Collectively enhanced high-power and high-capacity charging of quantum batteries via quantum heat engines

We consider a three-level heat engine similar to the three-level maser heat engine Scovil and Schulz-DuBois (1959), which consists of two heat baths and a three-level system as the working substance (Fig. 1). The two heat baths have different respective inverse temperatures $\beta_{h}$ and $\beta_{c}$. In this paper, we set $\hbar=k_{B}=1$. $N$ engines are used to charge $N$ quantum batteries. The Hamiltonian of the working substance of each $j$-th engine is given by $H_{E_{j}}=\omega_{h}\outerproduct{2}{2}_{E_{j}}+\omega_{0}\outerproduct{1}{1}_{E_{j}}+0\times\outerproduct{0}{0}_{E_{j}}=\omega_{h}\outerproduct{2}{2}_{E_{j}}+\omega_{0}\outerproduct{1}{1}_{E_{j}}$, where $\{\ket{0}_{E_{j}},\ket{1}_{E_{j}},\ket{2}_{E_{j}}\}$ is the complete orthonormal basis of the Hilbert space of the three-level working substance. We do not explicitly treat the heat baths as quantum systems. Instead, we consider the following Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) master equation Lindblad (1976); Gorini et al. (1976) of the working substance as the resulting dynamics from the interaction with the two heat baths:

where $\omega_{c}:=\omega_{h}-\omega_{0}$ is the energy gap between $\ket{1}_{E_{j}}$ and $\ket{2}_{E_{j}}$, $\Gamma_{b}\;(b=h,c)$ is the relaxation rate with respect to each bath, ${a}_{h,j}:=\outerproduct{0}{2}_{E_{j}}$, ${a}_{c,j}:=\outerproduct{1}{2}_{E_{j}}$, and

In this model, the transition between the levels $\ket{0}_{E_{j}}$ and $\ket{2}_{E_{j}}$ is coupled with the hotter heat bath, and similarly, the transition between $\ket{1}_{E_{j}}$ and $\ket{2}_{E_{j}}$ is coupled with the colder bath. The steady state ${\rho}_{E_{j}}^{\mathrm{SS}}$of the engine under this dynamics is given as

To extract the work, we impose the condition $\beta_{h}/\beta_{c}<\omega_{c}/\omega_{h}<1$. Indeed, this parameter regime yields the population inversion $\bra{0}{\rho}_{E_{j}}^{\mathrm{SS}}\ket{0}_{E_{j}}<\bra{1}{\rho}_{E_{j}}^{\mathrm{SS}}\ket{1}_{E_{j}}$ between the ground state $\ket{0}_{E_{j}}$ and the first excited state $\ket{1}_{E_{j}}$. Then, we can extract the work to the battery through the coupling with the transition mode $\ket{0}_{E_{j}}\leftrightarrow\ket{1}_{E_{j}}$ as we explicitly do so later [Eq. (5)]. This model may be realized through the respective frequency filters passing $\omega_{h}$, $\omega_{c}$, and $\omega_{0}$ Scovil and Schulz-DuBois (1959).

As for the battery, we consider a two-level system with the resonance frequency $\omega_{0}$ as a unit quantum battery to be charged by the heat engine. We consider $N$ noninteracting quantum batteries, whose Hamiltonian is $H_{B}=\sum_{j=1}^{N}H_{B_{j}}=\sum_{j=1}^{N}\omega_{0}\outerproduct{1}{1}_{j}$, where $H_{B_{j}}=\omega_{0}\outerproduct{1}{1}_{j}$ is the Hamiltonian of $j$-th battery. The batteries are initialized in the empty state $\ket{\psi_{0}}:=\otimes_{j=1}^{N}\ket{0}_{j}$.

To investigate the collective effects, we compare the collective-charging protocol with the individual-charging protocol. Especially, the interaction between the batteries is neglected to focus on the effects purely come from the collectiveness of the charging operation itself. In the collective-charging protocol (Fig. 1 left), the interaction

is introduced between every $i$-th engine and $j$-th battery so that all $N$ batteries are collectively charged by each heat engine, where ${{S}}_{B_{k}}^{+}=\outerproduct{1}{0}_{k}$ and ${{S}}_{B_{k}}^{-}=\outerproduct{0}{1}_{k}$. Here, the coupling constant is normalized by $1/\sqrt{N}$ factor in accordance with the condition to have the well-defined thermodynamic limit for $N\rightarrow\infty$ Julià-Farré et al. (2020). The total Hamiltonian in this case is written as

where ${{S}}_{B}^{\pm}=\sum_{j=1}^{N}{{S}}_{B_{j}}^{\pm}$ and $H_{E}=\sum_{j=1}^{N}H_{E_{j}}$. In experiments, we can also take the convention without the normalization factor $1/\sqrt{N}$ up to some finite $N$ which is possible to put together while keeping a priori fixed coupling strength. Even if we use only single engine to charge $N$ batteries in the convention without the normalization factor $1/\sqrt{N}$, the main results below do not change. Note that this charging protocol only involves local two-body interactions, which is important for the experimental feasibility. In addition, the energy resource in our model is purely heat from the reservoir, requiring no prefilled charger quantum state or time-dependent control of the Hamiltonian like in conventional models. This protocol is cheap in this sense.

We compare the scaling of the performance of this collective-charging protocol with the individual-charging protocol (Fig. 1 right). In the individual-charging protocol, the interaction $K_{j,j}^{(1)}$ is only imposed between the engine and battery with the same index $j$. Hence, each single battery is charged by a single engine. Here, the coupling constant is not normalized by $1/{\sqrt{N}}$ in this case reflecting the fact that the pairs of an engine and a battery are independent from each other. The total Hamiltonian in this case reads

The total system of the individual-charging protocol is just $N$ independent and identical copies of a single pair of an engine and a battery, which is included in the collective-charging protocol with $N=1$. Thus, all we have to analyze is the performance of the collective-charging protocol with generic $N$. Hence, we focus on the collective-charging protocol.

If the time scale of the thermalization of the engine with each heat bath is much shorter than the time scale of the interaction with the battery, the engine keeps the population inversion and effectively works as a reservoir with negative temperature Levy et al. (2016). We can estimate the time scale of the transitions induced by the coupling between the engine and the battery as $\left(\frac{g}{\sqrt{N}}\sqrt{\langle{S}_{B}^{-}{S}_{B}^{+}\rangle+1}\right)^{-1}$, and the time scale associated with the coupling with each heat bath as $\Gamma_{h(c)}^{-1}[1+\exp(-\beta_{h(c)}\omega_{h(c)})]^{-1}$. Since $\langle{S}_{B}^{-}{S}_{B}^{+}\rangle$ is at most of order $O(N^{2})$, we have a criterion

for the validity of the ad-hoc master equation (ME) of the state $\rho_{B}(t)$ of the battery

where $\beta_{e}^{-1}=\omega_{0}(\beta_{h}\omega_{h}-\beta_{c}\omega_{c})^{-1}<0$ is the effective temperature, and $\Gamma_{e}=(2g)^{2}(\Gamma_{h}e^{-\beta_{h}\omega_{h}}+\Gamma_{c}e^{-\beta_{c}\omega_{c}})^{-1}(1+e^{-\beta_{h}\omega_{h}}+e^{-\beta_{e}\omega_{0}})^{-1}$ is the effective damping rate. Equation (8) can be derived by heuristically applying the Born-Markov-type approximation based on the time-scale separation (7) in a similar way to Appendix B of Levy et al. (2016). Especially, the same dynamics $(\Gamma_{e}/N)\left[\mathcal{D}[{{S}}_{B}^{-}]({\rho}_{B})+e^{-\beta_{e}\omega_{0}}\mathcal{D}[{{S}}_{B}^{+}]({\rho}_{B})\right]$ is induced by each of $N$ engines, and hence ME (8) holds in total.

We note that the form of ME (8) coincides with the dissipation due to the environment, replacing $\Gamma_{e}$ and $\beta_{e}$ with the damping rate $\Gamma$ and the inverse temperature $\beta$ of the environment, respectively. Hence, in the presence of the environmental dissipation, just the parameters $\Gamma_{e}$ and $\beta_{e}$ are modified while keeping the form of ME (8). The effective damping rate and the inverse temperature are modified to be $\tilde{\Gamma}_{e}=\Gamma_{e}+\Gamma$ and $\tilde{\beta}_{e}=\omega_{0}^{-1}[\log(\Gamma_{e}+\Gamma)-\log(\Gamma_{e}e^{-\beta_{e}\omega_{0}}+\Gamma e^{-\beta\omega})]$, respectively. Thus, as long as $(\Gamma_{e}e^{-\beta_{e}\omega_{0}}+\Gamma e^{-\beta\omega})/(\Gamma_{e}+\Gamma)>1$, we still have negative effective temperature $\tilde{\beta_{e}}<0$, which means that the charging is stable against the environmental dissipation. In the following, we investigate the charging performance of the battery initially in the totally empty state $\ket{\psi_{0}}$ based on ME (8), assuming $\tilde{\beta}_{e}<0$, where $\tilde{\Gamma}_{e}$ and $\tilde{\beta}_{e}$ will be denoted respectively as $\Gamma_{e}$ and $\beta_{e}$ for simplicity of the notation.

At first, let us look at the numerical results. We use the Python toolbox “QuTiP” Johansson et al. (2013) to solve ME (8). Figures 2(a) and 2(b) display the time evolution of the expectation value $\langle H_{B}\rangle:=\tr H_{B}\rho_{B}$ and the fluctuation $\sigma_{N}:=\sqrt{\langle H_{B}^{2}\rangle-\langle H_{B}\rangle^{2}}$ of the energy respectively, with $N=10$ for both of the collective and the individual-charging protocols. The parameters are set to $\beta_{e}\omega_{0}=-0.8$ and $\Gamma_{e}/\omega_{0}=0.10$. As can be seen from these figures, the charging ends in convergence to the steady state. In closed QBs, the charged energy fluctuates in general depending on when we stop the charging Rosa et al. (2019). Our protocol is intrinsically free from such instability owing to the convergence to the steady state, as in other open QBs Quach and Munro (2020); Çakmak (2020).

Figure 2(c) displays the energy density $e_{N}^{\mathrm{SS}}:=\tr H_{B}\rho_{B,N}^{\mathrm{SS}}/(\omega_{0}N)$ of the steady state $\rho_{B,N}^{\mathrm{SS}}$ of $N$ batteries, which stands for the capacity of the battery. Since each single battery is charged independently in the individual charging protocol, it is obvious that the energy density in this case does not change with the total number $N$ of the batteries. On the other hand, the energy density in the collective-charging protocol is approaching to the unity, which is the maximum possible value corresponding to the perfect excitation. Hence, the capacity of the collective $N$ batteries not only scales super-extensively but also approaches to the perfect one in terms of the energy density as $N$ gets large. Figure 2 (d) displays the energy fluctuation $\sigma_{N}^{\mathrm{SS}}$ of the steady state of $N$ batteries, which measures the quality of the charged energy. It is remarkable that $\sigma_{N}^{\mathrm{SS}}$ immediately converges to a constant around $N=10$ in the collective protocol, in contrast to the individual protocol, where the fluctuation scales as $\propto\sqrt{N}$. In fact, these collective enhancements result from emergent quantum statistics induced by the symmetry of ME (8), and hence are universal independently of the parameters. Below we rigorously show the detail of the collective enhancements.

In the individual protocol, we simply have $N$ independent copies of a battery whose time evolution follows ME (8) with $N=1$. Then, the state of each single battery will converge to the steady state $\rho_{B,1}^{\mathrm{SS}}=(\outerproduct{0}{0}_{1}+e^{-\beta_{e}\omega_{0}}\outerproduct{1}{1}_{1})/(1+e^{-\beta_{e}\omega_{0}})$. Hence, the total state becomes $(\rho_{B,1}^{\mathrm{SS}})^{\otimes N}$, and the expectation value and the variance of the energy are just given as $N$ times of $e_{1}^{\mathrm{SS}}$ and $(\sigma_{1}^{\mathrm{SS}})^{2}$ respectively. To investigate the collective protocol, we define collective spin operators ${S}^{Z}:=\sum_{i=1}^{N}({{S}}^{+}_{B_{i}}{{S}}^{-}_{B_{i}}-1/2)$ and ${S}^{2}:=\frac{1}{2}({{S}}^{+}_{B}{{S}}^{-}_{B}+{{S}}^{-}_{B}{{S}}^{+}_{B})+({S}^{Z})^{2}$, though they do not have to represent physical spin degrees of freedom. We denote the simultaneous eigenstate of $S^{2}$ and $S^{Z}$ with respective eigenvalues $J(J+1)$ and $M$ by $\ket{J,M}$. From the commutation relations $[S^{2},{S}_{B}^{\pm}]=0$ and $[S^{2},H_{B}]=0$, the quantum number $J$ is conserved by the dynamics (8). Therefore, because the initial state $\ket{\psi_{0}}$ is $\ket{J=N/2,M=-N/2}$, the time evolution of the battery state $\rho_{B}(t)$ is contained in the $J=N/2$ subspace $\mathcal{H}_{N/2}$ spanned by $\{\ket{N/2,m-N/2}|m=0,1,\cdots,N\}$, a completely symmetric subspace with respect to the permutation of the batteries. For the analysis of the dynamics within the subspace $\mathcal{H}_{N/2}$, it is convenient to apply the following linear mapping to the $(N+1)$-dimensional Hilbert space spanned by $\{\ket{m}|m=0,1,\cdots,N\}$ using the ladder operator $c:=\sum_{m=1}^{N}\sqrt{m}\outerproduct{m-1}{m}$:

which is a finite-dimensional analogy of the Holstein-Primakoff transformation Holstein and Primakoff (1940). In fact, it satisfies $\pi({S}_{B}^{\pm}\ket{N/2,m-N/2})=\pi({S}_{B}^{\pm})\pi(\ket{N/2,m-N/2})$, and hence we can identify the relevant operators and state vectors through this mapping. Thus, ME (8) with the initial state $\ket{\psi_{0}}$ is mapped to

Following this dynamics, the state converges to the steady state $\pi({\rho}_{B,N}^{\mathrm{SS}})=e^{-\beta_{e}\omega_{0}{c}^{\dagger}{c}}/\tr e^{-\beta_{e}\omega_{0}{c}^{\dagger}{c}}$, which corresponds to

where $Z_{N}=\sum_{m=0}^{N}e^{-\beta_{e}\omega_{0}m}$. Hence, the steady state turns out to be a mixture of the completely symmetric states $\ket{N/2,m-N/2}$, which is effectively bosonic. Such an emergent bosonic quantum statistics comes from the $J$ conserving symmetry of the dynamics (8) which cannot distinguish each battery. The significant enhancement in the capacity and the quality of the batteries may be understood as a kind of the Bose-Einstein condensation in this emergent bosonic state where the roles of the ground state and the excited state are now opposite because of the negative temperature.

Indeed, we have analytical expressions of the capacity $Ne_{N}^{\mathrm{SS}}$ and the fluctuation $\sigma_{N}^{\mathrm{SS}}$ of the collective protocol as follows:

Since $\beta_{e}<0$ and $\omega_{0}>0$, we have $\lim_{N\rightarrow\infty}e_{N}^{\mathrm{SS}}=1$ and $\lim_{N\rightarrow\infty}\sigma_{N}^{\mathrm{SS}}=(\omega_{0}/2)\csch\left(-\beta_{e}\omega_{0}/2\right)$. Therefore, the collective capacity asymptotically approaches the perfect excitation, and the numerical results are now rigorously proved. We also note that the speed of the convergence is estimated as $O(N^{-1})$ for $e_{N}^{\mathrm{SS}}$ and $O(Ne^{\frac{\beta_{e}\omega_{0}}{2}N})$ for $\sigma_{N}^{\mathrm{SS}}$, which agrees with Fig. 2(c) and 2(d).

In general, the average whole charged energy $Ne_{N}^{\mathrm{SS}}$ is not necessarily available depending on the possible operation. One reasonable measure of the available energy is the ergotorpy $\mathcal{E}_{N}:=Ne_{N}^{\mathrm{SS}}-\min_{U:\mathrm{unitary}}\tr H_{B}U\rho_{B,N}^{\mathrm{SS}}U^{\dagger}$ Allahverdyan et al. (2004), which characterizes the maximum extractable energy under a cyclic process where the energy difference is described by a unitary operator. The optimum unitary operator to extract the energy from $\rho_{B,N}^{\mathrm{SS}}$ is the one which maps $\ket{N/2,N/2}$ to $\ket{\psi_{0}}$ and $\ket{N/2,m-N/2}$ $(m\neq N)$ to $\ket{0}_{1}\otimes\cdots\otimes\ket{0}_{m-1}\otimes\ket{1}_{m}\otimes\ket{0}_{m+1}\otimes\cdots\otimes\ket{0}_{N}$, and hence it turns out that the minimum locked energy asymptotically converges to a constant as follows:

Thus, in the collective protocol, the asymptotic freedom “$\lim_{N\rightarrow\infty}\mathcal{E}_{N}/(Ne_{N}^{\mathrm{SS}})=1$” Andolina et al. (2019) holds with $O(N^{-1})$ of the speed of convergence. The asymptotic freedom of the collective protocol implies that almost whole average stored energy $Ne_{N}^{\mathrm{SS}}$ can be extracted if $N$ is sufficiently large. This is in contrast to the individual protocol, where the asymptotic freedom is not satisfied because the ergotropy is just extensive as $N\mathcal{E}_{1}=-N\omega_{0}\tanh(\beta_{e}\omega_{0}/2)$ and hence its ratio to the total energy remains to be a constant value $\mathcal{E}_{1}/e_{1}^{\mathrm{SS}}=1-e^{\beta_{e}\omega_{0}}<1$.

In summary, the capacity per single battery in the collective-charging protocol asymptotically reaches the perfect excitation. That is in contrast to the individual protocol, where the capacity is just extensive. Moreover, in the collective protocol, the whole average charged energy is actually extractable in the asymptotic sense that the ratio of the ergotropy to the average energy is convergent to unity as $N$ gets large. That is again in contrast to the individual protocol, where nonzero constant portion of the average energy remains locked regardless of $N$. These significant enhancements can be understood as a kind of Bose-Einstein condensation to the excited state due to the effective negative temperature and the emergent bosonic quantum statistics of the steady state. Notably, we have rigorously proved these results based on the analytic solvability of the steady state of our model.