecting an optimal site for single ion addressing, and our long-term goal of designing sites for high dipole moments. In the second approach, we engineer the crystal from the top down, fabricating whispering gallery mode resonators in calcium fluoride. I have fabricated five high Q resonators in doped and undoped CaF2, and commenced their characterisation. We have prototyped a coupler for use of these devices in a cryostat with no free-space optical access. This engineered crystal system - a high dipole erbium site coupled to a high Q resonator - provides a path toward competitive optical cooperativity for single erbium ions in the solid state.
We show that crystal-field calculations for C_{1} point-group symmetry are possible, and that such calculations can be performed with sufficient accuracy to have substantial utility for rare-earth based quantum information applications. In particular, we perform crystal-field fitting for a C_{1}-symmetry site in ^{167}Er^{3+}:Y_{2}SiO_{5}. The calculation simultaneously includes site-selective spectroscopic data up to 20 000 cm^{-1}, rotational Zeeman data, and ground- and excited-state hyperfine structure determined from high-resolution Raman-heterodyne spectroscopy on the 1.5 μm telecom transition. We achieve an agreement of better than 50 MHz for assigned hyperfine transitions. The success of this analysis opens the possibility of systematically evaluating the coherence properties, as well as transition energies and intensities, of any rare-earth ion doped into Y_{2}SiO_{5}.