Feedback and Rate Asymmetry of the Josephson Junction Noise Detector

Feedback and Rate Asymmetry of the Josephson Junction Noise Detector

Abstract

The Josephson junction noise detector measures the skewness of non-Gaussian noise via the asymmetry of the rate of escape from the zero-voltage state upon reversal of the bias current. The feedback of this detector on the noise generating device is investigated in detail. Concise predictions are made for a second Josephson junction as noise generating device. The strong nonlinearity of this component implies particularly strong feedback effects, including a change of sign of the rate asymmetry as the applied voltage approaches twice the superconducting gap.

pacs:
85.25.Cp, 72.70.+m, 73.23.-b

In the last decade there have been extensive theoretical studies of non-Gaussian noise generated by nonlinear electronic nanostructures(1); (2). This new field of full counting statistics (FCS) has put forward numerous predictions indicating that the higher order noise cumulants contain valuable information about the electronic transport mechanisms within the noise generating element that are not accessible from more standard measurements of the conductance and the noise power alone. A few years ago, first experimental observations of non-Gaussian current noise have been reported(3); (4), and more recently quantum point contacts(5); (6) and Josephson junctions(7); (8) have emerged as promising on-chip noise detectors.

The current experimental efforts mainly focus on reliable measurements of the third noise cumulant , the skewness of the noise. A Josephson junction (JJ) in the zero-voltage state can detect the skewness of a noise current passing the detector since is odd under time reversal and thus leads to an asymmetry of the rate of escape from the zero-voltage state of the JJ when the direction of the bias current is reversed.(7); (8) After some primary suggestions for JJ noise detectors(9); (10); (11); (12); (13), concrete theoretical studies for the setup used in experiments have started recently.(14); (15); (16)

Now, that the theoretical tools for predictions for JJ noise detectors with realistic parameters are available, we shall address here the feedback of the noise detector on the noise generating device and show that this feedback can modify the rate asymmetry, the quantity determined experimentally, quite considerably. A detailed understanding of this feedback is essential for reliable data analysis. But this issue is also of relevance for all on-chip detectors, since this very concept implies two mesoscopic devices, detector and measured device, that interact on the same chip and thus have to be treated on the same footing. To be concrete, we shall specifically investigate the detection of non-Gaussian noise generated by a another JJ biased by a voltage with close to twice the gap of the superconductor. Because of the strong nonlinearity of the noise generating device in this region, feedback of the detector leads to particularly pronounced modifications of the rate asymmetry.

A typical realization of a JJ noise detector is sketched in Fig. 1: A JJ with capacitance and critical current is connected by two branches which act as noise sources. A bias voltage is applied to the first branch with an Ohmic resistor in series with the junction. This branch allows to control the detector by fixing the average bias current, applying current pulses, and reading out voltage signals. A second voltage is applied to another branch which includes a non-linear noise generating conductor, again in series with the detector. This noise generating element can most generally be characterized by its charge transfer statistics, respectively its voltage dependent current cumulants . Current experimental set-ups are more sophisticated, but the circuit diagram in Fig. 1 captures the essentials of a JJ on-chip noise detector.

The state variables of the JJ detector are the charge on the junction capacitance and the phase difference between the order parameters of the superconductors on either side of the tunnel barrier. The phase dynamics of the detector can be described by classical physics, since the effective temperature determined by the strength of Gaussian noise from the two branches (see below) is typically much larger that the crossover temperature below which macroscopic quantum tunneling of the phase(17) becomes relevant. The stochastic dynamics of the setup can then be described in terms of a Hamiltonian (18) which depends on the state variables and the conjugate thermodynamic forces . We note that the Hamiltonian of the stochastic theory has to be distinguished from the Hamiltonian of the microscopic theory. depends only on gross variables of the system and is measured in the same units as the rate of change of entropy [] rather than energy []. For the setup in Fig. 1 , where

 HD=2eℏQCμ−Icsin(φ)λ (1)

describes plasma oscillations of the JJ detector, and

 HB = [1RB(VB−QC)+C1]λ (2) + [T2RB+C24kB]λ2+2kB∞∑j=3Cjj!(λ2kB)j

adds the two biasing branches. While the branch with Ohmic resistor generates Johnson-Nyquist noise and has only two nonvanishing cumulants and , the other branch is characterized by the cumulants that are taken at voltage .

The canonical equations of motion following from the Hamiltonian determine the most probable path between a given initial state and a final state , and the transition probability between these states may be written as a path integral(18)

 pi→f=∫D[Q,φ,λ,μ]exp{−12kBA[Q,φ,λ,μ]}, (3)

where the action functional is given by

 A[Q,φ,λ,μ]=∫t0ds[˙Qλ+˙φμ−H(Q,φ,λ,μ)]. (4)

During escape of the detector from the zero-voltage state, the dimensionless quantity can be shown(16) to be always much smaller than 1. Furthermore, the voltage across the JJ detector

 VJ=QC=ℏ2e˙φ (5)

that builds up under noise activation is proportional to . Keeping in Eq. (2) only terms that are at most of third order in and , we obtain , where

 H2=2eℏQCμ−Icsin(φ)λ+(It−1R||QC)λ+Teff2R||λ2 (6)

describes the stochastic dynamics in presence of Gaussian noise, while

 H3 = 124k2BCN3λ3−14kB∂CN2∂VNQCλ2+12∂2CN1∂V2NQ2C2λ (7)

includes the leading order effects of non-Gaussian noise. Here we have defined the total bias current through the detector

 It=VBRB+CN1 (8)

and the parallel (differential) resistance

 1R||=1RB+∂CN1∂V (9)

of the circuit. Moreover, we have introduced the effective temperature

 Teff = R||[TRB+CN22kB] (10)

characterizing the strength of Gaussian noise. The are the cumulants taken at voltage .

In terms of the effective parameters introduced above, the leading order Hamiltonian has the standard form for a biased JJ described by the resistively and capacitively shunted junction model.(19) The phase moves in the “tilted washboard” potential

 U(φ)=−ℏIc2e[cos(φ)+sφ] (11)

driven by Gaussian noise. Here is the dimensionless bias current. For , the potential has extrema in the phase interval at

 φwell,top=arcsin(s)=π2∓δ, (12)

with for . When the JJ is trapped in the state , the average voltage across the junction vanishes. This zero-voltage state is metastable, and to escape from the well, the junction needs to be activated to the barrier top at by noise forces. The weak non-Gaussian noise described by also gives a contribution to this process, and the rate of escape may be written as

 Γ=fe−(B2+B3), (13)

where is the prefactor of the rate, and the exponential factor is determined by the action of the most probable escape path, which is a solution of the canonical equations of motion following from the Hamiltonian with the boundary conditions and . The exponential rate factor has a dominant contribution

 B2=ΔUkBTeff (14)

arising from , where

 ΔU=U(φtop)−U(φwell) (15)

is the barrier height of the metastable well. The correction due to non-Gaussian noise may be evaluated by treating as a perturbation. Following the lines of reasoning of Ref. (16) one finds

 Missing or unrecognized delimiter for \right (16)

where

 C3,eff=CN3−3kBTeff∂CN2∂VN+3(kBTeff)2∂2CN1∂V2N (17)

is the effective third noise cumulant. The second and third terms in Eq. (17) arise from the feedback of the JJ on the noise generating device, which is a consequence of the finite voltage arising during escape. The quantity

 J=−16∫∞−∞dt˙φ3relax(t) (18)

is expressed in terms of the deterministic trajectory starting at at the barrier top and relaxing (in the absence of noise forces) to the well bottom reached at time .

The coefficient is even under time reversal, while is odd, like the third noise cumulant. Accordingly, when the direction of the bias current is reversed, the rate coefficient shows an asymmetry which is a signature of non-Gaussian noise. The deviation of the rate ratio

 Γ(It)Γ(−It)=exp[−2B3(It)]≈1−2B3(It), (19)

from 1 allows for an experimental determination of despite the fact that the corrections due to non-Gaussian noise are typically small.

In the following we will focus on a specific noise generating element, namely a JJ with normal state tunneling resistance and capacitance biased by a voltage which is tuned close to twice the gap of the superconductor. In view of the lead to the voltage source, this noise generating element is in series with an approximately Ohmic lead resistance in the range of a few . A corresponding setup is currently studied experimentally,(20) and is well suited to examine the feedback of the detector on the noise generating element, since the behavior of the JJ is highly nonlinear for .

A JJ with parameters around those indicated in Tab. 1 has a very low sub-gap current, and for the current is carried by tunneling quasiparticles. We can then employ the semiconductor model,(21) where the superconductor is described in terms of quasiparticles with a density of states (DOS) given by the familiar BCS expression

 NS(E)N0 = |E|√E2−Δ2, (20)

where is the normal state DOS at the Fermi level. The mean current through the JJ can be written as

 ⟨I⟩ = Cqp1=e(Γ→−Γ←), (21)

where the forward and backward quasiparticle tunneling rates across the junction interface are given by(22); (23)

 Γ→(V) = 1e2Rt∫dEdE′NS(E)NS(E′+eV)N20 (22) ×f(E)[1−f(E′+eV)]P(E−E′)

and , respectively. Here, is the Fermi distribution. The influence of the electromagnetic environment is described in terms of the probability that a tunneling quasiparticle looses the energy to the environment. This dynamical Coulomb blockade (DCB) effect leads to a smearing of the expected discontinuity of at . Since the phase dynamics of the JJ detector is very slow on the time scale of quasiparticle tunneling in the noise generating JJ, the impedance of the electromagnetic environment is essentially given by the resistance of the lead to the voltage source . For this case of an Ohmic environment one has(23)

 P(E) = ∫∞−∞dt2πℏeiEt/ℏ (23) ×exp[2∫∞−∞dωωReZeff(ω)RKe−iωt−11−e−ℏω/kBT],

where is the effective impedance and . For the set of parameters given in Tab. 1, the resulting --characteristics of the noise generating JJ is shown in Fig. 2. The inset shows the function .

The cumulants of the quasiparticle tunneling current are readily determined from the tunneling rates (22). One has (24)

 Cqpj = ej[Γ→+(−1)jΓ←] (24)

which implies, in particular, . Although has a significant effect on the noise cumulants via DCB, in the region , its influence on the voltage across the noise generating JJ can safely be neglected since . Hence, the modifications of this voltage are essentially due to the detector feedback of order discussed above. Accordingly, in Eqs. (10) and (17), we can put .

The highly nonlinear behavior of the cumulants for strongly affects the effective temperature and the effective third cumulant . Both quantities are shown in Fig. 3 as functions of the applied voltage, again for the representative set of parameters given in Tab. 1. While the skewness of the noise is always positive for positive , the rate asymmetry (19), which is proportional to , passes through zero as is decreased and it takes large negative values for voltages slightly above . This change of sign constitutes a pronounced feedback effect that should be easily observable experimentally.

In summary, we have shown that feedback of the JJ noise detector on the noise generating device can be very pronounced if the device under investigation is highly nonlinear. In particular, another JJ as noise source should allow for a detailed experimental study of this effect, which leads to a strongly enhanced asymmetry of the switching rate of the detector JJ and even a change of sign of the apparent noise skewness as the voltage applied to the noise generating junction approaches twice the superconducting gap. The size of the feedback corrections sensitively depends on DCB of quasiparticle tunnelling.

The authors wish to thank F.S. Bergeret, D. Esteve, Q. Le Masne, A. Levy-Yeyati, H. Pothier and C. Urbina for helpful discussions. Financial support was provided by the nanoscience program of the European Research Area (ERA).

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