Ultra-stable charging of fast-scrambling SYK quantum batteries

Collective behavior strongly influences the charging dynamics of quantum batteries (QBs). Here, we study the impact of nonlocal correlations on the energy stored in a system of N QBs. A unitary charging protocol based on a Sachdev-Ye-Kitaev (SYK) quench Hamiltonian is thus introduced and analyzed. SYK models describe strongly interacting systems with nonlocal correlations and fast thermalization properties. Here, we demonstrate that, once charged, the average energy stored in the QB is very stable, realizing an ultraprecise charging protocol. By studying fluctuations of the average energy stored, we show that temporal fluctuations are strongly suppressed by the presence of nonlocal correlations at all time scales. A comparison with other paradigmatic examples of many-body QBs shows that this is linked to the collective dynamics of the SYK model and its high level of entanglement. We argue that such feature relies on the fast scrambling property of the SYK Hamiltonian, and on its fast thermalization properties, promoting this as an ideal model for the ultimate temporal stability of a generic QB. Finally, we show that the temporal evolution of the ergotropy, a quantity that characterizes the amount of extractable work from a QB, can be a useful probe to infer the thermalization properties of a many-body quantum system.

JHEP11(2020)067 We consider an SYK system of N QBs under a unitary charging protocol, as pictorially sketched in figure 1. We argue that nonlocal and chaotic correlations, after the initial quench, lead to a very fast -and homogeneous -excitation of many energy levels, with huge creation of entanglement [58,59] and, more importantly, in a collective fashion. By characterizing fluctuations of the average energy stored, we show that this collective behavior is reflected in an exponential suppression of the temporal fluctuations at all time scales, leading to an ultra precise charging of the QB. To corroborate these results we perform extensive numerical simulations, based on exact diagonalization, showing also a systematic comparison with a prototipical quantum system with many-body local correlations, i.e. a one-dimensional spin chain in the Anderson, ergodic or many-body localized (MBL) phase [60]. This paper is organized as follows. In section 2 we introduce the unitary charging protocol, its model-independent features and some preliminary definitions. In section 3 and section 4 we introduce the SYK model under investigation and we compare its behavior to a spin chain in the MBL phase. We analyze energy fluctuations of different kinds, i.e. disorder, quantum, and temporal fluctuations, showing comparison between SYK and spin-chain (MBL or Anderson) based QBs. In particular, we demonstrate that SYKbased QBs result in exponentially suppressed temporal fluctuations at all times, a peculiar feature that can be linked to the collective and nonlocal nature of the system and to its chaotic, and fast thermalizing, property. In section 5 we inspect the role played by quantum chaos in suppressing the temporal fluctuations. We will argue that the suppression of the temporal fluctuations of a generic, chaotic, QB can be linked to the spectral rigidity of the corresponding quench Hamiltonian. On this respect, the very high degree of spectral rigidity of the SYK Hamiltonian, as observed in [61], explains the great performance of the corresponding SYK QBs. In section 6 we make a digression and we study the amount of extractable energy from a SYK-like QB, i.e. its ergotropy. We show that in general a SYK QB displays very low values of ergotropy, a feature that can be traced back to its highly entangling dynamics. Interestingly, we argue that by inspecting the time evolution of this quantity one can infer the thermalization time scale of a quantum system. Section 7 contains a summary of our main findings.

JHEP11(2020)067 2 Charging protocol and energy fluctuations
We study the charging mechanism of a QB, following a unitary protocol based on a doublesudden quench [16,[18][19][20]. The system is initially assumed to be in the ground state |0 of a given time-independent Hamiltonian,Ĥ 0 (empty battery). Subsequently, it evolves under the HamiltonianĤ (t) =Ĥ 0 + κ λ(t)Ĥ 1 , (2.1) whereĤ 1 is a time-independent driving Hamiltonian and the dimensionless parameter κ controls the relative strength betweenĤ 0 andĤ 1 . The function λ(t) describes the charging time interval and is defined by λ(t) = 0 , t < 0 and t > τ , with τ being the charging time. Denoting by |ψ(t) the evolved state under the total Hamiltonian (2.1) (in this work we set = 1), the averaged energy stored in the QB at the end of the charging time is where we defined Ĥ 0 τ ≡ ψ(τ )|Ĥ 0 |ψ(τ ) . Unless specified, we will always consider many-body QBs composed of N cells (in our case N qubits) whose static Hamiltonian is given bŷ

4)
σ α j (α = x, y, z) denoting the usual spin-1/2 Pauli operators corresponding to the jth qubit, and h being the QB energy scale. We will also indicate withĤ In a many-body QB, the stored energy E(τ ) may display some universal features as a function of τ : it undergoes an initial growth for small τ , while at larger times it fluctuates in time around an average value [30] E(τ 1 , τ 2 ) = 1 whose precise value depends on the specific model of QB considered. In order to analyze the speed and performance of the charging protocol, we define an optimal charging timeτ as the one at which the energy stored in the QB reaches a value equal to a fixed fraction of the average energy. Notice that, since temporal fluctuations are always present, the usual definition ofτ as the time at which the energy stored in the battery reaches its maximum value is not well defined. The charging precision of a QB is influenced by different and independent factors that may be responsible for temporal, disorder and quantum fluctuations. The first kind of fluctuations can be quantified by computing Hereafter we will use the symbol · in order to denote the average over different realizations of the charging HamiltonianĤ 1 , which may depend on some parameters drawn according to a given probability distribution. We also define the dimensionless quantity where ∆Ĥ 0 = N h is the bandwidth ofĤ 0 in eq. (2.4). More in general, the bandwidth of an Hermitian operatorÔ is defined as the norm ∆Ô ≡ µ Disorder fluctuations may be responsible for an indetermination in E(τ ) due to imperfections in the fabrication of the QB, which can be modeled as suitable random parameters entering the full HamiltonianĤ. These fluctuations are defined by On the other hand, quantum fluctuations are caused by quantum indetermination, which is intrinsically present in the charging process, since |ψ(τ ) is not an eigenstate ofĤ 0 . These can be quantified by σ As before, we introduce the dimensionless quantities It would be desirable to find models of QBs able to reach high values ofĒ and, at the same time, minimizing the various fluctuations. While an overall rescaling of the whole Hamiltonian (Ĥ → αĤ) only implies a redefinition of times, and thus can be easily taken into account, the role played by the relative strength κ of the charging Hamiltonian is less trivial [see eq. (2.1)]. A small value of κ makesĤ 1 to be a small perturbation of the global HamiltonianĤ, thus resulting in a low value ofĒ. In contrast, by increasing κ, the charging HamiltonianĤ 1 becomes a strong perturbation and may induce transitions from |0 to the highly excited states ofĤ 0 .

Models
To unveil the role played by the many-body character of a QB on its charging precision, we have studied a variety of Hamiltonians, which can be seen as one-dimensional spin-1/2 chains (each spin representing a quantum cell, as discussed above). These include interactions among the various spins and disorder in the coupling strengths: as we shall see below, the interplay between two such ingredients is crucial to stabilize the process of energy injection and thus to any reliable definition of many-body QB.
The first class of charging Hamiltonians is given bŷ

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The first term in the right-hand-side stands for a nearest-neighbor Ising coupling, where the coefficients J j are composed of a constant piece plus a static random fluctuation term, J j = J +δJ j , with δJ j sampled over a uniform distribution with support in [−δJ , δJ]. The second term describes a next-to-nearest-neighbor interaction with fixed coupling constant J 2 . The Hamiltonian in eq. (3.1), together with the static QB model (2.4), constitutes a many-body system that exhibits a variety of different quantum phases, ranging from the Anderson localized (AL) to the many-body localized (MBL), as well as to the ergodic phase. The phase diagram ofĤ 0 +Ĥ (MBL) 1 has been extensively studied, since it represents one of the prototypical models of many-body localization-delocalization transition [60,[62][63][64].
The second class of charging Hamiltonian that we analyze is a highly nonlocal model inspired by the so-called "Kourkoulou-Maldacena" SYK model [65]. In its original formulation, the SYK model describes a strongly interacting system of 2N Majorana fermions, coupled through a fully nonlocal, all-to-all, random interaction. Denoting withγ j the jth Majorana fermion (γ j =γ † j , with j = 1, . . . , 2N ), such that {γ i ,γ j } = δ ij , the SYK charging Hamiltonian writesĤ The couplings J ijkl of the quartic term are randomly Gaussian distributed, with null mean values and variance The above Hamiltonian can be cast in the spin-1/2 setting by employing a Jordan-Wigner transformation (JWT),γ Once rewritten in this language, the SYK model looks highly nonlocal and interacting, contrary to the charging Hamiltonian of the MBL spin chain [eq. (3.1)], which instead couples next-to-nearest neighbors, at most. On the opposite hand, the static "Kourkoulou-Maldacena" HamiltonianĤ 0 , which in the Majorana language readŝ is equivalent to the spin-1/2 Hamiltonian of eq. (2.4), after using the above JWT. Each quantum cell of the QB thus corresponds to a couple of neighboring Majorana fermions. As we shall see in the next sections, these two classes of models exhibit radically different behaviors in terms of QB charging stability. The reason intimately resides in the nonlocality of SYK, and on its highly chaotic nature, with respect to a nearest neighbor or next-to-nearest neighbor spin chain models. These particular values of the parameters, for the cases (i) and (ii), have been chosen to coincide with the values considered in [30]. However, we have extensively cheched that all the results shown and discussed hereafter are qualitatively the same with respect to the specific choice of the Hamiltonian parameters, within the same class of model.

Temporal fluctuations
We first focus on the temporal fluctuations of the average energy E(τ ) stored in the QB through the charging protocol. To give a hint on the importance of such kind of noise in the charging performance, we analyze the time behavior of the ratio R(τ ) between the energy stored in the QB and the bandwidth ofĤ 0 , (4.1) Figure 2 displays some representative results for the indicator R(τ ) as a function of the charging time τ , for a single ensemble realization, in the AL, the MBL, and the SYK cases. The numerical data have been obtained by exact diagonalization and discrete time evolution, using logarithmic time step intervals. We checked that the time step intervals were sufficiently small to ensure the convergence of the results. The various curves stand for different values of the dimensionless parameter κ in the charging Hamiltonian (2.1). We observe that all the curves grow as a function of time τ until they saturate to a value JHEP11(2020)067 corresponding toĒ, whose precise value depends on κ. Moreover, by increasing κ,Ē increases as well, up to the value R(τ ) ∼ 1/2, while any larger κ does not help in further increasingĒ, while it simply reduces the value ofτ . This means that, when κ is strong enough, the quench Hamiltonian induces a transition from |0 to a superposition involving several eigenstates ofĤ 0 , symmetrically distributed around the center of the bandwidth ofĤ 0 , thus ensuring thatĒ ∼ ∆Ĥ 0 /2. On the other hand, the temporal fluctuations are mostly unaffected by κ and one has to find smarter ways to reduce them. This set of fluctuations will be the main focus of the analysis, and we will show how they can be efficiently suppressed. As we will see, the internal structure ofĤ 1 will play a crucial role for this task.
In ref. [30], it was argued that the presence of interactions inĤ 1 can help in reducing the temporal fluctuations during the charging of the QB. Here we show that, while local interactions have only limited effects on the fluctuations, nonlocal correlations allow to build models of QBs with high temporal stability inĒ. These qualitative features are already visible from figure 2, where we clearly see that by increasing the non locality degree of the interactions (from left to right panel) temporal fluctuations are greatly reduced. One of the main goals of this paper will be to explain such behavior and to quantitatively describe it.
We now fix the value of κ and first comment on the optimal charging time of the QB, τ , evaluated as the time at which the energy stored in the QB reaches a value equal to 99% of the average energy (we have tested that the results are not affected by the arbitrary choice of this cutoff), as well as the corresponding energy Ē . We now refer to quantities averaged over several realizations ofĤ 1 . Our numerical simulations indicate (not shown) that, for all the three cases considered, the optimal charging time is a decreasing function with the number N of cells, while the averaged energy stored in the battery at the optimal time scales linearly with the number of sites. The last property easily follows from the fact that, as already pointed out in section 2, when κ is large enough, Ē is determined by the bandwidth ofĤ 0 , which scales linearly with N . These two observations agree with the results obtained in ref. [30] for the MBL model, and certify that all our models are indeed able to properly charge the battery.
To obtain a more quantitative assessment of the role of temporal fluctuations in the QB charging mechanism, we have performed further extensive simulations for the MBL and the SYK model, by fixing the constant κ. The corresponding analysis of the AL model is not reported in the main text, since already from figure 2 is clear that it shows huge temporal fluctuations at all times. For sake of completeness, this is reported in appendix B.
In order to make fair comparisons between these models, we have set the constant κ in the MBL QB equal to one, while for the SYK QB it has been fixed in such a way that the two quench Hamiltonians have the same bandwidth, ∆Ĥ MBL 1 = ∆Ĥ(SYK) 1 . The results of our analysis are reported in figure 3.
The upper part of panels 3(a) and 3(b) display the ratio R(τ ) of eq. (4.1), for a single ensemble realization of the MBL spin chain and the SYK battery, respectively. We immediately recognize that, compared to the analogous plot for the Anderson spin chain [cf. figure 2(a), the green curve], the MBL battery is able to partially reduce the JHEP11(2020)067  The functions R(τ ) (upper part) and σ k in the various energy sectors (lower part) as a function of time, for a single realization of the random variables. Shaded areas denote the time intervals for the early-and late-time window (light grey and dark grey, respectively) analyzed below -see text. Hereτ = 0.14 and 0.12, in the two panels. Panels (c) and (d): The early-and late-time temporal fluctuations, Σ (t) N (τ 1 , τ 2 ) of eq. (2.7), for the MBL and the SYK models, as a function of the number N of QB cells. Continuous lines correspond to the fits (4.10)-(4.11) and (4.12), respectively. Results in the two bottom panels have been obtained by averaging over 500 (c) and 100 (d) ensemble realizations. temporal fluctuations at late times, i.e. for times much larger than the optimal charging time τ τ . However, at early times, i.e. for times roughly included in the light grey areas in the panels, fluctuations are still very large. On the contrary, the plot clearly shows that the SYK battery is extremely precise and stable at any time scale: all the temporal fluctuations, after reaching τ , are completely removed. To analyze this fact we shall also define the JHEP11(2020)067 following time interval, identified by the dark grey areas in the panels, It should be stressed that the early-time window is very relevant for energy storage purposes, since one would like to have a great control of the charging precision immediately after reaching the saturation of the energy stored. We stress that the precise values of the early and late time windows are not important, and the details of these choices do not affect qualitatively the behaviors we are discussing. The two plots 3(a) and 3(b) clearly unveil the qualitative advantage of the SYK model, and confirm the intuition that nonlocal correlations play a crucial role in the charging dynamics. Thus, a strongly interacting, nonlocal, quench (like the SYK model) represents a perfect candidate to build models of very stable QBs with high charging precision. Moreover, in section 5 we show that nonlocality alone inĤ 1 is not enough, and that the highly chaotic dynamics of the SYK system is crucial in order to efficiently suppress the temporal fluctuations. In appendix A we analyze the role of the static HamiltonianĤ 0 , by making a comparison with another kind of SYK-like QB with a nonlocalĤ 0 and showing that the two models are qualitatively analogous. Therefore the precise form ofĤ 0 does not play a major role in the charging dynamics.

Fluctuations in terms of transition amplitudes
Let us now have a closer inspection at the microscopic origin of the improved efficiency and charging stability, in the presence of nonlocal correlations. In general, temporal energy fluctuations are caused by transitions of the probability amplitudes The first condition is immediate to understand: if two eigenstates have similar energies, the energy stored in the battery will not vary much after the transition (the extreme case would be a transition between degenerate eigenstates).
The second condition is more subtle: let us consider the limit case in which the evolved ket, |ψ(t) , can be written as a superposition of all the eigenstates ofĤ 0 with approximately the same probability amplitudes where D is the dimension of the system's Hilbert space. Since D = 2 N is exponentially large in the number N of cells, all the coefficients c k,i will be very small. In this case, a JHEP11(2020)067 transition between eigenstates, even with very different energies, will not be reflected in large fluctuations. Indeed, since the bandwidth ofĤ 0 scales linearly in N , the fluctuation will be, at most, exponentially small in N . In contrast, in the situation where only just few (of order N ) eigenstates ofĤ 0 are involved in the expansion of the evolved state, some of the coefficients c k,i can be relatively large and a transition including one of these states will cause a large fluctuation in the energy stored.
Given these considerations, we expect that in the MBL case the evolved state at early times should have non vanishing overlap with just few eigenstates ofĤ 0 (for each energy level), while involving more and more states at late times, thus reducing the associated fluctuations. On the contrary, for the SYK model the evolved state should involve a large portion of the Hilbert space ofĤ 0 from the very early times.
To corroborate this hypothesis, we first notice that the energy spectrum ofĤ 0 is formed by several lines, well separated from each other. Each line having energy E k has a degeneracy degree d k counted by the number of configurations with the right number of aligned spins. Hence, to estimate if, for a given energy eigenvalue, the evolved state has non vanishing overlap with just few or many eigenstates ofĤ 0 , we consider the quantity which expresses the probability of measuring the evolved state in one of the eigenstates |k, i associated to the level E k . We have taken into account all of such eigenstates, i.e. with fixed k and varying i = 1, . . . , d k , and computed the standard deviation associated to the corresponding quantities (4.8), divided by their average value. Namely, We can thus determine if the expansion of |ψ(τ ) in the degenerate eigenstates for a given energy level is involving many or few of the eigenstates, with the former case corresponding to small values σ k and the latter associated to large values of σ k . The results are reported in the lower parts of panels 3(a) and 3(b), for the MBL and the SYK model, and confirm our conjecture: the MBL system shows at early times huge values of σ k for each energy sector, and in correspondence with these peaks we can clearly trace a huge temporal fluctuation of the average energy stored in the battery (see the upper panel of the figure). This behavior gets reduced by increasing time and, after bouncing for a while, the system reaches low values for all the σ k s. On the other hand, from the very beginning, JHEP11(2020)067 the SYK model displays low values for σ k (around one order of magnitude smaller), clearly showing that in this model many more eigenstates ofĤ 0 , for each energy level, are rapidly involved in the expansion of the evolved state. Hence, the charging protocol turns out to be very stable, and this is reflected in the very small temporal fluctuations. It should be emphasized that the low values of all the σ k s, for the SYK model, are reached at a time scale which is even shorter than the optimal charging time, τ , thus ensuring the total absence of temporal fluctuations.
This microscopic argument confirms that temporal fluctuations get suppressed when an initially localized state (in the eigenbasis ofĤ 0 ) spreads and covers a very large portion of the eigenstates ofĤ 0 and, as such, we think that it could be naturally linked to the physics of scrambling and of thermalization. Indeed, the thermalization properties of the SYK model have been already investigated in refs. [66,67], where it has been demonstrated that this model shows thermalization, even without long time averaging. This fact corresponds to a quantum version of mixing, a much stronger phenomenon as compared to ergodicity [67]. On the other hand, a MBL system does not thermalize in the thermodynamic limit. Hence, we expect that the huge suppression of the temporal fluctuations at late times, in this case, should be a finite N effect: by increasing the size of the system, the time at which the fluctuations are highly suppressed should tend to infinity. This is consistent with the results, reported in figure 4(b), of ref. [30], where an increase in the temporal fluctuations of the MBL model (without separating early and late times) could be observed at the largest values of N .

Temporal fluctuations of the charging energy in the early-and late-time windows
So far we have argued that, in general, during the charging process of a generic QB, two different time windows can be identified: after reaching the optimal charging time, τ , we have an "early-time" window, in which the averaged energy stored in the battery, E(τ ), undergoes huge temporal fluctuations, the expansion of the evolved state |ψ(τ ) on the basis of the eigenstates ofĤ 0 involves just few eigenstates for each energy level. On much larger time scales, the dynamics turns to a "late-time" window, in which the energy E(τ ) displays suppressed temporal fluctuations, the evolved state has spread to cover a large portion of the eigenstates ofĤ 0 . We have also argued that the time of crossover, between the early time and the late time behavior, is connected with the thermalization properties of the system under investigation and, as such, it is model-dependent.
We now turn to a more explicit evaluation of the temporal fluctuations in the charging energy, [cf. eq. (2.6)], in the two time windows defined before. To be precise, we address the dimensionless quantity Σ (t) N (τ 1 , τ 2 ) of eq. (2.7) in the two time windows (early and late) over which we take the time integral. In figure 3(c) we plot the results for the MBL spin chain. The behavior for different N , of (2.7), at early and late times is qualitatively different: while at late times Σ is fastly decreasing with N , at early time Σ instead shows a much slower decrease. In figure 3(d) we plot the results for the SYK battery. Here, the situation is different: both the early and late time fluctuations are rapidly suppressed in N .

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These observations can be made quantitative: in the MBL case, the early time curve is greatly reproduced by the function with a and b being fitting parameters. On the other hand, the late time behavior is well reproduced by Σ which shows that the late time temporal fluctuations are exponentially suppressed with N . The numerical data for the SYK case instead can be reproduced by the function In summary, this shows that the temporal fluctuations, in the SYK model, both at early and late times, are exponentially suppressed by increasing the size of the battery. On the other hand, the MBL battery, shows this exponential suppression only at late times, while at early times it follows a 1/ √ N suppression factor, only. The exponential suppression, at early times, of SYK battery, makes clear that with this model it is possible to obtain very stable charging protocols, in which the average energy stored in the battery is essentially determined with very high precision, even with relatively small batteries.
It has been shown in [68] (in the similar context of work extraction) that precisely an exponential and a 1/ √ N suppression factors are associated to, respectively, collective processes, i.e. processes in which all the cells are collectively controlled in the protocol, and single cells protocols, in which each cell is individually processed. It is then natural to expect that, due to the nonlocal nature of its hamiltonian, the SYK has a genuine collective dynamics from the very early times, shorter than the optimal charging time τ , while the MBL battery needs a certain amount of time to start a collective dynamics, with an initial single-body behavior. On this respect, we conclude that an integrable Hamiltonian, like the AL spin chain in the left panel of figure 2, never reaches a collective dynamics. In section 5 we will confirm this intuition, by showing that the presence of quantum chaos in the quench Hamiltonian is necessary to ensure the exponential suppression of the fluctuations. This is in perfect agreement with the microscopic description of the fluctuations we have provided in section 4.1.1, and with the findinds of section 6, where we show that the MBL battery needs a large amount of time in order to involve a large portion of the eigenstates ofĤ 0 in the expansion of the evolved state |ψ(τ ) . Moreover, the absence of a crossover in the SYK system can be again understood in terms of the microscopic description of section 4, where we observed that for the SYK system |ψ(τ ) involves a large portion of the Hilbert space from times which are smaller than the optimal charging time, thus ensuring that all the temporal fluctuations are exponentially suppressed.

Disorder and quantum fluctuations
We now move to the discussion of the disorder and quantum fluctuations for all the three models considered. From (2.8) and (2.9), we see that both these quantities have to be JHEP11(2020)067  Moving to quantum fluctuations, Σ (q) N (τ ), cf. eq. (2.10), the results are reported in figure 5. Again, we find that the MBL model shows a crossover when moving from the early time window to the late time window, becoming more similar to the SYK behavior only at late times. We also see that the quantum fluctuations for the SYK model are larger than for all the other models we have considered. Furthermore, it is worth to notice that for all the three models, the values of the temporal and disorder fluctuations are quite similar, while the quantum fluctuations are always larger than the other sources of fluctuations, reaching the largest value of ∼ 0.50 for the SYK battery. This large value of quantum fluctuations for the SYK model could be put in relation with its high charging power, [69], since it has been recently observed in ref. [26] that high levels of quantum fluctuations are necessary in order to increase the charging power of a QB.

The role of quantum chaos on the charging stability
We now elucidate the role that quantum chaos plays in the suppression of the temporal fluctuations, i.e. in the charging stability of a QB.
To this end, it is instructive to consider two slightly different models of QBs. In both cases, the unitary charging protocol is given by: whereĤ 0 is the local constant Hamiltonian (2.4). The quench termsĤ s 2 with s = F, B, instead, are random mass Hamiltonians, that means quadratic in the field operators, the two differing for their fermionic/bosonic statistics (see below). The caseĤ F 2 is defined bŷ with the random couplings K ij having null mean values and variances On the other hand,Ĥ B 2 is built using the following, real, hard-core bosonic operators,χ i , with i = 1, . . . , 2N , which satisfy the following algebra:  Figure 6. The charging ratio R(τ ) -see eq. (4.1) -as a function of τ , for a QB described by the Hamiltonian in eq. (5.1). We study a single realization of the coupling constants K ij , for both the bosonic quadratic model, eq. (5.5), and the fermionic model, eq. (5.2), for N = 15.

The bosonic quench Hamiltonian is then given bŷ
with the same random coupling constants K ij as in (5.3). The factor s(i, j) reads and it ensures thatĤ B 2 is Hermitian. Although very similar, being both quadratic in the field operators, the two quench Hamiltonians (5.2) and (5.5) have very different properties: the fermionic system is indeed integrable, while the bosonic model is chaotic. This can be verified by inspecting simple quantum chaos diagnostics, like the so-called r-statistics [70], as we show in appendix C.1.
This difference has a direct consequence on the QB performance, as shown in figure 6 where the charging for a single realization of the coupling constants K ij is reported. As usual, we have fixed the constant κ to ensure thatĤ B 2 andĤ F 2 have the same bandwidth. From the figure, one can clearly see that, despite the striking similarity of the two models, the temporal stabilities are completely different, with the bosonic model, being chaotic, which efficiently truncates the fluctuations while the fermionic one shows large fluctuations even at late time. This shows that quantum chaos is needed to reach charging stability. Given this result, a natural question which arises is whether more conventional chaotic models, like spin chain models in the ergodic phase, show the same level of charging stability of the SYK quench defined in (3.2). As shown in appendix C.2, however, it turns out that the charging stability of the SYK QB is exceptional and definitely larger than the level of charging stability reached by an ergodic spin chain.
It thus remains to understand which particular feature of the chaotic SYK Hamiltonian is responsible for the extremely efficient suppression of the temporal fluctuations. While a detailed understanding of this point is beyond the scope of the paper, we observe here that the short-range chaos observables, like the r-statistics, simply tell us whether a certain Hamiltonian, at very small energy scales (or equivalently for very long times), shows the same spectral correlations as predicted by random matrix theory (RMT). However, they say nothing about how large in energy, or equivalently how short in time, the agreement with RMT persists.
To address this issue, which in a sense defines how strong is the chaotic nature of a system, one has to study the so-called long range chaos diagnostics, like the spectral rigidity or the associated spectral form factor [71]. On this respect, in [61] and [72], it has been observed that the Thouless time, which is the time scale at which the agreement with RMT becomes manifest, for the SYK model is very small, of order log N , while for more canonical spin chain models is parametrically larger, of order N or N 2 . This in turn implies that the SYK model has a much larger spectral rigidity than the most conventional spin chain ergodic models, i.e. it is strongly chaotic. We believe that the very large spectral rigidity of the SYK Hamiltonian is the key ingredient behind the excellent performance of the SYK QB. As a concluding remark on this aspect, to further corroborate the agreement with RMT, we can compare the SYK performance with the one of a QB in which the quench Hamiltonian is extracted directly from the Gaussian unitary ensemble (GUE), i.e. in which the quench Hamiltonian is a random hermitian matrix, although such a QB does not represent any physical model by itself. For the reasons just explained, i.e. for the role played by the spectral rigidity, the GUE based QB would represents the upper limit to the possible charging stability of a generic QB, since the spectral rigidity of a GUE matrix is, by definition, maximal. This comparison is reported in figure 7. Interestingly, we see that the performance of the SYK QB is very similar to the GUE QB, thus suggesting that the SYK QB, with its high level of spectral rigidity [76], is likely to reach the upper bound on the possible charging stability of a generic, physical, QB.

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6 The ergotropy as a measure of thermalization Another important quantity, which characterizes the performance of a QB, is the so-called ergotropy, E [10,27]. Let us recall that it quantifies the amount of extractable work from a QB after the charging protocol [10,16,27]. Indeed, if one assume to have access to just M < N cells of the full QB, part of the energy stored will be locked by internal correlations, thus reducing the efficiency of the QB itself. Given a density matrix ρ, representing the evolved state |ψ(τ ) after tracing out the useless N − M cells, the associated ergotropy is: It is known that the ergotropy is severely affected by the presence of entanglement, [27]: if the evolved state |ψ(τ ) is highly entangled, the resulting density matrix ρ will be highly mixed, and the corresponding levels of ergotropy will be very low, thus showing that, in this case, the amount of extractable energy from a subset of M cells is low. An interesting quantity to study is the following  figure 8(a) we see that X M,14 (τ ), for the SYK battery, is very low. This result follows from the fact that the SYK Hamiltonian is highly entangling, as discussed in refs. [58,59]. Moreover, from the lower panel of figure 8(a) we learn that the value of X M,N (τ ), at a given value of M , is highly affected by the size of the full battery, with X 3,N (τ ) which decreases by increasing the dimension of the battery.
Summarizing, the fact that eq. (4.12) signals an exponential suppression of fluctuations, together with the observation that the ergotropy decreases with N , suggests that it is not convenient to build big batteries based on the SYK protocol and just keep a small portion of them at the end of the charging, while it is much more convenient to work directly with small batteries.
Moving to the MBL case, the situation is very different: from the upper panel of figure 8(b) we see that the amount of extractable energy is by far higher in agreement with the results of ref. [30], where it was observed that the levels of ergotropy for the MBL system are generally very high, a feature that can be traced back to the low level of entanglement typical of the MBL phase. Much more interesting is the time behavior JHEP11(2020)067 This observation confirms the picture we outlined in the previous section: the dynamics of the MBL battery shows a clear change when passing from early times to late times. The behavior at early times is similar to the one expected for an integrable system, while at late times it becomes more similar to the behavior of a chaotic system. The crossover between the two behaviors is in correspondence with the thermalization of the system and, once again, we stress that it should tend to infinity in the thermodynamic limit for the MBL system contrary to the SYK, for which the thermalization properties have been studied in the large N limit, see refs. [66,67].

Summary and outlook
In this paper, we have introduced a new class of quantum batteries, in which the unitary charging protocol is realized via a sudden quench with a SYK-like Hamiltonian. We have argued, and shown via extensive numerical computations, that such a charging protocol is able to dramatically suppress the strength of the temporal fluctuations.
As a byproduct, we have found evidence that a new interesting time scale can be uncovered during the charging of a quantum battery; namely the time scale at which the charging protocol turns to be collective, which corresponds to the time at which one can observe a transition in the strength of the temporal fluctuations as a function of the size of JHEP11(2020)067 the system. We have also provided a microscopic understanding of this new time scale, as the one at which an initially localized state (in the eigenbasis of the constant Hamiltonian) has spread to cover a large portion of the eigenbasis of the constant Hamiltonian.
By making use of this last point of view, and using also the temporal evolution of the ergotropy as a further probe, we then conjecture that the high stability of the charging protocol based on the SYK model is just another manifestation of the fast scrambling (and fast thermalizing) property of the SYK Hamiltonian, thus suggesting that the stability reached by the SYK quench puts an upper bound on the level of stability that a QB can show.
Of course, there are many open points which would be worth to explore. It would be desirable to find further evidences for the conjecture that the charging stability of the SYK QBs is an upper bound for the charging stability of a generic QB. The results of our paper suggest an interesting connection between the charging stability and the degree of the spectral rigidity of a chaotic quench Hamiltonian, promoting the latter as another useful quantity to determine how strong is its chaotic behaviour and hence the related performance of a generic QB. Another promising line of research would be to study the charging protocol described in this paper from the holographic point of view, perhaps along the lines of [73]. Such an approach could be also relevant both to confirm the presence of an upper bound on the possible charging stability of a quantum battery and also to find its possible implications in the physics of the black holes. An interesting check to perform is to investigate the role of the local term in the Hamiltonian,Ĥ 0 , on the charging performance of a given QB. To this end, we can study a slightly different version of the SYK model, usually called "mass-deformed" SYK model (m-SYK), studied in [74][75][76]. In this model, the quench Hamiltonian,Ĥ 1 , is the usual quartic Hamiltonian of the SYK model, as defined in (3.2), while the constant term is given by the nonlocal random mass term, defined in (5.2).
We have compared, for the same realization of the disorder couplings J ijkl in both the models and for a realization of K ij , the function R(τ ) for both the SYK and the m-SYK batteries. We have renormalized the bandwidth ofĤ 2 such that the constraint ∆Ĥ  From figure 9 we clearly see that the two performances are almost the same, both in terms of the maximal value reached by R(τ ), and in terms of the strength of the fluctuations, with a small advantage for the SYK model. This shows that the role of the particularĤ 0 term on the charging performance is very limited, and that only the quench Hamiltonian really matters in the unitary charging protocol.

B Temporal fluctuations in the AL phase
For completeness, we report in figure 10 the behaviour of the temporal fluctuations for the Anderson model, analogous to the plots discussed in figures 3(c)-3(d). We see that, in this case, in both time windows (early and late time behaviours), the data are greatly JHEP11(2020)067 reproduced by a 1 √ N -like function. This result is in line with our expectation, since the Anderson model does not thermalize and, as discussed in figure 2, it shows huge temporal fluctuations at all the time scales, contrary to what happens for the other two models considered. This implies that in the AL phase a collective behaviour, with a corresponding exponential suppression of temporal fluctuations, is never reached even at late times.
C Chaotic properties and QB performance C.1 Chaotic properties of the quadratic SYK Hamiltonians In this section, we show that, despite being very similar, the two quadratic, SYK-like, Hamiltonians (5.2) and (5.5) present very different properties, with the fermionic Hamiltonian being integrable and the bosonic Hamiltonian being chaotic.
To this end, i.e. to show the chaotic/integrable nature of the two models, it is sufficient to focus on a short-range diagnostics of quantum chaos, i.e. testing the agreement with the RMT preditions for very small energy separations, of order of the mean level spacing. In particular, we consider the so-called r-statistics, also known as adjacent gap ratio [70]. This quantity, which is equivalent to the well-known nearest neighbor spacing distribution defined, for example, in [77], has the advantage that it does not require to rescale the spacings by the mean level density. In other words, it does not require an unfolding of the spectrum, that can be a delicate issue [70].
The r-statistics can be operatively defined as follows: For a given Hamiltonian spectrum, the distinct energy levels, E i , are listed in ascending order E 1 ≤ E 2 ≤ E3 . . . , (C.1) and the corresponding nearest level spacings are computed as Finally, one has the ratios r i ≡ min(r i , r i+1 ) max(r i , r i+1 ) . (C. 3) The ratios r i , once averaged over many ensemble realizations, can be used as a shortrange diagnostics of quantum chaos. Indeed, it can be shown [78] that for a non-chaotic model, the average values r i agree with the predictions for a Poissonian spectrum, r i ∼ 0.386. On the other hand, for a chaotic spectrum, the values of r i are larger than 0.5 and, more precisely, they agree with the predictions of RMT, which are r i ∼ 0.536, r i ∼ 0.603 and r i ∼ 0.676 for the Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE), respectively.
Given these preliminaries, we have computed for both the Hamiltonians (5.2) and (5.5) the r-statistics. The results are reported in figure 11, where we clearly see that the quadratic fermionic Hamiltonian shows integrable behavior, while the bosonic Hamiltonian is clearly chaotic.  Figure 12. The charging ratio R(τ ) -see eq. (4.1) -as a function of τ , for a single realization of the coupling constants, for both the quartic local SYK model, (3.2), and the spin chain model, (3.1), in the ergodic phase.

C.2 The charging stability for an ergodic spin chain
One may wonder whether a more conventional chaotic Hamiltonian, instead of the SYK model, like the one in (3.1) in the ergodic phase (obtained by setting J = δJ = 1.67h and J 2 = 0.5h), can be as efficient as the SYK Hamiltonian in reducing the temporal fluctuations.
In figure 12 we compare the charging protocol between the ergodic spin chain (3.1) and the SYK protocol, for N = 15 cells. It is immediate to see that the SYK QB is much more efficient in suppressing the temporal fluctuations, thus showing that quantum chaos, solely determined looking at the RMT predictions of the short-range diagnostics of chaos, does not guarantee the same quality of the SYK QB. We can study also for the JHEP11(2020)067 N , as done for the MBL and the SYK models in (4.10)-(4.12), respectively. As already noticed for the SYK QB, also in this case the fluctuations at early and at late time are equivalent, and more precisely, as we report in figure 13, we see that the fluctuations are again exponentially truncated by increasing the system size, but by comparing their strength with the SYK fluctuations, as in figure 3(d) (and reported here for convenience), we see that they are significantly larger than in the SYK case. This suggests that the exponential suppression of the temporal fluctuations is a generic feature of quantum chaos, as diagnosed by the short-range diagnostics, but that the strength of the suppression with N is controlled by the level of spectral rigidity.
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