Riemann zeros from Floquet engineering a trapped-ion qubit

Ran He,Guang-Can Guo,Yun-Feng Huang,C E Creffield,Jin-Ming Cui,Ming-Zhong Ai,Yong-Jian Han,G Sierra,Chuan-Feng Li

doi:10.1038/s41534-021-00446-7

Abstract

The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in random matrix theory and quantum chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped-ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm.

Highlights

The Riemann zeta function ζ(s) is the Rosseta stone for number theory
The ζ-function is expressed in three different “languages”: as the series ∑nn−s over the positive integers n, as the product prime numbers p, and as the product /
We have presented an experimental method for measuring the location of the zeros of the Riemann ζ-function, by using Floquet engineering to control the quasienergy levels of a periodically driven trapped ion

Summary

Introduction

The Riemann zeta function ζ(s) is the Rosseta stone for number theory. The stone, found by Napoleon’s troops in Egypt, contains the same text written in three different languages, which enabled the Egyptian hieroglyphics to be deciphered. The ζ-function is expressed in three different “languages”: as the series ∑nn−s over the positive integers n, as the product prime numbers p, and as the product /. Q∏pn1ð/1(1À−s=pρ−ns)Þeos=vρen r the over the Riemann zeros ρn[1]. Riemann conjectured in 1859 that these zeros would have a real part equal to a half, ρn 1⁄4 1 2 þ iEn, where. En is a real number[2]. This is the famous Riemann Hypothesis (RH), one of the six unsolved

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Conclusion