Comprehensive solutions to the Bloch equations and dynamical models for open two-level systems

The Bloch equation and its variants constitute the fundamental dynamical model for arbitrary two-level systems. Many important processes, including those in more complicated systems, can be modeled and understood through the two-level approximation. It is therefore of widespread relevance, especially as it relates to understanding dissipative processes in current cutting-edge applications of quantum mechanics. Although the Bloch equation has been the subject of considerable analysis in the 70 years since its inception, there is still, perhaps surprisingly, significant work that can be done. This paper extends the scope of previous analyses. It provides a framework for more fully understanding the dynamics of dissipative two-level systems. A solution is derived that is compact, tractable, and completely general, in contrast to previous results. Any solution of the Bloch equation depends on three roots of a cubic polynomial that are crucial to the time dependence of the system. The roots are typically only sketched out qualitatively, with no indication of their dependence on the physical parameters of the problem. Degenerate roots, which modify the solutions, have been ignored altogether. Here the roots are obtained explicitly in terms of a single real-valued root that is expressed as a simple function of the system parameters. For the conventional Bloch equation, a simple graphical representation of this root is presented that makes evident the explicit time dependence of the system for each point in the parameter space. Several intuitive, visual models of system dynamics are developed. A Euclidean coordinate system is identified in which any generalized Bloch equation is separable, i.e., the sum of commuting rotation and relaxation operators. The time evolution in this frame is simply a rotation followed by relaxation at modified rates that play a role similar to the standard longitudinal and transverse rates. These rates are functions of the applied field, which provides information towards control of the dissipative process. The Bloch equation also describes a system of three coupled harmonic oscillators, providing additional perspective on dissipative systems.

Figure 1

Parameter values of ${\omega }_{12}^2$ that give degenerate roots of the characteristic polynomial ($\gamma =1$) and critically damped solutions to the Bloch equation are plotted as a function of ${\omega }_3^2$, shown as red (solid) lines calculated using Eq. (48). The parameters are scaled to $R_{\delta }^2/3$ as in Eq. (47). In the interior of the region delineated by these curves (light red), there are three distinct real roots (${\tilde{c}}_1<0$ and $\gamma <1$) resulting in overdamped solutions. Outside this region (light blue), one real and two complex-conjugate roots produce oscillatory underdamped solutions, with ${\tilde{c}}_1>0$ above the overdamped region and ${\tilde{c}}_1>0$ and $\gamma >1$ below the overdamped region. In addition, underdamped solutions are obtained for any ${\omega }_3^2>1$ (normalized to the units shown in the figure).

Figure 2

Contours of the characteristic polynomial's guaranteed real root $z_1$, calculated according to Eqs. (C6) and normalized to $R_{\delta }$, are plotted as a function of ${\omega }_{12}^2$ and ${\omega }_3^2$ normalized as in Fig. 1. The root satisfies $-1\le z_1\le 2$, as expected from Eq. (51), with lines of constant $z_1$ as derived in Eqs. (53, 54, 55). The $z_1=0$ contour is shown as a dashed line. Contours of the frequency $\varpi$ from Eq. (22) that appears in the oscillatory underdamped solutions of the Bloch equation are also plotted in the rightmost panels. Within the overdamped region defined in Fig. 1 and expanded in the bottom panels, there is no oscillation or frequency $\varpi$, and only one of the three real roots is plotted.

Figure 3

The Bloch equation is shown in the text to model the displacements, from equilibrium positions $r_i=0$, of a system of three unit masses coupled by springs of stiffness $k_{ij}$. One model identifies velocity-dependent damping terms. An alternative model is expressed as an ideal frictionless system that is, nonetheless, damped. Asymmetric couplings $k_{ij}\ne k_{ji}$ provide a dissipative mechanism in both models. The mechanical springs depicted in the figure are therefore only an analogy.

Figure 4

Trajectories for initial vector ${\mathcal{M}}_0$ acted upon by propagator $e^{-\mathrm{\Gamma }t}$ are displayed in the $\{{\tilde{\mathit{s}}}_1,{\tilde{\mathit{s}}}_2,{\tilde{\mathit{s}}}_3\}$ coordinates developed as the natural system for describing propagator dynamics. The component of ${\mathcal{M}}_0$ along ${\tilde{\mathit{s}}}_1$ decays at the rate $\overline{R}-z_1$, while components in the $({\tilde{\mathit{s}}}_2,{\tilde{\mathit{s}}}_3)$ plane rotate in the plane and decay at the rate $\overline{R}+z_1/2$. The different panels represent different ${\mathcal{M}}_0$, fields ${\mathbf{\omega }}_e$, transverse relaxation rate $R_2$, and longitudinal relaxation rate $R_3$, with details of the predicted system evolution described in more detail in the text. Physical parameters are in units of inverse seconds. (a) Initial state ${\mathcal{M}}_0=(-1,1,1)$. Physical parameters ${\mathbf{\omega }}_e=(0,0,{10}^4), R_2=400$, and $R_3=200$ give coordinates ${\tilde{\mathit{s}}}_1=\widehat{\mathit{z}}, {\tilde{\mathit{s}}}_2=\widehat{\mathit{y}}$, and ${\tilde{\mathit{s}}}_3=\widehat{\mathit{x}}$ and the well-known rotation about ${\mathbf{\omega }}_e={\omega }_3$ followed by longitudinal and transverse relaxation. (b) Initial state ${\mathcal{M}}_0=(1,-1,0)$. Parameters ${\mathbf{\omega }}_e=(5000,0,0), R_2=400$, and $R_3=200$ lead to coordinates ${\tilde{\mathit{s}}}_1=\widehat{\mathit{x}}, {\tilde{\mathit{s}}}_2=(0,-1,.02)$, and ${\tilde{\mathit{s}}}_3=\widehat{\mathit{z}}$. Rotation is also about ${\mathbf{\omega }}_e$ for ${\omega }_3=0$ (on resonance), but now ${\tilde{\mathit{s}}}_2$ is not perpendicular to ${\tilde{\mathit{s}}}_3$, so the rotation in the plane transverse to ${\tilde{\mathit{s}}}_1$ is not at constant angular frequency. (c) Parameters ${\mathbf{\omega }}_e=(0,300,300), R_2=100$, and $R_3=1$ lead to nonorthogonal oblique coordinates ${\tilde{\mathit{s}}}_1=(0.12,0.69,0,71), {\tilde{\mathit{s}}}_2=(0.99,0.04,0.12)$, and ${\tilde{\mathit{s}}}_3=(0.,0.72,-0.70)$. Initial ${\mathcal{M}}_0=(-0.12,0.69,0,71)$ is normal to the $({\tilde{\mathit{s}}}_2,{\tilde{\mathit{s}}}_3)$ plane, but has components in the plane and along ${\tilde{\mathit{s}}}_1$ in the oblique coordinate system, so spirals about ${\tilde{\mathit{s}}}_1$ as shown. (d) Initial $\mathcal{M}=(-0.99,0.17,0)$ is orthogonal to ${\tilde{\mathit{s}}}_1$. Parameters ${\mathbf{\omega }}_e=(0,3000,3000), R_2=1000$, and $R_3=1$ lead to nearly identical coordinates as in (c). Here ${\mathcal{M}}_0$ projects onto ${\tilde{\mathit{s}}}_1$ in oblique coordinates and therefore decays along this direction, resulting in the spiral as shown.