Perturbative many-body transfer

Having sketched in the previous subsection the spectral decomposition of the Hamiltonian operator, one is able to express the dynamics of an arbitrary number of excitations in the initial state in the chain in terms of single-particle dynamics [23, 25, 26]. We will however restrict in the following to initial states featuring only one excitation residing on each of the sender sites, while the wire and the receiver sites are empty. Whereas the restriction of at most one excitation per site is a necessary conditions if equation (1) models spinless fermions, for bosons multiple excitation-occupancy per site would be allowed initial states. We will not consider the latter initial state conditions also in view of comparing the role of the particle statistics in the investigated dynamics. Clearly for bosonic excitations, the dynamics brings along multiple excitation-occupancy per site.

The transition amplitude for the transfer of n_s excitations, residing on the sender sites $\{{n}_{s}\}=\{{s}_{1},{s}_{2},\ldots ,\,{s}_{{n}_{s}}\}$, to the receiver sites r, residing on the receiver sites $\{{n}_{r}\}=\{{r}_{1},{r}_{2},\ldots ,\,{r}_{{n}_{r}}\}$, can be expressed in terms of the submatrix ${F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}(t)$ of the transition amplitude matrix F(t), where only the rows (columns) corresponding to the sites in the block {n_s} ({n_r}) are taken into account. The transition matrix F(t) itself is built from single-particle transition amplitudes

$$\begin{eqnarray}&&{f}_{i}^{j}(t)=\langle j| {{\rm{e}}}^{-{\rm{i}}{t}\hat{H}}| i\rangle =\sum _{k=1}^{N}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}\langle j| {\phi }_{k}\rangle \langle {\phi }_{k}| i\rangle =\sum _{k=1}^{N}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}{\phi }_{{jk}}{\phi }_{{ki}}^{* }\end{eqnarray} \tag{ 3 }$$

as follows

$$\begin{eqnarray}F(t)=\left(\begin{array}{cccc}{f}_{1}^{1}(t) & {f}_{1}^{2}(t) & \cdots & {f}_{1}^{N}(t)\\ {f}_{2}^{1}(t) & \cdots & \cdots & {f}_{2}^{N}(t)\\ \vdots & & \ddots & \vdots \\ {f}_{N}^{1}(t) & & \cdots & {f}_{N}^{N}(t)\end{array}\right),\end{eqnarray} \tag{ 4 }$$

where $\hat{H}$ is the Hamiltonian in equation (2) and ϕ_k its eigenvectors. Being F unitary,

$$\begin{eqnarray}&&\sum _{j=1}^{N}{\left|{f}_{i}^{j}(t)\right|}^{2}=1,\forall i\,\mathrm{and}\,\sum _{i=1}^{N}{\left|{f}_{i}^{j}(t)\right|}^{2}=1,\forall j,\end{eqnarray} \tag{ 5 }$$

embody the normalisation condition for the single-particle transition probability from a fixed site index i, or to a fixed site index j, as expected by excitation number conservation.

As depicted in figure 1, in the presence of the mirror symmetric Hamiltonian in equation (1), the eigenvectors of the tridiagonal matrix A are known to be either symmetric or antisymmetric [27]: ${\phi }_{{kn}}={\left(-1\right)}^{k+1}{\phi }_{k,N+1-n}$, with J_i > 0 and eigenvalues ω_k listed in decreasing order. This yields ${f}_{i}^{j}(t)={f}_{j}^{i}(t)$ and ${f}_{i}^{j}(t)={f}_{N+1-i}^{N+1-j}(t)$, resulting in both a persymmetric and centrosymmetric transition matrix F. Clearly, once sender and receiver blocks (of the same size) are chosen at each edge of the chain, the resulting submatrix will retain only its persymmetry. Furthermore, the effect of a uniform potential h on the eigenvalues ω_k in equation (2) equals only to a uniform shift of their values at zero potential. As a result of the mirror-symmetry, the eigenvalues are symmetric around their middle value. Thus, one has ${\omega }_{\tfrac{N}{2}+i}=-{\omega }_{\tfrac{N}{2}+1-i}$, where $i=1,2,\ldots ,\,\tfrac{N}{2}$ for even N and ${\omega }_{\tfrac{N+1}{2}+i}=-{\omega }_{\tfrac{N+1}{2}-i}$, where $i=0,1,2,\ldots ,\,\tfrac{N-1}{2}$ for 'odd N'. All these conditions translate in having ${f}_{i}^{j}(t)$ purely real (imaginary) for even (odd) i + j.

In view of the previous results, we now explicitly construct the submatrix ${F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}(t)$ for an arbitrary number of excitations n_s = n_r. For ${n}_{s}\ne {n}_{r}$, the transition amplitude is identically null because of the excitation number conserving nature of the Hamiltonian.

Let us assume, without loss of generality, that each of the n_s excitations resides on each site of the lattice at both edges, i.e. {n_s} = 1, 2, ..., n_s and {n_r} = N + 1 − n_r, N + 2−n_r, ..., N, see figure 1. Dropping henceforth the time-dependence, the relevant submatrix, ${F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}$ of F, is obtained by selecting the first n_s rows and the last n_r column,

$$\begin{eqnarray}{F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}=\left(\begin{array}{cccc}{f}_{1}^{N+1-{n}_{r}}(t) & {f}_{1}^{N+2-{n}_{r}}(t) & \cdots & {f}_{1}^{N}(t)\\ {f}_{2}^{N+1-{n}_{r}}(t) & \cdots & \cdots & {f}_{2}^{N}(t)\\ \vdots & & \ddots & \vdots \\ {f}_{{n}_{s}}^{N+1-{n}_{r}}(t) & & \cdots & {f}_{{n}_{s}}^{N}(t)\end{array}\right).\end{eqnarray} \tag{ 6 }$$

Finally, the transition probability for n_s excitations initially on the sender block to be retrieved on the receiver block is obtained from the square modulus of the determinant and the permanent of ${F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}$ for fermions and bosons, respectively [1]. In the determinant (permanent) expansion, each term represents an allowed many-body transition amplitude channel—in the form of a product of one-body transition amplitudes from a sender to a receiver site. Hence, the square modulus accounts for interference among all these channels.

It is interesting to stress that, although in general ${\left|\det ({F}_{\{{n}_{s}\}}^{\{{n}_{r}\}})\right|}^{2}\ne {\left|\mathrm{perm}({F}_{\{{n}_{s}\}}^{\{{n}_{r}\}})\right|}^{2}$, equality is retrieved whenever all non-vanishing terms in the determinant have the same signature. This results, as we will show in the following, that at specific times—including when ${\max }_{t}\left[{F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}\right]$ is achieved—the transition probability of n_s excitations between the edges of the chain is independent of its fermionic or bosonic nature.

In order to relax a bit the notation, hereafter we will label the n_r receiver sites starting from the edge, n_r = 1, 2, ..., n_s. This allows to highlight the persymmetry of the submatrix ${F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}$

$$\begin{eqnarray}{F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}=\left(\begin{array}{cccc}{f}_{1}^{{n}_{s}}(t) & {f}_{1}^{{n}_{s}-1}(t) & \cdots & {f}_{1}^{1}(t)\\ {f}_{2}^{{n}_{s}}(t) & \cdots & \cdots & {f}_{2}^{1}(t)\\ \vdots & & \ddots & \vdots \\ {f}_{{n}_{s}}^{{n}_{s}}(t) & & \cdots & {f}_{{n}_{s}}^{1}(t)\end{array}\right),\end{eqnarray} \tag{ 7 }$$

which now translates in ${f}_{i}^{j}(t)={f}_{j}^{i}(t)$. As a consequence, there are only $\tfrac{{n}_{s}\left({n}_{s}+1\right)}{2}$ distinct transition amplitudes in the submatrix in equation (7). Still, finding the conditions by which the transition probability approaches one is a formidable task. A determinant (permanent) of a n_s-dimensional square matrix is made up of a sum of n_s! terms, each given by a product of n_s transition amplitudes, of which, at most, $\lceil \tfrac{{n}_{s}}{2}\rceil$ terms are equal because of persymmetry, with $\lceil \bullet \rceil$ being the ceiling function. Therefore, at least $\lceil \tfrac{{n}_{s}}{2}\rceil$ transition amplitudes have to reach one at the same time. Notice also that both $\det \left({F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}\right)$ and $\mathrm{perm}\left({F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}\right)$ are purely real (imaginary) for odd (even) lengths of the chain. Because ${F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}$ is a corner submatrix of equation (4), it is not unitary, but ${\left|{F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}{F}_{{n}_{s}}^{{{n}_{r}}^{\dagger }}\right|}_{{ij}}\leqslant 1$ and

$$\begin{eqnarray}&&\sum _{j\in \{{n}_{r}\}}{\left|{f}_{i}^{j}(t)\right|}^{2}\leqslant 1,\forall i\in \{{n}_{s}\}\,\mathrm{and}\,\sum _{i\in \{{n}_{s}\}}{\left|{f}_{i}^{j}(t)\right|}^{2}\leqslant 1,\forall j\in \{{n}_{r}\},\end{eqnarray} \tag{ 8 }$$

hold as a consequence of the particle-number conservation.

In this work, we derive the conditions for which the transition probability, both for fermions and bosons, approaches one by weakly coupling the sender and receiver block to the wire. We dub this dynamical regime perturbative transfer. Notice that the transfer becomes perfect, i.e. $\det \left({F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}\right)\,\left[\mathrm{perm}\left({F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}\right)\right]\to 1$ in the limit ${J}_{0}\to 0$, which, however, implies also infinite transfer time. In the following we set J₀ = 0.01, although we checked that perturbative transfer does not depend on the specific value of J₀ insofar the weak-coupling condition J₀ ≪ J = 1 is satisfied.

Perturbative couplings have been used in several settings, from quantum-state transfer to entanglement generation. However, previous works focused mainly on one-excitation transfer [28–30], with some exceptions dealing with two-excitation transfer [24, 25, 31]. The case of $n\gt 2$ excitation transfer has not yet been addressed in the perturbative regime. Let us first recap a few results for the one- and two-excitation perturbative transfer which will be useful to describe the relevant dynamical features taking place also for n_s > 2.

For one-excitation transfer, the bosonic or fermionic nature of the particle does not play any role, as there is no statistics involved and the transfer amplitude, is given by equation (3). Because of the perturbative coupling, only the two (three) eigenvectors, lying in the middle of the single-particle spectrum, have non-negligible overlap with the initial and final state, see figure 2. This reduces the transition probability to

$$\begin{eqnarray}{| {f}_{1}^{N}(t)| }^{2}=\{\begin{array}{ll}{\left|{\sum }_{k=\tfrac{N}{2}}^{\tfrac{N}{2}+1}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}{\phi }_{{Nk}}{\phi }_{k1}^{* }\right|}^{2}={\left|{\left|{\phi }_{\tfrac{N}{2}1}\right|}^{2}(1-\cos {\omega }_{{e}}t)\right|}^{2} & \mathrm{for}\ N\ \mathrm{odd}\\ {\left|{\sum }_{k=\tfrac{N+1}{2}-1}^{\tfrac{N+1}{2}+1}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}{\phi }_{{Nk}}{\phi }_{k1}^{* }\right|}^{2}=\left|2{\rm{i}}{\left|{\phi }_{\tfrac{N+1}{2}-\mathrm{1,1}}\right|}^{2}\sin {\omega }_{o}t\right| & \mathrm{for}\ N\ \mathrm{even}\end{array},\end{eqnarray} \tag{ 9 }$$

where, for even N, ${\omega }_{e}=\tfrac{{E}_{\tfrac{N}{2}+1}-{E}_{\tfrac{N}{2}}}{2}\sim {J}_{0}^{2}$ and ${\phi }_{\tfrac{N}{2},1}\simeq \tfrac{1}{\sqrt{2}}$, and, for odd N, ${\omega }_{e}=\tfrac{{E}_{\tfrac{N+1}{2}}-{E}_{\tfrac{N+1}{2}-1}}{2}\sim {J}_{0}$, and $\left|{\phi }_{\tfrac{N+1}{2}-\mathrm{1,1}}\right|\simeq \tfrac{1}{\sqrt{2}}$ only for N ≫ 1. The approximate values for these coefficients can be obtained by a simple procedure, which we illustrate below for even N. When J₀ = 0, the sender and the receiver each have one eigenenergy state in the single-excitation sector with energy E = h, that $\left|1\right\rangle$ and $\left|N\right\rangle$, respectively. In the presence of a perturbative coupling, J₀ ≪ 1, the degeneracy between the sender and the receiver eigenstate is broken and, in the single-particle sector, the eigenstates are $\left|{{\rm{\Psi }}}^{\pm }\right\rangle \simeq \tfrac{1}{\sqrt{2}}\left(\left|1\right\rangle \pm \left|N\right\rangle \right)$, because of mirror-symmetry. From equation (9), we see that excitation transfer can be achieved with a probability perturbatively close to one. A similar procedure yields perturbative transfer of one excitation also for odd N. The main difference between the two cases lies in the fact that, for even N, there are no resonances between the sender (receiver) and the wire single-particle energy states, whereas, for odd N such a resonance occurs, see figure 2. Consequently, in the former case the energy splitting is a second-order perturbative effect, whereas, in the latter, it is a first-order one. This translates in shorter transfer times for the odd-length wire.

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The characteristic feature of 1-excitation transfer in equations (9) is the presence of a single frequency, which gives rise to Rabi-like oscillations of the excitation between the pair of two-level systems embodied by the sender and the receiver qubit. For n = 1 excitation transfer, this is a direct consequence of the weak-coupling which couples perturbatively only two (three) single-particle levels. On the other hand, for n > 1, there will be more levels entering the dynamics (the precise number will be given in the next subsection) and, therefore, more frequencies enter the sum of the transition amplitude in equation (3). As a consequence, Rabi-like oscillations are much harder to achieve. Nevertheless, if one of the frequencies is much smaller than every other, then it will dominate the dynamics, i.e. it will form the envelop of the transition amplitude in equation (3) and, therefore, unit probability is achievable in a Rabi-like dynamical scenario. Such a scenario is here defined as perturbative transfer. Let us also specify that here we are referring to perturbative transfer of excitations, that is, having the determinant (permanent) of equation (7) equal to one. Although every physical quantity in quadratic models can be expressed in terms of single-particle amplitude, perturbative transfer of excitations does not necessarily imply perturbative transfer of, say, an arbitrary quantum state. Nevertheless, as we will see in section 4, perturbative transfer of excitations implies perturbative transfer of energy and magnetisation, for instance.

In order to determine the number of eigenstates giving a non-negligible contribution to the transition amplitude in equation (3), it is necessary to identify which states of the sender (receiver) block exhibit resonances with the wire's eigenstates. In the weak-coupling regime, this identifies different lengths of the wire giving rise to resonances between its eigenenergies and those of the sender (receiver) block.

As mentioned in the previous subsection, for an excitation sitting initially on the first site, $\left|1\right\rangle$, only two or three terms are relevant in the wave packet in equation (3), depending whether N is even or odd, respectively [28]. In the former case there are only two eigenvectors $\left|{\phi }_{k}\right\rangle$ of the system having a non-negligible overlap with sites 1 and N, whilst in the latter they amount to three. This can be deduced by considering the number of resonant energy levels of the uncoupled system, sender, receiver, and wire. For J₀ = 0, there is only one single-particle energy eigenstate for the sender and the receiver, respectively, with energy E = h. The energy spectrum of the wire is given by ${E}_{k}=h+\cos \tfrac{k\pi }{{n}_{w}+1}$ [32]. Therefore, in order to have degeneracy between the sender (receiver) and the wire, N has to be odd as the condition $\cos \tfrac{k\pi }{{n}_{w}+1}=0$ has to hold. When J₀ is switched on in the weak-coupling limit, J₀ ≪ 1, the degeneracy is lifted by δ. For even N, it becomes a second-order perturbation effect, and the energy splitting is ${\rm{O}}({J}_{0}^{2})$, whereas, for odd N, the effect is of first order yielding an energy splitting ${\rm{O}}({J}_{0})$. Being the transfer time τ ∝ δ⁻¹, perturbative transfer in odd-length chains is faster than in even-length ones.

Now we consider the case of ${n}_{s}={n}_{r}\gt 1$. In order to have resonant energy levels with the wire, made of n_w sites, the following condition has to hold

$$\begin{eqnarray}&&\displaystyle \frac{k\pi }{{n}_{s}+1}=\displaystyle \frac{q\pi }{{n}_{w}+1},\,k=1,\,..,{n}_{s}\,\mathrm{and}\,q=1,\,..,{n}_{w}.\end{eqnarray} \tag{ 10 }$$

which, when put in the following form

$$\begin{eqnarray}&&q=\displaystyle \frac{{n}_{w}+1}{{n}_{s}+1}k,\end{eqnarray} \tag{ 11 }$$

shows that whenever two length of wire, n_w and m_w are congruent modulo n_s + 1, i.e. ${n}_{w}\equiv {m}_{w}\,\left(\mathrm{mod}\left({n}_{s}+1\right)\right)$, the two wires share the same number of resonant modes with the sender. As a consequence, different lengths of wire n_w, but belonging to the same equivalence class, will exhibit similar dynamical behaviour, in particular, with respect to perturbative excitation transfer. To find the number of resonant modes n_res, one has to solve equation (11) for each integer p in the least residue system $\mathrm{mod}\left({n}_{s}+1\right)$, i.e. p = 0, 1, ..., n_s. It turns out that the mode q of the wire is resonant with the mode k of the sender for

$$\begin{eqnarray}&&q=\displaystyle \frac{m\left({n}_{s}+1\right)+p+1}{{n}_{s}+1}k=\left(m+\displaystyle \frac{p+1}{{n}_{s}+1}\right)k,\end{eqnarray} \tag{ 12 }$$

where m is an integer and the length of the wire is ${n}_{w}=m\left({n}_{s}+1\right)+p$.

A few instances, relevant in the following, will be analysed. For p = 0, hence a wire of length ${n}_{w}=m\left({n}_{s}+1\right)$, equation (12) reads

$$\begin{eqnarray}&&q={mk}+\displaystyle \frac{1}{{n}_{s}+1}k.\end{eqnarray} \tag{ 13 }$$

This equation never holds as $k\lt {n}_{s}+1$ and therefore no resonances are present between the wire and the sender for arbitrary n_s. For p = 1 and ${n}_{w}=m\left({n}_{s}+1\right)+1$, one gets

$$\begin{eqnarray}&&q={mk}+\displaystyle \frac{2}{{n}_{s}+1}k,\end{eqnarray} \tag{ 14 }$$

which is satisfied only for n_s odd and brings about resonance between the mode $k=\tfrac{{n}_{s}+1}{2}$ and $q=\tfrac{{n}_{w}-1}{2}+1$ of the sender block and the wire, respectively. Furthermore, because of the reflection symmetry of the energy spectrum of both systems, this is the only resonance present. Finally, we consider the case p = n_s, corresponding to ${n}_{w}=m\left({n}_{s}+1\right)+{n}_{s}$. Equation (12) becomes $q=\left(m+1\right)k$, meaning that each sender energy eigenstate is resonant with one eigenstate of the wire. This is the maximum number of resonances in the system as the energy levels of the uncoupled blocks in figure 1 are non-degenerate.

Following such a procedure for each p we build the table in figure 3 for an arbitrary number of senders n_s.

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As can be seen from figure 4, the two-fermion transition probability is the envelop for the bosonic one. As a consequence, perturbative transfer is achieved at the same time for both classes of particles. This is a general feature of the model and can be explained by means of perturbation theory.

In the weak-coupling limit, and in the absence of resonances with the wire, we can approximate the perturbed eigenstates having non-zero overlap with the sender and the receiver sites as the symmetric and antisymmetric linear combination of the degenerate single-particle eigenstates of the sender and the receiver block

$$\begin{eqnarray}&&{\left|{{\rm{\Psi }}}_{k}\right\rangle }_{{sr}}=\displaystyle \frac{1}{\sqrt{2}}\left[\sqrt{\displaystyle \frac{2}{{n}_{s}+1}}\sum _{l=1}^{{n}_{s}}\sin \displaystyle \frac{k\pi l}{{n}_{s}+1}\left|l\right\rangle \pm \sqrt{\displaystyle \frac{2}{{n}_{s}+1}}\sum _{l=1}^{{n}_{s}}\sin \displaystyle \frac{k\pi l}{{n}_{s}+1}\left|N+1-l\right\rangle \right].\end{eqnarray} \tag{ 15 }$$

It turns out that the transition amplitude in equation (3) is bounded by

$$\begin{eqnarray}&&\mathop{{\rm{\max }}}\limits_{t}| {f}_{i}^{j}(t)| =\mathop{{\rm{\max }}}\limits_{t}\left|\sum _{k=1}^{N}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}{\phi }_{{jk}}{\phi }_{{ki}}^{* }\right|\leqslant \sum _{k=1}^{N}| {\phi }_{{jk}}{\phi }_{{ki}}^{* }| =\displaystyle \frac{2}{{n}_{s}+1}\sum _{k\in \{{n}_{s}\}}\left|\sin \displaystyle \frac{k\pi j}{{n}_{s}+1}\sin \displaystyle \frac{k\pi i}{{n}_{s}+1}\right|.\end{eqnarray} \tag{ 16 }$$

The last term on the rhs of equation (16) is equal to one only for i + j = n_s + 1 and i = j. This translates in a submatrix ${F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}$ (equation (7)) which can, at most, have unit single-particle amplitudes either on the main diagonal or on the skew-diagonal, respectively. Although, the determinant (permanent) of ${F}_{\{{n}_{s}\}}^{\{{n}_{r}\}}$ may become one also due the contribution of many terms in their respective expansion, it is highly improbable that such a fully constructive interference between wavepackets ${f}_{i}^{j}(t)$, for arbitrary i and j, will take place, especially in the presence of many frequencies entering the dynamics. As a result, also in view of the normalisation condition in equation (5), perturbative transfer is most likely to occur when all the terms either on the main, or on the skew-diagonal, will reach unit single-particle transition amplitude. It is now immediate to realise that perturbative transfer for an arbitrary number of excitations is independent from their bosonic or fermionic nature as the signature of the determinant of equation (7) does not play any role.

The result in equation (16) and the resonance conditions derived in section 2.3 allow us also to give a rule of thumb as to whether perturbative excitation transfer is achievable for an arbitrary number of excitations n_s in a wire of given length n_w by means of the protocol of figure 1. Building on the argument for one-particle transfer in section 2.2, the single-particle transition amplitudes entering the submatrix in equation (6) are given by equation (3), where only resonant modes have to be kept.

Let us consider the case where j = N + 1−i, i.e. mirror-symmetric sites in the sender and receiver block, respectively, corresponding to the elements on the diagonal of the matrix in equation (7). In the absence of resonant modes with the wire, equation (3) reads

$$\begin{eqnarray}\begin{array}{rcl}{f}_{i}^{j}(t) & = & \sum _{k=1}^{2{n}_{s}}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}{\phi }_{{jk}}{\phi }_{{ki}}^{* }=\sum _{k=1}^{2{n}_{s}}{\phi }_{{jk}}{\phi }_{{ki}}^{* }({{\rm{e}}}^{-{\rm{i}}{\delta }_{k}t}-{{\rm{e}}}^{{\rm{i}}{\delta }_{k}t}){{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}\\ & = & \sum _{k=1}^{{n}_{s}}{\phi }_{{jk}}{\phi }_{{ki}}^{* }({{\rm{e}}}^{-{\rm{i}}{\delta }_{k}t}-{{\rm{e}}}^{{\rm{i}}{\delta }_{k}t})({{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}+{{\rm{e}}}^{{\rm{i}}{\omega }_{k}t})=-{\rm{i}}\sum _{k=1}^{{n}_{s}}\displaystyle \frac{{\phi }_{{ik}}{\phi }_{{ki}}^{* }}{4}\sin {\delta }_{k}t\cos {\omega }_{k}t,\end{array}\end{eqnarray} \tag{ 17 }$$

where in the last line, without loss of generality, we have considered an instance of even i + j and mirror-symmetry of the energy spectrum has been exploited. The transition amplitude is hence given by a wavepacket of n_s travelling waves, each given by a product of harmonic functions, being sines or cosines depending on n_w, n_s, and k. The specific form of the harmonic function of ${\omega }_{k}$ not being relevant, we notice that the frequencies entering the functions satisfy δ_k ≪ ω_k, as the energy shift of the kth energy level is negligible with respect to its unperturbed value. As a consequence, $\sin {\delta }_{k}t$ shapes up the envelop of kth wave of ${f}_{i}^{j}(t)$. Therefore it is straightforward to conclude that, in order to have ${f}_{i}^{j}(t)=1$ at some specific time t = τ, the δ_k's should be all commensurate, which is a hard condition to fulfil, or only one ${\delta }_{k}^{* }$ should be much smaller than all the others. The latter condition defines the rule of thumb for perturbative excitation transfer:

$$\begin{eqnarray}&&\exists !\,{\delta }_{k}\ll {\delta }_{q},\end{eqnarray} \tag{ 18 }$$

where the δ's are the energy shifts of the corresponding energy levels entering equation (17). Equation (18) states that if in the wavepacket of equation (17) there is only one energy being corrected at a higher order in perturbation theory, perturbative excitation transfer is attainable. Indeed, being n-excitation transfer achievable by the product of the single-particle transfer on the (skew) diagonal, each evolving with the same eigenenergies as in equation (17), the transfer time is given by $\tau \simeq \tfrac{\pi }{2{\delta }_{k}^{* }}$.

An identical argument applies in the presence of resonances with the wire where the single-particle transition amplitude reads

$$\begin{eqnarray}&&{f}_{i}^{j}(t)=\sum _{k=1}^{2{n}_{s}+{n}_{w}}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}{\phi }_{{jk}}{\phi }_{{ki}}^{* }\end{eqnarray} \tag{ 19 }$$

and the energy shifts δ_k are evaluated taking into account the triple quasi-degenerate nature of the energy level(s).

Let us now address the case of n_s > 2. For three fermionic excitations, in order to have perturbative transfer

$$\begin{eqnarray}| {F}_{s}^{r}{| }^{2}={\left|\begin{array}{ccc}{f}_{1}^{N-2} & {f}_{1}^{N-1} & {f}_{1}^{N}\\ {f}_{2}^{N-2} & {f}_{2}^{N-1} & {f}_{1}^{N-1}\\ {f}_{3}^{N-2} & {f}_{2}^{N-2} & {f}_{1}^{N-2}\end{array}\right|}^{2}\simeq 1,\end{eqnarray} \tag{ 20 }$$

where, without ambiguity, we have labelled by s and r the sender and receiver sites, respectively. According to the arguments in the previous sections, we analyse the contribution of main diagonal to the determinant, with similar arguments holding for the skew diagonal contribution,

$$\begin{eqnarray}&&| {F}_{s}^{r}{| }^{2}={\left|{\left({f}_{1}^{N-2}\right)}^{2}{f}_{2}^{N-1}\right|}^{2}.\end{eqnarray} \tag{ 21 }$$

For the case of n_w = 4n + 1, the single-particle transition amplitude in equation (3) now reads

$$\begin{eqnarray}&&{f}_{i}^{j}(t)=\sum _{k=1}^{7}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}{\phi }_{{jk}}{\phi }_{{ki}}^{* }.\end{eqnarray} \tag{ 22 }$$

From figure 3 we notice that two double quasi-degenerate and one triple quasi-degenerate eigenstates have non-negligible overlap with the sender and receiver sites. As the former degeneracy is resolved at second-order in perturbation theory, and the latter at first-order, this implies that, for ${J}_{0}\to 0$, we may expect the rule of thumb in equation (18) to hold as 2nd-order energy shifts are ${\rm{O}}({J}_{0}^{2})$ wheareas 1st-order shifts are ${\rm{O}}({J}_{0})$.

Indeed, we see that perturbative transfer is ruled by the following term

$$\begin{eqnarray}&&| {F}_{s}^{r}(t)| \simeq {\left|{\sin }^{2}{\omega }_{76}^{-}t\right|}^{2},\end{eqnarray} \tag{ 23 }$$

where ${\omega }_{76}^{-}=\tfrac{{E}_{7}-{E}_{6}}{2}$ is the 2nd-order perturbation energy shift of the double quasi-degenerate energy eigenstate. The positions of E₆ and E₇ of equation (2) in the single-particle energy spectrum of the chain, ordered in increasing values, are given by $k=\lfloor \tfrac{N+1}{2}{\cos }^{-1}\tfrac{1}{\sqrt{2}}\rfloor -1$ and $\lfloor \tfrac{N+1}{2}{\cos }^{-1}\tfrac{1}{\sqrt{2}}\rfloor -2$, respectively.

Concerning the other lengths of wire n_w in figure 3, for n_s = 3, we notice that they all have exclusively 1st- or 2nd-order perturbation energy corrections. By the rule of thumb in equation (18), we do not expect perturbative transfer, being all the energy shifts of the same order of magnitude for a given n_w. In addition, we show that also non-perturbative transfer does not occur, being the energy shifts incommensurate.

Let us first analyse the non-resonant cases in figure 3 n_w = 4l, 4l + 2. As only six eigenstates take part in the dynamics, the single particle transition amplitude between a sender and a receiver site reduces to

$$\begin{eqnarray}&&{f}_{i}^{j}(t)=\sum _{k=1}^{6}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}{\phi }_{{jk}}{\phi }_{{ki}}^{* }.\end{eqnarray} \tag{ 24 }$$

From the perturbative expansion of equation (15), the envelop of the transition amplitude in equation (24) can be written as

$$\begin{eqnarray}&&| {F}_{s}^{r}(t)| =\left|\displaystyle \frac{1}{4}{\left(\sin {E}_{4}t+\sin \displaystyle \frac{{E}_{6}-{E}_{5}}{2}t\right)}^{2}\sin \displaystyle \frac{{E}_{6}-{E}_{5}}{2}t\right|,\end{eqnarray} \tag{ 25 }$$

where E₄ is given by the energy level labelled by $k=\tfrac{N}{2}+1$, E₆ and E₅ by $k=\lfloor \tfrac{N+1}{2}{\cos }^{-1}\tfrac{1}{\sqrt{2}}\rfloor$ and $\lfloor \tfrac{N+1}{2}{\cos }^{-1}\tfrac{1}{\sqrt{2}}\rfloor +1$, respectively. From equation (25), it is evident that, in order to achieve transfer of 3 excitations, E₄ and ${E}_{6}-{E}_{5}$ have to be commensurate. This implies that

$$\begin{eqnarray}\left\{\begin{array}{ll}{E}_{4}t & =\ \tfrac{\left(4n+1\right)\pi }{2}\\ \tfrac{{E}_{6}-{E}_{5}}{2}t & =\ \tfrac{\left(4m+1\right)\pi }{2}\end{array}\right.\,\mathrm{and}\,\left\{\begin{array}{ll}{E}_{4}t & =\ \tfrac{\left(4n+3\right)\pi }{2}\\ \tfrac{{E}_{6}-{E}_{5}}{2}t & =\ \tfrac{\left(4m+3\right)\pi }{2}\end{array}\right.,\end{eqnarray} \tag{ 26 }$$

have to hold with n and m integers, in order to have the oscillatory functions in equation (25) be 1 or −1 at the same time. Hence, one of the two following conditions has to be fulfilled

$$\begin{eqnarray}&&\displaystyle \frac{{E}_{6}-{E}_{5}}{2{E}_{4}}=\displaystyle \frac{4m+1}{4n+1}\,\mathrm{and}\,\displaystyle \frac{{E}_{6}-{E}_{5}}{2{E}_{4}}=\displaystyle \frac{4m+3}{4n+3}.\end{eqnarray} \tag{ 27 }$$

The impossibility of the transfer arises because, for ${J}_{0}\to 0$, we find numerically that the energy ratio $\tfrac{{E}_{6}-{E}_{5}}{2{E}_{4}}\to \tfrac{1}{2}$. Therefore, equations (27) can not be fulfilled by any integer pair n and m, as can be readily seen from the fact that they can be cast into

$$\begin{eqnarray}&&8m=4n-1\,\mathrm{and}\,8m=4n-3,\end{eqnarray} \tag{ 28 }$$

respectively. The same argument about incommensurability of the eigenfrequencies entering equation (24) applies for wires of length n_w = 4n + 3. Notice that, in the latter case, according to figure 3 there are 3 sets of triple quasi-degenerate eigenstates, all coming from 1st-order perturbation expansion. Nevertheless the same argument applies as the ratio of the energy shifts is found numerically to be $\tfrac{1}{2}$ for ${J}_{0}\to 0$.

Notice that, as we are reporting a limiting procedure, there may be instances of J₀ where the ratio becomes quasi-commensurate, and after a very large amount of time a transfer probability close to one may be achieved. Such fortuitous cases, however, are not the topic of our investigation, as we are considering the conditions to be fulfilled in order to achieve perturbative transfer in the generic limit of weak coupling instead of some specific values of J₀, which may eventually be a set of zero measure and hence extremely sensible to disorder.

To summarise, we have found that for n_s = 3 excitations, placed at one edge of a wire of length n_w and in the weak-coupling limit ${J}_{0}\to 0$, perturbative transfer is achievable only for n_w = 4l + 1 where the unique 2nd-order perturbation eigenenergy correction determines the transfer time. Other equivalence classes of the wire's length do not achieve unit transfer of three excitations because all the energy shifts belong to the same perturbation order and commensurability between frequencies is not achieved. In figure 5 we depict the results only for the case of successful perturbative transfer.

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Let us now address the case of n_s = 4. From figure 3 we see that all energy shifts, for a given n_w, are of the same order in perturbation theory, either 1st-order for n_w = 5l + 4 or 2nd-order in all the other case. Therefore, at variance with the case n_s = 3, the condition for perturbative excitation transfer given by equation (18) is not satisfied. Nevertheless, there are lengths of the wire n_w exhibiting successful n_s = 4 excitation transfer, whereas other lengths do not. The reason, as we will show, can be traced back to the fact that for some length of wires n_w, the energy splitting at 2nd-order in perturbation theory is almost one order of magnitude lower for some energy levels than it is for others. Let us analyse first the successful case.

For n_w = 5l + 2, the perturbated eigenstates are located at position $k=\lfloor \tfrac{N+1}{\pi }{\cos }^{-1}\tfrac{\sqrt{5}+1}{4}\rfloor -1$ and $k=\lfloor \tfrac{N+1}{\pi }{\cos }^{-1}\tfrac{\sqrt{5}+1}{4}\rfloor$ for the higher energy state, and $k=\lfloor \tfrac{N+1}{\pi }{\cos }^{-1}\left(\tfrac{\sqrt{5}-1}{4}\right)\rfloor$ and $k=\lfloor \tfrac{N+1}{\pi }{\cos }^{-1}\left(\tfrac{\sqrt{5}-1}{4}\right)\rfloor +1$ for the lower one. By numerical evaluation, we obtain that the ratio ${\omega }_{78}^{-}$ to ${\omega }_{56}^{-}$ goes to 0.14, for J₀ ≪ 1 and irrespective of l. The same situation occurs for n_w = 5l + 1. In these cases n_s = 4 excitation transfer occurs, although it is not ruled by a single frequency and hence, according to our definition, is is not perturbative excitation transfer. Indeed, in order to determine the transfer time, one has to find the maximum of two-single particle transition amplitudes entering the 4 excitation transition probability between the edges of the chain,

$$\begin{eqnarray}&&{\left|{F}_{s}^{r}\right|}^{2}={\left|{\left({f}_{1}^{N-3}\right)}^{2}{\left({f}_{2}^{N-2}\right)}^{2}\right|}^{2}.\end{eqnarray} \tag{ 29 }$$

The fact that one energy shift is almost one order of magnitude lower than the other entering the dynamics allows one to determine the order of magnitude of the transfer time as given by $\tau =\tfrac{\pi }{2{\omega }_{78}^{-}}$. In figure 6 a comparison of the latter with the exact numerical result for excitation transfer is shown in panel d. On the other hand, for n_w = 5l, 5l + 3, 5l + 4, one has ${\omega }_{56}^{-}\simeq {\omega }_{78}^{-}$, with the ratio going to 0.38. perturbative transfer does not occur and also non-perturbative transfer has not been found for several instances within time intervals related to the inverse of the energy splits. Clearly, this does not mean that the excitations may not be transferred at a certain time, being only two frequencies involved and occasional instances of commensurability may occur between the energy shifts, but this would hardly be robust against the length of wire and perturbations of J₀.

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Finally, we present an unified scenario for the shortest transfer time achievable via perturbative transfer for n_s = 1, 2, 3 excitations in the sender block in figure 6. We have also added the case n_s = 4 to highlight its qualitatively similar behaviour to perturbative transfer. Here we assume that the sender and receiver are connected by a wire able to transfer from one to four excitations by weakly coupling the respective blocks to the end to the wire. In order to have a wire able to perform such a task, its length n_w has to fall in all the equivalence classes allowing perturbative transfer for n_s = 1, 2, 3 and quasi-perturbative transfer for n_s = 4. Whereas n_w can be arbitrary for n_s = 1, 2, for n_s = 3, 4, the length of the wire has to be n_w = 4l + 1 and n_w = 5l + 1 or n_w = 5l + 2, respectively. This yields to wires length of n_w = 20l + 1 and n_w = 20l + 17, respectively. In figure 6, we report the transfer times for the former case, noticing its linear increase with the wire's length for n_s = 2, 3, 4. On the other hand, for n_s = 1, the increase is $\sqrt{{n}_{w}}$ as the frequency involved in the perturbative transfer is derived from resolving the degeneracy via first-order perturbation theory, as shown for odd N in section 2.2.

An arbitrary one-body observable in second quantisation is given by

$$\begin{eqnarray}&&\hat{O}=\sum _{{nm}}{a}_{{nm}}{\hat{c}}_{n}^{\dagger }{\hat{c}}_{m}+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ 30 }$$

where the $\hat{c}$'s are bosonic or fermionic operators acting on site n and m. Expressing average of the observable's dynamics in Heisenberg representation, where $\hat{H}$ is given by equation (1) yields

$$\begin{eqnarray}&&\left\langle \hat{O}(t)\right\rangle =\sum _{{nm}}{a}_{{nm}}\sum _{{kq}}{\phi }_{{kn}}{\phi }_{{qm}}^{* }{{\rm{e}}}^{{\rm{i}}\left({E}_{k}-{E}_{q}\right)t}\left\langle {\hat{c}}_{k}^{\dagger }{\hat{c}}_{q}\right\rangle .\end{eqnarray} \tag{ 31 }$$

As the single-particle spectrum of the Hamiltonian in equation (1) is identical both for fermions and bosons, the only difference between the dynamics of an observable on a fermionic or bosonic many-body system that can possibly arise has to come from the average on the initial state of the operators on the RHS of equation (31).

In our setting the initial state is given by one or zero excitations per site

$$\begin{eqnarray}&&\left|{\rm{\Psi }}(0)\right\rangle =\prod _{i=\{{n}_{s}\}}{\hat{c}}_{i}^{\dagger }\left|0\right\rangle ,\end{eqnarray} \tag{ 32 }$$

which is also the only initial state that fermions and bosons can have in common. Evaluating the average on the rhs of equation (31) on this initial state reads

$$\begin{eqnarray}&&\left\langle 0\right|{\hat{c}}_{1}{\hat{c}}_{2}...{\hat{c}}_{{n}_{s}}{\hat{c}}_{k}^{\dagger }{\hat{c}}_{q}{\hat{c}}_{{n}_{s}}^{\dagger }...{\hat{c}}_{2}^{\dagger }{\hat{c}}_{1}^{\dagger }\left|0\right\rangle .\end{eqnarray} \tag{ 33 }$$

Expressing all operators in the position basis reads

$$\begin{eqnarray}&&\sum _{{ij}}{\phi }_{{ki}}{\phi }_{{qj}}^{* }\left\langle 0\right|{\hat{c}}_{1}{\hat{c}}_{2}...{\hat{c}}_{{n}_{s}}{\hat{c}}_{i}^{\dagger }{\hat{c}}_{j}{\hat{c}}_{{n}_{s}}^{\dagger }...{\hat{c}}_{2}^{\dagger }{\hat{c}}_{1}^{\dagger }\left|0\right\rangle .\end{eqnarray} \tag{ 34 }$$

By using Wick's theorem, we notice that the non-zero fully-contracted terms are those having an even number of permutations. As a consequence, the dynamics of an arbitrary one-body observable, such as in equation (31), is independent of the bosonic or fermionic nature of the excitations. For instance, the average number of particles on a lattice site, $\left\langle \hat{n}(t)\right\rangle =\left\langle {\hat{c}}_{n}^{\dagger }(t){\hat{c}}_{n}(t)\right\rangle$ is the same whether the Hamiltonian in equation (1) refers to bosons or fermions, notwithstanding Pauli's exclusion principle holds for fermions whereas bosons allow for multiple occupation.

It is easy to show that the same holds for n-body observables of the form

$$\begin{eqnarray}&&\hat{O}=\sum _{{nmijrs}...}{\alpha }_{{nm}...}{\hat{c}}_{n}^{\dagger }{\hat{c}}_{m}{\hat{c}}_{i}^{\dagger }{\hat{c}}_{j}{\hat{c}}_{r}^{\dagger }{\hat{c}}_{s}...\,+\,{\rm{h}}{\rm{.}}{\rm{c}}.,\end{eqnarray} \tag{ 35 }$$

when the average is evaluated on an initial state of the form of equation (32) and the dynamics is ruled by a quadratic Hamiltonian such as in equation (1). A relevant example of a 2-body observable of the form of equation (35) independent from the statistics of the excitations is the density-density fluctuations $\left\langle {\hat{n}}_{i}(t){\hat{n}}_{j}(t)\right\rangle$.

As it is well known, the Hamiltonian in equation (1) models also a 1D spin-$\tfrac{1}{2}$ chain with isotropic interactions on the XY plane, i.e.,

$$\begin{eqnarray}&&\hat{H}=\sum _{i}^{N}{J}_{i}\left({\hat{S}}_{i}^{x}{\hat{S}}_{i+1}^{x}+{\hat{S}}_{i}^{y}{\hat{S}}_{i+1}^{y}\right)+{h}_{i}{\hat{S}}_{i}^{z}\end{eqnarray} \tag{ 36 }$$

when the standard Jordan–Wigner transformation is carried out [33]. Because of the Jordan–Wigner mapping, the (fermionic) Hamiltonian in equation (1) can model an XX spin-$\tfrac{1}{2}$ open chain, where the average total magnetisation (along the z-direction) of a set of spins residing on sites {i} is given by $\left\langle {\hat{S}}^{z}\right\rangle ={\sum }_{\{i\}}\left\langle {\hat{S}}_{i}^{z}\right\rangle ={\sum }_{\{i\}}\tfrac{2\left\langle {\hat{c}}_{i}^{\dagger }{\hat{c}}_{i}\right\rangle -1}{2}$. As a consequence, the magnetisation of the receiver block evolves as

$$\begin{eqnarray}&&\left\langle {\hat{S}}_{\{r\}}^{z}(t)\right\rangle =\sum _{\{r\}}\left\langle {\hat{c}}_{r}^{\dagger }(t){\hat{c}}_{r}(t)\right\rangle -\displaystyle \frac{{n}_{r}}{2},\end{eqnarray} \tag{ 37 }$$

where the average is evaluated over the initial state having all the spins in the sender block flipped. In the Heisenberg picture and using Wick's theorem, it is possible to express the receiver's block magnetisation as a function of single-particle transition amplitudes ${f}_{i}^{j}(t)$:

$$\begin{eqnarray}&&\left\langle {\hat{S}}_{\{r\}}^{z}(t)\right\rangle =\sum _{i=\{s\}}\sum _{j=\{r\}}| {f}_{i}^{j}{| }^{2}-\displaystyle \frac{{n}_{r}}{2}\equiv | | {F}_{s}^{r}(t)| {| }_{F}^{2}-\displaystyle \frac{{n}_{r}}{2},\end{eqnarray} \tag{ 38 }$$

where $| | \bullet | {| }_{F}$ is the Frobenius matrix norm and ${F}_{s}^{r}(t)$ is the submatrix defined in equation (7). The result for n_s = 3 is shown in figure 7. Notice that, although the transition probability oscillates, for bosons, between 0 and 1 on a timescale of J in the the corresponding scenario in figure 5, the average number of bosons on the receiver block varies only between 2 and 3. Therefore, on a large time interval, with respect to J, around the transfer time τ at least two excitations out of three are located on the receiver block irrespective of their bosonic of fermionic nature. Indeed, the dynamics of the occupation number $\left\langle {\hat{n}}_{i}(t)\right\rangle$ of site i entering equation (37) is identical for bosons and fermions, as by the argument of section 4.1. As an example, let us consider the case n_s = 2. Although the dynamics of the transition probability differs, as reported in figure 4, the subspace spanned by the two photons in the receiver block is composed by $\{\left|11\right\rangle ,\tfrac{1}{\sqrt{2}}\left(\left|02\right\rangle \pm \left|20\right\rangle \right)\}$, which are all states having the same $\left\langle {\hat{n}}_{i}(t)\right\rangle$.

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The transfer of energy from one spatial location to another has always been a central topic in physics. Recently, a lot of attention has been devoted to the so-called quantum batteries, i.e. quantum devices able to store energy and release it upon demand at specific times [34–37]. Devising a protocol to extract the maximum amount of energy from a charged battery, establishing a bound on its amount, and stabilising the battery's charge has been addressed in several works [38–42]. Another line of research is embodied by the investigation of the charging protocol of a quantum battery [43–45], and, apart from a few instances [46], mainly non-interacting systems embodying the quantum battery have been considered.

Our work can be immediately rephrased in terms of a charging protocol of a many-body quantum battery. Dubbing the sender block as charger, the receiver block as battery and the wire as a quantum cable connecting the charger to the battery, a quite natural set-up for charging a quantum battery is represented by figure 1.

Nevertheless, in order to reinterpret the excitations dynamics in section 2 as a charging protocol, a few precautions are in place. As shown in [45], the charging protocol should involve a time-dependent Hamiltonian

$$\begin{eqnarray}&&\hat{H}(t)={\hat{H}}_{0}+\lambda (t){\hat{H}}_{1},\end{eqnarray} \tag{ 39 }$$

where ${\hat{H}}_{0}={\hat{H}}_{C}+{\hat{H}}_{w}+{\hat{H}}_{B}$, are the time-independent Hamiltonians of the charger, the wire, and the battery, respectively. ${\hat{H}}_{1}$ is the Hamiltonian connecting the charger (battery) to the wire and λ(t) is the coupling constant responsible for switching on and off the interaction between the charger (battery) and the wire when the charging protocol starts and ends. Generally, it is assumed that λ(t) is given by a step-function having a value of 1 for $t\in [0,\tau ]$ and 0 otherwise. Because of that time dependence, energy may not be conserved and there could be some switching energy δE_sw injected or extracted from the system during the protocol. This can be evaluated [45] by

$$\begin{eqnarray}&&\delta {{\rm{E}}}_{{sw}}(\tau )={\rm{tr}}\left[{\hat{H}}_{1}\left(\rho (0)-\rho (\tau \right)\right],\end{eqnarray} \tag{ 40 }$$

where ρ is the density matrix of the full system. By definition of the switching energy in equation (40), for $\left[{\hat{H}}_{0},{\hat{H}}_{1}\right]=0$, one has $\delta {{\rm{E}}}_{{sw}}(\tau )=0$. However, this is not the case for our model, since the commutator does not vanish. Nevertheless, evaluating equation (40), we obtain a zero switching energy due to the mirror-symmetry of our model. The two terms entering equation (40) are equivalent to

$$\begin{eqnarray}&&{\rm{tr}}[{\hat{H}}_{1}\rho (0)]=\langle {\rm{\Psi }}(0)| {\hat{c}}_{{n}_{C}}^{\dagger }{\hat{c}}_{{w}_{1}}+{\hat{c}}_{{n}_{w}}^{\dagger }{\hat{c}}_{{1}_{B}}+{\rm{h}}.{\rm{c}}.| {\rm{\Psi }}(0)\rangle ,\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}&&{\rm{tr}}[{\hat{H}}_{1}\rho (\tau )]=\langle {\rm{\Psi }}(0)| {\hat{c}}_{{n}_{C}}^{\dagger }(\tau ){\hat{c}}_{{w}_{1}}(\tau )+{\hat{c}}_{{n}_{w}}^{\dagger }(\tau ){\hat{c}}_{{1}_{B}}(\tau )+{\rm{h}}.{\rm{c}}.| {\rm{\Psi }}(0)\rangle ,\end{eqnarray} \tag{ 42 }$$

where the last equation is written in the Heisenberg representation. Equation (41) is identically null because of the choice of the initial state of our system, whereas, expressing equation (42) in terms of single-particle transition amplitudes, results in

$$\begin{eqnarray}&&{\rm{tr}}\left[{\hat{H}}_{1}\rho (\tau )\right]=\sum _{n\in \{{n}_{C}\}}\left({\left({f}_{n}^{{n}_{C}}\right)}^{* }{f}_{n}^{{1}_{w}}+{\left({f}_{n}^{{n}_{w}}\right)}^{* }{f}_{n}^{{1}_{B}}+{\rm{c}}.{\rm{c}}.\right),\end{eqnarray} \tag{ 43 }$$

with c.c. denoting complex conjugation. The above expression is identically null as each ${\left({f}_{n}^{i}\right)}^{* }{f}_{n}^{i+1}$ results to be purely imaginary according to the conditions outlined in section 2.1 for mirror-symmetric matrices.

As a consequence, the figures of merit for the charging protocol of a quantum many-body system via a quantum wire, are those reported in [45]. The mean energy stored in the battery and the mean storing power are, respectively,

$$\begin{eqnarray}&&{{\rm{E}}}_{B}(\tau )={\rm{tr}}[{{ \mathcal H }}_{B}{\rho }_{B}(\tau )],\,{P}_{s}(\tau )=\displaystyle \frac{{{\rm{E}}}_{B}(\tau )}{\tau }.\end{eqnarray} \tag{ 44 }$$

Other useful quantities are the maximum energy stored and the maximum power,

$$\begin{eqnarray}&&{\bar{{\rm{E}}}}_{s}(\tau )\equiv \mathop{\max }\limits_{\tau }[{{\rm{E}}}_{s}(\tau )]\equiv {\rm{E}}(\bar{\tau }),\,{\tilde{P}}_{s}(\tau )\equiv \mathop{\max }\limits_{\tau }[{P}_{s}(\tau )]\end{eqnarray} \tag{ 45 }$$

and their corresponding optimal charging times,

$$\begin{eqnarray}&&\bar{\tau }\equiv \mathop{\min }\limits_{{\rm{E}}(\bar{\tau })={\bar{{\rm{E}}}}_{s}(\tau )}[\tau ],\,\tilde{\tau }\equiv \mathop{\min }\limits_{P(\tilde{\tau })={\tilde{P}}_{s}(\tau )}[\tau ].\end{eqnarray} \tag{ 46 }$$

Lastly, the charging power obtained at maximum energy is defined as,

$$\begin{eqnarray}&&{\bar{P}}_{s}(\tau )\equiv \displaystyle \frac{{\bar{{\rm{E}}}}_{s}(\tau )}{\tau },\end{eqnarray} \tag{ 47 }$$

which is generally different from the maximum power because the times at which maximum energy and maximum power are achieved, $\bar{\tau }$ and $\tilde{\tau }$ respectively, may not coincide.

In this subsection we choose h > 1, so that the charger state with all spin aligned in the positive z-direction is the highest energy eigenstate of ${\hat{H}}_{B}$, with energy $\tfrac{{n}_{B}h}{2}$. Applying the same magnetic field h to the rest of the system, wire and battery, allows us to use the formalism of section 2 to evaluate the above figures of merit, as a uniform magnetic field in equation (1) implies only an uniform shift by h of all single-particle eigenenergies, with the eigenvectors remaining unchanged. Such a uniform shift brings along only an irrelevant overall phase factor in the dynamics as it amounts to adding a constant to the Hamiltonian in equation (1).

Interestingly, only the one-body terms in ${\hat{H}}_{B}$ contribute to the mean energy ${{\rm{E}}}_{B}(\tau )$ in equation (44). This can be immediately seen as, at time τ, the density matrix of the battery ρ_B(τ) represents the state with all the spins flipped as perturbative transfer has occurred. In addition, it results also that the energy due to the inter-spin interaction term is vanishing at all times, as a result of the following equation

$$\begin{eqnarray}&&{{\rm{E}}}_{I}\equiv \sum _{i\in \{{n}_{B}\}}\displaystyle \frac{1}{2}\langle {\hat{c}}_{i+1}^{\dagger }{\hat{c}}_{i}+{\rm{h}}{\rm{.}}{\rm{c}}.\rangle =\sum _{\displaystyle \genfrac{}{}{0em}{}{i\in \{{n}_{B}\}}{n\in \{{n}_{C}\}}}({({f}_{n}^{i})}^{* }{f}_{n}^{i+1}+{\rm{c}}{\rm{.}}{\rm{c}}.),\end{eqnarray} \tag{ 48 }$$

again because of the conditions on ${f}_{i}^{j}(t)$ for mirror-symmetric matrices, as already derived for the switching energy δE_sw.

This allows us immediately to use our results to confirm that all the charger's energy is transferred to the battery and, remarkably, no energy is stored in two-body correlations at any time. This has several advantages: on the one hand, only single-qubit operation are necessary to extract the energy from the many-body battery and, on the other hand, the n_B spins embodying the battery can be split in independent, non-interacting partitions without any loss of the initially stored energy. An instance of the charging process of a quantum many-body battery is shown in figure 8 for the case of n_s = 4. Notice that the power at maximum energy as by equation (47) is obtained at a considerably larger time than the maximum power, equation (45).