Categories: Physics, Quantum mechanics, Two-level system.

Optical Bloch equations

For an electron in a two-level system with time-independent states ∣g⟩\ket{g} (ground) and ∣e⟩\ket{e} (excited), consider the following general solution to the time-dependent Schrödinger equation:

∣Ψ⟩=cg∣g⟩e−iεgt/ℏ+ce∣e⟩e−iεet/ℏ\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 introduces a time-dependent sinusoidal term H^1\hat{H}_1 to the Hamiltonian. In the electric dipole approximation, H^1\hat{H}_1 is given by:

H^1(t)=−p^⋅E(t)p^≡qx^E(t)=E0cos⁡(ωt)\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 E\vb{E} is an electric field, and p^\hat{\vb{p}} is the dipole moment operator. From Rabi oscillation, we know that the time-varying coefficients cgc_g and cec_e can then be described by:

dcgdt=iq⟨g∣x^∣e⟩⋅E02ℏ ei(ω−ω0)t cedcedt=iq⟨e∣x^∣g⟩⋅E02ℏ e−i(ω−ω0)t cg\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 ω0≡(εe ⁣− ⁣εg)/ℏ\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 E\vb{E} as follows, where the amplitudes E0−\vb{E}_0^{-} and E0+\vb{E}_0^{+} may be slowly-varying with respect to the carrier wave e±iωte^{\pm i \omega t}:

E(t)≡E−(t)+E+(t)≡E0−eiωt+E0+e−iωt\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 E\vb{E} is real, E0+=(E0−)∗\vb{E}_0^{+} = (\vb{E}_0^{-})^*. Similarly, we define the transition dipole moment p0−\vb{p}_0^{-}:

p0−≡q⟨e∣x^∣g⟩p0+≡(p0−)∗=q⟨g∣x^∣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}

With these, the equations for cgc_g and cec_e can be rewritten as shown below. Note that E−\vb{E}^{-} and E+\vb{E}^{+} include the driving plane wave, and the rotating wave approximation is still made:

dcgdt=iℏp0+⋅E−e−iω0t cedcedt=iℏp0−⋅E+eiω0t cg\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 ∣Ψ⟩\ket{\Psi} as defined above, the corresponding pure density operator ρ^\hat{\rho} is as follows:

ρ^=∣Ψ⟩⟨Ψ∣=[cece∗cecg∗e−iω0tcgce∗eiω0tcgcg∗]≡[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}

We take the tt-derivative of the matrix elements, and insert the equations for cgc_g and cec_e:

dρggdt=dcgdtcg∗+cgdcg∗dt=iℏp0+⋅E−cecg∗e−iω0t−iℏp0−⋅E+cgce∗eiω0tdρeedt=dcedtce∗+cedce∗dt=iℏp0−⋅E+cgce∗eiω0t−iℏp0+⋅E−cecg∗e−iω0tdρgedt=dcgdtce∗eiω0t+cgdce∗dteiω0t+iω0cgce∗eiω0t=iℏp0+⋅E−cece∗−iℏp0+⋅E−cgcg∗+iω0cgce∗eiω0tdρegdt=dcedtcg∗e−iω0t+cedcg∗dte−iω0t−iω0cecg∗e−iω0t=iℏp0−⋅E+cgcg∗−iℏp0−⋅E+cece∗−iω0cecg∗e−iω0t\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:

dρggdt=iℏ(p0+⋅E−ρeg−p0−⋅E+ρge)dρeedt=iℏ(p0−⋅E+ρge−p0+⋅E−ρeg)dρgedt=iω0ρge+iℏp0+⋅E−(ρee−ρgg)dρegdt=−iω0ρeg+iℏp0−⋅E+(ρgg−ρee)\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 ρ^\hat{\rho}. But in practice, these quantities decay due to various processes, e.g. spontaneous emission.

Suppose ρee\rho_{ee} decays with rate γe\gamma_e. Because the total probability ρee+ρgg=1\rho_{ee} + \rho_{gg} = 1, we have:

(dρeedt)e=−γeρee  ⟹  (dρggdt)e=γeρee\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 ρgg\rho_{gg} decay into ρee\rho_{ee} with rate γg\gamma_g:

(dρggdt)g=−γgρgg  ⟹  (dρggdt)g=γgρgg\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 γ⊥\gamma_\perp:

(dρegdt)⊥=−γ⊥ρeg(dρgedt)⊥=−γ⊥ρge\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 ρ^\hat{\rho}, which are the basis of the Maxwell-Bloch equations and by extension all laser theory:

dρggdt=γeρee−γgρgg+iℏ(p0+⋅E−ρeg−p0−⋅E+ρge)dρeedt=γgρgg−γeρee+iℏ(p0−⋅E+ρge−p0+⋅E−ρeg)dρgedt=−(γ⊥−iω0)ρge+iℏp0+⋅E−(ρee−ρgg)dρegdt=−(γ⊥+iω0)ρeg+iℏp0−⋅E+(ρgg−ρee)\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 γg=0\gamma_g = 0 and γ⊥=γe/2\gamma_\perp = \gamma_e / 2.

References

  1. F. Kärtner, Ultrafast optics: lecture notes, 2005, Massachusetts Institute of Technology.
  2. H.J. Metcalf, P. van der Straten, Laser cooling and trapping, 1999, Springer.