A new research study by SISSA, performed in partnership with the University of St Andrews and the University of Trieste, proposes an advanced technique that could help provide physics solutions using a kind of “quantum abacus.”
St Andrews University Laboratory. Image Credit: St Andrews University
By integrating theoretical and experimental studies, researchers could replicate a quantum potential with energy levels matching the first 15 prime numbers and the first 10 lucky numbers using holographic laser methods.
This result, reported in PNAS Nexus, paves the way to obtaining potentials with finite sequences of integers as arbitrary quantum energies, and to resolving mathematical questions associated with number theory with quantum mechanical experiments.
Every physical system is characterized by a certain set of energy levels, which basically make up its ID. In this work, we have reversed this line of reasoning: is it possible—starting from an arithmetic sequence, for example, that of prime numbers—to obtain a quantum system with those very numbers as energy levels?
Giuseppe Mussardo, Theoretical Physicist, International School for Advanced Studies, SISSA
The scientists applied a system of differential equations to calculate the potential appearance in the Schrödinger equation of a quantum system, whose energy states match a finite sequence of integers, removed from a specific series, such as prime numbers or close lucky numbers.
Using advanced holographic experimental methods, they could then develop light traps with intensity profiles matching the theoretically defined potential.
So far, we have calculated and created in the laboratory the quantum potential based on the first 15 prime numbers and the first 10 lucky numbers, but the same approach could be applied, in principle, to other—even infinitely long—number sequences, as long as the numbers don't grow too fast. More precisely, the growth of the infinite sequence needs to be less than N squared.
Giuseppe Mussardo, Theoretical Physicist, International School for Advanced Studies, SISSA
The experimental portion of the study needed the use of multifaceted holographic methods and was carried out by Donatella Cassettari’s team at the University of St Andrews.
“The realization of these quantum potentials is a stringent test of the holographic techniques that have recently been developed by my team,” said the scientist. “This study shows that complex intensity profiles can be realized with good accuracy. Many interesting directions can be taken based on these results, for example potentials with a greater number of energy levels. In the future, testing the limit of these techniques will also be interesting.”
“Our work shows the feasibility of this method and paves the way for the exploration of genuine mathematical issues and arithmetical manipulations through quantum mechanics experiments: a kind of ‘quantum abacus’,” observes Andrea Trombettoni, a physicist at the University of Trieste.
“The method used and its possible generalizations may also lead to new applications in the field of quantum technology, which is currently the focus of much work and interest. Indeed, it provides a possible basic ingredient for a new class of devices, in which new algorithms that are relevant to quantum technologies—particularly quantum computing—can be designed and tested,” concluded the scientist.
Journal Reference
Cassettari, D., et al. (2023) Holographic realization of the prime number quantum potential. PNAS Nexus. doi.org/10.1093/pnasnexus/pgac279.