[Paper Review] Formation and Dynamics of a Schrödinger-Cat State in Continuous Quantum Measurement

G. P. Berman, F. Borgonovi+6 more|ArXiv.org|Jan 7, 2001

Mechanical and Optical Resonators5 references3 citations

TL;DR

This paper investigates the formation of a Schrödinger-cat state in a continuous quantum measurement using magnetic resonance force microscopy (MRFM). By modeling a single spin coupled to a quasi-classical cantilever as a closed quantum system governed by the Schrödinger equation, the authors demonstrate through numerical simulations that the cantilever's position probability distribution evolves into a quasi-periodic, asymmetric two-peak state—characteristic of a Schrödinger-cat state—under cyclic adiabatic spin inversion.

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 Schrödinger 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 Schrödinger-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.

Motivation & Objective

• To investigate the dynamics of a truly continuous quantum measurement in a closed quantum system, using MRFM as a physical realization.
• To understand the formation of macroscopic superposition states (Schrödinger-cat states) in a measurement device during continuous monitoring of a single spin.
• To examine how environmental interactions might lead to wave function collapse and quantum jumps in such a system, without assuming pre-existing collapse mechanisms.
• To provide a rigorous quantum mechanical description of the spin-cantilever system without artificial approximations, contrasting with classical or semi-classical models.
• To explore the implications of wave function collapse on the measurement outcome, particularly whether the spin state collapses into a definite eigenstate or remains a superposition.

Proposed method

• Formulate a full quantum mechanical Hamiltonian for a single spin coupled to a cantilever modeled as a quantum harmonic oscillator in a coherent state.
• Solve the time-dependent Schrödinger equation numerically for the spin-cantilever system under cyclic adiabatic inversion of the spin.
• Use the Heisenberg representation to derive equations of motion that reduce to classical dynamics in the many-spin limit.
• Simulate the quantum dynamics using a Fock basis of 2000 states to accurately capture the evolution of the cantilever's position probability distribution.
• Model environmental decoherence phenomenologically by introducing random wave function collapses into the larger or smaller peak of the two-peak distribution, with probabilities proportional to peak areas.
• Repeat simulations with different sequences of collapse times and peak choices to simulate realistic experimental dynamics under decoherence.

Experimental results

Research questions

• RQ1 Can a truly continuous measurement lead to the formation of a Schrödinger-cat state in a macroscopic measurement device like a cantilever?
• RQ2 What is the nature of the wave function collapse in a continuous measurement scenario, and does it lead to a definite spin eigenstate?
• RQ3 How does the interaction with the environment affect the stability and dynamics of the Schrödinger-cat state in the spin-cantilever system?
• RQ4 Why does the spin not jump into a definite |↑⟩ or |↓⟩ state upon collapse, even when the cantilever's wave function collapses into one peak?
• RQ5 What is the role of the asymmetry in the two-peak probability distribution in determining the likelihood and outcome of quantum jumps?

Key findings

• The cantilever's position probability distribution evolves into a quasi-periodic, asymmetric two-peak structure under cyclic adiabatic spin inversion, characteristic of a Schrödinger-cat state.
• The two peaks have an area ratio of approximately 100:1, meaning that in 99 out of 100 collapses, the wave function collapses into the larger peak.
• Despite wave function collapse into one peak, the spin does not jump into a definite |↑⟩ or |↓⟩ eigenstate; instead, it remains a superposition, with the inequality between P11 and P22 preserved in 99% of cases.
• The collapse process does not reverse the relative populations of the spin states P11 and P22 in most cases—only in about 1% of collapses is the inequality reversed.
• The system exhibits a cycle: Schrödinger-cat state formation via unitary evolution, followed by collapse due to decoherence, then re-evolution into a new cat state.
• The model reduces to classical equations of motion in the many-spin limit, confirming consistency with conventional MRFM experiments.