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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Quantum Otto Cycle

#### 2.1. The Heat Cycle

- 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, $\mathrm{Tr}[\xb7]$ denotes the trace operation, and $\Delta {\rho}_{}={\rho}_{1}-{\rho}_{2}$. In this paper, we consider the free Hamiltonian of the form ${H}_{i}\equiv 2{B}_{i}{h}_{0}$ ($i=1,2$), 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. The Coupled Model

#### 3.1. Majorization

#### 3.2. Energy Level Ordering

## 4. Efficiency Enhancement and the Upper Bound

- 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).

## 5. Complete Otto Cycles

- $\mathbf{x}\ne \mathbf{0},\mathbf{y}=\mathbf{0}$: 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.$${\eta}_{}=\frac{{\eta}_{0}}{1+\frac{8yJ}{x{B}_{1}}}<{\eta}_{0}.$$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.$$\eta =\frac{{\eta}_{0}}{1-\frac{8\left|y\right|J}{x{B}_{1}}}>{\eta}_{0}.$$

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. PWC for the Coupled Model

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

## Appendix B. PWC for the Coupled Model

**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 $2{s}_{1}+1$.

${P}_{1}={e}^{2s{B}_{1}/{T}_{1}}/{Z}_{1}$ |
---|

${P}_{2}={e}^{2(s-1){B}_{1}/{T}_{1}+8sJ/{T}_{1}}/{Z}_{1}$ |

${P}_{3}={e}^{2(s-1){B}_{1}/{T}_{1}}/{Z}_{1}$ |

${P}_{4}={e}^{2(s-2){B}_{1}/{T}_{1}+8sJ/{T}_{1}+8J(s-1)/{T}_{1}}/{Z}_{1}$ |

${P}_{5}={e}^{2(s-2){B}_{1}/{T}_{1}+8sJ/{T}_{1}}/{Z}_{1}$ |

${P}_{6}={e}^{2(s-2){B}_{1}/{T}_{1}}/{Z}_{1}$ |

. |

. |

. |

${P}_{n/2-2{s}_{1}}={e}^{{B}_{1}/{T}_{1}+8sJ/{T}_{1}+...+8J(s-\left(\right)open="("\; close=")">2{s}_{1}-1)}$ |

. |

. |

. |

${P}_{n/2}={e}^{{B}_{1}/{T}_{1}}/{Z}_{1}$ |

${P}_{n/2+1}={e}^{-{B}_{1}/{T}_{1}+8sJ/{T}_{1}+...+8J(s-\left(\right)open="("\; close=")">2{s}_{1}-1)}$ |

. |

. |

. |

${P}_{n/2+2{s}_{1}+1}={e}^{-{B}_{1}/{T}_{1}}/{Z}_{1}$ |

. |

. |

. |

${P}_{n-5}={e}^{-2(s-2){B}_{1}/{T}_{1}+8sJ/{T}_{1}+8(s-1)J/{T}_{1}}/{Z}_{1}$ |

${P}_{n-4}={e}^{-2(s-2){B}_{1}/{T}_{1}+8sJ/{T}_{1}}/{Z}_{1}$ |

${P}_{n-3}={e}^{-2(s-2){B}_{1}/{T}_{1}}/{Z}_{1}$ |

${P}_{n-2}={e}^{-2(s-1){B}_{1}/{T}_{1}+8sJ/{T}_{1}}/{Z}_{1}$ |

${P}_{n-1}={e}^{-2(s-1){B}_{1}/{T}_{1}}/{Z}_{1}$ |

${P}_{n}={e}^{-2s{B}_{1}/{T}_{1}}/{Z}_{1}$ |

- (a)
- ${B}_{2}<{B}_{1}\theta $, which implies the following:
- 1
- ${L}_{X}\equiv ({P}_{1}^{{}^{\prime}}-{P}_{1})+({P}_{n}-{P}_{n}^{{}^{\prime}})<0$;
- 2
- $X<0$, thereby proving that under ${B}_{2}<{B}_{1}\theta $, it is not possible for the coupled system to work as an engine at all.

- (b)
- ${B}_{2}>{B}_{1}\theta $, which implies the following:
- 1
- ${L}_{X}$ does not bear a definite sign. Although the term (${P}_{n}-{P}_{n}^{{}^{\prime}}$) in ${L}_{X}$ is positive definite, yet the sign of the other term (${P}_{1}^{{}^{\prime}}-{P}_{1}$) is not definite;
- 2
- Under WCS, we are able to prove $X>0$ for ${B}_{2}>{B}_{1}\theta $, thereby implying that it is a necessary condition for ${W}_{av}>0$. However, WCS also demands ${P}_{1}^{{}^{\prime}}>{P}_{1}$ or $0<J<{J}_{c}$. Therefore, the latter constitutes a sufficient condition for ${W}_{av}>0$.
- 3
- When ${P}_{1}^{{}^{\prime}}>{P}_{1}$ does not hold, ${L}_{X}$ does not have definite sign. Thus, depending on the control parameters, other terms in X can be positive. In this case, we cannot predict the sign of ${W}_{av}$.

- a
- If ${L}_{X}<0$ (which happens for ${B}_{2}<{B}_{1}\theta $), then ${W}_{av}<0$.
- b
- If ${L}_{X}>0$ (which happens for ${B}_{2}>{B}_{1}\theta $ and $0<J<{J}_{c}$), then ${W}_{av}>0$.
- c
- If no definite sign can be assigned to ${L}_{X}$ (which may happen even when ${B}_{2}>{B}_{1}\theta $ holds, but with no condition on the range of J), the system may or may not work as an engine.

## Appendix C. Condition on J from W_{av} > 0

**Table A2.**Spin dependent factors ${m}_{1}$ and ${m}_{2}$ when the energy eigenvalues of the coupled system are expressed as the following: ${E}_{k}={m}_{1}B-8{m}_{2}J$. 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 |

$-2(s-1)$ | $s,0$ | $2,3$ |

$-2(s-2)$ | $[s+(s-1\left)\right],s,0$ | $4,5,6$ |

$-2(s-3)$ | $[s+(s-1)+(s-2\left)\right],[s+(s-1\left)\right],s,0$ | $7,..,10$ |

. | . | . |

. | . | . |

$-2(s-r)$ | $[s+(s-1)+...+(s-(2{s}_{1}-1))],...,0$ | $n/2-2{s}_{1},...,n/2$ |

$2(s-r)$ | $[s+(s-1)+...+(s-(2{s}_{1}-1))],...,0$ | $n/2+1,...,n/2+2{s}_{1}+1$ |

. | . | . |

. | . | . |

$2(s-3)$ | $[s+(s-1)+(s-2\left)\right],[s+(s-1\left)\right],s,0$ | $(n-9),..,(n-6)$ |

$2(s-2)$ | $[s+(s-1\left)\right],s,0$ | $(n-5),(n-4),(n-3)$ |

$2(s-1)$ | $s,0$ | $(n-2),(n-1)$ |

$2s$ | 0 | n |

## Appendix D. Proof for X > Y_{1}

- We first consider the lower half levels. With ${m}_{1}$ being negative for all $k=1,...,n/2$ (see Table A2), the total contribution from these levels to X takes the following form.$$\frac{1}{2}\sum _{k=1}^{n/2}\left|{m}_{1}\right|({P}_{k}^{{}^{\prime}}-{P}_{k})$$Now, we add these to obtain the coefficients of these terms in U, denoted by ${m}_{3}\equiv \frac{|{m}_{1}|}{2}+\frac{{m}_{2}}{s}$, which have been listed in Table A3. As can be observed, ${m}_{3}$ has a positive part given by "s" and a negative part, say ${m}_{4}$. The total contribution from the lower half levels to U is therefore written as follows.$$\sum _{k=1}^{n/2}\left(\right)open="("\; close=")">\frac{|{m}_{1}|}{2}+\frac{{m}_{2}}{s}$$$$=\sum _{k=1}^{n/2}(s+{m}_{4})({P}_{k}^{{}^{\prime}}-{P}_{k})=s\sum _{k=1}^{n/2}({P}_{k}^{{}^{\prime}}-{P}_{k})+\sum _{k=1}^{n/2}{m}_{4}({P}_{k}^{{}^{\prime}}-{P}_{k})$$With ${m}_{4}<0$, the second part is positive because of Equation (A29), and the first part is considered later on.

**Table A3.**Coefficients ${m}_{3}$ of the terms (${P}_{k}^{{}^{\prime}}-{P}_{k}$) in U with k varying from $1,2,...,n/2$.

k | ${\mathit{m}}_{3}=\mathit{s}+{\mathit{m}}_{4}$ |
---|---|

1 | s |

$2,3$ | $s,(s-1)$ |

$4,5,6$ | $(s-\frac{1}{s}),(s-1),(s-2)$ |

$7,8,9,10$ | $(s-\frac{1}{s}-\frac{2}{s}),(s-1-\frac{1}{s}),(s-2),(s-3)$ |

. | . |

. | . |

$(n/2-2{s}_{1}),...,n/2$ | $\left(\right)open="["\; close="]">s-(r-2{s}_{1})-\frac{1}{s}-...-\frac{\left(\right)}{2}s,...,(s-r)$ |

- 2.
- We now consider the upper half levels. The total contribution of these levels to X and $Y/s$ is considered separately. The former is given as the following.$$\sum _{k=n/2+1}^{n}\frac{{m}_{1}}{2}({P}_{k}^{}-{P}_{k}^{{}^{\prime}})$$With ${m}_{1}$ being positive (Table A2) for all $k=n/2+1,..,n$, the above expression is positive because of Equation (A29).As for these levels’ contribution to $Y/s$, it is given as the following.$$\sum _{k=n/2+1}^{n-2}\frac{{m}_{2}}{s}({P}_{k}^{{}^{\prime}}-{P}_{k})\equiv \sum _{k=n/2+1}^{n-2}({m}_{5}+{m}_{6})({P}_{k}^{{}^{\prime}}-{P}_{k}^{})$$$$=\sum _{k=n/2+1}^{n-2}{m}_{5}({P}_{k}^{{}^{\prime}}-{P}_{k}^{})+\sum _{k=n/2+1}^{n-2}{m}_{6}({P}_{k}^{{}^{\prime}}-{P}_{k}^{})$$Note that not all the levels contribute to Y because many levels do not explicitly depend on J. Here, ${m}_{5}$ and ${m}_{6}$ are the positive and negative parts of ${m}_{2}/s$, respectively (see Table A4). The second part in the above equation is positive because of (A29), and the first part is considered later on.

**Table A4.**Coefficients ${m}_{2}/s$ (obtained from Table A2) of the terms (${P}_{k}^{{}^{\prime}}-{P}_{k}$) in $Y/s$ with k running over all upper half energy levels i.e., $k=n/2+1,...,n$.

k | ${\mathit{m}}_{2}/\mathit{s}={\mathit{m}}_{5}+{\mathit{m}}_{6}$ | ${\mathit{m}}_{5}$ |
---|---|---|

$(n/2+1),...,(n/2+2{s}_{1}+1)$ | $2{s}_{1}-\frac{1}{s}-...-\frac{\left(\right)}{2}s$ | $2{s}_{1},...,0$ |

. | . | |

. | . | |

$(n-9),(n-8),(n-7),(n-6)$ | $(3-\frac{1}{s}-\frac{2}{s}),(2-\frac{1}{s}),1,0$ | $3,2,1,0$ |

$(n-5),(n-4),(n-3)$ | $(2-\frac{1}{s}),1,0$ | $2,1,0$ |

$(n-2),(n-1)$ | $1,0$ | $1,0$ |

n | 0 | 0 |

- 3.
- Adding up the total contribution to U from all the energy levels we have the following.$$\sum _{k=1}^{n/2}(s+{m}_{4})({P}_{k}^{{}^{\prime}}-{P}_{k})+\sum _{k=n/2+1}^{n}\frac{{m}_{1}}{2}({P}_{k}^{}-{P}_{k}^{{}^{\prime}})+\sum _{k=n/2+1}^{n-2}({m}_{5}+{m}_{6})({P}_{k}^{{}^{\prime}}-{P}_{k}^{})$$From the first and second points, we now have two parts which are yet to be proved positive. Their sum is given as follows.$$\sum _{k=1}^{n/2}s({P}_{k}^{{}^{\prime}}-{P}_{k})+\sum _{k=n/2+1}^{n-2}{m}_{5}({P}_{k}^{{}^{\prime}}-{P}_{k}^{})$$Using relations such as ${P}_{n}^{{}^{\prime}}<{P}_{n}$ and ${P}_{n-1}^{{}^{\prime}}<{P}_{n-1}$ (from Equation (A29)) and ${P}_{1}^{{}^{\prime}}>{P}_{1}$ in the normalization condition of the following probabilities:$$\sum _{k=1}^{n}({P}_{k}^{{}^{\prime}}-{P}_{k})=0$$$$\sum _{k=1}^{n/2}s({P}_{k}^{{}^{\prime}}-{P}_{k})+\sum _{k=n/2+1}^{n-2}s({P}_{k}^{{}^{\prime}}-{P}_{k}^{})>0.$$As shown below, ${m}_{5}<s$. Therefore, with ${P}_{k}^{{}^{\prime}}<{P}_{k}$, we can safely replace s by ${m}_{5}$ in the above inequality, thereby proving $U>0$.

#### Proof for **m**_{5} < **s**

_{5}

**Table A5.**Spin dependent factors ${m}_{1}$ and ${m}_{2}$ for the $(1/2,1)$ coupled system when the energy eigenvalues are expressed as: ${E}_{k}={m}_{1}B-8{m}_{2}J$.

${\mathit{m}}_{1}$ | ${\mathit{m}}_{2}$ | k |
---|---|---|

$-2s$ | 0 | 1 |

$-2(s-1)$ | $s,0$ | $2,3$ |

$2(s-1)$ | $s,0$ | $4,5$ |

$2s$ | 0 | 6 |

**Proof.**

- We first consider the lower half ($k=1,2,3$) of the levels. With ${m}_{1}$ being negative (see Table A5), the total contribution from these levels to X takes the form of the following.$$\frac{1}{2}\sum _{k=1}^{3}\left|{m}_{1}\right|({P}_{k}^{{}^{\prime}}-{P}_{k})=s({P}_{1}^{{}^{\prime}}-{P}_{1})+(s-1)({P}_{2}^{{}^{\prime}}-{P}_{2}+{P}_{3}^{{}^{\prime}}-{P}_{3})$$Similarly, contribution of lower half levels to $Y/s$ is written as follows.$$Y/s=\frac{{m}_{2}}{s}({P}_{2}^{{}^{\prime}}-{P}_{2})=({P}_{2}^{{}^{\prime}}-{P}_{2}).$$The total contribution of the lower half levels to U is as follows.$$U=X+Y/s=s({P}_{1}^{{}^{\prime}}-{P}_{1})+(s-1)({P}_{2}^{{}^{\prime}}-{P}_{2}+{P}_{3}^{{}^{\prime}}-{P}_{3})+({P}_{2}^{{}^{\prime}}-{P}_{2})$$$$U=s({P}_{1}^{{}^{\prime}}-{P}_{1})+s({P}_{2}^{{}^{\prime}}-{P}_{2})+(s-1)({P}_{3}^{{}^{\prime}}-{P}_{3})=\sum _{k=1}^{3}s({P}_{k}^{{}^{\prime}}-{P}_{k})+(-1)({P}_{3}^{{}^{\prime}}-{P}_{3})$$The second part is positive because of Equation (A29), and the first part is considered later on.
- We now consider the upper half levels. The total contribution of these levels to X and $Y/s$ is considered separately. The former is given as follows.$$\sum _{k=4}^{6}\frac{{m}_{1}}{2}({P}_{k}^{}-{P}_{k}^{{}^{\prime}})=(s-1)({P}_{4}-{P}_{4}^{{}^{\prime}}+{P}_{5}-{P}_{5}^{{}^{\prime}})+s({P}_{6}^{}-{P}_{6}^{{}^{\prime}})$$The above expression is positive because of Equation (A29).As for these levels’ contribution to $Y/s$, it is given as the following.$$Y/s=\frac{{m}_{2}}{s}({P}_{4}^{{}^{\prime}}-{P}_{4})=({P}_{4}^{{}^{\prime}}-{P}_{4})$$This part is negative and will be considered later on.
- From points one and two the following terms in U are yet to be shown positive.$$\sum _{k=1}^{3}s({P}_{k}^{{}^{\prime}}-{P}_{k})+({P}_{4}^{{}^{\prime}}-{P}_{4})$$Using relations such ${P}_{6}^{{}^{\prime}}<{P}_{6}$, ${P}_{5}^{{}^{\prime}}<{P}_{5}$ (from Equation (A29)) and ${P}_{1}^{{}^{\prime}}>{P}_{1}$ in the normalization condition of probabilities, given as the following:$$\sum _{k=1}^{6}({P}_{k}^{{}^{\prime}}-{P}_{k})=0$$$$\sum _{k=1}^{4}s({P}_{k}^{{}^{\prime}}-{P}_{k})>0\Rightarrow \sum _{k=1}^{3}s({P}_{k}^{{}^{\prime}}-{P}_{k})+s({P}_{4}^{{}^{\prime}}-{P}_{4})>0$$With ${P}_{4}^{{}^{\prime}}<{P}_{4}$ and $1<s$ ($s=3/2$ for the present case), we can safely replace s by 1 in the the above expression, thereby proving $U>0$.

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**Figure 1.**The above figure shows the energy levels of the two-spins (${s}_{1},{s}_{2}$) system for (

**a**) $J=0$ and (

**b**) $J>0$. The degeneracy in the energy levels is lifted as the interaction is switched on ($J\ne 0$). Here, ${s}_{1}<{s}_{2}$ and $s={s}_{1}+{s}_{2}$.

**Figure 2.**(

**a**) Variation of ${P}_{k}^{}-{P}_{k}^{{}^{\prime}}$ with the coupling factor J for $(1/2,1)$ system, with k values ranging from 2 to 6. The parameters are set at ${B}_{1}=4,{B}_{2}=3$, with temperatures (

**a**) ${T}_{1}=4,{T}_{2}=2$ and (

**b**) ${T}_{1}=6,{T}_{2}=3$. Here, ${J}_{c}=1/3$. The value of J for which ${P}_{2}^{}-{P}_{2}^{{}^{\prime}}$ (red curve) changes sign (from positive to negative) approaches ${J}_{c}$ for lower temperatures (see also Figure 3).

**Figure 3.**Majorization conditions shown by positivity of all quantities ${P}_{6}^{}-{P}_{6}^{{}^{\prime}}$ (purple), ${\sum}_{k=5}^{6}{P}_{k}^{}-{P}_{k}^{{}^{\prime}}$ (green), ${\sum}_{k=4}^{6}{P}_{k}^{}-{P}_{k}^{{}^{\prime}}$ (blue), ${\sum}_{k=3}^{6}{P}_{k}^{}-{P}_{k}^{{}^{\prime}}$ (brown) and ${\sum}_{k=2}^{6}{P}_{k}^{}-{P}_{k}^{{}^{\prime}}$ (red) as function of the coupling strength J for ($1/2,1$) system of Figure 2. The parameters are set at ${B}_{1}=4,{B}_{2}=3$, with temperatures (

**a**) ${T}_{1}=4,{T}_{2}=2$ and (

**b**) ${T}_{1}=6,{T}_{2}=3$. The point where the red curve intersects the lower curve is where ${P}_{2}^{{}^{\prime}}={P}_{2}$. It is observed that for higher bath temperatures (for a given ratio ${T}_{2}/{T}_{1}$), this point shifts to lower J values.

**Figure 4.**Schematic of a complete Otto cycle (COC) as an engine using two heat reservoirs (${T}_{1}>{T}_{2}$), involving two energy levels of the working medium. The heat absorbed from the hot reservoir is ${q}_{1}={e}_{f}-{e}_{i}$, while the heat rejected to the cold bath is ${q}_{2}={e}_{f}^{{}^{\prime}}-{e}_{i}^{{}^{\prime}}$. The work extracted per complete cycle is $w={q}_{1}-{q}_{2}$.

**Figure 5.**Variation of efficiency (solid lines) for different values of spin ${s}_{2}$, with ${s}_{1}=1/2,{B}_{1}=4,{B}_{2}=3,{T}_{1}=1$ and ${T}_{2}=0.5$. The corresponding upper bounds (${\eta}_{ub}$) have been shown by dashed lines. The uncoupled efficiency (${\eta}_{0}$) is shown by the horizontal black line.

**Figure 6.**Variation of extracted work with coupling strength J for different spin combinations (${s}_{1},{s}_{2}$). The fields are set at values ${B}_{1}=4,{B}_{2}=3$, and the bath temperatures are as follows: (

**a**) ${T}_{1}=1$,${T}_{2}=0.5$; (

**b**) ${T}_{1}=6,{T}_{2}=3$.

**Figure 7.**Variation of (

**a**) efficiency and (

**b**) work with coupling strength J for different spin combinations (${s}_{1},{s}_{2}$), where $s=7/2$ is held fixed. The solid pink, green and brown lines show the variation for $(1/2,3),(1,5/2)$ and $(3/2,2)$ cases, respectively. The parameters are set at values ${B}_{1}=4,{B}_{2}=3$ and ${T}_{1}=4,{T}_{2}=2$. Here, ${J}_{c}=0.142$. The upper bound and the uncoupled efficiency are shown by dashed blue and black lines in (

**a**), respectively. The Carnot efficiency is $0.5$.

**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 | $-2sB$ |

$2,3$ | $-2(s-1)B$ |

$4,5,6$ | $-2(s-2)B$ |

. | . |

. | . |

$(n/2-2{s}_{1}),...,n/2$ | $-2(s-r)B$ |

$(n/2+1),...,(n/2+2{s}_{1}+1)$ | $2(s-r)B$ |

. | . |

. | . |

$(n-5),(n-4),(n-3)$ | $2(s-2)B$ |

$(n-2),(n-1)$ | $2(s-1)B$ |

n | $2sB$ |

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Johal, R.S.; Mehta, V.
Quantum Heat Engines with Complex Working Media, Complete Otto Cycles and Heuristics. *Entropy* **2021**, *23*, 1149.
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Quantum Heat Engines with Complex Working Media, Complete Otto Cycles and Heuristics. *Entropy*. 2021; 23(9):1149.
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2021. "Quantum Heat Engines with Complex Working Media, Complete Otto Cycles and Heuristics" *Entropy* 23, no. 9: 1149.
https://doi.org/10.3390/e23091149