# Momentum-space entanglement after smooth quenches

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## Abstract

We compute the total amount of entanglement produced between momentum modes at late times after a smooth mass quench in free bosonic and fermionic quantum field theories. The entanglement and Rényi entropies are obtained in closed form as a function of the parameters characterizing the quench protocol. For bosons, we show that the entanglement production is more significant for light modes and for fast quenches. In particular, infinitely slow or adiabatic quenches do not produce any entanglement. Depending on the quench profile, the decrease as a function of the quench rate \(\delta t\) can be either monotonic or oscillating. In the fermionic case the situation is more subtle and there is a critical value for the quench amplitude above which this behavior is changed and the entropies become peaked at intermediate values of momentum and of the quench rate. We also show that the results agree with the predictions of a Generalized Gibbs Ensemble and obtain explicitly its parameters in terms of the quench data.

## 1 Introduction

A quantum quench is one of the simplest protocols to put a quantum system away from equilibrium. The typical setup is to prepare a state at some initial time slice (e.g., the ground state of a Hamiltonian \(H_0\)) and suddenly let it evolve in time with a different Hamiltonian \(H_1\) which acts on this state in a non-trivial way. The problem can be equivalently formulated as that of a time-dependent Hamiltonian *H*(*t*) that changes from \(H_0\) to \(H_1\) as one of its parameters change in time. This has the advantage of allowing smooth transitions that happen within a finite time scale \(\delta t\) rather than instantaneously. In any case, one is usually interested in the dynamics of various physical observables such as correlation functions and entanglement measures during the process. The study of quenches provides a window to explore a number of important questions concerning the non-equilibrium quantum dynamics, such as the mechanism underlying thermalization (or not) of isolated systems [1]. It has attracted increasingly more attention due to recent developments in cold atom physics that made possible to experimentally probe the real-time dynamics following a quantum quench [2].

*m*(

*t*) are amenable to analytical solution for any quench rate \(\delta t\). The main reason behind that is the observation that the problem can be equivalently understood as the one of a standard (constant mass) quantum scalar field placed in an cosmological background – more precisely the Friedmann–Lemâitre–Robertson–Walker (FLRW) spacetime. The latter has been studied back in the 70’s by Bernard and Duncan [3, 4], where mode solutions for some specific choices of mass profiles were obtained and its quantization was performed, so the results together with the intuition from quantum field theory in curved spacetime can be adapted to understand the mass quenches above. This approach has been used in [5, 6] to study the behaviour of 1-point functions following the quench (see also [7, 8, 9] for related work).

In the present paper, we focus instead on *momentum-space entanglement*. That is, we divide the Fock space in terms of positive and negative-momentum modes of the quantum field and calculate the entanglement production due to the quench between a single mode and all the others (though we shall see that only modes carrying opposite momenta are actually entangled). We investigate mass quenches in both free scalar and free fermion theories. The work follows the logic of [15] (see also [16]), which discussed the entanglement produced between momentum modes due to the expansion of the universe for a specific choice of FLRW scale factor. We will consider the same choice, which in our case translates to mass-increasing or mass-decreasing quenches, as well as another class of quench profiles that recovers the initial mass at late times. For previous work on momentum space entanglement in quantum field theory we refer the reader to [17, 18, 19]; see also [20, 21] for similar work with spin chains.

An important aspect is that, unlike the majority of interacting quantum systems which are known to thermalize after a quench in the sense of approaching a thermal (or Gibbs) ensemble at late times, integrable systems such as the free field models studied here do not reach thermal equilibrium in this usual sense. This is because they possess an infinite number of conserved charges that conspire to constrain the dynamics, a fact that was first observed experimentally in [22] using a system of Bose gases in one dimension. However, later in [23] it was proposed that integrable systems do thermalize in a broader sense to a new kind of equilibrium state called the Generalized Gibbs Ensemble (GGE) (see, e.g., [24] for a review and validity checks for a variety of 1D systems). Having that in mind, we will also show that our results for the entanglement production precisely match the GGE prediction and, moreover, that the parameters characterizing this ensemble can be expressed in closed form in terms of the parameters defining the quench protocol. This reinforces the relevance of considering momentum-space entanglement as an interesting probe of thermalization of quantum systems. Unlike its real-space counterpart, the entanglement between momentum modes is not tied to any particular spatial sub-region of the system, so one should expect it to capture different physics characterized by non-local correlations in real space. It is worth to recall that the most accepted explanation for the spreading of real-space entanglement after a quench relies on the quasi-particle picture introduced in [10], where it is assumed that pairs of quasi-particles with opposite momenta created within the correlation length are entangled, while those far apart from each other are not. As these particles travel through the system, real-space entanglement happens to grow ballistically. This intuition is able to correctly reproduce many of the qualitative features of the entanglement dynamics, and has been used to derive new results [25]. From our calculations we can make this concrete by computing exactly how entangled the different particles produced are as a function of the parameters controlling the quench, such as its amplitude and speed. Hence, understanding the dynamics of momentum-space entanglement may also shed light into our understanding of how real-space entanglement grows. The free field example is chosen for convenience, since it is amenable to exact analytical results while still allowing for interesting phenomena such as the approach to the GGE, but we hope that our study can be a useful benchmark in future studies of thermalization in more complicated interacting models.

The paper is organized as follows. In Sect. 2, we review the exact mode solutions and quantization of a free massive scalar field in FLRW spacetime and show how a conformal rescaling provides the solution to a mass quench in flat spacetime. In Sect. 3 we calculate the total amount of momentum space entanglement produced by different quenches at late times by following the same logic used in [15] for the curved space picture. Section 4 presents the generalization for a fermionic field, while Sect. 5 compares the results with the prediction using a generalized Gibbs ensemble and shows an explicit expression for it in terms of quench parameters. Finally, Sect. 6 contains the closing remarks.

## 2 Smooth quantum quenches from FLRW fields

*m*conformally coupled to a curved background metric \(g_{\mu \nu }(x)\) in any number

*d*of spacetime dimensions,

Such a symmetry is particularly useful for the purposes of quantization of the field when the curved background \(g_{\mu \nu }(x)\) is a conformally flat spacetime, i.e., \(g_{\mu \nu }(x)=\Omega (x)^2\eta _{\mu \nu }\), since one can use a Weyl rescaling to map to the Minkowski metric \(\eta _{\mu \nu }\) and then resort to intuition from canonical field quantization in flat spacetime.

*t*the metric can be written as \(g_{\mu \nu }(t)=a(t)^2\eta _{\mu \nu }\), i.e,

*a*(

*t*). Unlike the massless case discussed above, the massive scalar field action (4) does not remain invariant under Weyl rescalings. In fact, by introducing the new scalar field \(\phi \) according to \(\varphi (t,{{\varvec{x}}})\equiv a(t)^{-\Delta }\phi (t,{{\varvec{x}}})\) the action can be rewritten in terms of the flat spacetime metric as

*m*(

*t*) in the mass of a scalar field in flat spacetime. Hence we see that the problem of mass quenches in flat space can be recast as the “dual” or equivalent problem of a (constant mass) conformally coupled quantum scalar field \(\varphi \) living in a curved FLRW background.

^{1}

Of course such a relation is not a peculiarity of mass quenches. It is a straightforward exercise to check that a general interaction term \(\sim \lambda \varphi ^n\) (\(n\ge 2\)) in the action (4) in FLRW gets mapped under the same Weyl rescaling to a quench \(\sim \lambda (t)\phi ^n\) of the transformed scalar field that lives in flat space, with \(\lambda (t)\equiv \lambda \,a(t)^{n+(2-n)d/2}\). We are dealing here with the special case of \(n=2\) only for practical reasons, since the equation of motion is linear.

In flat spacetime, the canonical procedure for quantizing fields [26] begins with first finding the positive-frequency normal modes \(u_{{\varvec{k}}}(t,{{\varvec{x}}})\sim e^{\text {i}\,({{\varvec{k}}}\cdot {{\varvec{x}}}-\omega t)}\) that solve the Klein–Gordon equation in order to express the quantum field \(\phi \) in the basis formed by \(u_{{\varvec{k}}}\) and its complex conjugate. Such a meaningful classification into positive-frequency modes, however, is only possible because flat spacetime admits a timelike Killing vector field \(K\equiv \partial _t\) whose corresponding conservation law guarantees well-defined “energy” eigenvalues at any time, i.e., \(\text {i}\,\partial _tu_{{\varvec{k}}}=\,\omega \,u_{{\varvec{k}}}\) with \(\omega \ge 0\). In FLRW spacetime this timelike isometry *K* is clearly not present in general since the metric (9) depends explicitly on time, so the very first step of the quantization procedure seems to fail once we go beyond the flat space scenario. This is a general feature of curved spacetimes that can be boiled down to the diffeomorphism invariance of general relativity, namely the inexistence of a preferred time coordinate to which a meaningful notion of energy can be associated.

Fortunately, this problem can be worked around at least when the conformal factor \(a(t)^2\) in (9) asymptotically approaches constant values as *t* goes to \(\pm \infty \). In this case a timelike Killing vector emerges asymptotically in the past and future infinity and one can still make sense of the quantization procedure (i.e., asymptotic positive-frequency modes, the vacuum state, particle excitations, and so on). This particular subset of FLRW spacetimes will be the one of interest in the following. Besides this technical reason, incidentally it turns out to be a useful toy model from the point of view of mass quenches, where well-defined initial and final equilibrium states before and after the quench are required. It is important to stress though that it has little relevance from the perspective of cosmology, where none of the relevant scale factors *a*(*t*) happens to saturate to constant values.

We will be interested in two representative cases where the mass profile \(m(t)=m\,a(t)\) allows for exact solutions, but before particularizing to specific choices of *m*(*t*) let us first sketch the general strategy (see next section for explicit expressions in the cases of interest).

^{2}The mode functions \(\chi _{{\varvec{k}}}(t)\) are easily shown to satisfy a simple harmonic oscillator equation with a time-dependent fundamental frequency, namely

*not*annihilated by \(a^{\text {in}}_{{\varvec{k}}}\) (and vice-versa). Indeed, a no-particle state for one observer can look like a complicated particle bath as told by the other. For instance, the number of particle excitations carrying momentum \({{\varvec{k}}}\) counted by an asymptotic observer at past infinity (using the number operator \(N^\text {in}_{{\varvec{k}}}\equiv a^{\dagger \,\text {in}}_{{\varvec{k}}}a^{\text {in}}_{{\varvec{k}}}\)) obviously vanishes in the “in” vacuum, \(\langle 0_\text {in}|N^{\text {in}}_{{\varvec{k}}}|0_\text {in}\rangle =0\), while (by virtue of (15)) it is non-zero in the “out” vacuum,

*m*(

*t*), in addition to the usual particle number piece \(\sim a^\dagger _{{\varvec{k}}}a_{{\varvec{k}}}\) the Hamiltonian contains also interaction terms between opposite momentum modes \({{\varvec{k}}}\) and \(-{{\varvec{k}}}\). Such terms disappear when \(m(t)=\text {const}\), where the modes are simply \(\chi _{{\varvec{k}}}(t)=\frac{1}{\sqrt{2\omega _{{\varvec{k}}}}}e^{-\text {i}\,\omega _{{\varvec{k}}}t}\) and

*H*reduces to its standard form \(H=\int d^{d-1}{{\varvec{k}}}\,\omega _{{\varvec{k}}}a^\dagger _{{{\varvec{k}}}}a_{{\varvec{k}}}+E_0\).

Equation (20) suggests that it might be interesting to split the Fock space into \({{\varvec{k}}}\) and \(-{{\varvec{k}}}\) modes and calculate the entanglement properties of this bipartite system. This shall be done in Sect. 3, but first let us illustrate the results above explicitly for the two cases of interest.

### 2.1 Two specific quench profiles

#### 2.1.1 Tanh profile

*A*and

*B*are related to the quench data by

*adiabatic*quench, while \(\delta t\rightarrow 0\) would correspond to a step function or

*instantaneous*quench profile of the type discussed in [27, 28]. For \({m^2_\text {out}}_{}={m^2_\text {in}}_{}\) (or \(B=0\)) we recover the static case of no quench.

*z*into \(1-z\) (here \(z=\frac{1}{2}(1+\tanh \frac{t}{\delta t})\)), the Bogoliubov coefficients (14) for the present model have been obtained in [4], namely

#### 2.1.2 Sech profile

## 3 Entanglement production by the quench

The goal of the present section is to study the total amount of entanglement produced between scalar field modes by the mass quenching process introduced above. The discussion follows closely the one of [15]. In what follows, we will be working in the Heisenberg picture, so states are not supposed to change in time.

Let us first give a qualitative picture of what is going on here before delving into the calculations. We have seen in last section that there are two equally good bases in which one can express the field, one adapted to early and the other to late time observers. In the cosmological problem, both observers use the same time coordinate *t* (defined by the metric (9)) since its tangent vector provides a timelike Killing vector adapted to each of them. Given this time coordinate, both observers can then define a local notion of particle excitation and vacuum, which is valid only at either the asymptotic past or future. That is why, even by working in the Heisenberg picture, we are able to talk about particle and entanglement production after the expansion of the universe.

The same reasoning translates directly to the problem of quantum quenches, which does not involve curved spacetimes at all (it is defined in flat spacetime) but we know to be equivalent to a FLRW field. Now, both the pre-quench and post-quench observers use the same time coordinate *t* associated to some inertial reference system. But if an experimenter preparing the state at early times is to have a meaningful notion of particles, he or she must use the in-modes, while an observer who will analyze the system at late times, after the quench is finished, naturally picks the out-modes. As a result, we will now show that an initial product state (with respect to positive and negative-momentum bipartitioning of the Fock space) is seen by the post-quench observer as entangled. This entanglement production is obviously tied to our time-dependent Hamiltonian (even though the states do not evolve in time), since the in and out solutions are obtained from it.

^{3}

^{4}

^{5}

*with respect to the out-basis*the negative-momentum modes to obtain the reduced state

*n*essentially as a power law, i.e.,

^{6}

*n*thus becomes just a geometric series which is easily carried out to yield the following closed-form expression for the Renyi entropies

It is important to notice that tracing out the positive momentum modes instead of negative ones in (39) would yield the same Renyi and entanglement entropies as long as the full density matrix \(\rho \) is a pure state, since in this case the Schmidt form (38) guarantees the two reduced density matrices \(\rho _{{{\varvec{k}}}}\) and \(\rho _{-{{\varvec{k}}}}\) to have the same eigenvalues \(|c_n|^2\).

*m*(

*t*) can in principle be fully reconstructed only from entanglement data. The expression for the EE is too complicated to invert and find \(\gamma _{B}(S)\) analytically, but for the Renyi’s the situation is simple enough so that this can be done, namely, all we have to do is solve for \(\gamma _{B}\) the

*q*-th order equation

The physical interest, however, is on the \({{\varvec{k}}}\)- and \(\delta t\)-dependence of the entropies themselves, which we now analyze in detail.

### 3.1 Tanh profile

^{7}An interesting special case is that of \({m^2_\text {in}}_{}=0\), where the pre-quench Hamiltonian is that of a conformal field theory (the free massless boson). In the following we shall limit our numerical analysis to the entanglement entropy and the first few integer Rényi entropies (\(q=2,3,4,5\)).

*k*) modes than between UV (high

*k*) modes. The magnitude of the entropies is proportional to the mass difference \(\delta m^2={m^2_\text {out}}_{}-{m^2_\text {in}}_{}\) between the initial and the final states. The maximal value corresponds to quenches that start from the massless boson CFT, although this case is subtle since there is formally a divergent zero mode contribution \(\sim \log 1/k\) at \(k=0\). In practice we can simply ignore this fact since in this case there is no (\({{\varvec{k}}},-{{\varvec{k}}}\)) splitting of modes at all to begin with. Let us recall that the usual upper bound \(S_{EE}\le \log \text {dim}({\mathcal {H}})\) for the EE is infinite here since the Hilbert space for the reduced state \(\rho _{{\varvec{k}}}\) is infinite-dimensional, so there is nothing contradictory about the fact illustrated in the plot that the EE is not limited from above.

Figure 2 illustrates the dependence of the entropies on the time scale \(\delta t\) for a single mode *k*. It becomes clear that faster quenches produce more mode entanglement than slower ones. In particular, as \(\delta t\) grows all curves approach zero asymptotically, indicating that infinitely slow or *adiabatic quenches* (those with \(\delta t\rightarrow \infty \)) do not create any entanglement between field modes.

### 3.2 Sech profile

*k*. That is, light degrees of freedom are more entangled than heavy ones, similarly to the Tanh profile case. However,there is an interesting difference when fixing

*k*and exploring the \(\delta t\) dependence, as shown in Fig. 4. The entanglement does not decrease monotonically as in the Tanh profile case, but rather oscillates with an amplitude that decreases as \(\delta t\) increases. The origin of this oscillatory behavior can be seen in (45). Also, notice that again there is a divergence as

*k*approaches zero that we should not worry about as we commented above for the Tanh profile.

### 3.3 Entanglement per particle

*k*, and as a function of

*k*for fixed \(\delta t\). This is presented in the Figs. 5 and 6 below for both the Tanh and Sech profiles. It is important to note that this ratio is not constant, besides our earlier remarks that both quantities behave qualitatively similar as a function of the various quench parameters.

## 4 Fermionic case

*d*-dimensional flat spacetime, defined by the Clifford algebra \(\{\gamma ^\mu ,\gamma ^\nu \}=2\eta ^{\mu \nu }\). Just as in the bosonic case, the same equation appears when one analyzes a free fermion with constant mass

*m*placed on a curved FLRW spacetime. Specifically, the Dirac equation for such a fermion \(\Psi \) minimally coupled to gravity reads

*m*(

*t*), while in that case it was for \(m(t)^2\)).

^{8}Translational symmetry in the spacelike directions allows the ansatz

*d*components and this is a matrix equation. Writing

*d*even and \(2^{(d-3)/2}\) for

*d*odd, and the curved space spinor mode solutions

*k*). A similar analysis has been carried out in [16] where the idea was to quantify the entanglement production for a fermionic system due to the cosmological expansion of the FLRW spacetime.

*k*and \(-k\) modes one can repeat the steps done previously and express the initial vacuum in terms of the out basis. The result takes the form

*k*and an antiparticle with momentum \(-k\) as told by the late time observer. Therefore, we see that the initial vacuum is populated by particle-antiparticle pairs of the out type carrying opposite momenta. This should be contrasted with (68) for bosons, in which case there was an infinite tower of multiparticle excitations thanks to the absence of Pauli’s principle in that case.

*q*are immediately found to be

*k*, and the time scale \(\delta t\). Similarly to the free boson case, it is fair to state that the parameter \(\gamma _{F}\) encodes all the late-time entanglement properties between opposite momentum modes. The resemblance between the formulas (72), (73) and (43), (44) for the fermionic and bosonic entropy production is striking. In fact, the expressions for the Renyi entropies can be converted into minus one another under \(\gamma _{F}\leftrightarrow -\gamma _{B}\). Notice that this simple relation between the bosonic and fermionic Rényi entropies does not commute with the limit \(q\rightarrow 1\), i.e., it is not shared by the expressions for the EE.

The entropies are again monotonic functions of \(\gamma _{F}\) and can be inverted to determine \(\gamma _{F}(S)\), i.e., the information concerning mode entanglement is in principle enough to determine all the quench parameters. An important difference with respect to the bosonic result, however, is that the EE in the present case is limited from above by \(\log 2\approx 0.7\), since the reduced state for particles with momentum *k* lives in a two-dimensional Hilbert space. In particular, even when one of the masses vanishes (i.e., when the quench crosses a critical point) this upper bound prevents the occurrence of the divergent zero mode contribution that takes place for bosons. Another important difference is that now the expressions are not symmetric under \(\omega _\text {in}\leftrightarrow \omega _\text {out}\), so the behavior for \(m_\text {in}>m_\text {out}\) is distinct from \(m_\text {in}<m_\text {out}\) (or \(\delta m>0\) and \(<0\), respectively) and must be analyzed separately, as we shall see.

Figure 7 shows the *k*-dependence of the EE and the second Renyi entropy for a fixed quench rate \(\delta t\). The lefthand side figure corresponds to quenches that decrease the mass, while the other two to increasing-mass quenches. For the former the entropies are always monotonically decreasing functions of *k*, showing that entanglement production is more noticeable between IR modes. However, for the case of increasing-mass quenches this is only true up to a critical value of \(\delta m\) (see (b) and (c)). Above this critical value of \(\delta m\) this monotonic behavior is broken as shown in the right figure. Interestingly, this indicates that for large enough final masses the maximal entanglement production is not achieved at IR modes but rather at an intermediate momentum value \(k=k_\text {max}\).

Figure 8 illustrates the dependence of the entropies on the time scale \(\delta t\). Cases (a), (b), (c) show the result for a single mode *k* in a quench with \(\delta m<0\) (the former) and \(\delta m>0\) (the latter two), respectively, while (d), (e), (f) show the total entanglement produced in each of the two situations, obtained after integration over all \(k\ge 0\) (this is the left-right entropy studied, e.g., in [31]). We see that in mass-decreasing quenches the entanglement production is always bigger the faster the quench is done. For a single mode *k*, this is still true for mass-increasing quenches up to a limiting value \(\delta m^*>0\) ((a) and (b)) but fails to be true for deformations stronger than this (case (c)), where maximum production is then achieved at an intermediate value of \(\delta t\). This unusual feature disappears when one integrates over all modes *k*, as shown in (d). Notice also that again adiabatic quenches do not produce any mode entanglement, since all curves asymptote to zero as \(\delta t\) grows.

## 5 Connection with the generalized Gibbs ensemble

In this section, we will show that our above-mentioned results for the Renyi and entanglement entropies agree with the predictions of a Generalized Gibbs Ensemble, showing that this steady state correctly describes the late time dynamics after smooth quenches even from the point of view of entanglement measures (in addition to the well-understood case of correlation functions).

*k*via the parameter \(\gamma _{B}\). This gives support to the results of [33], where thermalization to a GGE was argued to hold at the level of two-point correlators.

A similar construction should hold also for the fermionic model, in which case the Lagrange multipliers characterizing the GGE would be fully determined by the fermionic parameter \(\gamma _{F}\) of (71).

## 6 Conclusions

In this work, we have calculated the late-time entanglement in momentum space produced after smooth mass quenches for free scalar (in *d* dimensions) and fermion theories (in 1+1 dimensions). The strategy is inspired by [15, 16], which studied the entanglement production for quantum fields in an expanding universe, and can be translated to the problem of a quench through a Weyl rescaling of the field. The initial state was taken to be the vacuum as defined by a pre-quench observer. As the quench is performed, particle excitations carrying all possible momentum modes are produced as told by a post-quench observer. We then calculated the entanglement and Rényi entropies between a single quantum field mode \({{\varvec{k}}}\) and its opposite mode carrying momentum \(-{{\varvec{k}}}\), which is the only non-trivial case in our exactly solvable model (i.e., no entanglement is produced between other pairs (\({{\varvec{k}}},{{\varvec{k}}}'\)) of modes other than this). The results have simple analytical formulas and are expressed in terms of a single parameter, \(\gamma _{B}\) for bosons and \(\gamma _{F}\) for fermions, that encodes all the late-time entanglement properties.

For (\(1+1\))-dimensional bosons, we have shown that our results match the predictions of a Generalized Gibbs Ensemble where the conserved charges are taken to be the mode number operators as defined by the post-quench observer, namely \(N^\text {out}_{k}=a^{\dagger \,\text {out}}_{k}a^\text {out}_{k}\) (for all *k*). We were able to precisely calculate the Lagrange multipliers that characterize the GGE in terms of the parameters \(m_\text {in},m_\text {out},\delta t\) defining the quench, namely \(\lambda _k=-\log \gamma _{B}\). Hence, having fully specified the GGE state, the long-time behavior of any local observable after the quench can now be calculated (not only the entanglement and Renyi entropies presented here).

In the bosonic case, for both the Tanh and the Sech mass profiles, we saw that at any fixed quench rate \(\delta t\) more entanglement is produced between light modes (the ones carrying small momentum *k*) than between heavy ones, as expected. The entanglement production is monotonically decreasing with *k* and its magnitude grows with the magnitude of the quench, characterized by the absolute value of the difference \(\delta m^2={m^2_\text {out}}_{}-{m^2_\text {in}}_{}\) of the initial and final mass in Tanh case and by \(m_{0}^2\) (the maximum value of the mass during the quench) in the Sech case. The picture is qualitatively similar to the particle production rate given by \(n_k=|\beta _k|^2\).

The dependence on \(\delta t\) for a given mode \({{\varvec{k}}}\) is more interesting: for both profiles it is true that more entanglement is produced for faster quenches (small \(\delta t\)) with respect to slow ones, while adiabatic quenches (\(\delta t\rightarrow \infty \)) do not produce entanglement at all. In particular, this means that, among all the quenches reaching some \({m^2_\text {out}}_{}\ne 0\), maximal entropy production is achieved for the one that started from a CFT (i.e., \({m^2_\text {in}}_{}=0\)). However, for the Tanh profile the entanglement is found to decrease monotonically as a function of \(\delta t\), while for the Sech profile the decrease is non-monotonic, being given by damped oscillations modulated by *k*.

In the fermionic case the results are more subtle. First we noted that the sign of \(\delta m=m_\text {out}-m_\text {in}\) becomes important, that is, the result here is not invariant under \(m_\text {in}\leftrightarrow m_\text {out}\) and, as a consequence, whether the quench increases or decreases the mass has a significant impact on the final result. For a fixed rate \(\delta t\), and for a quench decreasing the mass, we find a very similar result to that of the Tanh profile for bosons. For quenches that increase the mass, however, the results remain similar to that of bosons only up to a certain critical value \(\delta m^{*}\), above which the entanglement ceases to be a monotonic function of |*k*|. Instead, as we increase the mass beyond this critical value, we observe a peak at some intermediate value \(k_\text {max}\) that grows as \(\delta m\) grows. In other words, for quenches that increase too much the mass it fails to be true the statement that more entanglement is produced between light modes. Also, for a particular mode *k*, the behaviour of the entanglement and Rényi entropies as a function of \(\delta t\) has the same features discussed in the last paragraph. Namely, for quenches that decrease the mass the result is very similar to the bosonic one, while for quenches that increase the mass this holds true only up to a critical value \(\delta m^{*}\) above which the entropies cease to decrease monotonically with \(\delta t\), becoming instead peaked at an intermediate value. If we integrate over all modes *k* to get the total entanglement production the entropies remain monotonically decreasing, although for \(\delta m > \delta m^{*}\) we can still notice a small bump at an intermediate value of \(\delta t\).

We believe these results shall hold (at least qualitatively) for generic mass quenches in free field theories, that is, regardless of the precise form of the quench profile *m*(*t*). For instance, we expect the same behavior as in the Tanh case for any other quench profile that goes from the same \(m_\text {in}\) to the same \(m_\text {out}\) within a finite scale \(\delta t\). This expectation is supported even by comparing the qualitative results for the Tanh and the Sech profiles (which do not share the same initial and final masses), namely the fact that the entanglement production is more significant for light modes and faster quenches. Of course we have no formal proof of that, but it would be very surprising if by simply changing *m*(*t*) the *k* or \(\delta t\) dependence of the entropies happened to be qualitatively different.

As future prospects, one would like to perform similar calculations for weakly coupled quantum field theories in order to access how the presence of interactions modify the results. Another interesting generalization would be a geometric prescription for computing momentum-space entanglement in AdS/CFT, by generalizing the Ryu–Takayanagi prescription [34, 35]) which is appropriate to deal only with real-space entanglement entropy. This was already speculated on [36] but the problem remains open. In fact, as discussed in [36], one of the main difficulties for putting forward a holographic proposal on momentum-space entanglement is the fact that this quantity is not well studied on the field theory side (specially for interacting theories). Hence, we hope our work can contribute to this issue.

## Footnotes

- 1.
Actually in \(d=2\) the conformal coupling (6) vanishes, so the FLRW scalar field has the usual minimal coupling.

- 2.
Sometimes for a better handling of UV divergences it is convenient to restrict \({{\varvec{k}}}\) to a finite range such as a torus \({\mathbb {S}}^{d-1}\) of length

*L*(i.e., taking \(k_i\in [0,L]\) with periodic boundary conditions at the two extrema). We will be implicitly doing this in the following when writing for instance Kronecker deltas \(\delta _{{{\varvec{k}}}{{\varvec{k}}}'}\) instead of Dirac ones, which serves well our physical purposes here without unnecessary technical complications of dealing with infinities. - 3.
Strictly speaking, the product over momentum modes above is not well-defined since the momentum \({{\varvec{k}}}\) is a continuum variable. The way to make sense of it is by restricting \({{\varvec{k}}}\) to a compact space (e.g., using periodic boundary conditions) as mentioned previously. Anyway, none of the conclusions below are affected by this subtlety.

- 4.
Here we are using the shorthand notation \(|n_{\text {in}_{{\varvec{k}}}}\,n_{\text {in}_{-{{\varvec{k}}}}}\rangle \equiv |n_{\text {in}_{{\varvec{k}}}}\rangle \otimes |n_{\text {in}_{-{{\varvec{k}}}}}\rangle \) for tensor products between states belonging to subspaces with opposite momenta \(\pm {{\varvec{k}}}\) (and similarly for “out” states).

- 5.
- 6.
The subscript

*B*stands for “boson”, to be contrasted with the fermionic case in the next section. - 7.
When \({m^2_\text {in}}_{}={m^2_\text {out}}_{}\), \(\gamma _{B}\) vanishes and there is no entropy production, which is trivial since in this case there is no quenching at all (see (21)).

- 8.
Equivalently, from the curved spacetime point of view the bosonic mass profile (21) corresponds to a FLRW metric with scale factor \(a(t)=\sqrt{A+B\tanh \frac{t}{\delta t}}\) while the fermionic one (52) to \(a(t)=A+B\tanh \frac{t}{\delta t}\) (with the cosmological parameters

*A*,*B*related to the ratios \(\frac{m_\text {in}}{m},\frac{m_\text {out}}{m}\) as in (22)).

## Notes

### Acknowledgements

The authors are pleased to thank Veronika Hubeny, Mukund Rangamani, and Horatiu Nastase for discussions and comments on the draft. D.W.F.A. would like to acknowledge hospitality at the QMAP center of UC Davis, where part of this work was developed. D.W.F.A is supported by the PDSE Program scholarship of CAPES - Brazilian Federal Agency for Support and Evaluation of Graduate Education within the Ministry of Education of Brazil, and by the CNPq Grant 146086/2015-5. G.C. acknowledges financial support from the Brazilian ministries MCTI and MEC.

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