Dynamical Control of Quantum Heat Engines Using Exceptional Points

Quantum heat engines extract useful work from thermal reservoirs using quantum matter as their working substance. Contrary to their classical counterparts, which do not include coherence in its microscopic degrees of freedom and suffer from irreversible loss during a classical thermodynamic cycle, quantum heat engines are expected to benefit from quantum features to surpass the output power and efficiency that can be attained by an equivalent classical heat engine Quantumthermodynamics ; Science-299-862 ; parado ; PRL-122-110601 . The growing interest in quantum heat engines is also fueled by the interest in understanding the quantum-classical transition in energy-information and work-heat conversion. Additional motivations include the need to maximise the efficiency (the ratio of useful work to the input heat) and the output power while keeping power fluctuations minimal in micro- and nano-scale heat engines, in which quantum fluctuations and non-equilibrium dynamics play a crucial role PNAS-108-15097 ; PRL-112-030602 ; new1 ; Science-352-235 ; Natcommum-10-202 . Microscopic and nanoscopic heat engines with and without the involvement of quantum coherences have been implemented with single trapped ions Science-352-235 ; Natcommum-10-202 ; PRL-123-080602 , ensembles of nitrogen-vacancy centres in diamond PRL-122-110601 , magnetic resonance PRL-100-140501 ; PRL-123-240601 , a single electron box PNAS-111-13786 , and impurity electron spins in a silicon tunnel field-effect transistor PRL-125-166802 . Many interesting proposals have also been put forward for their realisation in superconducting circuits PRL-97-180402 ; Nonew2 and optomechanics PRL-112-150602 ; PRL-114-183602 .

Another field that has been attracting increasing interest is non-Hermiticity, including parity-time ($\mathcal{PT}$) symmetry  PRL-80-5243 in physical systems. In particular, non-Hermitian spectral degeneracies known as exceptional points (EPs) have been shown to have tremendous effects on the dynamics of physical systems, leading to many counterintuitive features which have led to the development of novel functionalities and classical devices  Science-363-42 ; Natmater-18-783 ; Natcommun-9-1308 ; Natcommun-11-1610 ; new2 ; new3 ; new4 ; new5 . Effects of non-Hermiticity have generally been studied by an effective non-Hermitian Hamiltonian and its spectral degeneracies. Recently, there is a growing interest to harness non-Hermiticity and EPs for quantum applications Science-364-878 ; PRL-125-240506 ; Natphys-13-1117 ; PRL-126-083604 ; PRA-103-L020201 ; arXiv:2108.05365 . In quantum systems, however, Hamiltonian EPs (known as HEPs) cannot capture the whole dynamics because these exclude quantum jumps and the associated noises. Instead, one should resort to the Liouvillian formalism which takes into account both coherent non-unitary evolution and quantum jumps LEP ; new6 ; new7 ; PRX-Quantum-2-040346 ; arXiv:2111.04754 . In this formalism, EPs are defined as the degenerate eigenvalues of Liouvillian superoperators. Thus they are referred to Liouvillian EPs (LEPs), whose properties and effects on quantum systems have remained largely unexplored except in some recent experiments in superconducting qubit systems  arXiv:2111.04754 ; PRL-127-140504 .

As open quantum systems which exchange energy with thermal reservoirs, quantum heat engines naturally exhibit non-Hermitian dynamics, which can be controlled by judiciously tuning parameters of the heat engine to operate it in the exact- or broken-phases separated by LEPs (i.e., LEPs correspond to the transition points between the exact- to the broken- phases). Here we report the experimental implementation of a quantum Otto engine using a single ⁴⁰Ca⁺ ion confined in a linear Paul trap PRL-120-210601 ; PRR-2-033082 ; PRL-127-030502 , and demonstrate the control of the engine efficiency and output power by harnessing LEPs and their associated dynamics. This constitutes an interesting observation of the signatures of LEPs in a quantum heat engine. We note that previous experiments studied non-Hermiticity in single-spin systems by considering only HEPs Science-364-878 ; PRL-126-083604 ; PRA-103-L020201 . In contrast, here we use LEPs and their ramifications to control the performance of a quantum heat engine. Thus, our study takes into account quantum jumps and the associated dynamics. As it will become clear below and discussed in PRX-Quantum-2-040346 ; arXiv:2111.04754 , the LEP in this system corresponds to the critical damping point which emerges in the parameter space as the system transits between the oscillatory and non-oscillatory dynamics, in analogy with a damped harmonic oscillator.

Results

Under proper laser irradiation as in Fig. 1B, we obtain an effective two-level model with engineered drive and decay (Supplementary Note 1), as shown in Fig. 1C, which can be described by the Lindblad master equation,

where $\mathcal{L}$ is the Liouvillian superoperator, $\rho$ denotes the density operator, $\gamma_{\mathrm{eff}}$ is the effective decay rate from the excited state $|e\rangle$ to the ground state $|g\rangle$ (Supplementary Note 1), and

with $\Delta$ denoting the frequency detuning between the driving laser and the resonance transition of the ion and $\Omega(t)$ representing the drive amplitude (i.e., Rabi frequency) of the laser PRL-127-030502 ; PRR-2-033082 ; PRL-120-210601 . The dynamics of the system is fully captured by the Liouvillian superoperator $\mathcal{L}$ whose eigenvalues for $\Delta=0$ are $\lambda_{1}=0$, $\lambda_{2}=-\gamma_{\mathrm{eff}}$, $\lambda_{3}=(-3\gamma_{\mathrm{eff}}-\xi)/4$, and $\lambda_{4}=(-3\gamma_{\mathrm{eff}}+\xi)/4$, with $\xi=\sqrt{\gamma_{\mathrm{eff}}^{2}-16\Omega^{2}}$ (Supplementary Note 2 and Supplementary Figure 1). When $\xi=0$, that is $\gamma_{\mathrm{eff}}=4\Omega$, the eigenvalues $\lambda_{3}$ and $\lambda_{4}$ merge, giving rise to a second order LEP at $\tilde{\lambda}=-3\gamma_{\mathrm{eff}}/4$. Clearly, for $\gamma_{\mathrm{eff}}>4\Omega$ (weak coupling), both $\lambda_{3}$ and $\lambda_{4}$ are real with a splitting amount $\xi$, corresponding to the broken phase characterised by a non-oscillatory dynamics with purely exponential decay huber ; naka . For $\gamma_{\mathrm{eff}}<4\Omega$ (strong coupling), on the other hand, $\lambda_{3}$ and $\lambda_{4}$ form a complex conjugate pair which splits in their imaginary parts by $\xi$, corresponding to the exact phase characterised by an oscillatory dynamics. Thus, the LEP 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, $\gamma_{\mathrm{eff}}>4\Omega$). As such, the LEP here is similar to the critical damping point of a damped harmonic oscillator and emerges in the transition between the regions of oscillatory (exact phase) and non-oscillatory (broken phase) dynamics.

In a classical isochoric process, the pressure $P$ and the temperature $T$ change but the volume $V$ remains unchanged, and the working substance reaches thermal equilibrium with the heat bath only at the end of this process. In a classical adiabatic stroke, all thermodynamic quantities $P$, $T$, and the volume $V$ vary (i.e., no invariant thermodynamic quantity) and there is no requirement that occupation probabilities remain unchanged. Therefore, work is done only during the classical adiabatic strokes (no work is done during classical isochoric strokes). Similarly, a quantum heat engine does work only during the quantum adiabatic strokes (i.e., no work is done during the quantum isochoric strokes) but with a different underlying mechanicsm than the classical adiabatic strokes: In a quantum adiabatic stroke, $P_{n}$ and $S$ should remain unchanged during the process (thus no heat exchange) but $E_{n}$ may shift. This change in $E_{n}$ leads to non-zero work.

As demonstrated later, we find that the coherence-enabled improvements in work and power output of a quantum heat engine arise if the coherence during the work strokes (i.e., quantum adiabatic strokes) induces a hump in the thermal strokes (i.e., quantum isochoric strokes). Therefore, to answer the question stated above and clarify the relation among LEPs, the surviving coherence after the thermal strokes, and the performance of a quantum heat engine, we have designed experiments implementing Otto cycles with (i) both isochoric strokes in the exact phase ($\gamma_{\mathrm{eff}}<4\Omega$, oscillatory dynamics), (ii) both isochoric strokes in the broken phase ($\gamma_{\mathrm{eff}}>4\Omega$, non-oscillatory dynamics), and (iii) isochoric heating stroke in the exact but isochoric cooling stroke in the broken phases. We note that a fourth case would be the opposite of (iii), corresponding to a quantum refrigerator, totally reversing the process of the heat engine under the setting (iii). This case is beyond the scope of the present study and thus we have not performed experiments under this setting.

In our system of a single trapped ion, we implement the ingredients of the Otto cycle as follows: Hot and cold heat baths are prepared by tuning $\Omega/\gamma_{\mathrm{eff}}$, which is the ratio of the driving laser beam strength $\Omega$ to the effective decay $\gamma_{\mathrm{eff}}$ of the qubit. This implies that laser irradiation together with the real environment constitutes the baths, where the hot and cold baths correspond to strong and weak drives, respectively. The qubit absorbs or releases heat by coupling to the hot or the cold baths, respectively. Here, we adjust $\gamma_{\mathrm{eff}}$ by varying the power of the laser with wavelength 854 nm, which is tuned to the $P_{3/2}$ - $D_{5/2}$ transition and $\Omega$ is adjusted by tuning the power of the 729 nm laser red-tuned to the $S_{1/2}$ - $D_{5/2}$ transition (Fig. 1B). We evaluate the performance of the heat engine, e.g., the work output to the cold bath and heat absorbed from the hot bath, by monitoring the variation in the populations of the two-level system.

In our treatment under the rotating frame with respect to the driving laser frequency, quantum adiabatic strokes are executed by tuning the frequency of the driving laser which helps vary the internal energy gap $\Delta$ (but without population change) and the temperature of the working substance. Similarly, quantum isochoric strokes are performed by tuning $\Omega/\gamma_{\mathrm{eff}}$ which controls the heat exchange between the working substance and the thermal baths. Thus, while the power of the 854 nm laser helps tune the qubit decay, the power and the frequency of the 729 nm laser helps implement the four strokes of the Otto cycle as follows (Fig. 1E, F and Supplementary Figure 2): Starting with the qubit at steady state with a small population in the excited state, we first carry out an adiabatic compression by increasing the detuning $\Delta$ linearly from $\Delta_{\mathrm{min}}$ to $\Delta_{\mathrm{max}}$, where $P_{e}$ remains in a small and constant value (Step 1 $\rightarrow$ Step 2). Next, we perform isochoric heating by rapidly increasing $\Omega/\gamma_{\mathrm{eff}}$ to a large value, during which the detuning remains equal to $\Delta_{\mathrm{max}}$ (Step 2 $\rightarrow$ Step 3). Then, we carry out an adiabatic expansion by linearly decreasing $\Delta$ from $\Delta_{\mathrm{max}}$ to $\Delta_{\mathrm{min}}$, with $P_{e}$ staying unchanged (Step 3 $\rightarrow$ Step 4). Finally, we perform isochoric cooling by rapidly decreasing $\Omega/\gamma_{\mathrm{eff}}$ to a small value with the $\Delta$ remaining unchanged as $\Delta_{\mathrm{min}}$ (Step 4 $\rightarrow$ Step 1). To accomplish a closed Otto cycle, we wait, after finishing the last stroke, for the system reaching the steady state and returning to the initial state.

Performances of the single-ion quantum heat engine. We evaluate the role of coherence in the performance of the quantum heat engine by monitoring the oscillations in the populations of the qubit states in the isochoric stroke and then assess the net work, the output power, and the efficiency of the heat engine as a function of the execution time $t_{2}$ of the second stroke (isochoric heating) while the execution times of other strokes are kept fixed (Supplementary Notes 4 and 5).

The strong coupling and the exact-phase regimes overlap for $\Omega/\gamma_{\mathrm{eff}}>1/4$ and similarly the weak coupling and the broken-phase regimes overlap for $\Omega/\gamma_{\mathrm{eff}}<1/4$, with the LEP at $\Omega/\gamma_{\mathrm{eff}}=1/4$ being the transition point. Our theoretical study and numerical simulations indicate that in the broken phase (weak coupling regime), the net work produced in an Otto cycle linearly increases with increasing $\Omega/\gamma_{\mathrm{eff}}$. As $\Omega/\gamma_{\mathrm{eff}}$ is increased to transit into the exact phase (strong coupling), the net work increases with $\Omega/\gamma_{\mathrm{eff}}$ in a slower and bifurcated fashion due to enhancement of the coherence in the isochoric strokes, as shown in Fig. 1D. Therefore, as we will show below, by tuning $\Omega/\gamma_{\mathrm{eff}}$, and hence the competition between the driving field strength and the qubit decay, one may elucidate the effects of LEPs, non-Hermiticity, and the associated changes in coherence on the net work, output power, and efficiency of the quantum heat engine.

First, we implement both of the isochoric strokes in the exact phase (i.e., the regime with complex conjugate eigenvalue pairs and hence with oscillatory dynamics). The time evolution of the excited state population $P_{e}$ during the full Otto cycle exhibits a hump in the second stroke (i.e., isochoric heating) and a ramp in the fourth stroke (i.e., isochoric cooling), which reflect the sufficient coherence involved due to the strong coupling effect (Supplementary Note 4 and Supplementary Figure 4), in full agreement with the results obtained from the simulations of the master equation (Fig. 2A₁). These features are also seen when the output power and the net amount of work are obtained as a function of the execution time $t_{2}$ of the second stroke, while the execution time of the other strokes are kept fixed (Fig. 2A₂). The clear hump at $t_{2}\approx$ 4 $\mu$s is an indication of the role of coherence in the second stroke on the output power and the net work. We evaluate the efficiency of the Otto cycles in our experiments by calculating both the conventional efficiency $\eta_{c}$ (defined as the ratio of the net work to the heat absorbed during a full Otto cycle) and the heat absorption efficiency $\eta_{q}$ of the second stroke (i.e., quantum engine efficiency), which corresponds to the energy difference between the initial and end time points of the second stroke (Supplementary Note 3). We find that (Fig. 2A₃): (i) both $\eta_{c}$ and $\eta_{q}$ exhibit humps with a trend similar to that observed in the net work and output power plots, implying the involvement of coherence in the process, (ii) $\eta_{c}$ is less than the efficiency $\eta_{O}=1$ of the ideal Otto cycle, that is $\eta_{c}<\eta_{O}$, and (iii) $\eta_{c}<\eta_{q}$, indicating heat absorption during both the isochoric heating and the isochoric cooling strokes (2nd and 4th strokes).

Heat absorption during the isochoric cooling stroke is expected only in quantum systems and is a result of coherence (i.e., see the ramp in Fig. 2A₁). This additional heat absorption in the cooling stroke might be the reason for $\eta_{c}<\eta_{O}$, which suggests that for improved $\eta_{c}$, one should prevent heat absorption during the cooling stroke by minimising, if not removing, the coherence during this cycle. We note that there is a slight reduction in the population of the excited state $|e\rangle$ during the first and third strokes (Fig. 2A₁). This deviation from the ideal theoretical expectation (i.e., no population change during the adiabatic compression and expansion strokes) can be attributed to the fact that in the experiments the frequency of the 729 nm laser was not tuned smoothly in a continuous fashion but instead we used a sequence of discrete steps using an AOM (Supplementary Note 6). When this imperfection is taken into account, we observe a good agreement between theory and our experimental results. So the slight unexpected deviation in the excited state population does not affect our identification of the Otto cycle physics.

Next, we implement both of the isochoric strokes in the broken phase (i.e., the regime with real eigenvalues: non-oscillatory exponentially decaying dynamics). In contrast to the previous case, in which both isochoric strokes were implemented in the exact phase (oscillatory dynamics), the time evolution of the excited state population $P_{e}$ in this case exhibits neither a hump nor a ramp during the isochoric strokes and $P_{e}$ stays small during the full Otto cycle (Fig. 2B₁) due to weak coupling, in agreement with the results obtained by solving the Master equation. This indicates that coherence is largely erased during the isochoric strokes implemented in the broken phase, and thus the net work and output power are much smaller (Fig. 2B₂). In this regime, we observe that $\eta_{c}$ stays almost constant and has a value very close to $\eta_{O}$ as $t_{2}$ is varied, whereas $\eta_{q}$ gradually increases and attains a value very close to $\eta_{O}$ after $t_{2}=4$ $\mu$s (Fig. 2A₃), indicating that the net work is nearly equivalent to the heat absorbed during the isochoric heating (i.e., 2nd stroke). This is a typical feature of classical Otto cycles, where quantum coherence is not involved.

We note that due to the cold reservoir at the effective temperature $T=0$ K with $\Delta=0$, the Otto cycle efficiency $\eta_{c}$ we observe here is $\eta_{c}\simeq$ 99.07$\%$, which is much higher than the efficiencies reported previously for heat engines implemented in a ${}^{13}C$ nucleus PRL-123-240601 , a trapped ion PRL-123-080602 , and a quantum dot NatNanotech-13-920 . Moreover, comparison of the different conditions in Figs. 2 and 3 suggests that larger reduction in coherence during an Otto cycle, by executing the isochoric strokes in the broken phase, results in higher $\eta_{c}$, but significantly reduced net work and output power.

The two sets of experiments discussed above suggest that coherence in the isochoric heating and cooling strokes of the Otto cycle enhances the net work and output power at the expense of the efficiency $\eta_{c}$, which approaches the ideal value when coherence involved during the isochoric strokes is not sufficient. These experiments also suggest that coherence-enabled heat absorption during the isochoric cooling cycle is the reason for the reduced $\eta_{c}$, when isochoric strokes are implemented in the exact phase (Supplementary Figure 4).

These results open a window of opportunity to achieve high $\eta_{c}$ simultaneously with high net work and output power: Execute the isochoric heating in the exact phase, but the isochoric cooling in the broken phase. Implementing the isochoric cooling in the broken phase will largely reduce the coherence and thus significantly decrease the amount of heat absorption during the cooling stroke. This will prevent the reduction in $\eta_{c}$ and at the same time generates more work and power. In order to confirm this prediction, we have designed the final experiments in such a way that the isochoric heating is executed in the exact phase, whereas the isochoric cooling is executed in the broken phase. The time evolution of the excited state population $P_{e}$ in this case exhibits a hump during the heating stroke which is executed in the exact phase and no ramp is seen during the cooling stroke, which is executed in the broken phase (Fig. 3A). This implies the existence of sufficient coherence only during the heating stroke but not during the cooling stroke. The maximum value of $P_{e}$ is as high as the value obtained for the case when both isochoric strokes are executed in the exact phase. The hump (which is the signature of the presence of sufficient coherence) is seen in the plots of: net work, output power, $\eta_{q}$ versus $t_{2}$, and $\eta_{c}$ versus $t_{2}$ (Fig. 3B, C). The net work and the output power obtained in this setting (Fig. 3B) are much higher than the previous two settings, whose results are given in Fig. 2A₂ and 2B₂. Moreover, we obtain $\eta_{c}\approx$1 for $t_{2}\leq$ 5 $\mu$s, implying that the net work is nearly equivalent to the absorbed heat during the heating stroke. This is the result of the absence of sufficient coherence and hence less heat absorption during the cooling stroke.

Note that $\eta_{c}$ experiences only a very small decrease down to $\eta_{c}\approx 0.9$ for longer $t_{2}$, which is much higher than the $\eta_{c}\sim 0.65$ obtained when both isochoric strokes are executed in the exact phase, and only slightly lower than the $\eta_{c}=1$ obtained when both strokes are executed in the broken phase. We note that the net work, output power, $\eta_{c}$, and $\eta_{q}$ achieved their maximum at $t_{2}=$ 4 $\mu$s. Our experiments clearly demonstrate that the exact phase (and hence more coherence) during the isochoric heating (second stroke) and broken phase (and hence less coherence) during the isochoric cooling (fourth stroke) lead to a better quantum heat engine performance.

Discussion

Further study would evolve in two directions. First, one of the vibrational modes of the trapped ion working as the heat engine can be used as the quantum load (i.e., optical states of the ion act as the working substance and the vibrational modes coupled to them act as the load) and study heating and cooling processes in the spirit of sideband heating or sideband cooling. Second, an additional ion confined in the same trap with the ion working as the heat engine can be used as the load. One can then rearrange the strokes of the engine cycle to perform heating or refrigeration. For example, performing the strokes of the Otto cycle in counterclockwise direction as shown in Fig. 1E with the stroke sequence as 1$\rightarrow$2$\rightarrow$3$\rightarrow$4$\rightarrow$1 will lead to heating whereas carrying out the Otto cycle in the clockwise direction with the stroke sequence as 1$\rightarrow$4$\rightarrow$3$\rightarrow$2$\rightarrow$1 will result in cooling. In these cases, the engine will be coupled to (decoupled from) the load during the adiabatic compression and expansion strokes (isochoric heating and cooling strokes). Performing measurements on the load after each engine cycle would then help understanding the cooling and heating process as a function of the number of engine cycles. One should however keep in mind that correlations may build up between the quantum engine and the quantum load during the adiabatic strokes (when they are coupled); therefore, one should be careful when interpreting heating/cooling, work, and other thermodynamic quantities. Further studies are needed to have a deeper physical insight into the role of Liouvillian exceptional points in the performance of quantum heat engines and to better quantify the heat, power, and efficiency of quantum heat engines coupled to quantum loads.

The authors are grateful to Yunlan Zuo for her help in plotting Figure 1. 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, 12074346, 12074390, 11835011, 11804375, 11804308, 91636220, by Postdoctoral Science Foundation of China under Grant No. 2022MT10881, by Natural Science Foundation of Henan Province under Grant No. 202300410481 and 212300410085, by K. C. Wong Education Foundation (GJTD-2019-15). S.K.O acknowledges the support from Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) Award No. FA9550-21-1-0202. F.N is supported in part by NTT Research, JST and JSPS.

Author contributions

HJ and MF conceived the idea and designed the experiments; JWZ and FZ performed the experiments with help from JCL, JTB and BW. Theoretical background and simulations were provided by JQZ and GYD. 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.

Competing interests

The authors declare no competing interests.

Supplementary materials for
“Dynamic Control of Quantum Heat Engines with Exceptional Points”

Defining $H_{0}=\omega_{l}\left|e\right\rangle\left\langle e\right|+\omega_{p}\left|p\right\rangle\left\langle p\right|$ and using the Baker-Hausdorff formula

we find

where we have $\Delta=\omega_{e}-\omega_{l}$ and $\omega_{p}=\omega_{d}+\omega_{l}$, $\omega_{d}=\omega_{p}-\omega_{l}=\omega_{p}-(\omega_{e}-\Delta)\simeq\omega_{p}-\omega_{e}$. The effective Hamiltonian $H_{{\rm eff}}$ is then given by PRA-85-032111

where the non-Hermitian Hamiltonian $H_{{\rm NH}}$ is given by $H_{{\rm NH}}=-\frac{i}{2}(\gamma_{e}+\gamma_{g})\left|p\right\rangle\left\langle p\right|=-\frac{i}{2}\gamma\left|p\right\rangle\left\langle p\right|$, with $\gamma_{i}$ corresponding to the decay rate from the level $|p\rangle$ to the level $|i\rangle$, while the perturbative excitations $V_{+}$ and de-excitations $V_{-}$ are defined as $V_{+}=\frac{\Omega_{p}}{2}|p\rangle\left\langle e\right|+\frac{\Omega}{2}\left|e\right\rangle\left\langle g\right|$ and $V_{-}=\frac{\Omega_{p}}{2}\left|e\right\rangle\left\langle p\right|+\frac{\Omega}{2}\left|g\right\rangle\left\langle e\right|$. We then write the effective Lindblad operator as PRA-85-032111

from which the effective decay rate $\gamma_{\mathrm{eff}}$ from the excited state $|e\rangle$ to the ground state $|g\rangle$ is found as

The same result by an alternative method can be found in PRR-2-033082 . We note that the expression for $H_{{\rm eff}}$ in Eq. (S7) is the same as Eq. (2) in the main text and $\gamma_{{\rm eff}}$ appears in Eq. (1) of the main text and in the expression of the LEP of the system.

Since these eigenvalues are related to dissipation terms, their real parts indicate decaying effects while their imaginary parts define the eigenenergies LEP . It is easily seen that the eigenvalues $\lambda_{3}$ and $\lambda_{4}$ become degenerate and an LEP emerges at $\lambda_{3}=\lambda_{4}=-3\gamma_{\mathrm{eff}}/4$, when $\gamma_{\mathrm{eff}}=4\Omega$. As explained in the main text, for $\gamma_{\mathrm{eff}}>4\Omega$, we have real $\lambda_{3}$ and $\lambda_{4}$ with a splitting amount given by $\xi=\sqrt{\gamma_{\mathrm{eff}}^{2}-16\Omega^{2}})$. Thus, when $\gamma_{\mathrm{eff}}>4\Omega$, the system is in the broken phase. For $\gamma_{\mathrm{eff}}<4\Omega$, on the other hand, $\lambda_{3}$ and $\lambda_{4}$ become complex conjugate pairs with a splitting of $\xi$ in their imaginary parts, corresponding to the exact phase.

Moreover, to justify the validity of our reduced matrix, i.e., Eq. (12), which comes from the effective Hamiltonian Eq. (8), we have calculated the LEPs for both Hamiltonians (S6) and (S7), and found that the LEPs share the same decay rate as written in (S9) in the case of $\Delta=0$. This is further confirmed in Supplementary Figure 1. The eigenvectors corresponding to the above eigenvalues are,

and

Clearly, at $\Delta=0$ and $\gamma_{{\rm eff}}=4\Omega$, the eigenstates $\rho_{3}$ and $\rho_{4}$ coalesce, confirming that a second-order LEP emerges at $\gamma_{{\rm eff}}=4\Omega$, separating the exact- and broken-phases.

Using the expression $C_{l_{1}}(\rho)=\sum_{i\neq j}|\rho_{i,j}|$ for the coherence PRL-113-140401 , we have calculated the coherence involved in an Otto cycle in our system and presented the results in Supplementary Figure 4. We see that the coherence remaining at the end of an Otto cycle is the lowest when the system is operated such that both of the isochoric strokes are in the broken phase and the highest when both of the isochoric strokes are implemented in the exact phase.

In Figs 2A₁, 2B₁ and 3A, we note a slight reduction ($\approx$0.1) in the population of the excited state $|e\rangle$ during the first and third strokes. This deviation from the ideal theoretical expectation (i.e., no population change during the adiabatic compression and expansion strokes) can be attributed to the fact that in the experiments the frequency of the 729 nm laser was not tuned smoothly in a continuous fashion but instead we used a sequence of discrete steps using an AOM. As shown in Supplementary Figure 6, although the required continuous change of the detuning $\Delta$ (blue line) can be equivalently accomplished by discrete steps of operation, as plotted by the red line. In our experiments, the variation of detuning is accomplished by a sequence of AOM which leads to a slightly different tuning curve (green) due to the switching time of the AOM. This leads to imperfect variation of the detuning, bringing in unexpected phases in the operation. Nevertheless, considering these unexpected phases in numerical simulation, we have fitted the experimental observations involving such cases by theory very well, see Figs. 2A₁, 2B₁ and 3A in the main text.