Quantum many-body thermal machines enabled by atom-atom correlations

R. S. Watson1 $\dagger$ and K. V. Kheruntsyan1$\star$

1 School of Mathematics and Physics, University of Queensland, Brisbane,
Queensland 4072, Australia

<EMAIL_ADDRESS>
⋆<EMAIL_ADDRESS>
## Abstract

Particle-particle correlations, characterized by Glauber’s second-order
correlation function, play an important role in the understanding of various
phenomena in radio and optical astronomy, quantum and atom optics, particle
physics, condensed matter physics, and quantum many-body theory. However, the
relevance of such correlations to quantum thermodynamics has so far remained
illusive. Here, we propose and investigate a class of quantum many-body
thermal machines whose operation is directly enabled by second-order atom-atom
correlations in an ultracold atomic gas. More specifically, we study quantum
thermal machines that operate in a sudden interaction-quench Otto cycle and
utilize a one-dimensional Lieb-Liniger gas of repulsively interacting bosons
as the working fluid. The atom-atom correlations in such a gas are different
to those of a classical ideal gas, and are a result of the interplay between
interparticle interactions, quantum statistics, and thermal fluctuations. We
show that operating these thermal machines in the intended regimes, such as a
heat engine, refrigerator, thermal accelerator, or heater, would be impossible
without such atom-atom correlations. Our results constitute a step forward in
the design of conceptually new quantum thermodynamic devices which take
advantage of uniquely quantum resources such as quantum coherence,
correlations, and entanglement.

###### Contents

  1. 1 Introduction
  2. 2 Interaction-driven Otto heat engine.
  3. 3 Work from second-order Glauber correlations.
  4. 4 Interaction-driven Otto accelerator, heater, and refrigerator.
    1. 4.1 Accelerator [A]
    2. 4.2 Heater [H]
    3. 4.3 Refrigerator [R]
  5. 5 Summary and outlook.
  6. A The Lieb-Liniger model for the 1D Bose gas
  7. B Transverse Otto cycle
  8. C Instantaneity of the sudden quench
  9. D Glauber’s second order correlation function
  10. E Exact thermodynamic Bethe ansatz results
  11. F Maximum efficiency and maximum work
  12. G Thermal operation regimes of other QTM’s

##  1 Introduction

Quantum thermal machines (QTM), such as quantum heat engines (QHE),
refrigerators, and quantum batteries, are central in the theoretical and
experimental development of the emerging field of quantum thermodynamics [1,
2]. Their primary utility is to explore the fundamental laws of thermodynamics
in the quantum realm and to demonstrate possible advantages gained by
utilising quantum resources. Accordingly, QTM’s are expected to play a similar
role in the development of quantum technologies as classical heat engines
played in fostering scientific advances during the Industrial Revolution. In
the past decade, progress in the control over quantum platforms, such as
single ions [3, 4], nitrogen vacancy centers [5], and single-atom impurities
immersed in an ultra-cold atomic bath [6], have led to the realization of
single-particle QHE’s. Such single-particle QHE’s represent the ultimate limit
in the realization of an ‘infinitesimal machine’ [7].

However, to utilize the breadth of quantum resources available, one must move
beyond single-particle systems—to engines that utilize interacting many-
particle systems. Such QHE’s are uniquely positioned to take advantage of
quantum resources, such as entanglement [8, 9, 10, 11], correlations [12, 13,
14], or quantum coherence [15, 16, 17, 18, 19, 20], to enhance the performance
of classical heat engines [21] or perform entirely new tasks that would be
impossible classically [22]. In particular, control over inter-particle
interactions allows for the creation of strictly many-body QHE’s [21, 23, 24,
25, 26], which have recently been realized in the laboratory [27, 28]. These
very recent experimental developments underscore the need for further studies
of thermodynamic processes in the context of interacting quantum many-body
thermal machines.

Here, we propose a quantum many-body Otto heat engine—as well as related
thermal machines including the Otto refrigerator, thermal accelerator, and
heater—using a uniform one-dimensional (1D) Bose gas with repulsive contact
interactions as the working fluid. In the proposed Otto cycles, the unitary
work strokes are driven by a sudden quench of the interaction strength. We
demonstrate how the thermodynamic performance, in particular net work and
efficiency, of these many-body QTM’s can be calculated through the
experimentally measurable atom-atom local pair correlation [29, 30, 31]. The
atom-atom local correlation, $g^{(2)}(0)$, is described by the second-order
Glauber correlation function $g^{(2)}(r)$ at zero interparticle separation
(i.e., when $r=0$, where $r=|x-x^{\prime}|$ is the distance between the two
particles with positions $x$ and $x^{\prime}$) and characterizes the
probability of pairs of atoms to be found at the same point, relative to
uncorrelated atoms.

The benefits of using the 1D Bose gas as the working fluid in the proposed
Otto cycles is that the underlying theoretical model—the Lieb-Liniger model—is
exactly solvable in the uniform limit via the Yang-Yang thermodynamic Bethe
ansatz (TBA) [32, 33, 34], in addition to being experimentally realizable
using ultracold atomic gases confined to highly anisotropic traps [35, 36,
37]. This offers unique opportunities for gaining physical insights into the
performance of such Otto QTM’s as a tractable and testable quantum many-body
problem. Additionally, the Lieb-Liniger gas has a rich phase diagram spanning
several nontrivial regimes [29, 30], from the weakly interacting
quasicondensate [38, 39] through to the strongly interacting Tonks-Girardeau
regime of fermionization [40, 41]. The atom-atom correlation within these
regimes takes on a wide range of values between $0<g^{(2)}(0)<2$ [29, 30, 42],
depending on the temperature and interaction strength, which aids the
operation of the proposed Otto cycles under a variety of conditions. We
evaluate the performance of the 1D Bose gas Otto QTM’s, but we emphasise that
the broad conclusions arrived at here are not limited to the Lieb-Liniger
model.

A related interaction-driven Otto engine cycle with a uniform 1D Bose gas as
the working fluid was previously studied by Chen et al. in Ref. [25] in the
opposite limit of a quasi-static, or isentropic (rather than sudden quench),
work strokes. In this case, the performance of the engine cannot be expressed
merely in terms of Glauber’s second-order correlation function, as both the
kinetic and interaction energies evolve and change during the interaction
quench. Nevertheless, the performance of such an isentropic Otto engine could
still be evaluated analytically in the low-temperature regime, owing to the
known thermodynamic properties of the system using the Tomonaga-Luttinger
liquid approach.

##  2 Interaction-driven Otto heat engine.

In a uniform 1D Bose gas, described by the integrable Lieb-Liniger model [32]
(see Appendix A), the interatomic interaction strength $\chi$ can be expressed
in terms of the harmonic trap frequency $\omega_{\perp}$ in the tightly
confined (transverse) direction and the 3D $s$-wave scattering length $a_{s}$
as $\chi\simeq 2\hbar\omega_{\perp}a_{s}$ [43]. Accordingly, changing the
interaction strength $\chi$ may be achieved by either tuning the external
trapping potential that controls the transverse confinement $\omega_{\perp}$
or by changing the scattering length $a_{s}$ by means of a magnetic Feshbach
resonance [44]. The former option leads to a volume change of the gas (i.e.
transverse expansion or compression), and hence can be thought of as analogous
to mechanical work in the conventional Otto cycle. However, changing $\chi$
via a change of the scattering length leads to identical results, which then
justifies our referral to the engine cycle as the Otto cycle [45, 46, 24, 25,
27, 6, 47, 48, 49, 50] regardless of the means of tuning the interaction
strength (for further discussion, see (see Appendix B)). We emphasize,
however, that the dynamics of the quantum Otto cycle explored here are
strictly longitudinal, with the gas always remaining in its transverse ground
state.

Figure 1: An interaction-driven quantum many-body Otto engine cycle, operating
between two interaction strengths, $\chi_{c}$ and $\chi_{h}$. Unitary work
strokes (BC and DA) are shown in black, while non-unitary thermalization
strokes (AB and CD) are color-coded to the cold (blue) and hot (red)
reservoirs at temperatures $T_{c}$ and $T_{h}$, respectively.

The interaction-driven Otto engine cycle, which we thus consider, consists of
four strokes (see Fig. 1):

  * (1)

Thermalization with hot reservoir, A$\to$B: the working fluid, consisting of
$N$ total atoms at interaction strength $\chi_{h}$, is connected to a hot
($h$) reservoir at temperature $T_{h}$, where it is left to equilibrate,
taking in heat
$Q_{1}\\!=\\!\langle\hat{H}\rangle_{\textbf{B}}\\!-\\!\langle\hat{H}\rangle_{\textbf{A}}\\!>\\!0$,
which is to be partially converted into beneficial work in the subsequent
stroke. Here $\hat{H}$ is the system Hamiltonian (see Appendix A), and
$\langle\hat{H}\rangle_{\textbf{j}}$ is its expectation value, i.e., the total
energy of the system, in state $\textbf{j}=\\{\textbf{A,B,C,D}\\}$.

  * (2)

Unitary expansion, B$\to$C: the working fluid, now in a thermal equilibrium
state at temperature $T_{h}$, is decoupled from the hot reservoir and has its
interaction strength quenched from $\chi_{h}$ to $\chi_{c}\\!<\\!\chi_{h}$,
generating beneficial work
$W_{1}\\!=\\!\langle\hat{H}\rangle_{\textbf{C}}\\!-\\!\langle\hat{H}\rangle_{\textbf{B}}\\!<\\!0$
done by the fluid.

  * (3)

Thermalization with cold reservoir, C$\to$D: the working fluid is connected to
a cold ($c$) reservoir at temperature $T_{c}<T_{h}$ and allowed to equilibrate
at constant interaction strength $\chi_{c}$ while dumping energy in the form
of heat
$Q_{2}\\!=\\!\langle\hat{H}\rangle_{\textbf{D}}\\!-\\!\langle\hat{H}\rangle_{\textbf{C}}\\!<\\!0$
into the cold reservoir.

  * (4)

Unitary compression, D$\to$A: disconnected from the cold reservoir, the
working fluid has its interaction strength quenched from
$\chi_{c}\\!\to\\!\chi_{h}$, with work
$W_{2}\\!=\\!\langle\hat{H}\rangle_{\textbf{A}}\\!-\\!\langle\hat{H}\rangle_{\textbf{D}}\\!>\\!0$
done on the fluid, and the system returning to the initial state of the
overall cycle.

Such an engine cycle generates net beneficial work (done by the fluid) if the
total work $W\\!=\\!W_{1}\\!+\\!W_{2}\\!<\\!0$, i.e., if $|W_{1}|>W_{2}$ (or
$Q_{1}>|Q_{2}|$), with efficiency

$\eta=-\frac{W}{Q_{1}}=1-\frac{|Q_{2}|}{Q_{1}},$ (1)

where we used the conservation of energy $W+Q=0$, with $Q=Q_{1}+Q_{2}$ being
the total heat [51].

##  3 Work from second-order Glauber correlations.

In this work, we specifically consider a sudden or instantaneous quench of the
interaction strength $\chi$ in the unitary strokes (2) and (4). (For a
discussion of “instantaneity” of the sudden quench, see Appendix C). Under a
sudden interaction quench, the initial ($i$) and final ($f$) expectation
values over field operators in the system Hamiltonian, i.e., the expectation
values before and after the quench, remain unchanged as the system did not
have sufficient time to evolve into a new state. Hence, the only contribution
to the difference in total energy before and after the quench,
$\langle\hat{H}\rangle_{f}-\langle\hat{H}\rangle_{i}$, comes from the
difference between the interaction terms, $\frac{1}{2}\chi_{f}\int
dz\langle\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}\rangle_{f}-\frac{1}{2}\chi_{i}\int
dz\langle\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}\rangle_{i}$,
where
$\langle\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}\rangle_{f}=\langle\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}\rangle_{i}$
in a sudden quench, and $\hat{\Psi}^{\dagger}(z)$ and $\hat{\Psi}(z)$
represent the field creation and annihilation operators. Accordingly, the
energy difference can be expressed as
$\langle\hat{H}\rangle_{f}-\langle\hat{H}\rangle_{i}\\!=\\!\frac{1}{2}(\chi_{f}\\!-\\!\chi_{i})\overline{G^{(2)}_{i}}$,
where we have defined the total (integrated) second-order correlation of the
thermal equilibrium state $\overline{G^{(2)}_{i}}\\!=\\!\int
dz\langle\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}\rangle_{i}$
[30].

Identifying the $i$ and $f$ states as points B (hot, $h$) and C, or as D
(cold, $c$) and A in the digram of Fig. 1, the net work of the Otto heat
engine can be expressed as

$W=-\frac{1}{2}(\chi_{h}-\chi_{c})\left(\overline{G^{(2)}_{h}}-\overline{G^{(2)}_{c}}\right),$
(2)

whereas the efficiency, Eq. (1), of the engine as

$\eta=1-\frac{\langle\hat{H}\rangle_{h}-\langle\hat{H}\rangle_{c}-\frac{1}{2}\left(\chi_{h}-\chi_{c}\right)\overline{G^{(2)}_{h}}}{\langle\hat{H}\rangle_{h}-\langle\hat{H}\rangle_{c}-\frac{1}{2}\left(\chi_{h}-\chi_{c}\right)\overline{G^{(2)}_{c}}}.$
(3)

These equations allow for evaluation of the performance of the interaction-
driven Otto engine under a sudden quench through solely the _equilibrium_
properties of the gas as all relevant expectation values here are with respect
to $h$ (B) or $c$ (D) states.

The equilibrium phase diagram of the uniform 1D Bose gas consists of several
distinct regimes (connected by smooth crossovers) that can be characterized by
the normalized same-point atom-atom correlation function $g^{(2)}(0)$ [29].
This local pair correlation is a thermodynamic quantity that can be calculated
from the exact TBA, as well as via approximate analytic methods in six
asymptotic regimes; it is shown in Fig. 5 in Appendix D. The $g^{(2)}(0)$
function and the different regimes of the 1D Bose gas can be parameterized by
dimensionless interaction strength, $\gamma\\!=\\!m\chi/\hbar^{2}\rho$, and
dimensionless temperature, $\tau\\!=\\!2mk_{B}T/\hbar^{2}\rho^{2}$, where $m$
is the boson mass and $\rho\\!=\\!N/L$ is the 1D density for $N$ atoms in a
system of length $L$.

For a uniform 1D Bose gas, the total correlation in the hot or cold thermal
equilibrium state may be expressed as $\overline{G^{(2)}_{h(c)}}\\!=\\!N\rho
g^{(2)}_{h(c)}(0)$ (see Appendix D). Combining this with Eq. (2), the net work
per particle can be expressed as

$\frac{W}{N}=-\frac{\hbar^{2}\rho^{2}}{2m}(\gamma_{h}-\gamma_{c})\left(g^{(2)}_{h}(0)-g^{(2)}_{c}(0)\right),$
(4)

meaning the net work is directly proportional to the difference between atom-
atom correlations of the 1D Bose gas in the hot and cold thermal equilibrium
states. This simple relationship between thermodynamic work and Glauber
second-order correlation function represents the key result of this work.

From Eq. (4), and given that $\gamma_{h}$ is always larger than $\gamma_{c}$
(as $\chi_{h}>\chi_{c}$; see Fig. 1), we see that if the local pair
correlations did not depend on the respective interaction strengths and
temperatures, i.e. if they were the same,
$g^{(2)}_{h}(0)\\!=\\!g^{(2)}_{c}(0)$, then the net work per particle would
vanish. We therefore conclude that extracting net work ($W\\!<\\!0$) from this
Otto cycle, and hence operating it as a heat engine, can only be enabled by
atom-atom correlations; more specifically, the only way to extract net work is
to have $g^{(2)}_{h}(0)\\!>\\!g^{(2)}_{c}(0)$ (see Fig. 5 (c) in Appendix D
for an illustration).

Figure 2: Performance of the interaction driven quantum Otto heat engine.
Columns (a)–(c) demonstrate net work, $|W|$, and efficiency, $\eta$, as a
function of the ratio of interaction strength, $\gamma_{h}/\gamma_{c}$, and
temperature, $\tau_{h}/\tau_{c}$, between the hot ($h$) and cold ($c$) thermal
equilibrium states. Regions corresponding to $W\\!>\\!0$, here colored grey,
are outside the engine operation regime and are considered further in Figs. 3
and 4 below. The example of panel (a) is for $\gamma_{c}\\!=\\!10^{-3}$ and
$\tau_{c}\\!=\\!10^{-2}$, where $\gamma_{h}/\gamma_{c}$ and
$\tau_{h}/\tau_{c}$ explores the parameter range within the region II of the
equilibrium phase diagram of Fig. 5 (b) of Appendix D . Similarly, panel (b)
explores region IV, with $\gamma_{c}\\!=\\!1$ and $\tau\\!=\\!10$, whereas (c)
explores region VI, with $\gamma_{c}\\!=\\!10$ and $\tau\\!=\\!1$.

Net work, Eq. (4), and efficiency, Eq. (3), of this quantum Otto engine,
calculated for simplicity using analytic approximations to the atom-atom
correlation function and total energy [29, 52], are shown in Fig. 2 as a
function of the ratio of interaction strengths, $\gamma_{h}/\gamma_{c}$, and
temperatures, $\tau_{h}/\tau_{c}$, for three of the six asymptotic regimes
(for results outside of the regimes of analytic approximation, see Appendix
E). Notably, in this Otto engine cycle, for any fixed value of the temperature
ratio, the interaction strength quench corresponding to maximum net work is
approximately the same as that providing maximum efficiency; this occurs as,
to first order, the heat intake $Q_{1}$ varies slowly with
$\gamma_{h}/\gamma_{c}$, meaning $\eta\\!\propto\\!W$ (see Appendix F). The
observed increase of net work and efficiency in all regimes under large
temperature ratio may be attributed to the fact that the local correlation of
the hot thermal state in Eq. (4) is always a monotonically increasing function
of $\tau$. However, as the correlation function is also monotonically
decreasing under $\gamma_{h}$, this results in no extractable net work under
sufficiently large interaction strength ratios for any given temperature
ratio.

At a glance, one may conclude that the net work, which is enabled through the
$g^{(2)}(0)$ correlation function, is maximized under the largest possible
difference in correlation function, i.e.
$g^{(2)}_{h}(0)\\!-\\!g^{(2)}_{c}(0)\\!\simeq\\!2$. However, to achieve this,
while also guaranteeing that $\gamma_{h}\\!>\\!\gamma_{c}$, would require an
unrealistically high (from practical point of view) temperature ratio to
operate between regimes VI and IV, shown in Fig. 5 (b) of Appendix D. Rather,
we observe that, while the $g^{(2)}(0)$ correlation function is responsible
for enabling operation as a heat engine, the _magnitude_ of net work is
governed more strongly by the difference in the interaction strengths,
$\gamma_{h}-\gamma_{c}$, which is more susceptible to large variations in
practice using, e.g., magnetic Feshbach tuning of the $s$-wave scattering
length.

Consequently, it is in the weakly interacting ($\gamma\ll 1$) region II, shown
in Fig. 2 (a), where $\gamma_{h}-\gamma_{c}$ is small, that we observe the
lowest magnitude of net work. In comparison, the magnitude of net work is
largest in regime IV, shown in Fig. 2 (b), where $\gamma\\!\sim\\!1$ and hence
the difference $\gamma_{h}\\!-\\!\gamma_{c}$ can also be on the order of one.
The same considerations apply to the strongly interacting ($\gamma\\!\gg\\!1$)
regime VI, where one can operate under the largest magnitudes of interaction
strengths; however, in this regime the net work is diminished due to the
vanishing of correlation itself ($g^{(2)}\\!\ll\\!1$) arising from the effect
of fermionization [40]. In contrast to these observations, the efficiency of
the engine, Eq. (3), is inversely dependent on the total energy of the thermal
states, which is minimal in the weakly interacting low temperature regime II,
which thus has the largest efficiency.

##  4 Interaction-driven Otto accelerator, heater, and refrigerator.

Glauber’s $g^{(2)}(0)$ correlation function is inherently dependent on the
interaction strength and temperature [29]. This implies that the condition for
the Otto cycle to operate as a heat engine,
$g^{(2)}_{c}(0)\\!<\\!g^{(2)}_{h}(0)$, where $\gamma_{c}\\!<\\!\gamma_{h}$,
cannot hold under large quenches of interaction strength as the gas becomes
increasingly fermionized (i.e., $g_{h}^{(2)}(0)\\!\to\\!0$) in the limit
$\gamma\\!\to\\!\infty$ [40].

However, beyond the heat engine [E] operation regime, a further three QTM’s
are thermodynamically allowed [53], namely, the accelerator [A], heater [H],
and refrigerator [R]. For these QTM’s, one may define a coefficient of
performance (CoP) according to [54]:

$\mathrm{CoP[QTM]}=\frac{\mathrm{benefit\,of\,operation}}{\mathrm{cost\,of\,operation}}.$
(5)

We now discuss the operating conditions and the CoP’s of these three
additional QTM’s in detail.

###  4.1 Accelerator [A]

Figure 3: Energy flow diagrams and schematics of the interaction driven Otto
cycles in different regimes: (a) heat engine [E], (b) accelerator [A], (c)
heater [H], and (d) refrigerator [R]. In all four regimes, the difference
between the energies of the hot and cold thermal state (B and D) are the same,
however the energies of the resulting state after the interaction driven work
strokes can be different, leading to these four different physically valid
outcomes. Panel (d) shows the layout of how the operating regimes of these
different QTM’s depend on the ratio of interaction strengths,
$\gamma_{h}/\gamma_{c}$, and temperatures, $\tau_{h}/\tau_{c}$, of the hot and
cold thermal states in same the three asymptotic regimes (II, IV, and VI) as
in Fig. 2. The respective coefficients of performance of these QTM’s within
these operating boundaries are shown in Fig. 4.  Figure 4: Coefficients of
performance (CoP) of QTMs in the accelerator [A], heater [H], and refrigerator
[R] regimes within the respective parameter ranges II, IV, and VI shown in
Fig. 3 (e).

The conditions of operating the Otto cycle as a thermal accelerator are given
by: $W\\!>\\!0$, $Q_{1}\\!>\\!0$, $Q_{2}\\!>\\!0$, with $|Q_{2}|>|Q_{1}|$,
where $W\\!=\\!0$ defines the border between the heat engine and the
accelerator, whereas $Q_{1}\\!=\\!0$ defines the border between the
accelerator and the next QTM, the heater; see Fig. 3, where we show the
simplified schematics of all additional QTM’s compared to the heat engine from
Fig. 1, which we repeat here in panel (a) for comparison. The thermal
accelerator [53], shown in panel (b) of Fig. 3, enhances the natural flow of
heat, taken into the working fluid from the hot reservoir, $Q_{1}$, and
transferred to the cold reservoir, $Q_{2}$, which is aided by the working
fluid through the investment of net mechanical work, $W>0$, in the process.
According to Eq. (5), its CoP is given by:

$\mathrm{CoP[A]}=-\frac{Q_{2}}{W}=1+\frac{Q_{1}}{W}>1,$ (6)

where,

$\displaystyle Q_{2}$
$\displaystyle=-\langle\hat{H}\rangle_{h}+\langle\hat{H}\rangle_{c}+\frac{N\hbar^{2}\rho^{2}}{2m}(\gamma_{h}-\gamma_{c})g^{(2)}_{h}(0),$
(7) $\displaystyle Q_{1}$
$\displaystyle=\langle\hat{H}\rangle_{h}-\langle\hat{H}\rangle_{c}-\frac{N\hbar^{2}\rho^{2}}{2m}(\gamma_{h}-\gamma_{c})g^{(2)}_{c}(0),$
(8)

whereas work $W$ is given by Eq. (4) as before. The magnitude of CoP[A] is
shown in Fig. 4 (a), where we note that, at the border between [E] and [A],
the coefficient of performance diverges due to its inverse dependence on $W$.
Operation of this QTM, enabled through $g^{(2)}_{h}(0)\\!<\\!g^{(2)}_{c}(0)$,
additionally requires that
$|W_{1}|=\langle\hat{H}\rangle_{\mathbf{B}}\\!-\\!\langle\hat{H}\rangle_{\mathbf{C}}>0$
(where $W_{1}<0$) and
$W_{2}=\langle\hat{H}\rangle_{\mathbf{A}}\\!-\\!\langle\hat{H}\rangle_{\mathbf{D}}>0$,
which are proportional to $g^{(2)}_{c}(0)$ and $g^{(2)}_{h}(0)$, respectively,
must individually remain smaller than the energy gap, $\Delta
E\\!=\\!\langle\hat{H}\rangle_{\mathbf{B}}\\!-\\!\langle\hat{H}\rangle_{\mathbf{D}}$,
between the hot and cold thermal states, as shown in the cycle diagram in Fig.
3 (b). The conditions $|W_{1}|<\Delta E$ and $W_{2}<\Delta E$ may be expressed
as

$\displaystyle
N\frac{\hbar^{2}\rho^{2}}{2m}(\gamma_{h}-\gamma_{c})g^{(2)}_{h}(0)<\Delta E,$
(9) $\displaystyle
N\frac{\hbar^{2}\rho^{2}}{2m}(\gamma_{h}-\gamma_{c})g^{(2)}_{c}(0)<\Delta E,$
(10)

and are equivalent to $Q_{1}\\!>\\!0$ and $Q_{2}\\!<\\!0$, respectively.

###  4.2 Heater [H]

Operating the Otto cycle in the heater regime requires:
$W>0,Q_{1}<0,\,Q_{2}<0$, and is shown schematically in Fig. 3 (c). This QTM
utilizes mechanical work to heat up both hot and cold thermal reservoirs. The
border with the accelerator regime is defined by $Q_{1}\\!=\\!0$, and the
lower border with the refrigerator QTM is defined by $Q_{2}\\!=\\!0$; see Fig.
3 (e). We note that the condition $Q_{1}\\!<\\!0$ corresponds to
$W_{2}\\!=\\!\langle\hat{H}\rangle_{\mathbf{A}}\\!-\\!\langle\hat{H}\rangle_{\mathbf{D}}\\!>\\!\Delta
E$, opposite to the condition for the accelerator regime [A]. Additionally,
for a fixed temperature ratio, $\tau_{h}/\tau_{c}$, since an arbitrarily large
interaction strength quench to $\gamma_{h}\gg 1$ incurs fermionization (i.e.
$g^{(2)}_{h}(0)\\!\to\\!0$), operation as a heater is inevitable as
$\gamma_{h}/\gamma_{c}\to\infty$ at any fixed value of $\tau_{h}/\tau_{c}$
(see Appendix G).

If one considers the benefit of operation of the heater to be heating of both
reservoirs, then its CoP is trivially $\mathrm{CoP[H]}\\!=\\!-Q/W\\!=\\!1$.
Instead, we define the benefit of operation of this QTM to be the heating of
the hot reservoir; thus

$\mathrm{CoP[H]}=-\frac{Q_{1}}{W}=1-\frac{|Q_{2}|}{W}<1,$ (11)

which shown in Fig. 4 (b).

One may alternatively define the benefit of the heater as heating the cold
reservoir, in which case the CoP would be given by:

$\mathrm{CoP[H]}^{\prime}=-\frac{Q_{2}}{W}=1-\mathrm{CoP[H]}<1.$ (12)

Both definitions of $\mathrm{CoP[H]}$, however, are limited to be less than or
equal to $1$ by energy conservation .

###  4.3 Refrigerator [R]

The conditions of operating the Otto cycle as a refrigerator are:
$W>0,\,Q_{1}<0,\,Q_{2}>0$, with $|Q_{2}|<|Q_{1}|$. The purpose of this thermal
machine is to cool down the cold reservoir by extracting heat and dumping it
into the hot reservoir, with the aid mechanical work done by the working
fluid. The boundary between the refrigerator and the heater is defined by
$Q_{2}\\!=\\!0$, see Fig. 3 (d). The CoP for the refrigerator is given by [54]

$\mathrm{CoP[R]}=\frac{Q_{2}}{W}=\frac{|Q_{1}|}{W}-1,$ (13)

and is shown in Fig. 4 (c); it diverges in the limit of infinitesimal quenches
in both interaction strength and temperature, because the benefit of
refrigeration, $Q_{2}$, vanishes slower than the cost, $W$, in these limits
(see Appendix G. Further, as noted in the section addressing the heater QTM,
large interaction strength quenches inevitably incur operation as a heater for
any fixed temperature ratio. This implies that refrigeration occurs only over
a finite parameter region, most clearly visible in regimes IV and VI of Fig. 3
(e).

The role of the atom-atom local correlation function $g^{(2)}(0)$ in the
thermal operation regimes and the respective boundaries of these additional
QTM’s is discussed further in Appendix G.

##  5 Summary and outlook.

We have proposed a sudden interaction-quench Otto cycle operating in a quantum
many-body regime using a repulsive 1D Bose gas as a working fluid. Extracting
net work from such a cycle in the heat engine regime was shown to be enabled
by atom-atom correlations. Such correlations are characterized by Glauber’s
second-order correlation function, $g^{(2)}(0)$, which is a thermodynamic
quantity that can be calculated from the exact TBA solution through
application of the Hellmann-Feynman theorem to the Helmholtz free energy [29].
Further, we have investigated other operational regimes of this Otto cycle,
such as the thermal accelerator, heater, and refrigerator cycles, defining and
examining their coefficients of performance.

Though our specific results for the net work and the efficiency were
calculated for a uniform 1D Bose as an example, the broad conclusions arrived
at here on the basis of Eqs. (2) and (3) are applicable to any other many-body
system with short-range contact interactions and should aid the tests of
quantum thermodynamic concepts and realization of novel QTMs in laboratory
settings.

## Acknowledgements

This work was supported through Australian Research Council Discovery Project
Grant No. DP190101515.

Appendices

In these Appendices, we briefly review the Lieb-Liniger model of the one-
dimensional (1D) Bose gas with contact interactions, and discuss how the
interaction quench can be achieved via changes of the strength of the
transverse confinement of the 1D Bose gas. We also discuss physical
considerations for the sudden quench to be effectively instantaneous. We next
present the relevant analytic results for the Glauber’s second-order
correlation $g^{(2)}(0)$ and the Hamiltonian energy $\langle\hat{H}\rangle$ in
six different asymptotic regimes of the 1D Bose gas, used in the main text for
evaluating the net work and efficiency of the Otto quantum heat engine. We
also present equivalent numerical results obtained using the exact
thermodynamic Bethe ansatz (TBA), and use these to explore a sudden quench
Otto cycle operation outside of the analytically tractable asymptotic regimes
of the 1D Bose gas. This is followed by a discussion of the maximum efficiency
at maximum work. Finally, we discuss the role of the pair correlation
$g^{(2)}(0)$ in the operation of other quantum thermal machines (QTMs)
presented in the main text, such as accelerator, heater, and refrigerator.

##  Appendix A The Lieb-Liniger model for the 1D Bose gas

The Lieb-Liniger model for the uniform 1D Bose gas with repulsive contact
interactions [32] is described by the follwoing second-quantized Hamiltonian

$\begin{split}\begin{aligned}
\hat{H}=\hat{H}^{kin}+\hat{H}^{int}=-\frac{\hbar^{2}}{2m}\int
dz\hat{\Psi}^{\dagger}\frac{\partial^{2}}{\partial
z^{2}}\hat{\Psi}+\frac{\chi}{2}\int
dz\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi},\end{aligned}\end{split}$
(14)

where $m$ is the atomic mass, $\chi$ is the strength of the contact
interactions (see main text), and $\hat{\Psi}^{\dagger}(z)$ and
$\hat{\Psi}(z)$ are the bosonic field creation and annihilation operators,
respectively. We additionally highlight the separation of the Lieb-Liniger
Hamiltonian into its kinetic energy, $\hat{H}^{kin}$, and interaction energy,
$\hat{H}^{int}$, components, to be referred to later.

Ground state solutions to this integrable model are dependent only on a single
dimensionless interaction strength, $\gamma\\!=\\!m\chi/\hbar^{2}\rho$, where
$\rho\\!=\\!N/L$ is the linear density for $N$ particles in a system of size
$L$. Finite temperature solutions, on the other hand, can be obtained using
the Yang-Yang thermodynamic Bethe ansatz (TBA) [33], and can be parameterized
by an additional dimensional parameter, the dimensionless temperature
$\tau\\!=\\!2mk_{B}T/\hbar^{2}\rho^{2}$ [29].

##  Appendix B Transverse Otto cycle

For a magnetically trapped ultracold 1D Bose gas, the work done via transverse
compression and expansion is ultimately magnetic: it is done by the magnetic
field on the atomic dipole moments when $\omega_{\perp}$ is increased, or vice
versa – by the atomic dipole moments on the magnetic field when
$\omega_{\perp}$ is decreased. Alternatively, the change in the interaction
strength $\chi$ is implemented through control over the $s$-wave scattering
length $a_{s}$ via a magnetic Feshbach resonance [44], in which case the
nature of the work is still magnetic.

We use the term Otto cycle in the same sense as used to describe, e.g., a
harmonic oscillator Otto engine [45, 46, 24, 25, 27, 6, 47, 48, 49, 50],
wherein the harmonic oscillator frequency (rather than the volume of the
system) is fixed as an external parameter during the thermalization strokes.
In our case, it is the interaction strength that is fixed, which itself is
proportional to the transverse harmonic confinement frequency of the 1D Bose
gas.

##  Appendix C Instantaneity of the sudden quench

Realistically, a sudden quench of interaction strength from $\chi_{h(c)}$ to
$\chi_{c(h)}$ would still occur over a finite duration $\Delta t$. The
“instantaneity” of the quench utilized in the main text refers to the
assumption that $\Delta t$ is much shorter than the characteristic time scale
for longitudinal dynamics, i.e. that $\Delta
t\\!\ll\\!ml_{\text{cor}}^{2}/\hbar$, where $l_{\text{cor}}$ is the
characteristic short-range correlation length in the system, given,
respectively, by: the healing length $l_{h}\\!=\\!\hbar/\sqrt{m\chi\rho}$ in
regimes I and II; thermal phase coherence length
$l_{\phi}\\!=\\!\hbar^{2}\rho/mk_{B}T$ in regime III; thermal de Broglie
wavelength $\lambda_{T}\\!=\\!\sqrt{2\pi\hbar^{2}/mk_{B}T}$ in regime IV;
absolute value of the 1D scattering length $|a_{1D}|=2\hbar^{2}/m\chi$ in the
regime of high-temperature fermionization V; and the Fermi wavelength
$\lambda_{F}\\!=\\!2/\rho$ in the Tonks-Girardeau regime of low-temperature
fermionization VI. Thus, it is with respect to the _longitudinal_ dynamics
that we refer to our quench as sudden. With respect to the _transverse_
dynamics, on the other hand, we are assuming that $\Delta t$ is sufficiently
long ($\Delta t\gg 2\pi/\omega_{\perp}$) compared to the characteristic
transverse timescale, $2\pi/\omega_{\perp}$, governed by the transverse
harmonic trap frequency $\omega_{\perp}$ [43]. As a result, the quench would
retain the system in the transverse ground state, and hence would not
compromise the 1D character of the system. As such, the work done on (or by)
the system during the unitary strokes can be regarded as transversely
quasistatic.

Figure 5: Atom-atom correlations, described by Glauber’s $g^{(2)}(0)$
correlation function, for the uniform 1D Bose gas evaluated using the exact
Yang-Yang TBA [29, 33]. Panel a shows $g^{(2)}(0)$ as a function of the
dimensionless interaction strength, $\gamma$, and temperature, $\tau$. In
panel b, this is translated into a contour diagram, in which we also show the
crossover boundaries (white solid and dashed lines) between the different
asymptotic analytic regimes [29]. Panel (c) shows line plots of $g^{(2)}(0)$
vs $\gamma$, at different fixed values of $\tau$, together with two possible
choices, D-B or $\textbf{D-B}^{\prime}$, of the thermal equilibrium operating
points of the Otto cycle from Fig. 1; as we see, according to Eq. (4),
operating the Otto cycle as an engine (with $W<0$) can be achieved between the
points D-B ($\gamma_{c}\longleftrightarrow\gamma_{h}$), where
$g^{(2)}_{c}(0)\\!<\\!g^{(2)}_{h}(0)$, but not between $\textbf{D-B}^{\prime}$
($\gamma_{c}\longleftrightarrow\gamma^{\prime}_{h}$), where
$g^{(2)}_{c}(0)\\!>\\!g^{(2)}_{h}(0)$ due to the stronger interaction quench,
even though the temperature at D is still lower than at $\textbf{B}^{\prime}$.

##  Appendix D Glauber’s second order correlation function

The two-point correlation function may be generally defined through

$g^{(2)}(z,z^{\prime})=\frac{\langle\hat{\Psi}^{\dagger}(z)\hat{\Psi}^{\dagger}(z^{\prime})\hat{\Psi}(z^{\prime})\hat{\Psi}(z)\rangle}{\rho(z)\rho(z^{\prime})}.$
(15)

For a uniform (translationally invariant) system with
$\rho(z^{\prime})\\!=\\!\rho(z)\\!=\\!\rho$, this $g^{(2)}(z,z^{\prime})$
depends only on the relative distance $|z-z^{\prime}|$, i.e.
$g^{(2)}(z,z^{\prime})\\!=\\!g^{(2)}(|z-z^{\prime}|)$. If one is interested in
the same point ($z\\!=\\!z^{\prime}$) correlation function, as utilized in the
main text for calculation of the net work and efficiency of the quantum Otto
cycle, this in turn becomes $g^{(2)}(0)$.

The 1D Bose gas can be characterized by six distinct asymptotic regimes
defined through the same-point correlation function [29], as shown in Fig. 5.
The weakly interacting ($\gamma\ll 1$), low temperature ($\tau\ll
2\sqrt{\gamma}$) quasicondensate regime can be treated using the Bogoliubov
theory for quasicondensates [39], and is characterised by suppressed density
fluctuations, but fluctuating phase. This may be subdivided into regions
dominated by quantum (I) and thermal (II) fluctuations [29]. At higher
temperatures, the gas becomes nearly ideal, and can be treated using
perturbation theory with respect to $\gamma$ [29, 42]. This asymptotic region
may in turn be subdivided into quantum degenerate (III) and non-degenerate
(IV) regimes. Finally, in the strongly interacting regime ($\gamma\gg 1$),
where the Fermi-Bose gas mapping applies, the 1D Bose gas can be well
approximated by a nearly ideal Fermi gas, and can be treated using
perturbation theory with respect to $1/\gamma$ [29, 42]. This regime can be
further subdivided into two regions corresponding to high-temperature (V) and
low-temperature (VI) fermionization.

In each of these asymptotic regimes, the pair correlation function
$g^{(2)}(0)$ can be derived in closed approximate analytic form, where we
additionally define the boundary of these regimes in terms of $\gamma$ and
$\tau$:

$\displaystyle\text{I}\\!:\,$ $\displaystyle
g^{(2)}(0)\\!\simeq\\!1-\frac{2}{\pi}\gamma^{1/2}+\frac{\pi\,\tau^{2}}{24\,\gamma^{3/2}},\,\left[\frac{\tau}{2}\\!\ll\\!\gamma\ll
1\right],$ (16) $\displaystyle\text{II}\\!:\,\,$ $\displaystyle
g^{(2)}(0)\simeq
1+\frac{\tau}{2\sqrt{\gamma}}\,\,,\quad\left[2\gamma\ll\tau\ll
2\sqrt{\gamma}\right],$ (17) $\displaystyle\text{III}\\!:\,\,$ $\displaystyle
g^{(2)}(0)\simeq
2-\frac{4\gamma}{\tau^{2}},\quad\left[2\sqrt{\gamma}\ll\tau\ll 1\right],$ (18)
$\displaystyle\text{IV}\\!:\,\,$ $\displaystyle g^{(2)}(0)\simeq
2-\gamma\sqrt{\frac{2\pi}{\tau}},\,\,\,\left[\tau\gg\text{max}\\{1,\gamma^{2}\\}\right],$
(19) $\displaystyle\text{V}\\!:\,\,$ $\displaystyle
g^{(2)}(0)\simeq\frac{2\tau}{\gamma^{2}},\,\,\left[\frac{\pi^{2}}{(1+2/\gamma)^{2}}\ll\tau\ll\gamma^{2}\right],$
(20) $\displaystyle\text{VI}\\!:\,$ $\displaystyle
g^{(2)}(0)\simeq\frac{4\pi^{2}}{3\gamma^{2}}\left(1\\!+\\!\frac{\tau^{2}}{4\pi^{2}}\right)\\!,\left[\tau\\!\ll\\!\frac{\pi^{2}}{(1+2/\gamma)^{2}},\,\gamma\\!\gg\\!1\right]\\!.$
(21)

Further, we may express the total energy of system in each asymptotic regime
as [52],

$\displaystyle\text{I}:\,\,\,$
$\displaystyle\langle\hat{H}\rangle\\!\simeq\\!N\frac{\hbar^{2}\rho^{2}}{2m}\left(\gamma-\frac{4}{3\pi}\gamma^{3/2}+\frac{\pi}{12}\frac{\tau^{2}}{\gamma^{1/2}}\right),$
(22) $\displaystyle\text{II}:\,\,\,$
$\displaystyle\langle\hat{H}\rangle\\!\simeq\\!N\frac{\hbar^{2}\rho^{2}}{2m}\left(\gamma+\frac{\zeta(3/2)}{4\sqrt{\pi}}\tau^{3/2}+\frac{\zeta(1/2)}{2\sqrt{\pi}}\tau^{1/2}\gamma\right),$
(23) $\displaystyle\text{III}:\,\,\,$
$\displaystyle\langle\hat{H}\rangle\\!\simeq\\!N\frac{\hbar^{2}\rho^{2}}{2m}\left(\frac{1}{2}\zeta(3/2)+2\gamma-\frac{6\gamma^{2}}{\tau^{2}}\right),$
(24) $\displaystyle\text{IV}:\,\,\,$
$\displaystyle\langle\hat{H}\rangle\\!\simeq\\!N\frac{\hbar^{2}\rho^{2}}{2m}\left(\frac{\tau}{2}+2\gamma-\frac{3}{2}\sqrt{\frac{\pi}{2}}\frac{\gamma^{2}}{\tau^{1/2}}\right),$
(25) $\displaystyle\text{V}:\,\,\,$
$\displaystyle\langle\hat{H}\rangle\\!\simeq\\!N\frac{\hbar^{2}\rho^{2}}{2m}\left(\frac{\tau}{2}+\frac{1}{2}\sqrt{\frac{\pi}{2}}\tau^{1/2}-\sqrt{\frac{\pi}{2}}\frac{\tau^{1/2}}{\gamma}\right),$
(26) $\displaystyle\text{VI}:\,\,\,$
$\displaystyle\langle\hat{H}\rangle\\!\simeq\\!N\frac{\hbar^{2}\rho^{2}}{2m}\left(\frac{\pi^{2}}{3}-\frac{4\pi^{2}}{3\gamma}+\frac{\tau^{2}}{3\gamma}\right),$
(27)

where $\zeta(s)$ is the Riemann zeta function of $s\\!\in\\!\mathbb{R}$.

As described in the main text, the normalized local second-order correlation
function may be rearranged and integrated for the total (integrated)
correlation function,

$\begin{split}\begin{aligned} \\!\overline{G^{(2)}}\equiv\\!\int
dz\langle\hat{\Psi}^{\dagger}(z)\hat{\Psi}^{\dagger}(z)\hat{\Psi}(z)\hat{\Psi}(z)\rangle\\!=\\!\int_{0}^{L}\\!dzg^{(2)}(0)\rho^{2}.\end{aligned}\end{split}$
(28)

Utilizing the linear density, $\rho\\!=\\!N/L$, this may be expressed as
$\overline{G^{(2)}}\\!=\\!N\rho g^{(2)}(0)$, which is used in Eq. (4) of the
main text for expressing the exact net work of the uniform 1D Bose gas in
terms of $g^{(2)}(0)$.

##  Appendix E Exact thermodynamic Bethe ansatz results

Experimental realization of a 1D Bose gas often falls outside the asymptotic
regimes where analytic approximations are applicable. In such situations, we
may utilize the exact Yang-Yang thermodynamic Bethe ansatz [33, 34] to
evaluate the equilibrium properties of the gas required for calculating net
work and efficiency via Eqs. (2) and (3) of the main text, respectively. This
is presented in Fig. 6 (a) for experimentally realistic set of system
parameters that inhabit the boundary between asymptotic parameter regimes II
and III (see Fig. 5). Further, one may utilize the exact TBA to confirm the
results derived via approximate analytics in the main text. This is
illustrated in Figs. 6 (b) and (c), where we see excellent agreement between
these results when the parameters $\gamma$ and $\tau$ are sufficiently deep
into the analytic asymptotic regimes.

Figure 6: Performance of the sudden interaction quench quantum Otto cycle,
numerically evaluated via the thermodynamic Bethe ansatz. Panel a demonstrates
numerically evaluated net work and efficiency for a system with a cold thermal
state defined by $\gamma_{c}\\!=\\!0.1$, $\tau_{c}\\!=\\!0.5$, lying on the
border of regimes II and III (see Fig. 5), and thus lying outside the range of
the analytic approximations utilized in the main text. Panel (b) is a copy of
Fig. 2 (a) of the main text shown here for comparison with the results of
numerical TBA evaluation of the same cycle shown in panel (c). Here, there is
excellent agreement in the net work between panels (b) and (c), with small
disagreement under large interaction strength and temperature ratios, as the
hot thermal state is approaching the edge of the asymptotic regime where the
analytic approximations become less applicable.

##  Appendix F Maximum efficiency and maximum work

For a fixed ratio of temperatures, it was noted in the main text that the
interaction strength ratio corresponding to maximum work approximately
coincides with that for maximum efficiency, which is uncommon for highly
nonequilibrium engine cycles [21, 46]. In the sudden interaction-quench Otto
cycle, such coincidence occurs due to the dependence of the total energy of
the hot and cold thermal equilibrium states on the interaction strength.

As shown in Eq. (14), the total energy may be separated into its kinetic
energy, which scales predominately with temperature, and interaction energy,
which scales predominately with interaction strength. Thus, for a fixed ratio
of temperatures, $\tau_{h}/\tau_{c}$, the difference between the total
energies of the hot and cold thermal state may be given as a sum of two terms:
the first is the kinetic energy difference, determined by the temperature
ratio and therefore approximately constant, the second given by the
interaction energy difference, which scales with the interaction strengths,
$\gamma_{h}$ and $\gamma_{c}$, of the hot and cold thermal states as

$\langle\hat{H}^{int}\rangle_{h}\\!-\\!\langle\hat{H}^{int}\rangle_{c}\\!=\\!N\frac{\hbar^{2}\rho^{2}}{2m}\left(\gamma_{h}g^{(2)}_{h}(0)\\!-\\!\gamma_{c}g^{(2)}_{c}(0)\right).$
(29)

However, when operating within a single asymptotic regime under a moderate
quench of interaction strength, the $g^{(2)}(0)$ correlation function is only
slowly varying with $\gamma$. This means, to first approximation,
$g^{(2)}_{h}(0)\\!\simeq\\!g^{(2)}_{c}(0)$, which in turn transforms the
interaction energy difference to

$\langle\hat{H}^{int}\rangle_{h}\\!-\\!\langle\hat{H}^{int}\rangle_{c}\\!\simeq\\!N\frac{\hbar^{2}\rho^{2}}{2m}\left(\gamma_{h}\\!-\\!\gamma_{c}\right)g^{(2)}_{c}(0).$
(30)

The heat intake, which is given by Eq. (8) of the main text, is therefore well
approximated by

$\displaystyle
Q_{1}=\langle\hat{H}\rangle_{h}-\langle\hat{H}\rangle_{c}-\frac{N\hbar^{2}\rho^{2}}{2m}(\gamma_{h}-\gamma_{c})g^{(2)}_{c}(0)\simeq\langle\hat{H}^{kin}\rangle_{h}-\langle\hat{H}^{kin}\rangle_{c},$
(31)

which is approximately constant for a fixed temperature ratio, as detailed
above. Therefore, the efficiency, which is given by $\eta\\!=\\!W/Q_{1}$,
scales predominately with $W$, hence $\eta\\!\propto\\!W$.

##  Appendix G Thermal operation regimes of other QTM’s

Under large interaction strength quenches, for fixed temperatures $\tau_{c}$
and $\tau_{h}$, it was noted in the main text that the heater is the
inevitable mode of operation. This scenario requires the fulfilment of two
conditions: first, we require $Q_{1}\\!<\\!0$, meaning the magnitude of the
work out,
$|W_{1}|\\!=\\!N\frac{\hbar^{2}\rho^{2}}{2m}(\gamma_{h}\\!-\\!\gamma_{c})g^{(2)}_{c}(0)$
(where $W_{1}<0$) exceeds the energy gap between the hot and cold thermal
states, given by $\langle\hat{H}\rangle_{h}\\!-\\!\langle\hat{H}\rangle_{c}$.
We further require $Q_{2}\\!>\\!0$, meaning that the work in,
$W_{2}\\!=\\!N\frac{\hbar^{2}\rho^{2}}{2m}(\gamma_{h}\\!-\\!\gamma_{c})g^{(2)}_{h}(0)$
(where $W_{2}>0$), must remain less than this same gap (see Fig. 3 (c) of the
main text).

As detailed above, in Appendix F on maximum efficiency at maximum work, the
total energy difference between the hot and cold thermal states for fixed
temperature ratios, $\tau_{h}/\tau_{c}$, is given by a sum of the kinetic
energy difference, which is approximately constant, and the interaction energy
difference, where the correlation function is approximately constant in a
single asymptotic regime, and hence given by Eq. (29). In contrast, for a
large quench in interaction strength, the correlation function is no longer
approximately constant, and $g^{(2)}_{h}(0)$ is strongly monotonically
decreasing as a function of $\gamma_{h}$, i.e.
$g^{(2)}_{h}(0)\\!<\\!g^{(2)}_{c}(0)$. We therefore find that the work input,
$W_{2}$, exceeds the interaction energy difference given in Eq. (29),

$W_{2}\propto(\gamma_{h}-\gamma_{c})g^{(2)}_{c}(0)>\gamma_{h}g^{(2)}_{h}(0)\\!-\\!\gamma_{c}g^{(2)}_{c}(0).$
(32)

Further, as the kinetic energy term is approximately constant, $W_{2}$
inevitably exceeds the difference in total energy between the hot and cold
thermal states due to its linear dependence on $\gamma_{h}$.

Similarly, since $g^{(2)}_{h}(0)$ monotonically decreases with $\gamma_{h}$
for large quenches of interaction strength, the magnitude of the work output,
$|W_{1}|\propto(\gamma_{h}-\gamma_{c})g^{(2)}_{h}(0)$, remains less than the
energy gap between the hot and cold thermal states:

$|W_{1}|\propto(\gamma_{h}-\gamma_{c})g^{(2)}_{h}(0)<\gamma_{h}g^{(2)}_{h}(0)\\!-\\!\gamma_{c}g^{(2)}_{c}(0).$
(33)

These two conditions, taken together, imply operation as a heater under large
interaction quenches.

In contrast, for any fixed value of $\gamma_{c}$ and $\gamma_{h}$, an
increasingly higher temperature of the hot thermal state, $\tau_{h}$, means
that the corresponding correlation function, $g^{(2)}_{h}(0)$, monotonically
increases towards its maximum value of $g^{(2)}_{h}(0)\\!\simeq\\!2$, which is
achieved in regime IV, defined in Eq. (19). Thus, there is always a value of
$\tau_{h}$ such that $g^{(2)}_{h}(0)\\!>\\!g^{(2)}_{c}(0)$, turning the Otto
cycle into the engine operation regime.

Finally, in the refrigerator thermal operation regime, for $\tau_{h}=\tau_{c}$
and an infinitesimal quench of interaction strength,
$\gamma_{h}-\gamma_{c}=\delta\gamma$, the net work vanishes as
$W\\!\propto(\gamma_{h}-\gamma_{c})$
$\times(g^{(2)}_{h}(0)-g^{(2)}_{c}(0))\propto\delta\gamma^{2}$. This occurs as
the zeroth order terms in the correlation function cancel when taking their
difference in a single asymptotic regime. In contrast, the heat intake,
$Q_{1}$, depends on the difference in total energy, which to first order
vanishes as $Q_{1}\\!\propto\\!\delta\gamma$. This results in
$\mathrm{CoP[R]}\\!=\\!|Q_{1}|/W\\!-\\!1\\!\propto\delta\gamma^{-1}$, which
diverges as $\delta\gamma\\!\to\\!0$, as noted in the main text.

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