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Phonon Random Walks: Using Classical Chains to Simulate Quantum Dynamics

Daxing Xiong, Eli Barkai

Published Jun 15, 2016• 0 citations
OpenAlex

Abstract

Inspired by quantum walks (QW), we propose the concept of classical phonon random walks (PRW). With this we show a correspondence between classical and quantum dynamics. To be specific, we explain how a classical chain of particles with linear interactions can be used to simulate a quantum like wave function $\psi_{m}(t)$. In particular, the real (imaginary) part of $\psi_{m}(t)$ is the normalized classical momentum (a new $\pi/2$ shifted momentum) correlation function of the thermalized many-body system. Energy and heat densities are directly analogous to wave function's modulus squared $\left|\psi_{m}(t)\right|^{2}$. We also propose a formula predicting the non-universal spreading of these energy packets in terms of the phonon dispersion of the system. Our equilibrium molecular dynamics simulations of several linear chains perfectly verify these relations. The extension of the concept to integrable Toda chain is also discussed.

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Inspired by quantum walks (QW), we propose the concept of classical phonon random walks (PRW). With this we show a correspondence between classical and quantum dynamics. To be specific, we explain how a classical chain of particles with linear interactions can be used to simulate a quantum like wave function ψm(t)\psi_{m}(t)ψm​(t). In particular, the real (imaginary) part of ψm(t)\psi_{m}(t)ψm​(t) is the normalized classical momentum (a new π/2\pi/2π/2 shifted momentum) correlation function of the thermalized many-body system. Energy and heat densities are directly analogous to wave function's modulus squared ∣ψm(t)∣2\left|\psi_{m}(t)\right|^{2}∣ψm​(t)∣2. We also propose a formula predicting the non-universal spreading of these energy packets in terms of the phonon dispersion of the system. Our equilibrium molecular dynamics simulations of several linear chains perfectly verify these relations. The extension of the concept to integrable Toda chain is also discussed.
We propose a simulation strategy which uses a classical device of linearly coupled chain of springs to simulate quantum dynamics, in particular the quantum walks. Through this strategy, we obtain the quantum wave function from classical evolution. Specially, this goal is achieved with the classical momenta of the particles on the chain and their Hilbert transform, from which we construct the many-body momentum and Hilbert transformed momentum pair correlation functions yielding the real and imaginary parts of the wave function, respectively. With such wave function, we show that the classical chain's energy and heat spreading densities can be related to the wave function's modulus square. This relation indicates a concept of phonon random walks, and thus it provides a new perspective to understand ballistic heat transport. The results here may give a definite answer to Feynman's idea of using a classical device to simulate quantum physics.
We propose a simulation strategy which uses a classical device of linearly coupled chain of springs to simulate quantum dynamics, in particular quantum walks. Through this strategy, we obtain the quantum wave function from the classical evolution. Specially, this goal is achieved with the classical momenta of the particles on the chain and their Hilbert transform, from which we construct the many-body momentum and Hilbert transformed momentum pair correlation functions yielding the real and imaginary parts of the wave function, respectively. With such a wave function, we show that the classical chain's energy and heat spreading densities can be related to the wave function's modulus square. This relation provides a new perspective to understand ballistic heat transport. The results here may give a definite answer to Feynman's idea of using a classical device to simulate quantum physics.
We have investigated the time evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson’s boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann’s entropy and quantum purity. In addition, we have studied Wigner’s function. The timedependent behavior of the quantum entanglement between a free particle—in the lattice—and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.
We have investigated the time evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson's boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann's entropy and quantum purity. In addition, we have studied Wigner's function. The time-dependent behavior of the quantum entanglement between a free particle—in the lattice—and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.
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.
We investigate quantum walks in multiple dimensions with different quantum coins. We augment the model by assuming that at each step the amplitudes of the coin state are multiplied by random phases. This model enables us to study in detail the role of decoherence in quantum walks and to investigate the quantum-to-classical transition. We also provide classical analog of the quantum random walks studied. Interestingly enough, it turns out that the classical counterparts of some quantum random walks are classical random walks with a memory and biased coin. In addition random phase shifts ``simplify'' the dynamics (the cross-interference terms of different paths vanish on average) and enable us to give a compact formula for the dispersion of such walks.