CERMIT adiabatic rapid passage

Overview

The Zeeman field \(B_z(\vec{r})\) experienced by spins at location \(\vec{r}\) is a sum of the \(z\) component of the tip field and the static field, which we take to be oriented along the \(z\) axis,

(1)\[B_z(\vec{r}) = B_0 + B_z^{\mathrm{tip}}(\vec{r})\]

Here we assume that \(B_0 \gg B_z^{\mathrm{tip}}\), which requires us to consider only the \(z\) component of the tip field and spin magnetization. The spin-dependent change in the spring constant is determined using

(2)\[\delta k_{\mathrm{spin}} = - \sum_j \delta M_z (\vec{r}_j) \frac{\partial^2 B_{z}^{\mathrm{tip}}({\vec{r}}_j)}{\partial z^2} \Delta V_j\]

where \(\delta M_z\) is the change in magnetization density and \(\Delta V_j\) is the volume of each voxel (or volume element) in the sample. We have drawn equation (2) (with its leading minus sign) from reference [2] (equation 8). We have taken care to implement the leading minus sign here.

Let us now derive the equations we will use to calculate the change in magnetization density \(\delta M_z (\vec{r})\) resulting from the adiabatic rapid passage. In a frame of reference rotating counter-clockwise about the \(z\) axis at frequency \(\omega\), the magnetization density evolves under the action of the effective field

(3)\[\vec{B}_{\mathrm{eff}} = (B_z(\vec{r}) - \frac{\omega}{\gamma}) \hat{z} + B_1 \hat{x}_{\mathrm{R}}\]

with \(\hat{z}\) a unit vector along \(z\) axis, \(\hat{x}_{\mathrm{R}}\) a transverse rotating basis vector, \(B_z(\vec{r})\) given by equation (1), \(\omega\) the frequency of the applied oscillating field, \(\gamma\) the gyromagnetic ratio, and \(2 B_1\) the amplitude of the oscillating magnetic field, assumed to be linearly polarized. We are making the rotating wave approximation in using equation (3) to describe the evolution of magnetization under linearly polarized irradiation. It is useful to write down a unit vector parallel to the effective field:

(4)\[\vec{b}_{\mathrm{eff}} = \frac{B_z(\vec{r}) - \omega/\gamma} {\sqrt{(B_z(\vec{r}) - \omega/\gamma)^2 + B_1^2}} \hat{z} + \frac{B_1} {\sqrt{(B_z(\vec{r}) - \omega/\gamma)^2 + B_1^2}} \hat{x}_{\mathrm{R}}\]

Dividing both the numerators and denominators in the equation (4) by \(B_1\), this unit vector can be written as

(5)\[\hat{b}_{\mathrm{eff}}(\Omega) = \frac{\Omega}{\sqrt{\Omega^2+1}} \hat{z} + \frac{1}{\sqrt{\Omega^2+1}} \hat{x}_{\mathrm{R}}\]

with

(6)\[\Omega = \frac{\gamma B_z(\vec{r}) - \omega}{\gamma B_1}\]

the (unitless) ratio of the resonance offset to the Rabi frequency; \(\Omega > 0\) and \(\hat{b}_{\mathrm{eff},z} > 0\) for spins at a field above the resonance field \(\omega/\gamma\) while \(\Omega < 0\) and \(\hat{b}_{\mathrm{eff},z} < 0\) for spins at a field below the resonance field \(\omega/\gamma\).

Now consider the evolution of sample magnetization during an adiabatic rapid passage through resonance. The magnetization is initially along the \(z\) axis. Just before time \(t = 0\),

(7)\[\vec{M}(0^{-}) = M_{z}(0) \: \hat{z}\]

At time \(t = 0\) the irradiation is turned on with an initial offset frequency of \(\Omega_{\mathrm{i}}\). Since this effective field is not quite parallel to the \(z\) axis in the rotating frame, the initial magnetization vector will precess around it. The component of the initial magnetization perpendicular to the initial effective field \(\hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}})\) will quickly dephase, within in a time \(T_2 \sim 5 \: \mu\mathrm{s}\). The component of the initial magnetization parallel to the initial effective field will survive this dephasing. The magnetization after this dephasing, at time \(t = 0^{+}\), is given by the projection of \(\vec{M}_{z}(0^{-})\) onto \(\hat{b}_{\mathrm{eff}}\),

(8)\[\vec{M}(0^{+}) = M_{z}(0) \left( \hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}}) \cdot \vec{M}_{z}(0^{-}) \right) \: \hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}})\]

The prefactor in parenthesis may be positive or negative, depending on whether \(\vec{M}_{z}(0^{-})\) and \(\hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}})\) are parallel (\(\Omega >0\)) or antiparallel (\(\Omega <0\)). Substituting equations (5) and (7) into equation (8)

(9)\[\begin{split}\begin{align} \vec{M}(0^{+}) & = M_{z}(0) \frac{\Omega_{\mathrm{i}}}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \left( \frac{\Omega_{\mathrm{i}}}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \hat{z} + \frac{1}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \hat{x}_{\mathrm{R}} \right) \\ & = M_{z}(0) \left( \frac{\Omega_{\mathrm{i}}^2}{\Omega_{\mathrm{i}}^2+1} \hat{z} + \frac{\Omega_{\mathrm{i}}}{\Omega^2_{\mathrm{i}}+1} \hat{x}_{\mathrm{R}} \right) \end{align}\end{split}\]

We can see that equation (9) captures \(\vec{M}_{z}(0^{+})\) correctly for spins initially above and below resonance when the irradiation is turned on. For example, when \(\Omega = +10\), \(\vec{M}_{z}(0^{+}) = 0.99 \: \hat{z} + 0.01 \: \hat{x}_{\mathrm{R}}\) while when \(\Omega = -10\), \(\vec{M}_{z}(0^{+}) = 0.99 \: \hat{z} - 0.01\: \hat{x}_{\mathrm{R}}\). In both cases, \(\vec{M}_{z}(0^{+})\) points up as it should. The magnitude of \(\vec{M}_{z}(0^{+})\) is

(10)\[\begin{split}\begin{align} \| \vec{M}(0^{+}) \| & = M_{z}(0) \left( \frac{\Omega_{\mathrm{i}}^4}{(\Omega_{\mathrm{i}}^2+1)^2} + \frac{\Omega_{\mathrm{i}}^2}{(\Omega_{\mathrm{i}}^2+1)^2} \right)^{1/2} \\ & = M_{z}(0) \left( \frac{\Omega_{\mathrm{i}}^2 (\Omega_{\mathrm{i}}^2 + 1) } {(\Omega_{\mathrm{i}}^2+1)^2} \right)^{1/2} \\ & = M_{z}(0) \frac{\| \Omega_{\mathrm{i}} \|}{\sqrt{\Omega_{\mathrm{i}}^2 + 1}} \end{align}\end{split}\]

At a time just after \(t = 0^+\), the adiabatic rapid passage is initiated and \(\Omega\) is swept from the initial offset \(\Omega_{\mathrm{i}}\) to a final offset \(\Omega_{\mathrm{f}}\). At the end of the sweep, at time \(t_{\mathrm{f}}\), the magnetization density vector will have the same magnitude as it did at time \(t = 0^+\), but will be oriented parallel or antiparallel to the final effective field, \(\hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{f}})\),

(11)\[\vec{M}(t_{\mathrm{f}}) = \| \vec{M}_{z}(0^{+}) \| \: \mathrm{sign}(\Omega_{\mathrm{i}}) \left( \frac{\Omega_{\mathrm{f}}}{\sqrt{\Omega_{\mathrm{f}}^2+1}} \hat{z} + \frac{1}{\sqrt{\Omega_{\mathrm{f}}^2+1}} \hat{x}_{\mathrm{R}} \right)\]

Here \(\mathrm{sign}(\Omega_{\mathrm{i}})\) accounts for the final magnetization being parallel (for positive initial offset, \(\mathrm{sign}(\Omega_{\mathrm{i}}) = +1\)) or antiparallel (for negative initial offset, \(\mathrm{sign}(\Omega_{\mathrm{i}}) = -1\)) to the final effective field. We are interested in the \(z\)-component of the final magnetic field vector. Substituting equation (10) into equation (11) and using

\[\mathrm{sign}(\Omega_{\mathrm{i}}) \: \| \Omega_{\mathrm{i}} \| = \Omega_{\mathrm{i}}\]

(12)\[M_{z}(t_{\mathrm{f}}) = M_{z}(0) \dfrac{\Omega_{\mathrm{i}}}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \dfrac{\Omega_{\mathrm{f}}}{\sqrt{\Omega_{\mathrm{f}}^2+1}}\]

At each point in the sample, the change, final minus initial, in \(z\) component of magnetization following the adiabatic rapid passage is given by

(13)\[\delta M_{z} = M_{z}(t_{\mathrm{f}}) - M_{z}(0) = M_{z}(0) \left( \dfrac{\Omega_{\mathrm{i}}}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \dfrac{\Omega_{\mathrm{f}}}{\sqrt{\Omega_{\mathrm{f}}^2+1}} -1 \right)\]

If we sweep from \(\Omega_{\mathrm{i}} \rightarrow +\infty\) (way above resonance) to \(\Omega_{\mathrm{f}} \rightarrow -\infty\) (way below resonance), then \(\delta M_{z} = -2 M_{z}(0)\). This is what we expect to see. For a swept-field or swept-tip experiment,

(14)\[\Omega_{\mathrm{i}} = \frac{B_z(\vec{r}_{\mathrm{i}}) - \omega/\gamma}{B_1} \: \: \: \mathrm{and} \: \: \: \Omega_{\mathrm{f}} = \frac{B_z(\vec{r}_{\mathrm{f}}) - \omega/\gamma}{B_1}\]

while for a swept-frequency experiment,

(15)\[\Omega_{\mathrm{i}} = \frac{B_z(\vec{r}) - \omega_{\mathrm{i}}/\gamma}{B_1} \: \: \: \mathrm{and} \: \: \: \Omega_{\mathrm{f}} = \frac{B_z(\vec{r}) - \omega_{\mathrm{f}}/\gamma}{B_1}\]

To compute the change in magnetization contributing to signal at each position, we will use the equation (13) and either equation (14) (for a swept-tip experiment) or equation (15) (for a swept-frequency experiment). In the swept-frequency calculation we need to compute the field at each point. In the swept-tip calculation, we need to compute at each position \(\vec{r}\) in the sample the \(z\) component of the magnetic field only at the beginning (\(\vec{r} = \vec{r}_{\mathrm{i}}\)) and end (\(\vec{r} = \vec{r}_{\mathrm{f}}\)) of the tip sweep.

Adiabaticity

We would also like to assess the adiabaticity of the sweep. With

\[M_{z}(t) = M_{z}(0) \: \cos{(\theta(t))},\]

the adiabaticity parameter is generally defined as

\[\alpha = \frac{\dot{\theta}}{\gamma B_1}.\]

In a cryogenic ESR-MRFM observing \(\mathrm{E}^{\prime}\) centers in quartz via cyclic adiabatic inversion, Wago and coworkers observed a peaking of signal when \(\alpha \sim 0.1\) (note that they define the adiabaticity parameter as \(1/\alpha\)) [3]. In a room temperature NMR-MRFM experiment observing proton magnetization in an ammonium nitrate crystal via cyclic adiabatic inversion and force detection, Klein and coworkers observed lossless inversion of magnetization when \(\alpha \leq 0.1\) [1]. In both of these experiments, a linear frequency sweep was used. We note that more efficient sweeps have been devised. [4] [5] For a linear sweep, the adiabaticity parameter is largest near resonance, where

\[\alpha_{\mathrm{res}} = \frac{1}{\gamma B_1^2} \frac{d B_{\mathrm{eff}}}{d t} = \frac{1}{\omega_1} \frac{d \Omega}{d t}\]

Here \(\Omega\) is given by equation (6) and \(\omega_1 = \gamma B_1\) is the Rabi frequency. In a swept-tip experiment, the field at position \(\vec{r}\) changes by an amount \(\delta B = B_{z}^{\mathrm{tip}}(\vec{r}_{\mathrm{f}}) - B_ {z}^{\mathrm{tip}}(\vec{r}_{\mathrm{i}})\) in a time equal to half of a cantilever period, \(\delta t = 1/(2 f_c)\). If we approximate the sweep as linear, then the adiabaticity parameter is given by

(16)\[\alpha_{\mathrm{res}}(\vec{r}) \approx \frac{1}{\gamma B_1^2} \frac{\delta B}{\delta t} = \frac{2 f_c}{\gamma B_1^2} \| B_{z}^{\mathrm{tip}}(\vec{r}_{\mathrm{f}}) - B_{z}^{\mathrm{tip}}(\vec{r}_{\mathrm{i}}) \|\]

We have introduced an absolute value sign to guarantee that \(\alpha\) is positive and independent of the sweep direction. We write :math:` alpha_{mathrm{res}}(vec{r})` to emphasize that the adiabaticity parameter should be evaluated at each position \(\vec{r}\) in the sample. Equation (16) is only strictly valid at resonance and, moreover, does not account for the sinusoidal time dependence of \(\vec{r}(t)\) during the cantilever motion. Nevertheless, we will use it to access the adiabaticity of the spin inversion in the swept-tip experiment. Since :math:` alpha` is smaller for sample spins that do not pass through resonance, equation (16) provides an upper-bound estimate for the adiabaticity parameter at any location. The spins that contribute most to the signal are those that pass through resonance; for these spins, equation :eq:` Eq:alpha-swept-tip` should be reasonably accurate.

In the swept-frequency experiment, the irradiation frequency is ramped from \(\omega_{\mathrm{i}}\) to \(\omega_{\mathrm{f}}\) in a time :math:` Delta T_{mathrm{sweep}}`; the period of the sweep \(\Delta T_ {\mathrm{sweep}}\) is not restricted to be half a cantilever period. The adiabaticity parameter is independent of position \(\vec{r}\) and, assuming a linear frequency sweep, equal to

(17)\[\alpha_{\mathrm{res}} = \frac{1}{\gamma^2 B_1^2} \frac{\| \omega_{\mathrm{f}} - \omega_{\mathrm{i}} \|} {\Delta T_{\mathrm{sweep}}}\]

As with equation (16), equation (17) is only strictly valid for spins that pass through resonance. Spins far away from the resonant slice will experience an \(\alpha\) even smaller and \(\alpha_{\mathrm{res}}\). We can therefore regard equation :eq:` Eq:alpha-swept-freq` as an upper bound for the adiabaticity parameter experienced by any spin in the sample.

[1]

Klein, O.; Naletov, V. & Alloul, H. “Mechanical Detection of Nuclear Spin Relaxation in a Micron-size Crystal”, Eur. Phys. J. B, 2000, 17, 57 - 68 [10.1007/s100510070160].

[2]

Lee, S.-G.; Moore, E. W. & Marohn, J. A. “A Unified Picture of Cantilever Frequency-Shift Measurements of Magnetic Resonance”, Phys. Rev. B, 2012, 85, 165447 [10.1103/PhysRevB.85.165447].

[3]

Wago, K.; Botkin, D.; Yannoni, C. & Rugar, D. “Force-detected Electron-spin Resonance: Adiabatic Inversion, Nutation, and Spin Echo”, Phys. Rev. B, 1998, 57, 1108 - 1114 [10.1103/PhysRevB.57.1108].

[4]

Baum, J.; Tycko, R. & Pines, A. “Broadband and Adiabatic Inversion of a Two-level System by Phase-modulated Pulses”, Phys. Rev. A , 1985, 32, 3435 - 3447 [10.1103/PhysRevA.32.3435].

[5]

Kupce, E. & Freeman, R. “Optimized Adiabatic Pulses for Wideband Spin Inversion”, Journal of Magnetic Resonance, Series A, 1996, 118, 299 - 303 [10.1006/jmra.1996.0042].

Experiment Summary

mrfmsim.experiment.CermitARPGroup	Simulates a Cornell-style frequency-shift magnetic resonance force microscope experiment in which a single frequency-sweep adiabatic rapid passage through resonance is used to invert the spins.
mrfmsim.formula.polarization.rel_dpol_arp(...)	Relative change in polarization for adiabatic rapid passage.

mrfmsim.experiment.CermitARPGroup = <mrfmsim.group.ExperimentGroup 'CermitARPGroup'>

Simulates a Cornell-style frequency-shift magnetic resonance force microscope experiment in which a single frequency-sweep adiabatic rapid passage through resonance is used to invert the spins.

• CermitARP

  CermitARP(B0, B1, df_fm, f_rf, grid, h, magnet, sample)
  returns: dk_spin
  group: CermitARPGroup
  graph: CermitARP_graph
  handler: MemHandler
  
  Simulate CERMIT ARP for a large tip.
  

• CermitARPSmallTip

  CermitARPSmallTip(B0, B1, df_fm, f_rf, grid, h, magnet, sample, trapz_pts, x_0p)
  returns: dk_spin
  group: CermitARPGroup
  graph: CermitARPSmallTip_graph
  handler: MemHandler
  
  Simulate CERMIT ARP for a small tip.