The Maxwell–Bloch equations, also called the optical Bloch equations[1] describe the dynamics of a two-state quantum system interacting with the electromagnetic mode of an optical resonator. They are analogous to (but not at all equivalent to) the Bloch equations which describe the motion of the nuclear magnetic moment in an electromagnetic field. The equations can be derived either semiclassically or with the field fully quantized when certain approximations are made.
The derivation of the semi-classical optical Bloch equations is nearly identical to solving the two-state quantum system (see the discussion there). However, usually one casts these equations into a density matrix form. The system we are dealing with can be described by the wave function:
\psi=cg\psig+ce\psie
2 | |
\left|c | |
g\right| |
+
2 | |
\left|c | |
e\right| |
=1
The density matrix is
\rho=\begin{bmatrix}\rhoee&\rhoeg\ \rhoge&\rhogg\end{bmatrix}=\begin{bmatrix}ce
* | |
c | |
e |
&ce
* | |
c | |
g |
\ cg
* | |
c | |
e |
&cg
* | |
c | |
g |
\end{bmatrix}
(other conventions are possible; this follows the derivation in Metcalf (1999)).[2] One can now solve the Heisenberg equation of motion, or translate the results from solving the Schrödinger equation into density matrix form. One arrives at the following equations, including spontaneous emission:
d\rhogg | |
dt |
=\gamma\rhoee+
i | |
2 |
(\Omega*\bar\rhoeg-\Omega\bar\rhoge)
d\rhoee | |
dt |
=-\gamma\rhoee+
i | |
2 |
(\Omega\bar\rhoge-\Omega*\bar\rhoeg)
d\bar\rhoge | |
dt |
=-\left(
\gamma | |
2 |
+i\delta\right)\bar\rhoge+
i | |
2 |
*(\rho | |
\Omega | |
ee |
-\rhogg)
d\bar\rhoeg | |
dt |
=-\left(
\gamma | |
2 |
-i\delta\right)\bar\rhoeg+
i | |
2 |
\Omega(\rhogg-\rhoee)
In the derivation of these formulae, we define
\bar\rhoge\equiv\rhogee-i\delta
\bar\rhoeg\equiv\rhoegei\delta
\rhoeg(t)
\gamma | |
2 |
\Omega
\Omega=\vec{dg,e
\delta=\omega-\omega0
\omega
\omega0
\vec{d}g,e
g → e
\vec{E}0=\hat{\epsilon}E0
\vec{E}=
\vec{E0 | |
Beginning with the Jaynes–Cummings Hamiltonian under coherent drive
H=\omegaca\daggera+\omegaa\sigma\dagger\sigma+ig(a\dagger\sigma-a\sigma\dagger)+iJ(a\dagger
-i\omegalt | |
e |
-a
i\omegalt | |
e |
)
where
a
\sigma= | 1 |
2 |
\left(\sigmax-i\sigmay\right)
|\psi\rangle →
-i\omegalt\left(a\daggera+\sigma\dagger\sigma\right) | |
\operatorname{e} |
|\psi\rangle
H=\Deltaca\daggera+\Deltaa\sigma\dagger\sigma+ig(a\dagger\sigma-a\sigma\dagger)+iJ(a\dagger-a)
where
\Deltai=\omegai-\omegal
J
2 | |
J=\sqrt{2P(\Delta | |
c |
+
2)/(\omega | |
\kappa | |
c |
\kappa)}
2\kappa
\rho |
=-i[H,\rho]+2\kappa\left(a\rhoa\dagger-
1 | |
2 |
\left(a\daggera\rho+\rhoa\daggera\right)\right)+2\gamma\left(\sigma\rho\sigma\dagger-
1 | |
2 |
\left(\sigma\dagger\sigma\rho+\rho\sigma\dagger\sigma\right)\right)
The equations of motion for the expectation values of the operators can be derived from the master equation by the formulas
\langleO\rangle=\operatorname{tr}\left(O\rho\right)
\langleO |
\rangle=\operatorname{tr}\left(O
\rho |
\right)
\langlea\rangle
\langle\sigma\rangle
\langle\sigmaz\rangle
d | |
dt |
\langlea\rangle=i\left(-\Deltac\langlea\rangle-ig\langle\sigma\rangle-iJ\right)-\kappa\langlea\rangle
d | |
dt |
\langle\sigma\rangle=i\left(-\Deltaa\langle\sigma\rangle-ig\langlea\sigmaz\rangle\right)-\gamma\langle\sigma\rangle
d | |
dt |
\langle\sigmaz\rangle=-2g\left(\langlea\dagger\sigma\rangle+\langlea\sigma\dagger\rangle\right)-2\gamma\langle\sigmaz\rangle-2\gamma
At this point, we have produced three of an infinite ladder of coupled equations. As can be seen from the third equation, higher order correlations are necessary. The differential equation for the time evolution of
\langlea\dagger\sigma\rangle
\langlea\rangle=(\gamma/\sqrt{2}g)x
\langle\sigma\rangle=-p/\sqrt{2}
\langle\sigmaz\rangle=-D
\Theta=\Deltac/\kappa
C=g2/2\kappa\gamma
y=\sqrt{2}gJ/\kappa\gamma
\Delta=\Deltaa/\gamma
And the Maxwell–Bloch equations can be written in their final form
x |
=\kappa\left(-2Cp+y-(i\Theta+1)x\right)
p |
=\gamma\left(-(1+i\Delta)p+xD\right)
D |
=\gamma\left(2(1-D)-(x*p+xp*)\right)
Within the dipole approximation and rotating-wave approximation, the dynamics of the atomic density matrix, when interacting with laser field, is described by optical Bloch equation, whose effect can be divided into two parts:[3] optical dipole force and scattering force.[4]