Hyperfine interactions of Er 3+ ions in Y 2 SiO 5 : electron paramagnetic resonance in a tunable microwave cavity
Yu-Hui Chen, Xavier Fernandez-Gonzalvo, Sebastian P. Horvath, Jelena V. Rakonjac, Jevon J. Longdell
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Hyperfine interactions of Er ions in Y SiO :electron paramagnetic resonance in a tunable microwave cavity Yu-Hui Chen, Xavier Fernandez-Gonzalvo, Sebastian P. Horvath, Jelena V. Rakonjac, and Jevon J. Longdell
The Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics,University of Otago, 730 Cumberland Street, Dunedin, New Zealand. (Dated: January 31, 2018)The hyperfine structure of the ground state of erbium doped yttrium orthosilicate is analyzedwith the use of electron paramagnetic resonance experiments in a tunable microwave resonator.This work was prompted by the disagreement between a recent measurement made in zero magneticfield and a previously published spin Hamiltonian which. The ability to vary magnetic field strength,resonator frequency, and the orientation of our sample enabled us to monitor how the frequencies ofhyperfine transitions change as a function of a vector magnetic field. We arrived at a different set ofspin Hamiltonian parameters, which are also broadly consistent with the existing data. We discussthe reliability of our new spin Hamiltonian parameters to make predictions outside the magneticfield and frequency regimes of our data. We also discuss why it proved to be difficult to determinespin Hamiltonian parameters for this material, and present data collection strategies that improvethe model reliability.
PACS numbers: 32.10.Fn, 76.30.Kg, 32.30.Dx.
I. INTRODUCTION
Due to long optical and hyperfine coherence times, rareearth doped crystals are considered to be a very promis-ing candidates in the quest for applications in quantummemories and quantum information. These materialshave been shown to have optical coherence times on theorder of milliseconds and allow for tailoring of the inho-mogeneous linewidth using external electric or magneticfields . By transferring the excitations to hyperfine spinenergy levels, it has been demonstrated that the coher-ence time of rare-earth doped crystals can be extended tosix hours . The physics behind this extension is to finda specific magnetic field so that the hyperfine transitionshave zero first-order Zeeman (ZEFOZ) shift and are onlysensitive to second order magnetic field fluctuations .To utilize this technique, one requires an accurate modelof the hyperfine structure for the rare-earth dopant.Among the rare earth ions, erbium (Er) has a uniqueoptical transition located at 1.5 µ m that makes it com-patible with optical fibres, and Er doped crystals haveboth optical life times and coherence times of the or-der of milliseconds . For this reason, Er based opticalquantum memories are strong candidates for inclusion ina future optical fiber based quantum network. Moreover,erbium has a stable isotope Er (natural abundance22.94%) with a nuclear spin of I = 7 /
2, which resultsin a hyperfine structure extending over a 5 GHz rangeat zero magnetic field. This has prospects for achievingultra-long coherence times , as well as enabling the devel-opment of quantum microwave-to-optical converters .For this reason, there has been considerable ef-fort devoted to understanding the hyperfine structureof Er doped yttrium orthosilicate ( Er :Y SiO ).This includes the identification of effective three-level Λsystems , as well as, more recently, the observation of aground-state coherent Raman process with a coherence time of 50 µ s , which was limited by electron spin-spininteractions. This limitation has been circumvented byexperiments employing strong external magnetic fields,yielding a coherence time of 1.3 seconds for hyperfinetransitions in Er :Y SiO , and thus demonstratingthe practical viability of quantum memories at 1.5 µ m. Inorder to guide such developments using accurate theoret-ical models, in addition to enabling the above outlinedZEFOZ technique, it is crucial to have an accurate under-standing of the hyperfine structure of Er doped yttriumorthosilicate .The hyperfine struncture of the ground state I / Z of Er doped Y SiO was first measured by standardelectron paramagnetic resonance (EPR) experiments at9.5 GHz by Guillot-N¨oel et al. . By analyzing the angu-lar variations of the eight allowed and some forbiddenhyperfine transitions the spin Hamiltonian parameterswere determined. However, the predicted ground-stateenergy levels of Er :Y SiO from this original set ofparameters resulted in some discrepancies with zero-fieldmeasurements .In this paper, the ground-state spin Hamiltonian pa-rameters of Er :Y SiO are determined by EPR ex-periments with a tunable loop-gap resonator. In contrastto previous EPR measurements using a fixed-frequencymicrowave resonator , both the resonator frequency aswell as the applied magnetic field were varied to yield twodimensional EPR scans, allowing one to monitor how thefrequency of a particular hyperfine transition varies withan applied magnetic field. Because of the large numberof parameters, simulated annealing was used to finda set of parameters that best fit the observed spectra.Published zero-field EPR data was then introduced intothe calculation to refine the fitted Hamiltonian. In orderto determine the uncertainties in the spin Hamiltonianparameters, the Markov-Chain-Monte-Carlo technique was used to sample the posterior probability distribution. II. ERBIUM DOPED Y SiO When using rare earth ion doped crystals for quan-tum information applications, ideally the host crystalwould be free of nuclear and electron spins so that thereis no magnetic pertabations in the local fields for thedoped ions. While no such host crystal has been demon-strated, Y SiO has low nuclear-spin fluctuations. Yt-trium (100% Y) has a nuclear magnetic moment ofonly -0.137 µ n , Si has one magnetic isotope of abun-dance of 4.6% with a moment of -0.554 µ n , and O hasone magnetic isotope of abundance of 0.04% with -1.89 µ n , where µ n is the nuclear magneton. Very longnuclear spin coherence times have been demonstrated forboth praseodymium and europium dopants in Y SiO and on the basis of previously published spin Hamilto-nian parameters long spin coherence times can alsobe expected for Er :Y SiO . Attempts to utilize Er :Y SiO in quantum information have been ham-pered by the low symmetry of the rare earth dopant, al-though the low symmetry does provide benefits for build-ing a Λ system. Y SiO has a monoclinic structure withC h space group and two crystallographic sites of C symmetry, site 1 and site 2, where Er ions can sub-stitute for yttrium ions. Here we follow the definition ofsite 1 and site 2 where I / Z ↔ I / Y transition isat 1536 nm for site 1 and 1539 nm for site 2 as in thereferences . For each crystallographic site, there aretwo magnetically inequivalent ion subclasses for a mag-netic field in an arbitrary direction, with the exceptionof when the magnetic field is aligned along the crystals C (or “ b ”) axis or perpendicular to it . As Er ionshave a nuclear spin of I = 7/2 and an effective electronicspin of S = 1/2, there are 16 hyperfine energy levels forthe ground state, even in the absence of an external mag-netic field. These hyperfine splittings can be represent bythe following spin Hamiltonian H = µ e B · g · S + I · A · S + I · Q · I − µ n g n B · I , (1)where µ e is the Bohr magneton, B the applied magneticfield, g the Zeeman g -matrix, A the hyperfine matrix, Q the electric quadrupole matrix, µ n the nuclear magneton,and g n = − . g factor. However, dueto the low point-group symmetry of the Y SiO crystal,the g , A , and Q matrices have noncoincident principalaxes. Therefore, not only their principal values, but alsotheir individual corresponding Euler angles need to bedetermined, which makes finding the spin Hamiltonianparameters difficult. III. EXPERIMENTAL SETUP
Our sample is a cylindrical Er :Y SiO crystalsupplied by Scientific Material Inc, which has a lengthof 12 mm and a diameter of 4.95 mm. In this sample, 50parts per million of the Y ions are substituted by iso-topically pure Er ions. The coordinate system we will use in this paper uses the crystallographic b axis,in addition to the two other optical extinction axes D , D , which are orthogonal . Orientation of the samplewas carried out by Scientific materials with the crystalcut so that the b axis was along the longitudinal axis ofthe cylinder, and the D axis was identified by a flat cutof the curved surface.The loop-gap resonator, which is made from oxygenfree cooper, is shown in FIG. 1. The Er :Y SiO sample sits in the central hole of the resonator. Thetunable resonator is based on a three-loop two-gap con-figuration, where the resonant frequency can be finelytuned by changing the width of the gap d , as illus-trated in FIG. 1(b) and further details can be found inreference . Our cavity and sample assembly were cooledto 4.3 K using a homebuilt cryostat (cooling head: Cry-omech PT405). At this temperature, the resonator typi-cally showed Q factors of 6 × and was tunable from3 GHz to 5.5 GHz.The magnetic field was supplied by a custom high tem-perature superconductor vector magnet from HTS-110Ltd. It could provide up to ±
312 mT in one direction, z , and ± . x and y . The y axisof the magnet is aligned along the b axis of our sample,allowing the D and D axes to be placed anywhere inthe x − z plane by rotating the sample inside the cylin-drical hole of the microwave resonator during the samplemounting process. Each orientation of the crystal canbe described by the angle θ between D and z , which isdefined as positive when the rotation from D to z alongthe b axis is anti-clockwise. Note that our measurementsof the magnetic field needed to make the two magneti-cally inequivalent subclasses degenerate showed a 1 . ◦ misalignment between the x − z plane and the D − D plane, which were taken into account during the experi-ment and the fitting process. The values for θ were firstmeasured with the help of a camera to an accuracy of ± ◦ , which were allowed to vary during the fitting pro-cess; the resulting values for θ were in a range of valuesconsistent with the photographic measurements.EPR spectra were taken with fixed x and y fields andsweeping the current applied to the z coils. For each ori-entation ( θ ) of the sample, and each cavity frequency,we swept B z from 0 to 300 mT with ( B x , B y ) either(0 , δ ( B z )) mT or one of ( ± . , ± .
3) mT, where δ ( B z )was introduced to address the small misalignment be-tween the x − z plane and the D − D plane. The ap-plied magnetic fields are illustrated in FIG. 1(c)The EPR transitions were detected using a frequencymodulation approach that we have reported earlier .Shown in FIG. 1(d) are EPR lines for three different res-onator frequencies. By moving the plunger, we were ableto gradually tune the frequency of our resonator. Wetherefore obtained five two-dimensional (2D) EPR scansfor each θ and were able to monitor how the frequenciesof particular hyperfine transitions changed as a functionof magnetic field. z xy (c) (b)(a) (d) 0 50 100 150 EP R I n t en s i t y ( a r b . un i t ) Magnetic field in z axis(mT) f res =3.386 GHz f res =3.794 GHz f res =4.120GHz Figure 1: Experimental setup. (a) Photo of our tunable loop-gap resonator. Left, body of our cavity; right, a cap anda plunger. y and z indicate the corresponding directions ofour magnetic field. (b) Schematic picture of the tunable res-onator. The resonant frequency can be tuned by moving theplunger to change the gap, d . Coordinate systems of our crys-tal and our magnetic field are also shown. (c) Schematic ofthe applied magnetic fields. Shown here is the scanning of B z for B x = 10 . B y = 0 mT. (d) Typical EPR lines atthree different resonator frequencies. The lines are verticallyshifted for clarity. IV. RESULTS
Figure 2 shows some representative EPR spectra fromour experiments. The readers are referred to the Supple-mental Material for a complete set of data. As the ab-solute strengths of the EPR signals differ widely accord-ing to the resonance frequency of the microwave cavity,which is a result of the frequency dependence of the ca-ble loss and the coupling strength of the antenna, everyslice of each color figure in FIG. 2 is an EPR spectrumnormalized to their individual maxima. Figures 2(a) and2(b) were the data for θ = 64 . ◦ , (c) and (d) are for θ = 104 . ◦ , and (c) and (d) are for θ = 85 . ◦ .In FIG. 2, sudden color changes from blue to yellow in-dicate EPR peaks with their dispersive lineshapes. Twosets of ‘parallel’ lines with different ‘slopes’ can be iden-tified in FIG. 2. They correspond to two different effec-tive g factors. Based on the previous knowledge of the g factor from literature , we can easily distinguishwhich line belongs to site 1 or site 2 and then input theminto the fitting procedure, as guided by the purple linesand the blue lines. In general, we observed more thaneight ‘allowed’ transitions in both site 1 and site 2, andthe strengths of the EPR signals are on the same orderof magnitude. This is because the hyperfine states arehighly mixed states at the this relatively low magneticfield.In FIG. 2(c) and (d), there are several lines whichcannot be ascribed to either site 1 or site 2. See, for example, the lines marked by the black arrows inFIG. 2(c). They also appear to have hyperfine transitionsand are anisotropic; we attribute them to impurities inthe Y SiO . Such impurities have also been previouslyobserved in Y SiO crystals .The fitting of the spin Hamiltonian parameters wasbased on our 2D EPR scans. In general, the measuredEPR spectra of Er are composed of EPR signalsfrom two inequivalent magnetic subclasses of both site 1and site 2, the spin Hamiltonian parameters of which arerelated by a 180 ◦ rotation along the b axis in Y SiO .That is to say, we normally have four sets of hyperfinelines in each of our 2D EPR scans. However, θ was cho-sen to give significant different effective g factors to site1 and site 2, and thus it was easy to categorize the mea-sured EPR lines to site 1 and site 2 prior to our fittingprocedure.The spin Hamiltonian parameters g , A , and Q werechosen to be symmetric as in previous measurements and parameterised by their principal values and Eulerangles to describe their directions. The Euler angles hereare chosen to represent an intrinsic rotation sequence of z − x ′ − z ′′ from the right-handed ( D , D , b ) system tothe right-handed coordinate system of the principal axes.For example, let matrix M be defined in ( D , D , b ) and M p is defined according to its principal axes. Followingthis definition, we have M = R T · M p · R (2)where the rotation matrix R = R z ( γ ) · R x ( β ) · R z ( α ),and R z ( γ ) , R x ( β ) , R z ( α ) represent rotations of α , β , and γ along z , x ′ (the x after the first elemental rotation),and z ′′ (the z after the second rotation) axes. In our fit-ting, the axes with the biggest principal values are chosento be along z , and the other principal axes are allowedto vary. For both site 1 and site 2, the principal axes of g , A , and Q are not coincident. So we had to determinesix parameters for g , six for A , and five for Q which isa traceless matrix. Together with four θ s (sample ori-entations), this results in 38 parameters in total to bedetermined in our fitting.From our 2D EPR scans, we can extract the frequenciesof hyperfine transitions as a function of B z , i.e., pointsas ( f, B z ), where the hyperfine transition frequencies andresonator frequencies satisfy f tran = f res = f . In prin-cipal, a misfit function can be defined by summing thevariances of either B z or f . Because f res can be measuredvery precisely (the precision can be up to kHz, and it islimited by the noise of the microwave detector), it is bet-ter to use f as arguments and B z as dependent variablesto define a misfit function. The misfit function is thendefined as misfit = N X i ( B cal z,i − B exp z,i ) (3)where B cal z,i are the calculated B z at one determined peak f i , B exp z,i are the corresponding experimental B z values, i (b) (c) (d) (e) (f) F r equen cy ( G H z ) Magnetic field in z axis (mT) (a) ϴ = 104 o Bx = 10.3 mT By = -10.3 mT ϴ = 85 o Bx = 0 By = -0.012*Bz ϴ = 64 o Bx = 0 By = -0.012*Bz
Figure 2: Experimental EPR data and calculated transition lines. In the color plots, each horizontal slice is a normalized EPRspectrum as a function of B z , while the applied B x and B y are indicated by the corresponding insets. (a) is the measured EPRspectra at θ = 64 . ◦ . The scanned resonator frequency range is 3388.6 MHz to 4975.6 MHz. (c) is the measured EPR spectraat θ = 104 . ◦ . The scanned resonator frequency range is 3960.6 MHz to 5058.8 MHz. (e) is the measured EPR spectra at θ = 84 . ◦ . The scanned resonator frequency range is 3811.3 MHz to 5184.8 MHz. The circle dots in (b), (d), and (f) are theexperimental EPR peaks extracted from (a), (c), and (e), respectively. On top of the experimental points, the pruple lines arethe EPR lines for site 1 from calculations using the parameters in Table I and blue lines are for site 2. is the index of our EPR peaks, and N is the total num-ber of EPR peaks that were used in the fitting. It iseasy to calculate the hyperfine transition frequencies fora given magnetic field (by diagonalization of the totalHamiltonian), but the inverse problem, getting B cal z,i fora given f i , is more difficult. Instead of doing that, weused interpolation to give B exp z,i at any given f i . In otherwords, we first set a B cal z,i to calculate a hyperfine transi-tion frequency f i , and then we want to compare this B cal z,i with a B exp z,i . But since there is no such a B exp z,i at thatparticular f i , an interpolation from the closest experi-mental points is used to find a ‘synthetic’ experimentalpoint B exp z,i , and then the misfit at f i was calculated us- ing Eq. (3). This method works for our case because theerror of interpolation is much less than the uncertaintiesof our experiment. Using frequency instead of magneticfield to define a misfit function similar to Eq. (3) could bean alternative, but this requires carefully choosing differ-ent weighted numbers for data sets of different effective g factors (approximately from 2 to 15 in our experiment).This is because different g factors can introduce signifi-cantly different shifts in frequency even for the same errorin magnetic field.Using trial spin Hamiltonian parameters, we calculatedpoints ( f i , B cal z,i ) by diagonalization of the spin Hamilto-nian. For each calculated magnetic field, the diagonal-ization of the spin Hamiltonian results in 120 possibletransitions, and only those strong transitions could beseen in our EPR experiments, therefore those f i witha transition strength above one seventh of their max-ima were considered to be measurable in our measure-ments (EPR signals are proportional to the squares ofthe transition strengths). The screening of weak transi-tions made the fitting process easier and faster. By usingthe least-squares method to minimize the misfit functionin Eq. (3), our spin Hamiltonian parameters, togetherwith θ s, were calculated. The obtained spin Hamiltonianparameters were then refined by another run of weightedleast-squares which took into account the zero field EPRdata in literature . The results are given in Table I.After the best fit parameters were obtained, the “tem-perature” of the simulated annealing algorithm wasraised to a level set by assuming each B exp z,i has an un-certainty of 0.5 mT. The result is a set of parametersthat sample the posterior probability distribution. Un-certainties shown in Table I were calculated from the ac-cepted 37652 samples. It is worth of noting that usingthe Markov-Chain-Monte-Carlo technique to find the un-certainties is difficult in our case due to the in total 38parameters in the fitting; therefore the uncertainties forthose numbers with big uncertainties might be underesti-mated. For calculation purposes, we keep two significantdigits for all the uncertainties and the fitted numbersare rounded accordingly. The experimental θ parametersgiven by fitting are θ = 64 . ◦ ± . ◦ , θ = 7 . ◦ ± . ◦ , θ = 104 . ◦ ± . ◦ , θ = 84 . ◦ ± . ◦ . The resultingvalues for θ were in a range of values consistent with thephotographic measurements.The spin Hamiltonian parameter matrices of site 1and site 2 in ( D , D , b ) obtained were: g = . − . − . − .
95 8 .
90 5 . − .
56 5 .
57 5 . (4) g = . − .
77 2 . − .
77 1 . − . . − .
43 1 . (5) A = . − . − . − . . . − . . . MHz (6) A = − . . − . . − . − . − . − . . MHz (7)and the quadrupole interaction parameters are: Q = . − . − . − . − . − . − . − . − . MHz (8) Q = − . − . − . − . − . − . − . − . . MHz (9)Our magnets were calibrated with an accuracy of ± ± ± O (20 MHz), which agrees withthe uncertainties of the principal values of A and Q inTable I. Another source of uncertainty is the errors in θ . The initial values of θ were measured by the use ofa camera, which has an accuracy of approximately 3 ◦ .These numbers were allowed to vary during the fittingprocesses, which yielded uncertainties of O (1 ◦ ).The ground state g factors have been measuredpreviously , . The values of the g factors determinedfrom our experiment are similar to their measurements.As listed in Table I, the g z values are the most accuratewhich have uncertainties less than 2%, and the g x valuescould not be well determined because the values them-selves are small. Similarly, the largest principal values A z of both site 1 and site 2 are then the most accurateamong the three principal values of the hyperfine inter-action A , which have uncertainties of less than 1%. Theuncertainties of the principal values of A , O (20 MHz),are limited by the error of our experiment, which makesthe small principal values, A y less accurate, as shown inTable I. As for the quadrupole interaction Q , their uncer-tainties are comparable to their principal values, as listedin Table I. As a result, Q was not as well determined as g and A in our experiment. V. RELIABILITY OF SPIN HAMILTONIANPARAMETERS
When deriving the physical parameters of a systemfrom fits to data, a potential problem is that the resultingfit is fortuitous. The fact that the derived parameters fitthe data doesn’t mean the “true” set of parameters havebeen found. This problem is exacerbated in a situationlike Er :Y SiO because the low symmetry meansthat there is a large number of parameters. Furthermorethe large nuclear spin of Er means that the spectraare in general very dense with spectral lines, and highprecision is required if the spin Hamiltonian is going tobe helpful in assigning lines.As a test of our new spin Hamiltonian we made a pre-diction of the zero-field transitions reported previously (Fig. 3). Before we added this data to the fit the pre-dicted spectra was consistent with the observed spectragiven the ∼
20 MHz uncertainty in the spin Hamiltonianparameters. When the zero-field transitions were addedto the dataset used for fitting, very good agreement was
Table I: Principal values and Euler angles of g , A , and Q of Er in Y SiO at 4.3K.Site 1 Site 2Principal values Principal valuesEuler angles (deg) Euler angles (deg) α β γ α β γg x = 2 . ± . g y = 0 . ± . g z = 14 . ± .
10 205 . ± .
43 55 . ± .
41 29 . ± . g x = 0 . ± . g y = 1 . ± . g z = 15 . ± .
20 261 . ± .
22 100 . ± . . ± . A x = 260 . ± A y = 22 ±
45 (MHz) A z = 1524 . ± . ± . . ± .
14 26 . ± . A x = 139 ± A y = 13 ±
98 (MHz) A z = − ±
20 262 . ± .
98 94 . ± . . ± . Q x = 9 . ± . Q y = 15 . ± . Q z = − ( Q x + Q y ) 151 . ± . . ± . . ± . Q x = 6 ± Q y = 36 . ± . Q z = − ( Q x + Q y ) 142 . ± . . ± . . ± . | S | m a x ( % ) S t r eng t h ( a . u . ) site 1 site1 (a)(c)(d)(b) 3.05 site 1site 2 Frequency (GHz)
Figure 3: (a) Experimental zero-field EPR spectra reproducedfrom the reference ; (b) Calculated zero-field EPR spectrumwith spin Hamiltonian parameters in Table I, which were ob-tained by fitting both the 2D EPR data and the zero-fieldEPR data. (c) Calculated zero-field EPR spectrum with spinHamiltonian parameters which are obtained by just fitting the2D EPR data. The transition strengths of site 2 in (b) and(c) are much smaller compared to site 1; therefore for claritythe transitions from site 2 are not shown in (b) and (c). (d)Calculated zero-field EPR spectrum with spin Hamiltonianparameters in the reference . M a gn e ti c F i e l d ( m T ) site 2, D D plane Figure 4: Angular variation in the ( D , D ) plane of eight al-lowed hyperfine transitions of site 2 at 9.5 GHz. Black spotsand black lines are experimental points and fit curves adaptedfrom the reference ; the coloured lines are transition linesbased on spin Hamiltonian parameters in Table I. Similarbehavior was observed when comparing our spin Hamiltonianwith other results from the paper, except for one field rota-tion in the ( D , b ′ ) plane where we were unsure of the exactorientation used. achieved. This indicates that while our spin Hamilto-nian parameters can be used to successfully describe lowmagnetic field behaviour more data near zero-field couldsignificantly improve the predictive power of the spinHamiltonian parameters. In FIG. 4, the angular vari-ation in the ( D , D ) plane of eight allowed hyperfinetransitions of site 2 predicted by the parameters in Ta-ble I are plot on top of the experimental data from theliterature . Although the parameters in Table I wereobtained in 3 ∼ .This leads to the question of why it is difficult to de-termine usable spin Hamiltonian parameters in this case.We believe it is related to the fact that the g tensorsin this case are very anisotropic, as shown in Table I.When making standard high-field EPR measurements aswas the case in the literature , the dominant term in thespin Hamiltonian is µ e B · g · S . This term splits the energylevels into two subspaces each of eight levels. One of the-ses subspaces has electron spin up and the other has spindown along a quantization direction determined by the“effective” magnetic field direction ˆn = B · g / ( | B · g | ). Ineach of these two subspaces S can be replaced with a clas-sical vector pointing along or opposite to ˆn . Because ofthis the next most significant term, which is the hyperfinesplitting I · A · S , is analogous to an anisotropic nuclearZeeman term with S being like the applied magnetic fieldand A being like the Zeeman tensor. The nuclear Zee-man component of the energy eigenstates are thereforequantized along the direction ˆm = A · ˆn / ( | A · ˆn | ). Theirsplittings are determined primarily by | A · ˆn | with addi-tional perturbations from the (real) nuclear Zeeman termand the nuclear quadrupole term. In order to properlydetermine the hyperfine tensor A , sufficient data must becollected such that the effective magnetic field ( ˆn ) suffi-ciently samples all directions. This is made difficult bythe fact that the g tensor is highly anisotropic. For mostmagnetic field directions, ˆn will end up close to pointingalong the principle axis of the g tensor with the biggestprinciple value. This is particularly an issue for site 1where one of the transverse g values is very close to zero.A similar argument says that a good coverage of ˆm isneeded to properly determine the Q tensor.This problem with sufficient coverage of magnetic fielddirections is illustrated in the results of Table I. Forhyperfine A , the biggest remaining uncertainty for site 1is A y that has its principle direction close to the principleaxis of g y whose value is close to zero. It can also be seenmanifested in Fig. 4 where predictions from our new spinHamiltonian parameters are overlaid on a figure from thereference . The experimental data agrees with both thenew and the old spin Hamiltonian predictions. Howeverthe old and new predictions differ significantly for thelargest magnetic field values, which is apparent becausethe large magnetic field values correspond to a small g value at fixed-frequency EPR experiments.For this work a tunable resonator was chosen to over-come these difficulties, by allowing measurements whereenergy levels are anti-crossing near zero magnetic field.This wasn’t entirely successful because the signals forboth standard EPR and the variant used here disappearin these situations.Almost all the data used here were straight lines onthe magnetic field versus frequency graphs (see Fig. 2).The improvement of our parameters over others can be mostly attributed as much to better coverage of effectivemagnetic field directions as the fact that energy levels arehighly mixed at low magnetic field. VI. DEGENERACY OF SPIN HAMILTONIANPARAMETERS
Another issue is that many different spin Hamiltonianslead to the same EPR behaviour. Given a vector of spinoperators S = ( S x , S y , S z ) applying a rotation U to thisvector of operators gives a set of spin operators with thesame algebra. For this reason given an arbitrary rotation U , Eq. (1) and H = µ e B · g · U · S + I · A · U · S + I · Q · I − µ n g n B · I , (10)are equivalent.Because the nuclear Zeeman term is very small givenanother arbitrary rotation V leads to another set of al-most equivalent Hamiltonians H = µ e B · g · U · S + I · V T · A · U · S + I · V T · Q · V · I − µ n g n B · V · I , (11)This degeneracy does mean that one has to be carefulbefore saying that two spin spin Hamiltonians are reallydifferent, and it is important if one is trying to under-stand the nature of the site at a deeper level. As the pre-dictions for energy levels and EPR transition strengthsunder these rotations are the same, it wasn’t the reasonfor the discrepancies addressed in this work. Further-more the standard practice of taking the matrices to besymmetric, means that this degeneracy only effects thesign of the principle values.The spin Hamiltonian Eq. (1) won’t be valid for ar-bitarily large magnetic field values, because at high fieldsthe upper electron spin manifold from the ground crystal-field level (Z ) will start to mix with the lower electronspin manifold from the second lowest crystal-field levels(Z ). We don’t think that this is the reason for the dis-crepancy between this work and the old spin parametersfor two reasons. Firstly, Z is 1260 GHz higher in energythan Z and the measurements in the reference weremade with only 9.5 GHz splittings so the mixing betweenthe levels should be small. Our crystal field calculationssupport this assertion. The second reason is that our newspin hamiltonian agrees with the measurements from thereference . It only disagrees with the spin Hamiltonianparameters that was derived from these measurements. VII. CONCLUSION
In conclusion, we have characterized the hyperfinestructure of the ground state of Er ions in Y SiO by measuring EPR spectra in a tunable microwave cavityfor different crystal orientations. Compared to conven-tional EPR, the ability to vary the resonator frequencyand magnetic field gives more details of how the fre-quency of one particular hyperfine transition depends onthe applied magnetic field in a 2D pattern. Based onthe 2D EPR data, the matrices of the Zeeman g factor,hyperfine interactions, and quadrupole interactions of Er :Y SiO are determined by least-squares fitting.The uncertainties of predicted hyperfine energy levels areapproximately 20 MHz. The spin Hamiltonian parame-ters not only agree with the 2D EPR scans in this paper,but are also consistent with the zero-field EPR data andthe EPR data at 9.5 GHz . The difficulty in characteriz-ing the hyperfine structure of Er :Y SiO is ascribedto the fact that g tensors are highly anisotropic whichmeans special attention must be given to the coverage ofmagnetic field directions. To further narrow down the spin Hamiltonian parameters, transition points at zeromagnetic field or anti-crossing points at low magneticfields would be useful. While standard microwave EPRspectroscopy is no longer suitable for anti-crossing points,the combination of both optical and microwave detection,e.g. Raman Heterodyne spectroscopy , is a possibility. VIII. ACKNOWLEDGEMENTS
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