*Lectures on Nonlinear Optics - Lecture 5*

# Quantum mechanics II: Formulation of nonlinear optical interactions

**lect5.pdf** [283 kB]
Lecture 5 in Portable Document Format.

### Contents

In the previous lecture, the quantum mechanical origin of the linear and
nonlinear susceptibilities was discussed. In particular, a perturbation
analysis of the density operator was performed, and the resulting system
of equations was solved recursively for the *n*:th order density operator
in terms of
, where the zeroth order
term (independent of the applied electric field of the light) is given by
the Boltzmann distribution at thermal equilibrium.

So far we have obtained the linear optical properties of the medium, in terms of the first order susceptibility tensor (of rank-two), and we will now proceed with the next order of interaction, giving the second order electric susceptibility tensor (of rank three).

## 1. The second order polarization density

For the second order interaction, the corresponding term in the perturbation
series of the density operator *in the interaction picture*
becomes [1]

In order to simplify the expression for the second order susceptibility, we will in the following analysis make use of a generalization of the cyclic perturbation of the terms in the commutator inside the trace, as

By inserting the expression for the second order term of the perturbation series for the density operator into the quantum mechanical trace of the second order electric polarization density of the medium, one obtains

In analogy with the results as obtained for the first order (linear) optical
properties, now express the term *E*_{α1}(τ_{1})*E*_{α2}(τ_{2})
in the frequency domain, by using the Fourier identity

which hence gives the second order polarization density expressed in terms of the electric field in the frequency domain as

where the second order (quadratic) electric susceptibility is defined as

This obtained expression for the second order electric susceptibility does not possess the property of intrinsic permutation symmetry. However, by using the same arguments as discussed in the analysis of the polarization response functions in lecture two, we can easily verify that this tensor can be cast into a symmetric and antisymmetric part as

and since the antisymmetric part, again following the arguments for the second order polarization response function, does not contribute to the polarization density, it is customary (in the Butcher and Cotter convention as well as all other conventions in nonlinear optics) to cast the second order susceptibility into the form

where **S**, commonly called the *symmetrizing operator*, denotes
that the expression that follows is to be summed over the 2! = 2
possible pairwise permutations of (α, ω_{1}) and
(β, ω_{1}), hence ensuring that the second order
susceptibility possesses the intrinsic permutation symmetry,

## 2. Higher order polarization densities

The previously described principle of deriving the susceptibilities
of first and second order are straightforward to extend to the *n*:th
order interaction. In this case, we will make use of the following
generalization of Eq. (2),

which, when applied in the evaluation of the expectation value of the electric
dipole operator of the ensemble, gives the *n*:th order electric
susceptibility as

where now the symmetrizing operator **S** indicates that the expression
following it should be summed over all the *n*! pairwise permutations of
(α_{1},ω_{1}), … ,
(α_{n}, ω_{n})$.

It should be emphasized the symmetrizing operator **S** always
implies summation over *all* the *n*! pairwise permutations of
(α_{1},ω_{1}), … ,
(α_{n}, ω_{n})$, *regardless of
whether the permutations are distinct or not*. This is due to that
eventually occuring degenerate permutations are taken care of in the degeneracy
coefficient *K*(-ω_{σ}; ω_{1}, …,
ω_{n}) in Butcher and Cotters convention, as described
in lecture three and in the
additional notes which were handed out during
lecture four.

## References

[1]
It should be noticed that the form given in
Eq. (1) not only applies to an ensemble of molecules, of arbitrary composition, but
also to *any* kind of level of approximation for the interaction, such
as the inclusion of magnetic dipolar interactions or electric quadrupolar
interactions as well. These interactions should (of course) be incorporated
in the expression for the interaction Hamiltonian
,
here described in the interaction picture.