Eigenfunction Images of a Wave Chaotic Microwave Cavity

We all know that chaos occurs when a particle’s trajectory, or evolution in time, is extremely sensitive to its initial conditions. But what happens when the particle is a single electron and it is trapped in a very small box, such as a two-dimensional semiconductor quantum dot? The answer is the subject of a field called "quantum chaos". An even deeper problem is to consider what happens when a magnetic field is applied to the electron and the symmetry of time-reversal is lost. Is there any perceptible change in the chaotic properties of this particle? The answer is yes, according to a series of experiments carried out in the laboratory of Prof. Steven Anlage of the Physics Department at the University of Maryland at College Park. Anlage’s group uses a microwave cavity to simulate the physics of a single electron trapped in a small two-dimensional box. Theyhave shown that the allowed energy levels of the electron are affected by the magnetic field [P. So, et al., Phys. Rev. Lett. vol. 74, p. 2662 (1995)] by carefully examining the resonant frequencies of the microwave cavity. They also observed very strong changes in the distribution of probability density of the electron by mapping out the standing wave patterns of the microwave resonant modes in the cavity [D. H. Wu, et al., Phys. Rev. Lett. vol. 81, p. 2890 (1998)]. For example, the number of localized "hot spots" where the electron is very likely to be found is strongly suppressed by the magnetic field. Now they have shown that a continuum of time-reversal symmetry-broken states of the electron exist in the box, in very good agreement with theory [S.-H. Chung, et al., Phys. Rev. Lett., vol. 85, p. 2482 (2000)]. These results are of relevance to the physics of sub-micrometer semiconductor quantum dots, and to understanding electromagnetic compatibility issues forimproved electronic component reliability.Shown below are some example images from our wave chaotic cavity.

The quarter bow tie cavity used in the experiment has dimensions as given in the figure. The depth of the cavity is 0.310". There is a flat lid covering the cavity from the top. In this geometry, the system is 2D for frequencies below 18GHz. The two circles seen in the figure are the coupling port locations where the cavity is excited and the measurements are made.

 

The eigen images shown above are produced by scanning a perturbation inside the cavity. Red corresponds to low electric field magnitude and blue corresponds to high electric field magnitude. The features in the images show |E|2 inside the cavity.

The resonant frequency of the cavity corresponding to the eigen image is indicated on the top left side of each image.

Click for more images from our collection (in GHz):
1. 3.60
2. 5.30
3. 5.37

The eigen functions below are imaged when there is a ferrite in the cavity. The ferrite is 0.2" thick and 8.2" long and it is placed adjacent to the left wall. Since the images are produced by scanning a metal perturbation inside the cavity using a magnet and the ferrite inside the cavity is sensitive to magnetic field, the first 4" of the cavity was not imaged.

White spots in the images below are the areas in the cavity which can not be imaged due to very small eigen energy separation.

Linear patterns of strong localization of electric field is observed in many of the images.

 

Click for more images from our collection (in GHz):
1. 11.73, 11.80, 11.94
2. 12.50, 12.53, 12.57, 12.71,12.73, 12.94

We wish to acknowledge the NSF Division of Materials Research for supporting this research under NSF DMR-9123198, NSF DMR-9258183 (NYI), NSF DMR-9624021 and a sub-contract from NSF DMI-9560360. We acknowledge the NSF Division of Electrical and Communication Systems for supporting this research under NSF ECS 9632811

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