Long-Range Order of Dilute Rare-Earth Spin Ensemble Revealed with Cavity QED

Warrick G. Farr,1 Maxim Goryachev,1 Jean-Michel le Floch,1 Pavel Bushev,2 and Michael E. Tobar1

1ARC Centre of Excellence for Engineered Quantum Systems,

University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia2Experimentalphysik, Universitat des Saarlandes, D-66123 Saarbrucken, Germany

(Dated: December 12, 2014)

This work demonstrates strong coupling regime between an erbium ion spin ensemble and HybridCavity-Whispering Gallery Modes in a Yttrium Aluminium Garnet dielectric crystal. Couplingstrengths of 220 MHz and mode quality factors in excess of 106 are demonstrated. The spin ensembleexhibits memory effects as well as remnant magnetisation. A qualitative change of system magneticfield response between 190 and 445mK is interpreted as a phase transition. This work is the firstobservation of the long range order in an ensemble of dilute Rare-Earth impurities similar to thephenomenon of dilute ferromagnetism in semiconductors.

Quantum Electrodynamics (QED) is considered to bethe forefront of modern physical science. It investigatesthe quantum weirdness of the ‘light-matter’ interactionsand attempts to apply it to such emergence areas asquantum computing and quantum communications aswell contemporary physics, attempting to answer funda-mental questions of nature. For these purposes, researchexperiments utilize different types of ‘light’ (representedby various circuit and cavity resonators) and ‘matter’ (forexample spin ensembles, single ions and artificial super-conducting qubits).

A particular example of a QED experiment is a ‘spins-in-solids’ system where a host for quantum matter isbound within a dielectric crystal serving as a 3D mi-crowave or optical cavity. Indeed, isolated spin impu-rities of such crystals constitute an ensemble of two-levelquantum systems, which can be probed with the elec-tromagnetic cavity field. For a number of such system,strong coupling regimes where cavity photons hybridisewith spin ensemble excitation has been demonstrated[1–7]. Although, much of this work deals with ensembles ofspins, which do not interact with each other, some resultshave demonstrated that spin-spin interactions will lead toextra observable effects [8, 9]. In such cases, where theconcentration of impurities is large enough to establishlong-range interactions between them, leads to implica-tions of the magnetic phase. Indeed, the answer mayvary between Curie or van Vleck paramagnetism in ex-tremely dilute systems[10] to completely magnetically or-dered phases in a ferromagnetic systems[11].

This work demonstrates an intermediate regime whichcan be qualified as ‘dilute ferromagnetism’[12–14] wherethe long range order is established by dilute randomlydistributed impurities in a non-magnetic crystal. More-over, unlike all of the works on dilute ferromagnetismwhere the phenomenon is created by Iron Group ions,we demonstrate an experiment with an ensemble of RareEarth ions, in particular erbium impurities in single crys-tal Yttrium Aluminium Garnet, Y3Al5O12 (YAG). Thiscombination of the impurity ensemble and the host is in-teresting due to simultaneous existence of anisotropic mi-

crowave (X-band[15] (8-12 GHz)) and optical (telecom-munication band[16]) transition that can be utilised forquantum frequency converters[17–22]. Due to the excel-lent optical properties of YAG[23], the system may bedesigned as quantum microwave-to-optical converter.

The experiment is based on Whispering Gallery Modes(WGM) of a single crystal YAG playing a role of a cylin-drical high Q-factor dielectric cavity (see Fig. 1 for aschematic view). The crystal is a host for Er3+ magneticimpurities substituting Y3+ ions during the growth pro-cess. The YAG cavity (14.95 mm in height, 15.00 mmin diameter, with crystal z-axis aligned with the cylinderaxis) is supplied by the manufacturer (Scientific Materi-als, Inc.) without a specified Er3+ concentration. Thecrystal is held by a pair of metallic posts and housedin a oxygen free copper cavity. For cavity transmissionmeasurements, coupling of microwave signal lines to thesystem modes is realised with two straight antennae, par-allel to the z-axis. Details of similar WGM spectroscopyis discussed previous works[8, 24, 25].

The demonstration is performed with an open QEDsystem composed of a photonic mode and rare-earth spinensemble (see Fig. 1, (A)) coupled between each otherwith strength g. Both subsystems are coupled to theenvironment as κ and Γ respectively. Coupling of thephotonic mode to the excitation and detection microwavetransmission lines are β1 and β2 respectively.

Dielectric cylindrical resonators are typically designedto exhibit WGMs (see Fig. 1, (B)). These modes areformed by standing or travelling waves formed by con-tinuous total internal reflection from the cylindrical bor-der between two media, i.e. dielectric and vacuum. Al-though, if the dielectric cylinder is enclosed into a metaliccavity with a central post, it may exhibit another typeof mode associated with the post itself. These CentralPost Modes (CPM) are closely related to the so-calledre-entrant cavity modes[26]. For the actual experimen-tal structure, CPMs have been modelled numerically andclassified (see Table I) with the good agreement betweenexperiment and simulation. Moreover, corresponding fill-ing factors correlate with the observed linewidths as the

2

(A) (B)(A)

fgfdfhdfPhotonic Mode

Environment

WGM

Central Post Modes (CPM)

Magnetic !eld Electric !eld

FIG. 1. (A) Couplings between the subsystems: photoniccavity mode, Er3+ ensemble, the environment as well the ex-ternal excitation and measurement apparatus. (B) A crystalWGM is created by the wave propagating along the cylindercircumference. (C) A Central Post Mode is characterised bystrong field associated with the central post.

latter is determined by the former. These cavity modesare expected to have lower Q-factors because the metal-lic walls of the cavity act as an imperfect mirror, result-ing in a considerable field close to metal surface or con-siderable current in the surface itself. In consequence,this structure allows cavity modes with complex fieldstructures[27].

CPM modes have lower filling factors within the di-electric crystal, and thus exhibit relatively low valuesof Q-factor (in comparison to WGM) due to the signifi-cant dissipation in the central post. This results in modelinewidths κ in the megahertz range which exceeds thelinewidths of typical WGMs in YAG by several orders ofmagnitude[28]. In the low order CPM case, κ > Γ forall the modes in the range 10.8 GHz and 12 GHz. Thecavity transmission results depicting the CPM modes fortwo different values of system temperatures are shown inFig. 2. Note that crystal WGMs with significantly higherQ-factors are not visible at this frequency scale.

The darker regions in Fig. 2 parallel to the magneticfield axis correspond to higher cavity transmission andrepresent CPM modes. When the DC magnetic fieldtunes the Er3+ ensemble to a CPM mode frequency,the system exhibits an avoided level crossing (ALCs).There are clearly two major transitions of Er3+, Sa andSb with g-factors of ga = 7.5 and gb = 5.16. Theselines correspond to the ESR |−1/2〉 → |+1/2〉 of Er3+

substituting for the Y3+ at a single site with D2 sym-metry, which has 2 subclasses of magnetically inequiva-lent sites [16, 29]. Coupling parameters of CPM modesin the 10 − 12 GHz range are given in Table I. De-spite comparatively large linewidths, several CPM modesexhibit strong coupling with both transitions of the er-

11.8

11.6

11.4

11.2

11.0

10.8

11.8

11.6

11.4

11.2

11.0

10.8

Fre

qu

en

cy

(G

Hz)

Po

we

r Tr

an

smis

sio

n (

a.u

.)

80DC Magnetic Field (mT)

120

11.5

160

11.6

80DC Magnetic Field (mT)

120 160

11.911.7Frequency (GHz)

11.8

FIG. 2. Transmission spectroscopy of the Er:YAG cavityshowing a series of CPM modes exhibiting multiple interac-tions with the erbium ensemble for two temperatures: (A)T = 19 mK, (B) T = 930 mK. (C) Mode splitting for a11.72GHz CPM mode induced by the Sa transition.

bium ensemble. This regime is characterised by the spin-photon coupling exceeding the cavity and spin ensemblelinewidths g > κ,Γ. The strong coupling regime is pri-marily limited by the spin linewidth Γ that is 2π × 54MHz.

The hyperfine structure of 167Er in both in Fig. 2 (A)and (B) is most clearly visible at magnetic fields lowerthan the Sa ensemble, since the lines otherwise overlapwith the less well resolved hyperfine lines of the Sb ensem-ble. Such hyperfine and quadrupole hyperfine structureshave been previously observed in Er3+ doped YttriumOrthosilicate crystals at millikelvin temperatures[4, 30,31].

Fig. 3 shows the coupling g/2π of two selected CPM(11.71 GHz and 11.32 GHz) to both erbium principaltransitions Sa and Sb as a function of environment tem-perature. The temperature dependence demonstratesa combination[32] of the Curie linear dependence andvan Vleck constant paramagnet dependencies respec-tively above and below 200mK. The transition is due tothe fact that below this temperature all electron spinshave condensed to the lowest energy level |−1/2〉. Thestrongest coupling occurs at the lowest temperatures be-tween the 11.71 GHz mode and the Sb transition.

Coupling between a spin ensemble and a microwavecavity depend on a square root of the number of spins.For the experiment discussed in this work, a priori con-centration of erbium impurities is not known. Although

3

f(GHz) Q g(Sa), MHz g(Sb), MHz ξ

10.899 16,000 117 133 0.1110.991 16,000 107 118 0.1111.066 8,500 174 196 0.3111.218 8,200 154 174 0.2511.422 22,000 125 145 0.3411.428 22,000 125 145 0.3411.437 44,000 148 156 0.2911.440 11,000 119 134 0.2911.716 13,000 178 220 0.3511.720 19,000 178 220 0.3511.734 34,000 58 67 0.1411.748 28,000 58 67 0.1411.920 76,000 131 151 0.2111.921 90,000 131 151 0.21

TABLE I. Parameters of CH modes between 10.8 and 12 GHz:coupling g between Sa and Sb at 19mK, ξ is the magneticfilling factor of the photonic mode.

11.714 GHz,

11.714 GHz,

11.230 GHz,

11.230 GHz,

FIG. 3. Temperature dependence of the coupling g/2π be-tween the selected CPM modes and electron spin transitionsSa, Sb. The dashed line shows g/2π ∼ 54 MHz threshold forthe strong coupling regime.

it can be deduced based on the knowledge of the mea-sured coupling strength g the photon filling factor ξ⊥using the following estimation:

n =4~

ω0µ0ξ⊥

(

g

gACµB

)2

, (1)

gAC is the AC g-factor, ω0/2π is the resonance frequencyand µB is the Bohr magnetron. Whereas g is determinedfrom the experiment, ξ⊥ is found with finite elementmodelling of the microwave cavity based on previouslymeasured YAG permittivity[33]. The resulting numberξ⊥ ≈ 0.35 describes the ratio of the magnetic field alongthe radial and axial directions within the crystal to thetotal magnetic field within the CPM mode. The densityof the Sa transition spins in the ground state is foundfrom the interaction with the ω0/2π = 11.71 GHz modeis ∼ 9 × 1019 cm−3. The corresponding density of Sb

transition spins approaches ∼ 2× 1020 cm−3. These con-

centrations are significantly larger than typical concen-trations used in other microwave QED experiments withrare-earth ion ensembles[4, 5, 30, 34].For QED experiments, it is imperative to have a low

number of excitations. This parameter could be esti-mated based on the cavity incident power Pinc, cavity-transmission line couplings β1 and β2 and the systemlinewidth[35]:

N =Pinc

~ω0κ

4 β1

(1 + β1 + β2)2. (2)

giving N = 13 for the 11.71 GHz mode and externalcouplings β1 = β2 = 10−4.Pure high-order WGMs exhibit a high confinement of

the microwave energy inside the dielectric crystal, this re-sults in significantly lower coupling to the environment,in particular to the metallic walls. Typical values of Q-factors for WGMs will exceed 105, this inverts the rela-tion between spin and photon couplings to the environ-ment κ ≪ Γ. The low linewidths of these modes allowsus to resolve so-called mode doublets, degenerate modeswhose reflection or time-reversal symmetries are brokendue to various imperfections of the system[32]. It shouldbe mentioned that some CPM modes are also capable ofdemonstrating large values of quality factor due to rela-tively good energy confinement. WGMs in 10− 20 GHzrange were characterised in the same manner as the lowQ-factor CPM modes, i.e. transmission measurementsare made as a function of the external DC magneticfield. The high Q-factor measurements demonstrate aparticular memory effect. This phenomenon reveals it-self in the considerable difference between negative andpositive DC magnetic fields scans as the magnetic fieldis progressively scanned and is dependent on the direc-tion of change in magnetic field, this is demonstratedin Fig. 4. The fractional frequencies δf = f−f0

f0of dou-

blets of two modes is plotted against the applied magneticfield. The memory phenomenon can be explained withthe hysteric effect, which typically arises during the mag-netisation of long-range-ordered magnetic materials suchas ferromagnets. It should be underlined that such hys-teretic effects have not been observed with other param-agnetic crystals in similar experimental set ups involvingWGMs[10, 24, 25, 32, 34].Another observation is that the minimal separation be-

tween a pair of resonances of a doublet (the extremum)is not exactly at B = 0 T, but it is shifted to some fi-nite negative field B0 of around -1 mT. This field cannotbe attributed simply to the magnet bias, since this effectestimated from previous experiments fluctuates arounda significantly lower value. The B0 offset occurs for alarge number of WGMs, which is demonstrated in Fig. 5(A). This figure also shows that the WGE3,0,0 mode(f0 = 10.420 GHz) shows the greatest response to themagnetic field. This can be explained by the fact thatthis mode interacts more strongly with the Sa erbium

4

8

6

4

2

0

-2

-4

-50 0 -5050 500

(a) (b)

FIG. 4. Asymmetry of WGM doublets in terms of fractionalfrequency δf for two direction of the field change: (a) de-creasing magnetic field (∆B < 0), (b) increasing magneticfield (∆B > 0).

spin flip transition, due to its proximity in frequency atzero field.In order to quantify the hysteresis effect we define η±

as a measure of the asymmetry of the WGM curve, for theupper and lower doublets respectively. They are definedas

η± =1

2Br

B0∫

−Br+B0

δf±dB −1

2Br

Br+B0∫

B0

δf±dB, (3)

where ± sign stands for the upper or lower resonance of adoublet, Br is the maximum absolute magnetic field forwhich the modes are compared. This parameter equalszero in the limiting case of a symmetric system. Es-timations for η± together with other parameters of themodes are given in Table II. The table demonstrates thatthe asymmetry for the higher frequency mode is alwayslarger in magnitude. Additionally, signs of the coefficientη± are opposite and depend on directionally of the mea-surements.The WGE3,0,0 mode was measured at a series of

increasingly higher temperatures between 19 mK and930 mK. The corresponding series of curves, shown inFig. 5 (B), indicates that the system response qualita-tively changes between 191mK and 445mK as confirmedby the η± factors listed in Table II. For example, theresponses at 445mK and 920mK demonstrate negligiblehysteresis and the bias field B0 cannot be detected. Thisfact suggests that the impurity spin ensemble exhibits aphase transition between 191 and 445mK.At lower temperatures the system demonstrates the

remnant magnetisation effect evident in the asymmetryof the WGM doublets. However, at higher temperatures,the spin interaction with cavity photons behaves in thefully memoryless manner. This phase transition occurswhen the average coupling between dilute impurity spins

0

5

10

-5-40 0-20 20

19.3 mK B →

191 mK ← B

445 mK B →

920 mK ← B

0

5

10

-5

15

-40 0-20 20 40

(A)

(B)

FIG. 5. Magnetic field response of WGM doublets in termsof the fractional frequency δf : (A) a set of modes measuredsimultaneously after crystal magnetisation to +5 T. Descrip-tion of the modes is given in Table. II, (B) WGE3,0,0 modedoublets measured at different temperatures. The qualitativechange of behaviour happens between 190 mK and 440 mK.

exceeds the thermal fluctuation energy leading to thelong range order. Using the estimation for the spin en-semble density, one can find the average coupling betweenimpurity ions[36] that, in the present case, approachesJ/~ ∼ 0.5 GHz. Next, using the mean field approach,the critical temperature can be estimated as T = zJ

kB

where kB is the Boltzmann constant and z is the num-ber of its nearest neighbours. So, in order to obtain thecritical temperature above 0.2K, that is experimentallyobserved, the latter parameter should exceed 8.5. Takinginto account the relatively dense packing of erbium im-purities, about one per 8.5 unit cells, this number can beviewed as realistic. Note that some signature of ferromag-netic coupling between rare earth ions has been observedin weakly doped YVO4, LiYF4 and YAG where the ef-fect of transition satellites is explained by dipole-dipoleinteraction between ion pairs[37]

In summary, we demonstrated a QED experiment with

5

TABLE II. Parameters of various WGMs. Bm is the maximum applied field prior to measurement. The measurements arepresented in chronological order.

Mode structure f0 (GHz) T(mK) Q · 105 B0(mT) Bm (mT) η+ · 10−4 η

−· 10−4 Direction

WGE4,0,2 15.145 19 3.8 -0.5 -300 -46 34 →

WGE3,0,0 10.420 19 30 -1.3 +5000 59 -21 ←

NC1b 16.856 19 2.8 -1.2 +5000 19 -6.7 ←

WGH5,0,0 17.211 19 2.5 -1.2 +5000 39 -26 ←

NC2b 17.711 19 3.3 -1.3 +5000 18 -5.7 ←

WGH5,0,1 18.117 19 5 -1.3 +5000 30 -12 ←

NC3b 18.833 19 16 -1.2 +5000 27 -16 ←

WGE3,0,0 10.420 39 3 -0.7 +200 7.0 -2.7 ←

WGE3,0,0 10.420 116 3 -0.4 -70 -7.4 3.6 →

WGE3,0,0 10.420 191 3 -0.7 +200 23 -11 ←

WGE3,0,0 10.420 445 3 -0.7 -70 -3.4 -1a ←

WGE3,0,0 10.420 920 3 -1.0 +200 0a 0a ←

a Implies the measured value is below sensitivity of measurement, η−

is less sensitive than η+.b Non-categorised modes.

strong coupling between photons and erbium spin ensem-bles in a YAG resonator. The spin ensemble demon-strated behaviour unusual for paramagnetic systems.This behaviour includes remnant magnetisation, hystere-sis and memory effects and sharp change of magneticfield response at some critical temperature in a dilutespin ensemble observed by virtue of WGMs. These factsdraw analogies with the phenomenon of dilute ferromag-netism studied in semiconductors doped with Iron Groupions[12].

This work was supported by Australian ResearchCouncil grants CE110001013 and FL0992016.

[1] Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques,D. Zheng, A. Dreau, J.-F. Roch, A. Auffeves, F. Jelezko,J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve,Phys. Rev. Lett. 105, 140502 (2010).

[2] D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo,L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs,B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf,Physical Review Letters 105, 140501 (2010).

[3] R. Amsuss, C. Koller, T. Nobauer, S. Putz, S. Rot-ter, K. Sandner, S. Schneider, M. Schrambock, G. Stein-hauser, H. Ritsch, J. Schmiedmayer, and J. Majer, Phys.Rev. Lett. 107, 060502 (2011).

[4] S. Probst, H. Rotzinger, S. Wunsch, P. Jung, M. Jerger,M. Siegel, A. V. Ustinov, and P. A. Bushev, Phys. Rev.Lett. 110, 157001 (2013).

[5] A. Tkalcec, S. Probst, D. Rieger, H. Rotzinger,S. Wunsch, N. Kukharchyk, A. D. Wieck, M. Siegel,A. V. Ustinov, and P. Bushev, Phys. Rev. B 90, 075112(2014).

[6] E. Abe, H.Wu, A. Ardavan, and J. J. L. Morton, AppliedPhysics Letters 98, 251108 (2011).

[7] V. Ranjan, G. de Lange, R. Schutjens, T. Debelhoir,J. P. Groen, D. Szombati, D. J. Thoen, T. M. Klapwijk,R. Hanson, and L. DiCarlo, Physical Review Letters110, 067004 (2013).

[8] W. G. Farr, M. Goryachev, D. L. Creedon, and M. E.Tobar, Phys. Rev. B 90, 054409 (2014).

[9] J. Bourhill, K. Benmessai, M. Goryachev, D. L. Creedon,W. Farr, and M. E. Tobar, Phys. Rev. B 88, 235104(2013).

[10] M. Goryachev, W. G. Farr, D. L. Creedon, and M. E.Tobar, Phys. Rev. A 89, 013810 (2014).

[11] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan,M. Kostylev, and M. E. Tobar, Phys. Rev. Applied 2,054002 (2014).

[12] T. Dietl and H. Ohno, Rev. Mod. Phys. 86, 187 (2014).[13] T. Dietl, Journal of Physics: Condensed Matter 19,

165204 (2007).[14] T. Dietl, Nat Mater 2, 646 (2003).[15] C. P. J. Poole and H. A. Farach, eds., Handbook of Elec-

tron Spin Resonance, Vol. 2 (Springer-Verlag, 1999).[16] A. Kaminskii, in Laser Crystals, Springer Series in Op-

tical Sciences, Vol. 14 (Springer Berlin Heidelberg, 1981)pp. 114–165.

[17] C. O’Brien, N. Lauk, S. Blum, G. Morigi, and M. Fleis-chhauer, Phys. Rev. Lett. 113, 063603 (2014).

[18] L. Williamson, Y.-H. Chen, and J. Longdell,arxiv.org/abs/1403.1608 (2014).

[19] L. Tian, P. Rabl, R. Blatt, and P. Zoller, Phys. Rev.Lett. 92, 247902 (2004).

[20] K. Stannigel, P. Rabl, A. S. Sørensen, P. Zoller, andM. D. Lukin, Phys. Rev. Lett. 105, 220501 (2010).

[21] A. M. Stephens, J. Huang, K. Nemoto, and W. J. Munro,Phys. Rev. A 87, 052333 (2013).

[22] D. V. Strekalov, H. G. L. Schwefel, A. A. Savchenkov,A. B. Matsko, L. J. Wang, and N. Yu, Phys. Rev. A 80,033810 (2009).

[23] G. Liu and B. Jacquier, eds., “Spectroscopic properties ofrare earths in optical materials,” (Springer and TsinghuaUniversity Press, 2007) Chap. 7.

[24] W. G. Farr, D. L. Creedon, M. Goryachev, K. Benmessaı,and M. E. Tobar, Phys. Rev. B 88, 224426 (2013).

[25] M. Goryachev, W. G. Farr, and M. E. Tobar, AppliedPhysics Letters 103, 262404 (2013).

[26] J.-M. Le Floch, Y. Fan, M. Aubourg, D. Cros, N. C. Car-valho, Q. Shan, J. Bourhill, E. N. Ivanov, G. Humbert,V. Madrangeas, and M. E. Tobar, Review of ScientificInstruments 84, (2013).

6

[27] M. Tobar, J. Hartnett, E. Ivanov, P. Blondy, andD. Cros, Instrumentation and Measurement, IEEETransactions on 50, 522 (2001).

[28] J. G. Hartnett, A. N. Luiten, J. Krupka, M. E. Tobar,and P. Bilski, Journal of Physics D: Applied Physics 35,1459 (2002).

[29] Y. Sun, T. Bottger, C. W. Thiel, and R. L. Cone, Phys.Rev. B 77, 085124 (2008).

[30] S. Probst, A. Tkalcec, H. Rotzinger, D. Rieger, J.-M.Le Floch, M. Goryachev, M. E. Tobar, A. V. Ustinov,and P. A. Bushev, Phys. Rev. B 90, 100404 (2014).

[31] P. Bushev, A. K. Feofanov, H. Rotzinger, I. Protopopov,J. H. Cole, C. M. Wilson, G. Fischer, A. Lukashenko,and A. V. Ustinov, Phys. Rev. B 84, 060501 (2011).

[32] M. Goryachev, W. G. Farr, D. L. Creedon, and M. E.Tobar, Phys. Rev. B 89, 224407 (2014).

[33] J. Krupka, K. Derzakowski, M. Tobar, J. Hartnett, andR. G. Geyer, Measurement Science and Technology 10,387 (1999).

[34] M. Goryachev, W. Farr, N. Carvalho, D. Creedon, J. L.Floch, S. Probst, A. Ustinov, P. Bushev, and M. Tobar,arXiv:1410.6578 (2014).

[35] J. G. Hartnett, J. Jaeckel, R. G. Povey, and M. E. Tobar,Physics Letters B 698, 346 (2011).

[36] E. Baibekov, I. Kurkin, M. Gafurov, B. Endeward,R. Rakhmatullin, and G. Mamin, Journal of MagneticResonance 209, 61 (2011).

[37] O. Guillot-Noel, V. Mehta, B. Viana, D. Gourier,M. Boukhris, and S. Jandl, Phys. Rev. B 61, 15338(2000).