Analytically Solvable Driven Time-Dependent Two-Level Quantum Systems

Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.

Figure 1

Pulse from Eq. (20) with (top panel from left to right) $a=0,\frac{2}{3},\frac{5}{3}$ and (bottom panel from left to right) $a=-1,-\frac{1}{4}$.

Figure 2

Pulse from Eq. (22) with (from top to bottom) $b=-\frac{1}{4},0,\frac{1}{2},1,2$ and the corresponding evolution operator components.

Figure 3

Pulse from Eq. (27) with (left panel, from top to bottom) $a=2\sqrt{2},2,\sqrt{2},1,1/\sqrt{2}$ and (right panel, from top to bottom) $a=1/\sqrt{2},0.6,0.5,0.4,0.3$.

Figure 4

Control from Eq. (28) with (left) $a=0.1$ and (right) $a=0.5$.