Classically Forbidden Signatures of Quantum Coherence in the Mesoscopic Lipkin-Meshkov-Glick Model

Abstract: We derive strict quantitative conditions under which a collective quantum system of N~370 spins exhibits classically forbidden temporal correlations in a spinor Bose-Einstein condensate (BEC). The Lipkin-Meshkov-Glick (LMG) model near its Z_2-breaking quantum critical point supports a mesoscopic superposition a|P> + b|R> of two macroscopic ordered phases (|P>: m_z ~ +m_; |R>: m_z ~ -m_) at the Goldilocks crossover N ~ N_c, where the tunnel splitting equals k_B T and macroscopic susceptibility chi ~ N coexists with finite quantum coherence. We establish two quantum-discriminating predictions. P4 (Landau-Zener crossover, proposed discriminator): quantum tunnelling drives P_error -> 0 exponentially with quench time, while the classically non-ergodic system remains kinetically frozen at P_error -> 1 - a parametrically large, computable separation that rests on a specific kinetic foil. P5 (Leggett--Garg inequality violation, strictly model-independent): K_3 > 1 is forbidden by macrorealism for all classical models satisfying non-invasive measurability. A five-level Lindblad simulation yields K_3 ~ 1.32 and dephasing threshold gamma_phi < 0.289 s^{-1} at the BEC target (gamma_phi = 0.05 s^{-1}, N = 370, Gamma/J = 0.95) - a margin of 8.8x above the optimal measurement interval, well within current experimental reach. The threshold has a precise physical origin in the emergent collective Z_2 symmetry: exact parity eliminates the dominant dephasing cross-term and renders the LGI correlator immune to T_1 population mixing without requiring ground-state preparation (2.35x improvement over the naive mean-field estimate); constructive dynamical phase alignment of higher odd-parity states at the benchmark parameters contributes a further 2.47x. All results are reproducible from the provided self-tested Python code.

• The paper identifies classically forbidden quantum signatures by analyzing LGI violation under realistic dephasing conditions.
• It employs multi-level Lindblad master equation solutions to track coherence in the Goldilocks mesoscopic regime with precise thresholds.
• The study confirms robust symmetry protection enabling experimental discrimination between quantum coherence and classical stochasticity.

Classically Forbidden Signatures of Quantum Coherence in the Mesoscopic Lipkin-Meshkov-Glick Model

Introduction and Context

This paper addresses the emergence and detection of strictly quantum phenomena in a mesoscopic realization of the collective Lipkin-Meshkov-Glick (LMG) spin model, with a focus on the experimental regime accessible via spinor Bose–Einstein condensates (BECs). The LMG model, characterized by collective all-to-all interactions and exact $\mathbb{Z}_2$ symmetry, possesses a quantum phase transition separating two ordered phases. The regime of interest is the "Goldilocks zone," where the system size and parameters are tuned such that quantum superpositions of macroscopically distinct phases ($P$, $R$) occur, while the tunnel splitting matches the environmental thermal energy ($\Delta \approx k_B T$).

The central contribution is a strict, quantitative analysis of whether such a mesoscopic system can exhibit dynamical and statistical signatures that are forbidden within classical physics, in the presence of realistic, Markovian dephasing noise characteristic of current BEC implementations.

Theoretical Framework and Model Analysis

LMG Hamiltonian and Phase Structure

The LMG Hamiltonian,

$$\hat{H} = -\frac{J}{2N} \left( \sum_{i=1}^N \hat{\sigma}_i^z \right)^2 - \Gamma \sum_i \hat{\sigma}_i^x - h \sum_i \hat{\sigma}_i^z,$$

admits two symmetry-broken spin-coherent states ($P$, $R$) in the ordered phase ($\Gamma < J$), corresponding to macroscopically distinct magnetizations. For large but finite $N$, the true eigenstates are symmetric and antisymmetric superpositions of $P$ and $P$0, separated by a tunnel splitting $P$1 that decays exponentially with system size. The Goldilocks regime is realized when $P$2, where the system is maximally susceptible ($P$3) yet retains nontrivial quantum coherence.

The collective $P$4 symmetry, which flips all spins, is a key structural feature. This symmetry not only defines the phase structure but, crucially, protects certain quantum coherence signatures against environmental decoherence and population noise.

Open System Dynamics and Noise

The system’s coupling to the Markovian environment is modelled via the Lindblad master equation, with the dominant decoherence channel arising from global magnetic field fluctuations, modelled as collective dephasing ($P$5). The distinction between local and collective dephasing baths is quantitatively assessed; collective noise is shown to set a much stricter size–coherence trade-off. The analysis is performed in three levels of approximation: mean-field two-level, analytic two-level in the energy eigenbasis, and full five-level Lindblad solution.

Signatures of Quantum Coherence: Discrimination from Classicality

P4: Landau–Zener Crossover as a Quantum–Classical Discriminator

The paper defines a kinetic protocol (P4) wherein the system is prepared in one ordered phase and swept via a longitudinal bias field ($P$6) through the critical point. The classical prediction, based on Model A (Fokker-Planck) dynamics in the presence of a macroscopic free-energy barrier, yields a transition error probability $P$7 for all realistic sweep durations ($P$8) since classical escape over the barrier is exponentially suppressed in $P$9.

In contrast, the quantum LMG solution predicts that $R$0 falls exponentially with the Landau–Zener sweep time, due to coherent tunneling between the phases. The quantum–classical distinction is thus parametrically large and, under the relevant regime of parameters, unambiguous.

P5: Leggett–Garg Inequality Violation

The paper's strongest result is a strict, model-independent violation of the Leggett–Garg inequality (LGI) for temporal correlation of a dichotomic order parameter,

$R$1

which is strictly forbidden in any macrorealist, noninvasive-measurability framework. The correlators are evaluated with the observable $R$2, and the evolution is traced via the Lindblad master equation using the Lüders instrument for measurement back-action, with no reliance on the specific interpretation of quantum mechanics.

Key Results: For system size $R$3 and coupling $R$4, the five-level Lindblad solution yields $R$5 at experimentally relevant dephasing rates ($R$6), a robust violation. The critical dephasing rate for LGI violation is found numerically as $R$7, more than an order of magnitude above current experimental noise floors.

The analysis carefully tracks the effects of dephasing and population noise. Three levels of coherence time estimation are presented: the traditional mean-field two-level (Level A), analytic two-level in eigenbasis (Level B, which incorporates parity symmetry and thus corrects the dephasing estimate by a factor of $R$8), and the full five-level (Level C, further corrected upward by $R$9 from constructive contributions at optimal parameters).

Parity symmetry is shown to be fundamental: It is responsible for both the insensitivity of $\Delta \approx k_B T$0 to $\Delta \approx k_B T$1-type mixing and for robust violation without ground-state preparation, as the LGI is strictly immune to thermal occupation of the odd-parity excited state.

Numerical Validation and Experimental Scaling

The authors provide exhaustive numerical validation. The five-level Lindblad model is cross-checked against a ten-level calculation, confirming convergence better than $\Delta \approx k_B T$2 throughout the regime of interest. The code is made available for experimental calibration, supporting all threshold and maximally violating values reported.

The Goldilocks regime is strictly identified: The optimal mesoscopic size range is $\Delta \approx k_B T$3--$\Delta \approx k_B T$4 for current BEC experiments, balancing susceptibility and coherence. The decoherence-only constraint allows $\Delta \approx k_B T$5, but practical operation is bounded by the simultaneous requirements for macroscopic response and finite tunneling.

Implications, Limitations, and Future Directions

Theoretical and Experimental Consequences

• Macrorealist Falsification: The observation of the predicted LGI violation will rule out all macrorealist, noninvasively measurable classical models for this many-body system, under strict, transparent experimental conditions.
• Robustness: The analysis is robust against explicit symmetry-breaking dephasing, measurement imperfections, and even against speculative non-unitary extensions to quantum mechanics (GRW, CSL).
• Protocol Generality: The four key mechanistic features enabling violation—symmetry protection, measurement theory, interpretation-independence, and decoherence-robustness—extend beyond the LMG model, informing the search for quantum phenomena in other mesoscopic platforms.

Limitations and Open Problems

• The LGI protocol relies critically on the collective $\Delta \approx k_B T$6 symmetry, and small beyond-LMG corrections (e.g., higher-mode scattering in experiment) may introduce minor residual sensitivity.
• Accurate control over particle number and system parameters is necessary in practice; atom-number fluctuations at the scale of $\Delta \approx k_B T$7 introduce ensemble averaging effects.
• The window for simultaneous quantum coherence and macroscopic response is intrinsically narrow due to quadratic $\Delta \approx k_B T$8-scaling of decoherence in collective baths.

Future Developments

• Extending these techniques to other collective quantum phase transitions (molecular magnets, polariton condensates) is a clear next step.
• Further studies will include more realistic noise modeling (e.g., non-Markovian $\Delta \approx k_B T$9 noise) and non-ideal symmetry cases.
• The formalism provides an actionable framework for experimental groups to establish, probe, and falsify macroscopic quantum coherence in ultracold atomic platforms.

Conclusion

The paper offers a rigorous, quantitatively detailed framework for distinguishing quantum coherence from classical stochasticity in the mesoscopic LMG model, focusing on regimes imminently accessible in modern BEC experiments. It provides strict, experimentally relevant thresholds for observing classically forbidden temporal correlations, identifies the precise mechanism for symmetry-enhanced coherence protection, and delivers the necessary analytical and numerical tools for experimental falsification of macrorealism. The approach establishes a paradigm for forthcoming investigations into mesoscopic quantum phenomena, enabling direct probing of the quantum–classical boundary in collective systems (2604.18638).