Interferometric and Bipartite OTOC for Non-Markovian Open Quantum Spin-Chains and Lipkin-Meshkov-Glick Model

The information scrambling phenomena in an open quantum system modeled by Ising spin chains coupled to Lipkin-Meshkov-Glick (LMG) baths are observed via an interferometric method for obtaining out-of-time-ordered correlators ($\mathcal{F}-$OTOC). We also use an anisotropic bath connecting to a system of tilted field Ising spin chain in order to confirm that such situations are suitable for the emergence of ballistic spreading of information manifested in the light cones in the $\mathcal{F}-$OTOC profiles. Bipartite OTOC is also calculated for a bipartite open system, and its behavior is compared with that of the $\mathcal{F}-$OTOC of a two-spin open system to get a picture of what these measures reveal about the nature of scrambling in different parameter regimes. Additionally, the presence of distinct phases in the LMG model motivated an independent analysis of its scrambling properties, where $\mathcal{F}-$OTOC diagnostics revealed that quantum chaos emerges exclusively in the symmetry-broken phase.

💡 Research Summary

This paper investigates information scrambling in non‑Markovian open quantum many‑body systems by employing two complementary out‑of‑time‑ordered correlator (OTOC) diagnostics: the interferometric F‑OTOC and the Haar‑averaged bipartite OTOC. The authors first review the theoretical background of quantum chaos, the role of OTOCs as probes of operator growth and commutator growth, and the distinction between regularized, physical, and interferometric OTOCs. They then describe in detail an interferometric protocol that uses a control qubit to implement forward and backward evolutions (ξ_f(t) and ξ_b(t)) of the system under a completely positive trace‑preserving (CPTP) map, allowing the extraction of the complex four‑point function F(t)=⟨A†(t)B†A(t)B⟩ from the expectation value of σ_x on the control. To isolate genuine scrambling from pure dissipative effects, they introduce a corrected F‑OTOC, F_c(t)=F(t,A,B)/F(t,I,B).

The bipartite OTOC is defined by Haar‑averaging over all unitary operators acting on two complementary subsystems A and B, assuming an initially maximally mixed state. The resulting quantity G(t)=1−Re⟨A†(t)B†A(t)B⟩ (up to normalization) grows exponentially for chaotic dynamics and remains small for integrable or weakly chaotic regimes.

Two concrete spin‑chain models are studied. First, an Ising chain (with a tilted transverse field) is coupled globally to a Lipkin‑Meshkov‑Glick (LMG) bath, which features all‑to‑all interactions among its spins. By varying the system‑bath coupling strength g, the authors explore both Markovian (weak coupling) and non‑Markovian (strong coupling) regimes. Numerical simulations show that for strong coupling the interferometric F‑OTOC decays rapidly to near zero, while the bipartite OTOC rises sharply, indicating fast scrambling and rapid entanglement generation between system and bath. In the weak‑coupling limit the decay is slower and G(t) grows modestly, reflecting partial information retention.

Second, a tilted‑field Ising model (TFIM) is attached at one end to an anisotropic spin‑chain bath. Here the focus is on the spatial propagation of scrambling. The F‑OTOC profiles display clear light‑cone structures: a perturbation applied at one site spreads ballistically, producing a sharp front beyond which F(t) remains large and inside which it collapses. Strong non‑Markovian coupling sharpens the cone and increases the front velocity, whereas weaker coupling leads to a diffusive‑like spread. This demonstrates that the interferometric OTOC can directly visualize ballistic information propagation in open systems.

Finally, the authors turn to the LMG model itself, analyzing its scrambling behavior across its quantum phase transition. In the symmetric (paramagnetic) phase the OTOCs show only mild decay and growth, indicating the absence of chaos. In the symmetry‑broken (ferromagnetic) phase, however, the corrected F‑OTOC exhibits a rapid drop while the bipartite OTOC grows exponentially, signalling the emergence of quantum chaos exclusively in the broken‑symmetry region. This result links the onset of chaos to the order parameter of the LMG model.

Throughout the work, the impact of non‑Markovian memory effects is highlighted: they suppress possible revivals of the F‑OTOC, enhance the growth rate of the bipartite OTOC, and modify the shape of the light‑cone in the TFIM‑bath setup. The paper also discusses experimental feasibility, suggesting that superconducting circuits or trapped‑ion platforms equipped with an ancillary control qubit could implement the interferometric scheme and test the theoretical predictions.

In summary, the study provides a comprehensive comparison of interferometric F‑OTOC and bipartite OTOC as diagnostics of scrambling in open quantum spin systems, elucidates how bath structure (global LMG vs. local anisotropic chain) and non‑Markovianity affect scrambling dynamics, and uncovers a clear connection between symmetry‑breaking in the LMG model and the appearance of quantum chaos.