Holographic Realization of the Prime Number Quantum Potential

We report the first experimental realization of the prime number quantum potential $V_N(x)$, defined as the potential entering the single-particle Schr\"{o}dinger Hamiltonian with eigenvalues given by the first $N$ prime numbers. We use holographic optical traps and, in particular, a spatial light modulator to tailor the potential to the desired shape. As a further application, we also implement a potential with lucky numbers, a sequence of integers generated by a different sieve than the familiar Eratosthenes's sieve used for the primes. Our results pave the way towards the realization of quantum potentials with arbitrary sequences of integers as energy levels and show, in perspective, the possibility to set up quantum systems for arithmetic manipulations or mathematical tests involving prime numbers.