Formation and Dynamics of a Schr\"odinger-Cat State in Continuous Quantum Measurement

TL;DR

This paper models the continuous measurement process of a single spin using MRFM, demonstrating the formation of a Schrödinger-cat state in the cantilever's quantum state through rigorous Schrödinger equation simulations.

Contribution

It provides a detailed quantum mechanical analysis of spin-cantilever dynamics during continuous measurement, revealing the emergence of Schrödinger-cat states without artificial assumptions.

Findings

Cantilever evolves into a Schrödinger-cat state with asymmetric peaks.

Simulation shows quasi-periodic appearance and vanishing of peaks.

Environmental interaction does not cause the spin to jump into an eigenstate.

Abstract

We consider the process of a single-spin measurement using magnetic resonance force microscopy (MRFM) as an example of a truly continuous measurement in quantum mechanics. This technique is also important for different applications, including a measurement of a qubit state in quantum computation. The measurement takes place through the interaction of a single spin with a quasi-classical cantilever, modeled by a quantum oscillator in a coherent state in a quasi-classical region of parameters. The entire system is treated rigorously within the framework of the Schr\"odinger equation, without any artificial assumptions. Computer simulations of the spin-cantilever dynamics, where the spin is continuously rotated by means of cyclic adiabatic inversion, show that the cantilever evolves into a Schr\"odinger-cat state: the probability distribution for the cantilever position develops two asymmetric peaks that quasi-periodically appear and vanish. For a many-spin system our equations reduce to the classical equations of motion, and we accurately describe conventional MRFM experiments involving cyclic adiabatic inversion of the spin system. We surmise that the interaction of the cantilever with the environment would lead to a collapse of the wave function; however, we show that in such a case the spin does not jump into a spin eigenstate.