Quantum Heat Engines with Complex Working Media, Complete Otto Cycles and Heuristics

Abstract

Quantum thermal machines make use of non-classical thermodynamic resources, one of which include interactions between elements of the quantum working medium. In this paper, we examine the performance of a quasi-static quantum Otto engine based on two spins of arbitrary magnitudes subject to an external magnetic field and coupled via an isotropic Heisenberg exchange interaction. It has been shown earlier that the said interaction provides an enhancement of cycle efficiency, with an upper bound that is tighter than the Carnot efficiency. However, the necessary conditions governing engine performance and the relevant upper bound for efficiency are unknown for the general case of arbitrary spin magnitudes. By analyzing extreme case scenarios, we formulate heuristics to infer the necessary conditions for an engine with uncoupled as well as coupled spin model. These conditions lead us to a connection between performance of quantum heat engines and the notion of majorization. Furthermore, the study of complete Otto cycles inherent in the average cycle also yields interesting insights into the average performance.

1. Introduction

Thermodynamics originated as an empirical study of steam engines, which blossomed into a framework of exceptional generality and simplicity. Quantum thermodynamics is an emerging research field that aims to extend classical thermodynamics and statistical physics into the quantum realm, offering new challenges and opportunities in the wake of a host of non-classical features. A dominant interest is to understand energy-conversion processes at length scales and temperatures where quantum effects become imperative. Inspired by our enhanced capabilities towards nanoscale design and control, this endeavour is being pursued by scientists from diverse backgrounds, such as statistical physics, quantum information, quantum optics, many-body physics and so on. In order to lay foundations for technological breakthroughs, a variety of fundamental questions are being addressed, ranging from issues of thermalisation of quantum systems to examining the validity of thermodynamic concepts, such as the definitions of work, heat, efficiency and power at the nanoscale. The accord between quantum mechanics and thermodynamics is yet to fully unfold [1,2,3]. Its fundamental implications have inspired numerous proposals for thermal machines based on quantum working media [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. Two major issues which are addressed in such proposals are as follows: What are the performance bounds of heat engines working in quantum regime and what are the thermodynamic properties of these quantum systems which control these bounds? The performance analysis of various quantum analogues of classical heat engines serve as a test bed to study different extensions of thermodynamic ideas in the quantum world. With the recent development of quantum information technology [51,52,53,54] and a number of interesting results, the study of quantum heat engines (QHEs) has drawn much interest. In fact, the past few years witnessed conducive studies exploring how quantum statistics, discreteness of energy levels, quantum adiabaticity, quantum coherence, quantum measurement and entanglement affect the operation of heat engines and cycles in various experimental set-ups, including trapped ions, transmon qubits and more [55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84].

Finite time thermodynamic cycles [31,33,85,86,87,88,89,90,91,92,93,94,95,96,97,98] and the study of open quantum systems [99,100,101,102,103,104,105,106] have drawn significant attention in the recent years. These studies aim to arrive at more practical estimates of the performance measures for these machines. However, the importance of quasi-static models of QHEs lies in the fact that they provide a benchmark against which we can compare the behaviour of finite time or more realistic models of heat engines. A range of quantum working substances has been used to model these QHEs. Amongst these, the study of simple, coupled quantum systems [6,18,29,30,107,108,109,110,111,112,113,114,115,116] can yield important insights into the role of quantum interactions in enhancing the performance of model thermal machines. In particular, an upper bound (${\eta }_{\mathrm{ub}}$) for quantum Otto efficiency using two coupled spin-$1/2$ particles has been obtained which is tighter than the Carnot bound (${\eta }_{\mathrm{C}}$) [6,18]. However, this upper bound was shown to be violated for a spin-1/2 particle coupled with an arbitrary spin [111].

In this paper, we generalise the above model and treat two arbitrary spins which are coupled through XXX interaction. We derive conditions ensuring the operation of Otto engine with an arbitrary spin (uncoupled model) as well as for the coupled model. We are able to analytically show that coupling can enhance thermal efficiency and derive a new spin-dependent upper bound to the Otto efficiency, which generalizes and obtains, as a special case, the previous bound of Reference [6]. The new bound is dependent on the magnitude of the total spin quantum number $s=s_1+s_2$. Given the arbitrary magnitudes of the spins and complex nature of the energy spectrum, we focus on the worst-case or best-case scenarios (WCS/BCS) to approach the problem. By proceeding in this manner, we observe that WCS implies a certain majorization relation between the canonical probability distributions of the working medium relative to hot and cold reservoirs. Thus, we discover a robust connection between the performance of our thermal machine and the property of majorization.

We also establish the consistency of our model with the second law. As a novel tool, we study complete Otto cycles (COCs) in order to characterise the performance of our engine. For a COC, the working medium starts and ends in the same state. We notice a subtle difference in the sense in which second law may be applied at the COC level compared to the average level. We show that COCs that follow the second law under a certain operation (say as an engine) suggest conditions applicable for the average performance of the machine. In this manner, COCs offer a potentially handy tool of estimating parameters for the operating regime of the machine.

It is apparent that as the quantum working medium becomes complex, an exact analysis becomes intractable. This is especially true when the working medium is neither a few-particles system having a simple energy spectrum nor a medium close to thermodynamic limit where some scaling law may aid in mathematical simplicity [117]. Thus, in order to target this intermediate regime, it is pragmatic to formulate heuristics. A significant motivation for our paper is to explore the use of heuristics in view of the complexity of the given problem. Heuristics have been employed in various disciplines such as cognitive science, behavioural economics and computer science to name a few. A heuristic is a rule of thumb providing insights into the behaviour of a system in the face of complexity or uncertainty [118,119,120]. It must be appreciated that the value of a heuristic lies in providing a shortcut method that requires a simpler analysis, thus trading accuracy and completeness for speed.

The paper is organized as follows. In Section 2, we introduce our model of two coupled spins ($s_1,s_2$) as the working substance of the Quantum Otto engine. In Section 2.1, various stages of the heat cycle are described, and the positive work condition for the uncoupled model is discussed. The proof for the same is sketched in Appendix Appendix A. In Section 3, the spins are coupled, and we find the coupling range in which positive work extraction is ensured (proofs are sketched in Appendix B and Appendix C), which is related to the notion of majorization in Section 3.1 and further used to order the system’s energy levels for $J\ne 0$ in Section 3.2. In Section 4, the conditions for maximum enhancement of coupled system’s efficiency over the uncoupled model are discussed. An upper bound to engine’s efficiency is also calculated in the considered domain of coupling. A detailed proof for the positive entropy production for the coupled system is sketched in Appendix D. In Section 5, an analysis is carried out by using the notion of complete Otto cycles. Finally, we discuss the results of our analysis in Section 6.

2. Quantum Otto Cycle

The working substance consists of two spins with arbitrary magnitudes, $s_1$ and $s_2$, coupled by 1-D isotropic Heisenberg exchange interaction in the presence of an externally applied magnetic field of magnitude B along the z-axis. The system Hamiltonian in the first Stage of the cycle can be written as follows:

$${\mathcal{H}}_1\equiv H_1+H_{int}=2B_1\left(s_1^{\left(z\right)}\otimes I+I\otimes s_2^{\left(z\right)}\right)+8J{\overrightarrow{s}}_1.{\overrightarrow{s}}_2$$

(1)

where $J>0$ is the strength of the anti-ferromagnetic coupling. ${\overrightarrow{s}}_1\equiv \{s_1^{\left(x\right)},s_1^{\left(y\right)},s_1^{\left(z\right)}\}$ and ${\overrightarrow{s}}_2\equiv \{s_2^{\left(x\right)},s_2^{\left(y\right)},s_2^{\left(z\right)}\}$ are the spin operators for the first and the second spin, respectively. $H_{int}$ is the interaction Hamiltonian, and $H_1$ is the free Hamiltonian. We have taken Bohr magneton ${\mu }_{\mathrm{B}}=1$, and the gyromagnetic ratio for both spins has been taken to be two [121].

Let n = $\left(2s_1+1\right)\left(2s_2+1\right)$ be the total number of energy levels with $\left|{\psi }_k\right.〉$ as the corresponding energy eigenstates. When the system is in thermodynamic equilibrium with a heat bath at temperature T, the density matrix ${\rho }_1$ for the working substance can be written as follows:

$${\rho }_1=\sum _{k=1}^n{P}_k\left|{\psi }_k\right.〉\left.〈{\psi }_k\right|,$$

(2)

where $P_k=e^{-E_k/T}/Z$ are the occupation probabilities of the energy levels, and $Z={\sum }_k{e}^{-E_k/T}$ is the partition function for the system. We have rendered the Boltzmann constant $k_{\mathrm{B}}$ equal to unity.

Let us consider the case where one spin is an integer, and the other is a half integer. Some examples of such spin combinations are $\left(\frac{3}{2},2\right),\left(\frac{1}{2},2\right)$ and $\left(\frac{5}{2},4\right)$. The energy eigenvalues of the Hamiltonian $\mathcal{H}$ for a general ($s_1,s_2$) coupling are shown in Figure 1. It is to be noted that a term $8s_1{s}_{2}J$ common in all the eigenvalues has been neglected as the physical properties of the system would be independent of it. The ordering of these energy levels would depend upon the conditions on the parameters which the positive work condition for the system would provide, which will be discussed in the coming sections.

2.1. The Heat Cycle

The four stages constituting the Otto cycle are as follows.

Stage 1: 

The system is at thermal equilibrium with a heat reservoir at temperature $T_1$ with energy $e_k$ for which its occupation probabilities are $p_k$, and the corresponding density matrix is ${\rho }_1$ (here, we are considering two non interacting spins with energy eigenvalues denoted by $e_k$ and occupation probabilities by $p_k$).

Stage 2: 

The system undergoes a quantum adiabatic process after it is isolated from the hot bath, and the magnetic field is changed from $B_1$ to a smaller value $B_2$. Here, the quantum adiabatic theorem is assumed to hold according to which the process should be slow enough so that no transitions are induced as the energy levels change from $e_k$ to $e_k^{{}^{\prime }}$.

Stage 3: 

Here, the system is brought in contact with a cold bath at temperature $T_2$ (<$T_1$). The energy eigenvalues remain at $e_k^{{}^{\prime }}$, and the occupation probabilities change from $p_k$ to $p_k^{{}^{\prime }}$ with the external magnetic field at $B=B_2$, and the density matrix of the system is ${\rho }_2$.

Stage 4: 

The system is detached from the cold bath, and the magnetic field is changed from $B_2$ to $B_1$ with occupation probabilities remaining unchanged at $p_k^{{}^{\prime }}$ and energy eigenvalues changing back from $e_k^{{}^{\prime }}$ to $e_k$ such that only work is performed on the system during this step. Finally, the system is attached to the hot bath again, and the cycle is completed such that the average heat absorbed is $q_{1,av}=\mathrm{Tr}[H_1\Delta {\rho }_{}]$, and the net work performed per cycle is $w_{av}=\mathrm{Tr}[(H_1-H_2)\Delta {\rho }_{}]$. Here, $\mathrm{Tr}[·]$ denotes the trace operation, and $\Delta {\rho }_{}={\rho }_1-{\rho }_2$. In this paper, we consider the free Hamiltonian of the form $H_i\equiv 2B_i{h}_0$ ($i=1,2$), where $h_0$ is an operator. We now have $w_{av}=2(B_1-B_2)\mathrm{Tr}[h_0\Delta {\rho }_{}]$; therefore, the efficiency in the absence of interaction is as follows.

$${\eta }_0=1-\frac{B_2}{B_1}.$$

(3)

Let us first discuss the positive work condition when $s_1$ and $s_2$ are non-interacting. The energy eigenvalues ($e_k$) of the free Hamiltonian, written in the order of increasing energy (if one spin is integer and the other is half integer), are listed in

Table 1. Levels indicating degeneracy and energy eigenvalues ($e_k$) for two uncoupled spins. Here, $s_1<s_2$ with $s=s_1+s_2$ and $r\equiv s-1/2$.

k	${\mathit{e}}_{\mathit{k}}$
1	$-2sB$
$2,3$	$-2(s-1)B$
$4,5,6$	$-2(s-2)B$
.	.
.	.
$(n/2-2s_1),...,n/2$	$-2(s-r)B$
$(n/2+1),...,(n/2+2s_1+1)$	$2(s-r)B$
.	.
.	.
$(n-5),(n-4),(n-3)$	$2(s-2)B$
$(n-2),(n-1)$	$2(s-1)B$
n	$2sB$

, and as it can be observed many energy levels for the non-interacting system are degenerate. There is only one level with energy proportional to $-s$ as well as s, two levels with energy proportional to $-(s-1)$ as well as $(s-1)$ and so on (the proportionality constant always being $2B$). Therefore, denoting the degeneracy by $“g”$, we have the following from Table 1:

$$g_{\left|s\right|}=1,g_{|s-1|}=2,g_{|s-2|}=3,...,g_{|s-r|}=2s_1+1$$

(4)

such that the total number of energy levels are as follows.

$$n=(2s_1+1)(2s_2+1)=2(g_{\left|s\right|}+g_{|s-1|}+g_{|s-2|}+...+g_{|s-r|}).$$

The Stage 1 occupation probabilities are written as $p_k=e^{-e_k/T_1}/z_1$, where $z_1={\sum }_{k=1}^n{e}^{-e_k/T_1}$ is the partition function of the system, which can be expressed as follows.

$$z_1=2\sum _{l=1}^{s+1/2}{g}_{|s-l+1|}.cosh\left[2(s-l+1)B_1/T_1\right].$$

(5)

The average heat exchanged with the hot reservoir is as follows:

$$q_{1,av}=\sum _{k=1}^n{e}_k\left(p_k-p_k^{{}^{\prime }}\right)=2B_{1}v,$$

(6)

where the primed probabilities are tabulated at $T=T_2$ and $B=B_2$. The average heat exchanged with the cold bath is as follows.

$$q_{2,av}=\sum _{k=1}^n{e}_k^{{}^{\prime }}\left(p_k^{}-p_k^{{}^{\prime }}\right)=2B_{2}v,$$

(7)

Thus, the work performed on average is the following.

$$w_{av}=q_{1,av}-q_{2,av}=2(B_1-B_2)v.$$

(8)

The explicit expression of v is given by Equation (A1). Since $B_1>B_2$ is assumed, the system works as an engine on average if and only if $v>0$. We prove in Appendix A that the condition required to satisfy $v>0$ is the following:

$$\frac{B_2}{T_2}>\frac{B_1}{T_1},\mathrm{or}{B}_2>B_1\theta ,$$

(9)

where $\theta =T_2/T_1$. Furthermore, as proved in Appendix A, Equation (9) implies $z_2>z_1$ as well as the following.

$$p_1^{{}^{\prime }}>p_1,\mathrm{and}{p}_n^{{}^{\prime }}<p_n,$$

(10)

From the above conditions, we can make the following inferences. Positive work extraction is favoured when the occupancy of ground (top) level is more (less) at the cold bath than at the hot bath, which suggests that heat is absorbed at the hot bath, decreasing (increasing) the occupancy of the ground (top) level, while heat is released at the cold bath, thus increasing (decreasing) the occupancy of the ground (top) level.

Since the working medium returns to its initial state (restoring the Hamiltonian as well as coming to be in equilibrium with the hot reservoir), the net change in entropy $\Delta S_{0,av}$ is due to the entropy changes only in the heat baths. The decrease in the entropy of the hot bath is $-q_{1,av}/T_1$ and increase in entropy of the cold bath is $q_{2,av}/T_2$. Thus, the net entropy change in one cycle is the following.

$$\Delta S_{0,av}=-\frac{q_{1,av}}{T_1}+\frac{q_{2,av}}{T_2}=\left(-\frac{B_1}{T_1}+\frac{B_2}{T_2}\right)v.$$

(11)

We have seen that $w_{av}>0$ or $v>0$ requires Equation (9) to hold. Under these conditions, it follows that $\Delta S_{0,av}>0$, and the consistency with the second law is established at the level of average performance as an engine. Similarly, we observe that the efficiency satisfies the following. ${\eta }_0<1-T_2/T_1={\eta }_C$.

3. The Coupled Model

Let us now couple the two spins, with $J>0$ being the anti-ferromagnetic coupling strength. The corresponding energy eigenvalues are shown in Figure 1b, where the ordering of the eigenvalues can be considered when the coupling parameter J is small. Moreover, as the coupling is switched on, the degeneracy of the previously degenerate levels is now lifted. Let us express an energy eigenvalue of the coupled system as $E_k=m_{1}B-8m_{2}J$, where $m_1=-2s,...,+2s$ and $m_2$ can only take positive values including zero, as shown in

Table A2. Spin dependent factors $m_1$ and $m_2$ when the energy eigenvalues of the coupled system are expressed as the following: $E_k=m_{1}B-8m_{2}J$. The energy levels “k” which fall within the same band (i.e., having same $m_1$) have also been specified.

${\mathit{m}}_1$	${\mathit{m}}_2$	k
$-2s$	0	1
$-2(s-1)$	$s,0$	$2,3$
$-2(s-2)$	$[s+(s-1\left)\right],s,0$	$4,5,6$
$-2(s-3)$	$[s+(s-1)+(s-2\left)\right],[s+(s-1\left)\right],s,0$	$7,..,10$
.	.	.
.	.	.
$-2(s-r)$	$[s+(s-1)+...+(s-(2s_1-1))],...,0$	$n/2-2s_1,...,n/2$
$2(s-r)$	$[s+(s-1)+...+(s-(2s_1-1))],...,0$	$n/2+1,...,n/2+2s_1+1$
.	.	.
.	.	.
$2(s-3)$	$[s+(s-1)+(s-2\left)\right],[s+(s-1\left)\right],s,0$	$(n-9),..,(n-6)$
$2(s-2)$	$[s+(s-1\left)\right],s,0$	$(n-5),(n-4),(n-3)$
$2(s-1)$	$s,0$	$(n-2),(n-1)$
$2s$	0	n

in Appendix C. The values $m_1$ and $m_2$ depend on the index k, but we have omitted it here for brevity of notation.

Now, the average heat absorbed from the hot bath ($Q_{1,av}$), the heat rejected to the cold bath ($Q_{2,av}$) and the average work performed in one cycle, $W_{av}=Q_{1,av}-Q_{2,av}$, are given as follows:

$$\begin{array}{ccc}{Q}_{1,av}& =& 2B_{1}X+8JY,\\ Q_{2,av}& =& 2B_{2}X+8JY,\\ W_{av}& =& 2(B_1-B_2)X,\end{array}$$

where the following is the case.

$$X=\frac{1}{2}\sum _{k=1}^n{m}_1(P_k^{}-P_k^{{}^{\prime }}),Y=\sum _{k=2}^{n-2}{m}_2(P_k^{{}^{\prime }}-P_k).$$

(12)

The spin dependent factors $m_1$ and $m_2$ are obtained from the expressions of the equilibrium occupation probabilities of the energy levels $E_k$ (shown in Figure 1), which in general are written as follows.

$$P_k=\frac{e^{-m_1{B}_1/T_1+8m_{2}J/T_1}}{Z_1}.$$

(13)

For explicit expressions of $P_k$, refer to

Table A1. Stage 1 occupation probabilities of the energy levels $E_k$ of the coupled spin system. $s_1$ is smaller of the two spins in the terms involving the factor $2s_1+1$.

$P_1=e^{2s{B}_1/T_1}/Z_1$
$P_2=e^{2(s-1)B_1/T_1+8sJ/T_1}/Z_1$
$P_3=e^{2(s-1)B_1/T_1}/Z_1$
$P_4=e^{2(s-2)B_1/T_1+8sJ/T_1+8J(s-1)/T_1}/Z_1$
$P_5=e^{2(s-2)B_1/T_1+8sJ/T_1}/Z_1$
$P_6=e^{2(s-2)B_1/T_1}/Z_1$
.
.
.
$P_{n/2-2s_1}=e^{B_1/T_1+8sJ/T_1+...+8J(s-\left(2s_1-1\right))/T_1}/Z_1$
.
.
.
$P_{n/2}=e^{B_1/T_1}/Z_1$
$P_{n/2+1}=e^{-B_1/T_1+8sJ/T_1+...+8J(s-\left(2s_1-1\right))/T_1}/Z_1$
.
.
.
$P_{n/2+2s_1+1}=e^{-B_1/T_1}/Z_1$
.
.
.
$P_{n-5}=e^{-2(s-2)B_1/T_1+8sJ/T_1+8(s-1)J/T_1}/Z_1$
$P_{n-4}=e^{-2(s-2)B_1/T_1+8sJ/T_1}/Z_1$
$P_{n-3}=e^{-2(s-2)B_1/T_1}/Z_1$
$P_{n-2}=e^{-2(s-1)B_1/T_1+8sJ/T_1}/Z_1$
$P_{n-1}=e^{-2(s-1)B_1/T_1}/Z_1$
$P_n=e^{-2s{B}_1/T_1}/Z_1$

in Appendix B. $Z_1$ is the Stage 1 partition function of the system for which its expression may be rewritten as follows:

$$Z_1=\left\{\begin{array}{c}{\mathcal{Z}}_1+2cosh\left[2(s-1)B_1/T_1\right].e^{8sJ/T_1}+\hfill \\ 2cosh\left[2(s-2)B_1/T_1\right].\left(e^{8sJ/T_1}+e^{8(s-1)J/T_1}\right)+...+\hfill \\ 2cosh\left[2(s-r)B_1/T_1\right].\left(e^{8sJ/T_1}+e^{8(s-1)J/T_1}+...+e^{8(s-(2s_1-1))J/T_1}\right),\hfill \end{array}\right.$$

(14)

where ${\mathcal{Z}}_1\equiv 2{\sum }_{k=1}^{s+1/2}cosh\left[2(s-k+1)B_1/T_1\right]$. Similarly, we can define $P_k^{{}^{\prime }}$, the canonical probabilities due to cold bath, by replacing $B_1\to B_2$ and $T_1\to T_2$ in the above expressions for $P_k^{}$.

For the proof of PWC for the coupled model (Appendix B), we show that for the so-called worst case scenario (WCS) is given by the following:

$$P_k^{{}^{\prime }}<P_k^{},k=2,3,...,n,\mathrm{and}{P}_1^{{}^{\prime }}>P_1,$$

(15)

along with Equation (9). It follows that $X>0$. Consistent with Equations (15) and (A25), we then calculate the strictest condition on the allowed range of J (Appendix C), which is given by the following.

$$0<J<\frac{B_2-B_1\theta }{4s\left(1-\theta \right)}\equiv J_c.$$

(16)

Therefore, we conclude that $X>0$ or PWC is satisfied under Equations (9) and (16), with the latter constituting the sufficient condition for the coupled system to work as an engine.

3.1. Majorization

Majorization [122] is a powerful mathematical concept that defines a preorder on the vectors of real numbers. It is particularly useful to compare two probability distributions. We will highlight its occurance in the context of the working regime of our engine by comparing the two equilibrium probability distributions.

Now, for the uncoupled model, the relevant probability distributions are the canonical probabilities $\left\{p_k\right\}$ and $\left\{p_k^{{}^{\prime }}\right\}$, which, at finite temperatures, are ordered as $p_n^{}<p_{n-1}^{}<\cdots <p_1^{}$ and $p_n^{{}^{\prime }}<p_{n-1}^{{}^{\prime }}<\cdots <p_1^{{}^{\prime }}$, respectively. In Lemma A2 of Appendix B, we proved that Equation (9) is a necessary condition that ensures $w_{av}>0$ in the regime of the so-called worst case scenario (WCS), given by the following:

$$p_k^{{}^{\prime }}\le p_k^{},k=2,3,...,n\mathrm{and}{p}_1^{{}^{\prime }}\ge p_1^{},$$

where the equality holds for $B_2/T_2=B_1/T_1$. Therefore, the above relations imply the following.

$$\begin{array}{cc}\hfill p_n^{{}^{\prime }}& \le p_n,\hfill \\ \hfill p_n^{{}^{\prime }}+p_{n-1}^{{}^{\prime }}& \le p_n^{}+p_{n-1}^{},\hfill \\ \hfill & ⋮\hfill \\ \hfill \sum _{k=1}^{n-1}{p}_k^{{}^{\prime }}& \le \sum _{k=1}^{n-1}{p}_k,\hfill \\ \hfill \sum _{k=1}^n{p}_k^{{}^{\prime }}& =\sum _{k=1}^n{p}_k.\hfill \end{array}$$

The above set of conditions (M) is summarised by stating that $\left\{p_k^{{}^{\prime }}\right\}$ majorizes $\left\{p_k^{}\right\}$ and denoted as $\left\{p_k\right\}\prec \left\{p_k^{{}^{\prime }}\right\}$. As a powerful tool, majorization can be used to prove other results. Intuitively, it indicates that the distribution $\left\{p_k^{}\right\}$ is more mixed than $\left\{p_k^{{}^{\prime }}\right\}$. Thus, as an important consequence, $\left\{p_k\right\}\prec \left\{p_k^{{}^{\prime }}\right\}$ implies that $S\left(p_k\right)\ge S\left(p_k^{{}^{\prime }}\right)$, where $S\left(p\right)$ is the Shannon entropy of the distribution $\left\{p\right\}$ (proportional to the thermodynamic entropy of the working medium in equilibrium with a reservoir). In fact, this is expected, since the flow of heat for the engine is on the average from hot to cold. Then, along with heat, thermodynamic entropy is also lost to the cold reservoir. However, the condition of majorization is more general than the above mentioned relation between the entropies.

Similarly for the coupled model, we have shown that Equations (9) and (16) ensure $W_{av}>0$ under the following conditions: $P_k^{{}^{\prime }}<P_k^{},\forall k=2,3,...,n$ and $P_1^{{}^{\prime }}>P_1^{}$. In general, we may write the following.

$$P_k^{{}^{\prime }}\le P_k^{};k=2,3,...,n\mathrm{and}{P}_1^{{}^{\prime }}\ge P_1^{}.$$

(17)

Thus, for the coupled model, we can also write down the set of conditions equivalent to Equation (M), and infer that $\left\{P_k\right\}\prec \left\{P_k^{{}^{\prime }}\right\}$, which implies $S\left(P_k\right)\ge S\left(P_k^{{}^{\prime }}\right)$. In other words, if the Stage 3 equilibrium distribution majorizes Stage 1 equilibrium distribution, then we have positive work extraction from the coupled system.

It is possible to find a range of parameter values which satisfy Equation (17). In Figure 2, we show the behaviour of $(P_k^{}-P_k^{{}^{\prime }})$ for ($1/2,1$) system. It is observed that $(P_2-P_2^{{}^{\prime }})$ changes sign within the range $[0,J_c]$, indicating that every condition of Equation (17) may not hold in this range, especially at high bath temperatures. However, we observe that the majorization conditions continue to hold and $\left\{P_k\right\}\prec \left\{P_k^{{}^{\prime }}\right\}$, even if $P_2^{{}^{\prime }}>P_2$ (see Figure 3).

3.2. Energy Level Ordering

The actual arrangement of the energy eigenvalues depends on the positive work conditions derived above. As for the relative position of $2sB$ energy level, it will not change, because it is the highest energy eigenvalue of the system regardless of the coupling strength J. The ground state or the minimum energy state will be decided as follows.

There are two energy levels $-2sB$ and $-2(s-1)B-8sJ$ which can possibly form the ground state of the coupled system, and their energy gap is $|2B-8sJ|$. Given that $B_1>B_2$ and $0<J<J_c$, we can check that the following is the case.

$$J<J_c<\frac{B_2}{4s}<\frac{B_1}{4s}.$$

(18)

The above implies that $2B-8sJ>0$, thereby making $-2sB$ the lowest energy of the system and $-2(s-1)B-8sJ$ the energy of the first excited state. Now, Equation (18) opens different possibilities for the arrangement of other energy levels. For example, the levels $-2(s-2)B-8sJ-8(s-1)J$ and $-2(s-1)B$ have an energy gap of $|-2B+8sJ+8(s-1\left)J\right|$, and either of them can be at higher energy state than the other, and both the arrangements are acceptable. For the sake of concreteness, we assume the condition that there is no level crossing when $B_1$ is changed to a lower value $B_2$. One method of arranging the energy levels, in accordance with Equation (18), is shown in Figure 1, which is assumed for the discussion that follows. The net entropy production in one cycle $\Delta S_{av}$ for the coupled system $\Delta S_{av}=-Q_{1,av}/T_1+Q_{2,av}/T_2$ can be written as follows.

$$\Delta S_{av}=2X\left(\frac{B_2}{T_2}-\frac{B_1}{T_1}\right)+8JY\left(\frac{1}{T_2}-\frac{1}{T_1}\right).$$

(19)

In the above expression, due to Equation (9), the first term is always positive, but since $T_1>T_2$, the sign of the second term depends on Y which may not be positive. We will consider the WCS whereby under Equation (15), all terms in the defining sum Y (Equation (12)) are negative, thus making Y negative definite (note that $m_2>0$ for all k). By defining the following:

$$Y_1=-Y/s,a=2\left(\frac{B_2}{T_2}-\frac{B_1}{T_1}\right)>0b=8sJ\left(\frac{1}{T_2}-\frac{1}{T_1}\right)>0,$$

we have $\Delta S_{av}=aX-b{Y}_1$. The condition, given by Equation (16), on the coupling strength which ensures $W_{av}>0$ implies that $a>b$. Then, for $Y_1>0$, we have shown in Appendix D that PWC for the coupled system encapsulated in Equations (9) and (16) suffice to prove $X>Y_1$ and hence $\Delta S_{av}>0$. This establishes the consistency of our engine with the second law in the considered domain.

4. Efficiency Enhancement and the Upper Bound

In the above, we have established conditions for work extraction in the quantum Otto cycle for the coupled system and verified consistency with the second law. In this section, we explore how the coupling between the spins may enhance the efficiency of the engine.

The heat absorbed from the hot reservoir is given by $Q_{1,av}=2B_{1}X+8JY$, where X and Y are as defined in Equation (12). From the energy levels diagram, it is clear that the contribution $8JY$ to the exchanged heat comes solely from levels which depend on parameter J apart from the field B. Now, since $Q_{2,av}=2B_{2}X+8JY$, this ’extra’ contribution to heat is not available for conversion into work, and it is wasted if $8JY>0$. However, it may be utilized to enhance the efficiency of the cycle if $8JY<0$, thus effectively decreasing the heat absorbed from the hot reservoir. Remarkably, the WCS considered earlier implies that all terms entering the sum for Y are negative, and so we have $Y\le 0$ with $J>0$. Thus, the WCS directly results in a regime where we can expect an enhancement of the efficiency. Thus, for the operational regime discussed in previous sections, we can rewrite the expression for efficiency, $\eta =1-Q_{2,av}/Q_{1,av}$ as follows:

$$\eta =\frac{{\eta }_0}{1+\frac{8JY}{2X{B}_1}}=\frac{{\eta }_0}{1-\frac{4sJ{Y}_1}{X{B}_1}}$$

(20)

where $Y_1=-Y/s>0$. We have proved in Appendix D that $X>Y_1$. With $B_1>4sJ$ (Equation (18)), we obtain the following:

$$\eta <\frac{{\eta }_0}{1-4sJ/B_1}<1-\frac{T_2}{T_1}={\eta }_C,$$

(21)

where the second inequality follows due to the permissible range of J (Equation (16)). Thus, the expression of the following:

$${\eta }_{ub}=\frac{{\eta }_0}{1-4sJ/B_1}$$

(22)

constitutes an upper bound to the system’s efficiency, which is tighter than the Carnot efficiency, and is within the coupling range $0<J<J_c$.

The above expression bounding the efficiency of the Otto cycle is our main result of the paper. This expression is validated with numerical calculations in the discussion section. Note that ${\eta }_{ub}$ given by Equation (22) is dependent solely on the field values and the total spin of the two particles, while it is independent of the bath temperatures. This expression generalizes the upper bound derived earlier in Reference [6] for the $(\frac{1}{2},\frac{1}{2})$ system.

We close this section with a remark on the three possible spin combinations for our ($s_1,s_2$) system:

• When one spin value is a half-integer and the other is an integer;
• When both values are half-integer or both are integers;
• When both are of the same magnitude (both as half-integer or integer).

In this paper, we have discussed the first case only. The only difference between the present case and the other two cases is that, for the latter, when the spins are uncoupled, an energy level with zero energy and $2s_1+1$-fold degeneracy occurs but that does not affect the performance of the system. The reason is that after the coupling is turned on between the spins, this energy state splits into $2s_1+1$ non-degenerate energy levels, which depend only on the coupling factor J. Since J is kept fixed during the cycle, these levels do not shift in a cycle and hence do not contribute to the average work resulting in the same PWC as already derived for the first case. Similarly, it can be observed that these levels do not change the condition for maximal efficiency enhancement, and same upper bound can be obtained whatever the spin combination may be.

5. Complete Otto Cycles

The working medium for the classical Otto cycle is usually a macroscopic system amenable to thermodynamic treatment. This medium may be a collection of statistically independent, non-interacting individual quantum systems or elements, such as spin-$1/2$ particles or harmonic oscillators and so on. In the adiabatic step of the Otto cycle, the thermodynamic entropy of the working medium stays constant. This implies that there is no intrinsic control on the transitions experienced by individual elements of the working medium.

On the other hand, the working medium of a quantum Otto engine consists of individual elements. In a quasi-static cycle, the isochoric steps are stochastic while the adiabatic steps are deterministic. The quantum adiabatic step is executed slowly enough such that no transition is induced between energy levels of the element, which continues to occupy its initial state throughout the process. Thus, at the level of the ensemble, the occupation probabilities do not change during this process. Thus, such a process imposes maximal control on the evolution of the isolated element, and it is described by a quantum unitary process.

Still, due to the stochastic nature of the contact with the reservoirs, the element may not return to its initial state after the four steps of the cycle. Usually, we are interested in the average properties of the cycle by which the quantities such as heat and work are defined at the ensemble level. In this section, we focus on the complete Otto cycles (COCs) inherent in the average Otto cycle considered in earlier sections. The reason that Otto cycle is so often studied in the quantum thermodynamics literature is that the contributions towards heat and work can be clearly separated into different steps, which helps in the analysis. This distinction also holds at the level of COCs; the interaction of the working medium with a reservoir involves only the exchange of heat with the reservoir, whereas the quantum adiabatic step involves only work.

Consider the COC shown as an engine in Figure 4. If the working medium starts at energy level $e_i$, then by the end of the four stages it is again found at level $e_i$. Such a cycle can either run forward as an engine or backwards as a refrigerator. Analysing the performance of COCs is much easier since we are dealing with only two levels at a time without invoking occupation probabilities of the levels and any average quantities.

Let us represent an energy eigenvalue of the uncoupled system as $e\equiv m_{1}B$, where $m_1$ varies from $m_1=-2s,...,+2s$. Based on the final (f) and initial (i) values of $m_1$, let us define the quantity $x=m_{1,f}-m_{1,i}$, ranging as $x=\pm 2,...,\pm 4s$. Let $q_1,q_2,w$ denote the heat exchanged with the hot bath, cold bath and the work performed, respectively.

$$\begin{array}{ccc}{q}_1& =& e_f-e_i=x{B}_1,\\ q_2& =& e_f^{{}^{\prime }}-e_i^{{}^{\prime }}=x{B}_2,\\ w& =& q_1-q_2=x(B_1-B_2).\end{array}$$

With $B_1>B_2>0$, we have $q_h,q_c>0$ and $w>0$ if $x>0$. It is clear that for $x>0$ ($x<0$), a COC runs as an engine (refrigerator). The net entropy change ($\Delta S_0$) is contributed only by the reservoirs. Thereby, we obtain the following.

$$\Delta S_0=-\frac{q_1}{T_1}+\frac{q_2}{T_2}=x\left(-\frac{B_1}{T_1}+\frac{B_2}{T_2}\right).$$

(23)

Now, for $x>0$, the condition $B_2/T_2>B_1/T_1$ ensures that $\Delta S_0>0$, or we may say that the second law is then satisfied at the level of COC. Note that there is a subtle difference in the statement about the second law at the level of a COC versus the average performance level. In the former case, $x>0$ guarantees the operation of an engine, whereas the additional condition (9), i.e., $B_2/T_2>B_1/T_1$ makes this operation consistent with the second law. On the other hand, for the average operation as an engine, we require $v>0$, which itself requires the condition (9). The latter then automatically ensures consistency with the second law at the level of average performance.

Moreover, note that we do not impose the second law at the level of a COC, and the net entropy change for a COC may be negative, as for instance with $x<0$ or a COC operating as a refrigerator if $B_2/T_2>B_1/T_1$. Thus, we do not imply that COCs with $\Delta S_0<0$ do not happen. These observations result in the following interesting conclusion about the uncoupled model. A consistency with the second law for the average performance as engine ensures consistency with the second law for a COC as engine and vice versa.

Let us next study the effect of coupling between the spins. Now, there are no degenerate levels. Let us express an energy eigenvalue of the coupled system as $E\equiv m_{1}B-8m_{2}J$, where the $m_1,m_2$ values are given in Table A2. The levels with same $m_1$ were originally degenerate in the uncoupled model. For the coupled model, energy levels belong to the same band if they have the same value of $m_1$ but have different values of $m_2$. Furthermore, note that in every band there is one level that stays at the same energy even after the coupling is switched on. Now, for a COC between any two energy levels of the coupled system, the general forms of heat exchanged with the reservoirs, $Q_1$ and $Q_2$ and the work performed, $W=Q_1-Q_2$, can be written as follows:

$$\begin{array}{ccc}{Q}_1& =& x{B}_1+8Jy,\\ Q_2& =& x{B}_2+8Jy,\\ W& =& x(B_1-B_2),\end{array}$$

with $x=m_{1,f}-m_{1,i}$ and $y=m_{2,f}-m_{2,i}$. The net entropy change in one cycle is the following.

$$\Delta S=-\frac{Q_1}{T_1}+\frac{Q_2}{T_2}=x\left(\frac{B_2}{T_2}-\frac{B_1}{T_1}\right)+8Jy\left(\frac{1}{T_2}-\frac{1}{T_1}\right).$$

(24)

We discuss the possible COCs as below stated below.

• $\mathbf{x}\ne \mathbf{0},\mathbf{y}=\mathbf{0}$: These cycles occur between any two different energy bands having the same $m_2$. Therefore, if such a cycle proceeds as an engine ($x>0$), its efficiency is $W/Q_1=1-B_2/B_1={\eta }_0$. From Equation (24), this COC is consistent with the second law for $B_2>B_1\theta$.
• $\mathbf{x}=\mathbf{0},\mathbf{y}\ne \mathbf{0}$: These cycles are possible between energy levels of the same band, i.e., having same $m_1$. The work performed is zero, and the heat exchanged is $Q_1=8Jy=Q_2$. Thus, for $y>0$, the corresponding efficiency is also zero.
• $\mathbf{x},\mathbf{y}\ne \mathbf{0}$with the same sign: These cycles are possible between different bands for levels with different $m_1$ and $m_2$. If such cycles proceed as engine, i.e., $x>0$ (and $y>0$), then the corresponding efficiency is the following.

  $${\eta }_{}=\frac{{\eta }_0}{1+\frac{8yJ}{x{B}_1}}<{\eta }_0.$$

  (25)
  From Equation (24), this type of COC is consistent with the second law for $B_2>B_1\theta$, without imposing any further condition on the coupling strength $J\ge 0$. Therefore, if the second law allows COCs with $\eta ={\eta }_0$, then it also allows COCs with $\eta <{\eta }_0$.
• $\mathbf{x},\mathbf{y}\ne \mathbf{0}$with opposite signs: These cycles occur between energy levels of different bands with different $m_1$ and $m_2$. If $x>0$ for such cycles (and $y<0$), the corresponding efficiency is as follows.

  $$\eta =\frac{{\eta }_0}{1-\frac{8\left|y\right|J}{x{B}_1}}>{\eta }_0.$$

  (26)

From Equation (24), this COC is allowed by the second law if $B_2>B_1\theta$ and the following is the case.

$$0<J<\frac{x(B_2-B_1\theta )}{8\left|y\right|(1-\theta )}\equiv J_a.$$

(27)

Now, we look for the values of x and y which place the most stringent condition on the second law (Equation (24)) or, in other words, render $\Delta S$ as the least positive. This will be the worst-case scenario (WCS) in this context, as other values of x and y would yield a larger upper bound $J_a$. Thus, the range imposed by the WCS will hold for all COCs, making all of them consistent with the second law.

The first term in Equation (24) takes the minimum value if $x=2$. For the second term, let $y_{\min }<0$ denote the minimum value of y. Then, we obtain $-y_{\min }=[s+(s-1)+...+(s-(2s_1-1))]=s_1(2s_2+1)$. Substituting the above values of x and y in Equation (27), we obtain the following range of J.

$$0<J<\frac{B_2-B_1\theta }{4s_1(2s_2+1)(1-\theta )}\equiv J_x.$$

(28)

Therefore, it follows that for $B_2>B_1\theta$ and within the range $0<J<J_x$, all the COCs perform as an engine and satisfy the second law.

Now, from the probabilistic or average analysis, we concluded that the conditions $B_2>B_1\theta$ and the coupling range $0<J<J_c$ ensure the average performance as an engine. In order to compare the two ranges for J, note that $s_1(2s_2+1)\ge s$, where the equality is obtained for $s_1=1/2$ implying that, in general, $J_x\le J_c$. This has the following important consequence. The range for the parameter J in which the machine behaves as an engine on average subsumes the range for J in which all COCs, performing as an engine, are also consistent with the second law. Conversely, if we restrict to the range $0<J<J_x$, allowing all COCs running as engine to follow the second law, then the average operation as an engine in that range of parameters is also consistent with the second law.

Moreover from Section 5, we learn that out of all the possible COCs with $\eta >{\eta }_0$, the maximum possible value of efficiency is obtained from Equation (26) for minimum x, i.e., $x=2$ and $|y_{min}|=s_1(2s_2+1)$, given by the following.

$${\eta }_{\max }=\frac{{\eta }_0}{1-\frac{4s_1(2s_2+1)J}{B_1}}.$$

(29)

This cycle is allowed by the second law for the condition $B_2>B_1\theta$ and in the $0<J<J_x$ range of coupling. Interestingly, the coupling range required for $W_{av}>0$ goes beyond $J=J_x$ since $J_x\le J_c$. The case of $J_a=J_c$ is obtained when we substitute $x=2,\left|y\right|=s$ in Equation (27), and out of all the COCs allowed in this range, the maximum efficiency is given as follows: ${\eta }_0/\left(1-4sJ/B_1\right)$. The latter value is same as the upper bound, ${\eta }_{ub}$, inferred by analysing the average performance of the system. As it can be observed, ${\eta }_{ub}\le {\eta }_{max}$. For the special case of $(1/2,s_2)$ working medium, $J_x$ and $J_c$ values coincide irrespective of the value of $s_2$, resulting in ${\eta }_{max}={\eta }_{ub}$.

6. Discussion

We have analyzed the performance of a quantum Otto engine based on a working medium with a complex energy spectrum. An insight into the possible operational regimes is hard to obtain analytically for such a system. Using a heuristic-based approach and employing techniques such as worst-case/best-case reasoning, we have highlighted a regime in which the machine definitely works as an engine on average. These set of conditions can be related to the concept of majorization for the given model. Thereby, we find that majorization serves as a more robust criterion for positive work extraction from our engine.

We also introduced an analysis based on complete Otto cycles (COCs). Compared to the probabilistic analysis, the COC approach is much simpler and straightforward. The latter utilizes much less information than the ’average’ analysis, and the conclusions so obtained may not be as general. However, as a starting point, the criteria for COCs may serve as a useful heuristic to gain insight into the average performance of the Otto machine. As we have observed, there is an interesting correspondence between the COCs and the average Otto cycle with regard to the validity of the second law. One of our main results is an explicit expression for the upper bound of Otto efficiency for the coupled system. This expression reduces to the one found for the ($1/2,1/2$) case, with $s=1$ [6], or to the case of coupled, effective two-level systems [18]. The dependence of the average efficiency on coupling factor J and validity of the upper bound is demonstrated in Figure 5.

In addition to the above analytic approaches, we may also numerically study the implications of using higher spins on the performance of thermal machines. In order to make a few observations, we note that the higher $“s”$ values shift the maximum of work to the weak coupling regimes as shown in Figure 6a. Thus, higher magnitudes of spin may be a useful resource to achieve more work output for weak coupling strengths. Numerical analysis also shows that increasing the bath temperatures may increase the work output by orders of magnitude (see Figure 6b). We also observe an extended regime of positive work extraction from the system at high temperatures and this effect is more pronounced for lower “s” values. Along these lines, variations of the efficiency and work output with the coupling factor J may be studied, where $s_1$ and $s_2$ are varied for a fixed s value. Figure 7 shows different cases for the case of $s=7/2$. Note that ${\eta }_{ub}$ and $J_c$ (which depend on s and not on the values of individual spins) are the same for a given s.

Finally, other possible domains of operation such as the refrigerator and accelerator may be addressed by using the techniques explored in this paper. The study of local thermodynamics of individual spins relative to the global performance, and other models of coupled spins featuring different interactions are some of the potential avenues of future inquiry.