Strong coupling of a single photon to a magnetic vortex
SStrong coupling of a single photon to a magnetic vortex
Mar´ıa Jos´e Mart´ınez-P´erez
1, 2, ∗ and David Zueco
1, 2, † Instituto de Ciencia de Materiales de Arag´on and Departamento de F´ısica de la Materia Condensada ,CSIC-Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain Fundaci´on ARAID, Avda. de Ranillas, 50018 Zaragoza, Spain
Strong light-matter coupling means that cavity photons and other type of matter excitations arecoherently exchanged. It is used to couple different qubits (matter) via a quantum bus (photons) orto communicate different type of excitations, e.g. , transducing light into phonons or magnons. Anunexplored, so far, interface is the coupling between light and topologically protected particle-likeexcitations as magnetic domain walls, skyrmions or vortices. Here, we show theoretically that asingle photon living in a superconducting cavity can be strongly coupled to the gyrotropic modeof a magnetic vortex in a nanodisc. We combine numerical and analytical calculations for a su-perconducting coplanar waveguide resonator and different realizations of the nanodisc (materialsand sizes). We show that, for enhancing the coupling, constrictions fabricated in the resonator arecrucial, allowing to reach the strong coupling in CoFe discs of radius 200 −
400 nm having resonancefrequencies of few GHz. The strong coupling regime permits to coherently exchange a single photonand quanta of vortex gyration. Thus, our calculations show that the device proposed here servesas a transducer between photons and gyrating vortices, opening the way to complement supercon-ducting qubits with topologically protected spin-excitations like vortices or skyrmions. We finishby discussing potential applications in quantum data processing based on the exploitation of thevortex as a short-wavelength magnon emitter.
I. INTRODUCTION
Nanoscopic magnetic systems are a natural playgroundfor the nucleation, manipulation and study of topologicalparticle-like solitons.[1] For instance, a domain wall in aneasy-axis magnet constitutes a two-dimensional topolog-ical soliton. Pure easy-plane spins in three dimensionscan be arranged to contain a one-dimensional topologi-cal defect in the form of a vortex line. Finally, isotropicspins can also exhibit a zero-dimension topological de-fect named Bloch point. Skyrmions constitute a differentkind of topologically protected defect where the magne-tization order parameter does not vanish.[2] Being topo-logical, these particle-like objects are extremely stableagainst thermal fluctuations or material defects. Addi-tionally, domain walls and skyrmions can be efficientlymoved using low-power spin currents or alternating fieldswhich makes them very attractive for their integrationinto the well-known racetrack magnetic memory.[3, 4]Magnetic vortices are stabilized in thin film confinedgeometries.[5] Here, the magnetization curls clockwise orcounter-clockwise (defining the vortex circulation: C) inorder to minimize the total magnetostatic energy. Inthe core, exchange interaction forces the magnetizationto point up or down (defining the vortex polarity: P)leading to an extremely stable four-state logic unit.[6]The vortex core gyrates around its equilibrium posi-tion at sub-GHz and GHz frequencies. This gyrotropicmode can be used to promote polarity inversion whensufficiently large vortex velocities are reached, similarly ∗ [email protected] † [email protected] to the well-known process of walker breakdown in do-main walls.[7, 8]. Interestingly, static spin-polarized cur-rents can be used to produce self-sustained vortex gy-ration leading to the development of spin-torque vortexnanoscillators.[9] Vortex gyration can also be used to gen-erate short-wavelength (sub-micrometric) incoherent[10]or coherent spin-waves.[11]Spin-waves (or their corresponding quasiparticles,magnons) carry information on the spin angular mo-mentum at GHz and sub-THz frequencies with negligi-ble heat dissipation. Due to the possibility of propa-gating short wavelength exchange spin-waves, magnon-based devices could be miniaturized well down to thesub-10 nm scale being very attractive for novel informa-tion technologies such as data processing and comput-ing wave-based architectures.[12] Since recently, the in-terest in magnonic-devices has also moved towards thefield of light-matter interaction for quantum informationand processing. Mastering hybrid magnon-photon statesopens the way for, e.g ., coherently coupling distant spin-waves between them[13, 14] or to different solid-state re-alizations of qubits.[15]So far, most theoretical[16] and experimentalstudies[17] have focused on small YIG Yttrium–iron–garnet spheres coupled to the quasi-uniform magneticfield region in 3D cavities, so to only excite theuniform Kittel mode (having infinite wavelength).Magnon-photon coupling in the quantum regime usingsuperconducting coplanar waveguide (CPW) resonatorshas been addressed only rarely.[18, 19] Also very fewworks deal with magnetostatic (long-wavelength) spin-wave modes arising in confined geometries[20–22] and,notably, individual magnetic solitons such as vortices insoft-magnetic discs still need to be explored in detail.[23]Here, our mission is to figure out if it is possible to a r X i v : . [ qu a n t - ph ] M a r coherently couple microwave photons and vortex motionin the quantum regime. In doing so, we open the pos-sibility to communicate two quantum buses: photons,that are easily coupled to, e.g. , superconducting qubits,and a topological particle-like soliton that can carry in-formation but can also be used to emit and manipulatesub-micron spin-waves. It is important to notice that, inthis work, topology stabilizes the vortex solution but notits gyrotropic motion, which is limited by the dampingin the material. Below, we show theoretically that it isindeed possible to strongly couple a single photon to thegyrotropic motion of a vortex in a disc of CoFe. Thanksto the fact that the vortex resonance frequency can bemodulated by means of an external DC magnetic field,the coupling can be switched on/off in the few ns -regimeusing standard electronics. Thus, we obtain a tunabledevice. Our simulations take realistic material parame-ters, both for the CPW resonator and the magnetic disc,making our proposal feasible in the lab. II. RESULTSA. Vortex description
Vortices living in ferromagnetic nanodiscs exhibit anumber of resonant modes. Among them, the gyrotropicmode corresponds to the vortex precessing at frequency f G around its equilibrium position and being only mini-mally distorted. The gyration sense, on the other hand, issolely given by the right-hand rule to the vortex polarity.For discs having small aspect ratio, i.e. , t/r (cid:28)
1, this fre-quency is approximately proportional to f G ∝ M s t/r ,[24]with M s the material’s saturation magnetization, r theradius and t the thickness of the disc (see left inset in Fig.1 c ). Our purpose is to couple this mode to microwavesuperconducting resonators. For this reason, it will beconvenient to use nanodiscs having large t/r aspect ratiomade out of ferromagnetic materials with large µ M s ∼
1T so that f G is well in the GHz range. We highlight thatthe resonant frequency can not be made arbitrary largesince f G decreases when the thickness reaches several tensof nm. The later can be appreciated in Table I. In ad-dition to that, increasing too much the aspect ratio willtend to stabilize a (quasi) uniform magnetization stateinstead of the vortex.The microscopic magnetic structure of a typical vor-tex state in a magnetic nanodisc with r = 200 nm and t = 30 nm is shown in Fig. 1 a . It has been estimatedby means of micromagnetic simulations (see Methods section). This configuration corresponds to a clockwise( C =+1) in-plane curling and a central vortex pointingalong the positive z direction ( P =+1). We highlightthat, the discussion made here is independent of thevortex circulation or polarity, that can be set as initialconditions. We choose a low-damping ferromagnet suchas Co x Fe − x alloy (see Supporting Information fora more detailed discussion on the material choice). Be-
TABLE I. Numerically calculated values of the gyrotropic fre-quency ( f G ), line-width (∆ f G ) and damping parameter [ α v ,defined in Eq. (6)] for discs with different radius ( r ) andthicknesses ( t ). r t f G ∆ f G α v [nm] [nm] [GHz] [MHz] [ × − ]100 15 1.402 3.5 5.0200 30 1.255 3.3 5.3400 60 1.093 3.0 5.5 ing metallic, this material exhibits record low values ofthe Gilbert damping parameter α LLG ∼ × − for x = 0 .
25. Conveniently, Co x Fe − x alloys also offer a rel-atively large saturation magnetization of µ M s ∼ . x = 0 . x direction: b x ( τ ) = A sinc(2 πf cutoff τ ). Such perturbation is equivalent to ex-citing all eigenmodes susceptible to in-plane magneticfields at frequencies below f cutoff = 50 GHz. We obtainthe resulting time-dependent spatially-averaged magne-tization projected along the x direction M x ( τ ) over atotal time τ = 3 µ s. Calculating the corresponding fastFourier transform (FFT) results in the excitation spec-trum shown in Fig. 1 b . The first mode obtained at f G = 1 .
255 GHz is the previously described gyrotropicmode. The second and third modes, visible at higher fre-quencies of 15 .
41 and 20 .
94 GHz, respectively, are high-order azimuthal modes that will not be discussed here.The line-width of the gyrotropic mode can be char-acterized by fitting the corresponding resonance to alorentzian function giving ∆ f G = 3 . c ).Interestingly, applying an uniform out-of-plane magneticfield ( B DC ) modifies slightly the vortex magnetizationprofile leading to a linear dependence of f G on B DC .[26]This is exemplified in the right inset of Fig. 1 c for B DC applied along the positive z direction. Details of our nu-merical simulations are given in the Methods section.The vortex can, therefore, be treated as a harmonicoscillator driven by a time-dependent in-plane magneticfield b x ( τ ) = b x mw cos(2 πf τ ).[27, 28] At resonance, i.e. , f = f G , the amplitude of the resulting oscillating mag-netization will be maximum. In the linear (low ampli-tude) regime, this is given by the vortex susceptibilityand the driving amplitude at the disc center position ( r c ):∆ M x = χ x ( f G ) b x mw ( r c ). Here, ∆ M x represents the max-imum of the spatially-averaged magnetization along the x direction. B. Vortex-CPW coupling
Coherent coupling between the vortex gyrotropic modeand resonant photons will be analyzed in the framework A m p lit ud e [ a . u . ] f [Hz] -200 0 2001.21.4 A m p lit ud e [ a . u . ] f [GHz] f G [ GH z ] B DC [mT] a bc d M x [ A / m ] B DC z = 0 b rms xz y x = 0 y xz f G t 2r FIG. 1. a : Numerically calculated spatial distribution of the M x component for a C =+1, P =+1 magnetic vortex stabilizedin a nanodisc with r = 200 nm and t = 30 nm. Arrows indicate the in-plane curling of the magnetization. b : Numericallycalculated first resonant modes peculiar to the nanodisc shown in a . c : Enlarged view of the gyrotropic mode. Solid line isa fitting to a lorentzian function. The left inset serves to define the nanodisc dimensions. The right inset shows the lineardependence of f G on the out-of-plane applied magnetic field B DC for the P = ± d : Schematic representation of theproposed experiment. The nanodisc lies perpendicular to the central conductor of a superconducting CPW resonator so thatthe microwave magnetic field b rms (externally applied field B DC ) is applied parallel (perpendicular) to the nanodisc plane. Thecoordinate axes ( x, y, z ) are defined as well with y = 0 corresponding to the top part of the central conductor. -4 -2 0 +2 +4024 -4 -2 0 +2 +4024 -20+2 b r m s [ n T ] x [ m] y [ m ] x [ m] a b w = 7 m w = 1 m x FIG. 2. Numerically calculated spatial distribution of the amplitude of the x component of the rms magnetic field ( b x rms ) atthe midpoint of the central conductor, i.e. , z = 0. Arrows represent the ( x, y )-component of b rms with size proportional to thetotal in-plane amplitude. w is the width of the central Nb conductor (black rectangle) being 150 nm-thick. a : w =7 µ m. b : w =1 µ m. of superconducting CPW resonators. Such devices canroutinely reach quality factors Q ∼ − and oper-ate under large static in-plane external magnetic fields offew 100 mT. Strong coupling between the vacuum fluc-tuations of the CPW and superconducting qubits[29] in- augurated the field of circuit Quantum ElectroDynamics(QED) allowing to process quantum information with su-perconducting circuits.[30] Photons in the resonator maycarry the information shared between different qubits ormay be used to readout the qubit(s) state. The strongcoupling regime is needed to exchange this informationin a coherent way.As sketched in Fig. 1 d , a CPW resonator consist of acentral planar conductor coupled to external feeding linesthrough gap capacitors. The resonant frequency ( f CPW )depends on the geometrical dimensions of the resonatorand can, therefore, be tuned so that f CPW = f G . Anelectric current flows through the central conductor andthe surrounding ground plates with opposite directions.For a half-wavelength CPW resonator, the produced mi-crowave magnetic field will be maximum at the midpointof the central conductor, i.e. , z = 0 in Fig. 1 d . Thespatial distribution of this field can be computed numer-ically (see Methods section for details). The width andthickness of the central conductor are set to w = 7 µ mand 150 nm, respectively. The root mean square (rms)of the zero point current fluctuations is given by: i rms = 2 πf CPW (cid:114) (cid:126) π Z . (1)In this work we consider Z ∼
50 Ω as the inductanceof the resonator.[31] Eq. (1) follows from equating thezero point energy fluctuations to the inductive energyin the resonator, being the only contribution at z = 0.This yields (cid:126) πf CPW / LI /
2. Using the fact thatthe resonator inductance is L = Z /π f CPW we arrive to(1).Fig. 2 a shows the calculated spatial distribution ofthe x -component of the resulting field b x rms ( x, y ) at z = 0obtained for i rms = 11 nA. The later corresponds to set-ting f CPW = 1 GHz in Eq. (1). (cid:126)b rms has only x and y components. It is maximum at the corners of thecentral conductor, where the distribution of supercur-rent is maximized. As sketched in Fig. 2 a , we will as-sume that the magnetic nanodisc is located on top ofthe central conductor, lying on the ( x, y ) plane at po-sition z = x = 0. Being conservative, we will also as-sume that the nanodisc bottom edge lies 10 nm abovethe upper layer of the superconducting wire. Therefore,the disc center is positioned at r c = (0 ,
10 + r,
0) nmwhere both the y and z components of b rms are negligi-ble, i.e. , (cid:126)b rms ( r c ) ≈ b x rms ( r c ) (cid:126)x . In this way, the zero pointrms (in-plane) field can be used to excite the vortex dy-namics whereas an external homogeneous (out-of-plane)magnetic field can be used to tune the gyrotropic reso-nant frequency (see the right inset in Fig. 1 c ).The vortex response ∆ M x can be increased by increas-ing the local driving field amplitude b x rms ( r c ). This canbe easily achieved by decreasing the width of the cen-tral conductor (see Fig. 2 b ). For instance, assuming i rms = 11 nA and reducing w from 7 down to 1 µ m leadsto a total increase of b x rms from 0 . . r c = (0 , ,
0) nm. Patterning nano-constrictionsin localized regions of superconducting CPW resonatorshas been indeed used as a convenient method to increasethe effective coupling between resonant photons and spinqubits.[32, 33] Reducing w down to, e.g. , ∼
300 nm leadsto a negligible variation in the resulting experimental g / ( f G ) w [nm] g / [ M H z ] ab
100 (15) r = 400 nm ( t = 60 nm)200 (30) 100 (15) r = 400 nm ( t = 60 nm)200 (30) FIG. 3. a : Numerically calculated coupling g as a functionof the central conductor width w for nanodiscs having dif-ferent sizes. b : Strong coupling condition (4 g/ π ∆ f G > a . resonance frequency of the resonator and less than 5%change in the quality factor.In Fig. 2, we see that decreasing w also increases thefield inhomogeneity across the disc volume. This doesnot affect neither the resulting gyrotropic resonance fre-quency of the vortex nor the line-width as checked outnumerically for different disc sizes. As a matter of fact,we have verified numerically that the relevant parame-ter, i.e. , the disc response ∆ M x to an oscillating low-amplitude driving field, depends only on the magneticfield at the disc center, i.e. , b x rms ( r c ). C. Quantization and vortex motion-photon strongcoupling
Cavity quantization together with the fact that thevortex, around resonance f G , can be described using aharmonic mode, yield the quantum cavity-vortex model: H/ (cid:126) = 2 πf CPW a † a + 2 πf G a † v a v + g ( a † a v + h . c . ) (2)Here, h.c. means hermitean conjugate and a ( a † ) is abosonic operator that annihilates (creates) cavity pho-tons with frequency f CPW . The vortex operators a † v and a v create and annihilate single vortex excitations in thegyrotropic mode. The coupling strength is given by (See Methods ) g = b x rms ( r c )2 (cid:114) V χ x ( f G ) ∆ f G (cid:126) , (3)where V = 2 πr t is the disc volume. Here, we recallthat b x rms ( r c ) is the field generated by the single photoncurrent i rms [Eq. (1)] at the center of the vortex, r c (Cf. Fig. 2). We highlight that Eq. (3) is based onrather general arguments as the description of the vortexdynamics with a collective variable.[34] Therefore, similararguments can be extended to other magnetic topologicalobjects as skyrmions.The expression for g can be understood as follows. Thecoupling must be proportional to the magnetic field gen-erated in the resonator. We are interested in the cou-pling to a single photon, thus, g ∼ b rms . The secondterm in (3) is analogous to the cavity-magnon couplingcase. There, the coupling is proportional to √ N with N the number of spins. This is so since the coupling occursvia the macroscopic spin operator: S = S + + S − with S ± | l, m (cid:105) = (cid:112) ( l ∓ m )( l ± m + 1) | l, m ± (cid:105) . Spin-wavesare in the maximal spin-state l = N/ m = ± N/ χ ∼ M s / ∆ f G and, therefore, g ∼ √ V M s .Numerically, we compute the spatially-dependent b rms ( x, y ) field and the resulting vortex response around f G = f CPW for different disc sizes with fixed aspect ratio t/r . The gyrotropic frequencies and line-widths obtainedfrom the simulations for each disc are given in Table I.In Fig. 3 a we plot the resulting g values obtained wheninserting these parameters into Eq. (3) vs. the con-striction width w . Our numerical simulations show thatincreasing the volume is beneficial for enhancing the vor-tex gyration-cavity coupling. The figure also shows that g saturates when the constriction width is reduced belowthe disc radius. Looking at Eq. (3) we notice that thecoupling g is a trade off between the amplitude of therms field and the disc volume. On the one hand, b x rms in-creases for decreasing radius as b x rms ( r c ) increases whenapproaching the superconducting central conductor. Onthe other, the coupling increases with √ V .In order to catch these dependences, we approximate(3) as follows. The numerically calculated field at the disccenter is fitted by the simple formula: b x rms ( r c ) = µ π i rms × u w ( r ) . Here, u w ( r ) is a geometrical function of the discradius depending also on the width of the central conduc-tor. This can be described by u w ( r ) = a /r + a /r α ( w ) with 0 < α ( w ) <
1. It satisfies u w ( r ) → /r for r/w suf-ficiently large, i.e. , we recover the electrostatic formulaof an infinite line current (see Supporting Informa-tion ). Besides, the vortex susceptibility can been ap-proximated as χ x = ∆ M x /b x rms ( r c ) = ( γ/ π ) M s ξ / ∆ f G with γ/ π = 28 GHz/T and ξ a geometrical factor. In the case of discs, ξ = 2 / f G depends on thedisc thickness as t ∼ = 9 / rf G π/ ( γ/ π ) µ M s .[24] This isonly valid for discs having small aspect ratio t/r (cid:28) r -dependence of the cou-pling strength. Putting altogether we can approximatethe coupling in the case of vortices as: g ∼ = ξ (cid:115) πµ f Z r / u w ( r ) (4)Therefore, fixed the gyrotropic frequency, the couplinggrows with r . The coupling saturates as ∼ r / for w < r ( u w ( r ) → /r ) in agreement with figure 3 a .The strong coupling regime occurs when vortex mo-tion and photons exchange populations coherently in theform of vacuum Rabi oscillations. The condition to ob-serve such oscillations is 4 g/ π > ∆ f G .[36] In fig. 3 b we show that the strong coupling regime can be indeedreached using the device sketched in Fig. 1 d . For radius200 and 400 nm strong coupling occurs in the full rangeof investigated constriction widths. For smaller discs,it is reached for constrictions below 300 nm-width. Inthe weak coupling regime (shaded region in Fig. 3 b ) nocoherent exchange excitations occur but an overdampeddecay of excitations. The weak coupling regime is inter-esting per s´e as it is used to, e .g., control the spontaneousemission of atoms inside cavities. D. Coupling spectroscopy
Experimentally, the coupling strength can be measuredvia a transmission experiment as sketched in Fig. 4 a . Alow-power coherent input signal ( a in ) of frequency f issent into the resonator. The transmitted signal ( a out = T a in with T the transmission) is measured with a vectornetwork analyzer. The input-output theory[37] gives thefollowing formula for the transmission: T ( f ) = (cid:12)(cid:12)(cid:12)(cid:12) κ/ f − f CPW ) + R + i Γ (cid:12)(cid:12)(cid:12)(cid:12) (5)with R = π g ( f G − f )( f G − f ) +∆ f / and Γ = κ/ π g ∆ f/ f G − f ) +∆ f / . R and Γ are the real and imaginarypart of the self energy of the resonator and κ ∼ Q − is the cavity leakage (see Supporting Information ).Fig. 4 b shows the results obtained for a P = +1 vor-tex in a r = 400 nm and t = 60 nm disc on a w = 500nm-wide constriction. The transmission is plotted as afunction of the input frequency and the out-of-plane ap-plied magnetic field B DC (see inset in Fig. 1 c ). B DC isused to tune the gyrotropic vortex mode on resonancewith the resonator frequency. The latter is chosen to be f CPW = 1 .
093 to coincide with f G at zero B DC . At thispoint, the transmission shows the characteristic doublepeak. This is a signature of the strong coupling regime(Cf. Fig. 3 b ). The two peaks are associated to the real 𝑎 (cid:2919)(cid:2924) 𝑎 (cid:2925)(cid:2931)(cid:2930) ab B DC [mT]1.0891.0951.097 f [ GH z ] -10 +101.0931.091 0.10.50.3 T FIG. 4. a : Schematic representation of a transmission ex-periment. The resonator-vortex system is driven through atransmission line using an input current a in . The output sig-nal a out is then measured. b : Transmission T ( f ) [Eq. (5)] asa function of the driving frequency f and B DC . The yellowdashed line stands for the resonator frequency f CPW = 1 . f G ( B DC ). Calculations correspond to a discof r = 400 nm and t = 60 nm on a w = 500 nm-wide con-striction. part of the zeros of ( f − f CPW ) + R + i Γ. At resonance,for 4 g/ π > | ∆ f − κ | , the real parts are f CPW ± g , whichmeans that coherent oscillations between a single photonand the vortex motion can be resolved. On the otherhand, if 4 g/ π < | ∆ f − κ | , the real part of both solu-tions is f CPW . In this case, light-matter coupling is notresolved and they are in the weak coupling regime.
III. DISCUSSION
We discuss now the feasibility of the proposed exper-iment. The fabrication of superconducting CPW res-onators with f CPW ∼ Q ∼ − provid-ing line-widths of few 0 . − .
01 MHz is well within thecurrent state-of-the-art. Patterning sub-micron constric-tions at the central part of the central conductor poses notechnical difficulties neither. As demonstrated in Ref. 33,the width of the central conductor can be reduced downto w ∼
50 nm by Focused Ion Beam (FIB) milling. Ad-ditionally, applying in-plane external DC magnetic fields of few ∼ mT will not affect the properties of resonatorsbased on typical low critical temperature superconduc-tors ( e.g. , Nb).Fabrication of an individual sub-micron magnetic disccan be easily achieved by, e.g. , conventional electronbeam lithography and lift-off or FIB milling of thin mag-netic films. For a given magnetic material, properly set-ting the geometrical dimensions of the nanodisc will en-sure the stabilization of the magnetic vortex. As dis-cussed previously, ferromagnetic materials with large sat-uration magnetization are needed. Besides, the aspectratio shall be made large enough to get f G ∼ f G = 2 α v f G (6)where α v = α LLG φ is the vortex damping and φ =[1 + 1 / r/r v )] is a geometrical factor depending alsoon the disc thickness through the vortex characteristicradius r v .[35] For the dimensions discussed here, φ lieswithin 2 < φ <
3. Therefore, we will search for materialsexhibiting low Gilbert damping α LLG (cid:46) − providingsmall line-widths of few MHz.Finally, a further complication might come from theimposed experimental geometry (see Fig. 1 d ). Suchgeometry can be easily achieved in the following way.The nanodisc can be patterned on a thin (few 100 nm-thick) Si N membrane serving as carrier. The car-rier+nanodisc can be then located perpendicularly to theCPW resonator plane with nanometric resolution using ananomanipulator in a scanning electron microscope. At-taching the carrier+nanodisc to the resonator can be eas-ily achieved by FIB induced deposition of Pt. A similarprocedure has been successfully applied to the depositionof micrometric magnetic nanowires over the surface of su-perconducting sensors achieving ∼
50 nm-resolution.[38]We discuss in the following the convenience of usingdifferent ferromagnetic materials (see
Supporting In-formation for further details). For this purpose we com-pare the strong coupling condition 4 g/ π ∆ f G calculatedusing Eq. (4) and (6) for a r = 400 nm and t = 60nm disc in a w = 500 nm constriction. Conventionalferromagnets, e.g. , Fe or Ni Fe (Py), provide large µ M s ∼ . µ M s ∼ . α LLG ∼ × − and α LLG ∼ × − , respectively).[39] These wouldyield much too low coupling factors with an estimated4 g/ π ∆ f G ∼ . .
1, respectively. A record lowGilbert damping parameter is exhibited by YIG with α LLG ∼ × − but much too low µ M s ∼ .
18 T leadingto f G ∼
100 MHz [40]. Promising values of α LLG ∼ − have been reported for Heusler alloys like, e.g. , NiMnSb( µ M s ∼ .
85 T).[41] However, the latter yields just4 g/ π ∆ f G ∼ µ M s ∼ . g/ π ∆ f G ∼
3. We high-light that the employed value of α LLG ∼ × − wasmeasured experimentally in Ref. 25 for a 10 nm-thickfilm at room-temperature. Interestingly, even lower val-ues of α LLG could result from thicker substrates at mKtemperatures, yielding larger values of 4 g/ π ∆ f G . CONCLUSIONS
Using both numerical and analytical calculations wehave shown that strong coupling between the gyrotropicmotion of a magnetic vortex in a nanodisc and a singlephoton in a superconducting resonator is feasible withincurrent technology. To meet the strong coupling con-dition we have explored different materials and CPWarchitectures. We have found that CoFe discs with ra-dius 200 < r <
400 nm and thicknesses 30 < t < e .g., phase-lock dis-tant magnon-emitters or nanoscillators.We recall that, both the photons in the resonator andthe vortex motion are linear excitations. If we want toexchange single photons or to exploit this device in quan-tum information protocols nonlinear elements are nec-essary. Circuit QED has already demonstrated strongcoupling between a CPW cavity and different types ofsuperconducting circuits. Thus, we can envision cou-pling a transmon qubit in one en of the CPW.[15] Thisopens the way to further experiments as, e .g., probingvortex-photon entanglement by measuring photon dy-namics through qubit dispersive measurements.[44] IV. SUPPORTING INFORMATION
Magnetic field generated by the current i rms at the disccenter, discussion on the material choice, input-outputtheory, Hamiltonian diagonalization, normal modes andthe rotating wave approximation. ACKNOWLEDGMENTS
We thank Fernando Luis for inspiring discussions.We acknowledge support by the Spanish Ministeriode Ciencia, Innovaci´on y Universidades within projectsMAT2015-73914-JIN, MAT2015-64083-R and MAT2017-88358-C3-1-R, the Arag´on Government project Q-MADand EU-QUANTERA project SUMO.
V. METHODSA. Phenomenological quantization
We discuss here how to obtain the coupling strength g in (2). Based on our numerical simulations (see maintext), we can approximate the field fluctuations by theirvalue at the center of the disc: b x rms ( r c ). The spin-cavitycoupling is of the Zeeman type: H = (cid:88) j µ xj b x rms ( r c ) (7)with µ xj the magnetic dipole ( x -component) of the j -spinin the disc. Quantization of the CPW yields:ˆ b rms = b x rms ( r c )( a † + a ) . (8)The collective variable description for the vortexprecession[34, 35, 45] allows to write a quantized versionof the vortex magnetization as:ˆ µ j = µ j ( a † v + a v ) (9)Using Eqs. (7), (8) and (9) we arrive to the couplingconstant: (cid:126) g = V b x rms ( r c ) ∆ M x (10)with ∆ M x the maximum vortex response in magnetiza-tion. The latter is obtained by noticing that the vortexitself is driven via the resonator magnetic field with av-erage (cid:126) g < a † + a > . In the single photon limit we canreplace < a † + a > = 2 cos( f CPW τ ) obtaining:∆ M x = 4 g ∆ f G b x rms ( r c ) . (11)Inserting the above in (10) and using χ x ( f G ) =∆ M x /b x rms ( r c ), we end up with Eq. (3).Finally, let us comment on the light-vortex motionHamiltonian, which is a direct consequence of the Zee-man coupling (7) together with the quantized operators(8) and (9): H/ (cid:126) = 2 πf CPW a † a + 2 πf G a † v a v + g ( a + a † )( a v + a † v ) . (12)For the couplings considered here, i.e. , g/f CPW ∼ − (see Fig. 3), this Hamiltonian can be approximated withthe Rotating Wave Approximation (RWA) that dismissesthe counter-rotating terms a † a † v + h . c. In doing so, wearrive to vortex-light Hamiltonian (2). See SupportingInformation for a discussion on the validity of the RWA.
B. Numerical simulations
Micromagentic simulations are performed using thefinite difference micromagnetic simulation packageMUMAX3[46]. This software solves the time-dependentLaudau-Lifshitz-Gilbert equation for a given sample ge-ometry and material parameters assuming zero temper-ature. The saturation magnetization, exchange stiffnessconstant and Gilbert damping are set to M s = 1 . × A/m, A = 2 . × − J/m and α LLG ∼ × − for CoFe,respectively. We use a box with dimensions 2 r × r × t containing a disc of radius r and thickness t . For the r = 100 nm ( t = 15 nm), 200 (30) and 400 (60) discs,the boxes are discretized into 64 × ×
8, 128 × × × ×
32 identical cells, respectively, each be-ing 3 . × . × .
87 nm . The vortex ground-energyconfiguration of the disc is found by relaxing the systemin the presence of an homogeneous external out-of-planemagnetic field ( B DC ).Vortex dynamics are characterized using an in-planeperturbation magnetic field b ( x, y, t ) = b x,y b t where wehave splitted the space- and time-varying parts. b x,y is created by the current i flowing through the centralconductor of the CPW resonator being, therefore, non- homogeneous along the ( x, y ) plane and having a negli-gible z component. b x,y is calculated numerically using3D-MLSI[47] which allows to obtain the spatial distribu-tion of supercurrents circulating through 2-dimensionalsheets in a superconducting wire. The input parametersare the current i flowing through the central conductorhaving width w , thickness 150 nm and London penetra-tion depth λ L = 90 nm (typical for Nb).The gyrotropic mode is characterized using i = 10 mAand b t = sinc(2 πf cutoff t ) and calculating numerically theFFT of the resulting time-dependent spatially-averagedmagnetization along the x direction M x ( t ). This allowsus to obtain the gyrotropic characteristic frequency ( f G )and line-width (∆ f G ) by fitting to a Lorentzian curve L ( f ) = Aπ (∆ f G ) / f − f G ) + (∆ f G / . The vortex response to quantum vacuum fluctuationsis found by using b x,y = b rms ( x, y ) and b t = cos(2 πf G t )where b rms ( x, y ) is the field resulting when i = i rms . Thelatter is obtained from Eq. (1) for f CPW = f G giving i rms ∼ −
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