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.
Highlights
One of the fundamental notions of quantum mechanics is that position and momentum operators do not commute.This results in the Heisenberg uncertainty principle: Δx Δp ≥ 1 2 jh1⁄2x ;p ij with 1⁄2x ; p 1⁄4iħ, restricts the possible states of a particle and limits the ability to perform simultaneous position and momentum measurements [1,2,3,4]
If nosignaling in time (NSIT) is observed for all possible input states, it follows that the underlying observables commute
A protocol for testing Leggett-Garg inequalities (LGI) using modular variable measurements has been proposed previously [8], where it was shown that violation of the LGI can be used to differentiate between an oscillator described by a classical variable and a quantum mechanical oscillator
Summary
One of the fundamental notions of quantum mechanics is that position and momentum operators do not commute. Iħ, restricts the possible states of a particle and limits the ability to perform simultaneous position and momentum measurements [1,2,3,4] This is different for measurements of the position and momentum operator modulo a characteristic length or momentum scale (i.e., Xmod lx, Pmod lp), which can commute [5]. Such operators were first discussed in the context of the seminal Aharonov-Bohm effect [6], and they provide new perspectives in the study of fundamental aspects of quantum mechanics. We analyze signaling in time between the measurements and violate a Leggett-Garg inequality Using both methods, we confirm the quantum nature of the motional states using a small number of measurements. The experiments demonstrate control and show coherence of multicomponent superpositions of up to eight coherent, squeezed, or Fock state wave packets
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