Factorization with a logarithmic energy spectrum

Abstract

We propose a method to factor numbers based on the quantum dynamics of two interacting bosonic atoms where the single-particle energy spectrum depends logarithmically on the quantum number. We show that two atoms initially prepared in the ground state are preferentially excited by a time-dependent interaction into a two-particle energy state characterized by the factors. Hence, a measurement of the energy of one of the two atoms yields the factors. The number to be factored is encoded in the frequency of a sinusoidally modulated interaction. We also discuss the influence of off-resonant transitions and the limitation of the number to be factored imposed by experimental conditions.

Dirichlet series as interfering probability amplitudes for quantum measurements

“…In this appendix we briefly outline two ideas of how to realize an anharmonic oscillator with a logarithmic energy spectrum. The first one relies on an atom moving in an appropriately tailored potential and has been discussed in great detail in [15] and the second one employs the analogy between the time-independent Schrödinger equation and the Helmholtz equation of classical electrodynamics. In both cases we do not go into details but focus on the key features.…”

“…with eigenvalue E. In [15] we have constructed a potential such that the resulting energy spectrum is of the form (3). For this purpose we have solved the inverse spectrum problem using semi-classical approximations as well as exact numerical techniques based on the Hellmann-Feynman theorem.…”

“…Hence, an atom moving in a potential provided by a far-detuned light field with a spatial distribution found from the inversion problem leads us via the time-independent Schrödinger equation (A.1) to the logarithmic energy spectrum (3). For a more detailed discussion, and in particular, the question of how to realize a one-dimensional motion we refer to [15].…”

“…where ω  is a unit of energy, and γ and α are dimensionless parameters. For a suggestion of how to realize such an unusual oscillator we refer to [15] and to appendix A.…”

“…Like in the case of a single Dirichlet sum d ( , ) σ τ the dependence of D ( , ) σ τ on the imaginary part τ is again solely governed by the time evolution. The normalization determined by the requirement (13) and the probability amplitudes in (15)…”