Signatures of Strong Momentum Localization via Entanglement of Translational and Internal States

We show that atoms or molecules subject to fields that couple their internal and translational (momentum) states may undergo a crossover from randomization (diffusion) to strong localization (sharpening) of their momentum distribution. The predicted crossover should be manifest by a drastic change of the interference pattern as a function of the coupling fields.

Figure 1

(a) Proposed setup: the internal states of a cold atom (or molecule) are mixed by pairs of Raman beams and become entangled with different momentum states. Full lines indicate an ordered lattice of energies, dashed lines—a disordered lattice. (b) Momentum-space dimensionalities 1D, 2D, $\infty \mathrm{D}$, illustrated by lattice graphs with different connectivities. (c) Numerical Raman coupling strengths in log scale for Li atoms in a 1D setup (for parameters at paper end).

Figure 2

Momentum distribution for random systems with $(\Delta /J)=1$ (dashed line) and $(\Delta /J)=10$ (solid line), in a 1D system with $N=10$ states, compared to a periodic (regular) system with $(\Delta /J)=10$ (dotted line). The random momentum distribution is delocalized for $(\Delta /J)=1$, while strong localization appears for $(\Delta /J)=10$. The periodic distribution is not localized for $(\Delta /J)=10$. A localization crossover is found even though a random 1D system should not exhibit a transition, due to finite-size effects.

Figure 3

(a) Visibility of the interference fringes as a function of $\Delta /J$, in a 3D system. (b) True disorder: flow of the crossover point to zero in a 1D system, calculated (dots) and exponentially fitted (solid line). Quasicrystal: flow of the crossover to $(\Delta /J{)}_C=2$, calculated (squares) and linearly fitted (dashed line), indicating a true transition. (c) True disorder for a 3D system. The crossover point flows to $(\Delta /J{)}_C\simeq 10$, indicating a true transition. (d) Same for an $\infty \mathrm{D}$ system: crossover flows to $\infty$.