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Diffstat (limited to 'source/know/concept/maxwell-bloch-equations/index.md')
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1 files changed, 13 insertions, 12 deletions
diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md index ba8a677..0252b5c 100644 --- a/source/know/concept/maxwell-bloch-equations/index.md +++ b/source/know/concept/maxwell-bloch-equations/index.md @@ -12,13 +12,13 @@ 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: +to the time-dependent Schrödinger equation: $$\begin{aligned} - \Ket{\Psi} - &= c_g \: \Ket{g} \exp(-i E_g t / \hbar) + c_e \: \Ket{e} \exp(-i E_e t / \hbar) + \ket{\Psi} + &= c_g \ket{g} \exp(-i E_g t / \hbar) + c_e \ket{e} \exp(-i E_e t / \hbar) \end{aligned}$$ Perturbing this system with @@ -87,15 +87,16 @@ $$\begin{aligned} \end{aligned}$$ + ## 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: $$\begin{aligned} \hat{\rho} - = \Ket{\Psi} \Bra{\Psi} + = \ket{\Psi} \bra{\Psi} = \begin{bmatrix} c_e c_e^* & c_e c_g^* \exp(-i \omega_0 t) \\ @@ -159,11 +160,10 @@ $$\begin{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 (see [Einstein coefficients](/know/concept/einstein-coefficients/)). +e.g. [spontaneous emission](/know/concept/einstein-coefficients/). -Let $$\rho_{ee}$$ decays with rate $$\gamma_e$$. -Since the total probability $$\rho_{ee} + \rho_{gg} = 1$$, -we thus have: +Suppose $$\rho_{ee}$$ decays with rate $$\gamma_e$$. +Because the total probability $$\rho_{ee} + \rho_{gg} = 1$$, we have: $$\begin{aligned} \Big( \dv{\rho_{ee}}{t} \Big)_{e} @@ -220,10 +220,11 @@ $$\begin{aligned} } \end{aligned}$$ -Many authors simplify these equations a bit by choosing +Some authors simplify these equations a bit by choosing $$\gamma_g = 0$$ and $$\gamma_\perp = \gamma_e / 2$$. + ## Including Maxwell's equations This two-level system has a dipole moment $$\vb{p}$$ as follows, @@ -286,7 +287,7 @@ $$\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 $$d_0$$: +towards an equilibrium $$d_0$$: $$\begin{aligned} 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} |