Magnetic Resonance Force Microscopy and a Single-Spin Measurement

The following sections are included:

• Preface

• Contents

Recent remarkable progress in magnetic resonance imaging (MRI) naturally raises the question about the ultimate possibility for atomic scale MRI resolution (single spin detection). However, the MRI techniques are based on the phenomenon of electromagnetic induction, which implies the detection of a macroscopic number of spins. The minimum number of nuclear spins detected by MRI techniques is about 10¹² [1], and the minimum number of electron spins detected using electron spin resonance (ESR) techniques is about 10⁷ [2]. To resolve the problem of a single spin detection Sidles [3, 4] suggested using the force detection techniques like those used in atomic force microscopy (AFM). However, the magnetic force is much smaller than the electric force detected in AFM. In order to overcome this difference Sidles proposed a combination of magnetic resonance techniques for a single spin with the mechanical resonance of an ultrasensitive cantilever.

While a spin is a quantum object, in many cases its dynamics can be successfully described using quasiclassical theory. The main property of an electron or nuclear spin is the following: the spin's magnetic moment is parallel to the spin . We can write , where γ is the magnitude of the gyromagnetic ratio. The positive sign in this equation corresponds to the proton's spin and many other nuclear spins. The negative sign corresponds to the electron's spin and also some nuclear spins. We will consider an electron spin with a negative gyromagnetic ratio. The direction of the electron magnetic moment is opposite to the direction of the spin…

The main distinction between the quantum and quasiclassical descriptions of the spin is the following: in quantum mechanics the component of the spin along any axis may take only two values ±1/2 (in units of the Planck's constant ħ). In the S_z-representation the operator corresponding to the z-component of the spin is a diagonal matrix with the matrix elements ±1/2:…

A cantilever used in MRFM is a tiny beam, which is fixed at one end and free to vibrate at the other end. A small ferromagnetic particle is attached to the cantilever tip (CT). The force produced by a single spin on the ferromagnetic particle affects the parameters of the mechanical vibrations of the CT, which are to be measured in MRFM experiments. The motion of the CT with a ferromagnetic particle with no magnetic force can be described as a simple harmonic motion. The Hamiltonian of the corresponding effective harmonic oscillator can be written in the usual form…

The following sections are included:

• Static displacement of the cantilever tip (CT)

• Decoherence time

The following sections are included:

• Hamiltonian and master equation for the spin-CT system

• Solution for spin diagonal matrix elements

• Solution for spin off-diagonal matrix elements

Now we transfer to the main topic of our book — the theory of MRFM. In this chapter we will consider a simple (from the theoretical point of view) MRFM technique: application of a sequence of resonant π-pulses, which drive the periodic reversals of the spin. In turn, the spin periodic reversals drive the cantilever vibrations, which are to be detected (Berman and Tsifrinovich [22])…

The following sections are included:

• Schrödinger dynamics of the CT-spin system

• Decoherence and thermal diffusion for the CT

The following sections are included:

• CT-spin dynamics: discussion and estimates

• Experimental detection of a single spin

The following sections are included:

• Quasiclassical theory: simple geometry

• Quantum theory of the OSCAR MRFM

• OSCAR frequency shift for a realistic setup

The following sections are included:

• OSCAR relaxation in a spin ensemble

• Reduction of magnetic noise

• Simple model for quantum jumps

• Reduction of the frequency shift due to the CT-spin entanglement

The following sections are included:

• MRFM measurement of an entangled spin state

• MRFM based spin quantum computer

The following sections are included:

• Spin diffusion in the presence of a nonuniform magnetic field

• Suppression of the spin diffusion in a spin quantum computer

The following sections are included:

• Abbreviations

• Prefixes

• Notations

• Bibliography

• Index