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††thanks: Co-first authors with equal contribution††thanks: Co-first authors
with equal contribution††thanks: Co-first authors with equal contribution

# Chiral quantum heating and cooling with an optically controlled ion

J.-T. Bu  State Key Laboratory of Magnetic Resonance and Atomic and Molecular
Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy of
Precision Measurement Science and Technology, Chinese Academy of Sciences,
Wuhan 430071, China  University of the Chinese Academy of Sciences, Beijing
100049, China  J.-Q. Zhang  State Key Laboratory of Magnetic Resonance and
Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,
Innovation Academy of Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  G.-Y. Ding  State Key Laboratory of
Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of
Physics and Mathematics, Innovation Academy of Precision Measurement Science
and Technology, Chinese Academy of Sciences, Wuhan 430071, China  University
of the Chinese Academy of Sciences, Beijing 100049, China  J.-C. Li  State Key
Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan
Institute of Physics and Mathematics, Innovation Academy of Precision
Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071,
China  University of the Chinese Academy of Sciences, Beijing 100049, China
J.-W. Zhang  Research Center for Quantum Precision Measurement, Guangzhou
Institute of Industry Technology, Guangzhou, 511458, China  B. Wang  State Key
Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan
Institute of Physics and Mathematics, Innovation Academy of Precision
Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071,
China  University of the Chinese Academy of Sciences, Beijing 100049, China
W.-Q. Ding  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation
Academy of Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, China  University of the Chinese Academy of Sciences,
Beijing 100049, China  W.-F. Yuan  State Key Laboratory of Magnetic Resonance
and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,
Innovation Academy of Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  University of the Chinese Academy of
Sciences, Beijing 100049, China  L. Chen  State Key Laboratory of Magnetic
Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and
Mathematics, Innovation Academy of Precision Measurement Science and
Technology, Chinese Academy of Sciences, Wuhan 430071, China  Research Center
for Quantum Precision Measurement, Guangzhou Institute of Industry Technology,
Guangzhou, 511458, China  Q. Zhong  Department of Engineering Science and
Mechanics, and Materials Research Institute, Pennsylvania State University,
University Park, State College, Pennsylvania 16802, USA  A. Kecebas
Department of Engineering Science and Mechanics, and Materials Research
Institute, Pennsylvania State University, University Park, State College,
Pennsylvania 16802, USA  Ş. K. Özdemir<EMAIL_ADDRESS>Department of Engineering
Science and Mechanics, and Materials Research Institute, Pennsylvania State
University, University Park, State College, Pennsylvania 16802, USA  F. Zhou
<EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation
Academy of Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, China  Research Center for Quantum Precision
Measurement, Guangzhou Institute of Industry Technology, Guangzhou, 511458,
China  H. Jing<EMAIL_ADDRESS>Key Laboratory of Low-Dimensional
Quantum Structures and Quantum Control of Ministry of Education, Department of
Physics and Synergetic Innovation Center for Quantum Effects and Applications,
Hunan Normal University, Changsha 410081, China  M. Feng<EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Innovation Academy of Precision
Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071,
China  Research Center for Quantum Precision Measurement, Guangzhou Institute
of Industry Technology, Guangzhou, 511458, China  Department of Physics,
Zhejiang Normal University, Jinhua 321004, China

Quantum heat engines and refrigerators are open quantum systems, whose
dynamics can be well understood using a non-Hermitian formalism. A prominent
feature of non-Hermiticity is the existence of exceptional points (EPs), which
has no counterpart in closed quantum systems. It has been shown in classical
systems that dynamical encirclement in the vicinity of an EP, whether the loop
includes the EP or not, could lead to chiral mode conversion. Here, we show
that this is valid also for quantum systems when dynamical encircling is
performed in the vicinity of their Liouvillian EPs (LEPs) which include the
effects of quantum jumps and associated noise – an important quantum feature
not present in previous works. We demonstrate, using a Paul-trapped ultracold
ion, the first chiral quantum heating and refrigeration by dynamically
encircling a closed loop in the vicinity of an LEP. We witness the cycling
direction to be associated with the chirality and heat release (absorption) of
the quantum heat engine (quantum refrigerator). Our experiments have revealed
that not only the adiabaticity-breakdown but also the Landau-Zener-Stückelberg
process play an essential role during dynamic encircling, resulting in chiral
thermodynamic cycles. Our observations contributes to further understanding of
chiral and topological features in non-Hermitian systems and pave a way to
exploring the relation between chirality and quantum thermodynamics.

##  I
Introduction

Quantum heat engines (QHEs), using quantum matter as their working substance,
convert the heat energy from thermal reservoirs into useful work. QHEs have
been implemented in various microscopic and nanoscopic systems, including
single trapped ions and spin ensembles [1, 2, 3, 4, 5, 6, 7]. Routes to
realize QHEs in superconducting circuits [8] and quantum optomechanical
systems [9, 10] have also been proposed. Quantum refrigerators (QRs) are
typically achieved by reversing the sequence of the strokes of QHEs [11, 12],
removing heat from the cold bath at the expense of external work performed on
the system. QRs have been realized with superconducting qubits [13, 14],
quantum dots [15], and trapped ions [16, 17].

Another emergent field attracting widespread interest is non-Hermitian
dynamics in classical and quantum systems and exotic features associated with
non-Hermitian spectral degeneracies known as exceptional points (EPs) [18, 19,
20, 21, 22, 23, 24, 25, 26]. In contrast to Hermitian spectral degeneracies
where eigenvectors associated with degenerate eigenvalues are orthogonal, at
an EP both the eigenvalues and the associated eigenvectors become degenerate.
The location of the EPs in the parameter space of a system is typically
calculated from the system’s Hamiltonian which describes the non-unitary
coherent evolution of the open system. Such EPs are thus referred to as
Hamiltonian EPs (or HEPs). However, HEPs do not take quantum jumps and the
associated noise into account, and therefore does not depict the whole
dynamics of an open quantum system. Instead, one should resort to Liouvillian
formalism to describe both the non-unitary evolution and the decoherence and
hence the quantum jumps. Consequently, the EPs of the system are defined as
the eigenvalue degeneracies of Liouvillian superoperators - and thus
Liouvillian EPs (or LEPs) [27, 28, 29, 30, 31].

Open quantum systems exchange energy with external thermal baths, leading to
quantum jumps. In the presence of quantum jumps, the dynamics of QHEs and QRs
can be fully described and well understood using a non-Hermitian framework
based on Liouvillian formalism and LEPs, especially for the QHEs based on
qubits. Moreover, LEPs introduce unique physical properties to QHEs such as
the presence of an LEP can optimise the dynamics of QHEs towards their steady
states [30], enhance the QHE efficiency [32], and endow topological properties
to the QHE [33]. Nevertheless, LEPs and effects associated with the presence
of LEPs have remained largely unexplored in quantum systems and in particular
in the field of quantum thermodynamics.

Dynamically encircling a HEP in a parametric loop has shown to give rise to
chiral state transfer due to non-Hermiticity induced non-adiabatic transitions
[34, 35, 36, 39, 40, 37, 38, 41]. However, recent experimental studies have
shown that chiral behaviour can be observed even without encircling an HEP:
Any dynamically formed parametric loop in the vicinity of an EP should result
in chiral features thanks to the eigenvalue landscape close to the EP [42, 43,
44, 45, 46].  Meanwhile, the eigenvalue landscape of an LEP exhibits similar
Riemann surfaces, leading to non-trivial state transfer dynamics (e.g.,
entangled state generation) when the LEP is encircled [31]([47]). Some
experimental studies illustrate the challenge of exploring the
counterintuitive chiral behavior in quantum systems without encircling the
LEPs [33, 48, 49, 50]. For example, the topology and landscape of the Riemann
surface, along with the trajectory and evolution speed of the dynamical
process, significantly influence the results of parametric loops [33]. These
influences result from various aspects, including the phases of the Landau-
Zener-Stückelberg (LZS) process [48, 49, 50], quantum coherence, net work, and
efficiency of the quantum heat engine.

In this Letter, we experimentally demonstrate chiral behaviour in a qubit
system without encircling LEPs. Namely, we show chiral operation induced by
parametric loops in the vicinity of an LEP (without encircling it) in the
parameter space of a single trapped ion configured as a quantum engine for
heating and refrigeration. Our work brings together quantum thermodynamics,
LEPs, and chiral state transfer due to breakdown of adiabaticity,
demonstrating that non-adiabacity and LZS process [48, 49, 50] are essential
to chiral thermodynamic cycles. Our experiment connects, for the first time,
the LZS process to chirality in association with LEP-related thermodynamic
effects.

##  II
Results and Discussions

Experimental setup. Our experiment is carried out in a single ultracold
${}^{40}\mathrm{Ca}^{+}$ ion confined in a linear Paul trap (Fig. 1(a) with
more details in [51, 52]) whose axial and radial frequencies are,
respectively, $\omega_{z}/2\pi=1.01$ MHz and $\omega_{r}/2\pi=1.2$ MHz under
the pseudo-potential approximation.  Under an external magnetic field 0.6 mT
directed in axial orientation, the ground state $|4^{2}S_{1/2}\rangle$ is
split into two hyperfine energy levels while the metastable state
$|3^{2}D_{5/2}\rangle$ split into six. As shown in Fig. 1(b), we label
$|4^{2}S_{1/2},m_{J}=+1/2\rangle$ as $|g\rangle$,
$|3^{2}D_{5/2},m_{J}=+5/2\rangle$ as $|e\rangle$ and
$|4^{2}P_{3/2},m_{J}=+3/2\rangle$ as $|p\rangle$. After Doppler and resolved
sideband cooling of the ion, we reduce the average phonon number of the
$z$-axis motional mode of the ion to be much smaller than 1 with the Lamb-
Dicke parameter $\sim$ 0.11, which is sufficient to avoid detrimental effects
of thermal phonons (e.g., Rabi oscillation offsets). Introducing the dipolar
transition $|e\rangle\rightarrow|p\rangle$ by switching on 854-nm laser as
explained in [53, 32], we reduce this three-level system to an effective two-
level system representing a qubit [54]. We employ this qubit as the working
substance of the QHE and QR.

Figure 1: Parametric loops in the vicinity of an LEP in the parameter space of
a single trapped ion quantum heat engine. (a) The linear Paul trap. (b) Level
scheme of the ion, where the solid arrows represent the transitions with Rabi
frequencies $\Omega$ and $\tilde{\Omega}$ driven by 729 nm and 854 nm lasers,
respectively, and $\Delta$ is the detuning between the energy level and 729 nm
laser. The wavy arrow denotes the spontaneous emission with decay rate
$\Gamma$. This three-level model can be simplified to an effective two-level
system with tunable drive and decay. (c-f) Trajectories without encircling the
LEP on the Riemann sheets when completing a clockwise or counterclockwise
encirclement in $\Delta\mbox{-}\gamma_{\mathrm{eff}}$ parametric space
starting from $|\psi_{+}\rangle$ or $|\psi_{-}\rangle$, where the parameters
take the values $\Delta_{\mathrm{min}}/2\pi=-400$ kHz,
$\Delta_{\mathrm{max}}/2\pi=400$ kHz, $\gamma_{\mathrm{min}}\approx 100$ kHz,
$\gamma_{\mathrm{max}}\approx 1.45$ MHz. The empty circles represent the
starting points. Five orange corner points A, B, C, D, E (A’, B’, C’, D’, E’)
are labeled for convenience of description in the text. The LEPs are labeled
by stars.

The dynamical evolution of this effective two-level model is governed by the
Lindblad master equation,
$\dot{\rho}=\mathcal{L}\rho=-i[H_{\mathrm{eff}},\rho]+\frac{\gamma_{\mathrm{eff}}}{2}(2\sigma_{-}\rho\sigma_{+}-\sigma_{+}\sigma_{-}\rho-\rho\sigma_{+}\sigma_{-})$,
where $\mathcal{L}$ is the Liouvillian superoperator, and $\rho$ and
$\gamma_{\mathrm{eff}}$ denote the density operator and the effective decay
rate from $|e\rangle$ to $|g\rangle$, respectively. Here the effective
Hamiltonian is $H_{\mathrm{eff}}=\Delta|e\rangle\langle
e|+\frac{\Omega}{2}(|e\rangle\langle g|+|g\rangle\langle e|)$, where $\Delta$
represents the frequency detuning between the resonance transition and the
driving laser while $\Omega$ denotes the Rabi frequency. The eigenvalues of
$\mathcal{L}$ at $\Delta=0$ are given by
$\lambda_{1}=0,\lambda_{2}=-\gamma_{\mathrm{eff}}/2,\lambda_{3}=(-3\gamma_{\mathrm{eff}}-\xi)/4$
and $\lambda_{4}=(-3\gamma_{\mathrm{eff}}+\xi)/4$, with
$\xi=\sqrt{\gamma_{\mathrm{eff}}^{2}-16\Omega^{2}}$. It is evident that the
eigenvalues $\lambda_{3}$ and $\lambda_{4}$ coalesce when
$\gamma_{\mathrm{eff}}=4\Omega$, giving rise to a second order LEP at
$\tilde{\lambda}=-3\gamma_{\mathrm{eff}}/4$ [55]. In the weak coupling
situation $\gamma_{\mathrm{eff}}>4\Omega$, both $\lambda_{3}$ and
$\lambda_{4}$ are real with a splitting of $\xi/2$. This regime corresponds to
the broken phase characterized by a non-oscillatory dynamics accompanied by
purely exponential decay [28, 29]. In the strong coupling regime
($\gamma_{\mathrm{eff}}<4\Omega$), $\lambda_{3}$ and $\lambda_{4}$ are a pair
of complex conjugates with the imaginary parts $-\xi/4$ and $\xi/4$,
respectively. This regime corresponds to the exact phase characterized by an
oscillatory dynamics. Thus LEP acts as a critical damping point of the damped
harmonic oscillator which divides the parameter space into a region of
oscillatory dynamics (exact phase, $\gamma_{\mathrm{eff}}<4\Omega$) and a
region of non-oscillatory dynamics (broken phase,
$\gamma_{\mathrm{eff}}>4\Omega$).

Our QHE and QR cycles differ from their classical counterparts in the
definition and implementation of the characteristic thermodynamic quantities.
The working substance in our QHE and QR is the qubit defined above. The
thermal baths consist of the 729-nm laser irradiation and the actual
environment. This working substance carries out work by varying the detuning
$\Delta$ in the effective Hamiltonian $H_{\mathrm{eff}}$, in which the
increase and decrease of the population in the excited state $|e\rangle$
correspond to heat absorption and heat release of the working substance,
respectively. As a result, the Rabi interactions in the QHE and QR cycles
represent the energy exchange between the working substance and thermal baths.
Variations in the effective Hamiltonian lead to LZS process, along with
performed work, heat absorption, and heat release [33]. In our experiments,
all thermodynamic quantities are acquired from population variations of the
qubit and the corresponding tunable parameters [55].

Figures 1(a,b) present the thermodynamic cycle we carry out, consisting of two
iso-decay and two isochoric strokes with a fixed Rabi frequency $\Omega$ as in
Ref. [33]. For the iso-decay strokes, we achieve positive (negative) work by
decreasing (increasing) the detuning $\Delta$ while keeping the decay rate
$\gamma_{\mathrm{eff}}$ constant. The iso-decay strokes shift the energy
difference between the two levels of the working substance and function as the
expansion and compression processes for the work done. In the isochoric
strokes, our system rapidly reaches the steady state. Then, with the constant
values of $\Delta$ and $\Omega$, we increase (or decrease) the population of
the excited state for heating (or cooling) by decreasing (increasing) the
decay rate $\gamma_{\mathrm{eff}}$. This indicates that the increase
(decrease) of the population in the excited state corresponds to the heat
absorption (release) from (to) the thermal baths.

To study the effect of the parametric loops in the vicinity of the LEP on the
thermodynamic cycle and the engine performance, we prepare the qubit in the
superposition state $|\psi_{+}\rangle=(|e\rangle+|g\rangle)/\sqrt{2}$ or
$|\psi_{-}\rangle=(|e\rangle-|g\rangle)/\sqrt{2}$, and then perform a
thermodynamic cycle, corresponding to a loop in the parameter space, by tuning
$\gamma_{\mathrm{eff}}$ and $\Delta$ such that the LEP of the system is not
encircled. The superposition states $|\psi_{+}\rangle$ and $|\psi_{-}\rangle$
correspond to two eigenstates of $H_{\mathrm{eff}}$ for $\Delta=0$ kHz at the
middle point of the expansion and compression processes in the iso-decay
strokes, respectively.Moreover, to prevent our operations from affecting the
vibrational modes of the ion, leading to unexpected heating noises, we
restrict $|\Delta|$ to be smaller than half of the vibrational frequency.

Starting from the mid-points of these strokes requires five strokes to execute
a clockwise (CW) or a counterclockwise (CCW) loop. After the loop (CW or CCW)
is completed, we compare the final state $\rho$ of the qubit with the initial
state $|\psi_{+}\rangle$ or $|\psi_{-}\rangle$ by calculating the fidelity
$\langle\psi_{+}|\rho|\psi_{+}\rangle$ or
$\langle\psi_{-}|\rho|\psi_{-}\rangle$. Figures 1(c) and 1(d) show the
evolution of the system for two counterclockwise loops that start at the
initial states $|\psi_{-}\rangle$ and $|\psi_{+}\rangle$, respectively, and
are completed without encircling the LEP. We first implement an iso-decay
compression stroke from A to B by increasing the detuning $\Delta$ linearly
from 0 to its maximum value $\Delta_{\mathrm{max}}$ while keeping
$\gamma_{\mathrm{eff}}$ at its minimum value of $\gamma_{\mathrm{min}}$. The
second stroke is an isochoric cooling stroke from B
$(\Delta_{\mathrm{max}},\gamma_{\mathrm{min}})$ to C
$(\Delta_{\mathrm{max}},\gamma_{\mathrm{max}})$, which is realized by
increasing $\gamma_{\mathrm{eff}}$ from $\gamma_{\mathrm{min}}$ to
$\gamma_{\mathrm{max}}$ with the detuning remaining constant at
$\Delta_{\mathrm{max}}$. Then we execute the third stroke, i.e., an iso-decay
expansion from C $(\Delta_{\mathrm{max}},\gamma_{\mathrm{max}})$ to D
$(\Delta_{\mathrm{min}},\gamma_{\mathrm{max}})$, by decreasing $\Delta$ from
$\Delta_{\mathrm{max}}$ to $\Delta_{\mathrm{min}}$ while keeping
$\gamma_{\mathrm{eff}}=\gamma_{\mathrm{max}}$. The fourth stroke is an
isochoric heating from D $(\Delta_{\mathrm{min}},\gamma_{\mathrm{max}})$ to E
$(\Delta_{\mathrm{min}},\gamma_{\mathrm{min}})$ executed by decreasing
$\gamma_{\mathrm{eff}}$ from $\gamma_{\mathrm{max}}$ to
$\gamma_{\mathrm{min}}$ with $\Delta$ fixed at $\Delta_{\mathrm{min}}$. The
final step is an iso-decay compression stroke from E
$(\Delta_{\mathrm{min}},\gamma_{\mathrm{min}})$ back to
A$(0,\gamma_{\mathrm{min}})$ executed by increasing $\Delta$ from
$\Delta_{\mathrm{min}}$ to 0 while $\gamma_{\mathrm{eff}}$ is kept fixed at
$\gamma_{\mathrm{min}}$. The clockwise loops depicted in Figs. 1(e) and 1(f)
are also executed in five strokes with the initial states prepared in
$|\psi_{+}\rangle$ and $|\psi_{-}\rangle$ but in reverse order of the process
described above for counterclockwise loops. It is clearly seen that regardless
of the initial state that the loops start from, the system always ends up at
the final state $|\psi_{+}\rangle$ for a clockwise loop and at the final state
$|\psi_{-}\rangle$ for a counterclockwise loop. This observation that the
execution direction of the loop determines the final state is a signature of
the chiral behavior in our system. This process is often referred to as
asymmetric mode conversion or chiral state transfer [36, 46, 48].

Figure 2: Closed loops that do not exhibit mode conversion. (a)
Counterclockwise loop starting from $|\psi_{-}\rangle$. (b) Clockwise loop
starting from $|\psi_{+}\rangle$. Evolution of the system’s state along the
trajectory of the closed loop is characterized by the fidelity
$\langle\psi_{-}|\rho(t)|\psi_{-}\rangle$ in (a) and the fidelity
$\langle\psi_{+}|\rho(t)|\psi_{+}\rangle$ in (b). The circles and error bars
denote, respectively, the average and standard deviations of 10000
measurements. The solid curves connecting the dots are obtained by simulating
master equations. (c) Blue and red solid curves represent, respectively, the
time evolution of the net work in the counterclockwise and clockwise loops,
where only three of the strokes associated with doing work are depicted.
Durations of the five strokes are $T_{1}=T_{5}=6$ $\mu$s, $T_{3}=12$ $\mu$s,
and $T_{2}=T_{4}=150$ $\mu$s. Other parameters are $\Omega/2\pi=$ 120 kHz,
$\Delta_{\mathrm{min}}/2\pi=-400$ kHz, $\Delta_{\mathrm{max}}/2\pi=400$ kHz,
$\gamma_{\mathrm{min}}\approx 0$ kHz, and $\gamma_{\mathrm{max}}\approx 1.45$
MHz.

Quantum heating and cooling. Figure 2 displays experimental results of a
counterclockwise loop starting at $|\psi_{-}\rangle$ and a clockwise loop
starting at $|\psi_{+}\rangle$, corresponding to the schemes in Fig. 1(c,e).
These loops do not encircle the LEP of the system. In Fig. 2(b), we prepare
the qubit in the initial state $|\psi_{+}\rangle$ with fidelity
$\langle\psi_{+}|\rho_{A}|\psi_{+}\rangle=0.985$ which is lower than 1 due to
imperfections during the state preparation. The system returns to
$|\psi_{+}\rangle$ after completion of the loop. Figure 2(a) exhibits the
results of the counterclockwise loop starting at $|\psi_{-}\rangle$. The state
of the system evolves back to the initial state $|\psi_{-}\rangle$ when the
loop is completed. The two situations correspond, respectively, to the QR and
QHE cycles, as elucidated later. For the counterclockwise encirclement in Fig.
2(a), the system is initialized to
$|\psi_{-}\rangle=(|e\rangle-|g\rangle)/\sqrt{2}$. The Landau-Zener transition
occurs in the first stroke with the detuning $\Delta$ varied from 0 to
$\Delta_{\mathrm{max}}/2\pi=400$ kHz, and then the system evolves to a nearly
steady state in the second stroke due to the increase of the decay rate with a
large detuning $\Delta/2\pi=400$ kHz. In the third stroke, the LZS process
occurs, resulting in the population oscillation. After the fourth stroke the
system evolves to another nearly steady state as a result of the large
detuning $\Delta/2\pi=-400$ kHz and slow-varying dissipation. Finally, a
Landau-Zener transition occurs again in the fifth stroke with the detuning
$\Delta$ tuned from $\Delta_{\mathrm{min}}/2\pi=-400$ kHz to 0, resulting in
the final state $|\psi_{-}\rangle=(|e\rangle-|g\rangle)/\sqrt{2}$ [49, 50]. In
contrast, for the clockwise loop from
$|\psi_{+}\rangle=(|e\rangle+|g\rangle)/\sqrt{2}$, the system evolves to a
steady state after the fourth stroke and experiences a LZS with the detuning
$\Delta$ tuned from $\Delta_{\mathrm{max}}/2\pi=400$ kHz to 0, accumulating a
Stückelberg phase [50] and thus leading to the final state
$|\psi_{+}\rangle=(|e\rangle+|g\rangle)/\sqrt{2}$.

Since the system evolves along a closed trajectory, we can evaluate the net
work using $W_{1}=\int_{0}^{t}\rho(t)dH(t)$, where $\rho(t)$ describes the
state of the two-level system governed by $H(t)=\Delta(t)|e\rangle\langle
e|/2$. For the clockwise encirclement in Fig. 2(b), since the isochoric
strokes produce no work, here we only consider the first, third, and fifth
strokes with the first and fifth strokes executing expansion and the third
stroke executing compression. Moreover, the bath performs work on the system
in the iso-decay compression stroke (third stroke) and the system produces
work during the two iso-decay expansion strokes. The mean population in
$|e\rangle$ in the two expansion strokes is higher than that in the
compression stroke, implying a positive net work, and thus the system behaves
as the QHE. On the contrary, the counterclockwise loop in Fig. 2(a) leads to
negative net work due to the higher mean population in $|e\rangle$ in the two
compression strokes than that in the expansion stroke. This makes the system
perform as a QR. Thus, we conclude that the loops that do not result in
asymmetric mode conversion (CW loop starting at $|\psi_{+}\rangle$ and CCW
loop starting at $|\psi_{-}\rangle$) performs as a QHE (Fig. 2(b)) or a QR
(Fig. 2(a)).

Figure 3: Closed loops that lead to mode conversion. (a) Counterclockwise loop
starting at $|\psi_{+}\rangle$ and ending at $|\psi_{-}\rangle$. (b) Clockwise
loop starting at $|\psi_{-}\rangle$ and ending at $|\psi_{+}\rangle$.
Evolution of the system’s state along the closed loop trajectory is
characterized by the fidelity $\langle\psi_{+}|\rho(t)|\psi_{+}\rangle$ in (a)
and the fidelity $\langle\psi_{-}|\rho(t)|\psi_{-}\rangle$ in (b). The circles
and error bars respectively denote the average and standard deviations of
10000 measurements. The solid curves are obtained by simulating master
equations. Durations of the five strokes are $T_{1}=T_{5}=6$ $\mu$s,
$T_{3}=12$ $\mu$s, and $T_{2}=T_{4}=150$ $\mu$s. Other parameters are
$\Omega/2\pi=$ 120 kHz, $\Delta_{\mathrm{min}}/2\pi=-400$ kHz,
$\Delta_{\mathrm{max}}/2\pi=400$ kHz, $\gamma_{\mathrm{min}}\approx 0$ kHz,
and $\gamma_{\mathrm{max}}\approx 1.45$ MHz.

We mention that Rabi frequency $\Omega$ remains constant in our implementation
of QHE and QR cycles, indicating that the working substance keeps interacting
with the 729-nm laser. The heating and cooling processes are accomplished by
modulating the detuning and/or the decay rate. For example, the working
substance reaches the thermal equilibrium with the thermal baths by changing
$\gamma_{\mathrm{eff}}$ while keeping a constant value of $\Delta$ [56, 57,
6].

Chiral dynamics. The chiral behavior and asymmetric mode conversion appear for
counterclockwise loop starting at $|\psi_{+}\rangle$ and the clockwise loop
starting at $|\psi_{-}\rangle$, as witnessed in Fig. 3. In both of these two
cases, the system cannot return to the starting state after the loop is
completed. Together with the observation depicted in Fig. 2(a), we conclude
that the final states (i.e., the end points of the loops) depend only on the
encircling direction, not on the initial state (i.e., the starting point of
the loops): A CW (or CCW) loop in the vicinity of the LEP but not encircling
it ends up at the final state $|\psi_{+}\rangle$ (or $|\psi_{-}\rangle$)
regardless of whether the initial state is $|\psi_{+}\rangle$ or
$|\psi_{-}\rangle$.

It is evident that, whether the system evolves back to the starting point or
not is essentially associated with the LZS process and the breakdown of the
adiabaticity when executing the closed loop trajectory. This is reflected in
the fifth stroke of the loop when a state transfer occurs after experiencing a
Landau-Zener transition (Figs. 2 and 3).For the clockwise encirclement
starting from $|\psi_{+}\rangle$ or the counterclockwise encirclement from
$|\psi_{-}\rangle$, the Landau-Zener transition in the fifth stroke determines
if the closed thermodynamic cycle works as the chiral QHE or QR. In contrast,
for the clockwise loop starting at $|\psi_{-}\rangle$ or the counterclockwise
loop at $|\psi_{+}\rangle$, this Landau-Zener transition determines the chiral
behavior. In other words, when the system starts at the state
$|\psi_{+}\rangle$, a clockwise loop in the vicinity of (not encircling) the
LEP ends in the same state whereas a counterclockwise loop ends at the
orthogonal state $|\psi_{-}\rangle$. So the system acts as a QHE for the
clockwise loop and as a mode converter for the counterclockwise loop.
Similarly, when the system starts at the state $|\psi_{-}\rangle$, a clockwise
loop in the vicinity of (not encircling) the LEP ends in the orthogonal state
$|\psi_{+}\rangle$ whereas a counterclockwise loop ends at the same state
$|\psi_{-}\rangle$. So the system acts as a QR for the counterclockwise loop
and as a mode converter for the clockwise loop.

Here we emphasize that, the LZ process in the fifth stroke is the most
essential to the chiral behavior and asymmetric mode conversion . When the
evolution time of the fifth stroke is too short, our system works in the non-
adiabatic regime and lacks sufficient time to follow the variation of the
detuning, resulting in non-adiabatic transitions between two eigenstates via
phase accumulation through the LZS process [48, 49, 50]. These transitions
would prevent our system from exhibiting chiral behavior and asymmetric mode
conversion. In contrast, when the evolution time of the fifth stroke is too
long, our system works in the adiabatic regime and smoothly follows the
variation of the detuning. Nevertheless, the evolution duration is restricted
by the decoherence time, as clarified with the numerical simulation results
shown in Fig. S.8.

Although the displayed loops exclude the LEP, our numerical simulations
suggest a strong relevance of the observed chiral behavior to the existence of
the LEP. If the loop is too far away from the LEP, i.e., $\gamma_{\rm{eff}}$
is too small, no chirality would appear [55]. However, if the system
parameters are chosen such that the loops are closer to the LEP, but not
encircling it, the populations of $|\psi_{+}\rangle$ and $|\psi_{-}\rangle$
interchange for the clockwise loop starting at $|\psi_{-}\rangle$ and the
counterclockwise loop starting at $|\psi_{+}\rangle$ [55]. Moreover, since the
loops do not encircle the LEP, no topological phase transition is involved in
system’s response. We note that an essential ingredient for the observed
chirality is the transition of the system between two Riemann sheets when the
parameters are tuned to form a closed loop. Considering the system state
evolves only within a single Riemann sheet, we see the encircling direction
determines whether the chirality exists or not. For example, Figs. S5 and S6
in [55] show that chiral behaviour appears only in the counterclockwise and
clockwise encirclements, respectively. Otherwise the final state neither
returns to the initial state nor has state convention between
$|\psi_{+}\rangle$ and $|\psi_{-}\rangle$. This phenomena can be called non-
reciprocal chirality, i.e., unidirectional chirality.

As our chiral behavior and asymmetric mode conversion result from the LEP
rather than the Hamiltionian EP [38, 41, 34, 35, 36, 39, 40, 37], we have
fully captured the quantum dynamics including the quantum jumps and associated
noises, significantly extending the realization conditions of chiral behavior
and asymmetric mode conversion, and considerably reducing the experimental
difficulties in quantum control and measurement.  Realizing chiral behavior
without encircling an LEP helps reduce the parameter space needed to steer the
system. For example, encircling the LEP required varying
$\gamma_{\mathrm{max}}$ to values larger than 4$\Omega$ [33]; however,
observing chiral behavior without encircling the LEP requires varying
$\gamma_{\mathrm{max}}$ to values less than 2$\Omega$. This helps keeping our
system in the quantum regime for achieving the chiral behavior and asymmetric
mode conversion, and also opens up possibilities for a more focused and
extended exploration of the physical properties associated with the LEP.
Moreover, the combination of the adiabaticity breakdown [38] and the Landau-
Zener-Stückelberg phase [49, 50] leads to the presented chirality, whose
physical mechanics is distinct from the one resulting from spontaneous chiral
symmetry breaking [59]. Furthermore, our observed chiral behavior and
asymmetric mode conversion strongly depend on the LEPs resulting from quantum
jumps. Characterized by quantum jumps, the above phenomena in our experimental
system are inherently quantum. In contrast, without quantum jumps, our
experimental system would return to Hamiltonian EPs [25, 58], whose chiral
behavior and asymmetric mode conversion have been theoretically predicted [42]
and experimentally observed with waveguides [44] and fibers [45].

##  III
Conclusion

In conclusion, we have experimentally demonstrated, for the first time, a
chiral behavior in a single trapped-ion system without dynamically encircling
its LEP. We show clearly that asymmetric mode conversion is directly related
to the topological landscape of the Riemann surfaces and not necessarily to
encircling an LEP of this quantum system, supporting previous reports for
classical systems. Our experiments may open up new avenues in understanding
the chiral and topological behaviors in non-Hermitian systems and bridging
chirality and quantum thermodynamics.

##  IV Data availability

The data illustrated in the figures within this paper are available from the
corresponding authors upon request.

##  V Code availability

The custom codes used to generate the results presented in this paper are
available from the corresponding authors upon request.

##  VI
Acknowledgments

This work was supported by Special Project for Research and Development in Key
Areas of Guangdong Province under Grant No. 2020B0303300001, by National
Natural Science Foundation of China under Grant Nos. U21A20434, 92265107,
12074390, 11835011, by Key Lab of Guangzhou for Quantum Precision Measurement
under Grant No.202201000010, by Postdoctoral Science Foundation of China under
Grant No. 2022MT10881, and by Nansha Senior Leading Talent Team Technology
Project under Grant No. 2021CXTD02. Ş.K.O acknowledges the support from Air
Force Office of Scientific Research (AFOSR) Multidisciplinary University
Research Initiative (MURI) Award No. FA9550-21-1-0202 and the AFOSR Award No.
FA9550-18-1-0235.

##  VII
Author contributions

HJ and MF conceived the idea and designed the experiments; JTB and FZ
performed the experiments with help from JCL,BW, WQD, WFY and JWZ; GYD and JQZ
analyzed the data with help form QZ, AK, and LC . Theoretical background and
simulations were provided by GYD and JQZ. All authors contributed to the
discussions and the interpretations of the experimental and theoretical
results. MF, HJ, and SKO wrote the manuscript with inputs from all authors.

##  VIII
Competing interests

The authors declare that they have no competing interests.

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##  IX Figures in the maintext

Maintext Fig. 1: Parametric loops in the vicinity of an LEP in the parameter
space of a single trapped ion quantum heat engine. (a) The linear Paul trap.
(b) Level scheme of the ion, where the solid arrows represent the transitions
with Rabi frequencies $\Omega$ and $\tilde{\Omega}$ driven by 729 nm and 854
nm lasers, respectively, and $\Delta$ is the detuning between the energy level
and 729 nm laser. The wavy arrow denotes the spontaneous emission with decay
rate $\Gamma$. This three-level model can be simplified to an effective two-
level system with tunable drive and decay. (c-f) Trajectories without
encircling the LEP on the Riemann sheets when completing a clockwise or
counterclockwise encirclement in $\Delta\mbox{-}\gamma_{\mathrm{eff}}$
parametric space starting from $|\psi_{+}\rangle$ or $|\psi_{-}\rangle$, where
the parameters take the values $\Delta_{\mathrm{min}}/2\pi=-400$ kHz,
$\Delta_{\mathrm{max}}/2\pi=400$ kHz, $\gamma_{\mathrm{min}}\approx 100$ kHz,
$\gamma_{\mathrm{max}}\approx 1.45$ MHz. The empty circles represent the
starting points. Five orange corner points A, B, C, D, E (A’, B’, C’, D’, E’)
are labeled for convenience of description in the text. The LEPs are labeled
by stars.

Maintext Fig. 2: Closed loops that do not exhibit mode conversion. (a)
Counterclockwise loop starting from $|\psi_{-}\rangle$. (b) Clockwise loop
starting from $|\psi_{+}\rangle$. Evolution of the system’s state along the
trajectory of the closed loop is characterized by the fidelity
$\langle\psi_{-}|\rho(t)|\psi_{-}\rangle$ in (a) and the fidelity
$\langle\psi_{+}|\rho(t)|\psi_{+}\rangle$ in (b). The circles and error bars
denote, respectively, the average and standard deviations of 10000
measurements. The solid curves connecting the dots are obtained by simulating
master equations. (c) Blue and red solid curves represent, respectively, the
time evolution of the net work in the counterclockwise and clockwise loops,
where only three of the strokes associated with doing work are depicted.
Durations of the five strokes are $T_{1}=T_{5}=6$ $\mu$s, $T_{3}=12$ $\mu$s,
and $T_{2}=T_{4}=150$ $\mu$s. Other parameters are $\Omega/2\pi=$ 120 kHz,
$\Delta_{\mathrm{min}}/2\pi=-400$ kHz, $\Delta_{\mathrm{max}}/2\pi=400$ kHz,
$\gamma_{\mathrm{min}}\approx 0$ kHz, and $\gamma_{\mathrm{max}}\approx 1.45$
MHz.

Maintext Fig. 3: Closed loops that lead to mode conversion. (a)
Counterclockwise loop starting at $|\psi_{+}\rangle$ and ending at
$|\psi_{-}\rangle$. (b) Clockwise loop starting at $|\psi_{-}\rangle$ and
ending at $|\psi_{+}\rangle$. Evolution of the system’s state along the closed
loop trajectory is characterized by the fidelity
$\langle\psi_{+}|\rho(t)|\psi_{+}\rangle$ in (a) and the fidelity
$\langle\psi_{-}|\rho(t)|\psi_{-}\rangle$ in (b). The circles and error bars
respectively denote the average and standard deviations of 10000 measurements.
The solid curves are obtained by simulating master equations. Durations of the
five strokes are $T_{1}=T_{5}=6$ $\mu$s, $T_{3}=12$ $\mu$s, and
$T_{2}=T_{4}=150$ $\mu$s. Other parameters are $\Omega/2\pi=$ 120 kHz,
$\Delta_{\mathrm{min}}/2\pi=-400$ kHz, $\Delta_{\mathrm{max}}/2\pi=400$ kHz,
$\gamma_{\mathrm{min}}\approx 0$ kHz, and $\gamma_{\mathrm{max}}\approx 1.45$
MHz.

##  X Supplementary Materials

We present details about the dynamics and chirality of quantum heat engine
(QHE) and quantum refrigerator (QR). We also discuss the encirclement within
single Riemann sheets, i.e., in half of the encirclement area as considered in
the main text.

### I. order parameter with respect to Liouvillian exceptional point

Our system is governed by the Lindblad master equation

$\dot{\rho}(t)=-i[H_{\mathrm{eff}},\rho]+\frac{\gamma_{\mathrm{eff}}}{2}(2\sigma_{-}\rho\sigma_{+}-\sigma_{+}\sigma_{-}\rho-\rho\sigma_{+}\sigma_{-})=\mathcal{L}\rho,$
(1)

where $\mathcal{L}$ is called Liouvillian superoperator and $\rho$ is the
density operator of the system. Here the density matrix $\rho$ is expressed by
$\left(\begin{array}[]{ccc}\rho_{ee}&\rho_{eg}\\\
\rho_{ge}&\rho_{gg}\end{array}\right)$, with the raising and lowering
operators defined as $\sigma_{+}=\left(\begin{array}[]{ccc}0&1\\\
0&0\end{array}\right)$ and $\sigma_{-}=\left(\begin{array}[]{ccc}0&0\\\
1&0\end{array}\right)$, respectively. According to the corresponding equations
in the main text, we can write $\mathcal{L}$ as

$\mathcal{L}=\left(\begin{array}[]{cccc}-\gamma_{\mathrm{eff}}&i\Omega/2&-i\Omega/2&0\\\
i\Omega/2&-(\gamma_{\mathrm{eff}}/2+i\Delta)&0&-i\Omega/2\\\
-i\Omega/2&0&-(\gamma_{\mathrm{eff}}/2-i\Delta)&i\Omega/2\\\
\gamma_{\mathrm{eff}}&-i\Omega/2&i\Omega/2&0\end{array}\right).$ (2)

By setting $\Delta=0$, we find the eigenvalues of $\mathcal{L}$ as
$\lambda_{1}=0$, $\lambda_{2}=-\gamma_{\mathrm{eff}}/2$,
$\lambda_{3}=\frac{1}{4}(-3\gamma_{\mathrm{eff}}-\sqrt{\gamma_{\mathrm{eff}}^{2}-16\Omega^{2}})$,
and
$\lambda_{4}=\frac{1}{4}(-3\gamma_{\mathrm{eff}}+\sqrt{\gamma_{\mathrm{eff}}^{2}-16\Omega^{2}})$.
It is evident that the eigenvalues $\lambda_{3}$ and $\lambda_{4}$ becomes
degenerate at $\gamma_{\mathrm{eff}}=4\Omega$, which corresponds to the
Liouvillian exceptional point (LEP). For $\gamma_{\mathrm{eff}}>4\Omega$, we
have real $\lambda_{3}$ and $\lambda_{4}$ with a splitting of
$|\lambda_{4}-\lambda_{3}|=\xi/2$ with
$\xi=\sqrt{\gamma_{\mathrm{eff}}^{2}-16\Omega^{2}}$. For
$\gamma_{\mathrm{eff}}<4\Omega$, on the other hand, $\lambda_{3}$ and
$\lambda_{4}$ form a complex conjugate pair with a splitting of $|\xi|/2$ in
their imaginary parts.

Fig.S. 1: Topological phase transition with respect to the Liouvillian
exceptional point (LEP) that is determined by the decay rate
$\gamma_{\mathrm{eff}}$ and the drive strength $\Omega$.

The topological phase transition occurs at the LEP, as shown in Fig.S. 1.
Since the LEP is excluded in our displayed encirclements, no topological phase
transition occurs in our experiment. Nevertheless, the influence from the LEP
is witnessed in our experiment. As shown in the following sections, we have
observed experimentally the strong relevance of the chirality to the LEP and
the associated Riemann sheets.

In contrast, with the quantum jumps ignored, the above master equation (1) is
reduced to

$\dot{\rho}(t)=-i[H_{\mathrm{eff}},\rho]-\frac{\gamma_{\mathrm{eff}}}{2}(\sigma_{+}\sigma_{-}\rho+\rho\sigma_{+}\sigma_{-})=-i(H_{\mathrm{NH}}\rho-\rho
H_{\mathrm{NH}}^{{\dagger}})=\mathcal{L}_{H}\rho,$ (3)

where
$H_{\mathrm{NH}}=H_{\mathrm{eff}}-i\frac{\gamma_{\mathrm{eff}}}{2}\sigma_{+}\sigma_{-}$
is the non-Hermitian Hamiltonian and

$\mathcal{L}_{H}=\left(\begin{array}[]{cccc}-\gamma_{\mathrm{eff}}&i\Omega/2&-i\Omega/2&0\\\
i\Omega/2&-(\gamma_{\mathrm{eff}}/2+i\Delta)&0&-i\Omega/2\\\
-i\Omega/2&0&-(\gamma_{\mathrm{eff}}/2-i\Delta)&i\Omega/2\\\
0&-i\Omega/2&i\Omega/2&0\end{array}\right).$ (4)

is the Liouvillian operator without quantum jumps. Both $H_{\mathrm{NH}}$ and
$\mathcal{L}_{H}$ have been utilized to experimentally verify the phase
transition of EP for $\Delta=0$ [58]. The above derivations illustrate,
without involving the quantum jump item $\sigma_{-}\rho\sigma_{+}$, the LEP
returns to the Hamiltonian EP (HEP). In other words, the HEP cannot describe
the entire dynamics of the open quantum system.

We find the eigenvalues of $\mathcal{L}_{H}$ as
$\tilde{\lambda}_{1}=\tilde{\lambda}_{2}=-\gamma_{\mathrm{eff}}/2$,
$\tilde{\lambda}_{3}=(-\gamma_{\mathrm{eff}}-\sqrt{\gamma_{\mathrm{eff}}^{2}-4\Omega^{2}})/2$
and
$\tilde{\lambda}_{4}=(-\gamma_{\mathrm{eff}}+\sqrt{\gamma_{\mathrm{eff}}^{2}-4\Omega^{2}})/2$.
The eigenvalues $\tilde{\lambda}_{3}$ and $\tilde{\lambda}_{4}$ become
degenerate at $\gamma_{\mathrm{eff}}=2\Omega$, corresponding to the HEP. Real
$\tilde{\lambda}_{3}$ and $\tilde{\lambda}_{4}$ present a splitting of
$|\tilde{\lambda}_{3}-\tilde{\lambda}_{4}|=\sqrt{\gamma_{\mathrm{eff}}^{2}-4\Omega^{2}}$
when $\gamma_{\mathrm{eff}}>2\Omega$, while $\tilde{\lambda}_{3}$ and
$\tilde{\lambda}_{4}$ become a complex conjugate pair with a splitting of
$\sqrt{\gamma_{\mathrm{eff}}^{2}-4\Omega^{2}}$ in their imaginary parts.

Considering the experimental parameters and the presence of quantum jumps, we
conclude that the EP observed in our system is an LEP rather than a HEP. LEP
and HEP have been observed in previous experiments [60] and [24],
respectively. These experiments illustrate that, when the decay rate is fixed,
the coupling strength for the LEP [60] is much smaller than that for the HEP
[24], and this difference results from quantum jumps. Additionally, in the
case of $\gamma_{\mathrm{eff}}=0$, the eigenenergies of the LEP and HEP are
identical, and we can obtain
$\lambda_{1}=\lambda_{2}=\tilde{\lambda}_{2}=\tilde{\lambda}_{1}=0$,
$\lambda_{3}=\tilde{\lambda}_{3}=i\Omega$, and
$\lambda_{4}=\tilde{\lambda}_{4}=-i\Omega$ for the identical Lindblad
operators (2) and (4). In this case, the dynamics of the LEP and HEP share
identical results with the same initial states.

Besides, in our experiments, the Rabi frequency is $\Omega/2\pi=120$ kHz.
Based on other experimental parameters, we have theoretically calculated the
decay rate for the LEP as $\gamma_{\mathrm{LEP}}=4\Omega\approx 3.0$ MHz, and
for the HEP as $\gamma_{\mathrm{HEP}}=2\Omega\approx 1.5$ MHz. None of these
decay rates is smaller than the maximum decay rate
$\gamma_{\mathrm{max}}\approx 1.45$ MHz in our experiment. As a result,
although there is no HEP in our experiments, we can say, neither the HEPs nor
the LEPs are encircled in our experimental loops (see Fig.S. 2).

Fig.S. 2: (a) Positions of the LEP and HEP in the parameter space for the
eigenenergy Riemann surface. It shows, on the eigenenergy Riemann surface, the
EPs presented correspond to the LEPs rather than the HEPs. (b) Projected decay
rates ($\gamma_{\mathrm{max}}$, $\gamma_{\mathrm{HEP}}$, and
$\gamma_{\mathrm{LEP}}$) in the $\Delta-\gamma_{\mathrm{eff}}$ plane with the
Rabi frequency of $\Omega=120$ kHz.

### II. Thermodynamic quantities

We present here the definitions of the thermodynamic quantities, that are the
net work $W$ and the efficiency $\eta$. For our trapped-ion system, we define
the internal energy of the quantum heat engine as $U=\mathrm{tr}(\rho H)$,
where $\rho$ and $H$ are the density matrix and Hamiltonian of the quantum
system, respectively. In our experiments, the Hamiltonian of the working
substance is given by $H=\Delta\left|e\right\rangle\left\langle e\right|$. The
coupling strength $\Omega$ and the effective dissipation rate
$\gamma_{\mathrm{eff}}$ are employed to tune the system such that the QHE
cycle and the QR cycle are performed, as explained in the main text.

In classical thermodynamics, the first law of thermodynamics is expressed as
$dU=dW+dQ$. In contrast, in quantum thermodynamics, it is expressed as $dU=$
$d(\mathrm{tr}(\rho H))=\mathrm{tr}(\rho dH)+\mathrm{tr}(Hd\rho)$, where the
work and heat in the differential form are defined as $dW=\rho dH$ and
$dQ=Hd\rho$, respectively. Then the work $W_{\mathrm{in}}$ done by the
environment and the one $W_{\mathrm{out}}$ by the working substance are given
by $W_{\mathrm{in}}=-\sum\limits_{i}\rho_{i}dH_{i}^{\prime}$ (for
$dH_{i}^{\prime}>0$) and
$W_{\mathrm{out}}=-\sum\limits_{i}\rho_{i}dH_{i}^{\prime}$ (for
$dH_{i}^{\prime}<0$), respectively. As discussed in the main text,
$W_{{}_{\mathrm{in}}}$ and $W_{\mathrm{out}}$ are calculated, respectively, in
the iso-decay compression and iso-decay expansion strokes of our trapped-ion
QHE and QR cycles. Thus, the net work acquired can be written as

$\begin{array}[]{ccc}W_{\mathrm{net}}&=&W_{{}_{\mathrm{in}}}+W_{\mathrm{out}}=-\sum\limits_{i}\rho_{i}dH_{i}^{\prime},\end{array}$
(5)

which is used to calculate the net work in the main text Figure 2.

### III. Dynamics of quantum heat engine and quantum refrigerator cycles

We execute the clockwise and counterclockwise encirclements without encircling
the LEPs as designed in the main text by elaborately controlling the 854-nm
laser (for tuning $\gamma_{\mathrm{eff}}$) and the 729-nm laser (for tuning
$\Delta$) with $\Omega/2\pi\approx 120$ kHz in the whole process. The
dynamical processes of the four encirclements are demonstrated by observing
the evolution of the fidelity $\langle\psi_{+}|\rho(t)|\psi_{+}\rangle$ or
$\langle\psi_{-}|\rho(t)|\psi_{-}\rangle$ as shown in Fig. S 3(a-d).

Fig.S. 3: Variation of fidelity with respect to the detuning and the effective
decay rate $\gamma_{\mathrm{eff}}$, where we consider the fidelity
$\langle\psi_{+}|\rho(t)|\psi_{+}\rangle$ or
$\langle\psi_{-}|\rho(t)|\psi_{-}\rangle$ when completing a clockwise or a
counterclockwise encirclement, employing $\Delta_{\mathrm{min}}/2\pi=-400$
kHz, $\Delta_{\mathrm{max}}/2\pi=400$ kHz, $\gamma_{\mathrm{min}}\approx 0$
kHz, and $\gamma_{\mathrm{max}}\approx 1.45$ MHz. The red dots represent the
starting points and the red lines show the evolution trends, instead of the
real evolution trajectories, with the red arrows denoting the encircling
directions. The green dashed curves are the projection of the solid red curves
on the bottom plane for guiding eyes. Five orange corner points A, B, C, D, E
(A’, B’, C’, D’, E’) are labeled for convenience of description in the text.
The LEPs are labeled by the purple stars.

For the clockwise QHE cycle in Fig.S. 3(c), we first prepare the state of the
system in $|g\rangle$ and then apply a $\pi/2$ pulse to prepare the initial
state $|\psi_{+}\rangle=(|e\rangle+|g\rangle)/\sqrt{2}$, corresponding to
$\rho_{A}=$$\left(\begin{smallmatrix}0.4925&0.4925\\\
0.4925&0.4925\end{smallmatrix}\right)$ experimentally (due to the imperfect
ground state preparation and subsequent $\pi/2$ pulse operation), and thus the
initial fidelity is $\langle\psi_{+}|\rho_{A}|\psi_{+}\rangle=0.985$. Then we
execute the first iso-decay expansion process by decreasing the detuning
$\Delta$ from $\Delta=0$ kHz to $\Delta_{\mathrm{min}}=-2\pi\times 400$ kHz
while $\gamma_{\mathrm{eff}}\approx 0$ kHz remains unchanged with $T_{1}=6$
$\mu$s. The system evolves to
$\rho_{B}=$$\left(\begin{smallmatrix}0.668&0.444-0.122i\\\
0.444+0.122i&0.317\end{smallmatrix}\right)$ as numerically calculated. Then we
carry out the isochoric cooling process by increasing the decay rate
$\gamma_{\mathrm{eff}}$ from $\gamma_{\mathrm{min}}\approx 0$ kHz to
$\gamma_{\mathrm{max}}\approx 1.43$ MHz while $\Delta=\Delta_{\mathrm{min}}$
is fixed. Experimentally, we separate the evolution into five steps with the
increasing decay rate, i.e. $T_{2-1}=30$ $\mu$s, $\gamma_{\mathrm{eff}}\approx
0$ kHz; $T_{2-2}=30$ $\mu$s, $\gamma_{\mathrm{eff}}\approx 286$ kHz;
$T_{2-3}=30$ $\mu$s, $\gamma_{\mathrm{eff}}\approx 572$ kHz; $T_{2-4}=30$
$\mu$s, $\gamma_{\mathrm{eff}}\approx 858$ kHz; $T_{2-5}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 1.14$ MHz, the system evolves to
$\rho_{C}=$$\left(\begin{smallmatrix}0.021&-0.107-0.089i\\\
-0.107+0.089i&0.964\end{smallmatrix}\right)$ after this process due to the
large decay and detuning. The next stroke is the iso-decay compression with
the detuning tuned from $\Delta_{\rm{min}}$ to $\Delta_{\mathrm{max}}$ and the
constant value of $\gamma_{\mathrm{eff}}=\gamma_{\mathrm{max}}$, the duration
of this stroke is $T_{3}=12$ $\mu$s, and thus the system evolves to
$\rho_{D}=$$\left(\begin{smallmatrix}0.104&0.284-0.008i\\\
0.284+0.008i&0.88\end{smallmatrix}\right)$. The fourth step is the isochoric
heating process with the decay rate $\gamma_{\mathrm{eff}}$ tuned from
$\gamma_{\mathrm{max}}\approx 1.43$ MHz to $\gamma_{\mathrm{min}}\approx 0$
kHz when $\Delta=\Delta_{\mathrm{max}}$ is fixed. During this process, our
experimental operation is to separate the evolution into five steps along with
the decay, i.e., $T_{4-1}=30$ $\mu$s, $\gamma_{\mathrm{eff}}\approx 1.43$ MHz;
$T_{4-2}=30$ $\mu$s, $\gamma_{\mathrm{eff}}\approx 1.14$ MHz; $T_{4-3}=30$
$\mu$s, $\gamma_{\mathrm{eff}}\approx 858$ kHz; $T_{4-4}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 572$ kHz; $T_{4-5}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 286$ kHz, the system evolves to a steady state
$\rho_{E}=$$\left(\begin{smallmatrix}0.021&-0.036-0.137i\\\
-0.036+0.137i&0.964\end{smallmatrix}\right)$ after this stroke with negligible
non-diagonal elements due to the large decay and the long duration time. The
final step is the iso-decay expansion stroke process by decreasing the
detuning from $\Delta=2\pi\times 400$ kHz to $\Delta=0$ kHz while
$\gamma_{\mathrm{eff}}=\gamma_{\mathrm{min}}\approx 0$ kHz remains unchanged
with $T_{5}=6$ $\mu$s, and thus the system experiences a Landau-Zener
transition and accumulate a Stückelberg phase, reaching the final state
$\left(\begin{smallmatrix}0.651&0.464-0.044i\\\
0.464+0.044i&0.334\end{smallmatrix}\right)$. The calculated final fidelity is
$\langle\psi_{+}|\rho(t)|\psi_{+}\rangle=0.957$, indicating that the system
finally returns to the initial state after the encirclement.

By contrast, for the counterclockwise QR cycle in Fig.S. 3(a), we prepare the
initial state
$|\psi_{0}\rangle=|\psi_{-}\rangle=(|e\rangle-|g\rangle)/\sqrt{2}$,
corresponding to
$\rho_{A^{\prime}}=$$\left(\begin{smallmatrix}0.4925&-0.4925\\\
-0.4925&0.4925\end{smallmatrix}\right)$ experimentally, and the initial
fidelity is $\langle\psi_{-}|\rho(A^{\prime})|\psi_{-}\rangle=0.985$. We first
execute the iso-decay compression stroke by increasing the detuning $\Delta$
from $\Delta=0$ kHz to $\Delta_{\rm{max}}=2\pi\times 400$ kHz while
$\gamma_{\mathrm{eff}}\approx 0$ kHz remains unchanged with $T_{1}=6$ $\mu$s,
and thus the system evolves to
$\rho_{B^{\prime}}=$$\left(\begin{smallmatrix}0.668&-0.444+0.122i\\\
-0.444-0.122i&0.317\end{smallmatrix}\right)$. Then we carry out the isochoric
cooling stroke by increasing the decay rate $\gamma_{\mathrm{eff}}$ from
$\gamma_{\mathrm{min}}\approx 0$ kHz to $\gamma_{\mathrm{max}}=1.43$ MHz while
$\Delta=\Delta_{\mathrm{max}}$ is fixed. Experimentally, we separate the
evolution into five steps with the increasing decay rate, i.e. $T_{2-1}=30$
$\mu$s, $\gamma_{\mathrm{eff}}\approx 0$ kHz; $T_{2-2}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 286$ kHz; $T_{2-3}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 572$ kHz; $T_{2-4}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 858$ kHz; $T_{2-5}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 1.14$ MHz, and thus the system evolves to
$\rho_{C^{\prime}}=$$\left(\begin{smallmatrix}0.021&0.128-0.055i\\\
-0.128+0.055i&0.964\end{smallmatrix}\right)$ after this stroke. The next
stroke is the iso-decay expansion with the detuning tuned from
$\Delta_{\mathrm{max}}$ to $\Delta_{\mathrm{min}}$ with the constant value of
$\gamma_{\mathrm{eff}}=\gamma_{\mathrm{max}}$ with $T_{3}=12$ $\mu$s, and thus
the system evolves to
$\rho_{D^{\prime}}=$$\left(\begin{smallmatrix}0.098&-0.112-0.26i\\\
-0.112+0.26i&0.886\end{smallmatrix}\right)$ after this stroke. The fourth step
is the isochoric heating process with decay rate $\gamma_{\mathrm{eff}}$ tuned
from $\gamma_{\mathrm{max}}\approx 1.43$ MHz to $\gamma_{\mathrm{min}}\approx
0$ kHz when $\Delta=\Delta_{{\mathrm{min}}}$ is fixed. During this process, we
divide the evolution into five stages of the decay rate, i.e., $T_{4-1}=30$
$\mu$s, $\gamma_{\mathrm{eff}}\approx 1.43$ MHz; $T_{4-2}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 1.14$ MHz; $T_{4-3}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 858$ kHz; $T_{4-4}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 572$ kHz; $T_{4-5}=30$ $\mu$s,
$\gamma_{\mathrm{eff}}\approx 286$ kHz, and thus the system evolves to a
steady state
$\rho_{E^{\prime}}=$$\left(\begin{smallmatrix}0.021&0.036-0.137i\\\
0.037+0.137i&0.964\end{smallmatrix}\right)$ after this stroke with negligible
non-diagonal elements. The final step is the iso-decay compression stroke by
increasing the detuning from $\Delta=-2\pi\times 400$ kHz to $\Delta=0$ kHz
while $\gamma_{\mathrm{eff}}=\gamma_{\mathrm{min}}\approx 0$ kHz remains
unchanged with $T_{5}=6$ $\mu$s. The system experiences a LZS process and
accumulates a Stückelberg phase contrary to the clockwise encirclements, thus
reaching the final state $\left(\begin{smallmatrix}0.651&-0.464-0.044i\\\
-0.464+0.044i&0.334\end{smallmatrix}\right)$. The final fidelity is
$\langle\psi_{-}|\rho(t)|\psi_{-}\rangle=0.957$, indicating that the system
almost evolves to the initial state at the end of the encirclement.

While for the clockwise encirclement depicted in Fig.S. 3(d), we prepare the
state initially in
$|\psi_{0}\rangle=|\psi_{-}\rangle=(|e\rangle-|g\rangle)/\sqrt{2}$,
corresponding to $\rho_{A}=$$\left(\begin{smallmatrix}0.4925&-0.4925\\\
-0.4925&0.4925\end{smallmatrix}\right)$ experimentally, the fidelity is
initially $\langle\psi_{-}|\rho(A)|\psi_{-}\rangle=0.985$. We repeat the
experimental sequence of the QHE cycle, then the system evolves to a steady
state $\rho_{E}=$$\left(\begin{smallmatrix}0.021&-0.036-0.137i\\\
-0.036+0.137i&0.964\end{smallmatrix}\right)$ after the fourth stroke (the same
as the QHE cycle). After experiencing a LZS in the final stroke, the system
evolves to $\left(\begin{smallmatrix}0.453&0.445+0.206i\\\
0.445-0.206i&0.532\end{smallmatrix}\right)$. The final fidelity is
$\langle\psi_{-}|\rho(t)|\psi_{-}\rangle=0.0487$, showing the system cannot
return to the initial state after the encirclement. For the counterclockwise
encirclement depicted in Fig.S. 3(b), we prepare the initial state
$|\psi_{0}\rangle=|\psi_{+}\rangle=(|e\rangle+|g\rangle)/\sqrt{2}$,
corresponding to
$\rho_{A^{\prime}}=$$\left(\begin{smallmatrix}0.4925&0.4925\\\
0.4925&0.4925\end{smallmatrix}\right)$ experimentally, the initial fidelity is
$\langle\psi_{+}|\rho(A^{\prime})|\psi_{+}\rangle=0.985$. We repeat the
experimental sequence of the QR cycle, then the system evolves to a steady
state $\rho_{E^{\prime}}=$$\left(\begin{smallmatrix}0.021&0.058-0.129i\\\
0.058+0.129i&0.964\end{smallmatrix}\right)$ after the fourth stroke (the same
as the QR cycle). After the fifth stroke, the system evolves to
$\left(\begin{smallmatrix}0.667&-0.454-0.075i\\\
-0.454+0.075i&0.318\end{smallmatrix}\right)$. The final fidelity is
$\langle\psi_{+}|\rho(t)|\psi_{+}\rangle=0.039$, showing that the system
cannot evolve back to the initial state after the encirclement.

As a consequence, we conclude that the chirality depends on both the initial
state and the encircling direction, which results in different thermodynamic
processes, i.e., either QHE or QR.

### IV. State evolution with respect to eigenstates

Fig.S. 4: State evolution with respect to the eigenstates of the effective
Hamiltonian. The evolution of the encircling state for clockwise
(counterclockwise) loops starting from $|\psi_{+}\rangle$ ($|\psi_{-}\rangle$)
is characterized by the projection
$\langle\psi_{m}(t)|\rho|\psi_{m}(t)\rangle$
($\langle\psi_{p}(t)|\rho|\psi_{p}(t)\rangle$), where $\psi_{p}(t)$ and
$\psi_{m}(t)$ correspond, respectively, to the upper and lower branches of the
Riemann surface.

We numerically calculate the state evolutions with respect to the eigenstates
of the time-dependent effective Hamiltonian
$H_{\mathrm{eff}}=(\Delta-i\gamma_{\mathrm{eff}})|e\rangle\langle
e|+\Omega/2(|e\rangle\langle g|+|g\rangle\langle e|)$. The encirclements
plotted in Figs.S. 4(a)-(d) correspond to the trajectories depicted in Figs.
1(c)-(f) in the main text, respectively.

###  X.1 V. Variation of the chirality when approaching LEP

In this section, we carry out numerical simulation for the fidelity after
completing the clockwise or counterclockwise encirclements. Considering the
initial states $|\psi_{+}\rangle$ and $|\psi_{-}\rangle$, respectively, we
sweep $\gamma_{\mathrm{max}}$ but fix other parameters, monitoring how the
population of the final state varies versus $\gamma_{\mathrm{max}}$.

Fig.S. 5: Populations of the final states with respect to
$\gamma_{\rm{max}}/4\Omega$ in different cases of the initial state and
encircling direction. Calculations are made by following parameter values:
$\Delta_{\mathrm{min}}/2\pi=-400$ kHz, $\Delta_{\mathrm{max}}/2\pi=400$ kHz,
$\gamma_{\mathrm{min}}\approx 0$ kHz, $T_{1}=T_{5}=6$ $\mu$s, $T_{2}=T_{4}=50$
$\mu$s, $T_{3}=12$ $\mu$s. Red and blue solid curves represent the populations
of $|\psi_{+}\rangle$ and $|\psi_{-}\rangle$, respectively. Pink lines mark
the positions of LEPs.

We find a transition from no chirality to chirality with the increasing decay
rate $\gamma_{\mathrm{max}}$. When the decay rate $\gamma_{\mathrm{max}}$ is
small enough, the Landau-Zener–Stückelberg (LZS) process ensures the final
state to be the same as the initial state and no asymmetric mode conversion
occurs. As plotted in Fig.S. 5, we see that both the clockwise and
counterclockwise encirclements starting from $|\psi_{+}\rangle$ evolve to the
same final state $|\psi_{+}\rangle$, and the system finally reaches
$|\psi_{-}\rangle$ when starting from $|\psi_{-}\rangle$. With the increase of
the decay rate $\gamma_{\mathrm{max}}$, however, we see asymmetric mode
conversion occurring as $\gamma_{\mathrm{max}}/4\Omega\geq$ 0.05. This is due
to the non-adiabatic transition induced by the enhanced decay. As a result,
the chirality appears. Taking the initial state $|\psi_{+}\rangle$ as an
example, we find the non-adiabatic transition leading to a steady state after
the fourth stroke when encircling along the clockwise direction. Then an LZS
process results in the final state $|\psi_{-}\rangle$. When a counterclockwise
encirclement is performed, the state evolves back to $|\psi_{+}\rangle$
ultimately.  As a result, the decay rate $\gamma_{\mathrm{max}}$ plays a key
role in our experiment. Combined with the Landau-Zener transition, it leads to
non-adiabatic transitions, resulting in chiral behavior and asymmetric mode
conversion (see the starting points for $\gamma_{\mathrm{max}}\approx 0$ in
Fig.S. 5).

We consider that it is possible to have an analytical estimate of the critical
decay rate for the transition, although writing a specifically analytical
relation of the chirality transition with the decay rate is challenging. The
simplest way for such an estimate is to consider the shortest time required
for accomplishing a chirality transition. To this end, we assume that the
dissipation occurs only in the iso-decay process of the 3rd stroke with the
decay rate $\gamma_{\mathrm{max}}$. Then the shortest time for the system to
achieve its steady state is $T_{\mathrm{min}}=1/\gamma_{\mathrm{max}}$. If the
evolution time of the 3rd stroke is $T=12$ $\mu$s, the minimum decay rate for
the chirality can be simply estimated as
$\gamma_{\mathrm{min}}/4\Omega=(1/T)/4\Omega\approx 0.0276$. This complies
with the numerical result in Fig.S. 5.

### VI. Nonreciprocal chirality

Fig.S. 6: Populations $P_{+}$ and $P_{-}$ versus $\Delta$ and
$\gamma_{\mathrm{eff}}$ in clockwise and counterclockwise encirclements, where
the initial states $|\psi_{+}\rangle$ and $|\psi_{-}\rangle$ are considered
respectively, and the typical parameters take the values of
$\Delta_{\mathrm{min}}/2\pi=-400$ kHz, $\Delta_{\mathrm{max}}/2\pi=0$ kHz,
$\gamma_{\mathrm{min}}\approx 0$ kHz, and $\gamma_{\mathrm{max}}\approx 1.45$
MHz. The empty circles represent the starting points.  Fig.S. 7: Population
$P_{+}$ and $P_{-}$ versus $\Delta$ and $\gamma_{\mathrm{eff}}$ in clockwise
and counterclockwise encirclements, where the initial states
$|\psi_{+}\rangle$ and $|\psi_{-}\rangle$ are considered respectively, and the
typical parameters take the values of $\Delta_{\mathrm{min}}/2\pi=0$ kHz,
$\Delta_{\mathrm{max}}/2\pi=400$ kHz, $\gamma_{\mathrm{min}}\approx 0$ kHz,
and $\gamma_{\mathrm{max}}\approx 1.45$ MHz. The empty circles represent the
starting points.

To further understand the relationship between the chirality and the topology,
here we consider the encirclement restricted within single Riemann sheets. As
such, we calculate the state evolution in clockwise and counterclockwise
encirclements starting from $\Delta=0$ and $\gamma=\gamma_{\mathrm{min}}$ as
depicted in Figs.S. 6 and 7. We choose $t_{d}=6$ $\mu$s and $t_{y}=150$ $\mu$s
to keep the change rate of the parameters consistent with our experimental
conditions (Here $t_{d}$ and $t_{y}$ represent the durations of iso-decay and
isochoric strokes, respectively).

For the clockwise encirclements in Figs.S. 6(a, b), the initial state
$|\psi_{+}\rangle$ or $|\psi_{-}\rangle$ evolves into a steady state that is
very similar to the mixture state $(|e\rangle\langle e|+|g\rangle\langle
g|)/2$ during the fourth stroke. So neither the closed encirclement nor
asymmetric mode convention would appear in this case. In contrast, for the
counterclockwise encirclements starting from either $|\psi_{+}\rangle$ or
$|\psi_{-}\rangle$ as depicted by Figs.S 6(c, d), the system evolves to a
steady state at the end of the third stroke, which is very close to the pure
state $|g\rangle$ under the condition of large detuning. This steady state
experiences a LZS with the detuning increasing from $\Delta/2\pi=-400$ kHz to
$\Delta=0$ kHz, reaching $|\psi_{+}\rangle$ finally. Therefore, in this case,
starting from $|\psi_{+}\rangle$ yields a closed encirclement, and starting
from $|\psi_{-}\rangle$ presents asymmetric mode convention.

For the clockwise encirclements in Figs.S 7(a, b), the initial state
$|\psi_{\pm}\rangle$ evolves into a steady state at the end of the third
stroke and finally reaches $|\psi_{-}\rangle$. So the initial state
$|\psi_{-}\rangle$ leads to a closed encirclement, and starting from
$|\psi_{+}\rangle$ presents asymmetric mode convention. In contrast, for the
counterclockwise encirclements in Figs.S. 7(c, d), the system from either
$|\psi_{+}\rangle$ or $|\psi_{-}\rangle$ finally evolves to a mixed state very
similar to $(|e\rangle\langle e|+|g\rangle\langle g|)/2$. So both the closed
encirclement and asymmetric mode convention would not appear in this case.

Therefore, when the encirclements are accomplished within single Riemann
sheets, the chirality is nonreciprocal and incomplete, which can be called
non-reciprocal chirality, i.e., unidirectional chirality.

### VII. More discussions about the chiral behavior and asymmetric mode
conversion

Fig.S. 8: Fidelity $\langle\psi_{\pm}|\rho(T_{5})|\psi_{\pm}\rangle$ versus
the total evolution time $T_{5}$ in the clockwise and counterclockwise
directions, with different initial states $|\psi_{\pm}\rangle$ and quantum
jumps. We set $\gamma_{\mathrm{min}}=0$ kHz for the red solid curve,
$\gamma_{\mathrm{min}}=25$ kHz for the grey dashed-dotted curve, and
$\gamma_{\mathrm{min}}=50$ kHz for the black dotted curve. Other parameters
are $\Delta_{\mathrm{min}}/2\pi=1.0$ MHz, $\Delta_{\mathrm{max}}/2\pi=1.0$
MHz, $\gamma_{\mathrm{min}}\approx 0$ kHz, and $\gamma_{\mathrm{max}}\approx
1.45$ MHz.

Now, we numerically explore the impacts of the evolution time and dissipation
on chiral behavior and asymmetric mode conversion. After a long-time evolution
of the previous four strokes, our system decays to its steady state. Then, the
evolution time of the fifth stroke will govern the following evolution and
behavior as the Landau-Zener transitions. To clarify the above phenomena, we
plot the fidelity $\langle\psi_{\pm}|\rho(T_{5})|\psi_{\pm}\rangle$ versus the
total evolution time of the fifth strokes $T_{5}$ as in Fig.S. 8.

In the absence of dissipations, the evolution time of the fifth stroke plays a
crucial role in determining the adiabaticity. In the non-adiabatic regime,
where the evolution time is very short, the system lacks enough time to follow
the variation of the detuning. In this situation, non-adiabatic transitions
between two eigenstates can occur, leading to variations in the accumulated
phase. These phase variations can generate unexpected quantum interference of
the Landau-Zener-Stückelberg process and weaken the effects of chiral behavior
and asymmetric mode conversion (see the shadow areas in green in Fig.S. 8).

In contrast, in the adiabatic regime with a long enough evolution time, the
system smoothly follows the detuning variation, and the accumulated phase
approaches a fixed value. Therefore, adiabatic evolution can suppress non-
adiabatic transitions and ensure the manifestation of chiral behavior and
asymmetric mode conversion (see the red curves for $\gamma_{\mathrm{eff}}=0$
kHz in Fig.S. 8).

However, all experimental results are inevitably associated with dissipations,
which lead to quantum jumps. The dissipations in the quantum system lead to a
loss of coherence and introduce additional non-adiabatic processes to the
ideal Landau-Zener transition. Consequently, dissipations of the system affect
the chiral behavior and asymmetric mode conversion. When we set a fixed
evolution time, the increased dissipations adjust the chiral behavior towards
non-chiral behavior by non-adiabatic transitions (see the gray dashed-dotted
cures for $\gamma_{\mathrm{eff}}=25$ kHz and black dotted cures for
$\gamma_{\mathrm{eff}}=50$ kHz in Fig.S. 8).

As a result, our experiments demonstrate chiral behavior and asymmetric mode
conversion in general cases, which involve a fixed time and fixed dissipation
in the iso-decay strokes. Additionally, the imperfect chiral behavior and
asymmetric mode conversion arise from the fact that the conditions for two
isochoric strokes cannot follow the conditions for the large detuning.

Fig.S. 9: Fidelity $\langle\psi_{\pm}|\rho(T_{5})|\psi_{\pm}\rangle$ versus
the total evolution time $T_{5}$ in the clockwise and counterclockwise
directions, with different initial states $|\psi_{\pm}\rangle$ and no quantum
jumps. We set $\gamma_{\mathrm{min}}=0$ kHz for the red solid curve,
$\gamma_{\mathrm{min}}=25$ kHz for the grey dashed-dotted curve, and
$\gamma_{\mathrm{min}}=50$ kHz for the black dotted curve. Other parameters
are $\Delta_{\mathrm{min}}/2\pi=1.0$ MHz, $\Delta_{\mathrm{max}}/2\pi=1.0$
MHz, $\gamma_{\mathrm{min}}\approx 0$ kHz, and $\gamma_{\mathrm{max}}\approx
1.45$ MHz.

On the other hand, with quantum jumps ignored, the above results for LEPs
return to those for HEPs. Then, we plot the corresponding results for HEP in
Fig.S. 9. These figures demonstrate that, without the influence of quantum
jumps, we can achieve perfect chiral behavior and asymmetric mode conversion
when the evolution time $T_{5}$ is long enough.

These phenomena illustrate that quantum jump is indeed a key quantum feature
of the LEP, and our observed chiral behavior and asymmetric mode conversion
result from the presence of the LEP.