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diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md index 8ba9a5b..b306c7d 100644 --- a/source/know/concept/maxwell-bloch-equations/index.md +++ b/source/know/concept/maxwell-bloch-equations/index.md @@ -12,7 +12,7 @@ layout: "concept" --- For an electron in a two-level system with time-independent states -$\Ket{g}$ (ground) and $\Ket{e}$ (excited), +$$\Ket{g}$$ (ground) and $$\Ket{e}$$ (excited), consider the following general solution to the full Schrödinger equation: @@ -23,9 +23,9 @@ $$\begin{aligned} Perturbing this system with an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) -introduces a time-dependent sinusoidal term $\hat{H}_1$ to the Hamiltonian. +introduces a time-dependent sinusoidal term $$\hat{H}_1$$ to the Hamiltonian. In the [electric dipole approximation](/know/concept/electric-dipole-approximation/), -$\hat{H}_1$ is given by: +$$\hat{H}_1$$ is given by: $$\begin{aligned} \hat{H}_1(t) @@ -38,10 +38,10 @@ $$\begin{aligned} = \vb{E}_0 \cos(\omega t) \end{aligned}$$ -Where $\vb{E}$ is an [electric field](/know/concept/electric-field/), -and $\hat{\vb{p}}$ is the dipole moment operator. +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 $c_e$ +we know that the time-varying coefficients $$c_g$$ and $$c_e$$ can then be described by: $$\begin{aligned} @@ -53,8 +53,8 @@ $$\begin{aligned} \end{aligned}$$ We want to rearrange these equations a bit. -Therefore, we split the electric field $\vb{E}$ like so, -where the amplitudes $\vb{E}_0^{-}$ and $\vb{E}_0^{+}$ may be slowly varying: +Therefore, we split the electric field $$\vb{E}$$ like so, +where the amplitudes $$\vb{E}_0^{-}$$ and $$\vb{E}_0^{+}$$ may be slowly varying: $$\begin{aligned} \vb{E}(t) @@ -62,8 +62,8 @@ $$\begin{aligned} = \vb{E}_0^{-} \exp(i \omega t) + \vb{E}_0^{+} \exp(-i \omega t) \end{aligned}$$ -Since $\vb{E}$ is real, $\vb{E}_0^{+} = (\vb{E}_0^{-})^*$. -Similarly, we define the transition dipole moment $\vb{p}_0^{-}$: +Since $$\vb{E}$$ is real, $$\vb{E}_0^{+} = (\vb{E}_0^{-})^*$$. +Similarly, we define the transition dipole moment $$\vb{p}_0^{-}$$: $$\begin{aligned} \vb{p}_0^{-} @@ -74,8 +74,8 @@ $$\begin{aligned} = q \matrixel{g}{\vu{x}}{e} \end{aligned}$$ -With these, the equations for $c_g$ and $c_e$ can be rewritten as shown below. -Note that $\vb{E}^{-}$ and $\vb{E}^{+}$ include the driving plane wave, and the +With these, the equations for $$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} @@ -89,9 +89,9 @@ $$\begin{aligned} ## Optical Bloch equations -For $\Ket{\Psi}$ as defined above, +For $$\Ket{\Psi}$$ as defined above, the corresponding pure [density operator](/know/concept/density-operator/) -$\hat{\rho}$ is as follows: +$$\hat{\rho}$$ is as follows: $$\begin{aligned} \hat{\rho} @@ -108,9 +108,9 @@ $$\begin{aligned} \end{bmatrix} \end{aligned}$$ -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 $c_e$: +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 $$c_e$$: $$\begin{aligned} \dv{\rho_{gg}}{t} @@ -157,12 +157,12 @@ $$\begin{aligned} &= - i \omega_0 \rho_{eg} + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \big( \rho_{gg} - \rho_{ee} \big) \end{aligned}$$ -These equations are correct if nothing else is affecting $\hat{\rho}$. +These equations are correct if nothing else is affecting $$\hat{\rho}$$. But in practice, these quantities decay due to various processes, e.g. spontaneous emission (see [Einstein coefficients](/know/concept/einstein-coefficients/)). -Let $\rho_{ee}$ decays with rate $\gamma_e$. -Since the total probability $\rho_{ee} + \rho_{gg} = 1$, +Let $$\rho_{ee}$$ decays with rate $$\gamma_e$$. +Since the total probability $$\rho_{ee} + \rho_{gg} = 1$$, we thus have: $$\begin{aligned} @@ -174,7 +174,7 @@ $$\begin{aligned} \end{aligned}$$ Meanwhile, for whatever reason, -let $\rho_{gg}$ decay into $\rho_{ee}$ with rate $\gamma_g$: +let $$\rho_{gg}$$ decay into $$\rho_{ee}$$ with rate $$\gamma_g$$: $$\begin{aligned} \Big( \dv{\rho_{gg}}{t} \Big)_{g} @@ -185,7 +185,7 @@ $$\begin{aligned} \end{aligned}$$ And finally, let the diagonal (perpendicular) matrix elements -both decay with rate $\gamma_\perp$: +both decay with rate $$\gamma_\perp$$: $$\begin{aligned} \Big( \dv{\rho_{eg}}{t} \Big)_{\perp} @@ -196,7 +196,7 @@ $$\begin{aligned} \end{aligned}$$ Putting everything together, -we arrive at the **optical Bloch equations** governing $\hat{\rho}$: +we arrive at the **optical Bloch equations** governing $$\hat{\rho}$$: $$\begin{aligned} \boxed{ @@ -221,12 +221,12 @@ $$\begin{aligned} \end{aligned}$$ Many authors simplify these equations a bit by choosing -$\gamma_g = 0$ and $\gamma_\perp = \gamma_e / 2$. +$$\gamma_g = 0$$ and $$\gamma_\perp = \gamma_e / 2$$. ## Including Maxwell's equations -This two-level system has a dipole moment $\vb{p}$ as follows, +This two-level system has a dipole moment $$\vb{p}$$ as follows, where we use [Laporte's selection rule](/know/concept/selection-rules/) to remove diagonal terms, by assuming that the electron's orbitals are odd or even: @@ -243,8 +243,8 @@ $$\begin{aligned} \equiv \vb{p}^{-}(t) + \vb{p}^{+}(t) \end{aligned}$$ -Where we have split $\vb{p}$ analogously to $\vb{E}$ -by defining $\vb{p}^{+} \equiv \vb{p}_0^{+} \rho_{eg}$. +Where we have split $$\vb{p}$$ analogously to $$\vb{E}$$ +by defining $$\vb{p}^{+} \equiv \vb{p}_0^{+} \rho_{eg}$$. Its equation of motion can then be found from the optical Bloch equations: $$\begin{aligned} @@ -254,7 +254,7 @@ $$\begin{aligned} + \frac{i}{\hbar} \vb{p}_0^{+} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \Big( \rho_{gg} - \rho_{ee} \Big) \end{aligned}$$ -Some authors do not bother multiplying $\rho_{ge}$ by $\vb{p}_0^{+}$. +Some authors do not bother multiplying $$\rho_{ge}$$ by $$\vb{p}_0^{+}$$. In any case, we arrive at: $$\begin{aligned} @@ -265,7 +265,7 @@ $$\begin{aligned} } \end{aligned}$$ -Where we have defined the **population inversion** $d \in [-1, 1]$ as follows, +Where we have defined the **population inversion** $$d \in [-1, 1]$$ as follows, which quantifies the electron's excitedness: $$\begin{aligned} @@ -285,8 +285,8 @@ $$\begin{aligned} We can rewrite the first two terms in the following intuitive form, which describes a decay with -rate $\gamma_\parallel \equiv \gamma_g + \gamma_e$ -towards an equilbrium $d_0$: +rate $$\gamma_\parallel \equiv \gamma_g + \gamma_e$$ +towards an equilbrium $$d_0$$: $$\begin{aligned} 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} @@ -313,8 +313,8 @@ $$\begin{aligned} + \gamma_g (\rho_{gg} - \rho_{ee}) + \gamma_e (\rho_{gg} - \rho_{ee}) \end{aligned}$$ -Since the total probability $\rho_{gg} + \rho_{ee} = 1$, -and $d \equiv \rho_{ee} - \rho_{gg}$, this reduces to: +Since the total probability $$\rho_{gg} + \rho_{ee} = 1$$, +and $$d \equiv \rho_{ee} - \rho_{gg}$$, this reduces to: $$\begin{aligned} 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} @@ -324,10 +324,11 @@ $$\begin{aligned} \\ &= \gamma_\parallel ( d_0 - d ) \end{aligned}$$ + </div> </div> -With this, the equation for the population inversion $d$ +With this, the equation for the population inversion $$d$$ takes the following final form: $$\begin{aligned} @@ -338,10 +339,10 @@ $$\begin{aligned} \end{aligned}$$ Finally, we would like a relation between the polarization -and the electric field $\vb{E}$, +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 $\vb{B} = \mu_0 (\vb{H} + \vb{M})$: +and split $$\vb{B} = \mu_0 (\vb{H} + \vb{M})$$: $$\begin{aligned} \nabla \cross \vb{E} @@ -349,9 +350,9 @@ $$\begin{aligned} = - \mu_0 \pdv{\vb{H}}{t} - \mu_0 \pdv{\vb{M}}{t} \end{aligned}$$ -We assume that there is no magnetization $\vb{M} = 0$. +We assume that there is no magnetization $$\vb{M} = 0$$. Then we we take the curl of both sides, -and replace $\nabla \cross \vb{H}$ with Ampère's circuital law: +and replace $$\nabla \cross \vb{H}$$ with Ampère's circuital law: $$\begin{aligned} \nabla \cross \big( \nabla \cross \vb{E} \big) @@ -359,23 +360,23 @@ $$\begin{aligned} = - \mu_0 \pdv{}{t} \Big( \vb{J}_\mathrm{free} + \pdv{\vb{D}}{t} \Big) \end{aligned}$$ -Inserting the definition $\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$ -together with Ohm's law $\vb{J}_\mathrm{free} = \sigma \vb{E}$ yields: +Inserting the definition $$\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$$ +together with Ohm's law $$\vb{J}_\mathrm{free} = \sigma \vb{E}$$ yields: $$\begin{aligned} \nabla \cross \big( \nabla \cross \vb{E} \big) = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdvn{2}{\vb{E}}{t} - \mu_0 \pdvn{2}{\vb{P}}{t} \end{aligned}$$ -Where $\sigma$ is the active material's conductivity, if any; -almost all authors assume $\sigma = 0$. +Where $$\sigma$$ is the active material's conductivity, if any; +almost all authors assume $$\sigma = 0$$. Recall that we are describing the dynamics of a two-level system. In reality, such a system (e.g. a quantum dot) is suspended in a passive background medium, -which reacts with a polarization $\vb{P}_\mathrm{med}$ -to the electric field $\vb{E}$. -If the medium is linear, i.e. $\vb{P}_\mathrm{med} = \varepsilon_0 \chi \vb{E}$, +which reacts with a polarization $$\vb{P}_\mathrm{med}$$ +to the electric field $$\vb{E}$$. +If the medium is linear, i.e. $$\vb{P}_\mathrm{med} = \varepsilon_0 \chi \vb{E}$$, then: $$\begin{aligned} @@ -390,10 +391,10 @@ $$\begin{aligned} - \mu_0 \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{E}}{t} \end{aligned}$$ -Where $\varepsilon_r \equiv 1 + \chi_e$ is the medium's relative permittivity. -The speed of light $c^2 = 1 / (\mu_0 \varepsilon_0)$, -and the refractive index $n^2 = \mu_r \varepsilon_r$, -where $\mu_r = 1$ due to our assumption that $\vb{M} = 0$, so: +Where $$\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$$, so: $$\begin{aligned} \boxed{ @@ -402,9 +403,9 @@ $$\begin{aligned} } \end{aligned}$$ -$\vb{E}$ and $\vb{P}$ can trivially be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$. -It is also simple to convert $\vb{p}^{+}$ and $d$ -into the macroscopic $\vb{P}^{+}$ and total $D$ +$$\vb{E}$$ and $$\vb{P}$$ can trivially be replaced by $$\vb{E}^{+}$$ and $$\vb{P}^{+}$$. +It is also simple to convert $$\vb{p}^{+}$$ and $$d$$ +into the macroscopic $$\vb{P}^{+}$$ and total $$D$$ by summing over all two-level systems in the medium: $$\begin{aligned} |