Experimental verification of topologically induced vortices inside a billiard

Petr Šeba · Ulrich Kuhl · Michael Barth · Hans-Jürgen Stöckmann

Topological defects in the structure of the quantum phase are responsible for many observable phenomena known mainly in macroscopic quantum systems like superconductors or superfluid helium. But the phase defects manifest itself also in the ordinary quantum mechanics. The fact that quantum probability current may exhibit vortices centered at the singular points of the quantum phase has been pointed out already by [Dirac 1931].

In the case of two dimensional quantum systems it is possible [Berggren, Ji 1993; Exner et al. 1998] to identify the configuration space with a region in the complex plane. In such a way the topology of the related quantum phase can be described with the help of a Riemannian surface. To be more explicit, let us write the probability current density as

(1)

and the wavefunction as

(2)

where ρ = |ψ|² is the probability density and S denotes the phase. It is clear that the phase S cannot be defined on the nodal points of the wavefunction ψ, which represent singularities of S. The nontrivial topological structure of the phase leads to the appearance of vortices in the related probability current. To see this we insert (2) into (1) and get

(3)

The current

can be understood as a flow of the probability density ρ with a velocity

. The vorticity of

evaluated along a closed curve Γ gives [Hirschfelder 1977]

(4)

where δS is the phase change when winding once around the curve.

If the phase S does not have a branching point inside Γ, the difference equals to zero and the vorticity nullifies. If, on the other hand, Γ encircles a branching point of S, the vorticity of the velocity field is not zero. In this situation the corresponding probability current exhibits a vortex centered at the nodal point.

There is an experimental possibility to verify the existence of the vortices which is based on the fact that the electromagnetic field inside flat electromagnetic resonators fulfills the same equations of motion as the quantum particle.
The transport of electromagnetic energy through the resonator is described by the Poynting vector

(5)

For its stationary part we get in quasi-twodimensional resonators:

(6)

This is in complete analogy with eq. (3), where the stationary part of the Poynting vector replaces the probability current density, and the z component of the electric field strength replaces the wave function ψ.

**Figure 1:** a) Transmission *T_ant = |t_ant|²* from the entrance antenna to the probing antenna.
b) Transmission *T_abs* from the entrance antenna to the absorber.

Our experiment allows a direct measurement of $E_z(\vec{r})$ as a function of position, including its phase, and to construct the related Riemannian surface.
The cavity has a shape of a rectangle. The electromagnetic energy has been transported from an entrance antenna to a circular absorber (see Fig.2). The position of the upper plate supporting the probing antenna is varied. The transmission amplitude t_ant between entrance and probing antenna has been measured, including the phase, by means of a vector network analyzer.
In Fig. 1a) the transmission T_ant = |t_ant|² is shown for an arbitrary position of the probing antenna. From the measured electric field strength the energy flow is determined by means of eq. (6).

**Figure 2:** Flow of the electromagnetic energy from the entrance antenna (small circle) to the absorber (large circle) at the frequency ν = 3.7 GHz (marked by dashed line in Fig. 1). At the solid lines the wavefunction has aquired a phase of 2π upon encircling a vortex. These lines correspond to cuts in the Riemannian surfaces (see Fig. 3).

For certain frequencies there is an enhanced transport of the electromagnetic energy from the antenna to the absorber. The transmission from the entrance antenna to the absorber is plotted in Fig. 1b) as a function of frequency. Fig. 2 shows the Poynting vector in the resonator for the frequency marked by a dashed line in Fig. 1. The vortices are clearly visible. We have evaluated also the corresponding phase of the related electromagnetic field and plotted its cuts into the same plot (thick lines). One can see - in full agreement with the theory - that the cuts start at the center of the vortices. This supports the theoretical findings that a vortex is winding around a branching point of the related Riemannian surface of the phase. All observed vortices have the vortex number ±1. Vortices with higher vortex numbers n have not been observed in the experiment. The phase of the field is plotted in Fig. 3. The positions of the entrance antenna and the absorber are marked by inserted cylinders. The complicated structure of the corresponding Riemannian surface together with the branching points and the cuts is clearly visible. Similar pictures can be plotted also for other frequencies.

**Figure 3:** Three-dimensional view of the phase of the flow pattern shown in Fig.2.

Fig. 1b) shows that there are a number of frequencies where the flow to the absorber is close to zero. But this means that at these frequencies there must be a node line at the surface of the absorber, or, in other words, these frequencies must correspond to eigenfrequencies of the Sinai billiard obtained by replacing the absorber surface by a reflecting wall. This interpretation is corroborated by Fig 4a), showing a map of |E|² in a gray scale at a frequency corresponding to the dotted line in Fig. 1. The figure exhibits clearly a bouncing ball eigenfunction of the Sinai billiard. Fig 4b) shows the corresponding flow pattern. Since at this frequency the flow to the absorber is suppressed, the otherwise negligible leak to the probing antenna becomes dominant (the absorption in the walls is too small to become observable in the flow pattern). The leak flow is maximized at the points of large field strengths as expected. The same findings can be obtained from a comparison of Figs. 1a) and 1b): the relative transmission from the entrance to the probing antenna shows a maximum whenever the transmission to the absorber exhibits a minimum.

**Figure 4:** *top*) Gray scale plot of *|E|²* at the frequency ν = 3.8 GHz (marked by a dotted line in Fig. 1). At this frequency there is no flow to the absorber.
*bottom*) Flow pattern at the same frequency. At the points of large field strength a maximal flow to the probing antenna is observed.

The rectangular shape of the resonator and the circular shape of the absorber have been chosen due to technical reasons. We expect that similar phenomena will be observed also in different configurations, provided there is a flow of the electromagnetic energy inside the system and the system is not integrable (transport through ideal integrable waveguide is not accompanied with phase defects). It has to be stressed however that the existence of phase defects and related vortices is of pure wave origin and is not related to the underlying classical dynamics. In such a way it cannot be seen a signature of "quantum chaos".

Literature:

P.A.M. Dirac, Proc. R. Soc. Lond. A 133, 60 (1931)
J.O. Hirschfelder, J. Chem. Phys. 67, 5477 (1977)
K.-F. Berggren, Z.-L. Ji, Phys. Rev. B 47, 6390 (1993)
P. Exner et al., Phys. Rev. Lett. 80, 1710 (1998)
P. Šeba, U. Kuhl, M. Barth, H.-J. Stöckmann, J. Phys. A 32, 8225 (1999)
M. Barth, Dissertation (2001)