Realization of discrete quantum billiards in a two-dimensional optical lattice

We propose a method for optical visualization of the Bose-Hubbard model with two interacting bosons in the form of two-dimensional (2D) optical lattices consisting of optical waveguides, where the waveguides at the diagonal are characterized by different refractive indices than others elsewhere, modeling the boson-boson interaction. We study the light intensity distribution function averaged over the direction of propagation for both ordered and disordered cases, exploring the sensitivity of the averaged picture with respect to the beam injection position. For our finite systems, the resulting patterns are reminiscent the ones set in billiards, and therefore we introduce a definition of discrete quantum billiards and discuss the possible relevance to its well-established continuous counterpart.

Figure 1

Geometry of setup: A beam enters the 2D optical lattice and propagates along the $z$ axis. The refractivity index is constant along the $z$ axis and is either periodic or disordered in transverse directions. The corresponding mapping to the dynamics of two interacting distinguishable bosons in a chain is also done (see the text for details). The interaction between bosons is introduced by taking the different from the rest refractive index for the diagonal waveguides. The injection of a beam to the diagonal waveguide mimics launching initially both bosons at the same site (upper inset), while injecting the beam into an off-diagonal waveguide corresponds to the two bosons located initially on different sites (lower inset).

Figure 2

Characteristic pictures for discrete quantum billiard realization for different injection points depicted by a red cross in the absence of disorder and interaction ($W=0$ and $U=0$). In the main graphs, the averaged PDFs [see Eq. (3)] are displayed. The lower insets show the numbers of initially injected waveguides (or numbers of sites at which the particles are initially located).

Figure 3

All parameters and quantities are the same as in Fig. 2, except the interaction constant $U=1$.

Figure 4

Characteristic pictures for discrete quantum billiard realization with a rigid square placed inside a system for the interaction constant $U=1$ and the same injection point depicted by a red cross. (a), (b) Symmetric and asymmetric situations, respectively, with the corresponding probability density functions of eigenvalue spacings $s$ shown in (c) and (d). (c) The Poisson distribution (6), with the average spacing $d=0.0085$. (d) The Wigner-Dyson distribution (7), with the average spacing $d=0.0075$.

Figure 5

Averaged PDFs both in time and over many disorder realizations (all parameters are shown on the figures). The injection point in all graphs is taken at the middle of 2D optical lattice, $m=n=N/2$.

Figure 6

PDFs of eigenvalue spacings $s$ for three different cases and a chain with $N=51$ (the events with $s=0$ due to degeneracy are not counted). Black curve: $W=0$, $U=2$ (disorder strength is zero only). Red curve (dark gray): $W=2$, $U=2$. The case with uncorrelated disorder is considered. Green dash-dotted curve: the Wigner-Dyson distribution (7), with the average spacing $d=0.0032$. Blue dashed curve: the Poisson distribution (6), with the average spacing $d=0.0032$.