Quantum Battery Based On Quantum Discord At Room Temperature

From Champion's League Wiki
Jump to: navigation, search

The study of advanced quantum devices for energy storage has attracted the attention of the scientific community in the past few years. Although several theoretical progresses have been achieved recently, experimental proposals of platforms operating as quantum batteries under ambient conditions are still lacking. In this context, this work presents a feasible realization of a quantum battery in a carboxylate-based metal complex, which can store a finite amount of extractable work under the form of quantum discord at room temperature, and recharge by thermalization with a reservoir. Moreover, the stored work can be evaluated through non-destructive measurements of the compound’s magnetic susceptibility. These results pave the way for the development of enhanced energy storage platforms through material engineering.



Quantum Battery; Quantum Discord; Ergotropy; Metal Complexes.



Introduction. Batteries are common components in many technological devices, storing different types of energy and converting it under the form of electric current Li et al. (2018). Over the past years, the quest for an efficient use of energy has boosted the development of reliable mechanisms for energy storage Liu et al. (2019); Manzano-Agugliaro et al. (2013). Recently, a novel class of batteries has attracted the attention of the scientific community, namely, quantum batteries (QBs). Chemical batteries convert chemical energy into electric one through reactions between two species with different chemical properties Li et al. (2018). Differently, QBs are constituted of quantum systems, and exploit the superposition principle of states and its quantum correlations. Based on these, QBs are able to store an amount of work from the quantum states, called ergotropy, which is extracted to power quantum devices Ferraro et al. (2018); Alicki and Fannes (2013); Binder et al. (2015); Santos et al. (2020); Giorgi and Campbell (2015); Campaioli et al. (2017); Andolina et al. (2019); Çakmak (2020); Santos et al. (2019); Çakmak (2020); Francica et al. (2020); Le et al. (2018)-not necessarily as electric power.



However, there are several challenges to overcome before QBs are put into practical use, such as storing ergotropy at room temperature (i.e., preserving quantum correlations or coherence), offering a non-destructive access to the amount of stored work, and a charge lifetime comparable with that of the conventional classical batteries. Recently, several devices have been proposed as QBs, based on different quantum systems, for example, using solid-state and quantum optical systems, such as superconducting devices Strambini et al. (2020), circuit-QED Andolina et al. (2019); Rossini et al. (2020), and two-level emitters into waveguides Monsel et al. (2020). However, the platforms proposed so far for QBs hardly fulfill the above requirements, and the possibility to place QBs into practical use is still under debate Monsel et al. (2020); Strambini et al. (2020).



In this context, low-dimensional metal complexes (LDMC), such as quantum antiferromagnets Cruz et al. (2016); Souza et al. (2009); Reis et al. (2012); Čenčariková and Strečka (2020); Cruz and Anka (2020); Souza et al. (2008); He et al. (2017, 2017); Breunig et al. (2017); Čenčariková and Strečka (2020), appear as promising platforms to implement QBs prototypes. Indeed, these complexes present a molecular structure that shields them from environment fluctuations. This includes fluctuations in temperature Reis et al. (2012); Čenčariková and Strečka (2020), magnetic field Čenčariková and Strečka (2020); Cruz and Anka (2020); Souza et al. (2008), and pressure Cruz et al. (2017); Cruz and Anka (2020). LDMCs effectively behave as a two-qubit system (see Fig.1(a)), which can hold stable quantum correlations above room temperature Cruz et al. (2016); Reis et al. (2012); Souza et al. (2009).



In this work, we propose metal complexes as room-temperature operating QBs, where the working substance is a dinuclear copper (II) carboxylate-based metal-organic complex Cruz et al. (2016). The ergotropy is stored in the quantum discord between the spins, and the available ergotropy can be measured non-destructively by monitoring the magnetic susceptibility of the compound. Finally, the carboxylate structure proves robust against self-discharging processes Santos (2021).



Metal complexes as effective two-qubit systems. In this work, we focus on a metal-organic dinuclear cooper (II) compound with chemical formula Cu22_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT(HCOO)44_4start_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPT(HCOOH)22_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT(C44_4start_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPTH1010_10start_FLOATSUBSCRIPT 10 end_FLOATSUBSCRIPTN22_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT), whose crystalline structure is obtained through single-crystal X-ray analysis ccd (see Fig. 1a). The system is prepared using dehydrated copper (II) nitrate (Cu(NO33_3start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT)22_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT), piperazine (C44_4start_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPTH1010_10start_FLOATSUBSCRIPT 10 end_FLOATSUBSCRIPTN22_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT), formic acid, ethanol and distilled water; and refluxed at 425425\leavevmode obreak\ 425425 K for 3 hours. The piperazine organic compound yields a porous environment which allows for the production of metal-organic frameworks with magnetically isolated dimers of Cu(II).



The reduced magnetic unit is then formed by two metallic centers of Cu(II), with electronic configuration d9superscript𝑑9d^9italic_d start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT and s=1/2𝑠12s=1/2italic_s = 1 / 2, in a dimeric tetraformate unit. The syn-syn bond between them, characteristic of carboxylate-based metal complex, yields a short intermolecular separation of 2.68(2)2.6822.68(2)2.68 ( 2 ) Åitalic-Å\AAitalic_Å Cruz et al. (2016); Souza et al. (2009). This provides a nearly-ideal realization of an isolated two-qubit system Yurishchev (2011); Gaita-Ariño et al. (2019); Moreno-Pineda et al. (2018). It is worth noting that, despite the specificity of the present LDMC, this behavior is actually encountered in a broad range of metal complexes, thanks to their large intramolecular interaction energy, as compared to the intermolecular ones Cruz and Anka (2020); Yurishchev (2011); Souza et al. (2009, 2008); Čenčariková and Strečka (2020); Chakraborty and Mitra (2019); Yurishchev (2011); Gaita-Ariño et al. (2019); Moreno-Pineda et al. (2018). The energy of the two coupled qubit system is provided by the following Hamiltonian:



H=E0(S1(z)+S2(z))+J(S→1⋅S→2).𝐻subscript𝐸0superscriptsubscript𝑆1𝑧superscriptsubscript𝑆2𝑧𝐽⋅subscript→𝑆1subscript→𝑆2\displaystyle H=E_0\left(S_1^(z)+S_2^(z)\right)+J\left(\vecS_1% \cdot\vecS_2\right).italic_H = italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_z ) end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_z ) end_POSTSUPERSCRIPT ) + italic_J ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) . (1) Above, Sn(k)=(ℏ/2)σnksuperscriptsubscript𝑆𝑛𝑘Planck-constant-over-2-pi2superscriptsubscript𝜎𝑛𝑘S_n^(k)\!=\!(\hbar/2)\sigma_n^kitalic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = ( roman_ℏ / 2 ) italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT, where σnksuperscriptsubscript𝜎𝑛𝑘\sigma_n^kitalic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT represent the Pauli matrices and k∈x,y,z𝑘𝑥𝑦𝑧k\!\in\!\x,y,z\italic_k ∈ italic_x , italic_y , italic_z . The first right-hand term is the Zeeman Hamiltonian Reis (2013), where E0=μBgzBzsubscript𝐸0subscript𝜇𝐵subscript𝑔𝑧subscript𝐵𝑧E_0\!=\!\mu_Bg_zB_zitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, gzsubscript𝑔𝑧g_zitalic_g start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT the isotropic Landé factor; μBsubscript𝜇𝐵\mu_Bitalic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT represents the Bohr magneton; and Bzsubscript𝐵𝑧B_zitalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT is the external (fixed) magnetic field intensity. Such term describes the energy levels of each cell of the battery (qubit): H0|ϵi⟩=ϵi|ϵi⟩subscript𝐻0ketsubscriptitalic-ϵ𝑖subscriptitalic-ϵ𝑖ketsubscriptitalic-ϵ𝑖H_0\ket\epsilon_i\!=\!\epsilon_i\ket\epsilon_iitalic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | start_ARG italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ = italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩, with ϵi∝E0proportional-tosubscriptitalic-ϵ𝑖subscript𝐸0\epsilon_i\!\propto\!E_0italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∝ italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and |ϵi⟩ketsubscriptitalic-ϵ𝑖\ket\epsilon_i| start_ARG italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ the eigenvalue and eigenstate, respectively. In particular, the energy scale E0subscript𝐸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (and, as we shall see later, the ergotropy of the system - Fig. 1b), is proportional to the fixed magnetic field Bzsubscript𝐵𝑧B_zitalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT. The Heisenberg Hamiltonian Hint=J(S→1⋅S→2)subscript𝐻int𝐽⋅subscript→𝑆1subscript→𝑆2H_\textint=J\left(\vecS_1\cdot\vecS_2\right)italic_H start_POSTSUBSCRIPT int end_POSTSUBSCRIPT = italic_J ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) corresponds to the internal interaction between Cu(II) ions (intra-cell interaction), while J𝐽Jitalic_J represents the magnetic coupling constant.



The syn-syn metal carboxylate conformation yields a very short metal-to-metal magnetic interaction, leading to a huge magnetic coupling J/kB=748𝐽subscript𝑘𝐵748J/k_B\!=\!748italic_J / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 748 K between Cu(II) cells. This allows for the existence of stable quantum correlations above room temperature Reis et al. (2012); Souza et al. (2009). Consequently, as sketched in Fig. 1a, the energy levels of the system is composed by the singlet state |β-⟩ketsubscript𝛽\ket\beta_-| start_ARG italic_β start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_ARG ⟩ with energy E-=0subscript𝐸0E_-=0italic_E start_POSTSUBSCRIPT - end_POSTSUBSCRIPT = 0 and the triply-degenerate subspace βt⟩=ketsubscript𝛽tketsubscript𝛽ket↓absent↓ket↑absent↑\\ket\beta_\textt\\!=\!\\ket\beta_+,\ket\downarrow\downarrow,% \ket\uparrow\uparrow\ = start_ARG italic_β start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ⟩ , , with energy Et=Jsubscript𝐸t𝐽E_\textt=Jitalic_E start_POSTSUBSCRIPT t end_POSTSUBSCRIPT = italic_J, where we have defined |β±⟩=(|↓↑⟩±|↑↓⟩)/2ketsubscript𝛽plus-or-minusplus-or-minusket↓absent↑ket↑absent↓2\ket\beta_\pm\!=\!(\ket\downarrow\uparrow\pm\ket\uparrow\downarrow)/% \sqrt2| start_ARG italic_β start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT end_ARG ⟩ = ( | start_ARG ↓ ↑ end_ARG ⟩ ± | start_ARG ↑ ↓ end_ARG ⟩ ) / square-root start_ARG 2 end_ARG. Thus, the gap ΔE>0Δ𝐸0\Delta E>0roman_Δ italic_E >0 between the ground and first excited ground state indicates that the system will be in a entangled singlet ground state with anti-parallel alignment (quantum antiferromagnet).



In particular, for this class of materials, the static magnetic field Bzsubscript𝐵𝑧B_zitalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT splits the energy levels of the compound, inducing a quantum level crossing Chakraborty and Mitra (2019); Cruz and Anka (2020); Breunig et al. (2017), changing its corresponding populations Cruz and Anka (2020) (for details, see ref. Sup ). However, the strong magnetic coupling, yielded by the syn-syn bond between the Cu(II) cells, leads to a crossing field of Bc∼556similar-tosubscript𝐵𝑐556B_c\sim\!556italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∼ 556 T. This very large value of the crossing field is due to the large gap (Jint/kB=748subscript𝐽intsubscript𝑘𝐵748J_\textint/k_B\!=\!748italic_J start_POSTSUBSCRIPT int end_POSTSUBSCRIPT / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 748 K) between the ground (singlet entangled state) and the first excited states (triplet separable state) yielded by the syn-syn bond between the Cu(II) cells (see Fig. 1a). Therefore, for any Bz≪Bcmuch-less-thansubscript𝐵𝑧subscript𝐵𝑐B_z\ll B_citalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≪ italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT (E0≪Jmuch-less-thansubscript𝐸0𝐽E_0\ll Jitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ italic_J), the system behaves as an effective two-level one, with cycle between the singlet ground state |β-⟩ketsubscript𝛽\ket\beta_-| start_ARG italic_β start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_ARG ⟩ and the excited triplet |βt⟩ketsubscript𝛽t\ket\beta_\textt| start_ARG italic_β start_POSTSUBSCRIPT t end_POSTSUBSCRIPT end_ARG ⟩ one.



The density matrix for the coupled system in thermal equilibrium can be written, in the energy basis, as an X-state:



ρ(T,Bz)𝜌𝑇subscript𝐵𝑧\displaystyle\rho(T,B_z)italic_ρ ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) =\displaystyle== eζ2Z[2eβE000001+e-4ζ1-e-4ζ001-e-4ζ1+e-4ζ00002e-βE0],superscript𝑒𝜁2𝑍delimited-[]matrix2superscript𝑒𝛽subscript𝐸000001superscript𝑒4𝜁1superscript𝑒4𝜁001superscript𝑒4𝜁1superscript𝑒4𝜁00002superscript𝑒𝛽subscript𝐸0\displaystyle\frace^\zeta2Z\left[\beginmatrix2e^\beta E_0&0&0&0% \\ 0&1+e^-4\zeta&1-e^-4\zeta&0\\ 0&1-e^-4\zeta&1+e^-4\zeta&0\\ 0&0&0&2e^-\beta E_0\endmatrix\right]\leavevmode obreak\ ,divide start_ARG italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_Z end_ARG [ start_ARG start_ROW start_CELL 2 italic_e start_POSTSUPERSCRIPT italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 + italic_e start_POSTSUPERSCRIPT - 4 italic_ζ end_POSTSUPERSCRIPT end_CELL start_CELL 1 - italic_e start_POSTSUPERSCRIPT - 4 italic_ζ end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 - italic_e start_POSTSUPERSCRIPT - 4 italic_ζ end_POSTSUPERSCRIPT end_CELL start_CELL 1 + italic_e start_POSTSUPERSCRIPT - 4 italic_ζ end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 2 italic_e start_POSTSUPERSCRIPT - italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] , (6) where Z(T,Bz)=eζ+e-3ζ+2eζcosh(βE0)𝑍𝑇subscript𝐵𝑧superscript𝑒𝜁superscript𝑒3𝜁2superscript𝑒𝜁𝛽subscript𝐸0Z(T,B_z)\!=\!e^\zeta+e^-3\zeta+2e^\zeta\cosh\left(\beta E_0\right)italic_Z ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) = italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT - 3 italic_ζ end_POSTSUPERSCRIPT + 2 italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT roman_cosh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) is the partition function, β=1/kBT𝛽1subscript𝑘𝐵𝑇\beta\!=\!1/k_BTitalic_β = 1 / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T and ζ=βJ/4𝜁𝛽𝐽4\zeta\!=\!\beta J/4italic_ζ = italic_β italic_J / 4.



Extractable work (ergotropy). Unlike classical batteries, a QB is characterized by a finite amount of work in a quantum system, which can be extracted via unitary process Allahverdyan et al. (2004). The maximum amount of available work extractable via unitary processes is called ergotropy Liu et al. (2019); Alicki and Fannes (2013); Çakmak (2020).



Given the system state ρ(T,Bz)𝜌𝑇subscript𝐵𝑧\rho(T,B_z)italic_ρ ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) and the QB internal spectrum of H0subscript𝐻0H_0italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, living in an N𝑁Nitalic_N-dimensional Hilbert space, the ergotropy is defined as the amount of work which can be extracted by unitary operations V𝑉Vitalic_V:



ℰ(T,Bz)=Tr[ρ(T,Bz)H0]-minV∈𝒱Tr[Vρ(T,Bz)V†H0],ℰ𝑇subscript𝐵𝑧Trdelimited-[]𝜌𝑇subscript𝐵𝑧subscript𝐻0subscript𝑉𝒱Trdelimited-[]𝑉𝜌𝑇subscript𝐵𝑧superscript𝑉†subscript𝐻0\mathcalE(T,B_z)\!=\!\mboxTr\left[\rho(T,B_z)H_0\right]-\min_V\in% \mathcalV\left\\mboxTr\left[V\rho(T,B_z)V^\daggerH_0\right]\right\,caligraphic_E ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) = Tr [ italic_ρ ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] - roman_min start_POSTSUBSCRIPT italic_V ∈ caligraphic_V end_POSTSUBSCRIPT Tr [ italic_V italic_ρ ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] , (7) where the minimization is taken over the set 𝒱𝒱\mathcalVcaligraphic_V of all unitary operators acting on the system Allahverdyan et al. (2004). The energy eigenvalues of the battery self-Hamiltonian H0subscript𝐻0H_0italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT are ϵ1≤ϵ2≤⋯≤ϵNsubscriptitalic-ϵ1subscriptitalic-ϵ2⋯subscriptitalic-ϵ𝑁\epsilon_1\!\leq\!\epsilon_2\!\leq\!\cdots\!\leq\!\epsilon_Nitalic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_ϵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ ⋯ ≤ italic_ϵ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT; and the eigenvalues of ρ(T,Bz)𝜌𝑇subscript𝐵𝑧\rho(T,B_z)italic_ρ ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) are ϱ1≥ϱ2≥⋯≥ϱNsubscriptitalic-ϱ1subscriptitalic-ϱ2⋯subscriptitalic-ϱ𝑁\varrho_1\!\geq\!\varrho_2\!\geq\!\cdots\!\geq\!\varrho_Nitalic_ϱ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_ϱ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ ⋯ ≥ italic_ϱ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT, associated to eigenvectors |ϱn⟩ketsubscriptitalic-ϱ𝑛\ket\varrho_n| start_ARG italic_ϱ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG ⟩ Sup . Then, the ergotropy in the Eq. (7) can be written as Allahverdyan et al. (2004):



ℰ(T,Bz)=∑i,nN,Nϱn(T,Bz)ϵi(|⟨ϱn|ϵi⟩|2-δni),ℰ𝑇subscript𝐵𝑧superscriptsubscript𝑖𝑛𝑁𝑁subscriptitalic-ϱ𝑛𝑇subscript𝐵𝑧subscriptitalic-ϵ𝑖superscriptinner-productsubscriptitalic-ϱ𝑛subscriptitalic-ϵ𝑖2subscript𝛿𝑛𝑖\displaystyle\mathcalE(T,B_z)=\sum olimits_i,n^N,N\varrho_n(T,B_z% )\epsilon_i\left(|\langle\varrho_n|\epsilon_i\rangle|^2-\delta_ni% \right),caligraphic_E ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N , italic_N end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( | ⟨ italic_ϱ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_δ start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT ) , (8) where we note that this definition is tied to a specific ordering of the eigenvalues ϱnsubscriptitalic-ϱ𝑛\varrho_nitalic_ϱ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT of Eq. (6) and H0subscript𝐻0H_0italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, due to the δnisubscript𝛿𝑛𝑖\delta_niitalic_δ start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT term. This leads to the following expression for the ergtropy per molecule, in thermal equilibrium:



ℰCu(II)=E01-e4ζ[cosh(βE0)-3sinh(βE0)]e4ζ[2cosh(βE0)+1]+1.subscriptℰCu(II)subscript𝐸01superscript𝑒4𝜁delimited-[]𝛽subscript𝐸03𝛽subscript𝐸0superscript𝑒4𝜁delimited-[]2𝛽subscript𝐸011\mathcalE_\textCu(II)=\ E_0\left\\frac1-e^4\zeta\left[\cosh\left% (\beta E_0\right)-3\sinh\left(\beta E_0\right)\right]e^4\zeta\left[2% \cosh\left(\beta E_0\right)+1\right]+1\right\.caligraphic_E start_POSTSUBSCRIPT Cu(II) end_POSTSUBSCRIPT = italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG 1 - italic_e start_POSTSUPERSCRIPT 4 italic_ζ end_POSTSUPERSCRIPT [ roman_cosh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) - 3 roman_sinh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ] end_ARG start_ARG italic_e start_POSTSUPERSCRIPT 4 italic_ζ end_POSTSUPERSCRIPT [ 2 roman_cosh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + 1 ] + 1 end_ARG . (9)



On the other hand, the enormous gap between the ground (singlet - entangled) and the first excited (triplet - separable) state provided by the syn-syn bound, allows us to address the system in the regime of magnetic susceptibility at E0≪kBTmuch-less-thansubscript𝐸0subscript𝑘𝐵𝑇E_0\ll k_BTitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T. From Eq. (9), it is possible to write the ergotropy ℰCu(II)subscriptℰCu(II)\mathcalE_\textCu(II)caligraphic_E start_POSTSUBSCRIPT Cu(II) end_POSTSUBSCRIPT, in terms of the Bleaney-Bowers magnetic susceptibility equation Bleaney and Bowers (1952) (for details, see ref. Sup )



ℰE0≪kBT(T)=E0kBTχ(T)4NAg2μB2[e-4ζ-1],subscriptℰmuch-less-thansubscript𝐸0subscript𝑘𝐵𝑇𝑇subscript𝐸0subscript𝑘𝐵𝑇𝜒𝑇4subscript𝑁𝐴superscript𝑔2superscriptsubscript𝜇𝐵2delimited-[]superscript𝑒4𝜁1\mathcalE_E_0\ll k_BT(T)=\ E_0\frack_BT\chi(T)4N_Ag^2\mu_B% ^2\left[e^-4\zeta-1\right]\leavevmode obreak\ ,caligraphic_E start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_T ) = italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T italic_χ ( italic_T ) end_ARG start_ARG 4 italic_N start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG [ italic_e start_POSTSUPERSCRIPT - 4 italic_ζ end_POSTSUPERSCRIPT - 1 ] , (10) where NAsubscript𝑁𝐴N_Aitalic_N start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is the Avogadro number. As detailed in Sup , this magnetic susceptibility regime is valid in the limit of small magnetic fields Bz≪Bcmuch-less-thansubscript𝐵𝑧subscript𝐵𝑐B_z\ll B_citalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≪ italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT compared to the crossing one (Bc∼556similar-tosubscript𝐵𝑐556B_c\!\sim\!556italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∼ 556 T) Chakraborty and Mitra (2019).



Therefore, Eq. (10) represents one of the main results of this letter, in which the evaluation of the amount of stored ergotropy in metal complexes are experimentally accessible, by measuring a macroscopic property of the system: the magnetic susceptibility.



Ergotropy measurement and quantumness of metal-carboxylate QBs. Up to date, proposals of QBs have been done through an ergotropy destructive readout, through state tomography Liu et al. (2019); Ferraro et al. (2018); Le et al. (2018); Strambini et al. (2020); Monsel et al. (2020). The QB proposed here provides a different approach to read the stored ergotropy, by measuring the available work of a carboxylate-based QB accessible without altering the stored energy, this is a strong request for designing realistic QBs. Fig. 2a presents the theoretical (black line) and experimental (open circles) behavior of the ergotropy as a function of the temperature, where the experimental data was obtained from the measurement of the magnetic susceptibility of the substance. The theoretical curve was plotted by Eq. (9), considering the Earth’s magnetic field of Bz=10-4subscript𝐵𝑧superscript104B_z\!=\!10^-4italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT T, and the experimental parameters g=2𝑔2g=2italic_g = 2 (d9superscript𝑑9d^9italic_d start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT ions) and J/kB=748𝐽subscript𝑘𝐵748J/k_B\!=\!748italic_J / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 748 K, in a good agreement with the laboratory environment in which batteries should operate. It is worth highlighting that by decreasing the temperature we get the maximum amount of ergotropy ℰCu(II)max≈1.12subscriptsuperscriptℰmaxCu(II)1.12\mathcalE^\textmax_\textCu(II)\!\approx\!1.12caligraphic_E start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT Cu(II) end_POSTSUBSCRIPT ≈ 1.12 mJ/mol, where values larger than 99.75%percent99.7599.75\%99.75 % of ℰCu(II)maxsubscriptsuperscriptℰmaxCu(II)\mathcalE^\textmax_\textCu(II)caligraphic_E start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT Cu(II) end_POSTSUBSCRIPT can be reached for temperatures Tmax≤100subscript𝑇max100T_\textmax\!\leq\!100italic_T start_POSTSUBSCRIPT max end_POSTSUBSCRIPT ≤ 100 K reaching the 100%percent100100\%100 % below 83838383 K. Moreover, as can be seen, at room temperature (∼293similar-toabsent293\sim 293∼ 293 K), the amount of stored energy is 75%percent7575\%75 % of the maximum one. This suggests that the carboxylate-based QB could operate under daily life conditions.



To explore the metal-carboxylate as a QB, we now present experimental results that give access to the ergotropy, which here corresponds to the quantum discord. The characterization of the quantumness of the battery is done by computing entanglement of formation (ℱEsubscriptℱ𝐸\mathcalF_Ecaligraphic_F start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT) Hill and Wootters (1997); Wootters (1998) and the quantum discord based on Schatten 1-norm 𝒟(T)𝒟𝑇\mathcalD(T)caligraphic_D ( italic_T ) Ciccarello et al. (2014), in terms of the magnetic susceptibility Sup . First, we find that the ergotropy as function of the temperature is given by the Schatten 1-norm quantum discord as Sup



ℰE0≪kBT(T)=2E0𝒟(T),∀ J>0.formulae-sequencesubscriptℰmuch-less-thansubscript𝐸0subscript𝑘𝐵𝑇𝑇2subscript𝐸0𝒟𝑇for-all𝐽0\mathcalE_E_0\ll k_BT(T)=2E_0\mathcalD(T),\leavevmode obreak\ % \leavevmode obreak\ \forall\leavevmode obreak\ \leavevmode obreak\ J>0.caligraphic_E start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_T ) = 2 italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT caligraphic_D ( italic_T ) , ∀ italic_J >0 . (11)



From above equation we observe that, for any dinuclear metal complex of spin-1/2 with an antiparallel alignment (J>0𝐽0J\!>\!0italic_J >0), the maximum amount of extracted work is related to the existence of genuinely quantum correlations beyond entanglement between the Cu(II) ions, which is quantified by quantum discord. The inset in Fig. 2a show experimental result for the discord, showing that the decay of the ergotropy with temperature corresponds to the decay of Schatten 1-norm quantum discord.



Figure 2: (a) Experimental measurement of ergotropy, in mili Joule per mole, as function of the environment temperature T𝑇Titalic_T (in Kelvin) for the metal-carboxylate QB proposed. The inset shows the experimental quantum discord as function of the temperature. Black solid line denote theoretical results and open circles show the experimental result, computed for the Earth’s magnetic field Bz=10-4subscript𝐵𝑧superscript104B_z\!=\!10^-4italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT T. Vertical dashed line indicates room temperature T=293𝑇293T\!=\!293italic_T = 293 K. (b) Theoretical (solid line curve) and experimental result (open circles) for entanglement of formation, compared with the theoretical result for normalized ergotropy (dot-dashed line).



To see how the ergotropy is stored as quantum correlations beyond entanglement, we also consider the entanglement of formation ℱEsubscriptℱ𝐸\mathcalF_Ecaligraphic_F start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT as shown in Fig. 2b. To this end, we normalized the ergotropy by its maximum value ℰCu(II)norm=ℰCu(II)/ℰCu(II)maxsubscriptsuperscriptℰnormCu(II)subscriptℰCu(II)subscriptsuperscriptℰmaxCu(II)\mathcalE^\textnorm_\textCu(II)\!=\!\mathcalE_\textCu(II)/% \mathcalE^\textmax_\textCu(II)caligraphic_E start_POSTSUPERSCRIPT norm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT Cu(II) end_POSTSUBSCRIPT = caligraphic_E start_POSTSUBSCRIPT Cu(II) end_POSTSUBSCRIPT / caligraphic_E start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT Cu(II) end_POSTSUBSCRIPT, and thus both quantities ℱEsubscriptℱ𝐸\mathcalF_Ecaligraphic_F start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT and ℰCu(II)normsubscriptsuperscriptℰnormCu(II)\mathcalE^\textnorm_\textCu(II)caligraphic_E start_POSTSUPERSCRIPT norm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT Cu(II) end_POSTSUBSCRIPT are defined in the same interval (i.e., [0,1]01[0,1][ 0 , 1 ]). Fig. 2b suggests that the entanglement is not adequate to explain the qualitative behavior of the ergotropy as function of the temperature. Thus, as expected from Eq. (11), there is ergotropy even without entanglement, but not without quantum discord. In addition, as theoretically detailed in Sup , by increasing the range of temperature, one finds that above the temperature of entanglement Te=J/(ln(3)kB)subscript𝑇e𝐽3subscript𝑘𝐵T_\texte=J/(\ln(3)k_B)italic_T start_POSTSUBSCRIPT e end_POSTSUBSCRIPT = italic_J / ( roman_ln ( start_ARG 3 end_ARG ) italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) in which ℱE(T≥Te)=0subscriptℱ𝐸𝑇subscript𝑇e0\mathcalF_E(T\!\geq\!T_\texte)\!=\!0caligraphic_F start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ( italic_T ≥ italic_T start_POSTSUBSCRIPT e end_POSTSUBSCRIPT ) = 0, the ergotropy is is 16.6%percent16.616.6\%16.6 % of the maximum one (ℰCu(II)norm(T≥Te)>0subscriptsuperscriptℰnormCu(II)𝑇subscript𝑇e0\mathcalE^\textnorm_\textCu(II)(T\!\geq\!T_\texte)\!>\!0caligraphic_E start_POSTSUPERSCRIPT norm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT Cu(II) end_POSTSUBSCRIPT ( italic_T ≥ italic_T start_POSTSUBSCRIPT e end_POSTSUBSCRIPT ) >0) and satisfies the Eq. (11). Therefore, these results lead to the conclusion that other quantum correlations beyond entanglement needs to taken into account to describe the amount of ergotropy stored in metal complexes.



Charging and discharging of metal-carboxylate QBs. Due to the high magnetic interaction obtained from the syn-syn bond in the metal-carboxylate compounds, the huge magnetic coupling (J=748𝐽748J\!=\!748italic_J = 748 K) supports the existence of stable quantum correlations at room temperature (T≈293𝑇293T\approx 293italic_T ≈ 293 K) Cruz et al. (2016). More specifically, it allows the studied carboxylate-based metal complex to store a finite amount of ergotropy as quantum discord up to a threshold temperature 513513513513 K (obtained through thermogravimetric analysis reported in ref. Cruz et al. (2016)) under which this material starts to degrade. Moreover, the strong magnetic coupling makes the system practically immune to the magnetic field variations in the experimentally feasible region of small magnetic fields Bz≪Bcmuch-less-thansubscript𝐵𝑧subscript𝐵𝑐B_z\!\ll\!B_citalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≪ italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. Hence, in this regime the role of magnetic field is to define the energy scale (E0subscript𝐸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT) of the battery, as it can be seen in Eq. (10). Therefore, this system does not need a time-dependent external field for the charging process, differently from the several kinds of QBs already proposed in literature Campaioli et al. (2017); Andolina et al. (2019); Santos et al. (2020); Le et al. (2018); Kwon et al. (2018); Monsel et al. (2020); Kamin et al. (2020).



Since there is no additional decoherence channel, the dimeric cells are magnetically isolated, and the material does not degrade below the threshold temperature. Thus, the system remains in the state given by Eq. (6) and it is able to store a finite amount of energy quantum discord stable at room temperature. Thus, the charge lifetime is defined by the existence of non-zero genuine quantum correlations between the cells of the battery. Since the ergrotopy increases as we decrease the temperature, one can charge the QB the submitting the battery to thermal contact with a cold reservoir, under presence of a static reference magnetic field Bz≪Bcmuch-less-thansubscript𝐵𝑧subscript𝐵𝑐B_z\ll B_citalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≪ italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. Therefore, the charging process of this carboxylate-based QB is given by a mechanism known in the literature as charging assisted by thermalization Hovhannisyan et al. (2020).



In this regard, a charging and discharging steps of a major cycle can be presented for this QB, as sketched in Fig. 1b: (I) an external stimulus (e.g., electromagnetic field pulses Moreno-Pineda et al. (2018); Gaita-Ariño et al. (2019), or pressure Cruz et al. (2017); Cruz and Anka (2020)), lowers the degree of quantum discord of the system, drives it to the triplet state subspace (more specifically the state |↓↓⟩ket↓absent↓\ket\downarrow\downarrow| start_ARG ↓ ↓ end_ARG ⟩), discharging the battery by consuming the stored work Francica et al. (2020); Monsel et al. (2020); Hovhannisyan et al. (2020); Çakmak (2020); (II) removing the stimulus, the material, in thermal equilibrium with a reservoir, returns to the singlet ground-state by increasing the population of |β-⟩ketsubscript𝛽\ket\beta_-| start_ARG italic_β start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_ARG ⟩, charging the battery (see top figure of the charging-discharging process). It is worth noting that these two steps belong to a major cycle and further discussions will be presented elsewhere.



Conclusions. In summary, this Letter shows that carboxylate-based magnetic systems are promising platforms for QBs. The available work is stored as quantum discord, i.e., genuine quantum correlations, at room temperature. Due to the syn-syn bound a strong magnetic interaction emerges,leading to a large gap between the singlet ground state and the first excited one (see Fig. 1), which makes this material suitable for engineering room temperature QBs. In this sense, an external (fixed) magnetic field Bzsubscript𝐵𝑧B_zitalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT can be used to tune the total available work for the battery. In addition, the ergotropy measurement of the QB proposed here is done by non-destructive experimental techniques associated with the measurement of the compounds magnetic susceptibility, and no energy is lost during the readout of the available work. The strong intra-cell interactions also make this system largely immune to the self-discharging process, which allows it to store energy as long as there is quantum discord in the system at room temperature and beyond, up to threshold temperature (513 K) in which the material degrades. Moreover, at room temperature, the efficiency of this complex is approximately 75%percent7575\%75 % of the maximum one (obtained at T<83𝑇83T\!italic_T <83 K), in which quantum correlations appear, once again, as a harvestable resource of great interest for quantum technologies Sapienza et al. (2019).



Since a realistic implementation of a quantum battery has not yet been settled in an LDMC, the role of its chemical aspects when operating it as a QB remains to be studied. Nevertheless, the results presented in this letter open a broad avenue for research of metal complexes as candidate platforms for QBs and the development of enhanced energy storage platforms through material engineering. For example, quantum correlations in metal complexes can be handled by controlling external parameters, as structural factors and thermodynamic properties Wasielewski et al. (2020); Čenčariková and Strečka (2020); Cruz et al. (2017); Cruz and Anka (2020). Thus, different kinds of control of the gap between the ground and excited state in dinuclear spin-1/2 metal complexes QBs can be done through material engineering techniques Gaita-Ariño et al. (2019); Moreno-Pineda et al. (2018); Wasielewski et al. (2020). In addition, quantum properties of solid-state systems are drastically affected by thermal decohering effects, hindering the development of feasible quantum batteries that operate at room temperature. This letter shows that the study on carboxylate-based materials can change this view and paves the way for the enhancement and stability of the charging and energy storage processes in QBs.



Acknowledgements.The authors would like to thank P. Brandão and A. M. dos Santos for the compound’s data. A. C. Santos is supported by São Paulo Research Foundation (FAPESP) (Grant No 2019/22685-1). R. B. benefited from Grants from São Paulo Research Foundation (FAPESP, Grants No. 2018/15554-5 and 2019/22685-1). R. B. benefited from grants from the National Council for Scientific and Technological Development (CNPq, Grant Nos. 313886/2020-2 and 409946/2018-4). M.S. Reis thanks FAPERJ for financial support. M.S. Reis belongs to the INCT of Refrigeração e Termofísica funding by the National Counsel of Technological and Scientific Development (CNPq), Grant No. 404023/2019-3. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - finance code 001.



Supplemental Material for:



Quantum battery based on quantum discord at room temperature C. Cruz,1,∗1∗^1,\color[rgb]0,0,1\aststart_FLOATSUPERSCRIPT 1 , ∗ end_FLOATSUPERSCRIPT M. F. Anka,2,†2†^2,\color[rgb]0,0,1\daggerstart_FLOATSUPERSCRIPT 2 , † end_FLOATSUPERSCRIPT M. S. Reis,2,‡2‡^2,\color[rgb]0,0,1\ddaggerstart_FLOATSUPERSCRIPT 2 , ‡ end_FLOATSUPERSCRIPT R. Bachelard,3,§3§^3,\color[rgb]0,0,1\lx@sectionsignstart_FLOATSUPERSCRIPT 3 , § end_FLOATSUPERSCRIPT and Alan C. Santos3,¶3¶^3,\color[rgb]0,0,1\lx@paragraphsignstart_FLOATSUPERSCRIPT 3 , ¶ end_FLOATSUPERSCRIPT 11^1start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPTGrupo de Informação Quântica, Centro de Ciências Exatas e das Tecnologias, Universidade Federal do Oeste da Bahia - Campus Reitor Edgard Santos. Rua Bertioga, 892, Morada Nobre I, 47810-059 Barreiras, Bahia, Brasil. 22^2start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPTInstituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, 24210-346 Niterói, Rio de Janeiro, Brasil. 33^3start_FLOATSUPERSCRIPT 3 end_FLOATSUPERSCRIPTDepartamento de Física, Universidade Federal de São Carlos, Rodovia Washington Luís, km 235 - SP-310, 13565-905 São Carlos, SP, Brasil. ∗∗^\color[rgb]0,0,1\aststart_FLOATSUPERSCRIPT ∗ [email protected], ††^\color[rgb]0,0,1\daggerstart_FLOATSUPERSCRIPT † [email protected], ‡‡^\color[rgb]0,0,1\ddaggerstart_FLOATSUPERSCRIPT ‡ [email protected]



§§^\color[rgb]0,0,1\lx@sectionsignstart_FLOATSUPERSCRIPT § [email protected], ¶¶^\color[rgb]0,0,1\lx@paragraphsignstart_FLOATSUPERSCRIPT ¶ [email protected]



Appendix A Quantum level-crossing in a metal complex



Quantum antiferromagnets typically present a quantum level-crossing in the presence of an external magnetic field Bzsubscript𝐵𝑧B_zitalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, splitting its energy levels and changing the corresponding populations Chakraborty and Mitra (2019); Cruz and Anka (2020); Breunig et al. (2017). In particular for the carboxylate based metal complex described in the main text, the magnetic field splits the degeneracy of the triply-degenerate subspace βt⟩=β+⟩,ketsubscript𝛽tketsubscript𝛽ket↓absent↓ket↑absent↑\\ket\beta_\textt\\!=\!\\ket\beta_+,\ket\downarrow\downarrow,% \ket\uparrow\uparrow\ start_ARG italic_β start_POSTSUBSCRIPT t end_POSTSUBSCRIPT end_ARG ⟩ = start_ARG italic_β start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ⟩ , , inducing a level-crossing between its singlet-ground state and the first excited one when the field reaches a critical value Chakraborty and Mitra (2019); Breunig et al. (2017). This critical value can be calculated through the evaluation of the populations. From the Hamiltonian:



H=E0(S1(z)+S2(z))+J(S→1⋅S→2),𝐻subscript𝐸0superscriptsubscript𝑆1𝑧superscriptsubscript𝑆2𝑧𝐽⋅subscript→𝑆1subscript→𝑆2\displaystyle H=E_0\left(S_1^(z)+S_2^(z)\right)+J\left(\vecS_1% \cdot\vecS_2\right),italic_H = italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_z ) end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_z ) end_POSTSUPERSCRIPT ) + italic_J ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , (S1) and the spectral decomposition of the state ρ(T,Bz)=e-βH/Z𝜌𝑇subscript𝐵𝑧superscript𝑒𝛽𝐻𝑍\rho(T,B_z)=e^-\beta H/Zitalic_ρ ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) = italic_e start_POSTSUPERSCRIPT - italic_β italic_H end_POSTSUPERSCRIPT / italic_Z, where Z=Tr[e-βH]𝑍Trdelimited-[]superscript𝑒𝛽𝐻Z=\mboxTr\left[e^-\betaH\right]italic_Z = Tr [ italic_e start_POSTSUPERSCRIPT - italic_β italic_H end_POSTSUPERSCRIPT ], the battery state eigenvalues ϱisubscriptitalic-ϱ𝑖\varrho_iitalic_ϱ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (population) and its corresponding eigenvectors can be written as:



ϱ1subscriptitalic-ϱ1\displaystyle\varrho_1italic_ϱ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =11+eζ(1+2cosh(βE0))→|β-⟩,absent11superscript𝑒𝜁12𝛽subscript𝐸0→ketsubscript𝛽\displaystyle=\frac11+e^\zeta(1+2\cosh\left(\beta E_0\right))% \rightarrow\ket\beta_-,= divide start_ARG 1 end_ARG start_ARG 1 + italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT ( 1 + 2 roman_cosh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) end_ARG → | start_ARG italic_β start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_ARG ⟩ , (S2)



ϱ2subscriptitalic-ϱ2\displaystyle\varrho_2italic_ϱ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =eζ+βE01+eζ(1+2cosh(βE0))→|↑↑⟩,absentsuperscript𝑒𝜁𝛽subscript𝐸01superscript𝑒𝜁12𝛽subscript𝐸0→ket↑absent↑\displaystyle=\frace^\zeta+\beta E_01+e^\zeta(1+2\cosh\left(\beta E_% 0\right))\rightarrow\ket\uparrow\uparrow,= divide start_ARG italic_e start_POSTSUPERSCRIPT italic_ζ + italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG 1 + italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT ( 1 + 2 roman_cosh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) end_ARG → | start_ARG ↑ ↑ end_ARG ⟩ , (S3)



ϱ3subscriptitalic-ϱ3\displaystyle\varrho_3italic_ϱ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT =eζ1+eζ(1+2cosh(βE0))→|β+⟩,absentsuperscript𝑒𝜁1superscript𝑒𝜁12𝛽subscript𝐸0→ketsubscript𝛽\displaystyle=\frace^\zeta1+e^\zeta(1+2\cosh\left(\beta E_0\right))% \rightarrow\ket\beta_+,= divide start_ARG italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT end_ARG start_ARG 1 + italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT ( 1 + 2 roman_cosh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) end_ARG → | start_ARG italic_β start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ⟩ , (S4)



ϱ4subscriptitalic-ϱ4\displaystyle\varrho_4italic_ϱ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT =eζ-βE01+eζ(1+2cosh(βE0))→|↓↓⟩,absentsuperscript𝑒𝜁𝛽subscript𝐸01superscript𝑒𝜁12𝛽subscript𝐸0→ket↓absent↓\displaystyle=\frace^\zeta-\beta E_01+e^\zeta(1+2\cosh\left(\beta E_% 0\right))\rightarrow\ket\downarrow\downarrow,= divide start_ARG italic_e start_POSTSUPERSCRIPT italic_ζ - italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG 1 + italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT ( 1 + 2 roman_cosh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) end_ARG → | start_ARG ↓ ↓ end_ARG ⟩ , (S5) where |β±⟩=(|↓↑⟩±|↑↓⟩)/2ketsubscript𝛽plus-or-minusplus-or-minusket↓absent↑ket↑absent↓2\ket\beta_\pm\!=\!(\ket\downarrow\uparrow\pm\ket\uparrow\downarrow)/% \sqrt2| start_ARG italic_β start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT end_ARG ⟩ = ( | start_ARG ↓ ↑ end_ARG ⟩ ± | start_ARG ↑ ↓ end_ARG ⟩ ) / square-root start_ARG 2 end_ARG; β=1/kBT𝛽1subscript𝑘𝐵𝑇\beta=1/k_BTitalic_β = 1 / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T; ζ=-βJ𝜁𝛽𝐽\zeta\!=\!-\beta Jitalic_ζ = - italic_β italic_J; and E0=μBgBzsubscript𝐸0subscript𝜇𝐵𝑔subscript𝐵𝑧E_0\!=\!\mu_BgB_zitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_g italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, with g𝑔gitalic_g being the isotropic Landé factor and μBsubscript𝜇𝐵\mu_Bitalic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT the Bohr magneton. Since the quantum level crossing is temperature independent, Fig. S1 shows the populations ϱnsubscriptitalic-ϱ𝑛\varrho_nitalic_ϱ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT as function of the magnetic field at room temperature T=293𝑇293T=293italic_T = 293 K, obtained from the experimental parameters g=2𝑔2g=2italic_g = 2 (d9superscript𝑑9d^9italic_d start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT ions) and J/kB=748𝐽subscript𝑘𝐵748J/k_B\!=\!748italic_J / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 748 K Cruz et al. (2016). The inset shows the magnetic field dependence of the energy levels



E1subscript𝐸1\displaystyle E_1italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =0→|β-⟩,absent0→ketsubscript𝛽\displaystyle=0\rightarrow\ket\beta_-,= 0 → | start_ARG italic_β start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_ARG ⟩ , (S6)



E2subscript𝐸2\displaystyle E_2italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =J-E0,→|↑↑⟩,fragmentsJsubscript𝐸0,→ket↑absent↑,\displaystyle=J-E_0,\rightarrow\ket\uparrow\uparrow,= italic_J - italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , → | start_ARG ↑ ↑ end_ARG ⟩ , (S7)



E3subscript𝐸3\displaystyle E_3italic_E start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT =J→|β+⟩,absent𝐽→ketsubscript𝛽\displaystyle=J\rightarrow\ket\beta_+,= italic_J → | start_ARG italic_β start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ⟩ , (S8)



E4subscript𝐸4\displaystyle E_4italic_E start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT =J+E0→|↓↓⟩,absent𝐽subscript𝐸0→ket↓absent↓\displaystyle=J+E_0\rightarrow\ket\downarrow\downarrow,= italic_J + italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → | start_ARG ↓ ↓ end_ARG ⟩ , (S9) obtained from the spectral decomposition of the system Hamiltonian, Eq. (S1). As can be seem, the magnetic field over the populations is driven our system from a maximally entangled ground state |β-⟩=[|↓↑⟩-|↑↓⟩]/2ketsubscript𝛽delimited-[]ket↓absent↑ket↑absent↓2\ket\beta_-\!=\!\left[\ket\downarrow\uparrow-\ket\uparrow\downarrow% \right]/\sqrt2| start_ARG italic_β start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_ARG ⟩ = [ | start_ARG ↓ ↑ end_ARG ⟩ - | start_ARG ↑ ↓ end_ARG ⟩ ] / square-root start_ARG 2 end_ARG, to a separable ground state |↑↑⟩ket↑absent↑\ket\uparrow\uparrow| start_ARG ↑ ↑ end_ARG ⟩, when the magnetic field surpass the critical value Bc≈556subscript𝐵𝑐556B_c\approx 556italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ≈ 556 T (E0=Jsubscript𝐸0𝐽E_0\!=\!Jitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_J). This huge value of crossing field is a consequence of the strong magnetic coupling (J/kB=748𝐽subscript𝑘𝐵748J/k_B\!=\!748italic_J / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 748 K) yielded by the syn-syn𝑠𝑦𝑛𝑠𝑦𝑛syn-synitalic_s italic_y italic_n - italic_s italic_y italic_n bound, which shields the compound from environment fluctuations Reis et al. (2012); Čenčariková and Strečka (2020); Cruz and Anka (2020); Souza et al. (2008); Cruz et al. (2017). Thus, from Eq. (S6)-(S9), for any Bz≪Bcmuch-less-thansubscript𝐵𝑧subscript𝐵𝑐B_z\ll B_citalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≪ italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT (E0≪Jmuch-less-thansubscript𝐸0𝐽E_0\ll Jitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ italic_J), the compound behaves as an effective two-level system.



Appendix B Ergotropy of dinuclear spin-1/2 metal complexes



Quantum systems can be defined as a quantum batteries when a finite amount of work, stored in its quantum states can be extracted via unitary process Liu et al. (2019); Alicki and Fannes (2013); Çakmak (2020). Considering the carboxylate-based metal complex described by Eq. (S1), one can define the battery Hamiltonian as



H=H0+Hint,𝐻subscript𝐻0subscript𝐻𝑖𝑛𝑡H=H_0+H_int,italic_H = italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_H start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT , (S10) where H0=E0(S1(z)+S2(z))subscript𝐻0subscript𝐸0superscriptsubscript𝑆1𝑧superscriptsubscript𝑆2𝑧H_0=E_0\left(S_1^(z)+S_2^(z)\right)italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_z ) end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_z ) end_POSTSUPERSCRIPT ) is the battery self-Hamiltonian, which defines the energy scale ϵ1≤ϵ2≤⋯≤ϵNsubscriptitalic-ϵ1subscriptitalic-ϵ2⋯subscriptitalic-ϵ𝑁\epsilon_1\!\leq\!\epsilon_2\!\leq\!\cdots\!\leq\!\epsilon_Nitalic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_ϵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ ⋯ ≤ italic_ϵ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT and eigenstates |ϵi⟩ketsubscriptitalic-ϵ𝑖\ket\epsilon_i| start_ARG italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ of the quantum battery, satisfying the eigenvalue equation H0|ϵi⟩=ϵi|ϵi⟩subscript𝐻0ketsubscriptitalic-ϵ𝑖subscriptitalic-ϵ𝑖ketsubscriptitalic-ϵ𝑖H_0\ket\epsilon_i\! Gaming news =\!\epsilon_i\ket\epsilon_iitalic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | start_ARG italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ = italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩; and Hint=J(S→1⋅S→2)subscript𝐻𝑖𝑛𝑡𝐽⋅subscript→𝑆1subscript→𝑆2H_int=J\left(\vecS_1\cdot\vecS_2\right)italic_H start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT = italic_J ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) defines an internal interaction of the system. The system has a corresponding density matrix, defined in thermal equilibrium as:



ρ(T,Bz)𝜌𝑇subscript𝐵𝑧\displaystyle\rho(T,B_z)italic_ρ ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) =\displaystyle== eζ2Z[2eβE000001+e-4ζ1-e-4ζ001-e-4ζ1+e-4ζ00002e-βE0],superscript𝑒𝜁2𝑍delimited-[]matrix2superscript𝑒𝛽subscript𝐸000001superscript𝑒4𝜁1superscript𝑒4𝜁001superscript𝑒4𝜁1superscript𝑒4𝜁00002superscript𝑒𝛽subscript𝐸0\displaystyle\frace^\zeta2Z\left[\beginmatrix2e^\beta E_0&0&0&0% \\ 0&1+e^-4\zeta&1-e^-4\zeta&0\\ 0&1-e^-4\zeta&1+e^-4\zeta&0\\ 0&0&0&2e^-\beta E_0\endmatrix\right]\leavevmode obreak\ ,divide start_ARG italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_Z end_ARG [ start_ARG start_ROW start_CELL 2 italic_e start_POSTSUPERSCRIPT italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 + italic_e start_POSTSUPERSCRIPT - 4 italic_ζ end_POSTSUPERSCRIPT end_CELL start_CELL 1 - italic_e start_POSTSUPERSCRIPT - 4 italic_ζ end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 - italic_e start_POSTSUPERSCRIPT - 4 italic_ζ end_POSTSUPERSCRIPT end_CELL start_CELL 1 + italic_e start_POSTSUPERSCRIPT - 4 italic_ζ end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 2 italic_e start_POSTSUPERSCRIPT - italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] , (S15) where Z(T,Bz)=eζ+e-3ζ+2eζcosh(βE0)𝑍𝑇subscript𝐵𝑧superscript𝑒𝜁superscript𝑒3𝜁2superscript𝑒𝜁𝛽subscript𝐸0Z(T,B_z)\!=\!e^\zeta+e^-3\zeta+2e^\zeta\cosh\left(\beta E_0\right)italic_Z ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) = italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT - 3 italic_ζ end_POSTSUPERSCRIPT + 2 italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT roman_cosh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) is the partition function.



For non-pure states (Trρ2≠1𝑇𝑟superscript𝜌21Tr\\rho^2\\! eq\!1italic_T italic_r italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≠ 1), the amount of available work which can be extracted from unitary operations V𝑉Vitalic_V is not given by the internal energy of the system U=TrH0ρ𝑈𝑇𝑟subscript𝐻0𝜌U\!=\!Tr\H_0\rho\italic_U = italic_T italic_r italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_ρ , but rather by the maximum amount of available work extractable via unitary processes, defined by the ergotropy Allahverdyan et al. (2004); Liu et al. (2019); Alicki and Fannes (2013); Çakmak (2020):



ℰ(T,Bz)=Tr[ρ(T,Bz)H0]-minV∈𝒱Tr[Vρ(T,Bz)V†H0],ℰ𝑇subscript𝐵𝑧Trdelimited-[]𝜌𝑇subscript𝐵𝑧subscript𝐻0subscript𝑉𝒱Trdelimited-[]𝑉𝜌𝑇subscript𝐵𝑧superscript𝑉†subscript𝐻0\mathcalE(T,B_z)\!=\!\mboxTr\left[\rho(T,B_z)H_0\right]-\min_V\in% \mathcalV\left\\mboxTr\left[V\rho(T,B_z)V^\daggerH_0\right]\right\,caligraphic_E ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) = Tr [ italic_ρ ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] - roman_min start_POSTSUBSCRIPT italic_V ∈ caligraphic_V end_POSTSUBSCRIPT Tr [ italic_V italic_ρ ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] , (S16) where the minimization, taken over the set 𝒱𝒱\mathcalVcaligraphic_V of all unitary operators which acts on the system Allahverdyan et al. (2004) is obtained by writing Eq. (S15) in its spectral decomposition, with the populations ϱnsubscriptitalic-ϱ𝑛\varrho_nitalic_ϱ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Eq. (S6)-(S9), in decreasing order ϱ1≥ϱ2≥⋯≥ϱNsubscriptitalic-ϱ1subscriptitalic-ϱ2⋯subscriptitalic-ϱ𝑁\varrho_1\!\geq\!\varrho_2\!\geq\!\cdots\!\geq\!\varrho_Nitalic_ϱ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_ϱ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ ⋯ ≥ italic_ϱ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT, and the eigenvalues of the self-Hamiltonian in the increasing order ϵ1≤ϵ2≤⋯≤ϵNsubscriptitalic-ϵ1subscriptitalic-ϵ2⋯subscriptitalic-ϵ𝑁\epsilon_1\!\leq\!\epsilon_2\!\leq\!\cdots\!\leq\!\epsilon_Nitalic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_ϵ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ ⋯ ≤ italic_ϵ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT. Then, the maximum amount of available work which can be extracted from a quantum state, is given by Allahverdyan et al. (2004); Francica et al. (2017); Alicki and Fannes (2013):



ℰ=∑i,nN,Nϱnϵi(|⟨ϱn|ϵi⟩|2-δni).ℰsuperscriptsubscript𝑖𝑛𝑁𝑁subscriptitalic-ϱ𝑛subscriptitalic-ϵ𝑖superscriptinner-productsubscriptitalic-ϱ𝑛subscriptitalic-ϵ𝑖2subscript𝛿𝑛𝑖\displaystyle\mathcalE=\sum olimits_i,n^N,N\varrho_n\epsilon_i\left% (|\langle\varrho_n|\epsilon_i\rangle|^2-\delta_ni\right).caligraphic_E = ∑ start_POSTSUBSCRIPT italic_i , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N , italic_N end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( | ⟨ italic_ϱ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_ϵ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_δ start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT ) . (S17)



In particular, as mentioned before, due to the characteristics quantum level crossing presented by quantum antiferromagnets Chakraborty and Mitra (2019); Cruz and Anka (2020); Breunig et al. (2017), it is possible to establish two orderings of the quantities ϱnsubscriptitalic-ϱ𝑛\varrho_nitalic_ϱ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Consequently, for dinuclear spin-1/2 metal complexes, the ergotropy can defined in two physical regimes: one for E0<jsubscript𝐸0𝐽e_0\!italic_e start_postsubscript 0 end_postsubscript <italic_j, corresponding to bz<bcsubscript𝐵𝑧subscript𝐵𝑐b_zitalic_b italic_z <italic_b italic_c end_postsubscript; and other for e0≥jsubscript𝐸0𝐽e_0\!\geq\!jitalic_e ≥ italic_j, bz≥bcsubscript𝐵𝑧subscript𝐵𝑐b_z\geq b_citalic_b italic_b end_postsubscript. thus, from eq. (s17), we obtain the ergotropy per molecule in thermal equilibrium with a reservoir at temperature t𝑇titalic_t under presence of static reference magnetic field bzsubscript𝐵𝑧b_zitalic_b as< p>















</jsubscript𝐸0𝐽e_0\!italic_e>



ℰ=E01-eζ[cosh(βE0)-3sinh(βE0)]eζ[2cosh(βE0)+1]+1,E0<j4e0eζsinh(βe0)eζ(2cosh(βe0)+1)+1,e0≥j.fragmentsefragmentssubscript𝐸01superscript𝑒𝜁delimited-[]𝛽subscript𝐸03𝛽subscript𝐸0superscript𝑒𝜁delimited-[]2𝛽subscript𝐸011subscript𝐸0𝐽4subscript𝐸0superscript𝑒𝜁𝛽subscript𝐸0superscript𝑒𝜁2𝛽subscript𝐸011subscript𝐸0𝐽.\mathcale=\left\\beginaligned e_0\left\\frac1-e^\zeta\left[\cosh% \left(\beta e_0\right)-3\sinh\left(\beta e_0\right)\right]e^\zeta% \left[2\cosh\left(\beta e_0\right)+1\right]+1\right\,\leavevmode obreak\ % \leavevmode e_0<j\\ \frac4e_0e^\zeta\sinh\left(\beta e_0\right)e^\zeta\left(2\cosh% e_0\right)+1\right)+1,\leavevmode obreak% \ e_0\geqj\endaligned\right.\leavevmode .caligraphic_e="start_ROW" start_cell italic_e start_postsubscript 0 end_postsubscript divide start_arg 1 - start_postsuperscript italic_ζ end_postsuperscript [ roman_cosh ( italic_β ) 3 roman_sinh ] end_arg 2 + , <italic_j end_cell end_row start_row 4 ≥ italic_j . (s18)< p>















</j4e0eζsinh(βe0)eζ(2cosh(βe0)+1)+1,e0≥j.fragmentsefragmentssubscript𝐸01superscript𝑒𝜁delimited-[]𝛽subscript𝐸03𝛽subscript𝐸0superscript𝑒𝜁delimited-[]2𝛽subscript𝐸011subscript𝐸0𝐽4subscript𝐸0superscript𝑒𝜁𝛽subscript𝐸0superscript𝑒𝜁2𝛽subscript𝐸011subscript𝐸0𝐽.\mathcale=\left\\beginaligned>



In particular, for the carboxylate-based metal complex described in the main text, the regime Bz≥Bcsubscript𝐵𝑧subscript𝐵𝑐B_z\geq B_citalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≥ italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT are not accessible by usual experimental techniques, due to the huge value of crossing field (556556556556 T), consequence of the syn-syn bound. Therefore, the experimentally feasible available work, stored in the metal-carboxylate compound, can be found only in the regime E0<jintsubscript𝐸0subscript𝐽inte_0\!italic_e start_postsubscript 0 end_postsubscript <italic_j int end_postsubscript.< p>















</jintsubscript𝐸0subscript𝐽inte_0\!italic_e>



Appendix C Ergotropy and magnetic susceptibility



From the system Hamiltonian, Eq. (S1), we obtain the canonical partition function Z(T,Bz)=eζ+e-3ζ+2eζcosh(βE0)𝑍𝑇subscript𝐵𝑧superscript𝑒𝜁superscript𝑒3𝜁2superscript𝑒𝜁𝛽subscript𝐸0Z(T,B_z)\!=\!e^\zeta+e^-3\zeta+2e^\zeta\cosh\left(\beta E_0\right)italic_Z ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) = italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT - 3 italic_ζ end_POSTSUPERSCRIPT + 2 italic_e start_POSTSUPERSCRIPT italic_ζ end_POSTSUPERSCRIPT roman_cosh ( italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Then, one can obtain the magnetization of a dinuclear spin-1/2 metal complex as Reis (2013)



ℳz(T,Bz)subscriptℳ𝑧𝑇subscript𝐵𝑧\displaystyle\mathcalM_z(T,B_z)caligraphic_M start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) =NkBTZ(T,Bz)∂∂BzZ(T,Bz),absent𝑁subscript𝑘𝐵𝑇𝑍𝑇subscript𝐵𝑧subscript𝐵𝑧𝑍𝑇subscript𝐵𝑧\displaystyle=\fracNk_BTZ(T,B_z)\frac\partial\partial B_zZ(T,B_% z)\leavevmode obreak\ ,= divide start_ARG italic_N italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_ARG start_ARG italic_Z ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) end_ARG divide start_ARG ∂ end_ARG start_ARG ∂ italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG italic_Z ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) ,



=NgμB(e2βE0-1)1+eβE0(1+e-ζ+eβE0).absent𝑁𝑔subscript𝜇𝐵superscript𝑒2𝛽subscript𝐸011superscript𝑒𝛽subscript𝐸01superscript𝑒𝜁superscript𝑒𝛽subscript𝐸0\displaystyle=\fracNg\mu_B\left(e^2\beta E_0-1\right)1+e^\beta E_0% \left(1+e^-\zeta+e^\beta E_0\right)\leavevmode obreak\ .= divide start_ARG italic_N italic_g italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_e start_POSTSUPERSCRIPT 2 italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT - 1 ) end_ARG start_ARG 1 + italic_e start_POSTSUPERSCRIPT italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( 1 + italic_e start_POSTSUPERSCRIPT - italic_ζ end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT italic_β italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) end_ARG . (S19)



Thus, the regime of magnetic susceptibility of this system is reached under low values of magnetic field in which the magnetization has the linear dependence ℳz=Bzχsubscriptℳ𝑧subscript𝐵𝑧𝜒\mathcalM_z=B_z\chicaligraphic_M start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT italic_χ Reis (2013). In particular, for the described carboxylate-based metal complex, with the experimental parameters J/kB=748𝐽subscript𝑘𝐵748J/k_B\!=\!748italic_J / italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 748 K and g=2𝑔2g=2italic_g = 2 Cruz et al. (2016), we obtain the theoretical magnetization ℳz(T,Bz)subscriptℳ𝑧𝑇subscript𝐵𝑧\mathcalM_z(T,B_z)caligraphic_M start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_T , italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) as a function of the magnetic field for different temperatures (Fig. S2). As can be seen in Fig. S2, for the temperature regime under which the magnetic susceptibility is measured, the system remains in the limit of small magnetic fields even for values up to dozens of Teslas. In a general way, the enormous gap between the ground (singlet - entangled) and the first excited (triplet - separable) state provided by the syn-syn bound, allows us to address the system in the regime of magnetic susceptibility at E0≪kBTmuch-less-thansubscript𝐸0subscript𝑘𝐵𝑇E_0\ll k_BTitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T when Bz≪Bcmuch-less-thansubscript𝐵𝑧subscript𝐵𝑐B_z\ll B_citalic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≪ italic_B start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT.



Thus, applying the limit E0≪kBTmuch-less-thansubscript𝐸0subscript𝑘𝐵𝑇E_0\ll k_BTitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T in Eq. (S19) we can obtain the the Bleaney-Bowers equation for magnetic susceptibility for two spin-1/2121/21 / 2 interacting systems Bleaney and Bowers (1952)



χ(T)𝜒𝑇\displaystyle\chi(T)italic_χ ( italic_T ) =limE0≪kBTℳzBz=2N(gμB)2kBT(13+e-ζ).absentsubscriptmuch-less-thansubscript𝐸0subscript𝑘𝐵𝑇subscriptℳ𝑧subscript𝐵𝑧2𝑁superscript𝑔subscript𝜇𝐵2subscript𝑘𝐵𝑇13superscript𝑒𝜁\displaystyle=\lim_E_0\ll k_BT\frac\mathcalM_zB_z=\frac2N(g% \mu_B)^2k_BT\left(\frac13+e^-\zeta\right).= roman_lim start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT divide start_ARG caligraphic_M start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG = divide start_ARG 2 italic_N ( italic_g italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_ARG ( divide start_ARG 1 end_ARG start_ARG 3 + italic_e start_POSTSUPERSCRIPT - italic_ζ end_POSTSUPERSCRIPT end_ARG ) . (S20)



From Eq. (S18) is possible to write the ergotropy in the regime of magnetic susceptibility (E0≪kBTmuch-less-thansubscript𝐸0subscript𝑘𝐵𝑇E_0\ll k_BTitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≪ italic_k start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T ) as



ℰE0≪kBT=E0(1-eζ3eζ+1),E0<j0,e0≥j.fragmentssubscriptℰmuch-less-thansubscript𝐸0subscript𝑘𝐵𝑇fragmentssubscript𝐸01superscript𝑒𝜁3superscript𝑒𝜁1subscript𝐸0𝐽0subscript𝐸0𝐽.\mathcale_e_0\ll k_bt="\left\\beginaligned" e_0\left(\frac1-e^% \zeta3e^\zeta+1\right),\leavevmode obreak\ \leavevmode e_0% <j\\ 0,\leavevmode e_0\geqj\endaligned\right.% .caligraphic_e start_postsubscript italic_e 0 end_postsubscript ≪ italic_k italic_b italic_t></j0,e0≥j.fragmentssubscriptℰmuch-less-thansubscript𝐸0subscript𝑘𝐵𝑇fragmentssubscript𝐸01superscript𝑒𝜁3superscript𝑒𝜁1subscript𝐸0𝐽0subscript𝐸0𝐽.\mathcale_e_0\ll>