Laser Spectroscopy Topics 1 through 10

Laser Spectroscopy (Classes 21 through 28)

Class 21

Correspondence between the equations of motion for an optically-driven two-level atom and the equations for a spin-½ particle in a rotating B field: identification of frequencies, and “co-rotating” and “counter-rotating” terms.
Rotating-wave approximation (RWA): justify dropping the counter-rotating terms from the two-level atom equations by estimating the order of magnitude contribution to the transition probability from these terms.
Equations of motion in the RWA: equivalent forms (Steck 5.23 -- see exercise, and Demtröder 2.85); transformation of c₂ to remove explicit time-dependence in the equations (Steck 5.25); on-resonance (Δ = 0) equations of motion (Steck 5.43).
Solutions for the on-resonance equations of motion (Steck 5.48, see exercise); time-dependent probabilities of finding the atom in the two eigenstates of H₀ (Demtröder 2.92); plot of probabilities vs time for the specific initial conditions, c₁(0) = 1, c₂(0) = 0 (Demtröder, figure 2.20); Rabi flopping and the Rabi frequency.
Optical pulse duration and the resulting state of the two-level atom: inversion of the atomic state using a "π pulse"; creating an equal-amplitude superposition state using a "π/2 pulse"; difference in the phase of the atomic state for a π/2 pulse and a 3π/2 pulse.
"Semi-quantum" picture of interaction of light with atoms: the single photon absorption probability and the definition of absorption cross section; ideal experiment to determine the absorption cross section by photon counting.

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

Overview of physical models of laser-atom interactions, and some physical phenomena described by these models:
- classical Lorentz model
- semi-classical model
- semi-quantum model
- full quantum model (quantized field modes and atom as quantum dynamical system)
Description of the resonance fluorescence spectrum of a two-level atom driven by a resonant, strong laser field.
Rate of excitation/absorption by a two-level atom in the semi-quantum model: probability of absorption after N photons interact with the atom in its ground state; example for rate of excitation for atom interacting with 1 photon per second; relation between the photon flux and the beam intensity; expression for the resonant excitation rate (Steck 3.33).
Rate of excitation in the semi-classical model: transition probability as a function of time; weak-field approximation (Demtröder 2.72) and the quadratic time-dependence of | c₂(t) |², and the linear time-dependence of the excitation rate.
Discrepancy between the semi-quantum model (constant excitation rate) and the weak-field semi-classical model (linear time-dependent rate) --- the absence of the damping rate in the semi-classical model.
Phenomenological inclusion of damping in the semi-classical model: uncertainty in the transition frequency (energy), and the association of the natural lineshape of radiative decay with the probability distribution of the transition frequency; the effective transition probability as a weighted average of the transition probability at all detunings; the normalized Lorentzian lineshape weighting function.

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⁻ : 3²S_1/2 | F=2, m_F = −2 ⟩ ⟶ 3²P_3/2 | F'=3, m_F' = −3 ⟩

σ⁺ : 3²S_1/2 | F=2, m_F = +2 ⟩ ⟶ 3²P_3/2 | F'=3, m_F' = +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

Review of HW 11: precession and nutation of the ⟨ S ⟩ vector at near and far detunings.
Solutions of the coupled equations of motion for the two-level atom at arbitrary detunings: transcribe solutions of spin-½ particle in rotating B field by identifying corresponding quantities in the two-level atom problem; the generalized Rabi frequency (Demtröder 2.90); transition probability for arbitrary detuning, P_1,2(t, Δ) (Demtröder 2.89, Steck 5.60).
Effective transition probability for the two level atom with uncertainty in the transition frequency: evaluate integral to obtain weighted average of transition probabilities over all detunings (ref. Gradshteyn and Ryzhik, 3.826); limit of integral for t ≫ 1/γ and the linear time-dependence of the effective transition probability; constant time excitation rate, R, (Allen and Eberly, 6.9) in the modified semi-classical model; the constant rate R as an example of Fermi's golden rule.
Quantum-mechanical expression for the absorption cross section, obtained by equating the semi-classical effective rate of excitation with the rate from the semi-quantum model.

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

⟨ S_p ⟩ = ( ⟨ S_x ⟩² + ⟨ S_y ⟩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

Spontaneous emission decay: Weisskopf-Wigner result for the decay rate of the excited state (Steck, 11.29).
Derivation of the spontaneous emission decay rate via the Einstein rate equations for atoms in a thermal radiation field; equilibrium formula for the spectral energy density, and its comparison to the blackbody formula; relations between the Einstein A and B coefficients; validity of the relations between A and B coefficients for non-thermal radiation fields; the spectral energy density on resonance, and its relation to the intensity of a monochromatic beam and the decay rate; solution of the decay rate by equating the Einstein excitation rate and the semi-classical excitation rate with damping; relation between | ⟨ 1 | d | 2 ⟩ |² and | ⟨ 1 | d·ε | 2 ⟩ |² for a spherically symmetric atom; spontaneous decay rate expression (Demtröder, Table 2.2).
Oscillator strength: definition as the ratio of the spontaneous emission decay rate to the classical dipole decay rate (formula in Demtröder, Table 2.2, and Merzbacher, 19.39); applicability of the oscillator strength to absorption of radiation; the Thomas-Reiche-Kuhn sum rule for oscillator strengths in a multi-electron atom (see Merzbacher, section 19.3).

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

HW13:

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

Comparison of classical energy loss rate from oscillating dipole with quantum mechanical energy loss rate (due to spontaneous emission): dipole moment amplitude vs dipole moment matrix element.
The two-level atom absorption cross section: λ² / 2π.
Relation between absorption cross-section and the absorption coefficient: derive on-resonant absorption coefficient, α₀, by including both excitation and stimulated emission (Demtröder 2.99); high-intensity limit of α₀ (medium becomes transparent); extension of the absorption cross-section to allow for homogeneous broadening (Steck, 3.18) ; inclusion of Doppler broadening in the absorption coefficient via velocity distributions of the population densities; low-intensity limit expression for the homogeneous and Doppler broadened absorption coefficient.
Solution of the Einstein rate equations in the steady-state limit (Demtröder 3.67); High intensity and low intensity limits; the steady-state population difference (Demtröder 3.69); the saturation parameter, S (Demtröder 3.71), and the saturation intensity, I_sat.

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

Review of absorption phenomena explained by the rate equation model, and limitations of this model.
The optical Bloch equations for the two-level atom:
- Definition of the pseudo-spin vector, and its components for the two-level atom (Allen and Eberly, 2.25).
- Length of the pseudo-spin vector (see exercise).
- Equations of motion for the pseudo-spin vector, derived from the equations of motion for c₁ and c₂ (Steck, 5.23).
- Motion of the pseudo-spin vector in the absence of a driving optical field (Ω = 0): precession of the vector at the resonance frequency; the direction of precession.
- The rotating-frame transformation: rotation of the coordinate system about the z-axis at the driving field frequency; the rotation matrix and equations of transformation from pseudo-spin components to the Bloch vector components, u, v, and w (Allen and Eberly, 2.35).
- Equations of motion for the Bloch vector: the optical Bloch equations without spontaneous emission decay (Steck, 5.108); see exercise.
- Demonstrate that the Bloch vector is stationary for Ω = 0; Length of the Bloch vector.
- Connection of the Bloch vector components (u, v) with the in-phase and in-quadrature amplitudes of the dipole moment for a driven dipole oscillator; Comparison of the classical equations of motion for u, and v with the Bloch equations.

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

Collimated atomic beam sources for nearly Doppler-free spectroscopy.
Expectation value of the dipole moment for the quantum state of a two-level atom (Steck, 5.114): connection to Bloch vector components (u, v); chain of physical quantities describing the optical properties of a medium: u, v → dipole moment → polarization of medium → susceptibility → index of refraction and absorption.
Addition of spontaneous emission decay to the optical Bloch equations: incorporating the exponential decay of | c₂ |² into the inversion, w(t); derivation of the equation of motion for w(t) in the absence of a driving field; the Bloch equations with decay terms (Steck, 5.127).

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

Consequences of decay terms in the Bloch equations: possibility of steady-state solutions with a constant intensity driving field; Bloch vector length, u² + v² + w², no longer always equal to 1; analytic solutions of the equations by H.C. Torrey (see journal reference).
Evolution of the Bloch vector, for any initial condition, in the absence of a driving field: decay from {u(0), v(0), w(0)} to {0, 0, -1}.
Relationship between the inversion and the probability of the atom being in the upper level.
Bloch equations on resonance: steady-state solution; time-dependent behavior of the inversion --- comparison to the semi-classical solution for | c₂(t) |² without decay; low intensity and high intensity limits for the steady-state inversion.
Steady-state solutions for arbitrary detuning: profile of steady-state inversion vs. detuning; width of the steady-state inversion profile (Allen and Eberly, 6.23; Demtröder, 3.78).
Relation between the absorption coefficient and the steady-state inversion; expression for the absorption profile for arbitrary intensity --- power-broadened line shape; homogeneous width of the power-broadened absorption line (Demtröder, 3.78); comparison of line shapes at different intensities (Demtröder, Fig. 3.24).

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) = −n w(S, Δ) σ(Δ)

where S is the saturation parameter (I / I_sat), 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