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
- 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 c_{2} 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 H_{0} (Demtröder 2.92); plot of
probabilities vs time for the specific initial conditions,
c_{1}(0) = 1,
c_{2}(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 | c_{2}(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^{−} : 3^{2}S_{1/2}
| F=2, m_{F} = −2 ⟩ ⟶
3^{2}P_{3/2}
| F'=3, m_{F'} = −3 ⟩
σ^{+} : 3^{2}S_{1/2}
| F=2, m_{F} = +2 ⟩
⟶
3^{2}P_{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 ea_{0}
a) For beam intensities of I = 1, 10, 100, and 1000 mW/cm^{2},
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/cm^{2} (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 B_{21}, and verify
that A_{21} has units of s^{−1}, from the relation
between A_{21} and B_{21}.
HW13:
a) Estimate the order of magnitude of A_{21} 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 A_{21}, 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:
λ^{2} / 2π.
- Relation between absorption cross-section and the absorption coefficient:
derive on-resonant absorption coefficient, α_{0},
by including both
excitation and stimulated emission (Demtröder 2.99); high-intensity
limit of α_{0} (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,
n_{2} / n_{1}, 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 ^{87}Rb has a wavelength of 795 nm, and the
magnitude of the dipole matrix element is 3.0 ea_{0}.
In the two-level atom approximation, determine the resonant absorption
cross-section (cm^{2}), the spontaneous emission decay rate
(s^{−1}), and the saturation intensity (mW/cm^{2}) 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 c_{1} and
c_{2} (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
| c_{2} |^{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 ε_{1}, i.e.
γ_{31} = ε_{1}γ
With the numbers given below, this problem is a model for optical hyperfine
pumping in ^{87}Rb.
a) Write down the rate equations for the open two-level system, in terms of
the normalized populations, n_{1}, n_{2},
n_{3}.
b) Assume a resonant cross section, σ_{13} = 8.1×10^{−10} cm^{2},
an intensity of 1.4 mW/cm^{2}, wavelength for the 1 → 3
transition of 780 nm, an upper level lifetime of 26.2 ns, and a branching ratio,
ε_{1} = ½.
With initial normalized poulations,
{ n_{1}, n_{2}, n_{3} }
(t = 0) = { 1, 0, 0 }
solve the rate equations from a), numerically, and make a graph showing
n_{1}(t), n_{2}(t),
n_{3}(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, u^{2} + v^{2} +
w^{2}, 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 | c_{2}(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