Single-Time Selective Weight Redistribution in T-Symmetry-Broken BEC Bright Solitons via Itō Field Reversal: Density, Norm, and ABL Probability Diagnostics

Technical Summary: Single-Time Selective Weight Redistribution in T-Symmetry-Broken BEC Bright Solitons via Itō Field Reversal

Problem Statement
The paper addresses the challenge of testing quantum macroscopic irreversibility within objective collapse models. Existing models often fail to reconfigure system boundary conditions from first principles, leaving the phenomenology of "collapse asymmetry" (specifically T-symmetry breaking) poorly understood. While companion work established the dynamical equations for T-symmetry breaking via Itō field reversal in a stochastic Cubic-Quintic Nonlinear Schrödinger Equation (CQ-NLSE), it remained silent on the specific phenomenological signatures of this asymmetry. The central problem is distinguishing whether collapse asymmetry manifests as genuine decoherence or as a redistribution of probability weights between forward and backward dynamical branches.

Methodology
The study employs a theoretical diagnostic framework applied to attractive Bose-Einstein Condensate (BEC) bright solitons (specifically modeling $^7$Li or $^{85}$Rb). The methodology involves:

1. Dynamical Framework: Utilizing a stochastic CQ-NLSE with Itō noise, where forward ($\Phi_f$) and backward ($\Phi_b$) order parameters evolve under distinct quintic nonlinearity coefficients ($\omega/8$ and $5\omega/8$, respectively) driven by a frequency $\omega$ and wave-number $\kappa$.
2. Three-Pronged Diagnostic Approach: The paper derives and analyzes three complementary observables to probe T-symmetry breaking:
   • Particle Density ($\rho$): Analyzing the spatial distribution $|\Phi|^2$ for forward and backward branches.
   • Norm ($N$): Calculating the total particle number to assess probability conservation and redistribution.
   • Aharonov-Bergmann-Lebowitz (ABL) Probability: Applying the ABL formalism to assign transition probabilities conditioned on both pre-selected (forward) and post-selected (backward) boundary states. This serves as a model-independent, two-time diagnostic.
3. Analytical and Numerical Analysis: The authors derive closed-form analytical expressions for densities, norms, and ABL probabilities using bright soliton ansatzes. These are then evaluated across Low, Intermediate, and Strong confinement regimes (varying $\omega$ and $\kappa$) to observe asymptotic behaviors and asymmetry gaps.

Key Contributions

• Formulation of a Diagnostic Framework: The paper establishes a unified framework using density, norm, and ABL probability to diagnose T-symmetry breaking in stochastic soliton systems.
• Derivation of Asymmetry Signatures: It analytically demonstrates that the backward branch is consistently suppressed relative to the forward branch across all three diagnostics.
• Distinction Between Decoherence and Redistribution: A primary theoretical contribution is the finding that the observed asymmetry is not a decay of coherence (decoherence) but a redistribution of probability weights. The total norm is conserved but redistributed between branches.
• Parameter Invariance Properties: The study identifies specific invariances:
  • The joint ABL probability depends solely on the wave-number $\kappa$ and is exactly invariant under the driving frequency $\omega$.
  • The redistribution ratio between branches depends solely on $\omega$ and is invariant under $\kappa$.
  • In the strong-confinement limit ($\omega \to \infty$), both the norm ratio and the ABL probability ratio converge to a fixed value of $\sqrt{5}$.

Results

• Density Evolution: At $t=0$, forward and backward densities differ only in peak magnitude. By $t=2.0$, the forward density ($\rho_f$) flattens into a near-uniform plateau, losing sensitivity to $(\omega, \kappa)$, while the backward density ($\rho_b$) remains dependent on these parameters. The forward-to-backward density ratio grows from $\approx 10$ (Low regime) to $\approx 1100$ (Strong regime).
• Norm Conservation: The norms for both branches are conserved ($dN/dt = 0$). However, the ratio $N_f/N_b$ depends only on $\omega$, approaching the asymptotic limit of $\sqrt{5}$ as $\omega$ increases.
• ABL Probability Asymmetry: The forward ABL probability ($P_f^{NORM}$) is strictly greater than the backward probability ($P_b^{NORM}$) across all tested parameters, with no crossover. The asymmetry gap $|\Delta P_{ABL}|$ grows monotonically with $\kappa$ at fixed $\omega$.
• Joint Probability Behavior: The joint probability $P_{f/b}^{NORM}$ is found to be independent of $\omega$, depending only on $\kappa$. This indicates that the driving frequency determines which branch acquires more weight, but not the degree of coherence or overlap between the forward and backward histories.

Significance and Claims
The paper claims that the collapse asymmetry in this system manifests as a single-time selective weight redistribution rather than a genuine two-time decoherence process.

• Mechanism: The T-symmetry breaking is driven by the redistribution of probability weights between forward and backward channels, a mechanism intrinsic to the closed soliton system's stochastic dynamics.
• Experimental Proposal: The authors propose that a controllable measurement of ABL probability branches in BEC bright solitons, specifically targeting a null asymmetry gap ($|\Delta P_{ABL}| = 0$), would constitute a direct experimental test of this redistribution mechanism.
• Limitations: The framework is noted to exclude environmental coupling, acting as a closed-system mechanism. The results are derived from analytical expressions and numerical evaluations of these expressions, not from new experimental data generation.

The work concludes that while density and norm provide single-time signatures of asymmetry, the ABL formalism extends this to a genuine two-time structure, revealing that the "collapse" in this model is a re-weighting of existing probability amplitudes rather than a loss of quantum coherence.