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arXiv:0709.1986 · Physical Review A

Quantum phase transition using quantum walks in an optical lattice

C. M. Chandrashekar, Raymond Laflamme

Published Aug 11, 2008• 55 citations
arXivHTMLPDF

Abstract

We present an approach using quantum walks (QWs) to redistribute ultracold atoms in an optical lattice. Different density profiles of atoms can be obtained by exploiting the controllable properties of QWs, such as the variance and the probability distribution in position space using quantum coin parameters and engineered noise. The QW evolves the density profile of atoms in a superposition of position space, resulting in a quadratic speedup of the process of quantum phase transition. We also discuss implementation in presently available setups of ultracold atoms in optical lattices.

DOI
10.1103/physreva.78.022314

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We present an approach using quantum walks (QWs) to redistribute ultracold atoms in an optical lattice. Different density profiles of atoms can be obtained by exploiting the controllable properties of QWs, such as the variance and the probability distribution in position space using quantum coin parameters and engineered noise. The QW evolves the density profile of atoms in a superposition of position space, resulting in a quadratic speedup of the process of quantum phase transition. We also discuss implementation in presently available setups of ultracold atoms in optical lattices.
We present an approach to induce localization of a Bose-Einstein condensate in a one-dimensional lattice under the influence of unitary quantum-walk evolution using disordered quantum coin operation. We introduce a discrete-time quantum-walk model in which the interference effect is modified to diffuse or strongly localize the probability distribution of the particle by assigning a different set of coin parameters picked randomly for each step of the walk, respectively. Spatial localization of the particle or state is explained by comparing the variance of the probability distribution of the quantum walk in position space using disordered coin operation to that of the walk using an identical coin operation for each step. Due to the high degree of control over quantum coin operation and most of the system parameters, ultracold atoms in an optical lattice offer opportunities to implement a disordered quantum walk that is unitary and induces localization. Here we present a scheme to use a Bose-Einstein condensate that can be evolved to the superposition of its internal states in an optical lattice and control the dynamics of atoms to observe localization. This approach can be adopted to any other physical system in which controlled disordered quantum walk can be implemented.
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This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. \n \nIn the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. \n \nWe investigate symmetry operations and environmental effects on dynamics of the walk on a line and an n−n-n−cycle highlighting the interplay between noise and topology. \n \nUsing the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.
This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. In the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. We investigate symmetry operations and environmental effects on dynamics of the walk on a line and an n-cycle highlighting the interplay between noise and topology. Using the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.
We propose a different implementation of discrete-time quantum walks for a neutral atom in an array of optical microtraps or an optical lattice. We analyze a one-dimensional walk in position space, with the coin, the additional qubit degree of freedom that controls the displacement of the quantum walker, implemented as a spatially delocalized qubit, i.e., the coin is also encoded in position space. We analyze the dependence of the quantum walk on temperature and experimental imperfections such as shaking in the trap positions. Finally, combining a spatially delocalized qubit and a hyperfine qubit, we also give a scheme to realize a quantum walk on a two-dimensional square lattice with the possibility of implementing different coin operators.