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A study of the tunnelling-charging Hamiltonian of a Cooper pair pump M. Aunola Dept. of Physics, University of Jyväskylä, P.O. Box 35 (Y5), FIN-40351 Jyväskylä, Finland Email: [email protected].fi (February 6, 2008) 1 0 0 2 [ (c−1 1)2/N]1/2. The model Hamiltonian is diago- Generalpropertiesofthetunnelling-chargingHamiltonian nalkwikth −respect to φ, which is assumed to be fixed, but n of a Cooper pair pump are examined with emphasis on the P could be controlled by an external bias voltage accord- a J symmetries of the model. An efficient block-diagonalisation ing to dφ/dt = 2eV/~ and is subject to any voltage 1 schemeandacompatibleFourierexpansionoftheeigenstates fluctuations.7,8 T−he conjugate variable Mˆ, the average is constructed and applied in order to gather information on 1 number of tunnelled Cooper pairs, is undetermined in importantobservables. Systematicsoftheadiabaticpumping with respect toall ofthemodelparameters areobtained and the present model. ] n thelink to thegeometrical Berry’s phase is identified. o c Adiabatic transport of single electrons in arraysof small - r metallic tunnel junctions has been widely studied in re- p cent years.1,2 In the Coulomb blockade regime phase- u s shifted gate voltages have been used to induce a dc cur- . rent I = nef, where n is the number of carried elec- t a tronsand−f isthegatingfrequency. Normal-statepumps m transporting single electrons have reached accuracy that - can be considered for metrological applications.2 Pump- d ing of Cooper pairs has gained interest due to new ideas n in quantum measuring and computing.3,4 o c A quantitative theory of pumping Cooper pairs in [ gated arrays of Josephson junctions when the environ- ment has negligible impedance has been presented.5,6 2 v The leading order pumped current is I 2ef[1 FIG.1. (a)AschematicdrawingofagatedJosephsonarray 8 aN(εJ)N−2cosφ], where aN is a constant≈a−nd εj :−= of N(≥ 3) junctions. The Ck and Cg,k are the capacitances 0 E /E is the coupling strength. Here E and E := ofthejunctionsandgates,respectively. b)Theoptimaloper- J C J C 4 (2e)2/2C are the Josephson coupling energy and the ation ofthegatecharges~q. Each cyclecarries approximately 1 charging energy, respectively. According to recent cal- one Cooper pair through thearray, when εJ≪1. 1 culations the cosφ-dependent inaccuracy should be ex- 0 perimentally observable at least in a certain frequency The matrix elements of the charging Hamiltonian H 0 C / range.7 In this article the model for adiabatic pumping are given by the capacitive charging energy t a from Refs. 5 and 6 is examined in a thorough manner, 2 m concentrating on the symmetries and general features of N v2 1 N v d- tshtreonsigmcpoluifipelidngssysatnedm,lotnhguasrerxaytesn.dTinhge othveeratrlleabtemhaevnitoutor h~n|HC(~q)|~niφ =ECk=1 ckk − N k=1 ckk! , (2) X X n ofthepumpedcurrentisexplainedandsystematised,but   o wherethenumberofCooperpairsoneachislandisgiven not rigorously proven. c by ~n. The quantities v , k =1,...,N, are a solution of AschematicviewofaCooperpairpumpandtheideal k : v operation of the gate voltages V are shown in Fig. 1. Xi OnanyoftheN legsofacycleatg,mk osttwogatevoltages vk−vk+1 =nk−qk. (3) are changed. The tunnelling-charging Hamiltonian Tunnelling of one Cooper pair through the kth junction r a changes ~n by~δ ,wherethenon-zerocomponentsare(if H =HC+HJ, (1) applicab|le)i(~δ )k=1 and (~δ ) = 1. The tunnelling k k k k−1 − neglectsthequasiparticletunnellingandotherdegreesof Hamiltonian then reads freedom. The full set of model parameters are ε , total J N phasedifferenceoverthearrayφ,therelativejunctionca- H = ckEJ(~n+~δ ~neiφ/N + H.c.). (4) pacitances~c, where c := C /C and N C−1 = N/C, J − 2 | kih | k k k=1 k ~n,k=1 and the (normalised) gate charges ~q := q ,...,q , X 1 N−1 P { } where q := C V /2e. For homogeneous arrays Thesupercurrentflowingthroughthearrayisdetermined k g,k g,k − c := 1 and the inhomogeneity is quantified by X = by the supercurrent operator k inh 1 I =( 2e/~)(∂H/∂φ), (5) becomes degenerate for φ = π + 2lπ, l Z. For ho- S − ∈ mogeneous arrays resonance points are located at ~q = aGâteauxderivative9 ofthefullHamiltonian. Bychang- ~n (1,...,1)/N, where ~n is arbitrary. The corresponding the gate voltages adiabatically along a closed path ing±poles in the Berry’s phase give raise to Q , and ε p J Γ, a charge transfer Qtot := Qs +Qp is induced. The determines which poles are important. pumped charge, Qp, depends only on the chosen path, Due to symmetry of the representationH˜ the original while the charge due to direct supercurrent, Qs, also de- amplitudes in 2π-periodic states are given by pends on how the gate voltages are operated. If the system remains in a adiabatically evolving state m , the ∞ total transferred charge,Qtot, in units of 2e re|adis5,6 a~qn~,φ = a~qn~,leiφ(l+Yn~/N), (10) − l=−∞ X 1 τ ∂E (t) mI l l dm m S ~Z0 ∂φ dt−2IΓl(X6=m)Im(cid:20)h E| l|−ihE|m i(cid:21), (6) wcohnedrietiroenalaF~qn~,o′φu:r=ier|ac~qn~o,′eφffi|eciφieYnn~′t/sNa~qn~fo,lratrheeficxheadrgbeystthaetega|~nu′gie. where dm is the change in m due to a differential The averagednumber of tunnelled Cooper pairs, | i | i change of the gate voltages d~q. Intermediate states |li and energy denominators can M:= ~n,l(l+Y~n/N)(a~qn~,l)2, (11) be removedby rewritingthe off-diagonalmatrix element isuniqueuptoagPauge-dependentinteger. Theapparent between stationary states as contradiction between sharp phase difference combined m∂H/∂φl =(E E )[∂( m)/∂φ]l , (7) withsharpvalueof isanartefactdue togauge-fixing. h | | i l− m h | | i StrongercouplinginMcreasesthe varianceof . Disconti- M and using the identity l l = ˆ1. The canonical rep- nuities in gauges certainly occur on any closed path en- l| ih | resentationMˆ = i∂/∂φconnectsthepumpedchargeto circling an odd number of resonance points. A gauge is the average numb−er of tPunnelled Cooper pairs by unstable near a discontinuity,but everywhereawayfrom resonance points many valid gauges exist. Especially on Q =2 Re mMˆ dm . (8) the gating path depicted in Fig. 1, the dominant charge p h | | i states on each leg give very stable gauges. IΓ h i The pumped charge, Q , can be evaluated using a p A single eigenstate for phase differences φ and φ+ dφ gauge-independent differential expression is required for each integration point. This expression iBwdaeersnrtgyiifi’vseespnhMwâistaeh,sotγuhmte(aΓlnin)ekx=pbleiitcwiΓtehiemdne|nQdtmpifiiac.1na0tdioTtnhheiengceRooenmfn.ee5tcrtaiicnoandl dQp(φ)=l′∞=0~n,l=∞−∞(cid:20)2(l1++Yδ~nl′/0N)d(a~n,la~n,l+l′) X X wasmentionedinRef.11. FHorεJ →0onlytwochargeare +l′(a~n,lda~n,l+l′ −a~n,l+l′da~n,l)]cos(l′φ), (12) of importance during each leg. From the wave function where the φ-independent average is simply d . Due to m =[(1 a2)1/2, aeiφ/N]T,wherea:0 1,oneobtains M | i − → the normalisation of m , the coefficients are orthonor- hmA|Mˆre|dfemrein=ceds(taa2t)e/2N~n aninddQucpe=s a1cfoornvaenfuiellnctylcalbe.elling mal, i.e. ~n,la~n,la~n,l|+l′iφ= δl′0, which cancels the terms | 0i multiplying the fulldifferentialbyl′. Expression(12)in- of the charge states. Each state is denoted by in- P dicates that the averaged charge transfer for a full cycle tegers y N , 0 Y := y < N, such that { k}k=1 ≤ ~n k k is is exactly 2e, regardless of the inhomogeneity of the ~n = ~n + y ~δ . The numbers y tabulate the num- − 0 k k k P k array or reasonable deformations of the gating path. ber and direction of tunnellings from ~n to ~n. The dis- 0 The tunnelling-charging Hamiltonian can often be tance betwPeen charge states is defined by d˜(~n ,~n ) := 1 2 block diagonalised with the following transformation. min Nk=1|yk(1)−yk(2)+l|:l ∈Z . If d˜(~n1,~n2) = 1 Letanorthonormalbasis{|si} spanthe Hilbert spaceH (=Al)(cid:16),cPhthaennge|~no1fi abnadsis|~nU2i a~nre (lt=h)(cid:17)neeaiφrYen~st/Nn~neighbyoieulrdss. a asndHtsh′e.maCtrhioxoesleempernotjescotifoanHoapmerialttoonrsianPHi bbeyhsPsi′ :== HamiltonianmatrixH˜ ={U| Hi}U−1{w|ithpropeir}tiesH˜(φ+ h |kdi=(|<1∞i)|ikihik|andrequirethat ijPiP{j =} iPi =ˆ1. 2π) = H˜(φ) and H˜( φ) = (H˜(φ))∗. Thus eigenstates If all of the row sums − P P P and eigenvalues of H˜ are periodic under 2π, or state la- W := dj h , k =1,...,d , (13) bels change cyclically. Usually the former happens, so ij,k k′=1 ikjk′ i the supercurrent in a stationary state is described by a are independentPof k, the Hamiltonian commutes with Fourier sine series the projection operator P := ψ ψ , where ψ = i| iih i| | ii hISi(N,εJ,φ,~c,q~) := ∞l=1αlsin(lφ). (9) dHi−1=/2H dki=1H|iki. Twhiuths HmactarinxbPeelewmreintttesnofasHa dgirivecetnsausm The ground state supercurrPent behaves differently only PP ⊕ ˆ1−P P attheso-calledresonancepoints,wherethegroundstate h := ψ H ψ =W (d /d )1/2 =W∗(d /d )1/2. (14) ij h i| | ji ij i j ji j i 2 The block-diagonalisingtransformation is not specific to the present problem, as it amounts to extracting states TABLEI. Thenumberofsubspacelabels(sl)andincluded with a specific symmetry from all of the system’s eigen- chargestates(ch)forbasesB(l) withseveralN andexamples 0 states. The ground state of H +H is always an eigen- of bases B(l) and B(l). C J 1 2 state of H , because aq~,φ=0 > 0 for all ~q and ~n. The P ~n Fourierexpansion(10)canbeusedsimultaneouslyifeach N sl ch sl ch 0/1/2 l l subspace label, i, corresponds to a single value of Y . ~n 4 52 182 1872 9572 For homogeneous arrays at ~q = ~n all junctions 0 2 14 0 5 82 626 3082 46400 are indistinguishable in terms of the charging energy. 0 2 11 6 114 1918 3468 1.56·105 The subspace labels are of the form m¯ with d = 0 2 9 N!/( jjm=a1xmj!). This subspace containsz¯charge em¯igz¯en- 7800 11476822 154442988 3396441887 51..2320··110056 states carrying the label 9 210 37082 2899 2.44·106 Q 0 2 6 8 69 12331 14727 6.15·107 ~y = z1(1),...,z1(m1),z2(1),...,z2(m2),...,zj(mmajxmax) , (15) 712 24855 9452 128421194 2.19·106 (cid:16) (cid:17) wherez < <z andm + +m =N,orany 1 ··· jmax 1 ··· jmax distinct permutation of ~y. Thus only the number and multiplicity of tunnellings is of importance. As a concrete example, take basis B(l) and the sub- On the gating path the gate charges can be written 1 spacelabelsN ,(N 1,1) and(1,N 1) , stand- (afisn~qal=) o(p1ti−mxal)~nc0ha+rgxe~ns1t,awtehfeorret|h~ne0ig(iv|~nen1i)legis.tThehuinsiotinael ing for 2N +10charg−e eige(n0s,1t)ates at ~q =−~n0.(T0,1h)e block- diagonalised Hamiltonian obtained from Eqs. (14) and junctionbecomesdistinguishableandsubspacelabelscan (16) is given by bechosenasz ;m¯ ,wherem¯ referstotheremainingN 0 z¯ z¯ 1 components, 0 ≤ z0+ Ni=−11mizi < N and dz0;m¯z¯ −= 0 KN∗ KN (NT−he1)d!/ia(gojjnm=aa1lxmanjd!).nonP-zero off-diagonal matrix ele- Hbd =KKN∗ 2δ(N K−1)//√NN 2δ(NN3KN∗1)//√NN (17) ments of HQ are the common charging energies and N N3 N − P   −(εJmj/2)e±iφ/N(d(z0;)m¯z¯/d(z0′;)m¯z′¯′)1/2 (16) iwnhgeorerdKerNsu:p=er−cu(rεrJe/n2t)efoiφr/εN√N1. iFsoarlrNead=y r3eptrhoedulecaedd-. J ≪ with m = 1 when applicable, respectively. The block- Note that labels (N 1,1) and (1,N 1) canbe 0 (0,1) (0,1) − − diagonalised matrices are sparser than the original ma- joined, if φ is a multiple of π. trices and the amplitudes in eigenvectors are multiplied The discussion has now lead to the main results of by (d )1/2 due to combining of several amplitudes thepaper,thegeneralsystematicsofthepumpedcharge. (z0;)m¯z¯ into one. A further block diagonalisation is possible at For ideal gating sequences d = 1/N for each leg due M ~q =~n and ~q =(~n +~n )/2 if φ is a multiple of π. to symmetry. Corresponding integrated pumped charge 0 0 1 A truncation picks the charge states required for reli- reads able evaluation of eigenstates and observables. First an initial truncation, a set of states Bi = ~nj , is chosen. Qp(N,εJ,φ,~c,leg):=1/N + ∞l=1bl,legcos(lφ), (18) {| i} Its extensions, bases B(l), contain all neighbours up to i which is always positive. ReplaPce 1/N by d for non- and including lth nearest neighbours of each and every M ideal cycles. When performing numerical calculations, ~n . For non-ideal cycles and/or inhomogeneous arrays | ji the basis mustcontainall ofthe important chargestates the initial truncation is the “b basis” of Ref. 6 which re- andthe number of anglesused infast Fouriertransform, produces leading order supercurrent and inaccuracy. A 2j, must be large enough not to allow misidentification “c basis” is just the first order extension of a “b basis”, of non-negligible components a and a as a single i.e. “b basis(1)”. ~n,l ~n,l+2j component. Failure to satisfy these requirements causes For ideal cycles and homogeneous arrays the optimal systematicalerrorduetoincorrectamplitudesandspuri- B truncation corresponds to 3N 2 labels, that is 0 − ousinterferencebetweenFouriercomponentsinEq.(10), 0;(j,N 1 j) ,“1;(j,N 1 j) ”,j =0,...,N − − (0,1) − − (−1,0) − respectively. The integration points should be chosen so 1, and 1;(j,N 1 j) , j = 1,...,N 2. These − − (0,1) − that the magnitude of norms dm is neither too large 3 2N−1 2chargestatesactuallycontributetotheleading k| ik nor varies too much. · − ordersupercurrentandinaccuracy. Morerestricted,few- The rate of convergence for each cos(l′φ)-dependent state truncations B1 = ~n0 and B2 = ~n0 , ~n1 are term in Eq. (18) behaves as 1/(#steps)2. For a reason- {| i} {| i | i} ofuse especiallywhenthe groundstate energyissought. able choice of integration points 300–500 steps per leg, The efficiency of the block diagonalisationfor bases B0(l) muchlessthanusedinRefs.5and6,usuallygivesrelative is shown in Table I where the number of labels is com- precision of the order of 10−5 (disregarding the system- paredagainstthe numberofincludedchargeeigenstates. aticalerror)for coefficients with magnitude greaterthan Examples of bases B(l) and B(l) are also given. 10−8. This greatly diminishes convergenceproblems due 1 2 3 to smallness of da for large bases and large number of ferred charge Q = Q for inhomogeneous ar- ~n,l p leg p,leg angles. The expression 2l′a~n,lda~n,l+l′ has been used as rays does not have to satisfy βl >βl+1, although always P the latter part of Eq. (12) for many of the data points. b /b <1. Thissymmetryisalsobrokenbynon-ideal l+1 l − Inthiscasetherateofconvergencebehavesas1/#steps. gating sequences, even for single legs. Bases B(l) have also been used when calculating Q . In order to conclude, the properties of the tunnelling- 2 p charging Hamiltonian of a Cooper pair pump have been TABLE II. The first four ratios β and the limiting value examinedusinganefficientblock-diagonalisationscheme l β˜for ε =0.4 as function of N and for N =4 as function of and a compatible Fourier expansion of the eigenstates. J ε . Subscriptof β˜inthelast columnindicates thecoefficient Explicit enforcement of the model symmetries produces J β used when estimating thelimiting value. strong systematics of the pumped charge, even if the l structureoftheFouriercoefficients b ∞ wasnotex- { l,leg}l=0 (ε ) β β β β (β˜) haustively proven. These properties are possibly related J N 1 2 3 4 l tothedynamicalalgebraofsingleandcoupledJosephson 0.4 0.8802 0.7782 0.7470 0.7347 0.722 3 60 junctionsdescribedinRef.14,whichoffersacomplemen- 0.4 0.7250 0.5706 0.5354 0.5225 0.509 4 25 tary view of the present problem in case of a supercon- 0.4 0.5494 0.3842 0.3551 0.3451 0.335 5 20 ducting loop. 0.4 0.3844 0.2393 0.2188 0.2121 0.206 6 15 0.4 0.2516 0.1401 0.1272 0.1231 0.120 7 11 0.48 0.1563 0.0785 0.0708 0.0685 0.0679 ACKNOWLEDGMENTS 0.4 0.0934 0.0427 0.0383 0.0370 0.036 9 7 0.4 0.0542 0.0227 0.0203 0.0196 0.019 10 7 ThisworkhasbeensupportedbytheAcademyofFin- 0.04 0.0188 0.0096 0.0095 0.0095 0.0095 4 4 land under the Finnish Centre of Excellence Programme 0.2 0.3444 0.2164 0.2049 0.2025 0.201 4 14 0.6 0.8853 0.7833 0.7469 0.7296 0.701 2000-2005 (Project No. 44875, Nuclear and Condensed 4 40 0.8 0.9475 0.8876 0.8598 0.8444 0.808 Matter Programme at JYFL). The author thanks Mr. 4 80 1.0 0.9738 0.9390 0.9198 0.9079 0.870 J.J. Toppari for insightful comments. 4 120 1.2 0.9860 0.9654 0.9526 0.9441 0.908 4 160 The sequence of the Fourier coefficients b ∞ , { l,leg}l=0 where b =2/N, is alternating and decreasing in magni- 0 tude. Furthermore, the ratios β := b /b , l=1,2,... 1H. Pothier, P. Lafarge, C. Urbina, D. Esteve and M.H. l l l−1 form a decreasing sequence with l|imiting|value β˜ := Devoret,Europhys. Lett 17, 249 (1992). lim β b β .12 Because ∂Q /∂φ > 0 in the range 2M.W.Keller,J.M.Martinis,N.M.ZimmermanandA.H. φ l→(0∞,π)l,≥Q 0is1bounded from abpove by [2(1 β )−1 Steinbach,Appl.Phys.Lett.69,1804(1996);M.W.Keller, ∈ p − 1 − J.M. Martinis and R.L. Kautz, Phys. Rev. Lett. 80, 4530 1]/N and from below by 1/N 2β /(N +2β ). For fi- 1 1 − (1998); M. W. Keller, A. L. Eichenberger, J. M. Martinis nitevaluesofεJ theratiosβl andβ˜/β1 aremonotonously and N. M. Zimmerman, Science 285, 1716 (1999). increasingfunctionsofεJ withlimitingvalueofunity,be- 3D.V. Averin,Solid StateCommun. 105, 659 (1998). cause Qp(φ = 0,εJ ) 0, which enforces a stricter 4A.ShnirmanandG.Schön,Phys.Rev.B57,15400(1998). →∞ ց limit for the ratio β˜/β1. This behaviour is explained by 5J. P. Pekola, J. J. Toppari, M. Aunola, M. T. Savolainen increasinglong-range(high-l′)correlationsinEq.(12)or and D. V.Averin,Phys. Rev.B 60, R9931 (1999). actuallyinthe state itself. Onthe otherhand, the ratios 6M. Aunola, J. J. Toppari and J. P. Pekola, Phys. Rev. B are monotonously decreasing functions of N for fixed ε 62, 1296 (2000). J as correlations are weakened in longer arrays. All of the 7J. P. Pekola and J. J. Toppari, submitted to Phys. Rev. above-mentioned features are clearly shown in Table II, Lett. (2000). where β for the homogeneous case are depicted as func- 8G.-L. Ingold and Yu. V. Nazarov, in Single Charge Tun- l tion of N at ε =0.4 and as function of ε for N =4.13 nelling, Coulomb Blockade Phenomena in Nanostructures, J J Thesepropertiesofthe pumpedchargeareratherrobust eds. H. Grabert and M.L. Devoret, (Plenum Press, New against relatively small systematical errors . York,1992). 9W. Rudin,Functional Analysis, (McGraw Hill, 1973). For small values of ε 1 one has β b β and β (Nε /2)N−2[N(NJ ≪1)/2(N 2)!].lT>1he≈effe0ct1s due 10M. V. Berry,Proc. R. Soc. Lond.A 392, 45 (1984). 1 ∼ J − − 11G.Falci, R.Fazio, G.M.Palma, J.Siewert,andV.Vedral, to inhomogeneity, that is the relative sizes of b , can 1,leg Nature407, 355 (2000). estimated by fixing the the leg index r in the inhomo- 12To some degree already in matrix elements hm|Mˆ|dmi. geneity prediction, Eq. (37) of Ref. 6. The correspond- 13The full data set and some tools for reproducing the data ing results work quite well, only slightly overestimating is available at http://www.jyu.fi/∼mimaau/Cooper/ the ratio between largest and and smallest coefficients, 14E. Celeghini, L. Faoro, and M. Rasetti, Phys. Rev. B 62, even for large inhomogeneities X . The total trans- inh 3054 (2000). 4