Interacting particles in phase space

Start date: 01.01.2000
Funded by: Universität Augsburg, from 2000-2005 funded via the SFB484
Local head of project: Gert-Ludwig Ingold Peter Hänggi (until the end of 2005)
Publications: Publication list


Husimi functions are employed in this project to study localization transitions as well as entangled spin-½ states. As a positive definite phase-space distribution, the Husimi function allows to quantitatively describe how strongly a quantum state is localized in phase space. The inverse participation ratio is particularly well suited to perform a disorder average in the case of localization transitions in disordered systems. For entangled states, a relation with the length of the concurrence vector is found.


Localization transitions

Husimi function

The phase-space description is particularly well suited to describe a localization transition because it allows to simultaneously account for real space and momentum space properties of a quantum state. The transition occurs as function of a parameter, e.g. the potential strength, from a state extended in real space and localized in momentum space to a state localized in real space and extended in momentum space. As the Husimi function, for which an example is shown in the figure, is a positive definite phase-space distribution, it allows to study the transition by means of the Wehrl entropy (WKIH99).

It is numerically very costly to determine the Wehrl entropy. Therefore, an extension to two- and three-dimensional systems only becomes possible if one restricts oneself to the inverse participation ratio as shown in WIHW02. An introduction to this topic is given in IWAH03.

Further insight into the mechanism of the Anderson transition in disordered systems as a function of the spatial dimension can be obtained by analyzing the Aubry-André model. For this one-dimensional model, a localization transition occurs as a function of the strength of a quasiperiodic potential. Compared with the Anderson transition in one dimension, a completely different phase-space scenario results. While the phase-space distribution contracts at the one-dimensional Anderson transition, for the Aubry-André model it first expands when approaching the transition with increasing potential strength before localizing when moving through the critical potential strength (IWAH02). The difference between the two models therefore lies in the delocalized regime. Its origin is the different coupling of momentum eigenstates in the two one-dimensional models.

A detailed numerical study of the Anderson model in one, two, and three dimensions together with perturbative results was published in WIHW03. It turns out that the inverse participation ratio for the Anderson model in two and three dimensions display the same behavior as function of potential strength as the one-dimensional Aubry-André model. However, in two dimensions, the Anderson transition with increasing system size shifts towards vanishing potential strength as expected. In AWIHV04, the Anderson and the Aubry-André model are confronted. The phase space portrait of selected states is shown in animations as a function of the potential strength. These animations are freely available as Supplementary Data of the original publication.

Entangled states

Phase space representation of a Bell state

While the Husimi function employed in the analysis of localization transitions is defined on the basis of the coherent states of the harmonic oscillator, for the study of entangled states of spin-½ particles spin-coherent states on the Bloch sphere are used. The figure depicts such a Husimi function for a Bell state in which the product states |01> and |10> are superposed. The bridge in the middle is a consequence of the phase-coherent superposition and shifts if the two product states are superposed with another relative phase.

Already this figure indicates that the entanglement influences the distribution in phase space and therefore the inverse participation ratio. In fact, for pure states of an arbitrary number of spin-½ particles one can relate it to the length of the concurrence vector (SI07). Furthermore, this phase-space measure can be used to study the entanglement at a phase transition as was demonstrated for the Ising model in a tilted magnetic field. The graphical representation of the Husimi function for more than two particles is difficult due to the high dimensionality of phase space. Despite the need to resort to cross sections it was demonstrated in Ingold07 that the phase-space structures of GHZ- and W-states are fundamentally different in agreement with the different kind of entanglement found in these states.