Six images, stitched together, of chaos emerging in the kicked rotor. (Credit: E. Rozenbaum and E. Edwards/JQI)
Weather forecasters have a notoriously difficult job as small changes that take place in temperature or air pressure, which eventually drives weather systems and winds, can have major consequences on a global level. This sensitivity to small changes is commonly known as the butterfly effect, which creates difficulty and chaos in predicting weather patterns.
Chaos also occurs in several other places, and for over a century scientists have analyzed its role in physics. Only from the 1980s have physicists examined the connections between quantum mechanics and chaos, considered to be the most basic theory on the building blocks of the universe.
Quantum physics itself seems to forbid chaotic behavior and this is indeed one wrinkle that prevails in studying quantum chaos. The rules by which the quantum world is governed are in fact extremely simple to give rise to a similar type of unpredictability as the weather.
This encouraged the researchers to closely study the differences between ordinary chaotic systems and their quantum counterparts, a task that has been slowed down as scientists do not have the mathematical tools that will help quantify chaos in a quantum setting.
A promising diagnostic tool has been recently used by researchers from the
Joint Quantum Institute (JQI) and the Condensed Matter Theory Center (CMTC) at the University of Maryland in order to characterize one of the simplest systems used by physicists to study chaos.
The emergence of quantum interference effects is tracked by this new diagnostic, which further shows that they ultimately destroy ordinary chaotic behavior. The Feb. 21 issue of
Physical Review Letters features the work carried out by JQI and CMTC graduate student Efim Rozenbaum and two collaborators.
The new diagnostic was applied by the researchers to a standard example of chaos that has been under study for decades. It is known as the kicked rotor, basically referring to a spinning rod, attached to its center, that gets flipped at standard intervals, similar to a coin that is made to spin on a tabletop.
For weak flicks, smooth trajectories are traced out by the rod’s orientation and rotation over time. The trajectories are not altered much by the slightly different initial orientations and speeds. However, the system becomes chaotic for strong flicks and minute changes lead to extremely different trajectories.
"The important question has been how this picture manifests itself, or whether it even exists, in quantum mechanics," says JQI Fellow Victor Galitski, a coauthor of the new study.
The Lyapunov exponent is one number that captures the level of chaos in the ordinary kicked rotor. It is bigger for stronger kicks, explaining how rapidly individual trajectories diverge. However, quantum physics does not have sharply defined trajectories due to a basic limit called the Heisenberg uncertainty principle. All quantum systems are infested by this uncertainty, which prevents a perfect knowledge of the initial orientation and rotation speed of the kicked rotor.
Galitski and his collaborators observed an intriguing quantity recently employed in another area of physics, the quantum behavior of black holes, that seemed to comprise of all the features required for a quantum analog of the Lyapunov exponent. These researchers are currently calculating this new quantity for the simplest possible system.
The quantum kicked rotor was studied by the team at differing levels of “quantumness”, thus permitting them to alter, numerically, the strength of the Heisenberg uncertainty principle. Uncertainty was not found in one limit corresponding to the ordinary kicked rotor. The other limit however corresponded to a fully quantum rotor.
The consistent and rapid manner in which chaos was destroyed by quantum effects is considered to be an immensely striking finding. A handful of kicks enabled quantum effects to win out and chaos to disappear even for a rotor that was only weakly quantum.
A new diagnostic tool, very much related to the quantum Lyapunov exponent, helped to capture this transition. The quantum Lyapunov exponent is expected to grow quickly before stopping the explosive growth suggestive of chaos. The researchers highlight that the time for this sharp cutoff corresponds to the time taken for the building up of quantum interference effects.
The team hopes to use this new diagnostic in various other settings, like developing connections between thermodynamics and the quantum phenomenon of many-body localization that was recently discovered.