# Measurement-based quantum Otto engine with a two-spin system coupled by
anisotropic interaction: enhanced efficiency at finite times

Chayan Purkait<EMAIL_ADDRESS>Asoka Biswas  Department of Physics,
Indian Institute of Technology Ropar, Rupnagar, Punjab 140001, India

###### Abstract

We have studied the performance of a measurement-based quantum Otto engine
(QOE) in a working system of two spins coupled by Heisenberg anisotropic
interaction. A non-selective quantum measurement fuels the engine. We have
calculated thermodynamic quantities of the cycle in terms of the transition
probabilities between the instantaneous energy eigenstates, and also between
the instantaneous energy eigenstates and the basis states of the measurement,
when the unitary stages of the cycle operate for a finite time $\tau$. The
efficiency attains a large value in the limit of $\tau\rightarrow 0$ and then
gradually reaches the adiabatic value in a long time limit
$\tau\rightarrow\infty$. For finite values of $\tau$ and for anisotropic
interaction, an oscillatory behaviour of the efficiency of the engine is
observed. This oscillation can be interpreted in terms of interference between
the relevant transition amplitudes in the unitary stages of the engine cycle.
Therefore, for a suitable choice of timing of the unitary processes in the
short time regime, the engine can have a higher work output and less heat
absorption, such that it works more efficiently than a quasi-static engine. In
the case of an always-on heat bath, in a very short time the bath has a
negligible effect on its performance.

##  I Introduction

The laws of classical thermodynamics are known to be applicable to the
thermodynamic limit. It is quite interesting to study whether these laws are
valid also in the quantum limit. In this regard, thermal machines (e.g. heat
engines and refrigerators) can be considered as a suitable platform to explore
this issue in quantum systems. Deviations from the classical limit of
efficiency of these machines can be an important marker to understand the
effect of quantum mechanical properties of the system.

In fact, it is rather crucial to explore whether it is possible to enhance the
efficiency of thermal machines by harnessing quantum features such as
coherence, many-body correlations, and non-thermal population distributions.
There have been several studies in different types of quantum heat engine
(QHE) models to show that quantum coherence is indeed beneficial to achieve an
enhanced performance of the QHEsLatune et al. (2021); Mitchison et al. (2015);
Scully et al. (2003); Shi et al. (2020); Latune et al. (2019); Brandner et al.
(2015); Uzdin et al. (2015); Uzdin (2016); Brandner et al. (2017); Scully et
al. (2011); Rahav et al. (2012). Roles of quantum correlation and entanglement
Brunner et al. (2014); Altintas et al. (2014); Barrios et al. (2017); Altintas
et al. (2015); Hewgill et al. (2018); Zhang et al. (2007), the interaction
within a coupled system Das and Ghosh (2019); Thomas and Johal (2011); Çakmak
et al. (2016); Çakmak et al. (2017a); Altintas and Müstecaplıoğlu (2015);
Ivanchenko (2015); Huang et al. (2020a, 2014), and the non-thermal heat baths
De Chiara and Antezza (2020); Huang et al. (2012); Roßnagel et al. (2014);
Alicki and Gelbwaser-Klimovsky (2015); de Assis et al. (2019); Scully (2001);
Scully et al. (2003) in the performance of quantum thermal machines have been
investigated. It was shown that the efficiency of QHEs can be improved beyond
the Carnot limit using squeezed thermal baths Huang et al. (2012); Roßnagel et
al. (2014); Klaers et al. (2017). But, their efficiency is bounded by a
generalised efficiency limit for QHEs energised by non-thermal heat baths
Niedenzu et al. (2018).

From the time of Maxwell, it was known that work could be extracted from a
single-temperature heat bath using information gained from measurements. This
type of engine is known as Szilard’s engine, in which results of selective
measurement are used to provide feedback on engine operation Maruyama et al.
(2009); Li et al. (2012); Jordan et al. (2019). Recently, it was also shown
that projective measurement of the ground state can be used to mimic the
release of heat from a system to a cold bath during an isochoric process Chand
and Biswas (2017a, b); Chand et al. (2021) in an ion-based QHE. In later
works, quantum measurement has been used to fuel the working system in a QHE,
in which the isochoric heating stage in a standard quantum Otto engine (QOE)
is replaced by a non-selective quantum measurement Yi et al. (2017); Jordan et
al. (2019); Das and Ghosh (2019); Huang et al. (2020b). Therefore, the engine
works with a single heat bath as a heat shrink and non-selective quantum
measurement as a heat source.

A finite-time analysis is also an important aspect of studying QHE, as for
practical applications we need a finite amount of power. Moreover, a QHE in
finite time may show true quantum nature in its performance which may not
possible to observe in the quasistatic performance. Standard QHEs are operated
by Hamiltonians who do not commute at different times Denzler and Lutz (2020);
Dann et al. (2020); Rezek and Kosloff (2006); Lee et al. (2020); Camati et al.
(2019); Çakmak et al. (2017b); Türkpençe and Altintas (2019); Çakmak and
Müstecaplıoğlu (2019); Plastina et al. (2014); Rezek (2010a). Consequently,
quantum internal friction arises when a quantum system is driven unitarily by
an external control parameter in finite time. This induces nonadiabatic
transitions between the instantaneous eigenstates of the Hamiltonian, and also
generates coherence in the energy eigenbasis. As a result, a larger amount of
entropy is produced and irreversibility is increased in engine operation,
which degrades the performance of QHEs Çakmak et al. (2017b); Türkpençe and
Altintas (2019); Çakmak and Müstecaplıoğlu (2019); Plastina et al. (2014);
Rezek (2010a). On the other hand, in the presence of quantum coherence, QHEs
can produce more power output than the classical ones Uzdin et al. (2015);
Uzdin (2016); Brandner et al. (2017); Scully et al. (2011); Rahav et al.
(2012); Dodonov et al. (2018). Power output can be improved by not only the
finite-time unitary stages, but also via the non-Markovian effects during
finite-time bath interaction Abiuso and Giovannetti (2019). In fact, in a very
recent study on a finite-time QOE, it has been shown that coherence can act
like a dynamical quantum lubricant Camati et al. (2019). This can be
interpreted as a dynamical interference effect, which takes place between the
residual coherence after incomplete thermalization and the coherence generated
in the subsequent finite-time unitary driving process.

While the measurement processes and finite-time operation can individually
have substantial effects on the performance of the QHEs, there have been very
few studies when both protocols are used together. It is recently shown that
it is possible to improve the performance of a single-qubit QHE by suitably
choosing the measurement basis such that the degradation effect due to
coherence production in a standard QOE can be overcome Lin et al. (2021). In
this work, we will investigate the finite-time performance of a two-spin QOE
by using a non-selective quantum measurement to fuel the engine.

In our model, two spins are coupled with each other by Heisenberg anisotropic
XY interaction, in presence of external time-varying homogeneous magnetic
fields. By changing the anisotropy parameter one can have the Heisenberg XX or
Ising spin Hamiltonian as a limiting case. The free part and the driving part
of the Hamiltonian do not commute, leading to the non-commuting nature of the
Hamiltonian at two different times. Consequently, it initiates transitions
between the instantaneous energy eigenstates, and introduces quantum features
into the performance of a QHE through unitary driving processes in finite
time. In this paper, we aim to investigate if a measurement-based engine
operating in finite time performs better than when operated quasistatically
and we show that it is indeed so. The corresponding enhancement of engine
efficiency can be attributed to the anisotropy in the system and the use of
the measurement protocol. Such an enhancement could not be achieved, if the
engine was fueled by a heat bath, instead of a non-selective measurement. We
will also show that even if the spins remain coupled with a heat bath
throughout the cycle (including the stages, when the magnetic fields are
varied), it has a negligible effect on the engine’s performance, for faster
unitary stages. However, for a longer duration of these stages, coupling to
the bath dominates and the performance of the engine degrades.

The paper is organized as follows. In Sec. II, we introduce the two-spin model
of the working system. We describe different stages of the quantum Otto cycle
and the relevant thermodynamic quantities. Next in Sec. III, we describe the
finite time performance of the cycle. We provide a theoretical analysis of the
thermodynamic quantities in terms of the transition probabilities. We also
compare them with the quasistatic and the sudden limit of work and efficiency.
In Sec. IV, we study the case when the thermal bath continuously interacts
with the spins, even when the magnetic field is changed. We conclude the paper
in Sec. V.

##  II Quantum heat engine model

In this section, we will briefly introduce our model of the QOE.

###  II.1 System model

We consider a system of two spins coupled by Heisenberg anisotropic XY
interaction in a transverse magnetic field [$B(t)\geq 0$]. The Hamiltonian is
represented by Çakmak and Müstecaplıoğlu (2019); Cherubim et al. (2022);
Suzuki et al. (2012)

$\hat{H(t)}=\hat{H_{0}}(t)+\hat{H_{I}},$ (1)

where,

$\displaystyle\hat{H_{0}}$ $\displaystyle=$ $\displaystyle
B(t)(\hat{\sigma}_{1}^{z}+\hat{\sigma}_{2}^{z})$ (2)
$\displaystyle\hat{H_{I}}$ $\displaystyle=$ $\displaystyle
J[(1+\gamma)~{}\hat{\sigma}_{1}^{x}\hat{\sigma}_{2}^{x}+(1-\gamma)~{}\hat{\sigma}_{1}^{y}\hat{\sigma}_{2}^{y}].$

Here $\hat{H}_{0}$ is the free part of the Hamiltonian, and $\hat{H}_{I}$
represents the interaction between two spins with $\gamma\in[-1,1]$ Kamta and
Starace (2002); Yeo et al. (2005) as the anisotropy parameter, and $J$ is the
coupling constant between the spins. The operators $\hat{\sigma}_{i}^{x,y,z}$
are the standard Pauli matrices for the $i$th ($i\in 1,2$) spin. If
$\gamma=0$, the Hamiltonian becomes Heisenberg isotropic XX type, and for
$\gamma=\pm 1$, this becomes the Ising spin Hamiltonian. For $\gamma\neq 0$,
$\hat{H}_{I}$ and $\hat{H}_{0}$ do not commute, which in turn gives rise to
$[\hat{H}(t),\hat{H}(t^{\prime})]\neq 0$. This further indicates that we may
see quantum behaviour in finite-time engine operation Rezek (2010b).

The eigenvectors and the corresponding eigenvalues of the Hamiltonian
$\hat{H}(t)$ are given by

$\begin{array}[]{l}\ket{\psi_{0}}=\frac{1}{\sqrt{2}}(\frac{B-K}{\sqrt{K^{2}-BK}}\ket{11}+\frac{\gamma
J}{\sqrt{K^{2}-BK}}\ket{00}),~{}E_{0}=-2K\\\
\ket{\psi_{1}}=\frac{1}{\sqrt{2}}(-\ket{10}+\ket{01}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E_{1}=-2J\\\
\ket{\psi_{2}}=\frac{1}{\sqrt{2}}(\ket{10}+\ket{01}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E_{2}=2J\\\
\ket{\psi_{3}}=\frac{1}{\sqrt{2}}(\frac{B+K}{\sqrt{K^{2}+BK}}\ket{11}+\frac{\gamma
J}{\sqrt{K^{2}+BK}}\ket{00}),~{}E_{3}=2K\end{array}$ (3)

where $K=\sqrt{B^{2}+\gamma^{2}J^{2}}$. These energy eigenstates can be
divided into two categories. The state that are dependent on the system
parameters $B(t)$ and $J$, namely, $\left|\psi_{0}\right\rangle$ and
$\left|\psi_{3}\right\rangle$, evolve with time. The other ones which are
independent of the system parameters, namely $\left|\psi_{1}\right\rangle$ and
$\left|\psi_{2}\right\rangle$ are the standard Bell states that remain
unchanged with time. We will show in this work that the former ones play a
fundamental role in the behaviour of the measurement-based cycle. Note that in
the limit of $\gamma=0$, the eigenstates $|\psi_{0,3}\rangle$ take the form of
product (i.e., disentangled) states, with the respective eigenvalues $\mp 2B$.
[see App. A].

Figure 1: Schematic diagram of the Otto cycle

###  II.2 Quantum Otto cycle and thermodynamic quantities

We consider that the working system undergoes an Otto cycle. The schematic
diagram of the cycle is shown in the Fig. 1. The strokes of the cycle are
described below.

Unitary expansion (A to B): The working system is initially prepared in a
thermal state $\hat{\rho}_{A}=e^{-\beta\hat{H}_{1}}/Z$ at inverse temperature
$\beta=1/T\left(k_{B}=1\right)$, with $\hat{H}_{1}=\hat{H}(0)$ and
$Z=\operatorname{Tr}(e^{-\beta\hat{H}_{1}})$. During this stage of the cycle,
the system is decoupled from the heat bath and the external magnetic field is
changed from $B_{1}$ to $B_{2}$ during a finite time-interval $\tau$. We
choose a linear ramp for this change: $B(t)=B_{1}+(B_{2}-B_{1})(t/\tau)$,
where $0\leqslant t\leqslant\tau$. The state of the working system at the end
of this stage changes to
$\hat{\rho}_{B}=\hat{U}(\tau)\hat{\rho}_{A}\hat{U}^{\dagger}(\tau)$, where
$\hat{U}(\tau)=\mathcal{T}\exp[-\iota\int_{0}^{\tau}dt\hat{H}(t)]$ is the
relevant time evolution operator, with $\mathcal{T}$ indicating the time-
ordering. Also, a certain amount of work, $W_{1}$ is done by the system, which
can be calculated as $W_{1}=\langle E_{B}\rangle-\langle E_{A}\rangle$, where
$\langle E_{A}\rangle=\operatorname{Tr}(\hat{\rho}_{A}\hat{H}_{1})$ and
$\langle E_{B}\rangle=\operatorname{Tr}(\hat{\rho}_{B}\hat{H}_{2})$ indicate
the expectation values of the internal energies of the system at the start and
the end of this stage. Note that $\hat{H}_{2}=\hat{H}(\tau)$.

Isochoric heating (B to C): The heating of a system can be generally
understood to be associated with an increase in its entropy. Usually, a system
is heated using a heat bath. This can be alternatively achieved by applying a
non-selective quantum measurement on the working system. In order to ensure
that the energy supplied by this measurement is nonzero, the measurement
operator $\hat{M}$ should not commute with the Hamiltonian, i.e.
$[\hat{H}\left(B_{2}\right),\hat{M}]\neq 0$. If $\hat{\rho}$ is the state
before the measurement, the post-measurement state is usually written as
$\sum_{\alpha}\hat{M}_{\alpha}\hat{\rho}\hat{M}_{\alpha}$, where
$\hat{M}_{\alpha}=|M_{\alpha}\rangle\langle M_{\alpha}|$ is the projection
operator associated to the non-degenerate eigenvalues of the observable $M$
with eigenstates $|M_{\alpha}\rangle$, satisfying
$\hat{M}_{\alpha}^{\dagger}=\hat{M}_{\alpha}$ and
$\sum_{\alpha}\hat{M}_{\alpha}^{2}=\mathds{1}$. In the present case, we
perform a global measurement of the state of the system Huang et al. (2020b),
in the Bell basis
$\\{\left|\psi_{\pm}\right\rangle=\frac{1}{\sqrt{2}}(|00\rangle\pm|11\rangle)$,
$\left|\phi_{\pm}\right\rangle=\frac{1}{\sqrt{2}}(|01\rangle\pm|10\rangle)\\}$.
The leads to a post-measurement state given by
$\hat{\rho}_{C}=\sum_{\alpha=1}^{4}\hat{M}_{\alpha}\hat{\rho}_{B}\hat{M}_{\alpha}$.
where $\hat{M}_{\alpha}$ describes the relevant projection operators as
follows:
$\hat{M}_{1,2}=\left|\psi_{\pm}\right\rangle\left\langle\psi_{\pm}\right|$ and
$\hat{M}_{3,4}=\left|\phi_{\pm}\right\rangle\left\langle\phi_{\pm}\right|$.
During this stage, the entropy of the system increases due to its interaction
with the measurement apparatus and this increase can be considered equivalent
to heating. The corresponding heat ‘absorbed’ can be calculated as
$Q_{M}=\langle E_{C}\rangle-\langle E_{B}\rangle$, where the internal energy
$\langle E_{C}\rangle=\operatorname{Tr}(\hat{\rho}_{C}\hat{H}_{2})$.

Unitary compression (C to D): The working system remains decoupled from the
heat bath in this stage. The magnetic field is driven from $B_{2}$ to $B_{1}$
in a finite time $\tau$ using the protocol $B(\tau-t)$. The state of the
working system at the end of this stage becomes
$\hat{\rho}_{D}=\hat{V}(\tau)\hat{\rho}_{C}\hat{V}^{\dagger}(\tau)$, where
$\hat{V}(\tau)=\mathcal{T}\exp[-\iota\int_{0}^{\tau}dt\hat{H}(\tau-t)]$ is the
time evolution operator. A certain amount of work, $W_{2}$, is done on the
system, which can be calculated as $W_{2}=\langle E_{D}\rangle-\langle
E_{C}\rangle$, where the internal energy $\langle
E_{D}\rangle=\operatorname{Tr}(\hat{\rho}_{D}\hat{H}_{1})$.

Isochoric cooling (D to A): During this final stage of the cycle, the system
is now coupled with a heat bath at the temperature $T$, whereas the magnetic
field remains fixed at $B_{1}$. The system releases some amount of heat
$Q_{L}$ to the bath, which can be calculated as $Q_{L}=\langle
E_{A}\rangle-\langle E_{D}\rangle$. We assume that this process is carried out
over a long time so that the system reaches thermal equilibrium with the bath.

Total work done in a complete cycle can be calculated as
$W=(W_{1}+W_{2})=-(Q_{M}+Q_{L})$. If $W<0$, then the total work in a complete
cycle is done by the working system. Also, the working system absorbs some
amount of heat in the measurement process, if $Q_{M}>0$. Then, the working
system in a complete cycle works as a heat engine. So, the efficiency of the
engine is given by $\eta=|W|/Q_{M}$.

##  III Finite time operation of the engine

Usually, quantum heat engines are studied quasistatically. If we allow
different stages of the engine cycle only for finite times, the performance of
the engine is expected to deviate substantially from the steady state. We show
in Fig. 2, how the efficiency varies with respect to the duration of the
unitary stages. We assume that each of these stages (unitary expansion and
compression) occurs for the same duration $\tau$. All simulations are done
using QuTip Johansson et al. (2012) software package.

Figure 2: Efficiency as a function of duration $\tau$ of the unitary stages,
for different values of the anisotropy parameter $\gamma=0$ (dash-dotted red
line), $\gamma=0.3$ (point-marked magenta line), $\gamma=0.6$ (dotted green
line), $\gamma=1$ (solid blue line). The other parameters are $B_{1}=1$,
$B_{2}=2$, $T=1$. All the quantities are dimensionless with respect to $J$ and
also we have used $k_{B}=\hbar=1$.

As seen in the Fig. 2, the efficiency oscillates at the transient time-scale
for $\gamma\neq 0$. This means that if the unitary stages are executed for a
very short time $\tau\gtrsim 0$, the efficiency can be larger or smaller than
that obtained for a large value of $\tau$. If the unitary processes are
prolonged, the oscillation in efficiency disappears. Thus, a finite-time
measurement-based engine can perform better than the same engine operating for
a longer duration, for a suitable selection of the duration $\tau$ for unitary
processes.

Note that, if one would use a local measurement, instead of global ones,
similar oscillatory behaviour in the efficiency of the engine could be seen,
for finite-time operation Das and Ghosh (2019); Huang et al. (2020a). Also in
these cases, the engine performs better than its quasistatic counterpart at
finite times for specific choices of the local basis.

###  III.1 Thermodynamic quantities in terms of transition probabilities

The results as mentioned above can further be analyzed in terms of the
transition probabilities between the instantaneous eigenstates of the
Hamiltonian. The internal energies (derivations are given in the App B) of the
system at four vertices of the QHE diagram in Fig. 1, are given by

$\displaystyle\langle E_{A}\rangle=-4K_{1}\frac{\sinh
2K_{1}\beta}{Z}-4J\frac{\sinh 2J\beta}{Z}\;,$ (4) $\displaystyle\langle
E_{B}\rangle=-4K_{2}(1-2\xi)\frac{\sinh 2K_{1}\beta}{Z}-4J\frac{\sinh
2J\beta}{Z}\;,$ $\displaystyle\langle
E_{C}\rangle=-4K_{2}(1-2\delta)(1-2\chi)\frac{\sinh
2K_{1}\beta}{Z}-4J\frac{\sinh 2J\beta}{Z}\;,$ $\displaystyle\langle
E_{D}\rangle=-4K_{1}(1-2\delta)(1-2\lambda)\frac{\sinh
2K_{1}\beta}{Z}-4J\frac{\sinh 2J\beta}{Z}\;,$

where $Z=2\cosh(2K_{1}\beta)+2\cosh(2J\beta)$ is the partition function,
$K_{1}=\sqrt{B_{1}^{2}+\gamma^{2}J^{2}}$,
$K_{2}=\sqrt{B_{2}^{2}+\gamma^{2}J^{2}}$,
$\xi=|\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}$,
$\delta=|\langle\psi_{+}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle|^{2}$,
$\chi=|\langle\psi_{0}^{(2)}|\psi_{+}\rangle|^{2}$, and
$\lambda=|\langle\psi_{3}^{(1)}|\hat{V}(\tau)|\psi_{-}\rangle|^{2}$. Clearly,
$\xi$ accounts for the transition probability between two different
eigenstates $|\psi_{3}\rangle$ and $|\psi_{0}\rangle$ during the unitary
expansion. Also, because the instantaneous energy eigenstates
$|\psi_{0,3}\rangle$ do not truly coincide with the measurement basis states
$|\psi_{\pm}\rangle$, their nonzero overlap gives rise to certain transition
between them during measurement and unitary compression stages of the cycle.
This can be seen by rewriting the states $|\psi_{\pm}\rangle$ in terms of the
instantaneous energy eigenstates, as

$\displaystyle|\psi_{+}\rangle=-\frac{c_{2}-d_{2}}{a_{2}d_{2}-b_{2}c_{2}}|\psi_{0}^{(2)}\rangle+\frac{a_{2}-b_{2}}{a_{2}d_{2}-b_{2}c_{2}}|\psi_{3}^{(2)}\rangle$
(5)
$\displaystyle|\psi_{-}\rangle=-\frac{c_{2}+d_{2}}{a_{2}d_{2}-b_{2}c_{2}}|\psi_{0}^{(2)}\rangle+\frac{a_{2}+b_{2}}{a_{2}d_{2}-b_{2}c_{2}}|\psi_{3}^{(2)}\rangle,$

where

$\displaystyle
a_{2}=\frac{B_{2}-K_{2}}{\sqrt{K_{2}^{2}-B_{2}K_{2}}},b_{2}=\frac{\gamma
J}{\sqrt{K_{2}^{2}-B_{2}K_{2}}},c_{2}=\frac{B_{2}+K_{2}}{\sqrt{K_{2}^{2}+B_{2}K_{2}}},d_{2}=\frac{\gamma
J}{\sqrt{K_{2}^{2}+B_{2}K_{2}}}\;.$ (6)

Then the relevant transition probabilities can be written as

$\displaystyle\delta=\left|-\frac{c_{2}-d_{2}}{a_{2}d_{2}-b_{2}c_{2}}\langle\psi_{0}^{(2)}|U(\tau)|\psi_{0}^{(1)}\rangle+\frac{a_{2}-b_{2}}{a_{2}d_{2}-b_{2}c_{2}}~{}\langle\psi_{3}^{(2)}|U(\tau)|\psi_{0}^{(1)}\rangle\right|^{2}\;,$
(7)
$\displaystyle\lambda=\left|-\frac{c_{2}+d_{2}}{a_{2}d_{2}-b_{2}c_{2}}\langle\psi_{3}^{(1)}|V(\tau)|\psi_{0}^{(2)}\rangle+\frac{a_{2}+b_{2}}{a_{2}d_{2}-b_{2}c_{2}}\langle\psi_{3}^{(1)}|V(\tau)|\psi_{3}^{(2)}\rangle\right|^{2}\;,$

and

$\chi=\left|\frac{c_{2}-d_{2}}{a_{2}d_{2}-b_{2}c_{2}}\right|^{2}.$ (8)

The expressions of the work can be obtained using the Eq. 4 as

$W_{1}=4[K_{1}-K_{2}(1-2\xi)]\frac{\sinh 2K_{1}\beta}{Z}$

and

$W_{2}=4(K_{2}-K_{1})(1-2\delta)(1-2\lambda)\frac{\sinh 2K_{1}\beta}{Z}.$

Thus, the total work in a complete cycle is given by

$W=W_{1}+W_{2}=-4[K_{2}\\{(1-2\xi)-(1-2\delta)(1-2\chi)\\}-K_{1}\\{1-(1-2\delta)(1-2\lambda)\\}]\frac{\sinh
2K_{1}\beta}{Z}.$ (9)

Also, the heat ‘absorption’ during the measurement stroke is given by

$Q_{M}=4K_{2}[(1-2\xi)-(1-2\delta)(1-2\chi)]\frac{\sinh 2K_{1}\beta}{Z}.$ (10)

The efficiency of the cycle is therefore given by

$\eta=\frac{|W|}{Q_{M}}=1-\frac{K_{1}[1-(1-2\delta)(1-2\lambda)]}{K_{2}[(1-2\xi)-(1-2\delta)(1-2\chi)]}\;.$
(11)

The plot of the transition probabilities $\xi$, $\delta$, $\chi$, and
$\lambda$ with respect to the duration $\tau$ of the individual unitary
processes are shown in the Fig. 3. Note that $\delta$ and $\lambda$ exhibit
oscillatory dependence on $\tau$, $\chi$ remains constant, while $\xi$
displays a monotonic decay, as $\tau$ increases. Though the transition
probabilities $\delta$ and $\lambda$ are the same, for our choice of the
measurement basis and the eigenstates of the Hamiltonian, it is, generally
speaking, not a universal feature Lin et al. (2021).

The oscillation in the finite time efficiency is primarily due to the
oscillation in the transition probabilities $\delta$ and $\lambda$. The
oscillation in the transition probabilities $\delta$ can be attributed to the
interference between the probability amplitudes for the transitions
$|\psi_{0}^{(1)}\rangle\rightarrow|\psi_{0}^{(2)}\rangle$ and
$|\psi_{0}^{(1)}\rangle\rightarrow|\psi_{3}^{(2)}\rangle$ (see Eq. 7).
Similarly, oscillation in $\lambda$ is due to the transition
$|\psi_{-}\rangle\leftrightarrow|\psi_{3}^{(1)}\rangle$. The other transition
probability $\xi$, also known as quantum internal friction Jiao et al. (2021);
Rezek (2010a), is negligible with respect to the $\delta$, $\chi$ and
$\lambda$ in a measurement-based QHE. As the unitary stages are prolonged,
oscillation in the finite time efficiency disappears and the efficiency
approaches to the quasistatic limit (see Sec. III.2).

Note that the two energy eigenstates $\left|\psi_{1}\right\rangle$ and
$\left|\psi_{2}\right\rangle$ are the same as two Bell states
$\left|\phi_{\pm}\right\rangle$. As these states have been used in measurement
basis, their occupation probabilities do not change in the measurement
process, and therefore, these states do not contribute to the calculation of
heat [see Eq. (7), (8), and (10)]. Moreover, the eigenvalues of these
eigenstates are independent of the external control parameter $B(t)$, and thus
the contribution of these states to work is also zero [see Eq. (9)]. The only
contribution to the engine performance arises from the two other eigenstates
$\left|\psi_{0}\right\rangle$ and $\left|\psi_{3}\right\rangle$.

Figure 3: Transition probabilities as a function of the duration $\tau$ of
each unitary stage. We have used the left y-axis for transition probabilities
$\delta$ and $\lambda$ (solid black line), and $\chi$ (dash-dotted black line)
and the right y-axis for transition probability $\xi$ (point-marked red line).
The other parameters are $B_{1}=1$, $B_{2}=2$, $\gamma=1$.

###  III.2 Quasistatic (adiabatic) limit of the thermodynamic quantities

In order to calculate the quasistatic value of the efficiency, we consider
that the unitary processes are performed quasistatically, i.e., for an
infinite time interval. Therefore, there are no nonadiabatic transitions
between two instantaneous energy eigenstates, and the unitary processes become
adiabatic. So, in such limit, we can write,
$\xi=|\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}\stackrel{{\scriptstyle\tau\rightarrow\infty}}{{=}}0$
(Fig. 3). Also, the transition probabilities between the instantaneous energy
eigenstates and the basis states of measurement take the following forms for
very large $\tau$:

$\delta\stackrel{{\scriptstyle\tau\rightarrow\infty}}{{=}}\left|\frac{c_{2}-d_{2}}{a_{2}d_{2}-b_{2}c_{2}}\right|^{2}\;\;,\;\;\;\lambda\stackrel{{\scriptstyle\tau\rightarrow\infty}}{{=}}\left|\frac{a_{2}+b_{2}}{a_{2}d_{2}-b_{2}c_{2}}\right|^{2}.$

Using the expressions of $a_{2},b_{2},c_{2},d_{2}$ (see Eq. 6) we can indeed
find that $\delta=\chi=\lambda=\frac{1}{2}-\gamma J/2K_{2}$, at the
quasistatic limit (see also Fig. 3).

Thus, the expressions of the work and heat absorption of the cycle can be
obtained as

$\displaystyle W_{q}=-16(K_{2}-K_{1})\chi(1-\chi)\frac{\sinh 2K_{1}\beta}{Z},$
(12) $\displaystyle Q_{Mq}=16K_{2}\chi(1-\chi)\frac{\sinh 2K_{1}\beta}{Z}\;,$

and the quasistatic value of the efficiency is given by

$\eta_{q}=|W_{q}|/Q_{Mq}=1-\frac{K_{1}}{K_{2}}\;.$ (13)

Clearly, the expression of this efficiency is independent of the temperature
of the heat bath used in the cold isochoric process. This indicates that the
performance of the engine does not depend upon the temperature of the heat
bath in the case of global measurement, which we have used in the isochoric
heating stage. We emphasize that this is unlike the case for a local
measurement where the performance of an engine depends upon the temperature of
the heat bath Das and Ghosh (2019). Also, it can be seen from the Eq. 13 that
for nonzero $\gamma$, the expression of the efficiency is very much similar to
the efficiency of a single-spin QHE with two heat baths Dann et al. (2020) or
a single heat bath and a non-selective quantum measurement at the isochoric
heating stage Yi et al. (2017). This similarity arises as only two
intermediate energy levels ($|\psi_{0,3}\rangle$, as mentioned in Sec. II.1)
contribute to the engine performance, due to our specific choice of the
measurement basis. Therefore, a measurement-based heat engine with a coupled
two-spin working system for global measurement acts like a two-level (single-
spin) heat engine, which is evident in the expression of the efficiency (Eq.
13). Interestingly, even a two-stroke QHE made up of two different working
systems with two different frequencies can lead to the same form of efficiency
Piccione et al. (2021). However, the expression of efficiency will differ if
one uses a coupled two-spin working system along with two heat baths or with a
single bath plus local measurement instead of global measurement.

###  III.3 Sudden (quench) limit of the thermodynamic quantities

In order to calculate the sudden limit of the thermodynamic quantities, we
consider that the external magnetic field is changed suddenly
($\tau\rightarrow 0$) from $B_{1}$ to $B_{2}$ or vice versa. In this case
$\hat{U}(\tau),\hat{V}(\tau)\rightarrow\mathds{1}$, therefore the state of the
system does not change over unitary processes. So, in this limit, the
transition probabilities can be written as

$\displaystyle\delta$ $\displaystyle\stackrel{{\scriptstyle\tau\rightarrow
0}}{{=}}$
$\displaystyle\left|-\frac{c_{2}-d_{2}}{a_{2}d_{2}-b_{2}c_{2}}{}~{}\langle\psi_{0}^{(2)}|\psi_{0}^{(1)}\rangle+\frac{a_{2}-b_{2}}{a_{2}d_{2}-b_{2}c_{2}}~{}\langle\psi_{3}^{(2)}|\psi_{0}^{(1)}\rangle\right|^{2}=-\frac{(B_{1}-K_{1}+\gamma
J)^{2}}{4K_{1}(B_{1}-K_{1})},$ $\displaystyle\lambda$
$\displaystyle\stackrel{{\scriptstyle\tau\rightarrow 0}}{{=}}$
$\displaystyle\left|-\frac{c_{2}+d_{2}}{a_{2}d_{2}-b_{2}c_{2}}~{}\langle\psi_{3}^{(1)}|\psi_{0}^{(2)}\rangle+\frac{a_{2}+b_{2}}{a_{2}d_{2}-b_{2}c_{2}}~{}\langle\psi_{3}^{(1)}|\psi_{3}^{(2)}\rangle\right|^{2}=\frac{(B_{1}+K_{1}-\gamma
J)^{2}}{4K_{1}(B_{1}+K_{1})},$ (14) $\displaystyle\xi$
$\displaystyle\stackrel{{\scriptstyle\tau\rightarrow 0}}{{=}}$
$\displaystyle\left|\langle\psi_{0}^{(2)}|\psi_{3}^{(1)}\rangle\right|^{2}=-\frac{\gamma^{2}J^{2}+(B_{1}+K_{1})(B_{2}-K_{2})}{4K_{1}K_{2}(B_{2}-B_{1})(B_{1}+K_{1})}\;.$

Also from the Eq. 8, we get

$\chi=\frac{(B_{2}-K_{2})(B_{2}+K_{2}-\gamma J)^{2}}{4\gamma J^{2}K_{2}}\;.$
(15)

Using (14) and (15) in the expressions of work (Eq. 9) and heat absorption
(Eq. 10), we obtain the sudden limits of work and heat absorption which are
given by

$\displaystyle W_{s}=-4\frac{B_{1}(B_{2}-B_{1})}{K_{1}}\frac{\sinh
2K_{1}\beta}{Z},$ (16) $\displaystyle
Q_{Ms}=4\frac{B_{1}B_{2}}{K_{1}}\frac{\sinh 2K_{1}\beta}{Z}\;.$

Therefore, the efficiency at this limit is given by

$\eta_{s}=\frac{|W_{s}|}{Q_{Ms}}=1-\frac{B_{1}}{B_{2}}.$ (17)

Interestingly, the efficiency does not depend on the anisotropy parameter in
the sudden limit. This means that the QOE has the same efficiency for all
$\gamma$ when $\tau\rightarrow 0$. This can be seen in the Fig. 2 that for our
choices of $B_{1}=1$ and $B_{2}=2$, this is equal to 0.5, irrespective of the
values of $\gamma$, whereas for large $\tau$, the efficiency saturates to a
lower value. Therefore, with two spins coupled by anisotropic interaction as
the working system, a measurement-based QHE operating in the sudden limit
performs better than an engine operating in the adiabatic limit.

###  III.4 Analysis of the heat engine performance

For the quasistatic operation of the cycle, we show the variation of the
efficiency as a function of work in the Fig. 5. It is clear from this plot
that the engine performance degrades with the increase of the anisotropy
parameter $\gamma$. This is because, as $\gamma$ increases, the heat
absorption in the measurement process increases and the work output decreases
after a slow increase in the lower range of $\gamma$. We also observed that
there exists a certain value of $\gamma\sim 0.46$, for which the work output
gets maximized.

Figure 4: Variation of the absolute value of the work done $W_{t}$ (solid
black line, labelled on the left y-axis) and heat absorbed $Q_{Mt}$ (point-
marked red line, labelled on the right y-axis) as a function of duration
$\tau$ of the unitary stage. The other parameters are $B_{1}=1$, $B_{2}=2$,
$T=1$, and $\gamma=1$.  Figure 5: The parametric plot of the variable $\gamma$
on the work-efficiency plane. We have taken the absolute value of the work.
Here the point-marked red line represents the finite-time value (for
$\tau=0.1$) and the solid black line represents the quasistatic value. The
anisotropy parameter $\gamma$ varies from $0$ to $1$. The point $0.5$ on the
solid black line corresponds to $\gamma=0$, while the left end of the plot
corresponds to $\gamma=1$. The other parameters are $B_{1}=1$, $B_{2}=2$, and
$T=1$.

The variation of the work and heat absorption with respect to the duration
$\tau$ of the unitary processes are shown in the Fig. 4. Also, the plots of
the efficiency with respect to work are shown in the Fig. 5. From these plots,
we can see that a finite-time engine can deliver more work than the same
engine operating in the quasistatic limit with a proper choice of the time
interval $\tau$ of the unitary processes. In addition to that, the finite-time
engine absorbs less amount of heat in the measurement process than the same
engine operating in the quasistatic limit. Consequently, when operated for
finite times, the engine requires less energy resource, and still can perform
better than its quasistatic counterpart.

Interestingly, this outperforming is further improved for larger anisotropy
parameter $\gamma$. When $\gamma=0$, no transition takes place between two
instantaneous energy eigenstates, and the unitary stages remain adiabatic,
irrespective of their duration. Also, there is no interference-like effect
between two transition probability amplitudes as for the anisotropic case,
which can be seen in App. A. Thus, the efficiency does not change with respect
to $\tau$, as displayed in the Fig. 2. Therefore, operating the engine even
for a finite time would lead to the same efficiency as for the case when
operated quasistatically.

##  IV Always-on coupling to the heat bath

It may not always be possible to decouple a quantum system from its bath,
which acts as a heat bath for the HE operation, depending upon the
architecture of the working system and the bath. Also, there is a cost
associated with coupling and decoupling the working system from a heat bath
Kurizki and Kofman (2022); Bhattacharjee and Dutta (2021). In the previous
section, we assumed that the working system is completely isolated from its
bath during the work-delivering stages, so that the stages AB and CD remain
unitary. We consider here that the HE operation is implemented in a type of
realistic architecture in which the working system cannot be decoupled from
its bath Kurizki and Kofman (2022). It is therefore necessary to take into
account the dissipation of energy from the working system to the bath during
the stages AB and CD. This requires solving the master equation, which is
given below, with a time-dependent Hamiltonian under a dissipative bath. Here,
we assume that the remaining two isochoric stages of the cycle are identical
to those mentioned in the Sec. II.2. Furthermore, since the measurement
process is assumed to be instantaneous, the bath will not have any effect on
the system during measurement.

We consider that the temperature of the heat bath is $T$ and a single spin
decay to the bath. Then the master equation in the interaction picture for two
spins can therefore be written as Huang et al. (2012)

$\displaystyle\frac{\partial\hat{\rho}}{\partial t}=$
$\displaystyle\iota[\hat{\rho},\hat{H}]+\sum_{i=1,2}\left[\Gamma_{i}(t)\left\\{n(\omega_{i}(t))+1\right\\}\left(\hat{X}_{i}\hat{\rho}\hat{X}_{i}^{{\dagger}}-\frac{1}{2}\hat{X}_{i}^{{\dagger}}\hat{X}_{i}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{X}_{i}^{{\dagger}}\hat{X}_{i}\right)\right.$
(18)
$\displaystyle\left.+~{}\Gamma_{i}(t)n\left(\omega_{i}(t)\right)\left(\hat{X}_{i}^{{\dagger}}\hat{\rho}\hat{X}_{i}-\frac{1}{2}\hat{X}_{i}\hat{X}_{i}^{{\dagger}}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{X}_{i}\hat{X}_{i}^{{\dagger}}\right)\right]\;,$

where $\Gamma_{i}(t)\left\\{n\left(\omega_{i}(t)\right)+1\right\\}$ and
$\Gamma_{i}(t)n\left(\omega_{i}(t)\right)$ are time-dependent dissipation
rates,
$n\left(\omega_{i}(t)\right)=\left[\exp\left(\frac{\hbar\omega_{i}(t)}{kT}\right)-1\right]^{-1}$
is the average number of photons in the bath at the transition frequencies
$\omega_{i}(t)$, $\Gamma_{i}(t)$ =
$0.1\omega_{i}(t)e^{-\omega_{i}(t)/\omega_{c}}$ is the time-dependent
dissipation rate of the spin to an Ohmic type bath Dann et al. (2018); Leggett
et al. (1987), and $\omega_{c}$ is the cut-off frequency of the bath spectral
density. These rates are time-dependent because of the time-dependence of the
Hamiltonian $\hat{H}(t)$ Dann et al. (2018); Albash et al. (2012). Note that
we are assuming that the Eq. 18 remains valid for the time scales involved
with the system and the bath dynamics. This is possible if the bath time scale
$1/\omega_{c}$ is much smaller than the system time-scale $1/{\rm
min}_{j}(E_{j})$ [where $E_{j}$ is given by Eq. 3] and the duration $\tau$
during which the magnetic field is changed Dann et al. (2018).

Note that the jump operators associated to a system operator $\hat{X}$ are
given by Breuer et al. (2002)

$\hat{X}(\omega)=\sum_{\epsilon^{\prime}-\epsilon=\omega}|\epsilon\rangle\langle\epsilon|\hat{X}|\epsilon^{\prime}\rangle\langle\epsilon^{\prime}|,$

where $\\{|\epsilon\rangle\\}$ is the basis of the eigenvectors of the system
Hamiltonian $\hat{H}$. In the present case, the first spin is assumed to
interact with the heat bath via $\hat{\sigma}^{x}$ operator, and the
corresponding jump operators $\hat{X}_{i}$ and the respective transition
frequencies are given by

$\begin{array}[]{ll}\hat{X}_{1}=\frac{1}{2}\left(\frac{B+K-\gamma
J}{\sqrt{K^{2}+BK}}~{}\left|\psi_{1}\right\rangle\left\langle\psi_{3}\right|+\frac{B-K+\gamma
J}{\sqrt{K^{2}-BK}}~{}\left|\psi_{0}\right\rangle\left\langle\psi_{2}\right|\right),\quad\hbar\omega_{1}=2K+2J\\\
\hat{X}_{2}=\frac{1}{2}\left(\frac{B+K+\gamma
J}{\sqrt{K^{2}+BK}}~{}\left|\psi_{2}\right\rangle\left\langle\psi_{3}\right|+\frac{B-K-\gamma
J}{\sqrt{K^{2}-BK}}~{}\left|\psi_{0}\right\rangle\left\langle\psi_{1}\right|\right),\quad\hbar\omega_{2}=2K-2J.\end{array}$
(19)

These operators satisfy the relations
$[\hat{H},\hat{X}_{i}]=-\omega_{i}\hat{X}_{i}$ and
$[\hat{H},\hat{X}_{i}^{{\dagger}}]=\omega_{i}\hat{X}_{i}^{{\dagger}}$. Note
that $K$ and $|\psi_{0,3}\rangle$ are functions of $B(t)$, and are therefore
time-dependent, which also gives rise to the time dependence of the jump
operators in Eq. 19.

The heat and work in an open quantum system in presence of an external derive
are defined as Kurizki and Kofman (2022)

$\begin{array}[]{ll}Q(t)=\int_{0}^{t}\operatorname{Tr}\left[\dot{\hat{\rho}}\left(t^{\prime}\right)\hat{H}\left(t^{\prime}\right)\right]dt^{\prime}\;,\\\
W(t)=\int_{0}^{t}\operatorname{Tr}\left[\hat{\rho}\left(t^{\prime}\right)\dot{\hat{H}}\left(t^{\prime}\right)\right]dt^{\prime}.\end{array}$
(20)

The total change in the average energy of the system in a process is given by
$\Delta E(t)=E(t)-E(0)$, where
$E(t)=\operatorname{Tr}[\hat{\rho}(t)\hat{H}(t)]$ is the average energy at a
time $t$. This change in energy can be written as contributions from two
separate thermodynamic quantities as

$\Delta E(t)=W(t)+Q(t).$ (21)

Figure 6: Efficiency $\eta$ as a function of duration $\tau$ of the unitary
stages for $\Gamma_{i}(t)=0$ (for the isolated system: solid blue line) and
$\Gamma_{i}(t)$ = $0.1\omega_{i}(t)e^{-\omega_{i}(t)/\omega_{c}}$ (for open
system: dashed red line). The other parameters are $B_{1}=1$, $B_{2}=2$,
$T=1$, $\omega_{c}=1000$, and $\gamma=1$.

The work in the AB and CD processes can then be represented as follows.

$\begin{array}[]{ll}W_{1}(\tau)=\int_{0}^{\tau}\operatorname{Tr}\left[\hat{\rho}_{A\rightarrow
B}\left(t^{\prime}\right)\dot{\hat{H}}_{A\rightarrow
B}\left(t^{\prime}\right)\right]dt^{\prime}\\\
W_{2}(\tau)=\int_{0}^{\tau}\operatorname{Tr}\left[\hat{\rho}_{C\rightarrow
D}\left(t^{\prime}\right)\dot{\hat{H}}_{C\rightarrow
D}\left(t^{\prime}\right)\right]dt^{\prime}.\end{array}$ (22)

So, the total work is given by $W=W_{1}+W_{2}$. We calculated the heat
transfer between the system and the heat bath using the Eq. 21. The numerical
solution of the master equation has been done using the 4th-order adaptive
Runge-Kutta method and the numerical integration to calculate the work is done
using the Trapezoidal rule. The heat absorption in the measurement process is
calculated as discussed in the Sec. II.2.

We show in the Fig. 6 how the efficiency varies with the duration $\tau$ of
the unitary stages. It is clear from this plot that the presence of the bath
has a negligible effect on its performance in a very short time. We can,
therefore, employ such a measurement-based heat engine model whenever one
requires a finite amount of power. One does not have to decouple the working
system ever to obtain a finite amount of power, if the engine runs for a
finite duration. However, the longer the duration $\tau$, the engine
efficiency decreases, due to the dominant effects of the bath over the
external control parameter. The dissipative part of the master equation
dominates over the unitary part and therefore, the system releases more energy
to the bath as heat than it releases as the work. Also, the spins absorb more
energy during the measurement stage as $\tau$ increases. We must emphasize
that if both the spins are considered to individually interact with the heat
bath Hu et al. (2018); Liao et al. (2011), the main results of this paper will
remain the same.

In this section, we have considered the QOE with an always-on single bath
along with a measurement protocol. On the other hand, it is possible for a QOE
that operates with two heat baths to maintain such an always-on coupling,
while still achieving a reciprocating cycle by periodically changing the
interaction strength with the baths. However, as the performance of these QOEs
deteriorates in a finite time, such an always-on interaction cannot give us
operational advantages over a measurement-based engine.

###  IV.1 Power analysis of the engine

The isochoric cooling process of the system with a thermal bath is not an
instantaneous process, and ideally takes infinite time. To make a power
analysis, we assume that the state of the system becomes very close to a
thermal state $\rho_{A}$ at a finite time $t_{c}$. In order to make an
estimate of this closeness we have calculated the trace distance between two
states $\hat{\rho}$ and $\hat{\rho}_{A}$, defined as
$D(\hat{\rho},\hat{\rho}_{A})=\frac{1}{2}\operatorname{Tr}\left|\hat{\rho}-\hat{\rho}_{A}\right|$
Camati et al. (2019), where the state $\hat{\rho}$ is obtained by solving by
the master equation [Eq. (18], with time-independent dissipation rate
coefficients, as the magnetic field is kept constant at $B=B_{1}$ during this
cooling process. Also, as defined in Sec. II.2, $\rho_{A}$ represents the
thermal state at a temperature $T$ (that of the cold bath) when the magnetic
field is maintained at $B=B_{1}$ (also see Fig. (1). We estimated $t_{c}$, as
a time when the trace distance $D(\hat{\rho},\hat{\rho_{A}})$ becomes $\sim
10^{-2}$. Such a finite-time analysis of the cooling process helps us in
defining the power of the engine as $P=W/(2\tau+t_{c})$, where $\tau$ is the
duration of each unitary stage, and we have assumed that the measurement is an
instantaneous process. The plot of the power as a function of the anisotropy
is shown in Fig. 7.

Figure 7: Variation of the absolute value of work (black solid line, labelled
on the left y-axis) and power (point-marked red line, labelled on the right
y-axis) as a function of anisotropy parameter $\gamma$ in the limit of
$\tau\to 0$. A moderate thermalization time ranges from $61$ to $191$ for
anisotropy $0\leq\gamma\leq 1$ when the trace distance
$D(\hat{\rho},\hat{\rho}_{A})\sim 10^{-2}$. The other parameters are
$B_{1}=1$, $B_{2}=2$, $T=1$.

We found that in the limit of $\tau\to 0$, the work does not change
substantially with respect to $\gamma$, as compared to the quasistatic limit,
which is discussed in the Sec. III.4. However, the thermalization time
increases with the increase in anisotropy, leading to a reduction of power.
Further, if we consider that both spins interact with the bath, thermalization
of the system during the isochoric cooling process can be achieved at a much
faster pace, and hence more power would be generated by the engine.

##  V Conclusions

We have studied the performance of a measurement-based QOE in a two-spin
working system coupled by the Heisenberg anisotropic XY interaction. A non-
selective quantum measurement is used to fuel the engine. The non-commuting
nature of the Hamiltonian at two different times initiates transitions between
the instantaneous energy eigenstates at finite time unitary processes.
Furthermore, the instantaneous energy eigenstates do not coincide with the
measurement basis states which causes some transition between them. The
relevant thermodynamic quantities are calculated in terms of these transition
probabilities. We found that the efficiency oscillates largely at short times
when the two-spin system is coupled by an anisotropic interaction, while for
isotropic interaction there is no oscillation. This oscillation in efficiency
is explained in terms of interference between different transition
probabilities at finite times. It is observed that the oscillation in
efficiency dies out as the unitary processes extend for a longer time and
eventually the efficiency approaches the quasistatic limit. Thus, proper
control of the duration of unitary processes during transient times can lead
to higher work output and less heat absorption. As a result, a finite-time
engine can be more efficient than a quasi-static engine. The efficiency
further increases with increasing anisotropy while in the quasistatic limit,
it is observed that the performance deteriorates with an increase in
anisotropy.

Also, we studied the performance of the HE under the condition of always-on
coupling to the heat bath. We found that the presence of the bath has a
negligible effect on its performance in a very short time limit. However, for
a longer duration of the stages AB and CD, its performance degrades. This is
primarily due to the dominance of the bath interaction over external control
during these stages.

##  Appendix A Eigenvectors and eigenvalues for $\gamma=0$ and the
corresponding transition probabilities

In the limit of $\gamma\to 0$, the eigenstates and corresponding eigenvectors
of the Hamiltonian $\hat{H}(t)$ take the following forms:

$\begin{array}[]{l}\ket{\psi_{0}}=\ket{00},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E_{0}=-2B\\\
\ket{\psi_{1}}=\frac{1}{\sqrt{2}}(-\ket{10}+\ket{01}),~{}~{}~{}~{}~{}E_{1}=-2J\\\
\ket{\psi_{2}}=\frac{1}{\sqrt{2}}(\ket{10}+\ket{01}),~{}~{}~{}~{}~{}~{}~{}E_{2}=2J\\\
\ket{\psi_{3}}=\ket{11},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}E_{3}=2B.\end{array}$
(23)

Clearly, the states $|\psi_{0,3}\rangle$ are no longer entangled, though they
differ from the Bell states $|\psi_{\pm}\rangle$. Also in this limit, it can
be shown using the above eigenstates, that the transition probabilities as
mentioned in the Sec. III.1, are reduced to $\delta=\lambda=\chi=1/2$ and
$\xi=0$, where we used:
$\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle=$
$\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle=$
$\langle\psi_{0}^{(1)}|\hat{V}(\tau)|\psi_{3}^{(2)}\rangle=$
$\langle\psi_{3}^{(1)}|\hat{V}(\tau)|\psi_{0}^{(2)}\rangle=0$, and
$\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle=$
$\langle\psi_{3}^{(1)}|\hat{V}(\tau)|\psi_{3}^{(2)}\rangle=1$. Therefore, for
the isotropic case, the efficiency becomes $\eta=1-B_{1}/B_{2}$, which does
not depend on $\tau$.

##  Appendix B Derivation of the internal energies:

###  B.1 At A -

The Hamiltonian at point A of the cycle can be expressed as

$\hat{H}_{A}=\hat{H}_{1}=\sum_{i=0}^{3}E_{i}^{(1)}|\psi_{i}^{(1)}\rangle\langle\psi_{i}^{(1)}|$

where $\\{|\psi_{i}^{(1)}\rangle\\}$ are the eigenstates of the Hamiltonian
$\hat{H}_{1}$. As we consider that the system at A is in thermal equilibrium
with the heat bath, the thermal density matrix is given by

$\hat{\rho}_{A}=\frac{e^{-\beta\hat{H}_{1}}}{Z}=\sum_{i=0}^{3}P_{i}|\psi_{i}^{(1)}\rangle\langle\psi_{i}^{(1)}|$
(24)

where ${P_{i}=e^{-\beta E_{i}^{(1)}}/Z}$ is the thermal occupation probability
of the $i$th eigenstate and $Z$ is the relevant partition function. So, the
average internal energy at point A is given by

$\langle
E_{A}\rangle=\Tr(\hat{H}_{1}\hat{\rho}_{A})=\sum_{i=0}^{3}P_{i}E_{i}^{(1)}=-4K_{1}\frac{\sinh
2K_{1}\beta}{Z}-2J\frac{\sinh 2J\beta}{Z}.$ (25)

###  B.2 At B -

The Hamiltonian at the point B of the cycle can be expressed as

$\hat{H}_{B}=\hat{H}_{2}=\sum_{i=0}^{3}E_{i}^{(2)}|\psi_{i}^{(2)}\rangle\langle\psi_{i}^{(2)}|,$

where $\\{|\psi_{i}^{(2)}\rangle\\}$ are the eigenstates of the Hamiltonian
$\hat{H}_{2}$. The density matrix at the point B after the unitary process AB
can be obtained as

$\hat{\rho}_{B}=\hat{U}(\tau)\hat{\rho}_{A}\hat{U}^{{\dagger}}(\tau)=\sum_{i=0}^{3}P_{i}\hat{U}(\tau)|\psi_{i}^{(1)}\rangle\langle\psi_{i}^{(1)}|\hat{U}^{{\dagger}}(\tau)\;.$
(26)

The average internal energy at the point B can be obtained as

$\displaystyle\langle E_{B}\rangle=\Tr(\hat{H}_{2}\hat{\rho}_{B})$ (27)
$\displaystyle=P_{0}E_{0}^{(2)}~{}\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle\langle\psi_{0}^{(1)}|\hat{U}^{\dagger}(\tau)|\psi_{0}^{(2)}\rangle~{}+~{}P_{3}E_{0}^{(2)}~{}\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle\langle\psi_{3}^{(1)}|\hat{U}^{\dagger}(\tau)|\psi_{0}^{(2)}\rangle$
$\displaystyle+~{}P_{1}E_{1}^{(2)}~{}+P_{2}E_{2}^{(2)}~{}+~{}P_{0}E_{3}^{(2)}~{}{}\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle\langle\psi_{0}^{(1)}|\hat{U}^{\dagger}(\tau)|\psi_{3}^{(2)}\rangle$
$\displaystyle~{}+~{}P_{3}E_{3}^{(2)}~{}\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle\langle\psi_{3}^{(1)}|\hat{U}^{\dagger}(\tau)~{}|\psi_{3}^{(2)}\rangle$
$\displaystyle=P_{0}E_{0}^{(2)}(1-\xi)~{}+~{}P_{3}E_{0}^{(2)}\xi~{}+~{}P_{1}E_{1}^{(2)}~{}+~{}P_{2}E_{2}^{(2)}~{}+~{}P_{0}E_{3}^{(2)}\xi~{}+~{}P_{3}E_{3}^{(2)}(1-\xi)$
$\displaystyle=-4K_{2}(1-2\xi)\frac{\sinh 2K_{1}\beta}{Z}-4J\frac{\sinh
2J\beta}{Z}\;,$

where we have used the microreversibility condition
$|\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}=|\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle|^{2}=\xi$
(proof is given in the App. C) and
$|\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle|^{2}=|\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}=1-\xi$.
In unitary stages for a short time interval $\tau$, nonadiabatic transitions
occur between energy eigenstates that are coupled Cherubim et al. (2022). In
the present case, such transitions will be induced between the levels
$|\psi_{0}\rangle$ and $|\psi_{3}\rangle$. So, the terms like
$\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{1}^{(1)}\rangle$,
$\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{2}^{(1)}\rangle$,
$\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{1}^{(1)}\rangle$ etc. become zero.

###  B.3 At C -

The density matrix after the measurement stage can be written as

$\displaystyle\hat{\rho}_{C}=\sum_{\alpha=1}^{4}\hat{M}_{\alpha}\hat{\rho}_{B}\hat{M}_{\alpha}$
(28)
$\displaystyle=P_{0}\delta|\psi_{+}\rangle\langle\psi_{+}|~{}+~{}P_{0}(1-\delta)|\psi_{-}\rangle\langle\psi_{-}|~{}+~{}P_{1}|\psi_{1}^{(1)}\rangle\langle\psi_{1}^{(1)}|~{}+~{}P_{2}|\psi_{2}^{(1)}\rangle\langle\psi_{2}^{(1)}|$
$\displaystyle+P_{3}(1-\delta)|\psi_{+}\rangle\langle\psi_{+}|~{}+~{}P_{3}\delta|\psi_{-}\rangle\langle\psi_{-}|,$

where $\hat{M}_{\alpha}^{{\dagger}}=\hat{M}_{\alpha}$ and we have used the
microreversibility condition
$|\langle\psi_{+}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle|^{2}=|\langle\psi_{-}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}=\delta$
(proof can be found in the App. C)) and
$|\langle\psi_{-}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle|^{2}=|\langle\psi_{+}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}=1-\delta$.
The average internal energy at the point C can be obtained as

$\displaystyle\langle E_{C}\rangle=\Tr(\hat{H}_{2}\hat{\rho}_{C})$ (29)
$\displaystyle=E_{0}^{(2)}P_{0}\delta|\langle\psi_{0}^{(2)}|\psi_{+}\rangle|^{2}~{}+~{}E_{0}^{(2)}P_{0}(1-\delta)|\langle\psi_{0}^{(2)}|\psi_{-}\rangle|^{2}~{}+~{}E_{0}^{(2)}P_{3}(1-\delta)|\langle\psi_{0}^{(2)}|\psi_{+}\rangle|^{2}$
$\displaystyle+E_{0}^{(2)}P_{3}\delta|\langle\psi_{0}^{(2)}|\psi_{-}\rangle|^{2}~{}+~{}E_{1}^{(2)}P_{1}~{}+~{}E_{2}^{(2)}P_{2}~{}+~{}E_{3}^{(2)}P_{0}\delta|\langle\psi_{3}^{(2)}|\psi_{+}\rangle|^{2}~{}$
$\displaystyle+~{}E_{3}^{(2)}P_{3}(1-\delta)|\langle\psi_{3}^{(2)}|\psi_{-}\rangle|^{2}+~{}E_{3}^{(2)}P_{3}(1-\delta)|\langle\psi_{3}^{(2)}|\psi_{+}\rangle|^{2}~{}+~{}E_{3}^{(2)}P_{3}\delta|\langle\psi_{3}^{(2)}|\psi_{-}\rangle|^{2}$
$\displaystyle=-4K_{2}(1-2\delta)(1-2\chi)\frac{\sinh
2K_{1}\beta}{Z}-4J\frac{\sinh 2J\beta}{Z}\;,$

where we have used
$|\langle\psi_{0}^{(2)}|\psi_{+}\rangle|^{2}=|\langle\psi_{3}^{(2)}|\psi_{-}\rangle|^{2}=\chi$
and
$|\langle\psi_{0}^{(2)}|\psi_{-}\rangle|^{2}=|\langle\psi_{3}^{(2)}|\psi_{+}\rangle|^{2}=1-\chi$,
which can be proved using the conservation of probability

$|\langle\psi_{0}^{(2)}\mid\psi_{-}\rangle|^{2}+|\langle\psi_{3}^{(2)}\mid\psi_{-}\rangle|^{2}=1.$

###  B.4 At D -

The density matrix at point D after the unitary process CD is given by

$\hat{\rho}_{D}=\hat{V}(\tau)\hat{\rho}_{C}\hat{V}^{{\dagger}}(\tau)\;.$ (30)

Similarly to points A, B, and C, we can derive the average internal energy at
point D which is given by

$\langle
E_{D}\rangle=\Tr(\hat{H}_{1}\hat{\rho}_{D})=-4K_{1}(1-2\delta)(1-2\lambda)\frac{\sinh
2K_{1}\beta}{Z}-4J\frac{\sinh 2J\beta}{Z}\;,$ (31)

where we have used the microreversibility condition
$|\langle\psi_{0}^{(1)}|\hat{V}(\tau)|\psi_{+}\rangle|^{2}=|\langle\psi_{3}^{(1)}|\hat{V}(\tau)|\psi_{-}\rangle|^{2}=\lambda$
(proof is given in the App. C) and
$|\langle\psi_{3}^{(1)}|\hat{V}(\tau)|\psi_{+}\rangle|^{2}=|\langle\psi_{0}^{(1)}|\hat{V}(\tau)|\psi_{-}\rangle|^{2}=1-\lambda.$

##  Appendix C Proof of the microreversibility conditions:

We show below in details the proof of the relation
$|\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}=|\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle|^{2}$.

$\displaystyle|\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle|^{2}=\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle\langle\psi_{0}^{(1)}|\hat{U}^{{\dagger}}(\tau)|\psi_{3}^{(2)}\rangle$
$\displaystyle=\langle\psi_{3}^{(2)}|\hat{U}(\tau)(\mathbb{I}-|\psi_{1}^{(1)}\rangle\langle\psi_{1}^{(1)}|~{}-~{}|\psi_{2}^{(1)}\rangle\langle\psi_{2}^{(1)}|~{}-~{}|\psi_{3}^{(1)}\rangle\langle\psi_{3}^{(1)}|)\hat{U}^{{\dagger}}(\tau)|\psi_{3}^{2}\rangle\;,$
(32)

where we have used the completeness relation
$\sum_{i=0}^{3}|\psi_{i}^{(1)}\rangle\langle\psi_{i}^{(1)}|=\mathbb{I}$. The
above relation can then be rewritten as

$\displaystyle\langle\psi_{3}^{(2)}|\hat{U}(\tau)\hat{U}^{{\dagger}}(\tau)|\psi_{3}^{(2)}\rangle-|\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}=1-\left(1-|\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}\right)$
$\displaystyle=|\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}\;.$
(33)

In the last step of the above derivation, we have used the conservation of the
probability:

$|\langle\psi_{0}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}~{}+~{}|\langle\psi_{3}^{(2)}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}=1.$

Similarly, we can prove the other microreversibility conditions, namely,
$|\langle\psi_{+}|\hat{U}(\tau)|\psi_{0}^{(1)}\rangle|^{2}=|\langle\psi_{-}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}$,
and
$|\langle\psi_{0}^{(1)}|\hat{V}(\tau)|\psi_{+}\rangle|^{2}=|\langle\psi_{3}^{(1)}|\hat{V}(\tau)|\psi_{-}\rangle|^{2}$
by using $\sum_{\alpha=1}^{4}\hat{M}_{\alpha}^{2}=\mathds{1}$ and the
conservation of probability, respectively

$\begin{gathered}|\langle\psi_{+}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}~{}+~{}|\langle\psi_{-}|\hat{U}(\tau)|\psi_{3}^{(1)}\rangle|^{2}=1,\\\
|\langle\psi_{0}^{(1)}|\hat{V}(\tau)|\psi_{-}\rangle|^{2}~{}+~{}|\langle\psi_{3}^{(1)}|\hat{V}(\tau)|\psi_{-}\rangle|^{2}=1.\end{gathered}$

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