[SOLVED] nonlinear optics hamiltonian

[sperelat@hanmail.net]

I'm studying squeezing now ,and I have question about the
hamiltonian,
I understood the appropriate form of Hamiltonian for squeezing,but I
don't understand
how nonlinear optical interaction can provide such a Hamiltonian.,
What I know is just classical explanation of nonlinear optical
process,and I can't get
the quantum optics books that has explanation(like Milburn's)
How can we derive the Hamiltonian of second harmonic generation or
OPA or Kerr effect?
Hans Bachor's book shows it by canonical quantization, but I couldn't
understand even
classical Hamiltonian form ther wrote.

( form of ...beta*alpha^2...beta=second harmonic
amplitude,alpha=fundamental frequency amplitude)

[p.kinsler@imperial.ac.uk]

sperelat@hanmail.net wrote:
> What I know is just classical explanation of nonlinear optical
> process,and I can't get
> the quantum optics books that has explanation(like Milburn's)
> How can we derive the Hamiltonian of second harmonic generation or
> OPA or Kerr effect?
> Hans Bachor's book shows it by canonical quantization, but I couldn't
> understand even
> classical Hamiltonian form ther wrote.

> ( form of ...beta*alpha^2...beta=second harmonic
> amplitude,alpha=fundamental frequency amplitude)

Here's a quick and dirty explanation.

Hillery and Mlodinow Phys Rev A 1984 have, a perturbative
second-order nonlinear interaction (ignoring some prefactors) is

H_2 = chi_2 E^3

Now since for mode operators a_i, a_i^\dagger, we can write

E = \sum_i ( a_i exp(-i w_i t) + a_i^\dagger exp(+i w_i t))

H_2 = chi_2 \sum_i \sum_j \sum_k
( a_i exp(-i w_i t) + a_i^\dagger exp(+i w_i t) )
( a_j exp(-i w_j t) + a_j^\dagger exp(+i w_j t) )
( a_k exp(-i w_k t) + a_k^\dagger exp(+i w_k t) )

This is quite a long expansion, and contains all kinds of terms,
most of which oscillate in time rather quickly.

So let's just pick out two modes, a_1 and a_2, whose mode
frequencies are w_1 and w_2; with 2 w_1 = w_2. The only
parts of the expansion for H_2 above which do not oscillate
are

chi_2 a_2 exp(-i w_k t) a_1^\dagger exp(+i w_1 t) a_1^\dagger exp(+i w_1 t)
+chi_2 a_2^\dagger exp(+i w_k t) a_1 exp(-i w_1 t) a_1 exp(-i w_1 t)

= chi_2 a_2 a_1^\dagger^2 + chi_2 a_2^\dagger a_1^2

which can turn into your beta*alpha^2 with an appropriate
change of notation.

--
---------------------------------+---------------------------------
Dr. Paul Kinsler
Blackett Laboratory (QOLS) (ph) +44-20-759-47520 (fax) 47714
Imperial College London, Dr.Paul.Kinsler@physics.org
SW7 2BW, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/