www.sciencemag.org/cgi/content/full/science.aaa3693/DC1

Supplementary Materials for

Coherent coupling between a ferromagnetic magnon and a superconducting qubit

Yutaka Tabuchi,* Seiichiro Ishino, Atsushi Noguchi, Toyofumi Ishikawa, Rekishu

Yamazaki, Koji Usami, Yasunobu Nakamura

*Corresponding author. E-mail: tabuchi@qc.rcast.u-tokyo.ac.jp

Published 9 July 2015 on Science Express DOI: 10.1126/science.aaa3693

This PDF file includes:

Materials and Methods Figs. S1 to S4 References

Supplementary Materials for“Coherent coupling between ferromagnetic magnon

and superconducting qubit”

Yutaka Tabuchi,1∗ Seiichiro Ishino,1 Atsushi Noguchi,1 Toyofumi Ishikawa,1

Rekishu Yamazaki,1 Koji Usami,1 Yasunobu Nakamura1,21Research 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

∗To whom correspondence should be addressed; E-mail: tabuchi@qc.rcast.u-tokyo.ac.jp

Experimental apparatusAn illustrative drawing and a photo of the experimental apparatus are shown in Fig. S1. Thewiring, circuit components and instruments used in the experiments are shown in Fig. S2. Eachof the microwave powers presented in the paper refers to the one at the relevant cavity port.

Qubit and cavity modesWe use a transmon-type qubit with the bare qubit frequency ωq,bare/2π and the anharmonic-ity α/2π of 8.204 GHz and −158 MHz, respectively. The anharmonicity α is defined as thedifference ωef − ωge between the bare transition frequencies of the ground state to the first ex-cited state ωge (≡ ωq,bare) and the first excited state to the second excited state of the qubitωef . The qubit couples to the cavity modes TE101, TE102 and TE103, with coupling strengths of85 MHz, 121 MHz and 142 MHz, respectively. Owing to the coupling to the cavity field, thequbit frequency ωq/2π is shifted by −71 MHz (Lamb shift) from ωq,bare/2π.

The cavity has the bare mode frequencies (ω101,bare/2π, ω102,bare/2π and ω103,bare/2π) of6.993 GHz, 8.420 GHz and 10.452 GHz. They are subject to a shift due to coupling with thequbit by −6 MHz, +68 MHz, +9 MHz, respectively. We tune the coupling of the cavity modesto the input and output ports by adjusting the length of the centre pin protruding into the cavity.

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The input and output couplings for the modes TE102 and TE103 are κ102,in/2π = 0.68 MHz,κ102,out/2π = 0.53 MHz, κ103,in/2π = 0.15 MHz and κ103,out/2π = 2.72 MHz. The internalloss of the modes TE102 and TE103 are 1.26 MHz and 1.24 MHz, respectively.

Throughout the paper, we use the renormalized frequencies for the qubit ωq and the cavitymodes (ω101, ω102 ≡ ωc, and ω103), taking into account the frequency shifts due to the interac-tion. Namely, ωq/2π = 8.136 GHz, ω101/2π = 6.987 GHz, ω102/2π (≡ ωc/2π) = 8.488 GHzand ω103/2π = 10.461 GHz.

YIG sampleWe purchased the YIG single crystal from Ferrisphere Inc. (30). The spherical sample is pol-ished to the surface roughness below 50 nm and glued to an aluminum-oxide rod at the factory.We apply the static magnetic field Bstatic to the 〈100〉 crystal axis.

Cavity-magnon couplingMethod A static magnetic field Bstatic around 0.29 T is applied locally to the sphere with acompact magnetic circuit consisting of a pair of permanent magnets made of pure iron and asuperconducting coil (See Fig. S1). We use a probe power of −141 dBm corresponding to thesingle-photon average occupancy of the TE102 mode.

The color plot in Fig. 1B shows the transmission coefficient around the TE102 mode fre-quency as a function of the probe microwave frequency and the static field presented in the coilcurrent. The cavity and the Kittel mode frequencies shown in the dashed lines are obtained fromfitting based on the model where the cavity mode is coupling to the Kittel mode and anothermagnetostatic mode. The transmission coefficient t(ω) for the fitting is explicitly written as:

t(ω) =

√κ102,inκ102,out

i(ω − ω102)− κ102,in+κ102,out+κint2

+ |gm|2i(ω−ωFMR)−γm/2 + |gm′ |2

i(ω−ωFMR′ )−γm′/2

(S1)

Here, we introduce the frequency ωFMR′ , coupling strength gm′ and decay rate γm′ of the othermagnetstatic mode. For Fig. 1B, the phase of the transmission coefficient is rotated such that thereal part of the coefficient is pure absorption spectrum. The magnetostatic mode which appearsin the upper branch is detuned from the Kittel mode by (ωFMR′ − ωFMR)/2π = 4.3 MHz and iscoupled to the cavity mode with a strength of gm′/2π = 4.2 MHz.

If we look at Fig. 1B closer, we find that there is a faint replicated cavity line at around8.45 GHz. This is produced by a small but finite population of the qubit excited state. Owingto the small detuning between the qubit and the TE102 mode, the qubit gives a relatively largedispersive shift of the cavity, χ102, to the cavity mode. The effective interaction Hamiltonian iswritten as Hshift/h = 2χ102 a

†a σ+σ−, where a is the annihilation operator of the TE102 mode

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and σ− = [σ+]† is the lowering operator of the qubit. The offset between the bright and darkcavity lines corresponds to 2χ102/2π = −36.6 MHz.

Qubit-magnon couplingMethod As the qubit dispersively coupled to the TE103 mode with the qubit-state-dependentfrequency shift χ103/2π = 0.46 MHz, the qubit state can be measured through the change inthe TE103 mode response to the probe microwave. We measure the reflection coefficient witha probe power of −135 dBm corresponding to the single-photon average occupancy of theTE103 mode, and with a qubit excitation power of −130 dBm corresponding to the qubit Rabifrequency of 1.7 MHz.

The color plot in Fig. 2B shows the real part of the reflection coefficient at ω103/2π =10.461 GHz as a function of the excitation microwave frequency and the static field representedin the coil current. The phase of the reflection coefficient is rotated such that the change owingto the qubit excitation is maximum. Another magnetostatic mode that is seen in Fig. 1B weaklycouples to the qubit and gives rise to an additional splitting around 8.125 GHz.

Coupling strength The effective coupling between the qubit and the Kittel mode gqm,s =gqgm/∆ should be calculated using the bare qubit ωq,bare and cavity ω102,bare frequencies toavoid double-counting of the effect of interactions in the perturbative treatment. Therefore,we use the detuning ∆/2π = (ωq,bare − ω102,bare)/2π of 215 MHz, and evaluate the couplingto be gqm,s/2π = 11.8 MHz. For the additional splitting (Figs. 2B and 2C) due to the othermagnetostatic mode, the coupling strength gqm′,s/2π = gqgm′/∆ is calculated to be 2.4 MHz,which is consistent with the experimental value of 2.5 MHz.

Parametrically induced couplingSpectroscopy To find parametrically-induced transitions, we perform spectroscopy of the hy-brid system. Figure S3A illustrates the method. We use the TE103 mode for the qubit readout;when the drive microwave hits transitions related to the qubit, the cavity reflection coefficientat ω103 changes.

In Fig. S3B, the upper diagonal dashed line corresponds to the Kittel-mode frequency, whichlinearly moves along with the coil current and splits when the frequency coincides with theTE102-mode frequency. (Note that the Kittel mode is also observed by the direct measurementthrough the TE102 mode as shown in Fig. 1B.) On the other hand, the lower dashed line isascribed to the parametrically-induced transition. The transition frequency is equal to (ωq +ωFMR)/2. The transition frequency and the Kittel mode frequency ωFMR meet each other at thequbit frequency ωq.

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Method The Kittel mode is detuned from the qubit, so that the static beam-splitter-type cou-pling via the TE102 mode, which is described by the Eq. 1 in the main text, is negligible. TheKittel mode frequency at this condition is ωFMR/2π = 8.410 GHz. The frequecy is shiftedby +15 MHz from the bare Kittel-mode frequency ωFMR,bare/2π = 8.395 GHz, owing to thedispersive shift caused by the qubit. However, this shift is independent of the qubit states |g〉and |e〉.

The amplitude of the cavity reflection coefficient Re(r) around the Kittel mode frequency isplotted in the Fig. 3B as a function of the probe and the parametric-drive frequencies. Small butfinite mixing of the Kittel mode with the TE102 mode allows the direct Kittel-mode spectroscopyby the probe microwave. All the spectra in Fig. 3B are measured with the probe power of−135 dBm. Panels i-vi correspond to the drive power Pd of −104, −103, −99, −96.5, −93.5,and −90.5 dBm, respectively.

Coupling strength The parametrically-induced coupling strength gqm,p is derived from aHamiltonian taking into account the TE102 mode, the qubit and the Kittel mode, which is ex-pressed as

H/h = ω102,barea†a+ ωFMR,barec

†c

+(ωq,bare −

α

2

)b†b+

α

2

(b†b)2

+ gq(a†b+ ab†) + gm(a†c+ ac†)

+

√κ102

Pd

hωd

[a† exp (−iωd) + a exp (iωd)

], (S2)

where a and b are the annihilation operators of the photon in the TE102 mode and a transmonqubit, α is the anharmonicity of the qubit, and κ102 = κ102,out since we introduce the microwavedrive through the cavity output port. Here, we define the lowest two levels of the anharmonicoscillator as the qubit subspace. Provided that we explicitly write the lowering operator asb =

∑n≥0

√n+ 1|n〉〈n + 1|, the qubit lowering operator is defined as σ− = (σ+)† = |0〉〈1|.

By applying successive unitary transformations to diagonalize the Hamiltonian (31) and usingapproximation in which higher order terms are safely truncated, we obtain the effective Hamil-tonian

Heff = Hqm,p + Hres, (S3)

where Hqm,p and Hres represent a parametrically-induced tunable coupling and a residual staticcoupling between the qubit and the Kittel mode, respectively. They are written as

Hqm,p/h = gqm,p σ+ c† exp [i(ωq + ωFMR + 2ξ − 2ωd)t]

+ g∗qm,p σ− c exp [−i(ωq + ωFMR + 2ξ − 2ωd)t] (S4)

Hres/h = 2ξ c†c σ+σ−. (S5)

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The coupling strength gqm,p is expressed as

gqm,p ' 2gqg∗m

(κ102Pd

hωd

)(gq

∆d

)2

×[

5/6

(∆q −∆m)2∆m

− 1

(∆q −∆m)(∆q −∆m − α)∆m

+2/3

(∆q −∆m)(∆q + ∆m)∆m

+4/3

(∆q −∆m)2(∆q + ∆m)

+1/3

∆q(∆q + ∆m)∆m

+1/2

(∆q −∆m − α)(∆q −∆m − 2α)∆m

+1/6

∆q(∆q −∆m)∆m

+1/2

∆q(∆q −∆m)2

+2/3

∆q(∆q −∆m)(∆q + ∆m)− 1/6

(∆q −∆m)(∆q −∆m − 2α)∆m

− 1/2

(∆q − α)(∆q −∆m)(∆q −∆m − α)

− 1/12

(∆q − α)(∆q −∆m)∆m

+1/12

(∆q − α)(∆q −∆m − 2α)∆m

− 1/6

(∆q − α)(∆q −∆m)(∆q −∆m − 2α)

], (S6)

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and the residual coupling strength ξ is denoted as

ξ ' −|gq|2|gm|2

×[

5/12

∆q(∆q −∆m)∆m

+5/12

(∆q −∆m)∆2m

+1/4

∆q∆2m

+1/12

∆2q∆m

+1/12

∆2q(∆q −∆m)

+5/12

(∆q −∆m)2∆m

− 1/12

(∆q−α)∆2m

− 5/12

(∆q − α)(∆q −∆m − α)∆m

− 5/12

(∆q −∆m − α)∆2m

− 1/12

(∆q − α)2∆m

− 1/12

(∆q − α)2(∆q −∆m − α)

− 5/12

(∆q −∆m − α)2∆m

]. (S7)

Here, we define ∆q = ω102,bare − ωq,bare + 2gq/(ω102,bare − ωq,bare), ∆m = ω102,bare − ωFMR +gq/(ω102,bare−ωq,bare), and ∆d = ω102,bare−ωd + gq/(ω102,bare−ωq,bare). The conversion ratiogqm,p/Pd between the drive power and the coupling strength in our experiment is estimated to be4.9 MHz/pW, which is indicated in the inset of Fig. 3C as a green dashed line. The discrepancybetween the theoretical (4.9 MHz/pW) and the experimental (4.0 ± 0.4 MHz/pW) values isattributed to higher-order terms dropped in the secular approximation.

Residual coupling In the tunable-coupling experiment, we suppress the static coupling be-tween the qubit and the Kittel mode, gqm,s, by setting a relatively large detuning of 274 MHz.However, the residual coupling Hres remains finite, which shifts the Kittel-mode frequencydepending on the qubit state (|g〉 or |e〉). We obtain the residual coupling strength ξ/2π of−2.5 MHz from the observation of the Kittel-mode doublet in the presence of qubit excitation,whereas the calculated value from Eq. (S7) is −1.2 MHz.

In order to understand the effect of the residual coupling, we perform a numerical simulationof the Kittel-mode spectrum under the parametric drive. Parameters are chosen such that theyreproduce the result shown in panel v of Fig. 3B: the residual coupling ξ/2π = −2.5 MHz,the coupling strength gqm,p/2π = 2.2 MHz, the Kittel mode linewidth γm/2π = 1.3 MHz, thequbit relaxation rate γq/2π = 0.13 MHz and the qubit dephasing rate γφ/2π = 0.3 MHz.

Figure S4 shows the simulated Kittel-mode spectrum as a function of the probe frequencyand the drive frequency. The simulation reveals that the residual coupling Hres shifts the Kittel-mode frequency depending on the qubit state. As the parametric drive excites the qubit andthe Kittel mode simultaneously, the Kittel-mode frequency is shifted by 2ξ. At the resonance itfurther shows two-photon Rabi splitting. The simulation reproduces the characteristic featuresin Fig. 3B pretty well.

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a

gb

d

10 mm

e

h

fc

A Bi

b

d

e

a

i

Fig. S1: Experimental apparatus. (A) Cut model of the cavity. The elongated rectangularcavity (a) made of oxygen-free copper has dimensions of 24 × 3 × 53 mm. Two SMA con-nectors (b; another is on the opposite side) are attached for the microwave transmission andreflection spectroscopy. Couplings of the ports to cavity modes are tuned appropriately by ad-justing the length of the connector center pin protruding into the cavity. A pair of disc-shapeneodymium permanent magnets (c; another is behind the cavity), with a diameter of 10 mmand a thickness of 1.0 mm each, are placed at the ends of a magnetic yoke (d) made of pureiron. The magnets produce a static field of about 0.29 T at the 4-mm gap in between. The mag-netic field can be additionally tuned by the current along a 104-turn superconducting coil (e).The field-to-current conversion ratio is approximately 1.7 T/A. An YIG sphere (f) glued to analuminum-oxide rod along the crystal axis 〈110〉 is mounted in the cavity at the center of thegap between the magnets. The static field is applied in parallel with the crystal axis 〈100〉.A transmon-type superconducting qubit (g), consisting of two large-area aluminum pads anda single Josephson junction (Al/Al2O3/Al), is fabricated on a silicon substrate (See the insetof Fig. 1A) and is mounted inside the cavity. The qubit and the YIG sphere are separated by35 mm in the horizontal direction. A double-layer magnetic shield made of aluminum (h) andpure iron (i) covers a half of the cavity to protect the qubit from the stray field of the magnet.(B) Picture of the apparatus hanging beneath the mixing-chamber plate of a dilution refrigerator.The cavity is cooled down to 10 mK.

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VectorNetworkAnalyzer

300 K

4 K

10 mK

26d

B

1dB

Cry

ogen

icam

plifi

erAm

plifi

er

NbT

i wire

Atte

nura

tors Cu

wire

Magnet& Coil

MWcavity

Microwave source

Isol

ator

s

Qubit YIG Yoke

16dB

Current source

20dB

Low-pass filter

Fig. S2: Measurement setup. The attenuation on the input lines, which at 10 GHz amounts to49 dB including losses at cables (phosphor-bronze coaxial cables; Coax Corp. SC-119/50-PBC-PBC) and connectors, is enough to prevent the room-temperature thermal noise from reachingthe sample space. To further diminish noise above 12 GHz at input port, we introduce a low-pass filter (RLC F-30-12.4-R). For the output line, we place a circulator and isolators (QuinstarXTE0812KCS and XTE0812KC), which give the isolation ratio of more than 60 dB. All mea-surements in this Report are done with a vector network analyzer (VNA; Agilent E5071C).For the static- and parametric-coupling experiments, the excitation microwave generated by amicrowave source (Agilent E8247C) is combined with the probe microwave from VNA in adirectional coupler (Krytar 120420) and is introduced to the input port. The reflected or trans-mitted signal from the cavity is amplified in a series of amplifiers at 4 K (Caltech CRYO4-12)and at room temperature (MITEQ AFS4-08001200-09-10P-4). A current source (YokogawaGS200) is used to tune the static field applied to the YIG sphere.

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Driv

e fre

quen

cy (G

Hz)

Coil current δI (mA)

Ref

lect

ion

coef

ficie

nt

Re(

)

∆r

BA

Ene

rgy

Ground state

TE102

Cavity

Kittel mode

TE102

Qubit

TE103

Qub

it R

eado

ut

Cavity

Drive microwave

Kitte

l mod

e

Parametric co

upling

4 8q12 16

8.2

8.3

8.4

8.5

8.6

0.0

0.1

0.2

0.3

Fig. S3: Spectroscopy of parametrically-induced transitions. (A) Method of the spec-troscopy. The qubit dispersively couples to the cavity TE103 mode, giving rise to the qubit-state-dependent frequency shift χ103. To investigate the energy-level structure of the hybridsystem consisting of the qubit, the TE102 mode and the Kittel mode, we search for transitionsby driving the system and by simultaneously monitoring the qubit excitation via the frequencyshift of the TE103 mode. A few allowed transitions are observed in the low-drive-power limit,while many dipole-forbidden transitions appear when the system is driven strongly. (B) Spec-trum taken with a strong drive power. Change in the cavity reflection ∆r at ω103 is measuredas a function of the drive frequency and the static magnetic field represented by the coil currentδIq. For a reference, δIq is defined to be zero when the Kittel-mode frequency ωFMR coincideswith the qubit frequency ωq. We use the probe and drive microwave power of −135 dBm and−96.7 dBm, respectively. The line observed at 8.47 GHz, broadened and shifted due to thehigh-power drive, corresponds to the TE102 mode. Note that the Kittel mode is also detected inthe indirect measurement via the qubit because of the residual interaction. The two-photon para-metric transition is observed at the frequency (ωFMR +ωq)/2. Other qubit-related multi-photontransitions, e.g., the qubit-TE102-mode parametric transition at [(ωq + ωc)/2]/2π = 8.29 GHz,are seen as horizontal lines.

9

-12 -10 -8 -6 -4 -2 0 2

-12

-8

-4

0

4

Drive frequency (MHz)

Pro

be fr

eque

ncy

(MH

z)

0.0

1.0

0.5

Spe

ctra

l den

sity

(arb

. uni

t)

Fig. S4: Numerical simulation of parametrically-induced coupling. A simulated Kittel-mode spectrum is plotted as a function of the probe frequency and the parametric-drive fre-quency. Parameters used for the simulation correspond to those in panel v of Fig. 3B (SeeSupplementary Material). The residual static coupling ξ shifts the Kittel-mode frequency de-pending on the qubit states (|g〉 and |e〉). The diagonal dashed line depicts the detuning of theparametric drive. Away from the two-photon parametric resonance |g, 0〉 ⇔ |e, 1〉, only theKittel-mode excitation |g, 0〉 ↔ |g, 1〉 (upper horizontal dashed line) appears since the qubit isin the ground state. As the drive frequency approaches the resonance, parametric drive excitesthe qubit together with a Kittel-mode magnon, leading to a shift of the Kittel-mode spectrumto the line |e, 0〉 ↔ |e, 1〉 (lower horizontal dashed line). The dressed states between |g, 0〉 and|e, 1〉 give rise to an avoided crossing at the resonance, indicating the parametrically-inducedcoherent coupling (Inset of Fig. 3).

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