Quantum heat engines (QHEs) and refrigerators (QRs) are open quantum systems whose dynamics are well-described using a non-Hermitian formalism. A key feature of non-Hermiticity is the existence of exceptional points (EPs), absent in closed systems. Classical systems show chiral mode conversion via dynamic encirclement near an EP, regardless of whether the loop includes the EP. This paper extends this to quantum systems by dynamically encircling Liouvillian exceptional points (LEPs), which account for quantum jumps and noise. LEPs offer unique properties to QHEs, such as optimizing dynamics, enhancing efficiency, and imparting topological properties. However, LEPs and their effects remain largely unexplored in quantum thermodynamics. Dynamically encircling a Hamiltonian EP (HEP) causes chiral state transfer due to non-adiabatic transitions. However, recent studies show chiral behavior even without encircling an HEP, as any dynamic loop near an EP displays chiral features. LEPs exhibit similar Riemann surfaces, leading to non-trivial state transfer dynamics. This paper experimentally demonstrates chiral behavior in a qubit system without encircling LEPs, using a single trapped ion as a quantum engine for heating and refrigeration. The experiment connects, for the first time, the Landau-Zener-Stückelberg (LZS) process to chirality and LEP-related thermodynamic effects. This research bridges quantum thermodynamics, LEPs, and chiral state transfer.

The paper reviews existing literature on quantum heat engines and refrigerators, highlighting their implementations in various systems like trapped ions, spin ensembles, superconducting circuits, and quantum optomechanical systems. It also discusses the emerging field of non-Hermitian dynamics and exceptional points (EPs) in classical and quantum systems. The literature on Hamiltonian EPs (HEPs) and their role in chiral state transfer is reviewed, contrasting it with the lesser-explored field of Liouvillian EPs (LEPs) and their implications for quantum thermodynamics. The authors note the challenges in exploring chiral behavior in quantum systems without encircling LEPs, citing influences such as the Riemann surface topology, dynamic process trajectory, and speed, along with the phases of the Landau-Zener-Stückelberg (LZS) process, quantum coherence, network, and quantum heat engine efficiency.

The experiment uses a single ultracold 40Ca+ ion confined in a linear Paul trap. The ground state |2S1/2⟩ is split into two hyperfine levels, and the metastable state |2D5/2⟩ into six, by an external magnetic field. A three-level system, simplified to an effective two-level qubit, is created using 729-nm and 854-nm lasers, with tunable Rabi frequency Ω and effective decay rate γeff. The dynamics are governed by a Lindblad master equation, including the effective Hamiltonian Heff = Δ|e⟩⟨e| + Ω(|e⟩⟨g| + |g⟩⟨e|), where Δ is the detuning. The eigenvalues of the Liouvillian superoperator reveal an LEP when γeff = 4Ω. The experiment uses this qubit as the working substance of a QHE and QR. Thermodynamic cycles are implemented with two iso-decay and two isochoric strokes. Iso-decay strokes involve varying detuning Δ while keeping γeff constant, performing work. Isochoric strokes involve rapidly reaching steady states by changing γeff with constant Δ, corresponding to heating or cooling. The qubit is initialized in superposition states |+⟩ or |−⟩. Clockwise (CW) and counterclockwise (CCW) loops in the Δ-γeff parameter space, encircling or not encircling the LEP, are performed. The fidelity of the final state compared to the initial state quantifies the chiral behavior. The net work is calculated from the time evolution of the system's state. The Landau-Zener-Stückelberg (LZS) process and the breakdown of adiabaticity during the loops are analyzed.

The experiment demonstrates chiral quantum heating and cooling. Loops near the LEP, without encircling it, show chiral behavior: CW loops end at |+⟩ and CCW loops at |−⟩, irrespective of the initial state. Loops not exhibiting asymmetric mode conversion act as QHEs (positive net work) or QRs (negative net work). The Landau-Zener-Stückelberg (LZS) process and adiabaticity breakdown are crucial for the chiral thermodynamic cycles. The net work (Wnet) is calculated from the time evolution of the Hamiltonian, considering the contributions of the isochoric and iso-decay strokes. Specifically, a counterclockwise loop starting at |−⟩ (Fig 2a) resulted in quantum refrigeration (QR), while a clockwise loop beginning at |+⟩ (Fig 2b) acted as a quantum heat engine (QHE). Figure 3 shows the chiral dynamics, highlighting that the final state depends only on the loop direction, not the starting state. The experiment quantitatively demonstrates the connection between the LZS process and chirality within the context of LEP-related thermodynamic effects.

The findings demonstrate the chiral nature of quantum thermodynamic cycles near an LEP, even without explicitly encircling the LEP. The interplay between the LZS process and adiabaticity breakdown is essential to the observed chiral behavior. The ability to control quantum heating and cooling through the manipulation of parameters near the LEP opens possibilities for designing efficient quantum thermal devices. This work connects the previously separate fields of non-Hermitian physics and quantum thermodynamics, deepening our understanding of topological effects in open quantum systems and their potential applications in quantum technologies.

This work experimentally demonstrates chiral quantum heating and cooling using a single trapped ion qubit. The chiral behavior is achieved by performing closed-loop trajectories in the parameter space near a Liouvillian exceptional point (LEP), without explicitly encircling the LEP. The Landau-Zener-Stückelberg (LZS) interference and the breakdown of adiabaticity are identified as key mechanisms driving this chiral behavior. The results advance our understanding of the interplay between non-Hermitian physics and quantum thermodynamics. Future research could investigate the effects of different loop shapes, the optimization of the thermodynamic cycles, and the exploration of more complex quantum systems.

The study focuses on a specific type of thermodynamic cycle and a single trapped ion system. The generalizability of the findings to other systems or different cycle configurations needs further investigation. Imperfections in state preparation and measurement could slightly affect the precision of the results. A more detailed analysis of the role of decoherence and quantum noise could be undertaken.