Fabrication and characterization of aluminum SQUID transmission lines

Luca Planat,1 Ekaterina Al-Tavil,1 Javier Puertas Martínez,1 Rémy Dassonneville,1

Farshad Foroughi,1 Sébastien Léger,1 Karthik Bharadwaj,1 Jovian Delaforce,1 VladimirMilchakov,1 Cécile Naud,1 Olivier Buisson,1 Wiebke Hasch-Guichard,1 and Nicolas Roch1

1Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France∗

We report on the fabrication and characterization of 50 Ω, flux-tunable, low-loss, SQUID-basedtransmission lines. The fabrication process relies on the deposition of a thin dielectric layer (few tensof nanometers) via Atomic Layer Deposition (ALD) on top of a SQUID array, the whole structure isthe covered by a non-superconducting metallic top ground plane. We present experimental resultsfrom five different samples. We systematically characterize their microscopic parameters by measur-ing the propagating phase in these structures. We also investigate losses and discriminate conductorfrom dielectric losses. This fabrication method offers several advantages. First, the SQUID arrayfabrication does not rely on a Niobium tri-layer process but on a simpler double angle evaporationtechnique. Second, ALD provides high quality dielectric leading to low-loss devices. Further, theSQUID array fabrication is based on a standard, all-aluminum process, allowing direct integrationwith superconducting qubits. Moreover, our devices are in-situ flux tunable, allowing mitigationof incertitude inherent to any fabrication process. Finally, the unit cell being a single SQUID (noextra ground capacitance is needed), it is straightforward to modulate the size of the unit cell pe-riodically, allowing band-engineering. This fabrication process can be directly applied to travelingwave parametric amplifiers.

I. INTRODUCTION

Being able to reproduce the rich physics of non-linearfiber optics [1] in the microwave domain would be amajor milestone in microwave physics, since the non-linearities at stake in this frequency range are ordersof magnitude larger than in the optical domain. Non-linear optical fibers have brought fiber amplifiers, a keytechnology for communication systems but are also veryappealing to quantum optics since their dispersion canbe tailored to create photonic crystals, allowing the in-vestigation of phenomena such as frequency translationof single photons [2]. In the microwave domain, elec-trical signals propagate in transmission lines [3]. Thequantum nature of these microwave photons can usuallybe discarded, unless the circuits in which they propa-gate are cooled down to very low temperatures (below100 mK). Under these conditions, we speak about cir-cuit Quantum Electrodynamics (cQED) [4]. While inthe optical domain non-linearity can be enhanced bydoping fibers with rare-earth materials, microwave su-perconducting quantum circuits can be made stronglynon-linear and low-loss by combining superconductingmaterials and Josephson junctions [5] or by taking ad-vantage of the self-non-linearity of disordered supercon-ductors [6]. This approach has been very successful andlead to the observation of strong light-matter coupling [7],resonance fluorescence with extinction as high as 94% [8]or near quantum-limited parametric amplifiers [9] since itcombines both dissipationless and very nonlinear charac-teristics. However, all these experiments rely on resonantstructures – the microwave equivalent of optical cavities.

∗ nicolas.roch@neel.cnrs.fr

So far, only few experiments used nonlinear transmis-sion lines – the microwave equivalent of nonlinear opticalfibers – in the quantum regime. One notable exceptionis the demonstration of Traveling Wave Parametric Am-plifiers based on either Josephson junctions [10, 11] ordisordered superconductors [12–15]. The small numberof experimental implementations is explained by the factthat fabricating a long, low-loss, impedance matched,nonlinear transmission line is very demanding. In thecase of a Josephson junction transmission line (JJ-TL),nonlinearity is strong [16], sparing the need for long struc-tures. The challenge here is to lower the naturally largeimpedance of Josephson junction arrays [17] compared to50 Ω – by increasing their capacitive effect to the ground,which asks for very concentrated capacitors – while main-taining sufficient low losses. Combining all these require-ments was demonstrated using a complex niobium tri-layer fabrication process [18]. Disordered superconductortransmission lines, on the other hand, are mainly limitedby their relatively weak nonlinearity. Obtaining sizablenonlinear quantum effects requires meter-long structures,which are strongly prone to fabrication defects [13, 19].

In this work, we present a SQUID transmission line(S-TL) fabricated using a simple, aluminium-based pro-cess. Impedance matching is obtained via a top groundplane separated from the SQUID array by a very thinalumina layer. These S-TL show losses on-par with previ-ously reported values [10] and a characteristic impedanceclose to 50 Ω. This impedance can be adjusted in-situ,owing to the flux tunability of the structure [20]. Fur-thermore, thanks to the simple architecture of our S-TL,the impedance of each unit cell can be tailored at willto create photonic-crystal-like transmission lines [21, 22].In this article two types of devices, based on the exactsame fabrication process, are presented: 50 Ω matchedS-TL and resonant structures made of shorter S-TL. The

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FIG. 1. Fabrication flow for SQUID-based transmis-sion lines and resonators. Step 1: fabrication of longSQUID arrays using electron-beam lithography and doubleangle evaporation of aluminium. Step 2: Deposition of a con-formal alumina layer via Atomic Layer Deposition (ALD).Step 3: Evaporation of a thick metallic layer (gold or cop-per) acting as an electrical ground. This layer is patternedusing a combination of electron-beam lithography and lift-of.(Inset:)

latter are used as test structures to characterise our fab-rication process in the single microwave photon regime.

This article is organized as follows. In Section II weintroduce the fabrication flow and the microwave designof the S-TL. Section III presents the low temperaturemicrowave properties of five different devices. Section IVand Section V focus on the microscopic origins of S-TLlosses and their strong power dependance respectively.In the last section magnetic flux response of the S-TL ispresented.

II. FABRICATION PROCESS

In this section we detail the fabrication process ofSQUID-based transmission lines. Four different batcheswere obtained using this recipe. Five different deviceswere then characterized at very low temperatures, asreported in Table I. Devices are fabricated on high-resistivity silicon substrates (thickness 275 µm).

The back side of the silicon wafer is covered by a thinlayer of titanium (10 nm, for adhesive purpose) and athick layer of gold (200 nm) to ensure good thermal andelectrical contact to the sample holder. The fabrica-tion process relies on three simple steps summarised inFig. 1. First a SQUID array is fabricated using doubleangle evaporation of aluminium (evaporator MEB550Sfrom Plassys), separated by an in-situ oxidation to growthe tunnel barrier. For batch 1, 2 and 3, the oxida-

100μm 100μm

100μm120μm

(d)

(b)

(c)

(a)CPW Microstrip 31

33

FIG. 2. Pictures of samples from the first batch.Pictures a and b display SQUID-based transmission lineswhereas c and d are showing details of resonant structuresused as control samples. Every picture is labeled by a numbercorresponding to a fabrication step as shown in Fig. 1. (a)Input of the SQUID-based transmission line after the firststep. Left side of the picture: bonding pad with taperedshape. Right side of the picture: few dozens of SQUID. (b)Same structure but after step 3. The gold layer is depositedeverywhere but on the bonding pad. SQUID are still distin-guishable from below the alumina and gold layers. (c) Pictureof the resonant structure after step 3. The feed-line is visiblein the middle. On the right side: inter-digital capacitor cou-pling the feed-line to a section of SQUID-based transmissionline (623 unit cells). (d) Zoom in.

tion pressure is 4 torr, for batch 4 it is 1 torr. The re-sist mask is patterned with a 100 keV electron beamwriter (model nB5 from NanoBeam) to allow bridge freefabrication [23]. The typical size of a single Josephsonjunction is ten microns high and half a micron wide (seeAppendix A). A unit cell (two Josephson junctions plusconnecting wires in a loop) is about 3 µm wide. This tech-nique allows fabrication of low disorder arrays combiningup to 2000 unit cells [24]. However, one of the main diffi-culty in fabricating such long arrays are stitching errors,coming from a wrong focus of the e-beam due to unavoid-able tilt in the substrate. To overcome them, we use thefocus map feature of our electron beam writer, allow-ing to readjust dynamically the focus during the writingprocess. This is achieved by fitting, prior to the writingprocess, the surface of the chip (8.1 mm × 8.1 mm) by atilted plane. The fit is done by measuring the four cor-ners’ height where gold marks were previously deposited.The array is terminated by tapered bonding pads (respec-tively coupling capacitance) as shown in Fig. 2b (resp.d). During step 2, a thin film of alumina is depositedvia Atomic Layer Deposition (ALD) using the Savannah

3

system, from Cambridge Nanotech. The sample is in-serted inside a chamber pumped down to 0.29 mbar. Wetried two different deposition temperatures, 150 C and200 C, to infer the effect of temperature on the dielec-tric quality. Thicknesses of the various films are reportedin Table I. ALD films combine low microwave losses andconformal deposition. This latter property is crucial toguarantee electrical isolation between the SQUID and thetop ground plane deposited during step 3. Access to thebonding pads is guaranteed by windows in this metalliclayer. These openings are obtained via a second lithog-raphy step and lift-off of the metallic film. We checkedthat the thickness of the alumina layer is not affectedby this subsequent step. Finally, we could easily con-tact the bonding pads through the alumina layer usingmicro-bonding given the thickness of this layer (less than40 nm).

Device A B C D EBatch 1 2 2 3 4Type TL TL Res TL TL

ALDtemperature

(C) 150 150 150 200 150Dielectric

thickness (nm) 38 38 38 28 28Ground

thickness (nm) 200 400 400 400 1000GroundMaterial Au Au Au Au Cu

JJ sizeH(µm)xW(µm)

12.0 x0.45

10.5 x0.40

10.5 x0.40

12.0 x0.45

12.0 x0.45

LJ(pH) 83 125 125 135 51CJ(fF) 490 380 380 485 485Cg(fF) 34.0 26.0 31.0 46.0 50.5

Phase velocity(106 m.s−1) 1.96 1.76 1.62 1.32 2.06

tan δ (10−3) 6.5 6.0 5.0 4.0 6.5Lc (10−4)

(m−1.Hz−12 ) 2.5 1.0 x 0.8

notmeasurable

Phase calib.Method Switch Switch N/A Thru Thru

TABLE I. Summary of the five samples presented in this ar-ticle. Four are transmission lines (TL) and one is a resonator(Res). Main fabrication characteristics, SQUID characteris-tics and loss coefficients are summarized. Calibration meth-ods are also reported

Regarding microwave engineering, we designed theSQUID arrays as a 50 Ω microstrip transmission line.Since our SQUID display a typical inductance L ≈100 pH, and a characteristic impedance Zc =

√L/Cg ≈

50 Ω is being sought, it requires a shunt capacitanceCg ≈ 40 fF. We model Cg as a planar capacitanceCg = ε0εrS/t (where ε0 is the vacuum permittivity andS the SQUID’s area). Given we are using alumina as theinsulator (dielectric constant εr ≈ 9.8 ) it translates into

a dielectric thickness of t ≈ 30 nm. These numbers leadto a transmission line with a microstrip geometry withunique features, such as wave velocity below 1% of thelight velocity while being matched to 50 Ω environment.We also designed a smooth transition from microstripgeometry to Coplanar Waveguide (CPW) geometry toallow for large bonding pads. This transition is shownin Fig. 2.b. We used a conventional tapered shape tokeep the impedance constant between the bonding padand the CPW-microstrip transition. Strictly speakingthis is not a CPW geometry since the aluminum layerand the top ground are not exactly in the same plane asthey are separated by the alumina layer. Nevertheless,electromagnetic simulations show that the electric fieldprofile is not altered, and that approximating this geom-etry as a CPW is correct. Indeed alumina and siliconhave close dielectric constants and a step of a few tenthof nanometers is negligible compared to the lateral dis-tance between the aluminum and the ground of few hun-dreds of micron. Regarding resonators, the section of theSQUID-based transmission line (length is approximately600 unit cells) is capacitively coupled to a feed-line asshown in Fig. 1 and Fig. 2.c and d.

III. CRYOGENIC MICROWAVE PROPERTIES

We now turn to the microwave characterization ofthese samples at very low temperature (T = 20 mK).Measurements were carried out using a standard cryo-genic setup. First we measure the dispersion relation(angular frequency versus wave-vector) of the samples.The experimental protocol to access such quantities isnot the same for the transmission lines and the reso-nant structures. The latter is obtained via two-tone spec-troscopy [25, 26]. The former follows the procedure ex-plained in [10]. The idea is to measure the propagatingphase φ of a microwave tone along the device under test(DUT). A proper calibration is needed to remove thecontribution of the cryogenic measurement setup. To doso, we used two calibration techniques. For samples Aand B, a cryogenic microwave switch (model R577433000from Radiall) would shunt the sample at 20 mK to infer,during the same cooldown, both the contribution of thesetup and of the DUT. For samples D and E we useda simple thru to calibrate the contribution of the setup.We placed a thru in CPW geometry instead of the actualchip containing the SQUID-based transmission line. Thistechnique requires two different cool-downs but gives abetter estimate of the setup contribution (wire bonds aresimilar to the DUT and there are no extra cables con-necting to the switch to the DUT). The last calibrationstep is to ensure that at zero frequency, this propagatingphase is zero. Indeed the phase measured with a VectorNetwork Analyser (VNA) is defined modulo 2π. We usea procedure similar to Macklin et al. [10] to adjust itproperly. The propagating phase φ can then be linked to

4

the wave-vector k using:

k(ω) =φ(ω)

L, (1)

where L is the length of the SQUID array and is per-fectly known since both the unit cell size a and the totalnumber of unit cells NJ are well known. Two measureddispersion relations are plotted in Fig. 3. Grey points inFig. 3.a were discarded from the fitting procedure sincethey are out of the operating frequency range (3 GHz to14 GHz) of our cryogenic HEMT amplifier (model LNF-LNC1-12A from Low Noise Factory).

To infer the microscopic parameters of the samples LJand Cg (see Appendix A), we fitted their dispersion re-lations using the following model [26]

ωk = ωplasma(

√1− cos ka

1− cos ka+ Cg/2CJ), (2)

where the Josephson capacitance CJ is a fixed param-eter and taken as 45 fF per micro meter square [27] andωplasma is the plasma frequency of a SQUID and readsωplasma = 1/

√LJCJ.

Mode number0 25 50 75 100

Wavevector ( )

Freq

uen

cy (G

Hz)

0.0 0.2 0.40.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

0.1 0.3

(b)(a)

SAMPLE CSAMPLE B

FIG. 3. Dispersion relations (a) Dispersion relation of aSQUID-based transmission line (sample B). Black line is a fitof the experimental data to Eq. (2). (b) Dispersion relationof a resonant structure (sample C). Data were acquired viatwo tone spectroscopy and fitted using Eq. (2) as well.

The values found for Cg throughout the differentbatches follow the correct trend: the bigger the junctionsare, the bigger Cg is; the thinner the dielectric is, thebigger Cg is. Regarding LJ, we experienced a drift frombatch 1 to batch 3 towards larger inductance. Junctionsfrom batch 4 were oxidized with a pressure of 1 torr in-stead of 4 torr in order to compensate this drift. Also,sample B and C should display the same microscopicparameters since they were fabricated within the samebatch (see Table I). The size of sample B unit cells was

modulated periodically while sample C has uniform cells.This modulation is very small (6% amplitude) and onlyopens a small photonic gap over a band of few hundredsof megahertz in the dispersion relation. This does notaffect its overall shape as shown in Fig. 2, where onlya small kink can be seen around 7 GHz. We find thesame values for the Josephson inductance but there is asmall discrepancy in the ground capacitance which mightbe due to the fitting procedure or to uniformity in thedielectric layer. Finally, we can define a low frequency(ω << ωplasma) characteristic impedance Zc =

√LJ/Cg

for both samples, leading to Zc between 65 Ω and 70 Ω.

IV. CONDUCTOR AND DIELECTRIC LOSSES

Losses of the transmission line is another very im-portant parameter. To understand loss mechanisms weprobed the calibrated transmission of S-TL, using thecalibration technique presented in the previous section.Fig. 4 shows this calibrated transmissions for samples A,B, D and E. They show losses on the order of 5 dB/cmat 6 GHz. Although these losses are significant, theyare comparable to previously reported structure basedon Josephson junctions [10]. A more thorough studycan explain the different loss mechanisms of our devices.Seen as microstrip lines, our samples suffer from two wellknown phenomena in microwave engineering: dielectricand conductor losses [28]. Within the superconductingcommunity, the latter form of losses have been rightfullyneglected since there are no significant losses inside thesuperconductors nor in thick metallic grounds (supercon-ducting or not). Such assumptions are not valid any-more in our geometry. Although the central conductoris superconducting, the top ground plane is made of anon superconducting metal (gold or copper, for the lastversion), with a finite thickness (hundreds of nanome-ters), comparable to the skin depth. This configurationcan lead to non negligible conductor losses as we will seelater. To determine the origin of losses, we fit the cali-brated transmission of the S-TL with a simple model [3],where αc and αd represent the conductor and dielectriclosses respectively. The total attenuation of the line canthen be expressed as

A = e(αc+αd)L, (3)

Considering our microstrip geometry with very largeshunt capacitance, it is safe to assume that the electricalfield is mainly confined within the top alumina substrate.Thus, we model the dielectric loss as αd as [3]:

αd =k tan δ

2(4)

We thus have a direct relation between the angular fre-quency ωk of the signal and the dielectric loss througha free parameter tan δ. In microwave engineering, it isdefined as the ratio of the imaginary part over the realpart of the complex permittivity of the medium.

5

Regarding αc, as mentioned before, we will only takeinto account top ground loss.

Given the microstrip line geometry of our system, weconsider [29]:

αc =R

Z0, (5)

where Z0 is the characteristic impedance of the line, andR is the radio resistance per unit length. R is given by

R =Rs

µ0

δl

δn, (6)

where Rs =√ωµ0/2σ is the skin resistance of the

conductor, µ0 the vacuum permeability and the ratioδl/δn characterises the change of inductance per unitlength δl when the conductor walls recede by a distanceδn. In a metal the current flows within the skin depthδs =

√2/ωµ0σ. If the conductor is much thicker than δs,

the change of inductance is negligible. This amounts toconsidering the ratio Lδl/µ0δn in the order of unity.

On the other hand if the conductor thickness is com-parable or smaller than δs, then the change of δl is large,the ratio is much larger than one and conductor loss in-creases. Obtaining a quantitative value of Lδl/µ0δn re-quires 3D electromagnetic simulations. It is beyond thescope of this paper. However this formula gives the cor-rect order of magnitude for αc when the top ground planeis thicker than δs

SAMPLE A

4 6 83 5 7 4 6 83 5 76

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3

2

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7

Frequency (GHz)

6

4

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2

5

7SAMPLE B

SAMPLE D SAMPLE E

(d)

(b)

(c)

(a)

|S21|

(dB)

LC= 0LC= 1.0x10-4LC= 2.5x10-4

LC= 0.8x10-4

LC= 2.5x10-4

LC= 0

LC= 1.0x10-4

LC= 2.5x10-4

LC= 0

LC= 2.5x10-4

LC= 1.0x10-4

LC= 0

FIG. 4. Calibrated, low temperature transmissionof S-TL. Solid (dashed) lines are experimental data (fitsto Eq. (3)). Values of fitting parameters are reported in Ta-ble I. Shaded blue areas represent 5.10−4 incertitude on tan δ.Black dotted and grey dash-dotted lines are obtained usingEq. (3) with the same tan δ but with different conductor loss(values indicated in each panel in m−1Hz−

12 ). All data were

taken with input power Pin = −106 dBm± 3dBm except forsample A where Pin = −101dBm± 3dBm.

Another important feature is that αc should scale asthe square root of the frequency. Then we model αc as

αc = Lc√f, (7)

where f is the signal frequency and Lc is a prefactor ac-counting for the conductor losses, which should decreaseas the thickness of the top ground metal increases. Thenwe fit the attenuation of the various S-TL characterizedin this work using two parameters: tan δ and Lc. Al-though S-TL attenuation shows a monotonous trend ver-sus frequency (see Fig. 4), we managed to fit indepen-dently these two parameters. Indeed surface resistivitydepends on the square root of the signal frequency anddielectric loss varies linearly with frequency. In otherwords, dielectric loss sets the slope of the insertion lossat high frequencies, while conductor loss affects the lowerfrequencies. All the obtained fitting parameters are pre-sented in Table I.

We now turn to the discussion of these parameters.First we measured that tan δ remains between 6.5.10−3±5 × 10−4 and 4 × 10−3 ± 5 × 10−4 for low-power mea-surements, which is on par with previously demonstratedJJ-TL [10]. Table I summarises the different depositionparameters of alumina. We observe that a higher deposi-tion temperature seems to improve dielectric loss. How-ever, given the incertitude over the loss tangent in thisstudy, we do not claim that deposition temperature isthe most important parameters to reduce dielectric loss.However, such values of loss tangent are very promising,since already comparable to state-of-the-art. Moreover,loss tangents as low as 2.45× 10−3, in the single photonregime, was reported for alumina with similar thickness,at very low temperatures [30].

Regarding conductor loss, we observe that as the top-ground thickness increases, Lc drops. We have plottedfor what we consider to be the best conductors loss (indashed blue) for each version as well as other values ofconductors (in dashed grey) loss to give insights on theconfidence interval (see Fig. 4). As a reminder, batch 2and 3 should have the same conductor loss since theirtop-ground thickness is the same. Another interestingpoint is that a gold conductivity of 100 MS m−1 was re-ported at very low temperature [31]. This translates intoa skin depth δs = 600 nm at 7 GHz. We can expect thesame order for copper ground as gold and copper haveclose conductivity at room temperature. Batch 1, wherethe top-ground thickness was well below δs, shows strongconductor loss. In the other batches, with thicker groundplanes, losses decreased. Until batch 4 (thickness 1 µm)where conductor loss is below what we can measure [32].

V. POWER-DEPENDENT LOSSES

We now report on the power-dependence of theselosses. In both the matched transmission lines and the

6

130 120 110 100

250

300

350

400

450

10 010 110 210 3-1 10

Power (dBm)

Qualit

y fa

ctor

ExternalInternal

Photon number

6

4

3

2

5

4 6 83 5 7

Frequency (GHz)

SAMPLE E SAMPLE C

Pin=-106 dBm

tan(δ)=7.5x10-3

Pin =-128 dBm

|S21|

(dB)

tan(δ)=6.5x10-3

(b)(a)

FIG. 5. Power dependence of losses. For both panels,power is referred to the input of the device. (a) S-TL cali-brated transmission (sample E). Input power and fitted losstangent are reported directly on the figure. (b) Quality fac-tors (internal and external) of a given resonant mode of sam-ple C (angular frequency ω0 = 2π × 7.47 GHz). Fit is ob-tained for the same resonance at various input powers (seeAppendix B). Photon number (top axis) is obtained usingEq. (B2).

resonant structures we observe that losses decrease whenpower increases (Fig. 5). The fitted loss tangent of the S-TL presented in Fig. 5.a depends on input power and sat-urates at low power, close to tan δ = 7.5×10−3±5×10−4.Fitted external and internal quality factors of a given res-onant mode are plotted in Fig. 5b for various input pow-ers in the feedline. The external quality factor remainsconstant while the internal quality factors increases withinput power. This observation is in agreement with whatwas reported in a JJ-TL [10]. At very low powers, closeto the single photon regime, losses saturate. This is con-sistent with the presence of Two-Level-Systems withinthe dielectric layer [30]. The internal quality factorand loss tangent are linked via Qi = 1/ tan δ [3]. Wecan compare quality factors from sample C, which isfrom the same batch as sample B (calibrated transmis-sion is shown in Fig. 4.b). For sample B, at low sig-nal power, close to the loss tangent saturation, we findtan δ = 6.0 × 10−3 ± 5 × 10−4. We compare this valueto low power quality factors measured for sample C. AtPinput = −136dBm [33] (n ≈ 0.1) we find, depending onthe resonant mode, an internal quality factor betweenQi,min = 125 and Qi,max = 300, with a mean valuearound 200 (see Appendix C). This translates into a losstangent of tan δmean = 1/200 = 5.0×10−3, in good agree-ment with the loss tangent found for sample B.

Frequency

(G

Hz)

|S21|

(dB)

DC current (mA)

90

80

70

60

65

4

6

8

10

12

14

0.50 0.25 0.00 0.25 0.50

55

75

85

95

16

18

0.75 0.75

20 10 0 10 2030 30

FIG. 6. Transmission of a S-TL versus flux and fre-quency (sample E).

VI. FLUX MODULATION

Finally, we describe the flux tunability of a S-TL. Fig. 6shows the flux dependence of the transmission of sam-ple E. At low frequencies, below 8 GHz, the transmis-sion is mostly flat with no visible ripples. At higher fre-quencies one can observe flux-dependent standing waves.This is explained by spurious reflections between the S-TL and its microwave environment. These reflectionscould be caused by wire-bonds or a non-optimized PCB-to-connector transition. We emphasize the smooth be-havior of the device during flux tuning, despite the largenumber of SQUID. This stability can be attributed to ourchoice of having a non-superconducting ground plane ontop of the SQUID, which prevents flux trapping and ef-fects due to Meissner currents. Interestingly the plasmafrequency of the SQUID can be directly observed as adrop in transmission. At zero flux it is above 20 GHz butdrops almost down to zero for magnetic fluxes close tohalf a fluxoid.

VII. CONCLUSION

We have introduced a process to fabricate SQUIDtransmission lines. It is simple, low-loss and offers in-situ flux tunability. The impedance matching to 50 Ωrelies on a simple yet effective idea: a metallic electri-cal ground deposited on top of the SQUID array, whichare separated by a thin alumina layer, guaranteeing elec-trical isolation. Impedance matching was demonstratedat very low temperatures (20 mK) and very low power(single photon regime). Two loss sources were identified;conductor losses due to the finite thickness of the metallicground compared to the skin depth and dielectric lossesconsistent with the presence of two-level systems. We

7

showed that increasing the thickness of the ground planimproves conductor losses. Regarding dielectric losses,we measured loss tangent down to 5.0×10−3. This valueis comparable to what was reported in the literature [10]but could be improved given the values previously ob-tained for alumina [30]. Finally, we demonstrated in-situ flux tunability of these S-TL. These devices couldbe used as Josephson Traveling Wave Parametric Ampli-fiers, based on four-wave mixing [10, 11, 22]. Their flux-tunability offers interesting perspectives regarding three-wave mixing [34, 35] or Kerr-free [36, 37] J-TWPA. Thesedevices could also be used to perform new quantum op-tics experiments in the microwave domain [38].

ACKNOWLEDGMENTS

Very fruitful discussions with D. Basko, K. R. Amin,and A. Ranadive are acknowledged. We also thank M.Selvanayagam for critical reading of the manuscript. Thesamples were fabricated in the Nanofab clean room. Thisresearch was supported by the ANR under contractsCLOUD (project number ANR-16-CE24-0005) and theEuropean Union’s Horizon 2020 Research and Innova-tion Programme, under grant agreement No. 824109,the European Microkelvin Platform (EMP). J.P.M.acknowledges support from the Laboratoire d’excellenceLANEF in Grenoble (ANR-10-LABX-51-01). R.D. andS.L. acknowledge support from the CFM foundationand the ’Investisements d’avenir’ (ANR-15-IDEX-02)programs of the French National Research Agency.

Appendix A: Electrical models

In this section, we present the electrical models usedto describe the samples we measured. For the S-TL, weused a standard telegrapher model where one SQUID is acell composed of a nonlinear Josephson inductance LJ, aground capacitance Cg and a Josephson capacitance CJ.The electrical sketch is shown in Fig. 7.a. The SQUID ar-ray is connected to 50 Ω pads via wire bonding. For theSQUID-based resonator, we use the same model. Theonly difference is that the line is capacitively coupledto the 50 Ω feedline. An electrical sketch is drawn inFig. 7.b.

Appendix B: S-Res measurements

We present here low temperature measurements ofsample C. It is 600 SQUID long with a free spectral rangeof 350 MHz. Transmission close to resonance is fitted us-ing the formula [39]:

2 μm

WL

a

(b)

(a)

FIG. 7. Electrical sketches of the two measured structures.(a) Electrical model of the S-TL. (b) Electrical model of the S-Res. Inset SEM picture of 3 SQUID. Highlighted blue regionsare Josephson junctions.

S21 =Z0

Z0 + iXe

1 + 2iQi(ω−ω0ω0

)

1 + QiQeZ0(Z0+iXe

) + 2iQi(ω−ω0ω0

), (B1)

where Z0 is the feedline characteristic impedance, Xethe reactance from the bonding wires, ω0 the resonancefrequency of the resonator, Qi and Qe the internal andexternal quality factors, respectively. In Fig. 8 we present5 resonances from 6.00 GHz to 7.75 GHz (amplitude andphase) from the S-Res at very low input power. Blacklines are fits to Eq. (B1). From this equation we extractinternal and external quality factors (see Fig. 5.b) thatwe plot as a function of input power and photon numberinside the cavity. The photon number was calibratedusing input/output theory [39]:

n =2 ω0Qe

~ω0( ω0Qe

+ ω0Qi

)2Pinput (B2)

where ω0 is the resonant frequency of the cavity andPinput the signal input power.

8

52

54

56

3

2

6.75 7.00 7.25 7.50 7.756.506.256.00

|S21|

(dB)

arg(S

21)

(rad

)

Frequency(GHz)

FitExperimental data

(a)

(b)

FIG. 8. Transmission |S21| of the hanger resonators fittedclose to the resonances

Appendix C: Frequency dependency of the internalquality factor

In this section, we present extracted quality factorsfor very low input power, Pinput = −136dBm ± 3dBm,that translates into 0.1 photon (see Appendix B for thephoton number calibration). At very low photon number,

close to the saturation of the losses, we observe a internalquality factor of 200 with a spread between 125 and 300for different resonant frequencies as shown in Fig. 9.

Internal quality factor100 150 200 250 300

0

1

2

3

4

5

6

7

FIG. 9. Internal quality factors extracted from the fit of 14modes in sample C between 3.86 GHz and 8.46 GHz at an in-put power Pin = −136dBm±3dBm, corresponding to the sin-gle photons level. We observe a spread of the internal qualityfactor around 200 at this power.

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