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Quantum random walks in waveguide lattices

Robert rer. nat. Keil

Published Feb 11, 2014• 1 citation
OpenAlex

Abstract

Random processes are ubiquitous in the natural world as well as in man-made environments. While classical random walks may behave highly complex or even chaotic, there outcome can, in principle, always be predicted from the parameters of the system and the initial conditions. In the realm of quantum mechanics, however, this is not possible as the underlying wave mechanics leads to intrinsically indeterministic outcomes. Moreover, if multiple indistinguishable particles are subjected to such a quantum random walk, their exchange symmetry causes quantum interference, thereby enriching the dynamics of the system even further. \nIn this work, quantum random walks of pairs of indistinguishable photons, the quanta of light, are investigated. Networks of coupled optical waveguides are chosen as the experimental platform of choice, offering a high degree of coherence and versatility. In these photonic lattices, light propagates along one spatial dimension, whereas the individual waveguides are connected by evanescent coupling in the transverse dimensions. \nIn particular, it is investigated how the various degrees of freedom, which are available in such photonic lattices, affect the trajectories in the quantum walks and their complexity. It is shown how the facilitation of both transverse dimensions allows for much richer quantum walks with properties unencountered in planar arrangements. But even if just a single transverse dimension is available, the coupling properties of the lattice along this dimension are a potent degree of freedom, which can be used to manipulate the quantum walk. Finally, an experimental technique is developed which enables a convenient characterisation of the expected quantum walk in an arbitrary waveguide lattice by classical light. \nThe thesis concludes with a summary of the results and an outlook onto further developments in the field.

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Random processes are ubiquitous in the natural world as well as in man-made environments. While classical random walks may behave highly complex or even chaotic, there outcome can, in principle, always be predicted from the parameters of the system and the initial conditions. In the realm of quantum mechanics, however, this is not possible as the underlying wave mechanics leads to intrinsically indeterministic outcomes. Moreover, if multiple indistinguishable particles are subjected to such a quantum random walk, their exchange symmetry causes quantum interference, thereby enriching the dynamics of the system even further. \nIn this work, quantum random walks of pairs of indistinguishable photons, the quanta of light, are investigated. Networks of coupled optical waveguides are chosen as the experimental platform of choice, offering a high degree of coherence and versatility. In these photonic lattices, light propagates along one spatial dimension, whereas the individual waveguides are connected by evanescent coupling in the transverse dimensions. \nIn particular, it is investigated how the various degrees of freedom, which are available in such photonic lattices, affect the trajectories in the quantum walks and their complexity. It is shown how the facilitation of both transverse dimensions allows for much richer quantum walks with properties unencountered in planar arrangements. But even if just a single transverse dimension is available, the coupling properties of the lattice along this dimension are a potent degree of freedom, which can be used to manipulate the quantum walk. Finally, an experimental technique is developed which enables a convenient characterisation of the expected quantum walk in an arbitrary waveguide lattice by classical light. \nThe thesis concludes with a summary of the results and an outlook onto further developments in the field.
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 walks are the quantum mechanical analog to classical random walks. However, due to the wavelike nature of quantum particles, they reveal drastically different dynamics compared to their classical counterpart. These unique properties make quantum walks a new promising platform for quantum algorithms and a powerful tool to understand complex transport phenomena. This thesis presents three experiments specifically designed to send photons on a quantum walk. The experiments are based on a fiber network spreading the photonic wave packets over discrete time bins. With these Quantum Walks in Time we simulate the propagation of quantum particles in various different environments, revealing diverse dynamics such as a fast ballistic spread and an Anderson-like localization effect. Furthermore we use our concept to implement the first quantum walk on a two-dimensional lattice. The underlying topology allows for investigations of the intriguing dynamics of two strongly interacting quantum particles, which reveals unique effects such as the creation of bound states.
Abstract Over the last 20 years quantum walks (QWs) have gained increasing interest in the field of quantum information science and processing. In contrast to classical walkers, quantum objects exhibit intrinsic properties like non-locality and non-classical many-particle correlations, which renders QWs a versatile tool for quantum simulation and computation as well as for a deeper understanding of genuine quantum mechanics. Since they are highly controllable and hardly interact with their environment, photons seem to be ideally suited quantum walkers. In order to study and exploit photonic QWs, lattice structures that allow low loss coherent evolution of quantum states are demanded. Such requirements are perfectly met by integrated optical waveguide devices that additionally allow a substantial miniaturization of experimental settings. Moreover, by utilizing the femtosecond direct laser writing technique three-dimensional waveguide structures are capable of analyzing QWs also on higher dimensional geometries. In this context, advances and findings of photonic QWs are discussed in this review. Various concepts and experimental results are presented covering, such as different quantum transport regimes, the Boson sampling problem, and the discrete fractional quantum Fourier transform.
The manipulation of the spatial structure of a light beam has many application in both classical and quantum physics. The possibility to exploit high dimensional degrees of freedom carried by a light beam can be employed, among the various applications, for simulating the dynamics on quantum particles in multidimensional spaces. Coupling these degrees of freedom, like the orbital angular momentum or the transverse linear momentum, with the polarization of light (also associated with the spin) allows to implement quantum walks on one and two dimensional lattices. This work presents a series of experiments where these implementations of photonic quantum walks were realized exploiting patterned liquid crystal devices. In particular, the experimental setup allows to investigate interesting effects of the non-trivial topological features of the simulated processes, and test methods for the experimental measurement of topological invariants. This research has also led to the introduction of a novel method of characterizing unknown structured light beams and to the study of the stability of polarization singularities in laser beams.
Quantum walks, in virtue of the coherent superposition and quantum interference, have exponential superiority over their classical counterpart in applications of quantum searching and quantum simulation. The quantum-enhanced power is highly related to the state space of quantum walks, which can be expanded by enlarging the photon number and/or the dimensions of the evolution network, but the former is considerably challenging due to probabilistic generation of single photons and multiplicative loss. We demonstrate a two-dimensional continuous-time quantum walk by using the external geometry of photonic waveguide arrays, rather than the inner degree of freedoms of photons. Using femtosecond laser direct writing, we construct a large-scale three-dimensional structure that forms a two-dimensional lattice with up to 49 × 49 nodes on a photonic chip. We demonstrate spatial two-dimensional quantum walks using heralded single photons and single photon-level imaging. We analyze the quantum transport properties via observing the ballistic evolution pattern and the variance profile, which agree well with simulation results. We further reveal the transient nature that is the unique feature for quantum walks of beyond one dimension. An architecture that allows a quantum walk to freely evolve in all directions and at a large scale, combining with defect and disorder control, may bring up powerful and versatile quantum walk machines for classically intractable problems.
Quantum random walks are the quantum counterpart of classical random walks, and were recently studied in the context of quantum computation. Physical implementations of quantum walks have only been made in very small scale systems severely limited by decoherence. Here we show that the propagation of photons in waveguide lattices, which have been studied extensively in recent years, are essentially an implementation of quantum walks. Since waveguide lattices are easily constructed at large scales and display negligible decoherence, they can serve as an ideal and versatile experimental playground for the study of quantum walks and quantum algorithms. We experimentally observe quantum walks in large systems ( approximately 100 sites) and confirm quantum walks effects which were studied theoretically, including ballistic propagation, disorder, and boundary related effects.