Entanglement propagation and dynamics in non-additive quantum systems

The prominent collective character of long-range interacting quantum systems makes them promising candidates for quantum technological applications. Yet, lack of additivity overthrows the traditional picture for entanglement scaling and transport, due to the breakdown of the common mechanism based on excitations propagation and confinement. Here, we describe the dynamics of the entanglement entropy in many-body quantum systems with a diverging contribution to the internal energy from the long-range two body potential. While in the strict thermodynamic limit entanglement dynamics is shown to be suppressed, a rich mosaic of novel scaling regimes is observed at intermediate system sizes, due to the possibility to trigger multiple resonant modes in the global dynamics. Quantitative predictions on the shape and timescales of entanglement propagation are made, paving the way to the observation of these phases in current quantum simulators. This picture is connected and contrasted with the case of local many body systems subject to Floquet driving.

Here, the spreading of entanglement entropy a er a sudden quench is investigated in a prototypical quantum long-range system.e entanglement features evidenced in previous studies is justi ed in terms of the peculiar spectral properties of strong long-range systems, which have been recently connected with the appearance of metastable states, whose lifetime grow with the system size 18 .3][44] , while it yields solid indications of the existence of various novel phases at intermediate system sizes, i.e. the proper regime for quantum technological applications.
It is convenient to frame our studies in the context of quantum O(n) rotor models, which constitute the prototypical tool in the context of quantum many-body physics.We consider a set of oscillator like variables [ a j , a j ] = iδ j,j δ a,a ( a = 1, . . ., n ), on a d-dimensional square lattice with N = L d sites, in presence of a generic quadratic interaction among di erent sites In the following we will assume periodic boundary conditions.
An appropriate solution for the model dynamics a er a global quench on any of the Hamiltonian parameters can be achieved, in the limit n → ∞ .ere, the quartic term in Eq. (1) can be decoupled via the self-consistent relation Formally, this means that the correlation functions involving a nite number of components, at any time t, can be obtained via the decoupled theory up to O(1/n) terms.In conclusion, the Hamiltonian in Eq. (1) at n → ∞ can be replaced by its quadratic counterpart with a self-consistent e ective mass e range of the interaction is encoded in the exact form of t j−j .e dynamical properties of the model in Eq. (1) have been deeply investigated for local and weak long-range couplings, presenting evidences of both dynamical phase transitions [48][49][50] and aging phenomena [51][52][53] , which also occur at higher order in 1/n 54 .Here, we are instead interested in the strong-long-range case, t j−j = 1 Nα |j − j | −α where α < d and N α = j |j| −α .e Kac scal- ing term N α is needed 55 to ensure energy extensivity.In the following, it is convenient to focus on the d = 1 case, as the qualitative features of the evolution shall not depend on the dimension d.
e solution of the dynamics of the model passes through the introduction of the Fourier modes, labeled by the momentum k = 2π m N with m = −N/2 + 1, . . ., N/2 .At variance with the traditional result in the nearest neighbours case (and with the weak long-range regime α > d ) the spectrum of the kinetic term remains discrete as the thermodynamic limit N → ∞ is approached, see the Methods section "Spectrum".As a result, the dynam- ics of the system is reduced to the one of the e ective model with frequency and c −1 α = 2 1−α (1 − α) so that ω 0 = 0 .For |m| 1 instead the spectral levels accumulate around ω = 1.e Hamiltonian in Eq. ( 4), as well as its dynamics, is readily diagonalized in terms of the ladder operators a k , a † k , which represent the creation and annihilation operators of spin-waves of momentum k e evolution equations of the amplitude functions f k (t) can be obtained via the Heisenberg equations of motion and read fk + (µ + ω 2 k )f k = 0 .While the energy of any single mode is not conserved, we can de ne a conserved energy per particle, , such that > µ 2 /2 (see Methods section "Equations of motion" for further details).
e entire model dynamics is crucially dependent on the evolution of the e ective mass µ(t) .As shown in the Supplementary Material, Sec. 1, one can take µ(t) as an independent degree of freedom, and describe its evolution, together with the one of the f k , in terms of the classical Hamiltonian (2) where P µ and p k are the conjugate momenta of µ and f k , respectively.e potential, which regulates the evolu- tion of the e ective mass µ(t) , reads . e e ective masses m k are given by m , n k being the initial occupation number of each mode.See Ref. 56 for an analogous picture in the classical case.Let us notice that the minus sign in Eq. ( 8) does a ect the equation of motions for f k .e corresponding equation of motion for µ can instead be written as the last term, that we denotes with g(t), encodes the contribution of the other modes.Because of the properties of the spectrum, Nm k → 0 for k 1 and g(t) is negligeble in the thermodynamic limit as long as n k , f k ∼ O(1) .More precisely, as shown in the Methods section "Spectrum", g(t) ∼ N −ζ , with ζ = max(1, 2 − 2α) so that, in the short time regime, µ(t) evolves as the position of a ctitious particle in the potential V (µ) , with the corre- sponding energy conservation E = 1 2 μ2 + V (µ) .Despite the cubic nature of the potential V (µ) one can prove that E < 0 for any viable initial condition in the system, see Supplementary Material Sec. 1.As a consequence, the motion of the e ective mass in the accessible region is a periodic oscillation around the only minimum of the potential V (µ).
e aforementioned classical dynamics only applies as long as the external force g(t) remains negligible, this condition may be violated even at short timescales in two cases: (1) e initial state at t = 0 contains at least one macroscopically populated mode leading to some extensive n k .
(2) e external drive g(t) may become extensive due to one (or more) k modes undergoing a parametric resonance.
Scenario (1) occurs for any dynamics beginning in the magnetized ground-states at r < − /2 , where the k = 0 mode acquires a macroscopic occupation, see Supplementary Material Sec. 2.Even for an initially neg- ligible external drive g(t), − , the oscillations of the e ective mass $\mu(t)$ may act as a drive for one or more modes, leading to scenario (2).Indeed, the equation of motion for each of the f k is in the form of a Hill equation, and the Floquet theorem implies that one can always nd a pair of oscillatory independent solutions, with κ > 0 (see Methods sec- tion "Floquet theorem for the Hill equation" for a derivation of this result).In the latter case, the corresponding excitation undergoes a resonance and the mode occupation ∼ e 2κt will scale as ∼ O(N) , signaling the rise of many-body correlations in the system.us, the time-scale for the external drive g(t) to become relevant is reduced to t q ∼ log N , which has to be compared with t q ∼ N ζ in the classical phase.In summary, while suppressed in the strict thermodynamic limit, for any nite size N the spreading of quantum correlations may still occur on a time-scale of order N or log N in the classical and resonant phase respectively, see Fig. 1.
Given the importance that long-range interactions are ought to play in quantum information and technology applications, it is natural to focus on regime (2), where quantum correlations can arise even for initial states in the disordered phase.e actual boundaries of the mesoscopic quantum phases in the phase space of the initial parameters cannot be simply determined analytically, so that it is convenient to resort to numerical analysis.As the Hamiltonian ( 8) is an integral of motion, the k-th mode cannot resonate as long as µ(t) + ω 2 k > 0 for every t (see also Supplementary Material Sec. 1).In particular, for r > 0 , no resonance can occur, in agreement with the fact that the initial model in Eq. (1) lies in the symmetric phase.For r < 0 , resonant phases can actually be observed and a rich phase diagram arises, as shown in Fig. 2 for the case r = −1 , α = 0.5 (see Methods sec- tion "Numerical methods" for details on the numerics).By decreasing further , large oscillations begin triggering the resonance in the k = 0 mode, then the m = 1 mode start resonating as well as the corresponding µ(t) + ω 2 k is no longer positive de nite, followed by the m > 1 ones.Let us notice that the presence of a gapped spectrum is crucial in order to have regions in parameter space with only a nite number of resonances.
As a consequence, we get an e ective description of the system in terms of a nite number of degrees of freedom by neglecting the non-resonant excitations in Hamiltonian (8). is is the case shown in Fig. 3, where the single particle picture (orange line) is shown to fail on a time-scale t q ∼ log N , while a two mode approxima- tion (cyan line) faithfully reproduces the numerical results (dotted blue line).On a timescale t > t q the resonant mode grows large, and then becomes again negligible, leading to periodic oscillations, punctured with periodic bursts, corresponding to the resonances.
As the number of resonant modes increases, the behavior of µ(t) becomes increasingly involved.Let us notice, however, that for thermal-like initial conditions, N k , f k (0) , p k (0) will depend only on the ω k , so that the m 1 modes will behave as a single particle.

Entanglement growth
Our overall scenario can be con rmed by the study of entanglement scaling a er a quench in the bare mass parameter r ( r − → r + ) at t = 0 , assuming the system is in the ground state for t < 0 in the disordered phase, where the above analysis holds.As a measure of the entanglement it is convenient to choose the Von Neumann entropy relative to the partition of the chain in two intervals.Due to the factorisation of the rotor models interaction term at large-n, and to the fact that the initial state is Gaussian, one can apply the formalism developed in Ref. 57 for computing the entanglement entropy (see Methods section (Entanglement Entropy for the nite interval)) starting from the momentum and position two-point functions.
Let us, then, consider the growth of entanglement entropy in an interval of length .For simplicity, we rst restrict to the case in which only the k = 0 mode may be resonant.Given that the single particle spectrum for long-range interactions accumulates at high-energy in the N → ∞ limit 18 at order 1/N one can discard the k dependence of all modes at k > 0 .erefore, we can replace f k (t) and ḟk (t) with their high energy limit f π (t) , Le : the classical phase, in which the classical dynamics of µ(t) is not in uenced by the presence of the quantum bath up until t q ∼ N ζ , so that there is no production of entanglement.Center: the resonant phase, in which the m = 0 mode becomes resonant and a ect the dynamics on the timescale t q ∼ log N , causing periodic bursts in S(t).Right: the multi-resonant phase, in which a larger number of modes resonate, causing a more complex oscillatory behavior with a nite production of entanglement.ḟπ (t) .As a result, the dynamical theory reduces to the one of a classical particle coupled to the resonant k = 0 mode plus a ∼ N times degenerate high energy mode, leading to the simpli ed quantum correlations It is worth noting that the latter expressions approximate the actual correlations of the system up to O(N −1 ) terms, which may become relevant if the length of the considered interval becomes of order N. e resulting expression for the Von Neumann entropy of an interval of length N (see Supplementary Material Sec. 3) where As expected, the 1/N factor included in the de nition of (t) suppresses the propagation of entanglement unless the k = 0 mode is resonant.us, the parametric resonance results in the production of entanglement on an intermediate time-scale t ∼ log N , which shall be accessible even for large many-particle experiments.e behaviour of S(t) in the resonant phase is shown in Fig. 4. S(t) is characterized by periodic bursts on a time-scale t q ∼ log N , a er which it returns to its initial value.Apart from lack of equilibration, which is signalled by the S(t) function not saturating, the periodic generation and fading of entanglement is a peculiar feature of longrange interactions, which is evidenced here for the rst time.Moreover, Eq. ( 11) implies that, the larger is, the faster quantum correlations will be established.is is in antithesis with the case of local and weak long-range systems, where the short-time growth of entanglement is independent on , due to the light-cone like structure of the Lieb-Robinson bound 58 .
e impact of the number of resonant modes (and thus of e ective degrees of freedom) on the evolution of S(t) is apparent if we consider the case multiple resonant modes beyond the k = 0 one.In this case, the analytic expression for the correlation functions in Eq. ( 11) does not apply.However, following Ref. 57one can derive the evolution of S(t) in the multi-resonant phase, see Fig. 4.
In summary, while in the N → ∞ limit, non-additive interaction hinders the spreading of quantum excita- tions, for large but nite N, the out-of-equilibrium phase diagram of long-range interacting quantum systems (10)    for the initial conditions µ(0) = −0.7 , μ(0) = 0 , = 1.2 (dotted blue line) compared with the corresponding single-particle classical picture (continuous orange) and the two-particle one (continuous cyan line).Since for this initial conditions only the k = 0 mode is resonant, the latter reproduces the right evolution, while the former fails on a nite timescale.Inset: di erence between µ(t) and the single-particle picture for di erent values of N, showing how the time-scale on which the approximation breaks down grows as log N.
presents a rich mosaic of phases.In the prototypical case of O(n) eld theories, we argued that such phases may be characterized by the scaling of the entanglement entropy.In particular, for sudden quenches from any ground state in the disordered equilibrium-phase, we identify three phases: (a) Classical phase: in the thermodynamic limit, the many-body dynamics corresponds to the one of a single classical particle, representing the e ective mass µ(t) .Correspondingly, at nite sizes, no entanglement emerges in the system up to a scale t q ∼ N ζ .(b) Resonant phase: for larger initial energies the oscillations of the e ective mass µ(t) trigger a resonance in the lowest k = 0 mode.As a consequence, the entanglement entropy grows in quasi-periodic bursts, the rst of which occurs at a time-scale t q ∼ log N .Yet, following each burst, the amount of entanglement vanishes and classical dynamics is restored.(c) Multi-resonant phase: at high energy, the e ective mass oscillations generated by the instantaneous quench are strong enough to generate multiple resonances, leading to a mosaic of di erent phases, each characterised by a number of active modes in the dynamics.While the phenomenology of all these phases is analogous to the single resonance one, the entanglement does no longer come back to zero.Indeed, as a large number of resonance is triggered, see Fig. 4, a nite amount of entanglement is generated at longtimes, which is however smaller than its short-range counterpart.In spite of such apparent saturation of entanglement entropy, the quasi-periodic oscillations persist at any large time, so that no actual equilibration occurs even in this case.
e present picture can be traced back to the discreteness of the single particle spectrum evidenced in Ref. 18 .en, the appearance of resonant phases belongs to the rich landscape of peculiar phenomena caused by strong long-range interactions, including metastable states 16,17 , ensemble inequivalence 59 or lack of adiabaticity 60,61 .In this perspective, our picture shall be regarded as universal and will be observed in any systems where longrange couplings generate a discrete spectrum, including cavity QED and trapped ions experiments.As a further evidence of the universal nature of our ndings, the slow growth of entanglement entropy, which we explicitly connected to the presence of a discrete spectrum, has also been found in numerical simulations of the ferromagnetic long-range Ising model a er a sudden quench 44 .While this slow growth of entanglement may be confused with the one found in nite range and weak long-range systems 41 , the latter is generated by quasi-particle con nement 34 or Bloch oscillations 62 and, due to its prethermal character, can only be observed at short times.On the contrary, the current picture follows from the collective character of strong long-range interactions, which produce an e ective global coupling between the degrees of freedom in the thermodynamic limit, see Fig. 1, stabilising the aforementioned phases even at large times.
Along the same lines, the collective character of long-range interactions with α < d prevents the divergence of low-energy excitations and stabilises the 1/n approach employed to study the dynamics, making it trustworthy also in the thermodynamic limit.As a result, the e ective mass µ(t) acts in all respects as a global drive and generates the resonant phases in analogy with previous studies of bosonic theories subject to an external periodic force 63 .Nevertheless, in the present case, the back action of the resonant modes on the drive prevents the linear growth of the entropy with time. is is consistent with the isolated nature of the system and, consequently, with the absence of any external energy source.
Our investigations belong to a long strand of literature that studies universal dynamical phenomena in at interacting systems, starting from the celebrated Lipkin-Meshkov-Glick and Sachdev-Ye-Kitaev models [64][65][66] 11) (dashed yellow line).Right panel: Production of entanglement for larger times in the multi-resonant phase, compared with the correspondent limiting value S for the same quench in the short-range limit ( α → ∞ ).See Methods section "Numerical methods" for a more detailed explanation of the numerical procedure.
and extending up to astrophysical scales 67 .Our ndings represent a substantial step forward in this direction as they encompass in a single framework the cases of both at ( α = 0 ) and of fully-connected power-law decaying interactions ( 0 < α < d ).us, we have presented a general framework capable to predict and justify a large variety of experimental and numerical observations, arising in experimental realisation of strong long-range interactions 1-3 .e present description exploits two main traits of strong long-range systems, namely the discreteness of the spectrum 18 and the global collective coupling emerging from their collective character, see the role of the global e ective mass µ in Eq. ( 8).
Here, a rst application of this general framework to sudden quenches and entanglement propagation has been presented, resulting in an explicit mapping to periodically driven systems (see the Hill equation in the Methods section "Floquet theorem for the Hill equation") which has no counterpart in short-or weak long-range interacting systems. is previously unknown relation us to fully characterise the dynamical phase diagram of strong long-range quantum systems and produce several analytical estimations over the scaling of the proper time of each phase, which may be used to justify and guide experimental endeavours.
In conclusion, the entanglement propagation in strong long-range interacting quantum systems is suppressed in the thermodynamic limit, but a rich phenomenology appears at intermediate system sizes.e strong collective character of long-range interactions, see Fig. 1, explicitly relates the quench dynamics with the one of a periodically driven quantum system, yielding a robust description for entanglement propagation and dynamics.In particular, the resonant phases (b) and (c) display a logarithmic scaling of time with the size t q ∼ log N , allowing their observation also in mesoscopically and, possibly, macroscopically large quantum simulators.In contrast with the known picture for local and, in general, additive systems, these phases do not have pre-thermal character, but they persist at all times, preventing decoherence and, possibly, allowing more robust technological applications.

Spectrum.
Let us now derive Eq. ( 5).If we write down the kinetic part of the Hamiltonian in terms of the Fourier modes we have: where Now, being t r = N −1 α |r| −α , with N α = r =0 |r| −α , we can write ω 2 k = 1 − tk / t0 where Since k = 2π N m , we notice that, as N → ∞ , we get the expression (5) up to O(N α−1 ) corrections.Let us now consider the spectral average of a regular function G(x), namely Since, from Eq. ( 14), at the lowest non trivial order Depending whether 1/2 < α < 1 or 0 < α < 1/2 the correction is then of order N −1 or N −2+2α respectively.Let us notice how this is completely di erent to what happens for the nearest-neighbours case, in which the gap between any two adjacent energy levels vanishes as N → ∞ .On the contrary, the result in Eq. ( 16) is reminis- cent of the α = 0 case, where, due to full permutational symmetry, discreteness of the well known α = 0 case, but the nite excitations energy allow for the appearance of multiple resonances in the out-of-equilibrium dynamics.
Equations of motion.From Eq. ( 4) one can derive the Heisenberg equations of motion for the operators k , k In particular, being the evolution equation linear, is convenient to expand k and k as where a k , a † k and a −k , a † −k are two independent sets of ladder operators which do not depend on time.e canonical commutation relations introduce the constraints Let us notice that, in terms of the f k , we can express the equal-time correlators of the model as k a k being the initial occupation number of the corresponding mode.If we use the de nitions of Eq. ( 6) into Eqs.(20) we get for the f k Once written in terms of the real and imaginary part of f k , one can interpret this as the equations of motion for a set of two-dimensional isotropic harmonic oscillators.e frequency varies in time, and it is determined self consistently by Eq. ( 3). e evolution respects the constraints Im(f * k ḟk ) = 1 , as they correspond to the conserved angular momenta of the oscillators.e single oscillator energy is instead not conserved.However, one can verify that the energy per particle is a constant of motion.Let us notice that, by de nition µ > r and µ 2 < 2 must hold.
Floquet theorem for the Hill equation.We remind brie y the basics of the Floquet theory.Let us notice that, in case µ(t) is periodic, the equation of motion (23) for the single mode has the form of an Hill equation with a(t) = a(t + T) .We may now consider the two independent solutions f 1 (t) and f 2 (t) such that f 1 (0) = 1 , ḟ1 (0) = 0 and f 2 (0) = 0 , ḟ2 (0) = 1 respectively.Let us notice that the Wronskian of the solutions W = f 1 ḟ2 − f 2 ḟ1 of the system is such that Ẇ = 0 .Now we notice that, being a(t) periodic of period T, f 1 (t + T) , f 2 (t + T) can be seen as a new pair of independent solutions, and thus expressed as a linear combination of f 1 (t) , f 2 (t) namely where C is a constant square matrix of order 2. In particular, imposing W(t) = W(t + T) we nd det C = 1 .Let us now consider two independent linear combinations f ± (t) , of f 1,2 (t) such that f ± (t + T) = ± f ± (t) , ± being the eigenvalues of C. On the other hand, since C is real and det C = 1 we have that only two cases are possible, either ± = e ±κT or ± = e ±iµT for some real (positive) κ , µ .While in the former case we have a quasi-periodic solution, in the latter we have a parametric resonance.
Entanglement entropy for the finite interval.Following the procedure of 57,68 , we introduce the matrix of the correlations (20) x (t) x (t) .
e Von Neumann entropy S(t) can be expressed in terms of the symplectic spectrum {σ n } of γ red , i.e. the matrix reduced to the subspace x, x ∈ (0, ) .e symplectic spectrum is de ned such that {σ 2 n } is the spectrum of −(Jγ red ) 2 , J being the symplectic unity In terms of the σ n , we have where: Numerical methods.In order to preserve the symplectic structure of the dynamics, the equation of motions for µ(t) and for the f k have been integrated with a second-order, volume-preserving, algorithm for separable e time-step chosen wad t = 0.05 .In Fig. 2 we have numerically determined the matrix C in Eq. ( 26) for each k, and veri ed the stability conditions Tr C < 2 , which guarantees imaginary eigen- values.In Fig. 4, the procedure of section "Floquet theorem for the Hill equation" was carried out numerically, by computing the matrix γ and its symplectic spectrum for di erent values of t.
(28) J = 0 I −I 0 . ( Let us now derive the equation of motion for the effective mass µ(t).By tacking the time derivative of both sides of Eq. ( 3) of the main text from which, exploiting the conservation of energy and Eq. ( 3) of the main text, we can write where we introduced and Let us notice that this equation, along with the equation of motion for the f k , can be derived from the Hamiltonian (9) of the main text , so that its contribution to the equations of motion of µ(t) is negligible.In this limit, we can thus set f k = 0 in the above Hamiltonian and consider the single particle dynamics: Within the same approximation up to N −ζ correction.Let us notice that, the motion of this effective particle takes place within the bounded region of the potential, since the corresponding (conserved) energy E = μ2 2 + V (µ) can only be negative.Indeed, by exploiting the Cauchy-Schwartz inequality, we find Now, putting together Eq. ( 6) and Eq. ( 3) of the main text and we find the constraint from which the condition E < 0 follows.
Let us now consider the modes f k = O(1).In this case the conserved energy becomes which differs from the single-particle energy for a quantity of order N −ζ .Now, if along the single-particle trajectory µ + ω 2 k > 0 for every k, then the curve defined in the space of parameters f k , ḟk , µ, μ remains close to the f k = 0 trajectory: in this case then, no resonance is possible.In particular, since µ(t) > r, this ensures that no resonance actually occurs in the r > 0 case.The same reasoning leads to the the condition r + ω 2 k > 0 to prevent the resonance of modes with k > 0.
Let us now derive the expression for µ gs and gs .Since each oscillator Φ k is now in the ground state, we have from which, exploiting Eqs.(22) of the Methods, we find valid up to an immaterial phase factor.Since µ gs is now a positive constant, the solution of Eqs. ( 23) of the Methods is given by Finally, from Eq. ( 3), In our case we can replace ω 2 k + µ gs with 1 + µ gs , up to O(N −ζ ) corrections, obtaining: This always has a unique solution.Since we are implicitly assuming µ gs > 0, in order to have an oscillatory behavior for the k = 0 mode, we have to require r > −λ/2.For r < −λ/2 the k = 0 mode acquires a non-zero occupation number, signalling the emergence of a finite magnetization.The fact that the system undergoes a phase transition even in one dimension is not surprising, since the Mermin-Wanger theorem no longer holds in presence of long-range interactions.The corresponding energy per particle is, up to O(N This allows for a simple physical interpretation: indeed gs is such that V (µ gs ) = 0, so that the ground state corresponds to the stable equilibrium for the fictitious particle in the potential V (µ).

Von Neumann entropy for a single resonance
We now apply the procedure exposed in Methods.In our case, from Eq. ( 10) of the Methods we have where Q(t),P (t),R(t) are by matrices defined as with P j,k = 1 , ∀ j, k = 1, . . . .Since [P, R] = 0 and [Q, R] = 0 we find − (Jγ red ) 2 = 1 4

Figure 1 .
Figure 1.Schematic depiction of the phases of our model along with the behavior of the bipartite Von Neumann entropy S(t).Le : the classical phase, in which the classical dynamics of µ(t) is not in uenced by the presence of the quantum bath up until t q ∼ N ζ , so that there is no production of entanglement.Center: the resonant phase, in which the m = 0 mode becomes resonant and a ect the dynamics on the timescale t q ∼ log N , causing periodic bursts in S(t).Right: the multi-resonant phase, in which a larger number of modes resonate, causing a more complex oscillatory behavior with a nite production of entanglement.

Figure 2 .
Figure 2. Phase diagram of the model for r = −1 , α = 0.5 as a function of the energy per particle and the initial e ective mass µ(0) , assuming μ(0) = 0 .Only the region with µ(0) between r and the minimum of V (µ) is shown, being all the other initial conditions nonphysical or redundant.In blue we have the resonance-free region (in which µ(t) is periodic); in orange the region in which the mode k = 0 is resonant; in green the region, where multiple modes above k = 0 are resonant as well.Inset: the potential V (µ) for = 2.25 , in which the values of µ(0) corresponding to di erent phases and the relative values of the classical energy E , see Eq. (8), are outlined.

6 Figure 3 .
Figure3.Behavior of µ(t) with r = −1 , = 1.24 , N = 10 6 for the initial conditions µ(0) = −0.7 , μ(0) = 0 , = 1.2 (dotted blue line) compared with the corresponding single-particle classical picture (continuous orange) and the two-particle one (continuous cyan line).Since for this initial conditions only the k = 0 mode is resonant, the latter reproduces the right evolution, while the former fails on a nite timescale.Inset: di erence between µ(t) and the single-particle picture for di erent values of N, showing how the time-scale on which the approximation breaks down grows as log N.

SFigure 4 .
Figure 4. Le panel: behavior of the Von Neumann entropy S(t) relative to a = 10 interval a er a ground state quench of r from 1 to −1 , for N = 10 4. e dark red, red and orange curves correspond to di erent values of ( = 1.24 , = 0.028 , = 0.0028 ), which in turn corresponds to have 1, 116 and 221 resonant modes respectively.All curves refer to the case α = 0.5 .e dotted dark blue and blue curves correspond to the short- range counterpart ( α = ∞ ) of the = 0.028 , = 0.0028 cases.Central panel: Behavior of S(t) in the single- resonance phase (red), characterized by periodic bursts, compared with the analytical prediction of Eq. (11) (dashed yellow line).Right panel: Production of entanglement for larger times in the multi-resonant phase, compared with the correspondent limiting value S for the same quench in the short-range limit ( α → ∞ ).See Methods section "Numerical methods" for a more detailed explanation of the numerical procedure.