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This "accordion lattice" was able to vary the lattice periodicity from 1.30 to 9.3 μm. More recently, a different method of real-time control of the lattice periodicity was demonstrated, [9] in which the center fringe moved less than 2.7 μm while the lattice periodicity was changed from 0.96 to 11.2 μm. Keeping atoms (or other particles ...
When talking about solid materials, the discussion is mainly around crystals – periodic lattices. Here we will discuss a 1D lattice of positive ions. Assuming the spacing between two ions is a, the potential in the lattice will look something like this: The mathematical representation of the potential is a periodic function with a period a.
Here is the amplitude of the variation of the onsite energies, is a relative phase, and is the period of the onsite potential modulation in units of the lattice constant. This Hamiltonian is self-dual as it retains the same form after a Fourier transformation interchanging the roles of position and momentum.
The schematic structure of a periodic superlattice is shown below, where A and B are two semiconductor materials of respective layer thickness a and b (period: = +). When a and b are not too small compared with the interatomic spacing, an adequate approximation is obtained by replacing these fast varying potentials by an effective potential ...
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of natural crystals gives rise to X-ray diffraction and that the atomic lattices (crystal structure) of semiconductors affect their conductivity of electrons.
The phase modulation of the Bloch state = is the same as that of a free particle with momentum , i.e. gives the state's periodicity, which is not the same as that of the lattice. This modulation contributes to the kinetic energy of the particle (whereas the modulation is entirely responsible for the kinetic energy of a free particle).
The angles that Bragg's law predicts are still approximately right, but in general there is a lattice of spots which are close to projections of the reciprocal lattice that is at right angles to the direction of the electron beam. (In contrast, Bragg's law predicts that only one or perhaps two would be present, not simultaneously tens to hundreds.)
[14] [5] The biggest downside with optical lattice technology is the trap lifetime, with atoms typically trapped for only a few tens of seconds. To see why ultracold atoms offer such a convenient realization of Bose–Hubbard physics, the Bose–Hubbard Hamiltonian can be derived starting from the second quantized Hamiltonian that describes a ...