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Journal of Physics B Atomic Molecular and Optical Physics

Bloch–Zener quantum walk

Stefano Longhi

Published Nov 8, 2012• 10 citations
Publisher

Abstract

Coherent transport of quantum particles in binary superlattices driven by an ac force can realize a kind of quantum walk (QW) that can be referred to as Bloch–Zener QW. In this regime, the particle wavepacket undergoes a sequence of Zener tunnelling events that mimic a discrete QW on a lattice, provided that the impulse of the ac force over one semi-cycle of oscillation is properly adjusted. In the Bloch–Zener QW, the coin states are related to the particle occupation of the two minibands, and thus it does not require any internal degrees of freedom of the particle.

DOI
10.1088/0953-4075/45/22/225504

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Coherent transport of quantum particles in binary superlattices driven by an ac force can realize a kind of quantum walk (QW) that can be referred to as Bloch–Zener QW. In this regime, the particle wavepacket undergoes a sequence of Zener tunnelling events that mimic a discrete QW on a lattice, provided that the impulse of the ac force over one semi-cycle of oscillation is properly adjusted. In the Bloch–Zener QW, the coin states are related to the particle occupation of the two minibands, and thus it does not require any internal degrees of freedom of the particle.
It is well known that an electron in a single-band tight-binding lattice coherently driven by an external dc electric field undergoes Bloch oscillations. In the framework of the extended Hubbard model, here it is shown theoretically that two strongly interacting electrons with on-site and nearest-neighbor site repulsion undergo a sequence of Bloch oscillations and Zener tunneling, the so-called Bloch-Zener oscillations (BZOs), among different narrow bands formed by bound particle states. Such a kind of BZOs arise because of nearest-neighbor site particle interaction and are thus rather distinct than BZOs observed for a single electron in biperiodic lattices supporting two single-particle minibands.
It is well known that a particle in a periodic potential with an additional constant force performs Bloch oscillations. Modulating every second period of the potential, the original Bloch band splits into two sub-bands. The dynamics of quantum particles shows a coherent superposition of Bloch oscillations and Zener tunnelling between the sub-bands, a Bloch–Zener oscillation. Such a system is modelled by a tight-binding Hamiltonian which is a system of two minibands with an easily controllable gap. The dynamics of the system is investigated by using an algebraic ansatz leading to a differential equation of Whittaker–Hill type. It is shown that the parameters of the system can be tuned to generate a periodic reconstruction of the wave packet and thus of the occupation probability. As an application, the construction of a matter wave beam splitter and a Mach–Zehnder interferometer is briefly discussed.
A scheme for directed transport of quantum particles in a static bipartite lattice driven by an ac force, based on a sequence of Landau-Zener transitions, is proposed theoretically. As compared to directed transport in bipartite lattices exploiting the phenomenon of coherent destruction of tunneling recently suggested by Creffield [Phys. Rev. Lett. 99, 110501 (2007)], the present scheme realizes a robust quantum state transport owing to the adiabaticity of the particle transfer. The transport is realized by an external force with a sawtooth profile and does not require modulation of lattice parameters (intersite couplings and energy levels). The effects of lattice disorder and particle interaction on the directed quantum transport are discussed.
It is well known that a particle in a periodic potential with an additional constant force performs Bloch oscillations. Modulating every second period of the potential, the original Bloch band splits into two subbands. The dynamics of quantum particles shows a coherent superposition of Bloch oscillations and Zener tunneling between the subbands, a Bloch-Zener oscillation. Such a system is modelled by a tight-binding Hamiltonian, a system of two minibands with an easily controllable gap. The dynamics of the system is investigated by using an algebraic ansatz leading to a differential equation of Whittaker-Hill type. It is shown that the parameters of the system can be tuned to generate a periodic reconstruction of the wave packet and thus of the occupation probability. As an application, the construction of a matter wave beam splitter and a Mach-Zehnder interferometer is briefly discussed. PACS numbers: 03.65.-w, 03.75.Be, 03.75.LmBloch-Zener oscillations 2 1.
A quantum theory of optical Bloch oscillations and Zener tunneling (ZT) in arrays of coupled waveguides is theoretically presented, and the particlelike behavior of photons undergoing ZT is highlighted. In singly-periodic arrays excited by a photon-number-state input beam, each photon behaves as a classical particle which independently undergoes a coin-toss ZT event with a probability described by classical Zener theory. In binary arrays, excitation with two tilted beams enables us to observe the Hong-Ou-Mandel interference for two photons undergoing Bloch-Zener oscillations.