3.9 Spectrum and Autocorrelation Function
103
of constant ˜
E 0 (Δω i,G ) is not quite valid, at the end of the pulse the S(ω) function
will be similar to the standard spectrum, but somewhat altered. Equations (3.85) and
(3.86) are then relationships between the absorption spectrum of state |G and the
autocorrelation function of the excited state created by a light pulse. This result is in
principle more accurate, the shorter is the pulse.
To illustrate how the autocorrelation function can be connected with measurable
dynamic properties of the system, we discuss a very simple model. Suppose that
only two states close in energy, |B and |D, are within the frequency range covered
by the radiation pulse. |B and |D are not eigenstates of the Hamiltonian: they
are coupled by a small interaction V = |V |e
iγ and their energy expectation values
are ε B and ε D . |B is a “bright state,” i.e., it is coupled to the ground state by a
nonvanishing transition dipole µ B,G . |D is instead a “dark state,” i.e., µ D,G = 0.
The exact eigenstates (see Appendix D) are
|ψ 1 = cos θ |B + sin θ e
−iγ
|D
|ψ 2 = − sin θ |B + cos θ e
−iγ
|D
(3.93)
where
tg θ =
(ε D − ε B ) −
(ε D − ε B ) 2 + 4 |V |
2
2|V |
.
(3.94)
The corresponding eigenenergies are
E ± =
1
2
ε D + ε B ±
(ε D − ε B ) 2 + 4 |V |
2
.
(3.95)
where the − for ψ 1 and the + sign for ψ 2 . The relevant transition dipoles are then
μ 1,G = ψ 1 |µ| G = cos θ µ B,G
μ 2,G = ψ 2 |µ| G = − sin θ µ B,G
(3.96)
and the excited wavefunction is
|Ψ exc (0) = cos θ |ψ 1 − sin θ |ψ 2 = |B .
(3.97)
Here we have dropped the inessential constant factors of Eq. (3.91), so that |Ψ exc (0)
is normalized. The excitation by a very short pulse only populates the bright state
and not the dark one. We see that we are just in the conditions of the two-state
Rabi problem. The population of state |B, which is the squared module of the
autocorrelation function, will oscillate as
ization, i.e. all the µ i,G vectors are parallel, because then we can replace
˜
E 0 (0) · µ i,G
2
with
˜
E 0 (0)
2
µ i,G
2 cos 2 α, where α is the angle between the light polarization and the µ i,G vectors.
In this case, the molecular orientation only affects the common factor cos 2 α and not the relative
weights of the spectral lines.
103
of constant ˜
E 0 (Δω i,G ) is not quite valid, at the end of the pulse the S(ω) function
will be similar to the standard spectrum, but somewhat altered. Equations (3.85) and
(3.86) are then relationships between the absorption spectrum of state |G and the
autocorrelation function of the excited state created by a light pulse. This result is in
principle more accurate, the shorter is the pulse.
To illustrate how the autocorrelation function can be connected with measurable
dynamic properties of the system, we discuss a very simple model. Suppose that
only two states close in energy, |B and |D, are within the frequency range covered
by the radiation pulse. |B and |D are not eigenstates of the Hamiltonian: they
are coupled by a small interaction V = |V |e
iγ and their energy expectation values
are ε B and ε D . |B is a “bright state,” i.e., it is coupled to the ground state by a
nonvanishing transition dipole µ B,G . |D is instead a “dark state,” i.e., µ D,G = 0.
The exact eigenstates (see Appendix D) are
|ψ 1 = cos θ |B + sin θ e
−iγ
|D
|ψ 2 = − sin θ |B + cos θ e
−iγ
|D
(3.93)
where
tg θ =
(ε D − ε B ) −
(ε D − ε B ) 2 + 4 |V |
2
2|V |
.
(3.94)
The corresponding eigenenergies are
E ± =
1
2
ε D + ε B ±
(ε D − ε B ) 2 + 4 |V |
2
.
(3.95)
where the − for ψ 1 and the + sign for ψ 2 . The relevant transition dipoles are then
μ 1,G = ψ 1 |µ| G = cos θ µ B,G
μ 2,G = ψ 2 |µ| G = − sin θ µ B,G
(3.96)
and the excited wavefunction is
|Ψ exc (0) = cos θ |ψ 1 − sin θ |ψ 2 = |B .
(3.97)
Here we have dropped the inessential constant factors of Eq. (3.91), so that |Ψ exc (0)
is normalized. The excitation by a very short pulse only populates the bright state
and not the dark one. We see that we are just in the conditions of the two-state
Rabi problem. The population of state |B, which is the squared module of the
autocorrelation function, will oscillate as
ization, i.e. all the µ i,G vectors are parallel, because then we can replace
˜
E 0 (0) · µ i,G
2
with
˜
E 0 (0)
2
µ i,G
2 cos 2 α, where α is the angle between the light polarization and the µ i,G vectors.
In this case, the molecular orientation only affects the common factor cos 2 α and not the relative
weights of the spectral lines.