Nonlinear Optics
Homework 5
due Tuesday, 20 Feb 2007

Problem 1: 
The Hamiltonian for a 3-D harmonic oscillator can be written as
H = ((p^2)/(2  m)) + 
            V(r) = ((ℏ^2)/(2  m)) (∇^2)+(1/2) 
            m (ω^2) (r^2) 
a) Writing this in Cartesian coordinates, and using separation of variables, show that the eigenfunctions can be written as 
(Subscript[u, n1])(x) (Subscript[u, n2])(y) (Subscript[u, n3])(
          z), with energy eigenvalues (Subscript[E, n1, n2, n3]) = (n1  + n2 + n3 +  
              3/2) ℏω
where the n's are the quantum numbers corresponding to the different directions, and the u's satisfy the standard 1D harmonic oscillator equation. For example, (Subscript[u, n1])(x) satisfies 
((ℏ^2)/(2  
                  m)) ((∂^2)/∂)(x^2) (Subscript[u, n1])(
              x) + (1/2) 
            m (ω^2) (x^2) (Subscript[u, n1])(
              x) = (Subscript[E, n1]) (Subscript[u, n1])(x), with energy eigenvalues  (Subscript[E, n1]) = (n1  +  1/2) ℏω .

Problem 2:
There are two ways to evaluate matrix elements such as that for the dipole operator.

a) Consider the 1D SHO. Calculate the matrix elements by direct integration: 
(Subscript[x, mn]) = <m | 
            x | n>  =  ∫(Subscript[u, m])[x] x (Subscript[u, n])[
                x] dx,
showing that you get a non-zero value only if the selection rule m = n±1 is satisfied. 

The 1-D SHO wavefunctions can be written in Mathematica as:

u[x_, 
          n_] := ((((1/(π (x0^2)))))^(1/
                4)) (1/(Sqrt[(2^n) n!])) HermiteH[n, x/x0] 
          Exp[-((x^2)/(2 (x0^2)))];

\(where\ x0\  = \ \@\(ℏ/m\ ω\)\ is\ the\ natural\ scale\ length\ for\ the\ \(\(SHO\)\(.\)\(\ \)\)\)

b) The alternate, more elegant, approach is to represent the position operator x as: 
x = \(TraditionalForm\`\(x0\/\@2\) \((\(a\_+\) + \(a\_-\))\)\), where the "creation" and "annihilation" operators \(TraditionalForm\`\(a\_+\)\) and \(TraditionalForm\`\(a\_-\)\) have the properties:
\(TraditionalForm\`\(a\_+\) | \(\(n\)\(>\)\)\  = \ \@\(n + 1\) | \(\(n + 1\)\(>\)\)\)
\(TraditionalForm\`\(a\_-\) | \(\(n\)\(>\)\)\  = \ \@n | \(\(n - 
              1\)\(>\)\)\)
Repeat the calculation of <m|x|n> this way to show you get the same result. 
In 3D, the \(TraditionalForm\`\(a\_+\), \ \(a\_-\)\) operators change only the quantum numbers associated with one direction, so for example, 
(Subscript[a, y-]) | n1, 
      n2, n3>  =  (Sqrt[n2]) | n1, n2 - 1, n3>

Problem 3
Calculation of χ's. This is too cumbersome to do with integrals. Use the relations derived in the previous problem to evaluate the matrix elements, most of which will be zero.

a) Calculate the linear term (χ^((1))) for an anisotropic SHO, where there are two resonant frequencies: 
ωo in the x and y directions and ωe in the z-direction. Show that you end up with a diagonal tensor for the dielectric constant.

b) Calculate (χ^((2)))for the isotropic SHO in the nonresonant case (3.2.28) and show that it is zero as you would expect from the symmetry.

c) Calculate (Subscript[(((χ^((3))))), 1111]) for the isotropic SHO in the nonresonant case (3.2.33), and show that it also is equal to zero. At first look, it is not obvious that this should be zero, but the permutation operation leads to cancellation. The permutation opererator can be done in Mathematica. Here is how to do it for one of the denominators. You set up a list of the terms to permute:

denom1[ωLst_] := (ω0 + ω(Subscript[Lst, [[1]]])) (-ω(Subscript[Lst, [[3]]]) - ω(Subscript[Lst, [[4]]])) (ω0 - ω(Subscript[Lst, [[4]]]));
ωs = {-(ωr + ωq + ωp), ωr, ωq, ωp};
ωsPerm = Permutations[ωs];

Next, you make a loop to sum up all the denominators, then FullSimplify. 
Finally you do the same operation for the second term and add them. 

When I first put this together, it seemed that there was a nonzero χ3 for a SHO without any nonlinear term added to the potential. Although this would not be expected classically, I rationalized that in fact the dipole term is in fact a change to the potential, and that in higher order there could be an effect. However, it does turn out the classical and quantum pictures agree as they should.

Note that if we were using any other potential, for example a square well, we will expect that this calculation will give a nonzero χ3. For the SHO, we'd add a quartic term, do time-independent perturbation theory to get the wavefunctions before we apply the electric field, then go through this procedure.

Problem 4
Boyd, problem 4.1