Beyond the Carnot Limit in the Internal Cycles of a Quantum Heat Engine under Finite Heat Reservoirs

As sketched in Fig. 1, our quantum Carnot cycle consists of two process, i.e., internal cycle process and external cycle process. The internal cycle process consisting of four consecutive steps, i.e., expansion-cooling-compression-heating. The system (i.e., the WS), governed by a time-dependent Hamiltonian, is coupled to two finite heat reservoirs at the initial temperatures $T_{h}^{(1)}$ and $T_{c}^{(1)}$ with the superscript (n) implying the $n$th round and $T_{h}^{(n)}>T_{c}^{(n)}$ (For simplicity of description, we omit the superscripts below for the first round, i.e., $T_{h,c}^{(1)}=T_{h,c}$). The internal cycle gets started from the starting point $A$ with the WS in thermal equilibrium with the high-temperature reservoir, where the WS is described by the Hamiltonian $H_{0}$. Each round of the cycle takes following four steps: (i) The system is kept in isolation from the reservoirs, undergoing an adiabatically isentropic expansion process, in which the system’s Hamiltonian $H(t)$ varying from $H_{0}$ to $H_{1}$ commutes with the initial state at any time $t$. Thus this step is accomplished by a Hamiltonian quench during which the system’ state is unaffected. The end of the step (i) is the system in thermal equilibrium with the low-temperature reservoir, implying $H_{1}=H_{0}T_{c}/T_{h}$. (ii) Contacted with the low-temperature reservoir, the Hamiltonian of the system is isothermally modulated from $H_{1}$ to $H_{2}$ with a duration $\tau_{c}$. In this case, the temperature of the low-temperature reservoir is also changed, which can be labeled as $T_{c}^{(2)}$, i.e., the beginning temperature of the low-temperature reservoir when interacting with the WS in the second round. (iii) Similar to step (i), a second quench is carried out, followed by a thermal equilibrium with the high-temperature reservoir. Thus, the Hamiltonian of the WS varies from $H_{2}$ to $H_{3}=H_{2}T_{h}/T_{c}^{(2)}$ at the end of this step. (iv) Coupled to the high-temperature reservoir again, the system is isothermally back to the Hamiltonian $H_{0}$ with a duration $\tau_{h}$, along with the temperature change of the high-temperature reservoir to be $T_{h}^{(2)}$ at the end of this step. Then the second round of the internal cycle begins, as labeled in Fig. 1 as the step (v) and so on. The internal cycle generally needs to go on for several rounds until some termination conditions are satisfied, as clarified later.

After the internal cycle is accomplished, the periodical reset process, to avoid the engine’s idling, is executed by the external cycle of the engine. In this process, we couple these two finite heat reservoires of internal cycles to two baths with infinite capacities, respectively, where the internal low-temperature finite heat reservoir delivers heat into the infinite cold bath with temperature $T_{c}$ and the internal high-temperature finite heat reservoir absorbs heat from the infinite hot bath with temperature $T_{h}$. After this process, the internal heat reservoirs will be in equilibrium with the corresponding external heat reservoires.

For our purpose, we first scrutinize the heat exchange in the internal cycle, i.e., in the isothermal processes. For the finite heat reservoirs, we define the heat capacities of the reservoirs as $\mathcal{C}_{k}=dU_{k}/dT_{k}$ with $k=h,c$ (also in the following) and the internal energy $U_{k}$ being finite. Considering the first round, the dynamics of the thermodynamic quantities in the isothermal process (in units of $k_{B}=\hbar=1$ throughout the paper) can be written as

and the work $W_{k}=\Delta U_{K}-Q_{k}$ with $Q_{k}$, $W_{k}^{ir}$, $S_{k}$ and $U_{k}$ denoting the heat exchange, irreversible work dissipation, entropy and internal energy of the WS, respectively. Then, the heat exchange can be obtained as (see Supplementary Information),

where $\Delta S_{k}=S(\pi_{k}(\tau_{k}))-S(\pi_{k}(0))$ with the von Neumann entropy $S(\cdot)$ and the Gibbs state $\pi_{k}(t)=e^{-\beta_{k}(t)H_{k}(t)}/\text{Tr}[e^{-\beta_{k}(t)H_{k}(t)}]$ at temperature $\beta_{k}(t)=1/T_{k}(t)$. In our case, $\Delta S_{c}+\Delta S_{h}\neq 0$ in the internal cycle. The second term comes from the temperature variation of the heat reservoir due to the finite capacity. The irreversible work dissipation in the last term is given by $W_{k}^{ir}=T_{k}V_{k}/\tau_{k}$ with $V_{k}$ relevant to the thermodynamic length at finite duration PRL-124-110606 ; PRL-51-1127 ; ZPB-59-449 ; JCP-83-334 ; PRL-99-100602 ; PRL-908-190602 ; JCP-140-244119 ; Q-3-197 . Then, the engine’s efficiency is given by,

where the last term originates from the misalignment of the end of the step (iv) from the starting point of the round. This term is always positive due to the temperature decrease of the high-temperature reservoir, ensuring breakthrough of the Carnot limit.

For a large heat reservoir, i.e., $\mathcal{C}_{k}\gg|\Delta S_{k}|$ and keeping the leading terms of $\mathcal{C}_{k}$, Eq. (1) is reduced to

and the temperature $T_{k}^{(2)}=T_{k}-\Delta S_{k}\tilde{T}_{k}/\mathcal{C}_{k}$ where the finite duration creates an effective temperature $\tilde{T}_{k}=(1-\iota_{k}/\tau_{k})T_{k}$ with the time parameter of the internal cycle $\iota_{k}:=V_{k}/S_{k}$. Furthermore, we can obtain the heat capacity of the WS for Hamiltonian $H_{0}$ as $\mathcal{C}_{S}=\text{Tr}[(\pi_{c}(0)-\pi_{h}(\tau_{h}))H_{0}]/(T_{h}-T_{h}^{(2)})$. Therefore, Eq. (2) is reduced to $\eta=\eta_{C}+\zeta$ with the Carnot limit $\eta_{C}=1-T_{c}/T_{h}$, and the increment $\zeta$ induced by the finite heat reservoirs, i.e.,

where $\delta S=\Delta S_{c}+\Delta S_{h}$ with $\delta S\ll\Delta S_{h}$, and the factor caused by the finite duration is defined as $g_{\tau}=\tau_{h}(\tau_{c}-\iota_{c})/\tau_{c}(\tau_{h}-\iota_{h})$.

Equation (4), showing the potential to surpass the Carnot limit (see Fig. 2(a,b), is one of the main results in the present work. The first term in $\zeta$ represents the efficiency increase due to the finite heat capacities of the high-temperature reservoir and the WS. In contrast, the second term, which is induced by the capacities of the heat reservoirs and the duration of the isothermal process, generally indicates the efficiency decrease. In the large reservoir limit, i.e., $\mathcal{C}_{k}\rightarrow\infty$, Eq. (4) returns to the Carnot efficiency with $\zeta=(1-g_{\tau})T_{c}/T_{h}$. Besides, since the conditions of $\iota_{c}\leq 0$ and $\iota_{h}\geq 0$ make $g_{\tau}\geq 1$, the maximal efficiency appears always in the slowly dynamical limit, i.e., $\tau_{c,h}\rightarrow\infty$. However, the engine working at the maximal efficiency will lose its power output due to an infinite duration of the isothermal process. As a result, the key problem here is the maximum power $P=-W/(\tau_{c}+\tau_{h})$ at the finite time, which can be solved by optimizing the durations of the two isothermal processes satisfying the condition $\partial P/\partial\tau_{k}=0$. This corresponds to the optimal ratio $\tau_{h}/\tau_{c}=\sqrt{T_{h}\epsilon/\mathcal{C}}$ with the amendment factor $\epsilon$ induced by the finite heat reservoirs (see Supplementary Information).

For the two reservoirs with the same heat capacity ($\mathcal{C}\equiv\mathcal{C}_{h}=\mathcal{C}_{c}$), Eq. (4) in the slowly dynamical limit is reduced to $\zeta=(\mathcal{C}_{S}T_{h}\Delta S_{h}-T_{c}\Delta S_{h}^{2}+T_{c}\mathcal{C}\delta S)/\mathcal{C}T_{h}\Delta S_{h}$. Keeping the first-order term of $\mathcal{C}$, we acquire the maximum power output as $P_{m}=P_{C}(1+\varsigma)$ in the high-temperature limit, where the maximum power output of the Carnot cycle is $P_{C}=\Delta S_{h}(\sqrt{T_{h}}-\sqrt{T_{c}})^{2}/4\iota$ (assuming $\iota:=\iota_{h}=-\iota_{c}$) EPL-81-20003 ; PRL-124-110606 ; PRL-105-150603 . The improvement induced by the effect of the finite heat reservoir is given by (see Methods for details)

Eq. (5) is another main result of the present work, which indicates the maximum power output surpassing the power limit of the Carnot cycle if the heat capacities, the entropy change and the temperature ratio $T_{c}/T_{h}$ make $\varsigma\geq 0$ (see Fig. 2(c,d)). At the maximum power output, the corresponding efficiency is reduced to $\eta_{m}=\eta_{CA}+\zeta$ with the Curzon-Ahlborn efficiency $\eta_{CA}=1-\sqrt{T_{c}/T_{h}}$ AJP-43-22 and the nonzero term $\zeta$ caused by the finite heat reservoir

which is always positive and inversely proportional to the heat capacity of the reservoir (see Fig. 2(e)), implying $\eta_{m}>\eta_{CA}$.

Since the temperature of the finite heat reservoir varies with the cycling, we have the following iterative relation,

Despite the shrink of the entropy change as the increase of the internal cycling, Eq. (7) shows an exponential variation tendency of the temperature (see Supplementary Information), which leads to exponential decays of the efficiency and the power output as $n$ increases. As shown in Fig. 2(f,g), both the maximum efficiency and the power output can surpass the corresponding Carnot limit in other rounds except the first. The smaller temperature difference between the two reservoirs favors the surpassing. However, the smaller temperature difference between the two reservoirs would increase the duration of single rounds in the isothermal process, which lowers the power output. Therefore, the temperature difference between the two reservoirs should be set appropriately. Otherwise, the decreasing efficiency and power output of the engine would finally disable the engine with the internal cycle proceeding.

In this process, the built-in low-temperature finite heat reservoir delivering heat into an infinite cold bath is cooled down to $T_{c}$ with the heat exchange of $Q^{r}_{c}=\mathcal{C}_{c}(T_{c}^{(m+1)}-T_{c})$. Meanwhile, an infinite hot bath injects heat into the built-in high-temperature finite heat reservoir and raises the latter’s temperature to $T_{h}$. This also implies the system’s temperature reaching $T_{h}$ due to the fact that the system is coupled to the high-temperature finite heat reservoir at the end of the $m$th round of the internal cycle. Thus, the heat absorption is assessed as $Q^{r}_{h}=(\mathcal{C}_{h}+\gamma\mathcal{C}_{S})(T_{h}-T_{h}^{(m+1)})$ with $\gamma>0$. In the slowly dynamical limit, we obtain the maximum efficiency of the external cycle as

with $\alpha\geq 1$ (see Supplementary Information for details). As plotted in Fig. 3(a), the efficiency of the external cycle can approach the Carnot limit in the large reservoir limit, but cannot surpass the Carnot limit PRE-96-012128 ; PRE-100-012122 although the analytical result based on some approximations shows the possibility of surpassing for the reservoirs with small heat capacity.

For a finite duration of the isothermal process, the average power output depends on the summed duration of the internal cycle and the reset time. Therefore, obtaining the maximal average power output is a multi-parameter optimization problem in a multi-cycle process, which can be stated as

In principle, maximizing the power output in each round of the internal cycle could produce a large average power output, even though not the maximal one, of the external cycle. A higher power output always favors a shorter reset time due to no power output during the reset process. Therefore, to acquire a large average power output, we may terminate the internal cycle at the end of the $m$th round for a given reset time $\tau_{e}$. As demonstrated in Fig. 3(b), for the case of two rounds of the internal cycle, the power output $P^{e}_{2}$ is closer to the maximal average power output $P^{e}$ than others. For $\tau_{e}<4\tau^{(1)}$, $m=2$ is an optimal solution for the internal cycling due to the trade-off between the cycling number and the reset time $\tau_{e}$.

We mention the influence from the possible dissipation, which was neglected above. The influence lies in two aspects. One is from the irreversible work dissipation caused by the finite duration in the cooling and heating process. The other is from the extra heat dissipation of the WS and the reservoirs due to the coupling with the external environment. In the slowly dynamical limit that the duration is infinitely long, the work dissipation caused by the first factor is zero but the second factor would yield the efficiency of the heat engine to be vanishing. In the case of the maximum power output, however, the duration is finite. Hence the main influence is caused by the first factor which restricts the efficiency of the Carnot cycle within the Curzon-Ahlborn (CA) efficiency, whereas the second factor is negligible. Nevertheless, this consideration of dissipation does not change the main results in our work.

We have explored the possibility going beyond the Carnot limit using a microscopic engine coupled to two finite heat reservoirs. The favorable feature of our proposal lies in the experimental relevance: No extra quantum resource necessary, no special system required and availability using current laboratory techniques. Our finding paves the way to employing finite-sized reservoirs in microscopic quantum thermodynamics and clarifies the high efficiency of the nano-engines coupled to two same-sized finite heat reservoirs with different temperatures.

The authors declare that they have no conflict of interest.

This work is supported by the National Key Research and Development Program of China under Grant No. 2022YFA1404500, by Cross-disciplinary Innovative Research Group Project of Henan Province under Grant No. 232300421004, National Natural Science Foundation of China under Grant Nos. 1232410, U21A20434, 12074346, 12274376, 12074232, 12125406, 12374466, by Natural Science Foundation of Henan Province under Grant Nos. 232300421075, 242300421212, by Major science and technology project of Henan Province under Grant No. 221100210400, by K. C. Wong Education Foundation (GJTD-2019-15), and by Nansha senior leading talent team technology project under Grant No. 2021CXTD02.

Y.Y.L., S.L.S. and M.F. proposed the idea and made the main deductions; Y.Y.L. carried out numerical calculation; Y.Y.L., M.F. and G. Chen wrote the paper; All the authors contributed to the interpretation of results and improvement of the manuscript.

Supplementary materials to this article can be found online.