Rescaling of applied oscillating voltages in small Josephson junctions

The standard theory of dynamical Coulomb blockade [P(E) theory] in ultra-small tunnel junctions has been formulated on the basis of phase-phase correlations by several authors. It was recently extended by several experimental and theoretical works to account for novel features such as electromagnetic environment-based renormalization effects. Despite this progress, aspects of the theory remain elusive especially in the case of linear arrays. Here, we apply path integral formalism to re-derive the Cooper-pair current and the BCS quasi-particle current in single small Josephson junctions and extend it to include long Josephson junction arrays as effective single junctions. We consider renormalization effects of applied oscillating voltages due to the impedance environment of a single junction as well as its implication to the array. As is the case in the single junction, we find that the amplitude of applied oscillating electromagnetic fields is renormalized by the same complex-valued weight Ξ(ω)=∣Ξ(ω)∣expiη(ω) that rescales the environmental impedance in the P(E) function. This weight acts as a linear response function for applied oscillating electromagnetic fields driving the quantum circuit, leading to a mass gap in the thermal spectrum of the electromagnetic field. The mass gap can be modeled as a pair of exotic ‘particle’ excitation with quantum statistics determined by the argument η(ω). In the case of the array, this pair corresponds to a bosonic charge soliton/anti-soliton pair injected into the array by the electromagnetic field. Possible application of these results is in dynamical Coulomb blockade experiments where long arrays are used as electromagnetic power detectors.


Introduction
Since the pioneering theoretical work by Likharev et al, [1,2] small Josephson junctions have been thought of as a dual system to large Josephson junctions-the roles of current and voltage are interchanged. In the case of large Josephson junctions, their effective interaction with oscillating electromagnetic fields has been intensively studied, demonstrating their unique suitability for microwave-based applications such as the metrological standard for the Volt (in terms of voltage Shapiro steps) and other microwave-based devices [3]. The dual system, on the other hand, holds enormous promise for complementary applications such as a metrological standard for the Ampere in terms of the current Shapiro steps [4].
However, observation of dual phenomena in small junctions faces daunting experimental and theoretical challenges due, in part, to the lack of an approach that consistently covers both regimes [5]. In particular, small tunnel junctions are prone to quantum and thermal fluctuations-their characteristics cannot be analyzed separate from their dissipative environment [6,7]. As a consequence of Heisenberg uncertainty principle, their current-voltage I−V characteristics is highly sensitive to energy changes in the environment [8]. Heuristically, tunneling of a single charge e across a tunnel junction of capacitance C and conductance 1/R is restricted unless the maximum energy it can absorb from zero-point fluctuations in the vacuum, h/RC, is sufficient to offset its own charging energy = e C E 2 Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
In the case of Josephson junctions, quantum tunneling is due to paired electrons (Cooper pairs). Thus, two distinct ratios parameterize the Josephson junction at zero temperature: (1) The phase/charge regime characterized by the ratio E E J c where E J is the Josephson coupling energy and E c the charging energy, which are simply the coefficients of the potential energy and kinetic energy respectively in the Hamiltonian of the Josephson junction; and (2) the superconducting/Coulomb blockade regime characterized by the aforementioned ratio R Q /R where R is the real part of the environmental impedance, Z(ω). In this picture, the small junction is defined in the regime c above the Coulomb blockade threshold voltage, thus resulting in current-voltage characteristics highly dependent on these environmental parameters. Formally, this implies that the action for large junctions ( ) phase correlations [9][10][11][12] where tunneling across the barrier is influenced by a high impedance environment treated within the Caldeira-Leggett model [13].
P(E) theory has successfully been tested to a great degree of accuracy in a myriad of experiments [14][15][16][17][18]. This has lead to its widespread application in describing progressively complex tunneling processes such as dynamical Coulomb blockade in small Josephson junctions and quantum dots [19,20]. Moreover, owing to significant improvement in microwave precision measurement technology such as near-quantum-limited amplification [21,22] and progress in theory, recently published works suggest novel features in the P(E) framework ranging from time reversal symmetry violation [23] and Tomonaga-Luttinger Liquid (TLL) physics [24], to renormalization of electromagnetic quantities appearing in the P(E) function [25][26][27][28]. Despite this progress, aspects of the theory remain elusive especially in the case of linear arrays.
Here, we apply path integral formalism to derive the Cooper-pair current and the BCS quasi-particle current in single small Josephson junctions. We consider renormalization effects of applied oscillating voltages due to wavefunction renormalization/Lehmann weights [29] that rescale the environmental impedance of the single junction as well as the array. The array is treated as infinitely long [30] and transformed into an effective circuit. As is the case for the single junction, we show that the Lehmann  also acts as a linear response function for oscillating electromagnetic fields, and can be interpreted as the probability amplitude of exciting a 'particle' of mass M from the junction ground state by the radio-frequency (RF) field [31]. The quantum statistics of this 'particle' are determined by the argument be m where ε m is identified as the Matsubara frequency [32]. In the case of the infinite array, this 'particle' corresponds to a bosonic charge soliton injected into the array. Possible application of these results is in accurately determining the absorbed RF power in dynamical Coulomb blockade experiments especially where long arrays are used as on-site electromagnetic power detectors [33,34].
The paper is organized as follows: Section 2 deals with the rescaling of oscillating voltages applied on single junctions. In the sections, 2.1 explains the basis for this rescaling, 2.3 introduces the finite temperature propagator and the environmental impedance as Green's function, 2.4 interprets the impedance Lehmann weight as a complex-valued probability amplitude for applied oscillating voltages (RF field) exciting a 'particle' with quantum statistics given by the argument of the factor, or equivalently as a linear response function and the RF field as the external force leading the full expression for the current-voltage characteristics that includes the RF field and renormalization effects.
Finally, section 3 considers the Lehmann weight in infinitely long arrays. The difference from the single junction is an additional Lehmann weight ( ) -Lexp 1 representing a finite range of the electromagnetic field due to the presence of a charge soliton of length Λ injected into the infinitely long array.
Note that, units where Planck's constant, Swihart velocity [35] and Boltzman constant are set to unity (= = =  c k 1 B ) and Einstein summation convention are used through out unless otherwise stated with where e is the electric charge, where f J , f x , f z and eΦ are the phases associated with the the voltage drop at the junction V J , voltage source (external voltage) V x and the flux stored by the circuit respectively, f n is the bath degrees of freedom represented by k coupled (via f) L C n n oscillators constituting the environment. Note that f¢ is the phase defined for convenience, to shorten the expression of the Lagrangian, avoiding always writing the full expression . The effective action, requires the impedance Green's function in the P(E) function to be modified by a wavefunction renormalization (Lehmann) weight, [29]), It is known-at least since the work of Callen and Welton [7]-that the (causal) response function 1 ( ) for a system driven by oscillating electromagnetic fields appears as the coefficient 2 of the black body spectrum. Consequently, this requires that the response ( ) ¢ V t RF as seen by the junction J in figure 1 be a weighted function of χ(t) and the applied oscillating voltage . Therefore, to accurately describe the I-V characteristics of J driven by an applied oscillating voltage V RF (t), it is not enough to simply rescale the impedance Z(ω) in the P(E) function: the amplitude and phase of the applied oscillating voltage V RF has to be renormalized accordingly.
In subsequent sections, we first consider tracing our steps from standard quantum electrodynamics (QED) and ease our way into circuit-QED and hence P(E) theory. We then proceed to introduce the finite temperature propagator for the junction and consider how the Lehmann weight arises for the impedance in P(E) theory, and its implications for single junctions and long arrays driven by V RF (t). We find that, a finite time varying flux Φ(t) stored by the circuit consistently implements the aforementioned wavefunction renormalization by guaranteeing the circuit responds linearly to V RF (t).

Connection of P(E) theory to quantum electrodynamics (QED).
Here, we shall connect the Caldeira-Leggett model to QED. We shall find out that circuit-QED is merely the 1 dimensional space time version of QED. This also allows us to link the propagator introduced in equation (B42) with the photon propagator in QED.
The Fourier transform of the summed terms in Caldeira-Leggett action given in equation , 3 n n n n n n n n 0 2 2 2 2 2 ( ) f¢ O 2 is a term with f¢ 2 that we initially neglect, f n are the Caldeira-Leggett phases of L C n n circuits in figure 1, w = L C 1 n n n and the interaction term S int is given by, where I F is the fluctuation current which we shall later set, I F =0.
Integrating out the fluctuating degrees of freedom, f n , we arrive at, where we have converted the Green's function G(ω), from the f n degrees of freedom, to the environmental impedance Z(ω). Proceeding to combine the two actions yields, By defining the electromagnetic vector potential Aand the fluctuation current density m J F , it can be seen that the interaction term given by S int above is actually Maxwellʼs action in disguise, 1, 0, 0 is the normal vector to the junction barrier, l≡d eff the effective barrier thickness and  the junction area. The last term is the Fourier transform of the action where ( ) Integrating out ( ) f w ¢ degrees of freedom in + S S 0 i n t given by equation (6) as in equation (5a) leads to a circuit-QED term, where a p ee = e 4 2 0 r is the fine structure constant. The QED term is obtained in a similar fashion by integrating out m A instead of f¢. Nonetheless, both expressions are essentially describing the same process. The difference is the dimensionality of the theory: QED is in 1+2 space-time dimensions whereas circuit-QED is solely in the time dimension (circuit-QED). Thus, we have showed that w C 1 2 and hence ( ) w G both have an interpretation as photon propagator in circuit-QED.
However, a question still remains: is there any significance of this trivial Fourier space transformation given by ? We notice that we can define a factor ( ) ( ) w w X = G G eff and claim that this factor renormalizes the propagator ( ) w G (of the f n degrees of freedom) to ( ) w G eff due to the presence of the Maxwell term, S int . Since this renormalization takes a photon propagator into a different photon propagator, the Feynman rules to calculate it resemble photon self-energy interactions.
In particular, in self-energy interactions, the photon polarizes the QED vacuum by creating electronpositron pairs which subsequently annihilate. Such pairs can be created an infinite number of times, thus the contribution to the amplitude of all the processes takes the form: , where G is the photon propagator and U the vacuum polarization energy (interaction term) [36]. Bearing this in mind, we formulate the following Feynman rules for the propagator: 2. The vacuum polarization energy term, ( ) w w = -U e C 2 2 is represented by: ; 3. Therefore, the leading order interaction term, where time increases from left to right. Note that diagram 1 reads as follows: A photon of energy ω is created, propagates with a probability amplitude ( ) w G and annihilates at a later time. Thus the amplitude must be assigned a photon creation operator and an annihilation operator, † a and a respectively and, in mathematical form, it should be written as ( ) † w aG a . Likewise, diagram 3 represents an interaction whereby a photon of frequency ω is produced by acting on the vacuum state with † a , it propagates with an amplitude given by ( ) ( ) ( ) w w w G U G then it annihilates by acting on the vacuum with ( ) w a . We emphasize that reversing ω reverses the aforementioned processes. This implies that the operators themselves should also be defined accordingly as, where ( ) q w is the Heaviside function. This clearly displays the roles of the positive and negative frequencies. Note that negative frequencies are allowed since we are interested in energy differences due to single photon emission and absorption processes. For instance, processes where a photon is created before annihilation are related to processes where a photon is annihilated before creation by reversing the sign of the frequency ω and reordering the † a a , operators appropriately. 3

Connection to P(E) theory: The finite Temperature Green's Function and Propagator
Observe that a straightforward regularization procedure verifies ( ) w Z eff plays the role of effective Green's function of the P(E) function, analogous to the renormalization of the propagator in QED which often leads to a (Lehmann) factor [29,37] analogous to ( ) w X . To elucidate this, consider the finite temperature propagator for ¢ S 0 , given by ( ), where k = e e e 1 , 2 is the Cooper pair, BCS quasi-particle charge. The prefactor k pi 2 2 included for ease of comparison later with the P(E) theory. The processes with are forbidden since represent actual negative frequency photons. We proceed to include the effect of S int from a finite tunnel junction impedance by introducing the potential where we have restored the ( ) f¢ O 2 term from equation (3). The effective propagator is given by the sum of all the single photon interactions at the junction due to U(ω) and is proportional to ( ) ( ) w w á -ñ a a for positive and ( ) ( ) w w á -ñ a a for negative frequencies (equation (13)), we find The term proportional to [ ( ) ( )] w w   U G n is the finite temperature propagator for the photon interacting n times with the junction impedance and the perturbation series is computed by analytic continuation of the series , where x is given by the diagram, . Thus, the effective Green's function G eff (ω) becomes, 1 . Consequently, the effective action is given by (13) and (15), where the analytic continuation, has been used to regularize the divergent sum when calculating averages,  á ñ for negative frequencies.

Vacuum excitation and photon amplitudes
Notice that in equation (14a), the factor [ ) be i m either rescales G(ω) or the photon number thermal averages  á ñ, implying that the photon number states are modified by the junction impedance. Rearranging, we find where sgn(ω) is the sign function, is a gap in the electromagnetic energy spectrum.
Thus, M is the energy of a 'particle' excited from the vacuum by the electromagnetic field where the probability that the vacuum will be excited is given by the Boltzmann distribution Since this excitation carries electromagnetic energy, | ( )| w X 2 gives the fraction of electromagnetic power absorbed via excitations at the capacitor C by the junction [31]. We note that the complex nature of is not a concern since we are already accustomed to shifting our frequencies or energies by an infinitesimal (e.g. ω to w e + i in equation (B37b)). Taking in equation (14b) the impedance Z(ω)=R to be real and y(ω)=iωC, We introduce quantum statistics of the 'particle' by identifying as the fermionic or bosonic Matsubara frequency [32] respectively where Î  m is an integer. This requires the oscillation period p w 2 of the electromagnetic field to greatly exceed the relaxation time RC of the circuit,  p w pRC 2 2 . Thus, when this condition is not satisfied, it leaves the possibility for 'anyons' [38] with exotic statistics. Consequently, we can take ( ) w X as the amplitude 5 [39] that a photon of frequency ω is absorbed by the junction creating a 'particle' of mass M and statistics according to the Matsubara frequency e m . This realization, together with the fact that the field theory introduced in appendix B.2.2 lives in 1+2 dimensions, suggests that anyonic excitations cannot be ignored [38].
Finally, notice that the exponent can be re-written conveniently as, m is the Green's function of the 'particle'. Thus, its excitation statistics can be computed as, is the inverse thermal Green's function of the 'particle' with mass M. Whether these 'particles' in the single junction are anything more than a tool to implement the renormalization scheme above remains to be investigated.

Applied alternating voltages
In section 2.4, we have established that the fraction of photons absorbed by a tunnel junction is where T is the oscillation period. Whether the ac power is efficiently transferred to the junction from this ac source ought to depend on ( ) w X . We proceed to formally express the explicit form of the effective alternating voltage at the junction.
This can be done by substituting is an alternating voltage corresponding to the effect of the RF field given by ( ) and Ω is the spectrum, the amplitude and frequency of the RF field respectively and the significance of the + sign is that + 0 is experimentally, not computationally equal to 0; it is the limit º  + t 0 0. Proceeding, the applied power P of the RF field is the mean-square value of ( ) V t RF given by, where the proportionality factor is the admittance of the junction.

Lehmann weight and linear response
However, we are interested in the power absorbed by the junction P J instead since it modifies the I−V characteristics. In section 2.4, we argued that this power P J is proportional to | ( )| X W 2 in the presence of bosonic, anyonic or fermionic excitations. Within the context of linear response theory [40], this means that the applied voltage ( ) V t RF acts as an external force, and the effective voltage as a linear response ( ) . Thus, the RF spectrum above gets modified to and P J is given by where we have used equation (20), (21) and (23), as expected.

Unitarity and the topological potential
This section displays the unitary nature of the renormalization effect. In particular, we show that the renormalization effect can be split into two: The fraction of ac voltage drop at the junction and the fraction of ac drop at the environment. This entails taking the quantum states of the environment and the junction as two orthornormal states | y = ñ 1 J and | y = ñ 0 z respectively (figure 2) undergoing a time dependent unitary transformation.
In particular, defining matrices  x and  RF and two quantum states y J and y z of the junction and the environment respectively, where s 0 is the 2×2 identity matrix, we find that these states . This requires the topological flux ( ) DF ¹ t 0 in equation (B40) not vanish in the presence of oscillating electromagnetic fields. Solving for the topological potential A(t), we find with equation (26c) relating the impedance Lehmann weight ( ) X W to the topological potential amplitude . Thus, the purpose of the topological potential is to implement the renormalization scheme above. By equation (26), we find that the topological flux ( ) ( ) both oscillate rapidly without converging. Nonetheless, this poses no problem since it is the flux that appears in the correlation function in equation (B40) rendering the I-V characteristics in equation (B44) perfectly well-defined.
Moreover, by equation (14b), we find that ) and thus we should expect power renormalization for virtually all applied frequencies.
ac ac eff , it should be substituted into equation (B44) to determine the Cooper-pair and quasi-particle tunneling current in the presence of microwaves. Thus, substituting ¢ V x into equation (B44) and using the identities, is the Bessel function of the first kind, x, y are arbitrary functions and Î  n is an integer, the I-V characteristics of the irradiated junction can be expressed in terms of the I-V characteristics I 1 and I 2 of the unirradiated junction, Here, I 1,2 are the quasi-particle, Cooper pair RF-free I−V characteristics given in equation (B44), J n (x) are Bessel functions of the first kind where the order n is the number of actual photons absorbed by the junction, V ac is the amplitude of the alternating voltage, Ω is the energy quantum of individual photons, V and I respectively are the applied dc voltage and tunneling current of the junction. The total current is shifted by the number of photons reflecting energy conservation and is proportional to the square of the Bessel function reflecting the modification of density of states. This equation neglects higher harmonics derived in [25] which were shown to be largely suppressed 6 .

Renormalization factor and soliton length
We consider the action for N 0 number of Josephson junction elements where the action resembles that for a single junction given in equation (1b) where all the admittance elements in the expression are replaced by is the effective admittance matrix. It is now straightforward to determine the phase-phase correlation function by recalling that its Fourier transform is equal to the imaginary part of the Green's function of the action read-off from equation This procedure yields, . This suggests that the Lehmann/wavefuntion renormalization weight is a matrix of the form, . The amplitude of the applied ac voltage will be modified by its determinant j k. This is akin to setting all the phase-phase interaction terms to zero. However, this is not the case since the islands will effectively interact when a charge soliton propagates along the array constituting a current. The injection of a soliton/anti-soliton pair into the array depends on the electrostatic potential at the junction at the edge and the one at the center of the array labeled 1 and 2. We shall approximate the array as infinite with  N 1 0 junctions, where each junction has a capacitance C and 6 The amplitude of the higher (current) harmonics has been shown by Grabert in [25], to correspond to | ( )| | ( ) ( )| X W = W W n Z n Z n eff where n is the n-th harmonic (assumed to be positive). Using the impedances Z(Ω)=R and ( ) ( , the renormalization factor | ( )| X W n is a rapidly decreasing function of n. 7 This action has been considered before within the context of one dimensional XY model of topological phase transitions, e.g. in . This reference can be consulted for introduction on how to approach dynamics and phase transitions in such a linear array. environmental impedance ( ) w = Z R, leading to the effective circuit depicted in figure 4. Thus, the capacitance of the rest of the array is computed by recognizing that for an infinite array, neglecting the capacitance of the first junction C and the self-capacitance of the first island C 0 does not alter the capacitance C r of the rest of the array, The total capacitance of the infinite array C A (excluding the first junction) is given by, We thus set the capacitance of half the array as = C C 2 3 A . Taking the junction phases as f 1 and f 2 , we can write the exact action of the effective circuit as, eff . This will affect the amplitude of the applied oscillating voltage by a Lehmann weight ( ) X det jk . When the angular frequency ω is much larger than the inverse of the time constant RC, we find, where Λ is the soliton length of the array. When an alternating voltage is applied across the array, the amplitude of the oscillating voltage will be renormalized by ( ) X~-Lexp A 1 . This represents a damping of the applied power of applied oscillating voltage.

Soliton field theory origin of the Lehmann weight in an infinite array
Consider the charge soliton lagrangian of the array, Observing that since the integrand is quadratic, we can apply the Bogomol'nyi inequality [36] ( ) to evaluate the mass, M given by, The total energy is given by,

Application: optimization of linear arrays for classical RF field power detection
We treat the array of Josephson junctions as an effective single junction based on the arguments presented in appendix C. This entails using the I-V characteristics given in equation (C1) under irradiation of the RF field, together with the renormalized external voltage ( ) in equation (C1), which simply results in equation (29), P t P t N 0 . This is the celebrated Tien-Gordon equation [41] describing photon-assisted tunneling of Cooper-pairs or quasi-particles of charge ke across a barrier. However, in the case of the array, current is predominantly generated by charge solitons. Thus, equation (37) ought to correspond to charge solitons/anti-solitons injected into the array by the influence of the RF field.
The universality of equation (29) and hence equation (37) is apparent when we observe that the necessary and sufficient conditions for reproducing it are: 1. The unirradiated characteristics of the sample take the form ( ) where the structure coefficients ( ) a k t depend on the density of states of the tunneling particles; 2. The effect of the RF field be to shift V to ( ) Thus, the aforementioned rescaling of ( ) k P t in the case of the array merely corresponds to a modification of the structure coefficients ( ) a k t in condition 1 by a multiplicative factor [ ( )] k á ñ-P t N 1 0 . Moreover, the renormalization effect highlighted in this paper concerns only condition 2. As a consequence, it is sufficient to work in the classical limit  k W eV ac of the RF field, corresponding to multi-photon absorption by the sample (single junction or array). Setting k q= W eV n sin ac , the sum over photon number n can be approximated by an integral formula corresponding to the classical detection of radiation [42], is the sum of the quasi-particle and Cooper-pair currents. For the simulation, we used the measured characteristics, ( ) of a linear array of 10 small Josephson junctions exhibiting distinct Coulomb blockade characteristics, thus by-passing simulating condition 1 which merely corresponds to standard P(E) theory [12,43].
The target parameters of the linear array per junction displayed in table 1 were determined by design during electron-beam lithography and oxidation during shadow evaporation. The tunnel resistance R T and E c are calculated respectively from the offset voltage and the differential conductance ( ) [44,45]. The differential conductance, alongside the measured Δ-H dependence determine the superconducting gap Δ. The Josephson coupling energy, E J is then determined by the Ambegaokar-Baratoff relation [46]  characteristics of the linear array (given by the black bold curve in figure 5) was measured via the well-known r-bias method [47]. It entails incorporating a fixed resistance given by = r 1 W M or 5 W M serially connected to the array and biasing both (the resistor and the array) with a voltage, and measuring the current and voltage values employing differential amplifiers with high input impedance. Noise reduction was achieved by applying half the dc voltage in each terminal with opposite polarity (-V 2 and V 2) relative to the ground.
The simulated curves given in figure 5 were numerically produced using equation (38), by applying Simpson's rule to approximate the integral,   [33,45,48] for both quasi-particle and Cooper-pair tunneling. This phenomenon is dual to microwave-enhanced phase diffusion [33,49,50]. The simulated results for | | X < 1 exhibit exactly the same Coulomb blockade lifting behaviour.
The lifting of Coulomb blockade characteristics with applied RF power can be exploited to design a microwave power detector by defining the Coulomb blockade threshold voltage V cb at a given threshold current I th and tracking its value in the presence of RF power. To illustrate this, we have plotted the V cb -V ac dependence at = I 3 th pA in figure 6 for simulated curves using equation (38) with selected values of the renormalization factor, | | X  1. The sensitivity of such as detector is given by . However, observe that detection range for | | X = 1 given by 0 mV   V 0.26 ac mV is significantly smaller than that for | | X < 1. For instance, the viable microwave power detection range for | | X = 0.1 is 0 mV   V 1.2 ac mV. This implies that the optimum microwave detector, depending on use, should lie between | | < X < 0.1 1. Since the renormalization factor for a linear array is predicted to be | | ( ) X~-Lexp 1 , this corresponds to an optimization condition for the soliton length of a suitable linear array for microwave detection. Such a linear array with soliton length  i C is the impedance of the junction. Likewise, when an infinitely long array [30,45] is modeled as half the infinite array interacting with two junctions, one at one edge of the the array and the other at the center, as illustrated in figure 4, we find an additional Lehmann weight ( ) X = -Lexp A 1 . This requires that applied oscillating electric fields are damped by the same factor over a finite range of electric fields along the array. This is dual to the Meissner effect where the Cooper-pair order parameter leads to a finite range of the magnetic field.
A Josephson junction circuit that exhibits a large Coulomb blockade voltage is ideal for the observation of the renormalization effect. In particular, for the single junction, power renormalization is negligible ( ( )  X W 1) only for extremely low microwave frequencies satisfying  W RC 1 . However, for samples exhibiting Coulomb blockade that also satisfy the Lorentzian-delta function approximation ) and thus we should expect power renormalization for virtually all applied frequencies. In the case of long arrays (  L N 0 ) with  X 1, RF amplitude renormalization should be readily observed due to the additional factor | ( )| ( ) Finally, a list of the electromagnetic quantities and their rescaled formulae is displayed in table 2.

Backaction considerations
The form of the admittance ( ) w y given in equation (B41e) neglects the back-action of the Josephson junctions on the environment (with the bath and the junction becoming entangled) which has been reported to dramatically change the predictions of the P(E) theory [28,51,52]. This back-action manifests through the non-linear inductive response of the junction where the Josephson coupling energy is renormalized, In our work, we have made an implicit assumption that, whenever the Josephson coupling energy E J is considerably small compared to all relevant energy scales such as the charging energy E c , and the current-voltage characteristics of the junction do not exhibit a superconducting branch, this back-reaction can be taken to be small. Nonetheless, showing only half the array as a black box where the effective capacitance of the whole array is given by considering this back-reaction in our theoretical framework especially in the case of a single Josephson junction is certainly warranted since the back-reaction has been suggested to dramatically alter the superconductor-insulator transition conditions for the Josephson junction [52].
Environmental Impedance (long array: figure 4)   Within our path integral approach, considering such effects entails performing the path integral for the rescaled P(E) function given by which is challenging to carry out successfully to all orders of perturbation. Typically, the exponent is linearized as which becomes the f 4 theory [53]. In turn, at order f 2 , the renormalized E J is expected to enter the usual Caldeira-Leggett expression as an inductance, is beyond the scope of this work.

Summary
We have employed path integral formalism to derive the Cooper-pair current and the BCS quasi-particle current in small Josephson junctions and introduced a model which transforms the infinitely long array [30] into an effective circuit with a rescaled environmental impedance, is the environmental impedance as seen by a single junction in the array. As is the case for the single junction, we expect that ( ) also acts as a linear response function for oscillating electromagnetic fields, and can be interpreted as the probability amplitude of exciting a 'particle' of mass b + L --M 1 1 from the junction ground state by the electromagnetic field [31], with the quantum statistics of this 'particle' determined by the complex phase e m identified as the Matsubara frequency [32]. In the case of the infinite array, this 'particle' corresponds to a bosonic charge soliton injected into the array 8 This analysis does not take into account random offset charges which are known to act as static or dynamical background charges in the islands of the array, resulting in shifting of the threshold voltage V th and noise generation affecting the soliton flow along the array [54,55].

Application
Since the quasi-particle current naturally reduces to the normal current and the supercurrent vanishes when the superconducting gap goes to D = 0, the final expression of the tunnel current equation (29) is essentially the time averaged current result previously proposed in [25]. In the classical limit when the RF frequency Ω is small compared to the amplitude of the alternating voltage (  k W eV 1 ac eff ), multi-photon absorption occurs. Setting, k q= W eV n sin ac , the sum over photon number can be approximated by an integral formula that corresponds to a classical detection of the RF field, [42] is given by equation (B44). This result offers a way to measure the magnitude of the Lehmann weight ( ) X W , where | ( )| X W is proportional to the sensitivity of the detector to RF power [48]. Conversely, this implies that our results are indispensable in dynamical Coulomb blockade experiments where long arrays are used as detectors of oscillating electromagnetic fields [33,34].
Within Linear Response Theory, the response˜( ) R t of a system is related to the driving force,˜( ) F t by the central causal relation˜(

R t t s F s ds,
where ( ) c t is the response function. The system variable,˜( ) R t obeys some equation of motion, t a function of ¶ ¶t . Introducing the Green's function of the system, ( ) G t R , satisfying, with ( ) q t the Heaviside function. Thus, we can equate the Green's function to the response function: acts as the response function to the applied oscillating electromagnetic field is to be understood as the result of the arguments in section 2.4, and not necessarily the converse. This leaves the possibility that linear response is violated in complicated circuits, where novel physics may lurk.

Appendix B. The Electromagnetic Environment in Large and Small Josephson Junctions
Despite the existence of excellent reviews on the subject and techniques [11,12,56], the authors found much of the techniques and prior concepts useful in following the arguments in this thesis scattered in various literature [13,36,40,[57][58][59]. In particular, the techniques used in the subsequent chapters include path integral formalism [60] and Green's functions [36,59] to calculate phase-phase correlation functions and four-vector notation [61] where Maxwell's equations appear for compactness. Thus, we include this section as a preamble for completeness and/or compactness. Hopefully it offers a more nuanced understanding of the Caldeira-Leggett model and P(E) theory in the context of Green's functions and generally a path integral framework.
Here, ( ) f t 2 x denotes the phase difference across the junction, where the subscript x distinguishes it from quantum phases of other circuit elements defined later in the manuscript, and E J is the Josephson coupling energy [63]. The simplest derivation of equation (B1) follows from the real and imaginary parts of these two coupled Schrodinger equations Here, m 1 and m 2 and the chemical potentials of the left (1) and right (2) junction respectively, y 1 and y 2 are the Cooper-pair wavefunctions of the left and right superconductors respectively and m 0 is a coupling energy term characterizing magnitude of overlap for the two wavefunctions across the insulator. When a potential difference (voltage) V x is applied across the junction, the two chemical potentials shift relative to each other in order to accomodate this change. This means that we can set m m -= eV From this, it is clear μ is simply the common chemical potential relative to which the voltage drop is measured. This observation implies we can set it to zero without loss of generality, m = 0. and, x . There is another advantage of setting m = 0. In particular, equation (B3) becomes a spinor equation, . These operators form tunneling matrix elements with the spinor and a transpose conjugate spinor defined as, For instance, tunneling from left to right requires replacing y 1 with y 2 and annihilating y 1 , which corresponds to s y + . The inverse process process corresponds to s y -. These matrices will be useful when calculating Cooperpair tunneling rates for small junctions (See equation (B34)).

B.2.2.
Sources of the electromagnetic field as the Josephson junction environment. Equation (B1) only considers the superconducting current and thus neglects the environment that lead to effects such as Coulomb blockade. The environment consists of all sources of the electromagnetic field (including the field itself) which couple to the Cooper-pair wavefunction via the phase difference thus determining the I−V characteristics satisfying equation (B1). Specifically, the environment arises from processes such as the alternating currents and voltages, thermal fluctuations in the form of Johnson-Nyquist noise and coupled high impedance circuit environments [6,49].
Using equation (B1), one can define a conserved energy by treating the junction as a capacitance where C is the capacitance of the junction. Modifying the last equation in (B9) to ( ) Furthermore, taking the limit for small junctions which corresponds to taking the area of the barrier  to be small such that the phase neither varies with y nor z, we arrive at equation (B10) describing the macroscopic physics of the microscopic degrees of freedom of the system undergoing Brownian motion due to a heat bath comprising k harmonic oscillators [57], where H B is the Hamiltonian of the heat bath consisting of L C n n circuits in parallel where w = L C 1 n n n , Q n , f n are the charges stored by and the phases of the elements and K(t) is referred to as the Kernel representing the dissipative nature of the circuit. The generalized Lagrangian for the system is given by where the fluctuation current is given by, , where the subscript a=J, x or z corresponds to the junction, voltage source and environment impedance and k = e e e 2 , corresponds to Cooper pair, quasiparticle charge respectively. That the effect of the environmental impedance ( ) w Z can be represented by a single quantum phase f z defined by the voltage drop over ( ) w Z is not at all obvious. At this stage, we treat it as an ansatz. It will not appear in the equations until we impose the topological constraint where d ab is the Kroneker delta. Operators, O(t) in the Heisenberg picture are related to the ones in the Schrödinger picture, with the unitary evolution operator U 0 (t) given by in the absence of tunneling. In what follows, we assume the Cooper-pair ground state energy m = 0, as we did in equation (B3). The tunneling current I(V ) at the junction is given by  is the time ordering operator with the property given by The tunneling current operator is J J J and the average á ñ ... is over, the quasi-particle equilibrium states, whose density matrix is given by   where we have introduced the so called P(E) function [12] ( ) ( ) ( ) with E some arbitrary energy. It gives the probability that the junction will absorb energy E from the environment. Note that equation (B44) reduces to the normal junction I−V characteristics    we discover that switching on electromagnetic interactions leads to a rescaled impedance and a rescaled response function given by ( ) ( ) ( ) ( ) ( ) w w w w X  X X = -L X This result is not surprising, since we have determined the I-V characteristics of the array by treating it as a single junction with the rest of the array acting as its environment. This means that the -N 1 0 junctions themselves act as bosonic excitations whose (average) number á ñ -= N N 1 0 c determines the electromagnetic cut-off, which is also the effective number of junctions that can be approximated as the environment of the effective single junction.

Appendix D. Path integral formalism with gaussian functional integral
For completeness, this section summarizes how to compute correlation functions with Gaussian functional integrals such as the ones used in section B.4.3 in the derivation of the propagator ( ) +¥ D t in equation (B42). Our approach differs from typical procedures with imaginary time [59]. We work with real time instead since the finite temperature propagator is trivially related to the zero temperature propagator (equation (B42b)).
Consider a quadratic action ( ) S X Y , with X as the coordinate variable, Y as a fluctuation force, a as a mass term and g a coupling constant. The computation procedure is then as follows: