Figure 1

Dynamics of a single quantum particle placed at $t=0$ on a single site. (a) In a periodic lattice, the particle-density distribution expands ballistically, with high density at the edges of the distribution. (b) Cross section of the density distribution after some evolution. (c),(d) In a disordered lattice, the expansion is limited to a finite region. (e) The density distribution, averaged over 1000 realizations of disorder. The expansion starts ballistically. After some propagation a localized component emerges around the initial position, [red arrow, red cross section in (f)]. The ballistic component decays in time, leaving the distribution exponentially localized [blue arrow, blue cross section in (f)]. Inset shows the distribution in semilog scale.Reuse & Permissions

Figure 2

Two-particle correlation function in disordered lattices. At $t=0$, two particles are placed at two neighboring sites. The correlation function is calculated after a short evolution, where remnants of the ballistic component are still present [see Figs. 1e, 1f], averaged over a 1000 disorder realizations. The matrices represent the probability to detect one particle at site $q$ and one particle at site $r$. (a) Bosons either scatter and become localized, or remain ballistic. However, if they both remain ballistic, they bunch to the same side of the distribution. (b) Magnification of the localized component at the center of the correlation map. (c) Fermions show similar behavior, only that if they both remain ballistic they always separate to different sides of the distribution. (d) Magnification of the correlation map, showing a checkered pattern. (e) The particle-density distribution, which is identical for fermions and for bosons, computed by tracing out the position of one of the particles (summing over the columns of the correlation matrix). (f) The interparticle distance probability for bosons (blue) and fermions (red). Bosons tend to appear at the same site, while fermions are more likely to be separated by an odd number of sites.Reuse & Permissions

Figure 3

The two-particle correlation function when at $t=0$, the two particles are placed at two nonadjacent sites. (a) Bosons show two possible effects—they are either both localized, or both remain ballistic. (b) Close-up on the correlation inside the localized component. (c) Fermions tend to split—one fermion becomes localized and the other remains ballistic. (d) Close-up on the correlation inside the localized component. (e) The particle-density distribution, identical for fermions and bosons. (f) The interparticle distance probability within the localized regime for bosons (blue line) and fermions (red line). The fermions exhibit a flat interparticle distance distribution (except at zero separation), while the bosons show an oscillating probability.Reuse & Permissions

Figure 4

Correlation maps for path-entangled bosons evolving in disordered lattices. (a) Correlation map for the initial condition $|\varphi ⟩=\frac{1}{2}[(a_0^{†}{)}^2+(a_1^{†}{)}^2]|0⟩$. (b) Magnified view of the localized component. (c) For $|\varphi ⟩=\frac{1}{2}[(a_{-1}^{†}{)}^2-(a_1^{†}{)}^2]|0⟩$. (d) Magnified view. These correlations are similar to the fermionic results in Figs. 2c, 2d, 3c, 3d (see text).Reuse & Permissions