Qubit-Controlled Displacements in Markovian Environments

TL;DR

This paper explores qubit-controlled displacement operations in harmonic oscillators, analyzing their realizations across various platforms and their dynamics under decoherence, with applications in entanglement and non-classical state preparation.

Contribution

It introduces a detailed theoretical framework for qubit-controlled displacements in Markovian environments, including analytical solutions and practical estimates for implementation.

Findings

Analytical solutions for the master equation in phase space

Feasibility estimates for various experimental setups

Demonstration of entanglement dynamics and non-classical state preparation

Abstract

We study a particular form of interaction Hamiltonian between qubits and quantum harmonic oscillators, whose closed system dynamics results in qubit controlled displacement operations. We show how this interaction is realizable in many setups, including nanomechanical systems, ion traps, cavity QED and circuit QED, and in each context we provide quantitative estimates for the relevant parameters. The dynamics of the system is investigated through a master equation, including typical decoherence mechanisms resulting from the coupling of the qubit and oscillator to a thermal Markovian environment. We show how to solve the master equation by adopting a phase-space representation for the oscillator, and derive analytical and approximate solutions for many special cases of interest. Finally, our techniques are applied to a relevant example by studying the dynamics of qubit-oscillator entanglement and the preparation of oscillator states with negative Wigner function.