### Experimental Violation of Leggett-Garg Type Inequalities Using Quantum Memories

**(From left to right) Zong-Quan Zhou, Chuan-Feng Li, Guang-Can Guo**

**Authors: Zong-Quan Zhou, Chuan-Feng Li, Guang-Can Guo**

**Affiliation: Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, China.**

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Ever since the birth of quantum mechanics we faced the difficulty to reconcile the superposition behavior of quantum particles and our intuitive experience in dealing with macroscopic objects which should occupy definite states at all times and independently of the observers. Leggett and Garg subsequently formulated the question qualitatively by means of the derivation of a series of inequalities based upon the premises of macroscopic realism and noninvasive measurability [1]. In practice, the main experimental challenge comes from the implementation of truly noninvasive measurements [2]. To avoid the requirement of performing noninvasive measurements, a different type of Leggett-Garg inequalities (LGtI) has been derived from the stationary assumption [3,4], which includes the time translational invariance of the probabilities and Markovianity of the system evolution [2]. If stationarity does hold for the considered system, these inequalities provide a quantitative way to witness the persistence of coherent effects, that is, they allow for benchmarking ‘quantumness’.

**Figure 1: Single photons excited the superposition states of two pieces of Nd:YVO**

_{4}crystals. The horizontally (H) and vertically (V) polarized components of a single photon are converted to collective excitation of billions of atoms in the first crystal and second crystal, respectively. The two crystals have thicknesses of 3 mm and are separated by a distance of 2 mm.Zong-Quan Zhou, Chuan-Feng Li and Guang-Can Guo from the University of Science and Technology of China, and Susana Huelga from the Ulm University have reported in Physical Review Letters [5] that they are able to violate the LGtI in a light-matter interfaced system. By separately benchmarking the Markovian character of the evolution and the translational invariance of the conditional probabilities, the observed violation of the LGtI is attributed to the quantum coherent character of the process.

In the experiment, narrowband single photons are generated from nonlinear crystal through the spontaneous parametric down-conversion process. Then the H+V-polarized single photons are employed to excite the superposition states of the collective excitation of two solid-state quantum memories. The group developed a novel technique, the polarization-dependent atomic-frequency-comb (AFC) technique, to control the dynamical evolution of the collective atomic states. A frequency detuning of δ is introduced between the H-polarized and V-polarized AFC, which determines the state evolution speed. Finally, the atomic states are read out through the AFC echo emission and analyzed with polarization-dependent single-photon detections. The recorded state evolution is shown in Figure 2(a) which is nearly perfect unitary evolution. The calculated LGtI is shown in Figure 2(b) and shows a violation of the classical bound by 6.9 standard errors. The stationary assumption was independently verified for the experimental setup. Therefore, the experimental violation of LGtI demonstrates that the dynamics of a collective excitation which is distributed across two macroscopically separated crystals can be well described by quantum mechanics, while a classical description is excluded.

**Figure 2: (a) Time evolution of the probabilities to find the system in atomic state D and A for an AFC excitation initially prepared in state D and with AFC detuning δ of 5 MHz. Here D is defined as an atomic state of “first crystal excited + second crystal excited” and A is the orthogonal state, “first crystal excited - second crystal excited”. (b). The envelope evolution of LGtI. The solid lines are the ideal quantum mechanical predictions. The blue dashed line represents the lower bound of -1, as predicted by classical incoherent theories.**

This is a coherent evolution of microscopic excitation in macroscopic objects and represents an interesting exploration of the crossover between quantum and classical regimes. These results provide a general procedure to benchmark

*quantumness*when temporal correlations can be independently assessed and confirm the persistence of quantum coherence effects in systems of increasing complexity. Further efforts to combine the multi-photon entanglement [6,7] and the atomic memory could lead to a test of LGtI with quantum superposition states of larger macroscopicity [8].

**Reference:**

**[1]**A. J. Leggett, Anupam Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?”, Physical Review Letters, 54, 857 (1985). Abstract.

**[2]**Clive Emary, Neill Lambert, Franco Nori, “Leggett-Garg Inequality”, Reports on Progress in Physics, 77, 016001 (2014). Abstract.

**[3]**S. F. Huelga, T. W. Marshall, E. Santos , “Proposed test for realist theories using Rydberg atoms coupled to a high-Q resonator”, Physical Review A, 52, R2497 (1995). Abstract.

**[4]**Susana F. Huelga, Trevor W. Marshall, Emilio Santos, “Temporal Bell-type inequalities for two-level Rydberg atoms coupled to a high-Q resonator”, Physical Review A, 54, 1798 (1996). Abstract.

**[5]**Zong-Quan Zhou, Susana, F. Huelga, Chuan-Feng Li, Guang-Can Guo, “Experimental detection of quantum coherent evolution through the violation of Leggett-Garg-type inequalities”, Physical Review Letters, 115, 113002 (2015). Abstract.

**[6]**A. I. Lvovsky, R. Ghobadi, A. Chandra, A. S. Prasad and C. Simon,, “Observation of micro-macro entanglement of light”, Nature Physics, 9, 541 (2013). Abstract.

**[7]**N. Bruno, A. Martin, P. Sekatski, N. Sangouard, R. T. Thew, N. Gisin, “Displacement of entanglement back and forth between the micro and macro domains”, Nature Physics, 9, 545 (2013). Abstract.

**[8]**A. J. Leggett, “Macroscopic Quantum Systems and the Quantum Theory of Measurement”, Progress of Theoretical Physics Supplements, 69, 80 (1980). Full Article.