Quantum Walk in Momentum Space with a Bose-Einstein Condensate

Abstract

We present a discrete-time, one-dimensional quantum walk based on the entanglement between the momentum of ultracold rubidium atoms (the walk space) and two internal atomic states (the "coin" degree of freedom). Our scheme is highly flexible and can provide a platform for a wide range of applications such as quantum search algorithms, the observation of topological phases, and the realization of walks with higher dimensionality. Along with the investigation of the quantum-to-classical transition, we demonstrate the distinctive features of a quantum walk and contrast them to those of its classical counterpart. Also, by manipulating either the walk or coin operator, we show how the walk dynamics can be steered or even reversed.

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

“…Altogether, we can state that the limits for observing a coherent quantum walk over a substantial number of kicks are defined by the upper bounds p SE ≤ 0.02 and ∆ β ≤ 0.02. Both ranges are already within experimental reach with p SE < 0.01 from [35] and ∆ β ≈ 0.02 in the more recent walk experiment [15], for which a similar SE rate probably applies.…”

“…In the experiment, the microwaves that induce the coin toss and the compensation of the light shift phase and the dynamic phase are not instantaneous but rather work for a finite time, actually the entire period in-between two kicks [15]. Since during the free evolution time the internal degrees are not directly coupled that should, however, not be a problem.…”

“…Here, the last two terms represent an effective coupling between the ground states that can be ignored as they scale with the difference of the two detunings (∆ 2 − ∆ 1 ) which is small. Indeed, most of the experimental data was taken for equal detunings ∆ 2 = ∆ 1 that produces a perfectly symmetric walk [15]. Biased walks are also possible with slightly different detunings, and hence different kicking strengths [14,15].…”

Light-cone and local front dynamics of a single-particle extended quantum walk

“…They have been used to model a variety of diverse physical processes like that of coherent transport of excitations between different potential sites [4,5], relativistic effects like Zitterberwegung and Klein tunnelling [6,7], Anderson localization [8], topological phases [9], strongly correlated many body systems [10,11], etc. These studies have gained further impetus due to experimental realizations of quantum walks in systems of trapped ions [12][13][14], cavity QED [15], photon waveguide arrays [16], ultra-cold atoms in optical lattices [17][18][19], etc.…”

Dynamics of solitons for nonlinear quantum walks

“…Nonlinear quantum walks (NLQWs), which are nonlinear versions of QWs, have been recently proposed by several authors [15][16][17] and in particular related to some nonlinear differential equations such as nonlinear Dirac equations [16]. For experimental realization of QWs by Bose-Einstein condensation, which can realize the nonlinearity in principle, see [18,19]. In [20], we have initiated an analytical study of NLQWs using the methods developed for the study of nonlinear dispersive equations.…”