Josephson junction detector of non-Gaussian noise
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Josephson junction detector of non-Gaussian noise
Hermann Grabert
Physikalisches Institut and Freiburg Institute of Advanced Studies,Albert-Ludwigs-Universit¨at, 79104 Freiburg, Germany (Dated: February 22, 2008)The measurement of higher order cumulants of the current noise generated by a nonlinear meso-scopic conductor using a Josephson junction as on-chip detector is investigated theoretically. Thepaper addresses the regime where the noise of the mesoscopic conductor initiates activated escapeof the Josephson detector out of the zero-voltage state, which can be observed as a voltage rise. Itis shown that the deviations from Johnson-Nyquist noise can mostly be accounted for by an effec-tive temperature which depends on the second noise cumulant of the conductor. The deviationsfrom Gaussian statistics lead to rather weak effects and essentially only the third cumulant can bemeasured exploiting the dependence of the corrections to the rate of escape from the zero-voltagestate on the direction of the bias current. These corrections vanish as the bias current approachesthe critical current. The theory is based on a description of irreversible processes and fluctuationsin terms of state variables and conjugate forces. This approach, going back to work by Onsager andMachlup, is extended to account for non-Gaussian noise, and it is shown that the thermodynamicallyconjugate force to the electric charge plays a role similar to the counting field introduced in morerecent approaches to describe non-Gaussian noise statistics. The theory allows to obtain asymptoti-cally exact results for the rate of escape in the weak noise limit for all values of the damping strengthof the Josephson detector. Also the feedback of the detector on the noise generating conductor isfully taken into account by treating both coupled mesoscopic devices on an equal footing.
PACS numbers: 72.70.+m, 73.23.-b, 05.70.Ln, 85.25.Cp
I. INTRODUCTION
Traditional nonequilibrium thermodynamics assumesGaussian fluctuations of the gross variables about theirmean values. This assumption is a natural consequenceof the central limit theorem implying small fluctuationsof additive variables distributed in a Gaussian way. Inthe last decade there have been extensive theoreticalefforts to calculate deviations from Gaussian statisticsfor electronic current fluctuations of mesoscopic devices.The complete knowledge of the number of charges trans-ferred through the device in a given interval of time isreferred to as full counting statistics (FCS). It has turnedout that FCS reveals details on microscopic processes inthe device that are not available through mere measure-ments of the mean current and the noise variance. Thiscan already be seen from a simple example known sincea long time. The FCS of a tunnel junction is Poissonianwhen the applied voltage is large compared to the tem-perature ( eV ≫ k B T ). In this case charges essentiallyonly tunnel from source to drain, and the Poissonianstatistics points to statistically independent transfers ofdiscrete charges.In contrast to the substantial literature on theo-retical predictions for FCS there are only rather fewexperiments that have measured deviationsfrom Gaussian noise. This is a consequence of the factthat these deviations are typically small and require so-phisticated experimental techniques to be detected. Thepioneering work by Reulet et al. , has measured thethird cumulant of the noise produced by a tunnel junc-tion. Since the noise was measured by room tempera-ture electronics, the signal had to be transmitted from the cryostat to the amplifier by coaxial cables. There-fore, in view of impedance matching, this set-up workswell only for noise generating devices with resistancesof order 50Ω. The more recent experiments employ on-chip noise detectors, either quantum point contacts orJosephson junctions. A first suggestion to use Joseph-son junctions as threshold noise detectors was made byTobiska and Nazarov in 2004, and since then vari-ous aspects of this idea have been analyzed by severalauthors. Two recent experiments have studied the noise gen-erated by a tunnel junction through measurements of theswitching rate of an on-chip Josephson junction out ofthe zero-voltage state. The skewness of the noise canbe extracted from the asymmetry of the switching ratewith respect to the direction of the bias current. In theregion of noise activated escape, relevant for the experi-ments, the switching of a Josephson junction noise detec-tor has been investigated in two recent papers. The workby Ankerhold describes the dynamics of the Joseph-son junction in terms of a Fokker-Planck equation drivenby external noise. An approximate analytical expressionfor the switching rate is obtained for the entire range ofdamping parameters. The subsequent work by Sokho-rukov and Jordan employs a path integral formalismand accounts for the feedback of the noise detector onthe noise generating device. The authors also deriveasymptotically exact results for the switching rate in theweak noise limit, however, only for the cases of vanishingdamping and strong overdamping. In these limiting casesthe problem simplifies considerably, since the number ofrelevant state variables is halved. The experimentallysignificant parameter range is at intermediate damping.The aim of the present work is to provide, for theregion of activated escape in the weak noise limit, anasymptotically exact solution for the switching rate ofa Josephson junction in presence of a device that gener-ates non-Gaussian noise. The mutual influence of the twomesoscopic devices, Josephson noise detector and noisegenerator, will fully be taken into account by treatingthem on an equal footing. Furthermore, the entire rangea damping parameters of the Josephson junction will becovered.The article is organized as follows. Sec. II briefly re-views a simplified version, sufficient to the present pur-poses, of the path integral representation of nonequi-librium thermodynamics in terms of thermodynamicallyconjugate variables. This approach was introduced morethan fifty years ago by Onsager and Machlup for thelinear range near equilibrium and was then extended tothe nonlinear range by Grabert, Graham, and Green. The method, which is based on the conventional conceptof Gaussian fluctuations, will then be applied in Sec. IIIto the thermal escape of a Josephson junction drivenby Johnson-Nyquist noise. These two introductory sec-tions will also serve to introduce the relevant notation.The model described in Sec. III will then be extendedin Sec. IV to account for non-Gaussian noise generatedby a nonlinear device. Finally, Sec. V discusses concreteresults for the experimentally relevant range of parame-ters and presents our conclusions. Some more technicaldetails are moved to appendices.
II. PATH INTEGRAL REPRESENTATION OFFLUCTUATIONS IN NONLINEARIRREVERSIBLE PROCESSES
Einstein and Onsager have related the stochas-tic theory of spontaneous fluctuations about equilibriumwith the deterministic theory of irreversible processes.Perhaps the most seminal expression of this relation be-tween irreversible processes and fluctuations is the pathintegral representation for the transition probability be-tween two macroscopic states. This functional, whichgives a generalization of the Boltzmann probability dis-tribution to the time domain, was introduced by Onsagerand Machlup for the linear range near equilibrium andextended to nonlinear processes by Grabert, Graham,and Green. Originally, the theory was formulated for closed sys-tems where the entropy is the appropriate thermody-namic potential. Here we want to apply the method todescribe mesoscopic systems that exchange energy witha cryostat. The modifications are, of course, well-known.The entire closed mega-system is divided into the systemof interest and the heat bath at constant temperature T ,and the Helmholtz free energy F becomes the relevantthermodynamic potential to characterize the system ofinterest. When the state of this system is described interms of the state variables a = ( a , . . . , a N ), the Onsager transport equations take the form˙ a I = f I = X J L IJ λ J , (1)where the L IJ are the Onsager transport coefficients,while the λ I = − T ∂F∂a I (2)are the thermodynamic forces. The transport equationsare nonlinear, if the thermodynamic forces are nonlinearfunctions of the state variables a or if the transport coef-ficients L IJ depend on the state variables. As will be seenbelow, for the problem addressed here, the state depen-dence of the transport coefficients is not relevant, and itwill therefore be assumed that the L IJ are constant, theymay depend on temperature and other external param-eters though. This simplifies the general theory treatedin Refs. 20,21 quite considerably. The state variables a I can be chosen to be either evenor odd under time reversal˜ a I = ε I a I , ε I = , for even variables − , for odd variables . (3)The Helmholtz free energy is an even variable F (˜ a ) = F ( a ) , (4)and the transport coefficients obey the reciprocal rela-tions L IJ (˜ a ) = ε I ε J L IJ ( a ) . (5)The matrix L IJ may be split into a symmetric part D IJ = 12 [ L IJ + L JI ] (6)and an antisymmetric part A IJ = 12 [ L IJ − L JI ] . (7)This implies a decomposition of the deterministic fluxes f I into a reversible drift r I = X J A IJ λ J , (8)with the symmetry r I (˜ a ) = − ε I r I ( a ), and an irreversibledrift d I = X J D IJ λ J , (9)with d I (˜ a ) = ε I d I ( a ). Only the irreversible drift con-tributes to the time rate of change of the free energy˙ F = − T X I λ I ˙ a I = − T X I λ I d I = − T X I , J D IJ λ I λ J . (10)Often, and in particular for the systems treated below,some of the state variables do not couple directly to mi-croscopic degrees of freedom, and their fluxes are purelyreversible. We then chose the set of state variables a so that the first n variables ( a , . . . , a α , . . . , a n ) are thosewith purely reversible fluxes˙ a α = f α = r α . (11)These variables will be distinguished by Greek indices α, β , while the remaining variables ( a n+1 , . . . , a i , . . . , a N )with partly irreversible fluxes will be marked by smallroman indices i,j. As previously, large roman indices I,Jrun through the complete set from 1 to N. Since the firstn transport equations of the set (1) take the form (11),the symmetric parts of some of the transport coefficientsvanish D α,β = D α, i = D i ,α = 0 . (12)In the stochastic theory of irreversible processes theirreversible drift is intimately connected with sponta-neous fluctuations about the deterministic motion. These fluctuations can be accounted for by random con-tributions η I to the thermodynamic forces λ I . Follow-ing the approach by Grabert, Graham, and Green, the stochastic theory can be described in terms of aHamiltonian H ( a, η ) = 12 X I , J D IJ η I η J + X I f I ( a ) η I , (13)which implies equations of motion of canonical form˙ a I = ∂H∂η I = f I + X J D IJ η J ˙ η I = − ∂H∂a I = − X J ∂f J ∂a I η J . (14)Note that the deterministic transport equations (1) arespecial solutions of (14) with η I = 0.The canonical equations can be interpreted as Euler-Lagrange equations and constraints (for the purely re-versible fluxes) of an action principle. The action de-termines the probability of a fluctuation path, and thetransition probability from an initial state a (0) = a to afinal state a ( t ) = a ′ may be written as a path integral p t ( a ′ | a ) = Z D [ a, η ] exp (cid:26) − k B A [ a, η ] (cid:27) , (15)with the action functional A [ a, η ] = Z t ds X I η I ˙ a I − H ( a, η ) . (16)Since in view of Eq. (12) the Hamiltonian (13) hasquadratic terms for the η i only, the action functional islinear in the η α which act as Lagrange parameters enforc-ing the constraints (11). The η i , on the other hand, are V I c CR FIG. 1: Circuit diagram of a Josephson junction with criticalcurrent I c and capacitance C biased by a voltage source V via a resistor R . random forces describing fluctuations away from the de-terministic motion. The Hamiltonian is quadratic in the η i because of the underlying assumption of Gaussian fluc-tuations. For mesoscopic systems this assumption maynot be sufficient and an appropriate extension of the ap-proach to incorporate non-Gaussian noise will be givenin Sec. IV. III. THERMAL ESCAPE OF A JOSEPHSONJUNCTION FROM THE ZERO-VOLTAGE STATE
In this section the thermally activated escape of aJosephson junction form the zero-voltage state is re-viewed utilizing the approach outlined in the previoussection. A. Transport Equations of a Biased JosephsonJunction
The state variables of the Josephson junction are thecharge Q on the junction capacitance C and the phasedifference ϕ between the order parameters of the super-conductors on either side of the tunnel barrier. Thetime rate of change of the phase is related to the volt-age V J across the Josephson junction by the Josephsonrelation V J = ~ e ˙ ϕ . (17)When a voltage V is applied to a Josephson junction inseries with an Ohmic resistor R , as depicted in the circuitdiagram Fig. 1, the electrical current I flowing throughresistor and junction reads I = 1 R ( V − V J ) = ˙ Q + I c sin( ϕ ) , (18)where the second equality follows with the help of Joseph-son’s relation I s = I c sin( ϕ ) for the supercurrent I s acrossthe junction. Combining Eqs. (17), (18) with V J = Q/C ,we readily find the deterministic equations of motion˙ ϕ = 2 e ~ QC ˙ Q = 1 R (cid:18) V − QC (cid:19) − I c sin( ϕ ) . (19)Clearly, ϕ is a variable with purely reversible flux.Let us introduce the free energy F ( Q, ϕ ) = F ( T, V ) + Q C − ~ e (cid:20) I c cos( ϕ ) + VR ϕ (cid:21) , (20)and the thermodynamic forces λ ϕ = − T ∂F∂ϕ = − T ~ e I c (cid:20) sin( ϕ ) − VR (cid:21) λ Q = − T ∂F∂Q = − T QC . (21)The equations of motion (19) can be then written in On-sager form ˙ ϕ ˙ Q = − eT ~ eT ~ TR λ ϕ λ Q . (22)Following the approach outlined in the previous section,and denoting the conjugate variables to ( ϕ, Q ) by ( µ, λ ),the Hamiltonian of the system is found to read H ( ϕ, Q, η, λ ) = T R λ + 1 R (cid:18) V − QC (cid:19) λ − I c sin( ϕ ) λ + 2 e ~ QC µ , (23)leading to the canonical equations˙ ϕ = ∂H∂µ = 2 e ~ QC ˙ Q = ∂H∂λ = 1 R (cid:18) V − QC (cid:19) − I c sin( ϕ ) + TR λ ˙ µ = − ∂H∂ϕ = I c cos( ϕ ) λ ˙ λ = − ∂H∂Q = 1 RC λ − e ~ µC . (24)While the purely reversible flux ˙ ϕ remains unchanged inthe stochastic theory, the flux ˙ Q is now supplemented bya current ( T /R ) λ describing Gaussian Johnson-Nyquistnoise from the Ohmic resistor. B. Decay of the Zero-Voltage State
As is apparent from Eq. (20), the Josephson junctionmoves in the effective “tilted washboard” potential U ( ϕ ) = − ~ e (cid:20) I c cos( ϕ ) + VR ϕ (cid:21) . (25) It is convenient to introduce the dimensionless bias cur-rent s = VRI c . (26)Then, for 0 < s <
1, the potential has extrema in thephase interval [0 , π ] at ϕ well , top = arcsin( s ) = π ∓ δ , (27)where for 1 − s ≪ δ ≈ p − s ) . (28)When the Josephson junction is trapped in the state ϕ well = π − δ , the average voltage V J across the junctionvanishes. However, this zero-voltage state is metastable,since the well is only a local minimum of the potential(25). To escape from the well, the junction needs to bethermally activated to the barrier top at ϕ top = π + δ .This process will be observed with large probability, whenthe barrier height is small, which is the case when thedimensionless bias current s is close to 1. We shall notdiscuss here escape by macroscopic quantum tunneling, which occurs at very low temperatures.The decay rate follows from the transition probabil-ity from ϕ well to ϕ top as governed by the path integral(15). The dominant contribution to the functional inte-gral comes from the minimal action path satisfying thecanonical equations (24). Let us first consider the reverseprocess, the relaxation from the barrier top ϕ top to thewell minimum ϕ well . In this case the most probable pathis the deterministic path, that is a solution of the evolu-tion equations (24) with µ = λ = 0. The two remainingequations of motion can be combined to read ~ e C ¨ ϕ + ~ e R ˙ ϕ + I c sin( ϕ ) = VR . (29)There is a solution ϕ relax ( t ) of (29) satisfying ϕ relax ( −∞ ) = ϕ top , ϕ relax (+ ∞ ) = ϕ well , (30)which describes the relaxation from the barrier top tothe well bottom. Since µ and λ vanish, this deterministictrajectory has vanishing action (16).The minimal action trajectory for thermally activatedescape from the zero-voltage state ϕ well is a solution ϕ esc ( t ) of the canonical equations (24) with ϕ esc ( −∞ ) = ϕ well , ϕ esc (+ ∞ ) = ϕ top . (31)The first two of the canonical equations (24) combine togive ~ e C ¨ ϕ + ~ e R ˙ ϕ + I c sin( ϕ ) = V + T λR . (32)Now, the ansatz ϕ esc ( t ) = ϕ relax ( − t ) satisfies the bound-ary conditions (31) and also the evolution equation (32)provided λ esc ( t ) = − ~ eT ˙ ϕ relax ( − t ) = ~ eT ˙ ϕ esc ( t ) , (33)where we have used the fact that ϕ relax ( t ) is a solu-tion of Eq. (29) with boundary conditions (30), and that˙ ϕ esc ( t ) = − ˙ ϕ relax ( − t ), ¨ ϕ esc ( t ) = ¨ ϕ relax ( − t ). The lastequation of the set (24) then gives µ esc ( t ) = − ~ C e (cid:20) ˙ λ esc ( t ) − RC λ esc ( t ) (cid:21) = − ~ eT (cid:20) ~ e C ¨ ϕ relax ( − t ) + ~ e R ˙ ϕ relax ( − t ) (cid:21) = − ~ eT (cid:20) VR − I c sin ( ϕ relax ( − t )) (cid:21) , (34)where we have again used the equation of motion (29) sat-isfied by ϕ relax ( t ) to derive the last line. Now, Eqs. (33)and (34) combine to give˙ µ esc ( t ) = I c cos ( ϕ esc ( t )) λ esc ( t ) , (35)so that the remaining equation of the canonical set ofequations (24) is also satisfied, and the ansatz ϕ esc ( t ) = ϕ relax ( − t ) gives indeed the minimal action escape path.To determine the action (16) of the escape path, wefirst note that the Hamiltonian (23), which is conservedalong a solution of the canonical equations, vanishes onthe escape path, since λ esc ( ±∞ ) = µ esc ( ±∞ ) = 0, as canbe inferred from Eqs. (33) and (34). Thus A esc = Z ∞−∞ dt h λ esc ( t ) ˙ Q esc ( t ) + µ esc ( t ) ˙ ϕ esc ( t ) i = Z ∞−∞ dt (cid:26) ~ eT ˙ ϕ esc ( t ) ~ e C ¨ ϕ esc ( t ) − ~ eT (cid:20) VR − I c sin ( ϕ esc ( t )) (cid:21) ˙ ϕ esc ( t ) (cid:27) , (36)where we have used the first of the canonical equations(24) as well as Eqs. (33) and (34) to express ˙ Q esc ( t ), λ esc ( t ), and µ esc ( t ) in terms of ϕ esc ( t ). The result (36)may now be transformed to read A esc = 1 T Z ∞−∞ dt ((cid:18) ~ e (cid:19) C ∂∂t ˙ ϕ + 2 U ′ ( ϕ esc ) ˙ ϕ esc ) = 2 T [ U ( ϕ top ) − U ( ϕ well )] , (37)where the last expression follows from the boundary con-ditions (31) obeyed by ϕ esc ( t ) for t → ±∞ .The rate of escape Γ from the metastable well may bewritten as Γ = f e − B , (38) where the exponential factor B is determined by the ac-tion of the most probable escape path ϕ esc of the pathintegral. Introducing the barrier height∆ U = U ( ϕ top ) − U ( ϕ well ) , (39)we obtain from Eqs. (15) and (37) for the exponentialfactor B = ∆ Uk B T , (40)which is just the standard Arrhenius factor for thermallyactivated decay. The pre-exponential factor f requiresan analysis of the fluctuations about the minimal actionpath and will not be addressed here. IV. JOSEPHSON JUNCTION DRIVEN BYNON-GAUSSIAN NOISE
So far we have studied a biased Josephson junctiondriven by Gaussian thermal noise. We now address thequestion how the rate of escape Γ from the zero-voltagestate is modified by the presence of non-Gaussian noise.To be specific, we shall consider the shot noise generatedby a normal state tunnel junction, since this case hasbeen examined in recent experiments.
However, thetheory likewise applies to other noise generating deviceswith short noise correlation times.
A. Hamiltonian for Non-Gaussian Noise
Let us consider a Josephson junction with capacitance C and critical current I c driven by two noise sources, seeFig. 2. A bias voltage V B is applied to one branch with anOhmic resistor R B in series with the junction. This partof the set-up corresponds to the model treated in the pre-vious section. A second voltage V N is applied to anotherbranch with a tunnel junction of resistance R N again inseries with the Josephson junction. Experimental set-upsare typically more sophisticated, but the circuit diagramin Fig. 2 captures the essentials of a Josephson junctionon-chip noise detector. The current I flowing throughthe Josephson junction is given by I = V B − V J R B + V N − V J R N = ˙ Q + I c sin( ϕ ) . (41)Proceeding as in Sec. III, one readily obtains the deter-ministic equations of motion˙ ϕ = 2 e ~ QC (42)˙ Q = 1 R B (cid:18) V B − QC (cid:19) + 1 R N (cid:18) V N − QC (cid:19) − I c sin( ϕ ) . Since the flux ˙ ϕ is purely reversible, the Hamiltonian H ( ϕ, Q, η, λ ) will depend on the conjugate variable η only C c V N B VR N R B I FIG. 2: Circuit diagram of a Josephson junction with criticalcurrent I c and capacitance C biased in a twofold way. Thebranch to the right puts an Ohmic resistor R B is series withthe junction and is biased by the voltage V B . The branch tothe left is biased by a voltage V N and R N is a noise generatingnonlinear element, specifically a normal state tunnel junctionwith tunnelling resistance R N . linearly, while the dependence on λ comprises linear andnonlinear terms. In contrast to the case studied in theprevious section, the nonlinear terms in λ will not bejust quadratic, since the noise generated by the normalstate tunnel junction is non-Gaussian. As the voltage V ′ N = V N − V J across the tunnel junctions grows relativeto k B T /e , the noise generated by the tunnel junctioncrosses over from Gaussian to Poissonian statistics. Forthe current I N through the tunnel junction one has h I N i = V ′ N R N h δI N ( t ) δI N ( t ′ ) i = C δ ( t − t ′ ) (43) h δI N ( t ) δI N ( t ′ ) δI N ( t ′′ ) i = C δ ( t − t ′ ) δ ( t ′ − t ′′ ) , where δI N ( t ) = I N − h I N i and C = eV ′ N R N coth (cid:18) eV ′ N k B T (cid:19) C = e V ′ N R N . (44)There are of course higher order noise cumulants, but,as we shall see, these are not important in the region ofnoise activated switching of the Josephson noise detector.The skewness of the noise described by C leads to acubic term in λ . Neglecting terms of fourth order, theHamiltonian takes the form H ( ϕ, Q, η, λ ) = T R B λ + 1 R B (cid:18) V B − QC (cid:19) λ + e (cid:16) V N − QC (cid:17) k B R N coth e (cid:16) V N − QC (cid:17) k B T λ + 1 R N (cid:18) V N − QC (cid:19) " λ + 124 (cid:18) ek B (cid:19) λ − I c sin( ϕ ) λ + 2 e ~ QC µ + O (cid:0) λ (cid:1) . (45)An expansion of the Hamiltonian in powers of λ is jus-tified, provided the dimensionless quantity eλ/k B ≪ λ causing the escape is proportional to the size ofthe fluctuations of the voltage V J across the Josephsonjunction, and eλ/k B is in fact very small, if the decayof the zero-voltage state occurs in the region of noise ac-tivated escape. Since V J = Q/C and λ are effectivelyproportional to each other, it does not make sense tokeep higher order terms in Q/C , rather, the two smallparameters, eλ/k B and Q/CV N , should be treated onan equal footing. Hence, the term in the second line ofEq. (45), which is already of second order in λ , can beexpanded to first order in Q/CV N . Likewise the Q/CV N dependence of the term of order λ can be dropped. Wethen find H ( ϕ, Q, η, λ ) = H ( ϕ, Q, η, λ ) + H ( ϕ, Q, η, λ ) , (46)where H ( ϕ, Q, η, λ ) = (cid:18) T R B + C ,N k B (cid:19) λ + (cid:18) I bias − R || QC (cid:19) λ − I c sin( ϕ ) λ + 2 e ~ QC µ , (47)describes Gaussian noise. Here we have introduced thebias current I bias = V B R B + V N R N , (48)the second noise cumulant C ,N = eV N R N coth (cid:18) eV N k B T (cid:19) , (49)and the parallel resistance1 R || = 1 R B + 1 R N . (50)The term H ( ϕ, Q, η, λ ) = 124 k B C ,N λ − k B ∂C ,N ∂V N QC λ (51)with the third noise cumulant C ,N = e V N R N (52)includes the leading order effects of non-Gaussian noise. B. Minimal Action Escape Path in the NearlyGaussian Regime
In the range of parameters studied here, the third or-der Hamiltonian (51) will describe weak corrections tothe dynamics governed by the Hamiltonian (47). In fact,this latter Hamiltonian is precisely of the form of theHamiltonian (23) studied in Sec. III for a Josephson junc-tion in parallel with on Ohmic conductor, provided wereplace R by the parallel resistance R || , the current V /R by the proper bias current I bias , and T by the effectivetemperature T eff = R || (cid:20) TR B + C ,N k B (cid:21) = R || (cid:20) TR B + eV N k B R N coth (cid:18) eV N k B T (cid:19)(cid:21) . (53)For eV N ≪ k B T the tunnel junction generates approx-imately Johnson-Nyquist noise and the effective tem-perature coincides with the cryostat temperature. Onthe other hand, for eV N ≫ k B T , the tunnel junctionis a source of shot noise with a noise power propor-tional to V N . The Josephson junction reacts to the ad-ditional Gaussian noise in the same way as to an ele-vated temperature. Approximate expressions for T eff have been presented previously. Experimentally, T eff can be substantially larger than T .The rate of escape Γ from the zero-voltage state of theJosephson junction will again be of the form (38), wherethe exponent B now takes the form B = B + B (54)with B = ∆ Uk B T eff . (55)The exponential factor B is determined by the action ofthe approximate escape path ϕ ( t ) that solves the canon-ical equations of motion resulting from the second orderHamiltonian (47). The second cumulant (49) of the noisegenerated by the normal state tunnel junction is takeninto account in terms of the effective temperature T eff .To include the effects of the third cumulant C ,N , oneneeds to determine the deviation ϕ ( t ) of the escape pathfrom ϕ ( t ). To this purpose we start with the canonicalequations that follow from Eqs. (46), (47) and (51). Wefind ˙ ϕ = ∂H∂µ = 2 e ~ QC ˙ Q = ∂H∂λ = I bias − R || QC − I c sin( ϕ ) + T eff R || λ + 18 k B C ,N λ − k B ∂C ,N ∂V N QC λ , (56) where we have made use of Eq. (53), and˙ µ = − ∂H∂ϕ = I c cos( ϕ ) λ ˙ λ = − ∂H∂Q = 1 R || C λ − e ~ µC + 14 k B ∂C ,N ∂V N λ C . (57)Now, the two differential equations (56) of first order canbe combined to one second order differential equation ~ e C ¨ ϕ + ~ e R || ˙ ϕ + I c sin( ϕ ) = I bias + T eff R || λ + I , (58)where I = 18 k B C ,N λ − ~ ek B ∂C ,N ∂V N ˙ ϕλ (59)is the additional noise current arising from H . Likewise,the Eqs. (57) combine to give ~ e C ¨ λ − ~ e R || ˙ λ + I c cos( ϕ ) λ = I ′ λ , (60)where I ′ = ~ ek B ∂C ,N ∂V N ˙ λ (61)again results from H .We now make the ansatz ϕ esc ( t ) = ϕ ( t ) + ϕ ( t ) λ esc ( t ) = λ ( t ) + λ ( t ) , (62)where ϕ ( t ) and λ ( t ) are the solutions of (58) and (60)for I = I ′ = 0, while ϕ ( t ) and λ ( t ) describe the mod-ifications of the path arising for finite I and I ′ . For I = 0, the equation of motion (58) is of the form of theevolution equation (32) studied in Sec. III, and we canproceed as there. Provided s = I bias /I c <
1, the po-tential U ( ϕ ) = − ( ~ / e ) I c [cos( ϕ ) + sϕ ] has a minimum ϕ well and a maximum ϕ top in the phase interval [0 , π ].From a solution ϕ relax ( t ) satisfying ~ e C ¨ ϕ + ~ e R || ˙ ϕ + I c sin( ϕ ) = I bias , (63)and the boundary conditions (30), we obtain an escapepath satisfying the Eqs. (58) and (60) for I = I ′ = 0and the boundary conditions ϕ ( −∞ ) = ϕ well , ϕ (+ ∞ ) = ϕ top λ ( −∞ ) = 0 , λ (+ ∞ ) = 0 (64)by putting ϕ ( t ) = ϕ relax ( − t ) λ ( t ) = ~ eT eff ˙ ϕ ( t ) . (65)Next, we insert the ansatz (62) into the evolution equa-tions (58) and (60) and keep only terms that are linearin the quantities ϕ , λ , I , and I ′ which describe cor-rections to the Gaussian case. Taking advantage of theequations of motion satisfied by ϕ and λ , we obtain ~ e C ¨ ϕ + ~ e R || ˙ ϕ + I c cos( ϕ ) ϕ = T eff R || λ + I , (66)and ~ e C ¨ λ − ~ e R || ˙ λ + I c cos( ϕ ) λ = ~ eT eff ˙ ϕ [ I c sin( ϕ ) ϕ + I ′ ] , (67)where I and I ′ defined in (59) and (61) are now evalu-ated with the leading order solutions (65). Hence I = 12 ( k B T eff ) (cid:18) ~ e (cid:19) (cid:18) C ,N − k B T eff ∂C ,N ∂V N (cid:19) ˙ ϕ , (68)and I ′ = 1 k B T eff (cid:18) ~ e (cid:19) ∂C ,N ∂V N ¨ ϕ . (69)We shall see that an explicit solution of these evolutionequations is not required to determine the action. C. Action of Escape Path
Since the Hamiltonian (46) vanishes along the escapepath, the action may be written A esc = Z ∞−∞ dt h λ esc ˙ Q esc − ˙ µ esc ϕ esc i , (70)where we have made a partial integration with respectto the first line of Eq. (36). From Eq. (56), we have˙ Q esc = ( ~ / e ) C ¨ ϕ esc , while Eq. (57) implies ˙ µ esc = I c cos( ϕ esc ) λ esc . Inserting this as well as the ansatz (62)into the action (70), we find after disregarding terms ofsecond order in ϕ and λ A esc = A + A , (71)where A = Z ∞−∞ dt (cid:20) ~ e Cλ ¨ ϕ − I c cos( ϕ ) λ ϕ (cid:21) , (72) and A = Z ∞−∞ dt (cid:20) ~ e C ( λ ¨ ϕ + λ ¨ ϕ ) (73) − I c cos( ϕ ) ( λ ϕ + λ ϕ ) + I c sin( ϕ ) λ ϕ ϕ (cid:21) . Now, the deviations ϕ and λ from the path of the Gaus-sian model are caused by the currents I and I ′ given inEqs. (68) and (69). These currents depend on the thirdnoise cumulant C ,N and on the derivative ∂C ,N /∂V N of the second cumulant. The detailed evaluation of theaction in App. B shows, that these two factors influencethe action A only in the combination C = C ,N − k B T eff ∂C ,N ∂V N . (74)A corresponding reduction of the effective third cumulantwas already noted by Sokhurokov and Jordan for thelimiting cases of weak and strong damping. The secondterm in Eq. (74) arises from the feedback of the Joseph-son junction on the noise generating junction, which is aconsequence of the finite voltage V J that builds up dur-ing escape. Experiments are usually done in the regime eV N ≫ k B T , where C ≈ C ,N (cid:18) − k B T eff eV N (cid:19) ≈ C ,N − R B + k B TeV N R N R B + R N ! , (75)so that the feedback becomes negligible for R N ≫ R B .In the opposite limit the feedback even changes the signof C .As shown in App. B, repeated use of the equationsof motion satisfied by ϕ , λ , ϕ , and λ allows oneto express A entirely in terms of ϕ ( t ). By virtue ofEq. (65), ϕ ( t ) is time reversed to the deterministic tra-jectory ϕ relax ( t ) describing the relaxation from the bar-rier top. Accordingly, the result (B20) in App. B may bewritten as A = − k B ( k B T eff ) (cid:18) ~ e (cid:19) C J . (76)where J = − Z ∞−∞ dt ˙ ϕ ( t ) . (77)Thus, the non-Gaussian correction to the rate exponent(54) reads B = 1( k B T eff ) (cid:18) ~ e (cid:19) C J . (78)What remains to be determined is the quantity J , whichdescribes a property of the system in the absence of noise.Let us introduce the energy function E ( ϕ, ˙ ϕ ) = 12 (cid:18) ~ e (cid:19) C ˙ ϕ + U ( ϕ ) , (79)where U ( ϕ ) is the potential (25) with V /R replaced by I bias = sI c . The time rate of change of E reads ddt E = (cid:18) ~ e (cid:19) C ˙ ϕ ¨ ϕ + ~ e [ I c sin( ϕ ) − I bias ] ˙ ϕ , (80)which, using the equation of motion (63) satisfied by ϕ relax ( t ), may be written as ddt E = − (cid:18) ~ e (cid:19) R || ˙ ϕ . (81)Along the deterministic trajectory ϕ relax ( t ) we may lookupon E as a function of ϕ . Then dEdϕ = 1˙ ϕ dEdt = − (cid:18) ~ e (cid:19) R || ˙ ϕ , (82)and from Eq. (79) we have˙ ϕ = ± e ~ r C ( E − U ) , (83)which combines with Eq. (82) to yield dEdϕ = ± ~ e R || r C ( E − U ) , (84)where the sign is determined by the fact that E decreasesalong the trajectory.The function E ( ϕ ) can easily be determined by numer-ical integration of Eq. (84). One starts from ϕ = ϕ top with energy E ( ϕ top ) = U ( ϕ top ) and integrates towardssmaller ϕ with the + sign of Eq. (84) until the first turn-ing point with E ( ϕ ) = U ( ϕ ) is reached. There, the inte-gration continues towards larger values of ϕ with the − sign of Eq. (84) up to the second turning point and soon, until the trajectory ends at E ( ϕ well ) = U ( ϕ well ).By virtue of Eq. (83) the formula (77) may be writtenas J = − C (cid:18) e ~ (cid:19) Z ∞−∞ dt ˙ ϕ ( E − U ) . (85)Changing from an integration over time to one overphase, we get J = − C (cid:18) e ~ (cid:19) Z ϕ well ϕ top dϕ ( E − U ) , (86)where the integration starts at ϕ top and goes back andforth between the turning points until it ends in ϕ well .The determination of the effect of non-Gaussian noise onthe rate of escape is thus reduced to an integration of thefirst order differential equation (84) and the evaluationof the integral (86). V. DISCUSSION
In this section we will give some concrete results in theexperimentally relevant range of parameters.
A. Dimensionless Quantities
It is convenient to formulate the theory in terms ofdimensionless quantities. Introducing the plasma fre-quency of the Josephson junction at vanishing bias cur-rent ω p = r e ~ I c C , (87)the result (86) may be written as J = ω p j , (88)where j = − Z ϕ well ϕ top dϕ ( e − u ) (89)is a dimensionless integral given in terms of the dimen-sionless energy e = 2 e ~ EI c = 12 ω p ˙ ϕ + u (90)and the dimensionless potential u = 2 e ~ UI c = − cos( ϕ ) − sϕ . (91)From Eq. (78), the correction B to the exponential fac-tor of the rate may then be written as B = (cid:18) ~ e (cid:19) ω p ( k B T eff ) C j . (92)To determine j from Eq. (89), one needs to solve thedimensionless form of Eq. (84), which reads dedϕ = ± γ p e − u ) , (93)where γ = 1 R || Cω p (94)is the dimensionless damping coefficient, which coincideswith the inverse quality factor Q = R || Cω p at vanishingbias current.0 B. Strong Damping
Let us first discuss the limit of strong damping γ ≫ e = u + κ (95)and find dκdϕ = − dudϕ ± γ √ κ . (96)This gives √ κ = ± √ γ (cid:18) dudϕ + dκdϕ (cid:19) , (97)so that the dimensionless kinetic energy κ is of order 1 /γ for large γ . The leading order solution κ = 12 γ (cid:18) dudϕ (cid:19) (98)satisfies the boundary conditions e = u , i.e. κ = 0, for ϕ = ϕ top and ϕ = ϕ well . Inserting Eq. (98) into Eq. (89),we obtain j = − γ Z ϕ well ϕ top dϕ (cid:18) dudϕ (cid:19) . (99)In the overdamped limit, there are no turning points, butthe phase gradually slides down from ϕ top to ϕ well . UsingEqs. (27) and (91), Eq. (99) is readily evaluated with theresult j = (cid:0) s (cid:1) arccos( s ) − s √ − s γ . (100)Now, the observed escape events occur typically for val-ues of the bias current I bias close to the critical current I c . Then, 1 − s ≪ j = 8 √
245 (1 − s ) / γ . (101)This latter formula is in accordance with the result bySukhorukov and Jordan in this limit. C. Very Weak Damping
Next we consider the case of a very weakly dampedJosephson junction, i.e., γ ≪
1. Then the trajectory ϕ ( t ) oscillates back and forth in the potential well andlooses energy only very gradually. Let us consider a seg-ment of the trajectory starting at a turning point ϕ + onthe barrier side of the potential, oscillating through thepotential well to a turning point ϕ − on the opposite side,and traversing the potential well again to a turning point ϕ ′ + . From Eq. (93) we find for the energy along this pathsegment e ( ϕ ) = e ( ϕ + ) + γ Z ϕ − ϕ + dϕ p e − u ) ± γ Z ϕϕ − dϕ p e − u ) , (102)where the + sign holds for the oscillation form ϕ + to ϕ − , and the − sign on the way back from ϕ − to ϕ ′ + . For γ ≪
1, this gives e ( ϕ ) = e + + γ Z ϕ − ϕ + dϕ p e + − u ) ± γ Z ϕϕ − dϕ p e + − u ) + O ( γ ) , (103)where e + = e ( ϕ + ) = u ( ϕ + ). This result can now beinserted into Eq. (89), to find for a segment of the ϕ -integral form ϕ + over ϕ − to ϕ ′ + ∆ j = − (Z ϕ − ϕ + dϕ ( e − u ) + Z ϕ ′ + ϕ − dϕ ( e − u ) ) (104)= 23 γ Z ϕ + ϕ − dϕ Z ϕϕ − dϕ ′ p e + − u ( ϕ ′ )] + O ( γ ) , where we have taken into account that the difference be-tween ϕ + and ϕ ′ + is of order γ .On the other hand, Eq. (103) gives for the change ∆ e of the energy during one oscillation period∆ e = − γ Z ϕ + ϕ − dϕ p e + − u ( ϕ )] + O ( γ ) . (105)Eqs. (104) and (105) combine to yield∆ j ∆ e = − f ( e ) + O ( γ ) , (106)where f ( e ) = 13 R ϕ + ϕ − dϕ ( ϕ + − ϕ ) p e − u ( ϕ ) R ϕ + ϕ − dϕ p e − u ( ϕ ) . (107)Dividing the integral (89) into segments of the form(104), we can transform the integral over ϕ into an inte-gral over energy. Using Eq. (106), we then obtain j = Z u ( ϕ top ) u ( ϕ well ) def ( e ) . (108)1Let us again study specifically the experimentally im-portant range 1 − s ≪
1. Then, the relevant range of ϕ values lies in the vicinity of π . Putting ϕ = π p − s ) ψ , (109)we find for the potential (91) u = − π s + √ − s ) / ς , (110)where ς = ψ − ψ . (111)With the scaled dimensionless energy e = − π s + √ − s ) / ǫ , (112)the result (108) with (107) can be transformed to read j = 23 (1 − s ) × Z − dǫ R ψ + ψ − dψ ( ψ + − ψ ) p ǫ − ς ( ψ ) R ψ + ψ − dψ p ǫ − ς ( ψ ) , (113)where ψ − and ψ + are the negative and smallest positiveroots of ς ( ψ ) = ψ − ψ = ǫ , respectively. The remainingintegral is just a numerical factor independent of s , anda numerical evaluation gives j = a (1 − s ) , with a = 0 . . . . . (114)This result is in accordance with the findings by Sukho-rukov and Jordan in the limit of vanishing damping. D. Intermediate Damping
In experiments typical values of the dimensionlessdamping coefficient γ are small but nonvanishing. Thefactor j in formula (92) for B must then be determinedfrom Eq. (89) using the solution of the differential equa-tion (93). While a numerical evaluation is straightfor-ward for arbitrary values of s , we shall focus on the exper-imentally relevant range 1 − s ≪
1. In terms of the scaledquantities introduced in Eqs. (109) – (112), Eq. (93) reads dǫdψ = ± ˜ γ p ǫ − ς ) , (115)where ˜ γ = (cid:18) − s (cid:19) γ . (116)This differential equation has to be solved with initialcondition ǫ (1) = ς (1) = , and integrated with the FIG. 3: The scaled dimensionless energy ǫ is shown as a func-tion of ψ for γ = 0 .
25. The energy decreases as the trajectorymoves back and forth in the potential ς ( ψ ) depicted as a greyline. proper sign back and forth between the turning pointsuntil the integration ends at ǫ ( −
1) = ς ( −
1) = − . Atypical solution is depicted in Fig. 3. In scaled unitsEq. (89) takes the form j = −
23 (1 − s ) Z − dψ ( ǫ − ς ) , (117)where the integral follows the ψ -path back and forth be-tween the turning points. Since the differential equation(115) depends on s and γ only in the combination ˜ γ , weput j = 23 (1 − s ) W (˜ γ ) , (118)where W (˜ γ ) = − Z − dψ ( ǫ − ς ) . (119)The function W (˜ γ ) determines the correction B of theexponential factor of the rate for arbitrary dampingstrength in the range of bias currents close to the criticalcurrent.From Eq. (114), we obtain W (0) = 1 . . . . , (120)while Eq. (101) gives for ˜ γ ≫ W (˜ γ ) ≈
815 1˜ γ , (121)where we have made use of Eq. (116). In between theselimiting results, the function needs to be determined nu-merically. A list of data points is provided in Table I, andthe function W (˜ γ ) is depicted in Fig. 4 together with thefindings of previous works. This should facilitate thecomparison with experimental results.2 ˜ γ W γ W W as a function of ˜ γ . γ ∼ W FIG. 4: W is depicted as a function of ˜ γ (straight line).Also shown are the results of Ref. 17 for vanishing damping,Eq. (120), (dot), and in the strong damping limit, Eq. (121),(dotted line). The approximate result of Ref. 16 is depictedas a dashed line. E. Conclusions
We have presented a theory for a Josephson junc-tion detecting non-Gaussian fluctuations by means of thenoise driven escape out of the zero-voltage state of thejunction. It has been assumed that the device is op-erated in a regime where the barrier of the washboardpotential is overcome by activated processes. This is al-ways the case if the temperature is not too low and/orthe junction capacitance is not too small. The study wasbased on the theory of irreversible processes and fluctua-tions developed by Onsager and Machlup and Grabert,Graham, and Green. An extension of the method toaccount for non-Gaussian fluctuations was outlined. Inthis approach the random motion of the system is de-scribed in terms of the state variables and the conjugateforces. The force λ conjugate to the electric charge Q ,which appears naturally in this approach, plays a rolesimilar to the counting field introduced in the more re-cent approaches to determine the full counting statisticsof electronic devices. A nonlinear noise generating element in series with theJosephson detector modifies the rate of escape out of thezero-voltage state. The main effect comes from the sec-ond noise cumulant C ,N . However, this Gaussian part of the noise is detected by the Josephson junction in thesame way as Johnson-Nyquist noise. Therefore, as wasshown explicitly, the second noise cumulant can be de-scribed in terms of an effective temperature T eff . Devi-ations from the accordingly modified Arrhenius law arethus due to higher order noise cumulants. The fluctua-tions causing the escape from the metastable well lead tofluctuations of the voltage V J across the Josephson junc-tion. It has been shown that these voltage fluctuationsare small compared to k B T eff /e , which implies that thedimensionless random force eλ/k B causing these fluctua-tions is always small compared to 1. Since the n th ordernoise cumulant gives rise to terms of order ( eλ/k B ) n , de-viations from the modified Arrhenius law essentially onlyarise from the third noise cumulant C ,N , and these cor-rections are typically small. However, the third cumulantis odd under time reversal and the sign of the effect de-pends on the direction of the bias current. Comparingrates for pulses tilting the potential to the right and theleft, respectively, the correction B can be extracted. A Josephson junction threshold detector operating in theregime of noise activated escape thus can measure thethird cumulant, the skewness of the noise, only. Anothereffect of the fluctuations of the voltage V J is a feedback ofthe Josephson detector on the noise generating device asdescribed by the effective third noise cumulant C definedin Eq. (74).The modification of the rate exponent due to the skew-ness of the noise has been determined for arbitrary damp-ing strength of the Josephson junction detector. Thereby,the theory developed goes considerably beyond the re-sults of previous works, that were restricted to lim-iting values of the damping strength or based on approx-imations. Explicit results where given for the case whenthe bias current is close to the critical current, which im-plies that the relevant part of the washboard potentialcan be described by a cubic potential. The effect of theskewness of the noise on the rate is, however, larger forsmaller values of the bias current. Experimentally, therange of relevant bias currents can be influenced by theform of the applied current pulses. The theory presentedhere can readily also be evaluated for the exact form ofthe washboard potential allowing for results for any valueof the bias current and all damping strengths.To be explicit, we have presented the theory using theexample of a normal state tunnel junction as noise gen-erating device. However, the theory readily also appliesto other noise generating elements, provided the corre-lation time of the noise is much smaller than the periodof plasma oscillations of the detector. Finally, in thisarticle, only the exponential factor of the rate has beendetermined. The corrections due to the skewness of thenoise were found to be rather small, and they need sophis-ticated experimental techniques to be detected reliably.Corrections to the pre-exponential factor of the same or-der of magnitude are entirely negligible, so that safelythe prefactor of the standard Gaussian noise theory canbe employed.3 Acknowledgments
This work was carried out in the summer of 2007 dur-ing a sabbatical visit to CEA-Saclay. The warm hos-pitality of the Quantronics group and the enlighteningdiscussions with the group members, in particular withD. Est`eve and H. Pothier, are gratefully acknowledged.The author also wishes to thank J. Ankerhold and J.Pekola for a number of interesting discussions, as well asD. Bercioux for discussions and providing the figures. Fi-nancial support was allocated by the European NanoSci-ERA Programme.
APPENDIX A: VALIDITY OF NEARLYGAUSSIAN APPROXIMATION
In this appendix we investigate the range of validity ofthe nearly Gaussian approximation used in Sec. IV. Sincethe leading order term ϕ ( t ) of the most probable escapepath is the time reversed relaxation path ϕ relax ( t ), theorder of magnitude of the phase velocity ˙ ϕ during escapecoincides with that during relaxation.Let us first consider the case of weak damping. Thetrajectory ϕ relax ( t ) starts with vanishing phase veloc-ity at the barrier top. The largest kinetic energy ( ~ / e ) C ˙ ϕ arises when the potential minimum ϕ well is reached for the first time. For weak damping the ki-netic energy then almost equals the potential energy dif-ference ∆ U . Accordingly, the voltage V J = ( ~ / e ) ˙ ϕ sat-isfies V J ≤ r UC . (A1)As damping increases the phase velocity and, accord-ingly, the maximal voltage across the Josephson junctiondecreases, so that V J will never exceed the estimate (A1)in the entire range of parameters.The plasma frequency of the Josephson junction at fi-nite bias current ω p ( s ) = ω p p sin( δ ) = r e ~ I c C sin( δ ) (A2)is the frequency of small undamped oscillations about theminimum ϕ well of the potential (25). For δ ≪ π , whichis the case for 1 − s ≪
1, Eqs. (25) - (28) yield for thebarrier height (39) of the potential∆ U ≈ ~ I c e δ . (A3)This can be combined with Eq. (A2) to give ~ ω p ( s ) ≈ r e ~ I c δC ≈ eδ r UC . (A4)The bound (A1) for the size of the fluctuations of V J maythus be written as eV J ≤ δ √ ~ ω p ( s ) . (A5) In the region of thermally activated escape one has ~ ω p ( s ) ≪ k B T eff . In view of Eq. (A5) this implies eV J k B T eff ≪ δ √ ≪ , (A6)so that eV J /k B T eff is a small dimensionless parameteralong the most probable escape path.Now, the leading order contribution λ to the force λ causing the escape is determined by Eq. (65), entailingthe estimate λ ≈ ~ eT eff ˙ ϕ ≈ V J T eff , (A7)which combines with the inequality (A6) to give eλk B ≪ . (A8)This shows that an expansion of the Hamiltonian in termsof λ , as done in Eq. (45), is indeed justified. The terms ofthird order in λ are then small, so that ϕ and λ describein fact small corrections to ϕ and λ , respectively.Because of the weak effects of non-Gaussian statistics,the correction B to the exponent of the rate is also small.From Eqs. (55) and (92), we find B B = ~ (2 e ) (cid:18) ~ ω p k B T eff (cid:19) C ∆ U j . (A9)For δ ≪ π we can insert Eqs. (118) and (A3). UsingEq. (28), we then find B B ≈ (cid:18) ~ ω p k B T eff (cid:19) C e I c W δ . (A10)Hence, the effect of the skewness of the noise vanishesproportional to (1 − s ) / as the bias current approaches I c . The ratio B /B can be seen as a product of threefactors B B ≈ (cid:18) ~ ω p ( s ) k B T eff (cid:19) × C e I c × W , (A11)where we have made use of Eq. (A2). Now, in the regimeof activated decay the first factor (1 / ~ ω p ( s ) /k B T eff ) is very small, while the last factor W is of order 1 for weakto moderate damping. Hence, one needs a large factor C /e I c to get observable effects from the skewness of thenoise. Since C ,N is proportional to V N , this means large V N , in particular, eV N ≫ k B T , so that the estimate (75)for C applies. To minimize the reduction of C ,N viathe feedback effects described by Eq. (75), one needs tochoose a bias resistor R B well below R N . Then the factor C e I c ≈ C ,N e I c = V N R N I c . (A12)This means that the current V N /R N should be large com-pared to I c and thus needs to be largely compensated by acurrent V B /R B in the opposite direction to keep the junc-tion biasing current (48) below I c . Experimentally, thiscompensation problem is addressed by employing moresophisticated set-ups. APPENDIX B: EVALUATION OF ACTION OFESCAPE PATH
In this Appendix we evaluate the expressions (72) and(73) for the action of the escape path in the nearly Gaus-sian approximation. Inserting the result (65) for λ , oneobtains from (72) A = 2 T eff Z ∞−∞ dt "(cid:18) ~ e (cid:19) C ˙ ϕ ¨ ϕ − ~ e I c cos( ϕ ) ϕ ˙ ϕ = 2 T eff Z ∞−∞ dt (cid:26) ∂∂t (cid:18) ~ e (cid:19) C ˙ ϕ − ∂∂t ~ e I c [cos( ϕ ) + ϕ sin( ϕ )] (cid:27) . (B1)Now, ˙ ϕ vanishes at the integration boundaries and − ( ~ / e ) I c [cos( ϕ ) + ϕ sin( ϕ )] coincides there with U ( ϕ top ) and U ( ϕ well ), respectively, since sin( ϕ well ) =sin( ϕ top ) = s . Accordingly, Eq. (B1) yields A = 2∆ UT eff , (B2)which gives the exponential factor (55) of the escape rate.After expressing λ in terms of ϕ and putting λ = ~ eT eff Λ , (B3)we obtain from Eq. (73) for the leading order non-Gaussian part of the action A = 1 T eff Z ∞−∞ dt (cid:26) (cid:18) ~ e (cid:19) C ( ˙ ϕ ¨ ϕ + ¨ ϕ Λ ) (B4) − ~ e I c [cos( ϕ ) ( ˙ ϕ ϕ + ϕ Λ ) − sin( ϕ ) ϕ ˙ ϕ ϕ ] (cid:27) . The integral in the first line gives after partial integration A , part I = 1 T eff Z ∞−∞ dt (cid:18) ~ e (cid:19) C (cid:16) ˙ ϕ ¨ ϕ + ϕ ¨Λ (cid:17) . (B5)In this expression we can eliminate the second orderderivatives ¨ ϕ and ¨Λ by means of the equations of mo-tion (66) and (67). Taking the definition (B3) into ac-count, we get A , part I = ~ eT eff Z ∞−∞ dt (cid:26) ˙ ϕ (cid:20) − ~ e R || ˙ ϕ − I c cos( ϕ ) ϕ + ~ eR || Λ + I (cid:21) + ϕ (cid:20) ~ e R || ˙Λ − I c cos( ϕ )Λ + ˙ ϕ ( I c sin( ϕ ) ϕ + I ′ ) (cid:21)(cid:27) . (B6) This result can now be inserted into (B4). After a partialintegration of the ϕ ˙Λ term and a further partial inte-gration along the lines I c [sin( ϕ ) ϕ ˙ ϕ − cos( ϕ ) ˙ ϕ ] ϕ = I c [ − ( ∂/∂t ) cos( ϕ ) ϕ ] ϕ → I c cos( ϕ ) ϕ ˙ ϕ , one obtains A = ~ eT eff Z ∞−∞ dt (cid:26) ˙ ϕ ( I + ϕ I ′ ) (B7)+ (cid:18) ~ e R || ˙ ϕ − I c cos( ϕ ) ϕ (cid:19) (Λ − ˙ ϕ ) (cid:27) . From Eqs. (68) and (69), we see that˙ ϕ ( I + ϕ I ′ ) = 12 (cid:18) ~ e (cid:19) k B T eff ) C ,N ˙ ϕ (B8) − k B T eff (cid:18) ~ e (cid:19) ∂C ,N ∂V N (cid:0) ˙ ϕ − ϕ ˙ ϕ ¨ ϕ (cid:1) . Now, under the integral ϕ ˙ ϕ ¨ ϕ = ϕ ( ∂/∂t ) ˙ ϕ →− ˙ ϕ , so that ˙ ϕ ( I + ϕ I ′ ) can be replaced by˙ ϕ ( I + ϕ I ′ ) → (cid:18) ~ e (cid:19) k B T eff ) C ˙ ϕ (B9)where C = C ,N − k B T eff ∂C ,N ∂V N . (B10)Since the action (B7) depends on Λ − ˙ ϕ only, is is nat-ural to make the ansatzΛ = ˙ ϕ + Λ ′ . (B11)From Eq. (B3) and the equations of motion (66) and (67)one then finds ~ e C ¨Λ ′ + ~ e R || ˙Λ ′ + I c cos( ϕ )Λ ′ = ˙ ϕ I ′ − ˙ I . (B12)Using Eqs. (68) and (69), the right hand side may bewritten as˙ ϕ I ′ − ˙ I = − (cid:18) ~ e (cid:19) k B T eff ) C ˙ ϕ ¨ ϕ , (B13)where again the cumulants appear only in the combina-tion (B10).We can now employ the evolution equation (B12) toexpress the term proportional to I c in the action (B7) infavor of terms with a purely polynomial dependence on ϕ . Using also Eqs. (B9), (B11), and (B13), we find A = ~ eT eff Z ∞−∞ dt (cid:26) (cid:18) ~ e (cid:19) k B T eff ) C ˙ ϕ + ~ e R || ˙ ϕ Λ ′ + 2 ϕ (cid:20) ~ e C ¨Λ ′ + ~ e R || ˙Λ ′ + (cid:18) ~ e (cid:19) (cid:18) k B T eff (cid:19) C ˙ ϕ ¨ ϕ (cid:21)(cid:27) . (B14)5After partial integrations along the lines ϕ ¨Λ ′ → ¨ ϕ Λ ′ , ϕ ˙Λ ′ → − ˙ ϕ Λ ′ , and ϕ ˙ ϕ ¨ ϕ = ϕ ( ∂/∂t ) ˙ ϕ → − ˙ ϕ ,this simplifies to read A = ~ eT eff Z ∞−∞ dt (cid:26) − (cid:18) ~ e (cid:19) k B T eff ) C ˙ ϕ + (cid:20) ~ e C ¨ ϕ − ~ e R || ˙ ϕ (cid:21) Λ ′ (cid:27) . (B15)Comparing the form of the evolution equation (B12)with the one satisfied by ϕ , namely Eq. (58) for I = 0,we are led to the ansatzΛ ′ ( t ) = A ( t ) ˙ ϕ ( t ) . (B16)Inserting this into Eq. (B12) and using the evolutionequation for ϕ as well as Eq. (B13), we find that A ( t )obeys the differential equation ~ e C (cid:20) (cid:18) ˙ A + 1 R || C A (cid:19) ¨ ϕ + (cid:18) ¨ A + 1 R || C ˙ A (cid:19) ˙ ϕ (cid:21) = − (cid:18) k B T eff (cid:19) (cid:18) ~ e (cid:19) C ˙ ϕ ¨ ϕ , (B17) which is satisfied, provided˙ A + 1 R || C A = − C ~ e k B T eff ) C ˙ ϕ . (B18)When the ansatz (B16) is plugged into (B15), we obtaina term proportional to A ˙ ϕ ¨ ϕ , which under the integralcan be replaced by − ˙ A ˙ ϕ . Accordingly, we find A = ~ eT eff Z ∞−∞ dt (cid:26) − (cid:18) ~ e (cid:19) k B T eff ) C ˙ ϕ − (cid:20) ~ e C ˙ A − ~ e R || A (cid:21) ˙ ϕ (cid:27) . (B19)Finally, in the integrand, the expression between squaredbrackets can be transformed by means of Eq. (B18) toyield for A the compact result A = − k B (cid:18) ~ e (cid:19) k B T eff ) C Z ∞−∞ dt ˙ ϕ . (B20) L. D. Landau and E. M. Lifshitz,
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