Optics, electromagnetic theory Definition

Maxwell–Bloch equations

Builds on Maxwell's equations Polarization density Solution of an ordinary differential equation Laser rate equations

Definition

Consider a single-mode laser cavity filled with identical two-level atoms whose transition is resonant with the mode. Let $E(t)$ be the real amplitude of the field, $P(t)$ the real amplitude of the atomic polarization, and $D(t)$ the population inversion, in suitable units. Let $\kappa > 0$ , $\gamma_\perp > 0$ and $\gamma_\parallel > 0$ be the decay rates of the field, the polarization and the inversion. Let $D_0$ be the inversion set by the pump, and $g > 0$ the coupling constant. In one common normalization, the Maxwell–Bloch equations for this laser are the system of ordinary differential equations $\frac{dE}{dt} = -\kappa E + g P, \qquad \frac{dP}{dt} = -\gamma_\perp P + g E D, \qquad \frac{dD}{dt} = -\gamma_\parallel (D - D_0) - 4 g E P.$

Remarks

The first equation comes from Maxwell's equations in the cavity, with the polarization as source. The other two are the optical Bloch equations of the atoms driven by the field. Without damping and pumping, the quantity $P^2 + D^2/4$ is conserved, which is the conservation of the length of the Bloch vector. In travelling-wave problems, the field equation is a partial differential equation in space and time instead.

Example: $E = P = 0$ , $D = D_0$ is always a steady state, the laser off. For $D_0 > \kappa \gamma_\perp / g^2$ there are also steady states with $E \ne 0$ and $D = \kappa \gamma_\perp / g^2$ , the laser on.

Non-example: the laser rate equations are not Maxwell–Bloch equations, since they contain no equation for the polarization $P$ . They follow from the Maxwell–Bloch equations when $\gamma_\perp$ is much larger than the other rates and $P$ is eliminated.

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Laser rate equations concept
Maxwell's equations concept
Polarization density concept
Solution of an ordinary differential equation concept
Maxwell–Bloch equations definition

Tag	`FMF327F0MV`
URI	`m2://FMF327F0MV`
Kind	definition
Writing	W0 stub
MSC	78A60, 81V80
Defines	Maxwell–Bloch equations
Uses	Maxwell's equations, Polarization density, Solution of an ordinary differential equation, Laser rate equations
Changed	3 November 2026

Maxwell–Bloch equations

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