Phase evolution in spatial dark states

Adiabatic techniques using multilevel systems have recently been generalized from the optical case to settings in atom optics, solid state physics, and even classical electrodynamics. The most well known example of these is the so-called stimulated Raman adiabatic passage (STIRAP) process, which allows transfer of a particle between different states with large fidelity. Here we generalize and examine this process for an atomic center-of-mass state with a nontrivial phase distribution and show that even though dark state dynamics can be achieved for the atomic density, the phase dynamics will still have to be considered as a dynamical process. In particular we show that the combination of adiabatic and nonadiabatic behavior can be used to engineer phase superposition states.

Figure 1

Schematic setup for CTAP for an atomic state carrying one quantum of angular momentum. The atom is initially located in the trap on the left-hand side and all other traps are considered empty. Tunneling only occurs between nearest neighbor traps.

Figure 2

(a) Distance between the left-middle and the middle-right trap. (b) Tunneling strength between neighboring traps derived from the analytical model described in the text. (c) Mixing angle $\theta$ and (d) time-dependent eigenvalues of the Hamiltonian (1). Time is measured in units of $1/\omega$, energy in units of $\hslash \omega ,$ and distance in units of $a_0$.

Figure 3

Angular momentum as a function of the overall duration of the process (upper plot) and the final states in the rightmost trap at points A, B, C, and D (lower plots). All simulations are in the adiabatic regime where 100% transfer is achieved. Time is measured in units of $1/\omega$ and angular momentum in units of $\hslash$.

Figure 4

Evolution of ${\epsilon }_{0y}$ (blue dashed), ${\epsilon }_{1y}$ (green dot-dashed), and ${\epsilon }_{0y}-{\epsilon }_{1y}$ (red solid) over the course of the CTAP process. Both energies are initially separated by $\omega =1$, but they do not maintain this separation due to the modification of the potential as the traps move closer together. Time is measured in units of $1/\omega$ and energy in units of $\hslash \omega$.

Figure 5

Final angular momentum as a function of time for different minimum distances between the traps: $d_{\min }=2.0$ (full line), $d_{\min }=2.2$ (dashed line), and $d_{\min }=2.4$ (dotted line). The offset of the different curves has been shifted for clarity. The inset shows the increase of the period with increasing $d_{\min }$. Time is measured in units of $1/\omega$ and angular momentum in units of $\hslash$.

Figure 6

Angular momentum as a function of the overall time the CTAP process takes for different nonlinearities. Time is measured in units of $1/\omega$ and angular momentum in units of $\hslash$.