1. Introduction to Coherence Spectroscopy Lecture 1 Coherence: “A term that's applied to electromagnetic waves. When they "wiggle" up and down together they are said to be coherent.” (http://www2.slac.stanford.edu/vvc/glossary.html)http://www2.slac.stanford.edu/vvc/glossary.html Spatial, temporal coherence.

2. Fourier Transform Microwave Spectrometer Frequency range: 4 – 26 GHz (ca. 0.1 – 1 cm -1 )

3. microwave cavity nozzle Fourier Transform Microwave Spectrometer

4. Two-Level Approximation Energy E b ; M b (q) / * b, E a ; M a (q) / * a, ε = ε o cos(ωt) SωSω SωoSωo Energies E a, E b and wavefunctions Ф a,Ф b from time-independent Schrödinger equation: H Ф a,b = E a,b Ф a,b

5. To understand how the system gets from Ф a to Ф b, we must consider the time-dependent Schrödinger equation: Hψ = i S ( dψ / dt) q With the solution: ψ a,b = M a,b exp (-iE a,b t / S ) Note that the probability density ψ * ψ for finding the system at the coordinates q is independent of time for states ψ a and ψ b (stationary states). Time-Dependent Schrödinger Equation

6. Superposition States Any linear combination of the eigenfunctions that represent the eigenstates of an operator R also represents a possible state: Ψ = a ψ a + b ψ b (superposition state) The probability to find the system in states ψ a and ψ b is a * a and b * b, respectively. In a superposition state, the property represented by R is not a constant of the motion.

7. Superposition states Ψ = a ψ a + b ψ b ; ψ a,b = Ф a,b exp (-i E a,b t / S ) The probability distribution: * Ψ * 2 = * a * 2 * Ф a * 2 + * b * 2 * Ф b * 2 + 2Re{a * b Ф a * Ф b exp[-i (E b -E a )t/ S ]} changes with time, and the expectation value of R will be time dependent. Non-stationary state { ωotωot

8. Eigenfunctions for the Rotational Hamiltonian Wavefunction for rotational state J=0: Wavefunction for rotational state J=1:

9. Superposition State Ψ = 0.454 ψ J=0 + 0.891 ψ J=1 Normalization of Ψ requires that * a * 2 + * b * 2 = 1

10. Molecule – Radiation Interaction Application of radiation with ω = ω 0 Electromagnetic field and dipole moment oscillate at the same frequency (‘Resonance’). The electromagnetic field exerts a torque on the oscillatory dipole moment. The expansion coefficients will now be functions of time: a(t), b(t)

11. If the system is initially in stat e ψ a : Application of Radiation Pulse with the Rabi frequency x = μ ε 0 / S (‘strength of radiation – molecule interaction’)

12. Ensemble Averages We now consider an ensemble of N 2-level systems. The ensemble average of an observable O, for example the dipole moment, is given by:

13. Ensemble Averages with the ‘density matrix’: and the ‘matrix representation’ of O:

14. Density Matrix Elements Diagonal elements ρ aa and ρ bb : population probabilities of the energy levels E a and E b Off-diagonal elements ρ ab and ρ ba (= ρ ab * ): coherence terms (fixed phase relation between the time-dependent wavefunctions of the two-level systems)

15. Time-Dependence of Density Matrix The time-dependent Schrödinger equation can be re-written: H : matrix representation of the Hamiltonian : time independent Hamiltonian (eigenvalues E a and E b ) : interaction operator (molecule- coherent radiation field) e.g. electric dipole interaction :

16. Time-Dependence of Density Matrix Elements with the Rabi frequency x = ε 0 μ / S (‘strength of radiation – molecule interaction’)

17. Rotating Coordinate System Transformation into coordinate system that rotates with the frequency ω of the coherent external field: Neglect fast oscillating terms (rotating wave approximation):

18. Bloch Variables Coherence terms: Population terms: (population sum) (population difference)

19. Bloch Equations without relaxation effects ∆ω = ω 0 -ω off-resonance frequency ω 0 : transition frequency ω: frequency of external radiation x = ε 0 μ / S Rabi frequency, ‘strength of radiation – molecule interaction’

20. Pulse Solutions Assume (hard pulses) Initial conditions (t=0): Coherence terms: u(0)=v(0)=0 Population difference: w(0)=w 0

21. π/2 Excitation Pulse A maximum polarization is achieved for a pulse length of t p = (π/2x) = (h/4μ ab ε 0 )

22. Time Evolution of Bloch Variables u v w time ε 0 = 0 ε 0 ≠ 0 π/2 pulse

23. Representation in Bloch Vector Diagram at t=0 (u=v=0, w=w 0 ) w u v w v w v after π/2 - pulse after π - pulse u u

24. Field-free Solutions Perturbation switched off: x = 0 Initial conditions (t = t p ): u(t p )=0; v(t p )=-w 0, w(t p )=0

25. Evolution after π/2 Excitation Pulse u v w time ε 0 = 0 ε 0 ≠ 0 ε 0 = 0 Δω=0, on resonance

26. Observable Time-Domain Signal The average dipole moment can be calculated: In terms of Bloch variables: 1 N: number density of two-level systems The observable signal is proportional to the "polarization" of the sample (macroscopic dipole moment/unit volume):

27. Emission after π/2 Excitation Pulse u v w time ε 0 = 0 ε 0 ≠ 0 ε 0 = 0

28. Representation of Signal in Frequency Domain Fourier transformation of time-domain signal S(t): Usually represented as power spectrum: |F(ω)| 2 = Re 2 {F(ω)} + Im 2 {F(ω)}

29. Pulse Length Dependence of Emission Signal w u -v w u Sample: OC 34 S

30. The End