Modern Spectroscopy: a graduate course.
Chapter 1. Light-Matter Interaction
1.1 Semiclassical description of the light-matter interaction.
Generally speaking, in spectroscopy we need to describe the light and matter as
one complete system. Fully quantum description therefore would start with writing down
a Hamiltonian for this system, which we can partition into the Hamiltonian for the
material system, Hamiltonian for the light field, and the light-matter interaction
Hamiltonian:
(1.1)
Hˆ  Hˆ M  Hˆ L  Hˆ int
Solving the full quantum problem would mean solving the coupled equations of
motion of quantum electrodynamics (QED) for the light field and quantum dynamics
(e.g., time-dependent Schrodinger equation) for the material system. This is generally a
difficult task even for simple systems.
In this course, we will adopt instead the semiclassical description of the lightmatter interaction, which obeys much simpler equations. That is, the light will be treated
as classical electromagnetic (EM) field described by Maxwell’s equations (instead of the
QED equations), while the matter will be described by quantum mechanics. Moreover,
this will allow another simplification: instead of having to simultaneously solve the
coupled equations of motion for the light and matter variables (e.g., Maxwell equations
coupled with the time-dependent Schrodinger equation or with quantum Liouville-von
Neumann equation), we can break up the light-matter interactions into “one-way” steps:a
Step 1. The light perturbs the quantum dynamics of the system such that the lightmatter interaction serves as perturbation to the material Hamiltonian
(1.2)
Hˆ  Hˆ M  Hˆ int
At this point, only the material EOMs need to be solved, we do not care about the
changes in the state of the light resulting from the interaction. The light affects the
material system, but the material system does not affect the light.
a
This is because in the semiclassical description, we replace the quantum operators representing the light
Hˆ L and Ĥ int of the total Hamiltonian
ˆ
ˆ
with their expectation values, E (r , t )  E (r , t )  E (r , t ) ,
field observables (e.g., its E-field of vector potential – vide infra) in
Hˆ  Hˆ M  Hˆ L  Hˆ int
ˆ
ˆ
A(r , t )  A(r , t )  A(r , t ) , etc. E (r , t ) and A(r , t ) are numbers, not operators, and therefore
commute with other operators in
material system.
Ĥ . This allows to separate out the field variables from the EOM for the
1
We go ahead and solve the quantum dynamics of the material system under
Hˆ  Hˆ M  Hˆ int (note that since light is an oscillatory EM field, Ĥ int is time-dependent).
This will allow us to calculate (time-dependent) expectation values of physical
observables characterizing motion of charged particles in the material, e.g. polarization.
Step 2. Using the quantum dynamics of the system calculated in Step 1, we
calculate time-dependent charge densities and currents in the material, plug them into the
Maxwell equations as source terms, and calculate the EM field radiated by the moving
charges in the material. This is our spectroscopic signal.
Scheme 1.
A. Full quantum description of the
light and matter.
B. Semiclassical two-step description
of the light-matter system.
As any approximation scheme, the semi-classical description of the light-matter
interaction has advantages and disadvantages/limitations.
Pros:
1. The semiclassical equations of motion are much easier to solve than the fully
quantum treatment.
2. Classical description of the EM field is naturally connected to the conventional
experimental methods for characterization/detection of the light signals: measurements of
light intensity, frequency, polarization state, pulse width, amplitude/phase, etc.
3. The semiclassical descrtiption allows a physically intuitive picture of how the
physical characteristics of the incoming light field affect the quantum dynamics of the
2
system. This will in turn lead to an understanding of how to design a spectroscopic
measurement to probe a particular aspect of the dynamics (e.g., how to use time delays
between short laser pulses to measure relaxation phenomena, how to use polarization to
measure molecule’s orientation and rotational motion, etc.)
4. Multipolar expansion of the classical EM field – matter interaction leads to a
convenient classification of the interactions into electric dipole, magnetic dipole, electric
quadrupole, etc., in order of diminishing interaction strength, and convenient (and
physically transparent) selection rules based on symmetry.
5. As we shall see, the semiclassical field-dipole interaction allows perturbative
expansion of the quantum dynamics of the material system that naturally yields a
hierarchy of linear and nonlinear optical processes, and shows what type of dynamics
contributes to a particular order nonlinear signal.
Cons:
1. Because we break the system-light interactions into two steps, the concept of
the energy conservation does not naturally arise form the semiclassical equations.
Energy conservation is something that needs to be ‘manually enforced’ in this treatment.
For example, in Step 1, the energy of the material system will not be conserved since it
may gain/lose energy from/to the light field, but we do not treat the state of the light in
this step, so we do not account where the energy comes from. It is possible to construct
the theory to be consistent with the energy conservation, as it should, of course (e.g., we
would get the ‘missing’ energy back into the light field in Step 2), but it needs to be done
“by hand”. (In a fully quantum picture, we would say that the light-matter interaction
leads to annihilation/creation of a photon of the light field, and corresponding
raising/lowering of the quantum state of the material system, so the energy conservation
would naturally arise from the coupled light-matter equations of motion).
2. Classical description of the light field is valid only for high photon population
numbers, i.e. for light of moderate intensity. For extremely low light intensities such as
for single-photon spectroscopy, quantum effects such as photon statistics, quantum noise,
and photon entanglement become important and require fully quantum treatment. These
effects are important in single-molecule spectroscopy and quantum optics, both emerging
directions in molecular spectroscopy.
3. Separating the light variables from the system EOMs leaves out some
retardation effects (that is, different parts of the material system interact with the light at
different moments in time because of the finite speed of light). Such effects arise in
extended chromophore systems such as molecular aggregates and assemblies, and
crystals. There are ways to included these effects in a more rigorous version of the
semiclassical treatment, but they will not be considered in this course (See S. Mukamel,
Ch. 16, 17)
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1.2 Classical EM field.
We begin with a brief review of the classical electrodynamics (it is recommended
that you read a proper EM text for a complete coverage of this material – e.g., Classical
Electrodynamics by J. D. Jackson). The classical EM field is completely described by
two vector fields, the electric field E (r , t ) and the magnetic field B(r , t ) , which obey the
Maxwell’s equations (here and below, we use the SI units)

0
(1.3.a)
E 
(1.3.b)
 B  0
(1.3.c)
 E  
(1.3.d)
  B  0 J   0 0
Gauss’s Law
Absence of magnetic monopoles
B
t
Faraday’s Law
E
t
Ampere’s Law with Maxwell’s correction
where  (r ) is the total charge density (free and bound), J (r ) is the total current density
(free and bound). ε0 and μ0 are the electric permittivity and the magnetic permeability of
1
vacuum. Note that the speed of light is c 
.
 0 0
Since E (r , t ) and B(r , t ) are not independent (connected by the Maxwell’s
equations), specifying all 6 vector components of the electric and magnetic field
overdetermines the state of the EM field. To (partially) alleviate this, we introduce the
vector potential A(r , t ) and the scalar potential (r , t ) (a total of 4 parameters: Ax, Ay,
Az, and Φ) such that
A(r , t )
 (r , t )
t
B(r , t )    A(r , t )
E (r , t )  
(1.4)
where c is the speed of light. (Note that writing the magnetic field B(r , t ) as a curl of a
vector field A(r , t ) automatically satisfies the second Maxwell’s equation   B  0 ,


since    A  0 for any A(r , t ) ).
However, specifying the 4 components of the potential is still overdetermined,
and thus the vector and scalar potentials are not uniquely defined. Indeed, if we add
gradient of an arbitrary function (r , t ) to A(r , t ) and at the same time subtract its time
derivative form (r , t ) ,
4
 A '  A  


 ,
 '   
t

(1.5)
the electric and magnetic fields as defined by (1.4) will remain the same. Indeed,
E' 
A '
A


A
  '    
   
     E
t
t
t
t
t
B '   A '(r , t )   A(r , t )     A(r , t )  B
(since   0 for any  ).
The choice of (r , t ) allows on to simplify the EM equations for a particular
situation. This is known as gauge fixing. For example, we can choose (r , t ) such that
 A  0
(1.6)
This is known as the Coulomb gauge (also known as the radiation or transverse gauge).
As we demonstrate below, in vacuum (in the absence of charges and currents), this choice
of gauge leads to only transverse electric and magnetic fields of the plane EM waves.
Also note that Coulomb gauge implies that in the absence of charges, (r , t )  0 (from
the Gauss’s law, Eq. (1.3.a)).
For EM field in vacuum, assuming Coulomb gauge, Eqs. (1.4) become
A
t
B   A
E
(1.6)
The first three of Maxwell’s equations (1.3 a-c) are automatically satisfied, and
substituting (1.6) into the last one (1.3d):
 B 
1 E
c 2 t
yields the equation for A(r , t ) :
   A  
1 2 A
c 2 t 2


which, after transforming  A     A  2 A  2 A because of the Coulomb
gauge choice, becomes the familiar wave equation
(1.7)
2 A 
1 2 A
0
c 2 t 2
The solution is of course the transverse plane wave
(1.8)
A(r , t )   A0ei ( k r t )  c.c.
5
where c.c. indicate complex conjugate (remember, the fields and potentials are physical
observables and thus must have real values!).

2
The wavevector k defines the propagation direction and the wavelength,
. The frequency ω is connected to the wavevector through the dispersion relation
k
  c k . The (scalar) amplitude of the vector potential is A0, and its direction
(polarization state) is specified by a unit vector  .
The electric and magnetic fields of the plane EM wave calculated according to
(1.6) are
(1.9)
E (r , t )   i A0ei ( k r t )  c.c.   E0ei ( k r t )  c.c.
B(r , t )  (k   )iA0ei ( k r t )  c.c.  bB0ei ( k r t )  c.c
Here, the scalar amplitude and polarization direction of the electric field are E0 and  ,
and for the magnetic field B0 and b .
Note that, due to the Coulomb gauge choice,  A  i(  k ) A0ei ( k r t )  c.c.  0 ,
therefore   k  0 . That is, the polarization  of the vector potential and of the electric
field is perpendicular to the wave propagation direction k (transverse wave).
Furthermore, the magnetic field polarization is b  k   , i.e. the magnetic field is
perpendicular to both the propagation direction and the electric field (Fig. 2). Lastly, the
E
E

amplitudes of the electric and magnetic fields are related by 0 
 c , i.e. B0  0 ,
c
B0 k
which gives you an idea why magnetic interactions are much weaker than the electric
interactions.
Figure 2.
1.3 Charged particle interacting with EM field.
A particle with charge e moving with velocity v in an EM field is subject to
Lorentz force (again, written in SI units)
(1.10)
F  e( E  v  B)
6
After a somewhat lengthy derivation (given, e.g., in W.S. Struve), it can be shown
that the effect of the Lorentz force can be fully described by replacing the particle’s
momentum in the equations of motion with
(1.11)
p  p '  p  eA
This is true for both classical equations of motion and for quantum mechanics where the
momentum is an operator p̂  i  . In the EM field, the momentum operator is
replaced by pˆ '  i   eA .
Thus, if the Hamiltonian for a particle of mass m moving in some potential V (r )
is
(1.12)
pˆ 2
ˆ
H0 
 V (r ) ,
2m
applying EM field with vector potential A(r , t ) will change the Hamiltonian to
(1.13)
Hˆ field on
pˆ  eA


2
 V (r )  Hˆ 0  Hˆ int .
2m
(Once again, we use Coulomb gauge, so (r , t )  0 for the light field and thus we do not
have the e term due to the EM field. There may be other electric charges affecting the
motion of our particle, their potential is included in V (r ) . We are only interested in the
effect of the EM field on the motion of the particle).
Thus, the particle-field interaction Hamiltonian can be written as




2
2
1  ˆ
e ˆ
e2
p  eA  pˆ 2   
p  A  A  pˆ 
A
(1.14) Hˆ int  Hˆ field on  Hˆ 0 


2m 

2m
2m
In this course, we will consider weak field-matter interactions that can be treated
perturbatively. In this case, the last term in (1.14), which is quadratic in the field, is
much smaller than the first two terms that are linear in the field. We therefore neglect the
last term.b
Substituting p̂  i  into (1.14),

i e
Hˆ int 
  A  A 
2m

Note that, in general, p̂ does not commute with A . However, working in the Coulomb
gauge, we can easily prove (Homework problem) that   A  A  , and thus
b
This can be done for most of the realistic cases encountered in spectroscopy, with the exception of strong
laser field-matter interactions when the interaction energy becomes comparable to the internal energy of
atoms/molecules, i.e. when the electric field of the light wave is comparable to the internal electric field.
Such light intensities can be achieved with femtosecond lasers. For example, a 100 femtosecond laser
pulse of 100 μJ focused to a 50 μm spot size produces electric field of 1.2 V/Å, i.e. comparable to the intraatomic fields.
7
(1.15)
i e
e
Hˆ int 
A    A  pˆ
m
m
It is straightforward to generalize result for a collection of N charged particles:
(1.16)
N
e
Hˆ int    i A  pˆ i
mi
i 1
This is the field-matter interaction Hamiltonian in the semiclassical approximation.
1.4 The electric dipole approximation
In solving our time-dependent quantum dynamics for the system under the
perturbed Hamiltonian Hˆ 0  Hˆ int , we will need to evaluate the matrix elements of the
perturbation operator Ĥ , f Hˆ i in some chosen basis set (usually, in the stationary
int
int
states of the unperturbed Hamiltonian Hˆ 0 n  En n ). In order to do this, we need to
consider the spatial dependence of Ĥ int .
For a plane wave, A(r , t ) is given by Eq. (1.8), which we can modify as
1
1


(1.17) A(r , t )   A0ei ( k r t )  c.c.   A0eit 1  ik  r  (ik  r )2  (ik  r )3  ...   c.c.
2!
3!


So that
i e
f Hˆ int i 
f A  i
m
(1.18)
i e
1



A0eit    f  i  f (ik  r ) i 
f (ik  r ) 2  i  ... 
m
2!


Now recall that k 
2

, where the wavelength of light λ is of the order of 1000-4000 Å
for UV, 4000-8000 Å for the visible, and >104 Å for IR light. On the other hand, the
integration f ... i is over the region of space where the molecule’s wavefunction is
non-zero, which is about the size of the molecule, 1-10 Å. Thus, (k  r ) is a small
number, of the order of 10-2 or less (unless it’s a really deep-UV light and a really large
molecule!). Thus, the series in powers of (k  r ) in Eq. (1.18) is rapidly converging.
Leaving only the first (0th-order) term in (1.18) is referred to as the electric dipole
approximation (for reasons that will become apparent soon). This yields
(1.19)
i e
e
dipole
f Hˆ int
i 
A0eit   f  i   A0eit   f pˆ i
m
m
8
m
The matrix element f pˆ i can be evaluated easily by using pˆ  i  Hˆ 0 , r 
(Homework 1, Problem 1). The result is
 fi
dipole
f Hˆ int
i 
f E i
(1.20)

which is the familiar expression for the field-dipole interaction, where   er is the
dipole moment operator, the E-field of the light is E (r , t )   E0eit  c.c , E0  i A0 ,
E  Ef
and  fi  f
is the transition frequency between the initial state i and the final
state f .
For a collection of particles, we obtain the same expression by simply summing
over the particles, with    ei ri now being the collective dipole moment of the system.
i
Although we will not consider the higher multipole terms in this course, for your
reference, the second term in the expansion (1.18) yields the magnetic dipole and electric
quadrupole interactions (see derivation in W. S. Struve).
Physically, the electric dipole approximation means that we neglect the spatial
variation of the EM field’s amplitude over the region of space occupied by the molecule.
In other words, we assume that different parts of the molecule experience the same
electric field of the light wave. The dipole approximation is therefore also known as the
local approximation, meaning that the light interaction with a molecule at position r is
determined only by the E-field at that point in space. This approximation may thus break
down for extended systems, e.g. metals, semiconductors, or molecular assemblies, where
charge carriers and/or elementary excitations can travel over much larger distances than
the typical molecular size. In such systems, nonlocal interactions such as electric
quadrupole may become important.
1.5 The transition dipole and selection rules
We can also re-write Eq. (1.20) as
(1.21)
 fi
dipole
f Hˆ int
i 
E f  i

As we shall see in Chapter 3, for a system initially existing in a stationary state i
and exposed to the perturbation Hˆ dipole , this matrix element determines the probability of
int
the transition to the final state f via the Fermi Golden rule expression,
dipole
Pf i  f Hˆ int
i
2
We therefore see that the transition probability (for a given light field strength) is
governed by the matrix element
9
 fi  f  i
(1.22)
known as the transition dipole moment.
Using symmetry considerations, it is sometimes possible to determine that the
transition dipole between two states is zero, that is the integral in (1.22) vanishes, without
even performing the integration. For example, for a one-dimensional case, μ=ex is an
odd function of position. If i and f wavefunctions are both even or both odd, the
integrand in f  i is odd and the integral vanishes. In these cases, we say that the
f  i transition is forbidden in the electric dipole approximation. If the transition dipole
does not strictly vanish by symmetry, the transition is dipole-allowed.
Here are some examples of the electric dipole selection rules.
1. Particle in a box
 0,
V ( x)  
,
0 xa
x  0, x  a
Energy levels and eigenstates:
En 
2
2
2ma
2
n2 ,
2
 n x 
sin 
 , n  1, 2,...
a
 a 
n 
f  i  0 for even←even and odd←odd transitions. (Homework Problem)
2. Harmonic oscillator V ( x) 
1
m 2 x 2
2
f  i  0 only for f=i±1, i.e. the selection rule is Δv=±1. (Homework Problem)
3. Anharmonic oscillator. Introducing anharmonicity, e.g. V '( x)   x3   x 4
relaxes the harmonic oscillator selection rules and allows overtone transitions Δv=±2,
etc. (depending on the anharmonicity – Homework Problem).
4. Hydrogen atom V (r )  
1 e
.
4 0 r
The stationary states are spherical harmonics times the radial functions (associated
Laguerre polynomials of r). Stationary states are defined by 3 quantum numbers: n, l, m.
For light E-field polarized along z-axis,
n f , l f , m f  ni , li , mi 
2


0
0
0
2
*
*
 d  sin  d  r drRn f l f Yl f m f er cos Rnili Ylimi
10
By representing cosθ in terms of spherical harmonics, the angular part of the integral
yields the following selection rules:
l  1
.
m  0
f  i  0 only if
5. Gerade-ungerade (g-u) symmetry. For atoms and molecules that have inversion
symmetry, their stationary states are classified as either “g” (symmetric) or “u”
(antisymmetric) with respect to inversion operation. Because   er has “u” symmetry,
the transition dipole vanishes for g←g and u←u transitions. Only u←g and g←u
transitions are allowed.
1.6 Oscillator strength
We can get a sense of the absolute magnitudes of the transition dipoles for
different systems (e.g., atoms and molecules, electronic, vibrational, or rotational
transitions) by comparing them to the transition dipole of one agreed upon reference
system. By convention, such reference system is the 1←0 transition of an electron in a
3D harmonic potential. The transition dipole is
(1.23)
3he2
1  3
8 me
(the oscillator force constant is chosen to give transition frequency ν – this way we have a
consistent reference for transitions at different frequencies). This transition is said to
have oscillator strength of 1.
Any transition dipole can be compared to μ1 (calculated at the same frequency),
and the ratio is called the oscillator strength for that transition:
(1.24)
f f i 
 f i
1
It can be shown that any transition involving one electron has oscillator strength
1. In fact, there is a strict sum rule for single-electron transitions (i.e. transition where
only one electron in the system changes its quantum state): the sum of oscillator
strengths for transitions originating in one particular state i and ending in all possible
final states f is 1,
(1.25)
f
f i
1
f
Strong electronic transitions (e.g., chromophore dye molecules) have oscillator strengths
close to 1. Forbidden electronic transitions such as singlet-triplet (e.g., phosphorescence)
have oscillator strength 10-2 or lower.
The transition dipole and the oscillator strength are of course connected to the
phenomenological Einstein coefficients for the transition:
11
4 2 e 2
B12 
f12
me hc
(1.26)
B21 
4 2 e 2 g1
f12
me hc g 2
A21 
8 2 e 2 2 g1
f12
me c 3 g 2
(where g1 and g2 are the degeneracies of the two states).
12