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Fluctuation-dissipation theorem and flux noise in overdamped Josephson junction arrays

The form of the fluctuation-dissipation theorem for a resistively shunted Josephson juction array is derived with the help of the method which explicitely takes into acoount screening effects. This result is used to express the flux noise power spectrum in terms of frequency dependent sheet impedance of the array. The relation between noise amplitude and parameters of the detection coil is analysed for the simplest case of a single-loop coil.

The first of them is based on measurement of the voltage induced in the detection coil by the currents flowing in the sample under the action of ac electric field produced by the current in the other (driving) coil. For the given geometry of the coils the measured signal can be used to extract [10] the complex frequency dependent sheet impedance Z ✷ (ω) of the sample on the assumption that for wave-lengths larger than the characteristic dimensions of the detection coil Z ✷ (ω) is not wave-length dependent.
In the case of the flux noise spectrum analysis the approaches to interpretation of the experimental data are much more varied. The theoretical predictions of the flux noise spectrum used for comparison with experimental data are found by relating it with [7,12] or (no less often) by replacing it by [3,13,14] a correlation function describing the vortex distribution and, naturally, turn out to be dependent on the particular choice of assumptions concerning the form of this distribution. Numerical simulations also demonstrate a clear tendency towards studying the vortex number noise [13,15,16] rather then the flux noise. The only attempt to achieve a description of the flux noise power spectrum in terms of the sheet impedance of the sample taking into account the actual geometry of the detection coil has been undertaken by Kim and Minnhagen [17]. However, this calculation is also based on expressing all quantities in terms of the vortex gas correlation functions and, therefore, can be trusted only in a limited range of parameters.
In the present work we argue that in the case of a resistively shunted Josephson junction array the general expression for the flux noise spectrum can be found without artificial decomposition of all fluctuations into the vortex part (which is usually assumed to be responsible for the flux noise) and the remaining so-called "spin-wave" part (which is traditionally neglected). Although in semi-phenomenological treatment [12][13][14]18] of twodimensional superconductors such decomposition seems to be inevitable, the case of an overdamped Josephson junction array allows for application of a more universal approach to calculation of the flux noise power spectrum. It is based on the direct relation (discussed in Sec. 2) of the flux noise with current correlations, which, on the other hand, can be expressed in terms of the complex frequency dependent sheet impedance with the help of the fluctuation-dissipation theorem.
The additional advantage of such approach is that it allows to include into consideration in systematic way the mutual influence between magnetic field fluctuations and current fluctuations (the screening effects), which insofar has been neglected in theoretical works [12][13][14][15][16][17] devoted to flux noise spectrum analysis. The form of the Hamiltonian, which should be used for the description of a resistively shunted array in presence of self-induced magnetic fields, is discussed in Sec. 3, and the corresponding dynamic equations in Sec. 4.
The explicit form of the fluctuation-dissipation theorem for resistively shunted Josephson junction array is derived in Sec. 5. It shows that the current correlations in the array are determined by the response of the current to the external electric field and not directly by the sheet impedance of the array (which is defined as a response to the total electric field) The nature of the expression for currents correlation function, related with the peculiarities of the two-dimensional geometry, allows to expect the same expression to be applicable for arbitrary two-dimensional systems in which capacitive effects can be neglected.
Our main result, the relation between the flux noise power spectrum and the frequency dependent sheet impedance of a two-dimensional superconductor, is presented in Sec. 6, which includes also the discusssion of the noise spectrum dependence on the parameters of the detection coil and comparison of the results with those of other authors.

II. FLUX NOISE AND CURRENT CORRELATIONS
In a flux noise experiment one measures and analyses the time dependence of a voltage created in a detection coil by fluctuations of currents in some conducting (or superconducting) object. This voltage is determined by the time derivative of the magnetic flux penetrating the coil, and the value of the flux can be expressed in terms of the current density distribution j(r) [r = (x 1 , x 2 , x 3 )] inside the object with the help of the Biot-Savart's law, which in the Coulomb gauge (divA = 0) can be written as where A(r) is the vector potential defining the distribution of the magnetic field (magnetic induction) B(r) = rot A(r) created by j(r), is the three-dimensional Laplacian and is the transverse part of j(r). Magnetic fields produced by the longitudinal part of j(r) cancel each other.
In the case of a system which can be considered as effectively two-dimensional and situated (for simplicity) in the plane x 3 = 0 the three-dimensional current density j(r) is reduced to where x ≡ x α (α = 1, 2) is the two-dimensional vector defining the position of a point in the plane x 3 = 0 and i ≡ i α is the two-dimensional vector describing the twodimensional current density. Substitution of Eq. (4) into Eq. (1) allows then to find that where is the (two-dimensional) Fourier transform of i(x), q = |q| and is the transverse part of i(q),q ≡ q/q being the unit vector parallel to q.
In the simplest case a coil can be approximated by a closed circular ring. Integration of A(r) over the perimeter of the ring x 2 1 + x 2 2 = r 2 situated at the distance h from the plane x 3 = 0 gives where is the amplitude of i t (q), ǫ αβ is the unit antisymmetric tensor, is the geometrical factor depending on the parameters of the coil and J 1 (z) is the first order Bessel function. In the case when the coil can be considered as consisting of N turns separated by the distance b from each other, the expression for F (q) should also include an additional factor obtained by summation of contributions from different turns [10]: The power spectrum of the flux noise is given by the flux-flux correlation function and with the help of Eq. (8) can be expressed in terms of the current density correlation function

III. THE HAMILTONIAN OF A JOSEPHSON JUNCTION ARRAY
When self-induced magnetic field is taken into account a square Josephson junction array can be described by the Hamiltonian [19,20] where ϕ n is the phase of the order parameter on the nth superconducting island, the variables A nα (defined on the bonds of the lattice) are determined by the integral of the vector potential A(r) over the line connecting the geometrical centers of the neighboring superconducting islands: and a is period of the lattice. The first term in Eq. (15) describes the Josephson energy of the junctions in the array. The coupling constant J entering this term is determined by the critical current I c of a single junction: which is assumed to be the same for all junctions, whereas ∇ α ϕ n denotes the difference of ϕ n between the neighboring sites of the lattice: Here e 1 = (1, 0) and e 2 = (0, 1). Notice that the combination which enters as the argument of the Josephson energy E J (θ) = −J cos θ is a gauge-invariant quantity. The second term in Eq. (15) is the energy of the magnetic field expressed in terms of the variables A nα . The matrix M nk ≡ M (n − k) is usually called the mutual inductance matrix [19][20][21] and is the directed sum of the variables A jα along the perimeter of a lattice plaquette (the lattice equivalent of rot A) and is proportional to the magnetic flux penetrating the plaquette. Variation of Eq. (15) with respect to A nα gives the equation which relates the value of the superconducting current through a junction with the vector potential of the magnetic field induced by the presence of the currents in the array. Here [like in Eq. (21)] ∇× stands for β ǫ βα ∇ β , whereas ∇ β designates the lattice difference, analogous to the one defined by Eq. (18), but shifted in the negative direction: On the other hand variation of Eq. (15) with respect to ϕ n gives the current conservation equation where is the lattice equivalent of divergence. Eq. (25) can be alternatively obtained by application of the operator ∇ α to Eq. (22). Therefore Eq. (22) and Eq. (25) [both obtained by variation of Eq. (15)] are not independent of each other. The vector potential of the magnetic field created by the currents flowing in the array can be chosen purely transverse (divA = 0), which in terms of A nα corresponds to In that case Eq. (22) is reduced to where ∆ L ≡ β ∇ β ∇ β is the two-dimensional lattice analog of the Laplacian: Comparison of Eq. (28) with Eq. (5) allows to find that for |n − k| ≫ 1 whereas for |n − k| ∼ 1 the form of M −1 nk depends on the particular shape of superconducting islands [21].
Linearization of Eqs. (22) and their solution allows to show that when the magnetic fields of the currents in the array are taken into account, the logarithmic interaction of vortices becomes screened [19] at so-called magnetic field penetration length Λ, exactly as it happens in superconducting films [22]. When screening is relatively weak (that is when Λ ≫ a), the value of Λ is given by and does not depend on the shape of superconducting islands forming the array [19].
Instead of considering Hamiltonian (15) as dependent on two different types of variables defined on the sites (ϕ n ) and on the bonds (A nα ) of the lattice, it is convenient to use a single variable, namely the gauge invariant phase difference θ nα defined by Eq. (19). In terms of θ nα the Hamiltonian (15) can be rewritten as variation of which with respect to θ nα reproduces Eq. (22) in the form where the expression for the superconducting current is naturally consistent with Eq. (23). As previously, the current conservation equation (25) can be obtained by application of the operator ∇ α to Eq. (33).

IV. DYNAMIC FLUCTUATIONS IN ARRAY OF RESISTIVELY SHUNTED JUNCTIONS
The dynamic description of the same system requires to complement the Hamiltonian H by the dissipative function W (we assume that the array is overdamped and therefore its dynamics is purely relaxational). In the case of the array formed by SNS (superconductor -normal metal -superconductor) junctions one can describe dissipation in terms of the effective resistance shunting each junction (so-called RSJ-model). This corresponds to W {θ} of the form where the effective viscosity is determined by the value of the shunting resistance R, which is assumed to be the same for all junctions. For W of the form (35) the conservation of energy is achieved when the time evolution of the variables θ nα is governed by the standard equations of relaxational dynamics: On the other hand, Eq. (37) can be rewritten in the form (33), where the expression for the current should be replaced by The time derivative of θ nα being proportional to the voltage, the second term in Eq. (38) can be easily identified as the normal current flowing through the junction. Consideration of purely relaxational dynamics means that we are neglecting capacitive effects and currents have to be conserved on each site of the lattice (in other words, only transverse current are allowed). This is ensured by the form of Eq. (33), substitution of which into Eq. (25) automatically leads to its fulfillment for any form of I nα .
In presence of thermal fluctuations the right-hand side of Eq. (37) should be complemented with the random force term ξ nα (t): the correlations of which are Gaussian and satisfy where T is the temperature expressed in energy units (that is multiplied by the Boltzmann constant k B ). We also have included in the right-hand side of Eq. (39) the non-random external force f nα (to be discussed later). In terms of the expression for the current the inclusion into Eq. (39) of the random term ξ nα corresponds to appearance in Eq. (38) of the additional (fluctuating) contribution to normal current δI nα : where δI nα ≡ −(2e/h)ξ nα . Note that since Eq. (33) describes the relation between the currents and the magnetic field induced by them, it has to remain fulfilled also when fluctuations of currents are taken into account. The validity of the current conservation equations (25) remains ensured by the form of the right-hand side of Eq.
(33). The suggestion to describe the dynamics of a resistively shunted Josephson junction array by Eqs. (41) has been put forward by Shenoy [23], who did not consider fluctuations of the magnetic field, that is assumed θ nα ≡ ϕ n+eα − ϕ n . In that case substitution of Eqs. (41) into current conservation equations (25) leads to dynamic equations for ϕ n with non-local effective viscosity [23]. Quite remarkably the inclusion into consideration of the magnetic field fluctuations leads to simplification of the dynamic equations which (in terms of θ nα ) become local. The idea that in presence of magnetic field fluctuations a resistively shunted Josephson junction array can be described by Eqs. (39), where θ nα is the gauge invariant phase difference, has been introduced by Domínguez and José [20].
It is well known that when the time evolution of some variables {θ} is governed by the standard Langevin equations of the form (39), the equilibrium (that is calculated for f nα = 0) correlation function is related with the response function by the fluctuation-dissipation theorem: However, in practical situation one is interested not in the response of the gauge invariant phase difference θ nα to the (unspecified) conjugate force f nα , but rather in more readily observable quantities such as conductivity, which is the response of a current to application of electric field. In situation when electric field E(r) is created due to presence of ac magnetic field it is given by where A(r) is the vector potential defining the (total) magnetic field B(r) = rot A(r).
In presence of the external magnetic field B e (r) = rot A e (r) the expression (20) describing the magnetic field energy should be replaced by the expression for the Gibbs free energy variation of which in absence of any other terms gives B(r) = B e (r). This leads to appearance in the Hamiltonian (32), describing the array, of the additional term Here the variables defined on lattice bonds, are related to the vector potential A e (r) of the external magnetic field exactly in the same way as earlier introduced variables A nα are related to the total vector potential A(r).
The form of the correction to the Hamiltonian given by Eq. (47) corresponds to the presence in Eq. (39) of the external force term Comparison of Eq. (49) with Eq. (33) shows that (up to the factor of 2e/h) f nα is related to A e nα exactly in the same way as I nα is related to θ nα . This allows to conclude that the correlation functions of the currents in the array and the response function have to satisfy the relation completely analogous to Eq. (44). Here (A e kβ ) t is the transverse part of A e kβ . As can be seen from the righthand side of Eq. (49), the longitudinal part of A e kβ is completely decoupled from fluctuations of θ nα .
In terms of effective conductivity g eff nα,kβ , defined as the response function of the current with respect to external voltage Eq. (52) can be rewritten as For wavevectors small in comparison with a −1 , the variables I nα can be identified with ai(x), whereas V e nα with aE e (x), where E e (x) is the projection of the external electric field E e (x) on the plane x 3 = 0 (here and below we assume that E e (q, ω) is transverse). This allows to rewrite Eq. (55) as where the Fourier transform g eff (q, ω) of the effective conductivity is the coefficient of proportionality in the relation One should distinguish between g eff (q, ω) and (also momentum and frequency dependent) sheet conductance g ✷ (q, ω), which is defined as the coefficient of proportionality between i(q, ω) and total electric field E (q, ω): The form of the current induced correction to E e (q, ω) can be easily found with the help of Eq. (5) and Eq. (45), which lead to Substitution of Eq. (58) into Eq. (59) then gives and, accordingly, where Z ✷ (q, ω) ≡ 1/g ✷ (q, ω) is momentum and frequency dependent sheet impedance. The form of Eq. (61) suggests that the response of a current in a two-dimensional system to external electric field is the same as if the proper sheet impedance of a system Z ✷ (q, ω) has been connected in series with the other contribution, which can be considered as the effective impedance of the empty space surrounding this system. This additional contribution is purely inductive and corresponds to In the case of a superconducting system in the low frequency limit Z ✷ (q, ω) ≈ −iωL ✷ , where L ✷ is the effective sheet inductance, substitution of which into Eq. (60) allows to rewrite it as is the (two-dimensional) magnetic field penetration length [22] already discussed in Sec. 3. Since in the considered system electric field appears only due to presence of ac magnetic field, the same length describes as well the screening of the electric field. The form of the fluctuation-dissipation theorem obtained after substitution of Eq. (61) into Eq. (56) being completely independent of the details of the structure of the particular system used for its derivation, one can expect it also to be valid for other two-dimensional superconducting (or simply conducting) systems, in particular superconducting films.

VI. RESULTS AND DISCUSSION
Substitution of Eq. (65) into Eq. (14) gives the expression for the flux noise power spectrum which is the central result of this work. It allows instead of constructing special theories explaining frequency dependence of S(ω) in different regimes to relate it with already known properties of Z ✷ (q, ω).
It is of interest to compare Eq. (66) with the expression for the quantity [the correction to frequency dependent mutual impedance Z m (ω)], which is measured in the framework of the two coil method and defined as the ratio of the measured voltage to the driving current. In our notation this expression [10] can be rewritten as where F (q) is the geometrical factor analogous to F (q), but describing the driving coil. Note that applicability of Eq. (66) and Eq. (67) for description of a Josephson junction array requires the lattice constant of the array to be much smaller then the geometrical parameters of the coil(s) (r and h) entering the expressions for F (q) [and F (q)]. Comparison of Eq. (67) with Eq. (66) shows that the real part of δZ m (ω) is determined by the same (real) component of g eff (q, ω) as the noise spectrum, so in absence of a difference between F (q) and F (q) the noise spectrum S(ω) would be simply proportional to Re[δZ m (ω)]: In the case of the simplest coil (a single circular loop of radius r) Eq. (66) is reduced to The analogous equation derived by Kim and Minnhagen [17] differs from Eq. (69) basically (i) by the absence of the term L s (q) and (ii) by the presence in the integrand of the additional factor q 4 . The former of the two discrepancies is rather natural, since in Ref. [17] the screening effects have not been taken into account, whereas the latter we believe to be the consequence of the incorrect calculation of the magnetic field produced by currents in the array.
In Ref. [17] this magnetic field is calculated as the sum of magnetic fields produced by current loops associated with lattice plaquettes, whereas the magnitude of a current in each loop is assumed to be given the directed sum of the currents in the junctions I nα along the perimeter of a plaquette (in the present work this sum is denoted ∇ × I). In such procedure the current of each junction is counted twice (as giving contributions to the loop currents associated with the two neighboring lattice plaquettes) but with opposite signs, so these two contributions almost cancel each other, which leads to appearance of the additional q-dependent factor in comparison with our Eq. (69). In a more consistent implementation of this approach the values of loop currents associated with arrays plaquettes should be chosen in such a way that the value of the current on each junction is given by the difference of the loop currents associated with the two neighboring plaquettes. These so called mesh currents [21,24] I m n are related with the currents on the junctions I nα as which explains the appearance of the additional factor of q 4 in calculation which uses (∇ × I) n instead of I m n . The main contribution to the integral in Eq. (69) is coming from the region so in the situation when h exceeds all the microscopic scales responsible for the q dependence of Z ✷ (q, ω), one can replace in Eq. (69) Z ✷ (q, ω) by and use S(ω) to extract information about L ✷ (ω) and R ✷ (ω). Let us first discuss the limit when the effects of screening can be neglected. This is possible if in the essential part of the interval (71) one can neglect L s (q) in comparison with L ✷ (ω) and therefore requires h ≪ Λ (and not h ≫ Λ as has been claimed in Ref. [4] and Ref. [17]). In that case Eq. (69) is reduced to This means that with increase of h the amplitude of the noise has first to change very slowly and then to decay proportionally to 1/h 2 . The experimental results of Festin et al. [4], obtained on thin YBCO film at h/r ∼ 1, are compatible with 1/h 2 dependence even better than with 1/h 3 dependence predicted by Kim and Minnhagen [17].
In the opposite limit of strong screening (Λ ≪ h) L ✷ (ω) is negligibly small in comparison with L s (ω) and S(ω) depends only on R ✷ (ω), but not on L ✷ (ω). In particular, for h ≪ r integration in Eq. (69) gives Note that for Λ ≪ h ≪ r and small ω one obtains the dependence S(ω) ∝ 1/ω, which is observed in noise measurements in Josephson junction arrays [7,8] over four decades in frequency. The idea that this particular dependence appears because SQIUD integrates contributions from wide interval of wavevectors has been put forward by Wagenblast and Fazio [14]. Our analysis has shown that in order to obtain the 1/ω dependence over four decades in frequency one should have r/h > ∼ 10 4 , which in the experiments of Refs. [7,8] definitely has not been fulfilled. Thus the origin of the 1/ω dependence of the flux noise power spectrum observed experimentally in Josephson junction arrays (in particular, at tempratures lower then the expected temperature of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition [25,26]) still remains to be elucidated.
Substitution of the results of Ambegaokar et al. [27] for vortex pair contribution to impedance into Eq. (66) or its simplified analogs allows to reproduce the dependence S(ω) ∝ 1/ω only exactly at the temperature of the BKT transition. On the other hand, the analogous frequency dependence of the noise spectrum observed in thin YBCO films [5] is ascribed to vortex hopping between neighboring pinning centers, a finite vortex concentration being associated with presence of a residual magnetic field. The same mechanism may be responsible for the results of experiments [6][7][8] on Josephson junction arrays.