Experimental violation of the Leggett-Garg inequality in a three-level trapped-ion system

Leggett-Garg inequality (LGI) is used to detect whether the macroscopic superposition state exists. Classical systems obey the LGI, but quantum systems may violate it. In the multilevel system, different state update rules determine different upper bounds of LGI. In this work, we experimentally test the LGI in a three-level trapped-ion system under the Lüders and the von Neumann state update rules, respectively. The maximum observed value of the Leggett-Garg correlator under the von Neumann state update rule is $K_3=1.739\pm 0.014$, which demonstrates a violation of the Lüders bound by 17 standard deviations and is by far the most significant violation under the specific model. The method used in our experiment could also be used in other multilevel experiments in the trapped-ion system.

Figure 1

Scheme for the LGI test. ${\rho }_0$ represents the initial density operator. Three separate experiments need to be performed to obtain $C_{21}$, $C_{32}$, and $C_{31}$. The $Q$ values are measured at the corresponding two moments, and each experiment is performed multiple times to obtain the joint probability $P_{ij}(Q_i,Q_j)$.

Figure 2

Energy level diagram of ${}^{40}{\mathrm{Ca}}^{+}$. The three levels $4S_{1/2}(m_J=-1/2)$, $3D_{5/2}(m_J=-1/2)$, and $3D_{5/2}(m_J=+1/2)$ form a qutrit and $4S_{1/2}(m_J=+1/2)$ is the auxiliary energy level.

Figure 3

Schemes for two different measurement methods. (a) The measurement scheme using the LSUR. (b) The measurement scheme using the VSUR. The laser sequence used in the four-step operations of the measurement is shown by serial numbers I, II, III, and IV.

Figure 4

Experimental results of $K_3$ under 31 different settings of $\mathrm{\Omega }\tau \in [0,2\pi ]$. (a) The experimentally determined result under the LSUR. (b) The experimentally determined result under the VSUR. The black curve is the theoretical result, the red dots are the experimental results, the orange horizontal line is the classical bound, and the violet horizontal line is the Lüders bound. The error bars are not visible as the statistical error is about ${10}^{-2}$.