Abstract
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.
Highlights
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT where |n is the Wannier state localized in the nth potential well
We present an algebraic ansatz for the time evolution operator and analyse the time evolution by general arguments
The possibility of constructing a Mach–Zehnder interferometer based on Bloch–Zener oscillations is briefly discussed
Summary
We discuss the spectrum of the period-doubled Hamiltonian (1) for the field-free case F = 0, i.e. Bloch bands and Bloch waves. A straightforward calculation yields the dispersion relation. With the miniband index β = 0, 1 and γ = sgn(δ), which is illustrated in figure 1. For δ = 0, the Bloch band splits into two minibands with band gap δ. In the Bloch basis, the time-independent Schrödinger equation with Hamiltonian (1) reads. Where the coupling of the two bands is given by the transition matrix element χ1,κ |H ZB|χ0,κ. The modulus of Mκ is shown in figure 2. The band transitions mainly take place at the edge of the Brillouin zone, and because of the delta-function in (8), only direct interband transitions are possible
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