Engineering Arbitrary Hamiltonians in Phase Space

TL;DR

This paper presents a general method to engineer arbitrary Hamiltonians in the Floquet phase space of a driven oscillator using non-commutative Fourier transformation, enabling new quantum state control.

Contribution

It introduces a novel protocol linking real-space driving potentials to desired phase space Hamiltonians, with analytical expressions for complex target Hamiltonians.

Findings

Analytical expressions for real-space potentials generating target Hamiltonians

Ability to create rotational lattices and sharp-boundary wells in phase space

Applicable to various experimental platforms for quantum state engineering

Abstract

We introduce a general method to engineer arbitrary Hamiltonians in the Floquet phase space of a periodically driven oscillator, based on the non-commutative Fourier transformation (NcFT) technique. We establish the relationship between an arbitrary target Floquet Hamiltonian in phase space and the periodic driving potential in real space. We obtain analytical expressions for the driving potentials in real space that can generate novel Hamiltonians in phase space, e.g., rotational lattices and sharp-boundary well. Our protocol can be realised in a range of experimental platforms for nonclassical states generation and bosonic quantum computation.