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New Journal of Physics

Factorization with a logarithmic energy spectrum

F. Gleisberg, R. Mack, Karl Vogel, Wolfgang P. Schleich

Published Feb 25, 2013• 11 citations
Publisher

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.

DOI
10.1088/1367-2630/15/2/023037

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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.
We propose a method to factor numbers based on two interacting bosonic atoms in a central potential, where the single-particle spectrum depends logarithmically on the radial quantum numbers of states corresponding to zero angular momentum. Bosons initially prepared in the ground state are excited by a sinusoidally time-dependent interaction into a state characterized by quantum numbers, which represent the factors of the number encoded in the frequency of the perturbation. We also discuss the complete single-particle spectrum as well as the limitations of our method caused by decoherence.
Abstract We propose an iterative scheme to factor numbers based on the quantum dynamics of an ensemble of interacting bosonic atoms stored in a trap where the single-particle energy spectrum depends logarithmically on the quantum number. When excited by a time-dependent interaction these atoms perform Rabi oscillations between the ground state and an energy state characteristic of the factors. The number to be factored is encoded into the frequency of the sinusoidally modulated interaction. We show that a measurement of the energy of the atoms at a time chosen at random yields the factors with probability one half. We conclude by discussing a protocol to obtain the desired prime factors employing a logarithmic energy spectrum which consists of prime numbers only.
In this paper we study transitions of atoms between energy levels of several number-theory-inspired trapping potentials under the effect of time-dependent perturbations. First, we simulate in detail the case of a trap whose single-particle spectrum is given by the prime numbers. We investigate one-body Rabi oscillations and the excitation line shape for two resonantly coupled energy levels, and we show that quantum control is a faster method for state preparation than periodic perturbation. Next, we investigate cascades of such transitions, particularly whether one can construct a quantum system where the existence of a continuous resonant cascade from a given initial energy eigenstate is predicated by the validity of a given statement in number theory. We find that such resonance cascades, in a suitably designed one-body system, can be used to illustrate that the sequence of natural numbers is closed under multiplication. We further present ideas for two more resonance cascade experiments designed to illustrate certain statements within the Diophantus-Brahmagupta-Fibonacci identity and the Goldbach conjecture.
We investigate the quantum dynamics of two interacting bosonic atoms confined in a one-dimensional anharmonic trap. The tunneling rate, an experimentally measurable parameter of the system, is calculated as a function of the effective coupling interatomic constant ggg from the ground (n=N=0n=N=0n=N=0) and first excited atomic states in the trap with respect to relative (n=2,N=0n=2,N=0n=2,N=0) and center-of-mass (n=0,N=2n=0,N=2n=0,N=2) atomic motion. This allows us to investigate the initial population and pair correlation, as well as the effective coupling constant ggg, of the system by comparing the calculated tunneling rate with the experimental one. We observe that the only possible tunneling scenario is a sequential particle tunneling in the cases we consider. We also analyze a rearrangement $(0,2)\ensuremath{\rightleftarrows}(2,0)$ of the spectrum in the limit $g\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}0$ of noninteracting atoms.
We propose an approach to analytically solve the quantum dynamics of bosonic systems. The method is based on reconstructing the quantum state of the system from the moments of its annihilation operators, dynamics of which is solved in the Heisenberg picture. The proposed dynamical reconstruction method is general in the sense that it does not require assumptions on the initial conditions of the system such as separability, or the structure of the system such as linearity. It is an alternative to the standard master-equation approaches, which are analytically demanding especially for large multipartite quantum systems. To demonstrate the proposed technique, we apply it to a system consisting of two coupled damped quantum harmonic oscillators.
We propose a new approach for the study of the time evolution of a factorized N-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schrödinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the N-particle Schrödinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.