Light-matter interaction. From spectroscopy to coherent control
Tamar Seideman Chemistry Department Northwestern University Tel.
847-467-4979 seideman @ chem.northwestern.edu
http://www.chem.northwestern.edu/~seideman/ Slide 2 There is a
variety of (good) sources of light (see previous page) but lasers
have nice properties, including monochromaticity, perfect
coherence, high intensity and variable polarization. In what
follows we will make good use of all these features. Outline
Preliminaries Elecromagnetic Radiation Field matter interaction The
electric dipole Hamiltonian Energy-domain spectroscopy
(monochromaticity) Electric dipole selection rules Atomic spectra
Molecular spectra Time-domain spectroscopy (coherence) Forming
superposition states Electronic wavepackets Vibrational wavepackets
Rotational wavepackets (intensity, polarization) Optical control
Using the coherence property Using the intensity property Using the
polarization property Slide 3 $ NSF CHE/DMR $ NSF PHY $ DOE $ NATO
$ HGF-NRC $ Guggenheim Foundation $ Humboldt Foundation Thanks to:
Slide 4 A (very!) qualitative picture of light-matter interaction:
Imagine a spring that we drive by tapping on the weight at the
natural spring frequency (on resonance). Curve (a) represents the
springs oscillation, curve (b) is the applied force that is
required to make it oscillate (absorption) and curve (c) is the
applied force that is required to stop it from oscillating
(stimulated emission) Consider next the response of an atom or a
molecule. The resonant driving force polarizes (distorts) the
system by mixing the initial with an excited state wavefunction. In
this cartoon we imagine polarizing the 1s orbital of the H-atom by
applying a field at resonance with the 1s 2p transition. As time
progresses from t 1 to t 4, the 2p character is increased, then the
superposition returns to the initial composition. Subsequently it
will oscillate. Fig. 1 Fig. 2 Slide 5 I. Preliminaries : I-1. A
(Very) Brief Introduction to Classical Electrodynamics The use of
classical mechanics for description of the field and quantum
mechanics for description of the material system is a good
approximation in all cases we will consider, since the number of
photons in the field will be large throughout the intensity range
of interest: Maxwells equations provide a complete description of
the electric and magnetic fields associated with electromagnetic
waves, but to construct a Hamiltonian we need a potential.
Introducing a vector potential and a scalar potential we are left
with an over-determined problem and hence freedom to choose a
constraint a gauge within which the wave is uniquely described. The
Coulomb gauge ( =0) leads to a plane-wave description of the
electric and magnetic fields: Light sourceIntensity (Wcm -2 )photon
density (cm -3 ) Mercury lamp 110 8 Continuous laser~10 - 10 6 10 9
-10 14 Moderate pulsed laser~10 9 -10 11 10 17 -10 19 Slide 6
Since,, where is the polarization direction of the vector
potential. (1) (2) (3) (4) (5) { We choose the time origin such
that A 0 is purely imaginary, Slide 7 It follows that We define
electric and magnetic field amplitudes as, in terms of which, Fig.
3 (6) (7) Slide 8 with components, Although the Lorentz force is
not derived from a potential energy, a Lagrangian for the problem
can be found, and can be seen to yield Newtons equations of motion
corresponding to the force (8) by substitution in Lagranges
equations, Given a Lagrangian, we can calculate the conjugate
momenta of the Cartesian coordinates of the particle, e.g., with
which the classical Hamiltonian is, I-2. Hamiltonian for a Charged
Particle Interacting with a Radiation Field The Lorentz force on a
particle of charge q moving at velocity is, (8) (9) (11) (12) (13)
(10) Slide 9 Relating the velocity to the momentum through, we
re-express the Hamiltonian as, Here we consider a collection of
mutually interacting charges (e.g., a molecule). Under field- free
conditions, the Hamiltonian describing the system is, whereas in
the presence of the field it can be written as, (14) (15) (16) (17)
Slide 10 Expanding the square one finds, small(18) (19) With the
quadratic term in A assumed small as compared to the linear term,
the Hamiltonian is cast in the anticipated form, where, Slide 11
Given the classical Hamiltonian in the form (19), we are in
position to substitute quantum operators for the classical momenta,
whereby, Here, [i.e., ], but since we are working in the Coulomb
gauge and one finds, (20) (21) Slide 12 I-3. The Electric Dipole
Approximation Recall, where is the wavelength of the
electromagnetic field in Fig. 3. In the UV regime ~10-100 nm, in
the visible ~400-700 nm and in the IR l=1 m-1 mm, whereas the
molecular system is of sub nm size. We thus expand the exponent in
V(t) as, where is the center of mass of the molecule. In the limit,
we neglect all terms but the first. Setting we have, The
approximation we derived (the electric dipole approximation)
applies to the limit. It holds for UV and visible light but not for
X-ray radiation, for instance. Higher order terms need be retained
also in many other instances, e.g., when the electric dipole
interaction is forbidden (and hence the zero order term in the
expansion vanishes when matrix elements of the interaction are
taken) or when one is interested in propagation effects. Inserting
the plane-wave solution for the vector potential we find, (22)
Slide 13 Inserting, we express the electric dipole Hamiltonian as,
(the equivalence of this and the more familiar form can be shown by
a gauge transformation) But what does it mean ? Consider an
electron interacting by Coulomb forces with an atomic core and
subject to a radiation field. Within the electric dipole
approximation, Ehrenfests theorem gives, (23) (24) (25) Slide 14
Eliminating, we find, We see that (within the electric dipole
approximation) the center of the wavepacket associated with the
electron evolves like a particle of mass m and charge q that is
subject to the central force of the core and a spatially uniform
electric field. I-4. Transition Rates and the Golden Rule
Approximation: Consider a molecular system that evolves subject to
a Hamiltonian of the form, Denoting the solutions of the
time-dependent Schroedinger equation corresponding to the
stationary Hamiltonian H 0 by j (t), we have, where j are the
eigenstates of H 0, (26) (27) (28) (29) Slide 15 The system
wavefunction can now be expanded in the complete basis of
stationary eigenstates as, Substituting (t) into the Schroedinger
equation we have, and from which or (30) (31) (32) (33) (34) (35)
Slide 16 Using the orthonormality of the expansion eigenstates we
have, or Formally Eq. (37) can be solved to give, The probability
of transition from an initial state to a final state of the zero
order Hamiltonian is then, Note that the interaction appears (here
and more generally) in the form of matrix elements. In this form it
can be readily transformed as, (36) (37) (38) (39) (40) Slide 17
(41) (42) (43) It follows that the matrix elements of V(t) can be
written, A nonperturbative solution to the set of coupled
differential equations (37) can be obtained numerically, by
propagating the set of coupled equations for sufficiently long time
for the time- dependent interaction to have decayed. Such solution
will be needed for most applications to be discussed later in the
afternoon. In cases where the coupling is small with respect to H 0
(as, for instance, energy-domain spectroscopy), first order
perturbation theory is applicable. In this framework it is assumed
Slide 18 that the amplitude of the initially prepared state remains
constant throughout, Using Eq. (44) in Eq. (38) we have, Eq. (45)
is general but we will focus on the case where the perturbation is
the dipole interaction of the material system with a laser pulse,
where E 0 (t) is the pulse envelope. Here the amplitude of interest
is proportional to the (incomplete) Fourier transform of the pulse
envelope, Our interest is generally in the form of the amplitudes
after turn off of the pulse. In that case the amplitude is, up to a
constant, the (complete) Fourier transform of the pulse envelope.
Assuming a Gaussian profile in time and energy, (44) (45) (46) (47)
Slide 19 With Eq. (48), the amplitudes in Eq. (47) take the form,
The Rotating Wave Approximation amounts to neglect of the second
term in the brackets as compared to the first one. In the combined
limits where and E m 0 such that the pulse energy is constant
(relevant to high-resolution spectroscopy), the Gaussian reduces to
a -function of energy and, This is the Golden Rule limit. (48) (49)
(50) (51) pulse energy = E p { small Slide 20 II. Energy Domain
Spectroscopy II-1. Atomic Spectra Atomic spectra contain a good
deal of physics and have had an important role in the development
of quantum mechanics and the history of science. Electric dipole
selection rules follow from the requirement that the integrand in
will be symmetric under all symmetry operations of the point group
(here the full rotation group, R 3 ). Under inversion, r -r, an
atomic wavefunction with angular momentum l has parity (-1) l. The
parity of the integrand is therefore, Hence, the only allowed
electric-dipole transitions are those involving a change in parity.
The atomic orbitals are angular momentum eigenstates and span the
irreducible representations and. The electric dipole moment
operator behaves as a translation and is therefore described as a
superposition of spherical harmonic with angular momentum 1, (52)
Slide 21 Therefore, from which follows the selection rule, To
develop selection rules for m l, the magnetic quantum number, we
note that the photon has an intrinsic helicity the projection of
its angular momentum on the quantization axis. For circularly
polarized field, it is conventional to define the quantization axis
as the line of flight, with which absorption of left-circularly
polarized light results in m l =1 and emission m l =-1, vice versa
for right circularly polarized light. In the general case Fig. 4:
Photon absorption can result in either l=1 or l=-1.The basis for
this selection rule is conservation of angular momentum and the
fact that the photon has helicity. l l 1 l+1 l l 1 (53) (54) (55)
Slide 22 the selection rule is, Algebraically the rotational
selection rules are easy to derive. We express the atomic
wavefunction as a product of a radial function by an eigenstate of
the angular momentum operator, finding for the angular part of the
dipole operator matrix element, For the first 3-j symbol on the
right to be non-zero m f need be m i +q, where q takes values 0 and
1. For the second to be nonzero l f need be l i 1. Recall we
invoked the electric dipole approximation electric dipole forbidden
transitions may well be, e.g., magnetic dipole or electric
quadrupole allowed. (56) (57) (58) Slide 23 II-2. Molecular Spectra
II-2-A. Pure Rotational Spectra We assume that (owing to the
time-scale disparity) the wavefunction can be separated into an
electronic, a vibrational and a rotational part, and hence the
electric dipole matrix element between ro-vibronic states takes the
form of a matrix element of the permanent electric dipole in a
given vibronic state between rotational functions. Linear molecules
rotate about two angles and are eigenfunctions of two operators,
the material angular momentum squared,, and the space-fixed
z-projection of the operator J z, corresponding to two conserved
quantum numbers - J and M J. Fig. 5: The Euler angles relate the
space- and body-fixed coordinate systems, where is the polar angle
between the space- and body-fixed z- axes, and and are the
azimuthal angles of rotation about the space- and body-fixed
z-axes, respectively. Slide 24 where nv is the permanent dipole
moment operator in state n,v. It follows that: Only polar molecules
can have a pure rotational spectrum If the dipole operator is
expressed in space-fixed Cartesian coordinates, = x, y, z, Eq. (59)
is given as where the Cartesian components of the permanent dipole
moment are readily expressed in terms of the spherical harmonics
as, (59) (60) (61) (62) Slide 25 The selection rules are therefore
determined by the same integral we discussed in the context of
atoms, which has a component that spans the totally symmetric
irreducible representation only if i.e., if J=J, J1 (excluding J=J=
0) and M J +q-M J = 0. Observable transitions therefore obey,
Symmetric top rotational functions are eigenstates of the total
angular momentum squared operator and its space- and body-z
projection operators with eigenvalues J, M and K, E J =BJ(J+1) (63)
(64) (65) (66) Slide 26 Since, however, the permanent dipole
operator must be independent of it is described by a combination
of, which are proportional to the spherical harmonics. It follows
that The corresponding eigenvalues are where the rotational
constants are given in energy unites, A = 2 /2I aa etc, centrifugal
distortion is neglected, and the I kk, k=a,b,c, are principal
moments of inertia, evaluated at the equilibrium configuration. The
pure rotational spectra of both linear and symmetric rotors
therefore consists of series of lines separated by 2B (in the
absence of centrifugal distortion). E JK =CJ(J+1)+(A-C)K 2 (prolate
symmetric top) =AJ(J+1)+(C-A)K 2 (oblate symmetric top) (67) (68)
(69) Slide 27 The rotational eigenstates of asymmetric tops are not
analytically expressible in general, and are typically written as
superpositions of symmetric top functions with coefficients
determined by diagonalization of the field-free rotational
Hamiltonian. The electric dipole matrix element is thus a multiple
sum of integrals of the form, [c.f. Eqs. (58) and (64)]. 2E J
=(A+C)J(J+1)+(A-C)E J ( ) (70) Slide 28 II-2-B. Vibrational Spectra
Assuming validity of the Born Oppenheimer approximation, the
transition matrix element of relevance is a vibrational matrix
element of the dipole moment in electronic state n, For a diatomic
molecule the dependence of on the vibrational coordinate is often
expressed as a power series in the displacement from equilibrium x,
To show a vibrational spectrum a diatomic molecule must have a
dipole moment that varies with extension For small displacements
from equilibrium, (71) (72) (73) (74) Slide 29 In the harmonic
limit this approximation leads to a weak selection rule: where m is
the reduced mass, is the vibrational frequency and a, a + are the
annihilation and creation operators, (75) (76) (77) (78) Slide 30
where is a Franck-Condon factor. II-2-C. Electronic Spectra The
classical basis of the Franck-Condon principle, wherein the
electronic system undergoes a vertical transition terminating at
the turning point of the excited state while the nuclei neither
move nor accelerate: The quantum mechanical version, where the
molecule makes a transition from the initial vibrational level into
the level with a vibrational wavefunction that best overlaps the
initial function: (79) Fig. 6 Slide 31 10 -13 10 -12 10 -11 10 -10
sec 22 III. Time-Domain Spectroscopy Why wavepackets? Molecules are
complicated creatures: Slide 32 III-1. Forming Superposition States
In the limit of High resolution spectroscopy: The bandwidth is
narrow with respect to the level spacing of the spectrum considered
(i.e., the pulse is long with respect to the system timescale) The
interaction is small compared to the field-free Hamiltonian where,
in the general case, the set of coupled equations is solved
numerically during the pulse and propagated analytically
subsequently. (80) Slide 33 Also simple is the (strongly quantum
mechanical) case of a Two-Level System, After the end of the pulse
the expectation value of the position operator is, where the
interference term oscillates at 2 / 12 (and likewise observables
that probe the probability density). In practice this is a Young
2-slits experiment: E2E2 E1E1 E2E2 E1E1 (81) (82) Fig. 7 Slide 34
After the pulse turn-off, the wavepacket evolution is determined
solely by the energy phases (and hence conveys information about
the excited manifold Hamiltonian), To understand the structure of
the short- and long-time evolution, we consider quantum systems
that exhibit regular periodic motion in the classical limit and
expand the eigenvalue E j about as a power series in j-j *, where j
* is the center of the wavepacket in quantum number space, Here,
j+1,j approaches the inverse period of the classical motion, cl =2
/T cl, in the 0 limit (where the spectrum is quasi-equidistant) but
in general, Intensive research of the broad superposition case was
initially motivated by the thought that wavepackets should buy us a
route to the classical limit. (83) (84) Slide 35 where energy is
measured with respect to the wavepacket center energy. From Eq.
(86), we have that (in nonlinear quantum systems that execute
regular periodic motion in the classical limit): If the wavepacket
is spatially localized (i.e., its spatial dimension x is small as
compared to the dimension L of the classical orbit corresponding to
the wavepacket center energy) at t = 0, its evolution at times t T
rev will mimic the evolution of a classical particle. At times t
non-negligible with respect to T rev, the nonlinearity of the
spectrum will become observable and the wavepacket will start
dephasing at a rate that scales as j 2, where j is the width of the
distribution in quantum number space. At multiples of T rev, the t
= 0 wavepacket will be precisely reconstructed and for a while (for
a period t T rev ) will again evolve nearly classically more so the
smaller x/L. (85) (86) With the definition the wavepacket of Eq.
(83) takes the form, Slide 36 As a numerical approach, the basis
set expansion [see Eqs. (80)] is simple in concept and essentially
universal, only technical details depend on the type of motion
considered: In the case of rotational wavepackets (superpositions
of J, J and M, or J,M and K eigenstates) the matrix elements of the
interaction are analytically expressible, see Eqs. (58) and (70).
In the case of electronic wavepakets (wavepackets of Rydberg
states) quasi-classical solutions are useful. [See. e.g., L.A.
Bureeva, Sov.Phys. Astronomy 12, 962 (1969); V.A. Davidkin and B.A.
Zon, Opt.Spectrosc.(USSR) 51, 13 (1981); N.B. Delone, S.P.
Goreslavsky and V.P. Krainov, J.Phys.B 27, 4403 (1994).] In the
case of vibrational wavepackets, it is often convenient to expand
the Hamiltonian on a grid and propagate the wavepacket in time by
application of the evolution operator, Slide 37 Expansion in terms
of the eigenstates of H 0 is advantageous if the Hamiltonian can be
stored in the core memory. In cases where the field-free
Hamiltonian is too complex to that end, an expansion in terms of an
approximate time-independent Hamiltonian is typically more
economical. A good example is nonradiative transitions in
polyatomic molecules: S 0 ( 1 A g ) laser internal conversion S 1 (
1 B 2u ) S 2 ( 1 B 3u ) Phys.Rev.Lett. 93, 093004 (2004) An Optimal
Control Approach to Suppression of Radiationless Transitions
Phys.Rev.Lett. 89, 233002 (2002) Theory of Time-Resolved
Photoelectron Imaging. Nonperturbative Calculation for an
Internally Converting Polyatomic Molecule Fig. 8 Slide 38 III-2.
Electronic Wavepackets In a classical world the atom can be thought
of as a point electron moving in an orbit around the nucleus as a
planet orbits the sun: One uses the term orbits for quantum
mechanical H-like wavefunctions but here it means something
qualitatively different: Fig. 9 Slide 39 The classical picture of
an electron orbiting a nucleus following Keplers laws for planetary
systems is wrong but sufficiently attractive that tremendous effort
has been devoted to making quantum states that will approach it. An
electronic wavepacket can do the job (in principle). We require a
state that is localized (within Heisenbergs restrictions) and
nonstationary: A classical electron An idealized wavepacket The
most common way of producing an electronic wavepacket with the
desired properties is to excite an atom from the ground state into
a superposition of high Rydberg states with a laser pulse of few to
few tens of picoseconds duration: [Rydberg, 1889 ] Fig. 10 Slide 40
The simplest electronic wavepacket is a radial wavepacket,
basically a radial shell of probability which breathes in and out
with exactly the period of a classical electron of the same energy:
A radial wavepacket An ensemble of classical electrons moving
together A quantum mechanical simulation At the instance the
wavepacket is formed After of the classical orbital period After of
the classical orbital period Time delay (ps) Ionization signal
Experimental results The wavepacket is ionized periodically, when
it revisits the core Yeazell et al., Phys.Rev. A 40, 5040 (1989)
thats not quite it though classical period Fig. 11 Slide 41 A
classical limit state (A Rydberg wavepacket that is localized in 3D
and travels along a classical orbit) Gaeta et al, Phys.Rev.Lett.
73, 636 (1994) Qualitative idea: (T k is the classical orbital
period) 0 T k 1.45 T k 2.9 T k 3.4 T k 3.15 T k 3.65 T k Wavepacket
simulation: Fig. 12 Slide 42 III-3. Vibrational Wavepackets To make
them in the lab short pulses are necessary: Consider the B state of
I 2. The vibrational level spacing around v =15-20 is about 69 cm
-1. The bandwidth of a 1 ns pulse is ca. 0.03 cm -1. That of a 20
fs pulse is ca. 700 cm -1. Fig. 13 Slide 43 In the harmonic limit,
If all coefficients are equal (c v = c for all v ) and v is large
with respect to unity, (87) (88) Slide 44 In this approximation the
distance traveled in a round trip is then, The center of the
wavepacket oscillates sinusoidally at the harmonic frequency, much
like a classical particle: Consider I 2 molecules, excited from the
ground (X) into the excited (B) state with a 565 nm, 20 fs long
pulse, where about 10 levels are coherently excited, m=1.05 x10 -22
g, and =1.3 x 10 13 Hz. In the approximation of a harmonic
potential and equal coefficients, c v = c for all v, c 2 =0.1,
yielding a distance of about 1.06. x V (89) Fig. 14 Slide 45
Fischer et al, Chem.Phys. 207, 331 (1996) 2.0 2.5 3.0 3.5 4.0 R ()
A simulation of the probability density of the wavepacket at t = 0:
The relevant I 2 potential energy curves: Fig. 15 Slide 46 -0.5 0
0.5 1.0 1.5 2.0 Time in picoseconds measured with respect to the
center of the pump pulse 340 fs Fig. 16 Slide 47 -2 4 10 Time in
picoseconds, measured with respect to the center of the pump pulse
Wavepacket dephasing: -2 4 10 16 22 Wavepacket revivals: At
multiples of T rev =2 / the t = 0 wavepacket is reconstructed.
Fractional revivals appear at shorter times (the half- revival,
corresponding to T rev /2 18 ps for I 2, is seen in Fig. 17) As t
becomes non-negligible with respect to 1/ the wavepacket components
step out of phase. Fig. 17 Slide 48 Whereas electronic and
vibrational wavepackets have been around for several decades,
rotational wavepackets are a newer concept. They have properties in
common with the electronic and vibrational analogs but: Rotational
spectra are dense (for I 2, for instance, the vibrational level
spacing is 80-200 cm -1 whereas the rotational level spacing is
2JB, where B is 0.037 cm -1 ) Rotational spectra are highly
nonlinear Electric dipole selection rules forbid the excitation of
rotational wavepackets in the weak- field (one-photon) limit The c
j vs j distribution is a real arithmetic, simple (essentially
Gaussian) function for most types of wavepackets (e.g., j=n, v),
whereas for the rotational case the c j exhibit an asymmetric
distribution with a j-dependent phase The revival pattern of
rotational wavepackets is radically different from the common case
III-4. Rotational Wavepackets J.Chem.Phys. 103, 7887 (1995) Slide
49 J0J0 J 0 +2 J 0 -2 J 0 +1 J 0 -1 J 0 +3 J 0 -3 J 0 +4 J 0 - 4
etc... J 0 +5 J 0 -5 Assume first that the field is linearly
polarized and tuned to resonance, (notice several allowed
transitions are omitted, for graphical convenience) Slide 50 What
terminates the rotational excitation ? Or R ~ (J max ) : [ (J) ~
BJ(J+1) ] ... Either J max ~ R : [ R = Rabi coupling field strength
] * There are several other good ways of coherently exciting
rotations Slide 51 |J|2|J|2 J | )| 2 J 1 > ~ But in principle
alignment is not guaranteed Slide 52 t I(t) J= 0 J= 1 t /2 | (t, )|
2 t Static (adiabatic) alignment The interaction turns on slowly on
the natural system time scale, >> rot Each eigenstate of the
unperturbed Hamiltonian evolves adiabatically into the
corresponding state of the complete Hamiltonian In the case
considered, these are spatially aligned. Zon & Katsnelson, JETP
42, 595 (1976) Friedrich & Herschbach, Phys.Rev.Lett. 74, 4623
(1995) Slide 53 In the long pulse limit laser alignment converges
to alignment in a strong DC field, an old and well-established
alignment method of stereodynamics. See, e.g, D. R. Herschbach and
coworkers, Z.Phys. D Atoms, Molecules and Clusters 18, 153 (1991)
H. J. Loesch and coworkers, J.Chem.Phys. 93, 4779 (1990) R. E.
Miller and coworkers, J.chem.Phys. 101, 9447 (1994) Slide 54
Dynamical alignment during a =200 fs pulse | (t, )| 2 - /2 /2 0.6
1.2 t(ps) | (t, )| 2 - /2 /2 30 60 t(ps) Alignment is enhanced
after the turn off Phys.Rev.Lett. 83, 4971 (1999) T rot = 0 Slide
55 Theory: A first experimental demonstration: F. Rosca-Pruna &
M.J.J. Vrakking, Phys.Rev.Lett 87, 153902 (2002) Phys.Rev.Lett. 83,
4971 (1999) Time Alignment Time Alignment T rev = / B T rot > 0
Slide 56 In the impulse (
