Decay of plasmonic waves in Josephson junction chains
We study the damping of plasma waves in linear Josephson junction chains as well as in two capacitively coupled chains. In the parameter regime where the ground capacitance can be neglected, the theory of the antisymmetric mode in the double chain can be mapped onto the theory of a single chain. We consider two sources of relaxation: the scattering from quantum phase slips (QPS) and the interaction among plasmons related to the nonlinearity of the Josephson potential. The contribution to the relaxation rate from the nonlinearity scales with the fourth power of frequency , while the phase-slip contribution behaves as a power law with a non-universal exponent. In the parameter regime where the charging energy related to the junction capacitance is much smaller than the Josephson energy, the amplitude of QPS is strongly suppressed. This makes the relaxation mechanism related to QPS efficient only at very low frequencies. As a result, for chains that are in the infrared limit on the insulating side of the superconductor-insulator transition, the quality factor shows a strongly non-monotonic dependence on frequency, as was observed in a recent experiment.
Josephson-junction (JJ) chains constitute an ideal playground to study a wealth of fascinating physical effects. Parameters of these systems can be engineered in a controllable way, leading to the emergence of various physical regimes. In chains with the charging energy dominating over the Josephson energy, the Coulomb blockade is observedHaviland and Delsing (1996) and a thermally activated conductance is foundZimmer et al. (2013) at low bias. Moreover, the critical voltage at which the conduction sets in, is governed by the depinning physics Vogt et al. (2015); Cedergren et al. (2017). In the opposite limit, where the Josephson energy is the dominant energy scale, superconducting behavior in the current-voltage characteristics is observed Chow et al. (1998); Ergül et al. (2013). Deep in the superconducting regime, plasmonic waves (small collective oscillations of the superconducting phase) are well-defined excitations above the classical superconducting ground state. The non-perturbative processes in which the phase difference across one of the junctions changes by —the so called quantum phase slipsBradley and Doniach (1984); Matveev et al. (2002); Pop I. M. et al. (2010); Rastelli et al. (2013); Ergül et al. (2013) (QPS)—are exponentially rare. Upon lowering the Josephson energy, QPS proliferate and eventually lead to the superconductor-insulator transition Bradley and Doniach (1984); Choi et al. (1998a); Chow et al. (1998); Haviland et al. (2000); Kuo and Chen (2001); Miyazaki et al. (2002); Takahide et al. (2006); Rastelli et al. (2013); Bard et al. (2017); Cedergren et al. (2017) (SIT) that occurs when the charging and Josephson energies are of the same order.
Disorder plays an important role in JJ chains. The effect of disorder was discussed in the context of the persistent current in closed JJ rings in Ref. Matveev et al., 2002. More recently, the impact of various types of disorder on the SIT was studied in Ref. Bard et al., 2017. Remarkably, the most common type of disorder, random off-set charges, works to enhance superconducting correlations. The mechanism behind this effect is the loss of coherence of QPS due to a disorder-induced random phase, see also Ref. Svetogorov and Basko, 2018.
In recent years, properties of JJ chains under microwave irradiation have attracted considerable interest. Microwave radiation leads to quantized current steps in the current-voltage characteristics that were argued to be promising for metrological applications Guichard and Hekking (2010). Another interesting direction in this context is the field of circuit quantum electrodynamics where novel regimes can be reached Zhang et al. (2014); Puertas Martinez et al. (2018). JJ chains can be further employed to provide a tunable ohmic environment Rastelli and Pop (2018). This environment is realized by two parallel chains that are coupled capacitively to each other, and inductively to transmission lines.
A similar setup was used in Ref. Kuzmin et al., 2018 to probe the reflection coefficient of a JJ double chain under microwave irradiation. Two parallel chains are short-circuited at one end while being coupled at the other end to a dipole antenna that can excite antisymmetric plasma waves (i.e., those with opposite amplitudes in the two chains). The whole sample is placed in a metallic waveguide which reduces the influence of external disturbances. Resonances corresponding to individual plasmonic modes at quantized momenta are clearly observed. This enables the reconstruction of the energy spectrum of the plasma waves. Because of finite damping, the resonances in the reflection coefficient acquire a finite width. By measuring the modulus and the phase of the reflection coefficient, the internal damping could be disentangled from the external losses such as the leakiness of the waveguide or the damping of the transmission line. For chains with a large Josephson energy, the experimentally found quality factor increased with lowering frequency of the microwave radiation. When the Josephson energy was reduced, the curves became flat and eventually showed a tendency to drop at low frequencies. This behavior was interpreted in Ref. Kuzmin et al., 2018 as a signature of the SIT. It was noted, however, that, in contrast to theoretical predictions, the observed behavior is controlled by the short-wavelength rather than the long-wavelength part of the Coulomb interaction in the chain. In particular, the “superconducting” behavior with the quality factor growing at low frequencies was observed in the range of parameters corresponding to the insulating phase of the chain.
The purpose of this work is to provide theoretical understanding of the effects related to the internal damping of plasma waves in JJ chains. We study two models: (i) a single linear chain, and (ii) a double chain of JJs, as in the experiment of Ref. Kuzmin et al., 2018. It is shown that the effective theory for the antisymmetric mode of the double chain can be mapped onto a theory for a single chain if the capacitance to the ground can be neglected. We identify two sources that lead to the decay of plasmons: (i) the scattering of plasmons induced by QPS and (ii) the interaction of plasmon modes via “gradient” anharmonicities. We find the contribution to the relaxation rate of a plasma wave for both kinds of damping mechanisms. The “gradient” nonlinearities are always irrelevant in the renormalization-group sense and the corresponding relaxation rate vanishes as at low frequencies. From the SIT point of view, this behavior can be viewed as “superconducting”. On the other hand, the contribution of QPS processes can show both “superconducting” and “insulating” trends depending on the parameters of the model. The QPS contribution is, however, strongly suppressed if the Josephson energy is much larger than the charging energy associated with the junction capacitance that controls the short-wavelength part of the Coulomb interaction. The combination of the two mechanisms (QPS and “gradient” anharmonicities) can thus lead to a change of the trend from “insulating” to “superconducting” at intermediate frequencies. This mimics a SIT in the intermediate frequency range, although the system is in fact deeply in the insulating phase from the point of view of its infrared behavior.
The paper is structured as follows. In Sec. II we introduce lattice models for a single JJ chain and for two capacitively coupled chains, and derive the effective low-energy field theory. Sec. III discusses two mechanisms contributing to the finite lifetime of the plasmonic waves in JJ chains. The scattering off QPS is studied in Sec III.1, and the decay because of interactions between plasmonic waves is analyzed in Sec. III.2. We analyze the interplay of both mechanisms in Sec. III.3. In Sec. IV we summarize the main results of the paper and compare them to experimental findings. Technical details can be found in the appendix.
Ii Lattice models and low-energy theory
In this work, we consider two closely related systems: a single linear chain of Josephson junctions depicted in Fig. 1a and a device consisting of two capacitively coupled chains shown in Fig. 1b. We are interested in their effective low-energy description. For a single chain of Josephson junctions with Coulomb interaction and disorder, Fig. 1a, the field theory was constructed previously in Ref. Bard et al., 2017. We briefly recall this derivation below and extend the theory by including the terms accounting for gradient nonlinearities. We then show that, up to numerical coefficients, the same effective description applies to the antisymmetric mode of the double JJ chain of Fig. 1b, provided that .
In the case of a single chain, we denote by and the junction capacitance and the capacitance to the ground, respectively. Tunnel barriers between the islands allow for hopping of Cooper pairs along the chain. The number of Cooper pairs and the superconducting phase of the -th island satisfy the canonical commutation relation, . Besides the Josephson energy that quantifies the hopping strength of Cooper pairs, there are the two charging energy scales and , where denotes the elementary charge. The charging energy quantifies the strength of the Coulomb interaction at short scales, while the energy controls the Coulomb-interaction strength in the infrared and, in particular, determines the position of the SITChoi et al. (1998a); Bard et al. (2017). The lattice Hamiltonian for this system has the form
is the dimensionless capacitance matrix and is the screening length for the 1D Coulomb interaction.
In the low-energy limit, it is legitimate to replace the lattice Hamiltonian (1) by an effective continuum model. The latter is conveniently writtenBard et al. (2017) in terms of the field related to the density of Cooper pairs by . The action of the model readsBard et al. (2017)
The quadratic part of the action (in the imaginary-time representation, with temperature and Matsubara frequencies )
describes the plasma waves with the energy spectrum
To facilitate our future discussion of the effective theory for the double chain setup of Fig. 1b, we have introduced here a numerical coefficient ; in the present case of a single chain we have . The parameters and in Eq. (4) are given by
Here is the velocity of low-energy plasmons with momentum and is the corresponding Luttinger constant. Note that we measure all distances in units of the lattice spacing and set , so that the velocity has the dimension of energy.
where is the imaginary time and is the (ultraviolet) value of the QPS amplitude that is usually called “fugacity”. This terminology is related to the fact that QPS can be considered as vortices in the Euclidean version of the -dimensional quantum theory. The fugacity for phase slips is exponentially small in the regime where the superconducting correlations are (at least locally) well developed,
Here plays the role of the Luttinger constant for the ultraviolet plasmons (with ), and is a numerical factor that depends on the screening length and also on details of the ultraviolet cutoff scheme. Estimates for in several limiting cases can be found in Refs.Bradley and Doniach, 1984; Choi et al., 1998a; Matveev et al., 2002; Rastelli et al., 2013; Svetogorov and Basko, 2018.
Among various types of disorder that are present in experimental realizations of the JJ chains, the strongest and the most important one is the random stray charges. Random stray charges modify the kinetic energy term in the lattice Hamiltonian, Eq. (1), according to
The wave function of the system accumulates then an extra phase in the course of a QPS due to the Aharnonov-Casher effectAharonov and Casher (1984); Matveev et al. (2002); Pop et al. (2012). Accordingly, the QPS action in the effective theory in the presence of charge disorder takes the form
For simplicity we assume Gaussian white noise disorder,
In the Hamiltonian language, the action (3) corresponds to
where the quadratic part is of the form
and the QPS contribution reads
Equations (3), (4) and (10) [the latter one reduces to Eq. (7) in the clean case] constitute the low-energy description of a JJ chain as derived in Ref. Bard et al., 2017. Several remarks are in order here. First, in this work we will be interested in the physics at moderate wavelengths and will approximate the dispersion relation (5) by its expansion at small :
Here the length (that differs from only by a numerical coefficient) sets the scale for the bending of the plasmonic dispersion relation.
Second, the only nonlinearity in our effective action at this stage is due to the QPS. In the ultimate infrared limit (or, in fact, for all in the case of short-range charge-charge interaction, ) the effective action Eqs. (3), (4) and (7) reduces to the standard sine-Gordon theory and describes the superconductor-insulator transition (SIT) that occurs Choi et al. (1998a) at . The SIT is driven by the QPS and in this sense they constitute the most important anharmonicity in the system. Other non-harmonic terms are, however, also possible. For example, the expansion of the Josephson coupling in Eq. (1) to the next-to-leading order provides the following contribution to the effective Hamiltonian:
which is associated with the action
In contrast to the QPS term (15), the nonlinearity (18) and all other non-linear terms that can be added to the effective Hamiltonian are built out of the local charge and current densities. Therefore, they always contain high powers of gradients and are irrelevant in the renormalization-group sense. We will occasionally refer to the anharmonicities of these type as to “gradient” anharmonicities to distinguish them from the QPS. The gradient anharmonicities are always unimportant at lowest energies. Yet, as we will see in Sec. III.2, they can control the decay of plasmons at sufficiently high frequencies if the bare value of the QPS amplitude is small.
We thus conclude that the effective action describing a chain of Josephson junctions is given by
Here and are given by Eqs. (4) and (10), respectively. As for the “gradient” anharmonicities represented by , we will use Eq. (19) as a specific form. We argue in Sec. III.2 that our main conclusions are insensitive to this particular choice.
Let us now turn to the discussion of the double-chain setup of Fig. 1b. In this case we denote by and the capacitance to the ground and the interchain capacitance, respectively. The lattice Hamiltonian for the double chain reads
Here, the indices refer to the two chains.
In the Gaussian approximation the spectrum of the Hamiltonian (21) consists of two modes, symmetric and antisymmetric, that are analogous to the charge and spin modes in a spinful Luttinger liquid. In this work we are interested in the physics of the antisymmetric mode that can be excited in the system by coupling to the dipole antenna Kuzmin et al. (2018). To simplify the analysis, we assume further that . It is not clear to us how well is this assumption satisfied in the experiments of Ref. Kuzmin et al., 2018; we believe it to be, however, of minor importance for our results. Specifically, our analysis should remain applicable, up to modifications in numerical coefficients of order unity, also for . The condition corresponds to sufficiently well coupled chains, with large splitting between the symmetric and antisymmetric modes111The opposite case, , corresponds to nearly decoupled chains, in which case the physics is more naturally described in terms of modes corresponding to individual chains rather than in terms of symmetric and antisymmetric modes. In the limit the decoupling is complete, and the analysis for a single chain applies, with playing the role of ..
In full analogy to the spin-charge separation in quantum wires, the (low-momentum) velocity of the symmetric (“charge”) mode, greatly exceeds, under the assumption , the velocity of the antisymmetric (“spin”) mode, . This observation allows one to integrate out the charge mode and formulate the effective description of the system in terms of the antisymmetric mode alone. Details of this derivation are presented in appendix A. It turns out that, just as in the case of a single JJ chain, the effective theory is given by Eqs. (20), (4), (10) and (19), with the parameters given by Eqs. (16), (17) and (6). The only difference is the value of the numerical factor that should now be set to .
Iii Relaxation of plasmonic waves
Plasmonic waves, which are long-wavelength excitations above the superconducting ground state, are subjected to interaction. As a result, once excited by, e.g., a microwave, a plasma wave can decay into several plasmons of lower energy. The two anharmonic terms in the action (20) provide two mechanisms for the decay of plasmons: interaction with QPS and “gradient” anharmonicities. We analyze these channels of plasmon decay one by one is Secs. III.1 and III.2, respectively. The interplay of the two mechanisms is discussed in Sec. III.3.
iii.1 Relaxation due to phase slips
We start with the discussion of the relaxation processes related to the scattering off QPS. Our analysis follows closely the one of Refs. Oshikawa and Affleck, 2002; Rosenow et al., 2007. The curvature of the plasmonic spectrum, as quantified by the length in Eq. (16) is of minor importance here and for the purpose of this section we approximate the plasmonic spectrum by
Correspondingly, the Gaussian action takes the form
A formal expansion of the QPS action, Eq. (7), in powers of shows that the plasmon can decay into an arbitrary large number of low-energy plasmons. We will determine directly the sum of all those contributions. This decay rate can be conveniently calculated from the imaginary part of the self-energy (of the Fourier transform) of the correlation function
where and denotes the average with respect to the full action, . On the Gaussian level, the imaginary-time Green function reads, in Fourier space,
where is the Matsubara frequency. With the help of the self-energy, the full Green function can be expressed as
The inverse lifetime of an excitation with energy is related to the imaginary part of the retarded self-energy on the mass shell:
In the following, we calculate the self-energy perturbatively in the QPS fugacity . The self-energy in the Matsubara space-time, , can be extracted from the perturbative expansion of the Green function, Eq. (26),
with the following result:
Here, the exponential contains the correlation function
denotes the number of junctions, and is the inverse temperature. The result for the self-energy in the imaginary time should be analytically continued to real time and then Fourier-transformed. The -dependence of the first term in Eq. (31) is determined by the following Matsubara time-ordered correlation function
The retarded version of this correlation function can be obtained in the standard way Giamarchi (2004). We find
where denotes the Heaviside step function. In order to extract the lifetime, we need to know the imaginary part of the self-energy in Fourier space. The second term on the RHS of Eq. (31) does not contribute to the imaginary part of . The imaginary part of the self-energy in Fourier space can thus be obtained via
It is convenient to switch to the light-cone variables :
iii.1.1 Clean case
In the regime we can set , and the integrations decouple. Performing the integrations, we obtain
is the Euler Beta function. Making use of Eq. (29), we extract the relaxation rate,
iii.1.2 Disordered case
In the limit of strong disorder, , the main contribution of the integrations in Eq. (38) comes from the region close to . We find
for the imaginary part of the self-energy, which is independent of momentum. This leads to the following scaling of the relaxation rate in the case of strong disorder, :
For a moderate disorder strength, we need to consider two cases. If , the clean result given by the first line of Eq. (41) remains valid. For the integration over in Eq. (38) is cut at the upper limit by . We can further neglect in the exponential function related to . This leads to the following behavior of the relaxation rate:
Equations (41), (43) and (44) give the relaxation rate of plasma waves due to QPS in different regimes and constitute the main result of this section. They are summarized in Fig. 2. The relaxation rate exhibits power-law scaling with frequency, temperature and the disorder. The corresponding exponents are non-universal and are determined by the value of the Luttinger parameter . Deep in the superconducting phase of the JJ chain, , the relaxation rate vanishes at low frequencies, while the opposite trend is predicted in the insulating phase with sufficiently small .
iii.2 Relaxation due to “gradient” nonlinearities
Let us now turn to the analysis of “gradient” anharmonicities described by the term in the effective action (20). They are irrelevant from the point of view of the renormalization group. However, at intermediate energy scales they contribute to the decay of the plasma waves on equal footing with QPS. We consider here the nonlinearity (18) corresponding to the correction (19) of the action, which arises as the quartic term of the expansion of the Josephson potential.
Perturbative treatment of the decay of plasmons caused by “gradient” anharmonicities was discussed in other contexts in Refs. Apostolov et al., 2013; Lin et al., 2013; Protopopov et al., 2014. The perturbation theory turns out to be ill-defined in the case of a linear plasmonic spectrum and in this section we use the dispersion relation (16) taking into account its finite curvature.
In order to calculate the relevant matrix element for the relaxation process, it is convenient to express the superconducting phase through bosonic creation () and annihilation operators (), that obey the standard bosonic commutation relations. This decomposition is of the formGiamarchi (2004)
where is the ultraviolet cutoff that can be send to zero in this calculation, and is the number of junctions per chain. Our analysis below largely follows the approach of Ref. Protopopov et al., 2014. The relaxation rate can be calculated using the diagonal part of the linearized collision integral,
where the transition probability is
and denotes the total energy of the initial (final) states. The modulus of the matrix element is given by
A right moving plasmon with momentum can relax via this nonlinearity by the scattering off a left moving thermal plasmon with momentum (see Fig. 3). According to the conservation laws, the momentum of the left moving particle
is much smaller than the momentum .
With the help of the momentum conservation we can perform the sum over . The delta function related to the energy conservation can be written in the form
Requiring to be real restricts the range of to the interval
In the continuum (long-chain) limit, the remaining summations over momenta transform into integrations, . Performing the integration over , we find
The -dependence in the denominator of the integrand can be neglected compared to . We assume now that the energy of the particle with momentum is much larger than temperature but not too large such that . In this case, the Bose function of the particle with momentum can be replaced by , and the Bose function related to the particle with momentum can be neglected. The Bose function related to the particle with momentum can be replaced by for and neglected for . The main contribution to the integral in Eq. (53) originates from the latter range of . After performing the integration, we find (under the assumptions , , and ) the following behavior of the relaxation rate:
Equation (54) constitutes the main result of this section. It predicts scaling of the relaxation rate of plasmons due to the “gradient” anharmonicity. The relaxation rate vanishes at low frequencies reflecting the irrelevant character of the gradient anharmonicities.
Before closing this section let us discuss the universality of the result (54) with respect to the particular form of the Hamiltonian given by Eq. (18). On phenomenological grounds various terms of the form are allowed in the effective Hamiltonian. For such terms are less relevant than the term considered here and, thus, contribute less to the lifetime of plasmons. On the other hand a cubic-in-density interaction, , is more relevant in terms of the scaling dimension. However, the energy and momentum conservations forbid the decay of a single plasmon into two particles. Correspondingly, a cubic non-linearity should be taken in the second order perturbation theory to produce a finite decay rate. The resulting process is again the one of Fig. 3 and leads to the same scaling of the relaxation rateApostolov et al. (2013); Lin et al. (2013); Protopopov et al. (2014).
iii.3 Interplay of QPS and “gradient” anharmonicities
In Secs. III.1 and III.2 we have analyzed the decay of plasmon excitations due to QPS and the “gradient” anharmonicities, respectively. While the relaxation rate due to “gradient” anharmonicities follows universal scaling, the QPS contribution is characterized by a non-universal exponent and reflects the SIT controlled by the value of . Let us now discuss the interplay of the two relaxation channels. We assume for definiteness that frequencies of interest are larger than temperature.
It is convenient to characterize the strength of the plasmon decay by a dimensionless parameter . This parameter is expected to be proportional to the quality factor studied in Ref. Kuzmin et al., 2018. Deep in the superconducting regime, , the relaxation of plasmons is always dominated by the “gradient” anharmonicities and the quality factor scales as , see Eqs. (41), (43), (44) and (54). Upon decreasing the Luttinger parameter , the QPS start to be visible in the quality factor. Specifically, the QPS dominate the low-frequency behavior of the quality factor under the condition (respectively, ), yielding its (respectively ) scaling in the cases of weak (respectively, strong) disorder. Furthermore, as a result of QPS, for sufficiently small ( for weak and for strong disorder), the quality factor goes down as frequency decreases. The resulting frequency dependence of will then be non-monotonic with a maximum around a crossover frequency where the QPS set in, as illustrated in Fig. 4.
In the discussion of the overall frequency dependence of the quality factor, it is important to keep in mind the exponential smallness of the fugacity , Eq. (8). Due to this fact, the frequency below which QPS dominate over “gradient” non-linearities is exponentially small for . As a result, even deep in the insulating regime, , the quality factor is dominated by the gradient non-linearities and thus grows with lowering the frequency in a wide frequency range if . Only at exponentially small frequencies this “superconducting” behavior crosses over to the decrease of the quality factor reflecting the insulating character of the system in the infrared limit.
Iv Summary and discussion
We have studied the decay of plasmonic waves in JJ chains. Motivated by a recent experimentKuzmin et al. (2018), we have considered, besides a single one-dimensional chain, also a model of two capacitively coupled linear chains. It has been shown that in the parameter regime where the capacitance to the ground () can be neglected, the theory for the antisymmetric mode in the double chain can be mapped onto a theory for a single chain. This was possible because the symmetric mode acquired a fast velocity due to the strong Coulomb interaction.
Two sources for the relaxation of plasma waves have been considered. First, the damping originating from the scattering generated by QPS leads to a relaxation rate that scales with frequency as a power law with a nonuniversal exponent that depends on the parameter . The scaling behavior of the relaxation rate related to QPS in different parameter regimes is summarized in Fig. 2. Since the QPS amplitude is exponentially small in the parameter , the rate is very sensitive to this parameter. The second mechanism for the relaxation of plasma waves is the interaction of them mediated by other nonlinear terms. As an example, we have considered the lowest-order nonlinearity coming from the Josephson potential. This term leads to a relaxation rate that scales as the fourth power of frequency. The vanishing of the relaxation rate at low frequencies reflects the irrelevance of this term in the renormalization group sense. Nevertheless, for a small phase-slip amplitude (fugacity), the contribution originating from this nonlinearity may be dominating in a wide range of frequencies.
Comparing our findings to the experiment of Ref. Kuzmin et al., 2018, we find a very good qualitative agreement between our theory and experimental observations. All of the samples shown in Fig. 3b of Ref. Kuzmin et al., 2018 are nominally in the insulating regime. Specifically, values of the Luttinger constant that are extracted from the measured values of the impedance (proportional to ) make one to expect the insulating behavior. However, the samples with a large ratio of show an increase of the quality factor when lowering the frequency. This behavior suggests that the systems are in the superconducting regime. This apparent contradiction is resolved by noting that the crossover scale below which the QPS effects show up is exponentially small in the square root of . As a result, the downturn of the quality factor indicating insulating behavior occurs below the lowest measured frequencies. For devices with a lower value of both and , the authors of Ref. Kuzmin et al., 2018 observe a flat behavior at intermediate frequencies with a tendency to drop at lowest measured frequencies. This behavior is qualitatively consistent with our prediction on the frequency dependence of the quality factor that is dominated by QPS in the insulating regime at low frequencies. For a more quantitative comparison, the extension of the experimental measurement method to lower frequencies would be beneficial.
Let us discuss in more detail experimental observations on dependences of the quality factor on various input parameters. We consider first the more insulating chains. The authors of Ref. Kuzmin et al., 2018 point out a stronger sensitivity of the quality factor to the parameter compared to the parameter for their weakest junctions (large and low )—an observation that is immediately understood within our theory. In these devices, the parameter is very small such that the exponent for the power law of the phase-slip contribution to the quality factor is only slightly modified when changing . Even relatively large changes of the order of (as in the experiment) have only a small effect, since the value of is still small and modifies the exponent only weakly. On the other hand, the fugacity of QPS depends exponentially on the square root of , which explains the observed strong dependence of the quality factor on this parameter.
Further, we compare the scaling predicted in our work to the experimental observations in low-impedance chains shown in Fig. S 4 in the Supplementary Material of Ref. Kuzmin et al., 2018. All these samples are characterized by a large ratio of over such that the QPS effects should be negligibly small in the range of measured frequencies. Indeed, the curves show an increasing behavior when lowering the frequency. More specifically, the corresponding frequency scaling of the quality factor is consistent with the theoretical expectation from the decay due to the nonlinearity. Discussing the dependence on other parameters, we notice that the charging experiences a particularly strong variation in the experiment (within a factor of ), while the variation of other device parameters is smaller. All experimental curves appear to collapse reasonably well when plotted as a function of the rescaled frequency . On the other hand, our prediction shows a strong power-law dependence () on the charging energy . We speculate that a different kind of nonlinearity may be responsible for the explanation of this discrepancy. It might originate from some kind of nonlinear capacitances and result in a different prefactor in the frequency dependence of the quality factor that does not depend so strongly on the charging energy . The identification and analysis of other types of nonlinearities constitutes an interesting prospect for future research.
While preparing this paper for publication, we learnt about a related unpublished work Houzet and Glazman .
Acknowledgements.We thank D. Abanin, L. Glazman, R. Kuzmin, V. E. Manucharyan, and A. Shnirman for fruitful discussions. This work was supported by the Russian Science Foundation under Grant No. 14-42-00044 and by the Deutsche Forschungsgemeinschaft. IP acknowledges support by the Swiss National Science Foundation.
Appendix A Derivation of low-energy field theory
This appendix is devoted to the derivation of the low-energy field theory for the antisymmetric mode of the double-chain system. Our starting point is the lattice Hamiltonian (21). We denote by , and the charging energies associated with the capacitance , and , respectively, and .
The basic idea is that in the limit of small capacitance the associated charging energy suppresses the charge fluctuations in the symmetric mode (at least at long scales) leaving us with the antiymmetric mode as the only dynamical degree of freedom. This observation was previously employed in the literature to obtain the low-energy theory of the antisymmetric mode, see Refs. Choi et al., 1998b, 2000. Here we generalize the results of Refs Choi et al., 1998b, 2000 to the case when the Coulomb interaction is long-ranged () and charge disorder is present in the system. We show that the effective theory takes the form of the sine-Gordon model, Eqs. (3), (4) and (10) supplemented by a “gradient” non-linearity term, Eq. (19).
The posed goal can be achieved in two different ways. In Sec. A.1, we present a semi-quantitative derivation of our results from the field-theory description of the symmetric and antisymmetric modes in the double chain. A more microscopic analysis of the initial lattice model (leading to the same results) is carried out in Secs. A.2 and A.3 for the cases of short-range () and long-range () Coulomb interaction, respectively.
a.1 Heuristic derivation from the continuum field theory
We start our discussion of the effective theory for the antisymmetric mode in the double chain from a heuristic derivation based on the filed-theory description of the lattice model (21). The latter is derived in full analogy to the case of a single JJ chain. To this end, we introduce two fields and related to the charge in the lower and upper chain via , as well as their combinations
In terms of these fields, the quadratic part of the action corresponding to the lattice model (21) reads
The QPS can be accounted for by
and is the random charge in the upper (lower) chain. Note that in Eq. (57) we consider QPS as happening independently in the upper and lower chains. This is justified provided that . The fugacity is then exponentially small in the parameter .
In the long wave-length limit, , the quadratic action (56) reduces to the Luttinger liquid form
Let us now consider the perturbative expansion of the partition function in the fugacity . The lowest non-vanishing correction arises in the second order and reads
Here, . In Eq. (61) we have performed explicit averaging over the symmetric mode but kept the correlation functions of in the unevaluated form. Introducing
Assuming that we are in the regime , we can approximate by unity. If the charge disorder is weak, we can further replace