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arXiv:quant-ph/0110070 · Physical Review A

Magnetic-resonance force microscopy measurement of entangled spin states

G. P. Berman, F. Borgonovi, George Chapline, P. C. Hammel, V. I. Tsifrinovich

Published Sep 18, 2002• 3 citations
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Abstract

We simulate magnetic-resonance force microscopy measurements of an entangled spin state. One of the entangled spins drives the resonant cantilever vibrations, while the other remote spin does not interact directly with the quasiclassical cantilever. The Schr\"odinger cat state of the cantilever (i.e., two trajectories of the quasiclassical cantilever) reveals two possible outcomes of the measurement for both entangled spins.

DOI
10.1103/physreva.66.032106

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We simulate magnetic-resonance force microscopy measurements of an entangled spin state. One of the entangled spins drives the resonant cantilever vibrations, while the other remote spin does not interact directly with the quasiclassical cantilever. The Schr"odinger cat state of the cantilever (i.e., two trajectories of the quasiclassical cantilever) reveals two possible outcomes of the measurement for both entangled spins.
We consider the process of a single-spin measurement using magnetic resonance force microscopy (MRFM) with a cyclic adiabatic inversion (CAI). 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 cantilever modelled by a quantum oscillator in a coherent state in a quasi-classical range of parameters. The entire system is treated rigorously within the framework of the Schrödinger equation. For a many-spin system our equations accurately describe conventional MRFM experiments involving CAI of the spin system. Our computer simulations of the quantum spin–cantilever dynamics show that the probability distribution for the cantilever position develops two asymmetric peaks with the total relative probabilities mainly dependent on the initial angle between the directions of the average spin and the effective magnetic field, in the rotating frame. We show that each of the peaks is correlated with the direction of the average spin (being along or opposite to the direction of the effective magnetic field). This generates two possible outcomes of a single-spin measurement, similar to the Stern–Gerlach effect. We demonstrate that the generation of the second peak can be significantly suppressed by turning on adiabatically the amplitude of the rf magnetic field. We also show that MRFM CAI can be used both for detecting a signal from a single spin, and for measuring the single-spin state by measuring the phase of the cantilever driving oscillations.
We consider a simple version of a cyclic adiabatic inversion (CAI) technique in magnetic-resonance force microscopy (MRFM). We study the problem: What component of the spin is measured in the CAI MRFM? We show that the nondestructive detection of the cantilever vibrations provides a measurement of the spin component along the effective magnetic field. This result is based on numerical simulations of the Hamiltonian dynamics (the Schr\"odinger equation) and the numerical solution of the master equation.
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.
We simulated the quantum dynamics for magnetic resonance force microscopy (MRFM) in the oscillating cantilever-driven adiabatic reversals (OSCAR) technique. We estimated the frequency shift of the cantilever vibrations and demonstrated that this shift causes the formation of a Schr\"odinger cat state which has some similarities and differences from the conventional MRFM technique which uses cyclic adiabatic reversals of spins. The interaction of the cantilever with the environment is shown to quickly destroy the coherence between the two possible cantilever trajectories. We have shown that using partial adiabatic reversals, one can produce a significant increase in the OSCAR signal.
A promising technique for measuring single electron spins is magnetic resonance force microscopy (MRFM), in which a microcantilever with a permanent magnetic tip is resonantly driven by a single oscillating spin. The most effective experimental technique is the oscillating cantilever-driven adiabatic reversals (OSCAR) protocol, in which the signal takes the form of a frequency shift. If the quality factor of the cantilever is high enough, this signal will be amplified over time to the point where it can be detected by optical or other techniques. An important requirement, however, is that this measurement process occurs on a time scale that is short compared to any noise which disturbs the orientation of the measured spin. We describe a model of spin noise for the MRFM system and show how this noise is transformed to become time dependent in going to the usual rotating frame. We simplify the description of the cantilever-spin system by approximating the cantilever wave function as a Gaussian wave packet and show that the resulting approximation closely matches the full quantum behavior. We then examine the problem of detecting the signal for a cantilever with thermal noise and spin with spin noise, deriving a condition for this to be a useful measurement.