The End of the Quantum Tunnel: Precise Instanton Level Crossing in Quantum Mechanics

In the quantum world, processes can be divided into two different categories. A class of so-called “perturbation” phenomena are relatively easy to detect in experiments and mathematical calculations. Examples are many: the light emitted by atoms, the energy produced by solar cells, the state of qubits in quantum computers.

These quantum phenomena depend on Planck’s constant, a fundamental constant of nature that determines how the quantum world differs from our large-scale world, but in a simple way. Although this constant is ridiculously small—its values, expressed in everyday units like kilograms, meters, and seconds, start at the 34th decimal place after the comma—the fact that Planck’s constant is not exactly zero is enough to calculate this Quantum effects.

Then, there are the “non-perturbative” phenomena. The most famous of these is radioactive decay: a process in which, thanks to quantum effects, elementary particles can escape the attractive forces that bind them to atomic nuclei. If the world were “classical”—that is, if Planck’s constant was exactly zero—then this attraction would be impossible to overcome.

In the quantum world, decay does happen, but only occasionally. For example, it takes an average of four billion years for a uranium atom to decay. The collective name for this rare quantum event is “tunneling”: in order for a particle to escape, it must “dig a tunnel” through the energy barrier connecting it to the nucleus. A tunnel that would take billions of years to dig makes “The Shawshank Redemption” look like child’s play.

Math to the rescue

Mathematically speaking, non-perturbative quantum effects are more difficult to describe than perturbative quantum effects. Still, in the century that quantum mechanics has been around, physicists have found many ways to deal with these effects and to accurately describe and predict them.

“Nonetheless, there is still much work to be done on this century-old problem,” says Alexander van Spaendonck, one of the authors of the new publication. “The description of tunneling phenomena in quantum mechanics needs further unification – —a framework within which all such phenomena can be described and studied using a single mathematical structure.”

Surprisingly, such a structure was discovered in mathematics 40 years ago. In the 1980s, the French mathematician Jean Écalle established what he called a renaissance framework with exactly the goal of providing structure to non-perturbative phenomena.

So why did it take 40 years for the natural combination of Écalle’s formalism and the application of tunneling phenomena to reach its logical conclusion?

Marcel Vonk, another author of the publication, explains: “Écalle’s original papers were long (more than 1000 pages in total), highly technical, and published only in French. Therefore, it was not until the mid-2000s that a large number of papers were published. Physics Scientists are becoming familiar with this revival’s “toolbox.”

“Initially, it was mainly applied to simple ‘toy models’, but of course the tools were also tried on real-life quantum mechanics. Our work takes these developments to their logical conclusion.”

Beautiful structure

The conclusion is that one of the tools in the Écalle toolbox, the “cross-series”, is very suitable for describing tunneling phenomena in basically any quantum mechanical problem, and always in the same way. By illuminating the mathematical details, the authors discovered that not only could all tunneling phenomena be unified into a single mathematical object, but also that certain “jumps” in how powerful these phenomena could be described could be described – an effect known as the Stokes effect’ Phenomenon.

Van Spaendonck shares: “Through our description of the Stokes phenomenon, we were able to show that some of the ambiguities that plague the ‘classical’ approach to calculating non-perturbative effects – infinitesimal, in fact – are in our All disappeared in the method.

“The cross-series results describing quantum tunneling split – or ‘factored’ – in a surprising way: into a ‘minimal’ cross-series that describes the fundamental tunneling that exists in essentially any quantum mechanical problem. phenomenon, and one we call ‘median cross-series’ describes more specific problem details, depending for example on the degree of symmetry of a certain quantum setup.

When this mathematical structure is fully figured out, the next question is of course where the new lessons can be applied and what physicists can learn from them. For example, in the case of radioactivity, some atoms are stable while others decay. In other physics models, the list of stable and unstable particles may change with slight changes in settings, a phenomenon known as “wall penetration.”

The researchers’ next idea is to use the same technique to illustrate the concept of wall penetration. Once again, many groups are working on this conundrum in many different ways, but now a similar unified structure may be just around the corner. There is definitely light at the end of the tunnel.

Works published in journals science post physics.