Measuring fluctuations with Josephson junctions

Transcription

1 Helsinki University of Technology Department of Engineering Physics and Mathematics Special assignment Tfy Physics 3th September 4 Measuring fluctuations with Josephson junctions Pauli Virtanen 5758F

2 Contents Introduction Theoretical basis 3. Quantum mechanics of electrical circuits Fluctuations Stationary fluctuations Equilibrium fluctuations Langevin equations Intrinsic fluctuations of current Tunneling through a Josephson junction Cumulant expansion Separating contributions to fluctuations Measuring fluctuations with Josephson junctions 3 3. Fluctuations at equilibrium RC circuit RCR circuit Driven fluctuations Langevin equations Second-order correlator Third-order correlator Convergence of expansions Tunneling current Antisymmetric part Symmetric part Analytical estimates Measuring fluctuations Discussion 4 A Dimensionless parameters 43 B Fourier transforms 44 C Linear response; fluctuations at equilibrium 44 C. Quantum fluctuation dissipation theorem C. Response and fluctuation in electrical circuits

3 Introduction Despite the passage of time, the passage of charge still intrigues physicists. In mesoscopic physics, concerned with small structures at low temperatures, much attention has been given to studying how the current through a conductor fluctuates around its average value. This is due to the fact that at low temperatures, these fluctuations are not purely unwanted noise, but reveal information about the microscopic processes giving rise to them. (See Ref. ] for an example.) Certain characteristic quantities, e.g. the second moments, of the fluctuations arising in mesoscopic electrical circuits have now been studied for many years, both experimentally and theoretically. Although other properties, e.g. the higher moments, may be evaluated theoretically, measuring them is often difficult: the effects may be small, installing a detector may disturb them significantly, and measuring enough data so that averages are accurate may take a very long time., 3] Hence, it is still interesting to study different ways of detecting noise. It turns out that one possible way to measure fluctuations is to monitor the tunneling of charges, in our case through a Josephson junction. 4 7] The Josephson junction (JJ) is a commonly occurring element in superconducting devices, with many applications. It consists of two superconductors separated by a thin layer of less superconducting substance, through which electrons may pass either as Cooper pairs (giving rise to supercurrent) or as quasiparticles (normal current). To act as a detector, the Josephson junction should be aware of its electromagnetic environment. This is the case in ultra-small metal insulator metal Josephson junctions. 8] Namely, in such junctions the transfer of a charge across the junction (through tunneling) may increase the electrostatic capacitive energy by amount large compared to the thermal fluctuations in the junction. At small voltages, this makes charge transfer energetically unfavorable and prevents current from flowing, giving rise to the so-called Coulomb blockade. However, fluctuations due to the environment may contribute enough energy to overcome this barrier. Hence, by examining the tunneling (super-)current through a Josephson junction biased at small voltages, one obtains information about the fluctuations in the environment. One may then consider a system consisting of a Josephson junction embedded in a detector circuit, connected to a source circuit. Through theoretical analysis, the fluctuations arising in the microscopical process of charge transfer in the source may be separated from the thermal and quantum fluctuations in the combined circuit. Hence, by examining the results, one may use a Josephson junction for measuring fluctuations due to non-equilibrium processes in the source. 9] In this work, we discuss a way to relate fluctuations of voltage and current to the tunneling current through a Josephson junction. The first part concerns the general theoretical basis of Josephson junctions in Coulomb blockade and fluctuations in electrical circuits, while the second part concentrates on a specific measurement device. The impatient reader should grasp the idea behind the methods in Sections.3 and.4, check the proposed setup in the beginning of Section 3, and finally proceed to the results in Section 3.3.

4 Theoretical basis This section outlines the theoretical tools we use for describing mesoscopic circuits, Josephson junctions and fluctuations. Since the problem here calls for application of both quantum mechanics and circuit theory, we first discuss how they fit together. Then, we consider what fluctuations are and how they may be characterized. Finally, we present the standard way of calculating tunneling rates through an ultra-small Josephson junction and show how it may be extended to handle additional fluctuations and their higher-order statistics.. Quantum mechanics of electrical circuits At low temperatures, fluctuations in electrical circuits are not well described by classical circuit theory, as thermal fluctuations become replaced with zeropoint quantum fluctuations. An established method for handling this is to cast the Kirchoff circuit equations in the language of quantum mechanics. Although considering macroscopic quantities such as the current as quantum mechanical observables may seem strange, estimates provided by the theory have proved to be accurate. Transition to quantum mechanics is usually made through Hamiltonian dynamics. This is also the way in the case of electrical circuits: one simply needs to find a suitable Hamiltonian that produces the Kirchoff equations. Then, one may identify the pairs of canonical conjugate variables, perform operator substitution and finally assert suitable commutator relations. However, here we have in fact no need to write out the Hamiltonian explicitly, so we do not discuss how to derive it. A detailed explanation of Hamiltonian mechanics for LC-oscillator circuits may be read for example in Refs. 8, ]. For electrical circuits the conjugate variables turn out to be the phase φ and the charge q at nodes of the circuit. The phase is defined in terms of the voltage V at the node, φ(t) e t dt V (t ), () where the factor in the front is chosen to be convenient in the sequel (we consider Cooper pairs). These variables also satisfy φ(t), q(t)] = ie, () which is the canonical commutator relation. An important point to make is that for all linear elements (e.g. capacitance, inductance,... ), the quantum-mechanical equations of motion coincide with the classical equations of circuit theory. This is due to the correspondence principle: if the equations of motion are linear, they are trivially the same for both the expectation values and the operators. In fact, this applies also to dissipative elements (resistors), which may be modeled with the Caldeira Leggett Hamiltonian 3]. Hence we may write (in the Fourier transformed form) I(ω) = Z (ω)v (ω), (3) where Z(ω) is the impedance (scalar) over a certain element, I(ω) the current (operator) through it and V (ω) the voltage (also an operator) over it. Thus, the 3

5 usual calculation rules of circuit theory remain unchanged by the introduction of quantum mechanics.. Fluctuations In general, fluctuations of some quantity, say, current I, may be characterized by its central moments (correlators), δi(t), δi(t)δi(t ), δi(t)δi(t )δi(t ),..., (4) where δi(t) I(t) I(t). For classical variables the order of time indices does not matter, but for quantum mechanical variables it is important. Often one also studies symmetrized correlators, linear combinations of correlators with some time ordering, for which interchanging time indices does not change the linear combination itself. Still, physical observables may be associated with many types of different linear combinations and time orderings. 4] The corresponding Fourier transforms F t δi(t) ] (ω), Ft,t δi(t)δi(t ) ] (ω, ω ),..., (5) are also quantities of interest, and they by definition describe the spectrum of the fluctuations. Also these quantities may be symmetrized in various ways. Analogous to central moments, other quantities that characterize fluctuations are the cumulants C n, the cumulant expansion of one variable being defined as e λi = exp ( λc + λ C +... ). (6) The first three cumulants C,..., C 3 are simply the first three central moments, others may be found out by doing the above expansion. Cumulants are also called irreducible correlators, for example fluctuations with a Gaussian distribution have non-vanishing cumulants only up to C, the rest vanish, whereas all the central moments do not. More generally, the cumulants measure the difference from a Gaussian distribution. One may generalize the above to obtain cumulant-like correlators of a timedependent variable I(t), by expanding ln e λi(t)+...+λni(tn) =... + λ λ λ n C n (t,..., t n ) (7) The (simple) joint cumulant C n (t,..., t n ) so defined is a correlation function with properties of cumulants. For quantum-mechanical variables, one must choose some time ordering in the expectation value. The joint cumulant can be Fourier transformed to give the so-called cumulant spectrum or polyspectrum of the variable I(t). More details on cumulants and higher-order statistics may be read starting for example from Ref. 5]. In this work we discuss mainly two important types of fluctuations: stationary fluctuations and equilibrium fluctuations. In the following, we briefly summarize some of their special properties... Stationary fluctuations The density matrix ρ of a stationary system is constant, and commutes with the time development operator U. Hence the correlators in such a system have 4

6 the property N N δi(t i ) = δi(t i + t), t, (8) i= i= which means that nothing fixes the origin of time conversely, correlators with this property are called stationary. This property has an immediate consequence for the Fourier transforms: F t n ] δi(t i ) ( ω) = i= ( n ) = πδ ω j j= n n d s e iωs+...+iωn sn e isn j= ωj n F t,...,t n i= ( n ) πδ ω j S n,i (ω,..., ω n ). j= δi(t i ) ] δi() ] i= (ω,..., ω n ) δi(t i ) If a relation such as F t δi(t) ] (ω) = G(ω)Ft δv (t) ] (ω) applies, it implies that F t n ] δi(t i ) ( ω) = i= by which we have S n,i (ω,..., ω n ) = n n i= i= ] ( n n ] G(ω i ) G ω j )F t δv (t i ) ( ω), j= i= ] δi() (9) () ] ( n G(ω i ) G ω j )S n,δv (ω,..., ω n ). () This relation is used extensively in the sequel, where we assume that the system is kept in a stationary state... Equilibrium fluctuations Systems in thermal equilibrium are often characterized by the canonical density matrix e βh, where H is the Hamiltonian of the system and β (k B T ) is the inverse temperature. If the Hamiltonian is known, one may in principle calculate all the correlators of equilibrium fluctuations. In the calculation of the second-order correlators, one often exploits the symmetries that arise from the fact that the equilibrium density is of the same form as the time development operator e iht/, which leads to the detailed balance relation. A more deep connection is established in the fluctuation dissipation theorem (F D), which relates the equilibrium fluctuations to the response of a system under perturbation. This is especially useful in circuit theory, since the quantities characterizing the response of the system are simply impedances, which are easy to calculate. Found by applying the unitarity of U(t) and the cyclic invariance of the trace in the expectation value. j= 5

7 Detailed balance The detailed balance relates Fourier transforms of correlators at different frequencies. The statement is simple and its proof are as follows: ] F (t)g() = Tr e βh e iht/ F ()e iht/ G() ] = Tr F ()e βh e ih( t i β)/ G()e ih( t i β)/ () = G( t i β)f (), where the cyclic invariance of the trace was exploited. For the Fourier transforms this symmetry implies ] F t F (t)g() (ω) = dt e iωt G( t i β)f () = ds e iωs e β ω G(s)F () = e β ω ] F t G(t)F () ( ω), which is the detailed balance identity. (3) Fluctuation dissipation theorem The fluctuation dissipation theorem is a general result concerning fluctuations at equilibrium. Its proof is lengthy, so it is deferred until appendix C.. For a circuit element with an impedance Z, the theorem predicts that the equilibrium fluctuations of current satisfy F t δi(t)δi() ] (ω) = ω ReZ (ω)](coth(β ω/) + ). (4) These may then be connected to the fluctuations φ of the phase at another point in the circuit by use of the Langevin equations, discussed below...3 Langevin equations We wish now to describe how fluctuations of voltage and current propagate through a circuit. The basic idea is that the current flowing through an element has intrinsic fluctuations, which arise due to the microscopic process of charge transfer, or due to equilibrium fluctuations. These fluctuations occur on a small timescale, and they drive the slower fluctuations of voltages at the nodes. Hence we write 6] I(ω) = Z (ω)( V (ω) + V (ω)) + δi(ω), (5) where V are the fluctuations of the voltage, and δi are the intrinsic fluctuations of current. Writing such equations for each element in the circuit and requiring the conservation of current (valid at least at low frequencies), one may then relate a certain V j to the fluctuations δi j of current in all elements in the circuit. Another way to do this is to apply the F D theorem directly to the fluctuations in the whole circuit by finding out the right response function (impedance). The results turn out to be identical. 6

8 If we now assume that δi j through different elements are independent of each other (since they are of microscopic origin), and that they do not depend on the fluctuations of voltage, we may write all the correlators V j (ω) V j (ω ) of V j as linear combinations of terms proportional to δi j (ω)δi j (ω ). This may also be done to higher-order correlators (the principle of minimal correlations 7]). 3 However, in reality fluctuations δi j do depend on the corresponding fluctuation V j of voltage, if the system is out of equilibrium. This may be taken into account by adding correction terms to the minimal correlation values, as is done in Ref. 8]. The form of these corrections may be found out for example by applying the Keldysh path-integral formulation of counting statistics 9]. However, the second correlator needs no corrections...4 Intrinsic fluctuations of current As described above, circuit elements in general exhibit intrinsic equilibrium fluctuations, the second moment characterized by the impedance. Out of equilibrium, for example when there is a potential difference across the element, there may also be additional fluctuations: driven noise, i.e., shot noise. This arises from the microscopic process of charge transfer, and depends on the details of the junction in question which are not visible in the impedance. At frequencies low enough, it turns out 6,] that the properties of (normalmetal) junctions may be characterized with the impedance and a set of numbers, the Fano factors. 4 In this work, the Fano factors F (n) are defined as C n,i ( ω = ) = F (n) e n I sgn( I ) n, at T, (6) where C n,i is the n:th cumulant of the current at zero frequency, and e is the charge of the electron. Here, F () is usually known as the Fano factor F in the literature, and it corresponds to the second correlator (shot noise). In the sequel, the Fano factors appear as parameters describing properties of fluctuation sources. There are many theoretical methods for evaluating the correlators of current fluctuations for many different types of mesoscopic circuit elements (see for example Refs. 6,, ]). In this work, we rely on the predictions detailed in Refs. 4,6] for the second correlator, and those detailed in Refs. 3,4] for the third. The results are applied in the second part of the work, where we examine the effect of driven fluctuations on the tunneling current through a Josephson junction. A few words on the experimental side. The second correlator of current fluctuations, the noise, has been studied experimentally for a long time, see Ref. 6]. However, the third correlator has been measured only very recently ], although the theory has already been taken quite far. Measurement of the higher-order correlators is generally more difficult. 3] 3 Principle of minimal correlations is here the assumption of complete conservation of the fluctuating current. 4 That is, one applies the scattering matrix approach 6], where properties of the junction are described by probabilities of transmission T n, which Fano factors depend on. 7

9 .3 Tunneling through a Josephson junction A standard way to calculate the tunneling supercurrent flowing through a Josephson junction is to apply perturbation theory. For this purpose, the phenomenological Hamiltonian for the Josephson junction in a circuit may be written as H = H env E J cos(φ) = H env E J (e iφ + e iφ ), (7) where H env is the Hamiltonian of the electromagnetical environment, i.e., the rest of the circuit. Applying the commutator relation (), we note e iφ, q] = ±e e iφ, by which e iφ q = q e. Hence, the Josephson terms in the Hamiltonian transfer charge through the junction. For small E J, we may apply Fermi s Golden Rule to calculate the tunneling rate for charge, following Ref. 8]. According to it, the transition rate from some initial state i to a final state f is given by Γ f i = π f E J ( e iφ + e iφ) i δ(e f E i ). (8) To calculate the forward tunneling, we neglect the e iφ part of the matrix element (i.e. consider only forward charge transfer), as is done in Ref. 8]. 5 To obtain the total rate of transitions, one needs to average over the initial states and sum over the final states: Γ = p i Γ f i = E J 4 p i dt i e ieit/ e iφ e ie f t/ f f e iφ i, (9) f,i f,i where the δ function was expanded to an integral, and p i is the probability of the initial state i. We now assume f and i be eigenstates of energy (for the nonperturbed time-independent Hamiltonian), whence i e ieit/ = i e ihenvt/ and e ie f t/ f = e ihenvt/ f. We then apply the time-development operators e ihenvt/ on the phases φ and enter the Heisenberg picture: Γ = E J 4 i dt i e iφ(t) ] f f e iφ() i = E J 4 dt e iφ(t) e iφ(), f () where we noted that f form a complete basis and that i p i i i is the quantum-statistical expectation value. Separating the stationary part, φ(t) = φ(t) + φ(t) = ev t/ + φ(t), we obtain E Γ = J 4 dt e iev t/ e i φ(t) e i φ(). () Similar calculation for the backward tunneling rate yields E Γ = J 4 dt e iev t/ e i φ(t) e i φ(). () For convenience, we now define P (E) dt e iet/ e i φ(t) e i φ() = π π F t e i φ(t) e i φ() ] (E/ ), (3) 5 This is not completely rigorous; one should apply linear response theory instead (see App. C) and find out that certain cross terms give no contribution. 8

10 and similarly P (E) = π F t e i φ(t) e i φ() ] (E/ ). (4) Thus, the tunneling rates may be written as Γ (V ) = πee J P (ev ) and Γ (V ) = πee J P ( ev ). Using the tunneling rates above, the total current through the Josephson junction may be expressed as I J (V ) = e( Γ (V ) Γ (V )) = πee J ( P (ev ) P ( ev )), (5) where the charge e of the Cooper pairs is taken into account. The above expression gives the supercurrent flowing through the Josephson junction, and it is valid provided that E J P (ev ) 8], which in effect sets limits for the impedance of the environment (P (E) curves for different impedances are evaluated in Subs. 3.). Moreover, the range of voltages V is limited, for details see the discussion on the Coulomb blockade limit in Ref. 5]. The exact V max depends on the environment and the value of E J, but should usually reach at least near the point where I J obtains its peak value. In addition to the supercurrent discussed above, there is also the current carried by quasiparticles. However, it tends to be suppressed at bias voltages smaller than /e 5 µv, where is the energy gap of the superconductor. 8, 6] In this work we hence neglect effects due to quasiparticles..3. Cumulant expansion The phase-phase correlators appearing in Eqs. () and () are not easy to evaluate directly in the general case. Hence we make a cumulant expansion, κ(λ, t) = e λ φ(t) e λ φ() = e C(t)+λC(t)+! λ C (t)+..., (6) where C j are now the cumulants of the fluctuating quantity ( φ(t) φ()), ordered so that φ(t) occurs always before φ() in products. The nth cumulant in the expansion may easily be resolved by differentiating both sides n times with respect to λ and finally letting λ =. This way (and applying stationarity of fluctuations), we obtain C =, C =, C = ( φ(t) φ()) φ(), (7) C 3 = 3 φ(t)( φ(t) φ()) φ(), (8) C 4 = ( 4 φ(t) φ(t) φ() 4 φ(t) φ() + φ() 3 ) φ() 3C, (9) This leads to the expansions C 5 =..., (3) e i φ(t) e i φ() = e J (t), e i φ(t) e i φ() = e J (t), (3) i J (t) = ( φ(t) φ()) φ() + φ(t)( φ(t) φ()) φ() +... = ( S φ(t) S φ()) + i (3) K φ(t) +..., 9

11 where K φ(t) S 3 φ(t, t) S 3 φ(t, ). The terms appearing in J (t) are the same as in J (t), but differ in sign for odd cumulants. As an aside, note that since the cumulant expansion is valid for all imaginary λ, the cumulants inherit symmetry properties of the original correlator. More explicitly, this results in J( t) = J(t), and furthermore for even cumulants we have C n ( t) = C n (t) and for the odd C n+ ( t) = C n+ (t). In terms of the expansion above, we may now write P (E) = π F t e J (t) ] (E/ ), P (E) = π F t e J (t) ] (E/ ), (33) which may be used for writing a simpler expression for the tunneling current (5). In the sequel, the form above will turn out to be very useful. The cumulant expansion is in fact an extension to the standard way of evaluating the phase-phase correlator. Assuming that the environmental Hamiltonian H env is quadratic (such as arising from the use of Caldeira Leggett model), and invoking stationarity, one can tediously show that 8] e ±i φ(t) e i φ() = e ( φ(t) φ()) φ(). (34) That is, the cumulant expansion terminates after the second cumulant, which is equivalent to the fluctuations in φ due to the environment being Gaussian. This then implies that for Gaussian fluctuations, J = J and hence Γ (V ) = Γ ( V ). One must here note that microscopical processes may yield non-gaussian fluctuations, i.e., non-vanishing cumulants of order higher than two, even at equilibrium. The fluctuation dissipation theorem clearly does not handle these. Still, for macroscopic systems, the fluctuations usually average towards a Gaussian distribution (compare with the central limit theorem). Convergence of expansion (3) is not obvious. However, if the matrix elements of φ(t) are small, the series is at least a power series in the corresponding small parameter. Moreover, convergence may occur if the fluctuations are of a type (e.g. Gaussian) for which the series terminates. The convergence for an example setup is discussed in Subs The cumulants appearing in J may be evaluated for a given system. In equilibrium one may apply the fluctuation-dissipation theorem to find out C (t) and include only it, i.e., consider the system macroscopic enough. In nonequilibrium, for example when additional noise is generated by a flowing current, one may apply the Langevin approach described above to find out the relation between the phase and current correlators..4 Separating contributions to fluctuations Consider now the case where we may separate J into two parts, J = J D + J S, where J D is due only to the Gaussian fluctuations at equilibrium, and J S describes some additional fluctuations due to driving. We show here how the effect of the latter may be separated from the former. As found out above in Eqs. (5) and (33), the tunneling current is I J (V ) = ee J F t e J D(t) e J S(t) e JD( t) e J ] S( t) (ev/ ) = πee J P D (E) P S (E) P D ( E) ] P S ( E) (ev ), (35)

12 where we used the fact that the fluctuations at equilibrium are Gaussian, and that Fourier transforms of products resolve to convolutions (denoted by ). We may now use the detailed balance relation P D ( E) = e βe P D (E) and properties of convolutions to find I J (V ) = πee J P D (E) ( P S (E) e βev e βe ] P S ( E)) (ev ). (36) In the absence of additional fluctuations we have I J, (V ) = πee J P D(eV )( e βev ) = πee J P D(e V )( e βe V ) sgn(v ), (37) from which we can solve P D to obtain, after some algebra, IJ, ( E e I J (V ) = ) e βe P S (E) + I J,( E e ) ] e βe P S ( E) (ev ) IJ, ( E e = ) ] e βe IJ, ( E e P S (E) (ev ) ) ] (38) e βe P S (E) ( ev ). The second line follows from the fact that I J, is an antisymmetric function of voltage. What makes Eq. (38) advantageous is the fact that I J, is a quantity that can usually be measured directly. This allows in principle one to find out properties of P S by doing two measurements, one without the additional noise and one with it. In the numerical evaluation of the above, one must note that lim t J S (t) is finite (see Subs. 3..), and P S (E) has δ-function contributions. We have P S (E) = δ(e)e JS( ) + π ejs( ) F t e J (t) JS( ) ] (E/ ), (39) and a similar expression applies for P S. The limiting value of J S is independent of odd cumulants, since in C n the terms φ(t) n and φ() n differ in sign and cancel due to stationarity. The remaining terms are of the form φ(t)... φ() and should vanish as t, at least if the correlation time of fluctuations is finite. Thus J S ( ) lim t J S (t) = lim t J S (t) = J S ( ). A way to proceed from here is to assume the additional fluctuations being small, and to expand P S as follows: 4] π P S (E) = F t e J (t) ] (E/ ) Ft + S φ(t) S φ() + i K φ(t) ] (E/ ) = π δ(e)( S φ()) + S φ(e/ ) + i K φ(e/ ) +..., (4a) π P S (E) π δ(e)( S φ()) + S φ(e/ ) i K φ(e/ ) +..., (4b) where S φ(t) = φ(t) φ(), S φ(ω) = F t S φ(t) ] (ω), (4a) K φ(t) = φ(t)( φ(t) φ()) φ() = S 3 φ(t, t) S 3 φ(t, ), (4b) S 3 φ(t, t ) φ(t) φ(t ) φ(), S 3 φ(ω, ω ) F t,t S3 φ(t, t ) ] (ω, ω ) (4c)

13 and K φ(ω) F ] t K φ(t) (ω) = dω (S π 3 φ(ω ω, ω ) S 3 φ(ω, ω )) = dω ImS iπ 3 φ(ω, ω )]. (4d) At the last step we applied the symmetry S 3 φ(ω, ω ) = S 3 φ(ω +ω, ω ), which arises from the symmetry φ(t) φ(t ) φ() = φ( t) φ(t t) φ() of stationary correlators of hermitian operators. Substituting the expansion above to the expression for the tunneling current, we obtain I J (V ) = ( S φ())i J, (V ) + ( E ) I J, (S s π e φ (E/ ) + i Kã(E/ )) φ ( E ) ( βe ) +I J, coth (S a e φ (E/ ) + i ] Ks φ(e/ )) (ev ), (4) where we split S s/a φ (ω) (S φ (ω) ± S φ ( ω)) and Ks/a φ (ω) (K φ(ω) ± K φ( ω)). 6 This then yields I J (V ) = I A (V ) + I S (V ), I S/A (I J(V ) ± I J ( V )), (43a) I A (V ) = I J, (V ) + ( E ) I J, I J, (V )] S s (E/ ) π e φ (43b) ( E ) ( βe ) ] + I J, coth S a e φ (E/ ) (ev ), ] I S (V ) = (ev ), i ( E ) I J, K ã 4π e (E/ ) + I φ J, ( E e) coth( βe ) Ks φ(e/ ) (43c) for the symmetric part I S and antisymmetric part I A of the current in the presence of additional fluctuations, neatly separating the contributions from the even and odd cumulants, and from the equilibrium fluctuations. From this result we see that properties of the spectrum of fluctuations appear directly in the tunneling current. If the change between equilibrium fluctuations and additional (driven) fluctuations is measured, one may deduce frequency-dependent properties in the additional fluctuations. Finally, as an aside, we note that Eq. (35) allows the symmetric and antisymmetric parts of the tunneling current be written as I A (V m ) = E J I S (V m ) = E J ( ev t dt sin ( ev t dt cos ) ] Im e JD(t) e JS,e(t) cosh(j S,o (t)), (44) ) ] Im e JD(t) e JS,e(t) i sinh(j S,o (t)), (45) where J S,e ( J S + J S ) contains even cumulants and J S,o ( J S J S ) the odd. These equations are easier to approach analytically than the convolutions, but the experimentally measurable current I J, does not appear in them. 6 In fact it turns out that S a φ (ω) = generally, and Ks φ(ω) = for classical third cumulant.

14 I R m R, F C m I m V D A R, F I sm B C J, E J V m I I J Figure : Considered setup for fluctuation measurements. The equations above also illustrate a point worth notice: I S is of first order in J S,o and approximately constant in J S,e, while δi A is of first order in J S,e and of second order in J S,o. Hence, provided J S is small, effects due to even and odd cumulants may easily be separated: even cumulants affect mostly the odd part of the current, while odd cumulants change mainly the even part of the current. In the sequel we concentrate on a particular setup where this theory may be applied. 3 Measuring fluctuations with Josephson junctions As presented above, fluctuations of phase across the Josephson junction affect the current flowing through it, in a way which makes it possible to probe the spectrum of the fluctuations simply by measuring the current. In this section, we calculate the current for fluctuations due to different sources in a specific example setup. The setup we consider is shown in Fig.. 7] It naturally separates into two parts, the detector (right) and the source (left). The detector consists of a Josephson junction with a capacitance C J and Josephson energy E J, in series with a resistor of resistance R m, biased with the non-fluctuating voltage V m. The detector is coupled to the source via capacitance C m. Here we take the source to consist of two resistors, and, with resistances R / and Fano factors F /. When driven with a finite voltage V D, the source generates additional noise, which we intend to measure. Of course, as is seen below, the detector is not restricted to measuring only this type of a source, but can measure arbitrary fluctuations of voltage at point A in the setup. Next we discuss the equilibrium (quantum) phase fluctuations in the setup, after which we show how different types of driven noise affect the tunneling current. 3. Fluctuations at equilibrium Our aim is now to evaluate J(t) = ( φ(t) φ()) φ() for equilibrium fluctuations. Before considering the full setup, let us first take a look how the fluctuations are evaluated in a simple RC circuit. This is made also, for example, in Ref. 8]. 3

15 φ R C E J C R (a) Circuit (b) Impedance at JJ Figure : RC Josephson circuit. 3.. RC circuit Consider the circuit in Fig. where we have written out the capacitance of the Josephson junction parallel to it (as in the RCSJ model 6]). The Langevin equations now read, for an insulating Josephson junction, I(ω) = iωcv (ω) = V (ω)/r + δi(ω) φ(ω) = e Z t (ω) δi(ω), (46) iω where Z t (ω) = ( R iωc) is the impedance seen at the point of the junction. Now, by Eq. (), ( ) ] e Z t (ω) F t φ(t) φ() ] (ω) = ω F t δi(t)δi() (ω) ( ) e Z t (ω) = ω ω (coth(β ω/) + ) R = π ReZ t(ω) (coth(β ω/) + ), R Q ω (47) where we applied the fluctuation dissipation theorem and the mathematical identity Z ReZ ] = ReZ. Moreover, we introduced the quantum resistance R Q h 4e for Cooper pairs. The result is in fact more general than it appears: the impedance Z t (ω) seen at the point of the contact may be arbitrary, and the F D theorem still yields correct results. Proceeding with the calculation, we obtain J(t) = dω ReZ t(ω) ( β ω ) ] coth (cos(ωt) ) i sin(ωt) R Q ω d ω ReZ t ( ω/τ) ( ω ) ] (48) = ρ coth (cos( ωt/τ) ) i sin( ωt/τ), ω R Q z where we used the fact that the real part of the impedance is always an even function of ω. Moreover, we introduced the RC time τ RC, the dimensionless resistance ρ R/R Q, and the dimensionless temperature z τ/( β). 4

16 C m R C J R m Figure 3: RCR impedance. The integral is tricky, but may be evaluated analytically. The result is 7] ( ) J RC (τ t) = ρ πz( + t ) + γ + ψ + π ( ) e t cot πz z n= e πz t n n (πzn) ] ] iρπ( e t ) sgn( t), (49) where γ is Euler s constant and ψ the digamma function. 7 corresponding zero-temperature limit is 5] J RC (τ t) = ρ(e t Ei( t) + e t Ei( t) ln t γ iπ( e t ) sgn t) ρ(ln t + γ) iρπ. t The (5) One should note that for time scales long enough, the linear asymptotic behavior from the finite temperature result becomes more relevant than the logarithmic behavior at zero temperature. 3.. RCR circuit The setup we consider is not an RC circuit, but in the following we show that the equilibrium fluctuations in our system are related to those of an RC circuit. Examining the layout in Fig., one may deduce that the impedance at the point of the Josephson junction is of the form appearing in Fig. 3. This is easy to evaluate: ( ) Z t (ω) = + R iωcj + ], (5) iωc m R m where, for our choice of the source, R = R + R. The impedance has the real part ReZ t (ω) = R m + (τ ω) + (τ ω) + (τ 3 ω) 4, τ C mr (R + R m ), (5) τ R m(c J + C m ) + C mr (R + R m ), τ 3 R m R C m C J, (53) which is not of the same form as in an RC circuit. However, we may separate this into partial fractions: ReZ t (ω)/r m = τ 5 τ τ 5 τ 4 + (τ 5 ω) + τ τ 4 τ 5 τ 4 + (τ 4 ω), (54) 7 Numerical evaluation of Eq. (49) is straightforward for πz, but on the other limit the sum becomes problematic, and we prefer to evaluate J RC there from integral (48). 5

17 where we introduced the time constants τ 4 τ τ 4 4 τ 4 3, τ 5 τ τ 4. (55) One can show that the discriminant under the square root is always positive, hence the separation into partial fractions is physically valid for all choices of parameters. We may now apply the results obtained for the RC circuit to our partial fractions, which yields J RCR (τ t) = τ 5 τ τ5 τ 4 J RC (t = τ 5 t, z = τ 5 β ) + τ τ4 τ5 τ 4 J RC (t = τ 4 t, z = τ 4 β ). (56) This naturally reduces to the result for an RC circuit in the limits R, R, C m or C m. What the resulting RC circuit exactly is, may be seen by examining the impedance in Eq. (5): we have Z t ( iωc+r ), where. For R C m R m C J : R = R m, C = C J + C m.. For R R m : R = R m, C = C J. 3. For C m C J and R C m R m C J : R = (R m + R ), C = C J. In the following, we call limit weak coupling consider the other configurations as strong coupling. Since J(t) is known, we may evaluate the P D (E) function as P D (E) = π F t e J(t) ] (E/ ) = E τ π d t e i te/e τ e J(τ t), (57) where we introduced the RC energy scale E τ τ, which we henceforth associate with the RC time τ R m (C J + C m ). However, below we also use other energy scales, the charging energies for Cooper pairs, E CC e C m + C J = πρe τ, E CJ e C J = ( + C m /C J )E CC, (58) corresponding to capacitances C m + C J and C J, respectively. By examining Eq. (49) one finds that the charging energy is the characteristic energy scale for J RC. By the discussion above, E CC should hence be the characteristic energy scale in the weak-coupling limit (case above), while in some of the other cases it is given by E CJ. Examples of P (E) functions in RC and RCR circuits with different parameters are shown in Fig. 4, scaled by E CC and E CJ. These illustrate the behavior of the P (E) functions between the three different limits. Figure 4a shows how P (E) changes as C m is varied between the limits of weak and strong coupling, keeping R fixed. Figure 4b in turn shows how P (E) varies for fixed C m, if R is changed from weak coupling to strong coupling. In the figures, the amount of variation seen depends on R m /R and C m /C J, respectively, as they give the differences between the limits. 6

18 PSfrag replacements PSfrag replacements P (E)ECJ C m /C J E/E CJ (a) Varying C m P (E)ECC R /R m E/E CC (b) Varying R PSfrag replacementsfigure 4: P (E) functions PSfragfor replacements different RCR circuits at a vanishing temperature for ρ = 3. Left: R =.5 R m while C m =.... C J. Gray curves show the RC circuit case with R = R m, C = C J + C m (dashed) and R = (Rm + R ), C = C J (dotted). Right: C m =.5 C J while R =.... C J. Gray curves show the RC circuit case with R = R m, C = C J + C m (dashed) and R = R m, C = C J (dotted). In every case, P (E) vanishes for E <. P (E)ECJ R C m /R m C J E/E CJ (a) P (E), large C m IS ECC/(eE J ) R (kω) V m (C J + C m )/e (b) Tunneling current Figure 5: Left: P (E) functions for large C m = 5C J and varying R, for ρ = 3. Gray curves show the RC circuit case with R = R m, C = C J + C m (dashed) and R = R m, C = C J (dotted). Right: I-V curves for a Coulomb-blockaded Josephson junction in an RC circuit. (Identical with Fig. in Ref. 8].) 7

19 We also note that de P (E) =, since J() =. Moreover, Eq. (3) implies that for a vanishing temperature, i.e., β, we have P (E) = for E <. This is in agreement with the numerical results. For C m C J, the two energy scales are far apart, E CC E CJ, and the difference between strong and weak coupling is large. Thus, already for small R R m C J /C m, we should see the long tail anticipating the transition from scale E CC to the larger scale E CJ. Due to the normalization, this then implies that the peak in P (E) should decrease as C m increases. This transition between the two energy scales is shown in Fig. 5a. As P (E) functions are known, we may evaluate the tunneling supercurrent from Eqs. (5). Some typical I J, (V ) curves for an RC circuit are shown in Fig. 5b. For these, we find agreement with the literature 8]. We note a characteristic feature: for ρ <, the P (E) functions are nonzero or diverge at E =, while for ρ >, P () =. As the validity of the theory requires P (E)E J, we find that for all ρ < this cannot be satisfied, and hence we consider only the case ρ > in this work. Above, we introduced a number of different dimensionless quantities. Refer to Appendix A for their typical values. 3. Driven fluctuations Let us now consider how excess fluctuations from the source are transmitted to the detector. For voltage fluctuations the result is not difficult to obtain. Assume now that we know the spectrum S A nv (ω,..., ω n ) of all nth order voltage fluctuation correlators at point A (see Fig. ). Simple circuit analysis then shows that φ B (ω) = e iω Ṽ B (ω) = e R m C J iωr m C Ṽ A (ω). (59) By applying Eq. (), we find the correlators ( ) n S B n φ ( ω) = Cm S A ( ω) ] (πρ)n nṽ D n ( ω) C J + C m (e/(c J + C m )) n, (6) where ω (ω,..., ω n ), and the dimensionless D n is D n ( ω) n ]( n ) ( iτω j ) + iτ ω j. (6) j= Taking the inverse Fourier transform of Eq. (6) then gives the time-dependent correlators φ(t ) φ(t n ) φ() = F ω S B n φ ( ω)] (t,..., t n ), (6) which may be used to compose the cumulants appearing in expansion (3). The remaining problem now is to evaluate the voltage fluctuations at point A, for a given type of a source. We might for example consider two tunnel junctions, positioned as R and R in Fig., and find out what kind of voltage fluctuations the current flowing through them generates. Here, one must note that the presence of the detector affects the source and alters these fluctuations. For the j= 8

20 second-order correlator, the Langevin equations with the minimal correlation principle discussed above give the correct result. For higher-order correlators, the corrections necessary become numerous. In the following, we calculate the second and third-order correlators of the driven noise, with the structure of the source as in Fig Langevin equations As the structure of the source is known, we prefer to relate the fluctuations of current directly to the fluctuations of phase at point B. We begin with writing the Langevin equations (see Fig. ) I = ( V A + V A V D )/R + δi, I = ( V A + V A )/R + δi, I m = ( V B + V B V m )/R m + δi m, I J = iωc J ( V B + V B ), I sm = iωc m ( V B + V B V A V A ), (63a) (63b) (63c) (63d) (63e) I = I + I sm, I m + I sm = I J. (63f) Here we neglect the tunneling through the Josephson junction, as it is treated with perturbation theory, and assume that capacitors produce no driven noise. Remembering that average values obey the usual circuit theory relations, we may solve the Langevin equations to obtain V B (ω) = R m( δi m (ω) iωc m R (δi δi δi m )) G(ω), (64) G(ω) iω(r m (C J + C m ) + R C m ) ω R m C J R C m (65) where again R = R + R. We now assume that there are no driven fluctuations in R m (the case for macroscopic resistors), so the contributions due to δi m are already included in the quantum fluctuations. What is left, gives the correlators (applying the minimal-correlation principle and Eq. ()): ( ) n e S B n φ ( ω) = R n mrc n m n S G n ( ω) n,i ( ω) + ( ) n Sn,I( ω) ] = ] n πr C m er Q τ n Sn,I ( ω) + ( )n Sn,I ( ω) π R Q (C J + C m ) E τ e n, G n ( ω) (66) where ω = (ω,..., ω n ) and G n ( ω) = G(ω ) G(ω n )G( n j= ω j). The frequency scale of S B ( ω) is given by the smallest of /τ and the frequency n φ scales of the intrinsic current correlators. The latter form above has the front factor scaled dimensionless, using /τ as the characteristic frequency scale. The corresponding time-dependent correlator may now be written as S nφ( t) B = 4 ] n R C m er Q Sn,I n F ( w/τ) + ( )n Sn,I ( w/τ) ] w R Q (C J + C m ) E τ e n ( t/τ), G n ( w/τ) (67) where the Fourier transform in w is dimensionless. 9

21 Usually the correlators S n,i depend on F (n), the Fano factors of the two noise source elements, their resistances R and R, and the voltage drops V and V over them. These are given by V = R R V D, V = V D, (68) R + R R + R as obtained through simple circuit analysis. Moreover, the (average) noisegenerating current through the two elements is I N = V D /(R + R ). The above calculation gives the minimal-correlation values for the various correlators. As described above, these are sufficient for the second correlator, but not for the third and higher correlators. Still, in this calculation, we evaluate only the minimal correlation results for the sake of obtaining at least order-of-magnitude estimates. The actual form of the corrections necessary may probably be found out via the Keldysh technique 9]. Finally, note that Eq. (66) takes into account the effects that the impedance of the detector has on the voltage fluctuations at the source. If this were neglected (weak coupling limit), we had G(ω) = iτω. In the general case discussed above, we have instead G(ω) = i ω( + rc/( + c)) ( ω rc/( + c) ) (69) where ω τω, r R /R m, and c C m /C J. Two limiting cases may now be identified:. Weak coupling: rc. Time constant is τ = R m (C m + C J ).. Renormalized weak coupling: C m C J. Time constant is τ = (R m + R )(C J + C m ). The weak coupling results derived below may be applied to both cases by scaling τ appropriately. Note however, that this does not apply to quantum fluctuations, which depend on the whole impedance, not just G. For discussion on this, see Subs Second-order correlator For the second-order phase correlator, we have ] S B φ (ω) = π R C m R Q τ ŜI (ω) + Ŝ I (ω) R Q (C J + C m ) + (τ ω) + (τ 3 ω) 4 (7) where τ and τ 3 are given in Eq. (5). This shows that the phase noise is connected to the noise in the current, with the high-frequency (ω τ ) part filtered out. Moreover, since we are now interested in driven fluctuations, the correlators ŜI(ω) here have the part described by the fluctuation dissipation relation (4) subtracted from them. There is a general result relating the second-order correlator of current fluctuations to the temperature and the voltage difference over the element. 6] The unsymmetrized noise is evaluated for example in Ref. 4], and when the equilibrium fluctuations are subtracted, one obtains S F e I Ŝ,I( ω/τ) () sinh(ṽ /z) ( ω/ṽ ) coth( ω/z) sinh(ṽ /z) ( ω, Ṽ ) = () cosh(ṽ /z) cosh( ω/z), E τ (7)

22 S () k B T/eV ω/ev Figure 6: Shot noise frequency dependence S () (ω), at different temperatures. Area under the curve is constant, and S () (ω = ) ev/(6k B T ) for large T. (See Eqs. (78) and (79).) where I = V/R is the current through the element, Ṽ ev/e τ the dimensionless voltage drop over the element, z k B T/E τ the dimensionless temperature, F () the Fano factor, and R the resistance of the element. The frequency and temperature dependence in S () is illustrated in Fig. 6. Using the equations above, and the definition (3), we may find out the Gaussian part J S, (t) = S φ(t) S φ() due to driven fluctuations: J S, (τ t) = C () ( t, Ṽ ) R C m R Q (C J + C m ) ] F () ] C () ( t, Ṽ) + F () C () ( t, Ṽ) RQ e I N, E τ e i ω t d ω + ( τ τ ω) + ( τ3 τ ω)4 S() ( ω, Ṽ ). (7) Fourier transforming e JS,(t) then yields P S (E) for Gaussian fluctuations, which we are aiming at. Moreover, I N V D /(R + R ) is the current flowing through the noise generators. Examples of the P S (E) functions are plotted in Figs. 7 and 8, for some typical values of parameters, and corresponding J S, (t) functions are plotted in Fig. 8. These correspond to R m 3 kω, and for C J 5 F the energy scales are µev. The voltages chosen are large enough to produce visible changes ( %) in the I V curves. Moreover, the figure shows how the approximation (4) compares with the exact numerical results. It appears that the P S function retains here the qualitative shape of the fluctuation spectrum, the deviation being the largest at zero frequency. There also seems to be a quantitative agreement up to reasonably large voltages. Finally, Fig. 8 shows how increasing temperature rounds and dampens the P S (E) function. Accuracy of the expansion. Quantitative estimates about the accuracy of the expansion (4) may be made by considering the quantity S φ(). We immediately see that it gives the order of magnitude for the J S, (t) function, since J,S () = and J,S (t) S φ() as t. This is due to the fact that S φ(t) as t since the integrand in the inverse Fourier transform

23 PSfrag replacements P (E)ECC V D(C J +C m) e.5, Sφ () =.87, Sφ () =.78 4, Sφ () =.43 8, Sφ () = E/E CC Figure 7: P S (E) calculated from shot noise with the parameters ρ = 5, PSfrag replacementsr, R = 3 R m, C m = 8 C J, z =.4 and F (), F () = (tunnel junctions). Corresponding approximations PSfrag replacements from Eq. (4) are plotted with dashed gray lines. The higher the peak, the larger V D. The δ(e) contribution in the exact result is given by e S φ() δ(e) and in the approximation by ( S φ())δ(e). JS,(t) V D(C J +C m) -.5 e t/τ P (E)ECC E/E CC z Figure 8: Left: J S, (t) calculated from shot noise at z =. Right: P S (E) calculated from shot noise, with V D (C J +C m )/e =.78 and varying z (resulting in S φ().3). Gray lines show the approximation (4). The values for z shown correspond to T... K for C J 5. The higher z, the lower the peak. Other parameters are the same as in Fig. 7.

24 is well-behaved (unlike for quantum fluctuations). Usually J,S (t) is also nearly monotonic, so S φ() gives directly its order of magnitude. At a vanishing temperature, the spectrum of additional fluctuations simplifies: S () ( ω, Ṽ/) ( ω/ṽ/ )θ( Ṽ/ ω ). (73) z Now, the integral of the non-oscillating part (corresponding to S φ()) in the integrand in Eq. (7) may be evaluated analytically: ] ) S φ() R C m (F () R Q = A(Ṽ) + F () R Q A(Ṽ), (74) R Q (C J + C m ) R R A(Ṽ ) τ τ ) 5 τ )] 4 τ5 τ 4 B (Ṽ B (Ṽ, (75) τ τ { B(x) x arctan(x) ln( + x ) x for x, π x ln x for x. (76) Since B(x) π x, we have an exact upper-bound estimate S φ() π R C m R Q (C J + C m ) ] τ τ 5 + τ 4 (F () + F () ) R Qe I N E τ, (77) valid again at a vanishing temperature. Although this works well for strong coupling or large voltages, it may not be very strict for small Ṽ, or more importantly, for τ 4 /τ R C m /(R m C J ) Ṽ (i.e. weak coupling at low voltages). For small voltages, one may either use the other limit for B or the full result. Increasing the temperature tends to suppress the non-equilibrium noise. This is due to the fact that we have 8 d ω S () ( ω, Ṽ ) = Ṽ, (78) which is independent of the temperature, but for increasing z the noise spectrum widens. The widening is basically given by ω z, and we also have S () Ṽ (, Ṽ ) = coth(ṽ /z) z/ṽ, for z. (79) 6z Since the detector sees the noise only at a fixed temperature-independent bandwidth, at high temperatures z Ṽ we have S φ() /z. For example multiplying Eq. (77) or (74) by the factor ( + z) provides results with a roughly correct asymptotic temperature dependence. (See Fig. 9b.) Accuracy of the expansion (4) and the approximations derived above are shown in Fig. 9a, for various values of parameters. Since the gray dots are above the dashed line, the figure shows that Eq. (77) indeed gives an upper bound estimate. More importantly, it also shows that the relative deviation (black dots) of the smooth part of P S (E) from π S φ (E/ ) is proportional to S φ() for S φ() <. This is what one would expect, as the next term in expansion (4) has the magnitude of S φ(), while the terms included in 8 The integral may be evaluated analytically. 3

25 PSfrag replacements PSfrag replacements π S φ() PS() /PS() S φ () (a) Expansion accuracy ( + z) Sφ(, z)/ Sφ(, z = ) z (b) Temperature dependence Figure 9: Left: Order-of-magnitude error estimates for the expansion (4), for parameters in the ranges C m /C J., ]; ρ., 8]; z {, 3}; R /R m, R /R m., 3]; F () = ; F () {, }; and Ṽ, ]. Gray dots indicate the approximations to S φ() obtained from Eq. (77), and black dots show the relative error π S φ () P S() /P S (). Dashed line indicates the curve x = y. Right: Temperature dependence of S φ() for different parameters. the approximation for the smooth part of P S (E) have the magnitude of S φ(). However, as seen in the figure, the relative numerical accuracy does not always improve beyond 3, even if S φ() decreases. Hence, the consistency of the expansion may be assessed simply by checking that S φ() applies. For vanishing temperature (and probably also for finite temperatures), this is guaranteed if (C m + C J ) V D e ṼCC,max R Q (C m + C J ) (R + R ) π R C mr m (F () + F () ) ( π + C ) ( J RQ + R Q C m R R τ 5 + τ 4 τ ) R + R, R m F () + F () (8) where the approximation is valid for R C m R m C J (then τ 5 τ, τ 4 ). For R, /R m = / and F (), =, the upper bound becomes 8( + C J/C m ) /ρ, which is often reasonably large. For the parameters in Fig. 7, the estimate yields 7.9, which is in a good agreement with the picture. For the currents I N, Eq. (8) implies I N I N,max.5 na ( 5 ) ( F + C ) ( J RQ + R ) Q. ρ C m C m R R F () + F () (8) With plausible values for the parameters, these currents should fall into the experimentally accessible range. Moreover, if we wish to obtain the smooth part of P S with a certain relative 4