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arXiv:2411.02910 · Physical Review Research

Observation of quantum-classical transition behavior of LGI in a dissipative quantum gas

Qinxuan Peng, Bolong Jiao, Hang Yu, Liao Sun, Haoyi Zhang, Jiaming Li, Le Luo

Published Apr 11, 2025• 3 citations
arXivHTMLPDF

Abstract

The Leggett-Garg inequality (LGI) is a powerful tool for distinguishing between quantum and classical properties in studies of macroscopic systems. Applying the LGI to non-Hermitian systems with dissipation presents a fascinating opportunity, as competing mechanisms can either strengthen or weaken LGI violations. On one hand, dissipation-induced nonlinear interactions amplify LGI violations compared to Hermitian systems; on the other hand, dissipation leads to decoherence, which could weaken the LGI violation. In this paper, we investigate a non-Hermitian system of ultracold Fermi gas with dissipation. Our experiments reveal that as dissipation increases, the upper bound of the third-order LGI parameter K 3 initially rises, reaching its maximum at the exceptional point, where K 3 = C 21 + C 32 − C 31 , encompassing three two-time correlation functions. Beyond a certain dissipation threshold, the LGI violation weakens, approaching the classical limit, indicating a quantum-to-classical transition (QCT). Furthermore, we observe that the LGI violation decreases with increasing evolution time, reinforcing the QCT in the time domain. This study provides a crucial stepping stone for using the LGI to explore the QCT in many-body open quantum systems.

DOI
10.1103/physrevresearch.7.023045

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The Leggett-Garg inequality (LGI) is a powerful tool for distinguishing between quantum and classical properties in studies of macroscopic systems. Applying the LGI to non-Hermitian systems with dissipation presents a fascinating opportunity, as competing mechanisms can either strengthen or weaken LGI violations. On one hand, dissipation-induced nonlinear interactions amplify LGI violations compared to Hermitian systems; on the other hand, dissipation leads to decoherence, which could weaken the LGI violation. In this paper, we investigate a non-Hermitian system of ultracold Fermi gas with dissipation. Our experiments reveal that as dissipation increases, the upper bound of the third-order LGI parameter K 3 initially rises, reaching its maximum at the exceptional point, where K 3 = C 21 + C 32 − C 31 , encompassing three two-time correlation functions. Beyond a certain dissipation threshold, the LGI violation weakens, approaching the classical limit, indicating a quantum-to-classical transition (QCT). Furthermore, we observe that the LGI violation decreases with increasing evolution time, reinforcing the QCT in the time domain. This study provides a crucial stepping stone for using the LGI to explore the QCT in many-body open quantum systems.
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 (PT\mathcal{PT}PT)-symmetric trapped-ion qubit. The upper bounds of both the third-order parameter K3{K}_{3}K3​ and fourth-order parameter K4{K}_{4}K4​ increase with dissipation and can reach the maximum value when the system approaches the exceptional point, where ${K}_{3}={C}_{21}+{C}_{32}\ensuremath{-}{C}_{31}$ is comprising three two-time correlations and ${K}_{4}={C}_{21}+{C}_{32}+{C}_{43}\ensuremath{-}{C}_{41}$ is comprising four correlations. We also find that the lower bounds of K3{K}_{3}K3​ and K4{K}_{4}K4​ exhibit distinctive behaviors, where the lower bound of K3{K}_{3}K3​ remains constant, but that of K4{K}_{4}K4​ 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.
We study the Leggett-Garg inequality (LGI) of a two-level system (TLS) undergoing coherent dynamics described by a non-Hermitian Hamiltonian and Lindblad equation with no quantum jumps. The nonlinear Bloch equation for the TLS density matrix predicts violations of LGI above the TLS L\"uders bound of 3/23/23/2, approaching the extremal case of LGI parameter K3=3{K}_{3}=3K3​=3 in the PT\mathcal{PT}PT-symmetric region and the PT\mathcal{PT}PT-broken region. We show that these findings are reproduced by using postselection to remove instantaneous quantum jumps from a three-level system described by the Lindblad equation with a single spontaneous emission dissipator. We trace the K3{K}_{3}K3​ excesses beyond the standard quantum limit of 3/23/23/2 to a nonuniform speed of evolution on the Bloch sphere. Finally, we consider the effects of competing Lindblad dissipators on the postselected non-Hermitian dynamics and the viability of observing K3{K}_{3}K3​ exceeding the L\"uder bound and approaching its algebraic maximum of three in current experimental setups.
Abstract We investigate the nonclassicality of an open quantum system using Leggett–Garg inequalities (LGI) which test the correlations of a single system measured at different times. Violation of LGI implies nonclassical behavior of the open system. We investigate the violation of the Leggett–Garg inequality for a two level system (qubit) spontaneously decaying under a general non-Markovian dissipative environment. A robust nonclassical behavior is shown through the violation of LGI in the non-Markovian regime of the environment. Our results are exact as we have evaluated the two-time correlation functions exactly without using quantum regression theorem. We probe quantum signatures of the open system through the exact dynamics of Leggett–Garg inequalities for a wide range of system-environment parameters beyond Born–Markov regime.
Violations of the Leggett-Garg inequality (LGI) up to its algebraic bound under non-Hermitian dynamics are well established theoretically. Here, we demonstrate that such extreme violations are intrinsically fragile when realistic measurement processes are taken into account. We consider an open two-level system described by a time-local hybrid Liouvillian, with a continuous parameter q∈[0,1]q \in [0,1]q∈[0,1], representing detector efficiency, i.e., the fraction of quantum jump trajectories that are retained in the ensemble. This parameter interpolates between trace-preserving Lindblad dynamics (q=1q=1q=1) and non-Hermitian ``no-jump" evolution (q=0q=0q=0). While K3K_3K3​ approaches its algebraic maximum of 3 in the null-efficiency limit, even an infinitesimal increase in detector efficiency induces a rapid, highly nonlinear suppression toward the classical bound. This logarithmic sensitivity reveals that maximal LGI violations are not robust physical features but rather singular limits of idealized measurement conditions. Our results have direct experimental implications: achieving algebraic LGI violations in systems undergoing continuous time evolution requires near-perfect suppression of detected quantum jumps (i.e., effective post-selection), placing stringent constraints on detector performance. In contrast to discrete protocols based on time-non-divisible dynamics, our framework shows that extreme violations arising within continuous, divisible quantum trajectory evolution constitute a fundamentally fragile regime.
Abstract Leggett Garg inequalities (LGIs) provides an elegant way for probing the incompatibility between the notion of macrorealism and quantum mechanics. For unitary dynamics the optimal quantum violation of a LGI is constrained by the L u ¨ ders bound. In this paper, we have studied two formulations of LGIs in three-time LG scenario, viz, the standard LGIs and third-order LGIs both for unbiased and biased measurement settings. We show that if system evolves under non-unitary quantum channel between two measurements, the quantum violations of both forms of LGIs exceed their respective L u ¨ ders bounds and can even reach their algebraic maximum in sharp measurement settings. We found that when the measurement is unsharp the quantum violations of both standard and third-order LGIs for non-unitary quantum channel can be obtained for lower value of the unsharpness parameter compared to the unitary dynamics. We critically examined the violation of L u ¨ ders bound of LGIs and its relation to the violation of various no-signaling in time conditions, another formalism for testing macrorealism. It is shown that mere violations of no-signaling conditions are not enough to warrant the violation of standard LGIs, an interplay between the violations of various NSIT condition along with a threshold value play an important role. On the other hand violation of third-order LGI is obtained when the degree of violation of a specific no-signaling in time condition reaches a different threshold value.
We study Leggett-Garg inequality (LGI) of a two level system (TLS) undergoing non-Hermitian dynamics governed by a non-linear Bloch equation (derived in J. Phys. A: Math. Theor. 54, 115301 (2021)) across a PT-transition. We present an algebraic identification of the parameter space for the maximum violation of LGI (in particular K3K_{3}K3​). In the PT-symmetric regime the maximum allowed value for K3K_{3}K3​ is always found to be greater than the quantum bound (Lüders bound) of 3/23/23/2 but it does not reach the algebraic maximum of K3=3K_{3}=3K3​=3 in general. However, in the limit where PT-symmetry breaking parameter approaches the exceptional point from the PT-symmetric side, K3K_{3}K3​ is found to asymptotically approach its algebraic maximum of 3. In contrast, the maximum value of K3K_{3}K3​ always reaches its algebraic maximum in the PT-broken phase i.e.i.e.i.e. K3→3K_{3}\rightarrow 3K3​→3. We find that (i) the speed of evolution (SOE) must reach its maximum value (in the parameter space of initial state and the time interval between successive measurements) to facilitate the value of K3→3K_{3} \rightarrow 3K3​→3, (ii) together with the constraint that its minimum value must run into SOE equals to zero during the evolution of the state. In fact we show that the minimum speed of evolution can serve as an order parameter which is finite on the PT-symmetric side and identically zero on the PT-broken side. Finally, we discuss a possible experimental realization of this dynamics by quantum measurement followed by post-selection procedure in a three level atom coupled to cavity mode undergoing a Lindbladian dynamics.