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Decoherence in a Cooper pair shuttle Alessandro Romito, Francesco Plastina, and Rosario Fazio NEST-INFM & Scuola Normale Superiore, I-56126 Pisa, Italy (Dated: January 20, 2009) 9 We examine decoherence effects in the Josephson current of a Cooper pair shuttle. Dephasing 0 duetogate voltage fluctuationscan eithersuppressorenhancethecritical currentand also change 0 itssign. Thecurrentnoisespectrum displaysapeak attheJosephson couplingenergyand showsa 2 phasedependence. n PACSnumbers: a J 0 TheJosephsoneffect1consistsinadissipation-lesscur- where E is the charging energy, E(L,R) is the Joseph- C J 2 rent between two superconducting electrodes connected son coupling to the left or right lead respectively, and ] through a weak link2,3. The origin of the effect stems ng is the gate charge. The variable nˆ is the number of l from the macroscopic coherence of the superconducting excess Cooper pairs in the grain and ϕˆ is its conjugate l a condensate. Since its discovery in 1962, the research on phase, [nˆ,ϕˆ]=−i. The system operates in the Coulomb h devicesbasedonthe Josephsoneffecthasbeenachieving blockade regime, E (b) ≪ E (b = R,L) so that only s- a number of importantbreakthroughsboth in pure2 and the two charge statJes {|n=0Ci,|n=1i} are important. e applied physics3. One of the most recent exciting de- Their relative energy (∆E ) is controlled by the gate C m velopments is probably the implementation of supercon- charge n (t). The superconducting gap is assumed to g . ducting nano-circuits for quantum information process- be the largestenergyscaleinthe problem,sothatquasi- t a ing4, whichrequiresthe ability to coherentlymanipulate particletunnellingcanbeneglected. BothE andE(L,R) m these devices. By now, this has been shown in several C J aretimedependent: whenthegrainisclosetooneofthe - experiments in systems of small Josephson junctions5. leads, the corresponding Josephson coupling is non-zero d Veryrecently,Goreliket al.6,7 proposedaveryappeal- (withvalueE )andthetwochargestatesaredegenerate n J o ing setup, the Cooper pair shuttle, able to create and (positions L and R in Fig.1). In the intermediate region c maintainphasecoherencebetweentwodistantsupercon- (position C), E(L) =E(R) =0. As in Ref.7 we employ a [ ductors. In its simplest realization, shown in Fig.1, the sudden approxiJmationJ(which requires a switching time 3 systemismadeupofasuperconductinggrain,externally ∆t ≪1/E ) and suppose E(L,R)(t) to be step functions v forcedtomoveperiodicallybetweentwosuperconducting J J 4 electrodes. Despite the fact that the grain is in contact 1 with only one lead at a time, the shuttle does not only 4 carry charge, as in the normal metal case8,9,10, but it (L) 12 also establishes phase coherence between the supercon- L φ 2 e φ EJ (R) ductors. L R EJ = 0 2 0 Aim ofthis workis to analyzehow the presence ofthe t t/ environmentaffectsthecoherenttransportintheCooper ng=1/2 ∆ tJ a pair shuttle. The interplay11 between the periodic driv- m EC (R) ing and the environmentaldephasing leads to severalin- C φ φ EJ = 0 - teresting results. By increasing the coupling to the envi- L R d n ronment it may result in an enhancement of the super- t o currentas well as in a change of its sign (π-junction). In ng=0 (R) tC c thelastpartofthis Letterweproposeaneffectiveimple- 2 e EJ : mentationof the shuttle mechanismwhere the switching R φ φ (L) iv of the Josephson couplings is controlled by an external L R EJ = 0 X magnetic field. The shuttle consists of a small super- t r conducting island coupled to two macroscopic leads and ng=1/2 tJ a forced to change its position periodically in time, with period T, from the Right (R) to the Left (L) electrode Figure 1: Time dependence of the Josephson and charging and back (see Fig.1). The grain is small enough so that energies in the Cooper pair shuttle.The three intervals L, C charging effects are important, while the two leads are and R, within the period T = 2tJ +2tC, correspond to the mislaacnrdosicsodpeicscarnibdedhabvyetdheefinHiatempilhtoanseiasnφL,R. The moving s(iitnuteartaiocntiso:nLti)mree)p;rCes)enrtesprEesJeLnt=sEEJJ(L,)n=g 0=,n1g/2=, 0E,JE(RJ)(R=) =0 0(freeevolution time);R)EJ(L) =0,ng =1/2, EJR =EJ - (interaction time). On the left hand side, the corresponding H =E [nˆ−n (t)]2− E(b)(t)cos(φˆ−φ ) (1) position of the shuttle with respect to the leads is shown for 0 C g J b b=XL,R each time interval. 2 ineachregion(seeFig.1). Wefurtherassume,asinRef.6, with w~† = tanh(E /T ) cosφ , sinφ , 0 ,w~† = L,R J b L,R L,R C thatthesystemisatthechargedegeneracyaslongasthe 0, 0, 0 , (cid:0) (cid:1) island is in contact with one of the electrodes. In the in- (cid:0) (cid:1) termediate region (C) it is not necessary to specify the −2γ (T ) 0 −E sinφ J b J L,R exactvariationofng(t), only the time integralof the en- GL,R = 0 −2γJ(Tb) −EJcosφL,R , ergy difference between the two charge states will enter E sinφ E cosφ 0 J L,R J L,R in the results.   (8) The shuttle is coupled via the charge operatornˆ to an environment described by the Caldeira–Leggettmodel12 −γ (T ) −E 0 C b C Hint =nˆ λi(ai+a†i)+Hbath . (2) GC = E0C −γC0(Tb) 10 , (9) Xi   where the bath is taken in thermal equilibrium at tem- InEq.(2),H isthebathHamiltonian,withbosonop- bath perature T . eratorsa ,a†foritsi−thmode. Theformofthecoupling b i i Here, γJ(Tb) and γC(Tb) are the temperature-dependent inEq.(2)candescribeeithergatevoltagefluctuations4or, dephasing rates in the regions L, R and C, respectively, in some limits, random switching of background charges obtained in the Born-Markov approximation. As an ex- in the substrate13. ample, for an ohmic bath with coupling to the environ- In order to analyze the transport process, we evaluate mentα≪1,onehas12γ (T )=(π/2)αE coth(E /2T ) J b J J b the time averagedsupercurrent at steady state and γ (T ) = 2παT . This treatment is valid provided C b b that γ ≪ T ,E , and that the time interval t 1 T J,C b J J(C) I =hIî≡ dthIˆ(t)i, (3) is much longer than both T−1 and E−1 . In the cou- T Z b J(C) 0 pling regionsL andR,the onlyenergyscaleissetby the and the power spectrum of the current fluctuations Josephson energy, while, during the free evolution time, therelevantscaleistheenergydifferencebetweencharge +∞ states. Correspondingly, all the physical quantities de- S(ω)= dτS˜(τ)e−iωτ (4) pend onthe phases 2θ =E t and 2χ= dt∆E (t)16. Z−∞ J J C C The other important variable is the pRhase difference where φ = φ − φ . The effect of damping is characterized L R by the two dimensionless quantities γ t and γ t 17. 1 J J C C S˜(τ)= h Iˆ(t+τ),Iˆ(t) i−hIˆ(t+τ)ihIˆ(t)i. (5) Average current-InthelimitingcaseconsideredbyGore- 2 h i+ lik et al.6,the Josephsoncurrentdoesnotdependonthe In the Schrödinger picture, the current operator is (~ = dephasing rates. One expects, however, that this can- 1) not be always the case. If, for example, the period T is muchlargerthantheinversedephasingrates,theshuttle Iˆ(t)=2eE sin(ϕˆ−φ ) Θ (t) (6) mechanism is expected to be inefficient and the critical J L L current should be strongly suppressed. In fact, we find corresponding to the exchange of Cooper pairs with the a quite rich scenario, depending on the relative value of left lead. The function Θ (t) is defined as follows: the various time scales and phase shifts. L Θ (t)=1whenthegrainisintheLregion,andΘ (t)= TheexpressionofthecurrentI(φ,θ,χ,γ t ,γ t )can L L J J C C 0 otherwise (the functions Θ (t) and Θ (t) are defined be obtained analytically from Eqs.(3,7). However, it is R C analogously). In order to evaluate Eqs.(3, 4), we need rather cumbersome and not instructive, so we prefer to to compute the reduced density matrix of the grainρ(t). present it in some limiting cases. A typical plot of I After one period the evolution of ρ(t) can be computed as a function of θ and φ is shown in Fig.2. Depending through a map M defined by ρ(t+T) = M ρ(t). The on the value of θ (a similar behaviour is observed as a t t stationarylimitisobtainedbystudyingthefixedpointof function of χ), the critical current can be negative, i.e. M 14. Since only two charge states in the grainare rel- the system can behave as a π-junction. The phase shifts t evant, the reduced density matrix can be parametrized, accumulated in the time intervals L,C and R, leading to in the charge basis, as ρ(t)=1/2[1I +~σ·~r(t)], where σ thecurrent-phaserelationshowninFig.2,areaffectedby i (i=x,y,z) are the Pauli matrices and r (t)=hσ i. The the dephasing rates in a complicated way. By changing i i assumption of a stepwise varying Hamiltonian consider- γ t andγ t ,certaininterferencepathsaresuppressed, J J C C ably simplifies the form of the map M , obtained as a resulting in a shift of the interference pattern and ulti- t composition of the time evolutions of ρ in the intervals mately in a change of the sign of the current, as shown L,C,R(seeFig. 1). Separatelyforeachoftheseintervals, in Fig.3. the master equation for~r(t) has the form15 An analysis of the critical current as a function of the dephasing rates reveals another interesting aspect: the ~r˙(t)= [G (t)~r(t)+2γ (T )w~ ] Θ (t) (7) Josephsoncurrentisanon-monotonousfunctionofγ t , k J b k k J J k∈{XL,R,C} i.e. by increasing the damping, the Josephson current 3 γ t ≪117), 2 p 0.67 C C 2e E γ t (cosφ+cos2χ)tanθsinφ J J J I ∼ tanh . 3 p weak T (cid:18)T (cid:19)γ t 1+cosφcos2χ b C C 2 (11) q The current tends to zero if the coupling with the bath p is negligible during the interactiontime. Inthis case,indeed, the time evolution in the intervals L,R is almost unitary, while, in the region C, pure dephasing leads to p a suppression of the off-diagonal terms of the reduced 2 density matrix ρ(t). As a result, in the stationary limit -0.67 thesystemisdescribedbyacompletemixturewithequal 0 weights. The current then tends to zero in both limit- 0 p f 2 p 3 p ing cases of large and small γ t . Therefore one should J J expect an optimal coupling to the environment where the Josephson current is maximum. A regime where the crossover between the strong and weak damping cases Figure 2: Supercurrent (in units of e/T) as a function of canbe describedinsimple terms is the limit γ →0, for C thesuperconductorphasedifferenceφandofthephaseaccu- a fixed value of θ. For example, at θ = π/4 the current mulated during the contact to one of the electrodes θ. The reads eo−thγeCrtCpa=ra4m/5e.teTrshearpelofitxisedobatsa:inχed=for5Tπb/6≪, eE−JγJtJ = 3/4, I= 2Tetanh„ETJ« b 2e−γJtJ[2e−2γJtJcosφ+(1+e−4γJtJ)cos2χ]sinφ × (12) (1+e−2γJtJ)(1+e−2γJtJcosφcos2χ+e−4γJtJ) Ι In the limit of vanishing γJtJ, Eq.(12) corresponds to the situation discussed in6. Indeed, both expressionsare T/e independentofthedephasingrates. Thedifferenceinthe 0.2 detailsofthecurrent-phase(s)relationshipareduetothe 0 0 different environment. In all the three cases presented here, Eqs.(10,11,12), -0.2 1 0.5 γ the change of sign of the current as a function of the phase shifts θ or χ is evident. 0.5γ e- CtC Current Noise -Cooperpairshuttlingisaresultofanon equilibrium steady state process. Therefore, to better - JtJ characterizethe transport,we analyzesupercurrentfluc- e 0 1 tuations as defined in Eq.(4). This should be contrasted withthestandardJosephsoneffect2,wherethesupercur- Figure 3: Average current (Tb ≪ EJ) as a function of the rent is an equilibrium property of the system. We first dephasing rates, with φ =−3π/4, θ = 7π/10, χ= 5π/6. As consider the zero frequency noise in the two regimes of a function of γJtJ, the supercurrent has a not-monotonous strong and weak dephasing. behavior. Note thechange of sign in thecurrent obtained by varyingdecoherence rates in each time intervalseparately. Whenthedephasingisstrong,correlationsontimescales larger than T are suppressed and the noise spectrum reads can increase. The behavior as a function of the dephas- 4e2 1 ing rates is presented in Fig.3. The presence of a maxi- S(0)strong ∼ −e−γJtJ cos(2θ)+e−2γJtJf(θ,φ,χ) , T (cid:26)2 (cid:27) mum Josephson current at a finite value of γ t can be J J (13) understoodbyanalyzingtheasymptoticbehaviorsinthe where f(θ,φ,χ) = cos2(2θ)−e−γCtCcosφcosχsin2(2θ). strong and weak damping limits, where simple analytic TheleadingterminEq.(13)isduetothedampedoscilla- expressions are available (in the following we do not ex- tions in the contact regions (L,R). The phase dependent plicitely write the temperature dependence in γ , γ ). J C contribution is exponentially suppressed since it comes i)Ifthedephasingisstrong,I canbeexpandedinpowers from correlations over times larger than the period. For of e−γJtJ and e−γCtC and, to leading order weak dephasing (same limits of Eq. (11)), we find Istrong ∼ 2Tetanh(cid:18)ETJb (cid:19)e−(γJtJ+γCtC)cos(2χ)sin(2θ) sinφ. S(0)weak ∼ 4Te2γC1tC 2(1t+anc2oθssφinco2sφ2χ), (14) (10) Strong dephasing is reflected in the simple (i.e. ∝ sinφ) which shows a much richer structure as a function of current-phase relationship and in the exponential sup- the phases θ and χ18. Finally, we briefly discuss the pression of the current itself. finitefrequencyspectruminthecaseofstrongdephasing ii)In the opposite limit of weak damping (γ t ≪ (see Fig.4). Superimposed to the peak at the Josephson J J 4 S(w ) T/4 e2 Josephson coupling are equal, EJ(L) =EJ(R). This implementation has several advantages. It allows to control 50 thecouplingwiththeenvironmentbysimplyvaryingthe time dependence ofthe appliedmagneticfield. The time 1500 25 scaleforthevariationofthemagneticfieldshouldbecon- trolledat the same levelas it is done in the implementa- 0.6 0.8 900 tionofJosephsonnanocircuitsforquantumcomputation (seeRef.4 foranextensivediscussion). Foraquantitative 300 comparisonwiththeresultsdescribedhere,themagnetic fieldshouldvaryonatime scaleshorterthan~/E ,tipi- J 0.85 1 1.15 w /EJ cally a few nanoseconds with the parameters of the first article in Ref.5. This is possible with present day tech- nology19. Ataqualitiveleveltheresultspresentedinthis Figure 4: Current noise spectrum as a function of ω for T =4tJ in strong dephasing limit (γJtJ =1.2 and γJ/EJ = paper(π-junctionbehavior,non-monotonousbehaviorin 0.008). Intheinset,weplotthespectruminarestrictedrange the damping)do notrely onthe step-changeapproxima- of frequencies to betterresolve the oscillations. tion of the Josephson couplings (which leads to Eq.(7)). AX AX B/B0 C C energy, there are oscillations of frequency of the order AX B(t) X 1 of T−1. The presence of these oscillations is related to L A AC /AL R the periodicityofthe islandmotion. Themodificationof these fringes as a function of the phases is a signature of AX AX AC /AR C C t the coherence in the Cooper pair shuttle. X X tJ tC tJ tC Weconcludebysuggestingapossibleexperimentaltest of our results which does not require any mechanically Figure 5: left: Sketch of the implementation of the shuttle processbymeansofatime-dependentmagneticfield. Crosses movingpart. ThetimedependenceoftheJosephsoncou- represent Josephson junctions. right: Plot of the time vari- plingsandn is regulatedbyatime dependent magnetic field and gatge voltage, respectively. The setup consists ation of the applied field (in unity of B0 = Φ0/(2AC),Φ0 is the flux quantum) in order to realize the Cooper pair shut- of a superconducting nanocircuit in a uniform magnetic tle. Thedifferentloop areascan bechosenin ordertoobtain field as sketched in Fig.5. By substituting the Joseph- E(L) =E(R). son junction by SQUID loops, it is possible to control J J the E by tuning the applied magnetic field piercing the J loop. The presence of three type of loops with differ- entarea,A ,A ,A allowstoachieveindipendently the Those effects are observable even if the magnetic field L R C three cases, where one of the two EJ’s is zero (regions changes on time-scales comparable or slower than EJ. L,R) or both of them are zero (region C), by means of a The only strict requirement is that only one Josephson uniform magnetic field. If the applied field is such that coupling at the time is switched on. a half-flux quantum pierces the areas A ,A or A , the We gratefully acknowledge many helpful discussions L R C Josephson couplings will be those of regions R,L and C, with G. Falci, Yu. Galperin and Yu.V. Nazarov. This respectivelyandtheHamiltonianofthesystemcanbeex- work was supported by the EU (IST-SQUBIT, HPRN- actlymappedontothatofEq.(1). 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