We have designed, fabricated and measured high-Q $\lambda/2$ coplanar
waveguide microwave resonators whose resonance frequency is made tunable with
magnetic field by inserting a DC-SQUID array (including 1 or 7 SQUIDs) inside.
Their tunability range is 30% of the zero field frequency. Their quality factor
reaches up to 3$\times10^4$. We present a model based on thermal fluctuations
that accounts for the dependance of the quality factor with magnetic field.Comment: subm. to JLTP (Proc. of LTD12 conference

A. Blais

A. Lupascu

A. Palacios-Laloy

A. Wallraff

D. Esteve

D. Vion

F. Mallet

F. Nguyen

I. Chiorescu

I. Siddiqi

J. Majer

J.M. Martinis

K.D. Osborn

L. Frunzio

L.D. Landau

M. Wallquist

M.A. Castellanos-Beltran

M.A. Sillanpaa

P. Bertet

P. Day

R. Barends

V.E. Manucharyan

Y. Nakamura

English

arXiv

Crossref

Tunable Resonators for Quantum Circuits

JLTP 3, 151 (2008)We have designed, fabricated and measured high-Q $\lambda/2$ coplanar waveguide microwave resonators whose resonance frequency is made tunable with magnetic field by inserting a DC-SQUID array (including $1$ or $7$ SQUIDs) inside. Their tunability range is $30\%$ of the zero field frequency. Their quality factor reaches up to 3$\times10^4$. We present a model based on thermal fluctuations that accounts for the dependance of the quality factor with magnetic field

Palacios-Laloy, A.

Nguyen, F.

Mallet, F.

Bertet, Patrice

Vion, Denis

Esteve, Daniel

HAL-CEA

Tunable resonators for quantum circuitsA. Palacios-Laloy, F. Nguyen, F. Mallet, P. Bertet, D. Vion, D. EsteveTo cite this version:A. Palacios-Laloy, F. Nguyen, F. Mallet, P. Bertet, D. Vion, et al.. Tunable resonators forquantum circuits. JLTP 3, 151 (2008). 2007. <hal-00193151>HAL Id: hal-00193151https://hal.archives-ouvertes.fr/hal-00193151Submitted on 30 Nov 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.hal-00193151, version 1 - 30 Nov 2007Noname manuscript No.(will be inserted by the editor)Tunable resonators for quantum circuitsA. Palacios-Laloy · F. Nguyen · F. Mallet ·P. Bertet · D. Vion · D. EsteveReceived: date / Accepted: dateAbstract We have designed, fabricated and measured high-Q λ/2 coplanar waveguidemicrowave resonators whose resonance frequency is made tunable with magnetic fieldby inserting a DC-SQUID array (including 1 or 7 SQUIDs) inside. Their tunabilityrange is 30% of the zero field frequency. Their quality factor reaches up to 3×104. Wepresent a model based on thermal fluctuations that accounts for the dependance of thequality factor with magnetic field.PACS 74.78.-w, 84.40.Dc, 85.25.Am, 85.25.Dq1 IntroductionOn-chip high quality factor superconducting resonators have been extensively studiedin the past years due to their potential interest for ultra-high sensitivity multi-pixeldetection of radiation in the X-ray, optical and infrared domains [1,2]. They consist ofa stripline waveguide of well-defined length, coupled to measuring lines through inputand output capacitors. The TEM modes they sustain have quality factors defined bythe coupling capacitors and reaching in the best cases the 106 range [2].It has also been demonstrated recently [3] that superconducting resonators providevery interesting tools for superconducting quantum bit circuits [4]. Indeed, a resonatorcan be used to measure the quantum state of a qubit [3,5,6,7]. Moreover, another res-onator may serve as a quantum bus and mediate a coherent interaction between qubitsto which it is coupled. The use of resonators might thus lead to a scalable quantumcomputer architecture [5]. The coupling of two qubits mediated by a coplanar waveg-uide (CPW) resonator has already been demonstrated [8,9]. In experiment [9], eachqubit needs to be tuned in and out of resonance with the resonator for the coupling tobe effective. Reference [10] proposed an alternative solution that consists in tuning theresonator in and out of resonance with each qubit. Here we report on the measurementof high quality factor resonators whose frequency can be tuned. Measurements similarQuantronics Group, Service de Physique de l’Etat Condens (CNRS URA 2464), DRECAM,CEA-Saclay,91191 Gif-sur-Yvette, France2Fig. 1 a: Tunable resonator scheme : a DC SQUID array is inserted between two λ/4 waveg-uides coupled to a 50 Ω measurement line through input and output capacitors Cc. b: Op-tical micrograph of a CPW niobium resonator. c: Typical coupling capacitor (design value :Cc = 27fF). d: Gap in the middle of the resonator, before SQUID patterning and deposition.e: Electron micrograph of an aluminum SQUID (sample A), fabricated using electron-beamlithography and double-angle evaporation. f: Electron micrograph of a 7-SQUID array (sampleB).to ours have been reported by other groups on lumped element [11] and distributed[12] resonators.2 Tunable resonator with DC SQUID: modelOur tunable resonators consist of λ/2 coplanar waveguides with an array of N DC-SQUIDs in series inserted in the middle of the central strip (see Fig. 1a). Each DCSQUID is a superconducting loop with self-inductance Ll intersected by two nominallyidentical Josephson junctions of critical current Ic0 ; the loop is threaded by a magneticflux Φ. The SQUID array behaves as a lumped non-linear inductance that depends onΦ, which allows to tune the resonance frequency.A λ/2 CPW resonator without any SQUID consists of a transmission line of lengthl, capacitance and inductance per unit length C and L, and characteristic impedanceZ0 =√L/C. We consider here only the first resonance mode that happens when3Fig. 2 A DC SQUID with two junctions of critical current Ic0 and loop inductance Ll, biasedby a magnetic flux Φ and by a current i, is equivalent to a lumped flux-dependent non-linearinductance LJ (Φ, i) that can be decomposed in an inductance LJ0(Φ) and a non-linear elementSNL(Φ) in series.l = λ/2 at a frequency ωr = pi/√LC, where L = Ll and C = Cl are the totalinductance and capacitance of the resonator. The quality factor Q results from thecoupling of the resonator to the R0 = 50Ω measurement lines through the input andoutput capacitors Cc leading toQc =pi4Z0R0C2cω2r, (1)from internal losses (Qint), and from possible inhomogeneous broadening mechanisms(Qinh). These combined mechanisms yieldQ−1 = Q−1c +Q−1int+Q−1inh. (2)As shown in Fig. 2, we model a SQUID as a non-linear inductance LJ (Φ, i) thatdepends on Φ and on the current i passing through it, so that the voltage across theSQUID isV = LJ (Φ, i)didt. (3)All SQUID properties are periodic in Φ with a period Φ0 = h/2e, the supercon-ducting flux quantum. Introducing the reduced flux quantum ϕ0 = Φ0/2pi, the SQUIDfrustration f = piΦ/Φ0, the effective critical current Ic(Φ) = 2Ic0| cos f | of the SQUIDat zero loop inductance, and the parameter β = LlIc0/ϕ0, our calculation of LJ (Φ, i)to first order in β and to second order in i/Ic(Φ) yields for f ∈ ]−pi/2, pi/2[LJ (Φ, i) = LJ0(Φ) +A(Φ)i2, (4)withLJ0(Φ) =ϕ0Ic(Φ)(1 + βcos 2f2 cos f), (5)A(Φ) =ϕ02I3c (Φ). (6)Equation 4 shows that the SQUID can be modelled as the series combination of alumped inductance LJ0(Φ) and of a non-linear device SNL(Φ) [13] (see Fig. 2).In the linear regime i ≪ Ic(Φ) corresponding to low intra-cavity powers, one canneglect the non-linear term in Eq. 4. The N-SQUID array then simply behaves as a4lumped inductance NLJ0(Φ). The device works in that case as a tunable harmonic os-cillator. Introducing the ratio ε(Φ) = LJ0(Φ)/L between the total effective inductanceof the SQUID and the resonator inductance, the frequency and quality factor areω0(Φ) = ωr11 +Nε(Φ), (7)Qext(Φ) = Qc [1 + 4Nε(Φ)] . (8)At larger peak current in the resonator i . Ic(Φ), the non-linear element SNL(Φ)has to be taken into account. The equation of motion of the oscillator acquires a cubicterm, similar to that of a Duffing oscillator [14]. This leads to a small additional shiftof the resonance frequency δω0(E) proportional to the total electromagnetic energy Estored in the resonator. Retaining first order terms in ε(Φ), we findδω0(Φ,E)ω0(Φ)= −N{2ω0(Φ)piR0[1 + 2Nε(Φ)]}2ϕ08I3c (Φ)E. (9)As shown by Eq. 7, a resonator including an array of N SQUIDs of critical currentNIc0 has approximately the same resonant frequency and same tunability range asa resonator including one SQUID of critical current Ic0. However, an interesting ad-vantage of using an array is to obtain a linear regime that extends to larger currents,allowing measurements at larger powers and therefore higher signal-to-noise ratios.3 Sample fabricationThe design and fabrication of our resonators closely followed Ref. [15]. The couplingcapacitors were simulated using an electromagnetic solver. Test niobium resonatorswithout any SQUIDs were first fabricated. They were patterned using optical lithog-raphy on a 200 nm thick niobium film sputtered on a high-resistivity (> 1000 Ω cm)oxidized 2-inch silicon wafer. The niobium was etched away using either dry or wetetching. Dry etching was done in a plasma of pure SF6 at a pressure of 0.3mbar andat a power such that the self-bias voltage was 30V and the etching rate 1.3 nm/s. Weobserved that adding oxygen to the plasma gave consistently lower quality factors.Wet etching was done in a solution of HF, H2O, and FeCl3 having an etching rateof approximately 1 nm/s at room-temperature. A typical resonator and its couplingcapacitor are shown in panels b and c of Fig. 1. Its 3.2 cm length yields a resonancefrequency around 1.8GHz.In addition to these test structures, some resonators had a gap in the middle (seeFig. 1d) used in a later step to fabricate a SQUID array by e-beam lithography anddouble-angle aluminum deposition (see panels e and f in Fig. 1). Before depositingthe aluminum, the niobium surface was cleaned by argon ion-milling (dose . 1018neutralized 500 eV ions per square centimeter). The Nb/Al contact resistance was foundto be in the ohm range, yielding tunnel junctions of negligible inductance compared tothat of the SQUID.5Fig. 3 Experimental setup. The sample is thermally anchored at the mixing chamber of adilution refrigerator with temperature 40−60mK. It is connected to a vector network analyzer(VNA) at room-temperature that measures the amplitude and phase of the S21 coefficient.The input line (top) is strongly attenuated (120 to 160 dB in total) with cold attenuators toprotect the sample from external and thermal noise, and filtered above 2GHz. The output line(bottom) includes a cryogenic amplifier with a 3K noise temperature and 3 cryogenic isolators.Design MeasurementsCc Qc Ll N Ic0 ωr/2pi Q(Φ = 0)Test 2 fF 6×105 0 1.906GHz 2×105Sample A 27 fF 3.4×103 40± 10 pH 1 330 nA 1.805GHz 3.5×103Sample B 2 fF 6×105 20± 10 pH 7 2.2µA 1.85GHz 3×104Table 1 Summary of sample parameters. See text for definitions.4 Experimental setupThe chips were glued on a TMM10 printed-circuit board (PCB). The input and outputport of the resonator were wire-bonded to coplanar waveguides on the PCB, connectedto coaxial cables via mini-SMP microwave launchers. The PCB was mounted in acopper box. The S21 coefficient (amplitude and phase) of the scattering matrix wasmeasured as a function of frequency using a vector network analyzer. Test resonatorswere measured in a pumped 4He cryostat reaching temperatures of 1.3K, with typicalinput power of −50 dBm and using room-temperature amplifiers. We measured internalquality factors up to 2×105 with both etching methods.The tunable resonators were measured in a dilution refrigerator operated at 40 −60 mK, using the microwave setup shown in Fig. 3. The input line includes room-temperature and cold attenuators. The output line includes 3 cryogenic isolators, acryogenic amplifier (from Berkshire) operated at 4K with a noise temperature of 3K,and additional room-temperature amplifiers. The attenuators and isolators protect thesample from external and thermal noise. This setup allowed to measure the samplewith intra-cavity energies as small as a few photons in order to operate in the linearregime, corresponding to typical input powers of −140 dBm at the sample level.6Fig. 4 (color online) a: Measured (thin line) amplitude (top) and phase (bottom) transmis-sion of sample A for Φ = 0 and fit (bold line) yielding a quality factor Q = 3300. b: Measuredresonance frequency of sample A (squares) as a function of applied magnetic flux and corre-sponding fit (full line) according to Eq. 7.5 Experimental resultsTwo tunable resonators were measured: sample A has only one SQUID (see Fig. 1e)and large coupling capacitors (27 fF) so that its total quality factor is determined byQc = 3.4 ×103. Sample B has an array of 7 SQUIDs in series (see Fig. 1f) and smallercoupling capacitors (2 fF) so that its quality factor is likely to be dominated by internallosses or inhomogeneous broadening. Relevant sample parameters are listed in table 1.A typical resonance curve, obtained with sample A at Φ = 0 for an input powerof −143 dBm corresponding to a mean photon number in the cavity n ≈ 1.2, is shownin Fig. 4. The |S21| curve was normalized to the maximum measured value. By fittingboth the amplitude and the phase response of the resonator, we extract the resonancefrequency and the quality factor Q. When the flux through the SQUID is varied, theresonance frequency shifts periodically as shown in Fig. 4b, as expected.The resonance frequency f0(Φ) and quality factor Q(Φ) are shown for both samplesin Fig. 5 over one flux period. The f0(Φ) curves in panels a and c are fitted withEq. 7. The agreement is good over the whole frequency range, which extends from1.3 to 1.75GHz, yielding a tunability range of 30%. The small discrepancy observedfor sample B might be due to a dispersion in the various SQUID loop areas that isnot taken into account in our model. The parameters obtained by this procedure forboth samples are shown in table 1; they are in good agreement with design values andtest-structure measurements.The Q(Φ) dependance for both samples is shown in panels b and d of Fig. 5. Bothsamples show a similar behaviour: the quality factor depends weakly on Φ when theflux is close to an integer number of flux quanta, whereas it shows a pronounced diparound Φ0/2.The largest quality factors are 3.5×103 for sample A and 3×104 for sample B.This difference is due to the different coupling capacitors. For sample A, the maximumquality factor is the same as measured on test resonators with similar capacitors andcorresponds to the expected Qc for Cc = 27 fF. Therefore sample A quality factor islimited by the coupling to the 50Ω lines around integer values of Φ0. The situationis different for sample B: the measured value is one order of magnitude lower thanboth the quality factor Qc =6×105 expected for Cc = 2 fF and the measured Q oftest resonators with the same capacitors (see table 1). This unexplained broadening70.0 0.5 1.001x1032x1033x1031.41.61.81.41.61.80.0 0.5 1.001x1042x1043x104b0 Q   SAMPLE B  f 0 (GHz) SAMPLE Aa c  d 0 Fig. 5 (color online) a and c:Measured resonance frequency f0 as a function of Φ/Φ0 (squares)for samples A and B, respectively, and fit according to Eq. 7 (solid line). b and d: Measuredquality factor Q (disks) as a function of Φ/Φ0. The solid line is calculated according to themodel (see text) for a temperature T = 60mK.of the resonance in presence of a SQUID array might be due either to the presenceof low-frequency noise in the sample, or to a dissipation source specifically associatedwith the SQUIDs. We note that flux-noise is not plausible since our measurementsshow no clear correlation with the sensitivity of the resonator to flux-noise. However,critical-current noise could produce such effect. Another possibility could be dielectriclosses in the tunnel barriers.We now turn to the discussion of the dip in Q(Φ) observed around Φ0/2. Weattribute it to thermal noise. Indeed, as discussed in section 2, the resonance frequencydepends on the energy stored in the resonator. At thermal equilibrium, fluctuations inthe photon number translate into a fluctuation of the resonance frequency and cause aninhomogeneous broadening. At temperature T , the resonator stores an average energygiven by Planck’s formula E = ~ω0(Φ)n, n = 1/{exp[~ω0(Φ)/kT ]−1} being the averagephoton number. The photon number and energy fluctuations are n2 − n2 = n(n + 1)and√δE2 =√E2+ ~ω0(Φ)E. (10)The characteristic time of these energy fluctuations being given by the cavity damp-ing time Q/ω0 with Q ≫ 1, a quasi-static analysis is valid and leads to an inhomoge-neous broadening δωinh = |dω0/dE|√δE2. Using Eq. 9, we getQ−1inh(Φ) =δωinh(Φ)ω0(Φ)= N{2ω0(Φ)piR0[1 + 2Nε(Φ)]}2ϕ08I3c (Φ)√δE2. (11)The resulting quality factor is Q−1 = Q−1inh+Q−1ext, which is plotted as full curves inpanels b and d of Fig. 5, for T = 60mK. The agreement is good, although Eq. 11 results8from a first-order expansion that is no longer valid in the close vicinity of Φ0/2. We havealso observed that Q values significantly degrade around Φ0/2 when the samples areheated, while remaining unchanged around integer numbers of Φ0. These observationssuggest that thermal noise is the dominant contribution to the drop of Q. Note that ourmodel does not take into account flux-noise, which evidently contributes to Qinh andcould account for the residual discrepancy between experimental data and theoreticalcurves in panels b and d of Fig. 5.6 ConclusionWe have designed and measured SQUID-based stripline resonators that can be tunedbetween 1.3GHz and 1.75GHz, with a maximum Q =3×104 limited by an unknownmechanism. The quality factor degrades due to thermal noise around Φ0/2. This lim-itation would be actually lifted with higher frequency resonators matching typicalJosephson qubit frequencies. Their tunability range at high Q would then be wideenough to couple a large number of qubits.Acknowledgements This work has been supported by the European project EuroSQIP. Weacknowledge technical support from P. Se´nat, P.F. Orfila and J.C. Tack, and fruitful discussionswithin the Quantronics group and with A. Lupascu, A. Wallraff, M. Devoret, and P. Delsing.References1. P. Day et al., Nature 425, 817 (2003) ; R. Barends et al., IEEE Trans. on Appl. Super-conductivity 17, 263 (2007).2. B. Mazin, PhD thesis, California Institute of Technology (2004).3. A. Wallraff et al., Nature 431, 162 (2004).4. Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, Nature (London) 398, 786 (1999); D. Vionet al., Science 296, 886 (2002); J. M. Martinis et al., Phys. Rev. Lett. 89, 117901 (2002);I. Chiorescu et al., Science 299, 1869 (2003).5. A. Blais et al., Phys. Rev. A 69, 062320 (2004).6. A. Lupascu et al., Nature Phys. 3, 119 (2007)7. I. Siddiqi et al., Phys. Rev. B 73, 054510 (2006).8. J. Majer et al., Nature 449, 443 (2007).9. M.A. Sillanpaa, J.I. Parks, R.W. Simmonds, Nature 449, 438 (2007).10. M. Wallquist, V. S. Shumeiko, and G. Wendin, Phys. Rev. B 74, 224506 (2006).11. K.D. Osborn et al., IEEE Trans. on Appl. Superc. 17 (2), 166 (2007).12. M. Sandberg et al., to be published.13. V. E. Manucharyan et al., Phys. Rev. B 76, 014524 (2007).14. L.D. Landau and E.M. Lifshitz, “Mechanics”, Pergamon, Oxford (1969).15. L. Frunzio et al., IEEE Trans. on Appl. Superc. 15, 860 (2005).

Tunable resonators for quantum circuits

Tunable resonators for quantum circuits

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