3. Quadrupolar Hyperpolarizabilities

1. 3. Quadrupolar Hyperpolarizabilities Two parameters in above theory for the effective nonlinear susceptibilties Parameters about material properties for interface system Parameters about the geometry of the optical measurement We summarize the sum-over-state expressions of these quantities, and the derive the expressions in the static limit in relation to the Kleinman symmetry.

2. Calculation of nonlinear optical susceptibility through use of the density matrix formulation of quantum mechanics Second-order susceptibility Density matrix can be formulated from the basic laws of quantum mechanics

3. Density matrix formulation Suppose that a quantum-mechanical system is known to be in a particular quantum-mechanical state s . This wavefunction obeys the Schrodinger equation Where Let write the solution to 3.3.1 as follow Where gives the probability amplitude that the atom, which is known to be in state s, is in energy eigenstate n at time t . Put (3.3.3) in (3.3.1) we obtain, Exactly known to be in the quantum state s For the case of exact state is not known, energy eigenstates Classical probability

4. There are certain interactions(such as those resulting from collisions between atoms) a change in the state of the system Let determine how density matrix itself evolves in time non zero Physical assumption Thermal excitation(incoherent process) cannot produce any coherent superposition of atomic state. Eq. 3.3.22 cannot be solved directly. Let try Perturbation method.

5. Let take the steady-state solution to 3.4.10a as Changing variable Put two eq. above in 3.4.10b Let perturb

6. We derived Assume atomic number density

8. 3.1 General Expression Dipole susceptibility Quadrupolar susceptibilities

10. The second expression has been derived by using the following equation, From [ref44], Christine Neipert and Brian Space, Alfred B. Roney

11. The quadrupolar hyperpolarizabilities are given in the following forms:

12. In the static limit the magnetic field is decoupled from the electric field, and thus we focus on the response to the electric field.

13. From eq.72, the following symmetry relations can be obtained,