Experimental demonstration of enhanced violations of Leggett-Garg inequalities in a $\mathcal{PT}$-symmetric trapped-ion qubit

The Leggett-Garg inequality (LGI) delineates a boundary between quantum and classical systems. While the temporal quantum correlations represented by LGIs have been extensively investigated in the Hermitian domain, the exploration of LGIs under non-Hermitian conditions remains scarce. Theoretical conjectures posit that nonunitary dynamics could transcend the boundary of LGIs set by Hermitian quantum mechanics, yet its empirical validation is lacking. Here, we demonstrate the enhanced violation of LGIs in a non-Hermitian system of a parity-time ($\mathcal{PT}$)-symmetric trapped-ion qubit. The upper bounds of both the third-order parameter $K_3$ and fourth-order parameter $K_4$ increase with dissipation and can reach the maximum value when the system approaches the exceptional point, where $K_3=C_{21}+C_{32}-C_{31}$ is comprising three two-time correlations and $K_4=C_{21}+C_{32}+C_{43}-C_{41}$ is comprising four correlations. We also find that the lower bounds of $K_3$ and $K_4$ exhibit distinctive behaviors, where the lower bound of $K_3$ remains constant, but that of $K_4$ depends on the target states and measurement operators. These findings reveal a pronounced correlation between the dissipative nature of a quantum system and its temporal correlations.

Figure 1

Schemes of testing third-order ($K_3$) and fourth-order ($K_4$) LGIs. (a) The experiment involves measuring three two-time temporal correlations ($C_{21}, C_{32}$, and $C_{31}$) to determine the parameter $K_3$. (b) The experiment requires measuring four two-time temporal correlations ($C_{21}, C_{32}, C_{43}$, and $C_{41}$) to determine the parameter $K_4$. Here, $|{\psi }_t\rangle$ denotes the target state, the time intervals are defined as $t_j-t_i=(j-i)\tau$, and $t_1=0$. The preparation of $|{\psi }_t\rangle$ can be regarded as the measurement at time $t_1$.

Figure 2

Experiment setup for testing LGIs. (a) The trap confines a single ${}^{171}\mathrm{Yb}^{+}$ ion through the application of radio frequency (rf) signals and direct current (dc) voltages to two rf electrodes (RF1 and RF2) and two sets of dc electrodes (DC1-5 and DC6-10), respectively. Additional dc voltages (DC11 and DC12) applied to the rf electrodes are utilized to displace the ion to the rf null position. The magnetic field (B) is oriented along the $\mathbf{Z}$ axis. The microwave signal used to drive the transition between the states $|0\rangle$ and $|1\rangle$ is generated through mixing a standard rf source with either a direct digital synthesizer (DDS) or an arbitrary waveform generator (AWG). The energy level diagram of the ${}^{171}\mathrm{Yb}^{+}$ ion is depicted within the dashed box; the involved four levels in the gray shaded region can be simplified to a dissipative two-level system. (b) The timing sequences for initialization (including cooling and optical pumping), dissipation beam, and detection beam are controlled by acoustic-optic modulators (AOMs) equipped with rf switches, which receive TTL (transistor transistor logic) sequences from the ARTIQ (advanced real-time infrastructure for quantum physics) device. The synchronization of the microwave and the dissipation laser is precisely controlled by simultaneously triggering and passing through the same length of rf cables. The intensity of the dissipation beam can be finely adjusted by varying the rf power applied on the AOM.

Figure 3

Experimental results for the LGI parameter $K_3$. (a) Measurement results of $K_3$ under Hermitian condition ($\gamma /J=0$). (b)–(d) Measurement results of $K_3$ under non-Hermitian condition, with the ratios of $\gamma /J$ being 0.472 (b), 0.669 (c), and 0.942 (d), respectively. The red solid lines in (a)–(d) are the theoretical results calculated by Eq. (6) [Eq. (7)]. The blue circles depict the experimentally measured results and the dashed line indicates the upper bound (3/2) of $K_3$ in the quantum system. Error bars for the experimental results are estimated using the standard deviation ($1\sigma$) from multiple rounds of experiments. To discern the population information of qubits effectively, each set of experiments is repeated 500 times. The dissipation rates in (a)–(d) correspond to the four dashed lines in Fig. 9. The coupling strength $J$ is $2\pi \times 10.4\mathrm{kHz}$.

Figure 4

Experimental results of the LGI parameter $K_4$. (a) Measurement result of $K_4$ under Hermitian condition ($\gamma /J=0$). (b)–(d) Measurement result of $K_4$ under non-Hermitian condition, with the ratios of $\gamma /J$ being 0.708 (b), 0.857 (c), and 0.915 (d), respectively. The red solid lines in (a)–(d) are the theoretical results calculated by Eq. (6) [Eq. (8)]. The blue circles depict the experimentally measured results and the black (gray) dashed line is the upper (lower) bound $2\sqrt{2}$ ($-2\sqrt{2}$) of $K_4$ in the quantum system. The error bars for the experimental results are estimated by the standard deviation ($1\sigma$) of multiple rounds of experiments. To discern the population information of qubits effectively, each set of experiments is repeated 500 times. The dissipation rate in (b)–(d) corresponds to three dashed lines in Fig. 9. The coupling strengths $J$ in (a) and (b)–(d) are $2\pi \times 11.5\mathrm{kHz}$ and $2\pi \times 10.4\mathrm{kHz}$, respectively.

Figure 5

Experimental results of state evolution dynamic. The three solid lines are theoretical results and different colors represent different dissipation strengths.

Figure 6

Numerical results for the maximum and minimum value of $K_3$ (a),(b) and $K_4$ (c),(d) by optimizing the parameters $\theta$ and $\varphi$, where the ratio of the coupling strength and dissipation strength approaches the unity ($\gamma /J\to 1$). Here, the observable operator is ${\sigma }_y$.

Figure 7

Experimental results of correlation functions $C_{ji}$ in LG parameter $K_3$. The dissipation intensity $\gamma$ in four pictures is (a) 0, (b) $0.472J$, (c) $0.669J$, and (d) $0.942J$, respectively, where the coupling strength $J$ is $2\pi \times 10.4\mathrm{kHz}$.

Figure 8

Experimental results of correlation functions $C_{ji}$ in LG parameter $K_4$. The dissipation intensity $\gamma$ in four pictures are (a) 0, (b) $0.708J$, (c) $0.857J$, and (d) $0.915J$, respectively. The coupling strength in (a) is set to $2\pi \times 11.5\mathrm{kHz}$ and the coupling strength in (b)–(d) is set to $2\pi \times 10.4\mathrm{kHz}$.

Figure 9

Theoretical plots of the third-order $K_3$ (a) and fourth-order $K_4$ (b) as a function of $\gamma$ and $\tau$. Here, the target state is $|+\rangle$ and the observable operator is ${\sigma }_y$. The black, blue, green, and red dashed lines in (a) represent $\gamma =2\pi \times 0\mathrm{kHz}, 2\pi \times 4.9\mathrm{kHz}, 2\pi \times 7\mathrm{kHz}$, and $2\pi \times 9.7\mathrm{kHz}$, respectively. Similarly, the black, blue, green, and red dashed lines in (b) represent $\gamma =2\pi \times 0\mathrm{kHz}, 2\pi \times 7.3\mathrm{kHz}, 2\pi \times 8.9\mathrm{kHz}$, and $2\pi \times 9.4\mathrm{kHz}$, respectively. The coupling strength $J$ in (a) and (b) is $2\pi \times 10.4\mathrm{kHz}$.

Figure 10

Numerical results of the higher-order LGI $K_4$. The lower (a) and upper (b) bounds of $K_4$ exhibit variations with the dissipation rate under different measurement operators and target states. The parameters for the black solid lines in (a),(b) are ${\theta }_m={\varphi }_m=\pi /2, \theta =\pi /2$, and $\varphi =\pi$. Meanwhile, the parameters for the blue dashed lines in (a),(b) are ${\theta }_m=\theta =0.51\pi , {\varphi }_m=\pi /2$, and $\varphi =\pi$. The coupling strength $J$ in (a) and (b) is $2\pi \times 10.4\mathrm{kHz}$. The exceptional point (EP) is marked by the blue pentagram.