Enhancement of Quantum Heat Engine by Encircling a Liouvillian Exceptional Point

Quantum heat engines are expected to outperform the classical counterparts due to quantum coherences involved. Here we experimentally execute a single-ion quantum heat engine and demonstrate, for the first time, the dynamics and the enhanced performance of the heat engine originating from the Liouvillian exceptional points (LEPs). In addition to the topological effects related to LEPs, we focus on thermodynamic effects, which can be understood by the Landau-Zener-Stückelberg process under decoherence. We witness a positive net work from the quantum heat engine if the heat engine cycle dynamically encircles a LEP. Further investigation reveals that a larger net work is done when the system is operated closer to the LEP. We attribute the enhanced performance of the quantum heat engine to the Landau-Zener-Stückelberg process, enabled by the eigenenergy landscape in the vicinity of the LEP, and the exceptional point-induced topological transition. Therefore, our results open new possibilities toward LEP-enabled control of quantum heat engines and of thermodynamic processes in open quantum systems.

Figure 1

(a) Level scheme of ${}^{40}{\mathrm{Ca}}^{+}$ ion, where the solid arrows represent the transitions driven by lasers with Rabi frequencies $\mathrm{\Omega }$ ($\tilde{\mathrm{\Omega }}$) for the 729-nm (854-nm) laser and detuning $\mathrm{\Delta }$ for the 729-nm laser. The wavy arrow represents spontaneous emission with decay rate $\mathrm{\Gamma }$. (b) Simulations of the populations in $|2⟩$ with respect to the detuning $\mathrm{\Delta }$ and the effective decay rate ${\gamma }_{\mathrm{eff}}$, where the red (green) solid curve represents the evolution of the population with (without) the LEP encircled. The dashed curves are the projection of the solid curves with the same color on the bottom plane for guiding eyes. The four corner points $A$, $B$, $C$, and $D$ are labeled for convenience of description in the text. The LEP is labeled by the purple star. (c) Illustration of trajectories with and without encircling the LEP on the Riemann surface, corresponding to the QHE cycles plotted by the solid lines with the same color in (b). In contrast to the green curve, representing the trajectory without encircling the LEP and localized just in one branch of the Riemann surface, the red curve encircling the LEP passes through the two branches (the dashed part denotes the track in the other branch), leading to the topological properties that are relevant to the thermodynamic effects observed in the present experiment.

Figure 2

Measured populations of the excited state, i.e., $P_2$, in isodecay strokes, where (a) and (b) are for the first and third strokes, respectively, in the case of encircling the LEP (${\mathrm{\Delta }}_{\min }/2\pi =-290\mathrm{kHz}$, ${\gamma }_{\max }=800\mathrm{kHz}$, ${\mathrm{\Delta }}_{\max }/2\pi =10\mathrm{kHz}$, ${\gamma }_{\min }=130\mathrm{kHz}$). The time is set as $T_1=T_3=30\mathrm{\mu }\mathrm{s}$. (c) and (d) represent the first and third strokes, respectively, in the case of not encircling the LEP (${\mathrm{\Delta }}_{\min }/2\pi =-223\mathrm{kHz}$, ${\gamma }_{\max }=800\mathrm{kHz}$, ${\mathrm{\Delta }}_{\max }/2\pi =-43\mathrm{kHz}$, ${\gamma }_{\min }=200\mathrm{kHz}$). The time is $T_1=T_3=18\mathrm{\mu }\mathrm{s}$. (e),(f),(g),(h) Net work corresponding to the panels (a),(b),(c),(d), respectively, is defined and explained in the text. In all the panels, $\mathrm{\Omega }/2\pi$ is a constant and fixed to be 29 kHz.

Figure 3

Numerical simulation showing the dependence of the net work on ${\mathrm{\Delta }}_{\max }$ of the QHE cycles. (a) The situation covering the big cycle of our experiment, where the QHE cycle starts from the point $A$ with ${\mathrm{\Delta }}_{\min }/2\pi =-290\mathrm{kHz}$ and ${\gamma }_{\max }=800\mathrm{kHz}$, sweeping the frequency to ${\mathrm{\Delta }}_{\max }/2\pi ⩽210\mathrm{kHz}$. So the point $C$ is set with $-289.9\mathrm{kHz}⩽{\mathrm{\Delta }}_{\max }/2\pi ⩽210\mathrm{kHz}$ and ${\gamma }_{\min }=130\mathrm{kHz}$. (b) The situation involving the small cycle of our experiment, where the QHE cycle starts from $A$ with ${\mathrm{\Delta }}_{\min }/2\pi =-223\mathrm{kHz}$ and ${\gamma }_{\max }=800\mathrm{kHz}$, sweeping the frequency to ${\mathrm{\Delta }}_{\max }/2\pi ⩽277\mathrm{kHz}$. So the point $C$ is set with $-222.9\mathrm{kHz}⩽{\mathrm{\Delta }}_{\max }/2\pi ⩽277\mathrm{kHz}$ and ${\gamma }_{\min }=200\mathrm{kHz}$. In both panels, we consider 804 points for ${\mathrm{\Delta }}_{\max }$. Black dots indicate the maximal detuning of the two cycles experimentally accomplished. The green-shaded and pink-shaded regions represent the cases with the LEP encircled, and white regions mean the cases without encircling the LEP. The points $A$ and $C$ as shown in Fig. 1, but are not labeled here.