Categories: Optics, Physics, Quantum mechanics, Two-level system.

Rabi oscillation

In quantum mechanics, we know from the amplitude rate equations that a time-dependent term H^1\hat{H}_1 in the Hamiltonian affects the state as follows, where cn(t)c_n(t) are the coefficients of the linear combination of unperturbed basis states neiEnt/\ket{n} e^{-i E_n t / \hbar}:

idcmdt=ncn(t)mH^1neiωmnt\begin{aligned} i \hbar \dv{c_m}{t} = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1}{n} e^{i \omega_{mn} t} \end{aligned}

Where ωmn(Em ⁣ ⁣En)/\omega_{mn} \equiv (E_m \!-\! E_n) / \hbar for energies EmE_m and EnE_n. Consider a two-level system {a,b}\{\ket{a}, \ket{b}\} with Ea<EbE_a < E_b, in which case the above equation can be written out as:

dcadt=iaH^1beiω0tcbiaH^1acadcbdt=ibH^1aeiω0tcaibH^1bcb\begin{aligned} \dv{c_a}{t} &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} e^{-i \omega_0 t} \: c_b - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{a} c_a \\ \dv{c_b}{t} &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} e^{i \omega_0 t} \: c_a - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{b} c_b \end{aligned}

Where ω0ωba\omega_0 \equiv \omega_{ba} is positive. We assume that H^1\hat{H}_1 has odd spatial parity, in which case Laporte’s selection rule states that the diagonal matrix elements vanish, leaving:

dcadt=iaH^1beiω0tcbdcbdt=ibH^1aeiω0tca\begin{aligned} \dv{c_a}{t} &= - \frac{i}{\hbar} \matrixel{a}{\hat{H}_1}{b} e^{-i \omega_0 t} \: c_b \\ \dv{c_b}{t} &= - \frac{i}{\hbar} \matrixel{b}{\hat{H}_1}{a} e^{i \omega_0 t} \: c_a \end{aligned}

We now choose H^1\hat{H}_1 to be as follows, sinusoidally oscillating with a spatially odd V(r)V(\vec{r}):

H^1(t)=Vcos(ωt)=V2(eiωt+eiωt)\begin{aligned} \hat{H}_1(t) = V \cos(\omega t) = \frac{V}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big) \end{aligned}

We insert this into the equations for cac_a and cbc_b, and define VabaVbV_{ab} \equiv \matrixel{a}{V}{b}, leading us to:

dcadt=iVab2(ei(ωω0)t+ei(ω+ω0)t)cbdcbdt=iVab2(ei(ω+ω0)t+ei(ωω0)t)ca\begin{aligned} \dv{c_a}{t} &= - i \frac{V_{ab}}{2 \hbar} \Big( e^{i (\omega - \omega_0) t} + e^{-i (\omega + \omega_0) t} \Big) \: c_b \\ \dv{c_b}{t} &= - i \frac{V_{ab}}{2 \hbar} \Big( e^{i (\omega + \omega_0) t} + e^{-i (\omega - \omega_0) t} \Big) \: c_a \end{aligned}

Here, we make the rotating wave approximation: assuming we are close to resonance ωω0\omega \approx \omega_0, we argue that ei(ω+ω0)te^{i (\omega + \omega_0) t} oscillates so fast that its effect is negligible when the system is observed over a reasonable time interval. Dropping those terms leaves us with:

dcadt=iVab2ei(ωω0)tcbdcbdt=iVba2ei(ωω0)tca\begin{aligned} \boxed{ \begin{aligned} \dv{c_a}{t} &= - i \frac{V_{ab}}{2 \hbar} \: e^{i (\omega - \omega_0) t} \: c_b \\ \dv{c_b}{t} &= - i \frac{V_{ba}}{2 \hbar} \: e^{-i (\omega - \omega_0) t} \: c_a \end{aligned} } \end{aligned}

Now we can solve this system of coupled equations exactly. We differentiate the first equation with respect to tt, and then substitute dcb/dt\idv{c_b}{t} for the second equation:

d2cadt2=iVab2(i(ωω0)cb+dcbdt)ei(ωω0)t=iVab2(i(ωω0)cbiVba2ei(ωω0)tca)ei(ωω0)t=Vab2(ωω0)ei(ωω0)tcbVab2(2)2ca\begin{aligned} \dvn{2}{c_a}{t} &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b + \dv{c_b}{t} \bigg) e^{i (\omega - \omega_0) t} \\ &= - i \frac{V_{ab}}{2 \hbar} \bigg( i (\omega - \omega_0) \: c_b - i \frac{V_{ba}}{2 \hbar} \: e^{-i (\omega - \omega_0) t} \: c_a \bigg) e^{i (\omega - \omega_0) t} \\ &= \frac{V_{ab}}{2 \hbar} (\omega - \omega_0) \: e^{i (\omega - \omega_0) t} \: c_b - \frac{|V_{ab}|^2}{(2 \hbar)^2} \: c_a \end{aligned}

In the first term, we recognize dca/dt\idv{c_a}{t}, which we insert to arrive at an equation for ca(t)c_a(t):

0=d2cadt2i(ωω0)dcadt+Vab2(2)2ca\begin{aligned} 0 = \dvn{2}{c_a}{t} - i (\omega - \omega_0) \dv{c_a}{t} + \frac{|V_{ab}|^2}{(2 \hbar)^2} \: c_a \end{aligned}

To solve this, we make the ansatz ca(t)=eλtc_a(t) = e^{\lambda t}, which, upon insertion, gives us:

0=λ2i(ωω0)λ+Vab2(2)2\begin{aligned} 0 = \lambda^2 - i (\omega - \omega_0) \lambda + \frac{|V_{ab}|^2}{(2 \hbar)^2} \end{aligned}

This quadratic equation has two complex roots λ1\lambda_1 and λ2\lambda_2, which are found to be:

λ1=iωω0+Ω~2λ2=iωω0Ω~2\begin{aligned} \lambda_1 = i \frac{\omega - \omega_0 + \tilde{\Omega}}{2} \qquad \quad \lambda_2 = i \frac{\omega - \omega_0 - \tilde{\Omega}}{2} \end{aligned}

Where we have defined the generalized Rabi frequency Ω~\tilde{\Omega} to be given by:

Ω~(ωω0)2+Vab22\begin{aligned} \boxed{ \tilde{\Omega} \equiv \sqrt{(\omega - \omega_0)^2 + \frac{|V_{ab}|^2}{\hbar^2}} } \end{aligned}

So that the general solution ca(t)c_a(t) is as follows, where AA and BB are arbitrary constants, to be determined from initial conditions (and normalization):

ca(t)=(Asin(Ω~t/2)+Bcos(Ω~t/2))ei(ωω0)t/2\begin{aligned} \boxed{ c_a(t) = \Big( A \sin(\tilde{\Omega} t / 2) + B \cos(\tilde{\Omega} t / 2) \Big) e^{i (\omega - \omega_0) t / 2} } \end{aligned}

And then the corresponding cb(t)c_b(t) can be found from the coupled equation we started at, or, if we only care about the probability density ca2|c_a|^2, we can use cb2=1ca2|c_b|^2 = 1 - |c_a|^2. For example, if A=0A = 0 and B=1B = 1, we get the following probabilities

ca(t)2=cos2(Ω~t/2)=12(1+cos(Ω~t))cb(t)2=sin2(Ω~t/2)=12(1cos(Ω~t))\begin{aligned} |c_a(t)|^2 &= \cos^2(\tilde{\Omega} t / 2) = \frac{1}{2} \Big( 1 + \cos(\tilde{\Omega} t) \Big) \\ |c_b(t)|^2 &= \sin^2(\tilde{\Omega} t / 2) = \frac{1}{2} \Big( 1 - \cos(\tilde{\Omega} t) \Big) \end{aligned}

Note that the period was halved by squaring. This periodic “flopping” of the particle between a\ket{a} and b\ket{b} is known as Rabi oscillation, Rabi flopping or the Rabi cycle. This is a more accurate treatment of the flopping found from first-order time-dependent perturbation theory.

The name generalized Rabi frequency suggests that there is a non-general version. Indeed, the Rabi frequency Ω\Omega is based on the special case of exact resonance ω=ω0\omega = \omega_0:

ΩVba\begin{aligned} \Omega \equiv \frac{V_{ba}}{\hbar} \end{aligned}

Some authors use Vba|V_{ba}| instead, but not doing that lets us use Ω\Omega as a nice abbreviation. As an example, Rabi oscillation arises in the electric dipole approximation, where H^1\hat{H}_1 is:

H^1(t)=qrE0cos(ωt)\begin{aligned} \hat{H}_1(t) = - q \vec{r} \cdot \vec{E}_0 \cos(\omega t) \end{aligned}

After making the rotating wave approximation, the resulting Rabi frequency is given by:

Ω=dE0\begin{aligned} \Omega = - \frac{\vec{d} \cdot \vec{E}_0}{\hbar} \end{aligned}

Where E0\vec{E}_0 is the electric field amplitude, and dqbra\vec{d} \equiv q \matrixel{b}{\vec{r}}{a} is the transition dipole moment of the electron between orbitals a\ket{a} and b\ket{b}. Apparently, some authors define d\vec{d} with the opposite sign, thereby departing from its classical interpretation.


  1. D.J. Griffiths, D.F. Schroeter, Introduction to quantum mechanics, 3rd edition, Cambridge.