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arXiv:2605.20464 · Physica Scripta

Self-Trapping Bounds for Continuous-Time Nonlinear Quantum Walks on Path and Cycle Graphs

Yujia Shi, Thomas G. Wong

quant-phSubmitted May 19, 2026• Announced May 21, 2026• 0 citations• Revised Aug 13, 2026 (v2)

v1 · May 19, 2026 v2 · Aug 13, 2026

Abstract

We explore a continuous-time quantum walk starting at a single vertex on the discrete path and cycle with a cubic nonlinearity. Such nonlinearities arise in Bose-Einstein condensates described by the Gross-Pitaevskii equation or by nonlinear optical waveguide arrays. When the nonlinearity is sufficiently strong, the walker remains localized at its initial vertex, a phenomenon known as self-trapping. This contrasts with linear quantum walks, which are known for spreading quickly in one dimension. While self-trapping has been known numerically, we introduce an analytical method that proves self-trapping and yields a quantitative relationship between the nonlinearity coefficient and the trapping probability. We propose that this trapping can be used for timing in quantum state transfer, where a qubit is held at a node until it is ready to be transferred, and it can also be held again at the receiving node. This scheme can also be interpreted as a form of quantum memory, with the trap and transfer corresponding to the storage and release of quantum information.

Journal reference: Phys. Scr. 101, 365101 (2026)
DOI: 10.1088/1402-4896/aea02d
Comments: 11 pages, 6 figures

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In the present paper, we analytically and numerically investigate the dynamics of Bose-Einstein condensates (BECs) loaded into deep optical lattices of one dimension (1D), 2D, and 3D. We focus on the self-trapped state and the effect of the lattice dimension. Under the tight-binding approximation, we obtain an analytical criterion for the self-trapped state of BEC using the time-dependent variational method. The phase diagram for self-trapping, soliton, breather, or diffusion of a BEC cloud is obtained accordingly and verified by directly solving a discrete Gross-Pitaevskii equation numerically. In particular, we find that the criterion and the phase diagrams are modified dramatically by the dimension of the optical lattices.

Abstract Mean field and beyond mean field model calculations of Bose–Einstein condensates trapped in optical lattices have shown that initially homogeneous condensates can evolve into self-trapped, strongly localized states in the presence of weak boundary dissipation, a phenomenon called self-localization . A dynamical phase transition from extended to localized states can be observed when the effective nonlinearity exceeds a critical threshold Λ eff c . We investigate this phase transition to self-localization in the mean field approximation of the discrete nonlinear Schrödinger equation. We quantitatively characterize the properties of the discrete breathers , i.e. the nonlinear localized solutions, at the phase transition. This leads us to propose and numerically verify an analytical lower bound Λ eff L for the critical nonlinearity based on the idea of self-induced Anderson localization.

We report the first experimental observation of nonlinear self-trapping of Bose-condensed 87Rb atoms in a one-dimensional waveguide with a superimposed deep periodic potential . The trapping effect is confirmed directly by imaging the atomic spatial distribution. Increasing the nonlinearity we move the system from the diffusive regime, characterized by an expansion of the condensate, to the nonlinearity dominated self-trapping regime, where the initial expansion stops and the width remains finite. The data are in quantitative agreement with the solutions of the corresponding discrete nonlinear equation. Our results reveal that the effect of nonlinear self-trapping is of local nature, and is closely related to the macroscopic self-trapping phenomenon already predicted for double-well systems.

Many-body quantum systems with effective nonlinearities have been shown to speed up quantum search on the complete graph, \textit{i.e.}, the combinatorial version of Grover's algorithm, at the expense of the number of particles needed for the effective nonlinearity to hold. Physically, however, data may not be arranged in an all-to-all network, and the task of searching incomplete graphs is the spatial search problem. We explore spatial search using a continuous-time nonlinear quantum walk on a variety of graphs. First, we consider incomplete graphs that are ``sufficiently complete'' so as to asymptotically search like the complete graph under a continuous-time (linear) quantum walk, which includes strongly regular graphs such as Paley graphs, regular graphs such as hypercubes, and irregular graphs such as complete bipartite graphs. For these sufficiently complete graphs, we analytically prove nonlinear speedups for Paley graphs and for complete bipartite graphs whose two partite sets both have size

Θ(N)

, for suitable cubic and cubic-quintic nonlinearities, and we give numerical evidence for stronger nonlinearities and for hypercubes. Second, we explore arbitrary-dimensional cubic lattices, and we numerically show that certain nonlinearities speed up search on sufficiently high dimensional lattices. Thus, nonlinear quantum search can remain viable even when the underlying graph is incomplete.