Categories: Electromagnetism, Laser theory, Physics, Quantum mechanics, Two-level system.

Maxwell-Bloch equations

For an electron in a two-orbital system {g,e}\{\ket{g}, \ket{e}\}, the Schrödinger equation has the following general solution, where εg\varepsilon_g and εe\varepsilon_e are the time-independent eigenenergies, and the weights cgc_g and cgc_g are functions of tt:

Ψ=cggeiεgt/+ceeeiεet/\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}

This system is being perturbed by an electromagnetic wave with electric field E\vb{E} given by:

E(t)E(t)+E+(t)\begin{aligned} \vb{E}(t) &\equiv \vb{E}^{-}(t) + \vb{E}^{+}(t) \end{aligned}

Where the forward-propagating component E+\vb{E}^{+} is a modulated plane wave E0+eiωt\vb{E}_0^{+} e^{-i \omega t} with slowly-varying amplitude E0+(t)\vb{E}_0^{+}(t), and similarly E(t)E0(t)eiωt\vb{E}^{-}(t) \equiv \vb{E}_0^{-}(t) e^{i \omega t}; since E\vb{E} is real, E0+ ⁣= ⁣(E0)\vb{E}_0^{+} \!=\! (\vb{E}_0^{-})^*.

For Ψ\ket{\Psi} as defined above, the pure density operator ρ^\hat{\rho} is as follows, with ω0(εe ⁣ ⁣εg)/\omega_0 \equiv (\varepsilon_e \!-\! \varepsilon_g) / \hbar being the transition’s resonance frequency:

ρ^=ΨΨ=[cecececgeiω0tcgceeiω0tcgcg][ρeeρegρgeρgg]\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}

Under the electric dipole approximation and rotating wave approximation, it can be shown that ρ^\hat{\rho} is governed by the optical Bloch equations:

dρggdt=γeρeeγgρgg+i(p0+Eρegp0E+ρge)dρeedt=γgρggγeρee+i(p0E+ρgep0+Eρeg)dρgedt=(γiω0)ρge+ip0+E(ρeeρgg)dρegdt=(γ+iω0)ρeg+ip0E+(ρggρee)\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}

Where we have defined the transition dipole moment p0\vb{p}_0^{-}, with q<0q < 0 the electron charge:

p0qex^gp0+(p0)=qgx^e\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}

However, the light wave affects the electron, so the actual electromagnetic dipole moment p\vb{p} is as follows, using Laporte’s selection rule to remove diagonal terms by assuming that the electron’s orbitals are spatially odd or even:

p=qΨx^Ψ=q(cgcggx^g+ceceex^e+cgceex^geiω0t+cecggx^eeiω0t)=q(ρgeex^g+ρeggx^e)=p0ρge(t)+p0+ρeg(t)p(t)+p+(t)\begin{aligned} \vb{p} &= 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} 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) \end{aligned}

Where we have split p\vb{p} analogously to E\vb{E} by defining p+p0+ρeg\vb{p}^{+} \equiv \vb{p}_0^{+} \rho_{eg}. Its equation of motion can then be found from the optical Bloch equations:

dp+dt=p0+dρegdt=p0+(γ+iω0)ρeg+ip0+(p0E+)(ρggρee)\begin{aligned} \dv{\vb{p}^{+}}{t} = \vb{p}_0^{+} \dv{\rho_{eg}}{t} = - \vb{p}_0^{+} \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg} + \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 ρge\rho_{ge} by p0+\vb{p}_0^{+}. In our case, we arrive at a prototype of the first of three Maxwell-Bloch equations:

dp+dt=(γ+iω0)p+i(p0E+)p0+d\begin{aligned} \boxed{ \dv{\vb{p}^{+}}{t} = - \Big( \gamma_\perp + i \omega_0 \Big) \vb{p}^{+} - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} d } \end{aligned}

Where we have defined the population inversion d[1,1]d \in [-1, 1] like so, to quantify the electron’s “excitedness” i.e. its localization to e\ket{e}:

dρeeρgg\begin{aligned} d \equiv \rho_{ee} - \rho_{gg} \end{aligned}

From the optical Bloch equations, we find its equation of motion to be:

dddt=dρeedtdρggdt=2γgρgg2γeρee+i2(pE+p+E)\begin{aligned} \dv{d}{t} &= \dv{\rho_{ee}}{t} - \dv{\rho_{gg}}{t} = 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} + \frac{i 2}{\hbar} \Big( \vb{p}^{-} \cdot \vb{E}^{+} - \vb{p}^{+} \cdot \vb{E}^{-} \Big) \end{aligned}

We can rewrite the first two terms in the following intuitive form, which describes a decay with rate γγg+γe\gamma_\parallel \equiv \gamma_g + \gamma_e towards an equilibrium d0d_0:

2γgρgg2γeρee=γ(d0d)d0γgγeγg+γe\begin{aligned} 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} = \gamma_\parallel (d_0 - d) \qquad \qquad d_0 \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} \end{aligned}

We introduce some new terms, and reorganize the expression:

2γgρgg2γeρee=2γgρgg2γeρee+γgρeeγgρee+γeρggγeρgg=γg(ρgg+ρee)γe(ρgg+ρee)+γg(ρggρee)+γe(ρggρee)\begin{aligned} 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} &= 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} + \gamma_g \rho_{ee} - \gamma_g \rho_{ee} + \gamma_e \rho_{gg} - \gamma_e \rho_{gg} \\ &= \gamma_g (\rho_{gg} + \rho_{ee}) - \gamma_e (\rho_{gg} + \rho_{ee}) + \gamma_g (\rho_{gg} - \rho_{ee}) + \gamma_e (\rho_{gg} - \rho_{ee}) \end{aligned}

Since the total probability ρgg+ρee=1\rho_{gg} + \rho_{ee} = 1, and dρeeρggd \equiv \rho_{ee} - \rho_{gg}, this reduces to:

2γgρgg2γeρee=γgγe(γg+γe)d=(γg+γe)(γgγeγg+γed)=γ(d0d)\begin{aligned} 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} &= \gamma_g - \gamma_e - (\gamma_g + \gamma_e) d \\ &= (\gamma_g + \gamma_e) \Big( \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} - d \Big) \\ &= \gamma_\parallel ( d_0 - d ) \end{aligned}

With this, the equation for the population inversion dd takes the form below, namely the second Maxwell-Bloch equation’s prototype:

dddt=γ(d0d)+i2(pE+p+E)\begin{aligned} \boxed{ \dv{d}{t} = \gamma_\parallel (d_0 - d) + \frac{i 2}{\hbar} \Big( \vb{p}^{-} \cdot \vb{E}^{+} - \vb{p}^{+} \cdot \vb{E}^{-} \Big) } \end{aligned}

Finally, we would like a relation between the polarization and the electric field E\vb{E}, for which we turn to Maxwell’s equations; we will effectively derive a modified form of the electromagnetic wave equation. Starting from Faraday’s law and splitting B=μ0(H+M)\vb{B} = \mu_0 (\vb{H} + \vb{M}):

×E=Bt=μ0Htμ0Mt\begin{aligned} \nabla \cross \vb{E} = - \pdv{\vb{B}}{t} = - \mu_0 \pdv{\vb{H}}{t} - \mu_0 \pdv{\vb{M}}{t} \end{aligned}

We assume that there is no magnetization M=0\vb{M} = 0. Then we we take the curl of both sides, and replace ×H\nabla \cross \vb{H} with Ampère’s circuital law:

×(×E)=μ0t(×H)=μ0t(Jfree+Dt)\begin{aligned} \nabla \cross \big( \nabla \cross \vb{E} \big) = - \mu_0 \pdv{}{t} \big( \nabla \cross \vb{H} \big) = - \mu_0 \pdv{}{t} \Big( \vb{J}_\mathrm{free} + \pdv{\vb{D}}{t} \Big) \end{aligned}

Inserting the definition D=ε0E+P\vb{D} = \varepsilon_0 \vb{E} + \vb{P} together with Ohm’s law Jfree=σE\vb{J}_\mathrm{free} = \sigma \vb{E} yields:

×(×E)=μ0σEtμ0ε02Et2μ02Pt2\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 σ\sigma is the active material’s conductivity, if any; almost all authors assume σ=0\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 Pmed\vb{P}_\mathrm{med} to the electric field E\vb{E}. If the medium is linear, i.e. Pmed=ε0χE\vb{P}_\mathrm{med} = \varepsilon_0 \chi \vb{E}, then:

μ02Pt2=×(×E)μ0σEtμ0ε02Et2μ02Pmedt2=×(×E)μ0σEtμ02t2(ε0E+ε0χE)=×(×E)μ0σEtμ0ε0εr2Et2\begin{aligned} \mu_0 \pdvn{2}{\vb{P}}{t} &= - \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}_\mathrm{med}}{t} \\ &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \pdvn{2}{}{t}\Big( \varepsilon_0 \vb{E} + \varepsilon_0 \chi \vb{E} \Big) \\ &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{E}}{t} \end{aligned}

Where εr1+χe\varepsilon_r \equiv 1 + \chi_e is the medium’s relative permittivity. The speed of light c2=1/(μ0ε0)c^2 = 1 / (\mu_0 \varepsilon_0), and the refractive index n2=μrεrn^2 = \mu_r \varepsilon_r, where μr=1\mu_r = 1 due to our assumption that M=0\vb{M} = 0, so the third Maxwell-Bloch equation’s prototype is:

μ02Pt2=×(×E)μ0σEtn2c22Et2\begin{aligned} \boxed{ \mu_0 \pdvn{2}{\vb{P}}{t} = - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \frac{n^2}{c^2} \pdvn{2}{\vb{E}}{t} } \end{aligned}

E\vb{E} and P\vb{P} can trivially be replaced by E+\vb{E}^{+} and P+\vb{P}^{+}. It is also simple to convert p+\vb{p}^{+} and dd into the macroscopic P+\vb{P}^{+} and total DD by summing over all two-level systems in the medium:

P+(x,t)=νpν+δ(xxν)D(x,t)=νdνδ(xxν)\begin{aligned} \vb{P}^{+}(\vb{x}, t) &= \sum_{\nu} \vb{p}^{+}_\nu \: \delta(\vb{x} - \vb{x}_\nu) \\ D(\vb{x}, t) &= \sum_{\nu} d_\nu \: \delta(\vb{x} - \vb{x}_\nu) \end{aligned}

We thus arrive at the Maxwell-Bloch equations, which are the foundation of laser theory:

μ02P+t2=××E+μ0σE+tn2c22E+t2P+t=(γ+iω0)P+i(p0E+)p0+DDt=γ(D0D)+i2(PE+P+E)\begin{aligned} \boxed{ \begin{aligned} \mu_0 \pdvn{2}{\vb{P}^{+}}{t} &= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \frac{n^2}{c^2} \pdvn{2}{\vb{E}^{+}}{t} \\ \pdv{\vb{P}^{+}}{t} &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+} - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} D \\ \pdv{D}{t} &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big) \end{aligned} } \end{aligned}


  1. F. Kärtner, Ultrafast optics: lecture notes, 2005, Massachusetts Institute of Technology.
  2. H. Haken, Light: volume 2: laser light dynamics, 1985, North-Holland.