Coherent optical control of a superconducting microwave cavity via electro-optical dynamical back-action Liu QiuRishabh Sahu William Hease Georg Arnold and Johannes M. Fink

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Coherent optical control of a superconducting microwave cavity

via electro-optical dynamical back-action

Liu Qiu⋆,†Rishabh Sahu⋆, William Hease, Georg Arnold, and Johannes M. Fink‡

Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria

(Dated: June 27, 2023)

Recent quantum technologies have established precise quantum control of various microscopic systems using electromagnetic

waves. Interfaces based on cryogenic cavity electro-optic systems are particularly promising, due to the direct interaction

between microwave and optical ﬁelds in the quantum regime. Quantum optical control of superconducting microwave circuits

has been precluded so far due to the weak electro-optical coupling as well as quasi-particles induced by the pump laser. Here

we report the coherent control of a superconducting microwave cavity using laser pulses in a multimode electro-optical device

at millikelvin temperature with near-unity cooperativity. Both the stationary and instantaneous responses of the microwave

and optical modes comply with the coherent electro-optical interaction, and reveal only minuscule amount of excess back-action

with an unanticipated time delay. Our demonstration enables wide ranges of applications beyond quantum transductions, from

squeezing and quantum non-demolition measurements of microwave ﬁelds, to entanglement generation and hybrid quantum

networks.

Microwave superconducting quantum technologies

have facilitated the electronic readout and control of su-

perconducting circuits and quantum dot spin qubits [1,

2], which holds the promise for quantum-enhanced sens-

ing [3] and scalable quantum computing [4]. Emerging

challenges include interfacing the superconducting cir-

cuits to complex electrical lines, which introduces excess

heat load and complexity beyond traditional cryogenic

systems. Photonic ﬁber links, due to the low propa-

gation loss and passive heating, can be adopted to de-

liver microwave signals for quantum circuits readout and

control at millikelvin temperatures, e.g. using photo-

diodes [5], mechanical transducers [6,7], or microwave

photonics [8,9]. Despite the ubiquitous electro-optic de-

vices in modern telecommunication networks with ultra-

high speed translation between electronic and optical

ﬁelds [10–12], their operations in the quantum regime

have been impeded so far due to the weak electro-optical

coupling, even at cryogenic temperatures [9].

Cavity electro-optics (CEO) employs resonantly-

enhanced electro-optic interaction with optimized spatial

overlap of microwave and optical modes [13,14]. It holds

great promises for general quantum measurement and

control of superconducting microwave circuits with op-

tical laser light [14–17], ranging from microwave-optical

entanglement generation [18–20], coherent microwave or

optical signal synthesis [14], to laser cooling of the mi-

crowave mode [21], and bidirectional microwave-optical

quantum transduction with near unity eﬃciency and low

added noise [21–24]. A multimode CEO system allows

for quantum thermometry [25,26] and quantum non-

demolition measurements of the microwave ﬁeld beyond

the standard quantum limit with signiﬁcantly reduced

probing powers [19,27–30]. One particularly promising

application of CEO is to build a complex optical quantum

network connecting hybrid superconducting microwave

†liu.qiu@ist.ac.at

‡jﬁnk@ist.ac.at

⋆These authors contributed equally to this work.

quantum circuits [31,32], with alternative approaches us-

ing electro- or piezo-optomechanical devices [6,7,33,34],

trapped atoms [35,36], rare-earth ions doped crystals

[37] and optomagnonic devices [38,39].

Such prospects rely on the optical coherent dynami-

cal control of the superconducting microwave cavity, i.e.

via the electro-optical dynamical back-action (DBA) [14].

This has been impeded so far due to the typically weak

electro-optical coupling, or the signiﬁcant excess back-

action, i.e. unwanted perturbations that are not due

to the electro-optic eﬀect, as a result of the required

strong optical pump. Despite the steady progress in

the last years, primarily on quantum transductions [21–

24], most CEO systems suﬀer from limited cooperativ-

ity C[14,40], a measure for coherent coupling versus

the microwave and optical dissipation. An endeavor to-

wards coherent electro-optical interaction at unitary co-

operativity has started in the last years, including explo-

rations in various electro-optic materials and fabrication

processes, e.g. based on aluminum nitride [22,41], bulk

and thin-ﬁlm lithium niobate (LN) [16,21,23,24,42,43],

barium titanate [44] and organic polymers [45]. How-

ever, excess dissipation [46,47] and back-action still

remain in optical and microwave resonators, originat-

ing from, e.g. piezoelectric [42,43], photorefractive ef-

fects [48,49], absorption [47], dissipative feedback [50],

quasi-particles [45,51], etc.

Pulsed operation in CEO devices reduces the inte-

grated optical power while maintaining the cooperativ-

ity, and has recently enabled demonstrations of quantum

transduction in the microwave ground state [21,41]. The

compatibility of CEO devices to superconducting mi-

crowave circuits calls for resolving and controlling pulsed

microwave signals in the time domain in a nondestruc-

tive manner [4–7]. However, the coherent optical dynam-

ical control of superconducting microwave cavity has re-

mained elusive.

In this work we demonstrate coherent electro-optic dy-

namical back-action in a multimode cavity electro-optic

device. Our results demonstrate coherent stationary and

instantaneous electro-optic DBA to the microwave mode,

arXiv:2210.12443v2 [quant-ph] 25 Jun 2023

such as the optical spring eﬀect and microwave linewidth

narrowing or broadening, with negligible excess back-

action. We observe electro-optically induced absorption

or transparency of the optical probing ﬁeld [22,52–54],

which opens up the possibility for dispersion engineer-

ing of propagating optical and microwave pulses. The

observed coherent electro-optical response conﬁrms the

feasibility of our multimode CEO system for the direct

quantum optical control and sensing of microwave ﬁelds

in the quantum back-action (QBA) dominant regime [14],

and provides important insights into the complex time-

dependence of pulsed quantum protocols, e.g. electro-

optic entanglement generation [20].

Results

Theoretical Model and Experiment We realize this

experiment in a multimode cavity electro-optical de-

vice [16] as depicted in Fig. 1(a), where a crystalline

lithium niobate whispering gallery mode (WGM) opti-

cal resonator is coupled to the azimuthal number m= 1

mode of a superconducting aluminum microwave cav-

ity inside a dilution refrigerator at ∼10 mK [21,23].

As shown in Fig. 1(b), we consider a series of optical

transverse-electric (TE) modes of the WGM resonator

with the same loss rate κo, i.e. the Stokes, pump and anti-

Stokes mode with frequencies ωs,ωp, and ωas. When the

optical free spectral range (FSR) matches the microwave

frequency Ωe, resonant three-wave mixing between the

microwave and adjacent optical modes arises via the cav-

ity enhanced electro-optic interaction, with the interac-

tion Hamiltonian

Heo/ℏ=g0ˆa†

pˆasˆ

b+g0ˆa†

pˆaasˆ

b†+h.c., (1)

where ˆas, ˆap, ˆaas and ˆ

bare the annihilation operators

for the Stokes, pump and anti-Stokes optical and mi-

crowave modes, and g0is the vacuum electro-optical cou-

pling rate. A on-resonance optical pump enhances the

electro-optic interaction given by g=√¯npg0, where ¯npis

the mean intra-cavity photon number of the pump mode.

This includes the two-mode-squeezing (TMS) interaction

between the Stokes and microwave mode [cf. ﬁrst term

in right-hand side of Eq. 1] and the beam-splitter (BS)

interaction between the anti-Stokes mode and microwave

mode [cf. second term in right-hand side of Eq. 1]. One

ﬁgure of merit of the CEO device is the multiphoton co-

operativity C= 4¯npg2

0/(κoκe), with κoand κethe loss

rates of the optical and microwave modes. The TMS or

BS interaction can be chosen by selectively suppressing

the counterpart via mode engineering, i.e. by coupling

the anti-Stokes or Stokes mode to an optical transverse-

magnetic (TM) mode of diﬀerent polarization at rate of

Jas or Js[16]. The interaction Hamiltonian is given by

HJ/ℏ=Jsˆa†

sˆas,tm +Jasˆa†

asˆaas,tm +h.c., (2)

with ˆas,tm and ˆaas,tm the annihilation operators for the

TM modes of frequency ωsand ωas.

Figure 1(c) shows the optical reﬂection characteriza-

tion of one TE mode family of our EO device around

1550 nm with similar total loss rate κo/2π≈26 MHz.

We note that, all modes are re-centered to the indi-

vidual TE mode resonance. The TE modes are para-

metrically coupled to a microwave mode with loss rate

κe/2π∼10 MHz, whose frequency is adjusted to match

the FSR. Mode 4 is strongly coupled to a TM mode

of similar frequency with rate J/2π∼26 MHz, which

manifests as a split mode for anti-Stokes or Stokes scat-

tering suppression when pumping mode 3 or 5 respec-

tively. More details regarding mode characterizations

are in Supplementary Information (SI), including opti-

cal losses and mode separations.

In the following we present temporal and spectral co-

herent dynamical response measurements in the pulsed

regime. As shown in Fig. 1(d), a strong optical pump

pulse of duration τis sent to the EO device, together

with a weak continuous probing ﬁeld around the mi-

crowave or optical (Stokes or anti-Stokes) resonance, to

probe the dynamical back-action during the pulse. We

introduce the normalized probing ﬁeld reﬂection between

pump pulse on and oﬀ

Rj(ω) = |Sjj (ω)/Sjj,oﬀ (ω)|2,(3)

with the reﬂection scattering parameters Sjj (ω), i.e. the

output and input ﬁeld amplitude ratio for mode j∈(e, o).

In Fig. 1(e), we show a typical normalized reﬂection

coeﬃcient over time with on-resonance probing in dif-

ferent mode conﬁgurations for a pump pulse of duration

τ= 250 ns and peak power of ∼500 mW. In the symmet-

ric case, i.e. mode 2 as pump mode with Js/as = 0, the

electro-optical dynamical back-action to the microwave

mode is in principle evaded. Due to balanced Stokes

and anti-Stokes scattering, the microwave susceptibility

remains the same,

χe(Ω) = 1/(κe/2−iΩ).(4)

Interestingly, the optical susceptibilities around the

Stokes and anti-Stokes mode frequencies are modiﬁed,

χo,s/as(Ω) = 1

χo(Ω)−1∓g2/(χe(Ω)−1±g2χo(Ω)),(5)

with χo(Ω) = 1/(κo/2−iΩ) the optical susceptibility.

The constructive and destructive interferences between

the probing ﬁeld and the electro-optical interaction result

in electro-optically induced absorption (EOIA) around

the Stokes mode and electro-optically induced trans-

parency (EOIT) around the anti-Stokes mode. Similar

dynamics has been reported previously in cavity optome-

chanics [53,54] and magnomechanics [55], which however

only arises in the presence of dynamical back-action [56].

As shown in Fig. 1(e) (upper left), the microwave on-

resonance reﬂection responds instantaneously to the ar-

riving pump pulse, and continues to drift even after the

pulse is oﬀ (t > 250 ns). Such excess back-action is negli-

gible, with less than 3% deviation in Re(Ωe). In Fig. 1(e)

(lower left), the optical on-resonance Stokes (anti-Stokes)

ωp

ωsωas

JsJas

Entangle t

Probe

Pump

(a)

(b)

(c)

(d)

ωoΩe

Ωe

ΩeΩe

(e)

Pulse O

Pulse On

FIG. 1. Multimode cavity electro-optical system in the pulsed regime. a, Schematic representation of the cavity

electro-optic device. A millimeter-sized lithium niobate optical resonator (light blue) is placed in the capacitor of the LC

circuit realized as an aluminum 3D microwave cavity (purple). Optical light is fed to the EO device via an antireﬂection-coated

diamond prism. b, Mode conﬁgurations of the CEO device, with one microwave mode (purple) coupled to three optical TE

modes, i.e. the Stokes (red), pump (black) and anti-Stokes mode (blue). A strong optical pump of frequency ωpgenerates

pump enhanced Stokes and anti-Stokes scattering at a rate g, which can be selectively suppressed by coupling to an optical

TM mode (dashed curve). c, Measured optical reﬂection (dots) of a series of modes with ﬁtting curves (lines), with mode 2

as the pump mode for the symmetric case (Js/as = 0) while mode 3 and 5 for the Stokes (Js= 0) and anti-Stokes (Jas = 0)

case respectively. All resonances are re-centered to the individual TE mode resonance frequency. Mode splitting in mode 4

indicates strong TE-TM mode coupling. d, Coherent dynamical response probing scheme, with a short optical pump pulse of

duration τand a weak continuous probing ﬁeld around the microwave or optical (Stokes or anti-Stokes) mode frequency. e,

Temporal on-resonance response R(ω)[cf. Eq. 3], i.e. the normalized probing ﬁeld reﬂection between pulse on and oﬀ (pump

peak power ∼500 mW). Left panel shows the symmetric case (ωp=ω2), with on-resonance microwave response (green curve)

in the upper panel and optical Stokes (ωs=ω1, orange curve) and anti-Stokes (ωas =ω3, green curve) responses in the lower

panel. Right panel shows the two asymmetric cases, i.e. the on-resonance microwave and optical Stokes responses (ωs=ω2)

in the Stokes case (ωp=ω3, red curves), and the on-resonance microwave and optical anti-Stoke responses (ωas =ω6) in the

anti-Stokes case (ωp=ω5, blue curves).

reﬂection decreases (increases) when the optical pulse ar-

rives and restores instantaneously after the pulse is oﬀ.

In addition, we consider the Stokes case with mode 3 as

pump mode (Js= 0), and the anti-Stokes case with mode

5 as pump mode (Jas = 0). Coherent electro-optical

DBA results in a modiﬁed microwave susceptibility,

χe,s/as(Ω) = 1

χe(Ω)−1∓g2χo(Ω).(6)

DBA on the Stokes (Stokes case) or the anti-Stokes (anti-

Stokes case) mode results in the modiﬁed susceptibility,

χo,s/as(Ω) = 1

χo(Ω)−1∓g2χe(Ω),(7)

assuming 4J2

as/s ≫κoκo,tm, with κo,tm the TM mode

loss rate. In both cases, Eq. 6and Eq. 7are symmetric

under interchange of microwave and the optical probing

mode, which enables mutual probing of the optical and

microwave ﬁeld with its counterpart. In the normal dissi-

pation regime, i.e. κo≫κe, the microwave mode under-

goes eﬀective narrowing (broadening) in the Stokes (anti-

Stokes) case, while the Stokes (anti-Stokes) probing ﬁeld

undergoes EOIA (EOIT), due to the constructive (de-

structive) interference between the probe ﬁeld and the

electro-optical interaction. In the reversed dissipation

regime, i.e. κo≪κe, the microwave mode experiences

EOIA (or EOIT), while the optical Stokes (anti-Stokes)

mode linewidth is eﬀectively narrowed (broadened). The

temporal on-resonance dynamics in the Stokes and anti-

Stokes cases are shown in the right panel of Fig. 1(e).

Similar to the symmetric case, the Stokes mode under-

goes EOIA in the Stokes case, while the anti-Stokes mode

undergoes EOIT in the anti-Stokes case.

Stationary Dynamical Back-action As shown in

Fig. 1(e), the on-resonance normalized reﬂections remain

stationary before and in the middle (t∼200 ns) of the

pulse. We reconstruct the coherent stationary spectral

response by sweeping the probe tone frequency around

the probing mode resonance, and perform a pump pulse

(a) (b)

FIG. 2. Stationary dynamical back-action to the microwave mode. A power sweep is conducted in each pump

conﬁguration. A joint ﬁt of the stationary Re(Ω) is performed with the original microwave linewidth as a shared parameter,

and the linewidth and frequency change for each power as remaining ﬁtting parameters. a, Microwave response measurements

with the same pump power as in Fig. 1(e). The upper panel shows the stationary Re(Ω) as dotted lines, with ﬁtting curves

as solid lines. The lower panel shows the reconstructed microwave reﬂection |See(Ω)|2with the pump on (solid curve) and

oﬀ (dashed curve) using obtained parameters from the joint ﬁt. b, Fitted microwave frequency shift and linewidth change

versus cooperativity C. Dashed lines are theoretical curves incorporating the full dynamical back-action model, using ﬁtting

parameters from the corresponding coherent optical response [cf. Fig. 3], including imperfect frequency detunings. Error bars

represent the 95% conﬁdence interval of the ﬁt.

power sweep in each conﬁguration.

To construct the microwave response, we perform a

joint ﬁt of the stationary Re(Ω) for diﬀerent powers, and

obtain the individual microwave linewidth and frequency

change. The upper panel of Fig. 2(a) shows the sta-

tionary spectral response Re(Ω) in three diﬀerent pump

conﬁgurations, with the same pump pulse power as in

Fig. 1(e). Re(Ω) remains unchanged due to the bal-

anced Stokes and anti-Stokes scattering in the symmetric

case (center), while it changes dramatically around the

mode resonance due to strong dynamical back-action in

the two asymmetric cases. The lower panel of Fig. 2(a)

shows the measured microwave reﬂection scattering pa-

rameter |See(Ω)|2with pulse on (oﬀ) as solid (dashed)

lines, indicating microwave linewidth narrowing and a

slight frequency increase in the Stokes case (ωp=ω3) and

linewidth broadening in the anti-Stokes case (ωp=ω5)

with an increased on-resonance reﬂection.

In Fig. 2(b), we show the extracted microwave fre-

quency (δΩe) and linewidth (δκe) change in the power

sweep, for each pump conﬁguration. The corresponding

microwave response ﬁtting curves are shown in Fig. S7

to S10 in the SI. In the symmetric case (ωp=ω2), no

evident frequency or linewidth change is observed due to

the evaded back-action. In the anti-Stokes case (ωp=ω5)

the microwave linewidth increases linearly with C, while

it decreases in the Stokes case (ωp=ω3). The theoretical

curves for both asymmetric cases match very well with

experimental results, using a full dynamical back-action

model incorporating optical response ﬁtting parameters

including imperfect frequency detunings [cf. Fig. 3(b)]. In

the anti-Stokes case, we observe a minuscule deviation in

the microwave frequency shift of ∼10−4Ωe. This can be

explained by the small detuning uncertainties (sub-MHz)

as discussed in the SI A, probably due to photorefrac-

tive [48,49] or quasi-particles eﬀects [6,51].

As shown in the upper panel of Fig. 3(a), we perform

a joint ﬁt of the stationary Ro(ω) in each probing con-

ﬁguration, i.e. Stokes mode probing in the Stokes and

symmetric cases while anti-Stokes mode probing in the

symmetric and anti-Stokes cases. This allows us to ex-

tract C,κo, and external coupling rate κo,ex in each prob-

ing conﬁguration. The detailed optical response ﬁtting

curves are show in Fig. S7 to S10. In the lower panel

of Fig. 3(a), we show the reconstructed optical reﬂection

eﬃciency |Soo(ω)|2with pulse on and oﬀ as solid and

dashed lines. The Stokes mode probing (left two pan-

els) reveals similar EOIA for the Stokes and symmetric

cases when the pump pulse is on, while the anti-Stokes

mode probing (right two panels) indicates similar EOIT

(a) (b)

FIG. 3. Stationary electro-optically induced absorption and transparency, with Stokes mode probing in the Stokes

and symmetric cases, while anti-Stokes mode probing in the symmetric and anti-Stokes cases. In each probing conﬁguration,

a pump power sweep is conducted, and a joint ﬁt of stationary Ro(ω) to the full dynamical back-action model is performed.

a, Measurements with same pump power as in Fig. 1(e), with the two left panels for Stokes mode probing and the two right

panels for anti-Stokes mode probing. The upper panel shows Ro(ω) (dotted lines) with ﬁtting curves (solid lines). The lower

panel shows reconstructed optical reﬂection |Soo(ω)|2with pulse on (solid curve) and oﬀ (dashed curve) in logarithmic scale,

which demonstrates EOIA in the Stokes case and EOIT in the anti-Stokes case. b, The upper panel shows |Soo(ωs)|2for

the two Stokes mode probing cases, while the lower panel shows |Soo(ωas)|2for the two anti-Stokes mode probing cases. The

corresponding theoretical curves are shown as dashed lines. Error bars indicate two standard deviations.

for the symmetric and anti-Stokes cases. In Fig. 3(b), we

show the on-resonance reﬂection eﬃciency versus Cin

diﬀerent probing conﬁgurations with theoretical curves

shown as dotted lines. In the upper panel, |Soo(ωs)|2

at the Stokes mode resonance ﬁrst approaches zero and

then increases with Cdue to EOIA. In the lower panel,

|Soo(ωas)|2at the anti-Stokes resonance increases slowly

as Cincreases due to EOIT. We note that, the diﬀerent

on-resonance |Soo|2at low Cis due to the slightly dif-

ferent external coupling eﬃciency of the optical modes.

To capture the stationary electro-optical dynamics, the

eﬀective Cis limited to ∼0.5 due to the Kerr nonlinear-

ity [21], which depends on the power and duration of the

applied pulse and results in optical parametric oscillation

in the optical resonator [57]. With further improvement

of κeand g0, the device can enable parametric ampli-

ﬁcation of the microwave and optical Stokes signal for

C≫1.

Transient Dynamical Back-action Emerging quan-

tum applications of CEO devices, such as ultra-low noise

microwave-optical quantum transduction and entangle-

ment generation, require strong optical pump pulses to

reach near unity C[21,24]. A detailed understanding of

the transient response of CEO devices is therefore cru-

cial for complex measurement protocols in the quantum

limit.

In Fig. 4(a), we show the transient response of the mi-

crowave mode in diﬀerent pump conﬁgurations with the

same power as in Fig. 1(e). Within each pump conﬁg-

uration, we perform a joint ﬁt of Ro(ω) over the pulse

incorporating the full DBA model, with C(t) and im-

perfect detunings as free parameters, as explained in

SI D. When the optical pump pulse arrives, the ﬁtted

C(t) increases smoothly in the beginning, reaches sta-

tionary value in the middle, and slowly decreases to zero

after the pulse. In the middle and lower panel, we show

the obtained microwave frequency and linewidth change

over the pulse as dotted lines, with theoretical curves as

dashed lines. The small blue shift of the microwave mode

in the two asymmetric mode conﬁgurations is due to im-

perfect detunings (sub-MHz) as explained in SI A. The

linewidth change follows closely the predicted coherent

electro-optical dynamical back-action, i.e. narrowing in

the Stokes case while broadening in the anti-Stokes case.

In the symmetric case, a very slight excess frequency drift

(∼10−5Ωe) and linewidth change (∼10−2κe) indicate a

ﬁnite amount of instantaneous excess back-action to the

microwave mode in the beginning and at the end of the

pulse, due to the loading and unloading of the optical

pump ﬁeld. We note that, similar instantaneous excess

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Coherentopticalcontrolofasuperconductingmicrowavecavityviaelectro-opticaldynamicalback-actionLiuQiu⋆,†RishabhSahu⋆,WilliamHease,GeorgArnold,andJohannesM.Fink‡InstituteofScienceandTechnologyAustria,AmCampus1,3400Klosterneuburg,Austria(Dated:June27,2023)Recentquantumtechnologieshaveestablishedpreciseq...

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Coherent optical control of a superconducting microwave cavity via electro-optical dynamical back-action Liu QiuRishabh Sahu William Hease Georg Arnold and Johannes M. Fink

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