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Optica

Coherent random walks in free space

Toni Eichelkraut, Christian Vetter, Armando Pérez-Leija, H. M. Moya-Cessa, Demetrios N. Christodoulides, Alexander Szameit

Published Oct 17, 2014• 20 citations
Publisher

Abstract

Two-dimensional continuous time quantum random walks (CTQRW) are physical processes where quantum particles simultaneously evolve in different permissible directions within discrete graphs. In order to force the quantum walkers (QWs) to evolve in such a fashion, one generally requires periodic potentials. Here, we demonstrate that two-dimensional CTQRW can be generated in free space by properly tailoring the initial wave functions. We analytically show that within a certain spatial region the arising probability distribution quantitatively resembles the probability pattern exhibited by a QW traversing a periodic lattice potential. These theoretical predictions were experimentally verified using classical laser light, appropriately shaped by a spatial light modulator. Expanding the presented results to the case of multiple walkers may open new possibilities in quantum information technology using bulk optics. (C) 2014 Optical Society of America

DOI
10.1364/optica.1.000268

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Two-dimensional continuous time quantum random walks (CTQRW) are physical processes where quantum particles simultaneously evolve in different permissible directions within discrete graphs. In order to force the quantum walkers (QWs) to evolve in such a fashion, one generally requires periodic potentials. Here, we demonstrate that two-dimensional CTQRW can be generated in free space by properly tailoring the initial wave functions. We analytically show that within a certain spatial region the arising probability distribution quantitatively resembles the probability pattern exhibited by a QW traversing a periodic lattice potential. These theoretical predictions were experimentally verified using classical laser light, appropriately shaped by a spatial light modulator. Expanding the presented results to the case of multiple walkers may open new possibilities in quantum information technology using bulk optics. (C) 2014 Optical Society of America
With the help of quantum-scattering-theory methods and the approximation of the stationary phase, we propose a one-dimensional quantum-random-walk (QRW) model, which describes for both tunneling and scattering above the potential, the coherent diffusion of independent particles described by wave packets in a periodic one-dimensional lattice. The QRW model describes for each lattice cell the time evolution of modulating amplitudes of two opposite-moving wave packets as they are scattered by periodic potential barriers. Since the QRW model is a coherent process, interference contributions in the probabilities bring about strong departures from classical results. For many identical free particles we obtain the theoretical and graphical Bose and Fermi two-body QRW probability distribution. The result is generalized to N identical free particles and we obtain the N-body Bose and Fermi QRW probability distribution.
Quantum random walks are shown to have non-intuitive dynamics which makes them an attractive area of study for devising quantum algorithms for long-standing open problems as well as those arising in the field of quantum computing. In the case of continuous-time quantum random walks, such peculiar dynamics can arise from simple evolution operators closely resembling the quantum free-wave propagator. We investigate the divergence of quantum walk dynamics from the free-wave evolution and show that, in order for continuous-time quantum walks to display their characteristic propagation, the state space must be discrete. This behavior rules out many continuous quantum systems as possible candidates for implementing continuous-time quantum random walks.
We report on our latest findings on photonic Quantum Walks (QW) of entangled particles in several quantum transport regimes as for instance in photonic Bloch lattices or disordered lattices. Furthermore, the discrete fractional Fourier transform of quantum states is presented in the context of QWs. The propagation of a classical single walker on a random graph is an important field in statistics and can be applied on many domains in nature reaching from a foraging animals over Brownian molecular motion towards the evolution of lead changes in competitive team sports. Its quantum mechanical counterpart, the Quantum Walk (QW), features genuine quantum objects to walk on a random graph and hence, leading to non-classical quantum interference, in particular when considering two or more walkers. Their great potential lies i.a. in their ability to exponentially speed up a multitude of algorithms, as for instance the data bank search algorithm from L. Grover. In this regard, photonic lattices of evanescently coupled waveguides represent an ideal platform to study QW dynamics, since they feature advantages in stability, robustness and miniaturization. Our waveguide lattices were fabricated by means the direct laser inscription technique. In doing so, three dimensional waveguide geometries are realizable, enabling the implementation of various QW graphs. Along these lines, we study QWs of entangled photon pairs in disordered lattices. There we observe Anderson co-localization, meaning that neither the individual photons nor the pair localizes, rather their distance does. It turns out that both photons tend to bunch as they will be found together in the same lattice site, implying that their original correlation remains while performing a QW in a disordered lattice. Further, we investigate QWs of non-local two-photon EPR-states in Bloch-lattices. The experimental results reveal the strong dependence of the correlation output statistics (bosonic-, anyonicor fermionic-like) on the input states' symmetry. Finally, the discrete fractional Fourier transform of quantum states is realized through a discrete QW lattice. The results demonstrate the enormous potential of waveguide lattices in the field of quantum photonics and give rise to further investigations like the Fourier suppression law.
Quantum random walks are shown to have non-intuitive dynamics which makes them an attractive area of study for devising quantum algorithms for long-standing open problems as well as those arising in the field of quantum computing. In the case of continuous-time quantum random walks, such peculiar dynamics can arise from simple evolution operators closely resembling the quantum free-wave propagator. We investigate the divergence of quantum walk dynamics from the free-wave evolution and show that in order for continuous-time quantum walks to display their characteristic propagation, the state space must be discrete. This behavior rules out many continuous quantum systems as possible candidates for implementing continuous-time quantum random walks. Furthermore our findings demonstrate that the properties of the quantum walk are critically dependant on its initialization which in turn impacts any subsequent quantum algorithms based upon it.