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Physical Review A

Composite von Neumann lattice

Itzhack Dana

Published Oct 1, 1983• 3 citations
Publisher

Abstract

A new general subset of coherent states, which is complete and overcomplete by just one element, is introduced. It is the union of $N$ arbitrary nonintersecting von Neumann lattices whose unit-cell areas have a harmonic mean of $\mathrm{Nh}$. First applications are given in the problems of crystal electrons in rational magnetic fields, incommensurate structures in a magnetic field, and quantum dynamics.

DOI
10.1103/physreva.28.2594

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von Neumann lattice coherent statesによる量子化された電磁場の古典極限

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A new general subset of coherent states, which is complete and overcomplete by just one element, is introduced. It is the union of NNN arbitrary nonintersecting von Neumann lattices whose unit-cell areas have a harmonic mean of Nh\mathrm{Nh}Nh. First applications are given in the problems of crystal electrons in rational magnetic fields, incommensurate structures in a magnetic field, and quantum dynamics.
This paper reports a novel type of vortex lattice, referred to as a bubble crystal, which was discovered in rapidly rotating Bose gases with long-range interactions. Bubble crystals differ from vortex lattices which possess a single quantum flux per unit cell, while atoms in bubble crystals are clustered periodically and surrounded by vortices. No existing model is able to describe the vortex structure of bubble crystals; however, we identified a mathematical lattice, which is a subset of coherent states and exists periodically in the physical space. This lattice is called a von Neumann lattice, and when it possesses a single vortex per unit cell, it presents the same geometrical structure as an Abrikosov lattice. In this report, we extend the von Neumann lattice to one with an integral number of flux quanta per unit cell and demonstrate that von Neumann lattices well reproduce the translational properties of bubble crystals. Numerical simulations confirm that, as a generalized vortex, a von Neumann lattice can be physically realized using vortex lattices in rapidly rotating Bose gases with dipole interatomic interactions.
A mathematical lattice, called the von Neumann lattice, is a subset of coherent states and exists periodically in the phase space. It is unlike solids or Abrikosov lattices that are observable in physical systems. Abrikosov lattices are vortices closely packed into a lattice with a flux quantum through a unit cell. Although Abrikosov lattices appear generally in various physical systems, vortex lattices with multiple-flux quantums through a unit cell are more stable than Abrikosov lattices in some physical regimes of the systems with non-local interactions between particles. No theory is able to describe these vortex lattices today. Here, we develop a theory for these vortex lattices by extending von Neumann lattices to the coordinate space with a unit cell of area that is proportional to flux quantums through a unit cell. The von Neumann lattices not only show the same physical properties as the Abrikosov lattice, but also describe vortex lattices with multiple-flux quantums through a unit cell. From numerical simulations of a rapidly rotating dipole-blockaded gas, we confirm that vortex lattices showed in our simulations are the representation of von Neumann lattices in the coordinate space. We anticipate our theory to be a starting point for developing more sophisticated vortex-lattice models. For example, the effect of Landau-level mixing on vortex lattice structures, vortices formed inside superfluid droplets and structural phase transitions of vortex matter in two-component Bose-Einstein condensates will be relevant for such developments.
An averaging procedure in phase plane is developed leading to orthonormality of coherent states on a von Neumann lattice. These states correspond to entire unit cells of area h (Planck constant) in the phase plane and they can be specified by ( mb , n 2πℏ/ b ), where m and n are integers and b is a constant related to the spread of the harmonic oscillator ground state. The product of uncertainties for the co-ordinate x and momentum p in these states is very close to ℏ. This is by a factor of 2π smaller than the unit cell area, and makes it, in principle, possible to measure simultaneously x and p for the location of a unit cell in the phase plane.