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Physical Review A

Quantum carpets and correlated dynamics of several fermions

Mathias Nest

Published Feb 17, 2006• 30 citations
Publisher

Abstract

We study theoretically the coherent and correlated motion of many fermions inside an infinite square well potential. We will look at electrons and $^{3}\mathrm{He}$ atoms, which behave very differently not only because of their masses, but also because of their different interaction potential. Also, the level of theory and the role of approximations in the solution of the time-dependent Schr\"odinger equation will be discussed.

DOI
10.1103/physreva.73.023613

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We study theoretically the coherent and correlated motion of many fermions inside an infinite square well potential. We will look at electrons and 3He^{3}\mathrm{He}3He atoms, which behave very differently not only because of their masses, but also because of their different interaction potential. Also, the level of theory and the role of approximations in the solution of the time-dependent Schr"odinger equation will be discussed.
We investigate wavepacket dynamics for a relativistic particle in a box evolving according to the relativistic Schrödinger (also known as the Salpeter) equation. We derive the solutions for an infinite well -- which contrary to the standard relativistic wave equations (such as the Klein-Gordon or Dirac equations) -- are well defined, and use these solutions to construct wavepackets. We obtain expressions for the wavepacket revival times and explore the corresponding quantum carpets (the space-time probability density plots) for different dynamical regimes. We further analyze level spacing statistics as the dynamics goes from the non-relativistic regime to the ultra-relativistic limit.
A semianalytical approach to the quantum dynamics of a plane pendulum is developed, based on Mathieu functions which appear as stationary wave functions. The time-dependent Schr\"odinger equation is solved for pendular analogs of coherent and squeezed states of a harmonic oscillator, induced by instantaneous changes of the periodic potential energy function. Coherent pendular states are discussed between the harmonic limit for small displacements and the inverted pendulum limit, while squeezed pendular states are shown to interpolate between vibrational and free rotational motion. In the latter case, full and fractional revivals as well as spatiotemporal structures in the time evolution of the probability densities (quantum carpets) are quantitatively analyzed. Corresponding expressions for the mean orientation are derived in terms of Mathieu functions in time. For periodic double well potentials, different revival schemes, and different quantum carpets are found for the even and odd initial states forming the ground tunneling doublet. Time evolution of the mean alignment allows the separation of states with different parity. Implications for external (rotational) and internal (torsional) motion of molecules induced by intense laser fields are discussed.
Quantum transport of strongly correlated fermions is of central interest in condensed matter physics. While the stationary expansion dynamics have recently been measured with cold atoms in 2D optical lattices, ab initio simulations have been limited to 1D setups so far. Here, we present the first precise fermionic quantum dynamics simulations for 2D and 3D. The simulations are based on nonequilibrium Green functions and incorporate strong correlations via TTT-matrix self-energies. The simulations predict the short-time dynamics, and we discover a universal scaling of the expansion velocity with the particle number. Our predictions can be verified experimentally using the recently developed fermionic atom microscopes.
Fermionic quantum systems are well described by the linear many-body Schrödinger equation. For interacting systems the full Schrödinger theory is extremely complicated and theoretical as well numerical investigations are not feasible. In practice, macroscopic properties of large systems can therefore only be accessed by means of approximate theories. The intention of this workshop was to showcase the most recent advances in the mathematical study of many-body interacting fermionic systems and to stimulate discussions among different research groups.