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arXiv:2603.20439 · Журнал Экспериментальной и Теоретической Физики

Interference of a chain of Bose condensates in the Pitaevskii-Gross approximation

I. N. Mosaki, A. V. Turlapov

cond-mat.quant-gasphysics.atom-phquant-phSubmitted Mar 20, 2026• Announced Mar 24, 2026• 3 citations
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Abstract

A long chain of Bose condensates freely expands and interferes after being released from an optical lattice. The interference fringes are well resolved both in the case of equal phases of the condensates and in the case of fluctuating phases. In the second case the positions of the fringes also fluctuate. The spectrum of the spatial density distribution, however, is reproducible despite the fluctuations. Moreover two types of peaks are distinguishable in the spectrum. The first type arises due to the phase fluctuations, the second type is associated with the coherence between the condensates. In the framework of the Pitaevskii-Gross equation we calculate the interference of the condensates and compare the calculation with experiment [Phys. Rev. Lett. 122, 090403 (2019)]. The calculation reproduces the positions of the spectrum peaks, including the dependence on the interparticle interaction. The calculated heights of the peaks, however, in some cases differ with the experimental ones.

Journal reference
Zh. Eksp. Teor. Fiz. 166, 30 (2024)
DOI
10.31857/S0044451024070046

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A long chain of Bose condensates freely expands and interferes after being released from an optical lattice. The interference fringes are well resolved both in the case of equal phases of the condensates and in the case of fluctuating phases. In the second case the positions of the fringes also fluctuate. The spectrum of the spatial density distribution, however, is reproducible despite the fluctuations. Moreover two types of peaks are distinguishable in the spectrum. The first type arises due to the phase fluctuations, the second type is associated with the coherence between the condensates. In the framework of the Pitaevskii-Gross equation we calculate the interference of the condensates and compare the calculation with experiment [Phys. Rev. Lett. 122, 090403 (2019)]. The calculation reproduces the positions of the spectrum peaks, including the dependence on the interparticle interaction. The calculated heights of the peaks, however, in some cases differ with the experimental ones.
The visibility of the density interference pattern of two Bose-Einstein condensates, which are produced in traps and overlap after the trapping potential is switched off, is investigated. Coherent wave packets are used to describe the order parameter in a second-quantization formalism. This results in a decrease of the visibility of the interference fringes with increasing time delay between the formation of the condensates and the observation of interference. In the two limiting cases of ideal and very dense gases the correlation time increases, τ →∞, and the result is identical to that obtained using an approach based on the Gross-Pitaevskii equation. Under the conditions of the experiment performed by M. R. Andrews, C. G. Townsend, H. J. Miesner et al. , Science 275 , 6367 (1997), the computed correlation time τ ≈0.2 s is much longer than the confinement time of the condensate, and it is possible to observe the predicted decrease of visibility of the interference fringes of the density of the atoms.
For a long periodic chain of Bose condensates prepared in the free space, the subsequent evolution and interference dramatically depend on the difference between the phases of the adjacent and more distant condensates. If the phases are equal, the initial periodic density distribution reappears at later times, which is known as the Talbot effect. For randomly-related phases, we have found that a spatial order also appears in the interference, while the evolution of the fringes differs with the Talbot effect qualitatively. Even a small phase disorder is sufficient for qualitatively altering the interference, though maybe at long evolution times. This effect may be used for measuring the amount of coherence between adjacent condensates and the correlation length along the chain.
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.
For the Bose-condensed gas in a combined potential consisting of an axially-symmetric harmonic magnetic trap and one-dimensional (1D) optical lattice, using the mean-field Gross-Pitaevskii (G-P) equation and the propagator method, we study the formation of matter-wave interference upon releasing the condensate from the combined potential. The analytical result of the wave function for matter wave interference at any time t is given, and the relation between the interference fringe period and the optical lattice period is obtained. The interference pattern of Bose-condensed gas in the optical lattices is analogous to the multiple slit interference pattern in optics, and coincides with the results of experiment performed by Andrews et al. and Peil et al. and the numerical solution of the G-P equation by Sadhan K. Adhikari.
In the last decades, the experimental research on Bose–Einstein interferometry has received much attention due to promising technological implications. This has thus motivated the development of numerical simulations aimed at solving the time-dependent Gross–Pitaevskii equation and its reduced one-dimensional version to better understand the development of interference-type features and the subsequent soliton dynamics. In this work, Bohmian mechanics is considered as an additional tool to further explore and analyze the formation and evolution in real time of the soliton arrays that follow the merging of two condensates. An alternative explanation is thus provided in terms of an underlying dynamical velocity field, directly linked to the local phase variations undergone by the condensate along its evolution. Although the reduced one-dimensional model is considered here, it still captures the essence of the phenomenon, rendering a neat picture of the full evolution without diminishing the generality of the description. To better appreciate the subtleties of free versus bound dynamics, two cases are discussed. First, the soliton dynamics exhibited by a coherent superposition of two freely released condensates is studied, discussing the peculiarities of the underlying velocity field and the corresponding flux trajectories in terms of both the peak-to-peak distance between the two initial clouds and the addition of a phase difference between them. In the latter case, an interesting correspondence with the well-known Aharonov-Bohm effect is found. Then, the recurrence dynamics displayed by the more general case of two condensates released from the two opposite turning points of a harmonic trap is considered in terms of the distance between such turning points. In both cases, it is presumed that the initial superposition state is generated by splitting adiabatically a single condensate with the aid of an optical lattice, which is then turned off. Nonetheless, although the lattice does not play any active role in the simulations, the parameters defining the initial states are in compliance with it, which helps in the interpretation and understanding of the results observed.