Quantitative knowledge of a system’s Hamiltonian enables researchers to
predict and control quantum systems.
The primary principle for coherent
control is manipulation of constructive
and destructive interference of quantum
pathways between the initial and target
states. 1, 2 The closed-loop method was
an important advance toward achieving
coherent control. With this method, an
algorithm iteratively learns the pathway
information by trial experiments and
suggests the next control design until it
converges on a targeted state. 3 In most
cases, the initial and target states are connected by many quantum pathways. Thus,
it is easier to gain control if we know
details about the contributing pathways
or the underlying Hamiltonian.
It is difficult to calculate a complex
system’s Hamiltonian, especially when
inter-particle interactions, incoherent
processes and environmental fluctuations must be taken into account. Thus,
the Hamiltonian is determined through
experiment. The greatest challenge of such
an experiment is how to completely isolate
each pathway so that the information can
be identified for each process.
We experimentally demonstrated that
quantum pathways can be isolated and
independently studied using optical 3-D
coherent spectroscopy (3DCS). 4 3DCS is
an extension of 2-D coherent spectroscopy
(2DCS), a powerful tool for studying the
coupling and dynamics in complex systems. With 2DCS, researchers disentangle
quantum pathways by unfolding a 1-D
spectrum onto a 2-D plane. 5 For many
systems, however, the quantum pathways
are only partially separated in 2DCS.
To completely isolate the pathways, we
extended 2DCS into 3-D to completely
unfold the spectrum.
As a proof-of-principle, we obtained
a 3-D spectrum of a potassium vapor.
Within the excitation laser bandwidth, the
3-D spectrum contains the full third-order
coherent response of the potassium.
From this, the spectral contributions
from different quantum pathways were
unambiguously isolated. Therefore, we
can determine the parameters describing
each quantum pathway—including the
transition energies, dipole moments and
relaxation rates—from the spectrum and
then construct the system’s Hamiltonian.
These measurements provide us with
the information we need for a control
scheme that uses the same bandwidth
pulses and is based on the third-order
optical response. Our experiment is
a critical step towards the complete
experimental determination of a system’s
(a) 3DCS is based on a transient four-wave mixing (TFWM) experiment. (b)
A typical 3-D spectrum of a potassium vapor. The projections on different
planes provide various 2-D spectra.
JILA, University of
Colorado and National
Institute of Standards
and Technology, Boulder,
Colo., U.S.A., and Florida
Miami, Fla., U.S.A.
Alan D. Bristow
JILA and West Virginia
University, W.Va., U.S.A.
Mark E. Siemens
JILA and University of
Denver, Colo., U.S.A.
Galan Moody and
Steven T. Cundiff
JILA and University
of Colorado, Boulder,
1. P. Brumer et al. Chem.
Phys. Lett. 126, 541
2. W. S. Warren et al. Science 259, 1581 (1993).
3. R. S. Judson et al.
with Optical 3-D Spectroscopy
Phys. Rev. Lett. 68,
4. H. Li et al. Nat. Com-
mun. 4, 1390 (2013).
5. S. T. Cundiff and S.
Mukamel. Phys. Today
66, 44 (2013).
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Determining the System Hamiltonian