Download Quantum Interference Effects In Atom-Atom And Ion-Atom

Quantum Interference Effects In Atom-Atom
And Ion-Atom Cold Collisions In The Presence
Of External Fields
THESIS SUBMITTED FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY (SCIENCE)
IN
CHEMISTRY (PHYSICAL CHEMISTRY)
BY
ARPITA RAKSHIT
Department of Chemistry
University of Calcutta
2014
To my parents . . . . . .
To Be An Achiever, You Got To Be A Believer First . . . A. Einstein
Acknowledgements
Before writing the thesis I want to take the opportunity to thank all the people who
provided enormous support, continuous encouragement and inspiration throughout
my Ph.D. research work.
First of all, I would like to express my earnest gratitude to my supervisor Professor Bimalendu Deb for his continuous support, encouragement and advices during
my study and research work at I.A.C.S. He has also taught me to aim high, to be
persevering, and above all, to embrace scientific challenges with passion and enthusiasm. I thank him for his insight, guidance and caring support, and for so much
I have learned from him. I will never forget his strong support during my difficult
times. I express my gratitude to Professor Deb Sankar Ray, Professor S. P. Bhattacharyya, Professor Debashis Mukherjee and Professor Kamal Bhattacharyya for
their kind help. I thank Sumantada ( Dr. Sumanta Kumar Das, post-Doctoral fellow, The Niels Bohr Institute) and Saikatda (Dr. Saikat Ghosh, I.I.T., Kanpur)
as collaborators for their valuable discussions, suggestions and help when I needed.
I thank Indian Association for the Cultivation of Science (IACS) for providing
all types facilities needed to carry out my work. I thank all the faculty and staff
members of Material Science Department for help and support. The fellowship
provided by the Council of Scientific and Industrial Research (CSIR) is gratefully
acknowledged.
My thanks also go to the current and former members of our lab, with whom
I have fond memories and from whom I have learned much and received a lot of
help. Jishadi, Debashree, Biswajit, Partha, Somnathda, Arpita, Somnath, Farzana
and Soham, thank you!
I would like to acknowledge my heartiest gratitude to my excellent teachers of my
undergraduate and master degree classes. I thank specially Dr. Rana Sen, Dr.
Anupa Saha, Dr. Pradipta Ghosh, Dr. Samrajnee Dutta and Dr. Durba Barik for
their care, love and continuous encouragement.
I thank Antaradi, Moumitadi, Shrabanidi, Namrata, Partha, Tuluma, Ananya,
Avijit and my other numerous friends who make my stay at IACS enjoyable. I
specially thank Shreetama and Shubhro for their valuable friendship, constant support and help in critical situations. In this joyful moment, I want to mention
specially Sohini, Kamalika and Anindita from B. Sc. batch of Scottish Church
College who have provided me constant motivation.
I would like to thank my college, Sidhu Kanhu Birsa Polytechnic, Keshiary and
Department of Technical Education and Training, Govt. of West Bengal for giving
me the permission for continuing my Ph.D. work after my joining as lecturer in
May, 2012. I would like to acknowledge Dr. Parijat De, Director of Technical
Education and Training, Govt. of West Bengal, Dr. Amit Ranjan Ghatak, exPrincipal and Mr. Asok Kumar Deb, Officer-In-Charge of my college for allowing
me to devote my off-time to research. I thank all my colleagues at I.C.V. Polytechnic, Jhargram and S. K. B. Polytechnic, Keshiary. I want to mention specially
Mr. Ushnish Sarkar, Mr. Arunanshu Das, Mr. Satyam Paul and Dr. Subrata
Kamilya for their constant encouragement.
Lastly I thank my parents. They have contributed much more than I can say. I am
overwhelmed having the scope to acknowledge them. I am thankful to my husband,
Dr. Uttam Kumar Das whose continuous moral support and love motivated me to
remain focused towards achieving this milestone of my journey.
IACS Kolkata
June, 2014
ARPITA RAKSHIT
Contents
Acknowledgements
1 Introduction
1.1 Preliminary Discussions . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
1
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2 Quantum Interference in Atoms and Molecules: A Review
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2.1 Fano Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Spontaneous Emission and Vacuum-Induced Coherence . . . . . . . 12
2.3 Quantum Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Fano-Feshbach Resonances
3.1 Ultracold Scattering and Resonances . . . . . . . .
3.1.1 Elastic Scattering . . . . . . . . . . . . . . .
3.1.2 Scattering Length in Ultracold Gases . . . .
3.1.3 Scattering Resonances at Low Energy . . . .
3.2 Single Channel Resonance . . . . . . . . . . . . . .
3.2.1 Shape Resonance . . . . . . . . . . . . . . .
3.2.2 Potential Resonance . . . . . . . . . . . . .
3.3 Feshbach Resonance : A Multichannel Resonance .
3.3.1 Magnetic Feshbach Resonance . . . . . . . .
3.3.2 Scattering Length and Feshbach Resonance .
3.4 Photoassociation and Optical Feshbach Resonance .
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4 Quantum Interference in Photoassociation in the Presence of Feshbach Resonance
4.1 Perspective of The Work . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Formulation of the Problem and Solution . . . . . . . . . . . . . . .
4.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Analytical Results . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Numerical Results and Discussion . . . . . . . . . . . . . . .
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
5 Vacuum- and Light-Induced
and Molecules
5.1 Perspective of The work .
5.2 Scheme 1 . . . . . . . . . .
5.3 Solution . . . . . . . . . .
5.4 Results and Discussions .
5.5 Scheme 2 . . . . . . . . . .
5.6 Master equation . . . . . .
5.7 Solution . . . . . . . . . .
5.8 Results and discussions . .
5.9 Conclusion . . . . . . . . .
Coherences in Cold Atoms
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6 Atom-Ion Cold Collisions: Formation of Cold
6.1 Theory of Atom-Ion Collision . . . . . . . . .
6.1.1 Atom-Ion Interaction Potential . . . .
6.1.2 Elastic Collisions . . . . . . . . . . . .
6.1.3 Radiative Transfer . . . . . . . . . . .
6.1.4 Spin Exchange Collision . . . . . . . .
6.2 Fomation of Cold Molecular Ion . . . . . . . .
6.2.1 Formulation of Problem . . . . . . . .
6.3 Results and Discussion . . . . . . . . . . . . .
6.4 Conclusions . . . . . . . . . . . . . . . . . . .
7 Conclusions and Outlook
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Molecular
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Ion
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Chapter 1
Introduction
1.1
Preliminary Discussions
The development of laser cooling and trapping of atoms (Nobel Prize 1997) [1] in
the 80’s and early 90’s, culminating in the realization of Bose-Einstein-Condensation
in dilute atomic gases (Nobel Prize 2001) [2] in 1995, has opened a new doorway
for atomic, molecular and optical sciences. Using laser, Doppler and evaporative
cooling cold atoms in quantum degenerate regime are now routinely obtained. Coherent control of external and internal degrees-of-freedom of cold atoms are made
possible by optical precision spectroscopy (Nobel Prize 2005) [3]. At sub-Kelvin
temperatures, atoms move extremely slowly, offering larger observation time required for high resolution spectroscopy.
Following the great success in cooling atoms and exploring physics and chemistry
with cold atoms, now the focus of interest has also encompased cold molecules.
Cold molecules with ro-vibrational structures can reveal many novel physics which
can not be observed in atomic systems. Now the problem is that producing cold
molecules in a straightforward way is difficult. Though most successful for atoms,
the method of laser cooling is not in general applicable to molecules due to their
complicated level structures. Laser cooling requires a cyclic transition which repeats absorption of cooling lasers and spontaneous emission. Laser cooling of
a molecule is extremely difficult since a large number of extra repumping lasers
would be required to avoid optical pumping to the ro-vibrational levels irrelevant to
the cooling transition. Different technologies like supersonic expansion, cryogenic
cooling technology etc. have been used to cool molecules with limited success.
1
Chapter 1. General Introduction
2
Alternatively, two indirect methods are now used for producing cold molecules by
associating cold atoms, namely photoassociation (PA) [4] and Feshbach Resonance
(FR) [5]. In both processes, translationally cold atoms are used as an initial source
and they are transferred to translationally cold molecules by association process
using external electromagnetic fields. Since collisions between cold atoms occur
for lower partial waves, the rotational temperature of formed molecules is also
low. Though this approach is limited to molecules whose constituent atoms can
be cooled by laser cooling, it may be regarded as one of the more promising ways
to achieve quantum degenerate molecular gases.
Cold atoms and molecules are important for exploring new physics and chemistry
[6]. Research areas such as molecular chemistry [7], condensed matter physics
[8] and astrochemistry [9] are revitalized with advancement in cold atom physics.
Cold atoms and molecules can serve as good candidates for observing quantum
interference and coherence. We are quite familiar with interference phenomena in
our everyday life. For example, if we throw two pebbles in a quiet pond, waves
produced by them interact and produce an intricate pattern of crests and troughs
as a result of constructive and destructive interferences between them. This can
be well described with help of classical physics. Idea of quantum interference
can be described similarly. When the atoms are driven by electromagnetic fields,
they undergo transitions from an initial to a final state. In some instances, these
transitions can occur through multiple indistinguishable but independent pathways
which may interfere leading to the destructive or constructive interference effects.
Interference effects are mostly studied in atomic systems and seldom in molecular
systems. But ultracold atom-molecule coupled systems obtained via free-bound
transitions have not yet been addressed adequately so far. Main objective of this
thesis is to explore different quantum interference phenomena in these systems.
1.2
Scope of the Thesis
Main objective of this thesis is to investigate quantum interference (QI) phenomena
in an ultracold atom-molecule coupled system in the presence of external fields
[10] within the framework of celebrated Fano theory [11]. Fano effect arises due to
interaction between configurations of discrete levels and continuum states. Two
excitation pathways leading to continuum states result in coherent interference
which leads to asymmetric absorption spectra.
Chapter 1. General Introduction
3
To show the effects of QI in the atom-molecule coupled system, we investigate
photoassociation (PA) in the presence of magnetic Feshbach resonance (MFR)
[12]. We work in dressed basis using Fano method. Solution of the problem gives
the analytical expression for linewidth and shift for any arbitrary laser intensity.
We show that power broadening of line width can be suppressed even at high
laser intensities by tuning the magnetic field close to Fano minimum. We also
demonstrate that Fano effect can lead to large light shift near Fano minimum.
These features arising from quantum interference are useful for efficient tuning of
scattering length by optical means.
We propose a novel PA scheme for realization of vacuum induced coherence (VIC)
[13] in atom-molecule coupled systems [20]. To the best of our knowledge, the
possibility of VIC in a PA system has not been discussed earlier. We consider
PA transitions from the collisional continuum of two atoms to two excited rovibrational levels belonging to same excited molecular electronic state and the
excited levels decay spontaneously due to interaction with the electromagnetic
vacuum. The two spontaneous emission pathways interfere resulting in VIC. We
take bosonic Ytterbium (174 Yb) atoms for numerical illustration of our theoretical
proposal. We solve our model using both Wigner-Weisskopf and master equation
approaches and demonstrate theoretically an interesting interplay between VIC
and PA. We show that VIC between two ro-vibrational levels arises due to the
quantum interference between spontaneous emission pathways from the rovibrational levels to the electronic ground molecular state. We also investigate this
system driven by two lasers using master equation approach. We study the temporal dynamics of the excited bound states and demonstrate quantum beats in
emission spectra as a signature of QI. We also demonstrate laser induced coherence (LIC) between two excited molecular states [15]. Thus it is possible to control
decoherences and spontaneous decay.
Finally, we investigate elastic and inelastic (charge transfer) collisions between
atoms and ions at low temperatures and discuss formation of cold molecular ions
by atom-ion photoassociation. Cold molecular ions may be obtained by various
processes such as photoassociative ionization, buffer gas and rotational cooling,
sympathetic cooling etc. Possibly, this is the first theoretical demonstration of
creating translationally and rotationally cold molecular ions by photoassociation
[16].
Chapter 1. General Introduction
1.3
4
Outline of the Thesis
Before studying in detail, we briefly provide a survey of the literature on quantum
interference in atoms and molecules in chapter 2. QI phenomena like electromagnetically induced transperancy, coherent population trapping, stimulated Raman
adiabatic Passage (STIRAP), slow light that appear in three level systems are
well known. Fano interference in ultracold atom-molecule coupled systems will
lead to analogous QI effects in new parameter regimes. We discuss in short Fano
interference, vacuum-induced coherence and quantum beating.
The continuum-bound spectra in atom-molecule coupled systems are discussed in
chapter 3. Description of different single- and multi-channel resonances are also
given. We mainly focus on Fano-Feshach resonance and discuss its tunability in
the presence of magnetic and optical field.
Photoassociation in the presence of magnetic Feshbach resonance is addressed in
chapter 4. We present a theoretical model and solve it analytically. Then we apply
it to a realistic system to verify the prediction of our theory and finally analyze
our numerical results.
In chapter 5 vacuum-induced and light-induced coherences in atom-molecule coupled systems are studied. We consider that two excited ro-vibrational levels in the
same electronic molecular potentials are coupled to the continuum of scattering
states of two ground-state atoms. We solve the problem in two ways. In the first
way we include the spontaneous decay terms using Wigner-Weisskopf approach.
In other case we solve it using master equation approach.
Chapter 6 is devoted to atom-ion scattering at cold and ultracold temperature
regimes. We also discuss different radiative and non-radiative processes. Here we
propose a new formalism for the formation of translationally and rotationally cold
molecular ions photoassociation.
Lastly, we conclude and give an outlook of our studies in chapter 7.
References
[1] C. N. Cohen-Tannoudji, Rev. Mod. Phys. 70, 707 (1998); W. D. Phillips, Rev.
Mod. Phys. 70, 721 (1998); S. Chu, Rev. Mod. Phys. 70, 685 (1998)
[2] W. Ketterle, Rev. Mod. Phys. 74, 1131 (2002); E. A. Cornell and C. E.
Wieman, Rev. Mod. Phys. 74, 875 (2002).
[3] J. L. Hall, Rev. Mod. Phys. 78, 1279 (2006); R. J. Glauber, Rev. Mod. Phys.
78, 1267 (2006); T. W. Hansch, Rev. Mod. Phys. 78, 1297 (2006).
[4] P. Fedichev, Y. Kagan, G. V. Shlyapnikov and J. T. M. Walraven, Phys. Rev.
lett. 77, 2913 (1996); K. M. Jones, E. Tiesinga, P. D. Lett and P. S. Julienne,
Rev. Mod. Phys. 78, 483 (2006).
[5] H. Feshbach, Ann. Phys. 5, 357 (1958); T. Köhler, K. Góral and P. S. Julienne,
Rev. Mod. Phys. 78, 1311 (2006).
[6] O. Dulieu, R. Krems, M. Weidemüller and S. Willitsch, Phys. Chem. Chem
Phys. 13, 18703 (2011); L. Carr, D. DeMille, R. V. Krems, J. Ye, New J.
Phys. 11, 055049 (2009) and references therein.
[7] R. Krems, Physics 3, 10 (2010) and references therein.
[8] G. Pupillo, A. Micheli, H. P. Büchler and P. Zoller, arXiv:0805.1896 (2008);
W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L.
Pollet and M. Greiner, Science 329, 547 (2010); M. Aidelsburger, M. Atala,
M. Lohse, J. T. Barrelro, B. Paredes and I. Bloch, Phys. Rev. Lett. 111,
185301 (2013).
[9] I. W. M. Smith, Low Temperature and Cold Molecules (Imperial College Press,
2008).
[10] B. Deb, A Rakshit, J Hazra and D. Chakraborty, Pramana J. Phys. 80, 3
(2013).
5
Chapter 1. General Introduction
6
[11] U. Fano, Phys. Rev. 124, 1866 (1961).
[12] B. Deb and A. Rakshit, J. Phys. B: At. Mol. Opt. Phys. 42, 195202 (2009).
[13] G. S. Agarwal, Springer Tracts in Mosern Physics:
Quantum Optics
(Springer-Verlag, Berlin, 1974).
[20] S. Das, A. Rakshit, and B. Deb, Phys. Rev. A 85, 011401(R) (2012)
[15] A. Rakshit, S. Ghosh and B. Deb, J. Phys. B: At. Mol. Opt. Phys. 47, 115303
(2014).
[16] A. Rakshit and B. Deb, Phys. Rev. A 83, 022703 (2011)
Chapter 2
Quantum Interference in Atoms
and Molecules: A Review
Quantum interference is one of the most profound effects of quantum mechanics.
Feynman referred it as ‘the only mystery’ of quantum mechanics. The field of
optical interference has a very old history back in early nineteenth century when
Thomas Young performed his famous double slit experiment[1–3]. In general,
interference means the superposition of two or more coherent waves resulting in
reinforcing or neutralizing effects. Atoms or molecules interacting with electromagnetic fields undergo transitions from initial to final states. Sometimes these may
occur through multiple indistinguishable pathways which may interfere enhancing
a desired process or suppressing the undesired one under suitable conditions. The
coherence signifies the correlation or matching between phases or amplitudes of
interfering pathways. Quantum interference has applications in devices such as
different types of interferometers [2], super computing quantum device(SQUID),
quantum cryptography, quantum computing [4] etc.
Light interacting with a two-level system is a primitive case and has been widely
studied [1]. When a multilevel system interacts with electromagnetic fields it can
display non-linear optical behaviour which is of great interest for quantum optics
researchers. The most simple system is a three-level atomic system. A three-level
system can be of three different types: 1) Vee , 2) Lambda and 3) Cascade which
are shown in Fig.2.1. Now let us consider a three-level system interacting with
two nearly-resonant coherent fields. Each field connects a separate transition, but
both transitions share a common energy state. These two pathways can interact
7
Chapter 2. Quantum Interference
(a) Vee-type
8
(b) Lambda-type
(c) Cascade-type
Figure 2.1: Different types of three-level systems.
with each other resulting in new and counter-intuitive observations. Interference
occurring within a single atom or molecule in the presence of electromagnetic fields
can lead to many interesting physical effects such as electromagnetically induced
transparency (EIT)[5], coherent population trapping (CPT)[6], lasing without inversion(LWI) [7], Rabi oscillation (RO)[8], Autler Townes splitting (AT) [9], slow
light [10], STIRAP [11] , vacuum-induced coherence (VIC)[12] , quantum beating
(QB) [13] etc.
Most of the studies on quantum interference have been done in case of atoms. With
the development of the fields of cold and ultra atoms and ions, continuum-bound
transitions in atom-molecule coupled system have started to draw attention. In
this thesis, our aim is to discuss the quantum interference effects in case atommolecule coupled systems. Our primary focus is on Fano interference. In 1961,
Ugo Fano studied the interference among the configurations of discrete level(s) to a
continuum [14]. Two ionization pathways interfere leading to an asymmetric peak
which is known as Fano profile. This method is useful to discuss the interference
effects in atom-molecule coupled system which involves at least one transition
from continuum to a bound state. Therefore, we can obtain dressed continuum as
discussed in Fano’s theory. In next section, Fano interference will be discussed in
Chapter 2. Quantum Interference
9
10
q=0
q=1
q=3
8
Sq(E)
6
4
2
0
-10
-5
0
ε
5
10
Figure 2.2: natural line shapes for different values of q [14].
short. Apart from Fano interference, another two interesting quantum interference
effects will play key role in chapter 5 and hence they need short introduction in
the perspective of atom-molecule coupled systems. They are vacuum induced
coherence (VIC) and quantum beating. In the following subsections, we shall
discuss these three effects in short.
2.1
Fano Interference
Fano interference [14] is named after famous scientist Ugo Fano. In 1961, Fano
studied the coupling of discrete configurations with a continuum of states in case
of autoionization. Let us consider a simple case of an autoionizing discrete level
interacting with a continuum. Two ionization pathways, one direct and another
through autoionizing state, interfere resulting in asymmetric spectral profile.
First, Let us consider an atomic system with discrete level φ of energy Eφ and
a continuum of states ψE ′ . Each of these states is assumed to be non degenerate. Next, the energy sub-matrix belonging to the subset of states φ, ψE ′ will be
Chapter 2. Quantum Interference
10
diagonalized. Its elements form a square sub-matrix:
hφ|H|φi = Eφ ,
(2.1)
hψE ′ |H|φi = VE ′ ,
(2.2)
hψE |H|ψE ′ i = E ′ δ(E − E ′ ).
(2.3)
The discrete energy level Eφ lies within the continuous range of values. Fano
treated the problem in dressed picture. The dressed eigenvector of the energy
matrix is assumed to have the form:
χE = aφ +
Z
dE ′ bE ′ ψE ′ ,
(2.4)
where the dressed coefficients a and bE ′ are the function of E. These coefficients
are obtained as solutions of the system of eqs. (2.1) to (2.3):
Eφ a +
Z
dE ′ VE∗′ bE ′ = Ea′ ,
VE ′ a + E ′ bE ′ = EbE ′ .
The formal solution can be written as
1
′
bE ′ =
+ z(E)δ(E − E ) VE ′ a.
E − E′
(2.5)
(2.6)
(2.7)
The asymptotic behaviour of χE is now compared to the continuum. If ψE ∝
sin(kr), where E =
Z
k 2 ~2
2m
and m is the mass of the system, then at asymptotic limit
dE ′ bE ′ ψE ′ ∝ −π cos(kr) + z(E) sin(kr) = sin(kr + ∆).
(2.8)
Here, ∆ = − arctan[π/z(E)] represents the phase shift due to the interaction
between the continuum of states ψE and discrete level φ. z(E) can be expressed
as
z(E) =
E − Eφ − F (E)
|VE |2
(2.9)
where,
F (E) = P
Z
dE ′
|VE ′ |2
.
E − E′
(2.10)
Chapter 2. Quantum Interference
11
P stands for the principal value of the integral. The phase shift ∆ varies by
∼ π as E covers the intervals ∼ |VE |2 about the resonance at E = Eφ + F (E).
Therefore, F (E) represents shift from resonance position of discrete level φ. With
proper normalization, the dressed amplitudes are given by
sin ∆
,
πVE
VE ′ sin ∆
=
− δ(E − E ′ ) cos ∆.
′
πVE E − E
a =
bE ′
(2.11)
(2.12)
Now if a transition takes place between a state |ii and the dressed state χE and
T be the transition operator, the transition probability amplitude is given by
hχE |T |ii =
sin ∆
hΦ|T |ii − hψE |T |ii cos ∆
πVE
(2.13)
where,
Φ=φ+P
Z
dE ′
|VE ′ ψE ′
E − E′
(2.14)
is an admixture of the discrete level and the states of the continuum. The sharp
variation of ∆ as E passes through resonance induces a sharp variation of hχE |T |ii.
Since sin ∆ is an even function and cos ∆ is an odd function of (E − Eφ − F (E)),
their contributions to hχE |T |ii by hΦ|T |ii and hψE |T |ii, respectively, ‘interfere
with opposite phase on the two sides of the resonance’ [14], which is a characteristic
feature of Fano resonances. The general line shape of of Fano resonance can be
written as
(ǫ + q)2
Sq (E) = 2
,
(ǫ + 1)
(2.15)
Here, the reduced energy ǫ is given by
ǫ = − cot ∆ =
E − Eφ − F (E)
Γ/2
(2.16)
and q is the ratio of indirect resonant scattering and background scattering. The
expression (2.15) shows that it gives rise to asymmetric line shapes as described in
Fig.2.2. This line shape profile is known as ‘Fano profile’. q describes the degree
of asymmetry in resonance.
Chapter 2. Quantum Interference
12
We will apply this well known Fano theory to treat the atom-molecule coupled
systems.
2.2
Spontaneous Emission and Vacuum-Induced
Coherence
Vacuum-induced coherence (VIC) takes place due to the interference between two
pathways of transitions in system-vacuum interaction [12]. Vacuum does not mean
some absolutely empty space. It is actually quantized three-dimensional multimode electromagnetic field. Normally, a vacuum state is denoted as |0ks i, where
k and s denote vacuum field wave vector and polarization. Let us consider a two
level atom interacting with the vacuum field. Just as light field drives an excited
atom to emit stimulated emission, interaction of an atom with the electromagnetic
vacuum results in spontaneous emission. As a result, state of the field changes from
|0ks i → |1ks i. 1 signifies the emission of photon and the system moves from an
excited to a ground state as shown in Fig.2.3. The overall state vector can be
written as [1]
|ψ(t)i = ca (t)|a, 0k i +
X
k
cb,k (t)|b, 1ks i,
(2.17)
where ca and cb,ks are the coefficients of corresponding states. The interaction
Hamiltonian is given by
Hint =
X
[gks Ŝ + âks ei(ω−νks )t + H.C.].
(2.18)
k
Here ω and νks is the atomic transition frequency and frequency of the field,
âks is the lowering operator of field and Ŝ + is the raising operator of the atom.
~E
~ vac |bi is the dipole coupling with d~ being the atomic dipole moment
gks = −ha|d·
p
~ vac = ~νks /2ǫ0 V ǫ~s . ~νks /2ǫ0 V is the amplitude of vacuum field, ǫ~s is the
and E
electric field polarization vector and V is the quantization volume. |gks |2 may be
written as
|gks|2 =
~νks
|d1 |2 cos2 θ
2ǫ0 V
(2.19)
Chapter 2. Quantum Interference
13
Figure 2.3: two-level system interacting with electromagnetic vacuum.
where, θ is the angle between d~ and ǫ~s . From the Schrödinger equation |ψ̇(t)i =
−(i/~)Hint |ψ(t)i, one can obtain
ċa (t) = −
X
ks
|gks |
2
Z
t
′
dt′ e(ω−νks )(t−t ) ca (t′ ).
(2.20)
0
The summation over the modes of vacuum fields can be converted to an integral
X
k
V
→2
(2π)3
∞
Z
dkk
0
2
Z
π
dθ sin θ
0
Z
2π
dφ.
(2.21)
0
Integrating over θ and φ and putting k = νks /c, the Eq. (2.20) becomes
4d2
ċa (t) = −
(2π)2 6~ǫ0 c3
Z
∞
3
dνks νks
0
Z
t
′
dt′ e(ω−νks )(t−t ) ca (t′ ).
(2.22)
0
Considering that the frequency of the vacuum field is centered around atomic
transition frequency, νk3 is replaced by ω 3 and the lower limit in the νks integration
by −∞. The dνks integral can be solved as
Z
∞
−∞
′
dνks ei(ω−νks )(t−t ) = 2πδ(t − t′ ).
(2.23)
Chapter 2. Quantum Interference
14
So, under Wigner-Weisskopf approximation one can write
γ
ċa (t) = − ca (t)
2
(2.24)
1 4ω 3 d2
γ=
4πǫ0 3~c3
(2.25)
where,
is the spontaneous decay constant. So this is the origin of spontaneous emission.
Now for demonstration of VIC, let us consider a Vee-type system consisting of
two excited states |2i and |3i and a ground state |1i (as depicted in Fig.2.1(a)).
The coupling of the system with the background vacuum fields causes spontaneous
decay from the excited states to the same ground state. Now these two spontaneous
emission pathways can interfere resulting in VIC. This effect can modify and even
quench the spontaneous emission. Several studies have suggested to control the
spontaneous emission by using external fields in atomic systems [12, 15–22]. Using
VIC, Hegerfeldt and Plenio showed periodic dark states and quantum beats in a
near-degenerate Vee-system [15]. VIC may lead to the modification and sometimes
suppression of resonance fluorescence [15–17]. Elimination of spectral line and even
the cancellation of spontaneous emission have been demonstrated as an application
of VIC [18]. It may lead to phase sensitive absorption and emission profile [23–25].
It also has been found effective for controlling decoherence in quantum information
processing [26].
All these schemes relies on two stringent conditions. First, The frequency splitting
between two excited states must not exceed the natural line width of transitions,
which means that the excited states must be near-degenerate. Second, the dipole
moments d~1 and d~2 associated with transitions |2i → |1i and |3i → |1i, respec-
tively must be non-orthogonal. If these two conditions are fulfilled, the resultant
interference term can be written as
γij =
√
γi γj
d~i .d~j
,
|d~i||d~j |
(2.26)
here i is not equal to j. This term changes the system dynamics. To meet up
both the requirements is very hard for atomic systems. A possible realization
of VIC for an excited atom interacting with an anisotropic vacuum [27–29] and
utilizing the j = 1/2 → j = 1/2 transition in
198
Hg+ and
139
Ba+ [30, 31] has been
Chapter 2. Quantum Interference
15
suggested. On the other hand molecules are the natural candidates for observing
VIC [32] . The required orthogonality criteria is easily satisfied if the two excited
states belong to same electronic configuration, but differs only in rotational or
vibrational quantum numbers. But only a few ventures have been made in case of
molecules [33].
Atom-molecule coupled system may provide itself as a better candidate. Using
photoassociation (PA) spectroscopy, low-lying rotational levels in an excited electronic level can be selectively populated. VIC will be significant if (i) the ground
state has no hyperfine interaction, (ii) these is no bound state close to ground
state dissociation threshold and (iii) excited molecular levels have a long lifetime.
To the best of our knowledge VIC in such PA systems has not been addressed. In
this thesis, VIC in the context of atom-molecule coupled systems will be discussed
for the first time by us [32]. In chapter 5, this will be demonstrated in detail.
2.3
Quantum Beats
The phenomena of quantum beats are important for studying the quantum interference in the multilevel atomic or molecular systems. Quantum beats in radiation
intensity arise from coherent superposition of two long-lived excited states. Such
state superpositions and their manipulations are of considerable recent interest to
create long-lived molecular-state qubits. The possibility of using quantum beats
as a spectroscopic measure for quantum superposition was discussed as early as
in 1933 [13]. Experimentally, spectroscopic study of quantum beats started since
1960s [34]. The use of lasers to create quantum superposition and detect resulting quantum beats in fluorescence started in early 1970s [35]. Forty years ago,
Haroche, Paisner and Schawlow [36] demonstrated quantum beats in florescence
light emitted from the excited hyperfine levels of a Cs atom as a signature of quantum superposition between the excited atomic states. Since then quantum beats
in fluorescence spectroscopy have been studied in a variety of physical situations
[1, 15]. These techniques open up new possibilities for studying excited state properties, state preparation and manipulation as well as collisional and spectroscopic
aspects of ultra-cold atoms and molecules.
Let us consider a Vee-type system, as shown in Fig.2.1(a). When the excited
states |2i and |3i being coherently excited by an external source decay to the
Chapter 2. Quantum Interference
16
same final state |1i with slightly different radiation frequencies, the interference
between these two transition pathways gives rise to a periodic modulation of the
intensity of emitted radiation with a modified frequency given by the difference
of two frequencies. This phenomena is known as quantum beating. Quantum
beats are manifested as oscillations in the emitted radiation intensity Iqb from two
correlated excited states as a function of time which is given by [2, 15, 37]
Iqb (t) = γ(ρ33 (t) + ρ22 (t) + 2Re[ρ23 (t)]).
(2.27)
Here ρnn is the population of the excited state | ni and ρ23 is the coherence between
| 2i and | 3i. We consider, γ2 = γ3 = γ, where γn is the spontaneous line width of
| nith excited state.
References
[1] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University
Press, Cambridge, 1997).
[2] Z. Ficek and S. Swain, Quantum Interference and Coherence (Springer, New
York, 2007)
[3] L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge
University Press, London, 1995)
[4] M. Keyl, Phys. Rep. 369, 431 (2002).
[5] O. Kocharovskaya, Y. I. Khanin, Sov. Phys. JETP 63, 945 (1986); S. Harris,
Phys. Today 50, 36 (1997).
[6] E. Arimondo and G. Orriols, Lettere al Nuovo Cimento 17, 333 (1976); E.
Arimondo, Prog. in Opt. 35, 257 (1937); H. R. Gray, R. M. Whitley and C.
R. Stroud, Jr. Opt. Lett. 3, 218 (1978).
[7] S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989), M. O. Scully, S. Y. Zhu and
A. Gavrielides, Phys. Rev. lett. 62, 2813 (1989).
[8] I. I. Rabi, Phys. Rev. 51, 652 (1937)
[9] S. H. Autler and C. H. Townes, Phys. Rev. 100, 703 (1955).
[10] W. J. Cromie, Physicists Slow Speed of Light (The Harvard University Gazette
Retrieved) (1999).
[11] N. V. Vitanov, M. Fleischhauer, B. W. Shore, and K. Bergmann, Adv. At.
Mol. Opt. Phys. 46, 55 (2001) 3
[12] G. S. Agarwal, Quantum Optics, Springer Tracts in Modern Physics, (Springer
Verlag, Berlin, 1974).
17
Chapter 2. Quantum Interference
18
[13] G. Breit, Rev. Mod. Phys. 5, 91 (1933).
[14] U. Fano, Phys. Rev. 124, 1866 (1961)
[15] G. C. Hegerfeldt and M. B. Plenio, Phys. Rev. A 46, 373; ibid 47, 2186 (1993).
[16] D. A. Cardimona, M. G. Raymer and J. C. R. Stroud, J. Phys. B. 15, 55
(1982).
[17] P. Zhou and S. Swain, Phys. Rev. Lett. 77, 3995 (1996); Z. Ficek and S.
Swain, Phys. Rev. A 69, 023401 (2004)
[18] S. Y. Zhu, R. C. F. Chan and C. P. Lee, Phys. Rev. A 52, 710 (1995); S. Y.
Zhu and M. O. Scully, Phys. Rev. lett. 76, 388 (1996); H. Huang, S. Y. Zhu
and M. S. Zubairy, Phys. Rev. A 55, 744 (1997); H. Lee, P. Polykin, M. O.
Scully and S. Y. Zhu, Phys. Rev. A 55, 4454 (1997).
[19] G. S. Agarwal, Phys. Rev. A 55, 2457 (1997).
[20] M. V. G. Dutt, J. Cheng, B. li, X. Xu, X. Li, P. R. Berman, D. G. Steel, A.
S. Bracker, D. Gammon, S. E. Economou, R. Liu and L. J. Sham, Phys. Rev.
Lett. 94, 227403 (2005).
[21] D. J. gauthier, Y. Zhu, T. W. Mossberg, Phys. Rev. Lett. 66, 2460 (1991).
[22] B. M. Garraway, M. S. Kim and P. L. Knight Opt. Commun. 117, 560 (1995).
[23] S. Menon and G. S. Agarwal, Phys. Rev. A 57, 4014 (1998).
[24] M. A. G. Martinez et al., Phys. Rev. A 55, 4483 (1997).
[25] E. Paspalakis and P. L. Knigh, Phys. Rev. Lett. 81, 293 (1998); E. Paspalakis,
C. H. Keitel and P. L. Knight, phys. Rev. A 58, 4868 (1998).
[26] S. Das and G. S. Agarwal, Phys. Rev. A 81, 052341 (2010).
[27] G. S. Agarwal, Phys. Rev. Lett. 84, 5500 (2000).
[28] Y. Yang, J. Xu, H. Chen and S. Zhu, Phys. Rev. Lett. 100, 043601 (2008); J.
P. Xu and Y. P. Yang, Phys. Rev. A. 81, 013816 (2010).
[29] S. Evangelou, V. Yanopapas and E. Paspalakis, Phys. Rev. A 83, 023819
(2011).
[30] M. Kiffner, J. Evers and C. H. Keitel, Phys. Rev. Lett. 96, 100403 (2006).
Chapter 2. Quantum Interference
19
[31] S. Das and G. S. Agarwal, Phys. Rev. A 77, 033850 (2008).
[32] S. Das, A. Rakshit and B. Deb, Phys. Rev. A 85, 011401(R) (2012).
[33] H. R. Xia, C. Y. Ye and S. Y. Zhu, Phys. Rev. Lett. 77, 1032 (1996), L. Li,
X. Wang, J. Yang, G. Lazarov, J. Qi and A. M. Lyyra, et al., ibid 84, 4016
(2000).
[34] A Corney and G. W. Series, Proc. Phys. Soc., 83, 213 (1964); J. N. Dodd, R.
D. Kaul, and D. M. Warrington, Proc. Phys. Soc., London 84, 176 (1964).
[35] T. W. Hansch, Appl. Opt. 11, 895 (1972); W. Gornik et al., Opt. Commun.
6, 327 (1972).
[36] S. Haroche, J. A. Paisner and A. L. Schawlow, Phys. Rev. Lett. 30 948 (1973).
[37] P. Zhou and S. Swain, J. Opt. Soc. Am. B 15, 2593 (1998).
Chapter 3
Fano-Feshbach Resonances
This chapter provides an overview of atom-atom scattering at low energy along
with atom-molecule transitions in the presence of external fields. At the outset,
a brief discussion on the ultracold scattering [5] is presented in section 3.1. The
next section provides discussion on different scattering resonances [6–8]. We focus
on Fano-Feshbach resonance. Both magnetic and optical Feshbach resonance are
discussed. Feshbach resonances are used for controlling interactomic interactions
[9], production of cold molecules [10, 11] and BEC-BCS crossover [12].
3.1
Ultracold Scattering and Resonances
Scattering theory provides the theoretical frame-work to describe collisions between particles. In our case we restrict our discussion to atom-atom scattering at
low energy only. In elastic scattering kinetic energy of the system remains constant
before and after the collision and thus the system remains in its initial state. This
case may be treated as single channel scattering. In inelastic scattering the kinetic
energy after a collision is not equal to that of the initial state and the system
changes its state. This can be treated by the methods of multichannel scattering.
3.1.1
Elastic Scattering
Let us consider two atoms colliding in the presence of an interaction potential
V (r). If V (r) is spherically symmetric, the Hamiltonian is decoupled into radial
20
Chapter 3. Resonances
21
part and angular part. Using partial-wave decomposition, the effective potential
can be written as
~2 ℓ(ℓ + 1)
Vef f (r) = V (r) +
.
2µr 2
(3.1)
where, r is separation between two atoms and µ is the reduced mass of the system.
ℓ denotes the quantum number corresponding to relative motion of two atoms or
partial wave. The centrifugal barrier VCB =
~2 ℓ(ℓ+1)
2µr 2
is in general significantly
higher than kinetic energy of colliding cold atoms. So, it suppresses the collisions
with ℓ > 0 at low energy. The time independent partial-wave Schrödinger equation
describing the system is given by
2 2
~ d
−
+ Vef f (r) ψk (r) = Eψk (r)
2µ dr 2
(3.2)
where, E = ~2 k 2 /(2µ) and k denotes the wavenumber of atoms.
For r → ∞, the total wave function Ψ(r) take the form
~
Ψ(r) ∼ eik·~r + f (k, θ)
eikr
.
r
(3.3)
The first term represents the incoming plane wave and second term represents
the scattered spherical wave. Here the scattering amplitude, f (k, θ) depends on
potential V . The angle θ is between the direction of incidence and the direction
of observation. For central potentials, the scattering amplitude can be expanded
in terms of partial waves as
f (k, θ) =
X
(2ℓ + 1)fℓ (k)Pℓ (cosθ)
(3.4)
ℓ
where, Pℓ (cos θ) is Legendre polynomial and represents the angular part and the
partial wave scattering amplitude is fℓ (k) =
2iδℓ (k)
element Sℓ (k) = e
1
(Sℓ (k) − 1)
2ik
with scattering matrix
. δℓ (k) is the phase shift in outgoing wave generated due
to scattering. The total cross section σ can be written as,
σ(k) =
Z
|f (k, θ)|2dΩ =
X
ℓ
4π(2ℓ + 1)|fℓ (k)|2 =
X 4π
ℓ
k
(2l + 1) sin2 (δℓ ) (3.5)
Chapter 3. Resonances
22
or,
σ(k) =
X
σℓ (k)
ℓ
So we can say that the total cross section is the sum of partial cross sections, σℓ .
3.1.2
Scattering Length in Ultracold Gases
As discussed before, at low energy only a few partial waves contribute to collision.
For ultracold atom-atom collisions only s-wave scattering is important. At near
zero energy the s-wave radial wave function at long range goes as
u0 ∼ sin(kr + δ0 ) ≈ sin k(r − a0 )
(3.6)
with δ0 being s-wave phase shift and
1
tan(δ0 )
k→0 k
a0 = − lim
(3.7)
is the s-wave scattering length. Scattering length contains all the physical information about the scattering process at low energy. A negative (positive) scattering
length implies attractive (repulsive) interaction. The scattering length depends on
the nature of potential and the position of highest bound molecular state in it.
The energy of highest bound state Eb may be related to s-wave scattering length
a0 by Eb = −~2 /2µa20 [7, 23]. If this bound state is just below the continuum, the
scattering length is large and positive. On the other hand, if the highest bound
state is deeply bound in such a way that a new bound state may be stemmed
in if the depth of the potential is slightly increased, then the value of scattering
length is large negative. When a bound state occurs just on or near the threshold
(zero energy bound state) of the continuum, the scattering length diverges. The
divergence of scattering length signifies the resonance condition.
3.1.3
Scattering Resonances at Low Energy
Occurrence of resonances is one of the most interesting phenomena of quantum
scattering. Scattering resonances are continuations of bound states in the continuum. In other word, resonance states may be defined as the quasi-bound state
Chapter 3. Resonances
23
with small finite lifetime. At resonance the two colliding particles at a given energy
spend some time in a virtual bound-like state, and then they get separated. The
imaginary part of resonance energy i.e the width of the resonance is inversely related to the life time of quasibound resonant state. Hence the resonance states are
basically the projections of real states in complex plane. In general the phase shifts
and hence scattering lengths are slowly varying functions of the collision energy
and the strength of the applied field. However, under resonance condition phase
shift goes through a rapid variation, from zero through the value of π/2, over a
small range of collision energy. As a result, the cross section changes dramatically
in that energy range.
If a simple resonance occurs, ℓ-th partial wave cross section can also be written as
σℓ =
4π(2ℓ + 1) 2
4π(2ℓ + 1)
(Γ/2)2
sin
(δ
)
=
.
ℓ
k2
k2
(E − ER )2 + Γ2 /4
(3.8)
This is known as Breit Wigner profile. Here, ER is the resonance energy and Γ
is the full width at half maximum, provided the resonance is reasonably narrow.
From Eq. (3.8), it is clear that the resonance cross section is Lorentzian in shape.
When the system reaches resonance condition, i.e E → ER , then the cross section
attains its maximum value. It is to be noted here that the enhancement of cross
section requires the phase shift δℓ = π/2.
Scattering resonances can be broadly divided into two categories. Two atoms may
collide with each other elastically remaining in the same channel as the incoming
or incident one, or they may undergo inelastic collision and go to other channels.
So the scattering in one channel may be modified by the effect of other channels and thereby may result in resonances. First one is single channel resonance
occurring during elastic single channel scattering. Second one is multichannel resonance [13, 14]. Feshbach resonance which involves at least two coupled channel
is the best known example of multichannel resonance. Again the Feshbach resonance (FR) in cold atoms can be classified into two categories –magnetic and
optical Feshbach resonance. As discussed in previous chapter, Fano’s Theory [15]
has the same essence as Feshbach resonance. Both methods deal with a continuum interacting with one (more than one) bound state(s). Therefore both the
methods are related, though the formalism and physical contexts in which they
are discussed are different. That’s why Feshbach resonance can be referred to as
Fano-Feshbach resonance. In the following sections, different types of resonances
Chapter 3. Resonances
24
will be discussed in detail with special emphasis on magnetic Feshbach resonance
(MFR) and photoassociation as optical Feshbach resonance (OFR).
3.2
Single Channel Resonance
Single channel resonances occurs when the particles scatter back into the incident
channel. In following sub sections shape resonance and potential resonance are
discussed in short as examples of single channel resonances.
3.2.1
Shape Resonance
Shape resonance [7, 8] is one of the most prominent example of single channel
resonance. Shape resonance is a metastable bound state trapped due to the shape
of a potential barrier of interparticle potential. If the potential barrier is infinitely
high, then a bound state can easily be accommodated behind it. As the barrier is
finite, the particles may tunnel through it though the presence of barrier helps to
form a quasi-bound state, in correspondence to the energy where the real bound
state were exist. This increases the scattering cross section implying a resonance.
Shape resonance ubiquitously occurs in case of nonzero partial wave scattering.
The long range potential in the expression (3.1) modifies due to the presence of
repulsive centrifugal barrier for nonzero angular momentum. Thus effective potential of single channel supports the resonance state. The phase shift passes through
the value of π/2 as the incident collision energy gradually changes and the partial
cross section σℓ pass through maximum value 4π(2ℓ + 1)/k 2 at resonance. For
s-wave, the effective potential becomes same as the normal long range potential,
hence there is no question of shape resonance to occur.
3.2.2
Potential Resonance
Potential Resonance [6, 7] occurs in the absence of any potential energy barrier
and is therefore a purely s-wave phenomenon. This resonance occurs due to the
presence of a bound state or a virtual bound state close to the collision threshold
of single channel. For an attractive potential the scattering length is negative. As
25
Potential Energy
Chapter 3. Resonances
Internuclear Separation (r)
Figure 3.1: Schematic diagram for Feshbach resonance.
the depth of the potential increases the phase shift increases and the scattering
length becomes more and more negative. Then the phase shift passes through π/2
and a new real bound state appears in the continuum and the scattering length
diverges to negative infinity. If the depth is again increased then the scattering
length changes sign and eventually decreases to finite positive value, then goes
through zero and again through negative infinity as another new bound state is
added to the potential.
3.3
Feshbach Resonance : A Multichannel Resonance
In late fifties Herman Feshbach (1917-2000) first introduced the concept of Feshbach resonance (FR) in the field of nuclear physics. He developed a new theory
of nuclear reactions [16] to treat resonant nucleon scattering that occurs when the
energy of initial scattering is equal to that of a bound state between nucleon and
nucleus [17]. In 1990s, physicists began to use this concept to realize resonant
scattering between ultracold alkali atoms. Such resonances in collisions of alkali
Chapter 3. Resonances
26
atoms were predicted for the first time by Tiesinga et al. [18, 19] and first experimentally realized in case of
23
Na and
85
Rb in the year 1998 [20–22]. Now FR has
been observed in most of alkali atoms and many heteronuclear mixtures.
Feshbach resonance is intrinsically multichannel scattering resonance. It occurs
when a pair of cold colliding atoms is coupled to a quasi bound state in higher
lying molecular potential and the scattering state is shifted to resonance with a
bound molecular level. A Fano resonance is essentially equivalent to a Feshbach
resonance. The term Fano resonance is usually associated with the asymmetric
line shape as a function of energy where as Feshbach resonance is normally associated with magnetic field tuning of the scattering length. But the origin of both
the resonance is same: the interference between a background and a resonant
scattering process.
3.3.1
Magnetic Feshbach Resonance
Feshbach Resonance, tuned by magnetic field, is called magnetic Feshbach resonance (MFR) [10, 11]. For demonstration, let us consider a simple two channel
model as depicted in Fig.3.1. Let us consider two molecular potentials Vbg (r) and
Vc (r) in different hyperfine states. At large separations Vbg (r) connects two free
colliding atoms. It is known as incident or open channel. In this channel motion is
unbound and the threshold energy of this channel is lower than the given energy
of the system. The wave function associated with this channel is continuum or
scattering wave function. On the other hand Vc (r) supports the molecular bound
states. The scattering energy of the two incident atoms lies below the dissociation
threshold of this channel.That is why it is called closed channel. For large internuclear separations, Vc (r) connects to a continuum that corresponds to atom pair
with higher internal states than that of Vbg (r) i.e atoms in higher hyperfine states
than that of Vbg (r). The origin of Feshbach resonance comes from the hyperfine
interaction Ehf , which mixes the singlet or triplet states . The hyperfine energy
Ehf is obtained by summing the hyperfine energy of individual atom. Hyperfine
enrgy of a single atom in the absence of magnetic field is given by
atom
Ehf
=
ahf
[fa (fa + 1) − sa (sa + 1) − ia (ia + 1)] .
~2
(3.9)
Chapter 3. Resonances
27
Now the highest lying vibrational level supported by the closed channel may lie
below or above the continuum of open channel A bound (or virtual )state of closed
channel just below (or above) the continuum of the open channel gives rise to a
large positive (or negative) scattering length. The molecular state has different
magnetic moment from that of the two free atoms. So both the potentials can
be tuned by applying an external magnetic field, provided the atoms must be
paramagnetic. Hence by varying applied magnetic field, a situation may appear
when the continuum state of the open channel coincides energetically with the
bound state of closed channel resulting in Feshbach resonance and the scattering
length then diverges.
Due to the coupling between the two channels, the effective collisional potential
gets modified. Zeeman effect allows mixing of the potential Vbg (R) and Vc (R)
changing the relative energies of the internal states. If the potential gets modified
then the scattering wave function is also modified as scattering wave function
also depends on the shape of the scattering potential. As a consequence the
scattering phase shift δ changes which in tern results in a varying scattering length
a. The varying scattering length can also be correlated to the varying position
of last bound state, as the binding energy EB of the last bound state is given by
EB = ~2 /(2µa2 ), where µ is the reduced mass of the system [7, 23].
3.3.2
Scattering Length and Feshbach Resonance
Feshbach resonance is the most appreciated tool for the tuning of scattering length
via external magnetic field in ultracold atomic collision. Near a Feshbach resonance
the scattering length a varies as [9]
a(B) = abg 1 −
∆B
B − B0
(3.10)
where, abg is the background scattering length at far-off resonant condition, ∆B
is the resonance width and B0 is the resonant magnetic field. ∆B depends on the
coupling between the two channels and the shift of two potentials as a function
of varying mangnetic field B. The entrance channel’s last bound vibrational level
determines abg .
Feshbach molecules can be formed from two free cold atoms by sweeping the
applied magnetic field adiabatically across a Feshbach resonance. In 1999, Abeelen
Chapter 3. Resonances
28
et al. [24] pointed out that it is possible to create ultracold molecules near FR. In
2002, Donley et al. were first to observe a signature of molecules created near FR
[25]. The molecules formed by FR tend to be in highly excited vibrational states,
though they are rotationally and translationally cold.
Ultracold molecules [26, 27] can be created from both bosonic [28–30] and fermionic
[2, 3] atoms. Boson-boson and boson-fermion molecules are very short lived. But
for the fermionic case, the molecules are quite long lived, because the inelastic
atom-molecule or molecule-molecule scattering are suppressed due to Pauli blocking [31, 32]. Such long lived molecules may allow us to study of ro-vibrational
states, relaxation processes, low temperature chemical reactivity, BEC-BCS crossover
and s-wave superfluidity etc.
3.4
Photoassociation and Optical Feshbach Resonance
Photoassociation (PA) [37] (depicted in Fig.3.2) is one of the most admired continuumbound process for the formation of ultracold molecule. The idea of using ultracold
and trapped atoms to make cold molecule through photoassociation was first suggested by Thorsheim et al. in 1987 [34]. PA is a process of resonant excitation in
which translationally cold colliding atom pair in the presence of a laser of appropriate frequency is transferred to a molecular bound level via free-bound electric
dipole transition with the aid of a single photon. The molecule is formed in a
ro-vibrational level near the threshold of excited electronic state and hence is
long-ranged as compared to normal diatomic molecule. Binding energy of such
loosely bound molecule lies in the range of sub mK to hundreds of mK. Therefore the initial temperature of the atomic gas should be cold enough. As the
molecule is being formed from initial translationally cold atom pair, hence the
molecule is translationally cold. Now at low energy, only collisions of the lowest
few partial waves are allowed. Hence through PA the excited molecules is formed
in low rotational levels. Hence the photoassociated molecule is translationally and
rotationally cold.
29
Potential Energy
Chapter 3. Resonances
Internuclear Separation (r)
Figure 3.2: Schematic diagram showing photoassociation.
The stimulated linewidth due to PA between the scattering state | ψ(E, ℓ)i and
the excited bound state | φ(v, J)i is given by the expression
Γ=
πI
|hφ(v, J) | D(r)) | ψ(E, ℓ)i|2
ǫ0 c
(3.11)
where, D(r) is the transition dipole moment, I is the intensity of applied laser and
c is the velocity of light. The rate of photoassociation can be written as [35]
KP A
Z ∞
1 X
(2ℓ + 1)
|SP A |2 e−E/kB T dE
=
hQT ℓ
0
(3.12)
where, kB is the Boltzmann constant, QT = (2πµkB T /h2 )3/2 is the translational
partition function with T being temperature of atomic gas. SP A , the scattering
S-matrix element, is given by
|SP A |2 =
γΓ/4
.
(E + ~ωℓ − Eb )2 + (Γ + γ)2
(3.13)
Here, ωℓ is frequency of applied laser, Eb is the bound state energy and γ is the
spontaneous line width of the excited state.
Chapter 3. Resonances
30
Though the molecule formed by the single photon PA is translationally and rotationally cold, but vibrationally it is very unstable. The excited molecule may
spontaneously decay to some ground bound state or to the ground continuum.
Now this photoassociated molecule may be driven by using a second laser of appropriate frequency to a low vibrational bound state of the ground electronic
configuration in stimulated manner. Thus in comparison with one-photon PA and
the problems with high vibrational quantum numbers, two-color photoassociation
is a big improvement.
PA can also be used to tune atom-atom interaction of cold atoms in the same
way as MFR. As PA uses optical field to couple the colliding atoms to excited
bound state, it can be referred as optical Feshbach resonance (OFR). In MFR, the
energy difference between the open and closed channels is controlled by external
magnetic field. The necessary criteria for this is that both channels must have
different magnetic moments and ground state should be degenerate in the absence
of any field. On the other hand, OFR tunes the atom-atom interaction by coupling
two-atom scattering state to excited bound state via photoassociation. Hence
OFR can be observed for all sorts of atoms where as MFR is observed for atoms
having permanent magnetic moment. Optical fields can be switched and controlled
much faster and can be focused much better than magnetic fields. OFR provides
a new way of research for alkaline earth metal atoms and similar systems with
nondegenerate ground state. It may be used to control the scattering wave function
to modify the PA rate and modulate the thermalization and loss rate.
The use of light fields to modify the atom-atom interaction and the scattering
length in atomic collisions has been first proposed by Fedichev et al. [36]. and
has been further explored by Bohn and Julienne [37, 38] using quantum defect approach. Optical s-wave scattering resonances have been first observed in sodium
vapour by Fatemi et al. [39] using one-color PA spectroscopy. Later several experiments have been designed to observe optical Feshbach resonance (OFR) in
one-color and two-color scheme [40–44]. To modify the scattering length by OFR,
in one photon PA scheme, laser is tuned close to PA resonance which couples
ground scattering state and excited molecular state. The optical field dresses the
excited bound state and by modulating the PA laser frequency the dressed state
can be tuned below, at or above the collisional threshold which results in dramatic
change in s-wave scattering length. But it also leads to atomic loss due to spontaneous decay via bound state. Hence the s-wave scattering length in the presence
Chapter 3. Resonances
31
of light field can be expressed as α − iβ. Here α denotes the modified scattering
length in the presence of OFR and the imaginary part β describes the inelastic
loss rate due to two body collision. For the weak coupling limit i.e for Γ ≤ γ, α
and β are given by [37, 38]
1
Γ
Γ∆
∆
α = abg −
= abg 1 −
2k ∆2 + (γ/2)2
2kabg (∆2 + (γ/2)2 )
1
Γγ
β =
2
k ∆ + (γ/2)2
(3.14)
(3.15)
where, ∆ is the laser detuning and abg is the non-resonant background scattering
length when there is no optical coupling.
Till now, it is found that OFR is not as efficient as MFR, as the excited photoassociated molecule may eventually decay leading to drastic loss of atoms from trap. If
an efficient all optical method could be devised, it would prove itself advantageous
over MFR, in particular to manipulate p or d partial wave interaction [45–47].
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Chapter 4
Quantum Interference in
Photoassociation in the Presence
of Feshbach Resonance
In previous two chapters, we have discussed about quantum interference effects
with an emphasis on Fano effect, magnetic Feshbach resonance and photoassociation as optical Feshbach resonance. This chapter is devoted for the discussion of
quantum interference effects in the context of photoassociation (PA) in the presence of magnetic Feshbach resonance (MFR) in the light of celebrated Fano effect.
Here we examine the effects of interference on PA spectrum and our aim is to
obtain large light shift along with an extremely narrow line width which will be
effective for the manipulation of atom-atom interactions and cold collisions.
The chapter is designed as follows: in section 4.1, we present the motivation and
perspective of present work. Formulation of the problem and its solution are
discussed in section 4.2. In the next section results and their interpretations are
discussed taking 7 Li as an example.
4.1
Perspective of The Work
In the previous chapter, we have seen the manipulation of atom-atom interaction
at low energy can be achieved by either magnetic Feshbach resonance (MFR)
or optical Feshbach resonance (OFR). Now it would be interesting to investigate
35
Chapter 4. PA in the presence of FR
36
what happens if PA occurs in the presence of MFR. Several experimental studies
on PA in the presence of MFR [1–4]) have been carried out over last 16 years. In
recent years, it has attracted a lot of interests both experimentally [5–10] as well as
theoretically [11–16] revealing significant effects of MFR on PA and cold collision
properties. According to of Franck Condon principle a continuum-bound or boundbound transition is most probable when the prominent anti-nodes of initial and
final states are located at a comparable internuclear separation. Probability for
such transition would be least when the anti-node of either of the states lies at
a separation close to the node of the other. Hence, in the presence of Feshbach
resonace, enhancement [7] and suppression in PA spectral intensity profile can take
place due to quantum interference between PA and Feshbach resonances resulting
in Fano-type [23] asymmetric spectral profile. Such quantum interferences can be
used for coherent control of cold atom-molecule conversion and ultracold collisions.
In this chapter we explore the possibility of suppression of power broadening in
strong-coupling PA by manipulation of continuum-bound coherences with a Feshbach resonance. We consider that two optically coupled bound states interacts
with a common continuum as shown in Fig.4.1. We solve the problem following
Fano’s Theory where the system is exactly diagonalised leading to a ‘dressed’ continuum state. We demonstrate that by tuning the magnetic field close to Fano
minimum where excitation probability vanishes, it is possible to obtain line narrowing in the PA spectrum with large shifts at high laser intensities which may be
useful in efficient tuning of elastic scattering length by optical means.
4.2
Formulation of the Problem and Solution
To model PA occurring in the presence of MFR, we first consider a simple three
channel model as depicted in Fig.4.1 [9]. It consists of two asymptotic hyperfine channels of which one is closed channel |1i having higher threshold energy
than the asymptotic collision energy and other one is open channel |2i having
lower threshold energy than that. In the presence of appropriate magnetic field
strength, scattering state associated with open channel can couple to a quasibound state supported by closed channel resulting in Feshbach molecular (FM)
state. So the presence of MFR modifies the continuum states of open channel
and vice versa. As the applied magnetic field is varied, this quasibound state can
move across the collision energy. Our model also consists a third channel |3i which
Chapter 4. PA in the presence of FR
37
Vex(r)
Excited bound state |3 〉
(S+P)
0
|3〉
LPA
|2〉
Bound−bound
|E〉
Continuum−bound
LPA
Vg(r)
Bound state |2〉
Closed channel 2
(S+S)
0
Continuum |E〉
Open channel 1
Inter−channel coupling V
r
Figure 4.1: A schematic diagram showing the coupling between the bound
state (magenta) | 2i and and continuum (green) | Ei with the excited bound
state (blue) | 3i via same laser LP A ( magenta and green double-arrow vertical
lines). Red dashed line indicates the hyperfine coupling between the open and
closed channel.
corresponds to the excited photoassociated molecular (PM) state in the excited
electronic state. Now when Photoassociation laser of appropriate frequency is applied, bound-bound dipole transition between FM state and PA state may occur
along with normal PA transition between open channel continuum and PM state.
We now assume that the energy spacing in FM states is much larger than the line
width of the PA laser, so PA laser effectively couples only one FM bound state
to PM state. We further assume that rotational spacing of PM states is much
larger than PA laser line width so that only one rotational level J of a particular
vibrational state v is coupled by the PA laser.
So, now there are three competing pathways. Two of these are continuum-bound
types and one is bound-bound dipole transition. So quantum interference may
naturally arise between any two of these three competing transition pathways.
Chapter 4. PA in the presence of FR
38
The dressed state of a system where PA of two colliding atoms is taking place in
the presence of MFR can be written as:
"
#
X
1
| ΨE i =
χ | 1i +
Φi (r) | ii
r
i=2,3
(4.1)
where, r is the relative coordinate of the two atoms. The continuum state has the
R
form χ = dE ′ bE ′ ψE ′ (r), where ψE ′ is an energy normalised scattering state of
collision energy E ′ and bE ′ is density of states of the unperturbed continuum. E
is the eigen energy of the dressed state, Φi s (i = 2, 3) are the wave functions of
perturbed bound states. The Hamiltonian for the system can be written as
H = Hkin + Helec + Hhf s + HB + HL
(4.2)
where,Hkin corresponds to total kinetic energy and Helec depends on electronic coordinate of two atoms, Hhf s is the hyperfine interaction term, HB and HL represent
respectively the magnetic and laser field interactions between atomic and molecular states. From the time-independent Schrödinger equation HΨE = EΨE under
Born-Oppenheimer approximation we obtain the coupled differential equations:
~2 d 2
−
+ BJ (r) Φ3 + [Ve (r) − ~δ − E − i~γ/2] Φ3
2µ dr 2
= −Ω1 χ − Ω2 Φ2 ,
(4.3)
2 2
~ d
−
+ V2 (r) − E Φ2 = −Ω∗2 Φ3 − V χ,
2µ dr 2
(4.4)
~2 d 2
+ V1 (r) − E χ = −Ω∗1 Φ3 − V ∗ Φ2 .
−
2
2µ dr
(4.5)
Here BJ = ~2 J(J + 1)/2µr 2 is the rotational term of the excited state and δ =
ωL − ωA is the laser-atom detuning. Vi (i = 1, 2) are the potentials including
hyperfine and Zeeman terms and Ve is the excited state molecular potential. The
hyperfine spin coupling between the cha5nnels 1 and 2 is denoted by V (r). Ω1 (r)
and Ω2 (r) represent the molecular Rabi couplings of the excited state | 3i with
the ground states | 1i and | 2i, respectively.
In the absence of couplings, let the unperturbed bound states be denoted by φ3 (r)
Chapter 4. PA in the presence of FR
39
and φ2 (r) with bound state energies E3 and E2 , respectively and the unperturbed
continuum states by ψE ′ with asymptotic collision energy E ′ . With the use of these
unperturbed solutions, we construct three Green’s functions GE (r, r ′ ), G2 (r, r ′ )
and G3 (r, r ′ ) which correspond to channels 1, 2 and 3, repectively. The continuum Green’s function GE (r, r ′) can be written as GE (r, r ′ ) = −πψEreg (r< )ψE+ (r> ),
where r<(>) implies either r or r ′ whichever is smaller (greater) than the other.
Here ψE+ (r) = ψEirr + iψEreg where ψEreg and ψEirr represent regular and irregular scattering functions, respectively. Asymptotically, ψE0,reg (r) ∼ j0 cos η0 − n0 sin η0 and
ψE0,irr (r) ∼ −(n0 cos η0 +j0 sin η0 ), where j0 and n0 are the spherical Bessel and Neu-
mann functions for partial wave ℓ = 0 (s-wave) and η0 is the s-wave phase shift in
the absence of laser and magnetic field couplings. The other two Green’s functions
correspond to bound states are of the form G3 (r, r ′ ) = − ~δ+E−E13 +i~γ/2 φ3 (r)φ3 (r ′ )
1
and G2 (r, r ′ ) = − E−E
φ2 (r)φ2 (r ′ ). Now using these above Green’s functions, we
2
can write down eqs (2) and (3) in the form
Φ3 = −
Φ2 = −
Z
Z
dr ′ [Ω1 (r ′ )χ(r ′ ) + Ω2 (r ′ )Φ2 (r ′ )] G3 (r, r ′ )
dr ′ [V ∗ (r ′ )χ(r ′ ) + Ω2 (r ′ )Φ3 (r ′)] G2 (r, r ′ ).
(4.6)
These equations may be also expressed as Φ3 = AP A φ3 and Φ2 = ACC φ2 , where
AP A and ACC is given by following expressions
dr ′ [Ω1 (r ′ )χ(r ′ ) + Ω2 (r ′ )Φ2 (r ′ )] φ3
~δ + E − E3 + i~γ/2
R ∗ ′
[V (r )χ(r ′ ) + Ω∗2 (r ′ )Φ3 (r ′ )] φ2 (r ′)
= −
E − E2
R ′
′
∗ ′
dr φ2 (r )V (r )χ(r ′ ) + Ω∗32 (r ′ )AP A )
= −
E − E2
AP A = −
ACC
R
(4.7)
(4.8)
dr ′ φ3 (r ′ )Ω2 (r ′ )φ2 (r ′ ) is bound-bound Rabi coupling. The continuum
R
state can be written as χ = dr ′ bE ′ ψE ′ (r). Thus we obtain
Here, Ω32 =
−
R
~2 d 2
ψE ′ (r) + [V1 (r) − E]ψE ′ (r) = −Ω∗1 (r)ÃP A φ3 (r) − ÃCC V ∗ (r)
2µ dr 2
(V ∗ + ÃP A Ω∗32 )
V (r)φ2(r)
= −Ω∗1 (r)ÃP A φ3 (r) − 2E
E − E2
(4.9)
where.
ÃP A =
Ω3E (E − E2 ) + Ω32 V2E
.
D(E − E2 ) − |Ω32 |2
(4.10)
Chapter 4. PA in the presence of FR
40
6
6
2
q = 1.0
q = 0.5
0
−5
0
4
q = −1.0
q = − 0.5
2
0
−5
5
0
−1
3
q = 0.5
∆E/ΓPA
∆E/ΓPA
(kares)
0
−1
q = 1.0
−2
(c)
q = 2.0
−3
−5
5
−1
(kares)
1
q = − 2.0
(b)
PA
4
(a)
Γ/Γ
Γ/Γ
PA
q = 2.0
0
2
1
(d)
q = − 0.5
0
q = −1.0
−1
−5
5
q = − 2.0
0
−1
5
−1
(kares)
(kares)
Figure 4.2: Subplots (a) and (b) show Γ/ΓP A Vs (kares )−1 and subplots
(c) and (d) exihibit ∆E/ΓP A Vs (kares )−1 . Plots are for different values q as
2 k2
indicated in the figure. k is defined as ~2m
= KB T
Here, D = (~δ + E − E3 + i~γ/2), V2E =
R
drφ3 (r)Ω1 (r)ψE ′ (r).
R
drφ2 (r)V (r)ψE ′ (r) and Ω3E =
Now Eq. (4.9) can be solved by constructing the Green’s function GE (r, r ′ ) with
the scattering solutions of homogeneous part. This Green’s function as discussed
earlier can be written as:
GE (r, r ′ ) = −π ψE0,reg (r)ψE0,irr (r ′ ) + iψE0,reg (r)ψE0,reg (r ′ ) (r ′ ≥ r)
GE (r, r ′ ) = −π ψE0,reg (r ′ )ψE0,irr (r) + iψE0,reg (r)ψE0,reg (r ′ ) (r ′ ≤ r).
(4.11)
Now, we can express the solution of Eq. (4.9) in the following form
ψE ′ =
+
exp(iη0 )ψE0.reg
′
+
Z
h
dr ′ GE (r, r ′ ) Ω∗1 (r ′ )ÃP A φ3 (r ′ )
#
∗
V2E
+ ÃP A Ω∗32
V (r ′ )φ2 (r ′ ) .
(E − E2 )
(4.12)
Chapter 4. PA in the presence of FR
41
Substituing ψE ′ in the expression of V2E , we obtain
V2E =
Ω32
0 2
0
0
P A V32
∆E2 − i πAE−E
|V2E
|
exp(iη0 )V2E
+ V32 AP A − iπV2E
Ω3E AP A + AP A E−E
2
2
0 2
(E − E2 − ∆E2 + iπ|V2E
| )/(E − E2 )
.
Here, V32 =
RR
(4.13)
dr ′ drφ2(r)V ∗ (r)Re[GE (r ′ , r)]V (r ′ )φ2 (r ′ ) represents continuum-
mediated an effective magneto-optical coupling between photoassociated and FeshRR ′
bach molecular bound states. ∆E2 =
dr drφ2(r)V ∗ (r)Re[GE (r ′ , r)]V (r ′ )φ2 (r ′ )
is the energy shift of the Feshbach molecular state due to its coupling with the con-
tinuum. The Stark energy shift due to laser coupling of Photoassociated molecular
RR ′
′
(PM) state with the continuum is given by ∆E3 =
dr drφ3 (r)Ω1 (r)Re[GE (r
, r)]
q
R
0
0
2
Ω∗1 (r ′ )φ3 (r ′ ). V2E
and Ω03E are defined as V2E
= drφ3(r)V (r)ψE0,reg (r) = hΓ
2π
q
R
hΓ3
and Ω03E = drφ3 (r)Ω1 (r)ψE0,reg (r) =
. Here Γ2 and Γ3 are the Feshbach
2π
resonance linewidth and stimulated PA linewidth, respectively.
Substituting Eqs. (4.13) and (4.12) in Eq. (4.10), we finally obtain
ÃP A =
0
exp(iη0 ) [Ω03E + GV2E
]
.
(4.14)
[Ω32 + V32 − iπV2E Ω3E ]
,
[E − E2 − ∆E2 + iπ|V2E |2 ]
(4.15)
D−
|Ω32 |2
E−E2
− (Bp + GBf )
Here,
G=
[−V32 + iπV2E Ω3E ]
(E − E2 )
(4.16)
[−∆E2 + iπ|V2E |2 ]
.
(E − E2 )
(4.17)
0
2
Bp = Eshif
t − iπ|Ω3E | − Ω32
and
Bf = V32 − iπV2E Ω3E − Ω32
Now, let us introduce a low-energy dimensionless interaction parameter β(k) which
is defined as β = (E − E2 − ∆E2 )/(Γ2 /2). It may be described as shifted energy of
the Feshbach molecule and related to the Feshbach resonance phase shift ηres (k)
as β(k) = − cot ηres (k) ∼ (kares )−1 + 21 re k. re is effective range and is related
to Feshbach resonance linewidth. ares related to the applied magnetic field B by
ares = −(abg ∆)/(B − B0 ) [19], where abg is the background scattering length,
Chapter 4. PA in the presence of FR
42
Γ/Γ
PA
30
weak
Γ
Γ
20
10
0
−30
−20
−10
0
10
20
30
−1
(kares)
3
weak
Γ/ΓPA
Γ
2
Γ
1
0
−4
−3
−2
−1
0
(ka
1
2
3
4
−1
)
res
Figure 4.3: Solid lines represent the linewidth Γ (in unit of ΓP A ) given by
the expression (4.20) as a function of (kares )−1 for q = −5.0 (upper panel) and
q = −0.1 (lower panel). Dotted lines represent the low coupling expression of
(4.27) for C1 = 6 and C2 = 1 (upper panel); and C1 = 0.1 and C2 = 0 (lower
panel). The values of C1 and C2 are so chosen such that in the limit ares → 0,
Γ → Γweak .
B0 is the resonance magnetic field, ∆ is resonance width. So, we can say that
β(k) ∼ −(B − B0 )/(kabg ∆). Let us also introduce another parameter analogous
to Fano asymmetry parameter q which is defined as
q=
Ω32 + V32
.
πΩ3E Ω2E
(4.18)
It is to be noted that q is independent of the laser intensity. In the limit k → 0,
both Ω32 and V32 become energy-independent while Ω3E V2E ∼ k. Thus at low
energy, q ∼ 1/k.
Chapter 4. PA in the presence of FR
43
Now substituting the expressions of G, Bf and Bp in Eq.(4.14) and using the
expressions β(k) and q, we finally obtain the expression of ÃP A as given below,
exp (iη0 )πΩ3E (β(k) + q)/(β(k) + i)
i
ÃP A = h
(q−i)2
0
~δ + E − E3 + Eshif
+
i~(γ
+
Γ
)/2
−
~Γ
3
3 (β(k)+i)
t
=
exp (iη0 )πΩ3E (β(k) + q)
.
(β(k) + i) [∆p + i~(γ + Γ3 )/2] − ~Γ3 (q − i)2
(4.19)
0
0
Here, ∆p = ~δ + E − (E3 + Eshif
t ), Eshif t is the energy shift in the absence of FeshRR ′
0
bach resonance and given by the expression Eshif
dr drφ3 (r)Ω∗1 (r)Re[GE (r ′ , r)]
t =
Ω1 (r ′ )φ3 (r ′ ).
4.3
4.3.1
Results and Discussions
Analytical Results
Since the rate of Photoassociation (PA) is proportional to |AP A |2 , from Eq. (4.19),
we obtain PA linewidth in the presence of MFR in the form of
Γ = f (q, β)Γ3 =
(β(k) + q)2
Γ3 .
β(k)2 + 1
(4.20)
Here Γ is the stimulated linewidth without MFR. From above equation, it is clear
that Γ depends on a nonlinear function f (q, β) of q and β. Note that, when the
magnetic field is far off resonant, i.e at β → ∞, Γ → Γ3 . On the other hand, when
β = −q, then Γ = 0 and AP A = 0. So by tuning β(k) close to −q, Γ may be made
arbtrarily small. Γ3 is proportional to laser intensity I. So it is possible to suppres
power broadening at increased laser intensities by tuning the magnetic field close
to Bmin where the PA rate is minimum. This point is given by β = −q. From the
Eq. (4.19), we also obtain the extra shift due to MFR in the form of
1 (q 2 − 1)β(k) − 2q
Γ3 .
∆E =
2
β(k)2 + 1
(4.21)
which again goes to zero as β → ∞. Again, for β ≃ −q, ∆E ≃ −qΓ3 /2, which is
proportional to I. Hence we can conclude that when line broadening is suppressed
0
by the tunability of MFR, total shift Etot = Eshif
t + ∆E still remains proportional
0
to I. Normally the shift Eshif
t is negative in the low energy regime. However, the
Chapter 4. PA in the presence of FR
44
total shift Etot in the presence of MFR can be positive or negative depending on
the values of q. So it may become possible to obtain arbitrarily small linewidth
with large shift at high intensities by appropriate tuning the magnetic field.
Next, we discuss the weak-coupling limit of (4.20) and (4.21) when laser intensity
is low. For this we first find the dressed continuum state in the limit Γ3 → 0.
From Eq. (4.8), we can write the expression of ÃCC as
ÃCC =
ÃP A Ω32 +
R
drψE (r)V ∗ (r)φ2 (r)
E − E2
(4.22)
When, Γ3 → 0, ÃP A → 0 and ∆E = 0, then ÃCC may be expressed as
ÃCC =
R
drψE (r)V ∗ (r)φ2 (r)
E − E2
(4.23)
Now substituting ψE (r) by Eq. (4.12) and considering ∆E = 0 as Γ3 → 0, ÃCC
may be written as
r
ÃCC = −
2
exp i(η0 + ηres ) sin ηres
πΓ2
(4.24)
and Eq. (4.12), at weak coupling limit, reduces to
ψE (r) = exp i(η0 + ηres )[ψE0,reg cos ηres + ψE0,irr sin ηres ]
(4.25)
The dressed continuum state in Eq. (4.1) reduces in the limit of Γ3 → 0 to
1
| ΨE i0 = [ÃCC φ2 (r) | 2i +
r
Z
bE ′ ψE ′ (r)dE ′ | 1i]
(4.26)
Taking bE ′ = δ(E − E ′ ), the stimulated linewidth Γweak in the weak coupling limit
is given by the Fermi golden rule expression
Γ
weak
2π
|
=
~
Z
rφvJ h3 | Ω1 (r) | ΨE i0 dr|2
r
2
2π 0 2 1
Ω32
Ωirr
3E
=
|Ω3E | 1 + −
sin ηres + 0 tan ηres 0
~
πΓ2 Ω3E
Ω3E
= Γ3 |1 + C1 tan ηres + C2 sin ηres |2
(4.27)
q
0,irr
2
0
)Ω32 /Ω03E . Here Ωirr
i. The
So, C1 = Ωirr
/Ω
and
C
=
(−
2
3E = h3 | Ω1 | ψE
3E
3E
πΓr
expression (4.27) is in agreement with (6) of [6]. When ηres → π/2, Γweak diverges
Chapter 4. PA in the presence of FR
45
−1
(cm sec )
0.2
0.1
0.05
K
PA
× 10
10
3
0.15
0
−100
0
δ
p
100
200
300
(MHz)
Figure 4.4: KP A in cm3 sec−1 Vs. detuning δp in MHz. Each pair of dashed
and solid curves are obtained for ΓP A = 1.0 MHz (left pair), ΓP A = 10.0 MHz
(middle pair) and ΓP A = 25.0 MHz (right pair) for the fixed q = −6.36. For
solid curves, magnetic fields are B = 705.00 G (left), B = 708.56 G (middle)
and B = 709.12 G (right). For dash-dotted curves, these are B = 713.19 G
(left), B = 711.09 G (middle) and B = 710.68 G (right). The magnetic fields
are so chosen such that the linewidth Γ remains fixed at 0.04 MHz.
and hence Eq. (4.27) is not valid near ηres = π/2. In other words, Eq. (4.27) is
not applicable close to Feshbach resonance. Finally we prove that line narrowing
in one-photon PA is not possible in the absence of coupling between the open and
closed channel. we analyze PA rate. It describes the loss of atoms due to the
decay of the excited state into decay channels. The expression is given by
KP A
1
=
hQT
Z
dEk
~2 γΓ exp(−Ek /KB T )
(Ek − ∆Es + ~δp )2 + ~(γ + Γ)2 /4
(4.28)
0
as a function of B and the detuning parameter δp = δ − (E3 + Eshif
t − Eth )/~,
where Eth is the threshold of the open channel. Here QT = (2πµKB T /h2 )(3/2) ,
Ek = E − Eth and KB is the Boltzmann constant. Now it can be noted that
the PA laser can be tuned either near continuum-bound frequency or near bound-
bound transition frequency. In first case, ~δp ≃ Ek and in second case −~δp =
Chapter 4. PA in the presence of FR
46
(E2 + ∆E2 − Eth ). In the limit Γ2 → 0,
~2 γΓP A
KP A ≃ n
0
(E − E3 − Eshif
t ) + ~δ −
~2 Ω232
E−E2
o2
(4.29)
+ i~2 (γ + ΓP A )2 /4
which is in agreement with the expression of |S1g |2 of [20], if we identify ∆1 and
∆2 of [20] with −~δ and E2 , respectively. In our case there is only one laser
coupling between the continuum and the excited bound state and also between
the two bound states. It is clear from the above expression that in the absence of
coupling between open and closed channels the narrowing of PA linewidth is not
be possible.
before going to numerical demonstration, finally, we discuss one important consequence of line narrowing. The asymptotic form of the perturbed scattering state
is given by
ΨE (r) = exp(iη0 )ψEreg′ (r) + T exp(−iη0 )ψE+′ (r).
(4.30)
From it, we can deduce the scattering T -matrix element in the following form
T = T0 + exp(2iη0 )Tf + exp[2i(η0 + ηres )]Tpf .
(4.31)
Here, T0 , back ground T -matrix element, and Tf , T -matrix element in the presence of magnetic field only, are expressed as follows: T0 = exp(iη0 ) sin η0 and
Tf = −1/(β + i). Tpf , T -matrix element due to photoassociation in the Fes-
hbach resonance is given by Tpf = −(Γ/2)/[(Ek + ~δp − ∆Es ) + i(γ + Γ)/2].
The terms T0 and Tf are independent of laser field while Tpf explicitly depends
on β, q and laser intensity. The amplitude of closed-channel bound state is
R
ÃCC = 2[ÃP A Ω∗32 + drψE (r)Ω1 (r)φ2(r)]/[Γ2 β + 2∆E2 ], where ∆Er is the shift of
the closed-channel bound state due to its coupling with the open channel. Note
that in the limit β → ±∞, ÃCC → 0, Γ → Γp . In the limit Γp → 0, ÃP A ≃ 0 and
the problem reduces to that of two coupled-channel Feshbach resonance [7]. In
the absence of spontaneous emission (γ = 0), we obtain T = [exp(2iηtot ) − 1]/2i,
where ηtot = η0 + ηres + ηpf with ηpf = − tan−1
Γ/2
.
Ek +~δp −∆E
It then follows that
the scattering matrix S = 1 + 2iT is unitary. The s-wave scattering amplitude
(0)
(f )
is f0 = T /k = f0 + exp(2iη0 )f0
(f )
f0
= Tf /k and
(pf )
f0
(q)
(0)
+ exp[2i(η0 + ηres )]f0 , where f0
= T0 /k,
= Tpf /k. In the presence of spontaneous emission, the real
Chapter 4. PA in the presence of FR
47
(nm)
720
20
10
700
710
720
B (Gauss)
710
∆ E (MHz)
B (Gauss)
a
res
715
30
705
150
100
50
0
700
0
10
0
10
20
ΓPA (MHz)
20
30
30
40
40
Figure 4.5: The locus of B and ΓP A for which the linewidth Γ remains fixed
at 0.04 MHz. The lower inset shows the variation of ∆E (in MHz) against ΓP A
(in MHz) and the upper inset exhibits the variation ares (in nm) against B (in
Gauss) at the fixed Γ = 0.04 MHz.
(pf )
and imaginary part of f0
are −Γ × (Ek + ~δp − ∆Es ) and Γ(Γ + γ), respec-
tively. In the limit k → 0 and δp → 0; and if Γ is suppressed and the shift ∆E is
enhanced such that Γ << γ and |∆E| >> γ, the elastic scattering will dominate
over the inelastic part. This is what exactly required for efficient manipulation of
scattering length by optical Feshbach resonance [21].
4.3.2
Numerical Results and Discussion
For numerical illustration, we consider a model system of two ground-state (S1/2 )
7
Li atoms undergoing PA from the ground molecular configuration 3 Σ+
u to the vi-
brational state v = 83 of the excited molecular configuration 1 3 Σ+
g which correlates
asymptotically to 2S1/2 + 2P1/2 free atoms [22, 23]. All the relevant parameters
0
γ, Eshif
t , ∆, abg and Γr are estimated from [22], [24] and [7]. In Fig.4.2, we have
plotted Γ/ΓP A and ∆E/ΓP A against 1/kares for positive and negative q values.
The maximum and minimum values of linewidth would be observed for β = 1/q
and β = −q, respectively. The magnitude of the change in shift due to PA in the
Chapter 4. PA in the presence of FR
48
presence of MFR is significant near β = −q. Fig.4.3 clearly shows that the stim-
ulated linewidth in the weak coupling limit, represented by dashed lines deviate
appreciably from nonperturbative results as shown by solid lines. The deviations
are the most prominent in the region (kares )−1 ≃ 0 (ηres ≃ π/2). Furthermore, for
lower q values these two results deviate most significantly. Fig.4.4 illustrates how
to suppress power-broadening by the appropriate tuning of magnetic field near
Bmin and thereby to keep the total linewidth close to the natural linewidth. There
are two values of β(B) and correspondingly two values of ares where the linewidth
Γ can be kept fixed at a small value at an increased laser intensity. In Fig.4.5, we
show how to vary ΓP A (or laser intensity) and the magnetic field in order to keep
Γ fixed at 0.04 MHz which is much smaller than the natural linewidth γ although
ΓP A can be many orders of magnitude higher than γ. The lower inset in Fig.4.5
shows that the extra shift ∆E can exceed γ by many orders of magnitude while
power-broadening is suppressed.
4.4
Conclusions
In this chapter, we have demonstrated that linewidth of photoassociation spectrum
can be narrowed down close to the natural linewidth by tuning the magnetic field
near Feshbach resonance along with a large shift. This enhencement of the life time
of excited molecular state may be beneficial for population transfer from ground
state collisional continuum to ground molecular state by two-photon Raman-type
PA. Furthermore, narrow linewidth with large shift will be useful for efficient
manipulation of scattering length by optical Feshbach resonance [21]. This will
be particularly important for altering scattering amplitude of higher partial waves
[26].
References
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943 (1999)
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Phys. 11, 055028 (2009)
[9] B. Deb and A. Rakshit, J. Phys. B: At. Mol. Opt. Phys. 42, 195202 (2009)
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[11] C. Chin, V. Vuletic, A. J. Kerman, S. Chu, E. Tiesinga, P. J. Leo and C. J.
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Rev. Lett. 98,043201 (2007).
49
Chapter 4. PA in the presence of FR
50
[14] K K Ni, S. Ospelkaus, M. H. G. de Miranda, A. Peer, B. Neyenheis, J. Zirbel,
S. Kotochigova, P. S. Julienne, D. S. Jin and J. Ye, Science 322, 231 (2008).
[15] D. M. Bauer, M. Lettner, C. Vo, G. Rempeand S. Durr, Nature Phys. 5, 339
(2009)
[16] D. M. Bauer, M. Lettner, C. Vo, G. Rempeand S. Dürr, Phys. Rev. A 79
062713 (2009)
[17] M. Gacesa, S. Ghosal, J. N. Boyd and R. Côté, Phys. Rev. A 88,063418 (2013)
[23] U. Fano, Phys. Rev. 124, 1866 (1961).
[19] A. J. Moerdjik, B. J. Verhaar and A. Axelsson, Phys. Rev. A 51, 4852 (1995).
[20] J. L. Bohn and P. S. Julienne, Phys. Rev. A 54, R4637 (1996).
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Rev. Lett. 77, 2913 (1996); F. K. Fatemi, K. M. Jones and P. D. Lett, Phys.
Rev. Lett. 85, 4462 (2000).
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Lett. 91 080402 (2003).
[23] E. R. I. Abraham, W. I. McAlexander, C. A. Sackett and R. G. Hulet, Phys.
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Chapter 5
Vacuum- and Light-Induced
Coherences in Cold Atoms
and Molecules
Most of the studies on quantum interferences and coherences have been done in
case of atomic systems and sometimes in case of molecular systems [1, 2]. But the
developments in the fields of free-bound spectra open up a new research area to
study quantum interferences in atom-molecule coupled systems. In this chapter,
we study vacuum-induced and light-induced coherences in case of ultracold photoassociation and propose novel photoassociation (PA) schemes for the realization
of vacuum-induced coherence in atom-molecule coupled system.
In the section 5.1, the perspective of our work is discussed. Then, we describe two
schemes that are proposed to discuss vacuum- and light-induced coherences in Veetype atom-molecule coupled system. The first scheme and its results have been
discussed in sections 5.2, 5.3 and 5.4. In first scheme, we employ Wigner-Weisskopf
approach to solve the problem where as in second scheme, master equation method
demonstrated by Agarwal et al. in [3] to treat spontaneous emission of continuumbound coupled autonizing Fano state, is adopted. It is discussed in sections 5.5,
5.6, 5.7 and 5.8. We demonstrate that it is possible to generate and manipulate
coherence between the excited states by PA lasers. We conclude in section 5.9.
51
Chapter 5. VIC and LIC
5.1
52
Perspective of The work
Vacuum-induced coherence (VIC) [3] results from the quantum interference between two spontaneous emission pathways. On the other hand, the light induced
coherence (LIC) occurs due to interplay of applied light fields. As we change the
relative intensity, relative phase shift and the detunings of both lasers, the coherence between the states changes. VIC can lead to the population trapping in the
excited state [2, 3]. This can be utilized in manipulating environment-induced
relaxation processes in a wide variety of systems, such as atoms, molecules, quantum dots [3–10]. It has also been found effective against decoherence in quantum
information processing [11]. In this chapter, our aim is to show how VIC can be
modulated by using LIC.
The necessary criteria for VIC to occur is the non-orthogonality of dipole transitions of two participating pathways.
For atomic systems, nonorthogonality
is a stringent condition to achieve for atoms. For possible realization of VIC
for an excited atom interacting with an anistopic vacuum [12–14], utilization of
j = 1/2 → j = 1/2 transition in
198
Hg + and
139
Ba+ ions has been suggested
[15, 16]. Recently, a proof-of-principle experiment verifying its presence has been
performed in quantum dots [17]. In spite of these attempts, a clear signature of
VIC in atomic systems has yet to be obtained. On the other hand, molecules
are the natural candidates for VIC as the non-orthogonality criteria is satisfied
naturally. Spontaneous transitions from two excited states, which belong to same
electronic state differing only in rotational or vibrational quantum numbers, to
same ground state have transition dipole moments which are nonorthogonal to
each other. The interference between these two spontaneous emission pathways
results in VIC.
Now, with the tremendous progress in the fields of free-bound PA spectroscopy
[18, 19], the low lying ro-vibrational levels in excited electronic states can be selectively populated. This occurs due to PA transitions from the collisional continuum
of two ground-state cold atoms. VIC would be significant in such an atom-molecule
interface system provided (i) there is no hyperfine interaction in the atoms that
are photoassociated, (ii) there is no bound state close to the dissociation continuum of the ground molecular state and (iii) excited molecular levels have a long
lifetime. To the best of our knowledge, the possibility of observing vacuum induced coherence in PA system has not been addressed so far. In this chapter, our
Chapter 5. VIC and LIC
53
J2
J1
Ve(r)
|φ2 !
ωL1
ωL2
|φ1 !
γ2
γ1
Vg(r)
|E!b
Continuum
Figure 5.1: A schematic diagram for creation of LIC and VIC. The twoatom continuum |Eb i is coupled to two rovibrational levels |v , J1 i(|φ1 i) and
|v , J2 i(|φ2 i) of an excited molecular state via the lasers of frequencies ωL1 and
ωL2 , respectively. γ1 and γ2 are the spontaneous decay rates of |φ1 i and |φ2 i,
respectively due to their vacuum coupling with continuum.
aim is to show VIC in atom-molecule interface system and to investigate whether
quenching of spontaneous emission is possible for such system or not.
5.2
Scheme 1
The basic idea of the first scheme is depicted in Fig.5.1 [20]. We consider as
our model a system which consists two excited molecular ro-vibrational levels
|v , J1i(|φ1 i) and |v , J2 i(|φ2i) , belonging to same electronic state, coupled to the
two-atom ground unperturbed continuum |Eib via lasers. Initially either |φ1 i or
|φ2 i is populated or partially both are populated via photoassociation of cold atoms
using two lasers L1 and L2 of frequencies ωL1 and ωL2 , tuned near | Eib →| φ1 i
and | Eib →| φ2 i transitions, respectively. Both the excited levels |φ1 i and |φ2 i
decay spontaneously to the same ground continuum with decay rates γ1 and γ2 ,
Chapter 5. VIC and LIC
54
respectively. We show that coherence between the excited ro-vibrational states
builds up due to their interaction with the background electromagnetic vacuum.
Moreover, we demonstrate that VIC would be affected by PA lasers.
The total Hamiltonian governing the dynamics of the system is given by H =
Hcoh + Hincoh where Hcoh is the coherent part involving PA couplings and is given
by,
Hcoh =
2
X
n=1
~ωbn | φn ihφn | +
Z
E ′ | E ′ ib b hE ′ | dE ′
Z X
2 n
o
+
ΛnE ′ e−iωLn t ŜE† ′ + H.C. dE ′ .
(5.1)
n=1
Here ~ωbn are the binding energies of the bound states |φn i(n = 1, 2); |E ′ ib is the
~n · E
~ Ln | E ′ ib is the laser coupling for the
bare continuum state and ΛnE ′ = hφn | D
~n
transition from the n-th bound state to the bare continuum |E ′ ib . The vectors D
~ Ln are the dipole moment and electric field of the laser associated with the
and E
n-th transition, respectively. The operator ŜE† ′ = |φn ib hE ′ | is a raising operator.
Hcoh is exactly diagonalizable [20–22] in the spirit of Fano’s theory [23] and we
obtain
Hcoh =
Z
E | Eid d hE | dE
(5.2)
where | Eid is the dressed continuum state with dressed energy E with the nor-
malization condition hE ′′ |Ei = δ(E − E ′′ ). It is given by
| Eid = A2E |φ2 i + A1E |φ1 i +
Z
CE ′ (E) | E ′ ib dE ′ .
(5.3)
AnE and CE ′ (E) are the dressed amplitudes. Solving time independent Schrödinger
equation HΨE = EΨE under Born-Oppenheimer approximation, we obtain the
coupled differential equations:
2 2
~ d
+ BJ (r) A1E φ1 + [Ve − (E − ~δ1 )] A1E φ1
−
2µ dr 2
Z
= −Λ1E CE ′ |E ′ ib dE,
(5.4)
Chapter 5. VIC and LIC
55
~2 d 2
+ BJ ′ (r) A2E φ2 + [Ve − (E − ~δ2 )] A2E φ2
−
2µ dr 2
Z
= −Λ2E CE ′ |E ′ ib dE,
(5.5)
Z
2 2
~ d
+ Vg (r) − E
CE ′ |E ′ ib dE = −Λ∗1E A1E φ1 − Λ∗2E A2E φ2 .
−
2
2µ dr
(5.6)
Here BJ = ~2 J(J + 1)/2µr 2 is the rotational term of the excited state and δ =
ωbn − ωLn is the laser-atom detuning. Ve and Vg are excited and ground molecular
potentials, respectively. These equations can be solved in a similar manner as
depicted in previous chapter using Fano’s method [23]. The dressed amplitudes
are given by
AnE =
(q + ǫn′ )
, n = 1, 2
πΛnE [(ǫ1 + i)(ǫ2 + i) − (q − i)2 ]
(5.7)
and
CE ′ (E) = δ(E − E ′ ) +
A1E Λ1E ′ + A2E Λ2E ′
.
(E − E ′ )
(5.8)
Here, n 6= n′ . ǫn = (ωE + ωLn − ωbn )/[Γn (E)/2]. The term Γn (E) = 2π|ΛnE |2 /~,
is the stimulated linewidth of the n-th bound state due to continuum-bound laser
coupling. Here q = V12 /(πΛ1E Λ2E ) is analogous to the well-known Fano’s q paR
rameter [23] with V12 = P dE ′ Λ1E ′ Λ2E ′ /(E − E ′ ) where ‘P’ stands for principal
value.
The incoherent part of the Hamiltonian Hincoh describes the interaction of vacuum
field with the system and is given by,
Hincoh =
Z
dE ′
" 2
XX
n=1 κ,σ
gn,σ (E ′ , κ)ŜE† ′ âκ,σ e−iωκ t + H.c.
#
(5.9)
~ vac and gn,σ (E ′ , κ) =
where âκ,σ is the annihilation operator of the vacuum field E
p
~i · E
~ vac (κ) | E ′ ib is the dipole coupling with E
~ vac (κ) =
~ωκ /2ǫ0 V ~εσ ,
−hφn | D
p
κ being the wave number, σ the polarization of the field and ~ωκ /2ǫ0 V the
amplitude of the vacuum field.
ρ22
ρ11
ρnn
Chapter 5. VIC and LIC
1
0.8
0.6
0.4
0.2
0
0
1
0.8
0.6
0.4
0.2
0
0
1
0.8
0.6
0.4
0.2
0
0
56
ρ11
ρ22
2
γt
4
6
8
10
δ2 = 0 MHz
δ2 = 1 MHz
δ2 = 3 Mhz
2
γt
4
6
8
10
δ2 = 0 MHz
δ2 = 1 Mhz
δ2 =3 MHz
2
4
γt
6
8
10
Figure 5.2: Uppermost Panel shows the plots of ρ11 and ρ22 in the absence
of laser which is the case of normal VIC. Both ρ11 and ρ22 go to zero in the
long time limit. In the middle and lower panel, the excited state population ρ11
and ρ22 are plotted as a function of dimensionless time γt for different values of
the detuning of the second laser, delta2 , keeping δ1 = 0. At δ2 = 1 MHz, the
population is trapped between the two excited states. The intensities of L1 and
L2 are 50 mW cm−2 and 0.1 mW cm−2 , respectively.
5.3
Solution
Let the joint state of the system-reservoir at a time t be expressed as,
| Ψ(t)i =
+
X
n
Z
an (t) | φn , {0}i
dE ′
X
κ,σ
bE ′ ,κ,σ (t) | E ′ ib | {1κ,σ }i
(5.10)
where an and bE ′ ,κ,σ are the amplitudes of n-th excited state and ground continuum,
respectively. The state |φn , {0}i corresponds to molecular excited state with field
in vacuum and |E ′ ib |{1κσ }i refers to ground bare continuum state with energy E ′
Chapter 5. VIC and LIC
57
1
I1 = 0.1 W/cm
ρ11
0.8
I1 = 2.0 W/cm
0.6
I1 = 5.0 W/cm
2
2
2
0.4
0.2
0
0
3
6
9
γt
12
15
12
15
1
ρ22
0.8
I1 = 0.1 W/cm
0.6
I1 = 2.0 W/cm
0.4
I1 = 5.0 W/cm
2
2
2
0.2
0
0
3
6
γt
9
Figure 5.3: Same as in Fig.5.2, but for different values of intensity of first
laser, I1 , keeping the intensity of second laser fixed at 0.1 mW/cm−2 , keeping
δ1 = δ2 = 0. The population is trapped at excited states for intensity of 2
W/cm−2 .
and one photon in mode κ of polarization σ. Now solving the time dependent of
Schrodinger equation
i
| Ψ̇(t)i = − H | Ψ(t)i
~
(5.11)
we obtain
ȧn (t) = −iω̃n an − i
Z
dE ′ gn,σ bE ′ ,κ,σ e−i(ωκ −ωLn )t
(5.12)
and
ḃE ′ ,κ,σ = −iω̃b (E ′ )bE ′ ,κ,σ − i
X
n
∗
gn,σ
(κ, E ′ )an ei(ωκ −ωLn )t
(5.13)
Chapter 5. VIC and LIC
58
where dressed frequencies are given by ω̃n =
1
~
R
|AnE |2 EdE and ω̃b =
1
~
R
|CE ′ (E)|2 EdE.
Putting the value of bE ′ ,κ,σ , obtained after integrating Eq.(5.13) in Eq. (5.12) and
using the standard Wigner-Weisskopf approach [2], we finally obtain
G2
G12 iδ12
L
)a1 −
e t a2
2
2
G1
G12 −iδ12
L t
ȧ2 = (−iω̃2 − )a2 −
e
a1 .
2
2
ȧ1 = (−iω̃1 −
(5.14)
(5.15)
Detuning δ12 in the above equation is given by δ12 = (ωL1 − ωL2 ) + (ω̃1 − ω̃2 ).
Let us consider, ãn is the modified amplitude related to an by transformations,
ã1 = a1 exp [i(ω̃1 − δ12 )t] and ã2 = a2 exp [iω̃2 t]. Hence the modified equations
become
ã˙ 1 = (−G1 − iδ12 ) ã1 − G12 ã2
ã˙ 2 = −G2 ã2 − G21 ã1
(5.16)
(5.17)
and Gn , the decay constant of the nth bound state, is given by
Gn
1
= 2
~
Z
dE
′
Z
dt′
X
κ,σ
|gnσ (E ′ , κ)|2
E′
′
exp i ωLn + ω̃n −
− ωκ (t − t )
~
(5.18)
and Gnn′ , cross damping term, reponsible for coupling between amplitudes can be
written as
Gnn′
1
≃ 2
~
Z
dE
′
Z
dt′
X
gnσ (E ′ , κ)gn∗ ′σ (E ′ , κ)
κ,σ
E′
′
exp i ωLn + ω̃n −
− ωκ (t − t )
~
(5.19)
where, n is not equal to n′ . For simplicity in writing the above equation we have
assumed (ωL1 + ω̃1 ) ≃ (ωL2 + ω̃2 ). It is important to understand that G12 arises
due to quantum interference of the spontaneous emission pathways resulting in
VIC between the excited states amplitudes. Summing over the vacuum modes
and then carrying out the time integral under Born Markov approximation, we
finally obtain
3 −1
Gn = (3πǫ0 ~c )
Z
~ n |E ′ ib |2
dE ′ (ωLn + ω̃n − E ′ /~)3 |hφn |D
(5.20)
Chapter 5. VIC and LIC
59
and
3
G12 = G21 = (3ǫ0 ~c π)
−1
Z
~ 1 |E ′ ihE ′ |D
~ 2 |φ2 i
dE ′ hφ1 |D
(ωL1 + ω̃1 − E ′ /~)3/2 (ωL2 + ω̃2 − E ′ /~)3/2 .
5.4
(5.21)
Results and Discussions
Solving the coupled Eqs. (5.16) and (5.17) analytically, we obtain
ã1 (t) = c1− ez− t + c1+ ez+ t
(5.22)
ã2 (t) = c2− ez− t + c2+ ez+ t
(5.23)
where, z± = 21 [−iδ12 −G+ ±Ω] and c1± = [±(−iδ12 −G− ±Ω)ã1 (0)∓2G12 ã2 (0)]/(2Ω)
p
2
and c2± = [±(iδ12 +G− ±Ω)ã2 (0)∓2G12 ã2 (0)]/(2Ω) with Ω = (G− + iδ12 )2 + 4G12
,
G− = G1 − G2 and G+ = G1 + G2 .
In the limit when laser intensities going to zero (weak coupling), we find ω˜n →
(ωbn − ωLn ), thus Gn reduces to the usual damping constant γn . Moreover, for low
energy we have
G12 ≃ γ12
p
(ωb1 ωb2 )3
~1 · D
~ 2 | φ2 i.
=
hφ1 | D
(3πǫ0 ~c3 )
(5.24)
Thus in the absence of lasers, the model reduces to normal VIC case in V-type
system [2]. The above equation shows that γ12 vanishes if the molecular transition
~1 and D
~2 are orthogonal. In our model D
~1 and D
~2 are the
dipole moments D
transition dipole moments between the same ground and excited electronic states,
therefore they are essentially nonorthogonal.
′
The excited state populations ρ11 = |ã1 |2 , ρ22 = |ã2 |2 and the coherence ρ12 = ã1 ã2∗
can be obtained from Eqs. (5.22) and (5.23). Explicitly
ρnn (t) =
e−2G1 t 2
2
2 AΩ − Bδ12
ρnn (0)
2
4Ω
2
−8BG12
ρn′ n′ (0) + 8BG12 δ12 Im[ρ12 (0)]
−8G12 ΩRe[ρ12 (0)] sinh(Ωt)] ,
(5.25)
Chapter 5. VIC and LIC
60
where n′ 6= n, A = [1 + cosh(Ωt)] and B = [1 − cosh(Ωt)] and we have consid2
2
ered G1 = G2 . At 4G12 = [4G12
− δ12
], it follows from above equation that ρ11
and ρ22 become time-independent in long time limit meaning coherent population
trapping in the excited states. When δ12 = 0, ρ11 (t → ∞) = ρ22 (t → ∞) =
[ρ11 (0) + ρ22 (0) − 2Re[ρ12 (0)]] /4 become exactly same as normal VIC case [2]. It
is worthwhile to emphasize that the results given in Eqs. (5.22), (5.23) and (5.25)
are general because they are applicable to any PA coupling regime.
For experimental realization of VIC, our model can be applied to the spin forbidden intercombination transition 1 S0 −3 P1 of bosonic
174
Yb [24–26] which has no
hyperfine interaction. The only molecular ground electronic state of
174
Yb is 1 Σg
which corresponds to 1 S0 +1 S0 at long separation and represents the only bare continuum |Eib of our model. The excited states |φn i can be chosen as ro-vibrational
levels in long range Ou+ state that can be populated by PA. For illustration, we
specifically consider excited ro-vibrational levels |φ1 i = |v = 118, J = 1i and
|φ2 i = |v = 118, J = 3i [25]. According to the selection rules of continuum-bound
transitions, the minimum partial wave (l) that be coupled to |φ2 i by PA is d wave
(l = 2). Usually at ultracold temperatures, d wave scattering amplitude becomes
insignificant due to large centrifugal barrier. But ground state scattering properties of
174
Yb are exceptional in the sense that it exhibits a prominent d-wave
shape resonance at temperatures as low as 25 µK [24, 27].
We now discuss our numerical results. In Fig.5.2, we show the dynamical behavior
of populations ρ11 and ρ22 as a function of scaled time γt. The upper most panel
shows populations in the absence of the lasers assuming γ1 = γ2 = γ12 = γ = 2.29
MHz [28]. The short time dynamics clearly shows exchange of population between
|φ1 i and |φ2 i due to VIC. In the lower two panels, we plot ρ11 and ρ22 of Eq.
(5.25) for different values of δ2 , keeping δ1 fixed. We find that, as δ2 increases
upto an optimum frequency, the lifetime of both the excited levels also increases.
Then at an optimum frequency , the population gets trapped in the excited state.
For the parameters of Fig.5.2, this optimum frequency is found to be 1 MHZ.
When δ2 increases beyond the optimum value, population falls off. Since the value
of dressed frequency ω̃n depends upon the PA laser intensities, we expect the
dynamics to be intensity dependent. Hence by varying the laser intensity of one
of the PA lasers while keeping all other parameters fixed, we can achieve excited
state population trapping for an optimum intensity of that laser. In Fig.5.3. we
show this explicitly for an optimized intensity I1 = 2 W cm−2 . Note that at this
61
0
0.06
-0.05
0.03
-0.1
Im[ρ12]
Re[ρ12]
Chapter 5. VIC and LIC
-0.15
0
-0.03
-0.2
-0.06
-0.25
0
2
4
γt
6
8
10
0
-0.09
0
2
4
γt
6
8
10
0
2
4
γt
6
8
10
0.06
Im[ρ12]
Re[ρ12]
-0.05
0.03
-0.1
-0.15
-0.2
0
-0.25
0
2
4
γt
6
8
10
-0.03
Figure 5.4: Plot of real and imaginary part of ρ12 against γt for different
detunings of second laser (two upper subplots ) and different intensities of the
first laser (two lower subplots). All other parameters of the upper subplots is
same as in Fig.5.2 while the other parameters of lower subplots are the same
as in Fig.5.3. In the two upper subplots, the detuning δ2 = 0 (black solid line),
δ2 = 1 MHz (red dotted line) and δ2 = 3 MHz (blue dashed lines) while in the
lower two subplots the intensity I1 = 0.1 W cm−2 (black solid line), I1 = 2.0 W
cm−2 (red dotted line) and I1 = 5.0 W cm−2 (blue dashed lines)
laser intensity, PA stimulated linewidth Γ1 is much larger than γ1 meaning that
the system is in the strong-coupling regime. In Fig.5.4, we plot the dynamical
behavior of the coherence ρ12 as a function of γt. It is clearly visible that the
imaginary part is much more smaller than the real part. The upper panel of
Fig.5.4 shows that Re[ρ12 ] becomes steady in the long time limit for an optimum
frequency. Lower panel of Fig.5.4 shows that Re[ρ12 ] becomes time-independent
in the long time limit for the optimum parameters for which population in Fig.5.3
becomes trapped.
Using scheme 1, we have shown that it is possible to generate and manipulate
coherence between two excited ro-vibrational states of a molecule by using the
Chapter 5. VIC and LIC
62
Figure 5.5: A schematic diagram showing the two-atom continuum |Eib
is coupled to two rovibrational levels |φ1 i and |φ2 i of an excited molecular
state via the lasers of frequencies ωL1 and ωL2 , respectively. γ1 and γ2 are
the spontaneous decay rates of |φ1 i and |φ2 i, respectively due to their vacuum
coupling with continuum to bound state |b0 i in ground molecular state.
technique of PA spectroscopy.
5.5
Scheme 2
Scheme 2 is depicted in Fig.5.5. Our model consists of two excited diatomic
molecular ro-vibrational states |φ1 i and |φ2 i (belonging to the same molecular
electronic state) coupled to the ground-state bare continuum |Eib of scattering
states, by the lasers 1 and 2, respectively. Initially either |φ1 i or |φ2 i or partially
both are populated due to two photoassociation lasers L1 and L2 of frequencies ωL1
and ωL2 , tuned near | Eib →| φ1 i and | Eib →| φ2 i transitions, respectively. The
ground continuum is assumed to have only one internal molecular state with only
Chapter 5. VIC and LIC
63
ρ11
ρ22
Re[ρ12]
Im[ρ12]
0.25
0.2
ρnn(0), Re[ρ12 (0)], Im[ρ12 (0)]
0.15
0.1
0.05
0
0
20
40 I (W/cm2) 60
1
80
100
20
40 I (W/cm2) 60
2
80
100
0.2
0.15
0.1
0.05
0
0
Figure 5.6: ρnn (0) (n = 1, 2) and the real and imaginary parts of ρ12 (0) are
plotted against I1 (upper panel) and I2 (lower panel) in unit of W cm−2 , keeping
the intensity of the other laser fixed at 1 W cm−2 . The other parameters are
φ = 0 and δ1 = δ2 = 0.
one threshold and no hyperfine interaction. We assume that the two free-bound PA
transitions between the ground-state continuum and the two excited ro-vibrational
states are strongly driven so that the spontaneous emissions from these two bound
states to the continuum are negligible as compared to the corresponding stimulated
ones. However, these two driven bound states can spontaneously decay to other
bound state |b0 i of binding energy ~ωb0 in the ground electronic configuration.
Our aim is to discuss the creation of laser-induced coherence and its implications
in decay dynamics within the framework of master equation approach. The work
shows exciting possibilities of manipulating excited state coherences using the
relative phase between two lasers.
The Hamiltonian governing the dynamics of this system can be written as H =
HS + HSR , where HS = Hcoh + ~ωb0 | b0 ihb0 | is the system Hamiltonian with
two parts: the first part Hcoh describes coherent dynamics with the two strong
Chapter 5. VIC and LIC
64
5
0
shift
/hγ
-5
En
-10
n=1
n=2
-15
-20
-25
10
1
1
10
10
2
10
3
4
10
10
5
6
10
7
10
8
10
9
10
10
10
11
10
0
Γn(E)/γ
10
10
-1
-2
10
-3
10
-4
10
1
10
10
2
10
3
4
10
10
5
6
10
E (Hz)
7
10
8
10
9
10
10
10
11
10
Figure 5.7: Light shifts (scaled by ~γ) and free-bound stimulated line widths
(scaled by γ) of the two excited bound states n = 1 (J = 1) (solid) and n = 2
(J = 3) (dashed) - both having the same vibrational quatum number v = 106
of 174 Yb2 (see text) are plotted as a function of collision energy E (in Hz) in
upper and lower panels, respectively; for I1 = I2 = 1 W cm−2 and the detunings
δ1 = δ2 = 0.
PA couplings. On the other hand, the second part HSR is the interaction part of
the system with a reservoir of vacuum electromagnetic modes. Explicitly, one can
write
Hcoh =
2
X
n=1
~(ωbn − ωLn ) | φn ihφn | +
Z
E ′ | E ′ ib b hE ′ | dE ′
Z X
2 n
o
†
+
ΛnE ′ ŜnE
+
H.C.
dE ′ ,
′
(5.26)
n=1
HSR =
XX
n=1,2 κ,σ
âκ,σ e−i(ωκ +ωLn )t Vn0 (κσ) | φn ihb0 | +H.c.
(5.27)
Chapter 5. VIC and LIC
65
Here ~ωbn are the binding energies of the bound states |φn i(n= 1, 2); |E ′ ib is the
bare continuum state. In deriving the above Hamiltonian, we have used rotating
wave approximation (RWA) [29]. In RWA, one works in a frame rotating with the
frequency of the sinusoidally oscillating field interacting with a two-level system
(TLS) and neglects the counter-rotating terms that oscillate with the sum of the
field and the system frequencies. It primarily relies on two conditions: (i) the
system relaxation time is much larger than the time period of oscillation of the
field and (ii) Rabi frequency or the system-field coupling is much smaller than the
transition frequency of TLS. These conditions are in general fulfilled in most cases
of a TLS interacting with a monochromatic optical field and therefore RWA can be
regarded as a cornerstone for studying quantum dynamics of TLS. Nevertheless,
RWA may break down in case of intense laser fields or short pulses when the
Rabi frequency or the coupling becomes comparable with the system frequency.
Generally, this may happen when the laser intensity is of the order of 1012 W
cm−2 or higher. In PA experiments the laser intensity is much lower, typically in
the W cm−2 or kW cm−2 . Strong-coupling regime in ultracold PA can be reached
with laser intensities higher than 1 kW cm−2 but much lower than 1 MW cm−2 .
For driven TLS, corrections beyond RWA and in terms of Bloch-Siegert shift [30]
have been discussed by Grifoni and Hanggi [31]. The corrections to RWA can be
formulated as a systematic expansion in terms of the ratio of Rabi frequency to the
field frequency [32]. In case of two coupled TLS, there exists a parameter regime
where leading order term in the expansion vanishes rendering the next higher order
term to be significant [32]. However, such situation does not arise in our case and
so RWA remains valid.
With electric dipole approximation, the laser coupling ΛnE ′ for the absorptive
transition from the bare continuum |E ′ ib to the nth excited bound state | φn i is
given by
~ n · ELn | E ′ ib
ΛnE ′ = ei(kLn ·R+φLn ) hφn | D
(5.28)
where, kLn , ELn and φLn are the wave vector, electric field and phase of the nth
laser, respectively; R is the center-of-mass position vector of the two atoms and
~ n is the free-bound molecular dipole moment associated with the nth bound
D
state. The electric dipole approximation here dictates that kLn r << 1, where r is
the separation between the two atoms. We have thus used exp(ikLn · r) ≃ 1 in
†
′
writing the above equation. The operator ŜnE
′ = |φn ib hE | is a raising operator,
Chapter 5. VIC and LIC
66
1.5
shift
E12
Γ12
0.5
E12
shift
/hγ, Γ12/γ
1
0
-0.5
-1
1
10
2
10
3
10
10
4
5
10
6
7
10
10
E (Hz)
10
8
9
10
10
10
10
11
12
10
shift in unit of hγ (black solid) and Γ
Figure 5.8: Plotted are E12
12 in unit of
hγ (red dashed) are plotted as a function of collision energy E (in Hz). Other
parameters are as same as in Fig.5.7.
~ vac and Vn0 (κσ) =
âκ,σ denotes the annihilation operator of the vacuum field E
p
~ n0 · E
~ vac (κ) | b0 i is the dipole coupling with E
~ vac (κ) =
−hφn | D
~ωκ /2ǫ0 V ~εσ ,
~ n0 the transition dipole moment between nth excited
ωκ being the wave number, D
bound state and the ground bound state | b0 i, σ the polarization of the field and
p
~ωκ /2ǫ0 V the amplitude of the vacuum field and ~ωb0 is the binding energy of
the bound state | b0 i. The Hamiltonian Hcoh is exactly diagonalizable [20, 33]
in the spirit of Fano’s theory [23]. The eigenstate of HS is a dressed continuum
expressed as
| Eidr =
2
X
n=1
AnE |φn i +
with the normalization condition
dr hE
and CE ′ (E) are derived in Ref [33].
′′
Z
CE ′ (E) | E ′ ib dE ′
(5.29)
|Eidr = δ(E − E ′′ ). The coefficients AnE
Chapter 5. VIC and LIC
67
0.2
Re[ρ12]
0.1
0
-0.1
-0.2
0
10
20
30
40
50
0.15
φ=0
φ=π
φ = π/2
Im[ρ12]
0.1
0.05
0
-0.05
0
10
20
30
40
50
2
I1 (W/cm )
Figure 5.9: Re[ρ12 (0)] and Im[ρ12 (0)] are plotted against I1 (in unit of W
cm−2 ) for different values φ of the difference between the phases of the two
lasers in upper and lower panels, respectively. The other parameters are I2 = 1
W cm−2 and δ1 = δ2 = 0
P
By using partial-wave decomposition of the bare continuum | E ′ ib =
ℓmℓ |
P
ℓmℓ
′
′
E ℓmℓ′ ibr , we have ΛnE ′ = exp[i(kLn · R + φLn )] ℓmℓ ΛJn Mn (E ) where Jn and
Mn are the rotational and the magnetic quantum number, respectively, of the
nth excited bound state in the space-fixed (laboratory) coordinate system. Note
ℓ
that Λℓm
Jn Mn (E) represents amplitude for free-bound transition from (ℓmℓ ) incident
partial-wave state to the nth bound state. To denote the amplitude for reverse
J n Mn
(E). Accordingly, we can write
(bound-free) transition, we use the symbol Λℓm
ℓ
′
P
P
P
ℓ mℓ′
ℓ′ mℓ′
AnE = ℓ′ mℓ′ AnE Yℓ,mℓ′ (k̂) and CE ′ (E) = ℓmℓ ℓ′ mℓ′ CE ′ ,ℓm
(E)Yℓ′ mℓ′ (k̂) where
ℓ
k̂ represents a unit vector along the incident relative momentum between the two
atoms. Explicitly,
ℓ′ m
ℓ′ m
AnE ℓ′
=
ℓ′ m
LL iθn′
ℓ′
(E)
eiθn ΛJn Mℓ′n (E) + ξn−1
ΛJn′ M
′ Knn′ e
n′
ξn −
LL
LL
ξn−1
′ Knn′ Kn′ n
, n′ 6= n
(5.30)
Chapter 5. VIC and LIC
68
and
′
ℓ mℓ′ Jn Mn
X AnE
Λℓmℓ (E ′ )
′
= δℓℓ′ δmℓ mℓ′ δ(E − E ) +
E − E′
n=1,2
ℓ′ mℓ′
(E)
CE ′,ℓm
ℓ
(5.31)
where θn = kLn · R + φLn , and
ξn (E) = ~(δnE + iΓn (E)/2),
(5.32)
~δnE = E + ~δLn − (En + Enshift )
(5.33)
with En being the binding energy of nth excited bound state measured from the
threshold of the excited state potential, Enshift is the light shift of the nth bound
state and δLn = ωLn − ωA with ωLn is the laser frequency of n-th laser and ωA the
atomic transition frequency. The two lasers interacting with the system results in
an effective coupling
LL
Knn
′
1
= Vnn′ − i ~Gnn′
2
(5.34)
between the two bound states where
Vnn′ = exp[i(θn − θn′ )]
Gnn′
X
ℓmℓ
P
Z
dE
′
J n Mn
′
ℓ
(E ′ )Λℓm
Λℓm
Jn′ Mn′ (E )
ℓ
E − E′
2π X JnMn
ℓ
(E)Λℓm
Λ
= exp[i(θn − θn′ )]
Jn′ Mn′ (E).
~ ℓm ℓmℓ
,
(5.35)
(5.36)
ℓ
The term Γn (E) = 2π|ΛnE |2 /~ = 2π
P
ℓ,mℓ
J n Mn
(E ′ )|2 /~, is the stimulated
|Λℓm
ℓ
linewidth of the n-th bound state due to continuum-bound laser coupling. Note
P shift
that the light shift Enshift = ℓ Enℓ
is the sum over all the partial light shifts
shift
Enℓ
=
X
mℓ
P
Z
dE ′
J n Mn
′
ℓ
(E ′ )Λℓm
Λℓm
Jn Mn (E )
ℓ
.
E − E′
Here, P stands for Principal value integral.
(5.37)
Chapter 5. VIC and LIC
69
0.1
Re[ρ12]
0.05
0
-0.05
-0.1
0
0.1
10
20
30
40
50
40
50
φ=0
φ=π
φ = π/2
Im[ρ12]
0.05
0
-0.05
0
10
20
30
2
I2 (W/cm )
Figure 5.10: Same as in Fig.5.9, but as a function of I2 keeping I1 = 1 W
cm−2
5.6
Master equation
The system Hamiltonian can be written in dressed basis as
H0 =
Z
EdE | Eidr dr hE | +~ωb0 | b0 ihb0 | .
(5.38)
To derive master equation we work in the dressed continuum basis of the system
Hamiltonian. We express bare basis in terms of dressed basis as follows
| bn i =
Z
dE | Eidr dr hE | bn i =
Z
dEA∗nE | Eidr
(5.39)
Z
dECE∗ ′ (E) | Eidr
(5.40)
and
′
| E ibr =
Z
′
dE | Eidr dr hE | E i =
Chapter 5. VIC and LIC
70
0.17
2
I1 = 1.30 W/cm
2
I1 = 39.70 W/cm
2
ρ11 (t)
0.16
I1 = 43.63 W/cm
0.15
ρ11 (t)
0.13
0.14
I1 = 1.3 W/cm
0.12
0.11
0.13
2
0
0
2
4
γt
8
6
2
4
h
10
6
γt
8
10
Figure 5.11: ρ11 as a function of dimensionless unit γt for different values I1
for I2 = 1 W cm−2 , δ1 = δ2 = 0 and φ = 0. The inset shows the plot of ρ11
when laser-2 is switched off and I1 = 1.3 W cm−2 , δ1 = 0 and θ1 = 0.
By substituting all bare basis states with there expansions in terms of dressed basis,
we can write system-reservoir interaction Hamiltonian in terms of dressed basis.
In the interaction picture, the effective system-reservoir interaction Hamiltonian
I
HSR
= eiH0 t/~HSR e−iH0 t/~ of the driven system interacting with a reservoir of
vacuum modes can be written as
I
HSR
=
X
κ,σ
+ H.c
−iωκ t
e
2
X
n=1
i(ωb0 −ωLn )t
e
âκ,σ
Z
†
dEA∗nE Vn0 (κσ)eiωE t Ŝ0E
(5.41)
where the superscript ‘I’ refers to interaction picture, Ŝ0E =| b0 i dr hE | and ωn0 =
ωbn − ωb0 Let ρS+R (t) denote the system-reservoir joint density matrix. Following
Agarwal [3], the projection operator P is defined by
PρS+R (t) = ρR (0)ρS (t)
(5.42)
Chapter 5. VIC and LIC
71
where ρR and ρS are the density matrices of vacuum and the dressed system (S)
system, respectively. With the use of this projection operator, Liouville equation
under Born approximation can be expressed [3] as
∂ PρIS+R (t) = −
∂t
t
Z
dτ PLIS (t)LIS (t − τ )PρIS+R (t − τ )
0
(5.43)
where
ρI = eiH0 t/~ρe−iH0 t/~
(5.44)
is the density matrix in the interaction picture. Here
LIS (t) · · ·
=
X
−iωκ t
e
κ,σ
h
i
+
âκσ Σ̂κσ (t), · · · + H.c.
(5.45)
where
Σ̂+
κσ (t)
=
2
X
−iωLn t
e
n=1
Z
†
dE Ŝ0E
ei(ωE −ωb0 )t A∗nE Vn0 (κσ).
(5.46)
Tracing over the vacuum states, we obtain
h
i
XZ t n
∂ I +
−
I
−iωκ τ
ρ (t) = −
Σ̂κσ (t), Σ̂κσ (t − τ )ρS (t − τ )
dτ e
∂t S
κ,σ 0
h
io
+
I
+ Σ̂−
(t),
Σ̂
(t
−
τ
)ρ
(t
−
τ
)
+ H.c.
(5.47)
κσ
κσ
S
From Eq. (5.47), making use of Markoff approximation, we derive the equations
of motion of reduced density matrix elements in dressed basis. These are
ρ̇EE ′
Z
dE ′ [AEE ′ ρEE ′ + C.c]
Z
Z
′
= −iωE0 ρE0 − dE AE ′ E ρE ′ 0 − dE ′ AE ′ E ′ ρE0
Z
= −iδEE ′ ρEE ′ − dE ′′ AE ′′ E ρE ′′ E ′ dE ′ ρEE ′
Z
−
dE ′′ AE ′ E ′′ ρEE ′′
ρ̇00 =
ρ̇E0
Z
dE
(5.48)
(5.49)
(5.50)
where δEE ′ = (E − E ′ )/~ and
AEE ′ ≃
1X
γnn′ (ωn − ωb0 ) exp[iδnn′ t]AnE A∗n′ E ′
2 nn′
(5.51)
Chapter 5. VIC and LIC
72
with δnn′ = ωLn − ωLn′ being the difference between n-th and n′ -th lasers and
γnn′ (ωn − ωb0 ) ≃
~ n0 D
~ 0n′ (ωn − ωb0 )3
D
.
3πǫ0 c3 ~
(5.52)
γnn‘ (x) is a function of x. γnn is the spontaneous linewidth of nth excited state
and γ12 = γ21 is the vacuum-induced coupling between the two excited states
[2, 20]. Note that in Eq. (5.52) we have neglected the light shift of the excited
levels in comparison to the transition frequency ωn0 = ωn − ωb0 which is in the
optical frequency domain while the typical light shifts as shown in Fig.5.7 are of
the order of MHz. The expression (5.52) is obtained in the following way: We
first substitute Eq. (5.46) into Eq. (5.47) and express the vacuum coupling Vn0
~ n0 . The sum
in terms of corresponding bound-bound transition dipole moment D
over κ and σ is replaced by an integral over the infinite vacuum modes. Using
standard Markoffian approximation, one can carry out first the integration over τ
and then over the vacuum modes to arrive at the expression for γnn′ as given in
Eq. (5.52). The normalization condition is
ρ00 +
Z
ρEE dE = 1
(5.53)
Eqs. (5.48)-(5.50) form a set of three integro-differential equations for the density
matrix elements expressed in the dressed continuum basis.
5.7
Solution
The density matrix elements can be expressed in bare basis by the transformation
ρnn′ =
Z
dE
Z
dE ′ AnE A∗n′ E ′ ρEE ′ .
(5.54)
In interaction picture, ρIEE ′ = exp(iδEE ′ t)ρEE ′ and the Eq. (5.50) can be rewritten
as
ρ̇IEE ′
= −
Z
dE
′′
AE ′′ E ρIE ′′ E ′ eiδEE ′′ t
−
Z
dE ′′ AE ′ E ′′ ρIEE ′′ eiδE ′′ E ′ t .
(5.55)
Chapter 5. VIC and LIC
73
The solution of the above equation can be formally expressed as
ρIEE ′ (t)
′
Z
t
Z
′
= δ(E − E ) −
dE ′′ AE ′′ E (t′ )eiδEE ′′ t ρIE ′′ E ′ (t′ )
dt
0
Z t Z
′
−
dt′ dE ′′ AE ′ E ′′ (t′ )ρIEE ′′ (t′ )eiδE ′′ E ′ t .
′
(5.56)
0
The delta function on the right hand side is the initial value ρIEE ′ (0). The quantity
AEE ′ (t) given in Eq. (5.51) is expressed in terms of the product AnE A∗n′ E of
the amplitudes of the nth and n′ th bound states in energy-normalized dressed
continuum of Eq. (5.29). If vacuum couplings are neglected, the bound-state
R
probability densities are given by ρnn = dE|AnE |dE and the coherence terms
R
ρnn′ = dEAnE A∗n′ E with n′ 6= n. It is important to note that, apart from
causing spontaneous decay of the nth bound-state probability with decay constant
γnn , vacuum couplings of the two excited bound states | b1 i and | b2 i with the
ground bound-state | b0 i effectively give rise to vacuum-induced coherence (VIC)
[3] between the two excited bound states with coupling constant γ12 . Recently,
atom-molecule coupled photoassociative systems are shown to be better suited
for realizing VIC [20]. Though the quantities γnn′ are calculable from Eq. (5.52)
when the molecular transition dipole moments Dn0 are given, for simplicity of our
model calculations, we have set γ11 = γ22 = γ12 = γ21 = γ. In fact, since we
consider that both the excited bound states belong to the same vibrational level
but differing only in rotational quantum number, the spontaneous linewidths γ11
√
and γ22 would not differ much. Furthermore, since γ12 = γ21 ≃ γ11 γ22 , we have
γ12 = γ21 = γ for the case considered here. The stimulated line width Γn (E) is a
function of the collision energy E for the ground state scattering between the two
ground state atoms. Both in the limits E → 0 and E → ∞, Γn vanishes. Let us
fix an energy Ē near which both Γ1 (Ē) and Γ2 (Ē) attain their maximum values.
It is then possible to write Eq. (5.51) in the form
AEE ′ (t) =
1X
γ̄nn′ exp[iδnn′ t]ĀnE Ā∗n′ E ′
~ nn′
(5.57)
p
p
where γ̄nn′ = γ/ Γn (Ē)Γn′ (Ē) and ĀnE = AnE ~Γn (Ē)/2 are the dimensionless
quantities. The absolute value of ĀnE is less than unity. Supposes, the intensities
of the two lasers are high enough so that Γn (Ē) >> γ for both the excited bound
states. In that case, using γ̄nn′ or the product γ̄nn′ ĀnE Ā∗n′ E ′ as a small parameter,
Chapter 5. VIC and LIC
74
we can expand Eq. (5.55) in a time-ordered series
ρIEE ′ (t)
Z
′
t
′
′
iδEE ′ t′
Z
t
′
= δ(E − E ) −
−
dt AE ′E (t )e
dt′ AE ′ E (t′ )eiδEE ′ t
0
0
Z t′
Z t Z
′
′′
+
dt′ dE ′′ AE ′′ E (t′ )eiδEE ′′ t ×
dt′′ AE ′ E ′′ (t′′ )eiδE ′′ E ′ t
0
0
Z t′
Z t Z
′
′′
+
dt′ dE ′′ AE ′′ E (t′ )eiδEE ′′ t
dt′′ AE ′ E ′′ (t′′ )eiδE ′′ E ′ t
0
0
Z t Z
Z t′
′′
′
dt′′ AE ′′ E (t′′ )eiδEE ′′ t
+
dt′ dE ′′ AE ′ E ′′ (t′ )eiδE ′′ E ′ t
0
0
Z t′
Z t Z
′
′′
+
dt′ dE ′′ AE ′ E ′′ (t′ )eiδE ′′ E ′ t
dt′′ AE ′′ E (t′′ )eiδEE ′′ t
0
0
+ ···
(5.58)
It is worthwhile to point out that this method of solution is similar in spirit
to that of time-dependent perturbation, however it differs in essence because we
have used dressed state amplitude as a small parameter and not the atom-field
coupling. If a large number of terms are taken, then the expansion essentially
provides solution for any time. However, numerically calculating higher order
terms becomes increasingly involved because of larger number of multiple integrals
in energy variable appearing in higher order terms. We therefore restrict our
numerical studies to a few leading order terms as described in the next section.
5.8
Results and discussions
Driven by the two strong lasers, the system is prepared in a dressed continuum
given by Eq. (5.29). Since this state is an admixture of the two excited bound
states, it is subjected to spontaneous emission. We include spontaneous emission
by considering the dressed levels to decay to a third bound level, thereby neglecting
the decay of the excited states to the ground-state continuum inside the dressedstate manifold.
To discuss the effects of the phase-difference φ = θ1 − θ2 between the two lasers,
the laser intensities I1 and I2 , and the detunings δ1 and δ2 on decay dynamics, we
ℓ
first rewrite the dressed-state amplitude Aℓm
nE of Eq. (5.30) in the form
ℓ
Aℓm
nE
ℓmℓ −i(θn −θn′ )
+ Ann
′ e
En + iGn /2
ℓmℓ
iθn ΛJn Mn (E)
=e
(5.59)
Chapter 5. VIC and LIC
75
0.155
0.13
2
I2 = 1.3 W/cm
I2 = 3.5 W/cm2
2
I2 = 6.05 W/cm
ρ22
0.15
2
I2 = 1.3 W/cm
0.12
0.11
0
2
4
ρ22 (t)
0.145
γt
6
8 10
0.14
0.135
0.13
0
1
2
3
4
5
γt
6
7
8
9
10
Figure 5.12: ρ22 (t) is plotted as a function of γt for different values of I2 for
I1 = 1 W cm−2 , δ1 = δ2 = 0 and φ = 0. The inset shows the plot of ρ22 when
only laser-2 is switched on at intensity I2 = 1.3 W cm−2 , δ2 = 0 and θ2 = 0.
ℓmℓ
−1 LL
where Ann
′ = ξn′ Knn′ and
′
shift
En = E + ~δn − (En + Enshift + Enn
′ ), n 6= n.
(5.60)
The additional shift for the nth excited bound state due to laser-induced cross
coupling with the other (n′ ) excited bound state is
−1 LL LL
shift
Enn
′ = Re[ξn′ Knn′ Kn′ n ].
(5.61)
LL LL
Here Gn = Γn + Γnn′ with Γnn′ = −2Im[Re[ξn−1
′ Knn′ Kn′ n ] being the contribution to
the total stimulated line width due to the cross coupling. In expression (5.59), the
first term in the numerator corresponds to single-photon transition amplitude due
to nth laser while the second term describe a net 3-photon transition amplitude
with 2 photons coming from the n′ th laser and the other one from nth laser.
Chapter 5. VIC and LIC
76
0.14
φ=0
φ=π
φ = π/2
ρ22
0.14
ρ11
0.13
0.13
0.12
0.11
0
0.1
1
2
3
4
0
1
2
-0.1
0
1
2
3
4
5
3
4
5
0.05
Im[ρ12]
0.05
Re[ρ12]
0.12
5
0
0
-0.05
-0.05
-0.1
0
1
2
Figure 5.13:
γt
3
4
5
γt
ρnn′ (t) are plotted against γt for different values of φ but for
fixed I1 = I2 = 1 W cm−2 and δ1 = δ2 = 0.
The foregoing discussion has so far remained quite general. Now, we apply our
method to ultracold
174
Yb atoms. For numerical illustration, we use realistic pa-
rameters following the recent experimental [24, 26–28] and theoretical [25] works
on PA with
174
Yb. We have chosen
174
Yb system because this offers some advan-
tages compared to other systems. For instance, it has no hyperfine structure and
the ground-state molecular potential of
174
Yb2 is spin-singlet only. Furthermore,
it has spin-forbidden inter-combination transitions. The total rotational quantum
number is given by J~ = J~e + ~ℓ where Je is the total electronic angular momentum.
For numerical work, we specifically consider a pair of
174
Yb atoms being acted
upon by two co-propagating linearly polarized cw PA lasers. The polarizations of
both lasers are assumed to be same. This geometry is the same as used in Ref [33]
for manipulation of d-wave atom-atom interactions. For our numerical work, we
consider that the two lasers drive transitions to the molecular bound states 1 and 2
characterized by the rotational quantum numbers J1 = 1 and J2 = 3, respectively;
of the same vibrational level v = 106. The two bound states belong to 0+
u (Hund’s
Chapter 5. VIC and LIC
77
0.5
0.4
0.3
Iqb
φ=0
φ=π
φ = π/2
0.2
0.1
0
0
1
2
3
4
5
0.5
I2 = 1.30 W/cm
0.45
I2 = 3.50 W/cm
Iqb
0.4
I2 = 6.05 W/cm
0.35
2
2
2
0.3
0.25
0.2
0
2
4
γt
6
8
10
Figure 5.14: In upper panel, we plot Iqb against γt for different values of φ
when other parameters are fixed as I1 = I2 = 1 W/cm2 and δ1 = δ2 = 0 MHz.
In lower panel Iqb is plotted as a function of γt for different values of I2 for I1 =
1 W/cm2 , φ = 0 and δ1 = δ2 = 0 MHz.
case c) molecular symmetry meaning that the projection of Je on the internuclear
axis being zero. Since
174
Yb atoms are bosons, only even partial waves are al-
lowed for the scattering between the two ground state atoms. Free-bound dipole
transition selection rules then dictate that the bound state 1 can be accessed from
s- and d-wave scattering states, while the bound state 2 is accessible from d- and
g-wave only. Thus d-wave ground scattering state is coupled to both the excited
LL
states by the two PA lasers resulting in the laser-induced coupling term Knn
. In
general, d-wave scattering amplitude is small at low energy. But, fortunately for
174
Yb atoms, there is a d-wave shape resonance [24, 27] in the µK temperature
regime leading to significant enhancement in d-wave scattering amplitude at relatively short separations where PA transitions are possible. In our calculations we
neglect g-wave contributions.
As we prepare the system in a desired dressed continuum, the populations of
Chapter 5. VIC and LIC
78
the two excited bound states and the coherence between them depend on the
relative intensity and phase between the two lasers. In the absence of spontaneous
emission (idealized situation), the dressed state properties correspond to the initial
conditions for our model. Fig.5.6 shows variation of the initial populations and the
coherence as a function of the intensity of either laser for the intensity of the other
laser being fixed at 1 W cm−2 . For all our numerical work, we set the spontaneous
line width γ = 2.29 MHz [28].
The variation of stimulated line widths and light shifts of the two bound states of
174
Yb2 as a function of collision energy E for the laser intensities I1 = I2 = 1 W
cm−2 and zero detunings are displayed in Fig.5.7. The shift E1shift (stimulated line
width Γ1 ) is a sum of s- and d-wave partial shifts ( stimulated line widths) while
the shift E2shift and the stimulated line width Γ2 are made of mainly d-wave partial
shift and width, respectively; with no contribution from s-wave. From Fig.5.7 we
notice that the shifts of both bound states as a function of energy change rapidly
from negative to positive value near E = Ē = 194 µK and the stimulated line
widths of both bound states exhibit prominent peaks at that energy. This can be
attributed to a d-wave shape resonance [24, 25]. We have found that the d-wave
partial stimulated line widths of both the bound states near shape resonance are
comparable. For the first bound state, the value of the d-wave partial stimulated
line width near the resonance is found to exceed the s-wave partial line width by
shift
about 2 orders of magnitude. In Fig.5.8, mutual light shift E12
and stimulated
line width Γ12 due to the coupling between the bound states are shown as a function
of collision energy E in Hz. The mutual shifts and widths arise from the coupling
of the d-wave scattering state with the two bound states by the two lasers. Owing
to the existence of the d-wave shape resonance, the laser couplings of the d-wave
scattering state to both bound states become significant, and so are the mutual
shifts and stimulated line widths.
Figs.5.9 and 5.10 exhibit intensity-dependence of the coherence ρ12 (0) for different
relative phases. The purpose of plotting these two figures is to assert that it is
possible to prepare the dressed system with a desirable coherence between the
two excited bound states by judiciously selecting relative intensities and phases
between the two lasers. It is interesting to note that d-wave shape resonance has
a drastic effect on the properties of dressed continuum. Because of this resonance,
the d-wave contributions to the amplitudes of transition to both the bound states
are large even at a low temperature allowing an appreciable cross coupling to
Chapter 5. VIC and LIC
79
develop between the two bound states. For very large laser intensities at δ1 =
δ2 = 0, light shifts would be so large that the system will be effectively far off
resonant and therefore ρnn′ (0) ≃ 0.
For calculating time-dependence of the density matrix elements for all the times,
we need to calculate a large number of terms appearing on the right hand side
of Eq. (5.58) order by order in γ̄nn′ . This is a laborious and time-consuming
exercise. Instead, to demonstrate the essential dynamical features arising from
quantum superposition of the two rotational states, we restrict our study of decay
dynamics to relatively short times. Inserting Eq. (5.51) in Eq. (5.58), retaining
the terms up to first order in γ̄nn′ , we have
h
i
t
ρnn (t) = Ãnn (0) − γ
dt′ |Ã12 (t′ )|2 + |B̃12 (t′ )|2 + |Ãnn (t′ )|2 + |B̃nn (t′ )|2
0
Z t
n
o
′
′
′
′
′
− 2γ
dt Re Ã12 (t )Ãnn (t ) + B̃nn (t )B̃12 (t ) cos(δ12 t′ ) + · · · (5.62)
Z
0
R
dEA∗nE An′ E cos(ωE t) and B̃nn′ (t) = dEA∗nE An′ E sin(ωE t) with
R
ωE = E/~. Here Ãnn (0) = dE|AnE |2 = ρnn (0). Similarly, the coherence term
where Ãnn′ (t) =
R
ρ12 can be calculated up to the first order in γ̄nn′ . These solutions hold good for
γt < |Ãnn (0)|−2 or equivalently, γt < |ρnn (0)|−2 for both n = 1, 2.
The decay dynamics of the populations ρ11 (t) and ρ22 (t) as a function of the scaled
time γt for different laser intensities are shown in Figs. 5.11 and 5.12, respectively.
These results clearly exhibit that, when the system is strongly driven by two
lasers, the decay is non-exponential and has small oscillations. The oscillation
are particularly prominent for short times. In the long time limit the oscillations
slowly die down. However, the oscillations can persist for long times if couplings
are stronger. We have chosen the values of the laser intensities I1 and I2 such that
the initial values of dressed population ρ11 (t = 0) or ρ22 (t = 0) are the same for
those intensities. We notice that, though the values ρ11 (0) (or ρ22 (0)) for a set of
I1 values for a fixed I2 value (or a set of I2 values for a fixed I1 ) are the same, their
time evolution is quite different and strongly influenced with the relative intensity
of the two lasers. That the population oscillations result from the laser-induced
coherence between the two bound states can be inferred by observing the decay
of the populations when either of the lasers is switched off. Plots of ρ11 and ρ22
against γt for only laser-1 and laser-2 switched on, respectively are illustrated in
the insets of Figs. 5.11 and 5.12 which show exponential decay of the populations
Chapter 5. VIC and LIC
80
0.52
δ1 = 0 MHz, δ2 = 0 MHz
δ1 = 1 MHz, δ2 = 1MHz
δ1 = 1 MHz, δ2 = -1 MHz
δ1 = -1 MHz, δ2 = 1 MHz
δ2 = -1 MHz, δ2 = -1 MHz
0.5
0.48
0.46
Iqb
0.44
0.42
0.4
0.38
0.36
0.34
0
2
4
γt
6
8
10
Figure 5.15: Iqb is plotted against γt for different values of detuning parameters. Other parameters are fixed as I1 = I2 = 1 W/cm2 and φ = 0.
ρ11 (t) and ρ22 (2) with no oscillations. When only one laser is tuned near the
resonance of either bound state, we do not have any coherence between the two
bound states. As the two excited bound states are about 57 MHz apart, one of the
bound states remains far off-resonant in case of single-laser driving. As a result,
the decay of the driven bound state occurs independent of the other bound state.
The laser-induced coherence between the two bound states is developed only when
we apply both the lasers.
The dynamical characteristics of population decay can be interpreted by analyzing
the time-dependence and relative contributions of the two expressions within the
third and second brackets on the RHS of Eq. (5.62). Since δ12 ≃ −57 MHz and the
free-bound couplings are most significant near E ≃ Ē ∼ 4 MHz as can be noticed
from Fig.5.7, we may perform the time integration on the terms associated with
Chapter 5. VIC and LIC
81
cos(δ12 t) in Eq. (5.62) in the slowly varying envelope approximation to obtain
n
o
− 2γRe Ã12 (t)Ãnn (t) + B̃nn (t)B̃12 (t) sin(δ12 t)/δ12 .
(5.63)
Further, since in energy integrations the major contributions will come from energies near E ≃ Ē, we may approximate
Ãnn′ (t) =
Z
dEA∗nE An′ E cos(ωE t) ≃ cos(ωĒ t)ρnn′ (0).
(5.64)
Similarly, B̃nn′ ≃ sin(ωĒ t)ρnn′ (0). Using these approximations, we get
ρnn (t) ∼ ρnn (0) − γt |ρ12 (0)|2 + ρnn (0)2
− 2γRe {ρ12 (0)ρnn (0)} sin(δ12 t)/δ12 + · · · .
(5.65)
This expression clearly shows that when the quantities (|ρ12 (0)|2 + ρnn (0)2 ) and
2Re(ρ12 (0)ρnn (0)) are of comparable magnitude, we expect oscillations in population decay with time period τosc =∼ 2π/|δ12 | ≃ 0.11 in unit of γ −1 . When the laser
intensities are not too high to induce large shifts, we would expect the qualitative
features of the oscillations will be largely governed by one time scale which is τosc .
In fact, the solid black curves in Figs. 5.11 and 5.12 clearly demonstrate oscillatory modulations with time scale τosc . However, when the laser intensities are high
enough so that the energy-dependent shifts and stimulated line widths are appreciable for a range of energies around E = Ē, then expression (5.65) would not be
useful to indicate correct qualitative features. In that case, we need to retain full
time dependence which will introduce another time scale 2π/ωĒ which is, in the
present context, roughly equal to 2π in unit of γ −1 . In such situations the net
result would be a competition between oscillations with the two time scales. The
plots in Figs. 5.11 and 5.12 at larger laser intensity or intensities clearly demonstrate such oscillatory modulations of the population decay with two time scales.
It is particularly important to note that for larger intensity and appropriate detunings, the early population decay can be made much slower for an appreciable
time duration. It is worthwhile to point out that, this analysis is done only to
gain insight into the physics of the decay dynamics of the system, all the results
presented here are obtained by numerically integrating over time t′ and the entire
range of energy.
Fig.5.13 shows the effects of φ on the temporal evolution of the populations ρnn
Chapter 5. VIC and LIC
82
and the coherence terms ρnn′ with n 6= n′ . Though φ does affect the behavior of
the oscillations in population decay, the magnitude of the populations at a time t
is not altered much with the change of φ. In contrast, the magnitudes of the real
and imaginary parts of the coherence term ρ12 are largely influenced by φ. When
φ is altered by π, the sign of both real and imaginary parts of ρ12 changes.
Finally, we discuss quantum beats by studying the temporal evolution of the intensity of light emitted from the two correlated excited bound states. Quantum beats
are manifested as oscillations in the emitted radiation intensity Iqb as a function
of time [2, 34], which is given by
Iqb (t) = γ(ρ11 (t) + ρ22 (t) + 2Re[ρ12 (t)]).
(5.66)
In lower panel of Fig.5.14 we show the effects of laser phase φ on quantum beats in
time-dependent fluorescent intensity. The effects of different intensities of laser-2
on quantum beats are illustrated in the lower panel of same figure. We demonstrate
the effects of different detunings on quantum beats in Fig.5.15. Because of the
shift
LL
mutual light shift E12
between the two bound states due to coupling term K12
,
the resonance conditions in case of two PA lasers are altered in comparison to those
in single PA laser case. This leads to the non-monotonous effects of detunings on
quantum beats as shown in Fig.5.15.
Before ending this section, we wish to make a few remarks on the possibility of
experimental demonstration of the physical effects discussed here. Our model can
be easily realisable with currently available technology of high precision PA spectroscopy. Ultracold bosonic Yb or Sr atoms appear to be most suitable for this
purpose. Because, they offer several advantages. First, their electronic ground
state is purely singlet and has no hyperfine structure. This means that the bare
continuum has no multiplet structure and so there is only one ground-state channel. Second, they have narrow line singlet-triplet inter-combination transitions.
Third, they have long-range excited bound states which are accessible via PA [25].
These bound states have relatively long life time (∼ microsecond). It is possible to
selectively drive two rotational levels as required for the model. Furthermore, since
both the excited bound states have the same vibrational quantum number, their
outer turning points will lie almost at the same separation. Because of long-range
nature of these excited bound states that are strongly driven by the two lasers
from the bare continuum, these two excited bound states are expected to have
the largest Franck-Condon overlap with the near-zero energy or the last bound
Chapter 5. VIC and LIC
83
state in the ground electronic potential. It is therefore quite natural that these
two driven bound states will predominantly spontaneously decay to the last bound
state. In fact, the scattering length of Yb atoms have been experimentally determined by detecting the last bound state via two-colour PA spectroscopy [26], since
the energy of the last bound state and the scattering length are closely related.
All these facts indicate that our model is a realistic one and the predicted excited
state coherence and the resulting quantum beats can be experimentally realisable.
5.9
Conclusion
We have demonstrated that it is possible to generate and manipulate coherence
between two excited ro-vibrational states of a molecule by using the technique of
PA spectroscopy. Though both the schemes seem to be same, but they have some
basic differences. In first scheme, the excited states populated by PA from ground
continuum state are assumed to decay to the same continuum only whereas in the
second scheme, the dressed continuum decays spontaneously outside the system,
to the last bound state of the ground configuration. In first case, we use WignerWeisskopf method and the spontaneous emission is considered phenomenologically,
while in second one, master equation approach is applied and the spontaneous decay is taken into account from first principle calculation. In first scheme, only
s-wave scattering is considered while both s- and d-wave are taken in the second scheme. The first scheme is quite simplified whereas second scheme is more
realistic as it considers decay of the system outside the dressed continuum. In
both cases we use
174
Yb for numerical demonstration though the excited states
chosen are different. Moreover the second scheme shows exciting possibilities of
manipulating excited state coherences using the relative phase between two lasers.
But from both the models, we have demonstrated that the spontaneous emission
from two correlated excited bound states are strongly influenced by the coherence
between them. We have also discussed the possibility of experimental realization
of our models. A promising candidate for exploring such excited state coherence
is the bosonic Yb atoms. Finally, our work may be useful in stimulating further
studies on laser manipulation of the continuum and the bound states between
ultracold atoms. Laser dressing of continuum-bound coupled systems in the context of autoionisation and photoionisation had been extensively studied earlier
demonstrating many interesting effects such as “confluence of coherences” [35],
Chapter 5. VIC and LIC
84
population trapping with laser-induced continuum structure [36], non-decaying
dressed-states in continuum [37], line-narrowing of autoionizing states [38], etc..
Our model can be extended to explore analogous effects in a new parameter regime
at the photoassociatve interface [39, 40] of ultracold atoms and molecules. The
manipulation of VIC with PA may be important for controlling decoherence and
dissipation in cold molecules.
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(2014)
Chapter 6
Atom-Ion Cold Collisions:
Formation of Cold Molecular Ion
In recent years, both the fields of cold and ultracold neutral atomic gases and cold
trapped atomic ions have developed greatly. Total control have been acquired
over both the fields down to the quantum level. The progresses in both the fields
provide us a new opportunity for exploring atom-ion quantum dynamics, charge
transfer and spin exchange reactions. Understanding ion-atom cold collision [1–12]
is important for realizing a charged quantum gas, studying charge transport [13]
at low energy, exploring polaron physics [14–16] and producing ion-atom bound
states [17] and cold molecular ions [18–20]. Molecular ions have lots of prospects
for exploring the new fields of physics and chemistry. Cold molecular ions may be
used as natural candidates for the measurement of electron dipole moment (EDM)
[21, 22]. The study of cold molecular ions has relevance in diverse areas such as
metrology [23, 24] and astrochemistry [25]. Recently, molecular ions have been
cooled into rovibrational ground states by all optical [20], laser and symphathetic
cooling methods [18, 19]. A large variety of diatomic and triatomic molecular
ions are also cooled by symphathetic methods [26–28]. Other methods such as
photoassociative ionization [29–33], buffer gas [34] and rotational cooling [35] have
been widely used for producing low-energy molecular ions. In this chapter, a
new method for preparing cold molecular ion is proposed. A new formalism to
prepare cold molecular ion from colliding cold atom-ion pair in a hybrid trap [1–
3, 36] by photoassociation (PA) [37] process is discussed. Both homonuclear and
heteronuclear collisions are studied with an emphasis on heteronuclear rediative
processes. It is shown that it is possible to form translationally and rotationally
88
Chapter 6. Atom-Ion Cold Collisions
89
Figure 6.1: Schematic diagram of possible physical processes which can take
place during atom-ion collision at low energy.
cold molecular ions by PA, starting from a cold alkaline earth metal ion and an
ultracold alkali-metal atoms.
In section 6.1, we outline the theoretical background for understanding of atom-ion
cold collision dynamics. Then we focus on elastic and inelastic processes in section
6.2. We propose new scheme for the formation of cold molecular ion. In section
6.3, we present results and discussion. In section 6.4, conclusions are drawn.
6.1
Theory of Atom-Ion Collision
In this section, a brief discussion is presented on theoretical framework for understanding atom-ion interaction in the cold and ultracold regimes where a full
Chapter 6. Atom-Ion Cold Collisions
90
E (a.u.)
-21.65
-21.7
1 +
2Σ
1 +
1Σ
-21.75
-21.8
-21.85
+
2
+
Σu
-14.55
E (a.u.)
2
Σg
-14.5
2
Πu
-14.6
-14.65
-14.7
-14.75
0
5
10
15
20
25
30
r (a.u.)
Figure 6.2: Upper panel shows 11 Σ+ (solid) and 21 Σ+ (dashed) model poten2 +
tials of (LiBe)+ system. Lower panel shows 2 Σ+
g (solid)), Σu (dashed-dotted)
+
and 2 Πu (dashed) potentials of Li2 .
quantum mechanical treatment becomes necessary. A point to be noted that we
are focusing on two body collisions and ions are singly charged.
Atom-ion two body collisions can generally be divided into two categories. First
one is the normal elastic collision where there is no change in internal structures
of colliding atom-ion. If the collision is taking place between atom A and ion B + ,
then elastic collision may be denoted as
A + B+ → A + B+.
(6.1)
But elastic collision is not the sole possible outcome of the collision. Due to the
interaction between two colliding particles, an electron may hop from atom A
to ion B + provided that colliding atom and ion are close enough. This type of
collision is known as inelastic collision in which the internal structure is changed.
Chapter 6. Atom-Ion Cold Collisions
91
This is given by
A + B + → A+ + B.
(6.2)
Now it is important to know that the inelastic collision can be divided into various
categories such as radiative and nonradiative charge transfer collision, photoassociative transfer or molecule formation [38–42] and spin-exchange collision. In
following subsections we discuss in detail about elastic and inelastic collisions.
But before that atom-ion interaction potential is discussed.
6.1.1
Atom-Ion Interaction Potential
A neutral atom placed in the electric field produced by an ion have an induced
dipole moment or higher induced multiple moments and the polarised neutral atom
is attracted towards the ion. To derive the corresponding interaction potential we
consider an atom with an induced dipole moment p. We can write p = αε, where
ε is the electric field of the ion and α is the atomic polarizability. The potential
energy of the atom in the presence of electric field of the ion ε is given by
1
1
U = − pε = − αε2.
2
2
(6.3)
Now using the well-known expression for the electric field of an ion, we get
V (r) = −
C4
2r 4
(6.4)
where C4 = q 2 α/4πǫ0 . This provides an attractive potential between the atom
and ion. Here r is the distance between the atom and ion and q is the charge of
ion. So the long range potential [10, 11] is given by the expression
1
V (r) = −
2
C4 C6 C8
+ 6 + 8 ···
r4
r
r
(6.5)
where, C6 , C8 correspond to quadrupole, octupole polarisabilities of atom concerned and C4 /2r 4 is dominating term. The polarisation interaction falls off much
more slowly than van der Waals interaction which represents the long range part
of interaction between neutral atoms. Hence collision between atom and ion is
dominated by the long range polarization interaction. The qualitative feature of
this long range interaction of atom-ion is governed by effective length β4 . It is
Chapter 6. Atom-Ion Cold Collisions
92
3.0
2.5
D(r) (a.u.)
2.0
1.5
1.0
0.5
0.0
0
5
10
15
20
25
30
r (a.u.)
Figure 6.3: Radial transition dipole matrix element as a function of separation
r for (LiBe)+ system.
calculated by equating centrifugal energy with the potential in (6.4). It is found
p
that β4 = µC4 /~2
6.1.2
Elastic Collisions
In case of purely elastic scattering the internal states of atom and ion donot change
as stated above. Continuum wave function for atom-ion cold collision can be
obtained by solving the partial wave Schrödinger equation given by
d2
2µ
ℓ(ℓ + 1)
2
ψℓ (kr) = 0
+ k − 2 V (r) −
dr 2
~
r2
(6.6)
where, r is the ion-atom separation. The wave function ψℓ (kr) has the asymptotic
form ψℓ (kr) ∼ sin [kr − ℓπ/2 + ηl ] with ηℓ (k) being the phase shift for ℓ-th partial
Chapter 6. Atom-Ion Cold Collisions
93
8
linear fit
log10σel (a.u.)
7
6
5
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
log10E (K)
Figure 6.4: Total elastic scattering cross-section σel for Li + Be+ (21 Σ+ )
collision is plotted against collision energy E in K. The dashed curve is a linear
fit for energies greater than 10−6 K.
wave. The total elastic scattering cross section is expressed as
∞
4π X
(2ℓ + 1) sin2 (ηℓ )
σel = 2
k ℓ=0
where k =
(6.7)
p
(2mE/~2 ). As the energy gradually increases more and more partial
waves start to contribute to total elastic scattering cross sections and the scattering
cross section at large energy is [10]
σel ∼ π
µC42
~2
13 1
π2
E− 3 .
1+
16
(6.8)
As k → 0, according to Wigner threshold laws ηℓ (k) ∼ k 2ℓ+1 if ℓ ≤ (n − 3)/2 with
n being the exponent of long-range potential behaving as ∼ 1/r n as r → ∞. If
ℓ > (n − 3)/2 then the threshold law is ηℓ (k) ∼ k n−2 . Since the long-range part of
Chapter 6. Atom-Ion Cold Collisions
94
6.5
linear fit
log10σel (a.u.)
6
5.5
5
4.5
4
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
log10E(K)
Figure 6.5: Same as in Fig.6.4 but for Li+ + Be (11 Σ+ ).
ground as well as excited ion-atom potentials goes as ∼ 1/r 4 as r → ∞, Wigner
threshold laws tell us that s-wave (ℓ = 0) ion-atom scattering cross section should
be independent of k while all the higher partial wave scattering cross sections
should go as ∼ k 2 in the limit k → 0.
6.1.3
Radiative Transfer
Charge Transfer process between atom A and ion B + may take place in following
ways:
(1) radiative charge transfer process
A + B + → A+ + B + ~ω,
(6.9)
Chapter 6. Atom-Ion Cold Collisions
95
4
10
σl (a.u)
10
3
10
l=0
l=2
1
0
10
-1
10
-3
10
-9
10
10
-8
10
-7
-6
10
10
-5
-4
10
-3
10
-2
10
-1
10
10
0
E (K)
Figure 6.6: Partial wave cross sections for Li+ + Be (11 Σ+ ) collision are plotted
as a function of E (in K) for ℓ = 0 (solid) and ℓ = 2 (dashed).
(2) radiative association process or photoassociative charge trasfer
A + B + → (AB)+ + ~ω,
(6.10)
(3) nonradiative charge transfer
A + B + → A+ + B.
(6.11)
Since at low energy nonradiative charge transfer is dominated, we are mainly
interested in the first two processes. The radiative processes occurs due to the
interaction of the system with the radiation field. The direct charge transfer
arises through the transition between different molecular states due to perturbation
caused by the nuclear motion. The radiative charge transfer cross section [38–40]
Chapter 6. Atom-Ion Cold Collisions
96
is given by
σct =
Z
ωmax
ωmin
dσct
dω
dω
(6.12)
where, ω is the angular frequency of emitted photon and
dσct
8ω 3 π 2 X 2
ℓMℓ,ℓ−1 (km , kn )
=
2
dω
3c3 km
l
2
+ (ℓ + 1)Mℓ,ℓ+1
(km , kn )
where,
Mℓ,ℓ′ (km , kn ) =
Z
0
(6.13)
∞
drψℓm (km r)D(r)ψℓn′ (kn r).
(6.14)
D(r) is the magnitude of the molecular transition dipole moment. Here km =
p
p
2µ [E − Vm (∞)] and kn = 2µ [E − Vn (∞) − ~ω] are the momentum of en-
trance and exit channels, respectively; and E is collision energy of entrance (m)
channel. Vm and Vn are the potential energies of the entrance (m) and exit (n)
channels, respectively. ψℓi (kir) is the wave function of ℓ-th partial wave for i-th
channel of momenum ki . The total radiative decay [39] from the upper state (m)
to the lower state (n) is given by
σrt =
∞
π X
(2ℓ + 1) [1 − exp(−4ζℓ )]
2
km
(6.15)
ℓ
where,
π
ζℓ =
2
Z
∞
0
|ψℓm (km r)|2Anm (r)dr
(6.16)
is a phase shift and
|Vn (r) − Vm (r)|3
4
Anm (r) = D 2 (r)
3
c3
(6.17)
is the transition probability. The difference between the total radiative transfer
cross section and the radiative chrage transfer cross section is the cross section is
the radiative association cross section.
Chapter 6. Atom-Ion Cold Collisions
10
97
5
4
10
σl (a.u.)
10
3
10
1
0
l=0
l=2
10
-1
10
-3
10
10
-4
-8
10
10
-7
-6
10
10
-5
-4
10
-3
10
-2
10
-1
10
10
0
E (K)
Figure 6.7: Same as in Fig.6.6 but for Li + Li+ ground state collision in 2 Σg
state.
6.1.4
Spin Exchange Collision
Another outcome of collisions between an alkali atom and an alkali type ion, both
in 2 S1/2 state is spin exchange collision. The interatomic potential is differnt for
the both spin parallel case (triplet state) than both spin anti parallel case (singlet
state). Hence this spin exchange interaction originates from the overall symmetry
requirement of the electron wave function. The quantum mechanical cross section
for spin exchange, in the degenarate internal state approximation [11, 43], is given
by
σ=
π l
Σl=0 (2l + 1) sin2 (δℓtriplet − δℓsinglet ).
2
k
(6.18)
where, δℓtriplet and δℓsinglet are the phase shifts due to scattering from triplet and
singlet state, respectively. A point to be noted that, though the validity of the
approximation is for the collision energies larger than the hyperfine splitting, it
Chapter 6. Atom-Ion Cold Collisions
98
8
7.5
2
+
2
+
log10 σel (a.u.)
Σu
Σg
7
6.5
6
5.5
5
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
log10 E (a.u.)
Figure 6.8: Same as in Fig.6.4 but for Li + Li+ collision in 2 Σ+
g (dashed) and
2 Σ+ (solid) potentials.
u
usually gives right order of magnitude at smaller energies [44]. It is also observed
that spin exchange collision rate increases as the temperature gradually decreases
.
All possible processes except the spin exchange collosion between an alkali atom
and an alkali like ion are shown in Fig.6.1 [45]
6.2
Fomation of Cold Molecular Ion
In our proposed mechanism for formation of cold molecular ion, cold alkali atom
and alkali-type ion are used as starting materials. Formation of ground molecular
ion requires a three-step radiative reaction process. in the first step, the ionatom pair in the excited elecronic state undergoes radiative charge transfer process
to form ground state ion-atom pair. In the second step, these ion-atom pair is
Chapter 6. Atom-Ion Cold Collisions
99
10
σct (a.u.)
8
6
4
2
-1
0
10
10
10
1
E (mK)
Figure 6.9: Charge transfer scattering crosssection σct (in a.u.) of (LiBe)+
system is plotted against collisional energy E (in mK).
photoassociated to the excited electronic bound state in the presence of laser of
suitable frequency. In the last step, the excited molecular ion is stimulated by
another lasser to deexcite to form ground molecular ion. since the molecular ion is
formed from translationally cold atom and ion, the molecular ion is translationally
and rotationally cold.
6.2.1
Formulation of Problem
To illustrate atom-ion radiative cold collisions, we consider a model system of 7 Li
+ Be+ undergoing elastic and radiative charge transfer collisions. The possible
experimental situation can be imagined as a single Be+ ion immersed in a BoseEinstein condensate of 7 Li atoms in a hybrid trap.
The molecular potentials 11 Σ+ (ground) and 21 Σ+ (excited) of (LiBe)+ system
as shown in Fig.6.2 (upper inset) asymptotically go to 1 S +1 S (Li+ + Be) and 2 S
Chapter 6. Atom-Ion Cold Collisions
100
2
7
|ηJl| x 10 (a.u.)
250
200
150
100
50
400
2
|ηJl| (a.u.)
600
200
0 -8
10
10
-7
-6
10
10
-5
10
-4
-3
10
-2
10
-1
10
0
10
1
10
E (K)
Figure 6.10: Square of Franck Condon overlap integral |ηJℓ |2 (in a.u.) for
Li-Li+ (upper) and (LiBe)+ (lower) is plotted against E (in K). In the upper
panel, |ηJℓ |2 is mutiplied by a factor of 107 .
+2 S (Be+ + Li), respectively. We construct model potentials 11 Σ+ (ground) and
21 Σ+ (excited) of (LiBe)+ system using spectroscopic constants given in Ref.[46].
Short range potential is approximated using Morse potential and the long range
potential [10, 11] is given by the expression (6.5).The short range and long range
parts of the potentials are smoothly joined by spline.
Since Li+ may be formed due to charge transfer collision between Be+ and Li,
we need to consider the interaction between this Li+ and other Li atoms present
+
2 +
2
in the condensate. The data for 2 Σ+
g , Σu and Πu potentials of Li2 are taken
from Ref.[47]. Dissociation energy De , equilibrium position re and effective range
β4 of the ground and excited state potentials of (LiBe)+ and LiLi+ systems are
given in Table 6.1. A comparison of potentials of these two systems reveals that
ground state potential 11 Σ+ of (LiBe)+ is much shallower than 2 Σ+
g potential of
Li+
2 . The equilibrium positions of both ground and excited state potentials of
Chapter 6. Atom-Ion Cold Collisions
101
Table 6.1: Dissociation energies De in a.u., equilibrium positions re and the
effective lengths β4 in Bohr radius for excited and ground state potentials (V (r))
of (LiBe)+ and (LiLi)+ systems.
system
(LiBe)+
(LiLi)+
(LiBe)+
(LiLi)+
V(r)
21 Σ+
2
Πu
1 +
1Σ
2 +
Σg
De
0.06
0.01
0.02
0.05
re
5.46
7.50
5.03
6.00
β4
1083.4
1019.8
515.9
1019.8
(LiBe)+ system lie almost at the same separation. Unlike the asymptotic behavior
of the excited 21 Σ+ potential of (LiBe)+ system, the excited state potential 2 Πu
+
of homonuclear Li+
2 molecular ion asymptotically corresponds to one Li ion in
the electronic ground S state and one neutral Li atom in the excited P state. The
equilibrium positions re of ground and excited state potentials of Li+
2 system are
shifted by 1.5 Bohr radius. For of (LiBe)+ system, we notice that β4 of excited
(21 Σ+ ) potential is almost twice that of the ground (11 Σ+ ) potential.
Let us first consider cold collision between Li and Be+ with both of them being
in 2 S electronic state. So, our initial system corresponds to the continuum of
21 Σ+ potential. Due to charge transfer collision neutal Be atom and Li+ ion are
generated. In the separated two-particle limit of this system, dipole transition to
ground state at the single particle level is forbidden. Furthermore, since at low
energy non-radiative charge transfer is suppressed, the dominant inelastic channel
is the radiative charge transfer transition that occurs at intermediate or short
separations. Electronic transition dipole moment between two ionic molecular
electronic states vanishes at large separation. Therefore, transitions occur at short
range where hyperfine interaction is negligible in comparison to central(Coulombic)
interaction.
The ground continuum atom-ion pair, formed by radiative charge transfer process,
can be photoassociated to form excited molecular ion. This process is basically
one photon PA process. The photoassociation rate coefficient is given by
KP A =
*
∞
πvr X
(2ℓ + 1)|SP A (E, ℓ, wL )|2
2
k
ℓ=0
+
(6.19)
where, vr = ~k/µ is the relative velocity of the two particles and h· · · i implies
averaging over thermal velcity distribution. Here SP A is S matrix element given
Chapter 6. Atom-Ion Cold Collisions
102
by
|SP A |2 =
δE2
γΓℓ
+ (Γℓ + γ)2 /4
(6.20)
where δE = E/~+δvJ , δvJ = ωL −ωvJ with EvJ = ~ωvJ being binding energy of the
excited ro-vibrational state, ωL being the laser frequency and γ the spontaneous
line width. Thus PA rate is primarily determined by partial wave stimulated line
width Γℓ given by
~Γℓ =
πI
h(J, ℓ)|DvJ,l |2
ǫ0 c
(6.21)
where,
DvJ,l = hφvJ | D(r) | ψℓ (kr)i
(6.22)
is the radial transition dipole matrix element between the continuum and bound
state wave functions ψℓ (kr) and φvJ (r), respectively. I is the intensity of laser, c is
the speed of light and ǫ0 is the vacuum permittivity. Here h(J, ℓ) is Hönl London
factor [48] which in the present context is given by
h(J, ℓ) = (1 + δΛ′ 0 + δΛ′′ 0 − 2δΛ′ 0 δΛ′′ 0 )
!2
J
1
ℓ
(2J + 1)(2ℓ + 1)
−Λ′ Λ′ − Λ′′ Λ′′
(6.23)
where Λ′ and Λ′′ are the projections of the total electronic orbital angular momentum of the excited and ground states, respectively, on molecular axis and (· · · ) is
the Wigner 3j symbol. The spontaneous line width γ of the excited state (v, J) is
given by
Z
1
~γ =
(∆E)3 |hφvJ | D(r) | ψE i|2 dE
3πǫ0 c3
#
X
∆3v′ J ′ |hφvJ | D(r) | φv′ J ′ i|2
+
(6.24)
v′ ,J ′
where ∆E = (EvJ − E)/~, ∆v′ J ′ = (EvJ − Ev′ ,J ′ )/~, ψE is the scattering wave
function and | φv′ J ′ i stands for all the final bound states to which the excited state
can decay spontaneously.
~ +L
~ + ~ℓ where S and
The total molecular angular momentum is given by J~ = S
Chapter 6. Atom-Ion Cold Collisions
103
L are the total electronic spin and orbital quantum number, respectively; and ℓ
stands for the angular quantum number of the relative motion of the two atoms.
For the particular model for (LiBe)+ system chosen here, we have L = 0 and S = 0
for both the ground and the excited electronic states. Thus here the total angular
momenta for both the ground and excited states are given by J = ℓ. However, it is
more appropriate to denote total angular quantum number of a molecular bound
state by J and that of the continuum or collisional state of this atom-ion system
by simply ℓ. The parity selection rule for the electric dipole transition between
the ground and excited states dictates ∆J = ±1.
In next step, a ground-state molecular ion is formed from excited molecular ion by
a stimulated Raman-type process by applying a second laser tuned near a boundbound transition between the excited and the ground potentials. Necessary criteria
is that the turning points of the two selected bound states should lie in almost
same distances, which results in large Franck Condon overlap between them. Rabi
frequency Ω for such bound-bound transition is given by
~Ω =
I
4πcǫ0
12
′
′
~
|hv, J | D(r).ǫ̂
L | v , J i|
(6.25)
where ǫ̂L is the unit vecot of laser polarization and | v, Ji and | v ′ , J ′ i are the two
bound states with hr | v, Ji = φvJ (r).
6.3
Results and Discussion
Standard renormalized Numerov-Cooley method [49] is used to calculate the bound
and scattering state wave functions. The molecular transition dipole matrix element of (LiBe)+ system is calculated using GAMESS. This matrix element strongly
depends upon separation and goes to zero at a large r as shown in Fig.6.3. In
Figs.6.4 and 6.5, we have plotted the excited and ground state elastic scattering
cross section σel as a function of energy E for Li + Be+ and Li+ + Be collisions,
respectively. We find that at least 35 partial waves are required to get converging results on elastic scattering at 1 µK. As the energy gradually increases larger
numbers of partial waves would be required to get converging results. At high
1
energies, for both the cases, σel decreases as E − 3 . The proportionality constant c
1
in the expression σel (E → ∞) = cE − 3 calculated using Eq. (6.8) for excited 21 Σ+
and ground 11 Σ+ potentials are 2936 and 1091 a.u., respectively, whereas linear
Chapter 6. Atom-Ion Cold Collisions
104
ψl (kr) (a.u.)
40
20
0
-20
-40
0
5
10
15
20
25
0
5
10
15
20
25
φvJ (r) (a.u.)
0.8
0.4
0
-0.4
r (a.u.)
Figure 6.11: Energy-normalized s-wave ground scattering (upper) and unit
normalized excited bound (lower) wave functions of (LiBe)+ system are plotted
as a function of separation r.
fit to σel vs. E curves provides c = 3548 and 1335 a.u., respectively. Figs.6.6 and
6.7 exhibit s- and d-wave partial scattering cross section as a function of energy
for Li+ +Be and Li+Li+ collisions, respectively. In Fig.6.8, we have plotted total
2 +
elastic scattering cross section for Li+Li+ collisions in 2 Σ+
g and Σu potentials.
Starting from the low energy continuum state of Li + Be+ collision in the 21 Σ+
potential, there arise two possible radiative transitions by which the system can
go to the ground electronic state 11 Σ+ . One is continuum-continuum and the
other is continuum-bound dipole transition. The transition dipole moment as a
function of separation as shown in Fig.6.3 shows that the dipole transition probability will vanish as the separation increases above 20a0 . So, a dipole trasition
has to take place at short separations. Let us consider radiative transfer processes from the upper (21 Σ+ ) to the lower (11 Σ+ ) state of (LiBe)+ . We then
need to apply the formulae (6.12) and (6.15) where m ≡ 21 Σ+ and n ≡ 11 Σ+ in
Chapter 6. Atom-Ion Cold Collisions
105
ψl (kr) (a.u.)
40
20
0
-20
-40
0
5
10
15
20
25
0
5
10
15
20
25
φvJ (r) (a.u.)
0.4
0.2
0
-0.2
-0.4
r (a.u)
Figure 6.12: Same as in Fig.6.11 but for Li-Li+ system.
our case. Continuum-continuum charge transfer cross section σct between 21 Σ+
and 11 Σ+ states of (LiBe)+ system is plotted against E in Fig.6.9. We evaluate
the photoassociative (continuum-bound) transfer cross section by subtracting σct
from the total radiative transfer cross section σrt calculated using the formula
(6.15). At energy E = 0.1 mK, σct and the photoassociative transfer cross section
are found to be 10.39 a.u. and 0.03 a.u., respectively. Thus we infer that the
continuum-continuum radiative charge transfer process dominates over the radiative association process. Also, we notice that σct is smaller than both the excited
and ground state elastic scattering cross sections σel (as given in Figs.6.4 and 6.5,
respectively) by several orders of magnitude.
Molecular dipole transitions between two ro-vibrational states or between continuum and bound states are governed by Franck-Condon principle. According to
this principle, for excited vibrational (bound) states, bound-bound or continuumbound transitions primarily occur near the turning points of bound states. In
general, highly excited vibrational state wave functions of diatomic molecules or
Chapter 6. Atom-Ion Cold Collisions
106
Table 6.2: Ro-vibrational energy (Ev J ), inner (ri ) and outer turning points
(ro ) of two selected bound states of (LiBe)+ molecular ion - one bound state in
excited (21 Σ+ ) and the other in ground (11 Σ+ ) potential. The energy Ev J is
measured from the threshold of the respective potential.
Potential
21 Σ+
11 Σ+
v
68
29
J
1
0
EvJ (a.u.)
-3.30×10−3
-0.25×10−3
ri (a.u.)
3.4
3.8
ro (a.u.)
16.3
16.6
molecular ions have their maximum amplitude near the outer turning points. Spectral intensity is proportional to the overlap integral.This means that the spectral
intensity for a continuum-bound transition would be significant when the continuum state has a prominent node near the outer turning point of the bound state.
For transitions between two highly excited bound states, Franck-Condon principle implies that the probability of such transitions would be significant when the
outer turning points of these two bound states lie nearly at the same separation.
The upper panel of Fig.6.10 shows the variation of the square of franck Condon
overlap integral |ηJℓ |2 between the ground s-wave (ℓ = 0) scattering and the ex-
cited ro-vibrational (v = 26, J = 1/2) states of Li-Li+ system as a function the
collision energy E. The lower panel of Fig.6.10 displays the same as in the upper
panel but for (LiBe)+ system with v = 68 and J = 1. The excited ro-vibrational
state v = 26, J = 1/2 of Li-Li+ is very close to dissociation threshold while the
excited ro-vibrational state v = 68, J = 1 of (LiBe)+ system is a deeper bound
state. Thses two excited states are so chosen such that free-bound Franck-Condon
overlap integral for both the systems become significant. Comparing these two
plots, we find that |ηJℓ |2 of Li-Li+ system is smaller than that of (LiBe)+ sys-
tem by seven orders of magnitude. To understand why the values |ηJℓ |2 for the
two systems are so different, we plot the the energy-normalized s-wave ground
scattering and the bound state wave functions of (LiBe)+ system in Fig.6.11 and
those of Li-Li+ system in Fig.6.12. A comparison of Figs.6.11 and 6.12 reveals
that, while in the case of (LiBe)+ the maximum of the excited bound state wave
function near the outer turning point coincides nearly with a prominent antinode
of the scattering wave fucntion, in the case of Li-Li+ the maximum of the bound
state wave function near the outer turning point almost coincides with a minimum
(node) of the scattering wave function. These results indicate that the possibility
of the formation of excited LiLi+ molecular ion via PA is much smaller than that
of (LiBe)+ ion. We henceforth concentrate on PA of (LiBe)+ system only.
We next explore the possibility of PA in Li+ -Be cold collision in the presence
Chapter 6. Atom-Ion Cold Collisions
107
7
l=0
l=2
6
4
2
| DvJ, l | (a.u.)
5
3
2
1
0 -8
10
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
-1
10
10
0
10
1
10
2
E (K)
Figure 6.13: Square of free-bound radial transition dipole moment (|DvJ,l |2 )
(in a.u.) for ground continuum states with ℓ = 0 (solid) and ℓ = 2 (dashed) and
excited bound ro-vibrational level with v = 68 and J = 1
of laser light. As discussed before, continuum-bound molecular dipole transition
matrix element depends on the degree of overlap between continuum and bound
states. PA rate (6.19) is proportional to the square of free-bound radial transition
dipole moment element |DvJ,l |2 . In Fig.6.13 we plot |DvJ,l |2 against E for s- (ℓ = 0)
and d-wave (ℓ = 2) ground scattering states and v = 68 , J = 1 excited molecular
state. It is clear from this figure that the contributions of both ℓ = 0
and ℓ = 2 partial waves are comparable above enegy corresponding to 0.1 mK. At
lower energy (E < 0.1 mK), only s-wave makes finite contribution to the the dipole
transition. Fig.6.14 exhibits |DvJ,l |2 as a fucntion of E for the transition from s-
wave (ℓ = 0) scattering state of the excited (21 Σ+ ) continuum to the ground
(11 Σ+ ) ro-vibrational state with v = 36 , J = 1. A comparison between the
Figs.6.13 and 6.14 reveals that the probability for the transition from the upper
continuum to the ground bound state is smaller by several orders of magnitude
Chapter 6. Atom-Ion Cold Collisions
108
-4
10
2
| DvJ, l | (a.u.)
-5
10
-6
10
10
-7
10
-8
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
E (K)
Figure 6.14: Same as in Fig.6.13 but for excited continuum state with ℓ = 0
and ground ro-vibrational state with v = 36 and J = 1.
than that from ground continuum to an excited bound state. In Fig.6.15, we have
plotted the rate of photoassociation KP A as a function of temperature T for laser
frequency tuned at PA resonance. The ion-atom PA rate as depicted in Fig.6.15
is comparable to the typical values of rate of neutral atom-atom PA at low laser
intesities. In Fig.6.16 we have plotted the rate of photoassociation as a function
of laser intensity at a fixed temperature T = 0.1 mK to show the saturation effect
that occurs around intensity I = 50 kW/cm2 . Thus the formation of excited
(LiBe)+ molecular ion by photoassociating colliding Li+ with Be with a laser of
moderate intensity appears to be a feasible process.
Now we discuss the possibility of formation of ground state molecular ion by
stimulated Raman-type process. Let us consider two selected bound states whose
salient features are given in Table 6.2. The outer turning points of these two
bound state almost coincide implying large Franck Condon overlap integral. Rabi
frequency corresponding to this bound-bound transition is found to be 285 MHz
Chapter 6. Atom-Ion Cold Collisions
109
-13
10
-14
10
3 -1
KPA (cm s )
-15
10
-16
10
10
10
-17
-18
-19
10
10
-5
-4
-3
10
10
10
-2
T (K)
Figure 6.15: Rate of photoassociation KP A (in cm3 s−1 ) of (LiBe)+ is plotted
against temperature (in K) at I = 1 W/cm2 and δvJ = ωL − ωvJ = 0
for laser intensity I = 1 kW/cm−2 . Comparing this value with the spontaneous
line width γ = 57 kHz of the excited bound state calculated using the formula
(6.24), we infer that even at a low laser intensity which is far below the saturation
limit, bound-bound Rabi frequency Ω exceeds γ by several orders of magnitude.
This indicates that it may be possible to form ground molecular ion by stimulated
Raman-type process with two lasers.
6.4
Conclusions
In this chapter, we have shown that alkaline earth metal ions immersed in an ultracold gas of alkali atoms can give rise to a variety of cold chemical reactions considering system of a Beryllium ion interacting with cold Lithium atoms. We have
predicted the formation of translationally and rotationally cold (LiBe)+ molecular
Chapter 6. Atom-Ion Cold Collisions
3 -1
KPA (cm s )
10
10
110
-11
-12
10
-13
-14
10 0
10
20
30
40
50
60
2
I (kW/cm )
Figure 6.16: KP A (in cm3 s−1 ) of (LiBe)+ is plotted as a function of laser
intensity I (in kW/cm2 ) at temperature T = 0.1 mK with laser tuned at PA
resonance.
ion by photoassociation. Theoretical understanding of low energy atom-ion scattering and reactions may be important for probing dynamics of quantum gases.
Since both Bose-Einstein condensation and fermionc superfluidity have been realized in atomic gases of Lithium, understanding cold collisions between Lithium
and Beryllium ion may be helpful in probing both bosonic and fermionic superfluidity. In particular, this may serve as an important precursor for generating and
probing vortex ring in Lithium quantum gases. It may also provide us new possibilities to study quantum interference phenomena like Stueckelberg oscillations
[50] and Landau-Zenner [51] transitions in atom-ion cold collisions which will be
briefly discussed in the next chapter.
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Chapter 7
Conclusions and Outlook
In this thesis, we have presented a study on ultracold collisions with an emphasis
on quantum interference, specially on Fano interference in atom-molecule coupled
systems. We have studied photoassociation in the presence of Feshbach resonance.
We have showed that line width of photoassociation spectrum can be narrowed
by Fano interference in Feshbach resonance along with large spectral shifts. We
have also suggested novel PA schemes for the realization of vacuum-induced and
light-induced coherences in atom-molecule coupled systems. We have showed that
it is possible to generate and manipulate coherence between excited ro-vibrational
states of a molecule by PA lasers. We have also analyzed atom-ion cold collisions
and predicted a new pathway for the formation of translationally and rotationally
cold molecular ion by cold collisions between atoms and ions in the presence of
laser light. Such studies may be beneficial for modification of decoherence and
dissipation in cold molecules, controlling the dynamics of quantum gas, creation
of artificial media with negative refractive index [1] (i.e with simultaneous negative
electric and magnetic permittivity) etc.
Now, we would like to mention some of our future plans. First of all, we wish
to study cold atom-ion collisions for different heteronuclear and homonuclear alkali atom and alkali ion pairs and alkali atom and alkaline earth metal ion pairs
and to check the possibility of formation of cold molecular ions in these systems.
We also have plans to study quantum interference phenomena like field induced
Stüeckelberg oscillations [2] and Landau-Zener [3] transitions in atom-ion cold
collisions. A transition between two potential curves at an avoided crossing due
to motional effects is known as Landau-Zener transition. Under certain simple
115
Chapter 7. Conclusion and Outlook
116
assumptions, the transition probability can be written as
P = exp −
2πV122
v(V11 − V22 )
(7.1)
Here V11 and V22 denote the slopes of the diabatic potential curves near pseudocrossing and V12 is the diabetic coupling potential. V12 is estimated from the
adiabatic energies Ei (R) as V12 = 21 ∆E = 12 |E1 (rx ) − E2 (rx )|. rx is the distance at
pseudo-crossing takes place and v is the velocity at the of the colliding particles.
Let us now consider a reaction process, in which system enters along the adiabatic
curve E2 (r) (diabetic curve V22 (r)) and leaves the reaction on curve E1 (r) (diabetic curve V11 (r)). Transition between these two curves may happen on either
the incoming or the outgoing side of the reaction. If both paths have the same
scattering angle, they are indistinguishable and therefore interfere. The resulting interference pattern has been first recognized by Stückelberg in 1932. We have
plans to study these effects in ultracold atom-ion collisional systems like in (SrX)+
and (MgX)+ , where X = Li, Na, K, Rb, Cs. Stückelberg oscillation may be measured as a function of scattering angle. These types of studies may be helpful for
solving problems in fast and slow collisions between atoms and ions without going
into numerical detail of multichannel close-coupling method [4].
Another project which we wish to do in near future is to study photoassociation
(PA) in the presence of Feshbach resonance (FR) in bosonic 7 Li and fermionic 6 Li
atoms as multichannel scattering. In chapter 4, we have already studied the same
problem in the framework of Fano’s theory. Now our aim is to check it using multichannel close-coupled scattering calculation [5]. The coupled equations introduce
a n × n interaction matrix. The diagonal elements of the interaction matrix define
the diabtic channel. The off-diagonal terms of the interaction matrix describe the
interaction between different channels. We plan to apply quantum defect theoretic approach [6] to solve multichannel scattering problem. On the other hand, a
multichannel problem may be simplified to a model of two-channel configuration
interaction in some cases [5]. The model consists of a single open-channel continuum state and a single isolated resonance closed-channel which incorporates the
exact closed-coupled interactions between multichannel closed states. For illustration, two ground state Li atoms undergoing collision in the presence of magnetic
field may be considered. For atomic 6 Li, electronic spin and nuclear spin quantum number ~s and ~i are 1/2 and 1, respectively. Hence the possible hyperfine
quantum numbers f~ are 3/2 and 1/2 [7]. In the absence of magnetic field their
Chapter 7. Conclusion and Outlook
117
projections mf s are degenerate. But the applied magnetic field splits different mf
states at asymptotic limit. So for two-atom collision, one threshold would split
into a number of channels in the long-range. Our objective is to calculate the
modified scattering wave functions at the magnetic field at FR with the help of
multichannel quantum defect theory and then to demonstrate PA in comparison
to the results obtained in chapter 4.
This work may help to develop insight into the research areas of atom-molecule
conversion and optical and magneto-optical Feshbach resonance at ultralow energy.
Our work may stimulate further studies on laser manipulation of continuum and
bound states. It may become useful for the studies of controlling decoherence
and dissipation in cold molecules. Hence we are hopeful that this thesis may
make a significant and timely contribution to the advancement of knowledge in
the emerging interdisciplinary fields of physics and chemistry.
References
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118