electromagnetism - Nonlinear optics as gauge theory

the widely used approach to nonlinear optics is a Taylor expansion of the dielectric displacement field $\mathbf{D} = \epsilon_0\cdot\mathbf{E} + \mathbf{P}$ in a Fourier representation of the polarization $\mathbf{P}$ in terms of the dielectric susceptibility $\mathcal{X}$:

$\mathbf{P} = \epsilon_0\cdot(\mathcal{X}^{(1)}(\mathbf{E}) + \mathcal{X}^{(2)}(\mathbf{E},\mathbf{E}) + \dots)$ .

This expansion does not work anymore if the excitation field has components close to the resonance of the medium. Then, one has to take the whole quantum mechanical situation into account by e.g. describing light/matter interaction by a two-level Hamiltonian.

But this approach is certainly not the most general one.

Intrinsically nonlinear formulations of electrodynamics

So, what kind of nonlinear formulations of electrodynamics given in a Lagrangian formulation are there?

One known ansatz is the Born-Infeld model as pointed out by Raskolnikov. There, the Lagrangian density is given by

$\mathcal{L} = b^2\cdot \left[ \sqrt{-\det (g_{\mu \nu})} - \sqrt{-\det(g_{\mu \nu} + F_{\mu \nu}/b)} \right]$

and the theory has some nice features as for example a maximum energy density and its relation to gauge fields in string theory. But as I see it, this model is an intrinsically nonlinear model for the free-space field itself and not usefull for describing nonlinear matter interaction.

The same holds for an ansatz of the form

$\mathcal{L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \lambda\cdot\left( F^{\mu\nu}F_{\mu\nu} \right)^2$

proposed by Mahzoon and Riazi. Of course, describing the system in Quantum Electrodynamics is intrinsically nonlinear and ... to my mind way to complicated for a macroscopical description for nonlinear optics. The question is: Can we still get a nice formulation of the theory, say, as a mean field theory via an effective Lagrangian?

I think a suitable ansatz could be

$\mathcal{L} = -\frac{1}{4}M^{\mu\nu}F_{\mu\nu}$

where $M$ now accounts for the matter reaction and depends in a nonlinear way on $\mathbf{E}$ and $\mathbf{B}$, say

$M^{\mu\nu} = T^{\mu\nu\alpha\beta}F_{\alpha\beta}$

where now $T$ is a nonlinear function of the field strength and might obey certain symmetries. The equation $T = T\left( F \right)$ remains unknown and depends on the material.

Metric vs. $T$ approach

As pointed out by space_cadet, one might ask the question why the nonlinearity is not better suited in the metric itself. I think this is a matter of taste. My point is that explicitly changing the metric might imply a non-stationary spacetime in which a Fourier transformation might not be well defined. It might be totally sufficient to treat spacetime as Lorentzian manifold.
Also, we might need a simple spacetime structure later on to explain the material interaction since the polarization $\mathbf{P}$ depends on the matter response generally in terms of an integration over the past, say

$\mathbf{P}(t) = \int_{-\infty}^{t}R\left[\mathbf{E}\right](\tau )d\tau$

with $R$ beeing some nonlinear response function(al) related to $T^{\mu\nu\alpha\beta}$.

Examples for $T$

To illustrate the idea of $T$, here are some examples.
For free space, $T$ it is given by $T^{\mu\nu\alpha\beta} = g^{\mu\alpha}g^{\nu\beta}$ resulting in the free-space Lagrangian $\mathcal{L} = -\frac{1}{4}T^{\mu\nu\alpha\beta}F_{\alpha\beta}F_{\mu\nu} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$ The Lagrangian of Mahzoon and Riazi can be reconstructed by
$T^{\mu\nu\alpha\beta} = \left( 1 + \lambda F^{\gamma\delta}F_{\gamma\delta} \right)\cdot g^{\mu\alpha}g^{\nu\beta}$.
One might be able to derive a Kerr nonlinearity using this Lagrangian.

So, is anyone familiar in a description of nonlinear optics/electrodynamics in terms of a gauge field theory or something similar to the thoughts outlined here?

Thank you in advance.

Sincerely,

Robert

Comments on the first Bounty

I want to thank everyone actively participating in the discussion, especially Greg Graviton, Marek, Raskolnikov, space_cadet and Willie Wong. I am enjoying the discussion relating to this question and thankfull for all the nice leads you gave. I decided to give the bounty to Willie since he gave the thread a new direction introducing the material manifold to us.
For now, I have to reconsider all the ideas and I hope I can come up with a new revision of the question that should be formulated in a clearer way as it is at the moment.
So, thank you again for your contributions and feel welcome to share new insights.

Answer

Just a few random thoughts.

There is something important in your observation that the Born-Infeld model is essentially a free-space model. It is known to Boillat and Plebanski (separately in 1970) that the Born-Infeld model is the only model of electromagnetism (as a connection on a $U(1)$ vector bundle) that satisfies the following conditions

1. Covariance under Lorentz transformations


2. Reduces to Maxwell's equation in the small-field strength limit



3. $U(1)$ gauge symmetry


4. Integrable energy density for a point-charge


5. No birefringence (speed of light independent of polarization).

(the linear Maxwell system fails condition 4.) (See Michael Kiessling, "Electromagnetic field theory without divergence problems", J. Stat. Phys. (2004) doi:10.1023/B:JOSS.0000037250.72634.2a for an exposition on this and related issues.)

Now, since you are interested in nonlinear optics inside a material, instead of in vacuum, I think conditions 1 and 5 can safely be dropped. (Though you may want to keep 5 as a matter of course.) Condition 4 is intuitively pleasing, but maybe not too important, at least not until you have some candidate theories in mind that you want to distinguish. Condition 3 you must keep. Condition 2, on the other hand, really depends on what kind of material you have in mind.

In any case, a small suggestion: personally I think it is better to, from the get-go, write your proposed Lagrangian as

$$ L = T^{abcd} F_{ab}F_{cd} $$

instead of $M^{ab}F_{cd}$. I think it is generally preferable to consider Lagrangian field theories of at least quadratic dependence on the field variables. A pure linear term suggests to me an external potential which I don't think should be built into the theory.

If you want something like condition 2, but with a dielectric constant or such, then you must have that $T^{abcd}$ admit a Taylor expansion looking something like

$$ T^{abcd} = \tilde{g}^{ac}\tilde{g}^{bd} + O(|F|) $$

where $\tilde{g}$ is some effective metric for the material. Birefringence, however, you don't have to insert in explicitly: most likely a generic (linear or nonlinear) $T^{abcd}$ you write down will have birefringence; it is only when you try to rule it out that you will bring in some constraints.

An interesting thing is to consider what it means to have an analogous notion to condition 1. In the free-space case, condition 1 implies that the Lagrangian should only be a function of the Lorentz invariant $B^2 - E^2$ (in natural units) and of the pseudo-scalar invariant $B\cdot E$. In terms of the Faraday tensor these two invariants are $F^{ab}F_{ab}$ and $F^{ab}{}^*F_{ab}$ respectively, where ${}^*$ denote the Hodge dual. The determination of the linear part of your theory (of electromagnetic waves in a material) is essentially by what you will use to replace condition 1. If you assume your material is isotropic and homogeneous, then some similar sort of scalar + pseudo-scalar invariants is probably a good bet.