Chiral quantum heating and cooling with an optically controlled ion

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 contribute 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.


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][37][38][39][40][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.

Results and Discussions
Experimental setup.Our experiment is carried out in a single ultracold 40 Ca + ion confined in a linear Paul trap (Fig. 1(a) with more details in [51,52]) whose axial and radial frequencies are, respectively, ω z /2π = 1.01 MHz and ω r /2π = 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 ⟩ is split into two hyperfine energy levels while the metastable state |3 2 D 5/2 ⟩ split into six.As shown in Fig. 1 as |g⟩, |3 2 D 5/2 , m J = +5/2⟩ as |e⟩ and |4 2 P 3/2 , m J = +3/2⟩ as |p⟩.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 ∼ 0.11, which is sufficient to avoid detrimental effects of thermal phonons (e.g., Rabi oscillation offsets).Introducing the dipolar transition |e⟩ → |p⟩ by switching on 854-nm laser as explained in [32,53], 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.
The dynamical evolution of this effective two-level model is governed by the Lindblad where L is the Liouvillian superoperator, and ρ and γ eff denote the density operator and the effective decay rate from |e⟩ to |g⟩, respectively.Here the effective Hamiltonian is H eff = ∆|e⟩⟨e|+ Ω 2 (|e⟩⟨g|+ |g⟩⟨e|), where ∆ represents the frequency detuning between the resonance transition and the driving laser while Ω denotes the Rabi frequency.The eigenvalues of L at ∆ = 0 are given by λ 1 = 0, λ 2 = −γ eff /2, λ 3 = (−3γ eff −ξ)/4 and λ 4 = (−3γ eff +ξ)/4, with ξ = γ 2 eff − 16Ω 2 .It is evident that the eigenvalues λ 3 and λ 4 coalesce when γ eff = 4Ω, giving rise to a second order LEP at λ = −3γ eff /4 [55].In the weak coupling situation γ eff > 4Ω, both λ 3 and λ 4 are real with a splitting of ξ/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 (γ eff < 4Ω), λ 3 and λ 4 are a pair of complex conjugates with the imaginary parts −ξ/4 and ξ/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, γ eff < 4Ω) and a region of non-oscillatory dynamics (broken phase, 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 ∆ in the effective Hamiltonian H eff , in which the increase and decrease of the population in the excited state |e⟩ 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].a,b) present the thermodynamic cycle we carry out, consisting of two iso-decay and two isochoric strokes with a fixed Rabi frequency Ω as in Ref. [33].For the iso-decay strokes, we achieve positive (negative) work by decreasing (increasing) the detuning ∆ while keeping the decay rate γ 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 ∆ and Ω, we increase (or decrease) the population of the excited state for heating (or cooling) by decreasing (increasing) the decay rate γ 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  Quantum heating and cooling.Figure 2 displays experimental results of a counterclockwise loop starting at |ψ − ⟩ and a clockwise loop starting at |ψ + ⟩, corresponding to the schemes in Fig. 1(c,e).These loops do not encircle the LEP of the system.In Fig.In contrast, for the clockwise loop from |ψ + ⟩ = (|e⟩ + |g⟩)/ √ 2, the system evolves to a steady state after the fourth stroke and experiences a LZS with the detuning ∆ tuned from ∆ max /2π = 400 kHz to 0, accumulating a Stückelberg phase [50] and thus leading to the Since the system evolves along a closed trajectory, we can evaluate the net work using , where ρ(t) describes the state of the two-level system governed by H(t) = ∆(t)|e⟩⟨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⟩ 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⟩ 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 |ψ + ⟩ and CCW loop starting at |ψ − ⟩) performs as a QHE (Fig. 2(b)) or a QR (Fig. 2(a)).
We mention that Rabi frequency Ω 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 γ eff while keeping a constant value of ∆ [6,56,57].
Chiral dynamics.The chiral behavior and asymmetric mode conversion appear for counterclockwise loop starting at |ψ + ⟩ and the clockwise loop starting at |ψ − ⟩, 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 |ψ + ⟩ (or |ψ − ⟩) regardless of whether the initial state is |ψ 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).In other words, when the system starts at the state |ψ + ⟩, 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 |ψ − ⟩.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 |ψ − ⟩, a clockwise loop in the vicinity of (not encircling) the LEP ends in the orthogonal state |ψ + ⟩ whereas a counterclockwise loop ends at the same state |ψ − ⟩.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 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., γ 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 |ψ + ⟩ and |ψ − ⟩ interchange for the clockwise loop starting at |ψ − ⟩ and the counterclockwise loop starting at |ψ + ⟩ [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 |ψ + ⟩ and |ψ − ⟩.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 [34][35][36][37][38][39][40][41], 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 γ max to values larger than 4Ω [33]; however, observing chiral behavior without encircling the LEP requires varying γ max to values less than 2Ω.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].

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

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 where L is called Liouvillian superoperator and ρ is the density operator of the system.Here By setting ∆ = 0, we find the eigenvalues of L as It is evident that the eigenvalues λ 3 and λ 4 becomes degenerate at γ eff = 4Ω, which corresponds to the Liouvillian exceptional point (LEP).For γ eff > 4Ω, we have real λ 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 where H NH = H eff − i γ eff 2 σ + σ − is the non-Hermitian Hamiltonian and is the Liouvillian operator without quantum jumps.Both H NH and L H have been utilized to experimentally verify the phase transition of EP for ∆ = 0 [58].The above derivations illustrate, without involving the quantum jump item σ − ρσ + , 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 The eigenvalues λ3 and λ4 become degenerate at γ eff = 2Ω, corresponding to the HEP.Real λ3 and λ4 present a splitting of | λ3 − λ4 | = γ 2 eff − 4Ω 2 when γ eff > 2Ω, while λ3 and λ4 become a complex conjugate pair with a splitting of γ 2 eff − 4Ω 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 γ eff = 0, the eigenenergies of the LEP and HEP are identical, and we can obtain λ 1 = λ 2 = λ2 = λ1 = 0, λ 3 = λ3 = iΩ, and λ 4 = λ4 = −iΩ 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.

II. Thermodynamic quantities
We present here the definitions of the thermodynamic quantities, that are the net work W and the efficiency η.For our trapped-ion system, we define the internal energy of the quantum heat engine as U = tr(ρH), where ρ 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 = ∆ |e⟩ ⟨e|.The coupling strength Ω and the effective dissipation rate γ 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(tr(ρH)) = tr(ρdH) + tr(Hdρ), where the work and heat in the differential form are defined as dW = ρdH and dQ = Hdρ, respectively.Then the work W in done by the environment and the one W out by the working substance are given by As discussed in the main text, W in and W 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 which is used to calculate the net work in the main text Figure 2. .Then we carry out the isochoric cooling stroke by increasing the decay rate γ eff from γ min ≈ 0 kHz to γ max = 1.43 MHz while ∆ = ∆ max is fixed.
Experimentally, we separate the evolution into five steps with the increasing decay rate, i.e.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
We numerically calculate the state evolutions with respect to the eigenstates of the time- the dissipation occurs only in the iso-decay process of the 3rd stroke with the decay rate γ max .Then the shortest time for the system to achieve its steady state is T min = 1/γ max .
If the evolution time of the 3rd stroke is T = 12 µs, the minimum decay rate for the chirality can be simply estimated as γ min /4Ω = (1/T )/4Ω ≈ 0.0276.This complies with the numerical result in represent the durations of iso-decay and isochoric strokes, respectively).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 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.
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.

FIG. 1 .
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 Ω and Ω driven by 729 nm and 854 nm lasers, respectively, and ∆ is the detuning between the energy level and 729 nm laser.The wavy arrow denotes the spontaneous emission with decay rate Γ.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 ∆-γ eff parametric space starting from |ψ + ⟩ or |ψ − ⟩, where the parameters take the values ∆ min /2π = −400 kHz, ∆ max /2π = 400 kHz, γ min ≈ 100 kHz, γ max ≈ 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.

Figures 1 (
Figures1(a,b) present the thermodynamic cycle we carry out, consisting of two iso-decay

FIG. 2 .
FIG. 2. Closed loops that do not exhibit mode conversion.(a) Counterclockwise loop starting from |ψ − ⟩.(b) Clockwise loop starting from |ψ + ⟩.Evolution of the system's state along the trajectory of the closed loop is characterized by the fidelity ⟨ψ − |ρ(t)|ψ − ⟩ in (a) and the fidelity ⟨ψ + |ρ(t)|ψ + ⟩ 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 µs, T 3 = 12 µs, and T 2 = T 4 = 150 µs.Other parameters are Ω/2π = 120 kHz, ∆ min /2π = −400 kHz, ∆ max /2π = 400 kHz, γ min ≈ 0 kHz, and γ max ≈ 1.45 MHz.

2 .
2(b), we prepare the qubit in the initial state |ψ + ⟩ with fidelity ⟨ψ + |ρ A |ψ + ⟩ = 0.985 which is lower than 1 due to imperfections during the state preparation.The system returns to |ψ + ⟩ after completion of the loop.Figure2(a) exhibits the results of the counterclockwise loop starting at |ψ − ⟩.The state of the system evolves back to the initial state |ψ − ⟩ 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 |ψ − ⟩ = (|e⟩ − |g⟩)/ √ The Landau-Zener transition occurs in the first stroke with the detuning ∆ varied from 0 to ∆ max /2π = 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 ∆/2π = 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 ∆/2π = −400 kHz and slow-varying dissipation.Finally, a Landau-Zener transition occurs again in the fifth stroke with the detuning ∆ tuned from ∆ min /2π = −400 kHz to 0, resulting in the final state |ψ − ⟩ = (|e⟩ − |g⟩)/ √ 2 [49, 50].
For the clockwise encirclement starting from |ψ + ⟩ or the counterclockwise encirclement from |ψ − ⟩, 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 |ψ − ⟩ or the counterclockwise loop at |ψ + ⟩, this Landau-Zener transition determines the chiral behavior.
evolution duration is restricted by the decoherence time, as clarified with the numerical simulation results shown in Fig. S.8.
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.

Figures in the maintext
Figures in the maintext

Maintext Fig. 1 . 2 .. 3 .
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 Ω and Ω driven by 729 nm and 854 nm lasers, respectively, and ∆ is the detuning between the energy level and 729 nm laser.The wavy arrow denotes the spontaneous emission with decay rate Γ.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 ∆-γ eff parametric space starting from |ψ + ⟩ or |ψ − ⟩, where the parameters take the values ∆ min /2π = −400 kHz, ∆ max /2π = 400 kHz, γ min ≈ 100 kHz, γ max ≈ 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.Closed loops that do not exhibit mode conversion.(a) Counterclockwise loop starting from |ψ − ⟩.(b) Clockwise loop starting from |ψ + ⟩.Evolution of the system's state along the trajectory of the closed loop is characterized by the fidelity ⟨ψ − |ρ(t)|ψ − ⟩ in (a) and the fidelity ⟨ψ + |ρ(t)|ψ + ⟩ 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 µs, T 3 = 12 µs, and T 2 = T 4 = 150 µs.Other parameters are Ω/2π = 120 kHz, ∆ min /2π = −400 kHz, ∆ max /2π = 400 kHz, γ min ≈ 0 kHz, and γ max ≈ 1.45 MHz.Closed loops that lead to mode conversion.(a) Counterclockwise loop starting at |ψ + ⟩ and ending at |ψ − ⟩.(b) Clockwise loop starting at |ψ − ⟩ and ending at |ψ + ⟩.Evolution of the system's state along the closed loop trajectory is characterized by the fidelity ⟨ψ + |ρ(t)|ψ + ⟩ in (a) and the fidelity ⟨ψ − |ρ(t)|ψ − ⟩ 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 µs, T 3 = 12 µs, and T 2 = T 4 = 150 µs.Other parameters are Ω/2π = 120 kHz, ∆ min /2π = −400 kHz, ∆ max /2π = 400 kHz, γ min ≈ 0 kHz, and γ max ≈ 1.45 MHz.

the density matrix ρ is expressed by   ρ ee ρ eg ρ ge ρ gg 
, with the raising and lowering operators defined as σ + = .According to the corresponding equations in the main text, we can write L as

3 and λ 4 Fig.S. 1 .
Fig.S. 1. Topological phase transition with respect to the Liouvillian exceptional point (LEP) that is determined by the decay rate γ eff and the drive strength Ω.

γ
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 (γ max , γ HEP , and γ LEP ) in the ∆ − γ eff plane with the Rabi frequency of Ω = 120 kHz.

III.Fig.S. 3 .. 2 ,
Fig.S. 3. Variation of fidelity with respect to the detuning and the effective decay rate γ eff , where we consider the fidelity ⟨ψ + |ρ(t)|ψ + ⟩ or ⟨ψ − |ρ(t)|ψ − ⟩ when completing a clockwise or a counterclockwise encirclement, employing ∆ min /2π = −400 kHz, ∆ max /2π = 400 kHz, γ min ≈ 0 kHz, and γ max ≈ 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 pointsA, 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.
value.Therefore, adiabatic evolution can suppress non-adiabatic transitions and ensure the manifestation of chiral behavior and asymmetric mode conversion (see the red curves forγ 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 dasheddotted cures for γ eff = 25 kHz and black dotted cures for γ eff = 50 kHz in Fig.S. 8).
and then perform a thermodynamic cycle, corresponding to a loop in the parameter space, by tuning γ eff and ∆ such that the LEP of the system is not encircled.The superposition states |ψ + ⟩ and |ψ − ⟩ correspond to two eigenstates of H eff for ∆ = 0 kHz at the middle point of the expansion and compression of the qubit with the initial state |ψ + ⟩ or |ψ − ⟩ by calculating the fidelity ⟨ψ + |ρ|ψ + ⟩ or ⟨ψ − |ρ|ψ − ⟩.Figures1(c) and 1(d) show the evolution of the system for two counterclockwise loops that start at the initial states |ψ − ⟩ and |ψ + ⟩, respectively, and are completed without encircling the LEP.We first implement an iso-decay compression stroke from A to B by increasing the detuning ∆ linearly from 0 to its maximum value ∆ max while keeping γ eff at its minimum value of γ min .The second stroke is an isochoric cooling stroke from B (∆ max , γ min ) to C (∆ max , γ max ), which is realized by increasing γ eff from γ min to γ max with the detuning remaining constant at ∆ max .Then we execute the third stroke, i.e., an iso-decay expansion from C (∆ max , γ max ) to D (∆ min , γ max ), by decreasing ∆ from ∆ max to ∆ min while keeping γ eff = γ max .The fourth stroke is an isochoric heating from D (∆ min , γ max ) to E (∆ min , γ min ) executed by decreasing γ eff from γ max to γ min with ∆ fixed at ∆ min .The final step is an iso-decay compression stroke from E (∆ min , γ min ) back to A(0, γ min ) executed by increasing ∆ from ∆ min to 0 while γ eff is kept fixed at γ min .The clockwise loops depicted in Figs.1(e) and 1(f) are also executed in five strokes with the initial states prepared in |ψ + ⟩ and |ψ − ⟩ but in reverse order of the process described above