[Paper Review] Quantum Walks of Two Interacting Particles in One Dimension

Xizhou Qin, Yongguan Ke+4 more|arXiv (Cornell University)|Feb 14, 2014

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper investigates continuous-time quantum walks of two indistinguishable particles (bosons, fermions, or hard-core bosons) in one-dimensional lattices with nearest-neighbor interactions. It analytically derives an effective single-particle model for strongly interacting particles, showing that they co-walk as a composite entity with a propagation speed three times faster than fermions or hard-core bosons, consistent with numerical simulations and recent experiments on ultracold atoms.

ABSTRACT

We investigate continuous-time quantum walks of two indistinguishable particles (bosons, fermions or hard-core bosons) in one-dimensional lattices with nearest-neighbour interactions. The two interacting particles can undergo independent- and/or co-walking dependent on both quantum statistics and interaction strength. We find that two strongly interacting particles may form a bound state and then co-walk like a single composite particle with statistics-dependent propagation speed. Such an effective single-particle picture of co-walking is analytically derived in the context of degenerate perturbation and the analytical results are well consistent with direct numerical simulation. In addition to implementing universal quantum computation and observing bound states, two-particle quantum walks offer a novel route to detecting quantum statistics. Our theoretical results can be examined in experiments of light propagations in two-dimensional waveguide arrays or spin-impurity dynamics of ultracold atoms in one-dimensional optical lattices.

Motivation & Objective

• To understand how quantum statistics and inter-particle interactions jointly influence two-particle quantum walks in one-dimensional lattices.
• To determine the conditions under which two interacting particles co-walk synchronously as a composite entity.
• To derive an effective single-particle description for co-walking in the strong interaction regime.
• To provide a theoretical framework consistent with experimental observations of two-magnon bound states in ultracold atoms.
• To propose feasible experimental realizations using photonic waveguide arrays and ultracold atoms in optical lattices.

Proposed method

• The study employs a Hamiltonian describing two indistinguishable particles on a 1D lattice with nearest-neighbor hopping (J) and on-site interaction (V), using periodic boundary conditions.
• Three types of quantum statistics are analyzed: bosonic, fermionic, and hard-core bosonic commutation relations.
• Degenerate perturbation theory is applied to derive the effective hopping amplitude for co-walking in the strong interaction limit.
• Numerical simulations of the time evolution of the many-body wavefunction are performed in both position and momentum spaces to analyze two-body correlations.
• The effective single-particle model is validated by comparing analytical predictions with direct numerical solutions of the full two-body problem.
• Experimental feasibility is demonstrated by mapping the quantum walk dynamics onto light propagation in two-dimensional waveguide arrays using quantum-optical analogues.

Experimental results

Research questions

• RQ1 How do quantum statistics and inter-particle interactions affect the co-walking behavior of two indistinguishable particles in a 1D lattice?
• RQ2 Under what conditions do two strongly interacting particles form a bound state and co-move as a single composite particle?
• RQ3 What is the effective propagation speed of two co-walking particles, and how does it depend on their quantum statistics?
• RQ4 Can the co-walking dynamics be described by an effective single-particle model in the strong interaction regime?
• RQ5 How do the results compare with recent experimental observations of two-magnon bound states in ultracold atomic systems?

Key findings

• Two strongly interacting particles form a bound state and co-walk as a composite particle with a propagation speed three times faster than that of two fermions or hard-core bosons.
• The effective single-particle hopping amplitude for co-walking bosons is exactly three times that for fermions or hard-core bosons, as derived from second-order degenerate perturbation theory.
• Numerical simulations confirm the analytical prediction, showing excellent agreement between the effective model and full two-body dynamics.
• Bosons exhibit bunching in both position and momentum space, while fermions show anti-bunching, and hard-core bosons display anti-bunching in position and bunching in momentum space.
• The theoretical results for hard-core bosons are consistent with the experimental observation of two-magnon bound states in ultracold atomic systems (Fukuhara et al., 2013a).
• The two-particle quantum walk model can be experimentally realized using light propagation in engineered two-dimensional waveguide arrays with tunable refractive index differences to simulate interaction strength.