Bloch equation

The mathematical model which is suitable for analyzing such problems is to consider an ensemble of spins, each spin being described by a magnetization vector $M = (M_X, M_Y, M_Z)$ in the laboratory frame whose evolution satisfies the so-called Bloch equation
\[ \dot{M}(\tau) = \bar{\gamma}\, M(\tau) \wedge B(\tau) + R(M(\tau)),\]
where $\tau$ is the time, $\bar{\gamma}$ is the gyromagnetic ratio of the considered nucleus, $B(\tau) = (B_X(\tau), B_Y(\tau), B_Z(\tau))$ is the total magnetic field applied to the system which decomposes into
\[ B(\tau) = B_0 + B_1(\tau),\]
where $B_0 = B_Z\, e_Z$ is a constant and strong polarizing field in the $Z$-direction, while the control RF-field $B_1(\tau) = B_X(\tau)\, e_X + B_Y(\tau)\, e_Y$ is in the transverse plane $(X,Y)$. The vectors $e_X$, $e_Y$ and $e_Z$ form the standard basis of $\R^3$. The $R(M)$ term represents the dissipation and is of the form:
\[ R(M) = \left( -\frac{M_X}{T_2}, -\frac{M_Y}{T_2}, -\frac{(M_Z-M_0)}{T_1} \right),\]
where $M_0$ is the equilibrium magnetization, and $T_1$, $T_2$ are the relaxation times which are the chemical signatures of the observed species and satisfy the physical constraints $0 < T_2 \le 2\, T_1$. The control is denoted $\omega(\tau) = (\omega_X(\tau),\omega_Y(\tau)) \coloneqq (-\bar{\gamma} B_X(\tau), -\bar{\gamma} B_Y(\tau))$ and is bounded by $\omega_\mathrm{max}$, i.e. $\Vert{\omega(\tau)}\Vert\le \omega_\mathrm{max}$, where $\omega_\mathrm{max}$ is the maximal experimental intensity of the experiments and $\Vert{\cdot}\Vert$ is the Euclidean norm.
The Bloch equation can be written in a rotating frame where the equilibrium is normalized introducing the state $q=(x,y,z)$, the matrices
\[ \Omega_z = \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\]
and $S(\tau) = \exp(\tau \, \bar{\omega}\, \Omega_z)$, and writting $M(\tau) = M_0 S(\tau) q(\tau)$. In the rotating frame the Bloch equation becomes:
\[ \dot{q}(\tau) = \begin{bmatrix} -{1}/{T_2} & -\Delta\omega & \omega_y \\ \Delta\omega & -{1}/{T_2} & -\omega_x \\ -\omega_y & \omega_x & -{1}/{T_1} \\ \end{bmatrix}\, q(\tau) + \begin{bmatrix} 0 \\ 0 \\ {1}/{T_1} \end{bmatrix},\]
where $\Delta \omega = \omega_0 - \bar{\omega}$ is the resonance offset, $\omega_0 = -\bar{\gamma} B_Z$ is the resonance frequency and the new control is
\[ \begin{aligned} \omega_x(\tau) \coloneqq \omega_X(\tau) \cos(\bar{\omega} \tau) + \omega_Y(\tau) \sin(\bar{\omega} \tau), \\ \omega_y(\tau) \coloneqq \omega_Y(\tau) \cos(\bar{\omega} \tau) - \omega_X(\tau) \sin(\bar{\omega} \tau), \end{aligned}\]
which preserves the control bound $\omega_x^2(\tau) + \omega_y^2(\tau) \le \omega_\mathrm{max}$. Finally, we introduce the normalized control and time:
\[ u \coloneqq \frac{u_\mathrm{max}}{\omega_\mathrm{max}}\, \omega, \quad t \coloneqq \frac{\omega_\mathrm{max}}{u_\mathrm{max}}\, \tau,\]
such that the normalized control satisfies $\Vert{u(t)}\Vert\le u_\mathrm{max}$. In the sequel, we fix $\bar{\omega} = \omega_0$ which gives $\Delta \omega = 0$ (it is called the resonant case) and which leads to the final normalized Bloch equation:
\[ \left\{ \begin{aligned} \dot{x}(t) &= \displaystyle -\Gamma\, x(t) + u_2(t)\, z(t), \\[0.2em] \dot{y}(t) &= \displaystyle -\Gamma\, y(t) - u_1(t)\, z(t), \\[0.2em] \dot{z}(t) &= \displaystyle \gamma\,(1-z(t)) + u_1(t)\, y(t) - u_2(t)\, x(t), \end{aligned} \right.\]
where
\[ \gamma \coloneqq \frac{u_\mathrm{max}}{w_\mathrm{max}\, T_1}, \quad \Gamma \coloneqq \frac{u_\mathrm{max}}{w_\mathrm{max}\, T_2}, \quad 0 < \gamma \le 2\Gamma.\]
Time parameterization
In this setting, the normalized Bloch equation has 3 positive parameters: $\gamma$, $\Gamma$ and $u_\mathrm{max}$. However, one may choose one parameter to fix since for any $\lambda > 0$ and any triplet $(\gamma, \Gamma, u_\mathrm{max})$, both systems, in coordinates $(x, y, z)$, with parameters $(\gamma, \Gamma, u_\mathrm{max})$ or $(\lambda\gamma, \lambda\Gamma, \lambda u_\mathrm{max})$ are equivalent, up to a time reparameterization.
Note that for the constant control $u(\cdot) =(u_\mathrm{max},0)$, starting from $x=0$ and denoting $q = (y,z)$, the Bloch equation becomes
\[ \dot{x}(t) = -\Gamma\, x(t), \quad \dot{q}(t) = u_\mathrm{max} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\, q(t),\]
and the solution is given by $x(t) = 0$, $q(t) = R(u_\mathrm{max}\, t) \, q(0)$ where $R(\theta)$ is the rotation matrix of angle $\theta$. Thus, this trajectory is periodic of period $T$ given by:
\[ T = \frac{2 \pi}{u_\mathrm{max}}.\]
Classically, one may fix the period to $T=1$ and so $u_\mathrm{max} = 2 \pi$, or on the contrary one may fix $u_\mathrm{max} = 1$ and choose $T = 2 \pi$. From now, we choose to fix $u_\mathrm{max} = 1$.
Spherical coordinates
The Bloch equation in the spherical coordinates
\[ \begin{aligned} x = \rho\, \sin\phi \cos\theta, \quad %\\[0.1em] y = \rho\, \sin\phi \sin\theta, \quad %\\[0.1em] z = \rho\, \cos\phi, \end{aligned}\]
with the feedback $u = R(\theta)^{-1} v$ becomes:
\[ \left\{ \begin{aligned} \dot{\rho}(t) &= \gamma \cos \phi(t)\, (1 - \rho(t) \cos \phi(t)) - \Gamma \rho(t) \sin^2 \phi(t), \\[1.0em] \dot{\phi}(t) &= \delta \sin \phi(t) \cos \phi(t) - \frac{\gamma}{\rho(t)} \sin \phi(t) + v_2(t), \\[0.0em] \dot{\theta}(t) &= - \cot \phi(t)\, v_1(t), \end{aligned} \right.\]
where $\delta \coloneqq \gamma - \Gamma$ and with the control constraint $v_1^2 + v_2^2 = u_1^2 + u_2^2 \le 1$.
The Bloch ball
Definition. The closed unit ball is called the Bloch Ball.
Proposition. The Bloch ball is invariant under the Bloch equations if and only if $0 \le \gamma \le 2 \Gamma$ and it is the smallest invariant closed ball centered at the origin if and only if $0 < \gamma \le 2\, \Gamma$.
Symmetry of revolution
Considering the Bloch equation in spherical coordinates, one can see that $\theta$ does not appear in the dynamics (it is a cyclic variable) so there exists a one dimensional symmetry group of translations on $\theta$. Hence, one may fix the initial value $\theta(0)$ to $\pi/2$ for instance (it implies $x(0) = 0$). Finally, imposing $\theta(0) = \pi/2$ and $u_2 = 0$, then, one can reduce (since $x(t)=0$ for any time $t$) the Bloch equation to the following two dimensional single-input control system:
\[ \left\{ \begin{aligned} \dot{y}(t) &= \displaystyle -\Gamma\, y(t) - u(t)\, z(t), \\[0.2em] \dot{z}(t) &= \displaystyle \gamma\,(1-z(t)) + u(t)\, y(t), \end{aligned} \right.\]
where $u \coloneqq u_1$ satisfies $|{u(t)}| \le 1$.
This control system is affine in the control and may be written in the form
\[ \dot{q}(t) = F_0(q(t)) + u(t)\, F_1(q(t)),\]
where $q \coloneqq (y,z)$ is the state and where the vector fields $F_0$ and $F_1$ are given by:
\[ F_0(q) = -\Gamma\, y \frac{\partial}{\partial y} + \gamma\, (1 - z) \frac{\partial}{\partial z}, \quad F_1(q) = -z \frac{\partial}{\partial y} + y \frac{\partial}{\partial z}.\]
The admissible control set $\mathcal{U}$ is then defined as
\[ \mathcal{U} \coloneqq \left\{ u(\cdot) \colon [0, \infty) \to [-1, 1] ~|~ u(\cdot) \text{ measurable} \right\}.\]