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 coefficients of a linear combination of unperturbed basis states ∣n⟩e−iEnt/ℏ\ket{n} e^{-i E_n t / \hbar}:

iℏdcmdt=∑ncn(t)⟨m∣H^1(t)∣n⟩eiωmnt\begin{aligned} i \hbar \dv{c_m}{t} = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} e^{i \omega_{mn} t} \end{aligned}

Where ωmn≡(Em ⁣− ⁣En)/ℏ\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=−iℏ⟨a∣H^1∣b⟩e−iω0t cb−iℏ⟨a∣H^1∣a⟩cadcbdt=−iℏ⟨b∣H^1∣a⟩eiω0t ca−iℏ⟨b∣H^1∣b⟩cb\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 ω0≡ωba\omega_0 \equiv \omega_{ba} is positive. It is realistic to 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=−iℏ⟨a∣H^1∣b⟩e−iω0t cbdcbdt=−iℏ⟨b∣H^1∣a⟩eiω0t ca\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^(r⃗)\hat{V}(\vec{r}):

H^1(t)=V^cos⁡(ωt)=V^2(eiωt+e−iωt)\begin{aligned} \hat{H}_1(t) = \hat{V} \cos(\omega t) = \frac{\hat{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 Vab≡⟨a∣V^∣b⟩V_{ab} \equiv \matrixel{a}{\hat{V}}{b}, leading us to:

dcadt=−iVab2ℏ(ei(ω−ω0)t+e−i(ω+ω0)t) cbdcbdt=−iVab2ℏ(ei(ω+ω0)t+e−i(ω−ω0)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 ω≈ω0\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=−iVab2ℏ ei(ω−ω0)t cbdcbdt=−iVba2ℏ e−i(ω−ω0)t ca\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=−iVab2ℏ(i(ω−ω0) cb+dcbdt)ei(ω−ω0)t=−iVab2ℏ(i(ω−ω0) cb−iVba2ℏ e−i(ω−ω0)t ca)ei(ω−ω0)t=Vab2ℏ(ω−ω0) ei(ω−ω0)t cb−∣Vab∣2(2ℏ)2 ca\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=d2cadt2−i(ω−ω0)dcadt+∣Vab∣2(2ℏ)2 ca\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=λ2−i(ω−ω0)λ+∣Vab∣2(2ℏ)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=iω−ω0+Ω~2λ2=iω−ω0−Ω~2\begin{aligned} \lambda_1 = i \frac{\omega - \omega_0 + \tilde{\Omega}}{2} \qquad \qquad \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~≡(ω−ω0)2+∣Vab∣2ℏ2\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, determined by the initial conditions and normalization:

ca(t)=(Asin⁡(Ω~t/2)+Bcos⁡(Ω~t/2))ei(ω−ω0)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 ∣ca∣2|c_a|^2, we can use ∣cb∣2=1−∣ca∣2|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=cos⁡2(Ω~t/2)=12(1+cos⁡(Ω~t))∣cb(t)∣2=sin⁡2(Ω~t/2)=12(1−cos⁡(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 called 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:

Ω≡Vbaℏ\begin{aligned} \Omega \equiv \frac{V_{ba}}{\hbar} \end{aligned}

Some authors write ∣Vba∣|V_{ba}| instead, but not doing so lets us use Ω\Omega as a nice abbreviation. For example, Rabi oscillation arises for electrons in an oscillating electric field (e.g. a light wave), in which case H^1\hat{H}_1 is as follows according to the electric dipole approximation, with charge q<0q < 0:

H^1(t)=−qr⃗⋅E⃗0cos⁡(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=−d⃗⋅E⃗0ℏ\begin{aligned} \Omega = - \frac{\vec{d} \cdot \vec{E}_0}{\hbar} \end{aligned}

Where E⃗0\vec{E}_0 is the electric field amplitude, and d⃗≡q⟨b∣r⃗∣a⟩\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 d⃗\vec{d} with the opposite sign, thereby departing from its classical interpretation.

References

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