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The standard Bloch oscillation normally refers to the oscillatory tunneling dynamics of quantum particles in a periodic lattice plus a linear gradient. In this work we theoretically investigate the generalized form of the Bloch oscillation in the presence of additional higher order gradients, and demonstrate that the higher order gradients can significantly modify the tunneling dynamics, particularly in the spectrum of the density oscillation. The spectrum of the standard Bloch oscillation is composed of a single prime frequency and its higher harmonics, while the higher-order gradients in the external potential give rise to fine structures in the spectrum around each of these Bloch frequencies, which are composed of serieses of frequency peaks. Our investigation leads to a twofold consequence to the applications of Bloch oscillations for measuring external forces: For one thing, under a limited resolution of the measured spectrum, the fine structures would manifest as a blur to the spectrum, and leads to intrinsic errors to the measurement. For another, given that the fine structures could be experimentally resolved, they can supply more information of the external force than the strength of the linear gradient, and be used to measure more complicated forces.

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