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Anderson localization is a profound phenomenon in condensed matter physics, representing a fundamental transition of eigenstates induced by disorder. The one-dimensional Aubry-André-Harper (AAH) model, an iconic quasiperiodic lattice model, is one of the simplest models that demonstrate the Anderson localization transition. Recently, with the growing interest in quantum lattice models in curved spacetime (CST), the AAH model in CST has been proposed as a way to explore the interplay between Anderson localization and CST physics. While a few CST lattice models have been realized in optical waveguide systems to date, significant challenges remain in the experimental preparation and measurement of states, primarily due to the difficulty of dynamically modulating lattices in such systems. In this study, we propose an experimental scheme using a momentum-state lattice (MSL) in an ultracold atom system to realize the AAH model in CST and study the Anderson localization in this context. Thanks to the individual controllability of the coupling between each pair of adjacent momentum states, the coupling amplitude in the MSL can be encoded as a power-law position-dependent form $J_n \propto n^{\sigma}$, facilitating effective simulation of CST. Numerical calculation results of the MSL Hamiltonian show an emergence of the phase separation in a 34-site AAH chain in CST, where wave packet dynamics exhibit localized behavior on one side of the critical site and extended behavior on the other. The phase separation critical site is observed by extracting turning points of the evolving fractal dimension and the wave packet width derived from evolution dynamic simulations. Furthermore, by modulating the spacetime curvature parameter $\sigma$, we propose a method for eigenstates preparation of the AAH chain in CST, and perform numerical simulations in the MSL. Through calculating the fractal dimension of eigenstates prepared following the aforementioned method, we analyze the localization properties of eigenstates under various quasiperiodic modulation phases, confirming the coexistence of localized phase, swing phase, and extended phase in the energy spectrum. Unlike traditional localized and extended phases, eigenstates in the swing phase of the AAH model in CST exhibit different localization properties under different modulation phases, indicating the prescence of a swing mobility edge. Our results provide a feasible experimental approach to study Anderson localization in CST and introduce a new platform for realizing quantum lattice models in curved spacetime.
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Keywords:
- Anderson localization /
- curved spacetime /
- ultracold atoms /
- momentum-state lattice
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