diff options
Diffstat (limited to 'source/know/concept/maxwell-bloch-equations')
| -rw-r--r-- | source/know/concept/maxwell-bloch-equations/index.md | 259 |
1 files changed, 63 insertions, 196 deletions
diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md index 0252b5c..1214703 100644 --- a/source/know/concept/maxwell-bloch-equations/index.md +++ b/source/know/concept/maxwell-bloch-equations/index.md @@ -11,96 +11,43 @@ categories: 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: +For an electron in a two-orbital system $$\{\ket{g}, \ket{e}\}$$, +the Schrödinger equation has the following general solution, +where $$\varepsilon_g$$ and $$\varepsilon_e$$ are the time-independent eigenenergies, +and the weights $$c_g$$ and $$c_g$$ are functions of $$t$$: $$\begin{aligned} \ket{\Psi} - &= c_g \ket{g} \exp(-i E_g t / \hbar) + c_e \ket{e} \exp(-i E_e t / \hbar) + &= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar} \end{aligned}$$ -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: +This system is being perturbed by an electromagnetic wave +with [electric field](/know/concept/electric-field/) $$\vb{E}$$ 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: - -$$\begin{aligned} - \dv{c_g}{t} - &= i \frac{q \matrixel{g}{\vu{x}}{e} \cdot \vb{E}_0}{2 \hbar} \exp\!\big( i \omega t \!-\! i \omega_0 t \big) \: c_e - \\ - \dv{c_e}{t} - &= i \frac{q \matrixel{e}{\vu{x}}{g} \cdot \vb{E}_0}{2 \hbar} \exp\!\big(\!-\! i \omega t \!+\! i \omega_0 t \big) \: c_g -\end{aligned}$$ - -We want to rearrange these equations a bit. -Therefore, we split the electric field $$\vb{E}$$ like so, -where the amplitudes $$\vb{E}_0^{-}$$ and $$\vb{E}_0^{+}$$ may be slowly varying: - -$$\begin{aligned} - \vb{E}(t) - = \vb{E}^{-}(t) + \vb{E}^{+}(t) - = \vb{E}_0^{-} \exp(i \omega t) + \vb{E}_0^{+} \exp(-i \omega t) -\end{aligned}$$ - -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: - -$$\begin{aligned} - \dv{c_g}{t} - &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e - \\ - \dv{c_e}{t} - &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g + &\equiv \vb{E}^{-}(t) + \vb{E}^{+}(t) \end{aligned}$$ - - -## Optical Bloch equations +Where the forward-propagating component $$\vb{E}^{+}$$ +is a modulated plane wave $$\vb{E}_0^{+} e^{-i \omega t}$$ +with slowly-varying amplitude $$\vb{E}_0^{+}(t)$$, +and similarly $$\vb{E}^{-}(t) \equiv \vb{E}_0^{-}(t) e^{i \omega t}$$; +since $$\vb{E}$$ is real, $$\vb{E}_0^{+} \!=\! (\vb{E}_0^{-})^*$$. For $$\ket{\Psi}$$ as defined above, -the corresponding pure [density operator](/know/concept/density-operator/) -$$\hat{\rho}$$ is as follows: +the pure [density operator](/know/concept/density-operator/) +$$\hat{\rho}$$ is as follows, +with $$\omega_0 \equiv (\varepsilon_e \!-\! \varepsilon_g) / \hbar$$ +being the transition's resonance frequency: $$\begin{aligned} \hat{\rho} = \ket{\Psi} \bra{\Psi} = \begin{bmatrix} - c_e c_e^* & c_e c_g^* \exp(-i \omega_0 t) \\ - c_g c_e^* \exp(i \omega_0 t) & c_g c_g^* + c_e c_e^* & c_e c_g^* e^{-i \omega_0 t} \\ + c_g c_e^* e^{i \omega_0 t} & c_g c_g^* \end{bmatrix} \equiv \begin{bmatrix} @@ -109,139 +56,59 @@ $$\begin{aligned} \end{bmatrix} \end{aligned}$$ -Where $$\omega_0 \equiv (E_e \!-\! E_g) / \hbar$$ is the resonance frequency. -We take the $$t$$-derivative of the matrix elements, -and insert the equations for $$c_g$$ and $$c_e$$: +Under the [electric dipole approximation](/know/concept/electric-dipole-approximation/) +and [rotating wave approximation](/know/concept/rotating-wave-approximation/), +it can be shown that $$\hat{\rho}$$ is governed by +the [optical Bloch equations](/know/concept/optical-bloch-equations/): $$\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}^{-} \exp(- i \omega_0 t) \: c_e c_g^* - - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g c_e^* + &= \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} - &= \dv{c_e}{t} c_e^* + c_e \dv{c_e^*}{t} - \\ - &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g c_e^* - - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e c_g^* + &= \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} - &= \dv{c_g}{t} c_e^* \exp(i \omega_0 t) + c_g \dv{c_e^*}{t} \exp(i \omega_0 t) + i \omega_0 c_g c_e^* \exp(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^* \exp(i \omega_0 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} - &= \dv{c_e}{t} c_g^* \exp(-i \omega_0 t) + c_e \dv{c_g^*}{t} \exp(-i \omega_0 t) - i \omega_0 c_e c_g^* \exp(- 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^* \: \exp(- i \omega_0 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}$$ -Recognizing the density matrix elements allows us -to reduce these equations to: +Where we have defined the transition dipole moment $$\vb{p}_0^{-}$$, +with $$q < 0$$ the electron charge: $$\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} + \vb{p}_0^{-} + \equiv q \matrixel{e}{\vu{x}}{g} \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} - } + \vb{p}_0^{+} + \equiv (\vb{p}_0^{-})^* + = q \matrixel{g}{\vu{x}}{e} \end{aligned}$$ -Some authors simplify these equations a bit by choosing -$$\gamma_g = 0$$ and $$\gamma_\perp = \gamma_e / 2$$. - - - -## Including Maxwell's equations - -This two-level system has a dipole moment $$\vb{p}$$ as follows, -where we use [Laporte's selection rule](/know/concept/selection-rules/) -to remove diagonal terms, by assuming that -the electron's orbitals are odd or even: +However, the light wave affects the electron, +so the actual electromagnetic dipole moment $$\vb{p}$$ is as follows, +using [Laporte's selection rule](/know/concept/selection-rules/) +to remove diagonal terms by assuming that +the electron's orbitals are spatially odd or even: $$\begin{aligned} \vb{p} - &= \matrixel{\Psi}{\hat{\vb{p}}}{\Psi} + &= q \matrixel{\Psi}{\vu{x}}{\Psi} \\ &= q \Big( c_g c_g^* \matrixel{g}{\vu{x}}{g} + c_e c_e^* \matrixel{e}{\vu{x}}{e} - + c_g c_e^* \matrixel{e}{\vu{x}}{g} \exp(i \omega_0 t) + c_e c_g^* \matrixel{g}{\vu{x}}{e} \exp(-i \omega_0 t) \Big) + + c_g c_e^* \matrixel{e}{\vu{x}}{g} e^{i \omega_0 t} + c_e c_g^* \matrixel{g}{\vu{x}}{e} e^{-i \omega_0 t} \Big) \\ &= q \Big( \rho_{ge} \matrixel{e}{\vu{x}}{g} + \rho_{eg} \matrixel{g}{\vu{x}}{e} \Big) - = \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t) - \equiv \vb{p}^{-}(t) + \vb{p}^{+}(t) + \\ + &= \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t) + \\ + &\equiv \vb{p}^{-}(t) + \vb{p}^{+}(t) \end{aligned}$$ Where we have split $$\vb{p}$$ analogously to $$\vb{E}$$ @@ -256,7 +123,7 @@ $$\begin{aligned} \end{aligned}$$ Some authors do not bother multiplying $$\rho_{ge}$$ by $$\vb{p}_0^{+}$$. -In any case, we arrive at: +In our case, we arrive at a prototype of the first of three Maxwell-Bloch equations: $$\begin{aligned} \boxed{ @@ -266,8 +133,8 @@ $$\begin{aligned} } \end{aligned}$$ -Where we have defined the **population inversion** $$d \in [-1, 1]$$ as follows, -which quantifies the electron's excitedness: +Where we have defined the **population inversion** $$d \in [-1, 1]$$ like so, +to quantify the electron's "excitedness" i.e. its localization to $$\ket{e}$$: $$\begin{aligned} d @@ -325,8 +192,8 @@ $$\begin{aligned} {% include proof/end.html id="proof-inversion-decay" %} -With this, the equation for the population inversion $$d$$ -takes the following final form: +With this, the equation for the population inversion $$d$$ takes the form below, +namely the second Maxwell-Bloch equation's prototype: $$\begin{aligned} \boxed{ @@ -337,9 +204,11 @@ $$\begin{aligned} Finally, we would like a relation between the polarization and the electric field $$\vb{E}$$, -for which we turn to [Maxwell's equations](/know/concept/maxwells-equations/). -We start from Faraday's law, -and split $$\vb{B} = \mu_0 (\vb{H} + \vb{M})$$: +for which we turn to [Maxwell's equations](/know/concept/maxwells-equations/); +we will effectively derive a modified form of +the [electromagnetic wave equation](/know/concept/electromagnetic-wave-equation/). +Starting from Faraday's law +and splitting $$\vb{B} = \mu_0 (\vb{H} + \vb{M})$$: $$\begin{aligned} \nabla \cross \vb{E} @@ -391,7 +260,8 @@ $$\begin{aligned} Where $$\varepsilon_r \equiv 1 + \chi_e$$ is the medium's relative permittivity. The speed of light $$c^2 = 1 / (\mu_0 \varepsilon_0)$$, and the refractive index $$n^2 = \mu_r \varepsilon_r$$, -where $$\mu_r = 1$$ due to our assumption that $$\vb{M} = 0$$, so: +where $$\mu_r = 1$$ due to our assumption that $$\vb{M} = 0$$, +so the third Maxwell-Bloch equation's prototype is: $$\begin{aligned} \boxed{ @@ -436,11 +306,8 @@ $$\begin{aligned} ## References 1. F. Kärtner, - [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-977-ultrafast-optics-spring-2005/lecture-notes/), - 2005, MIT. + [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. Haken, *Light: volume 2: laser light dynamics*, 1985, North-Holland. -3. H.J. Metcalf, P. van der Straten, - *Laser cooling and trapping*, - 1999, Springer. |