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vortices of persistent current in superconductors have been viewed for a long time as a testing ground for various models of classical and quantum phase transitions . this is due to relatively strong interactions between vortices coupled with a high degree of control over the vortex arrays . for example , inter - vortex interactions in the presence of a periodic external potential ( created experimentally by modulation of the superconducting film thickness @xcite ) made vortices a convenient target for investigation of commensurability transitions.@xcite theoretical studies of the effects of randomness on vortex structure and vortex dynamics have led to the notion of collective pinning,@xcite with importance stretching well beyond the physics of superconductivity.@xcite later , the discovery of high - temperature superconductors triggered studies of the vortex lattice melting transition and glassy behavior@xcite in layered superconductors as well as structural transitions of vortices interacting with columnar defects.@xcite because vortices in continuous superconductors have normal cores , vortex motion is dissipative@xcite and much of the above - mentioned work used classical statistical mechanics to address collective phenomena in vortex arrays.@xcite    the effects of quantum fluctuations of vortices in continuous films become important only close to the supeconductor - insulator transition , which requires special tuning of the films normal - state resistance.@xcite in arrays of josephson junctions , however , vortices do not have cores , allowing for the study of quantum fluctuations . an array of small superconducting islands connected by josephson junctions , where the island charging energies were sufficiently large for quantum effects to be important,@xcite was used in attempts to observe a quantum kosterlitz - thouless ( kt ) transition @xcite and mott transition@xcite in a 1d array . while these works advanced the nanofabrication techniques needed to produce highly regular arrays of small josephson junctions , the measurement results were ambiguous . the current - voltage ( @xmath0-@xmath1 ) characteristics gave inconclusive evidence for a kt transition in single - line 1d arrays.@xcite the observations of ref . were later interpreted to be consistent with a purely classical commensurability transition rather than the quantum mott transition.@xcite the suppression of quantum effects in these experiments stemmed from the low inductance of the continuous superconducting wires , which were necessary to make the josephson junction arrays . we should note , also , that in the majority of experiments the evidence for the various classical and quantum transitions mentioned above was based on signatures in @xmath0-@xmath1 characteristics . this method is limited to addressing highly averaged quantities , and relies on substantial deviations of the investigated system from equilibrium ( needed , for example , to overcome the static pinning of vortices ) . recent developments in superconducting qubit techniques offer the possibility of overcoming the described limitations of previous experimental studies of many - body physics of vortices . typical superconducting qubit experiments address the superconducting system using microwaves.@xcite the system is only weakly perturbed by the microwave excitation ; hence , this spectroscopic approach allows one to probe the system close to equilibrium . on the circuit element side , the development of the fluxonium qubit,@xcite which combines a josephson junction with a superinductor@xcite ( i.e. , an element exhibiting high inductance and low capacitance ) opens avenues for studying quantum effects in superconducting arrays .    in this work , we study theoretically a one - dimensional array consisting of superinductors and a chain of small josephson junctions as shown in fig . [ fig01 ] . two parameters characterize such an array : the ratios of the josephson ( @xmath2 ) and inductive ( @xmath3 ) energies ( see sec .  [ sec : model ] ) , combined into a characteristic length @xmath4 and the ratio  @xmath5 of charging and josephson energies , which controls the quantum fluctuations . despite its simplicity , this model allows for a variety of phase transitions of a classical or quantum nature ( depending on the ratio @xmath5 ) as a function of the applied magnetic flux  @xmath6 per plaquette of the array . classically , for fixed @xmath7 , there is a critical magnetic field @xmath8 at which the system undergoes a transition from a homogeneous state with no persistent currents through the inductors to a state with static persistent currents . as a function of the characteristic length  @xmath9 , eq . ( [ x_ell ] ) , we discuss two regimes , cf . [ fig01](b ) : a `` type - i '' regime ( @xmath10 ) featuring a second - order transition to a state of staggered persistent currents as shown in fig . [ fig_vortices](a ) , and a `` type - ii '' regime ( @xmath11 ) , in which the transition takes place by the sequential ( first - order ) introduction of localized vortices of persistent currents in the lattice plaquettes , fig . [ fig_vortices](b ) . these vortices correspond to _ kinks _ , meaning discontinuous jumps of height @xmath12 , in the node phase ( or node flux )  @xmath13 , see fig . [ fig01](a ) . repulsive interactions between kinks lead to a series of pinned commensurate phases with increasing @xmath6 . in the presence of quantum fluctuations , the number of kinks fluctuates and the initial transition at @xmath8 turns into a kt transition . this is followed by commensurate - incommensurate transitions between classical pinned phases and quantum liquid phases of floating crystalline cells of a kink lattice . each of the phases carries a `` fingerprint '' in the microwave absorption spectrum , as the nature of low - energy excitations is sensitive to the types of phases and the transitions between them . the microwave spectra also carry information about the crossover from classical to quantum critical behavior in the vicinity of the transition . yet another advantage of the spectroscopic approach is that it is a linear response to a weak perturbation . in this work , we will highlight the signatures of phase transitions that can be measured using microwave photons . the paper is organized as follows : in sec . [ sec : model ] , we formulate the mathematical model for the circuit under consideration . we also introduce and discuss two methods , capacitative and inductive , for coupling the circuit to an external resonator , and give general forms for the radiation absorption rate in the two cases .    in sec . [ sec : type - i ] , we describe the type - i regime , with @xmath9 comparable to a lattice spacing . in this limit , the low energy excitations are gapped plasmon oscillations , where the node phases  @xmath13 undergo small fluctuations . the plasmon excitations soften at the critical magnetic flux  @xmath8 , leading to a quantum ising transition between a phase with the magnetic fluxes @xmath14 ( cf . [ fig01 ] ) everywhere and a staggered phase  @xmath15 , with order parameter @xmath16 . section  [ sec : type - ii ] discusses the type - ii ( @xmath17 ) regime . here , the low - energy excitations are associated with the addition or removal of kinks . as the magnetic field is increased from  @xmath18 , the cost of creating a kink is reduced , leading to a visible peak in the absorption spectrum below the plasmon continuum . at a critical field , this energy cost vanishes , leading to a proliferation of kinks and the formation of a series of gapped and gapless crystalline phases of kinks . each of these has an observable signature in the excitation spectrum for adding or removing kinks . finally , quantum effects such as broadening of peaks in the excitation spectrum and the appearance of phases of incommensurate quantum liquids are discussed . a discussion of our analysis and its results is presented in sec . [ sec : discussion ] .    . ( b ) phase diagram for the classical ground state , cf . @xcite for the phase diagram of similar classical models . at inductances  @xmath19 such that @xmath20 , the system is in the `` type - i '' regime that features a single ising transition from a homogeneous phase into a phase of staggered persistent currents . for large @xmath19 or @xmath17 ( `` type - ii '' regime ) , the system , as a function of external flux  @xmath6 , develops subsequent phases of commensurate lattices of vortices or _ kinks_. the rational numbers describe the kink density in the node flux configurations  @xmath21 . darker shaded regions contain phases with higher denominators that complete the devil s staircase . the line separating the homogeneous and inhomogeneous phases is given by eq . ( [ phiec_ising ] ) in the type - i and by eq . ( [ x_phiec ] ) in the type - ii regime . ]    ) ordered phase in the type - i regime . ( b ) a vortex of static persistent currents decaying over the length  @xmath9 , stable in the type - ii regime . ] the circuit in fig .  [ fig01](a ) is described by the lagrangian @xmath22 & = \frac{\hbar^2}{2e_c}\sum_{j=0}^{n-1}\big(\dot{\phi}_{j}-\dot{\phi}_{j-1}\big)^2 - v[\phi ] \ , \label{x_lagrangian}\end{aligned}\ ] ] where  @xmath23 with @xmath24 the josephson junction capacitance , and we formally set @xmath25 , assuming periodic boundary conditions . the potential in  ( [ x_lagrangian ] ) takes the form @xmath26 & = \sum_{j=0}^{n-1}\big\ { \frac{e_l}{2 } \phi_j^2 - e_j\big[\cos\big(\theta_j-\phi_e\big)-1\big ] \big\ } \label{x_potential}\end{aligned}\ ] ] where @xmath27 , @xmath19 is the inductance of the inductors and @xmath28 with @xmath29 denoting the magnetic flux per plaquette and @xmath30 being the flux quantum . furthermore , we define @xmath31 , the phase difference across the josephson junction between nodes  @xmath32 and  @xmath33 , which in the following we denote as _ link _  @xmath33 . the potential @xmath34 $ ] is not invariant under @xmath35 , as would be expected for a superconducting system . this is because we neglect the phase slip processes that allow the inductors to relax to the true ground state of the system . superinductors such as those used in a fluxonium qubit are engineered so that this is a valid approximation.@xcite    at large  @xmath24 , the physics is dominated by the ( meta)stable configurations  @xmath21 that minimize the static potential  @xmath34 $ ] . the magnetic flux  @xmath6 and the characteristic length scale @xmath9 , defined in eq . ( [ x_ell ] ) , are the only parameters in this `` classical '' regime . throughout our theoretical analysis , we assume the limit of large systems ,  @xmath36 . note that , due to the inductive potential , a configuration with local flux  @xmath13 is physically distinct from a state with flux @xmath37 with integer  @xmath38 . in the ground state , each variable  @xmath13 will thus take values between  @xmath39 and  @xmath40 . furthermore , minimal configurations  @xmath21 necessarily satisfy  @xmath41 , which physically corresponds to current conservation .      by coupling the circuit in fig . [ fig01](a ) to microwaves , signatures of the various phases and phase transition become observable in absorption spectroscopy . additional interest in such experiments may arise because of well - defined peaks in the absorption spectrum below the plasma gap . we suggest two , in a certain way complementary , schemes of coupling@xcite the circuit to a microwave resonator : ( a ) inductive coupling and ( b ) capacitive coupling , see fig . [ fig_antennas ] . in the situation of fig . [ fig_antennas](a ) , we add an antenna capacitively coupled to two neighboring nodes  @xmath42 and  @xmath43 , which will couple the microwave mode to the josephson junction at link  @xmath43 . the hamiltonian for the the microwave mode ( with creation and annihilation operators  @xmath44 and  @xmath45 ) and its coupling to the circuit takes the form @xmath46 herein , @xmath47 is the frequency of the microwave mode and @xmath48 is the coupling constant , which is determined by the impedance of the microwave mode and the coupling capacitance.@xcite @xmath49 is the cooper pair number operator for the antenna josephson junction at link  @xmath43 . using fermi s golden rule , the absorption rate for driven radiation with frequencies  @xmath50 incident on the antenna has the general form @xmath51 with the sum being over the excited states @xmath52 with energy @xmath53 relative to the ground state .      for the inductive scheme , fig . [ fig_antennas](b ) , we add inductances  @xmath54 into the former `` ground line '' such that the system remains translationally symmetric . the antennas themselves may couple inductively to one or several of these inductances . here we suppose it solely couples to one link , say  @xmath43 .    the inductances  @xmath54 should be chosen small such that the length @xmath55 is not only large itself ( in comparison with unity ) but also @xmath56 ( implying @xmath57 ) , where @xmath9 is the characteristic length defined in eq . ( [ x_ell ] ) . in this limit , as far as the ground state and excitation energies are concerned , the inductances  @xmath54 merely lead to a renormalized characteristic length  @xmath58 . with good accuracy , we may thus set @xmath59 and ignore the effects of the inductances  @xmath54 in the analysis of the model  ( [ x_lagrangian ] ) .    using inductively coupled antennas , the microwave modes ( described by  @xmath45 and  @xmath44 ) effectively couple to the josephson phase differences  @xmath60 instead of cooper pair numbers  @xmath61 , cf . ( [ eq : antenna_hamiltonian ] ) . currents induced by the antenna at link  @xmath43 decay only over the ( large ) distance  @xmath62 , eq . ( [ ella ] ) . this leads to the following form of the absorption rate : @xmath63 with @xmath64 where  @xmath65 with @xmath66 and @xmath67 is the fourier transform of  @xmath60 . the last approximation is valid in the long - wavelength limit . the coupling constant @xmath68 in eq . ( [ absorptionrate_l ] ) is determined by the impedance of the microwave mode and the coupling inductances . in contrast to the capacitive coupling , which _ locally _ couples the microwave mode to one josephson link , the inductive scheme has a much _ longer range _ @xmath62 , eq .  ( [ ella ] ) . we will specify the spectral absorption rates for the capacitively coupled antenna ,  @xmath69 , and the inductively coupled antenna ,  @xmath70 , for the type - i regime in sec . [ sec : type - i ] and type - ii regime in sec . [ sec : type - ii ] . in the following , we will use units in which  @xmath71 . for  @xmath20 , as one increases the external flux  @xmath6 , the system undergoes a second - order phase transition from a low - flux homogeneous configuration @xmath72 , corresponding to zero persistent current through the inductors , into a staggered order  @xmath73 with an alternating persistent current through the inductors ; see fig . [ fig_vortices](a ) . this state is classically connected to the state with kink density @xmath74 at larger  @xmath9 in the type - ii regime , cf . the phase diagram in fig . [ fig01](b ) . in this type - i regime , close to  @xmath8 , the relevant low - energy excitations are plasmons , quantized small fluctuations in @xmath13 . mean - field theory , which assumes that these fluctuations are small , predicts that the transition happens at the critical flux  @xmath8 given by @xmath75 technically , it is more convenient to work with the field  @xmath31 , which represents the phase differences over the josephson links . for  @xmath76 , the field  @xmath60 fluctuates around the ordered configuration @xmath77 .      for  @xmath78 , mean - field theory for the phase differences  @xmath60 gives the plasmon spectrum : @xmath79 where , in the continuum approximation , the wave numbers are  @xmath80 . the lowest energy states occur for momentum  @xmath81 . for momenta  @xmath82 close to  @xmath40 , the spectrum is simplified to @xmath83 herein , @xmath84 is the plasmon gap with the second line showing that mean - field theory predicts the gap closing as a square root as  @xmath85 . at the transition , the softened low - energy plasmon modes have a linear dispersion relation @xmath86 , where @xmath87 is the sound velocity .    on the ordered side , for  @xmath76 , plasmons describe the fluctuations of the order parameter  @xmath16 . a similar mean - field theory calculation yields for momenta  @xmath82 with @xmath88 a soft - mode dispersion relation of the same form as  ( [ plasmon - spectrum - soft ] ) , with the gap larger by a factor of  @xmath89 , @xmath90 but otherwise behaves as a function of  @xmath91 in the same way as the gap  ( [ plasmon - gap ] ) on the disordered side .    in absorption spectroscopy , microwave photons excite plasmons so that we expect manifestations of the critical point at @xmath92 in the spectral absorption rate  @xmath69 , eq . ( [ absorptionrate ] ) , or @xmath70 , eq . ( [ absorptionrate_l ] ) , for capacitive or inductive coupling to the microwave resonator . specifically , we may write the cooper pair number operator  @xmath93 and its conjugate , the phase difference @xmath94 , in terms of the plasmon modes , @xmath95 where @xmath96 is the annihilation operator for a plasmon at wave number @xmath82 . inserting eq . ( [ eq : plasmon_number ] ) into eqs . ( [ absorptionrate ] ) , we find that at @xmath78 the spectral absorption rate from the ground state is given by @xmath97 for the capacitively coupled antenna , where @xmath98 is the plasmon density of states . for inductive coupling , inserting eq . ( [ eq : theta ] ) into eq . ( [ absorptionrate_l ] ) , we find @xmath99 for the softest plasmons with momentum  @xmath82 near  @xmath40 , i.e. @xmath88 . comparing eqs . ( [ fc_plasmons ] ) and  ( [ fl_plasmons ] ) , we infer that the inductive - coupling scheme leads to stronger response at low energies  @xmath100 and thus to higher - contrast results close to the transition at  @xmath8 . the plasmon spectrum  ( [ plasmon - spectrum - soft ] ) implies the density of states @xmath101 where @xmath102 denotes the heaviside step function . in the frequency region @xmath103 , the absorption rate features a square - root singularity , @xmath104 at the critical field  @xmath8 , where the gap closes , @xmath105 , the van - hove singularity disappears and eq . ( [ dos_plasmons ] ) becomes @xmath106 as plasmons have become soft acoustic modes . mean - field theory is valid as long as fluctuations are small . close to the critical flux  @xmath8 , plasmon fluctuations become significant as the modes at @xmath107 soften . introducing @xmath108 , we thus write an effective theory for the `` slow '' field  @xmath109 , keeping only the quadratic leading order in `` discrete gradients ''  @xmath110 so that @xmath111 $ ] . the euclidean action then reads @xmath112 \big\ } \label{action}\end{aligned}\ ] ] with  @xmath113 at zero temperature . outside a ginzburg region close to the critical field  @xmath8 , one can treat the action  ( [ action ] ) in the saddle - point approximation and the mean - field results from the preceding section become accurate .    in order to determine the ginzburg region , we expand the cosine - potential to fourth order in  @xmath109 and then employ a continuum approximation , @xmath114 and @xmath115 . rescaling coordinates and fields so that they become dimensionless and so the prefactors of the quadratic terms are @xmath116 , we find that close to  @xmath8 the prefactor of the @xmath117 term is small if @xmath118 for external fluxes  @xmath6 satisfying eq . ( [ ginzburg ] ) , the mean - field results of the preceding sections are valid .    very close to  @xmath8 , this ginzburg criterion breaks down as quantum fluctuations become strong . the quantum critical behavior , which is due to the non - linearity of the josephson current - phase relationship , leads to an excitation spectrum that is considerably different from mean - field theory and should correspond to a @xmath119-dimensional quantum ising chain.@xcite as a result , e.g. , the gap  @xmath120 in the plasmon spectrum , cf . ( [ plasmon - gap ] ) , is expected to close at  @xmath8 as @xmath121 i.e. with critical exponent  @xmath122 instead of  @xmath123 in the mean - field prediction  ( [ plasmon - gap ] ) . furthermore , the @xmath124-term in the action  ( [ action ] ) will become renormalized , effectively shifting the critical flux  @xmath8 to a higher value inside the ginzburg region . at criticality , @xmath92 , the system is a liquid with spectrum  @xmath125 and plasmon density of states of the form of eq . ( [ critical_f ] ) , but in the fluctuation regime , the renormalized sound velocity  @xmath126 has to be considered an effective phenomenological parameter . while parameters entering the observable quantities  @xmath127 and  @xmath128 , eqs . ( [ fc_plasmons ] ) and  ( [ fl_plasmons ] ) , will be effective ones , the qualitative threshold behavior should still be described in terms of eqs . ( [ dos_vh ] ) and  ( [ critical_f ] ) . observability of quantum critical behavior requires the system size  @xmath129 to be larger than the correlation length at the boundaries of the quantum critical region as given by eq . ( [ ginzburg ] ) . this leads to the condition @xmath130 for typical parameters@xcite for @xmath2 and @xmath131 ( and @xmath132 in the type - i regime ) , the right - hand side of this estimate is of order unity . here we consider the limit @xmath11 , corresponding to large inductances @xmath19 . this regime is realizable with superinductors as demonstrated in fluxonium qubits.@xcite the most interesting physical effects are due to the proliferation of _ kinks _ in the node phases  @xmath13 corresponding to current vortices as shown in fig .  [ fig_vortices](b ) . these vortices pick up currents over many plaquettes of the order of  @xmath9 and therefore are stable already at small external fluxes  @xmath6 . thus , phase transitions associated with kink proliferation preempt the instability driven by fluctuations of plasmons , which remain gapped for all external magnetic fluxes  @xmath6 . as a result , the ground state and excitations are fundamentally different from the type - i regime . we begin with the study of the phase diagram and the absorption spectrum in the `` classical '' limit . by `` classical '' , we mean that the capacitances are large enough ( and hence @xmath131 is small enough ) to make effects due to spontaneous quantum phase slips negligible but still allow for induced phase slips by microwave absorption . quantum effects due to finite capacitances alter the classical picture and excitation spectrum and will be investigated in sec . [ sec : quantum ] . the classical phase diagram in fig . [ fig01](b ) is obtained by finding the configurations  @xmath21 of node fluxes that minimize the potential  @xmath34 $ ] , eq . ( [ x_potential ] ) , cf .  refs .  . they are found from solving the set of equations given by @xmath133\ . \nonumber\end{aligned}\ ] ] summation over  @xmath33 of eq . ( [ dv ] ) yields the constraint  @xmath41 , which corresponds to zero net current to ground .    at zero external flux  @xmath6 , the ground state of the system is given by the homogeneous configuration  @xmath72 , which remains a _ local _ minimum of @xmath34 $ ] for non - zero  @xmath6 as long as @xmath134 . in the limit of large  @xmath17 , ( [ dv ] ) admits non - trivial solutions already at small  @xmath135 that contain _ kinks_,@xcite static local configurations of  @xmath13 with a jump of the order of  @xmath12 across one link . in order to study such soliton solutions , we employ villain s approximation@xcite , in which we expand  @xmath136 with integer  @xmath137 such that  @xmath138 is small .    for a _ single _ kink , with one jump of @xmath139 over the link between sites  @xmath140 and  @xmath141 and @xmath142 for all @xmath143 , villain s approach yields the configuration @xmath144 assuming that  @xmath145 . from eq . ( [ x_singlekink ] ) , we see that  @xmath9 , eq . ( [ x_ell ] ) , determines the scale of the width of a kink . inserting typical fluxonium values , this width is  @xmath146 , @xcite which sets the smallest system size necessary to observe the physics under discussion .    as the external flux  @xmath6 is increased , we may expect the kink solution  @xmath147 , eq . ( [ x_singlekink ] ) , to become more favorable than the homogeneous configuration  @xmath72 because of the josephson junction s preference for finite flux gradients at finite  @xmath6 , cf . ( [ x_potential ] ) . in fact , for the difference  @xmath148-v[0]$ ] between the potential energies of the single kink and homogeneous configuration we find @xmath149 where @xmath150 is the critical external flux in the type - ii regime , @xmath151 , cf . [ fig01](b ) . at flux @xmath8 , configurations  @xmath21 with kinks become energetically favorable over the homogeneous configuration . the critical flux  @xmath8 is analogous to the critical magnetic field  @xmath152 in type - ii superconductors , when vortices begin proliferating.@xcite its smallness in  @xmath153 reflects that the vortex is able to pick up currents over the ( large ) length scale  @xmath9 , cf . [ fig_vortices](b ) .    for @xmath76 , the density of kinks in the ground state grows continuously as a function of  @xmath6 . each kink , when nucleating individually , brings an energy gain of @xmath154 . on the other hand , this individual gain has to be balanced with the interaction energy between two kinks . for the repulsive potential for two kinks at links @xmath155 and @xmath33 we find @xmath156 which decays only at large distances on the scale of  @xmath9 , eq . ( [ x_ell ] ) .    the competition between kink - kink repulsion @xmath157 , eq . ( [ x_j ] ) , and the single - kink energy  @xmath158 , eq . ( [ delta ] ) , which plays the role of a chemical potential , completely determines the ground state as a function of the external flux  @xmath6 at @xmath17 . this interplay may be effectively described in terms of a classical spin chain model . specifically , we associate with each josephson link  @xmath33 a pseudo - spin that distinguishes whether there is a kink across it ( `` spin - up '' ) or not ( `` spin - down '' ) , see fig . [ fig04 ] . the effective spin hamiltonian then has the form @xmath159 where @xmath160 and we introduce pauli matrices  @xmath161 , @xmath162 , and @xmath163 .     configuration for a commensurate kink density  @xmath164 at @xmath165 . ( b ) the same state represented in the effective spin ladder model for the coupled josephson links . links with upward orientied spin feature a kink in  @xmath13 . ] the ground states of an ising chain with infinite - range convex interaction potentials were systematically studied as a function of the `` chemical potential ''  @xmath158 by bak and bruinsma@xcite and aubry@xcite . in the language of our model , they showed that in the limit of very large systems @xmath166 , the kink density  @xmath167 depends on  @xmath6 in the form of a devil s staircase:@xcite this means that @xmath168 only takes values in the rational numbers and rises monotonically and continuously with  @xmath6 such that for each rational  @xmath169 , there is a finite interval in  @xmath6 in which @xmath170 . for our effective model  ( [ x_clhamiltonian ] ) , this interval has the width @xmath171 figure  [ fig02](a ) illustrates the devil s staircase for fixed  @xmath9 , showing as a function of  @xmath6 the kink density up to commensurability order  @xmath172 . a ground state configuration  @xmath21 with kink density  @xmath173 is periodic,@xcite @xmath174 . for example the ground state configuration for  @xmath175 is @xmath176 , with `` primitive cell '' @xmath177 . for the non - unit fraction @xmath178 , a primitive cell contains a non - trivial basis : @xmath179 .     for the classical ground state as a function of the external flux  @xmath6 ( for @xmath180 fixed ) . the plot shows all densities with periodicity  @xmath181 ; higher - order commensurate phases are hidden behind the vertical dashed lines . the dashed curve is obtained in the continuum approximation  ( [ x_kinkdensity ] ) . ( b ) classical excitation levels by adding ( red , falling ) or removing ( blue , rising ) a kink . for fixed  @xmath6 , the lines mark the frequencies at which the spectral function exhibits a @xmath182-peak . the upper ( lower ) dashed curve shows the minimal energy for adding ( removing ) a kink in the continuum approximation , cf . ( [ minenergy ] ) and  ( [ e_annihilation ] ) . for excitations close to  @xmath8 , see fig . [ fig03 ] . ] figure  [ fig02](a ) also illustrates that close to  @xmath8 , the dependence of the kink density on  @xmath6 appears smooth so that we may try to effectively describe it using a continuum approximation . physically such a description seems reasonable because close to  @xmath8 , kinks are sparse such that the large distances between neighboring kinks `` wash out '' the discrete structure of the underlying lattice . thus , close to @xmath8 , the ground state density  @xmath168 of the classical model  ( [ x_clhamiltonian ] ) is related to the external flux  @xmath6 by @xmath183 } \big]\ . \label{x_kinkdensity}\end{aligned}\ ] ] solving this relation for  @xmath168 leads to the dashed curve in fig  [ fig02](a ) . for @xmath184 , eq .  ( [ x_kinkdensity ] ) would predict the relation  @xmath185 , which in particular implies @xmath74 at half flux quantum , @xmath186 , corresponding to the staggered order . however , discreteness effects are visibly superposed on the continuum model , mostly due to large intervals in @xmath6 that hold the same kink density , cf .  eq . ( [ x_width ] ) .    very close to  @xmath8 such that @xmath187\gg 1 $ ] , we explicitly find the dependence @xmath188 \right\}^{-1}\ , \label{x_kinkdensity_close}\end{aligned}\ ] ] which in particular shows that at @xmath85 the kink density grows with infinite slope . the transitions between the various commensurate phases of the model  ( [ x_clhamiltonian ] ) are first order.@xcite at zero temperature , quantum fluctuations due to the capacitative term in eq . ( [ x_lagrangian ] ) facilitate equilibration and prevent hysteresis effects if @xmath6 is varied sufficiently slowly in an experiment . realistic capacitive energies  @xmath189 lead to extended phases of incommensurate order , phases of `` floating primitive cells '' , around the transitions which we will discuss in sec . [ sec : quantum ] . in the regime of kink proliferation , the elementary excitations by photon absorption are the creation of an additional kink or the annihilation of a kink already in existence in the ground state . the typical energy scale associated with such excitations is given by  @xmath190 . this scale could possibly already be below the plasmon gap @xmath191 if @xmath192 . at the critical flux  @xmath8 , eq . ( [ x_phiec ] ) , when the first kinks nucleate in the ground state , the spectrum for kink excitations extends to zero energy . here , and in the vicinity of  @xmath8 , these are therefore the relevant low - energy excitations . on either side of  @xmath92 , the energies correspond to isolated low - frequency absorption peaks , see fig . [ fig03 ] .    in this section , we discuss the `` classical '' absorption spectrum , i.e.  we neglect spontaneous creation / annihilation of kinks due to quantum phase slips and only allow for such processes in the context of photon absorption . we discuss how quantum effects alter this picture in sec . [ sec : quantum ] .      in a setup with a single _ capacitively - coupled _ antenna at link  @xmath43 [ fig . [ fig_antennas](a ) ] , the system may be excited by _ locally _ introducing a kink at the link  @xmath43 of the antenna . in a single absorption act associated with kink creation or annihilation at link @xmath43 , the system responds by boosting the phase difference  @xmath193 over the @xmath43-th josephson junction from @xmath194 to  @xmath12 or vice versa , whereas phase differences at other links acquire negligible changes  @xmath195 . for this reason , we may estimate the matrix element in the absorption rate  @xmath127 , eq . ( [ absorptionrate ] ) , in the limit of a single fluxonium qubit@xcite , where the energy difference  @xmath196 between the excited and the ground state is determined by the external flux  @xmath6 and the effective potential a kink at  @xmath43 feels in the presence of kinks at other sites . the matrix element entering eq . ( [ absorptionrate ] ) is then given by @xmath197 herein , the parameter @xmath198 with dimensions of energy , is the amplitude of a quantum phase slip.@xcite in determining matrix elements and absorption rates , we assume the typical limit @xmath199 and @xmath200 for microwave frequencies  @xmath50 .    as a result , we obtain for the absorption rate in the case of capacitive coupling @xmath201 with @xmath202 . it features a single peak at frequency  @xmath196 , which corresponds to the energy cost for adding or removing a kink from the ground state at link  @xmath43 the spectrum of energies  @xmath203 is determined in secs . [ sec : addition ] and  [ sec : annihilation ] , see also fig . [ fig02](b ) . the _ inductive coupling _ setup [ fig . [ fig_antennas](b ) ] , as discussed in sec . [ sec : inductive ] , couples the single antenna to links over the long range of  @xmath62 . for realistic situations , we may assume that @xmath62 is much larger than the length  @xmath9 , the relevant scale for the statics of kinks . this allows the insertion of kinks at any link over a long range . if the antenna is situated at  @xmath43 , the absorption rate  @xmath128 , eq . ( [ absorptionrate_l ] ) , becomes @xmath204 where  @xmath205 , as the matrix element in eq . ( [ absorptionrate_l ] ) is given by latexmath:[\ ] ] close to  @xmath8 , and @xmath279 for @xmath280 . if this inequality is not satisfied , especially close to  @xmath8 , effects of the then strong effective hopping are more drastic and the single peak for kink annihilation fades away into the spectrum of a strongly - correlated quantum liquid , cf . the next section . away from  @xmath8 , however , the peak should remain intact , owing to the exponential smallness of  @xmath253 , eq . ( [ eq : x_phase_slip_amplitude ] ) .       for fixed  @xmath281 . quantum phase slips , in this illustration with amplitude @xmath282 , melt all commensurate kink densities  @xmath173 with @xmath283 , leading to extended incommensurate luttinger liquid ( ll ) phases , for which the spectral function has the form of eq . ( [ llspectrum ] ) . for general  @xmath253 , eq . ( [ cic ] ) determines the maximum  @xmath218 such that classical commensurate phases  @xmath173 survive the quantum fluctuations . ] quantum ising models beyond nearest neighbors , i.e. hamiltonians of the form of eq . ( [ x_hamiltonian ] ) , have recently @xcite been studied theoretically for power - law interactions , e.g. , in the context of rydberg atoms @xcite and experimentally using trapped ions.@xcite applying the theoretical results to our model  ( [ x_hamiltonian ] ) , we predict the phase diagram as a function of external flux  @xmath6 for fixed  @xmath17 and  @xmath253 to be the one shown in fig . [ fig_quantum ] .    below a critical flux  @xmath8 , the system is , as in the classical limit , in a homogeneous phase with gapped excitation spectrum , cf . [ sec : broadenedpeaks ] , but then undergoes a kosterlitz - thouless ( kt ) transition into a gapless phase of `` floating '' kinks . this phase is a luttinger liquid and the excitation spectrum for adding or removing kinks is analogous to the spectrum of adding or removing spinless fermions in a one - dimensional system with strong repulsion . upon further increasing  @xmath6 , the system undergoes a commensurate - incommensurate ( cic ) transition into a `` pinned '' phase featuring a rational kink density  @xmath173 and a gapped excitation spectrum as in fig . [ fig02](b ) . quantum effects broaden the `` classical '' peaks according to eq . ( [ broadenedpeak ] ) . after another cic transition , the system becomes again a luttinger liquid . an ( even ) number of cic transitions ( depending on  @xmath253 ) may follow before the system undergoes a second kt transition from a liquid into another gapped homogeneous phase , which in a final quantum ising ( qi ) transition turns into the staggered order corresponding to kink density  @xmath74 . let us have a closer look into the phases that feature a finite kink density . for this purpose , let us consider a classical ground state with kink density  @xmath173 and estimate its stability to quantum fluctuations . whereas the adjacent ground state for increasing ( decreasing ) @xmath6 has only one extra ( fewer ) kink , practically all the kinks need to be rearranged in order to minimize the potential energy.@xcite . we can interpret this state as the original state of period  @xmath218 with the addition of @xmath218 defects , in analogy to the two domain walls introduced in an antiferromagnet by flipping one spin.@xcite for the kink density  @xmath74 , this analogy is perfect . the quantum @xmath284 term introduces effective hopping of the defects by virtual quasi - simultaneous adding and removing of kinks , similar to the single - kink hopping discussed in sec . [ sec : broadenedpeaks ] . the classical energies of the virtually occupied intermediate states are of order @xmath285 , cf . [ fig02](b ) , giving a hopping amplitude @xmath286 . the kinetic energy gain from such hopping of @xmath218 defects , which is @xmath287 , then lowers the energy of the states with defects relative to the original period-@xmath218 ground state and , as a result , takes away a strip of width  @xmath288 from both ends of the interval the period-@xmath218 ground state has classically occupied in  @xmath6 . in fig . [ fig_quantum ] , e.g. , the @xmath289 phase had to cede almost one half of its classical interval in  @xmath6 , cf . ( [ x_width ] ) , to the strips to its left and its right .    in these strips , because of the mobility of the kinks , the commensurate configuration has melted and been replaced by an incommensurate state of floating `` defective '' primitive cells . this state is gapless and physically described by a luttinger liquid . creation of a kink at a link  @xmath33 is here analogous to injecting a spinless electron at a site  @xmath290 into a ( strongly - coupled ) quantum wire.@xcite therefore , using the well - known luttinger liquid results , we find for absorption at low energies the spectral function @xmath291 within the incommensurate phase , the luttinger parameter  @xmath292 varies@xcite as a function of  @xmath6 . at the cic transition involving a commensurate kink density  @xmath173 , it takes the value  @xmath293 , cf .  refs .  . if the width of the incommensurate strip exceeds the width of the interval of the classically commensurate state , i.e. if @xmath294 the classical state with period  @xmath218 is entirely unstable and merges into an incommensurate phase of floating defects between two commensurate phases of shorter period  @xmath218 . the estimate  ( [ cic ] ) shows explicitly that classical ground states of kink densities with smaller denominator  @xmath218 are more immune to quantum fluctuations . in fact , the classical phase  @xmath175 , the `` most immune '' pinned phase , would be destroyed entirely only if @xmath253 were so large that it would also destabilize the luttinger liquid into a gapped homogeneous quantum paramagnet.@xcite based on typical fluxonium parameters @xcite , we predict that classical phases of pinned densities  @xmath173 with  @xmath295 survive also in the presence the quantum fluctuations and systems of size larger than  @xmath296 would be needed to observe signatures of luttinger liquid physics such as an absorption spectrum according to eq . ( [ llspectrum ] ) .    upon decreasing of the external flux to a critical value  @xmath8 , the lowest-@xmath6 incommensurate phase destabilizes into a homogeneous phase in a kt transition@xcite . we note that the value of the critical flux  @xmath8 , because of quantum fluctuations , will be slightly larger than the classical value from eq . ( [ x_phiec ] ) . close to this transition , theory @xcite predicts a critical  @xmath297 . as the external flux  @xmath6 approaches half a flux quantum , the system undergoes a similar kt transition from the rightmost ( see fig . [ fig_quantum ] ) incommensurate phase ( with the same  @xmath298 ) into a homogeneous quantum paramagnet . eventually , there is a quantum ising transition@xcite to the antiferromagnet - like @xmath299 phase in the vicinity of @xmath186 . the model circuit , fig .  [ fig01](a ) , we have discussed has a surprisingly rich equilibrium phase diagram , cf .  figs .  [ fig01](b ) and  [ fig_quantum ] , despite its relatively simple structure . this arises due to the combination of the nonlinear properties of josephson junctions and the long - range interactions introduced by the coupling to a common ground . as a function of the circuit parameters @xmath300 , @xmath5 , and the external field @xmath6 , the model exhibits equilibrium phase transitions of the kosterlitz - thouless , commensurate - incommensurate , and ising classes . we have shown that circuit qed realizations of this model enable access to low - energy excitations . in particular , they allow to identify the quantum phase transitions in linear response by the absorption of microwaves using a capacitively ( c ) or inductively ( l ) coupled antenna . here we summarize our predictions for the absorption rate  @xmath301 in the various characteristic regimes of parameters , particularly @xmath9 and @xmath6 .      in this limit , the elementary excitations of the system are plasmons . the absorption rates  @xmath69 and  @xmath70 are given by eq . ( [ fc_plasmons ] ) and  ( [ fl_plasmons ] ) , respectively , with the density of states of plasmon excitations given by eq . ( [ dos_plasmons ] ) . at the critical external flux  @xmath92 , classically given by eq . ( [ phiec_ising ] ) , the spectrum is gapless with a uniform density of states [ eq . ( [ critical_f ] ) ] . in the proximity of  @xmath8 , where the ginzburg criterion  ( [ ginzburg ] ) is violated and quantum fluctuations are strong , the gap in the plasmon spectrum grows linearly as a function of the distance to  @xmath8 , [ eq .  ( [ gap_linear ] ) ] , while outside the ginzburg region , the classical square root dependence on the external flux [ eqs . ( [ plasmon - gap ] ) and  ( [ plasmon - gap - ordered ] ) ] sets in .      this regime , with @xmath302 , is closer to the the parameters realized in fluxonium qubits.@xcite the ground state and excitation spectrum at large flux , @xmath76 with  @xmath8 given by eq . ( [ x_phiec ] ) , are significantly different from the small - flux regime , @xmath78 , where the ground state is homogeneous . the elementary excitations are associated with the addition or removal of localized kinks ( or vortices ) . the absorption spectrum is different for the capacitive [ eq .  ( [ fckink ] ) ] or inductive [ eq .  ( [ fl ] ) ] coupling of the antenna , the former adding or removing kinks locally , the latter over an extended range . in the realistic quantum picture , the lowest excited states in the regime @xmath78 appear above a gap of order  @xmath303 and form a very flat quantum band [ eq . ( [ broadenedpeak ] ) ] . the band slightly broadens the classical peak in the absorption rate [ eq . ( [ spectrumleft ] ) and fig . [ fig03](b.1 ) ] . as @xmath85 , the gap closes and , upon undergoing a kosterlitz - thouless ( kt ) transition at @xmath8 , the system enters the phase of a `` floating '' crystal of kinks . the excitation spectrum here at @xmath76 is of luttinger - liquid type and at low energies is given by eq . ( [ llspectrum ] ) .    as @xmath6 is increased further , depending on the strength of quantum fluctuations ( phase slips ) , the system undergoes a commensurate - incommensurate ( cic ) transition , after which the ground state carries a pinned classical kink density . here , the excitation spectrum is essentially the classical spectrum of figs . [ fig02](b ) and [ fig03](b.2 ) , where quantum fluctuations slightly broaden the @xmath182-peaks according to eq . ( [ bandwidth_right ] ) . the classical spectral function for kink addition is given by eq . ( [ rho_continuum_close ] ) , for  @xmath6 not too far from  @xmath8 , and in general by eq . ( [ rho_discrete ] ) . for kink annihilation , the spectral function is displayed in eq . ( [ rho_annihilation ] ) . for kink densities  @xmath173 with @xmath304 or @xmath305 , which dominate the phase diagram , there is only a single peak in the absorption spectrum related to kink annihilation , which is the lowest excitation energy of the system , cf . ( [ e_annih ] ) and figs .  [ fig02](b ) and [ fig03](b.2 ) .        in our theoretical analysis , we built on the availability of a `` superinductance '' such as in fluxonium qubits@xcite when envisioning experimental realizations . as a result , the type - ii regime of small @xmath3 could be reached without considering the effects of additional parasitic capacitances to ground that would be unavoidable in a realization using ordinary electromagnetic inductance . such additional parasitic capacitances would suppress quantum fluctuations and thus enhance the classical behavior of the system . in the type - i regime , it decreases the effective @xmath131 for the low energy plasmon modes , narrowing the quantum critical region for the ising transition . in the type - ii regime , capacitance to ground decreases the quantum phase slip rate @xmath253,@xcite which helps to stabilize the classical pinned phases .    throughout , we have used periodic boundary conditions for theoretical convenience . realistically , it is easier to create an array with open boundary conditions and so for small system sizes there will be edge effects . these will extend over a length @xmath306 , eq . ( [ x_ell ] ) , as this is the scale for interactions in the system . yet for large systems whose size significantly exceeds at least the length  @xmath9 , our results should be directly applicable . we have also assumed the temperature is zero . both finite temperature and the finite size of the system prevent the formation of long - range order and in principle mean that there will not be a true phase transition . however , the equilibrium state and spectrum will retain signatures of the infinite system behavior on short enough length and timescales . we note that there is an intermediate regime @xmath307 , where the classical model has an incomplete staircase of first order transitions , cf . [ fig01](b ) and refs .  . in this parameter region , the type - ii regime commensurate and incommensurate phases of finite kink density for  @xmath76 turn into the single ising phase of the type - i regime . the classical theory@xcite predicts a sequence of `` superdegenerate''@xcite and multicritical points as the transition at  @xmath8 changes from first order at large  @xmath9 to second order in the type - i regime . the construction of an effective model for this region that also allows one to analytically study the effect of quantum fluctuations appears difficult . however , the study of the ground states and excitations may be amenable to numerical techniques such as quantum monte carlo as the problem is bosonic .    in the type - ii regime at large  @xmath9 , the low - energy behavior is well described in terms of localized kinks . using the locally - coupled capacitative antenna configuration , kinks can be selectively introduced or removed on individual sites . if multiple kinks are introduced , they will interact over the large length scale @xmath9 according the hamiltonian  ( [ x_clhamiltonian ] ) . this introduces the possibility of investigating the many - body non - equilibrium physics of the system . in the presence of dissipation , it may be possible to create a model system with a driven - dissipative steady - state of interacting kinks , an interesting addition to the set of non - equilbrium many body models that can be simulated with circuit qed systems.@xcite    we would like to thank g. rastelli , i. pop , m. h. devoret , j. keeling , yu chen , c. neill , s. hacohen - gourgy , and v. ramasesh for interesting and helpful discussions . this work was supported by nsf dmr1301798 and dmr1206612 , and aro w911nf1410011 . h. m. and r. t. b. acknowledge the yale prize postdoctoral fellowship . 99 o. daldini , p. martinoli , j. l. olsen , and g. berner , _ vortex - line pinning by thickness modulation of superconducting films _ , phys . . lett . * 32 * , 218 ( 1974 ) ; p. martinoli , o. daldini , c. leemann , and b. van den brandt , _ josephson oscillation of a moving vortex lattice _ , _ ibid . _ * 36*,382 ( 1976 ) ; o. daldini , c. leemann , and p. martinoli , _ coherent flux flow in superconducting films with periodically modulated thickness _ , solid state comm . * 16 * , 509 ( 1975 ) ; p. martinoli , o. daldini , c. leemann , and e. stocker , _ a.c . quantum interference in superconducting films with periodically modulated thickness _ , _ ibid . _ * 17 * , 205 ( 1975 ) . v. l. pokrovsky and a. l. talapov , _ ground state , spectrum , and phase diagram of two - dimensional incommensurate crystals _ , * 42 * , 65 ( 1979 ) and _ the theory of two - dimensional incommensurate crystals _ , jetp * 51 * , 134 ( 1980 ) . d. r. nelson and v. m. vinokur , _ boson localization and pinning by correlated disorder in high - temperature superconductors _ , phys . lett . * 68 * , 2398 ( 1992 ) and _ boson localization and correlated pinning of superconducting vortex arrays _ b * 48 * , 13060 ( 1993 ) . a. van oudenaarden and j. e. mooij , _ one - dimensional mott insulator formed by quantum vortices in josephson junction arrays _ , * 76 * , 4947 ( 1996 ) ; a. van oudenaarden , b. van leeuwen , m. p. m. robbens , and j. e. mooij , _ one - dimensional mott localization of quantum vortices in josephson - junction arrays _ , phys b * 57 * , 11684 ( 1998 ) . d. b. haviland , k. andersson , and p. gren , _ superconducting and insulating behavior in one - dimensional josephson junction arrays _ , j. low temp . phys . * 118 * , 733749 , ( 2000 ) ; k. andersson and d. b. haviland , _ escape from a zero - current state in a one - dimensional array of josephson junctions _ , phys . b * 67 * , 092507 ( 2003 ) . a. blais , r. s. huang , a. wallraff , s. m. girvin , and r. j. schoelkopf , _ cavity quantum electrodynamics for superconducting electrical circuits : an architecture for quantum computation _ , a * 69 * , 062320 ( 2004 ) . n. a. masluk , i. m. pop , a. kamal , z. k. minev , and m. h. devoret , _ microwave characterization of josephson junction arrays : implementing a low loss superinductance _ , lett . * 109 * , 137002 ( 2012 ) ; m. t. bell , i. a. sadovskyy , l. b. ioffe , a. yu . kitaev , and m. e. gershenson _ quantum superinductor with tunable nonlinearity _ , _ ibid . _ * 109 * , 137003 ( 2012 ) . m. marchand , k. hood , and a. caill , _ nonconvex interactions and the occurrence of modulated phases _ , phys . lett . * 58 * , 1660 ( 1987 ) and _ nonconvex interactions : a mechanism for the occurrence of modulated order in condensed matter _ , phys . b * 37 * , 1898 ( 1988 ) .                in order to avoid any confusion , we point out that although our interaction  @xmath157 , eq .  ( [ x_j ] ) , falls exponentially and so is certainly short - ranged in the sense of scaling , it is nevertheless non - zero at an arbitrarily large distance @xmath311 . this implies the theoretical completeness of the devil s staircase in our model , even though the `` stairs '' for kink densities @xmath173 with large  @xmath218 occupy exponentially small intervals in  @xmath6 , cf . ( [ x_width ] ) . p. richerme , c. senko , s. korenblit , j. smith , a. lee , r. islam , w. c. campbell , and c. monroe , _ quantum catalysis of magnetic phase transitions in a quantum simulator _ , phys . * 111 * , 100506 ( 2013 ) .
we evaluate the microwave admittance of a one - dimensional chain of fluxonium qubits coupled by shared inductors . despite its simplicity 
 , this system exhibits a rich phase diagram . 
 a critical applied magnetic flux separates a homogeneous ground state from a phase with a ground state exhibiting inhomogeneous persistent currents . depending on the parameters of the array 
 , the phase transition may be a conventional continuous one , or of a commensurate - incommensurate nature . 
 furthermore , quantum fluctuations affect the transition and possibly lead to the presence of gapless `` floating phases '' . 
 the signatures of the soft modes accompanying the transitions appear as a characteristic frequency dependence of the dissipative part of admittance .