Lattice modulation spectroscopy of one-dimensional quantum gases: Universal scaling of the absorbed energy

Lattice modulation spectroscopy is a powerful tool for probing low-energy excitations of interacting many-body systems. By means of bosonization we analyze the absorbed power in a one-dimensional interacting quantum gas of bosons or fermions, subjected to a periodic drive of the optical lattice. For these Tomonaga-Luttinger liquids we ﬁnd a universal ω 3 scaling of the absorbed power, which at very low-frequency turns into an ω 2 scaling when scattering processes at the boundary of the system are taken into account. We conﬁrm this behavior numerically by simulations based on time-dependent matrix product states. Furthermore, in the presence of impurities, the theory predicts an ω 2 bulk scaling. While typical response functions of Tomonaga-Luttinger liquids are characterized by exponents that depend on the interaction strength, modulation spectroscopy of cold atoms leads to a universal power-law exponent of the absorbed power. Our ﬁndings can be readily demonstrated in ultracold


I. INTRODUCTION
Cold atomic systems offer an unprecedented level of control on the properties of interacting quantum systems [1,2], and allowed for the realization of a plethora of novel phases and phenomena that were previously inaccessible in other experiments.They have given access to the "experimental solution" of certain models, and hence can be referred to as quantum simulators.Among those phenomena, one paradigmatic example for bosons on a lattice is the transition between a superfluid and Mott insulator state.Such a transition was successfully observed in three-dimension (3D) [3], 2D [4], and in 1D [5,6].For the latter, the transition is found to be in the universality class of the Berezinskii-Kosterlitz-Touless transitions.Cold atoms have thus provided a remarkable way for testing the universal properties of such models.
In order to analyze quantitatively the properties of the correlated phases and the transitions between them, it is important to develop a detailed understanding of different experimental probes.Among them is lattice modulation spectroscopy [7].This technique consists of modulating the amplitude [8][9][10] or the phase [11,12] of the optical lattice in which the atoms are trapped.After some time, the energy deposited in the system or the number of doubly occupied states are measured to characterize the underlying state [13,14].
For bosons this probe can determine energy gaps and thus to locate the Mott-to-superfluid transition [5][6][7].Moreover, specific modes of the superfluid could be excited, such as the Higgs mode of the superfluid [4] which is expected to occur in 3D and 2D superfluids.
Despite this effort, several questions remain to be investi-gated for bosons and fermions in 1D.For these systems, a symmetry-broken state cannot exist because of strong phase fluctuations even at zero temperature [15].Hence, only quasilong range order can exist as characterized by a powerlaw decay of the certain correlation functions.This result is part of the more general properties of Tomonaga-Luttinger liquids (TLL) that are to be expected to describe most of the interacting 1D quantum problems [15,16].Given the absence of true long-range superfluid order one may wonder whether the response to shaking, in a one dimensional bosonic system would also show traces of a Higgs mode as it did in higher dimensions [4].More generally this prompts for an analysis of the response to shaking of a one-dimensional TLL.
In the present paper we perform such an analysis, both in the gapless (superfluid) and gapped phase (Mott insulator).Using a combination of field theoretical and numerical Matrix Product State (MPS) techniques we obtain the response of the system to the shaking of the optical lattice.We find that this response is a powerlaw of the shaking frequency, with a universal exponent.This is quite remarkable in view of the fact that the TLL is normally characterized by responses which show nonuniversal powerlaw behavior, with exponents depending on the interaction strength.The choice of modulating the amplitude of the optical lattice is important.Would one modulate the phase instead, the conductivity is obtained [11] (as periodic phase modulation translates to a periodic force), which leads to non-universal exponents.
Our results also show that a well formed Higgs mode does not exist in one-dimension, as the response to shaking is monotonically increasing up to shaking frequencies of the order of the bandwidth of the system.In order to connect to experiments, we also analyze and discuss the effect of this response to open boundary conditions, as realized for example with a box potential, which constitute a relevant perturbation at low frequencies.In the presence of impurities similar effects are observed, provided that the concentration of impurities is small and one can simply add energy absorption due to different impurities.
While the focus of this paper is on one dimensional systems, we note that a similar approach can be used to describe modulation experiments in higher dimensional systems that allow for a hydrodynamic description.Time-periodic modulation of the interaction strength or the transverse confinement potential will result in the resonant parametric generation of excitation pairs [17][18][19][20], analogous to the 1D case discussed in this paper.The analysis of the absorbed energy can be done following the same approach that we discuss here, with the main difference being the phase space for collective modes.In the summary section we comment on the relevance of our analysis beyond ensembles of cold atoms and suggest possible applications of our results to chains of Josephson junctions and pump and probe experiments in electron systems.
The paper is organized as follows.In Sec.II, we introduce the models that we are investigating.In Sec.III we discuss the analytical calculation of the absorbed power of a one-dimensional gas subject to a lattice modulation within a Tomonaga-Luttinger liquid treatment and compare the lowenergy behavior with results obtained from time-dependent Matrix Product States.In Sec.IV we discuss the edge effects treated through an effective boundary potential that couples to the density of the fluid and also analyze the effect of a single impurity in the bulk of the system.In Sec.V, we consider modulation spectroscopy in the case of gapped systems such as they would occur for the Bose-and the Fermi-Hubbard model with repulsive interactions and commensurate filling.In Sec.VI, we present our summary and discuss our findings.

II. MODELS
We consider fermionic or bosonic ultracold atoms confined to a 1D tube subjected of a deep lattice potential.For deep enough potentials, such a system can be described in a tightbinding approximation by a Hubbard model [21].This leads to the Bose-Hubbard model for spinless bosons [1] H where b † l (b l ) creates (annihilates) a particle on site l and n l = b † l b l counts the particles on site l, and the Fermi-Hubbard model [2] for spin-1/2 fermions: where again c † l,σ and c l,σ are the creation and annihilation operators, respectively.
In modulation spectroscopy, the trapped atoms are probed by modulating the strength of the longitudinal periodic potential.The modulation lowers or raises the potential barrier between two consecutive minima, and thus to leading order modifies the strength of the tunneling J 0 , as well as the interaction within one well.Modulation of J is expected to be much larger, since it depends exponentially on the barrier height.We note that modulation of the Hamiltonian as a whole does not lead to energy absorption and what is important is the difference in the relative modulation strength of the two terms in the Hamiltonian.Therefore, we consider the time-dependent Hamiltonian, in which only the tunneling amplitude is modulated J 0 + δJ(t), giving rise to with ν ∈ b, f and: for bosons and for fermions.The label b, f refers to fermions and bosons, respectively.We work in the linear response limit, with δJ(t) = δJ cos(ωt) and δJ J 0 , U .In linear response, the absorbed power is given by [22] where: is the response function.In Sec.III, we calculate this response function in a low-energy/long wavelength limit.

III. BOSONIZATION
In the low energy/long wavelength limit, interacting bosons and fermions can be described within an effective continuum theory called the Tomonaga-Luttinger liquid [15,16,[23][24][25][26][27][28][29][30][31].The low energy excitations are phononic collective excitations with a linear dispersion, that describe density fluctuations (and when applicable spin fluctuations).In general, spin and density fluctuations propagate with different velocities, a phenomenon known as spin-charge separation [32,33].In that low energy limit, the original particles appear as coherent states of the collective modes.As a result, all the observables of the original system are expressible in terms of the collective modes.We will thus use in this section the bosonized representation to calculate the response function (7) and hence the resulting absorbed power.

A. Bosonized representation for fermions
In the fermionic case, away from commensurate filling, Hamiltonian (2) has the bosonized representation [15,27]: where u ρ , u σ are respectively the density and spin velocity, K ρ , K σ the density and spin Tomonaga-Luttinger exponents, and [φ ν (x), Π ν (x )] = iδ(x − x )δ ν,ν .For repulsive interactions, H σ is renormalized to a fixed point Hamiltonian H * σ with K * σ = 1 and g * 1⊥ = 0, yielding gapless excitations with linear dispersion ω = u σ |k|.For attractive interactions without external magnetic field the spin Hamiltonian H σ is gapped while the density Hamiltonian H ρ remains gapless [15].At half-filling, Umklapp processes are present [15].They contribute to the bosonized Hamiltonian (8) a term but, as shown in App.A, when Umklapp processes are irrelevant in the renormalization group sense [34], they add only subdominant contribution to the absorbed power at low frequency.When the Umklapp processes are relevant, they open a gap in the spectrum.
In the perturbative limit, since O f is proportional to the kinetic energy in Eq. (2), its bosonized form is simply the bosonized Hamiltonian of non-interacting spinless fermions divided by J 0 .Changing to the spin/charge fields, we find [15,27] Due to spin-charge separation, the absorbed power is the sum of a spin and a density contribution.To find an expression of O f applicable away from the perturbative limit, we note that O f is obtained by differentiating the Fermi-Hubbard Hamiltonian (2) with respect to J 0 .Assuming that the identity carries over to the bosonized description, we have and a similar approximation for O f,σ .As a further approximation, in the repulsive case, we take the fixed point values in H σ , and write Applying Eqs. ( 13)-( 14) in the perturbative case, Eq. ( 12) is recovered.It is important to note that the full expression of the fermion kinetic energy contains besides the linear dispersion valid near the Fermi points corrections coming from band curvature.So the expressions (8), ( 13) and ( 14) are really the most relevant terms in an expansion of the operator O f in a series of operators of increasing scaling dimensions.The contributions of operators of higher scaling dimensions are subdominant at low frequency as shown in App. A.

B. Bosonized representation for bosons
In this Section we turn to bosons.The Bose-Hubbard Hamiltonian with U > 0 has the bosonized representation: where Π(x) and φ(x) are conjugate operators that describe the boson density fluctuations, u is their velocity, and K the Tomonaga-Luttinger exponent [16,23,35].At integer filling, Umklapp processes contribute a term ∝ cos 2φ to the Hamiltonian (15).Similarly to the fermionic case, their contribution is subdominant as long as the system remains in a Tomonaga-Luttinger liquid ground state.The limit U → 0 of the Hamiltonian ( 15) is singular, with the velocity u vanishing to recover the quadratic dispersion of non-interacting bosons above a condensate, and the Tomonaga-Luttinger exponent going to +∞.Thus, in contrast to the fermionic case of Sec.III A, it is impossible to derive a bosonized representation of (4) by considering the non-interacting limit.However, assuming as in Sec.III A that the identity O b = ∂H b ∂J0 is applicable to the bosonized Hamiltonian (15), we find This expression is similar to (12).Moreover, in the hard core limit U → +∞, bosons can be mapped to non-interacting spinless fermions [36], and the fermionic expression ( 12) yields an explicit form of O b which fully agrees with (16).
As we discussed in the fermionic case, the expression ( 16) is only the first term in a series of operators of increasing scaling dimension that represent the various band curvature terms coming from the dispersion of the lattice model.

C. Response function in an infinite system
With repulsive interactions, both for fermions and for bosons, the calculation of the response function (7) reduces to the calculation of the response function of an operator of the form dx[AΠ 2 + B(∂ x φ) 2 ] for a Hamiltonian quadratic in Π and ∂ x φ.That calculation is further simplified by rewriting the bosonized form of the operator O b,f as linear combination of the Hamiltonian and an operator proportional to (∂ x φ) 2 .In the bosonic case, and in the fermionic case, for the perturbative limit, while in the non-perturbative limit, The Hamiltonian being time independent, the response function reduces up to a proportionality factor to the one of dx(∂ x φ) 2 .We note that this is the same response function as in the case where the on-site interaction is modulated.Furthermore, according to Eq. ( 19), the response function (7) vanishes for non-interacting fermions since K ρ = K σ = 1 for any J 0 in that case.This can be established more directly from the lattice Hamiltonian by noting that for U = 0, O f is proportional to the Hamiltonian.More importantly, Eq. ( 19) also shows that the contribution of the spin excitations calculated at the fixed point K * σ = 1 is vanishing.This indicates that for interacting fermions the dominant contribution comes from the density response.Due to the fact that the drive is coupling only to the density and not to the spin, this is expected to be the case on general grounds.Similarly, in the bosonic case, in the limit U → +∞, where K = 1 independent of J, the response function ( 7) is also vanishing.Again, this is more directly established by noting that O b is directly proportional to the hard core boson Hamiltonian in that limit.
We calculate χ(ω), by taking the analytic continuation For the sake of definiteness, we perform the calculation for bosons.Using translational invariance, we find that: Details on the evaluation of χ(ω) can be found in the App.B and for zero temperature the final result is where α is a short distance cutoff (of the order of the lattice spacing) and . Only the behavior for |ω| u/a ∼ J is reliably predicted by bosonization.For frequencies of order of the bandwidth, the linearized approximation for the dispersion certainly breaks down, and high energy excited states not described by bosonization can contribute as well to the energy absorption.
At finite temperature, Eq. ( 22) becomes so the response function behaves as ∼ ωT when ω T and as ω 2 when T ω, see Fig. 1.Thus the absorbed power is for bosons and for fermions.It has a universal power-law dependence on frequency, with an exponent independent of interactions.This universal behavior has to be contrasted with the conductivity [15,37] where the power-law exponent varies with the Tomonaga-Luttinger parameter, and thus depends on the microscopic interaction strength.Here, only the prefactor depends on the logarithmic derivative of the Tomonaga-Luttinger parameter with respect to the hopping amplitude.

D. Numerical results
In order to elucidate the universal frequency exponent of the absorbed power density P b /L ∼ δJ 2 ω 3 predicted from the Tomonaga Luttinger theory, we numerically evaluate the energy absorption in the Bose-Hubbard model, Eq. (D2) using matrix product states [38,39].In particular, we consider systems with 160 sites and non-integer boson density ρ = 1.2, to fully avoid Umklapp processes.We have checked the convergence of our results with the bond dimension of the matrix product state which ranges from χ = 400 to χ = 800.The absorbed power density, renormalized by the drive strength δJ 2 , shows a universal ω 3 scaling, as predicted from the Tomonaga Luttinger theory.Deviations at low frequencies are decreasing with increasing system size and are expected to arise from the residual contributions of system edges which add a ω 2 /L contribution that is leading in frequency but vanishing in the thermodynamic limit.
Our objective is to simulate an experimental protocol to measure the absorption.To this end, we first compute the ground state of our model and then apply a periodic modulation of the kinetic energy of the form J(t) = J 0 + δJ sin(ωt).We choose the driving strength δJ = 0.1J 0 , small enough, such that the absorbed energy increases linearly in time, as required from the linear response theory.We evolve the system for ten drive periods and extract the absorbed power density for a range of modulation frequencies, see Fig. 2. The power density scales as ω 3 in agreement with the Tomonaga Luttinger liquid prediction.At low frequencies there are small deviations from the predicted scaling, as expected from a contribution from boundaries (see Sec. IV A).

IV. BROKEN TRANSLATIONAL SYMMETRY
So far, our considerations have been restricted to an infinite system without defects.This section focuses instead on the case of systems with broken translational symmetry caused either by boundaries or impurities.

A. Effects of boundaries
Since trapped atoms are systems of finite length the effect of boundaries on their response must in principle be considered.We examine in this section the effect of edge potentials that pin the density and this can potentially modify the response to shaking.
The bosonized Hamiltonian in the presence of forward scattering edge potentials becomes [15,[40][41][42][43]: with the Dirichlet boundary conditions φ(0) = 0 and φ(L) = −πN .Those boundary conditions ensure that no current can leak through the edges of the system.As ρ(x) = −∂ x φ/π, the terms V simply represent a forward scattering in the vicinity of the system edges.Note that with the Dirichlet boundary conditions, a backscattering term −V b cos 2φ(0) can be reduced to a forward scattering term [15] so there is no loss of generality in Eq. (26).Since in bosonization, the particle-hole symmetry is φ → −φ and Π → −Π, V vanishes in a particlehole symmetric system [42].In the absence of such symmetry however, those terms can be nonzero.When one considers only the static properties, the edge potential can be eliminated by modifying the Dirichlet boundary conditions [43].However, when we modulate the lattice, the edge potential can be time dependent V = V (J(t)), and for that reason, it is better to retain the original boundary conditions.When we differentiate the Hamiltonian ( 26) with respect to J 0 , as in Eq. ( 16), the edge potential in (26) gives an extra edge contribution proportional to The response coming from the edge potential is calculated in App.C.It contributes to the absorbed power.The total absorbed power is therefore P tot.= P edge + P bulk ∼ ω 2 + Lω 3 .The edge response dominates below a crossover frequency ω * ∼ 1/L.The boundary potential V remains to be determined.A possible approach is to see how the Friedel oscillations are affected by these scattering potentials at the boundary.

B. Friedel oscillations and determination of the edge potentials
The edge potential V in Eq. ( 26) can be deduced from the Friedel oscillations [44][45][46] in the density profile of the ground state.The explicit calculation of the density profile in App.D leads us to the following expression valid sufficiently far from edges where 2k F = 2k F + 4KV uL and ϕ = 2KV u , with k F = πN 0 /L the nominal Fermi wavevector of the Friedel oscillations in a system of length L containing N 0 bosons.From (29), consecutive zeros of the Friedel oscillations are separated by the distance , revealing the presence of the edge potential.In the thermodynamic limit, 2k F reduces to 2k F .However, the phase shift ϕ = 2KV /u persists, and the fitted expression of the Friedel oscillations obtained by MPS reveals the presence or absence of a potential near the edge.

C. Effects of a single impurity
Let us finally consider a single impurity located at x 0 whose potential energy is given by H imp = V ρ(x)δ(x − x 0 ).Within the bosonization approach this term gives rise to two terms in the Luttinger liquid Hamiltonian: a term − 1 π ∂ x φ(x 0 ) which corresponds to a forward scattering process and a term proportional to cos(2φ(x 0 )) which corresponds to back scattering.Using the same treatment as above, the first term will be leading to a dominant ω 2 scaling function for the absorbed power, while the back scattering will contribute a term proportional to ω 2(K−1) with K > 1 and thus less relevant at low-frequency.Thus the presence of a single impurity would lead to a dominant ω 2 contribution to the absorbed power.

V. GAPPED SYSTEMS
In the case of fermions with attractive interactions, or in the case of fermions or bosons with repulsive interactions at commensurate filling, the spectrum can become gapped.The response in that gapped regime can be calculated either in the Luther-Emery limit [15,47] or in the more general case using the form factor expansion [48].Both methods predict a threshold in absorption power at the gap.
For the Bose-Hubbard model, below that gap the power absorption will be zero.In the Fermi-Hubbard case with repulsive interactions, we have seen that the spin response was suppressed, so that only the density response contributed.In the gapped state, the density response also does not contribute at frequency lower than the gap, making the threshold observable as well.In the case of the Fermi-Hubbard model with attractive interaction, since the density modes are gapless, the response at low frequency will be the ∼ |ω| 3 contribution.The threshold at the gap then appears as a cusp-shaped rapid increase of absorption.The physical interpretation of such threshold is quite simple.At frequencies lower than the binding energy of two fermions of opposite spins, the pairs of fermions behave as an interacting boson gas [25], yielding the ∼ |ω| 3 contribution to the absorbed power.As the frequency becomes comparable to the binding energy of the pair, another absorption channel from dissociation of the pairs becomes available, leading to the rapid increase of absorption.
For concreteness, let's first consider the Fermi-Hubbard model in the Mott insulating phase for the particular case of Luther-Emery limit where K ρ = 1/2.In that limit, the resulting absorbed power is given by (see App. E for a detailed calculation) where 2∆ < ω < 4∆ leading to a cusp singularity at ω = 2∆.This analysis can be extended away from the Luther-Emery point to any value of of the Luttinger parameter by the form factor expansion for the sine-Gordon model [48][49][50][51], as detailed in the Appendix F (see also [52]).The main result for the absorbed power (with 1/2 < K ρ < 1) is where we have found a threshold at twice the mass of the soliton M s and ν = K ρ /(1 − K ρ ) for the Fermi-Hubbard model.For K ρ < 1/2, besides the threshold behavior (31), discrete peaks coming from resonant absorption by soliton-antisoliton bound states become possible.This behavior could be readily observed in current experiments with cold atoms in the Mott insulating regime [6,53].
The same threshold behavior as in Eqs.(30,31) was also obtained in the opposite limit of a weak lattice [54] in which the depth of the periodic potential was modulated.One may thus speculate whether such threshold behavior is also observed for intermediate lattice strengths.

VI. SUMMARY AND OUTLOOK
We have analyzed in linear response the power absorbed by one-dimensional fermions and bosons in the Tomonaga-Luttinger liquid [23] or Luther-Emery liquid [47] phase, to the amplitude modulation of an optical lattice.In the Tomonaga-Luttinger liquid, we have found that the absorbed power possesses a universal ω 3 powerlaw onset, that has been confirmed by numerical simulations based on Matrix Product States.We have also shown that this power law crosses over to ∼ ω 2 , at low frequency in finite systems when edge effects are taken into account.A similar ω 2 behavior is found for systems with a single impurity located in the bulk.
Such universal behavior is surprising since in Tomonaga-Luttinger liquids theory, response functions usually show nonuniversal exponents determined by the interaction strength [55].The universal ω 3 scaling of the absorbed power can be readily measured for ultracold atoms in optical lattices confined to one-dimension by measuring the energy change over time.In Luther-Emery liquid phases, that can be obtained for commensurate densities, or with spin-1/2 fermions having attractive interaction, the absorbed power vanishes below a gap and shows a marked onset above, thus making, if possible, to identify this energy scale.
The discussion in this paper focused on experiments with spinless ultracold atoms.Before concluding this section we briefly review other systems in which ideas developed in this paper can be tested experimentally.Bosonic spin mixtures in optical lattices can be used to realize lattice spin Hamiltonians and spinor condensates [56][57][58].Recent experiments by Jepsen et al [59] used magnetic field dependence of the interspecies scattering length to realize XXZ spin chains with tunable anisotropy of interactions.In the regime of easy plane anisotropy XXZ chains are in the gapless regime, while the easy axis case corresponds to the gapped regime.Periodic modulation of J z /J ⊥ can be achieved in this system through periodic modulation of the magnetic field and should have an effect equivalent to modulation of the interaction strength for spinless bosons.These experiments have high local resolution, which will allow one to spatially resolve spin patterns induced by modulation of the interaction anisotropy.Hence predictions of our paper for both gapless and gapped regimes can be checked experimentally.
Recent progress in superconducting nanotechnology makes it possible to engineer arrays of coupled Josephson junctions whose parameters can be controlled dynamically.Lahteenmaki et al. [60] have demonstrated a dynamical Casimir effect in a one dimensional chain of Josephson junctions, in which the Josephson energy of the junctions has been modulated by periodically changing the background magnetic flux.Parametric generation of photons at half the modulation frequency observed in these experiments is the direct analogue of energy absorption in the Luttinger liquid discussed in our paper.Recent experiments by Kuzmin et al [61] demonstrated the possibility of tuning a chain of Josephson junctions through the superconductor to insulator transition and explored evolution of the collective phase mode across the transition.Hence 1D superconducting metamaterials make it possible to study modulation spectroscopy of 1D systems in both gapless and gapped phases.
Although we focussed in this paper to one dimensional system, a similar analysis can be applied to study periodic driving of higher dimensional systems provided that their lower energy excitations allow for an hydrodynamic description.In particular, we expect that our formalism should be useful for analyzing pump and probe experiments in interacting electron systems [62][63][64][65][66]. Recent experiments by von Hoegen et al. [67] have observed parametric excitation of Josephson plasmons in YBCO superconductors following resonant excitation of apical oxygen phonons.The microscopic mechanism of phonon-plasmon coupling is modulation of the superfluid density in copper-oxide planes by the phonon induced motion of oxygen atoms.Analogously to what we have discussed in this paper, resonant parametric excitation of plasmon pairs has been a crucial component of experiments by von Hoegen et al.One important difference, however, is that three wave mixing between phonons and plasmons involves two different types of plasmons, the so-called lower and upper Josephson plasmons.Formalism developed in our paper can be extended to the case of parametric instabilities involving different types of collective excitations.We expect that resonant parametric interactions between phonons and collective exci-tations of many-body electron systems should be a ubquitous phenomenon.Excited phonons can modulate several parameters of electron systems, including effective mass, interactions, and carrier density.Thus pump and probe experiments can be used to achieve parametric driving of a broad range of collective modes, including plasmons in superconductors, spin waves in magnets, and phasons in incommensurate CDW systems.
This expression correspond to Hartree and Fock diagrams.Explicitly one has: and similarly for the term with 1 → 2. In (A4) one can factorize the term which for K > 1 gives no power-law correction and recognize the following convolution integral Thus the Hartree correction of Φ(x, τ ) reduces to where G(q, ν) = uK ν 2 +(uq) 2 .One can easily show that this integral does not increase with ω so the Hartree correction can be neglected.
The Fock correction instead reduces to the integral Turning to the Fourier transform representation, we find where K is the Bessel function of second kind.It can be ex-panded for small q, ω and K − 1 non-integer: while the zeroth order term is q, ω independent, the first non analytic correction will be of the order (q 2 + ω 2 u 2 ) K−1 , which is subdominant compared to (q 2 + ω 2 u 2 ) when K > 1.In the integral (A11) we recognize (up to a factor g 2 ) the self energy-correction from a diagrammatic point of view and thus we can write that qualitatively neglecting holomorphic terms of order (q 2 + ω 2 u 2 )α 2 and higher.We can thus evaluate the Fock correction (A10) which is By simple power counting this integral behaves as ω 2(K−1) and for K > 2, it is subdominant compared to the term ω 2 as ω → 0. So, when K > 2 and the Umklapp scattering is irrelevant, the intensity of the modulation spectroscopy behaves as Appendix B: Evaluation of the retarded correlation function For the sake of definiteness, we present the calculation in the case of bosons.The fermionic case proceeds along the same line, with a simple change of prefactor.Using Wick's theorem, the correlator in Eq. ( 22) is rewritten leading to We need the integral and its analytic continuation.We write and obtain We find the analytic continuation iω → ω + i0 + of Eq. (B5) using the identity which gives This leads to Eq. ( 22) in the main text.
Appendix C: Calculation of the response function in the case of a system with boundaries In the case of a system with boundaries described by the Hamiltonian (26) with the operator O b given by (27), we first rewrite and as before we only have to calculate the response function of the bulk term proportional to (∂ x φ) 2 and the edge term proportional to ∂ x φ(0) + ∂ x φ(L).
To perform the calculation, one first rescales the fields, φ = √ K φ and Π = Π/ √ K and introduces the Fourier decomposition (D7)-(D8) to rewrite the Hamiltonian (26) in terms of shifted harmonic oscillators and the operator O b , without the contribution proportional to the Hamiltonian, We now introduce φn such that to have a Hamiltonian purely quadratic in φn .In terms of the new operators, The first line gives back the contribution calculated in App.B.
The second gives the contribution coming from the edge potential.The necessary Matsubara correlator is After taking the Fourier transform and making the analytic continuation, one finds In the limit of L → +∞, we end up with yielding the edge contribution (28), to be added to the bulk contribution.
note that ∂ x φ can still be non-vanishing as an operator, so we can a priori have edge scattering potentials V 1 and V 2 in (D3).Now, we introduce the Fourier decomposition which allows us to rewrite Until now, we have made no assumption concerning the symmetry of our bosonized Hamiltonian under parity.Using the Fourier expansion (D7), we can show that under a parity transformation, P φ n P † = (−1) n φ n .In the Hamiltonian (D9) V 1 and V 2 are exchanged by the parity transformation.So we recover V 1 = V 2 for a parity invariant Hamiltonian.To find the ground state, we have to minimize the first line with respect to N and determine the shift of oscillators imposed by the edge potentials.The minimization with respect to N yields while the shift of oscillators is The expectation value of φ(x) in the ground state is then φ We thus have Taking the limit of → 0, for x far enough from an edge, we find the simplified expression, and drop the integer part in Eq. (D14).Using Luttinger liquid theory and the expression (D6), we derive and we see that far from the edges, the Friedel oscillations behaves as if the number of particles was N tot.= N tot.+ K(V1+V2) πu . The expression (D15) applies only when α x and α L − x.It corresponds to the effective Dirichlet boundary conditions φ(0) = KV1 u and φ(L) = − KV2 u that result from the phase shift on φ(x) imposed by the edge potentials.When 0 < x < α = L, we cannot take the limit → 0 in Eq. (D13).There, φ(x) = O(x/α) → 0, ensuring that the original Dirichlet boundary conditions are satisfied.
The bosonized Hamiltonian for the fermions can also be written in terms of the rescaled fields and at the Luther-Emery point [47], K ρ = 1 2 , it becomes dx cos 2φ ρ .
(E2) That Hamiltonian is rewritten by introducing the pseudofermions in the form of a gapped non-interacting Hamiltonian where we called ∆ = 2g3 4πα .In terms of the pseudofermions we can rewrite the operator O ρ as where ρ L,R = Ψ † L,R Ψ L,R is the density operator of the L, R fermions.We have to evaluate the Matsubara correlator . This correlator can be expressed in terms of the creation and annihilation operators through the representation Ψ ν = 1 √ L k e ikx c , in terms of which the Hamiltonian (E6) is written as: (E10) This Hamiltonian can be diagonalized by standard Bogoliubov transformations and expressed in the form: with Then the calculations of the correlators (E9) proceeds by applying Wick's theorem once the single particle Green's function are known: The results for the correlators are: for 2∆ < ω < 4∆.This analysis can be extended away from the Luther-Emery point to any value of K = K ρ by using the form factor expansion for the sine-Gordon model [48][49][50][51], as detailed in the App.F (see also [52]).
Appendix F: The form factor approach In the present Appendix, we want to extend the results derived using the Luther-Emery limit to any value of K ρ .For K ρ > 1/2 the excitations are massive solitons and antisolitons of mass M s , while for K < 1/2 we also have breathers Kρ − 1 integer.Working in the vicinity of the Luther-Emery point the low-energy Hamiltonian is: where the operator ρa = Ψ † a Ψ a , a = L, R. It is the Hamiltonian of a massive Thirring model [68]: where v = u 2 2K + 1 2K , M = g 2πα and ḡ = πu ρ 2K − 1 2K .
The kinetic energy operator is related to the component T 11 of the momentum-energy tensor (see [69] p. 143), so that Since ρR ρL = ψ † R ψ † L ψR ψL , that operator can only have matrix elements between the ground state of the massive Thirring model and a state containing two solitons and two antisolitons, i.e., a state with energy at least 4M s .So that term will be not contribute for frequencies ω < 4M s , and we will have where according to [51], the form factor of the energy momentum tensor is: Now we have a threshold at twice the mass of the soliton.A similar threshold behavior was also obtained in the case of modulation of a weak optical lattice [54].Technically, this can be understood as follows.We can always substract an operator proportional to the Hamiltonian (F2) from the operator O.So we would obtain an equivalent result if T 11 was replaced by a term proportional to Ψ † R Ψ L + Ψ † L Ψ R , which is precisely the perturbing term in [54].

FIG. 2 :
FIG.2: Absorbed power density.Using matrix product states, we have evaluated the absorbed power in a periodically driven Bose-Hubbard model on an open chain of L = 160 sites and density ρ = 1.2 for two values of the interaction strength (see legend).The absorbed power density, renormalized by the drive strength δJ 2 , shows a universal ω 3 scaling, as predicted from the Tomonaga Luttinger theory.Deviations at low frequencies are decreasing with increasing system size and are expected to arise from the residual contributions of system edges which add a ω 2 /L contribution that is leading in frequency but vanishing in the thermodynamic limit.