Two-Level System with Periodic Perturbation #

This content is primarily based on Section 8, “Approximation Method,” of Chapter 2 in Physics of Atoms and Molecules (2nd Edition) by B. H. Bransden and C. J. Joachain¹.

1. Problem Setup #

Consider the time-dependent Schrödinger Equation

$$ \begin{align} \label{e1} i\hbar \frac{\partial}{\partial t}|\Psi\rangle = \hat{H}(t)|\Psi\rangle \end{align} $$

where $\hat{H}$ is expressed using the unperturbed Hamiltonian $\hat{H}_0$ and the perturbed Hamiltonian $\hat{H}’(t)$ as follows:

$$ \begin{align} \label{e2} \hat{H}(t) = \hat{H}_0 + \hat{H}’(t) \end{align} $$

Here, $\hat{H}_0$ describes a two-level system, meaning a system with only two eigenstates.

Assuming that the Schrödinger equation corresponding to $\hat{H}_0$ has already been solved, the eigenenergies and eigenstates satisfy the following equations:

$$ \begin{align} \hat{H}_0|b\rangle &= E_b |b\rangle \label{e3a}\\ \hat{H}_0|a\rangle &= E_a |a\rangle \label{e3b} \end{align} $$

Here, the two eigenenergies $E_a$ and $E_b$ are defined such that $E_b > E_a$.

Now, we define a periodic perturbation with respect to time as shown in Eq. \eqref{e4}.

$$ \begin{align} \label{e4} \hat{H}’(t) = \hat{A} e^{i\omega t} + \hat{A}^{\dagger} e^{-i\omega t} \end{align} $$

Here, $\hat{A}$ is a time-independent operator, and $\hat{A}^{\dagger}$ denotes its Hermitian conjugate. By constructing $\hat{H}’$ as in Eq. \eqref{e4}, $\hat{H}’$ becomes a Hermitian operator, as shown below:

$$ \begin{align} \label{e5} \hat{H}’(t) = {\hat{H}}^{’\dagger}(t) \end{align} $$

2. Solution #

2.1. Linear Combination of Eigenstates #

Considering the perturbation \eqref{e5} for the two-level system, the wavefunction can be expressed as a linear combination of the two eigenstates of $\hat{H}_0$:

$$ \begin{align} \label{e6} |\Psi(t)\rangle = c_a(t)|a\rangle e^{-iE_a t/\hbar} + c_b(t)|b\rangle e^{-iE_b t/\hbar} \end{align} $$

Substituting Eqs. \eqref{e2} and \eqref{e6} into Eq. \eqref{e1}, we obtain the following equations for the unknown coefficients $c_a$ and $c_b$:

$$ \begin{eqnarray} \left\{ \begin{aligned} i\hbar\dot{c}_a(t)&=\left[A_{aa} e^{i\omega t}+ A_{aa}^{\dagger} e^{-i\omega t}\right]c_a(t) + \left[A_{ab} e^{i\Delta \omega t} + A_{ab}^{\dagger} e^{-i(\omega+\omega_{ba}) t}\right]c_b(t) \\ i\hbar\dot{c}_b(t)&=\left[A_{ba} e^{i(\omega+\omega_{ba}) t} + A_{ba}^{\dagger} e^{-i\Delta \omega t}\right]c_a(t) + \left[A_{bb} e^{i\omega t} + A_{bb}^{\dagger} e^{-i\omega t}\right]c_b(t) \end{aligned} \right. \label{e7} \end{eqnarray} $$

Here, we set

$$ \begin{gather} \omega_{ba}=(E_b-E_a)/\hbar,~~\Delta \omega=\omega-\omega_{ba}, \label{e8a}\\ A_{ba}=\langle b|\hat{A}|a\rangle,~~A^{\dagger}_{ba}=A_{ab}^{*}. \label{e8b} \end{gather} $$

To solve the differential equation \eqref{e7}, we assume the initial condition

$$ \begin{align} \label{e9} c_a(t\leq 0)=1,~~c_b(t\leq 0)=0 \end{align} $$

where the system exists only in the state $|a\rangle$ for $t=0$.

2.2. Rotating Wave Approximation #

An interesting phenomenon occurs when the angular frequency of the periodic perturbation is close to the angular frequency corresponding to the energy gap between the two levels. We proceed by solving Eq. \eqref{e7} under the condition $\Delta \omega \ll \omega$.

The changes in the coefficients $c_a$ and $c_b$ are described by the right-hand side of Eq. \eqref{e7}. When solving the differential equation, integrating the rapidly oscillating terms over time on a long timescale results in negligible contributions. Thus, we apply an approximation (Rotating Wave Approximation) that neglects the rapidly oscillating terms.

Specifically, the term $e^{i\Delta \omega t}$ oscillates more slowly than $e^{\pm i(\omega+\omega_{ba})t}$, $e^{\pm i\omega t}$, and $e^{\pm i\omega_{ba} t}$, so we retain only this term. As a result:

$$ \begin{eqnarray} \left\{ \begin{aligned} i\hbar\dot{c}_a(t)&\approx A_{ab} e^{i\Delta \omega t} c_b(t) \\ i\hbar\dot{c}_b(t)&\approx A_{ba}^{\dagger} e^{-i\Delta \omega t} c_a(t) \end{aligned} \right.\label{e10} \end{eqnarray} $$

The equation \eqref{e10} can be solved exactly, yielding:

$$ \begin{eqnarray} \left\{ \begin{aligned} c_a(t) &= e^{i\Delta \omega t/2} \left[\cos\left(\frac{\omega_R}{2}t\right) - i \left(\frac{\Delta \omega}{\omega_R}\right)\sin\left(\frac{\omega_R}{2}t\right)\right] \\ c_b(t) &= \frac{2A_{ba}^{\dagger}}{i\hbar \omega_R} e^{-i\Delta \omega t/2} \sin\left(\frac{\omega_R}{2}t\right) \end{aligned} \right. \label{e11} \end{eqnarray} $$

Here,

$$ \begin{align}\label{e12} \omega_R=\left[{\Delta \omega}^2 + \frac{4|A_{ba}^{\dagger}|^2}{\hbar^2}\right]^{1/2} \end{align} $$

is called the Rabi flopping frequency.

The probability density of states $|a\rangle$ and $|b\rangle$ is expressed as:

$$ \begin{eqnarray} \left\{ \begin{aligned} |c_a(t)|^2 &= \cos^2\left(\frac{\omega_R}{2}t\right) + \left(\frac{\Delta \omega}{\omega_R}\right)^2\sin^2\left(\frac{\omega_R}{2}t\right) \\ P_{ba}(t)\equiv |c_b(t)|^2 &= \frac{4|A_{ba}^{\dagger}|^2}{\hbar^2 \omega_R^2} \sin^2\left(\frac{\omega_R}{2}t\right) \end{aligned} \right.\label{e13} \end{eqnarray} $$

Equation \eqref{e13} is an exact solution under the rotating wave approximation.

From Eq. \eqref{e13}, the probability density exhibits simple harmonic oscillations, with the oscillation period $T$ given by:

$$ \begin{align} \label{e14} T=\frac{2\pi}{\omega_R} \end{align} $$

です。

On the other hand, comparing this problem to the result derived using time-dependent first-order perturbation theory, we obtain the following result (derivation omitted):

$$ \begin{align} \label{e15} P_{ba}^{(1)}(t) = \frac{4|A_{ba}^{\dagger}|^2}{\hbar^2 {\Delta\omega}^2} \sin^2\left(\frac{\Delta\omega}{2}t\right) \end{align} $$

Comparing the results of the rotating wave approximation \eqref{e13} with the perturbation theory result \eqref{e15}, we find the following two points:

• When $\Delta \omega \ne 0$ and $|A_{ba}^{\dagger}|^2$ is sufficiently weak, the two results agree.
• When $\Delta = 0$, perturbation theory \eqref{e15} gives $P_{ba}^{(1)} \propto t^2$, so it is only valid for a very short time.

References #