Nano-Sized Drums Imply There's More to Know in Arena of Math, Physics, and Microelectronics

February 27, 2008

In an April 1966 article in the American Mathematical Monthly, Rockefeller University mathematician Mark Kac posed the question, "Can one hear the shape of a drum?"

If drums produce unique sound spectrums, then it might be possible to identify their shapes by identifying their spectrums--a procedure analogous to spectroscopy. But in the 1990s mathematicians proved that drums of different shapes could produce identical sounds.

The idea that you can't hear the shape of a drum-an outcome verified, for instance, with vibrations on the surface of soap bubbles-"Revolutionized our conception of the fundamental connections between shape and sound," said physicist Hari Manoharan (Stanford University). It "also had profound implications for spectroscopy in general, because it introduced an ambiguity."

Undeterred, Manoharan and his students investigated the drum question in the realm of quantum physics, where a breakthrough could have an effect on real nano-electronic systems. They discovered that, just as in the normal world, two nanostructures with different shapes can resonate in the same way, a phenomenon known as isospectrality.

The value of their discovery implies the creation of smaller computer chips. In fact, "Now your design palette is twice as big," Manoharan noted. In addition, their research has implications for string theory. "There is somehow embedded into the topology of our universe this bizarre spectral ambiguity," said Manoharan.

His team also devised a way to take measurements from two isospectral drums and then mathematically combine the information, via a process called quantum transplantation. "We discovered that this extra degree of freedom in geometry provides us with a method to 'cheat' quantum mechanics and obtain normally obscured quantum-mechanical phase information," Manoharan said.

Other ways to experimentally determine quantum phase information from atoms or molecules in gases, or from quantum dots and rings, rely on a process called interferometry. “The addition of a new method, ‘geometry over interferometry,’ ought to benefit researchers,” Manoharan said.

The team's paper "Quantum Phase Extraction in Isospectral Electronic Nanostructures," appeared in the journal Science (2/08/08). The researchers also created a video with two quantum drums beating with the same sound to complement their article.

Source: Stanford News