Self-consistent dynamics of a Josephson junction in presence of an arbitrary environment
SSelf-Consistent Dynamics of a Josephson Junction in the Presence of an ArbitraryEnvironment
Philippe Joyez
Quantronics Group, Service de Physique del’Etat Condens´e (CNRS URA 2464), IRAMIS,CEA-Saclay, 91191 Gif-sur-Yvette, France (Dated: May 28, 2013)We derive microscopically the dynamics associated with the d.c. Josephson effect in a super-conducting tunnel junction interacting with an arbitrary electromagnetic environment. To do so,we extend to superconducting junctions the so-called P ( E ) theory (see e.g. Ingold and Nazarov,arXiv:cond-mat/0508728) that accurately describes the interaction of a nonsuperconducting tunneljunction with its environment. We show the dynamics of this system is described by a small setof coupled correlation functions that take into account both Cooper pair and quasiparticle tunnel-ing. When the phase fluctuations are small the problem is fully solved self-consistently, using andproviding the exact linear admittance Y ( ω ) of the interacting junction. Fifty years ago Josephson stunned the community when he published [1] the equations that govern the behaviorof superconducting tunnel junctions. These Josephson relations, as they became known, link the voltage V and thesuperconducting phase difference ϕ across the junction, and the current I through it: I = I sin ϕ, V = (cid:126) e dϕdt . (1)If ϕ is static, V = 0, and a nondissipative current I flows through the junction, bounded by | I | (cid:54) I . This maximumsupercurrent I (or the corresponding Josephson coupling energy E J = I (cid:126) / e ) was originally predicted to be anintrinsic property of the tunnel junction, depending only on its resistance in the normal state and the superconductinggap of its electrodes [2], but not on other details such as the junction’s geometry, or its fabrication process. Along theyears, Josephson junctions (JJs) have proved invaluable electronic components forming exquisitely sensitive sensors(e.g., squid magnetometers, quantum-limited amplifiers), metrological Volt standard devices, or quantum bits andgates.It is important to note that the first Josephson relation was derived assuming that the phase ϕ has negligiblequantum fluctuations, and it is not obvious why it would be generally valid beyond this situation. Because theJosephson effect has, among others, metrological applications, the effect of phase fluctuations on Josephson tunnelingwere thoroughly investigated in the 1980s, mostly using path integral formalism [3–5]. It was concluded that inmost practical experimental situations a JJ can indeed be described using the effective Josephson Hamiltonian H J = − E J cos ϕ that directly corresponds to the first Josephson relation, with, however, small corrections due to phasefluctuations that originate in its electromagnetic environment (i.e., the circuit connected to the junction). Thiswas checked for instance in the so-called Macroscopic Quantum Tunneling experiments [4–6]. More recently, JJ-based quantum logic circuits were also shown to be accurately described using the effective Josephson Hamiltonian[7], with their electromagnetic environment partly responsible for their decoherence [8]. Note, however, that someenvironmental decoherence mechanisms in JJ qubits were recently identified that cannot be captured within only theeffective Josephson Hamiltonian model [9–11].On the other hand, the environment of a JJ can have a more dramatic effect: the phase fluctuations generated byan impedance larger than the resistance quantum R Q = h/ e ∼ . P ( E ) theory (PoET) [17–19]. This theory was developed in the 1990s to explain a reduction of differential conductanceat low voltage (also called “zero-bias anomaly”) in nonsuperconducting sub- µ m tunnel junctions, a phenomenon thatis now often referred to as dynamical Coulomb blockade. In its original form this theory evaluates the incoherenttunneling rate of electrons properly taking into account the probability P ( E ) that the environment absorbs an energy E during a tunnel event. While perturbative in tunneling, this theory is nonperturbative in the strength of the coupling a r X i v : . [ c ond - m a t . s up r- c on ] M a y to the environment and it can deal with an arbitrary frequency-dependent linear electromagnetic environment. Notethat it also applies to incoherent Cooper pair tunneling in JJs at finite sub-gap voltages. Its predictions were shown tobe quantitative in a number of experiments, in particular when the environment consists of resonators [20, 21]. Here,by generalizing PoET to the dc Josephson effect, a coherent flow of Cooper pairs through the junction, we obtaina unified nonperturbative treatment of arbitrary environmental effects in both normal and superconducting tunneljunctions. In this approach we show that one is lead naturally to introduce a self-consistent mean-field electrodynamicresponse of the junction, something that, as far as we know, has not been done explicitly previously for JJs. In thisformulation the junction is systematically and properly combined with the rest of the circuit, resulting in an intuitivepicture of the system. In the case when the phase fluctuations are small we work out the linear response of thejunction and a simple iterative scheme to evaluate a renormalized I and its admittance. As an illustrative example,we work out the self-consistency for a JJ in an Ohmic environment at zero temperature. In the conclusion we discussthe scope of our results and possible extensions. (a) (b) (c) Figure 1. (a) We consider a Josephson junction characterized by its normal state resistance R T and derive its effectivecritical current I eff0 taking into account both a static ( ϕ ) and a fluctuating phase difference ˜ δ ( t ) driven by the electromagneticenvironment. (b) As seen from an individual tunnel channel, the environment consists of the impedance Z ( ω ) of the connectingcircuit, of the junction’s own capacitance C and of the electromagnetic response due to tunneling in the other channels, heredescribed by a linear admittance Y ( ω ), but which in the general case is a nonlinear element. (c) We solve the problem in thecase of an Ohmic environment, retaining only the dominant inductive contribution in Y. The circuit we consider, shown in Fig. 1a, consists of a pure tunnel element connected in parallel with the junction’sgeometric capacitor and in series with an arbitrary linear electromagnetic environment with impedance Z ( ω ). TheHamiltonian of the circuit is H = H L + H R + H env + H T where H env describes the voltage source and Z ( ω ) in the manner of Caldeira and Legget [5] and H L,R are the BCSHamiltonians of the junction’s electrodes. For the left electrode, for instance, we have H L = (cid:88) (cid:96)σ ξ (cid:96) c + (cid:96)σ c (cid:96)σ − ∆ (cid:88) (cid:96) c + (cid:96) ↑ c +¯ (cid:96) ↓ + c ¯ (cid:96) ↓ c (cid:96) ↑ where σ is the spin index, (cid:96) is a composite channel and momentum index for the electrons in the leads and theoverbar denotes the opposite-momentum state ( H R has the same form, with states indexed by r instead of (cid:96) ). Finally H T = ˆ T + ˆ T † is the tunneling Hamiltonian treated as a perturbation, where the operator ˆ T = e i ˆ δ (cid:80) (cid:96),r,σ t (cid:96)r c + rσ c (cid:96)σ transfers an electron from the left to the right electrode. We work in a gauge where the electrodes have real BCSorder parameters ∆ (assumed identical in L and R ) and, consistently, the e i ˆ δ term here takes care of transferringthe electronic charge e between the electrodes [10, 19]. We restrict to zero dc voltage across the junction so thatˆ δ ( t ) = ϕ/ δ ( t ) with ϕ being the superconducting phase difference across the junction and ˜ δ ( t ) a zero-meanfluctuating phase operator driven by Z ( ω ). By introducing the standard Bogoliubons operators γ k = u k c k ↑ + v k c +¯ k ↓ ; γ k = − v k c k ↑ + u k c +¯ k ↓ ( k = (cid:96), r ) with the usual BCS coherence factors u k , v k we can diagonalize H L,R , whereas H T becomes H T = (cid:88) (cid:96),r t (cid:96)r [ γ +0 r γ (cid:96) ( − e i ˆ δ u (cid:96) v r − e − i ˆ δ u r v (cid:96) )+ γ +1 r γ (cid:96) ( − e − i ˆ δ u (cid:96) v r − e i ˆ δ u r v (cid:96) )+ γ +1 r γ (cid:96) ( e i ˆ δ u r u (cid:96) − e − i ˆ δ v r v (cid:96) )+ γ +0 r γ (cid:96) ( − e − i ˆ δ u r u (cid:96) + e i ˆ δ v r v (cid:96) )] + ( (cid:96) (cid:11) r ) † . In thermal equilibrium situations the supercurrent through the junction is given by the thermodynamic relation I = 2 e (cid:126) dFdϕ (2)where F is the free energy. To lowest order in perturbation theory the change of F due to H T can be cast as∆ F = 1 (cid:126) (cid:90) + ∞ dt Im S H T ( t ) (3)with S H T ( t ) = (cid:104) H T ( t ) H T (0) (cid:105) where the angular brackets denote averaging over the unperturbed quasiparticle andenvironment states that act as bath degrees of freedom whose time evolution is the unperturbed one. A straightforwardalgebraic calculation gives S H T ( t ) = (cid:88) (cid:96),r,η = ± | t (cid:96)r | [ u r v r u (cid:96) v (cid:96) ( A η ( t ) − B η ( t )) C ηη ( t ) e iηϕ +(( u (cid:96) v r + u r v (cid:96) ) A η ( t ) + ( u (cid:96) u r + v (cid:96) v r ) B η ( t )) C η − η ( t )] (4)with A η = ± ( t ) = (cid:104) γ η (cid:96) ( t ) γ − η (cid:96) γ − η r ( t ) γ η r + γ − η (cid:96) ( t ) γ η (cid:96) γ η r ( t ) γ − η r (cid:105) B η ( t ) = (cid:104) γ η (cid:96) ( t ) γ − η (cid:96) γ − η r ( t ) γ η r + γ − η (cid:96) ( t ) γ η (cid:96) γ η r ( t ) γ − η r (cid:105) C ηη (cid:48) ( t ) = (cid:104) e iη ˜ δ ( t ) e iη (cid:48) ˜ δ (0) (cid:105) where a fermion operator with a minus exponent means an annihilation operator. The e ± iϕ terms in Eq. (4) are eachrelated to the transfer of two spin-conjugate electrons in a given direction, i.e., a whole Cooper pair with charge 2 e ,they thus correspond to the Josephson effect. Note also that they come with the u r v r and u (cid:96) v (cid:96) factors that correspondto the anomalous Green’s function of the electrodes, carrying the essence of superconductivity. The ϕ -independentterms, on the contrary, are related to a back-and-forth transfer of an electron and correspond to ordinary quasiparticletunneling, the only processes remaining in the normal state. These processes do not transfer a net charge through thejunction but they still couple to the phase fluctuations and contribute to the dynamics of the JJ. While these processesare obviously disregarded when JJs are modeled using only the effective Josephson Hamiltonian (e.g. most JJ-basedqubit literature), the full Ambegaokar-Eckern-Sch¨on effective action for the JJ [3] [whose form is closely related to Eq.(4)] allows accounting for them in path integral formalism. In the present approach we handle these terms using onlytwo-point real-time correlators and sparing the use of path integrals. The correlators C + − ( t ) , C − + ( t ) that accompanyquasiparticle tunneling are those encountered in the standard PoET [specifically, (cid:104) e i ˜ δ ( t ) e − i ˜ δ (0) (cid:105) = (cid:82) d te − iEt/ (cid:126) P ( E ) isthe inverse Fourier transform of P ( E )], while the Cooper pair tunneling comes with distinct correlators C ++ ( t ) , C −− ( t ).For simplicity we here assume phase fluctuations are symmetric, i.e. C ++ = C −− and C + − = C − + (we discuss thelimit of validity of this assumption in the Supplemental Material [22]). Going to a continuum of states in the electrodes,from Eq. (4) we obtain the exact result at lowest order in tunneling S H T ( t ) = 2 R Q π R T [( p ( t ) − q ( t ) ) C + − ( t ) + m ( t ) C ++ ( t ) cos ϕ ] (5)where R T is the normal state tunnel resistance of the junction and m ( t ) , p ( t ) , q ( t ) are, respectively, the inverseFourier transforms of M ( ε ) = − ∆ f ( − ε ) ρ ( ε ) /ε , P ( ε ) = f ( − ε ) ρ ( ε ), Q ( ε ) = − f ( − ε ) θ (cid:0) ε − ∆ (cid:1) sgn( ε ) with ρ ( ε ) = | ε | Re (cid:0) ε − ∆ (cid:1) − / the BCS density of states, θ the Heavyside step and f ( ε ) the occupation probability of theBogoliubov quasiparticles, which need not be thermal. Here both electrodes are assumed identical but the generalcase could also be handled. Note that in principle the gap ∆ of the electrodes should be self-consistently evaluatedfrom f ( ε ), an effect which becomes important at temperatures comparable to the critical temperature or in strongnonequilibrium. If we first ignore a possible ϕ dependence of C + ± , then, by combining Eqs. (2), (3), and (5) oneobtains a generalization of the first Josephson relation with an effective critical current I eff0 = 2 πe (cid:126) R T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) + ∞ dt Im[ m ( t ) C ++ ( t )] (cid:12)(cid:12)(cid:12)(cid:12) (6)which remains valid beyond thermal equilibrium. This expression generalizes PoET in real-time formulation [23, 24].In the case where phase fluctuations are negligible C ++ ( t ) ≡
1, and one recovers all standard results on JJ, such as, e.g. , the temperature dependence of the critical current [2]. Hence C ++ is a kernel giving a renormalization of thecritical current with respect to the standard Ambegaokar-Baratoff value [2]. We will see below that C + ± should inprinciple depend on ϕ (albeit weakly in usual cases), thus yielding additional terms that cause a departure from thepurely sinusoidal current-phase relation predicted by Josephson.We now consider finite phase fluctuations and first assume that the degrees of freedom generating these fluctuationscan be regarded as a linear impedance Z eff as in the usual PoET [19]. Such fluctuations are then Gaussian andconsequently C + ± can be expressed in terms of only the two-point correlator S δ ( t ) = (cid:104) ˜ δ ( t )˜ δ (0) (cid:105) . As a consequenceof the fluctuation-dissipation theorem S δ ( t ) can in turn be evaluated from the spectral density of the environment.Namely C + − = e S δ ( t ) − S δ (0) = e J ( t ) C ++ = e − S δ ( t ) − S δ (0) = e − J ( t ) − S δ (0) (7) S δ ( t ) = (cid:90) + ∞−∞ dωω Re Z eff ( ω )2 R Q e − iωt − e − β (cid:126) ω . (8)Here we have also introduced the usual PoET notation J ( t ) = S δ ( t ) − S δ (0) [19]. Replacing C ++ in Eq. (6) we canpull out of the integral the renormalization factor λ = e − S δ (0) , which plays a major role in the following.Note that, unless Re Z eff ( ω ∼
0) = O ( ω ) or smaller, S δ (0) = ∞ (signaling thermal or quantum phase diffusion),yielding λ = 0 and thus I eff0 = 0 . This might seem surprising since in most cases when one measures a JJ, it isconnected to a circuit that contains normal metal at room temperature (with finite dc resistance), but its criticalcurrent is nevertheless measured finite. The apparent paradox is resolved when one considers the JJ as being part ofits own electromagnetic environment [see Fig. 1(b)] : a superconducting JJ perfectly shunts the rest of the circuit atzero frequency, preventing phase diffusion and the divergence of S δ (0). More importantly, doing so is actually the onlyway to enforce an amplitude and a dynamics of the phase fluctuations in the system that are actually consistent withthe presence of the junction, unlike in standard PoET [25]. This inclusion of the junction in its own environment canalso be justified microscopically: a typical metallic tunnel junction contains a very large number N of independentLandauer channels that only interact through their common phase. Thus, as seen from each individual channel,the other channels form a ( a priori nonlinear) bath whose response is that of the full junction (up to corrections oforder 1 /N ) and which are treated like the rest of the environment. Let us stress also that in typical tunnel junctionseven if the junction’s conductance is large, its individual channels remain very weakly transmissive. Hence, lowestorder perturbation in tunneling is sufficient and all the complications in the behavior of the JJ arise solely fromthe electromagnetic interaction among the channels and with the environment, which treat here in a self-consistentmean-field manner. Such a self-consistent mean-field approach of PoET has been successfully checked experimentallyin low-resistance normal-state junctions [26], and, in that case, when the junction is described as a linear element (seebelow), this was shown to correspond to a self-consistent harmonic approximation that minimizes the free energy in thepath integral description of the system [27]. Let us finally remark that in this mean-field approach the superconductingcharacter of the JJ gives rise to a chicken-and-egg situation that requires a self-consistent solution, much like for thevalue of ∆ in BCS theory itself.We now close the loop by working out the self-consistency in the linear regime assumed in this part. Within thishypothesis, the response of the junction can be obtained from a generalized fluctuation-dissipation relation [28] andis expressed as an admittance Y ( ω ) = cos ϕiL eff J ω + 2 (cid:90) ∞ dti Im S I ( t ) e iωt − (cid:126) ω (9)that is exact at lowest order in perturbation [22]. In this expression L eff J = (cid:0) e (cid:126) I eff0 (cid:1) − is the effective Josephsoninductance and S I ( t ) = (cid:104) ˆ I ( t ) ˆ I (0) (cid:105) is the correlator of the current operator ˆ I = e (cid:126) ∂H T ∂ϕ through the junction. Thislatter definition implies that S I ( t, ϕ ) = (cid:0) e (cid:126) (cid:1) S H T ( t, ϕ + π ), readily obtained from Eq. (5). In the self-consistentapproach we discuss here we shall then replace Z eff ( ω ) = [ Y ( ω ) + iCω + Z − ( ω )] − (10)in Eq. (8), where C is the junction capacitance and Z the impedance of the external circuit as seen from the junction[see Fig.1(b)]. Thus we are able to obtain the full dynamics of the system (and I eff0 as a by-product) by solving theself-consistency defined by Eqs. (5), (9), (10), (8), (7). This can, for instance, be done by iterating from an initialguess such as Y ( ω ) = cos ϕ/iL J ω , L J being the Josephson inductance in the absence of environment. In order to bevalid the iterated solution must be consistent with the assumption of linear behavior of the effective environment, i.e., (cid:112) S δ (0) (cid:28) π, (11)so that phase fluctuations do not feel the nonlinearity of the JJ. In practice this means I eff0 should not be reducedmore than a few percent with respect to I for this linear approach to be valid. If this later criterion if fulfilled, thenthe solution obtained is essentially the exact dynamics of the junction at lowest order in tunneling.Simplifying approximations can be made or not depending on the value of the “plasma frequency” ω p =(cos ϕ/L eff J C ) / defined as the resonance frequency of the purely inductive first term of Eq. (9) with the junction’scapacitance C . If ω p is significantly smaller than ω Gap = 2∆ / (cid:126) , then at low temperature it is a good approximationto keep in Y ( ω ) only the inductive term, that precisely suppresses the divergence of S δ (0). This is justified becausethe integral in Eq. (9) has only a slight capacitive contribution at frequencies ω (cid:46) ω Gap = 2∆ / (cid:126) with dissipationsetting in only at frequencies close to or above ω Gap . With this simplification Z eff reduces to the impedance of an LCoscillator resonating at ω p damped by the external impedance Z ( ω ). Furthermore, still in the case when ω p < ω Gap ,the characteristic time scale of phase fluctuations ( ω − p ) is significantly longer than that of m ( t ) which is ω − .Then, in Eqs. (6), (7) we can take the short-time limit J ( t →
0) = 0, yielding the simple renormalization I eff0 = λI .A similar renormalization of the Josephson coupling was obtained at ϕ = 0 in Refs. [4, 29]. We see here that thisis valid only when ω Gap is the fastest dynamics in the problem and that the opposite situation cannot be treatedcorrectly in approaches starting from the effective Josephson Hamiltonian.Let us now fully work out an example in the above simplifying assumption ω p < ω Gap , I eff0 = λI , and furtherrestricting to the “Ohmic” case where Z ( ω ) = R [Fig.1(c)] and zero temperature. Then the effective environmentreduces to an RLC circuit with impedance Z eff ( ω ) = ( λ cos ϕ/iωL J + iCω + R ) − for which S δ (0) can be calculatedanalytically and from which we derive the self-consistency equation λ = exp − R R Q tanh − (cid:18) − λq cos ϕ √ − λq cos ϕ (cid:19) + i π (cid:112) − λq cos ϕ (12)where q = R (cid:112) C/L J would be the quality factor of the plasma oscillation at ϕ = 0, in absence of renormalization.Again, valid solutions must satisfy Eq. (11), that is, 1 − λ (cid:28)
1. However this always fails at ϕ = π mod π where ω p vanishes and where a treatment beyond linear response is needed. When the approximation is valid (away from thepathological points) we predict that the renormalization of I is different at ϕ = 0 and ϕ = π , leading to a slightlyanharmonic current-phase relation. This anharmonicity is a generic feature in the self-consistent approach because itcauses C + ± (t) in Eq. (5) to have a ϕ dependence through the dynamical response of the JJ.In conclusion we have extended the framework of the PoET to address the effect of an arbitrary electromagneticenvironment on the Josephson effect in metallic tunnel junctions. Doing so we reached a self-consistent description ofthe Josephon effect, sheding new light on the interaction of a JJ with its environment, including its dynamics. Thisnotably predicts that the celebrated first Josephson relation generically departs from a sinusoid when the impedanceof its environment is increased, a fact that should be verifiable experimentally. For strictly dc Josephson effect andsmall phase fluctuations, the self-consistency is fully worked out using the exact linear admittance of the interactingJJ, a quantity that is accessible to measurements and that should be useful for quantum circuit engineering. We thinkmore work in this direction could extend this approach to non-dc situations and non-Gaussian phase fluctuations[22]. This would provide the general “circuit laws” for Josephson junctions, a quantum nonlinear generalization ofthe classical “impedance combination laws.”The author is grateful to all members of the Quantronics group for their constant interest and support and thankfullyacknowledges helpful discussions and input from C. Altimiras, H. Grabert, F. Hekking, M. Hofheinz, H. le Sueur, F.Portier, P. Roche and I. Safi. [1] B. D. Josephson, Phys. Lett. , 251 (1962).[2] V. Ambegaokar and A. Baratoff, Phys. Rev. Lett. , 486 (1963); , 104(E) (1963).[3] V. Ambegaokar, U. Eckern, and G. Sch¨on, Phys. Rev. Lett. , 1745 (1982).[4] For a review, see Gerd Sch¨on and A. D. Zaikin, Phys. Rep. , 237 (1990).[5] A. O. Caldeira and A. J. Leggett, Ann. of Phys. (N.Y.) , 374 (1983).[6] J. Clarke, A. N. Cleland, M. H. Devoret, D. Esteve, and J. M. Martinis, Science , 992 (1988).[7] For a review, see G. Wendin and V. S. Shumeiko, Low Temp. Phys. , 724 (2007).[8] G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, andG. Sch¨on, Phys. Rev. B , 134519 (2005).[9] John M. Martinis, M. Ansmann, and J. Aumentado, Phys. Rev. Lett. , 097002 (2009).[10] G. Catelani, J. Koch, L. Frunzio, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman, Phys. Rev. Lett. , 077002(2011). [11] M.H. Ansari, F.K. Wilhelm, U. Sinha, and A. Sinha, arXiv:1211.4745.[12] S. Corlevi, W. Guichard, F. W. J. Hekking, and D. B. Haviland, Phys. Rev. Lett. , 096802 (2006).[13] I. M. Pop, I. Protopopov, F. Lecocq, Z. Peng, B. Pannetier, O. Buisson, and W. Guichard, Nat. Phys. , 589 (2010).[14] V.E. Manucharyan, N.A. Masluk, A. Kamal, J. Koch, L.I. Glazman, and M.H. Devoret, Phys. Rev. B 85 , 024521 (2012).[15] N.A. Masluk, I.M. Pop, A. Kamal, Z. K. Minev, and M.H. Devoret, Phys. Rev. Lett. , 137002 (2012).[16] M. T. Bell, I. A. Sadovskyy, L. B. Ioffe, A. Yu. Kitaev, and M. E. Gershenson, Phys. Rev. Lett. , 137003 (2012).[17] M. H. Devoret, D. Esteve, H. Grabert, G.-L. Ingold, H. Pothier, and C. Urbina, Phys. Rev. Lett. , 1824 (1990).[18] S. M. Girvin, L. I. Glazman, M. Jonson, D. R. Penn, and M. D. Stiles, Phys. Rev. Lett. , 3183 (1990).[19] G.-L. Ingold and Y.V. Nazarov, in Single Charge Tunneling, edited by H. Grabert and M.H. Devoret, NATO ASI SeriesB (Plenum, New York, 1992), Vol. 294, pp. 21–107; G.-L. Ingold and Y.V. Nazarov, arXiv:cond-mat/0508728.[20] T. Holst, D. Esteve, C. Urbina and M. H. Devoret, Phys. Rev. Lett. , 3455 (1994).[21] M. Hofheinz, F. Portier, Q. Baudouin, P. Joyez, D. Vion, P. Bertet, P. Roche and D. Esteve, Phys. Rev. Lett. , 217005(2011).[22] See the Supplemental Material for a derivation of Eq. (9) and a discussion of possible extensions.[23] P. Joyez and D. Esteve, Phys. Rev. B 56 , 1848 (1997).[24] A. A. Odintsov, Zh. Eksp.Teor. Fiz. 94, 312 (1988) [Sov. Phys. JETP 67, 1265 (1988).][25] The PoET was originally developed for tunnel junctions having tunnel resistances much higher than the impedance oftheir environment. In this case the junction behaves nearly as an open circuit and phase fluctuations are determined onlyby the environment.[26] P. Joyez, D. Esteve and M. H. Devoret, Phys. Rev. Lett. , 1956 (1998).[27] G. G¨oppert and H. Grabert, C. R. Acad. Sci., Ser. IIb , 885 (1999).[28] I. Safi and P. Joyez, Phys. Rev. B 84 , 205129 (2011).[29] H. Grabert, G.-L. Ingold, and B. Paul, Europhys. Lett. , 360 (1998).[30] C. Flindt, T. Novotn´y, A. Braggio, and A.-P. Jauho, Phys. Rev. B 82, 155407 (2010)[31] C. Emary, J. Phys.: Condens. Matter SUPPLEMENTAL MATERIALDerivation of the JJ admittance
Here we evaluate the linear response of the junction to a vanishingly small ac excitation δV ( ω ) = iω (cid:126) e δϕ ( ω ) addedto the static phase difference ϕ of the junction. This can be done exactly, even in presence of the environment [28]. Atthe lowest order in the tunneling Hamiltonian and in the excitation, the time evolution of the current flowing throughthe junction under this perturbation is given by I ( t ) = i (cid:126) (cid:90) t −∞ d s (cid:104) [ ˆ I ( t ) , H T ( s )] (cid:105) = i (cid:126) (cid:90) t −∞ d s (cid:42)(cid:34) ˆ I ( t ) + δϕ ( t ) ∂ ˆ I∂ϕ ( t ) , H T ( s ) + δϕ ( s ) ∂H T ∂ϕ ( s ) (cid:35)(cid:43) = (cid:104) ˆ I (cid:105) + i (cid:126) δϕ ( t ) (cid:90) t −∞ d s (cid:42)(cid:34) ∂ ˆ I∂ϕ ( t − s ) , H T (cid:35)(cid:43) + i (cid:126) (cid:90) t −∞ d s (cid:28)(cid:20) ˆ I ( t − s ) , ∂H T ∂ϕ (cid:21)(cid:29) δϕ ( s )where, as in the body of the article, the angular brackets denote averaging over unperturbed states of the electrodeand the environment and the time evolution of operators is the unperturbed one. (cid:104) ˆ I (cid:105) is the dc supercurrent in absenceof the ac excitation. Using the identities : ∂H T ∂ϕ = (cid:126) e ˆ I ; ∂ (cid:98) I∂ϕ = − e (cid:126) H T S I ( t, ϕ ) = (cid:0) e (cid:126) (cid:1) S H T ( t, ϕ + π )and denoting δI ( t ) = I ( t ) − (cid:104) ˆ I (cid:105) , and S − I ( t ) the odd part of S I ( t ) we get δI ( t ) = − e (cid:126) i (cid:126) (cid:126) e δϕ ( t ) (cid:90) t −∞ d s S − I ( t − s, ϕ + π ) + i e (cid:90) t −∞ d s S − I ( t − s, ϕ ) δϕ ( s )= i e (cid:18) − δϕ ( t ) (cid:90) ∞−∞ d s S − I ( s, ϕ + π ) θ ( s ) + (cid:90) ∞−∞ d s S − I ( t − s, ϕ ) θ ( t − s ) δϕ ( s ) (cid:19) Going to the frequency domain δI ( ω ) = i e δϕ ( ω ) (cid:90) d tθ ( t )(2 S − I ( t, ϕ ) e iωt − S − I ( t, ϕ + π ))= i e δϕ ( ω ) (cid:18) − i eI eff0 cos ϕ + (cid:90) d tθ ( t )2 S − I ( t, ϕ )( e iωt − (cid:19) Finally we obtain the junction’s admittance as Y ( ω ) = δI ( ω ) δV ( ω ) = 2 e (cid:126) iω δI ( ω ) δϕ ( ω )= cos ϕiL eff J ω + 2 (cid:90) ∞ d tS − I ( t ) e iωt − (cid:126) ω This expression is a generalized fluctuation-dissipation relation [28]. Note that the integral contains contributionsfrom both Cooper pair and quasiparticle tunneling.
Beyond d.c. Josephson effect and Gaussian fluctuations
Could our mean-field approach be extended to address the full complexity of the dynamics of Josephson junction?In other words could it handle cases beyond the restrictions adopted above of (i) static phase difference ( i.e. strictlydc Josephson effect) and (ii) small fluctuations/linear response? When lifting restriction (i) the steady-state analysisconducted above is insufficient, and one needs to replace all translationally-invariant correlators introduced above bytwo-time correlators ( e.g. S H T ( τ ) = (cid:104) H T ( τ ) H T (0) (cid:105) → S H T ( t, t (cid:48) ) = (cid:104) H T ( t ) H T ( t (cid:48) ) (cid:105) (cid:54) = S H T ( t − t (cid:48) ) that follow non-Markovian dynamics. In a situation where the voltage across the JJ is finite and constant on average (a.c. Josephsoneffect) these time correlators are cyclostationnary. When the phase fluctuations become large (ii), because of thenon-linear response of the junction itself the time correlators also become non-Gaussian so that in Eq. 5 one shoulddistinguish and keep all four correlators of the charge transfer operator e : C ++ , C −− , C + − and C − + . Given theparenthood between the counting fields of Full Counting Statistics (FCS) and the charge transfer operator e i ˆ δδ