diff options
Diffstat (limited to 'source/know/concept/optical-bloch-equations/index.md')
| -rw-r--r-- | source/know/concept/optical-bloch-equations/index.md | 231 |
1 files changed, 231 insertions, 0 deletions
diff --git a/source/know/concept/optical-bloch-equations/index.md b/source/know/concept/optical-bloch-equations/index.md new file mode 100644 index 0000000..fe74b7e --- /dev/null +++ b/source/know/concept/optical-bloch-equations/index.md @@ -0,0 +1,231 @@ +--- +title: "Optical Bloch equations" +sort_title: "Optical Bloch equations" +date: 2023-01-19 +categories: +- Physics +- Quantum mechanics +- Two-level system +layout: "concept" +--- + +For an electron in a two-level system with time-independent states +$$\ket{g}$$ (ground) and $$\ket{e}$$ (excited), +consider the following general solution +to the time-dependent Schrödinger equation: + +$$\begin{aligned} + \ket{\Psi} + &= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar} +\end{aligned}$$ + +Perturbing this system with +an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) +introduces a time-dependent sinusoidal term $$\hat{H}_1$$ to the Hamiltonian. +In the [electric dipole approximation](/know/concept/electric-dipole-approximation/), +$$\hat{H}_1$$ is given by: + +$$\begin{aligned} + \hat{H}_1(t) + = - \hat{\vb{p}} \cdot \vb{E}(t) + \qquad \qquad + \vu{p} + \equiv q \vu{x} + \qquad \qquad + \vb{E}(t) + = \vb{E}_0 \cos(\omega t) +\end{aligned}$$ + +Where $$\vb{E}$$ is an [electric field](/know/concept/electric-field/), +and $$\hat{\vb{p}}$$ is the dipole moment operator. +From [Rabi oscillation](/know/concept/rabi-oscillation/), +we know that the time-varying coefficients $$c_g$$ and $$c_e$$ +can then be described by: + +$$\begin{aligned} + \dv{c_g}{t} + &= i \frac{q \matrixel{g}{\vu{x}}{e} \cdot \vb{E}_0}{2 \hbar} \: e^{i (\omega - \omega_0) t} \: c_e + \\ + \dv{c_e}{t} + &= i \frac{q \matrixel{e}{\vu{x}}{g} \cdot \vb{E}_0}{2 \hbar} \: e^{- i (\omega - \omega_0) t} \: c_g +\end{aligned}$$ + +Where $$\omega_0 \equiv (\varepsilon_e \!-\! \varepsilon_g) / \hbar$$ is the resonance frequency. +We want to rearrange these equations a bit, +so we split the field $$\vb{E}$$ as follows, +where the amplitudes $$\vb{E}_0^{-}$$ and $$\vb{E}_0^{+}$$ +may be slowly-varying with respect to the carrier wave $$e^{\pm i \omega t}$$: + +$$\begin{aligned} + \vb{E}(t) + &\equiv \vb{E}^{-}(t) + \vb{E}^{+}(t) + \\ + &\equiv \vb{E}_0^{-} e^{i \omega t} + \vb{E}_0^{+} e^{-i \omega t} +\end{aligned}$$ + +Since $$\vb{E}$$ is real, $$\vb{E}_0^{+} = (\vb{E}_0^{-})^*$$. +Similarly, we define the transition dipole moment $$\vb{p}_0^{-}$$: + +$$\begin{aligned} + \vb{p}_0^{-} + \equiv q \matrixel{e}{\vu{x}}{g} + \qquad \qquad + \vb{p}_0^{+} + \equiv (\vb{p}_0^{-})^* + = q \matrixel{g}{\vu{x}}{e} +\end{aligned}$$ + +With these, the equations for $$c_g$$ and $$c_e$$ can be rewritten as shown below. +Note that $$\vb{E}^{-}$$ and $$\vb{E}^{+}$$ include the driving plane wave, and the +[rotating wave approximation](/know/concept/rotating-wave-approximation/) is still made: + +$$\begin{aligned} + \dv{c_g}{t} + &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} e^{- i \omega_0 t} \: c_e + \\ + \dv{c_e}{t} + &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} e^{i \omega_0 t} \: c_g +\end{aligned}$$ + + +For $$\ket{\Psi}$$ as defined above, +the corresponding pure [density operator](/know/concept/density-operator/) +$$\hat{\rho}$$ is as follows: + +$$\begin{aligned} + \hat{\rho} + = \ket{\Psi} \bra{\Psi} + = + \begin{bmatrix} + c_e c_e^* & c_e c_g^* e^{-i \omega_0 t} \\ + c_g c_e^* e^{i \omega_0 t} & c_g c_g^* + \end{bmatrix} + \equiv + \begin{bmatrix} + \rho_{ee} & \rho_{eg} \\ + \rho_{ge} & \rho_{gg} + \end{bmatrix} +\end{aligned}$$ + +We take the $$t$$-derivative of the matrix elements, +and insert the equations for $$c_g$$ and $$c_e$$: + +$$\begin{aligned} + \dv{\rho_{gg}}{t} + &= \dv{c_g}{t} c_g^* + c_g \dv{c_g^*}{t} + \\ + &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_e c_g^* e^{- i \omega_0 t} + - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_g c_e^* e^{i \omega_0 t} + \\ + \dv{\rho_{ee}}{t} + &= \dv{c_e}{t} c_e^* + c_e \dv{c_e^*}{t} + \\ + &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_g c_e^* e^{i \omega_0 t} + - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_e c_g^* e^{- i \omega_0 t} + \\ + \dv{\rho_{ge}}{t} + &= \dv{c_g}{t} c_e^* e^{i \omega_0 t} + c_g \dv{c_e^*}{t} e^{i \omega_0 t} + i \omega_0 c_g c_e^* e^{i \omega_0 t} + \\ + &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_e c_e^* + - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_g c_g^* + + i \omega_0 c_g c_e^* e^{i \omega_0 t} + \\ + \dv{\rho_{eg}}{t} + &= \dv{c_e}{t} c_g^* e^{-i \omega_0 t} + c_e \dv{c_g^*}{t} e^{-i \omega_0 t} - i \omega_0 c_e c_g^* e^{- i \omega_0 t} + \\ + &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_g c_g^* + - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_e c_e^* + - i \omega_0 c_e c_g^* e^{- i \omega_0 t} +\end{aligned}$$ + +Recognizing the density matrix elements allows us +to reduce these equations to: + +$$\begin{aligned} + \dv{\rho_{gg}}{t} + &= \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big) + \\ + \dv{\rho_{ee}}{t} + &= \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big) + \\ + \dv{\rho_{ge}}{t} + &= i \omega_0 \rho_{ge} + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \big( \rho_{ee} - \rho_{gg} \big) + \\ + \dv{\rho_{eg}}{t} + &= - i \omega_0 \rho_{eg} + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \big( \rho_{gg} - \rho_{ee} \big) +\end{aligned}$$ + +These equations are correct if nothing else is affecting $$\hat{\rho}$$. +But in practice, these quantities decay due to various processes, +e.g. [spontaneous emission](/know/concept/einstein-coefficients/). + +Suppose $$\rho_{ee}$$ decays with rate $$\gamma_e$$. +Because the total probability $$\rho_{ee} + \rho_{gg} = 1$$, we have: + +$$\begin{aligned} + \Big( \dv{\rho_{ee}}{t} \Big)_{e} + = - \gamma_e \rho_{ee} + \quad \implies \quad + \Big( \dv{\rho_{gg}}{t} \Big)_{e} + = \gamma_e \rho_{ee} +\end{aligned}$$ + +Meanwhile, for whatever reason, +let $$\rho_{gg}$$ decay into $$\rho_{ee}$$ with rate $$\gamma_g$$: + +$$\begin{aligned} + \Big( \dv{\rho_{gg}}{t} \Big)_{g} + = - \gamma_g \rho_{gg} + \quad \implies \quad + \Big( \dv{\rho_{gg}}{t} \Big)_{g} + = \gamma_g \rho_{gg} +\end{aligned}$$ + +And finally, let the diagonal (perpendicular) matrix elements +both decay with rate $$\gamma_\perp$$: + +$$\begin{aligned} + \Big( \dv{\rho_{eg}}{t} \Big)_{\perp} + = - \gamma_\perp \rho_{eg} + \qquad \qquad + \Big( \dv{\rho_{ge}}{t} \Big)_{\perp} + = - \gamma_\perp \rho_{ge} +\end{aligned}$$ + +Putting everything together, +we arrive at the **optical Bloch equations** governing $$\hat{\rho}$$: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \dv{\rho_{gg}}{t} + &= \gamma_e \rho_{ee} - \gamma_g \rho_{gg} + + \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big) + \\ + \dv{\rho_{ee}}{t} + &= \gamma_g \rho_{gg} - \gamma_e \rho_{ee} + + \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big) + \\ + \dv{\rho_{ge}}{t} + &= - \Big( \gamma_\perp - i \omega_0 \Big) \rho_{ge} + + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \Big( \rho_{ee} - \rho_{gg} \Big) + \\ + \dv{\rho_{eg}}{t} + &= - \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg} + + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \Big( \rho_{gg} - \rho_{ee} \Big) + \end{aligned} + } +\end{aligned}$$ + +Some authors simplify these equations a bit by choosing +$$\gamma_g = 0$$ and $$\gamma_\perp = \gamma_e / 2$$. + + + +## References +1. F. Kärtner, + [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/pages/lecture-notes/), + 2005, Massachusetts Institute of Technology. +2. H.J. Metcalf, P. van der Straten, + *Laser cooling and trapping*, + 1999, Springer. |