16. Theory of Ultrafast Spectroscopy or Feynman Diagrams Made Simple

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16. Theory of Ultrafast Spectroscopy or
Feynman Diagrams Made Simple
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Nonlinear-Spectroscopic ExperimentsLimiting
Cases
Medium to be studied

Frequency Domain
or
cw monochro- matic beams
Time Domain
delta-function pulses
or
where
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Ultrashort laser pulses are an intermediate case.
time
Ultrashort laser pulses are really short, so they
appear to be time-domain experiments waiting to
happen. But, unlike true d-function pulses,
they have finite bandwidth. So they can be
resonant or nonresonant. This will be the key.
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Ultrafast-Spectroscopy Experiments An
Intermediate Case
Ultrashort pulses have large, but finite,
bandwidth. So experiments using them can be
resonant or nonresonant.
In addition, ultrashort-pulse experiments can
be nearly resonant. This involves much more
complex formulas. We wont treat this case.
Also, ultrashort-pulse experiments can be
nonresonant for some input pulses and resonant
for others. We can treat this case.
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Feynman Diagrams Made Simple
Quick quantum-mechanical derivation
Nonlinear-optical Feynman diagrams in the
frequency domaincw experiments
Example and tricks
Nonlinear-optical Feynman diagrams in the time
domaindelta-function-pulse experiments
Example and tricks
Feynman diagrams for experiments with
simultaneous time- and frequency-domain
characterultrashort-pulse experiments
The ultrashort-pulse domain Examples and tricks
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A Sneak Preview of Feynman Diagrams
Each diagram corresponds to a term in a complex
sum. We use such diagrams because theyre easier
to remember than the actual equation.
A Feynman diagram can be interpreted in the time
or frequency domains.
Time domain
Frequency domain
d
d
g
g
w3
a, b, g, and d represent states.
t3
b
t2
b
w2
w1
t1
a
a
a
a
delta-function pulse inputs at times, t1,
t2, and t3
cw beam inputs of frequencies,
In both domains, the particular ordering of the
pulse times or beam frequencies is referred to as
a time-ordering.
Many time-orderings contribute to the total
response/susceptibility.
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Semiclassical Nonlinear-OpticalPerturbation
Theory
Treat the medium quantum-mechanically and the
light classically. Assume negligible transfer
of population due to the light. Assume that
collisions are very frequent, but very weak
they yield exponential decay of any
coherence Use the density matrix to describe
the system. Effects that are not included in
this approach saturation, population of other
states by spontaneous emission, photon statistics.
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The density matrix
If the state of a single two-level atom is

?aa(t) or ?bb(t) are the population densities of
states a and b.
The density matrix, rij(t), is defined as
When laser beams with different k-vectors excite
the atom, rij(t) tends to have a spatially
sinusoidal variation.
A grating is said to exist if ?aa(t) or ?bb(t) is
spatially sinusoidal, A coherence is said to
exist if rab(t) or rba(t) is spatially sinusoidal.
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The density matrix
For a many-atom system, the density matrix,
rij(t), is defined as
where the sums are over all atoms or molecules in
the system.
Simplifying
The diagonal elements (gratings) are always
positive, while the off-diagonal elements
(coherences) can be negative or even complex.
So cancellations can occur in coherences.
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Why do coherences decay?
A coherence is the sum over all the atoms in the
medium.
The collisions "dephase" the emission,
causing cancellation of the total emitted light,
typically exponentially.
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Grating and coherence decay T1 and T2
A grating or coherence decays as excited states
decay back to ground. A coherence can also
cancel out if each atom has different phase. The
time-scales for these decays to occur are
Grating raa(t) or rbb(t) T1
relaxation time Coherence rab(t) or rba(t)
T2 dephasing time
Collisions dephase so, except in dilute gases,
T2 ltlt T1.
The measurement of these times is the goal of
much of nonlinear spectroscopy!
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Nonlinear-Optical Perturbation Theory
The Liouville equation for the density matrix
is
(in the interaction
picture) which can be formally integrated
which can be solved iteratively
Note that
i.e., a time ordering.
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Perturbation Theory (continued)
Expand the commutators in the integrand
Consider, for example, n 2
Thus, contains 2n terms.
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Perturbation Theory (continued)
Now, V is the perturbation potential energy due
to the light and is of the form, , where
E is the total light electric field.
But V is in the interaction picture, so we have
where
Note that U(t) U(t) U(t-t)
So a typical term looks like


is also in the interaction
picture Dividing out these U(t)s yields
Notice that time propagates from to to t along
two different paths.
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Perturbation Theory (continued)
So a typical term (in second order) is
But, in nth order, the E-field is typically the
sum of n input light fields
As a result, each of the above type of terms
expands into many terms. Allowing each field to
occur only once yields n! as many more. Thus, in
nth order, there are 2nn! terms!
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How do we remember all these terms?
Use diagrams!
Consider two input beams and this second-order
term, noting that time propagates from t0 to t
along two paths
time
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Perturbation Theory (contd)
Now expand in terms of the atomic
eigenstates For our second-order term, for
example
we find
Computation of the number of terms now is an
exercise left to the student
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Doing the integrals
Set t0 0
Dipole moment matrix elements at the ith beam
polarization
Now, to go further, well consider limiting cases.
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The Frequency Domain cw beams
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Including dephasing
Before we evaluate these integrals, we must
include dephasing. Every time a transition
frequency occurs, we must subtract off the
dephasing rate for that transition.
This is the usual method for adding width to a
transition. Thus
This addition comes from a complex analysis that
takes into account collisions. Now we can do the
integrals in the various cases.
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The Frequency Domain
Evaluating a single second-order term for
monochromatic fields yields
where
and where
The (-1) occurs in terms with an odd number of
V(t)s to the right of
The factor of is the population density
of the initial state. The factors of
are dipole-moment matrix elements between the
states a and b for the polarization of beam
k. The denominators contain the line shape--the
dynamical information.
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Drawing Feynman Diagrams in the cw Limit

• Draw two vertical line segments.
• 2. Draw a rightward-pointing diagonal arrow for
  each input field. Upward-pointing arrows
  correspond to absorbed photons, and
  downward-pointing photons correspond to emitted
  photons. Choose an ordering for these
  interactions, and also choose which side each
  should appear on. Label each interaction with a
  light frequency.
• 3. Write in states (a, b, c, d, e) at the base
  and just above each interaction.

Every possible diagram of this form corresponds
to a term in the expression for c(n), where n is
the number of interactions.
This diagram corresponds to
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Drawing Feynman Diagrams in the cw Limit
Include a factor of 1 if there are an odd number
of interactions on the right
Diagram
Include a factor of the initial population
density of the state at the base of the diagram
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Drawing Feynman Diagrams in the cw Limit
Piece of diagram
At each interaction, we write down a
dipole-moment matrix element
(1) means for the polarization of beam 1
After each interaction (reading upward), we write
a resonant denominator of the form
Diagram
where
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Interpreting Feynman Diagrams in the cw Limit
The contribution to c(n) is the product of all
factors shown below
Resonant denominator
Resonant denominator
Resonant denominator
Resonant denominator
Matrix elements
The population density of the state at the base
Two interactions on the right (a factor of 1 for
each) (-1)2
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Example Linear OpticsThe Absorption Coefficient
and Refractive Index vs. Frequency
Linear optical problems involve only one photon
Resonance frequency
Dephasing time
Light frequency
This is just the well-known complex Lorentzian
line shape, whose even (imaginary) component is
the absorption coefficient and whose odd (real)
component is the refractive index.
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How do you know which diagrams to include?
First consider the process, and include only the
most resonant, and hence strongest, terms.
For example, consider difference-frequency generat
ion,
Maximally resonant denominators
Anti-resonant denominators
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Example Higher-Order Wave Mixing
12-wave mixing
Signal frequency
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A 12-Wave-Mixing Feynman Diagram
All denomi-nators are maximally resonant.
Unfortunately, there are about 10,000 more such
diagrams to consider
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Drawing Feynman Diagrams in the Time Domain
Now, suppose that the input light is a sum of
delta-function pulses.
The relevant variables are now the pulse
relative delays,
t1
t
t2
We can now write Feynman diagrams for this class
of processes, but we must label the interactions
with times, rather than frequencies.
Every possible diagram of this form corresponds
to a term in the expression for the response,
R(n), where n is the number of interactions.
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The Time Domain d-function pulses
The integrals are now even easier.
Note that the result is a product of propagators.
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Interpreting Time-Domain Feynman diagrams
As before, include a factor of 1 if there are an
odd number of interactions on the right
As before, include a factor of the initial
population density of the state at the base of
the diagram.
Also, as before, at each interaction, we write
down a dipole-moment matrix element
Instead of resonant denominators, we write
simple exponential propagators
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Example The Linear Response
As before, linear optical problems involve only
one photon and use the same diagram
where . Dropping the
subscript, 1
Resonance frequency
Dephasing time
This is just the well-known fact that the
molecules oscillate at their own frequency,
emitting free-induction decay, and dephase
exponentially. The Fourier transform of this
response is the complex Lorentzian line shape,
whose even component is the absorption
coefficient and whose odd component is the
refractive index.
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Example The Excite-Probe Experiment
Excitation pulse
Observe change in probe- pulse energy vs. delay,
Probe pulse
The observed signal vs. delay is complex, with
three components
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The Excite-Probe Experiment (contd)
Signal pulse
The excite-probe experiment is a third-order
process, with the excitation pulse providing two
photons
Probe pulse
Excite Pulse(s)
Only state a is populated initially.
Coherence spike
Photon echo (PFID)
Excited- state decay
Delay,
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The intermediate domain ultrashort pulses
Ultrashort pulses have finite bandwidth and
finite pulse length. Can we define Feynman
diagrams for nonlinear-optical experiments with
them?
Yes!
All the integrals are of the form
For ultrashort pulses, two important cases yield
simple results.
Case 1. Resonant excitation
Set E(t) time
domain Case 2. Nonresonant excitation
Set E(t) constant
frequency domain
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The Ultrashort-Pulse Domain
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The Ultrashort Pulse Domain (contd)
Purely resonant ultrashort-pulse experiments are
pure time-domain experiments, and we use the
time-domain Feynman diagrams.
Purely nonresonant ultrashort-pulse experiments
are pure frequency- domain experiments, and we
use the frequency-domain Feynman diagrams.
What about experiments that are resonant at some
steps and nonresonant at others?
Resonant steps, so label with times
Nonresonant steps, So label with frequencies
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But can we define rules for interpreting this
time-frequency hybrid diagram that make sense?
Yes!
Doing the integrals, we see that time-domain
steps yield the same factors, But
frequency-domain (nonresonant) steps yield
slightly different denominators (we must take
into account the existing coherence from the
previous time-domain step). Also any resonant
pulse must be simultaneous with all nonresonant
pulses prior to it since the last resonant step.
Here, for example, pulses 1 and 2 must be
coincident in time (same for 3 and 4).
Propagator
Resonant denominator
Propagator
Resonant denominator
Matrix elements
The population density of the state at the base
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Example Femtosecond CARS
In fsec transient CARS, a two-photon Raman
resonance is excited, and its decay is probed vs.
delay, all by pulses.
The decay of the coherence, wga, is to be
measured by varying a delay.
Propagator
Propagator
Resonant denominator
Signal
Plotting the signal intensity vs. yields the
dephasing time, T2ag
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Conclusions
Nonlinear-optical Feynman diagrams better allow
us to remember the large number of terms in the
complex perturbation-theory expansion.
We can define Feynman diagrams for several cases
general input fields cw input fields (frequency
domain) delta-function input fields (time
domain) ultrashort-pulse input fields time or
frequency-domain or both
An understanding of nonlinear-optical Feynman
diagrams can make almost any nonlinear-optical or
nonlinear- spectroscopic problem (and even linear
ones!) relatively easy!