  We experimentally measured the ultra-narrow momentum width of an optical
trapped Bose-Einstein condensate (BEC) in situ based on matter-wave
interference, which validates our previous theoretical work [arXiv:
2205.02416]. By sweeping the interval of double stand-wave pulses, the BEC wave
packet was splitted into different diffraction orders and then we counted the
oscillation curve of the population of zero-momentum state to calibrate the
momentum width. Compared with our simplified theory, we observed an accelerated
evolution of interference fringes in time-domain. We evaluated this
interference process minutely by numerically calculating the Gross-Pitaevskii
equation and using Wigner function to intuitively demonstrate the influence of
the external potential and nonlinear term. We confirmed that the reduction of
interference fringe evolution period actually originates from the synergistic
cooperation of the mean-field interaction of the BEC and spatial density
modulation caused by the interference between different momentum states. Our
approach could be generalized to other ultra-cold atomic gases with different
momentum distributions, and in principle a single shot can obtain the result.
This quantum thermometry is particularly suitable for momentum width
calibration in practical deep cooling experiments, while for atomic samples at
pK level the mean-field interaction can be safely ignored.
