Table Of ContentBidirectional conversion between microwave and light via ferromagnetic magnons
R.Hisatomi,1,∗ A.Osada,1 Y.Tabuchi,1 T.Ishikawa,1 A.Noguchi,1 R.Yamazaki,1 K.Usami,1,† andY.Nakamura1,2
1Research Center for Advanced Science and Technology (RCAST),
The University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan
2Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan
(Dated: May 18, 2016)
Coherent conversion of microwave and optical photons in the single-quantum level can signifi-
cantly expand our ability to process signals in various fields. Efficient up-conversion of a feeble
signalinthemicrowavedomaintotheopticaldomainwillleadtoquantum-noise-limitedmicrowave
amplifiers. Coherent exchange between optical photons and microwave photons will also be a step-
pingstonetorealizelong-distancequantumcommunication. Herewedemonstratebidirectionaland
coherentconversionbetweenmicrowaveandlightusingcollectivespinexcitationsinaferromagnet.
6
The converter consists of two harmonic oscillator modes, a microwave cavity mode and a magne-
1
tostatic mode called Kittel mode, where microwave photons and magnons in the respective modes
0
are strongly coupled and hybridized. An itinerant microwave field and a traveling optical field can
2
be coupled through the hybrid system, where the microwave field is coupled to the hybrid system
y through the cavity mode, while the optical field addresses the hybrid system through the Kittel
a
mode via Faraday and inverse Faraday effects. The conversion efficiency is theoretically analyzed
M
andexperimentallyevaluated. Thepossibleschemesforimprovingtheefficiencyarealsodiscussed.
7 PACSnumbers: 03.67.Lx,42.50.Pq,75.30.Ds,76.50.+g
1
] I. INTRODUCTION astronomy, nuclear magnetic resonance, and magnetic
h
resonance imaging.
p
- Understanding and exploiting the interactions in well- Anyprocessthatconvertsfrequencyofanelectromag-
t
n controlled quantum systems are the key to build a neticfieldinherentlyrequiressomenonlinearinteraction.
a large-scaleartificialmany-bodyquantumsystem,suchas Thechallengefacedbythemicrowave-lightconversionin
u
quantum computers, quantum communication networks, the quantum regime is the weakness of such nonlinear-
q
[ and quantum simulators. By far the most important in- ity. Nevertheless, there are several attempts to realize
gredient of the artificial quantum system is the atom- such microwave-light conversions. Ferroelectric crystals
3
like anharmonic energy-level structures. Advances in suchaslithiumniobate(LN)andpotassiumtitanylphos-
v
superconducting quantum circuits, which provide such phate (KTP) have the large quadratic optical nonlinear-
8
energy-level structures with macroscopic circuitry [1], ity,χ(2),andarewidelyusedaselectro-opticmodulators.
0
9 make them one of the primary candidates [2]. The su- Usingahigh-qualityopticalwhispering-gallery-moderes-
3 perconducting artificial atoms can be exquisitely manip- onator made of LN, 10-GHz microwave photons are up-
0 ulatedbytheelectromagneticfieldsinthemicrowavedo- converted to optical sideband photons with the conver-
1. main [3, 4]. However, the quantum information carried sionefficiencyof1×10−3 [6]. Herethepolaritonmodesin
0 by microwave photons has to be imprisoned in low tem- the THz regime bring about the electro-optic effect and
6 perature environment to prevent them from being jeop- thus the microwave-light conversion is took place disper-
1 ardized by the thermal noise. Quasiparticle production sively.
v: in superconductors also hinders the direct optical access Instead of using nonlinearity in the dispersive regions
i which would enable the robust, fast, and long-distance
X of optically transparent materials, sharp resonances can
opticalcommunicationsbetweenthesuperconductingar-
be exploited for enhancing the nonlinearity. An exam-
r tificial atoms.
a ple is a spin resonance line in paramagnets. In partic-
Converting microwave to optical photons and vice
ular the sharp spin resonance lines of rare-earth ions in
versecould,however,remedytheaboveweaknessesofsu-
solids are successfully utilized for realizing the efficient
perconductingartificialatomsandconnectthetwovastly
quantum memories for light [7–10]. The paramagnet-
different worlds, i.e., low-temperature microwave quan-
based schemes may have a few concerns; one is the un-
tum processors and robust optical networks. The co-
wanted local spin-spin interaction when the spin den-
herent and efficient conversion can also open up a new
sity becomes large, the other is the difficulty of mode-
avenue for quantum-noise-limited amplification of mi-
matching between optical field and microwave field be-
crowave signals [5] in a variety of fields such as radio
cause of the absence of energetically well-separated spin-
wave modes. A magneto-optic modulator based on an
erbium-doped crystal placed in both an optical cavity
and a microwave cavity is suggested to overcome these
∗ [email protected] difficulties and expected to achieve a unit quantum effi-
† [email protected] ciency in the microwave-light conversion [11]. There are
2
attempts to implement this scheme [12, 13].
Microwave Light
Themostefficientconversionsofar(theconversionef-
ficiency 0.1) uses a nanomechanical resonator [14]. With
the deftly designed system where the optical and me-
chanicalresonators,aswellasmicrowaveandmechanical Microwave Kittel mode
cavity mode
resonators, are parametrically coupled with pump laser
andpumpmicrowave,respectively,coherentandefficient
conversion between microwave and light within a band-
width of 30 kHz is demonstrated [14]. Much broader-
bandwidth but less efficient microwave-light conversion
hasalsobeenreportedwithapiezoelectricoptomechani- FIG. 1. Architecture of the proposed microwave-light con-
calcrystal[15]. Themechanics-basedschemeshavesome verter. The converter consists of two strongly-coupled har-
advantages over the paramagnet-based schemes. First, monic oscillator modes, a microwave cavity mode aˆ, whose
the strong nearest-neighbor atom-atom coupling gives energy is specified by ¯hω , and a magnetostatic mode called
c
the system the rigidity, which makes the system insensi- Kittel mode cˆ, whose energy is specified by h¯ωm, and these
tive to the local perturbations. Second, the system with are strongly coupled at a rate g. An input (output) itiner-
ant microwave field mode aˆ (aˆ ) is coupled to the converter
rigidity,ingeneral,possessesrobustlong-wavelengthcol- i o
through the microwave cavity mode at a rate κ whereas an
lective excitation modes, which makes it easier to mode- c
input (output) traveling optical field modeˆb (ˆb ) is coupled
match between optical and microwave fields. i o
to the converter through the Kittel mode at a rate ζ. γ and
Hereweputforwardanideaofusingcollectivespinex- κ are rates of the intrinsic energy dissipation for the Kittel
citationsinaferromagnetnotonlytoresonantlyenhance modeandtheinternalenergylossforthecavity,respectively.
the microwave-light interaction but also to enjoy the ad-
vantages of having robust collective excitation modes.
We use a spatially-uniform magnetostatic mode, called cavity Q factor and sample dimensions [22].
Kittel mode, in yttrium iron garnet (YIG), which man-
The magnon-based microwave-light converter is even
ifests itself as a precessing large magnetic dipole. The
more attractive from the viewpoint of enlarging the po-
largeness of the dipole moment and the longevity of the
tential of the superconducting qubits. The microwave-
coherenceofthemagnonsintheKittelmodemakeitpos-
light converter based on ferromagnetic magnons is ex-
sibletocouplethemstronglytothemicrowavephotonsin
pected to have a broad bandwidth (around 1 MHz) and
a microwave cavity mode and to hybridize them [16, 17].
thus operates faster than the lifetime of a superconduct-
By exploiting the hybrid system formed by the Kittel
ing qubit currently available (around 100 µs [2]). More-
mode and the microwave cavity mode, we demonstrate
over, the ferromagnetic magnon has recently been co-
bidirectionalcoherentconversionbetweenmicrowaveand
herently coupled to a superconducting qubit [23, 24].
light, where the microwave field is coupled to the hybrid
The magnon-based microwave-light converter can then
system through the cavity mode while the optical field
beconsideredasoneofthecandidatesasatooltocoher-
addressesthehybridsystemthroughtheKittelmodevia
ently connect distant superconducting qubits via light.
Faradayandinverse Faradayeffect[18]. Notethatinre-
After a brief discussion of a theoretical model of
centyearstheinverseFaradayeffectattractsconsiderable
the microwave-light converter based on ferromagnetic
attention in the context of optical manipulation of mag-
magnons in Sec. II, we present experiments in which co-
netization [19]. Magnetization oscillations at microwave
herent and bidirectional conversion between microwave
frequencieshavebeensuccessfullyinducedbytheinverse
and light is demonstrated (Sec. III), followed by the dis-
Faraday effect with a single femtosecond laser [20, 21].
cussion of future prospects of magnon-based converters
Our approach to the inverse Faraday effect is distinct
(Sec. IV). We elaborate on the architecture of the con-
from such works; coherent magnon states are generated
verter in Appendix A. In Appendices B and C the cali-
by two phase-coherent continuous-wave (CW) lasers.
bration scheme used to infer the magnon-light coupling
We evaluate the conversion efficiency of the converter rate and that for evaluating the conversion efficiency are
theoretically and experimentally with a careful calibra- explained, respectively.
tionschemeandfindthattheconversionefficiencyofthe
orderof10−10andthatitislimitedbythesmallmagnon-
light coupling rate. We envisage, however, that the con-
versionefficiencycanbeimprovedbycombininganopti- II. THEORETICAL MODEL OF THE
CONVERTER
calcavityorreplacingYIGwithotherferromagnetspos-
sessing a narrow optical transition. Even with YIG, by
incorporating an optical cavity and arranging the cavity The architecture of the microwave-light converter is
in such a way that it supports two optical modes which showninFig.1. Theconverterisbuildonthethreecou-
are separated by Kittel-mode frequency (i.e., satisfying pling mechanisms with respective terms in the Hamilto-
triple resonances) the efficiency can be significantly im- nian, H , H , and H . Here H is the coupling between
I c p I
proved (up to 10−3) with realistic parameters such as a microwave cavity mode aˆ and a magnetostatic mode
3
called Kittel mode cˆ, given by On the other hand, when the angular frequency of inter-
est is ω =Ω−Ω , we have
H =h¯g(cid:0)aˆ†cˆ+cˆ†aˆ(cid:1), (1) b 0
I
cˆ(ω )=χ (ω )(cid:104)−(cid:112)ζˆb (Ω)−igaˆ(ω )(cid:105), (7)
withacouplingrateg. H describesthecouplingbetween b m b i b
c
an itinerant microwave mode aˆ (ω) and the microwave
i which is stemmed from the beam-splitter-type Hamilto-
cavity mode aˆ, given by,
nian appeared in Eq. (A14) and H in Eq. (1). Here the
I
H =−i¯h√κ (cid:90) ∞ dω (cid:16)aˆ†aˆ (ω)−aˆ†(ω)aˆ(cid:17), (2) susceptibility χm(ω) is defined as
c c −∞ 2π i i (cid:104) γ(cid:105)−1
χ (ω)= −i(ω−ω )+ . (8)
with a coupling rate κ . The parametric coupling be- m m 2
c
tween the Kittel mode cˆand a traveling optical photon Solving the algebraic equations (4) and (6) with the
mode ˆbi(Ω) can be brought about with a strong optical boundary conditi√ons aˆo(ω) = aˆi(ω) + √κcaˆ(ω) and
drive field (angular frequency Ω0). The term Hp is given ˆb†(Ω) = ˆb†(Ω)+ ζcˆ(ω ) [25] the amplitude conversion
by o i b
efficiency from microwave to light with the angular fre-
Hp =−i¯h(cid:112)ζ(cid:90) ∞ d2Ωπ (cid:0)cˆ+cˆ†(cid:1)(cid:16)ˆbi(Ω)eiΩ0t−ˆb†i(Ω)e−iΩ0t(cid:17), qasuency Ω = Ω0−ω (Stokes scattering) can be obtained
−∞ (cid:42) (cid:43) √
(3) ˆb†(Ω) ig κ ζχ (ω)χ (ω)
where ζ represent a parametric-coupling rate depending o = c m c . (9)
aˆ (ω) 1+g2χ (ω)χ (ω)
on the strength of the optical drive field. i m c
The conversion from microwave to light means that
On the other hand, solving the algebraic equations (4)
the input itinerant microwave photons in the mode des-
and (7) with the boundary conditions aˆ (ω) = aˆ (ω)+
ignatedbyaˆ areconvertedintoanoutputtravelingpho- √ √ o i
i κ aˆ(ω)andˆb (Ω)=ˆb (Ω)+ ζcˆ(ω )[25]theamplitude
tons in the modeˆb orˆb† in Fig. 1. The conversion from c o i a
o o conversion efficiency from microwave to light with the
lighttomicrowavemeansthereverseprocess,i.e.,thein-
angular frequency Ω = Ω +ω (anti-Stokes scattering)
put traveling photonsˆb orˆb† are converted into an out- 0
i i can be written as
put itinerant microwave photons aˆ . In Appendices A1,
o (cid:42) (cid:43) √
A2, and A3, we elaborate each element and their inter- ˆb (Ω) ig κ ζχ (ω)χ (ω)
o = c m c , (10)
actions. aˆ (ω) 1+g2χ (ω)χ (ω)
i m c
which is, in fact, equal to the anti-Stokes case shown in
A. Conversion efficiency
Eq. (9).
Theamplitudeconversionefficienciesfromlighttomi-
ThetotalinteractionHamiltonianHt =Hc+HI+Hp crowave can similarly be obtained. For microwave with
with the intrinsic dissipations represented by the rates the angular frequency ω =Ω −Ω it is
a 0
γ and κ, for the Kittel mode and the cavity mode, re-
(cid:42) (cid:43) √
spectively, defines the dynamics of the variables in the aˆ (ω ) ig κ ζχ (ω )χ (ω )
o a =− c m a c a . (11)
cfoolnlovweritnegr.FoFuorrietrh-edocmavaiitny rmeloadteionopferroamtorthaˆeweequhaatvioenthoef ˆb†i(Ω) 1+g2χm(ωa)χc(ωa)
motion [Eq. (A2)]: For microwave with the angular frequency ω = Ω−Ω
b 0
√ it is
aˆ(ω)=χ (ω)[− κ aˆ (ω)−igcˆ(ω)], (4)
c c i (cid:42) (cid:43) √
aˆ (ω ) ig κ ζχ (ω )χ (ω )
where the susceptibility χ (ω) is defined as o b = c m b c b . (12)
c ˆbi(Ω) 1+g2χm(ωb)χc(ωb)
(cid:20) κ+κ (cid:21)−1
χ (ω)= −i(ω−ω )+ c . (5)
c c 2
III. EXPERIMENTAL RESULTS
Hereandhereafterthethermalandquantumnoiseterms
areomitted. Forthe Kittelmodeoperator cˆtheFourier-
domain relation depends on the angular frequency of A. Characterizations
interest. When the angular frequency of interest is
ω =Ω −Ω we have Here we first summarize experimentally achieved pa-
a 0
rameters, which are relevant in the converter shown in
cˆ(ω )=χ (ω )(cid:104)(cid:112)ζˆb†(Ω)−igaˆ(ω )(cid:105), (6) Fig. 1, such as the coupling rates, g, κ , and ζ, appear-
a m a i a c
ing in Eqs. (1), (2), and (3), respectively, as well as the
whichisstemmedfromtheparametric-amplification-type intrinsic dissipations represented by the rates γ and κ,
Hamiltonian appeared in Eq. (A13) and H in Eq. (1). for the Kittel mode and the cavity mode.
I
4
YIG sphere
diameter = 0.75 mm
Photo
detector
Coil
Polarizer Microwave
amplifiers
1550 nm
laser
FIG. 2. Experimental setup for converting microwave to light. A spherical crystal (0.75-mm in diameter) made of yttrium
iron garnet (YIG) is placed in a microwave cavity to form a strongly-coupled hybrid system between the Kittel mode and the
microwavecavitymode. AstaticmagneticfieldisappliedtotheYIGsamplewithpermanentmagnets. Thefieldcanbevaried
with a coil through a magnetic circuit made of pure iron. A vector network analyzer is used to characterize the hybrid system
by measuring the microwave reflection coefficient from the system. To convert microwave to light, a 1550-nm continuous-wave
(CW) carrier laser is impinged on the YIG sample. Under the microwave drive, the polarization of the carrier laser oscillates
at the frequency of the induced magnetization oscillation producing the optical sideband field. The beat signal between the
carrierandthesidebandfieldismeasuredusingapolarizerandafastphotodetector,isamplifiedwithtwolow-noisemicrowave
amplifiers, and fed into the vector network analyzer.
1. Evaluation of κ, κ , g, and γ is thus in the strong coupling regime, i.e., g > γ,κ , at
c c
roomtemperature. Twopronounceddipsappearin|S |,
11
which is the signature of the hybridization between the
The experimental setup used in evaluating these pa-
Kittel mode and the cavity mode, that is, the normal-
rametersisshowninFig.2. Anitinerantmicrowavefield
generatedbyavectornetworkanalyzerdrivesthehybrid mode splitting with the cooperativity C = 4g2 =510
(κc+κ)γ
system consisting of a microwave cavity and the Kittel [Eq. (A6)] being a very large value.
mode. We use the fundamental mode (TE ) of the
101
rectangular cavity made of oxygen-free copper with the
volume V of 21×19×3 mm3 and the resonant frequency
ω /2π =10.45 GHz. By measuring the reflection coeffi-
c
cient from the cavity we obtain the scattering parame-
ter S (ω) and evaluate the cavity-related parameters as 2. Estimation of ζ
11
κ/2π =3.3MHz and κ /2π =25MHz.
c
Using a magnetic circuit consisting of a set of per-
For the Kittel mode, we can assume that the coupling
manent magnets, a yoke, and a coil, a static magnetic
constant G in Eq. (A7) is related to the Verdet constant
field B of 310 mT along z-axis is applied to the YIG
0 V asφ =Vl= 1Gnlwithφ beingtheFaradayrotation
sample across the cavity. The static field B0 can be F 4 F
angle, l being the length of the sample and n being the
varied with the current I through the coil (dB /dI =
0
spindensity. WithliteraturevaluesofV andnweobtain
50 mT/A), which in turn tunes the resonance angular
G = 7.2×10−26 m2 for YIG (see Appendix A1). With
frequency ω of the Kittel mode. Figure 3(a) shows a
m
this value of G we can estimate the coupling rate ζ from
two-dimensional spectrum of the measured power reflec-
tion coefficient |S |2 as a function of the frequency of the relation ζ = G2l2n P0 [Eq. (A12)]. With the follow-
11 16Vs h¯Ω0
the microwave drive (the angular frequency ω) and the ing parameters l = 0.75 mm, V = (4π/3)×0.383 mm3,
s
coil current. At the coil current I = 400 mA indicated P = 0.015 W, Ω /2π = 200 THz, we have ζ/2π =
0 0
by the dashed line in Fig. 3(a), the parameters γ and g 0.33 mHz. The coupling rate ζ is also independently
are deduced based on Eq. (A5). Blue points in Fig. 3(b) evaluatedbyasimplemagneto-opticalexperimentwhere
showthemeasuredamplitudeandphaseofS andtheir the shot noise is used as a calibrator as explained in Ap-
11
polarplotattheparticularcoilcurrent. Redcurvesshow pendix B. This procedure yields ζ/2π = 0.25 mHz, a
the fitting result based on Eq. (A5). From the fitting we reasonable agreement with the value obtained from the
obtainγ/2π =1.1MHzandg/2π =63MHz. Thesystem Verdet constant V above.
5
(a) (b)
1
m ode 1
Hz) 10.8 Kittel S|11 0.8 190°
G |
y (10.6 0.6
Frequenc10.4 Cmaovditey 0.511 |S| 2 S) 112 180° 0°
g( 0
10.2 ar
270°
-2
10.0 10.3 10.4 10.5 10.6
Frequency (GHz)
0
100 200 300 400 500 600 700
Coil current (mA)
(c) (d)
1
nit) 0.4
10.8 u 90°
Frequency (GHz) 111000...246 0.5LM |S| (arb. unit)2 |S| (arb. ) arg(SLMLM−0.200 180° 0.2370° 0°
10.0 10.3 10.4 10.5 10.6
0 Frequency (GHz)
100 200 300 400 500 600 700
Coil current (mA)
FIG. 3. (a) Power reflection coefficient |S |2 as a function of the microwave drive frequency (drive power: 0 dBm) and the
11
coil current. (b) Spectrum of |S |, arg(S ), and their polar plot, at a coil current I =400 mA indicated by the dashed line
11 11
in (a). Blue dots show the experimental data and red curves show the fitting results based on Eq. (A5). The two pronounced
dips in the reflection coefficient S (ω) in (b) are the signature of the hybridization between the Kittel mode and the cavity
11
mode. (c) |S |2 as a function of the microwave drive frequency and the coil current (carrier laser power: 450 µW). The data
LM
are simultaneously taken with |S |2 in (a). (d) Spectrum of |S |, arg(S ), and their polar plot, at I =400 mA indicated
11 LM LM
bythedashedlinein(c). BluedotsshowtheexperimentaldataandredcurvesshowthefittingresultsbasedonEq.(13). Two
tiny normal-mode splittings indicated by the arrows in (a) are caused by other magnetostatic modes. The dotted lines in (a)
and (b) indicate the coil current I =564 mA, where the maximum conversion is realized (see Sec. IIIC).
B. Conversion from microwave to light optical light is shown in Fig. 2. A 1550-nm CW laser
with the angular frequency of Ω (drive frequency) im-
0
pinges on the YIG sample, whose beam spot at the sam-
While the microwave absorption by the hybrid system
ple is roughly 0.15 mm in diameter. The polarization
can be measured in the microwave reflection measure-
of the laser before the sample is linear and along the
ment (S measurement), the accompanying magnetiza-
11 z-axis. After passing the sample the polarization os-
tionoscillationinducedintheYIGspherecanbeprobed
cillates at the frequency of the induced magnetization
bylight. Theprocesscanbeunderstoodasfollows. First,
oscillation by the Faraday effect, thus producing the
the itinerant microwave photons in the mode aˆ drive
i optical sideband at the angular frequency of Ω ±ω .
magnons coherently through the microwave cavity with 0 m
The beat signal between the drive field and the side-
the two interactions denoted by H and H in Eqs. (2)
c I band field is measured using a polarizer and a fast pho-
and(1). Thedrivenmagnonsthenscattersidebandpho-
todiode (New Focus 1554-B) with two low-noise mi-
tons ˆb through the Faraday interaction H in Eq. (3)
o p crowave amplifiers (MITEQ AFS4-08001200-09-10P-4)
withthestrongopticaldrivefield. Thesequantumtrans-
as shown in Fig. 2. This measurement culminates in
ferprocessesconstitutetheconversionfrommicrowaveto
the beat-down heterodyne signal, which corresponds to
light.
the measurement of the Stokes operator [see Eq. (A11)],
The experimental setup for converting microwave to
6
(cid:16) (cid:17)
sˆz(ω) ∝ ˆb†o(Ω0−ω)e−iΩ0t+ˆbo(Ω0+ω)eiΩ0t+h.c. . difference between the two fields has to be the Kittel
modefrequencyω /2π. Next, sincetheKittlemodehas
The microwave-to-light amplitude conversion coefficient, m
(cid:68) (cid:69) no linear momentum, the two laser field has to be co-
S (ω)∝ sˆz(ω) , can then be defined as
LM aˆi(ω) propagating to conserve the total momentum in the pro-
cess. Finally,onlythecombinationofthez-polarized(π-
√ (cid:32)(cid:42) (cid:43) (cid:42) (cid:43)(cid:33)
η ˆb†(Ω −ω) ˆb (Ω +ω) polarized) field and the y-polarized field can create and
S (ω)= o 0 + o 0
LM 2i aˆ (ω) aˆ (ω) annihilate magnons, where the two phase-coherent light
i i
√ fields interfere and create oscillating fictitious magnetic
= g ηκcζχm(ω)χc(ω), (13) field along x-axis. Here, among the oscillating fictitious
1+g2χ (ω)χ (ω) magnetic field, only the component co-rotating with the
m c
magnetization of the Kittel mode contributes to the cre-
where η is the amplification factor including the field
ation and annihilation of magnons, as in the standard
strength of the drive field (here acting as a local oscil-
magnetic resonance experiment (see Appendix A2) [26].
lator) and the gain of the photodetector and the mi-
crowaveamplifiers. Here(cid:68)ˆb†o(Ω0−ω)(cid:69)and(cid:68)ˆbo(Ω0+ω)(cid:69)are sepInarattheedebxypωerim=enΩt −thΩe itnwoanpghualasre-fcroehqeureennctyfiaesldshsoawrne
aˆi(ω) aˆi(ω) a 0
the amplitude conversion efficiency with the anti-Stokes in Fig. 4(c). Thus, only the parametric-amplification-
scattering [Eq. (10)] and that with the Stokes scattering typeinteraction(i.e.,theStokesscattering)inEq.(A13)
[Eq. (9)], respectively. is realized (see Appendix A3). Here the field with the
Figure 3(c) shows the two-dimensional spectrum of angular frequency Ω is polarized along z-axis while the
0
|S |2 as a function of the microwave drive frequency ω one with Ω is along y-axis. The created magnons pre-
LM
andthecoilcurrentI. Notethatthedataaresimultane- dominantly decay to the microwave cavity due the large
ouslytakenwiththespectrumof|S11|2 inFig.3(a). The cooperativity C = 4g2 [Eq. (A6)]. The coupled-
twospectraarecomplementaryinthesensethatthedips (κc+κ)γ
out microwave signal from the cavity is then amplified
in Fig. 3(a) appear as the peaks in Fig. 3(c), suggesting
and fed to a spectrum analyzer. Note here that, since
the faithful conversions from the microwave to the light
the microwave cavity acts as a very good microwave
quanta. Also plotted in Fig. 3(d) are the amplitude and
receiver, any stray microwave fields close to the reso-
thephaseofS andtheirpolarplot, atthecoilcurrent
LM nance frequency of the Kittel mode should be avoided.
I = 400 mA indicated by the dashed line in Fig. 3(c).
In order to have the driving angular frequency of an
Thefactthatthephasevaluesof S inFig.3(d)follow
LM electro-optic modulator (EOM) ω different from ω ,
E m
those of S in Fig. 3(b), except the scale factor of 2,
11 the y-polarized field is also frequency-shifted by ω /2π
A
clearly displays the coherent nature of the conversions.
= 80 MHz with an acousto-optic modulator (AOM) be-
From the fitting in Fig. 3(d) based on Eq. (13) we fore combined with the z-polarized field.
deduce g/2π = 63 MHz and γ/2π = 1.3 MHz, which
Figure 5(a) shows a measured noise power spectrum
are similar to those obtained from S in Sec. IIIA1.
11 recorded in the spectrum analyzer at a coil current I =
Here, the light-magnon coupling rate ζ is multiplied by
564 mA. The peak of the noise power corresponds to the
theuncalibratedamplificationfactorη,andηζ asawhole
lower-branch of the normal modes. Given the fact that
is used as a fitting parameter. In the inverse conversion
thethermalnoiseofthehybridizedsystemappearsabove
experiment, we shall provide the evaluation of ζ with a
the instrument noise level, our measurement is thermal-
careful calibration scheme explained in Appendix C.
noise-limitedatroomtemperature. TheinsetinFig.5(a)
shows the zoom-up of the peak region in Fig. 5(a) when
the YIG sample is illuminated by the two laser fields so
C. Conversion from light to microwave
as to bring about the inverse Faraday effect. The sharp
peak above the broad noise level indicates the presence
In Sec. IIIB we discussed the conversion from mi- ofcoherentmagnetizationoscillationsinducedbythetwo
crowavephotonstoopticalphotonsbasedonamagneto- phase-coherent optical fields.
optical effect, i.e., the Faraday effect. In this section,
The photon conversion efficiency from light to mi-
we shall discuss the inverse process; the conversion from (cid:12)(cid:68) (cid:69)(cid:12)2
crowave defined by |S+ |2 ≡ (cid:12) aˆo(ωa) (cid:12) , where the su-
optical photons to microwave photons based on an opto- ML (cid:12) ˆb†(Ω) (cid:12)
i
magnetic effect, i.e., inverse Faraday effect [18]. Our ap- perscript “+” emphasizes the fact that only the Stokes
proach to the inverse Faraday effect is to use two phase- scattering process is activated in the conversion process
coherent CW lasers as opposed to the impulsive method [seeEq.(11)], canthenbededucedfromthepowerspec-
commonly used [19]. trum by expressing the laser powers used for exciting
Figure 4(a) depicts the experimental setup. Two themagnonsandthemicrowavesignalpower(withinthe
phase-coherent laser fields generated from a monochro- bandwidth of the coherent magnon signal, which is lim-
maticCWlaseraresimultaneouslyimpingedontheYIG ited by the coherence of the two laser fields (∼ 10 Hz))
sphere to induce the inverse Faraday effect. To bring in terms of numbers of photons. For this purpose the
about the effect the following considerations have to be gain and loss of the microwave amplifiers and the inter-
taken. First, from the energy conservation the frequency vened coaxial cables have to be properly calibrated. The
7
(a)
YIG sphere
diameter = 0.75 mm
Microwave
amplifiers
Coil
1550 nm
PM fiber laser
(b) Optical fiber (c)
Laser Filter
1550 nm Splitter EOM cavity
AOM
Piezo PBS
PM fiber
Frequency
FIG. 4. (a) Experimental setup for converting light to microwave. The hybrid system consisting of the Kittel mode and the
microwave cavity mode is used for the conversion. Two phase-coherent laser fields generated from a monochromatic CW laser
are simultaneously impinged on the YIG sample to induce the inverse Faraday effect. The created magnons predominantly
decay to the microwave cavity, and the coupled-out microwave signal from the cavity is amplified and fed into a spectrum
analyzer. (b) Scheme to generate two phase-coherent laser fields. A monochromatic CW laser field with the wavelength of
1550nm(theangularfrequencyofΩ )issplitintotwopaths. Thefieldinoneofthepathsisphase-modulatedwithmodulation
c
angularfrequencyofω byanelectro-opticmodulator(EOM)andfilteredoutthecarrierandallothersidebandphotonsexcept
E
for the one of the first-order sidebands (the angular frequency of Ω=Ω +ω ) with a Fabry-P´erot filter cavity. The filtered
c E
fieldisthencombinedatapolarizingbeamsplitterwiththefieldintheotherpathwiththeangularfrequencyofΩ ,whichis
0
alsofrequency-shiftedbyω /2π =80MHzfromΩ withanacousto-opticmodulator(AOM).Apiezoelectricactuatorisused
A c
tocompensatethefluctuationoftheopticalpath-lengthdifferencebetweentwofieldsforstabilizingtherelativephasebetween
the two fields. The two resultant fields are separated by ω =Ω −Ω in angular frequency as shown in (c). Both of the fields
a 0
arecoupledtoapolarization-maintaining(PM)fiberbeforethesamplesoastomatchtheirspatialmodes. Thepowerisabout
15 mW for each field before entering the sample.
detailed calibration procedure is described in Appendix ativity C in Eq. (A6):
C. The calibrated photon conversion efficiency (cid:12)(cid:12)S+ (cid:12)(cid:12)2
ML
is plotted as the blue points in Fig. 5(b) as a function 4C κcζ
of the frequency difference between the two laser fields (cid:12)(cid:12)SM+L(cid:12)(cid:12)2 = (cid:16) ((cid:17)κc2+κ)(cid:16)γ (cid:17)2.
ω =Ω −Ω. The red curve is drawn based on Eq. (11) C+1−4 ∆c ∆m + 2 ∆c +2∆m
a 0 κc+κ γ κc+κ γ
with the light-magnon coupling rate ζ in Eq. (A14) mul- (14)
tipliedbythetransmittanceoflightT asawholebeinga The resonant condition ∆ =∆ =0 leads to
c m
fitting parameter. The maximum photon conversion effi-
ciency, (cid:12)(cid:12)S+ (cid:12)(cid:12)2 ∼ 10−10, is achieved at the upper-branch 4C κcζ
ofthenorMmLalmodewherethecoilcurrentisI =564mA (cid:12)(cid:12)S+ (cid:12)(cid:12)2 = (κc+κ)γ, (15)
ML (C+1)2
(seeFig.5(b)). Atthispointthedetuningfromthecavity
resonance ω is ∆ /2π ≡ (ω−ω )/2π = 320 MHz and
c c c and is not favourable since the large cooperativity C
that from the Kittel mode resonance ω is ∆ /2π ≡
m m worksadversely. Thenon-zerodetunings∆ and∆ ,on
(ω−ω )/2π =12 MHz. c m
m the other hand, counteract the adverse effect of C in the
To see why the maximum conversion can be achieved
denominator of Eq. (14) at the expense of the additional
at this particular detunings, let the photon conversion
(cid:16) (cid:17)2
efficiency, (cid:12)(cid:12)SM+L(cid:12)(cid:12)2, be represented in terms of the cooper- penalty term 2κ∆c+cκ +2∆γm . The optimal detunings
8
(a) (b)
−100 −90
−85 −100
50 Hz
−102 z) −90 −110
H
m/ −95 −120
dB −100 −130
Hz) −104 SD ( −105 −140
m/ P −110 10.2 10.4 10.6 10.8 11
B −106
d −115
D ( 10.4365
S Frequency (GHz) (c)
P
−108
−110
0
−112
10.4 10.42 10.44 10.46 10.48 0 5 10 15
Frequency (GHz) Time (s)
FIG. 5. (a) Measured noise power spectrum at a coil current I = 564 mA, indicated by the dotted lines in Figs. 3(a) and
(c). The peak of the noise power corresponds to the lower-branch of the normal modes. The inset shows the zoom-up of the
peak region in (a) under illumination of the two laser fields inducing the inverse Faraday effect. The sharp peak above the
broadnoiselevelindicatesthepresenceofthecoherentmagnetizationoscillations. (b)Calibratedphotonconversionefficiency
(cid:12)(cid:12)SM+L(cid:12)(cid:12)2 as a function of the frequency difference between the two laser fields ωa =Ω0−Ω. The blue points are experimentally
determined efficiencies while the red curve is drawn based on Eq. (11). (c) arg(S+ ) of the generated microwave output as a
ML
function of time, where the upper and the lower points represent the data with the relative phase shift by π.
are found by solving the coupled equations shifted by π. The result shows that the conversion from
light to microwave preserves phase coherence within the
∂ (cid:12)(cid:12)S+ (cid:12)(cid:12)2 =0 (16) time scale of several seconds.
∂∆ ML
c
∂ (cid:12)(cid:12)S+ (cid:12)(cid:12)2 =0 (17)
∂∆m ML IV. DISCUSSION
whose solutions corresponds to the extremal point of
(cid:12)(cid:12)SMW+Li(cid:12)(cid:12)t2hwitthheresinpdeecptetnodtehnetldyetumneinasgusr∆edc atnrdan∆smmi.ttance achTiheveedmiasxaimrouumndp1h0o−t1o0nascosnhvoewrnsioinnFeiffig.c5ie(nbc)yanwdeishparvie-
T ∼ 0.84 we deduce the light-magnon coupling marily limited by the small magnon-light coupling rate
rate ζ/2π =0.18 mHz, which is close to the value ζ. To realize a microwave-light converter in the quan-
ζ/2π =0.25 mHz independently obtained from a shot- tum regime the magnon-light coupling rate ζ has to be
noise-based calibration scheme (see Appendix B) as well improved by several orders of magnitude.
as the value ζ/2π =0.33 mHz evaluated from the Verdet There are several ways in which we could improve the
constant V, reinforcing the validity of the conversion ef- coupling rate ζ. First, an appropriately designed optical
ficiency we obtained. cavity can be incorporated in the converter architecture.
To see the conversion preserves the phase coherence, The converter then consists of three harmonic oscilla-
arg(S+ ), is also measured by replacing the spectrum tor modes, a microwave cavity mode, the Kittel mode,
ML
analyzer with a network analyzer. For this experiment and a optical cavity mode. A promising approach is to
the two laser fields are stabilized to have a definite rel- use whispering gallery modes (WGMs) supported by a
ative phase by actively compensating the fluctuation of spherical crystal of ferromagnet itself. There are some
the optical path-length difference between two fields by developments along this line [22, 27, 28]. With realistic
thepiezoelectricactuatorshowninFig.4(b). InFig.5(c) parametersofWGMsmadeofaYIGdisk,theconversion
arg(S+ ) of the generated microwave around 10.8 GHz efficiency (cid:12)(cid:12)S+ (cid:12)(cid:12)2 may improve up to around 10−3 [22].
ML ML
isshownasafunctionoftime,wheretheefficiencyofthe Second, other magnetic materials with a larger Verdet
microwave generation is the highest, as indicated by a constant than that of YIG can be used. For instance,
dashed line in Fig. 5(b). The upper and the lower points an ionic ferromagnetic crystal, chromium tribromide
in Fig. 5(c) represent the data with the relative phases (CrBr ),isknowntohaveanextremelylargeVerdetcon-
3
9
stant of the order of V = 8700 radians/cm at 1.5 K for Appendix A: Details of the architecture of the
the light at 500 nm [29]. It was demonstrated that the converter
conversion from microwave at 23 GHz to light was pos-
sible with magnons in a CrBr3 disk [30]. 1. Kittel mode
Given the fascinating developments of coherent light-
matter interfaces based on rare-earth ions [7–10], these The ferromagnetic sample we use in the microwave-
ions doped in a ferromagnetic crystal as spin impurities light converter is a spherical crystal made of yttrium
may be interesting. While the Kittel mode is used for a iron garnet (YIG). YIG is a ferri-magnetic insulator
microwave-matter interface, the spin impurities are used which possesses the following characteristics: (a) high
asalight-matterinterface. Whenthetemperatureissuf- Curie temperature of about T = 550 K; (b) high net
C
ficiently low these spin impurities would interact with spin density of n = 2.1×1022 cm−3 [32]; and (c) large
ferromagnetic magnons coherently. The doping, how- Verdet constant of V = 3.8 radians/cm at 1.55 µm [33].
ever, arouses the breaking of translational symmetry of Under a uniform static magnetic field the strong ex-
theferromagneticcrystal,whichwouldraisetheintrinsic change and dipolar interactions among iron spins define
magnon decay rate γ. It may therefore be beneficial to the low-lying energy levels of spin-wave excitations. For
replaceyttriumatomswithrare-earthatomscompletely, the modes with small wave-number, k, in small samples
which would also be good for boosting the optical den- (∼1 mm), the dipolar energy dominates and the electro-
sity. The Faraday rotation of erbium iron garnet (ErIG) magneticforcesareeffectivelymagnetostatic,resultingin
is, for instance, reported to be significantly larger than thesize-independentresonantfrequency[34,35]. Among
that of YIG around the absorption lines of Er [31]. these magnetostatic or Walker modes, we exploit for the
converter the Kittel mode with k = 0, i.e., uniformly-
precessing magnetization mode.
The magnons in the Kittel mode can be treated as
V. SUMMARY
quantainadampedharmonicoscillatormode. Theequa-
tion of motion is given by
We have demonstrated bidirectional coherent conver-
sion between microwave and light via ferromagnetic cˆ˙(t)=−iω cˆ(t)− γcˆ(t)−√γcˆ (t), (A1)
magnons. The converter is based on a hybrid system m 2 n
between a microwave cavity mode and the Kittel mode.
where cˆ(t) is the annihilation operator for the magnon,
Anitinerantmicrowavefieldiscoupledtothehybridsys-
tem through the microwave cavity, while a traveling op- ωm = 1+ωKα2 is the angular frequency of magnetization
oscillationwithω beingtheresonantangularfrequency
tical field addresses the hybrid system through the Kit- K
of the bare Kittel mode [34, 35] and α being the Gilbert
tel mode via Faraday or inverse Faraday effect. The
damping constant [36]. γ =2αω is the intrinsic energy
maximum photon conversion efficiency of the converter m
dissipationrate. Here,totakeintoaccountthenoiseterm
is around 10−10 and is limited by the small magnon-
accompanying the dissipation, the noise field operator
light coupling rate ζ. We have suggested some strategies
cˆ (t) is introduced [25].
for improving ζ. Given the fact that the ferromagnetic n
magnon can be coherently coupled to a superconduct-
ing qubit [23], pursuing the magnon-based microwave-
2. Purcell effect
light converter would make sense for realizing large-scale
quantum optical networks with superconducting qubits.
ForthecouplingbetweentheKittelmodeandtheitin-
erant microwave field the coupling rate is limited by the
intrinsic dissipation rate of the Kittel mode, γ. The cou-
ACKNOWLEDGMENTS pling rate beyond this can be achieved by using a mi-
crowave cavity to exploit the Purcell effect [37]. The use
We would like to thank Seiichiro Ishino, Hi- of a microwave cavity is also beneficial from the view-
roshi Kamimura, Jevon Longdell, Takuya Satoh, point of its magnetic field uniformity at the sample in-
Jake Taylor, and Matt Woolley for useful discussions. side. This makes highly selective excitation of the Kittel
This work is partly supported by the Project for De- mode possible.
veloping Innovation System of the Ministry of Educa- When the resonant frequency of the cavity mode co-
tion, Culture, Sports, Science and Technology, Japan incides with the Kittel mode frequency, i.e., ωc = ωm,
Society for the Promotion of Science KAKENHI (grant the coherent interaction results in hybridization of the
no. 26600071, 26220601, 15H05461), the Murata Science two modes. By writing the annihilationand creation op-
Foundation, the Inamori Foundation, Research Founda- erators for the cavity mode by aˆ and aˆ†, respectively,
tion for Opto-Science and Technology, and National In- the interaction Hamiltonian is given by Eq. (1), where
√
stitute of Information and Communications Technology g =g N isthecollectively-enhancedcoherentcoupling
0
(NICT). R.H., Y.T. and T.I. are supported in part by rate between the two modes with g being the single-
0
the Japan Society for the Promotion of Science. spin coupling rate, where N being the total number
10
of spins in the sample [16, 38]. With the zero-point-
amplitude of magnetic field B in a cavity of volume V
0
(cid:113) (cid:16) (cid:17)
is B0 = µ02h¯Vωc, g0 can be given by γe √B02 , where µ0
is the permeability of vacuum and γ is the electron gy-
e
romagnetic ratio. The factor √1 in the form of g0 comes Detuning
2
from the fact that among the intra-cavity field only the
componentco-rotatingwiththemagnetizationoftheKit-
tel mode contributes to the magnetic resonance [26].
Coupling between the hybrid mode and the itinerant
microwavefieldrequiresanadditionaldissipationchannel
associated with the cavity mode. Denoting the coupling
ratebetweenthemicrowavefieldoutof(into)a1Dtrans-
mission line aˆ (t) (aˆ (t)) and the cavity mode by κ , the
i o c
interaction Hamiltonian between the cavity mode aˆ and
theitinerantmicrowavemodeaˆ canbegivenbyEq.(2). FIG.6. EnergyleveldiagramrelevanttotheFaradayandthe
i
The equation of motion for the cavity mode is obtained inverse Faraday effects with YIG. The states describing the
electronic ground and excited states are specified by |g(cid:105) and
from Eqs. (1) and (2) as
|e(cid:105) and the magnon Fock states are denoted as |n(cid:105). Here de-
κ+κ √ pictedistheconfigurationinwhichtheinverseFaradayeffect
aˆ˙(t)=−iωcaˆ(t)−igcˆ(t)− 2 caˆ(t)− κcaˆi(t), (A2) with the parametric-amplification-type Hamiltonian (A13) is
inducedbytwophase-coherentfieldswithadetuningfromthe
where κ is the internal energy loss rate for the cavity |g(cid:105)↔|e(cid:105) transition being ∆. One of the two fields (blue ar-
addedintoEq.(A2)phenomenologically. HeretheKittel rows)havingtheangularfrequencyΩ0 andbeingz-polarized
(π-polarized) and the other field (red arrows) having the an-
modecˆ(t)manifestsitselfinthesecondtermintheright
gularfrequencyΩ(Ω<Ω )andbeingy-polarizedcoherently
hand side. 0
create and annihilate magnons (brown arrows). ˆb , ˆb , and
The equation of motion for the Kittel mode can simi- z i
cˆdenote the annihilation operators for the z-polarized field,
larly be obtained as
the y-polarized field, and the magnon, respectively.
γ √
cˆ˙(t)=−iω cˆ(t)−igaˆ(t)− cˆ(t)− γcˆ (t). (A3)
m 2 n
3. Faraday effect
Solvingthesetwocoupledequations(A2)and(A3)inthe
Fourier domain leads to the microwave reflection coeffi-
cient, S (ω) = aˆ (ω)/aˆ (ω). First, neglecting the noise Atravelingopticalfieldaddressesthehybridmodevia
√11 o i
term γcˆ (t) in Eq. (A3) we have an algebraic relation Faraday effect or spin-Raman effect [39]. The Faraday
n
between aˆ(ω) and cˆ(ω), that is, effect can be understood phenomenologically as that the
polarization of the linearly polarized light rotates due
ig to the circular birefringence of the transparent mate-
cˆ(ω)= aˆ(ω). (A4)
i(ω−ω )− γ rial. Any material showing circular birefringence pos-
c 2
sessesnon-zerovectorpolarizabilityandexhibitsthevec-
Thenbysubstitutingtherelation(A4),intoEq.(A2)and tor light shift in the ground-state Zeeman manifold [40–
√
usingtheboundaryconditionaˆ (t)=aˆ (t)+ γ aˆ(t)[25] 42], which leads to the Faraday effect.
o i c
we have Suppose that the light propagating along x-axis is lin-
early polarized along z-axis and interacts with a ferro-
i(ω−ω )− 1(κ−κ )+ g2
S (ω)= c 2 c i(ω−ωm)−γ2 . (A5) magnetic sample with a length l, which is magnetized
11 i(ω−ω )− 1(κ+κ )+ g2 along z-axis under a uniform static magnetic field. In
c 2 c i(ω−ωm)−γ2 this configuration the magnetization oscillation perpen-
dicular to z-axis is imparted to the polarization oscilla-
This implies that by measuring the microwave reflection
tions as a result of the Faraday effect. In this case the
coefficient S (ω) the parameters g, κ , γ, and κ can be
11 c interaction Hamiltonian Hˆ (t) can be given by [40–42]
evaluated. The dissipation hierarchy, g > κ > κ ∼ γ, F
c
would suggest that the energy stored in the Kittel mode
(cid:90) τ
is predominantly dissipated as the itinerant microwave
H (t)= dt ¯hG mˆ (t)sˆ (t)Ac, (A7)
F x x
photons. ThestrengthofthecouplingbetweentheKittel
0
modeandthemicrowavecavitycanthenbeevaluatedby
the cooperativity, where G is the coupling constant in the Faraday effect,
τ = l is the interaction time with c being the speed
4g2 of lighct in the material, and A is the cross section of
C = . (A6)
(κ +κ)γ the light beam. Here mˆ (t) is the x component of the
c x
magnetization density, which can be denoted in terms of
