# Quantum signatures in a quadratic optomechanical heat engine with an atom in
a tapered trap

Mohsen Izadyari  1,* Mehmet Öncü  2 Kadir Durak  2 and Özgür E.
Müstecaplıoğlu1,3  1Department of Physics, Koç University, Istanbul, 34450,
Turkey
2Department of Physics, Özyeğin University, Istanbul, 34794, Turkey
3TÜBİTAK Research Institute for Fundamental Sciences, Gebze, 41470, Turkey
<EMAIL_ADDRESS>
###### Abstract

We investigate how quantum signatures can emerge in a single atom heat engine
consisting of an atom confined in a tapered trap and subject to hot and cold
thermal reservoirs. A similar system was realized experimentally in Ref. [1].
We model such a system using a quadratic optomechanical model and identify an
effective Otto cycle in the system’s dynamics. We compare the engine’s
performance in the quantum and classical regimes by evaluating the power
dissipated. We find that lowering the temperature is insufficient to make the
single atom engine of Ref. [1] a genuine quantum-enhanced heat engine. We show
that it is necessary to make the trap more asymmetric and confined to ensure
that quantum correlations cause an enhancement in the power output.

††journal: optcon††articletype: Research Article

##  1 Introduction

Quantum heat engines (QHEs) are machines harnessing thermal energy using
quantum working substances [2, 3]. While the three-level maser, a quantum
device that amplifies microwave radiation, was recognized as a QHE several
decades ago [4], systematic studies of QHE have attracted much attention quite
recently [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. A pioneering
experiment was the demonstration of a piston-type (Otto) engine using a single
trapped ion [1]. Due to the relatively small energy gaps of the trapped ion at
the involved temperatures, genuine, profound quantum effects in the single-
atom piston were assumed to be non-existent. Here, we ask if it is sufficient
to lower the temperature to bring the experimental setup to the quantum regime
or if it is necessary to make more modifications. To answer this question, we
use a higher-order, so-called quadratic, optomechanical model of the
interaction between the motional degrees of freedom of the atom arising due to
the tapered trap geometry of the experiment.

It was intuitively expected that an optomechanical-type mutual interaction
effectively emerges when an atom is confined in a funnel-shaped trap [1].
Here, we explicitly verify this intuition, but surprisingly, we find an
additional degenerate three-wave mixing type coupling. It is a well-known
interaction for generating squeezed states [37, 38, 39]. After that, we
explore the signatures of the quantum squeezing in the system by
differentiating classical and quantum correlations via solving master and
classical stochastic Langevin equations separately. In a single atom system
trapped in a tapered trap, we find that quantum correlations arise by
increasing coupling strength between axial and radial modes. These quantum
correlations can significantly enhance the dissipated power relative to the
classical correlations. According to our model, the coupling strength depends
on the trap’s geometry and frequencies asymmetry. So, we indicate that the
dissipated power can be enhanced by increasing the trap asymmetry and reducing
the trap radius. The results show that lowering the temperature of the
experiment [1] per se is not sufficient to bring the single-ion engine to the
quantum regime. As a result, making the trap geometry more asymmetric and the
atom more confined at low temperatures make quantum squeezing type
correlations more significant for a remarkable performance increase than the
classical stochastic engine. Accordingly, we conclude that, in addition to
lowering the temperature, the original single-atom heat engine setup in Ref.
[1] can be converted into a genuine quantum-enhanced heat engine in a more
asymmetric and confined configuration.

This paper is organized as follows. In Sec. 2, we describe our Hamiltonian
model to describe the experimental ion in a tapered trap system in Ref. [1].
In Sec. 3, we investigate the dynamics of the system using the master equation
approach. We identify an effective Otto cycle for the radial mode in Sec. 4.
Sec. 5 compares the stochastic and quantum dissipated power outputs through
the axial mode. We conclude in Sec. 6.

##  2  Model System

Figure 1: Cross sections of the approximate (Left panels) and exact (Right
panels) funnel-shaped potential (in units of eV) of the tapered trap geometry.
The approximation requires $z\tan\theta\ll r_{0}$ where $\theta=30^{o}$ is the
tapered trap angle and $r_{0}=1.1$ mm is the trap radius at $z=0$. The
calculations are done for the Calcium ion of mass $m=40$u in atomic unit
$u=1.66\times 10^{-27}$ kg.

Single ion-heat engine realized in Ref. [1] consists of a Calcium ion of mass
$m$ in a tapered trap geometry. In the axial ($z$) direction the trap
frequency is $\omega_{z}$. In the directions, $x,y$, the trap frequencies are
given by

$\omega_{\alpha}=\frac{\omega_{0\alpha}}{(1+\epsilon z)^{2}},$ (1)

where $\epsilon:=(\tan\theta)/r_{0}$. Here, $r_{0}=1.1$ mm is the radius of
the trap at $z=0$ and $\theta=30^{o}$ is the trap angle. Trap frequencies
along x and y axes are given as $\omega_{0x}$ and $\omega_{0y}$, respectively.
It is found that during the heat engine operation, the ion makes axial
coherent oscillations with amplitude in the order of micrometers [1].
Accordingly, we can consider $\epsilon z\ll 1$ for the description of heat
engine dynamics. In this limit, the trap potential,

$U(x,y,z)=\frac{m}{2}(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2}),$
(2)

can be approximated by

$V(r,y)=\frac{m}{2}\omega_{r}^{2}r^{2}(1-gz)+\frac{m}{2}\omega_{z}^{2}z^{2},$
(3)

where $r^{2}=x^{2}+y^{2}$, and $g=4\epsilon\approx 2.1$ (1/mm). We neglected
the small difference between $\omega_{x}$ and $\omega_{y}$ and replace them
with the mean radial trap frequency, $\omega_{r}:=(\omega_{0x}+\omega_{0y})/2$
[1]. The trap potential $U(x,y,z)$ (using Eq. (1) to define $\omega_{x}$ and
$\omega_{y}$) and its approximation by $V(r,z)$ are compared in Fig. 1 (right
and left panels respectively). According to Fig. 1 (upper panels), the exact
potential can be superseded by approximated potential for small values of z.

The trapped ion is always subject to a cooling laser yielding the effect of a
cold thermal bath. An electrical white noise is applied to the radial degree
of freedom for periodical heating as illustrated in Fig. 2. The heat engine
system with two radial ($a$) and axial ($b$) vibrational modes (coupled by
$\beta$) is shown in Fig. 2. Accordingly, two thermal baths at temperatures
$T_{h}$ and $T_{a}$ couple to radial mode, and the axial mode is coupled to a
bath at temperature $T_{b}$. The coupling between system and thermal bath at
temperature $T_{h}$ is periodically turned on and off. The engine cycle is
examined classically in Ref. [1], which is justified for
$k_{B}T\gg\hbar\omega_{r}$, where $T$ is considered as the working
temperatures of the engine. Indeed, for all temperatures of interest,
including the high-temperature condition $T_{c}\gg\hbar\omega_{r}$ is
satisfied. On the other hand, the particular coupling of the axial and radial
degrees of freedom in Eq. (3) contains a quadratic nonlinearity in $r$. From
the quantum mechanical point of view, such a nonlinearity could yield a
quantum noise squeezing effect.

Let us express the quantum model of the center-of-mass (CM) motion, in the
form

$\displaystyle H_{0}$ $\displaystyle=\frac{P_{r}^{2}+P_{z}^{2}}{2M}+V(r,z).$
(4)

Here, $P_{r}$ and $P_{z}$ stand for the components of the CM momentum
operator, with $P_{r}^{2}=P_{x}^{2}+P_{y}^{2}$. We apply the harmonic-
oscillator quantization $r=(\hbar/m\omega_{r})^{1/2}\hat{q}_{r}$,
$z=(\hbar/m\omega_{z})^{1/2}\hat{q}_{z}$, $P_{r}=(\hbar
m\omega_{r})^{1/2}\hat{p}_{r}$, and $P_{z}=(\hbar
m\omega_{z})^{1/2}\hat{p}_{z}$ with
$\hat{p}_{r}=i(\hat{a}^{\dagger}-\hat{a})/\sqrt{2}$,
$\hat{p}_{z}=i(\hat{b}^{\dagger}-\hat{b})/\sqrt{2}$,
$\hat{q}_{r}=(\hat{a}^{\dagger}+\hat{a})/\sqrt{2}$, and
$\hat{q}_{z}=(\hat{b}^{\dagger}+\hat{b})/\sqrt{2}$ to write the Hamiltonian
(4) in terms of the vibrational phonon operators, (we take $\hbar=1$)

$\displaystyle\hat{H}_{\text{cm}}$
$\displaystyle=\omega_{r}\hat{n}_{r}+\omega_{z}\hat{n}_{z}-\frac{\beta}{4}(\hat{a}^{2}+\hat{a}^{\dagger
2}+2\hat{a}^{\dagger}\hat{a}+1)(\hat{b}+\hat{b}^{\dagger}),$ (5)

where we have dropped the constants $\omega_{r}/2$ and $\omega_{z}/2$.
$\beta\equiv g\omega_{r}\sqrt{\hbar/2m\omega_{z}}$ is the coupling coefficient
of $r$ and $z$ degrees of freedom; $\hat{n}_{r}=\hat{a}^{\dagger}\hat{a}$, and
$\hat{n}_{z}=\hat{b}^{\dagger}\hat{b}$ are the phonon number operators in $r$
and $z$ directions, respectively. We remark that the coupling coefficient
$\beta$ depends on trap geometry through the coupling constant g.

Figure 2: Diagram of the heat engine system based on the radial (a) and axial
(b) vibrational modes whose coupling is characterized by the parameter
$\beta$. Radial mode is coupled to two thermal baths at temperatures $T_{h}$
and $T_{a}$, where the coupling to the bath at temperature $T_{h}$ is
periodically turned on and off, and the axial mode is coupled to a bath at
temperature $T_{b}$. The background environment common to both modes is not
shown.

The Hamiltonian (5) can be compared to an optomechanical-like Hamiltonian, but
in addition to the occupation of mode $a$, quantum squeezing-type correlations
can induce additional pressure on mode-$b$ to produce coherent displacement or
useful work. To isolate the effects of different coupling mechanisms in Eq.
(5), we simulate each of them using the separate models,

$\displaystyle H_{\text{om}}$
$\displaystyle=\omega_{r}\hat{n}_{r}+\omega_{z}\hat{n}_{z}-\frac{\beta}{2}\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger}),$
(6) $\displaystyle H_{\text{class}}$
$\displaystyle=\omega_{r}\hat{n}_{r}+\omega_{z}\hat{n}_{z}-\frac{\beta}{4}(\hat{b}+\hat{b}^{\dagger}),$
(7) $\displaystyle H_{\text{sq}}$
$\displaystyle=\omega_{r}\hat{n}_{r}+\omega_{z}\hat{n}_{z}-\frac{\beta}{4}(\hat{a}^{2}+\hat{a}^{\dagger
2})(\hat{b}+\hat{b}^{\dagger}).$ (8)

Eq. (6) describes a pure optomechanical coupling where only the occupation of
mode-$a$ produces pressure on mode-$b$. Modes are uncoupled in Hamiltonian
(7), but there is still an effective classical drive on mode-$b$. Eq. (8)
injects coherence in mode-$b$ depending on the squeezing of mode $a$.

In the next section, we will simulate the system based on total Hamiltonian
Eq. (5) as system $A_{0}$, and each of the models in Eqs. (6-8) as systems
$A_{1}$, $A_{2}$ and $A_{3}$, respectively. Let’s emphasize that
optomechanical coupling can not emerge per se but is always accompanied by the
other classical and squeezed drive terms, and each coupling contributes with
comparable or the same strengths. Our separation is artificial and only for
the purpose of shedding light on the individual effects of each term on the
coherent dynamics of mode-b, in particular on the work output of the single
ion heat engine. We note that Ref. [24] assumes Gaussian thermal distribution
to model the spatial distribution of mode-a. This description would be closer
to the classical treatment of our optomechanical model of $A_{1}$ given by Eq.
(6).

##  3 Results

In this section, we introduce the parameters we used in our simulations and
describe the open quantum system dynamics of the system in terms of a master
equation.

###  3.1 Parameters

The experiment [1] has been performed at high temperature (in the order of
mK). As the usual route to the quantum regime is to reduce the temperature, we
consider the temperature in the order of the $\mu$K. However, we find that
lowering the temperature is insufficient to see the system’s quantum effects.
It is required to demonstrate the system can harness more power than its
stochastic counterpart to be classified as a genuine quantum (enhanced) heat
engine [40]. In addition to reducing the temperature, increasing $\beta$ plays
a crucial role in achieving clear signatures of quantum enhancement in the
heat engine operation with a single ion in a tapered trap. It is necessary to
exploit quantum squeezing type correlations to enhance power output relative
to the case where there are only classical correlations.

In the following, we take $T_{h}=166~{}\mu$K and $T_{a}=T_{b}=4~{}\mu$K while
we suppose the atom’s initial temperature is $T_{0}=10~{}\mu$K. Also, the
radial and axial frequencies are taken $\omega_{r}/2\pi=1$ MHz and
$\omega_{z}/2\pi=50$ kHz, respectively. According to the definition of the
coupling coefficient ($\beta\propto\tan\theta/r_{0}$), we consider high trap
asymmetry by taking $\theta=\pi/4$ and considering more confined traps with
$r_{0}\lesssim 20~{}\mu$m [41, 42] to make the coupling stronger. For example,
for $r_{0}\approx 2~{}\mu$m and using the aforementioned parameters, we find
$\beta/2\pi=100$ kHz. We emphasize that these parameters are not arbitrary but
taken as an example from a range of values for which the temperatures are
sufficiently low and squeezing type coupling is sufficiently large to get into
the quantum enhanced regime of engine operation.

The dissipation rates of mode $a$ to the cold and hot baths are considered to
be $\kappa_{a}/2\pi=\kappa_{h}/2\pi=200$ kHz, where $\kappa_{h}$ is taken as a
time-dependent square wave $\kappa_{h}(t):=\kappa_{h}u(t)$. The time-dependent
part is $u(t)=1$ for the heating and otherwise $u(t)=0$. Also, the dissipation
rate of mode $b$ to the cold bath is $\kappa_{b}/2\pi=50$ kHz.

###  3.2 The dynamics of the system

According to Eq. (5), the model effectively describes two nonlinearly coupled
single-mode resonators. We assume a microwave white noise and a cooling laser
are applied to the mode $a$, as shown in the experiment of Ref. [1]. This
continuous electrical noise and cooling laser plays the roles of two cold and
heat baths at temperatures $T_{a}$ and $T_{h}$ such that $T_{h}>T_{a}$. An
extra cooling laser beam is introduced to mode $b$ to keep the axial mode’s
oscillations in control by tuning its parameters on purpose to get a limit
cycle at temperature $T_{b}$. The background environment with temperature
$T_{0}$ has been ignored during the dynamics by assuming the coupling
coefficient of background environment with mode $b$ and $a$ less than
$\kappa_{j}$ where $j\in\\{h,a,b\\}$ [43]. The dynamics of the density matrix
$\rho$ of the two-resonator system is determined by the master equation [44,
45, 46]

$\displaystyle\frac{d\rho}{dt}$ $\displaystyle=-i[\hat{H}_{\lambda},\rho]$
$\displaystyle+\kappa_{a}(\bar{n}_{a}+1)D[\hat{a}]+\kappa_{a}\bar{n}_{a}D[\hat{a}^{\dagger}]$
$\displaystyle+\kappa_{h}(t)(\bar{n}_{h}+1)D[\hat{a}]+\kappa_{h}(t)\bar{n}_{h}D[\hat{a}^{\dagger}]$
$\displaystyle+\kappa_{b}(\bar{n}_{b}+1)D[\hat{b}]+\kappa_{b}\bar{n}_{b}D[\hat{b}^{\dagger}],$
(9)

where $\lambda=\text{cm, om, class, sq}$ represents different models and
$D[\hat{\alpha}]=(1/2)(2\hat{\alpha}\rho\hat{\alpha}^{\dagger}-\hat{\alpha}^{\dagger}\rho\hat{\alpha}-\rho\hat{\alpha}^{\dagger}\hat{\alpha})$
refers to the Lindblad dissipator superoperators with
$\hat{\alpha}=\hat{a},\hat{b}$. Mean number of phonons in mode $a$ and $b$ at
the bath temperature $T_{u}$ and $T_{b}$ is denoted by $\bar{n}_{u}$ with
$u\in\\{a,h\\}$ and $\bar{n}_{b}$, given by the Planck distribution (we take
$k_{B}=1$)

$\displaystyle\bar{n}_{u}$ $\displaystyle=\frac{1}{\exp(\omega_{r}/T_{u})-1},$
$\displaystyle\bar{n}_{b}$ $\displaystyle=\frac{1}{\exp(\omega_{z}/T_{b})-1}.$
(10)

We assume that before the cooling and heating lasers are applied, the ion and
its vibrational modes are in thermal equilibrium with the background thermal
environment at temperature $T_{0}$. Initial state then can be expressed as a
canonical thermal (Gibbs) state such that

$\displaystyle\rho(0)$
$\displaystyle=\frac{1}{\text{Tr}[e^{-\hat{H}_{\lambda}/T_{0}}]}e^{-\hat{H}_{\lambda}/T_{0}},$
(11)

and the initial states of the vibrational modes are determined by the reduced
density matrices $\rho_{b}(0)=\Tr_{b}(\rho(0))$ and
$\rho_{a}(0)=\Tr_{a}(\rho(0))$. We solve the master equation to determine
$\rho(t)$ for the given $\rho(0)$ using Python programming language and an
open source quantum optics package QuTiP [47]. Taking the partial trace of
$\rho(t)$ over the degrees of freedom of mode-$a$, we find the reduced state
of the mode-$b$ such that $\rho_{b}(t)=\Tr_{a}[\rho(t)]$. Time dependence of
the mean number of vibrational phonons in mode-$b$ is evaluated by
$\langle\hat{n}_{z}\rangle=\Tr(\rho_{b}(t)\hat{b}^{\dagger}\hat{b})$. Similar
calculations give dynamics of $\langle\hat{n}_{r}\rangle$, too.

Figure 3: Dynamics of the mean number of vibrational phonons $\langle
n_{z}\rangle$ in modes $b$ where time is scaled with $\omega_{z}$. The
parameters used in the plots are taken to be $\omega_{r}/2\pi=1$ MHz,
$\omega_{z}/2\pi=50$ kHz, $\bar{n}_{h}=3$, and $\beta/2\pi=100$ kHz. The solid
black, dotted blue, dash-dotted green, and dashed red lines show the evolution
of $\langle n_{z}\rangle$ under the models $A_{0}$, $A_{1}$, $A_{2}$, and
$A_{3}$, respectively.

The evolutions of $\langle\hat{n}_{z}\rangle$ for the four models $A_{0}$ to
$A_{3}$ are shown in Fig. 3. Since all the models $A_{0}$ to $A_{3}$ have
different Hamiltonians, the initial conditions differ. Nevertheless, the
differences in the initial value of $\langle\hat{n}_{z}\rangle$ are too small
to be visible in Fig. 3. Among all the terms in the overall interaction
$A_{0}$, optomechanical type coupling $A_{1}$ is the only mechanism to
increase the axial mode population. However, despite the qualitative
agreement, $A_{1}$ per se cannot quantitatively explain the
$\langle\hat{n}_{z}\rangle$ behavior. While the squeezing term $A_{3}$ cannot
directly populate $\langle\hat{n}_{z}\rangle$; it plays a subtle role to
increase $\langle\hat{n}_{z}\rangle$ beyond what is possible solely by
$A_{1}$.

Figure 4: The time evolutions of the variances of the quadratures
$(\langle\Delta\hat{q}_{r}\rangle)^{2}$ (red dotted line)
and$(\langle\Delta\hat{p}_{r}\rangle)^{2}$ (blue solid line) of the radial
mode. The inset magnifies the squeezing region where Variance $<1/4$.

Intuitively, the effect of squeezing-type terms $A_{3}$ in $A_{0}$ can be
interpreted as an enhancement of vibrational radial pressure on the axial
mode. When $A_{3}$ terms are present, the number of radial vibrational phonons
during the heating stage is expected to increase so that the optomechanical-
like coupling $A_{1}$ in $A_{0}$ can yield higher $\langle\hat{n}_{z}\rangle$.
The more radial phonons cause stronger pressure on the axial mode to transfer
more energy or power. To appreciate how the quantum character of the modes
enters into this picture, let us look at the dynamics of
$\langle\hat{n}_{z}\rangle$ according to the master equationn (3.2)

$\displaystyle\frac{d}{dt}\langle\hat{n}_{z}\rangle=\frac{\beta}{\sqrt{2}}(\langle\hat{q}_{r}^{2},\hat{p}_{z}\rangle+\langle\hat{q}_{r}^{2}\rangle\langle\hat{p}_{z}\rangle)-\kappa_{b}(\langle\hat{n}_{z}\rangle-\bar{n}_{b}),$
(12)

where
$\langle\hat{q}_{r}^{2},\hat{p}_{z}\rangle:=\langle\hat{q}_{r}^{2},\hat{p}_{z}\rangle-\langle\hat{q}_{r}^{2}\rangle\langle\hat{p}_{z}\rangle$
is a quantum correlation function between the radial and axial modes. The
dissipated power output of the engine is determined by
$P_{\text{dis}}=\omega_{z}\kappa_{b}(\langle\hat{n}_{z}\rangle-\bar{n}_{b})$,
which measures the net flux of energy dissipated into the environment [48]. In
order to produce more power than a stochastic engine that can only have
classical correlations, it is necessary for us to ensure that the system has
strong quantum correlations associated with the second moment of the
displacement of the radial mode. Quantum character of the
$\langle\hat{q}_{r}^{2}\rangle$ can be examined in terms of the quadrature
variances of the radial mode

$\displaystyle(\langle\Delta\hat{q}_{r}\rangle)^{2}$ $\displaystyle=$
$\displaystyle Tr[\rho_{a}\hat{q}_{r}^{2}]-(Tr[\rho_{a}\hat{q}_{r}])^{2},$
(13) $\displaystyle(\langle\Delta\hat{p}_{r}\rangle)^{2}$ $\displaystyle=$
$\displaystyle Tr[\rho_{a}\hat{p}_{r}^{2}]-(Tr[\rho_{a}\hat{p}_{r}])^{2},$
(14)

where $\rho_{a}(t)=Tr_{b}[\rho(t)]$ is the reduced density matrix of the
radial mode. When a quadrature variance is less than its vacuum value of
$1/4$, the state is said to be quadrature squeezed.

The time evolutions of $(\langle\Delta\hat{q}_{r}\rangle)^{2}$ (red dotted
line) and $(\langle\Delta\hat{p}_{r}\rangle)^{2}$ (blue solid line) are shown
in Fig. 4. The yellow shaded area, where Variance $<1/4$, in Fig. 4 indicates
that the radial mode starts each cycle in a squeezed thermal state. Fig. 5
plots the Wigner functions of the radial mode for different $\beta$, at the
same scaled time after thermalization of the system, when the system dynamics
is governed by the full model $A_{0}$. Wigner function of a thermal state in
the quadrature phase space has circular contours, which are changed to
ellipses due to quadrature squeezing. In the Fig. 5, we show that although the
quadrature phase space of radial mode has circular contours for small $\beta$
(Fig. 5 (a)), it becomes to ellipses contours by enhancing $\beta$ (figs. 5
(b,c)). So, for small $\beta$, the eccentricity is almost zero, and the
squeezing is negligible even at low temperatures. Accordingly, the dissipated
power of the engine will be dominated and determined by the classical
correlations only. On the other hand, for large values of $\beta$, second
order moments of the displacement operator should be treated quantum
mechanically in Eq. 12 and hence to the calculation of the dissipated power.
We present explicit calculation of $P_{\text{dis}}$ for both quantum and
classical stochastic treatments in Sec. 5.

(a)  (b)  (c)

Figure 5: Colour online)(a)-(c) The Wigner functions in the $p_{r}$, $q_{r}$
field quadrature phase space of the radial mode plotted for (a)
$\beta/2\pi=10$ kHz, (b) $\beta/2\pi=100$ kHz, (c) $\beta/2\pi=200$ kHz. The
other system parameters are $\omega_{r}/2\pi=1$ MHz, $\omega_{z}/2\pi=50$ kHz.

##  4 Effective Otto Cycle For the Radial Mode

For a single ion in an asymmetrical Paul trap [1], the hot bath is simulated
by the external white electric-field noise. The cold bath interaction is
realized by exposing the ion to the laser cooling beam. According to [1, 29],
the cooling laser is kept on during the process. Despite the continuous
coupling to the heat baths, we can identify an Otto engine cycle in our model.
We consider the mode $a$ as the working substance and treat the mode $b$ as
the engine’s flywheel where the work is stored. We replace the flywheel
displacement $q_{z}=\langle\hat{b}+\hat{b}^{\dagger}\rangle$ as a number in
model $A_{0}$. The reduced semi-quantum model describes a squeezed oscillator
Hamiltonian for mode $a$, whose diagonalization by the Bogoluibov
transformation (see A.) yields an effective frequency of
$\omega_{\text{eff}}$, and the effective mean energy can be defined as
$U=\langle H_{\text{cm}}^{{}^{\prime}}\rangle$.
The Otto engine is identified in Fig. 6 with four steps [29].

(A$\rightarrow$B) Hot isochore: the effective trapping frequency
($\omega_{\text{eff}}$) is kept constant at $\omega_{h}$ while contacting with
hot bath at temperature $T_{h}$. The thermalization will be done after a time
of $\tau_{h}$.

(B$\rightarrow$C) Adiabatic expansion: The atom evolves in the trapping
potential while isolated from the hot bath. After a time $\tau_{z}$, the
effective frequency will be changed from $\omega_{h}$ to $\omega_{c}$, and
work will be done by considering the displacement in the axial direction.

(C$\rightarrow$D) Cold isochore: the effective frequency remains in
$\omega_{c}$, and the cold bath with temperature $T_{c}$ is contacted with the
system for a time $\tau_{c}$.

(D$\rightarrow$A) Adiabatic compression: the effective frequency returns to
the initial frequency another time $\tau_{z}$ in the last step.
The time scales can be considered by $\tau_{h}\approx\tau_{c}<\tau_{z}$. An
effective, finite-time (non-ideal) Otto cycle (ABCD) is defined in the diagram
(Fig. 6) where we plot dependence of $U$ on $\omega_{\text{eff}}$ in the
steady state at $\bar{n}=3$.

Figure 6: Effective energy-frequency diagram of the radial mode $a$ by
considering the dependence of the mean effective energy $U$ to the effective
frequency $\omega_{\text{eff}}$ (in units of $\hbar\omega_{z}$).

###  4.1 Performance of the engine

We have seen that the engine operates by converting the thermal energy
harvested by the radial mode into coherent oscillations of the axial
vibrational mode. We can envision the vibrational mode as the work reservoir
storing coherent energy and examine the engine cycle for the radial mode. For
that aim, let’s consider a semi-classical model of the vibrational interaction
by replacing the coherent axial mode operators, in particular $q_{z}$ operator
in the vibrational coupling with a c-number, which is its expectation value.
After that, we can apply the Bogoluibov transformation (as described in A) to
the effective semi-classical model, which yields a simple uncoupled quantum
oscillator model. Accordingly, we conclude that the vibrational modes,
particularly the radial mode, are Gaussian states under the semi-classical
model. More specifically, the radial mode is a Gibbsian (thermal equilibrium)
Gaussian state for which a simple Von-Neumann entropy expression can be
derived in the form [43]

$S=(1+\langle\hat{n}_{c}\rangle)\ln(1+\langle\hat{n}_{c}\rangle)-\langle\hat{n}_{c}\rangle\ln(\langle\hat{n}_{c}\rangle).$
(15)

The effective temperature of the radial mode Gibbsian state is given by

$T_{\text{eff}}=\omega_{\text{eff}}/\ln(1+1/\langle\hat{n}_{c}\rangle).$ (16)

Here, the $\langle\hat{n}_{c}\rangle$ comes from the diagonalized form of
radial mode in Hamiltonian employing Bogoliubov transformation (A).

The temperature-entropy $T-S$ diagram is plotted in Fig. LABEL:fig_entropy-tem
which closely resembles an Otto engine. The potential work output can be
obtained by calculating the area of the T-S diagram. We approximate the work
from the T-S diagram, and the net work is estimated by $W\approx
0.22~{}\hbar\omega_{z}\simeq 7.3\crossproduct 10^{-30}$ joules. The Power can
be calculated by dividing work by the cycle time $T=20\crossproduct
2\pi/\omega_{z}$ as $P\simeq 2.9\crossproduct 10^{-27}$ Watts. The determined
heat intake of mode $a$ is $Q_{\text{in}}\simeq 3\crossproduct 10^{-28}$
joules, and therefore the efficiency is $\eta\simeq 2.4\%$.

Figure 7: T-S diagram of the mode $a$ which closely resembles that of an Otto
engine (in units of $\hbar\omega_{z}$)

.

The coupling coefficient $\beta$ in Hamiltonian (5) plays an essential role in
producing work and achieving higher efficiency, which can be done by revising
the trap’s geometry in the experiment [1]. Engine performance improves by
applying the quantum correlations, which are significant only for much higher
coupling regimes and at low temperatures. According to the definition of
$\beta\propto 1/r_{0}$, we can increase $\beta$ by lowering the trap’s radius
$r_{0}$. As we argued in Sec. 3.1, sufficiently large $\beta$ can be achieved
by reducing the trap’s radius to the micrometer scale. Fig. 8 shows the
dependence of the extracted work, determined by the area of the $T-S$ diagram,
on $r_{0}$. Here, $\theta$, radial, and axial frequencies are kept constant so
that $\beta$ increases with $r_{0}$. It can be pointed out that the work
output increases strongly in tightly confined tapered traps. We remark,
however, that increasing the coupling coefficient $\beta$ is limited in our
treatment where we use a local optomechanical master equation, which requires
$\beta$ to be smaller than the radial frequency [49].

Figure 8: Extracted work $W$ (in unit of $\hbar\omega_{z}$) versus trap’s
radius $r_{0}$ ($\mu$m) where radial and axial frequencies are constant
($\omega_{r}/2\pi=1$ MHz, and $\omega_{z}/2\pi=50$ kHz) and $\theta=\pi/4$.

##  5 Classical vs Quantum Heat Engine Regimes

The previous section identifies an effective Otto cycle by treating the radial
mode as a working fluid and assuming the coherent displacements of axial mode
are classical. This treatment allows us to use conventional thermodynamic
cycle diagrams in terms of mean effective energy, effective temperature,
effective frequency (spatial-like parameter), and entropy for our reduced
quantum oscillator model for the radial mode. In this case, the potential work
output of the effective Otto cycle is computed from the $T-S$ diagram. The
missing piece of this engine picture is the flywheel, which is the axial mode.
The work is translated to the piston (axial) mode, and it is determined in
terms of the dissipated power through the axial mode, which we will calculate
in the present section. Dissipated power is essentially determined with the
population of the axial mode (mean number of axial phonons). As we have seen
in Sec. 3, the dynamical equation of the mean number of axial phonons includes
a quantum correlator term, which can be significant on top of classical
correlations when quantum squeezing is present and can enhance the dissipated
power. When there are only classical correlations, we solve the Langevin
equations to determine the mean number of axial phonons. In addition, we
determine the mean number of axial phonons from the quantum master equation as
well. Using these two methods, we will determine the mean number of phonons to
be substituted to the same dissipated power formula to compare the classical
and quantum dissipated power outputs with each other.

To compare the quantum engine with its classical counterpart, we follow the
standard method of neglecting the non-commutative character of the quantum
operators (for example, position and momentum operators). In the
optomechanical engine models, they are replaced with c-numbers,
$\langle\hat{a}\rangle\rightarrow\upsilon_{r}$ and
$\langle\hat{b}\rangle\rightarrow\upsilon_{z}$ [43, 48]. The Heisenberg
equations of motion are then changed to classical Langevin equations, which
further include the stochastic noise terms describing the classical models of
the heat baths in the form

$\displaystyle\dot{\upsilon}_{r}$
$\displaystyle=-i\left(\omega_{r}\upsilon_{r}-\frac{\beta}{2}(\upsilon_{r}^{*}+\upsilon_{r})(\upsilon_{z}+\upsilon_{z}^{*})\right)$
$\displaystyle-\frac{\kappa_{r}+\kappa_{h}}{2}\upsilon_{r}+\xi_{h}(t)+\xi_{r}(t),$
$\displaystyle\dot{\upsilon}_{z}$
$\displaystyle=-i\left(\omega_{z}\upsilon_{z}-\frac{\beta\upsilon_{z}}{4}(\upsilon_{r}^{2}+\upsilon_{r}^{*2}+2\upsilon_{r}\upsilon_{r}^{*}+1)\right)$
$\displaystyle-\frac{\kappa_{b}}{2}\upsilon_{z}+\xi_{z}(t).$ (17)

Here $\xi_{k}(t)$ presents time dependent delta-correlated and temperature
dependent stochastic noise terms where $k=r,z,h$ [43, 48].

Defining the field quadratures $X_{i}$, $Y_{i}$ as
$\upsilon_{i}=1/\sqrt{2}(X_{i}+iY_{i})$, and writing the noise parameters as
$\xi_{i}=1/\sqrt{2}(\xi_{x}^{i}+i\xi_{y}^{i})$ where $i=r,z$ we can simulate
the equations as:

$\displaystyle dX_{r}$
$\displaystyle=\left(\omega_{r}Y_{r}-\frac{\kappa_{a}+\kappa_{h}}{2}X_{r}\right)dt+dW_{x}^{h}+dW_{x}^{a},$
$\displaystyle dY_{r}$ $\displaystyle=\left(-\omega_{r}X_{r}+\sqrt{2}\beta
X_{r}X_{z}-\frac{\kappa_{a}+\kappa_{h}}{2}Y_{r}\right)dt+dW_{y}^{h}+dW_{y}^{a},$
$\displaystyle dX_{z}$
$\displaystyle=\left(\omega_{z}Y_{z}-\frac{\kappa_{b}}{2}X_{z}\right)dt+dW_{x}^{b},$
$\displaystyle dY_{z}$
$\displaystyle=\left(-\omega_{z}X_{z}+\frac{\sqrt{2}\beta}{4}(4X_{r}^{2}+1)-\frac{1}{2}\kappa_{b}Y_{z}\right)dt+dW_{y}^{b}.$
(18)

Here, $dW^{i}=1/\sqrt{2}(dW_{x}^{i}+idW_{y}^{i})$ where
$\xi_{k}^{i}dt=dW_{k}^{i}$, with $k=x,y$, is the Wiener process with width
$\sqrt{\kappa_{i}\bar{n}_{i}dt}$.

We solve these equations to determine the mean number of phonons,

$\displaystyle\langle\hat{n}_{z}\rangle=\frac{1}{2}(X_{z}^{2}+Y_{z}^{2}),$
(19)

which is then used to determine the dissipated power. The calculation is then
repeated using the quantum master equation instead of the Langevin equations.
Fig. 9 shows the maximum dissipated power output of classical and quantum
calculations versus $\beta$. It reveals that the quantum model enables power
outputs that greatly exceed the power of stochastic engines for large $\beta$,
which identifies the regime in which the system can be classified as a genuine
quantum heat engine (according to the criteria in Ref. [40]).

The quantum correlation contributions to the engine performance are negligibly
small in the single-ion heat engine experiment [1], and it should be
classified as a classical heat engine even at low temperatures according to
the criteria of Ref. [40]. It is necessary to modify the setup by increasing
the asymmetry of the tapered trap and decreasing the trap’s radius (to
increase $\beta$) to bring the single-ion engine of Ref. [1] to the domain of
quantum heat engines. According to our calculations, the dissipated power
output of the engine overcomes the stochastic classical power output with the
help of quantum correlations when $\beta\geq 20$ kHz. As an example, to reach
this condition by using the parameters defined in Sec. 3 we can use
$\theta=45^{o}$ and $r_{0}\leq 10$ $\mu$m.

Figure 9: The maximum dissipated power (in units of $\hbar\omega_{z}$)
concerning coupling constant $\beta$ (in units of kHz). The black dashed and
solid blue curves show the results of classical and quantum simulations,
respectively.

##  6 Conclusion

In summary, we examined the single-atom heat engine using the asymmetric trap
configuration of experiment Ref. [1] employing a quadratic optomechanical
model. In our model, the working fluid mode (radial mode) was simultaneously
driven by the contribution of optomechanical, classical, and squeezed terms.
We clarified the effects of the coherent and squeezing couplings on the
engine’s operation beyond the standard optomechanical coupling. Furthermore,
it was found that the coupling coefficient $\beta$ plays a critical role in
increasing quantum correlations contribution and enables power outputs that
greatly exceed the power of stochastic engines. We showed that the engine
should perform in a high coupling and low-temperature regime to benefit from
the quantum correlations. According to our calculations, the quantum character
of the second order moments of the displacement operator emerges by increasing
$\beta$. In the strong coupling regime, the radial mode is in a thermal
squeezed state, and as a result, the system treats quantum mechanically. The
coupling coefficient depends on the trap’s geometry and the frequencies
asymmetry, so adjusting the trap’s geometry asymmetry and reducing the trap’s
radius can increase the efficiency and work of the engine. We concluded that a
single atom heat engine in a more confined and asymmetric trap operates in the
quantum regime where the dissipated power is more significant than its
classical counterpart with only classical stochastic correlation.

Acknowledgments This work was supported by the Scientific and Technological
Research Council of Turkey (TUBITAK) with project number 119F200.

##  Appendix A Effective entropy and temperature of the working mode $a$

To diagonalize the Hamiltonian (5), we first replace the $q_{z}$ operator with
its expectation value. The new effective semi-quantum model describes the
radial mode as the working fluid of the engine. The effective reduced model is
a quadratic form and it can be diagonalized using the Bogoliubov
transformations

$\displaystyle\hat{a}$ $\displaystyle=$
$\displaystyle\hat{c}\cosh{x}+\hat{c}^{\dagger}\sinh{x},$ (20)
$\displaystyle\hat{a}^{\dagger}$ $\displaystyle=$
$\displaystyle\hat{c}\sinh{x}+\hat{c}^{\dagger}\cosh{x}.$ (21)

Here, $\tanh{2x}=\beta q_{z}/(2\omega_{r}-\beta q_{z})$. The diagonalized
Hamiltonian becomes

$\displaystyle\hat{H}_{cm}^{{}^{\prime}}$
$\displaystyle=\omega_{\text{eff}}\hat{n}_{c}+\omega_{z}\hat{n}_{z}+\omega_{r}\sinh^{2}{x}$
$\displaystyle-\frac{\beta q_{z}}{2}\big{(}\sinh{2x}+\cosh{2x}\big{)},$ (22)

where an effective frequency $\omega_{\text{eff}}$ is defined to be

$\displaystyle\omega_{\text{eff}}=(\omega_{r}-\frac{\beta
q_{z}}{2})\cosh{2x}-\frac{\beta q_{z}}{2}\sinh{2x}.$ (23)

## References

  * [1] J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, “A single-atom heat engine,” Science 352, 325–329 (2016).
  * [2] J. E. Geusic, E. O. Schulz-DuBios, and H. E. D. Scovil, “Quantum equivalent of the _Carnot_ cycle,” Phys. Rev 156, 343 (1967).
  * [3] R. Kosloff and A. Levy, “Quantum heat engines and refrigerators: Continuous devices,” Annual Review of Physical Chemistry 65, 365–393 (2014).
  * [4] H. E. D. Scovil and E. O. Schulz-DuBois, “Three-level masers as heat engines,” Phys. Rev. Lett. 2, 262 (1959).
  * [5] S. T. Dawkins, O. Abah, K. Singer, and S. Deffner, _Single Atom Heat Engine in a Tapered Ion Trap_ (Springer, 2019).
  * [6] T. D. Kieu, “The second law, maxwell’s demon, and work derivable from quantum heat engines,” Phys. Rev. Lett. 93, 140403 (2004).
  * [7] H. T. Quan, P. Zhang, and C. P. Sun, “Quantum heat engine with multilevel quantum systems,” Phys. Rev. E 72, 056110 (2005).
  * [8] H. T. Quan, Y. xi Liu, C. P. Sun, and F. Nori, “Quantum thermodynamic cycles and quantum heat engines,” Phys. Rev. E 76, 031105 (2007).
  * [9] M. O. Scully, K. R. Chapin, K. E. Dorfman, M. B. Kim, and A. Svidzinsky, “Quantum heat engine power can be increased by noise-induced coherence,” Proc. Natl. Acad. Sci. 108, 15097 (2011).
  * [10] F. Altintas, A. U. C. Hardal, and O. E. Mustecaplıoglu, “Quantum correlated heat engine with spin squeezing,” Phys. Rev. E 90, 032102 (2014).
  * [11] O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, and E. Lutz, “Single-ion heat engine at maximum power,” Phys. Rev. Lett. 109, 203006 (2012).
  * [12] C. Bergenfeldt, P. Samuelsson, B. Sothmann, C. Flindt, and M. Buttiker, “Hybrid microwave-cavity heat engine,” Phys. Rev. Lett. 112, 076803 (2014).
  * [13] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. Lutz, “Nanoscale heat engine beyond the _Carnot_ limit,” Phys. Rev. Lett. 112, 030602 (2014).
  * [14] F. Altintas, A. U. C. Hardal, and O. E. Mustecaplıoglu, “Rabi model as a quantum coherent heat engine: From quantum biology to superconducting circuits,” Phys. Rev. A 91, 023816 (2015).
  * [15] K. Zhang and W. Zhang, “Quantum optomechanical straight-twin engine,” Phys. Rev. A 95, 053870 (2017).
  * [16] M. T. Naseem and O. E. Mustecaplıoglu, “Quantum heat engine with a quadratically coupled optomechanical system,” J. Opt. Soc. Am. B 36, 3000 (2019).
  * [17] G. Barontini and M. Paternostro, “Ultra-cold single-atom quantum heat engines,” New J. Phys. 21, 063019 (2019).
  * [18] J. Wang, Z. Wu, and J. He, “Quantum otto engine of a two-level atom with single-mode fields,” Phys. Rev. E 85, 041148 (2012).
  * [19] P. G. Steeneken, K. L. Phan, M. J. Goossens, G. E. J. Koops, G. J. A. M. Brom, C. van der Avoort, and J. T. M. van Beek, “Piezoresistive heat engine and refrigerator,” Nat. Phys. 7, 354 (2011).
  * [20] K. Kaur, V. Singh, J. Ghai, S. Jena, and Özgür E. Müstecaplıoğlu, “Unified trade-off optimization of a three-level quantum refrigerator,” Physica A 576, 125892 (2021).
  * [21] A. Ghosh, V. Mukherjee, W. Niedenzu, and G. Kurizki, “Are quantum thermodynamic machines better than their classical counterparts?” Eur. Phys. J. Spec. Top. 227, 2043–2051 (2019).
  * [22] J. Brantut, C. Grenier, J. Meineke, D. Stadler, S. Krinner, C. Kollath, T. Esslinger, and A. Georges, “A thermoelectric heat engine with ultracold atoms,” Science 342, 713 (2013).
  * [23] V. Blickle and C. Bechinger, “Realization of a micrometre-sized stochastic heat engine,” Nature Physics 8, 143 (2012).
  * [24] T. Hugel, N. B. Holland, A. Cattani, L. Moroder, M. Seitz, and H. E. Gaub, “Single-molecule optomechanical cycle,” Science 296, 1103 (2002).
  * [25] I. A. Martinez, E. Roldan, L. Dinis, D. Petrov, J. M. R. Parrondo, and R. A. Rica, “Brownian _Carnot_ engine,” Nature Physics 12, 67 (2016).
  * [26] Y. Zhang, “Optimization performance of quantum otto heat engines and refrigerators with squeezed thermal reservoirs,” Physica A 559, 125083 (2020).
  * [27] H. Naseri-Karimvand, B. Lari, and H. Hassanabadi, “Non-markovianity and efficiency of a q-deformed quantum heat engine,” Physica A 598, 127408 (2022).
  * [28] H. J. Metcalf and P. van der Straten, _Laser Cooling and Trapping_ (Springer, 1999).
  * [29] A. Levy, M. Göb, B. Deng, K. Singer, E. Torrontegui, and D. Wang, “Single-atom heat engine as a sensitive thermal probe,” New J. Phys. 22, 093020 (2020).
  * [30] A. Levy, L. Diósi, and R. Kosloff, “Quantum flywheel,” Phys. Rev. A 93, 052119 (2016).
  * [31] D. V. Lindenfels, O. Gräb, C. T. Schmiegelow, V. Kaushal, J. Schulz, M. T. Mitchison, J. Goold, F. Schmidt-Kaler, and U. G. Poschinger, “Spin heat engine coupled to a harmonic-oscillator flywheel,” Phys. Rev. Lett. 123, 080602 (2019).
  * [32] J. Sheng, C. Yang, and H. Wu, “Realization of a coupled-mode heat engine with cavity-mediated nanoresonators,” arXiv 2110.13022 (2021).
  * [33] M. Xu, J. T. Stockburger, G. Kurizki, and J. Ankerhold, “Mimimal model of a quantum heat engine in a bandgap environment,” arXiv 2109.12224 (2021).
  * [34] D. Gelbwaser-Klimovsky and G. Kurizki, “Work extraction from heat-powered quantized optomechanical setups,” Sci. Rep. 5, 7809 (2015).
  * [35] X. Huang, Q. Sun, D. Guo, and Q. Yu, “Quantum otto heat engine with three-qubit xxz model as working substance,” Physica A 491, 604–612 (2018).
  * [36] X. He, J. He, and J. Zheng, “Thermal entangled quantum heat engine,” Physica A 391, 6594–6600 (2012).
  * [37] G. Milburn and D. F. Walls, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 39, 401 (1981).
  * [38] L. A.Lugiato and G. Strini, “Production of squeezed states in a degenerate parametric amplifier,” Opt. Commun. 41, 67 (1982).
  * [39] B. Yurke, “Use of cavities in squeezed-state generation,” Phys. Rev. A 29, 408 (1984).
  * [40] R. Uzdin, A. Levy, and R. Kosloff, “Equivalence of quantum heat machines, and quantum-thermodynamic signatures,” Phys. Rev. X 5, 031044 (2015).
  * [41] T. Decker, Y. Zheng, and A. R. et al., “A microscale planar linear ion trap mass spectrometer,” Spectrometer. J. Am. Soc. Mass Spectrom 30, 482–488 (2019).
  * [42] P. F. H. et al, “Positioning of the rf potential minimum line of a linear paul trap with micrometer precision,” J. Phys. B: At. Mol. Opt. Phys. 42, 154008 (2009).
  * [43] A. U. C. Hardal, N. Aslan, C. M. Wilson, and O. E. Mustecaplioglu, “Quantum heat engine with coupled superconducting resonators,” Phys. Rev. E 96, 062120 (2017).
  * [44] H. Breuer and F. Petruccione, _The Theory of Open Quantum Systems_ (Oxford, 2007).
  * [45] M. O. Scully and M. S. Zubairy, _Quantum Optics_ (Cambridge University Press, 1997).
  * [46] J. M. Fink, L. Steffen, P. Studer, L. S. Bishop, M. Baur, R. Bianchetti, D. Bozyigit, C. Lang, S. Filipp, P. J. Leek, and A. Wallraff1, “Quantum-to-classical transition in cavity quantum electrodynamics,” Phys. Rev. Lett. 105, 163601 (2010).
  * [47] J. Johansson, P. Nation, and F. Nori, “Qutip 2: A python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
  * [48] A. Mari1, A. Farace1, and V. Giovannetti, “Quantum optomechanical piston engines powered by heat,” J. Phys. B: At. Mol. Opt. Phys. 48, 175501 (2015).
  * [49] M. T. Naseem, A. Xuereb, and Özgür E. Müstecaplıoğlu, “Thermodynamic consistency of the optomechanical master equation,” Phys. Rev. A 98, 052123 (2018).