Direct observation of Josephson capacitance M. A. Sillanp¨aa¨1, T. Lehtinen1, A. Paila1, Yu. Makhlin1,2, L. Roschier1, and P. J. Hakonen1 1Low Temperature Laboratory, Helsinki University of Technology, FIN-02015 HUT, Finland 2Landau Institute for Theoretical Physics, 119334 Moscow, Russia 6 TheeffectivecapacitancehasbeenmeasuredinthesplitCooperpairbox(CPB)overitsphase-gate 0 bias plane. Our low-frequency reactive measurement scheme allows to probe purely the capacitive 0 susceptibility due to the CPB band structure. The data are quantitatively explained using param- 2 eters determined independently by spectroscopic means. In addition, we show in practice that the n method offers an efficient way todo non-demolition readout of the CPB quantum state. a J PACSnumbers: 67.57.Fg,47.32.-y 3 1 Energy can be stored into Josephson junctions (JJ) a simple way to perform a non-destructive measurement ] according to E = E cos(ϕ), where ϕ is the phase dif- oftheCPBstateusingpurelytheCPBJosephsoncapac- J l − l ference across the junction, and the Josephson energy itance. a h EJ is related to the junction critical current IC through An SCPT (Fig. 1) consists of a mesoscopic island - IC = 2eEJ/~. By using the Josephson equations, this (total capacitance C = C + C + C ), two JJs, and s 1 2 g energy storing property translates into the well-known e of a nearby gate electrode used to polarize the island m fact that a single classical JJ behaves as a parametric with the (reduced) gate charge n = C V /e. The is- inductance L =~/(2eI ) for small values of ϕ. g g g . J C land has the charging energy E = e2/(2C), and the t C a Since the early 80’s, it has become understood that junctions have the generally unequal Josephson energies m ϕ itself can behave as a quantum-mechanical degree of E (1 d),wheretheasymmetryisgivenbyd. TheSCPT J ± - freedom [1]. In mesoscopic JJs, this is typically associ- HamiltonianisthenE (nˆ n )2 2E cos(ϕ/2)cos(θˆ)+ d C g J − − n ated with the competition between the Josephson and 2dE sin(ϕ/2)sin(θˆ) C V2/2. Here, the number nˆ of o Coulomb effects at a very low temperature. These fun- extrJa electron charges−ongthge island is conjugate to θˆ/2, c damentalphenomenatakeplaceifchargeonthejunction [ is localized by a large resistance R > R = h/(4e2) [2], where θˆis the superconducting phase on the island [10]. Q The SCPT is thus equivalent to a CPB (single JJ and 2 as well as in the Cooper-pair box (CPB), or the single- a capacitance in series with a gate voltage source) but v Cooper-pair transistor (SCPT), whose quantum coher- 7 ence is often considered macroscopic [3]. with a Josephsonenergy tunable by ϕ=2πΦ/Φ0, where 1 Φ0 =h/(2e) is the quantum of magnetic flux. 5 Inthefirsttheoreticallandmarkpapers[4,5]onquan- If d = 0 and E /E 1 the ground and excited 4 tum properties of ϕ it was already noticed that due to J C ≪ 0 localization of charge Q, the energy of a the JJ system state energies are (ng = 0...2): E0,1 = EC(n2g −2ng + 5 is similar to that of a non-linear capacitance. In spite of 2) (E cos(ϕ/2))2+(2E (1 n ))2 C V2/2, with 0 ∓ J C − g − g g at/ tShCePiTmpinorttahnecperoofmtihseinpghfienelodmoefnosunpeesrpceocniadlulyctiinngCqPuBbiotsr acolmarpqguetegathpetboahnidgshneurmleevreilcsa.llyFoinr tahegecnhearragleEstJa/tEeCb,aswise. m [6, 7], direct experimental verification of the Josephson Theeffective ”Josephson”capacitanceofthe CPBcan capacitance has been lacking, likely due to challenges - d posed by measuring small reactances, or by the extreme be related to the curvature of band k, similar to the ef- n fective mass of an electron in a crystal: sensitivity to noise. o c In this Letter, we present the first such direct experi- v: ment [8], where we determine the Josephsoncapacitance Ck = ∂2Ek(ϕ,ng) = Cg2∂2Ek(ϕ,ng). (1) i in the Cooper pair box. Related experiments have re- eff − ∂V2 −e2 ∂n2 X g g cently been performed by Wallraff et al. [9], but in their r case the key role is played by the transitions between a Usually, the system effective capacitance is obtained levels of a coupled system where the band gap between from a Lagrangian or Hamiltonian as ∂2 /∂V2 = the ground state and first excited state of the CPB, (∂2H/∂Q2)−1, without the minus sign. In EqL. (1),ghow- E E =∆E, is nearly at resonance with an oscillator 1 0 − ever, E ’s are, more precisely, the eigenvalues of the ofangularfrequencyω . Thus, detuning fully dominates k 0 Routhian = θ˙∂ [11], which serves as a Hamil- over the Josephson capacitance which can be clearly ob- tonian forHthe n,θθ˙Lde−grLee of freedom but as minus La- served in our experiments where we study directly the grangian for the phase α e V dt and V α˙, thus reactive response of the lowest band E0 at a frequency ≡ ~ g g ∝ ω0 ∆E/~. We determine the experimental parame- leading to Eq. (1). R ≪ ters independently using spectroscopy, and demonstrate Using the analytic formulas for E in the limit 0,1 2 capacitance, we used a large C > 0.5 fF. It was made g using an Al-AlO -Al overlay structure (see the image in x Fig. 1), with a prolongedoxidizationin 0.1 bar of O for 2 15min. Otherwise,ourCPBcircuitshavebeenprepared using rather standard e-beam lithography. The tunnel junctions having both an area of 60 nm 30 nm corre- × spond to an average capacitance of 0.17 fF each. The ∼ overlaygatehasC 0.7fFforanareaof180nm 120 g ≃ × nm. The main benefit of our method comes from the fact thatweworkataresonator(angular)frequencyω much 0 lower than the CPB level spacing ∆E. In Ref. [9] it is shown that ω depends on the resonator - CPB 0 (qubit) interaction because of two contributions. The frequency change is ∆ω = g2/δ, where the detuning 0 FIG. 1: Schematic view of the experiment. The resonant δ = ∆E ~ω , and the coupling coefficient g contains 0 frequencyoftheLC circuit (madeusinglumpedelements) is − the curvature of energy bands. In general, both δ and tunedbytheeffectivecapacitanceCeff oftheCooperpairbox the curvature depend on the (n , ϕ) point. Now, in shown in theSEM image. For details, see text. g our case everywhere ∆E ~ω , δ ∆E, and hence 0 ≫ ≃ ∆ω = g2/∆E = C ω /(2C ) has a contribution by 0 eff 0 S E /E 1 we get only the second derivative, not by the detuning. There- J C ≪ fore, we can resolve the reactive response due to purely 2C2E C(0,1) =C g C the bands of CPB, which has not been possible in previ- eff g − e2 × ous experiments. (2) E E2(1+cosϕ) When doing microwave spectroscopy, we have to con- 1 C J , × ∓ 4E2(n 1)2+ 1E2(1+cosϕ) 3/2! sider also the other side of the coin: ∆E increases C g− 2 J due to interaction with the resonator by [16] ε = which redu(cid:2)ces to the classical geometric(cid:3)capacitance ~ 2Ng2/∆E+g2/∆E , where N is the number of (1/C +1/(C +C ))−1 in the limit of vanishingly small quanta in the resonator. When driven by a gate am- g 1 2 (cid:0) (cid:1) E , except where Cooper-pair tunneling is degenerate plitude V , the resonator energy is E = V2C /2. At a J g R g S [e1r2a]l.ENJ/uEmCercicaanllybeevfoaulunadteindRgreaf.ph[1s3]o.f Ce(ff0,1) for a gen- eh/ig(h2Cegx)ciatantdiohnenamcepNlitu=deEnRg/≃(~1ω/02)w=eew2oCu/l(d8Chag2v~eωV0)g ≃∼ Our experimental scheme is illustrated in Fig. 1. We 4 103 which would yield ε ∆E. The data shown × ∼ perform low-dissipation microwave reflection measure- in this paper are, however, measured at a very low ex- ments [14, 15] ona series LC resonatorin which the box citation of ng 0.05 which corresponds to N 40 and ∼ ∼ effectivecapacitance,Eq.(1),isapartofthetotalcapac- ε 200 MHz which is an insignificant contribution to ∼ itance C +Ck . The resonator is formed by a surface ∆E. S eff mountinductorofL=160nH.Withastraycapacitance Fig. 2 (a) displays the measured phase shift Θ as a of C = 250 fF due to the fairly big lumped resonator, functionofthe twoexternalcontrolknobs(inthe follow- S the resonant frequency is f =800 MHz and the quality ing,n shouldbe understoodasbeing due to the control 0 g factorQ 16islimitedbytheexternalZ =50Ω. When gate, n =C V /e). The results show full 2e periodic- 0 g g0 g0 ≃ Ck varies,thephaseΘofthereflectedsignalV =ΓV ityasafunctionofn ,checkedbyincreasingtemperature eff out in g changes, which is measured by the reflection coefficient abovethe 2e ecrossoverat 300mK,andaΦ period 0 − ∼ Γ=(Z Z )/(Z+Z )=Γ eiΘ. Here,Z istheresonator with respect to Φ. The data was measured without any 0 0 0 − impedance seen at the point labelled ”in” in Fig. 1. In microwaveexcitation, and hence we expect to see effects all the measurements, the probing signal V was contin- duetothegroundbandC0 . Thecorrespondingtheoret- in eff uously applied. ical picture, obtained using Eq. (1) and straightforward Since we are rather far from matching conditions, the circuit formulas for Γ, is given in Fig. 2 (b). reflectionmagnitudeΓ remainsalwaysclosetoone. The As a vital step to get convinced of the measured ca- 0 variation in Θ due to modulation in Ck is up to 40◦ in pacitance modulation versus the calculation, we carried eff our measurements, corresponding to a shift of resonance out a detailed determination of the sample parameters frequency ∆f 6 MHz. In additionto bandpass filter- independently of the capacitance modulation by using 0 ≃ ing, we used two circulators at 20 mK. microwavespectroscopy (Fig. 3). To the weakly coupled As seen in Eq. (2), the modulation depth of Ck is control gate C of the SCPT, we applied continuous- eff g0 sensitive to C . Therefore, in order to faithfully demon- wave microwaves while slowly sweeping the CPB band g strate the Josephson capacitance in spite of the stray gap∆E withϕandn . Wheneverthemicrowaveenergy g 3 the microwave energy in Fig. 3 (c). Basedonthesurfacearea 0.022(µm)2 oftheoverlay ∼ gate, we estimate C 0.5 1 fF. The exact value was g ∼ − obtained by fitting to the modulation depth of C0 (see eff Eq. (2)), yielding C = 0.65 fF, corresponding to a spe- g cific capacitance very reasonable to a thick oxide 30 ∼ fF/µm2. FIG.2: (coloronline)(a)PhaseshiftΘmeasuredataprobing frequency 803 MHz ∼f0, and (b) Θ calculated using Eq. (1) with the ground band E0(ϕ,ng) evaluated numerically with parameters of Table I. FIG. 3: (color online) (a) Illustration of themicrowave spec- troscopy used to map the SCPT band gap ∆E = E1−E0. Thethreehorizontalplaneswhichintersectthebandgapcor- matchesthebandgap,thatis,~ωRF =∆E,theCPBbe- respond,from bottom to top,to themicrowave energy ~ωRF comes resonantly excited. Since typically the Josephson usedin(c)...(e),respectively. Whenever∆E =~ωRF (dashed corrections to the geometric capacitance are opposite in lines), thesystem experiencesresonant absorption; (b)peaks of resonant absorption (arrows) in the measured phase shift sign for the bands 0 and 1 (see Eq. (2)), band 1 would contribute an opposite phase shift signal. At resonance, at ωRF =11 GHz; (c) - (e) spectroscopy data represented as we would then expect to see mixture of C0 and C1 , surfaces in the ϕ, ng plane. The resonance conditions shown eff eff in (a) are plotted on top of the data; (f) T1 as a function of weighted by the state occupancies which depend on the measurement strength at ϕ=0, ng ∼1. microwave amplitude. We calculate that a high enough amplitude sufficient to saturate the populations into a ◦ 5so0rp−ti5o0n%peamkixitnurteh,ewmouealdsuyreiedldΘa. ∼Th3e erxespoencatantcieonabis- EJ (K) EC = 2eC2 (K) EJ/EC RT(kΩ) C (fF) d Cg (fF) 0.30 0.83 0.36 55 1.1 0.22 0.65 confirmed in Fig. 3 (b), where the resonance peaks are displayed at a few values of ϕ (when ϕ = 0, microwave TABLEI:Sampleparametersdeterminedbymicrowavespec- energy does not exceed the band gap, and for ϕ=π the troscopy. RT is theseries resistance of the two SCPT tunnel peak height is lower due to a smaller matrix element). junctions(otherparametersaredefinedintextandinFig.1). While slowly sweeping ϕ and n , the resonance condi- g tions correspond to contours (see Fig. 3 (a)), which ap- Fig. 4 illustrates the bare gate and flux modula- pearasannularridgesintheexperimentaldataofgraphs tions without microwave excitation in more detail, and 3 (c)-(e) around the minimum ∆E at (n = 1, ϕ=π). showsthecorrespondingnumericalcalculationsusingthe g − Since the band gap is sensitive to E as well as to the groundband. Asexpected,C reducestothegeometric J eff E /E ratio, the resonance contours allow for an accu- capacitancewhenCooper-pairtunnelingisblockedeither J C ratedeterminationoftheseparameters(TableI). Forex- by tuning the Josephson energy effectively to zero when ample, at (n = 1, ϕ=0), the band gap is 2E =12.5 ϕ is an odd multiple of π, or by gate voltage. At the g J − GHz, whereas at (n = 1, ϕ=π) ∆E has the absolute Coulomb resonance n = 1, however, the Josephson g g − ± minimum 2dE 3 GHz which was barely exceeded by capacitance is significant. In the special point n = 1, J g ≃ ± 4 ϕ = π, the most pronounced effect is observed, now times were limited to about 7 ns by parasitic reactances ± due to strong Cooper-pair fluctuations. The agreement in the somewhat uncontrolled high-frequency environ- between theory and experiment is good in Fig. 4 except ment, causing noise from Z to couple strongly due to 0 around n = 1 which we assign to intermittent poi- a large coupling κ = C /C 1. The result for T , g g 1 ± ∼ soning by energetic quasiparticles [17]. An estimate us- however, did not depend on the measurement strength ing C from Eq. (2), Θ = 2C √L/(C3/2Z ), falls to (Fig. 3 (f)), which supports the non-demolition charac- eff − eff S 0 within 15 % of the numericalresults except around inte- ter of this scheme. By fabricating the resonator on-chip ger n . it is straightforward to gain a full control of environ- g ment. Then, the impedance seen from the qubit gate Re(Z (ω =∆E/~)) 0.1mΩ,andaworst-caseestimate g yieldsT ~R /[4π≃κ2Re(Z (∆E/~))∆E] 1µs. Fora 1 K g ≈ ≫ dephasing time T averagedover n , we measured 0.5 2 g ∼ ns using Landau-Zener interferometry [20, 21]. This T 2 timeisonthesameorderasthespectroscopylinewidths in Fig. 3. In conclusion, using the phase of strongly reflected microwave signals, we have experimentally verified the Josephson capacitance in a mesoscopic Josephson junc- tion,i.e.,thequantitydualtotheJosephsonInductance. Good agreement is achieved with the theory on the Josephson capacitance. Implications for non-destructive readout of quantum state of Cooper-pair box using the capacitive susceptibility are investigated. We thank T. Heikkila¨, F. Hekking, G. Johansson, M. Paalanen, and R. Schoelkopf for comments and useful criticism. 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