Showcasing extractable work of quantum battery via entropic uncertainty relations

The storage and utilisation of energy have always been an important topic worldwide and can be achieved by devices, that is, so-called batteries. In contrast to the conversion of chemical energy into electric energy in the classical battery scheme, quantum batteries (QBs) review are regarded as a new and promising energy storage process, pioneeringly presented by Alicki and Fannes propose . The core principle of quantum batteries is using quantum resources (e.g., quantum entanglement and coherence) for energy storage resource1 ; resource2 ; resource3 . With the development of quantum information theory, quantum batteries have received considerable attention, and progress has been made in theory, including the construction of charging models model1 ; model2 ; model3 ; model4 ; model5 ; model6 ; model7 , open QB systems open1 ; open2 ; open3 ; open4 ; open5 ; open6 ; open7 ; open8 ; Dou-F-Q1 , many-body QBs (e.g., Dicke quantum battery and spin-based batteries) model4 ; r1 ; cells1 ; cells2 ; cells3 ; cells4 ; Dou-F-Q2 , and so on. Several promising experiments have been conducted for QBs in organic microcavities experiment1 , superconducting experiment2 , quantum dots experiment4 and NMR experiment3 platforms.

The great contribution of quantum resources is to allow QBs to have superextensive scaling of the charging power in many-body QB systems, such as entanglement of battery subspace resulting in quantum advantages over classical schemes review ; model1 ; model2 . However, many investigations review ; open8 ; Dou-F-Q2 ; x18 ; x18++ ; capacity ; measurement have revealed that quantum sources cannot independently and adequately predict the energy variation in QBs. This is due to several reasons, such as non-equilibrium thermodynamic processes in which multiple quantum resources alternately or simultaneously act on the battery. Therefore, it is argued that the analysis of QB’s performance needs more indicators in the specific charging protocol. Conversely, G$\rm\ddot{u}$hne and Lewenstein have studied the intrinsic relationship between entanglement and entropic uncertainty relations (EURs) ee . With this in mind, it is natural to ask whether EURs can be used to predict the change in quantum battery’s energy. Upon studying the relationship between QB’s energy and EURs, we found the answer to be positive.

The uncertainty relation, originally proposed by Heisenberg, is the cornerstone of quantum information theory. It proposes that one cannot accurately predict the measurement outcomes with respect to two arbitrary incompatible observables 1 . Afterwards, Kennard 2 and Robertson 3 generalized into the inequality for arbitrary two operators: $\Delta{\hat{M}_{1}}\cdot\Delta{\hat{M}_{2}}\geq|\langle{[{\hat{M}_{1}},{\hat{M}_{2}}]}\rangle|/2$ in terms of standard deviation, with $\Delta(\bullet)$ denoting the standard deviation and $\langle\circ\rangle$ denoting the expected value. Eversince, there have been various quantifications of uncertainty relations, including Shannon entropy 4 , Rényi entropy 6 , and von Neumann entropy JR . Furthermore, Berta et al. 7 presented quantum-memory-assisted entropic uncertainty relations (QMA-EURs). Liu $et$ $al.$ novelly derived an expression for QMA-EURs as multiple observables liu

herein, the von Neumann conditional entropy $S({A|B})=S({{{\hat{\rho}}_{AB}}})-S\left({{{\hat{\rho}}_{B}}}\right)$ m ; x ; wu-lin with $S\left({\hat{\rho}}\right)=-tr\left({\hat{\rho}{{\log}_{2}}\hat{\rho}}\right)$, and $f=\mathop{\max}\limits_{{i_{N}}}\left\{{\sum\limits_{{i_{2}}\sim{i_{n-1}}}{\mathop{\max}\limits_{{i_{1}}}}\left[{c\left({u_{{i_{1}}}^{1},u_{{i_{2}}}^{2}}\right)}\right]\prod\limits_{x=2}^{n-1}{\left[{c\left({u_{{i_{x}}}^{x},u_{{i_{x+1}}}^{x+1}}\right)}\right]}}\right\}$, where $u_{i_{x}}^{x}$ is the $i$th eigenvector of the operator ${{{\hat{M}}_{x}}}$ and ${c\left({u_{{i_{x}}}^{x},u_{{i_{x+1}}}^{x+1}}\right)}$ denotes the maximal overlap of the operators ${u_{{i_{x}}}^{x}}$ and $u_{{i_{x+1}}}^{x+1}$.

In this study, we focused on the effectiveness of EURs in discerning the energy variation for QBs modelled by a charger-mediated protocol with an external field and reservoir. We explored the relationship between the extractable work and the EUR of the QB. Compared with previous results reference1 ; x18 ; x18++ ; capacity , the merits of the current work can be summarised as follows: (i) Improving the extractable work (i.e., exergy and ergotropy) and conversion efficiency of the battery was revealed. (ii) We found that EUR tightness could serve as an excellent indictor for predicting extractable work and conversion efficiency. (iii) We revealed how the corresponding EURs contribute to the conversion efficiency of QB.

The remainder of this paper is organised as follows. Section II introduces a typical quantum-battery model with a battery-charger-field in the presence of reservoirs. In Section III, in addition to the performance of the QB, the relationship between extractable work and tightness is investigated. In Section IV, the mechanism for improving extractable work via EUR is provided. Finally, our article ends with a brief discussion and conclusions in Section V.

The diagrammatic charging model of a QB consists of a battery and a two-level charger. Simultaneously, the charger is powered by a driving laser field and coupled to a bosonic or fermionic reservoir. As illustrated in Fig. 1, the energy can be transferred to the battery owing to the interaction between the charger and the battery. Consequently, the energy storage of the battery was achieved. In the rotating frame, the Hamiltonian of the charging system ${{\cal H}_{s}}$ can be written as reference1 ; reference2 ; reference3

here, ${{\cal H}_{s}}$ denotes the Hamiltonian of the charger-battery system (the Boltzmann constant ${{k_{B}}=\hbar=1}$ hereafter, for simplicity). Moreover, $\hat{\sigma}_{j}^{A,B}\left({j=x,y,z}\right)$ are the Pauli operators and $\hat{\sigma}_{k}^{A,B}\left({k=+,-}\right)$ are the upper and down operators, ${\Delta_{l}}={\omega_{l}}-{\omega_{d}}\left({l=A,B}\right)$ is the detuning of a two-level system with a transition frequency of ${\omega_{l}}$ to the driving field with frequency ${\omega_{d}}$, assume the charger has the same frequency as the battery, i.e., ${\omega_{A}}={\omega_{B}}={\omega_{0}}$. ${g}$ is the coupling strength between the charger and battery, and $F$ is the amplitude of the driving field.

The Markovian master equation describing the charger-battery system can be obtained experiment1 ; reference1 ; reference2 ; measurement ; reference3

where ${D_{\Lambda}}\left[\hat{\rho}\right]=2\Lambda\hat{\rho}{\Lambda^{\dagger}}-{\Lambda^{\dagger}}\Lambda\hat{\rho}-\hat{\rho}{\Lambda^{\dagger}}\Lambda$ denotes the dissipator, and $J$ is the dissipative rate. Regarding the different types of reservoirs, the average particle number $n(T)$ is different at frequency ${\omega_{k}}$ and temperature $T$. Specifically, $n\left(T\right)={n_{b}}=1/\left[{\exp\left({\omega_{k}/T}\right)-1}\right]$ and $N\left(T\right)=1+n\left(T\right)$ in the bosonic reservoir. For the fermionic one, $n\left(T\right)={n_{f}}=1/\left[{\exp\left({\omega_{k}-\mu}\right)/T+1}\right]$ and $N\left(T\right)=1-n\left(T\right)$, where $\mu$ denotes the chemical potential of the reservoir.

Suppose that the initial state of the charger-battery is prepared in the ground state $\left|g\right\rangle$ (henceforth, the excited state is written as $\left|e\right\rangle$), that is,

Once the charging process begins $\left({t\geq 0}\right)$, the driving field continues to power the charger, and the charger is connected to the reservoir because it interacts with the charger, whereas energy is transferred through the charger to the battery by interaction. The internal energy of the charger ($A$) or battery ($B$) at any time $t$ can be expressed as

where ${{\cal H}_{0}^{i}}={\omega_{0}}{\hat{\sigma}_{+}^{i}}{\hat{\sigma}_{-}^{i}}$ and ${{\hat{\rho}_{i}}\left(t\right)}$ ($i=A$ or $B$) is the reduced density matrix of ${{\hat{\rho}_{AB}}\left(t\right)}$. In addition to the storage capacity of the QB, the extractable work is also an important factor in determining the QB’s performance. First, the extraction of useful work from a battery by thermalization (a nonunitary extraction process) can be quantified by the variations in the Helmholtz free energy of the battery. In other words, the battery is connected to a thermal bath with an inverse temperature $\beta$. Consequently, the work can be extracted as exergy x18++ ; exergy1 ; exergy2

where $\mathcal{F}\left({\hat{\rho}_{B}}\right)={\rm tr}\left({{\mathcal{H}_{0}}\hat{\rho}_{B}}\right)-S\left({\hat{\rho}_{B}}\right)/\beta$ is Helmholtz free energy, and ${{\hat{\rho}}_{\beta}}={e^{-\beta{\mathcal{H}_{0}^{B}}}}/{\rm tr}\left({{e^{-\beta{\mathcal{H}_{0}^{B}}}}}\right)$ is the thermal equilibrium state of the battery. To characterize the maximal amount of energy that can be extracted from a QB in the charging process under the cyclic unitary operations, the ergotropy reference4 ; reference5 is introduced as ${W_{e}\left(t\right)}={\rm tr}\left[{{{\cal H}_{0}^{B}}{\hat{\rho}_{B}}\left(t\right)}\right]-{\rm tr}\left[{{{\cal H}_{0}^{B}}{\hat{\rho}_{p}}}\right]$ where ${{\hat{\rho}_{p}}}$ is called the passive state of ${{\hat{\rho}_{B}}}(t)$ with the zero extractable energy by cyclic unitary operations. In spectral decomposition, the Hamiltonian $H$ and density matrix $\hat{\rho}$ are written as $H=\sum\nolimits_{j}{{e_{j}}}\left|{{e_{j}}}\right\rangle\left\langle{{e_{j}}}\right|$ and $\hat{\rho}=\sum\nolimits_{j}{{r_{j}}}\left|{{r_{j}}}\right\rangle\left\langle{{r_{j}}}\right|$ respectively, where ${{r_{j}}}$ and ${{e_{j}}}$ are the eigenvalues of $\hat{\rho}$ and $H$ with corresponding eigenstates $\left|{{r_{j}}}\right\rangle$ and $\left|{{e_{j}}}\right\rangle$. After ordering eigenvalues (${r_{0}}\geq{r_{1}}\geq{r_{2}}\geq\cdots$ and ${e_{0}}\leq{e_{1}}\leq{e_{2}}\cdots$), the passive state can be derived as ${\hat{\rho}_{p}}=\sum\nolimits_{j}{{r_{j}}}\left|{{e_{j}}}\right\rangle\left\langle{{e_{j}}}\right|$. The ergotropy can be reexpressed as

Furthermore, combining the density matrix of the charger-battery system ${\hat{\rho}_{AB}}\left(t\right)$, ${E_{B}}\left(t\right)$ and $W\left(t\right)$ can be described as follows:

and

where ${\rho_{ij}}\left(t\right)\left({i,j=1,2,3,4}\right)$ is the $i$-th row and $j$-th column matrix entry of ${\hat{\rho}_{AB}}\left(t\right)$ with $\kappa=2\left[{{\rho_{11}}\left(t\right)+{\rho_{33}}\left(t\right)}\right]-1$. Note that when charging for a sufficiently long period (i.e., $t\to\infty$), the charger-battery system becomes steady, and the system’s energy does not vary over time $t$. Hence, we have ${E_{i}}\left(t\right)\to{E_{i}}\left(\infty\right)$, ${W_{f}\left(t\right)}\to{W_{f}\left(\infty\right)}$ and ${W_{e}\left(t\right)}\to{W_{e}\left(\infty\right)}$ (${\dot{\hat{\rho}}_{AB}}=0$) in the limit. In fact, when $S\left({{{\hat{\rho}}_{\beta}}}\right)-S\left({{{\hat{\rho}}_{p}}}\right)>\beta\left[{{\rm tr}\left({{\mathcal{H}_{0}^{B}}{{\hat{\rho}}_{\beta}}}\right)-{\rm tr}\left({{\mathcal{H}_{0}^{B}}{{\hat{\rho}}_{p}}}\right)}\right]$, the battery always stores more exergy than ergotropy. The left of the above inequality is the entropy production of the battery during the thermalization process, and the right is the change in battery energy caused by thermalization exergy2 ; exergy3 . In addition, the work that can be extracted ${W_{f}\left(t\right)}\in\left[{0,\Delta{E_{B}}\left(t\right)}\right]$ is hold, where $\Delta{E_{B}}\left(t\right)={E_{B}}\left(t\right)-{E_{B}}\left(0\right)$ represents the stored energy of the battery.

As a matter of fact, the conversion rate of energy is termed as another nontrivial factor to evaluate the performance of QB, in addition to the amount of energy. Therefore, the conversion efficiency of the battery is given by

It quantifies the proportion of useful work in QB’s energy storage and deals with the energy cost review ; open1 ; cost3 of charging QBs (higher efficiency means lower energy costs). Therefore, a high-performance QB essentially needs to its higher efficiency to pursue more extractable work.

In general, quantum batteries with excellent performances have two significant advantages: stability and high power. In this consideration, we investigated the energy changes for the considered charging scenarios while the state of the charger-battery system was steady. Importantly, the role of the uncertainty’s tightness of QMA-EUR is discussed in detail while determining the acquirement and extraction of QB’s energy.

To probe the dynamics of entropic uncertainty, we use Pauli operators as the incompatible observables to be measured, which leads to post-measurement states $\left({i=x,y,z}\right)$

where $\mathbb{I}$ denotes an identity matrix. It can be observed that different measurement choices for $A$ lead to different post-measurement states. Subsequently, we explored the entropic uncertainty relationship of the system under different measurement choices. When two measurement operators $\left({{\hat{\sigma}_{x}},{\hat{\sigma}_{z}}}\right)$ are selected, the entropic uncertainty (i.e., left-hand side of Eq. (1)) is expressed as

and the corresponding lower bound (right-hand side of Eq.   (1)) can be expressed as

Consequently, the uncertainty’s tightness is

Incidentally, a smaller value of ${\Delta^{xz}}\left(t\right)$, i.e., stronger tightness-implies a tighter lower bound, and vice versa. For the steady state, $U_{l}^{xz}\left(t\right)\to U_{l}^{xz}\left(\infty\right)$ and $U_{r}^{xz}\left(t\right)\to U_{r}^{xz}\left(\infty\right)$; thus, we have ${\Delta^{xz}}\left(t\right)\to{\Delta^{xz}}\left(\infty\right)$ in terms of Eq. (14).

It was found that the tightness of the entropic uncertainty relation was closely associated with the battery’s conversion efficiency. However, the tightness is dependent on the uncertainty and its lower bound. Thus, the mechanism by which the EUR affects the efficiency of QB deserves further investigation. Motivated by this, we investigated the dynamics of the lower bound and the uncertainty.

Figs. 6(a)-(d) display the dynamics of uncertainty and lower bound. It can be seen that in the case of different parameters regimes, smaller uncertainty and lower bound often represent more efficient energy conversion. In particular, higher efficiency is accompanied by the fact that the distance of the uncertainty from the lower bound (i.e., ${\Delta^{xz}}\left(\infty\right)$) is always getting smaller (for Fig. 6(a) when $J/g>F/g$) or larger (for Figs. 6(b)-(d)). Therefore, the tightness can be an excellent indicator of efficiency.

Owing to the contribution of tightness to efficiency, it concludes that the entropic uncertainty relation is strongly related to the conversion efficiency of extractable work in QBs. The current result provides a new reliable indicator that predicts energy and efficiency of QBs.

In summary, we investigated the extractable work and the conversion efficiency of a QB model with a battery-charger-field in the presence of bosonic or fermionic reservoir using entropic uncertainty relations. We discussed in detail the effect of resonance and detuning between the charger-battery and driving field on the charging process of the QBs and examined how the different reservoirs coupling to the charger affect the extractable work and the efficiency. Several interesting results can be outlined: (1) resonance drive and higher-temperature reservoirs (both bosonic and fermionic) often induce battery to store more exergy. By contrast, lower-temperature bosonic reservoir favors ergotropy, and more ergotropy leads to more efficiency in most cases. Particularly, stronger dissipation or weaker drive will also improve efficiency. (2) On the one hand, in non-zero temperature reservoirs, strong tightness may increase exergy while suppressing ergotropy. In addition, energy variation induced by dissipation, drive strength and detuning complicates the relationship between tightness and extractable work. On the other hand, the tightness maintains a unified relationship with conversion efficiency compared with extractable work. The stronger tightness always indicates higher efficiency for dissipation greater than drive, when resonant drive and zero-temperature reservoir coexist. Meanwhile the weak tightness is often accompanied by higher efficiency when they do not coexist. (3) In various parametric regimes, smaller uncertainty and its bound often indicate higher efficiency. Therefore, the current investigations reveal that EURs can be applied to predict the QB’s extractable work and its conversion efficiency, which is of fundamental importance in the pursuit of high-performance practical QBs.