Quantum Walk in Momentum Space with a Bose-Einstein Condensate

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.

Figure 1

Experimental (a) and simulated (b) momentum distributions of our standard QW realized with $|k|=1.45$. Each time (kick #) represents one step of the walk, i.e., one realization of the experiment (or simulation). The amorphous population signal about the center of the momentum distribution in (a) is a residual atomic thermal cloud that, unlike the BEC, does not respond to the ratchet. The middle panels represent the quantum-to-classical transition: the standard QW (c) was conducted with a fixed coin toss phase. Signatures of a classical walk emerge at 8% phase noise (d). The walk is dominantly classical when randomizing the phase by 20% (e) and becomes fully classical when the phase is allowed to vary randomly within $2\pi$ (f). Panel (g) shows the evolution of momentum distribution pattern at the eighth step of the walk for the corresponding noise levels.

Figure 2

Experimental (a) and simulated (b) momentum distributions of the steered walk for a BC with $|k|=1.45$. Here the coin tosses were changed so as to produce the internal state $\sqrt{0.7}|1⟩+\sqrt{0.3}i|2⟩$ rather than $\sqrt{0.5}(|1⟩+i|2⟩)$. Panels (c) and (d) demonstrate the experimental and simulated BR steered walks with an unbiased coin and $k_1=-1.7$, $k_2=+1.0$ instead of $k_1=-1.45$, $k_2=+1.45$. (e) Shows the experimental (E.) and simulated (S.) variation of the mean momentum for BC and BR walks compared to the symmetric walk (SW).

Figure 3

Experimental (a) and simulated (b) momentum distributions of a reversed QW with $|k|=1.45$. After eight of the walk steps ${\widehat{\mathbf{U}}}_{\mathrm{step}}=\widehat{\mathbf{T}}\widehat{\mathbf{M}}$ are taken, applying their Hermitian conjugate ${\widehat{\mathbf{U}}}_{\mathrm{step}}^{†}={\widehat{\mathbf{M}}}^{†}{\widehat{\mathbf{T}}}^{†}$ for an additional eight steps reverts the system to its original state. These conjugates can be realized as ${\widehat{\mathbf{M}}}^{†}=\widehat{\mathbf{M}}(\pi /2,\pi /2)$ and ${\widehat{\mathbf{T}}}^{†}=\widehat{\mathbf{M}}(\pi ,\pi /2)\widehat{\mathbf{T}}\widehat{\mathbf{M}}(\pi ,-\pi /2)$ that when the intermediate $\widehat{\mathbf{M}}$ matrices multiply for successive steps, the reversal steps boil down to ${\widehat{\mathbf{U}}}_{\mathrm{step}}^{†}=\widehat{\mathbf{T}}\widehat{\mathbf{M}}(\pi /2,\pi /2)$ after a $\widehat{\mathbf{T}}\widehat{\mathbf{M}}(\pi ,-\pi /2)$ “reflection.” Panel (c) shows the experimental (E.) and simulated (S.) mean energy for a reversed walk with the minimum possible quasimomentum width and thermal cloud.