Order in the interference of a long chain of Bose condensates with unrestricted phases

Abstract

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.

Figures (8)

• Figure 1: (a) BECs in the lattice prior to the release and interference. The clouds of molecules shown in dark red, the standing-wave intensity shown in light purple. (b) The initial wave function along the lattice. The density is periodic, while phase $\varphi_j$ of the $j$th condensate is generally unrestricted with respect to the phases of other BECs.
• Figure 2: Interference of a chain with nearly phased adjacent condensates: images (left) and the respective Fourier transforms $|\tilde{n}_1(k)|$ (right). (a) At $t=0$, the onset of the expansion. (b) At $t=T_d$, the initial density distribution is nearly reestablished showing the Talbot effect. Each BEC overlaps with about 3 neighbors on the left and 3 on the right. (c) At $t=2T_d$, the interference is governed by the random phase relation between more distant neighbors which now overlap. In (b,c), the white bars show the full rms width along $z$ of a single condensate after the expansion.
• Figure 3: Interference of a chain where the adjacent condensates have nearly random relative phases: images (left) and the respective Fourier transforms $|\tilde{n}_1(k)|$ (right). (a) At $t=0$, the onset of the expansion. (b) At $t=T_d$, the principal harmonic in the density distribution is at about $k=\pi/d$ corresponding to period $2d$. (c) At $t=2T_d$, the principal harmonic corresponds to period $\simeq4d$.
• Figure 4: The absolute value of density spectrum (\ref{['eq:TalbotDensitySpectrum']}) at $t=T_d$. The principal harmonic lies at $k=\pi/d$ corresponding to spatial period $2d$.
• Figure 5: Interference at $t=T_d$ of a chain with a partial phase disagreement between the adjacent condensates: image $n_2(x,z)$ (left) and the respective Fourier transform $|\tilde{n}_1(k)|$ (right). Fit for finding the centers of the phase-randomization-related peaks is shown as the red curve.