[Paper Review] Experimental Demonstration of non-Markovian Dynamics via a Temporal Bell-like Inequality

Alexandre M. Souza, Jiayao Li+6 more|arXiv (Cornell University)|Aug 27, 2013

stochastic dynamics and bifurcation1 references8 citations

TL;DR

This paper experimentally demonstrates non-Markovian dynamics in a quantum open system using a temporal Bell-like inequality, specifically a generalized Leggett-Garg (LG) inequality, in a controlled nuclear magnetic resonance (NMR) setup. By violating the inequality under conditions where the system's dynamics are Markovian due to dephasing, the study reveals that the violation stems from quantum coherence rather than memory effects, establishing a clear link between non-Markovianity and quantum non-classicality in open systems.

ABSTRACT

We assess non-Markovianity of a quantum open-system dynamics through the violation of temporal bell-like inequalities in a controllable Nuclear Magnetic Resonance system. We investigate experimentally the connections between the violation of the addressed temporal Bell-like inequality and the non-divisibility of the effective evolution of our system, which we fully characterize in a broad range of experimentally controllable regimes. Finally, we investigate the link between our approach and Leggett-Garg-like inequalities based on time-translational two-time correlation functions. These results open up interesting perspectives for the effective, non-tomographic characterization of dynamical evolution by combining tools of different nature.

Motivation & Objective

• To experimentally test whether violations of a temporal Bell-like inequality can detect non-Markovian dynamics in open quantum systems.
• To disentangle the role of quantum coherence from genuine memory effects (non-Markovianity) in system-environment dynamics.
• To establish a connection between the violation of a generalized Leggett-Garg inequality and the non-divisibility of the system's dynamical map.
• To demonstrate that non-classical behavior detected via such inequalities can persist even in Markovian regimes dominated by dephasing.

Proposed method

• The experiment uses a nuclear magnetic resonance (NMR) system with a qubit (S) coupled to an environmental spin (E), allowing control over system-environment interaction strength via a tunable coupling angle θ.
• A time-local master equation is derived to model the system's reduced dynamics, with the dephasing rate γ and coupling strength J as key parameters.
• The generalized Leggett-Garg (LG) inequality is applied using the observable σₓ^S, with time-ordered two-time correlation functions ⟨σₓ^S(t)σₓ^S(0)⟩ forming the basis of the inequality.
• The inequality is tested under varying coupling strengths and dephasing rates, with the bound |L_Q(t)| ≤ Q_max⟨Q(0)⟩ serving as a classical Markovian benchmark.
• Experimental data for the correlation functions are measured via quantum state tomography and compared to theoretical predictions to assess violation.
• The analysis includes a reformulation of the information backflow measure σ_γ(t) to account for dephasing, enabling accurate assessment of divisibility in the presence of noise.

Experimental results

Research questions

• RQ1 Can violations of a temporal Bell-like inequality serve as a reliable witness for non-Markovian dynamics in a controlled quantum system?
• RQ2 To what extent can quantum coherence alone lead to violation of a generalized LG inequality, even when the system's dynamics are Markovian due to dephasing?
• RQ3 How does the interplay between system-environment coupling and dephasing affect the validity of classical Markovian models in open quantum systems?
• RQ4 Can the violation of the generalized LG inequality be distinguished from genuine non-Markovianity using a modified information backflow criterion?

Key findings

• The system exhibits violation of the generalized Leggett-Garg inequality even when the dynamics are Markovian due to dephasing, demonstrating that the violation is due to quantum coherence rather than memory effects.
• For J = 30 Hz and θ = π/18, the coupling strength remains within the Markovian regime, as confirmed by the non-negative σ_γ(t) function, which witnesses no information backflow.
• The theoretical threshold θ_M = √(γ csc(2πJt)/(2πJ)) defines a region where dynamics remain Markovian; θ = π/18 lies well within this region, ensuring classical Markovianity.
• Experimental data for f(t) and g(t) show excellent agreement with theoretical predictions, validating the model of the system's effective dynamics.
• Despite the absence of non-Markovianity, the system still violates the generalized LG inequality, indicating that such inequalities can detect non-classicality beyond memory effects.
• The study confirms that the violation of the generalized LG inequality cannot be attributed to non-divisibility of the dynamical map, but rather to the non-classical nature of the evolution under non-invasive measurement assumptions.