Designing molecular eigenstates in a four-level system
Teodora Kirova
1
and Frank C. Spano
2
1
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
2
Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122, USA
Received 5 November 2004; revised manuscript received 25 February 2005; published 23 June 2005
The interaction of single cw laser coupling field with a four-level system containing two closely spaced
upper levels a and b is studied as a means to attain quantum control. By adjusting the coupling field Rabi
frequency and detuning, it is possible to create target states of a prescribed admixture of states a and b within
a dressed state representation. An electromagnetically induced transparency of an additional probe beam is also
studied. The four-level system gives rise to generally two dark resonances for a given coupling field
detuning. When the resonances are degenerate EIT is enhanced.
DOI: 10.1103/PhysRevA.71.063816 PACS numbers: 42.50.Gy, 32.80.Qk, 33.80.b
I. INTRODUCTION
Coherent effects in systems consisting of just a handful of
energy levels continues to attract a great deal of interest from
diverse groups worldwide. Such processes include electro-
magnetically induced transparency EIT1–3, double dark
resonances 4–10, lasing without inversion 11, cancella-
tion of spontaneous emission 12–15, gross reduction in op-
tical pulse propagation velocity 16,17, as well as various
schemes with which to achieve quantum or coherent control
18. Many of these effects were originally demonstrated for
single atoms in the gas phase, although recently EIT has
been observed in small molecules such as Li
2
19 and NH
3
20, whereas the closely related phenomena of Autler-
Townes splitting 21 has been observed in Li
2
using cw
lasers 22 as well as H
2
23,N
2
24, CO 25, and benzene
26, using pulsed laser excitation to overcome Doppler
broadening. Generally, molecular systems are more challeng-
ing because of the additional degrees of freedom and higher
state density compared to single atoms.
In this paper we theoretically analyze quantum control
and EIT in a four-level system with two closely spaced
electronic excited states. Such states may arise in a molecular
system near an avoided crossing, for example, nearby singlet
and triplet states mixed by spin-orbit coupling. Coherent ef-
fects involving two nearby energy levels have been previ-
ously analyzed, primarily with respect to spontaneous emis-
sion cancellation 12–14. Berman 12 showed that the
cancellation can be understood in terms of two nearby states,
a and b, each composed of an admixture of a radiative state
b
0
and a metastable state a
0
. For example, b
0
and a
0
may
correspond to a pure singlet and pure triplet, respectively,
mixed by spin-orbit coupling to produce the mainly singlet
and mainly triplet states b and a. Application of an appropri-
ately detuned coupling laser, which optically connects state
b
0
but not a
0
to another metastable state c, creates a dressed
eigenstate involving only the unperturbed dark states a
0
and
c, with no admixture of the radiative state b
0
. Hence the state
is metastable and does not radiate.
The motivation for the present work involves a generali-
zation of the above, that of using the coupling field to mix
the states a
0
and b
0
in a controllable way. One of the appli-
cations is to optimally create admixed singlet-triplet states
that serve as gateway 27 states in alkali-earth dimers such
as Li
2
28–33, as a means to further excite high-energy trip-
lets which cannot otherwise be excited directly from the sin-
glet ground state. We show that controllable admixtures of a
0
and b
0
can be created with the use of a single continuous-
wave optical field with a prescribed detuning and Rabi fre-
quency related through a well-defined control equation. The
idea is similar in spirit to coherent control using stimulated
Raman adiabatic passage STIRAP34, in which eigen-
states are crafted in the dressed state basis set. In addition we
show the existence of generally two probe frequency detun-
ings at which the probe beam undergoes EIT. The two dark
resonances are compared to recent work on interacting dark
states 4–9.
II. LEVEL SCHEME AND HAMILTONIAN
We begin by considering the level scheme shown in Fig.
1, consisting of a four-level scheme with two closely
spaced upper levels a and b, with energies e
a
and e
b
,
respectively, under irradiation by two laser fields: a strong
coupling or control field with frequency
c
, which couples
the upper two states with upper ground level 2, and a weak
FIG. 1. Four-level system with probe and coupling laser de-
tunings and Rabi frequencies as indicated.
PHYSICAL REVIEW A 71, 063816 2005
1050-2947/2005/716/0638169/$23.00 ©2005 The American Physical Society 063816-1