Discrete time quantum walk of locally interacting walkers

Technical Summary: Discrete Time Quantum Walk of Locally Interacting Walkers

Problem Statement
The control of quantum correlations and spatial distributions in multi-particle systems is a fundamental challenge with implications for quantum control and computation. While discrete-time quantum walks (DTQW) are established paradigms for generating non-classical transport and quantum resources, the systematic understanding of how controllable, local interactions between walkers affect their spatial and entanglement dynamics remains incomplete. Previous theoretical work has addressed specific instances, such as directional correlations, bound state formation, and transitions between bosonic and fermionic behaviors, often within specific models or contexts. However, a unifying framework that encompasses these previously studied interaction models as special cases, particularly one that systematically analyzes phase-controlled interactions on both spatial distributions and quantum correlations, has been lacking.

Methodology
The authors introduce a general framework for local interactions between two quantum walkers on a one-dimensional lattice. The system is modeled as a discrete-time quantum walk where the total Hilbert space is the tensor product of coin and lattice spaces. The evolution is governed by a unitary operator $U = SC$, where $S$ is a conditional shift and $C$ is a coin flip (specifically the Hadamard coin).

To model interactions, the authors define a general interaction operator $V$ that acts conditionally when both walkers occupy the same lattice site. This operator applies an arbitrary unitary transformation $O$ to the joint coin space while leaving the position unchanged. The authors focus on a specific, physically motivated subclass of these interactions where the individual coin states are preserved, and the interaction introduces a relative phase shift. This is parameterized by two real angles, $\theta_+$ and $\theta_-$, derived from the Hamiltonian parameters $\alpha$ and $\beta$. The interaction operator is expressed as:
$$V(\theta_+, \theta_-) = I + z_+ P_+ + z_- P_-$$
where $z_\pm = e^{i\theta_\pm} - 1$, and $P_+$ ($P_-$) projects onto states where walkers are at the same site with the same (opposite) coin states.

The study primarily investigates a single-parameter case ($\theta_- = 0$) to isolate the effects of same-spin interactions, while also briefly exploring the two-parameter space to demonstrate the framework's versatility. The initial state consists of two non-correlated walkers localized at the center of the lattice with uncorrelated coin states. The dynamics are analyzed over $t=100$ time steps.

Key Contributions and Results

1. General Interaction Framework: The paper provides a versatile platform for engineering quantum correlations by introducing a general interaction operator that reduces to several previously studied models (e.g., position-only phase interactions and collision-phase models) as limiting cases. This establishes coin-state selectivity as a key organizing principle for multi-walker correlation dynamics.

2. Spatial Redistribution:

   • Non-interacting case ($\theta=0$): Walkers evolve independently, exhibiting ballistic spreading with a square-like symmetry and high probability at the lattice edges.
   • Interacting case: As the interaction parameter $\theta$ increases, the joint probability distribution $P(x_1, x_2)$ transitions from a square shape to a diagonal shape.
   • Localization: For $\theta \approx \pi/2$, there is a significant accumulation of probability along the diagonal, indicating a high likelihood of finding both walkers at the same position.
   • Marginal Distribution: The single-walker probability distribution $n(x; t)$ shows a structural change around $\theta = \pi$, characterized by the emergence of a central peak. For larger $\theta$, the distribution splits and pushes toward the edges, eventually recovering the non-interacting square shape for $\theta > 3\pi/2$.

3. Quantitative Analysis of Correlations:

   • Lattice Partitioning: The authors divide the lattice into interior ($I$), left ($L$), and right ($R$) regions. They define four probabilities ($P_A, P_B, P_C, P_D$) corresponding to finding walkers in specific configurations (e.g., both in the center, on opposite edges, on the same edge).
   • Findings: In the non-interacting limit, the probability of finding walkers on opposite sides ($P_B$) equals that of the same side ($P_C$). As interactions are introduced, $P_B$ initially increases. Around $\theta \approx \pi/4$, probability rapidly transfers to the central configuration ($P_A$), saturating around 45%. The probability of finding walkers on the same edge ($P_C$) remains low for most of the parameter range.

4. Entanglement Generation:

   • The authors quantify mutual correlations using the von Neumann entropy of the reduced density matrix of a single walker.
   • Results: For $\theta=0$, no entanglement is generated. For non-zero interactions, entanglement entropy exhibits small initial oscillations followed by saturation at long times. The entanglement is strongly dependent on $\theta$, showing a clear $2\pi$-periodic behavior. Entanglement builds up as the interaction is turned on, reaches a plateau, and then rapidly decreases after crossing a threshold around $\theta \approx 3\pi/2$.

5. Two-Parameter Dynamics: By varying both $\theta_+$ and $\theta_-$, the authors demonstrate that the interplay between same-spin and opposite-spin collision phases creates complex, non-monotonic interference patterns in the density distribution that cannot be captured by single-parameter models.

Significance and Claims
The paper claims that this general interaction framework provides a versatile platform for engineering quantum correlations. The results demonstrate that simple phase-controlled interactions can induce non-trivial dynamics, including the collapse of walker spread, strong spatial bunching, and the dynamic generation of entanglement.

The authors position their work as a tool for:

• Quantum Simulation: Providing a platform to simulate condensed matter systems.
• State Preparation: Enabling the engineering of specific spatial and entanglement correlations.
• Sensing Protocols: Offering potential for enhanced quantum sensing via controlled bunching.
• Scalability: Opening the possibility for analyzing many-body interactions for larger numbers of walkers.

The study is modest in its scope, focusing on a specific, simple yet profound interaction model to establish a baseline for understanding phase-controlled correlations, while noting that more complex interactions and spectral analyses (such as the explicit construction of bound-state wave functions) remain open questions for future work.