Magnetic resonance of a classical magnetic moment

Now, unlike in classical mechanics, most resonances studied in atomic physics are not harmonic oscillators, but two-level systems. Unlike harmonic oscillators, two-level systems show saturation. When a harmonic oscillator is driven longer or faster, higher and higher excited states are populated: the oscillator amplitude can become arbitrarily large. In contrast, the amplitude of oscillation in a two-level system is limited to one half in the appropriate dimensionless units.\footnote{An analogous dimensionless amplitude for the harmonic oscillator would be the amplitude measured in units of the oscillator ground state size.} Why, and under what conditions, do classical harmonic oscillator models of a two-level system work? A two-level system of energy difference $E$ can be approximated by a harmonic oscillator of frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_o=E/\hbar} when saturation effects in the two-level system are negligible, i.e. when the population of the second excited state in the harmonic oscillator problem is negligible, or equivalently when the population ratio of the first excited state to the the ground state is small: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1/P_0\ll 1} . This is the basis for the classical Lorentz model of the electron bound in the atom that describes many linear atomic properties (for instance the refractive index) very well. See "The origin of the refractive index" in chapter 31 of the Feynman Lectures on Physics \cite{Feynman1963}. When saturation comes into play, i.e. when the initial ground state is appreciably depleted, the harmonic oscillator ceases to be a good model. The limit on the oscillation amplitude in a two-level system suggests that a classical system with periodic evolution and a limit on the amplitude, namely rotation, could provide a better classical model of a two-level system. Indeed, the motion of a classical magnetic moment in a uniform field provides a model that captures almost all features of the quantum-mechanical two-level system, the exception beting the projection onto one of the two possible outcomes in a measurement. \section{Magnetic Resonance: The Classical Magnetic Moment in a Spatially Uniform Field}

Magnetic Moment in a Static Field

The interaction energy of a magnetic moment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu}} } with a magnetic field ${\bf {B}}$ is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W=- {\bf{\mu}} \cdot {\bf{B}} }

In a uniform field the force

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{F}} =- {\bf{\nabla}} W=0 }

vanishes, but the torque

${\bf {T}}={\bf {\mu }}\times {\bf {B}}$

does not. For an angular momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{L}} } the equation of motion is then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{ {\bf{L}} } = {\bf{T}} = {\bf{\mu}} \times {\bf{B}} =\gamma {\bf{L}} \times {\bf{B}} }

where we have introduced the gyromagnetic ratio $\gamma$ as the proportionality constant between angular momentum and magnetic moment, as shown in Figure \ref{fig:static_precession}. \begin{figure}

\caption{Precession of a the magnetic moment and associated angular momentum about a static field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} } .}

\end{figure} This describes the precession of the magnetic moment about the magnetic field with angular frequency

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_L=-\gamma B }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_L} is known as the Larmor frequency. For electrons we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_e=2\pi\times\unit{2.8}{\mega\hertz\per G}} , for protons Failed to parse (unknown function "\unit"): {\displaystyle \gamma_e=2\pi\times\unit{4.2}{\kilo\hertz\per G}} .

An Alternative Solution: Rotating Coordinate System

A vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{A}} } rotating at constant angular velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\Omega}} } is described by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{ {\bf{A}} } = {\bf{\Omega}} \times {\bf{A}} }

Then the rates of change of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{A}} } measured in a coordinate system rotating at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\Omega}} } and in an inertial system are related by

${\dot {\bf {A}}}_{\text{in}}={\dot {\bf {A}}}_{\text{rot}}+{\bf {\Omega }}\times {\bf {A}}_{\text{in}}$

This follows immediately from the following facts: \begin{itemize}

* If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  {\bf{A}} }
 is constant in the rotating system then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{ {\bf{A}} }_\text{in}= {\bf{\Omega}} \times {\bf{A}} _\text{in}} .

* If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  {\bf{\Omega}} =0}
 then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{ {\bf{A}} }_\text{in}=\dot{ {\bf{A}} }_\text{rot}} .

* Coordinate rotation is a linear transformation.

\end{itemize} This transformation is sometimes abbreviated as the schematic rule

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{d}{dt}\right)_\text{rot}= \left(\frac{d}{dt}\right)_\text{in}- {\bf{\Omega}} \times\big(\big)_\text{in} }

It follows that the angular momentum in a rotating frame obeys

${\dot {\bf {L}}}_{\text{rot}}={\dot {\bf {L}}}_{\text{in}}-{\bf {\Omega }}\times {\bf {L}}_{\text{in}}={\bf {L}}\times (\gamma {\bf {B}}+{\bf {\Omega }})$

If we choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\Omega}} = {\bf{\Omega_L}} =-\gamma {\bf{B}} } , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{L}} _\text{rot}} is constant in the rotating frame. Often it is useful to think of a fictitious magnetic field ${\bf {B}}_{\text{fict}}={\bf {\Omega }}/\gamma$ that appears in a rotating frame.

Larmor's Theorem for a Charged Particle in a Magnetic Field

The vanishing of the torque on a magnetic moment when viewed in the correct rotating frame is reminiscent of Larmor's theorm for the motion of a charged particle in a magnetic field, which we now present. If the Lorentz force acts in an inertial frame,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{F}} _\text{in}=q {\bf{v}} _\text{in}\times {\bf{B}} }

then in the rotating frame, according to the rule \ref{eqn:rot_frame_transformation} we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \dot{ {\bf{r}} }_\text{rot} &= \dot{ {\bf{r}} }_\text{in} - {\bf{\Omega}} \times {\bf{r}} _\text{in} \\ \ddot{ {\bf{r}} }_\text{rot} &= \ddot{ {\bf{r}} }_\text{in} - 2 {\bf{\Omega}} \times \dot{ {\bf{r}} }_\text{in} + {\bf{\Omega}} \times \left( {\bf{\Omega}} \times {\bf{r}} _\text{in} \right) \end{align}}

resulting in a force Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{F}} _\text{rot}=m\ddot{ {\bf{r}} }_\text{rot}} in the rotating frame given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} {\bf{F}} _\text{rot} &= q {\bf{v}} _\text{in} \times {\bf{B}} - 2 m {\bf{\Omega}} \times {\bf{v}} _\text{in} + m {\bf{\Omega}} \times \left( {\bf{\Omega}} \times {\bf{r}} _\text{in} \right) \\ &= q {\bf{v}} _\text{in} \times \left( {\bf{B}} + 2 \frac{m}{q} {\bf{\Omega}} \right) + m {\bf{\Omega}} \times \left( {\bf{\Omega}} \times {\bf{r}} _\text{in} \right) \end{align}}

where we have used Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ddot{r}_\text{in}= {\bf{F}} _\text{in}=q {\bf{v}} _\text{in}\times {\bf{B}} } . Choosing

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\frac{m}{q} {\bf{\Omega}} =- {\bf{B}} =-B {\bf{\hat e}} _z }

yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{F}} _\text{rot} =\frac{q^2}{4m}B^2 {\bf{\hat e}} _z\times\left( {\bf{\hat e}} _z\times {\bf{r}} _\text{in}\right)\approx 0 }

if the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} field is not too large. Thus the Lorentz force approximately disappears in the rotating frame.

Although Larmor's theorem is suggestive of the rotating co-ordinate transformation, Eq. 1.22, it is important to realize that the two transformations, though identical in form, apply to fundamentally di®erent systems. A magnetic moment is not necessarily charged- for example a neutral atom can have a net magnetic moment, and the neutron possesses a magnetic moment in spite of being neutral - and it experiences no net force in a uniform magnetic ¯eld. Furthermore, the rotating co-ordinate transformation is exact for a mag- netic moment, whereas Larmor's theorem for the motion of a charged particle is only valid when the B2 term is neglected.

Note that while the vanishing of the force is approximate, the vanishing of the torque on a magnetic moment in the rotating frame is an exact result.

Rotating Magnetic Field on Resonance

\begin{figure}

\caption{Field and moment vectors in the static and rotating frames for the case of resonant drive.}

\end{figure} Consider a magnetic moment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu}} } precessing about a field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} =B_0 {\bf{\hat e}} _z} with ${\bf {\mu }}=(\mu ,\theta ,\phi =-\omega _{0}t)$ in spherical coordinates, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0=-\gamma B_0} . Assume that we now apply an additional field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _1} , in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} -plane rotating at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0} . To solve the resulting problem it is simplest to go into the rotating frame (Figure \ref{fig:rotating_frame}). Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _1} is stationary, say along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\hat e}} _x} , and there is an additional fictitious field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _\text{fict}= {\bf{\omega_0}} /\gamma=- {\bf{B}} _0} which cancels the field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _0} . So in the rotating frame we are left just with the static field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _1} , and the magnetic moment precesses about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _1} at the Rabi frequency

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R=\gamma B_1 }

A magnetic moment initially along the $-{\bf {\hat {e}}}_{z}$ axis will point along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + {\bf{\hat e}} _z} axis at a time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_r T=\pi} , while a magnetic moment parallel or antiparallel to applied magnetic field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _1} does not precess in the rotating frame. \QU{ Assume the magnetic moment is initially pointing along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - {\bf{\hat e}} _z} axis. Assume that a rotating field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_1\ll B_0} is applied, but that it rotates at a frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_1>\omega_0} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0} is the Larmor frequency for the static field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} =B_0 {\bf{\hat e}} _z} . Compared to the on-resonant case, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_1=\omega_0} , is the oscillation frequency of the magnetic moment. \begin{enumerate}

* larger
* the same
* smaller

\end{enumerate} } \QU{ Same question as \ref{q:rabi_freq_blue_detuned} but for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_1<\omega_0} . }

Rotating Magnetic Field Off-Resonance

If the rotation frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _1} does not equal the Larmor frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0=\gamma B_0} associated with the static field ${\bf {B}}_{0}$, then in the frame rotating with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _1} at frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} the static field is not completely cancelled by the fictitious field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega/\gamma} , but a difference along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\hat e}} _z} remains, giving rise to a total effective field in the rotating frame

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _\text{eff}= B_1 {\bf{\hat e}} _x + \left(B_0-\frac{\omega}{\gamma}\right) {\bf{\hat e}} _z }

The effective field is static, lies at an angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} with the z

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan\theta=\frac{B_1}{B_0-\frac{\omega}{\gamma}} }

and is of magnitude

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| B_\text{eff}\right| =\sqrt{B_1^2+\left(B_0-\frac{\omega}{\gamma}\right)^2} }

The magnetic moment precesses around it with an effective (sometimes called generalized) Rabi frequency

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_R =\gamma B_\text{eff} =\sqrt{(\omega_0-\omega)^2+\omega_R^2}=\sqrt{\omega_R^2+\delta^2} }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R=\gamma B_1} is the Rabi frequency associated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_1} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=\omega-\omega_0} is the detuning from resonance with the Larmor frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0=\gamma B_0} . \subsection{Geometrical Solution for the Classical Magnetic Moment in Static and Rotating Fields} \begin{figure}

\caption{Geometrical relations for the spin in combined static and rotating magnetic fields, viewed in the frame co-rotating with the drive field ${\bf {B}}_{1}$. At lower right is a view looking straight down the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _\text{eff}} axis.}

\end{figure} Referring to Figure \ref{fig:rotating_coord_construction}, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi &= \Omega_R t \\ C &= \mu \sin \alpha \\ A^2 &= \mu^2 ( 1 - \cos \alpha )^2 + C^2 \\ &= \mu^2 ( 1 - 2 \cos \alpha + \cos^2 \alpha + \sin^2 \alpha ) \\ &= 2 \mu^2 (1 - \cos \alpha ) \\ \Rightarrow \cos \alpha &= 1 - \frac{A^2}{2 \mu^2} \end{align}}

On the other hand

${\begin{aligned}{\frac {A}{2}}&=\mu \sin \theta \sin {\frac {\phi }{2}}\\A&=2\mu \sin \theta \sin {\frac {\Omega _{R}t}{2}}\end{aligned}}$

so that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \cos \alpha &= 1 - \frac{4 \mu^2 \sin^2 \theta \sin^2 \frac{\Omega_R t}{2}}{2 \mu^2} \\ &= 1 - 2 \sin^2 \theta \sin^2 \frac{\Omega_R t}{2} \\ \mu_z(t) &= \mu \cos \alpha \\ &= \mu \left( 1 - 2 \frac{\omega_R^2}{\delta^2 + \omega_R^2} \sin^2 \frac{\Omega_R t}{2} \right) \\ \mu_z(t) &= \mu \left( 1 - 2 \frac{\omega_R^2}{\Omega_R^2} \sin^2 \frac{\Omega_R t}{2} \right) \end{align}}

With Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0 = \gamma B_0} the Larmor frequency of the static field, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=\omega-\omega_0} the detuning, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R=\gamma B_1} the resonant and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_R=\sqrt{\omega_R^2+\delta^2}} the generalized Rabi frequencies. Note that the precession is faster, but the amplitude smaller for an off-resonant field than for the resonant case. The above result is also the correct quantum-mechanical result.

"Rapid" Adiabiatic Passage

Rapid adiabatic passage is a technique for inverting a spin by (slowly) sweeping the detuning of a drive field through resonance. "Slowly" means slowly compared to the Larmor frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma B_\text{eff}} about the effective static field in the rotating frame for all times. The physical picture is as follows. Assume the detuning is initially negative (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta<0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \delta\gg\omega_R\right| } ). Since

$\tan \theta ={\frac {B_{1}}{B_{0}-{\frac {\omega }{\gamma }}}}={\frac {\omega _{R}}{\omega _{0}-\omega }}=-{\frac {\omega _{R}}{\delta }}$

the effective magnetic field initially points of a small angle relative to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\hat e}} _z} axis. If the detuning is increased slowly compared to the Larmor frequency, the spin will continue to precess tightly around Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _\text{eff}} , which for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=0} points along the x axis, and for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta\gg\omega_R} along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - {\bf{\hat e}} _z} axis (see Figure \ref{fig:rapid_adiabatic_passage}). \begin{figure}

\caption{Motion of the spin during rapid adiabatic passage, viewed in the frame rotating with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _1} . The spin's rapid precession locks it to the direction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _\text{eff}} and thus it is dragged through an angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} as the frequency is swept through resonance.}

\end{figure} Thus the magnetic moment, starting out along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _0=B_0 {\bf{\hat e}} _z} , ends up pointing along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - {\bf{B}} _0=B_0 {\bf{\hat e}} _z} . Note that in the rotating frame Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{\mu}} } remains always (almost) parallel to the effective field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _\text{eff}} . A similar precess is used in magnetic traps for atoms, but there Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\bf{B}} _\text{eff}} is a real, spatially dependent field constant in time. As the atom moves in this field, the fast precession of the magnetic moment about the local field keeps its direction locked to the local field, whose direction varies in the lab frame. Returning to rapid adiabatic passage, since the generalized Rabi frequency is smallest and equal to the resonant Rabi frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=0} , the adiabatic requirement is most severe there, i.e. near Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta=\pi/2} . Near Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta=\pi/2} we have, with $B_{z}=B_{0}-1/\gamma \omega (t)$,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \dot\theta\right| =\frac{ \left| \dot{B}_z\right| }{B_1}=\frac{ \left| \dot\omega\right| }{\gamma B_1} =\frac{ \left| \dot\omega\right| }{\omega_R}\overset{!}{\ll}\omega_R }

where the exclamation point in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overset{!}{\ll}} indicates a requirement which we impose. Consequently, if the evolution is to be adiabatic, we must have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \dot{\omega}\right| \ll\omega_R^2} . This means that the change $\Delta \omega$ of rotation frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} per Rabi period Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=2\pi/\omega_R} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \Delta\omega\right| = \left| \dot\omega\right| T= \left| \dot\omega\right| /\omega_R 2\pi} , must be small compared to the Rabi frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_R} . The quantum mechanical treatment yields a probability for non-adiabatic transition (probability for the magnetic moment not following the magnetic field) given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\text{na}=exp\left(-\frac{\pi}{2}\frac{\omega^2_R}{ \left| \dot\omega\right| }\right) }

in agreement with the above qualitative discussion.