Laser Spectroscopy (Classes 21 through 28)

Copyright © 2009 Krishna Myneni

Classes:  21  22  23  24  25  26  27  28  29

Homework:  12  13  14  15  16  Extra Credit

Class 21

Exercise: Show that our equations of motion for the two-level atom in the RWA, derived in class, are equivalent to Steck 5.23 if we shift the zero point of the energy to the ground state.

Exercise: Verify the solutions to the on-resonance equations of motion, given in Steck 5.48.

Class 22

HW12: When illuminated with either left-circular (σ) or right-circular (σ+) polarized light, on resonance, the following transitions in Na, and their associated levels, can represent an ideal two-level atom:

σ :  32S1/2  | F=2, mF = −2 ⟩  ⟶  32P3/2  | F'=3, mF' = −3 ⟩

σ+ :  32S1/2  | F=2, mF = +2 ⟩  ⟶  32P3/2  | F'=3, mF' = +3 ⟩

The magnitude of the dipole matrix element is the same for these two transitions, and has a value

| d | = 2.5 ea0

a) For beam intensities of I = 1, 10, 100, and 1000 mW/cm2, find the corresponding Rabi frequencies, assuming the laser frequency is resonant with the above transitions.

b) The lifetime of the upper level is 16.25 ns. For beam intensities of 1, 10, and 100 W/cm2 (note the units!), approximately how many times does the atom cycle between the upper and lower levels?

c) For a stationary atom in one of the two lower levels, determine the duration and intensity of an optical pulse needed to invert the atomic state. Use any reasonable intensity.

Class 23

Exercise: Determine the precession frequency of the ⟨ S ⟩ vector from the time-dependence of its azimuthal component:

⟨ Sp ⟩ = ( ⟨ Sx ⟩2 + ⟨ Sy ⟩2 )1/2

Exercise: Fill in the steps in the derivation of the effective transition probability of the two-level atom driven at the resonant frequency by a weak field, i.e. complete the integral and take the limit, t ≫ 1/γ.

Class 24

Exercise: Determine the units of B21, and verify that A21 has units of s−1, from the relation between A21 and B21.


a) Estimate the order of magnitude of A21 for an atom with a transition in the optical regime, e.g. use the dipole matrix element magnitude given in HW 12, and the corresponding transition frequency.

b) How does A21, computed above, compare with the classical estimate of the decay rate of a dipole oscillator at the same frequency?

Class 25

Exercise: Plot the ratio of populations, n2 / n1, vs. the saturation parameter, for S between 0 and 10. What is the ratio at S = 1? What fraction of the atoms are in the excited state for S = 1?

HW14: The D1 transition of 87Rb has a wavelength of 795 nm, and the magnitude of the dipole matrix element is 3.0 ea0. In the two-level atom approximation, determine the resonant absorption cross-section (cm2), the spontaneous emission decay rate (s−1), and the saturation intensity (mW/cm2) of the D1 transition. What is the natural line width (MHz), i.e. the absorption line width in the limit of zero longitudinal velocity, no collisions, and very low intensity?

Class 26

Exercise: Show that the pseudo-spin vector has unit length.

Exercise: Fill in the steps in the derivation of the equations of motion for the pseudo-spin vector.

Exercise: Derive the optical Bloch equations, starting from the transformation equations defining the Bloch vector components, {u, v, w}, differentiating these expressions w.r.t. time, carrying out the substitution of the pseudo-spin component equations of motion, and simplifying the resulting expressions.

HW15: Demtröder, 2.1: Determine the intensity and the peak spectral energy density for the given laser and beam geometry. Assume the laser has a Lorentzian spectral profile, with a FWHM of 1 MHz.

HW16: Demtröder, 2.4.

Class 27

Exercise: The initial state of a two-level atom is

| ψ(0) ⟩ = sqrt(½) ( | 1 ⟩ − i | 2 ⟩ )

Find the Bloch vector components, u(0), v(0), w(0). Does the Bloch vector change if the atom is undisturbed?

HW Extra Credit: The open two-level system.

Consider a three level system, with a resonant laser exciting the 1 → 3 transition. The upper level, 3, decays by spontaneous emission to both lower levels, 1 and 2. There is no decay from 2 to 1. Level 2 is sufficiently separated from both levels 1 and 3 that detuned driving of the 2 → 3 transition, from the laser tuned to 1 → 3, is negligible.

The total decay rate of upper level 3 is γ = 1/τ, where τ is the lifetime of the upper level. The branching ratio for spontaneous emission to level 1 is ε1, i.e.

γ31 = ε1γ

With the numbers given below, this problem is a model for optical hyperfine pumping in 87Rb.

a) Write down the rate equations for the open two-level system, in terms of the normalized populations, n1, n2, n3.

b) Assume a resonant cross section, σ13 = 8.1×10−10 cm2, an intensity of 1.4 mW/cm2, wavelength for the 1 → 3 transition of 780 nm, an upper level lifetime of 26.2 ns, and a branching ratio, ε1 = ½. With initial normalized poulations,

{ n1, n2, n3 } (t = 0) = { 1, 0, 0 }

solve the rate equations from a), numerically, and make a graph showing n1(t), n2(t), n3(t) vs t.

c) What are the populations at t = 1 microsecond? To what values do the populations go asymptotically in time, assuming infinite interaction time with the laser.

Class 28

Exercise: Either numerically, or analytically, find the solutions for the Bloch equations, on resonance (Δ = 0), and with the initial conditions,

{u(0), v(0), w(0)} = {0, 0 -1}

for a Rabi rate, Ω = 10γ. Plot the solutions v(t) and w(t) for the time interval t = (0, 20 / γ). Describe the motion of the Bloch vector.

Exercise: Derive the steady-state inversion for arbitrary detuning, w(Ω, Δ), and use it to determine the homogeneous, power-broadened absorption profile, from the expression,

α(S, Δ) = −nw(S, Δ) σ(Δ)

where S is the saturation parameter (I / Isat), Δ is the detuning from resonance, n is the number density of atoms, w is the steady-state inversion, and σ(Δ) is the cross-section profile. Plot the absorption profiles for a range of values for the saturation parameter, S; for example, S = 0, 1, 5, and 10.

Class 29

Final Exam