Maxwell-Bloch equations

The Maxwell-Bloch equations describe the interaction of an ensemble of quantum mechanical two-level systems with an oscillating electric field. They are used to describe the absorption and emission of light in solids and gases and play a central role in the theoretical understanding of amplification in lasers . The prerequisite is that the energy difference of the transition is close to the photon energy of the light and that the other transitions of the system have significantly different transition energies.

Equations

The Maxwell-Bloch equations are

${\displaystyle \partial _{t}{\mathcal {P}}=\left[i(\omega -\Omega )-{\frac {1}{\tau _{p}}}\right]{\mathcal {P}}+{\frac {|d_{12}|^{2}}{i\hbar }}\Delta n{\mathcal {E}}\quad ~}$ ${\displaystyle \partial _{t}\Delta n={\frac {1}{\hbar }}{\text{Im}}\left({{\mathcal {E}}^{*}{\mathcal {P}}}\right)-{\frac {n_{0}+\Delta n}{\tau }}}$ ${\displaystyle \left(\partial _{z}+{\frac {1}{c_{g}}}\partial _{t}\ \right){\vec {\mathcal {E}}}={\frac {i}{2}}\mu _{0}\omega c_{p}{\vec {\mathcal {P}}}}$

With:

• ${\displaystyle {\vec {\mathcal {E}}}}$: complex amplitude of the electric field
• ${\displaystyle {\vec {\mathcal {P}}}}$: complex amplitude of polarization
• ${\displaystyle \Delta n=n_{2}-n_{1}}$: Population inversion with and population density of levels 1 and 2 ${\displaystyle n_{2}}$ ${\displaystyle n_{1}}$
• ${\displaystyle n_{0}=n_{1}+n_{2}}$: Number of two-level systems per volume
• ${\displaystyle \omega }$: Frequency of the electric field
• ${\displaystyle \Omega }$: Frequency of transition with ${\displaystyle \Omega ={\frac {E_{2}-E_{1}}{\hbar }}}$
• ${\displaystyle \tau _{p}}$: Phase relaxation time, coherence time of polarization.
• ${\displaystyle \tau }$: Lifetime of the second state
• ${\displaystyle d_{12}=e\left\langle \varphi _{1}\right|{\vec {r}}\cdot {\vec {e_{E}}}\left|\varphi _{2}\right\rangle }$ Projection of the dipole junction matrix element onto the direction of the electric field
• ${\displaystyle c_{g}=\partial _{k}\omega }$: Group speed in the medium
• ${\displaystyle \mu _{0}}$: Magnetic field constant
• ${\displaystyle c_{p}={\frac {\omega }{k}}={\frac {c_{0}}{n}}}$: Phase velocity in the medium

Approximations

Coherent regime

In the coherent regime, one assumes that the typical time derivatives of and are much larger than the decay terms, i.e. ${\displaystyle \Delta n}$ ${\displaystyle {\mathcal {P}}}$

${\displaystyle \left|\partial _{t}{\mathcal {P}}\right|\gg \left|{\frac {\mathcal {P}}{\tau _{p}}}\right|\quad {\text{und}}\quad \left|\partial _{t}\Delta n\right|\gg \left|{\frac {\Delta n}{\tau _{p}}}\right|}$

applies. With this, the Maxwell-Bloch equations take the form

${\displaystyle \partial _{t}{\mathcal {P}}=\left[i(\omega -\Omega )\right]{\mathcal {P}}+{\frac {|d_{12}|^{2}}{i\hbar }}\Delta n{\mathcal {E}}}$

${\displaystyle \partial _{t}\Delta n={\frac {1}{\hbar }}{\text{Im}}\left({{\mathcal {E}}^{*}{\mathcal {P}}}\right)}$

${\displaystyle \left(\partial _{z}+{\frac {1}{c_{g}}}\partial _{t}\ \right){\vec {\mathcal {E}}}={\frac {i}{2}}\mu _{0}\omega c_{p}{\vec {\mathcal {P}}}}$

on. One can easily show that in this case

${\displaystyle \left|{\mathcal {P}}\right|^{2}+\left|d_{12}\right|^{2}\Delta n^{2}={\text{const}}=\left|d_{12}\right|^{2}n_{0}^{2}}$

applies. This is why the so-called Bloch vector was introduced

${\displaystyle {\vec {\mu }}={\frac {1}{|d_{12}|n_{0}}}{\begin{pmatrix}{\text{Re}}({\mathcal {P}})\\-{\text{Im}}({\mathcal {P}})\\\Delta n|d_{12}|\\\end{pmatrix}}}$

with near. The equation of motion applies to this ${\displaystyle |\mu |^{2}=1}$

${\displaystyle {\dot {\vec {\mu }}}={\vec {\mu }}\times {\vec {R}}\quad {\text{mit}}\quad {\vec {R}}={\begin{pmatrix}{\text{Re}}(\Omega _{R})\\-{\text{Im}}(\Omega _{R})\\-\Delta \\\end{pmatrix}}}$

with the so-called Rabi frequency and the detuning . ${\displaystyle \Omega _{R}(t)={\frac {|d_{12}|{\mathcal {E}}(t)}{\hbar }}}$ ${\displaystyle \Delta =\omega -\Omega }$

Solution of the Maxwell-Bloch equations in the coherent regime with resonant coupling for a Gaussian pulse ( ). So-called rabio oscillations can be seen here. ${\displaystyle \pi }$ ${\displaystyle \Theta (\infty )=9\pi }$

In the case of so-called resonant coupling, i. H. and the equations are real ${\displaystyle \omega =\Omega \Rightarrow \Delta =0}$ ${\displaystyle {\mathcal {E}}}$

${\displaystyle \partial _{t}{\mathcal {P}}=in_{0}\Omega _{R}|d_{12}|\Delta n}$

${\displaystyle \partial _{t}\Delta n=-n_{0}\Omega _{R}|d_{12}|{\mathcal {P}}.}$

The solutions to this system of differential equations are

${\displaystyle {\mathcal {P}}=in_{0}|d_{12}|\sin(\Theta (t))}$ ${\displaystyle \Delta n=-n_{0}\cos(\Theta (t))}$

with the so-called pulse area with ${\displaystyle \Theta }$

${\displaystyle \Theta (t)=\int \limits _{t_{0}}^{t}\underbrace {\frac {|d_{12}|{\mathcal {E}}(t')}{\hbar }} _{\Omega _{R}}\mathrm {d} t'}$

Thus lead and from vibrations, which are driven by the electric field. These are called Rabi oscillations . With the third Maxwell-Bloch equation, assuming a thin sample of length L, i. H. , for the re-emitted electric field ${\displaystyle \Delta n}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \tau _{p}\gg {\frac {L}{c_{g}}}}$

${\displaystyle {\mathcal {E}}_{\text{out}}(t)={\mathcal {E}}(t)-{\frac {n_{0}|d_{12}|}{2\varepsilon _{0}}}kL{\frac {c_{p}}{c}}\sin(\Theta (t)).}$

If you now prepare an incoming light pulse in such a way that with you can completely invert the medium . One then speaks of a pulse (see illustration). For the population inversion is zero and the polarization is maximal. With this method you can bring a material into a precisely defined state. ${\displaystyle \Theta (t\rightarrow \infty )=2\pi m+\pi }$ ${\displaystyle m\in \mathbb {Z} }$ ${\displaystyle \pi }$ ${\displaystyle \Theta (t\rightarrow \infty )=2\pi m+{\frac {\pi }{2}}}$

Derivation

To derive the Maxwell-Bloch equations one describes the interaction between the electric field and the atom in the so-called dipole approximation . The Hamilton operator of the system consists of two parts. The part that describes the atom without interaction with the electric field and the part that describes a dipole-like interaction between light and atom: ${\displaystyle {\hat {H}}_{0}}$ ${\displaystyle {\hat {H}}^{\text{ww}}}$

${\displaystyle {\hat {H}}={\hat {H}}_{0}+{\hat {H}}^{\text{ww}}}$

With

${\displaystyle {\hat {H}}^{\text{ww}}={\vec {E}}(t)\cdot {\vec {\hat {d}}}=-e\;{\vec {E}}\cdot {\vec {\hat {r}}}.}$

The wave function can be used in the basis of the undisturbed system ${\displaystyle |\Psi \rangle }$ ${\displaystyle |\varphi _{k}\rangle }$

${\displaystyle |\Psi \rangle =c_{1}(t)e^{-i\omega _{1}t}|\varphi _{1}\rangle +c_{2}(t)e^{-i\omega _{2}t}|\varphi _{2}\rangle }$

being represented. The Schrödinger equation is now

${\displaystyle i\hbar \partial _{t}|\Psi \rangle =\left({\hat {H}}_{0}+{\hat {H}}^{\text{ww}}\right)|\Psi \rangle .}$

By multiplying by and substituting the basic representation of follows ${\displaystyle \langle \varphi _{m}|e^{i\omega _{m}t}}$ ${\displaystyle |\Psi \rangle }$

${\displaystyle i\hbar {\dot {c}}_{1}=-d_{12}E(t)e^{-i\Omega t}c_{2}}$

${\displaystyle i\hbar {\dot {c}}_{2}=-d_{12}^{\star }E(t)e^{i\Omega t}c_{1}}$

It was exploited. The microscopic polarization of the system is now through ${\displaystyle d_{11}=d_{22}=0}$ ${\displaystyle {\vec {P}}}$

${\displaystyle {\vec {P}}=n_{0}\langle \Psi |-e{\vec {r}}|\Psi \rangle =\underbrace {-d_{12}e^{i\Omega t}c_{1}^{\star }c_{2}n_{0}{\vec {e}}_{\vec {E}}} _{={\vec {P}}^{+}}+\underbrace {\text{c.c.}} _{={\vec {P}}^{-}}}$

given. For the time derivatives of the polarization components and follows ${\displaystyle {\vec {P}}^{+}}$ ${\displaystyle {\vec {P}}^{-}}$

${\displaystyle {\begin{aligned}\partial _{t}{\vec {P}}^{+}&=-d_{12}e^{-i\Omega t}n_{0}{\vec {e}}_{\vec {E}}\left(-i\Omega +{\dot {c}}_{1}^{\star }c_{2}+c_{1}^{\star }{\dot {c}}_{2}\right)\\&=i\Omega {\vec {P}}^{+}-{\frac {i}{\hbar }}|d_{12}|^{2}E(t)\underbrace {n_{0}\left(|c_{1}|^{2}-|c_{2}|^{2}\right)} _{=-\Delta n}.\end{aligned}}}$

Thereby were the equations

${\displaystyle {\dot {c}}_{1}^{\star }c_{2}=-{\frac {i}{\hbar }}d_{12}^{\star }E(t)e^{i\Omega t}|c_{2}|^{2}}$

${\displaystyle c_{1}^{\star }{\dot {c}}_{2}={\frac {i}{\hbar }}d_{12}^{\star }E(t)e^{i\Omega t}|c_{1}|^{2}}$

used. The equation for is simply given by the complex conjugate equation. ${\displaystyle {\vec {P}}^{-}}$

${\displaystyle \partial _{t}{\vec {P}}^{-}=(\partial _{t}{\vec {P}}^{+})^{\star }}$

For the field-free case ( ) the polarization now oscillates harmoniously. In real systems, however, the polarization decays, which is why a decay term is added. The material constant is called the phase relaxation time. The so-called rotating wave approximation is also used . One sets ${\displaystyle E=0}$ ${\displaystyle {\frac {-{\vec {P}}}{\tau _{p}}}}$ ${\displaystyle \tau _{p}}$

${\displaystyle {\vec {E}}=\underbrace {{\mathcal {E}}(t,z)e^{i(kz-\omega t)}} _{{\vec {E}}^{+}}+\underbrace {{\mathcal {E}}^{\star }(t,z)e^{-i(kz-\omega t)}} _{{\vec {E}}^{-}}}$

and neglected in the equation for and correspondingly in the equation for , since the neglected terms also oscillate and are therefore small in comparison to the terms with . For the polarization it follows ${\displaystyle {\vec {E}}^{-}}$ ${\displaystyle {\vec {P}}^{+}}$ ${\displaystyle {\vec {E}}^{+}}$ ${\displaystyle {\vec {P}}^{-}}$ ${\displaystyle \omega +\Omega }$ ${\displaystyle \omega -\Omega }$

${\displaystyle \partial _{t}{\vec {P}}^{\pm }=\left(\mp i\Omega -{\frac {1}{\tau _{p}}}\right){\vec {P}}^{\pm }+{\frac {|d_{12}|^{2}}{i\hbar }}\Delta n\cdot {\vec {E}}^{\pm }}$

what through the approach still to ${\displaystyle {\vec {P}}^{+}={\mathcal {P}}(z,t)\cdot e^{i(kz-\omega t)}}$

${\displaystyle \partial _{t}{\mathcal {P}}=\left(i\Omega -{\frac {1}{\tau _{p}}}\right){\mathcal {P}}+{\frac {|d_{12}|^{2}}{i\hbar }}\Delta n\cdot {\mathcal {E}}}$

can be simplified. For the time derivative of the population inversion follows

${\displaystyle {\begin{aligned}\partial _{t}\Delta n&=2\partial _{t}n_{2}=2n_{0}\left({\dot {c}}_{2}^{\star }c_{2}+c_{2}^{\star }{\dot {c}}_{2}\right)\\&={\frac {4}{\hbar }}{\text{Im}}({\vec {E}}^{-}{\vec {P}}^{+})={\frac {4}{\hbar }}{\text{Im}}({\mathcal {E}}^{\star }{\mathcal {P}})\end{aligned}}}$

Here, too, the population inversion would remain constant in the field-free case , which is why a term is also added. ${\displaystyle (E=P=0)}$ ${\displaystyle -{\frac {n_{2}}{\tau }}=-{\frac {n_{0}+\Delta n}{\tau }}}$

${\displaystyle \partial _{t}\Delta n={\frac {4}{\hbar }}{\text{Im}}({\mathcal {E}}^{\star }{\mathcal {P}})-{\frac {n_{0}+\Delta n}{\tau }}}$

Here is the mean lifetime of the excited state. Finally, you need an equation for the electric field. This is based on the wave equation ${\displaystyle \tau }$

${\displaystyle \partial _{z}^{2}{\vec {E}}-{\frac {1}{c_{p}^{2}}}\partial _{t}^{2}\varepsilon {\vec {E}}=-\mu _{0}\partial _{t}^{2}{\vec {P}}}$

out. By inserting the connections and approaches that have already been obtained, this follows

${\displaystyle \partial _{z}^{2}{\vec {E}}^{+}=\left(\partial _{z}^{2}{\mathcal {E}}+2ik\partial _{z}{\mathcal {E}}-k^{2}{\mathcal {E}}\right)\cdot e^{i(kz-\omega t)}}$

${\displaystyle {\frac {1}{c^{2}}}\partial _{t}^{2}\varepsilon {\vec {E}}^{+}\approx -\left(k^{2}{\mathcal {E}}+{\frac {2ik}{c_{g}}}\partial _{t}{\mathcal {E}}\right)\cdot e^{i(kz-\omega t)}}$

${\displaystyle \partial _{t}^{2}{\vec {P}}^{+}=\left(-\omega ^{2}{\mathcal {P}}-2i\omega \partial _{t}{\mathcal {P}}+\partial _{t}^{2}{\mathcal {P}}\right)\cdot e^{i(kz-\omega t)}\approx -\omega ^{2}{\mathcal {P}}\cdot e^{i(kz-\omega t)}}$

and with it the last Maxwell-Bloch equation

${\displaystyle \left(\partial _{z}+{\frac {1}{c_{g}}}\partial _{t}\right){\mathcal {E}}={\frac {i}{2}}\mu _{0}c_{p}{\mathcal {P}}}$

literature

• Dieter Meschede: Optics, light and laser . Vieweg + Teubner Verlag; 3rd, through 2008 edition. ISBN 978-3-8351-0143-2