Cryogenic electro-optic interconnect for superconducting devices

Amir Youssefi,1, ∗ Itay Shomroni,1, ∗ Yash J. Joshi,1, 2 Nathan Bernier,1

Anton Lukashchuk,1 Philipp Uhrich,1 Liu Qiu,1 and Tobias J. Kippenberg1, †

1Institute of Physics, Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland2Indian Institute of Science Education and Research, Pune 411008, India

Encoding information onto optical fields us-ing electro-optical modulation is the backboneof modern telecommunication networks, offeringvast bandwidth and low-loss transport via opticalfibers [1]. For these reasons, optical fibers are alsoreplacing electrical cables for short range commu-nications within data centers [2]. Compared toelectrical coaxial cables, optical fibers also intro-duce two orders of magnitude smaller heat loadfrom room to milli-Kelvin temperatures, mak-ing optical interconnects based on electro-opticalmodulation an attractive candidate for interfac-ing superconducting quantum circuits [3–5] andhybrid superconducting devices [6]. Yet, lit-tle is known about optical modulation at cryo-genic temperatures. Here we demonstrate aproof-of-principle cryogenic electro-optical inter-connect, showing that currently employed Ti-doped lithium niobate phase modulators [7] arecompatible with operation down to 800 mK—below the typical operation temperature of con-ventional microwave amplifiers based on highelectron mobility transistors (HEMTs) [8, 9]—and maintain their room temperature Pockelscoefficient. We utilize cryogenic electro-opticalmodulation to perform spectroscopy of a super-conducting circuit optomechanical system, mea-suring optomechanically induced transparency(OMIT) [10–13]. In addition, we encode ther-momechanical sidebands from the microwave do-main onto an optical signal processed at roomtemperature. Although the currently achievednoise figure is significantly higher than that of atypical HEMT, substantial noise reduction shouldbe attainable by harnessing recent advances inintegrated modulators [14–17], by increasing themodulator length, or by using materials with ahigher electro-optic coefficient [18, 19], leading tonoise levels on par with HEMTs. Our work high-lights the potential of electro-optical modulatorsfor massively parallel readout for emerging quan-tum computing [3–6] or cryogenic classical com-puting [20] platforms.

Optical modulators are ubiquitous in our informationsociety and encode electrical signals in optical carriers

∗ These authors contributed equally.† tobias.kippenberg@epfl.ch

300K

3K

Ti+LiNbO3

Au

15mK

0.8K

3K

50K

Att.

DUT

HEMT

Pre-Amp.

300K fMW

Att.

DUT

PM

Pre-Amp.

fopt+fMWfopt−fMWfMW fMW

a b

c d

Coaxial cablePheat= 1mW

Optical fiberPheat= 10μW

e

Au pad

Coplanarwaveguide

LiNbO3 Chip

Ferrule

Fiber

Glue

MW port

FIG. 1. Principle of a cryogenic electro-optical intercon-nect for readout of superconducting devices. a, Simplifiedschematic of a conventional readout of a device under test (DUT)in a dilution fridge using a cryogenic HEMT amplifier. The dashedbox indicates an optional quantum-limited pre-amplifier not usedin this work. The devices are interrogated by input microwavesignals that are attenuated to reduce thermal noise, and ampli-fied using an HEMT amplifier at 3 K. b, Principle of a cryogenicelectro-optic readout scheme using an electro-optical phase modu-lator. The DUT is interrogated using the same microwave inputline, but the microwave signals are converted to the optical do-main at 3 K, reducing thermal load. c, Conducted heat througha typical cryogenic coaxial cable and optical fiber, between roomtemperature and 3 K. d, Schematic cross-section of a z-cut LiNbO3

electro-optic phase modulator. e, Microscope photo of the commer-cial phase modulator used in the experiment, showing the couplingregion between fiber and LiNbO3 chip.

that can be transported over fiber. Initially only usedfor long-haul communications, optical fiber links are nowalso replacing electrical cables for short range communi-

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11

Sep

2020

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cations within data centers [21–23]. This is motivated bythe high power consumption of electrical interconnectsthat spurred the development of optical interconnectsbased on silicon photonics [22]. Such interconnects mayalso be used in the future for on-board chip-to-chip com-munication [24, 25].

A similar challenge is foreseeable in superconductingquantum circuits, where recent advances [3–5] have high-lighted the potential associated with scaling supercon-ducting qubit technology [26]. Currently, significant ef-forts are underway to scale the number of qubits [27]. Asa result, one of the challenges that future progress in su-perconducting circuits will face is to massively increasethe number of microwave control and readout lines whilepreserving the base temperature and protecting qubitsfrom thermal noise.

Figure 1a shows a prototypical measurement chain ofa single superconducting device-under-test (DUT) thatoperates at the 15 mK stage of a dilution refrigerator.Coaxial cables are used to transmit output signals to theroom temperature as well as to send control signals tothe cold stages of the fridge. To read out GHz microwavesignals, a high electron mobility transistor (HEMT) am-plifier with low-added-noise [nadd ∼ 10 quanta/(s ·Hz)]is typically employed that operates at the 3 K stage andamplifies the DUT output signal for further processingoutside the cryostat. Although HEMTs are not quantum-limited [8, 9], the development of Josephson junction-based pre-amplifiers [28–31] that operate at the 15 mKstage have allowed near-quantum-limited microwave am-plification.

The presence of coaxial cables introduces additionalheat load from room temperature into the cold stagesof the refrigerator, which poses significant barrier tothe scalability of such systems [27]. In contrast, opti-cal fibers have superior thermal insulation, reducing theheat load per line by two orders of magnitude (Fig. 1c).Optical fibers additionally exhibit ultralow signal losses,∼0.2 dB/km, compared to ∼3 dB/m at GHz frequenciesfor coaxial lines (compensated by the HEMT amplifica-tion). Note also that thermal noise is completely negligi-ble at optical frequencies. Optical fibers could thereforeprovide a solution to scaling the number of lines withoutthe concomitant heating. For this approach, a criticalcomponent are transducers that convert input microwavesignals to the optical domain, which are compatible withlow temperature operation and are sufficiently efficient toensure low noise conversion of microwave to optical sig-nals. Indeed, substantial efforts are underway to createquantum-coherent interfaces between the microwave andoptical domains [32]. To date, quantum coherent conver-sion schemes based on piezo-electromechanical [33, 34],magneto-optical [35], and optomechanical [36–40] cou-pling have been developed. In addition, schemes basedon cavity electro-optics [41] have been demonstrated us-ing bulk [42–44] or integrated [45–47] microwave cavi-ties coupled via the Pockels effect to an optical cavitymode. Yet, all these schemes have in common that they

transduce narrowband microwave signals to the opticaldomain. While this ability is critical for future quan-tum networks, an optical replacement for the currentlyemployed HEMT amplifiers may be required for scalingcontrol lines. One route is therefore to use broadbandoptical modulators as already used today in telecommu-nication networks. While this approach may yield lowerconversion efficiency compared to systems employing nar-rowband resonant cavities, continued improvements indesign, and new material systems, can render it compet-itive, especially given its relative simplicity.

Here we explore this potential and replace the HEMTamplifier with a LiNbO3-based optical phase modulator(PM), the workhorse of modulator technology, in orderto directly transduce the DUT microwave output signalonto sidebands around the optical carrier field (Fig. 1b),detectable using standard homodyne or heterodyne de-tection schemes at ambient temperatures. To illustratethe principle of the readout, we consider the operatingprinciple of a PM. Optical PMs are based on the Pockelseffect (Fig. 1d) and induce a phase shift on the input op-tical field Ein(t), proportional to the voltage V (t) appliedon the input microwave port of the device,

Eout(t) = Ein(t)e−iπV (t)/Vπ ≈ Ein(t)[1− iπV (t)/Vπ],(1)

where the half-wave voltage Vπ is the voltage at which thephase shift is π, and typical V (t) Vπ is assumed. The

relation between microwave (field operator b) and optical(field operator a) photon flux spectral densities [48], Sbband Saa respectively, can be written as (see Appendix)

Saa[ωopt ± ωMW] = G× (Sbb[ωMW] + nadd) (2)

where ωMW and ωopt are the microwave signal and op-tical carrier frequencies, nadd is the added noise of thetransducer (referred to the input), and the transductiongain G is the number of transduced optical photons permicrowave input photon, given by (see Appendix):

G = PoptωMW

ωopt

π2Z0

2V 2π

(3)

where Popt is the power of the optical carrier at the out-put of the PM, and Z0 its input microwave impedance.In this experiment, we employ a commercial (ThorlabsLN65S-FC), z-cut traveling wave Ti-doped LiNbO3 PMwith specified bandwidth of 10 GHz and Vπ = 7.5 V at10 GHz (Fig. 1e). We use a 1555 nm fiber laser as theoptical source. The typical incident optical power onthe PM is 15 mW. The optical transmission of the PMwas reduced during the first cooldown, and measured at23%. Additional details on the cryogenic optical setupare given in the Appendix.

Previous works investigated the temperature depen-dence of the electro-optic coefficient and refractive indexof congruent LiNbO3 at low frequencies down to 7 K [49].Commercial x-cut LN modulators were also tested downto 10 K, showing a slight change in Vπ from its room tem-perature value [50, 51]. Ref. [52] discusses the behavior

3

of LiNbO3 modulators with superconducting electrodesdown to 4 K.

To date, however, such modulators have not been usedin a dilution refrigerator to directly read out a supercon-ducting device.

Characterization

To characterize the electro-optic behavior of the de-vice at cryogenic temperatures, we mount the PM on the800 mK flange of the dilution fridge. We directly drivethe microwave port of the PM at frequency ωMW usinga microwave source outside the fridge, generating side-bands around the optical carrier frequency (Fig. 2a). Thehalf-wave voltage Vπ is determined from the modulationdepth MD, defined as the ratio of the power in one ofsidebands to the power in the carrier,

Vπ = π

√Z0PMW

2 MD, (4)

where PMW is the power at the microwave input port ofthe PM. We measure MD by beating the output opticalsignal with a local oscillator (LO) with frequency ωopt +ωMW/2 + δ, generating two closely-spaced beatnotes atωMW/2 ± δ, due to the carrier and the high-frequencysideband. Using Eq. (4) we extract Vπ by sweeping themicrowave power and measuring MD. Figure 2c showsVπ at 5 GHz monitored as the fridge is cooled down fromroom temperature to 800 mK, and Fig. 2b shows Vπ atdifferent frequencies at 800 mK. Importantly, Vπ does notchange substantially from the room temperature value.

To investigate the effect of heating caused by opticaldissipation in the PM, we measured the steady state tem-perature of different flanges of the dilution fridge whenthe PM is mounted on the 800 mK flange. The resultsare shown in Fig. 2d. In the Appendix, by comparing toa calibrated heater, we show that these temperature in-creases can be attributed to optical power loss within thePM package (and not, e.g, light leakage into the fridgevolume). This allows quantitative comparison with, e.g.,heat dissipation due to a HEMT, and suggests reducedheat load in high optical transmission devices.

To further assess the performance of the PM at800 mK, we also performed a basic telecommunicationexperiment shown in Fig. 2e. An arbitrary waveformgenerator (AWG) directly drives the PM with a pseudo-random bit sequence at a rate of 5 GBaud. We beat theoptical phase-modulated carrier output with its referencearm, effectively forming a Mach-Zehnder interferometerwhose average transmission is tuned to the quadraturepoint by adjusting the laser frequency, and detect theelectrical signal on the oscilloscope. Figure 2f shows aneye diagram obtained from 8×105 samples. The open eyediagram features no error bits, hence the upper boundon bit error ratio is limited by total amount of mea-sured samples and can be estimated to be 5× 10−5 with95% confidence level [53]. These measurements clearly

demonstrate that the cryogenic modulator still functionsat 800 mK.

Optical Readout of Coherent MicrowaveSpectroscopy

Having established the cryogenic modulation proper-ties, we next carry out a cryogenic interconnect exper-iment, where the microwave output of a DUT is readout optically. As an example system we employ a su-perconducting electromechanical device in the form ofa mechanically-compliant vacuum gap capacitor para-metrically coupled to a superconducting microwave res-onator (Fig. 3a–e). These devices have been employedin a range of quantum electromechanical experiments,such as cooling the mechanical resonator to its quantumground state [54], strong coupling between mechanicaland microwave modes [13], squeezing of mechanical mo-tion [55], and demonstration of the quantum entangle-ment in the mechanical motion [56, 57], as well as imple-menting mechanically mediated tunable microwave non-reciprocity [58] and quantum reservoir engineering [59].The microwave resonance (frequency ωc ' 2π × 8.2 GHzand linewidth κ ' 2π×3 MHz) is coupled to the mechani-cal resonance (frequency Ωm ' 2π×6 MHz and linewidthΓm ' 2π × 10 Hz) of the capacitor via electromechanicalcoupling [60] (Fig. 3d). The electromechanical couplingrate is g = g0

√ncav, where g0 ' 2π × 150 Hz is inde-

pendently characterized [61] and ncav is intracavity mi-crowave photon number, proportional to the microwavepump power. The system is inductively coupled to a mi-crowave feed-line, enabling us to pump and read out themicrowave mode in reflection.

To demonstrate the electro-optical readout technique,we perform two-tone spectroscopy and measure optome-chanically induced transparency (OMIT) [10–12] (Fig. 3i)on the electromechanical sample, by applying a mi-crowave pump tone on the lower motional sideband (red-detuned by Ωm from the cavity resonance) and sweepinga second probe tone across the resonance. The strongpump damps the mechanical motion, resulting in a widereffective mechanical linewidth, Γeff = Γm + 4g2/κ. Themicrowave pump modifies the cavity response due to theelectromechanical coupling, resulting in a transparencywindow of width Γeff that appears on resonance, whichwe observe by the probe (Fig. 3g). We performed anOMIT experiment for different pump powers and ob-served the mechanical resonance via the transparencyfeature. In order to electro-optically read out the coher-ent response, the optical output is detected in a balancedheterodyne detector, using a frequency-shifted local os-cillator (Fig. 3g). Note that this scheme allows resolv-ing spectroscopic features finer than the laser linewidth(Fig. 3h). To compare the optical and HEMT readouts,the reflected signal is split and measured simultaneouslyusing both techniques (Fig. 3a). Figure 3f shows theOMIT results, with excellent agreement between the op-

4

0 2 4 60.013

0.014

0.015

0.80

0.82

0.84

2.83

2.84

2.85

Volta

ge (a

rb.u

.)

Time (ps)

Temperature (K)

800 mK

LOsetup190

THz

ω

190 THz

ω

190 THz

ω/2 + δ

MD

1555nm

Laser ESA

Frequency (GHz)

AWG

800 mK

OSC

a

b

c d

e f

Tem

pera

ture

(K)

15 mK flange

3 K flange

0.8 K flange

Optical input power to PM (mW)

FIG. 2. Cryogenic characterization of a LiNbO3 phase modulator. a, Experimental setup for low temperature characterizationof the phase modulator. b, plot of Vπ vs. frequency at 800 mK. c, Characterization of Vπ at 5 GHz vs. temperature from room temperatureto 800 mK. d, Measurement of heating due to optical dissipation when the phase modulator is mounted on the 800 mK flange. Plot ofthe steady state temperature vs. input laser power of the 3 K, 800 mK, and 15 mK flanges. e, Experimental setup for phase shift keyingdetection. RF signal from waveform generator is directly applied on a phase modulator. After homodyne detection the electrical signal isrecorded on a fast oscilloscope. f, Eye-diagram of an optical signal phase-modulated at a rate of 5 GBaud, the bit error ratio is 5× 10−5.

tical and HEMT readouts. At high pump powers, wheng ∼ κ, we observe mode splitting as a result of strongcoupling and mode-hybridization between the mechani-cal and microwave modes [13].

Optical Readout of an Incoherent MicrowaveSpectrum

Next, we employ our scheme to directly read out op-tically the power spectral density of a microwave signalemitted by the DUT. For this, we drive the mechanicaloscillator into self-oscillation by pumping the system onits upper motional sideband, ωpump = ωc + Ωm, induc-ing a parametric instability [60, 62–64]. The output mi-crowave spectrum features strong sidebands around themicrowave pump, at integer multiples of the mechanicalfrequency (Fig. 4a,b). Figure 4c shows these mechanicalsignals obtained simultaneously using both our opticalreadout and the HEMT amplifier. We use the knownproperties of the HEMT to estimate the transductiongain G [Eq. (2)] of our optical readout. The blue trace inFig. 4c shows the HEMT output referred back to its inputusing its known added noise, nHEMT

add ' 8 quanta/(s ·Hz),characterized independently. This calibration yields theHEMT input signal S, which is equal to the PM mi-crowave input. The noise in the optically detected spec-

trum, referred to the optical output of the PM, is domi-nated by the optical shot noise, 1 quanta/(s ·Hz), for ourG 1. In this calibration, we can obtain G from theoptical spectrum containing the transduced microwavesignal GS. The orange trace in Fig. 4c shows the opti-cal noise spectrum referred to the microwave input, andFig. 4d shows a zoom-in of a single sideband. In this cali-bration, the signal areas in both measurements are equalto S. Further explanation of this calibration is given inthe appendix. This yields G = 0.9×10−7, in good agree-ment with the theoretical value Gtheory = 3.5 × 10−7

obtained from Eq. (3) using the measured output opticalpower, Popt = 1.1 mW (optical efficiency of 5%, includ-ing losses in fiber connectors and heterodyne detectionsetup). We note that the frequency widening of the op-tically detected sidebands, observed in Fig. 4d, is due tofluctuations in the LO frequency, caused by the limitedbandwidth of the locking setup in conjunction with us-ing a minimal resolution bandwidth (RBW) of 1 Hz in thespectrum measurement. The integrated sideband power,however, is conserved. Improving the LO locking setupcan reduce this effect.

The added noise in the transduction process is (seeAppendix)

nadd =1

2G+ nMW

th +1

2, (5)

5

a b c

ed

f g

h i

100 μm

10 μm

DUT

ωcΩm

g

6 MHz 8 GHz

Гm in

out

κ

ℏωc

Prob

e

Pump

|np, nm⟩

|np+1, nm⟩

|np, nm+1⟩

ℏΩm

ωcωc - Ωm8 GHz

ωc - Ωm

Carrier

190 THz

HeterodynePump

Probe

δ

Cavity

LO

δ

20 dBDUT1 2

ΔΣ

6dB

20dB PM

HEMT Lockbox

Local oscillator

1555nm Laser

ESA

ESA

Heterodyne

15mK

0.8K

3K

50K

300K

3dB splitter

S 11 (d

B)

( − )/2 (kHz) ω ω π c

−303

−303

−303

−303

−303

−600 −400 −200 0 200 400 600

−303

-20 dBm

0 dBm

5 dBm

10 dBm

15 dBm

20 dBm

−10 −5 0 5 10

−3

0

3

( − )/2 (kHz) ω ω π c

S 11 (d

B)

FIG. 3. Electro-optic readout of a coherent microwave spectrum of a superconducting electromechanical system. a,Experimental setup. Left: dilution fridge, right: optical setup. b, Electromechanical system used as a DUT. c, optical micrograph of theLC resonator. d, Modal diagram of the electromechanical system. e, scanning electron micrograph of the mechanically compliant capacitor.f, coherent measurement of the electromechanical resonance for increasing microwave pump powers of (−20, 0, 5, 10, 15, 20) dBm at thesource, from bottom to top respectively. The probe power is −20 dBm at the source. By increasing the pump power, the optomechanicallyinduced transparency window emerges, and at stronger pump powers the modes get strongly coupled, leading to an avoided crossing effect.Blue lines correspond to HEMT readout and orange dots to optical readout. g, the frequency scheme for microwave tones, optical tones,and measured signal after heterodyning. h, high resolution measurement of the transparency window highlighted in f with the gray box.i, level scheme of the optomechanical system. The pump tone is tuned close to red sideband transitions, in which a mechanical excitationquantum is annihilated (mechanical occupation nm → nm − 1) when a photon is added to the cavity (optical occupation np → np + 1),therefore coupling the corresponding energy eigenstates. The probe tone probes reflection in which the mechanical oscillator occupationis unchanged. The pump tone modifies the response of the cavity and creates a transparency window appears on resonance (OMIT).

where nMWth is the average occupation of the thermal

photonic bath due to the microwave fields. This givesnadd ≈ 6 × 106. The noise floor of the optical measure-ment in Fig. 4c is 60 dB above the HEMT readout. Thisis due to the very small gain G ∼ 10−7, caused by thelarge Vπ and the limited optical power. However, there ismuch room for improvement in these parameters. Ref. 18reported a Vπ-length product of 0.45 V cm in a BaTiO3-based modulator, thus Vπ ∼ 50 mV can be realized in a∼ 10 cm device, possibly using low-loss superconductingelectrodes [47, 52]. The optical power can be increasedarbitrarily in principle, however one needs to consideroptical losses (mainly at the fiber-to-chip interfaces) thatlead to heating. Considering a device with an improvedoptical transmission of 66% [65] and incident power of15 mW, yields Popt ∼ 10 mW. This scenario achieves

G ∼ 5 × 10−2 [Eq. (3)], with nadd ≈ 20 at 3 K, com-petitive with HEMT performance, while the heat load of5 mW is half that of a typical cryogenic HEMT.

It is worth mentioning that many experiments utilizea near-quantum-limited pre-amplifier at the 15 mK stage(Fig. 1a,b). In this case, the noise added in the secondamplification stage, referred to the input, is ∼(GPAG)−1

(see Appendix), where GPA ∼ 103 is the pre-amplifiergain [28–31]. Thus, G & G−1

PA suffices to preserve near-quantum-limited amplification (See Appendix).

Conclusions

We have demonstrated the viability of LiNbO3 devices,currently-employed in the telecommunication market, as

6

(ω-ωpump)/2 (MHz) (ω-ωpump-Ωm)/2 (kHz)

Ωm/2

ωpump

Ωm

8 GHz

Carrier

190 THz

Pump

LOωpump

Pum

p

|np, nm⟩

|np+1, nm⟩

|np, nm+1⟩

(

) (p

hoto

ns/(

s H

z))

Sω

aa

−20 −10 0 10 20

100

105

1010

−1 0 1

a b

c d

FIG. 4. Electro-optic readout of an incoherent microwavespectrum of a superconducting electromechanical system.a, Frequency-domain picture: a microwave tone pumps the elec-tromechanical system on the upper motional sideband, inducing aparametric instability and generating mechanical sidebands equallyspaced around the tone by the mechanical resonance frequency Ωm.The phase modulator transfers the microwave spectrum on the opti-cal signal, which is subsequently mixed with a local oscillator (LO)and detected via heterodyne detection. b, Level scheme showingelectromechanically induced parametric instability. A blue-detunedpump photon scatters into an on resonance photon and generatesa phonon in the mechanical oscillator and causes anti-damping. Atpump power above a certain threshold induces instability in themechanical oscillator. c, Measured power spectral densities of themicrowave pump (central peak) and mechanical sidebands detectedby the HEMT (blue) and optical (orange) readouts. d, Enlarge-ment of the gray-shaded area in c, showing the power spectraldensities of the on-resonance mechanical sideband.

electro-optical interconnects in cryogenic platforms usedin superconducting quantum technologies, in particularas viable alternative to HEMT amplifiers with the po-tential of reduced heat load. By interfacing a commer-cial PM to a circuit-electromechanical system that waspreviously used to perform quantum experiments, we im-plemented an electro-optical readout of this system. Inaddition, we quantified the gap between conventional mi-crowave amplifiers and the electro-optical alternative. Itis feasible that this gap be closed in the near future,by improved devices with lower Vπ, resulting in a near-quantum-limited broadband microwave-to-optical inter-connect.

Appendix: Quantum mechanical model for a phasemodulator

In the following, we derive a simple quantum descrip-tion of the phase modulator to establish the quantumlimits in transducing the input microwaves. The cen-tral assumption is that the linear regime stays valid, forsufficiently low input microwave powers. As such, thescattering equations linking inputs to output should be

identical in both quantum and classical cases. We canuse the known classical regime as a starting point, withthe output optical field amplitude aout expressed as afunction of the input optical field ain as

aout = e−iπV/Vπ ain ≈ (1− iπV/Vπ)ain, (A.1)

where V is the classical voltage applied at the inputand the half-wave voltage Vπ is the voltage at whichthe phase modulator applies a phase shift of π. Forthe quantum model, the classical fields are replaced bytheir quantum equivalent. The microwave input becomes

V =√~ωMWZ0(b + b†)/

√2 with b the annihilation op-

erator for the microwave field at frequency ωMW travel-ing on a transmission line of impedance Z0. The opticalinput is ain = αe−iωoptt + δain, where α is the ampli-tude of the coherent carrier field of frequency ωopt, with|α|2 = Popt/~ωopt, and δain carries the quantum fluctua-tions of the input optical field. Inserting the expressionsin Eq. (A.1), we can compute δaout = aout−αe−iωoptt, thequantum fluctuations of the output optical field, given by

δaout = δain − i√Ge−iωoptt(b+ b†) (A.2)

with the transduction gain G given by Eq. (3).To understand the implications of Eq. (A.2) for the

quantum noise in the transduction, we compute thepower spectral density of the output optical field,

Soutδa†δa[ωopt + ωMW] =S in

δa†δa[ωopt + ωMW]

+G(Sb†b[ωMW] + Sbb† [−ωMW]).(A.3)

The first term corresponds to the added quantum noisedue to the input optical field. The second term con-tains contributions from the microwave frequency ωMW,including both the signal and noise. The third term con-tains added microwave noise at frequency −ωMW, com-posed of thermal and quantum noise components, respec-tively nMW

th + 1/2. Thus Eq. (??) can be simplified to

Soutδa†δa[ωopt +ωMW] = GSb†b[ωMW] +G

(nMW

th +1

2

)+

1

2.

(A.4)We emphasize two limiting cases. When G 1, as in ourexperiment, the added noise is dominated by the inputoptical quantum noise, the last term in Eq. (A.4). In theopposite limit, G 1, the added noise is dominated bythe microwave input noise, and the signal-to-noise ratiois independent of G. In any case, the added noise referredto the input is given by Eq. (5).

Appendix: Calibration of the transduction gain.

Figure 5 illustrates the procedure of experimentallycharacterizing the transduction gain of our electro-optictransducer. The microwave signal is split equally intotwo parts S, fed to the HEMT amplifier and the PM re-spectively (Fig. 5a). The HEMT added noise is charac-terized independently to be nHEMT

add = 8 quanta/(s ·Hz).

7

DUT

HEMT

G

nadd

EO transducer

GHEMT

12

qsu⋅aHnt

za

S GHEMT(S+ )

GS+G(nth +½)+½+½

S

q su ⋅a Hnt zaPS

D (

)

frequencynadd

HEMT

1GSq su ⋅a Hnt za

PSD

(

)

frequency

naddHEMT

naddHEMT

MWS

Heterodynedetection

a b

c

FIG. 5. Illustration of the gain characterization proce-dure. a, Propagation of the DUT signal through the system. b,Power spectral density of the HEMT output. c, Power spectraldensity of the optical heterodyne detector.

DUT

Pre-Amp.

GPA G

naddPA nadd

EO transducer

FIG. 6. Schematic signal flow when pre-amplifier is used.

We infer S from the spectrum of the HEMT amplified sig-nal, referring the noise floor to nHEMT

add (Fig. 5b). In thePM branch, the added noise of the transduction is givenby Eq. (5). The spectrum is detected using a balancedheterodyne detector, which adds 1/2 quanta/(s ·Hz) ofnoise (Fig. 5c). We can safely neglect G(nMW +1/2) andconsider the noise floor of the spectrum referred to theinput of the heterodyne detector, i.e. 1 quanta/(s ·Hz).This allows us to calculate GS, and finally obtain G withknowledge of S from the HEMT measurement.

When using a quantum-limited pre-amplifiers beforethe electro-optical transducer (not done in our exper-iment), we can model the readout chain as shown inFig. 6b. The total added noise of the readout chain is

ntotaladd = nPA

add +nadd

GPA' nPA

add +1

2GPAG(A.1)

Therefore when G ' 1/GPA, the total added noise willbe dominated by nPA

add ∼ 1 quanta/(s ·Hz) [28–31] andthe readout will be near-quantum-limited.

Appendix: Experimental details and heatingmeasurements.

We use a fiber-coupled lithium niobate PM from Thor-labs, model LN65S, used as-is with no modifications.Note that the minimum specified operating tempera-ture is 0 C. The device sustained several cooldown-warmup cycles with reversible behavior in its opticaltransmission. We measured 25% reduction in the op-tical transmission at cryogenic relative to room temper-ature. The PM metallic box was tightly clamped to theflange of the 800 mK or 3 K stage. We use a BlueforsLD-250 dilution refrigerator. The approxmiate availablecooling powers of the 15 mK, 800 mK, 3 K stages are12µW, 30 mW, 300 mW.

Figure 2c shows the variation of Vπ from room tem-perature to 800 mK, obtained during a cooldown of thedilution fridge and measured using the default thermome-ter on the 800 mK flange, located next to the heat ex-changer, about 10 cm from the PM. In order to rule outpossible temperature gradients, we mounted a calibratedthermometer next to the PM and monitored both ther-mometer readings during cooldown. Figure 7a shows themeasured relative temperature difference, which is lessthan ∼ 5% throughout the cooldown. Note that this ex-cludes pulse precooling and mixture condensation periodwhen the temperature is unstable (shown for complete-ness in Fig. 7a).

Figure 2d shows the temperature increase of the15 mK, 800 mK, and 3 mK stages of the dilution fridge asa function of the optical power incident on the PM, whichis mounted on the 800 mK stage. We performed a simplemeasurement to verify that this temperature increase canbe ascribed to light absorbed in the PM body (and not,e.g, light leakage into the fridge volume), correspondingto the optical transmission (insertion loss) of the PM. Weused the calibrated 120 Ω still heater built in the 800 mKstage to apply heat directly, we then repeated the mea-surement using optical input to the PM as the heatingsource (as in Fig. 2d). Figure 7a,b compares the resultsof this measurement, showing temperature increase inthe 3 K and 800 mK stages (the latter recorded with thetwo separated thermometers) vs. dissipated power. Inthe case of optical heating, the dissipated power is com-puted directly from the incident power on the PM andits measured transmission of 23%.

Figure 7b,c shows the result of this measurement.The optical heating shows a temperature increase of13.3 mK/mW (6.5 mK/mW) at the 800 mK (3 K) stage(Fig.7a), while the resistive heating shows a temperatureincrease of 14.1 mK/mW (8.3 mK/mW) at the 800 mK(3 K) stage (Fig.7b). Thus heating due to operation ofthe electro-optical interconnect is very similar to local-ized, resistive heating.

Data availability statement

The code and data used to produce the plots withinthis paper will be available at a Zenodo open-accessrepository. All other data used in this study are availablefrom the corresponding authors upon reasonable request.

ACKNOWLEDGMENTS

We thank Nils J. Engelsen for thorough reading ofthe manuscript. This work was supported by the Eu-ropean Union’s Horizon 2020 research and innovationprogramme under grant agreement No. 732894 (FETProactive HOT), and from the European Research Coun-cil (ERC) under the European Unions Horizon 2020 re-search and innovation programme (grant agreement No.

8

Optical lost power in PM (mW)

Tem

pera

ture

(K)

Resistive dissipated heat (mW)

a b c

0 1 2 3 4 50.78

0.80

0.82

0.84

13.3 mK/mWAt the heat exchangerAt the PM

Temperature (K)

ΔT/T

2.82

2.83

2.84

2.85

2.86

6.5 mK/mWAt the 3K flange

0 1 2 3 4

14.1 mK/mWAt the heat exchangerAt the PM

8.3 mK/mWAt the 3K flange

0 20 40 60 80

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

0.10

FIG. 7. Heat dissipation and temperature gradients. a, Relative temperature difference between PM box and heat exchanger,on 800 mK flange during a cooldown. The gray datapoints correpsond to specific periods of pulse precooling and mixture condensation,where the temperature is unstable. b, Measurement of heating due to optical dissipation when the phase modulator is mounted on the800 mK flange. c, Measurement of heating using a calibrated resistive heater mounted on the 800 mK flange.

835329). This work was supported by funding from theSwiss National Science Foundation under grant agree-ment NCCR-QSIT: 51NF40 185902 and Sinergia grant

no. 186364 (QuantEOM). The circuit electro-mechanicaldevice was fabricated in the Center of MicroNanoTech-nology (CMi) at EPFL.

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