diff options
Diffstat (limited to 'source/know/concept')
7 files changed, 304 insertions, 206 deletions
diff --git a/source/know/concept/central-limit-theorem/index.md b/source/know/concept/central-limit-theorem/index.md index e933ee7..42bc05b 100644 --- a/source/know/concept/central-limit-theorem/index.md +++ b/source/know/concept/central-limit-theorem/index.md @@ -9,7 +9,7 @@ layout: "concept" --- In statistics, the **central limit theorem** states that -the sum of many independent variables tends towards a normal distribution, +the sum of many independent variables tends to a normal distribution, even if the individual variables $$x_n$$ follow different distributions. For example, by taking $$M$$ samples of size $$N$$ from a population, diff --git a/source/know/concept/fredholm-alternative/index.md b/source/know/concept/fredholm-alternative/index.md index c954272..fdc90be 100644 --- a/source/know/concept/fredholm-alternative/index.md +++ b/source/know/concept/fredholm-alternative/index.md @@ -14,7 +14,7 @@ It is an *alternative* because it gives two mutually exclusive options, given here in [Dirac notation](/know/concept/dirac-notation/): 1. $$\hat{L} \Ket{u} = \Ket{f}$$ has a unique solution $$\Ket{u}$$ for every $$\Ket{f}$$. -2. $$\hat{L}^\dagger \Ket{w} = 0$$ has non-zero solutions. +2. $$\hat{L}^\dagger \Ket{w} = 0$$ has nonzero solutions. Then regarding $$\hat{L} \Ket{u} = \Ket{f}$$: 1. If $$\Inprod{w}{f} = 0$$ for all $$\Ket{w}$$, then it has infinitely many solutions $$\Ket{u}$$. 2. If $$\Inprod{w}{f} \neq 0$$ for any $$\Ket{w}$$, then it has no solutions $$\Ket{u}$$. @@ -31,7 +31,7 @@ this theorem can alternatively be stated as follows using the determinant: 1. If $$\mathrm{det}(\hat{L}) \neq 0$$, then $$\hat{L} \vec{u} = \vec{f}$$ has a unique solution $$\vec{u}$$ for every $$\vec{f}$$. 2. If $$\mathrm{det}(\hat{L}) = 0$$, - then $$\hat{L}^\dagger \vec{w} = \vec{0}$$ has non-zero solutions. + then $$\hat{L}^\dagger \vec{w} = \vec{0}$$ has nonzero solutions. Then regarding $$\hat{L} \vec{u} = \vec{f}$$: 1. If $$\vec{w} \cdot \vec{f} = 0$$ for all $$\vec{w}$$, then it has infinitely many solutions $$\vec{u}$$. @@ -48,7 +48,7 @@ Then for the equation $$\hat{M} \Ket{u} = \Ket{f}$$, we can say that: 1. If $$\lambda$$ is *not* an eigenvalue, then there is a unique solution $$\Ket{u}$$ for each $$\Ket{f}$$. 2. If $$\lambda$$ is an eigenvalue, then $$\hat{M}^\dagger \Ket{w} = 0$$ - has non-zero solutions. Then: + has nonzero solutions. Then: 1. If $$\Inprod{w}{f} = 0$$ for all $$\Ket{w}$$, then there are infinitely many solutions $$\Ket{u}$$. 2. If $$\Inprod{w}{f} \neq 0$$ for any $$\Ket{w}$$, then there are no diff --git a/source/know/concept/matsubara-greens-function/index.md b/source/know/concept/matsubara-greens-function/index.md index fd46abf..5e753db 100644 --- a/source/know/concept/matsubara-greens-function/index.md +++ b/source/know/concept/matsubara-greens-function/index.md @@ -11,7 +11,7 @@ layout: "concept" The **Matsubara Green's function** is an [imaginary-time](/know/concept/imaginary-time/) version of the real-time [Green's functions](/know/concept/greens-functions/). -We define as follows in the imaginary-time +We define it as follows in the imaginary-time [Heisenberg picture](/know/concept/heisenberg-picture/): $$\begin{aligned} diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md index 0252b5c..1214703 100644 --- a/source/know/concept/maxwell-bloch-equations/index.md +++ b/source/know/concept/maxwell-bloch-equations/index.md @@ -11,96 +11,43 @@ categories: layout: "concept" --- -For an electron in a two-level system with time-independent states -$$\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 @@ $$\begin{aligned} ## References 1. F. Kärtner, - [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-977-ultrafast-optics-spring-2005/lecture-notes/), - 2005, MIT. + [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/pages/lecture-notes/), + 2005, Massachusetts Institute of Technology. 2. H. Haken, *Light: volume 2: laser light dynamics*, 1985, North-Holland. -3. H.J. Metcalf, P. van der Straten, - *Laser cooling and trapping*, - 1999, Springer. diff --git a/source/know/concept/optical-bloch-equations/index.md b/source/know/concept/optical-bloch-equations/index.md new file mode 100644 index 0000000..fe74b7e --- /dev/null +++ b/source/know/concept/optical-bloch-equations/index.md @@ -0,0 +1,231 @@ +--- +title: "Optical Bloch equations" +sort_title: "Optical Bloch equations" +date: 2023-01-19 +categories: +- Physics +- Quantum mechanics +- Two-level system +layout: "concept" +--- + +For an electron in a two-level system with time-independent states +$$\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$$. + + + +## References +1. F. Kärtner, + [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/pages/lecture-notes/), + 2005, Massachusetts Institute of Technology. +2. H.J. Metcalf, P. van der Straten, + *Laser cooling and trapping*, + 1999, Springer. diff --git a/source/know/concept/parsevals-theorem/index.md b/source/know/concept/parsevals-theorem/index.md index 41e8fed..a7ce0bf 100644 --- a/source/know/concept/parsevals-theorem/index.md +++ b/source/know/concept/parsevals-theorem/index.md @@ -17,7 +17,7 @@ where $$A$$, $$B$$, and $$s$$ are constants from the FT's definition: $$\begin{aligned} \boxed{ \begin{aligned} - \Inprod{f(x)}{g(x)} &= \frac{2 \pi B^2}{|s|} \inprod{\tilde{f}(k)}{\tilde{g}(k)} + \inprod{f(x)}{g(x)} &= \frac{2 \pi B^2}{|s|} \inprod{\tilde{f}(k)}{\tilde{g}(k)} \\ \inprod{\tilde{f}(k)}{\tilde{g}(k)} &= \frac{2 \pi A^2}{|s|} \Inprod{f(x)}{g(x)} \end{aligned} @@ -29,7 +29,7 @@ $$\begin{aligned} We insert the inverse FT into the definition of the inner product: $$\begin{aligned} - \Inprod{f}{g} + \inprod{f}{g} &= \int_{-\infty}^\infty \big( \hat{\mathcal{F}}^{-1}\{\tilde{f}(k)\}\big)^* \: \hat{\mathcal{F}}^{-1}\{\tilde{g}(k)\} \dd{x} \\ &= B^2 \int @@ -65,7 +65,7 @@ $$\begin{aligned} &= 2 \pi A^2 \iint f^*(x') \: g(x) \: \delta\big(s (x \!-\! x')\big) \dd{x'} \dd{x} \\ &= \frac{2 \pi A^2}{|s|} \int_{-\infty}^\infty f^*(x) \: g(x) \dd{x} - = \frac{2 \pi A^2}{|s|} \Inprod{f}{g} + = \frac{2 \pi A^2}{|s|} \inprod{f}{g} \end{aligned}$$ {% include proof/end.html id="proof-fourier" %} diff --git a/source/know/concept/sturm-liouville-theory/index.md b/source/know/concept/sturm-liouville-theory/index.md index 75daae3..0ac7476 100644 --- a/source/know/concept/sturm-liouville-theory/index.md +++ b/source/know/concept/sturm-liouville-theory/index.md @@ -22,7 +22,7 @@ of eigenfunctions. Consider the most general form of a second-order linear differential operator $$\hat{L}$$, where $$p_0(x)$$, $$p_1(x)$$, and $$p_2(x)$$ -are real functions of $$x \in [a,b]$$ which are non-zero for all $$x \in ]a, b[$$: +are real functions of $$x \in [a,b]$$ which are nonzero for all $$x \in ]a, b[$$: $$\begin{aligned} \hat{L} \{u(x)\} = p_0(x) u''(x) + p_1(x) u'(x) + p_2(x) u(x) @@ -142,7 +142,7 @@ So even if $$p_0' \neq p_1$$, any second-order linear operator with $$p_0(x) \ne can easily be put in self-adjoint form. This general form is known as the **Sturm-Liouville operator** $$\hat{L}_{SL}$$, -where $$p(x)$$ and $$q(x)$$ are non-zero real functions of the variable $$x \in [a,b]$$: +where $$p(x)$$ and $$q(x)$$ are nonzero real functions of the variable $$x \in [a,b]$$: $$\begin{aligned} \boxed{ |