Quantum walks of kicked Bose–Einstein condensates

Abstract

We analytically investigate the recently proposed and implemented discrete-time quantum walk based on kicked ultra-cold atoms. We show how the internal level structure of the kicked atoms leads to the emergence of a relative light-shift phase immediately relevant for the experimental realization. Analytical solutions are provided for the momentum distribution for both the case of quantum resonance and the near-resonant quasimomenta.

Experimental realization of a momentum-space quantum walk

“…with J's being Bessel functions of the first kind, represents the "kick" by each pulse that leads to a symmetric diffraction of the wave function in the space spanned by the momentum eigenstates [48]. The prefactor e −ik , which originates from the dc component of the lattice potential, introduces a "global" phase to the atoms at each kick [49,50]. As will be shown later, we can compensate for this phase in our experiments.…”

Spontaneous emission in quantum walks of a kicked Bose-Einstein condensate

“…In our quantum walk, the situation is different since the two different hyperfine levels determine the different directions of the walker. Then SE crucially affects the time evolution of the walk, and, in contrast to the standard AOKR [16], analytical solutions like in the coherent closed case [19] seem impossible. Even the numerical evolution becomes more involved due to the more complex Lindblad operators to be introduced in the next section.…”

“…In earlier work [19], we derived the effective dynamics during the kick by adiabatically eliminating the excited state using the method of James and Jerke [20] leading to the effective Hamiltonian…”

“…As detailed in ref. [19], this leads to a conditional kick operator in the space spanned by the two internal states |1 and |2…”

“…2: Schematic representation of the system during the kick as an atomic three-level system in Λ-configuration (left) and of the effective system as an atomic two-level system (right). The adiabatic elimination of the excited state creates an additional light-shift energy difference between the levels [19] marked with the different kicking strengths ξ1k1 and ξ2k2.…”