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arXiv:2002.09139 · Journal of Physics B Atomic Molecular and Optical Physics

Quantum search with a continuous-time quantum walk in momentum space

Michele Delvecchio, Caspar Groiseau, Francesco Petiziol, Gil S Summy, Sandro Wimberger

Published Dec 18, 2019• 15 citations
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Abstract

Abstract The atom-optics kicked rotor can be used to prepare specific momentum distributions on a discrete basis set. We implement a continuous-time quantum walk (QW) and a quantum search protocol in this momentum basis. In particular we propose ways to identify a specific marked state from the final momentum distribution after the walker’s evolution. Our protocol is guided by current experimental possibilities making it accessible to experimentally implemented QWs with Bose–Einstein condensates.

DOI
10.1088/1361-6455/ab63ad

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Abstract The atom-optics kicked rotor can be used to prepare specific momentum distributions on a discrete basis set. We implement a continuous-time quantum walk (QW) and a quantum search protocol in this momentum basis. In particular we propose ways to identify a specific marked state from the final momentum distribution after the walker’s evolution. Our protocol is guided by current experimental possibilities making it accessible to experimentally implemented QWs with Bose–Einstein condensates.
Randomness is the essence of many processes in nature and human society. It can provide important insights into phenomena as diverse as disease transmission, financial markets, and signal processing. Quantum randomness is intrinsically different from classical stochasticity since it is affected by interference and entanglement. Quantum walks profit from the entanglement that is built up between a walker's position and the "coin" degree of freedom that determines the direction of a walker's motion. This entanglement makes quantum walks promising candidates for the implementation of quantum computational algorithms and as a detector of quantum behavior. While several proof-of-principle quantum walk experiments have recently been implemented, the walk scheme we demonstrate here involves the synthesis of the atom-optics kicked rotor with the dynamics of a state selective ratchet realized with a Bose-Einstein condensate. Here we present a discrete-time quantum walk that uses the momentum of ultra-cold rubidium-87 atoms as the walk space and two internal atomic states as the coin degree of freedom. To implement each step of the walk, i.e. a coin toss followed by a unitary shift operator, we use a microwave pulse, which produces a superposition of two internal states, and a resonant quantum ratchet, which entangles the internal and external degrees of freedom. Our quantum ratchet, whose direction at each step depends on the result of the coin toss, is realized by the atom-optics kicked rotor. We demonstrate the principle features of a standard quantum walk, contrasting them to the behavior of a classical walk. In addition, by manipulating either the walk or coin operator we show how the walk dynamics can be biased or reversed. Furthermore, we show the quantum-to-classical transition in our walk, via randomizing the mixing between the two internal states during each coin toss. Our walk offers distinct advantages arising from the robustness of its dynamics in momentum space, extendability to higher dimensions, and possible dependence on many-body correlations.
We introduce the driven discrete time quantum walk (QW), where walkers are added during the walk instead of only at the beginning. This leads to interference in walker number and very different dynamics when compared to the original QW. These dynamics have two regimes, which we illustrate using the one-dimensional line. Then, we explore a search application which has certain advantages over current search protocols, namely that it does not require a complicated initial state nor a specific measurement time to observe the marked state. Finally, we describe a potential experimental implementation using existing technology.
We analytically investigate the analogy between a standard continuous-time quantum walk in one dimension and the evolution of the quantum kicked rotor at quantum resonance conditions. We verify that the obtained probability distributions are equal for a suitable choice of the kick strength of the rotor. We further discuss how to engineer the evolution of the walk for dynamically preparing experimentally relevant states. These states are important for future applications of the atom-optics kicked rotor for the realization of ratchets and quantum search.
The kicking sequence of the atom-optics kicked rotor at quantum resonance can be interpreted as a quantum random walk in momentum space. We show how such a walk can become the basis for nontrivial classical walks by applying a random sequence of intensities and phases of the kicking lattice chosen according to a probability distribution. This distribution converts on average into the final momentum distribution of the kicked atoms. In particular, it is shown that a power-law distribution for the kicking strengths results in a L\'evy walk in momentum space and in a power law with the same exponent in the averaged momentum distribution. Furthermore, we investigate the stability of our predictions in the context of a realistic experiment with Bose-Einstein condensates.
We introduce a discrete-time quantum walk on a one-dimensional momentum space including both discrete jumps and continuous drift. Its time evolution has two diferent stages. Initially a Markovian diffusion develops during a characteristic time interval, after which dynamical localization sets in, as in the well known Quantum Kicked Rotor system. For some exceptional values of the model's parameter the system exhibits resonant behavior and the system model behaves as the standard discrete time quantum walker on the line.