Suspending superconducting qubits by silicon micromachining
Y. Chu, C. Axline, C. Wang, T. Brecht, Y. Y. Gao, L. Frunzio, R.J. Schoelkopf
SSuspending superconducting qubits by silicon micromachining
Y. Chu, a) C. Axline, C. Wang, T. Brecht, Y. Y. Gao, L. Frunzio, and R.J. Schoelkopf
Department of Applied Physics, Yale University, New Haven, Connecticut 06511,USA (Dated: 10 June 2016)
We present a method for relieving aluminum 3D transmon qubits from a silicon substrate using microma-chining. Our technique is a high yield, one-step deep reactive ion etch that requires no additional fabricationprocesses, and results in the suspension of the junction area and edges of the aluminum film. The drasticchange in the device geometry affects both the dielectric and flux noise environment experienced by the qubit.In particular, the participation ratios of various dielectric interfaces are significantly modified, and suspendedqubits exhibited longer T ’s than non-suspended ones. We also find that suspension increases the flux noiseexperienced by tunable SQUID-based qubits.The coherence times of superconducting qubits havesteadily increased over the past decade due to careful en-gineering of the electromagnetic environment, better ma-terials and fabrication methods, and improved device de-signs that minimize loss. State-of-the-art superconduct-ing qubits with the longest lifetimes ( T ) make use of verylow loss tangent dielectric substrates and have large sep-aration between planar conductors to decrease the effectof dielectric loss in the interfaces between materials .In particular, it has been shown that for aluminum 3Dtransmons on sapphire, T times are limited by the vari-ous interfaces between the dielectric substrate, the super-conducting metal, and vacuum . This effect can be at-tributed to the larger electric fields near metallic surfacesand the higher concentration of two-level systems (TLS)at disordered interfaces . At the same time, magneticimpurities at the surface of superconductors have beenproposed as the cause of 1 /f flux noise that limits theperformance of SQUID based qubits and sensors .In order to better understand these effects, one strat-egy is to drastically alter the geometry of materials andinterfaces that contribute to qubit loss and decoherence.In this Letter, we present a procedure for removing thesubstrate and suspending aluminum Josephson junctionson silicon by micromachining. Silicon is a low-loss dielec-tric that offers several advantages for implementing thenext generation of complex quantum circuits . Itsprevalent use in the semiconductor and MEMS indus-tries have led to a large variety of fabrication techniquesthat are not available for sapphire . Using silicon as asubstrate material enables the development of novel de-vices and architectures in circuit QED, such as multilayerquantum circuits that incorporate micromachined super-conducting enclosures and resonators . Substrate mi-cromachining has also been used to reduce dielectric lossand frequency noise in niobium titanium nitride copla-nar waveguide resonators on silicon . On the otherhand, silicon has a more complex surface chemistry thansapphire; for example, it forms an amorphous oxide layer a) Electronic mail: [email protected] that may be host to a large number of TLS’s and para-magnetic impurities .We suspend our qubits with a simple, one-step deepreactive ion etch (DRIE) using the BOSCH process thatdoes not require any additional steps to mask or protectthe devices . We begin with high resistivity (100)silicon wafers ( ρ > Ω · cm) and fabricate aluminum 3Dtransmon qubits using the standard Dolan bridge double-angle deposition technique . DRIE is then performed di-rectly on the fabricated qubits. This is possible becausealuminum itself is an excellent mask for the BOSCH pro-cess. We have performed this process on more than onehundred devices of various geometries, and found thatthe Josephson junctions were almost all unaffected ex-cept for a slight increase in the normal state resistance,possibly due to increased diffusion of the junction oxidewhen the devices are heated by the etching process.Figure 1 shows a schematic drawing and scanning elec-tron micrographs (SEM) of suspended 3D transmons. Wenote that all regions ∼
700 nm from the edge are under-cut, resulting in the junction region and the narrow 1 µ m wide leads on either side becoming completely sus-pended. This means that our process is compatible withaluminum based devices of any geometry, as long as sus-pended metal regions are supported by other larger fea-tures. The suspended single junction transmons are ro-bust against solvent cleaning, drying, and repeated ther-mal cycles. We observe, however, that more complex sus-pended structures such as SQUID loops are more easilydamaged, for example by surface tension during solventcleaning or wet etches.We first study the effect our process has on dielectricloss and qubit T . Following the analysis in Wang et al. ,we quantify the loss due to various dielectric materials us-ing their participation ratios and loss tangents (tan δ ). InFigure 2, we plot the measured T ’s of several types ofqubits with different designs and fabrication proceduresagainst the simulated participation ratios of their metal-substrate (MS) interfaces. Similar plots of T versus thesubstrate-air (SA) and metal-air (MA) participation ra-tios can be found in the supplementary materials . Inaddition to qubits of the design shown in Figure 1 (De-sign A), we also fabricated a set of qubit designs with a r X i v : . [ qu a n t - ph ] J un μ m10 μ m 2 μ mab c d FIG. 1.
Micromachined 3D transmons using DRIE(a)
A schematic drawing of a suspended 3D transmon on sil-icon. An overhang is created at the edges of the metal, whilethin features are suspended. (b)
SEM image of a BOSCHetched transmon showing the entire suspended leads aroundthe junction region (c)
Detailed view of the supporting siliconpedestals, showing the lighter-colored overhanging Al edgesand corrugated BOSCH profile. (e)
Detailed view of the com-pletely suspended Al-AlO x -Al Josephson junction. higher surface participation (Designs B and C). Design Cqubits have planar capacitors whose gaps can be varied tochange the surface participations. Drawings of all qubitdesigns can be found in the supplementary materials .The dielectric participation ratios were obtained throughelectromagnetic simulations that faithfully modeled thedevice geometries, including the undercut at the edge ofsuspended devices .Without any kind of surface preparation before or af-ter aluminum deposition, the typical T ’s of the Design Aqubits are only a few microseconds, which is more thanan order of magnitude worse than the same design onsapphire. The use of surface treatment techniques suchas buffered oxide etch (BOE) or oxygen plasma ashing(OPA) improves the lifetimes of the regular non-suspended qubits. DRIE further improves the T ’s ofthe Design A qubits. The highest T measured with thisprocedure was ∼ µ s for a etch depth of 60 µ m, whichis comparable to the T ’s of typical qubits of the samedesign on sapphire. We note that the qubit loss is likelyfrequency dependent because of coupling to resonant losschannels such as TLS’s. This can lead to low T ’s in ex-ceptional cases, which we have also included in Figure 2.We will explore this further in our later discussion of fluxtunable qubits.The results in Figure 2 indicate that the quality of in-terfaces of qubits on silicon are highly dependent on sur- other loss (e.g. bulk dielectric tan δ = 5e-7) M S t a n δ = - Design C, OPA Design B, OPA Design AOPA OPA or BOEsuspendedBOEnone
FIG. 2.
Lifetime of silicon transmons vs. metal-substrate participation ratio
Each point corresponds toone measured qubit. The MS participation of design C qubitswere varied by changing planar capacitor gap distance . Thelabel OPA signifies oxygen plasma ashed before and after de-position, while BOE signifies buffered oxide etch before de-position only. Errorbars are typical variation of T ’s overtime. Dashed lines are guides to the eye corresponding MSsurface loss and other effects that are independent of MS par-ticipation, such as bulk dielectric loss. Solid line indicates thecombinination of these loss mechanisms. face treatments and are generally higher loss than thoseon sapphire. The T ’s of non-suspended Design A qubitssuggest that surface treatment before and after deposi-tion is important on silicon. On the other hand, onlyOPA before deposition is needed to obtain > µ s T ’son sapphire . It is possible that, for example, the liftoffprocess leaves more resist residue on silicon than on sap-phire. In addition, any exposed silicon surface will forman oxide even after cleaning. Both resist and oxide arelikely to result in a higher loss SA interface, which hascomparable participation ratios as the MS interface fornon-suspended qubits. This may explain the observationthat while qubit T ’s are better after surface cleaning,they never reach the levels measured on sapphire.It is also evident from Figure 2 that it is insufficientto consider a simple model where qubit loss is dominatedby single dielectric interface. The qubits with high MSparticipation (Design C) have T ’s that follow a line ofconstant MS tan δ = 6 × − , consistent with being lim-ited by loss due to that interface. However, qubits withlower MS participation (Designs A and B) deviate fromthat line. One simple explanation for this trend is thatthe bulk dielectric loss for our silicon substrates becomessignificant once the MS participation has been sufficientlyreduced. The bulk dielectric participations of the mea-sured qubits, including the suspended ones, are all similarto within 10%. Therefore, we can indicate the T limitdue to bulk dielectric loss as a horizontal line in Figure2. We find that taking into account both bulk and MSsurface loss mechanisms results in a model that is consis-tent with data from both the suspended qubits and theregular qubits that underwent OPA.We emphasize, however, that other loss mechanismscan play a role as well. For example, the SA and MAparticipations scale similarly to the MS participation forthe regular, non-suspended qubits . However, unlike achange in qubit geometry, the DRIE process affects thethree interfaces differently. In particular, it increasesboth the SA and MA participations . Therefore, in-creased loss from the SA and MA interfaces could con-tribute to negating the T improvement expected fromthe reduced MS participation. We also cannot rule outmore complex effects of the DRIE, such as a change in theSA and MA tan δ ’s from damage or polymer depositionon these surfaces. Clearly, more investigations are neededto isolate and understand these different effects. Our re-sults indicate that micromachining is a new technique toalter the loss contributions of various materials in waysnot possible with changes in geometry alone. This canhelp us gain information about the roles of individualinterfaces in limiting qubit T ’s.In order to investigate the effects of DRIE on qubitbehavior in more detail, we also measured frequency de-pendence of the T and flux noise of regular and sus-pended tunable SQUID qubits. The qubit design is ex-actly the same as in Figure 1, except the single junctionis replaced by a 10 µ m × µ m SQUID loop, which iscompletely suspended after etching. A side-view SEM ofa suspended SQUID transmon is shown in Figure 3a. Wecompare this device with another that underwent BOEbefore deposition and no other surface cleaning or etch-ing after deposition. The two qubits were symmetricallyarranged inside the same copper cavity to ensure thatthey experienced a similar background electromagneticenvironment. Two separate solenoid coils mounted out-side the cavity and aligned with the location of the qubitsallowed us to individually control the frequency of eachdevice.We plot T as a function of qubit frequency for one pairof regular and suspended qubits in Figure 3b and 3c, re-spectively. For the regular qubit, we find that, in additionto a low overall T of < µ s, there are sharp dips in the T at a multitude of distinct frequencies. The suspendedqubit shows a higher overall T , but also exhibits a fewresonant features where the the T is drastically reduced.A second pair of qubits measured in the same manner ex-hibited similar behavior. These observations imply thatboth types of qubits are affected by the presence of res-onant loss channels such as TLS’s, as was observed inmany previous studies . We observe that resonant losschannels are less prevalent for the suspended qubit thanthe regular qubit. However, their presence may explainthe variability that we observe in the T measurementsof single junction qubits.The 3D SQUID transmons also allow us to investi-gate if and how suspension affects their magnetic en-vironments. In particular, many previous works haveobserved 1 /f flux noise experienced by several different types of superconducting qubits . It has been pro-posed that, similarly to dielectric loss, flux noise can becaused by defects in amorphous surface materials, suchas silicon oxide . However, typical measured flux noiselevels in SQUIDs have been orders of magnitude higherthan estimates based on known sources, and the origin ofthis important dephasing mechanism for superconduct-ing qubits remains uncertain .In order to measure the noise power spectral den-sity (PSD) of the qubits, we use the technique demon-strated in Bylander et al. . We measure the response ofthe qubits to a collection of Carr-Purcell-Meiboom-Gill(CPMG) dynamical decoupling sequences with varyingtime delays and number of pulses to filter out noise at dif-ferent frequencies. We can then use this in combinationwith the measured frequency-flux curves for the qubits toextract the PSD within a range of noise frequencies .The results are shown in Figure 3d. The PSD’s includedata from two pairs of regular and suspended qubits mea-sured in the same cavity on successive cooldowns. Thedata from the two pairs of qubits agree very well witheach other, indicating that observed differences betweenthe two types of qubits are not due to sample to samplevariations. The flux noise for both qubits exhibits a clearpower law dependence with an exponent of α = 0 . ± . α = 0 . ± . ∼ − Φ / Hz. Thissuggests that the measured noise in this frequency rangeis likely to be flux-related and not due to, for example,critical current fluctuations.While one might expect that removing the substratefrom underneath the SQUID loop would decrease the fluxnoise due to surface spins, our data indicates that theopposite effect occurs. It has been suggested that thedominant contributors to flux inside the SQUID loop arespins on the surface of the loop traces . Therefore, re-moving the silicon surface inside and outside the loopmay not have a large effect on the flux noise. On theother hand, DRIE also exposes the bottom surface ofthe aluminum loop, which forms a layer of amorphousAlO x in air. Our observations are consistent with the newAlO x layer having a higher concentration of spins thanthe aluminum-silicon interface, possibly because most ofthe SiO x was removed by BOE prior to deposition. Theobservation that the flux noise increased by more thana factor of two after etching could suggest that the newAlO x layer on the bottom surface contains more defectsthan the top surface. This might be the case given thatthe top oxide layer was grown in pure oxygen conditionsinside the evaporator rather than through exposure toair . We emphasize that while this explanation is con-sistent with our observations, further investigation would μ m ba c d FIG. 3.
Flux tunable suspended qubits (a)
Side view SEM image of a suspended SQUID transmon. (b, c) T vs qubitfrequency for regular (b) and suspended (c) qubits. See supplementary materials for zoomed-in view of a low- T feature in (b) . (d) PSD of flux noise extracted from dynamical decoupling of regular and suspended qubits, each including data fromtwo qubits. Solid lines are fits to the data. The extent of the lines indicate the frequency range in which the data was abovethe noise floor and therefore included in the fit. be needed to elucidate the microscopic origins of addi-tional flux noise in suspended qubits.We have demonstrated that micromachining of siliconsubstrates is compatible with aluminum Josephson junc-tion qubits. The process results in a reduction of themetal-substrate interface and an improvement of qubit T ’s. Our results seem to suggest that we are approachinga regime where qubit decay is dominated by other mecha-nisms such as dielectric loss of the bulk silicon substrate.The loss tangent of “undoped” high-resistivity silicon isnot very well known or understood, and is likely to be de-pendent on residual dopants and defects. We speculatethat the DRIE technique described here, in combinationwith higher quality substrates, can result in qubits witheven longer lifetimes. In addition, MS and bulk partici-pation ratios can be further reduced by redesigning thequbit so that the DRIE process suspends larger areas ofthe device. We emphasize, however, that dielectric losswill eventually become dominated by another material.Even in the limit of a qubit floating in vacuum, there willbe dielectric loss due to, for example, oxide on the sur-face of the metal. Beyond the reduction of dielectric loss,our measurements of flux noise with suspended SQUIDtransmons is another example of how qubit propertiescan be altered by changing the geometry of the substrateand the materials present in the environment. We expectthat other potential loss mechanisms for cQED devices,such as quasiparticles and phonon coupling , will alsobe affected. Therefore, our process provides a tool for un-derstanding and improving the various aspects involvedin the performance of superconducting qubits.We thank Luke Burkhart, Mollie Schwartz, and MichelDevoret for valuable discussions. Facilities use was sup-ported by the Yale SEAS cleanroom, YINQE and NSFMRSEC DMR-1119826. This research was supported bythe Army Research Office under Grant No. W911NF-14-1-0011. C.A. acknowledges support from the NSF Gradu-ate Research Fellowship under Grant No. DGE-1122492.Y.Y.G. acknowledges support from an A*STAR NSS Fel- lowship. Supplemental Materials for ”Suspending superconducting qubits by siliconmicromachining”
I. QUBIT DESIGNS AND FABRICATION PROCEDURES
Three types of 3D transmon qubit geometries were used to vary the surface participation in Figure 2 of the maintext. They were similar to the designs used in Wang et al. , and are shown in Figure S1. Design A, which had thelowest surface participations, was used to study the effects of DRIE etching. Design A Design B Design C µ m 977 µ m455 µ m60 µ m15 µ m300 µ m 500 µ m500 µ m 1 µ m10 µ m 940 µ md = 1.5 - 30 µ m120 µ m6 µ m 80 µ m FIG. S1.
Transmon qubit geometries used in this study
The surface participations of Design C were varied by changingthe distance d between the planar capacitor features. Table S1 shows the various combinations of fabrication steps performed on Design A qubits and the resultingbest T ’s. Fabrication of all qubits was performed using the Dolan bridge technique on 300-500 um thick (100) highresistivity ( ρ > Ωcm) silicon wafers (Crystec and SiliconQuest). After cleaning in acetone and methanol, the waferwas spun with a bilayer of e-beam resist consisting of 500 nm of MMA (8.5) MAA EL 13 and 70 nm of 950K PMMAA3, then baked at 175 ◦ C. Patterning of the qubit was done on a 100 kV VISTEC EBPG 5000+ e-beam writer, andthe wafer was subsequently developed for 55 seconds in 1:3 MIBK:IPA followed by a 5 second rinse in IPA. For someof the devices, the developed wafer was then either etched in 10:1 BOE for 10 s or in an oxygen plasma asher (GlowResearch AutoGlow) at 100 W for 10 s. We call this post-development step “clean 1” in Table S1. The wafer wasthen loaded into a Plassys e-beam evaporation system (MEB550S or UMS 300). In the case of BOE, the wafer waskept in a vacuum box during transport to the evaporator, which takes about five minutes. A bi-layer of aluminum(20 nm and 60 nm) was deposited using double-angle evaporation. In between the two layers, the junction barrierwas grown by thermal oxidation using using a 85:15 Ar:O mixture at 15 Torr for 12 minutes. Finally, the aluminumwas capped with another oxide layer grown at 3 Torr for 10 minutes. After deposition, liftoff was performed in 90 ◦ CNMP for several hours, then rinsed with acetone and methanol. Prior to dicing into individual chips with one qubiteach, a layer of photoresist was spun on the wafer to protect the qubits. After dicing in an ADT 7100 dicer, the resistwas removed by rinsing in solvent.The diced chips were then optionally oxygen plasma ashed (OPA) at 100 W for 3 minutes (clean 2) and/or DRIEetched. The DRIE was done using the BOSCH process with alternating SF inductively coupled plasma (ICP) etch(10 s, 35 mTorr, 700 W) and C F ICP passivation (3s, 35 mTorr, 700 W) steps. The plasma was turned off for 1minute after every 5 etch/passiviation cycles to prevent overheating of the sample. By changing the ratio of the etchand passivation step times, we can control the amount of undercut and the orientation of the sidewalls. A larger ratioresults in a larger undercut and a sidewall that slopes more inward with etch depth. Finally, the total number cyclescontrols the overall depth of the etch. We find that a 10s/3s etch/passivation cycle, resulting in a ∼
700 nm undercut,and a 60 um etch depth gives a significant modification of the participation ratios without jeopardizing the structuralintegrity of the devices.After DRIE, some of the chips were processed with OPA again at 100 W for 3 minutes to remove any depositedpolymer that might remain from the BOSCH process (clean 3).
TABLE S1.
Effect of cleaning procedures and etching on qubit T1s.
Parameters and placement of the clean and DRIEsteps in the fabrication procedure are described in the text. BOE: Buffered oxide etch. OPA: Oxygen plasma ashing.Clean 1 Clean 2 DRIE Clean 3 Max T ( µ s)None None None None 4BOE None None None 6BOE None Yes None 63BOE None Yes OPA 59BOE OPA None None 7BOE OPA Yes OPA 50OPA OPA None OPA 23OPA OPA Yes OPA 44 II. SIMULATIONS OF DIELECTRIC PARTICIPATION
Simulations for surface and bulk dielectric participation ratios were performed a method similar to that in Wang et al. . For the suspended qubits, the geometries of the various interfaces were modified accordingly, as shown inFigure S2.The DRIE suspension processes affects the substrate-air (SA) and metal-air (MA) interfaces in a qualitativelydifferent way than the metal-substrate (MS) interface. While it reduces the MS participation by about a factor of 5,it increases the SA participation by a factor of 4.5 and the MA participation by a factor of 3, as shown in Table S2and S3. As mentioned in the main text, the increase of these participation ratios could contribute to additional loss inthe suspended qubits, which would explain why their T ’s fall below the constant MS loss tangent line in Figure 2 ofthe main text. However, it is evident from Figure S3 that the unetched Design A qubits have shorter T ’s than wouldbe expected from a constant MS, SA, or MA loss tangent. Therefore, as proposed in the main text, an additionalloss mechanism is needed to explain this discrepancy. The simplest model assumes an effect that is independent ofsurface participations, such as bulk dielectric loss. TABLE S2.
Surface participation ratios for various qubits
The dimensions indicated for Design C are the gap sizes d for the planar capacitors, as shown in Figure S1.Qubit MS SA MADesign A, suspended 1.25e-5 2.90e-4 5.74e-6Design A, regular 6.16e-5 6.40e-5 1.90e-6Design B, 1.39e-4 1.64e-4 1.45e-5Design C, 30 µ m 3.32e-4 3.83e-4 3.53e-5Design C, 20 µ m 3.96e-4 4.55e-4 4.22e-5Design C, 10 µ m 5.63e-4 6.49e-4 6.19e-5Design C, 6 µ m 7.64e-4 8.85e-4 8.74e-5Design C, 3 µ m 1.25e-3 1.46e-3 1.55e-4Design C, 1.5 µ m 2.16e-3 2.36e-3 3.16e-4 III. RESONANT LOSS FEATURES
Both suspended and non-suspended flux tunable SQUID qubits exhibited dips in T at certain frequencies, consistentwith resonant loss mechanisms usually attributed to TLS’s in previous works . They appear to be more prevalent MS SA MA x z g
A B C (a) global 3D simulation (b) cross-sectional simulationx0yx z
FIG. S2.
Simulation geometry. (a)
An isometric overview of the transmon pads on a silicon substrate (light blue). Theleads near the junction, completely suspended in DRIE processed geometries, are shown in brown. The separation of the innerregion (orange) and exterior region (gray) of the pads is indicated by a green dashed line. A slice taken on the xz plane is shadedin red. (b)
The slice taken in (a) shows the etch profile, including the metal overhang (width v ) and deep substrate etch ( ∼ d deep). Interface surfaces are assumed to share a common thickness t , taken here to be 3 nm. The MS layer (red), which wouldnormally extend to the edge of the metal, is reduced by the etch. The MA layer (purple) also includes the underside of anymetal freshly exposed by the etch. The SA layer (dark blue) includes the sidewall surface of the remaining support substrate,increasing coverage significantly compared to unetched devices. The scalloped profile of the SA surface was approximated usinga smooth plane. In accordance with the procedure in Wang et al. , the perimeter region (gray) is divided into a cross-hatchedregion C (cid:13) , which can fail to converge in the global simulation if v is small compared to x . The region B (cid:13) is convergent in bothsimulations, and is used to bridge them in order to calculate the energy in the layers within region C (cid:13) . The division betweenA (cid:13) and B (cid:13) , at x = x , separates the inner and exterior regions. All dimensions are not to scale. in the non-suspended qubits, which exhibited several of such features of various widths. A detailed measurement of T versus frequency around one of these features is shown in S4. IV. FLUX NOISE ANALYSIS
The power spectral densities (PSD) of the flux noise experienced by suspended and regular SQUID transmonswere measured using dynamical decoupling techniques as described in Bylander et al. . We briefly summarize theprocedure here.We applied a series of Carr-Purcell-Meiboom-Gill (CPMG) sequences , given by (cid:16) π (cid:17) x − (cid:104) τ N − π y − τ N (cid:105) N − (cid:16) π (cid:17) x (S1)followed by dispersive measurement of the qubit state. The qubits were tuned to some flux point near Φ / T was relatively long. In our case, the measured signal is given by χ N ( τ ) = A + A (cid:18) ∂ω q ∂ Φ (cid:19) τ (cid:90) ∞ dωS ΦΦ ( ω ) g N ( ω, τ ) (S2)Where ω q is the qubit frequency, A and A are an overall offset and scaling determined by the qubit readoutparameters, and S ΦΦ ( ω ) is the PSD of the flux noise. By varying the length of the total sequence τ and the numberof echo pulses N , we can vary the center frequency and bandwidth of the filter function g N ( ω, τ ) that makes thedynamical decoupling sequence sensitive to a particular part of the noise spectrum . The expression for g N ( ω, τ )for the CPMG sequence can be found in the supplementary materials for Bylander et al. . Design C, OPA Design B, OPADesign A OPABOEnoneOPA or BOEsuspended Design C, OPA Design B, OPADesign A OPABOEnoneOPA or BOEsuspended ba S A t a n δ = - M A t a n δ = - FIG. S3.
Lifetime of silicon transmons vs. SA (a) and MA (b) participation ratios.
Dashed lines are guides to theeye corresponding to the indicated dielectric loss tangents.FIG. S4.
Resonant loss feature in non-suspended SQUID qubit
Zoom-in on one of the resonant features in Figure 3bof the main text.
Our goal is to invert equation S2 in order to determine S ΦΦ ( ω ) from the raw data. Each set of raw data consistsof varying τ for a fix number of echo pulses, an example of which is shown in Figure S5a. We then fit the data to afunctional form A + Ae − ( τ − τ T e − τ − τ T , (S3)which takes into account the independently measured T decay of the qubit and assumes a particular functional formfor the dephasing, which of course depends on the flux noise PSD that we would like to determine. Therefore, thefit is only used to extract A , A , and T , which are used to normalize the raw data and determine the range of τ forwhich the data is above the noise floor.Next, we simplify the problem by assuming that g N ( ω, τ ) is sharply peaked and replace it with a rectangular filterfunction with the same height and width. This becomes a better approximation for larger N and τ . Using thissimplified filter function, we can calculate the power spectral density of the frequency noise. An example is shown inFigure S5b after combining datasets with different N ’s for one of the suspended qubits. Note that each data point inthe raw data corresponds to one data point in the PSD. Finally, to convert this into flux noise, we determine ∂ω q /∂ Φby measuring the flux tuning spectrum of the qubits. a b
FIG. S5.
Determination of flux noise PSD (a)
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