Sequential modular position and momentum measurements of a trapped ion mechanical oscillator

TL;DR

This paper demonstrates sequential measurements of modular position and momentum variables in a trapped ion's mechanical oscillator, revealing quantum non-classicality through no-signaling-in-time and Leggett-Garg inequality violations.

Contribution

It implements and analyzes sequential modular variable measurements in a trapped ion system, showing their non-commutative nature and quantum interference effects.

Findings

No-signaling-in-time observed with classical states.

Signaling-in-time appears with quantum interference and squeezing.

Violation of Leggett-Garg inequality demonstrated.

Abstract

The non-commutativity of position and momentum observables is a hallmark feature of quantum physics. However this incompatibility does not extend to observables which are periodic in these base variables. Such modular-variable observables have been suggested as tools for fault-tolerant quantum computing and enhanced quantum sensing. Here we implement sequential measurements of modular variables in the oscillatory motion of a single trapped ion, using state-dependent displacements and a heralded non-destructive readout. We investigate the commutative nature of modular variable observables by demonstrating no-signaling-in-time between successive measurements, using a variety of input states. In the presence of quantum interference, which we enhance using squeezed input states, measurements of different periodicity show signaling-in-time. The sequential measurements allow us to extract two-time correlators for modular variables, which we use to violate a Leggett-Garg inequality. The experiments involve control and coherence of multi-component superpositions of up to 8 coherent, squeezed or Fock state wave-packets. Signaling-in-time as well as Leggett-Garg inequalities serve as efficient quantum witnesses which we probe here with a mechanical oscillator, a system which has a natural crossover from the quantum to the classical regime.