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Physical Review B

Analysis of magnetic interactions in rare-earth-doped crystals for quantum manipulation

O. Guillot-Noël, Philippe Goldner, E. Antic-Fidancev, J.-L. Le Gouët

Published May 12, 2005• 56 citations
Publisher

Abstract

The influence of magnetic interactions in rare-earth-doped crystals under an external magnetic field has been studied in order to obtain an efficient three-level $\ensuremath{\Lambda}$ system with the hyperfine levels of the rare earth. Nuclear Zeeman effect under the action of an external magnetic field removes the nuclear degeneracy. This interaction does not provide an efficient $\ensuremath{\Lambda}$ system because nuclear-spin flipping such as $\ensuremath{\mid}{M}_{I}⟩=\ifmmode\pm\else\textpm\fi{}\frac{1}{2}\ensuremath{\rightarrow}\ensuremath{\mid}{M}_{I}⟩=\ensuremath{\mp}\frac{1}{2}$ (${M}_{I}$ is the nuclear-spin projection) cannot be induced by an optical transition. However, this selection rule only applies to pure nuclear Zeeman effect. Indeed, it is shown that the coupling of the electronic Zeeman and of the hyperfine interactions releases the nuclear-spin selection rules $\ensuremath{\Delta}{M}_{I}=0$. This can be described in terms of a pseudonuclear Zeeman effect induced by an effective magnetic field. The relative strengths of the two optical transitions involved in the three-level system can be controlled by the orientation of the external magnetic field. The particular case of the ${\mathrm{Tm}}^{3+}$ ion in the ${\mathrm{Y}}_{3}{\mathrm{Al}}_{5}{\mathrm{O}}_{12}$ host (YAG) is discussed. ${\mathrm{Tm}}^{3+}$ hyperfine structure is determined using a complete Hamiltonian including free-ion, crystal-field, and magnetic interactions. A good three-level $\ensuremath{\Lambda}$ system is obtained in Tm:YAG with a transition strength ratio of 0.24 $(\ensuremath{\sim}1:4)$ between the two optical transitions. An analytical analysis based on a spin-Hamiltonian approach is proposed to explain the results of the complete crystal-field calculations. Finally, an experimental protocol that makes a crystal similar to the atomic samples used in previous quantum information investigations, with the additional benefits of absence of motion and long coherence time, is described.

DOI
10.1103/physrevb.71.174409

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The influence of magnetic interactions in rare-earth-doped crystals under an external magnetic field has been studied in order to obtain an efficient three-level \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ system with the hyperfine levels of the rare earth. Nuclear Zeeman effect under the action of an external magnetic field removes the nuclear degeneracy. This interaction does not provide an efficient \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ system because nuclear-spin flipping such as \ensuremath∣MI⟩=\ifmmode±\else\textpm\fi12\ensuremath→\ensuremath∣MI⟩=\ensuremath∓12\ensuremath{\mid}{M}_{I}⟩=\ifmmode\pm\else\textpm\fi{}\frac{1}{2}\ensuremath{\rightarrow}\ensuremath{\mid}{M}_{I}⟩=\ensuremath{\mp}\frac{1}{2}\ensuremath∣MI​⟩=\ifmmode±\else\textpm\fi21​\ensuremath→\ensuremath∣MI​⟩=\ensuremath∓21​ (MI{M}_{I}MI​ is the nuclear-spin projection) cannot be induced by an optical transition. However, this selection rule only applies to pure nuclear Zeeman effect. Indeed, it is shown that the coupling of the electronic Zeeman and of the hyperfine interactions releases the nuclear-spin selection rules \ensuremathΔMI=0\ensuremath{\Delta}{M}_{I}=0\ensuremathΔMI​=0. This can be described in terms of a pseudonuclear Zeeman effect induced by an effective magnetic field. The relative strengths of the two optical transitions involved in the three-level system can be controlled by the orientation of the external magnetic field. The particular case of the Tm3+{\mathrm{Tm}}^{3+}Tm3+ ion in the Y3Al5O12{\mathrm{Y}}_{3}{\mathrm{Al}}_{5}{\mathrm{O}}_{12}Y3​Al5​O12​ host (YAG) is discussed. Tm3+{\mathrm{Tm}}^{3+}Tm3+ hyperfine structure is determined using a complete Hamiltonian including free-ion, crystal-field, and magnetic interactions. A good three-level \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ system is obtained in Tm:YAG with a transition strength ratio of 0.24 (\ensuremath∼1:4)(\ensuremath{\sim}1:4)(\ensuremath∼1:4) between the two optical transitions. An analytical analysis based on a spin-Hamiltonian approach is proposed to explain the results of the complete crystal-field calculations. Finally, an experimental protocol that makes a crystal similar to the atomic samples used in previous quantum information investigations, with the additional benefits of absence of motion and long coherence time, is described.
The influence of an external magnetic field on the hyperfine structure of the 3 H4(0) and 1D2(0) crystal field states of Pr3+ in LiYF4 is studied in order to find an efficient three-level Λ system. Using an experimentally determined spin Hamiltonian, we show that three-level Λ systems can be obtained with equal strengths for the optically excited transitions under various magnetic field magnitudes and orientations. An analytical analysis based on two levels is proposed to find useful magnetic fields without extensive numerical calculations and to understand the general behaviour of the system. Pr3+ hyperfine structure has also been directly calculated using a complete Hamiltonian including free ion, crystal field and magnetic interactions. A good agreement with the spin Hamiltonian approach is found for the ground state whereas the excited state results poorly reproduce the experiment. This is attributed to the low accuracy of 1D2 crystal field wavefunctions. This suggests that transition strengths ratios could be calculated directly from the crystal field Hamiltonian with improved crystal field parameters.
Rare-earth ion-doped crystals are promising systems for quantum communication and quantum information processing. In particular, paramagnetic rare-earth centers can be utilized to realize quantum coherent interfaces simultaneously for optical and microwave photons. In this paper, we study hyperfine and magnetic properties of a Y2SiO5{\mathrm{Y}}_{2}{\mathrm{SiO}}_{5}Y2​SiO5​ crystal doped with 171Yb3+^{171}\mathrm{Yb}^{3+}171Yb3+ ions. This isotope is particularly interesting since it is the only rare--earth ion having electronic spin S=12S=\frac{1}{2}S=21​ and nuclear spin I=12I=\frac{1}{2}I=21​, which results in the simplest possible hyperfine level structure. In this work, we determine the hyperfine tensors for the ground and excited states on the optical $^{2}\mathrm{F}_{7/2}(0)\ensuremath{\longleftrightarrow}^{2}\mathrm{F}_{5/2}(0)$ transition by combining spectral hole burning and optically detected magnetic resonance techniques. The resulting spin Hamiltonians correctly predict the magnetic-field dependence of all observed optical-hyperfine transitions, from zero applied field up to fields where the Zeeman interaction is dominating the hyperfine interaction. Using the optical absorption spectrum, we can also determine the order of the hyperfine levels in both states. These results pave the way for realizing solid-state optical and microwave quantum memories based on a 171Yb3+:Y2SiO5^{171}\mathrm{Yb}^{3+}:{\mathrm{Y}}_{2}{\mathrm{SiO}}_{5}171Yb3+:Y2​SiO5​ crystal.
Rare-earth-doped crystals have been used for optical signal processing and storage applications. In this dissertation, their potential for quantum computing applications is explored. In one quantum computing scheme, information is stored in nuclear spin states and this information is then processed by using optical pulses through the coupling of these nuclear spin states to a common electronic level. To implement this scheme, nuclear spin states and coupling of these nuclear spin states to a common electronic level is required. Preliminary work in rare-earth materials like Pr3+ and Eu3+ has shown promising results regarding their suitability for quantum computing applications. One particular problem with these materials is that their transition wavelengths are only accessible with dye lasers. These lasers are inherently unstable, and currently few available systems exhibit the stability required for quantum computing applications. An alternative choice was to investigate other rare-earth ions like thulium. Thulium has a transition wavelength that can be accessed with diode lasers, which are commercially available, easy to stabilize, and compact. This dissertation is based on our investigations of Tm3+:YAG for quantum computing applications. Investigations involved a detailed characterization of the material. Nuclear spin states, in Tm3+:YAG, were obtained by applying an external magnetic field to the sample. First, interaction of an external magnetic field with the thulium ions at various sites in the crystal was analyzed. This analysis was used to measure the magnetic anisotropy in the material. These results show that it is possible, with the suitable choice of the magnetic orientation and the site in the crystal, to build a working 3-level quantum system. In the demonstration of single qubit operations in Tm3+:YAG, we first theoretically studied the effect of Gaussian spatial beam on the single qubit operations. Later on, we experimentally prepared a single isolated ensemble of ions in the inhomogeneously broadened absorption profile of the medium. This single isolated ensemble of ions was used as a test-bed to implement the single qubit operations. We also isolated two ensembles of ions in the inhomogeneous absorption profile of the medium. The interaction between these two isolated ensembles of ions was also studied.
Flip-flop processes due to magnetic dipole-dipole interaction between neighboring ions in crystals doped by rare-earth ions are one of the mechanisms of relaxation between hyperfine levels. Modeling of this mechanism has so far been macroscopic, characterized by an average rate describing the relaxation of all ions. Here, however, we present a microscopic model of flip-flop interactions between individual nuclear spins of dopant ions. Every ion is situated in a unique local environment in the crystal, where each ion has different distances and a unique orientation relative to its nearest neighbors, as determined by the lattice structure. Thus each ion has a unique flip-flop rate and the collective relaxation dynamics of all ions in a bulk crystal is a sum of many exponential decays, giving rise to a distribution of rates rather than a single average decay rate. We employ this model to calculate flip-flop rates in Pr3+{\mathrm{Pr}}^{3+}Pr3+:Y2SiO5{\mathrm{Y}}_{2}{\mathrm{SiO}}_{5}Y2​SiO5​ and show experimental measurements of population decay of the ground state hyperfine levels at $\ensuremath{\sim}2$ K. We also present a method to measure rates of individual transitions from hole burning spectra that requires significantly fewer fitting parameters in theoretical rate equations compared to earlier work. Furthermore, we measure the effect of external magnetic field on the flip-flop rates and observe that the rates slow down by two orders of magnitude in a field of 5--10 mT.
In many important situations, the dominant dephasing mechanism in cryogenic rare-earth-metal-ion--doped systems is due to magnetic field fluctuations from spins in the host crystal. Operating at a magnetic field where a transition has a zero first-order Zeeman (ZEFOZ) shift can greatly reduce this dephasing. Here we identify the location of transitions with a zero first-order Zeeman shift for optical transitions in Pr3+{}^{3+}3+:YAG and for spin transitions in Er3+{}^{3+}3+:Y2{}_{2}2​SiO5{}_{5}5​. The long coherence times that ZEFOZ can enable would make Pr3+{}^{3+}3+:YAG a strong candidate for achieving the strong-coupling regime of cavity QED, and would be an important step forward in creating long-lived telecommunications wavelength quantum memories in Er3+{}^{3+}3+:Y2{}_{2}2​SiO5{}_{5}5​. This work relies mostly on published spin-Hamiltonian parameters, but Raman heterodyne spectroscopy was performed on Pr3+{}^{3+}3+:YAG to measure the parameters for the excited state.
Microwave and optical transitions of rare-earth ions enable microwave-to-optical frequency transduction. This transduction is a coherent process; therefore, the narrow linewidths and long coherence times of rare-earth ions are crucial for achieving efficient frequency conversion. Particularly long coherence times can be achieved with small-spin materials, as unpaired electronic and nuclear spins introduce magnetic noise, serving as a decoherent source of the transitions. However, complete spin-free rare-earth materials cannot exist without charge compensation. Consequently, residual spins in existing rare-earth materials act as a bottleneck, limiting both their spin and optical coherence times. Rather than focusing on nearly spin-free materials, this thesis investigates the microwave and optical transitions of rare-earth ions in materials that are fully concentrated with electronic spins. Recent studies on rare-earth magnetic materials have demonstrated narrow optical linewidths resulting from magnetic ordering and a particularly strong coupling between microwave cavity modes and the collective excitations of rare-earth ions. Although these recent studies are noteworthy, the energy-level transitions, especially optical transitions, of rare-earth ions in magnetic materials remain largely unexamined. In this thesis, various spectroscopic studies are employed to characterise their transition properties and to develop a model of energy levels. The model presented is expected to be applicable to other rare-earth magnetic materials, thereby opening avenues for further investigation in this class of systems. Two types of rare-earth magnetic materials are investigated: neodymium gallate with a stoichiometric concentration of neodymium ions, and gadolinium vanadate doped with a low (ppm) concentration of erbium ions. Microwave and optical transmission spectroscopy are employed under externally applied magnetic fields to characterise the energy-state transitions of the rare-earth ions in both materials. The magnetic phases and corresponding spin configurations influence the observed spectra, making conventional approaches to spectral simulation inadequate. To address this challenge, mean field and magnon-coupled crystal-field models are developed in this thesis to predict magnon and optical transition spectra. These simulations exhibit good agreement with experimental results, allowing for the characterisation of transition properties. Following the characterisation of the material, a new transduction approach is introduced, developed, and implemented using doped rare-earth ions in a magnetic material. In this scheme, coherent conversion between microwave and optical modes is achieved by exciting an optical transition of the doped ions.