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Transformation of the Talbot effect in response to phase disorder

Ilia Mosaki, A. V. Turlapov

Abstract

Bose-Einstein condensates initially arranged in a long chain freely expand and interfere. If the initial phases of the condensates are identical, the initial density distribution is restored periodically during the expansion, giving rise to the Talbot effect. Even a slight disorder in the initial phases leads to a transformation of the interference pattern. In response to the phase disorder, the spectrum of the spatial density distribution acquires peaks that are absent in the case of identical phases. We derive an analytical expression for the spectrum of the spatial density distribution for an arbitrary phase disorder. We show that the new peaks emerging due to the phase disorder originate from pairwise interferences of the condensates. The positions of these peaks coincide with the wave vectors of the density modulations (wavelets) generated by such pairwise interferences. The absence of these peaks, when the initial phases are identical, is explained by the mutual destruction of the overlapping wavelets during their summation.

Transformation of the Talbot effect in response to phase disorder

Abstract

Bose-Einstein condensates initially arranged in a long chain freely expand and interfere. If the initial phases of the condensates are identical, the initial density distribution is restored periodically during the expansion, giving rise to the Talbot effect. Even a slight disorder in the initial phases leads to a transformation of the interference pattern. In response to the phase disorder, the spectrum of the spatial density distribution acquires peaks that are absent in the case of identical phases. We derive an analytical expression for the spectrum of the spatial density distribution for an arbitrary phase disorder. We show that the new peaks emerging due to the phase disorder originate from pairwise interferences of the condensates. The positions of these peaks coincide with the wave vectors of the density modulations (wavelets) generated by such pairwise interferences. The absence of these peaks, when the initial phases are identical, is explained by the mutual destruction of the overlapping wavelets during their summation.
Paper Structure (7 sections, 26 equations, 5 figures)

This paper contains 7 sections, 26 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Model of interference of a chain of Bose condensates
  3. Spectrum of the spatial density distribution
  4. Physical interpretation of the spectrum peaks
  5. Extension to an arbitrary condensate lattice
  6. Comparison with the Fraunhofer diffraction
  7. Summary

Figures (5)

  • Figure 1: One-dimensional density profile of a freely expanding chain of Bose--Einstein condensates at time $t=T_d$. The spatial region where the Talbot effect can occur is shown. The initial phases of the condensates are (a) identical, (b) completely disordered.
  • Figure 2: Amplitude of spatial density spectrum $\left| \tilde{n}(k,t) \right|$ for a chain of $M=100$ Bose--Einstein condensates at time $t = T_d$. The initial phases of the condensates are (a) identical, (b) completely disordered.
  • Figure 3: Average square of spectrum amplitude $\langle |\tilde{n}(k,t)|^2 \rangle$ for a chain of $M=100$ Bose--Einstein condensates at time $t = T_d$, plotted using equations \ref{['spectrum_abs']} and \ref{['f']}. The initial phases of the condensates are (a) identical, (b), (c) partially disordered, (d) completely disordered.
  • Figure 4: Interference of the square and hexagonal lattices of Bose--Einstein condensates. The initial lattice configurations are depicted in figures (a), (d); total sizes of the lattices are 40 $\times$ 40. For identical initial condensate phases, the amplitudes of the spatial density spectra at time $t=T_d$ are shown in figures (b), (e). For completely disordered phases, the amplitudes of the spatial density spectra at $t=T_d$ are displayed in figures (c), (f).
  • Figure 5: Average square of spectrum amplitude $\langle |\tilde{n}(k,t)|^2 \rangle$ for a chain of $M=100$ Bose--Einstein condensates at time $t = 500T_d$ in the Fraunhofer regime.