aa r X i v : . [ qu a n t - ph ] D ec Noname manuscript No. (will be inserted by the editor)
Tunable resonators for quantum circuits
A. Palacios-Laloy · F. Nguyen · F. Mallet · P. Bertet · D. Vion · D. Esteve
Received: date / Accepted: date
Abstract
We have designed, fabricated and measured high-Q λ/ × . Wepresent a model based on thermal fluctuations that accounts for the dependance of thequality factor with magnetic field. PACS
On-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 10 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 the resonator in and out of resonance with each qubit. Here we report on the measurementof high quality factor resonators whose frequency can be tuned. Measurements similar Quantronics Group, Service de Physique de l’Etat Condens (CNRS URA 2464), DRECAM,CEA-Saclay,91191 Gif-sur-Yvette, France
Fig. 1 a:
Tunable resonator scheme : a DC SQUID array is inserted between two λ/ C c . b: Op-tical micrograph of a CPW niobium resonator. c: Typical coupling capacitor (design value : C c = 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.
Our tunable resonators consist of λ/ 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 L l intersected by two nominallyidentical Josephson junctions of critical current I c ; the loop is threaded by a magneticflux Φ . The SQUID array behaves as a lumped non-linear inductance that depends on Φ , which allows to tune the resonance frequency.A λ/ l , capacitance and inductance per unit length C and L , and characteristic impedance Z = p L / C . We consider here only the first resonance mode that happens when Fig. 2
A DC SQUID with two junctions of critical current I c and loop inductance L l , biasedby a magnetic flux Φ and by a current i , is equivalent to a lumped flux-dependent non-linearinductance L J ( Φ, i ) that can be decomposed in an inductance L J ( Φ ) and a non-linear elementSNL( Φ ) in series. l = λ/ ω r = π/ √ LC , where L = L l and C = C l are the totalinductance and capacitance of the resonator. The quality factor Q results from thecoupling of the resonator to the R = 50 Ω measurement lines through the input andoutput capacitors C c leading to Q c = π Z R C c ω r , (1)from internal losses ( Q int ), and from possible inhomogeneous broadening mechanisms( Q inh ). These combined mechanisms yield Q − = Q − c + Q − int + Q − inh . (2)As shown in Fig. 2, we model a SQUID as a non-linear inductance L J ( Φ, i ) thatdepends on Φ and on the current i passing through it, so that the voltage across theSQUID is V = L J ( Φ, i ) didt . (3)All SQUID properties are periodic in Φ with a period Φ = h/ e , the supercon-ducting flux quantum. Introducing the reduced flux quantum ϕ = Φ / π , the SQUIDfrustration f = πΦ/Φ , the effective critical current I c ( Φ ) = 2 I c | cos f | of the SQUIDat zero loop inductance, and the parameter β = L l I c /ϕ , our calculation of L J ( Φ, i )to first order in β and to second order in i/I c ( Φ ) yields for f ∈ ] − π/ , π/ L J ( Φ, i ) = L J ( Φ ) + A ( Φ ) i , (4)with L J ( Φ ) = ϕ I c ( Φ ) „ β cos 2 f f « , (5) A ( Φ ) = ϕ I c ( Φ ) . (6)Equation 4 shows that the SQUID can be modelled as the series combination of alumped inductance L J ( Φ ) and of a non-linear device SNL( Φ ) [13] (see Fig. 2).In the linear regime i ≪ I c ( Φ ) corresponding to low intra-cavity powers, one canneglect the non-linear term in Eq. 4. The N-SQUID array then simply behaves as a lumped inductance NL J ( Φ ). The device works in that case as a tunable harmonic os-cillator. Introducing the ratio ε ( Φ ) = L J ( Φ ) /L between the total effective inductanceof the SQUID and the resonator inductance, the frequency and quality factor are ω ( Φ ) = ω r
11 + Nε ( Φ ) , (7) Q ext ( Φ ) = Q c [1 + 4 Nε ( Φ )] . (8)At larger peak current in the resonator i . I c ( 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 δω ( E ) proportional to the total electromagnetic energy E stored in the resonator. Retaining first order terms in ε ( Φ ), we find δω ( Φ, E ) ω ( Φ ) = − N ω ( Φ ) πR [1 + 2 Nε ( Φ )] ff ϕ I c ( Φ ) E. (9)As shown by Eq. 7, a resonator including an array of N SQUIDs of critical current NI c has approximately the same resonant frequency and same tunability range asa resonator including one SQUID of critical current I c . 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. The 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 ( > at a pressure of 0 . . O, and FeCl 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 . . . neutralized 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. Fig. 3
Experimental setup. The sample is thermally anchored at the mixing chamber of adilution refrigerator with temperature 40 −
60 mK. It is connected to a vector network analyzer(VNA) at room-temperature that measures the amplitude and phase of the S 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 2 GHz. The output line(bottom) includes a cryogenic amplifier with a 3 K noise temperature and 3 cryogenic isolators.Design Measurements C c Q c L l N I c ω r / π Q ( Φ = 0)Test 2 fF 6 × .
906 GHz 2 × Sample A 27 fF 3.4 × ±
10 pH 1 330 nA 1 .
805 GHz 3.5 × Sample B 2 fF 6 × ±
10 pH 7 2 . µ A 1 .
85 GHz 3 × Table 1
Summary of sample parameters. See text for definitions.
The 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 S 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 He cryostat reaching temperatures of 1 . −
50 dBm and using room-temperature amplifiers. We measured internalquality factors up to 2 × 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 4 K with a noise temperature of 3 K,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.
Fig. 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.
Two 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 by Q c = 3.4 × . 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 ≈ .
2, is shownin Fig. 4. The | S | 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 f ( Φ ) and quality factor Q ( Φ ) are shown for both samplesin Fig. 5 over one flux period. The f ( Φ ) curves in panels a and c are fitted withEq. 7. The agreement is good over the whole frequency range, which extends from1 . .
75 GHz, 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 Φ / × for sample A and 3 × 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 Q c for C c = 27 fF. Therefore sample A quality factor islimited by the coupling to the 50 Ω lines around integer values of Φ . The situationis different for sample B : the measured value is one order of magnitude lower thanboth the quality factor Q c =6 × expected for C c = 2 fF and the measured Q oftest resonators with the same capacitors (see table 1). This unexplained broadening b Q SAMPLE B f ( G H z ) SAMPLE A a c d Fig. 5 (color online) a and c: Measured resonance frequency f as a function of Φ/Φ (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/Φ . The solid line is calculated according to themodel (see text) for a temperature T = 60 mK. 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 Φ /
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 = ~ ω ( Φ ) n , n = 1 / { exp[ ~ ω ( Φ ) /kT ] − } being the averagephoton number. The photon number and energy fluctuations are n − n = n ( n + 1)and p δE = q E + ~ ω ( Φ ) E. (10)The characteristic time of these energy fluctuations being given by the cavity damp-ing time Q/ω with Q ≫
1, a quasi-static analysis is valid and leads to an inhomoge-neous broadening δω inh = | dω /dE | p δE . Using Eq. 9, we get Q − inh ( Φ ) = δω inh ( Φ ) ω ( Φ ) = N ω ( Φ ) πR [1 + 2 Nε ( Φ )] ff ϕ I c ( Φ ) p δE . (11)The resulting quality factor is Q − = Q − inh + Q − ext , which is plotted as full curves inpanels b and d of Fig. 5, for T = 60 mK. The agreement is good, although Eq. 11 results from a first-order expansion that is no longer valid in the close vicinity of Φ /
2. We havealso observed that Q values significantly degrade around Φ / Φ . 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 Q inh andcould account for the residual discrepancy between experimental data and theoreticalcurves in panels b and d of Fig. 5. We have designed and measured SQUID-based stripline resonators that can be tunedbetween 1 . .
75 GHz, with a maximum Q =3 × limited by an unknownmechanism. The quality factor degrades due to thermal noise around Φ /
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. S´enat, P.F. Orfila and J.C. Tack, and fruitful discussionswithin the Quantronics group and with A. Lupascu, A. Wallraff, M. Devoret, and P. Delsing.
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