Vibrational
energy flow is of fundamental importance as the initial step in
polyatomic molecule chemical reactions. In essence, molecules roughly
behave like a set of balls connected by anharmonic springs: tweaking
one of these springs and letting go will soon cause the whole
contraption to vibrate in an apparently random fashion. Or
is it as random as it seems? Basic theories of reaction dynamics (e.g.
the celebrated RRKM theory) assume that energy redistribution is
complete long before interesting chemical reactions happen. Yet there
are signs that the energy flow and reaction timescales are not
separated by orders of magnitude in time. Our goal is to slow down
energy flow further, until chemical reactions are no longer
"statistical." We investigate this problem
theorectically by developing tools to do computations on large and
highly excited molecules. This high energy regime offers two
challenges: 1) the potential surface of molecules is difficult to
obtain; 2) even with a potential surface or matrix elements in hand, it
is difficult to calculate molecular spectra or the time evolution of a
molecule's wavefunction because very large (10,000's to 1,000,000's) of
molecular energy levels are involved. We have developed
multidimensional quantum dynamics algorithms and realistic anharmonic
Hamiltonians. In particular, we developed a scaling model for
vibrational couplings based on the Born-Oppenheimer approximation. We
also study the problem experimentally by frequency- and time-resolved
stimulated emission pumping experiments on organic molecules. The key
in our experiments is that we study "skeletal vibration" involving
carbon, oxygen, sulfur, etc. atoms, not hydrogen stretching or bending
modes, which are not typical of the "bath" of vibrations into which
energy flows in large organic molecules. The calculations agree with
ours and other groups' experiments: vibrational energy redistribution
slows dramatically as the molecule explores more of the available
quantum states. At short times, the vibrational wavepacket decays
exponentially fast as predicted by the Golden Rule of time-dependent
perturbation theory. At longer times, the decay slows down and becomes
a polynomial in time. Another way of looking at
this problem is in terms of the dimensionality of the vibrational
wavepacket as it explores "state space." State space is a 3N-6
dimensional lattice of coordinates, one coordinate for each vibrational
mode of a molecule with N atoms. A laser initially excites a single
point or small group of states in this lattice. This group or
"vibrational wavepacket" then expands, until all states compatible with
energy conservation are covered in the statistical limit at long times.
Where our model differs from the Golden Rule is how the wavepacket gets
to the statistical limit: in the Golden Rule approximation, the
wavepacket uniformly expands in 3N-6 dimensions, leading to a nearly
exponential decay (in the limit where 3N-6 becomes very large). In our
model, and in the experiments, the wavepacket expands in only 1-4
dimensions, so it has a fractal dimension much smaller than 3N-6. The
nature of the coordinates along which the wavepacket expands changes
with time, so eventually the wavepacket does cover all eigenstates
allowed by energy conservation, but it takes much longer to do so.
As a result, the number of parameters to control the evolution of the
molecular wavepacket via coherent laser excitation grows more slowly in
time than expected. This can be exploited to allow coherent control of
the vibrational energy flow. Once the energy flow is controlled,
subsequent chemical reactions are no longer statistical. The
phenomenon of vibrational energy flow also provides a model laboratory
for molecular quantum computing. Because an isolated molecule obeys
energy conservation, energy flow is a pure dephasing process in terms
of the molecular eigenstates. Different vibrational modes, or different
combinations of energy levels, can have their populations and phases
switched around by laser pulses, in effect performing quantum
computations. For example, a set of 4 vibrational states arbitrarily
coupled by laser pulses can perform any 2-qubit calculation. These
computations occur on a femtosecond time scale, long before nanosecond
dephasing processes set in.
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