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From: Svetlana A. Malinovskaya, Tom Collins, and Vishesha Patel. Ultrafast
Manipulation of Raman Transitions and Prevention of Decoherence
Using Chirped Pulses and Optical Frequency Combs.
In John R. Sabin and Erkki J. Brändas, editors:
Advances in Quantum Chemistry, Vol.64,
Burlington: Academic Press, 2012, pp. 211-258.
ISBN: 978-0-12-396498-4
© Copyright 2012 Elsevier Inc.
Academic Press
Author's personal copy
CHAPTER
7
Ultrafast Manipulation of Raman
Transitions and Prevention of
Decoherence Using Chirped Pulses
and Optical Frequency Combs
Svetlana A. Malinovskaya, Tom Collins, and
Vishesha Patel
Contents
1. Introduction
2. Chirped Pulse Control in Coherent Anti-Stokes Raman Scattering
for Imaging of Biological Structure and Dynamics
2.1. Theory
2.2. On the role of the ac Stark shifts in the nature of
two-photon Raman transitions
2.3. The ‘‘roof’’ method
3. The Impact of Phase and Coupling Between the Vibrational
Modes on Selective Excitation in CARS Microscopy
3.1. Theory
3.2. Numerical results and discussion
4. Selective Excitation of Raman Transitions by Two Chirped Pulse
Trains in the Presence of Decoherence
4.1. Theoretical model
4.2. Numerical results: The effects of the vibrational energy
relaxation and collisional dephasing on coherence loss
4.3. Prevention of decoherence
5. Feshbach-to-Ultracold Molecular State Raman Transitions via a
Femtosecond Optical Frequency Comb
5.1. Theoretical model
212
216
217
219
223
227
227
229
238
239
242
243
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247
Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey, USA
E-mail address: [email protected]
Advances in Quantum Chemistry, Volume 64
ISSN 0065-3276, http://dx.doi.org/10.1016/B978-0-12-396498-4.00007-7
#
2012 Elsevier Inc.
All rights reserved.
211
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Svetlana A. Malinovskaya et al.
5.2. Numerical results: Creation of ultracold molecules using a
standard optical frequency comb and a phase-modulated
optical frequency comb
6. Conclusions
Acknowledgments
References
Abstract
249
254
255
255
Femtosecond, chirped laser pulse-based methods are presented to control
coherence in vibrational degrees of freedom in molecules via Raman transitions. The controllability of the methods is analyzed in the presence of fast
decoherence and the coupling between vibrational modes. Applications are
diverse and include the development of novel imaging techniques based on
the coherent anti-Stokes Raman scattering, and the implementation of the
optical frequency combs for internal state cooling from Feshbach molecules
aiming at creation of deeply bound ultracold polar molecules.
1. INTRODUCTION
Coherent optical light interacting with atoms and molecules is at the heart of
the atomic, molecular, and optical physics research that has been attracting
scientific minds already for a half-century. Multiphoton transitions that
result from light–matter interactions occur between discrete levels in
atoms and molecules and appeal to quantum properties of both the light
and matter. Atoms and molecules, when excited, in response modulate the
coherent light during its propagation through the medium; this modulation
may be significant and measurable. One of the exciting examples is twophoton resonances induced in molecules followed by a generation of Raman
fields whose frequencies differ from the carrier frequency of an incident light
by exactly the amount of vibrational frequency of a specific vibrational
mode. This phenomenon makes Raman spectroscopy an attractive and a
practical tool for investigations of the structure and ultrafast dynamics since
it possesses a signature of molecules or molecular groups involved. Studies
of the nature of Raman scattering include an investigation of fundamental
phenomena of light–matter interactions: light-induced population transfer
and coherence, as well as intrinsic effects taking place in molecules, such as
the coupling between vibrational modes via external fields. The amount of
coherence created between the vibrational levels serves as a source for a
generation of the Raman fields and provides with the Raman gain. Quantum
coherence between two states, which are, within our interests, the vibrational states of the ground electronic state, is measured by the quantity jai*ajj,
where ai and aj are the probability amplitudes of these states in the total wave
function of the system. The investigation of coherence between states
induced by the two-photon Raman transitions is one of the main topics of
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Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence
213
our interest and is discussed here. We also present our results on newly
developed quantum control methods aiming at creation of a maximum
coherence between predetermined states in atoms and molecules.1–4 A fundamental interest of creating a maximum coherence may be understood from
the Maxwell equations that show that the amplitude of the Raman fields is
proportional to the induced coherence in a system.5 We also look into the
effects of the coupling between vibrational modes via external fields on
controllability of the proposed methods.6 Nowadays, femtosecond pulse
techniques are admittedly prevailing over those using continuous-wave
beams to probe and control ultrafast dynamics in atoms and molecules.
Furthermore, the implementation of femtosecond, chirped laser pulses is
known to be advantageous for these purposes because of their robustness
in a broad range of pulse parameters, such as the pulse intensity, chirp rate,
and the pulse duration. Two-photon Raman transitions induced by chirped
femtosecond pulses have been implemented in solving a variety of physical,
chemical, and biomedical problems. One of the leading techniques for the
imaging of molecular specific structure is based on the induction of twophoton resonances in samples and is known as the coherent anti-Stokes
Raman scattering (CARS) microscopy. We have dedicated our goals to the
development of quantum control methods to advance bioimaging techniques
based on CARS microscopy. One of the critical issues related to coherence
property in condense phase is its loss. Decoherence is the scourge of many
processes that scientists are eager to take a control over within the research
areas that span from biomedicine to quantum computation. We present our
results of investigations on optimizing the quantum response of a system in
the presence of fast decoherence.7–9 We study decoherence phenomenologically with two main channels taken into account, the spontaneous emission
and collisional dephasing. We find that an application of the chirped pulse
trains may be very advantageous. One of the topics of our great interest is the
usefulness of Raman transitions in application to ultracold gases manipulation.10–12 We analyze a possibility of using the phase-locked pulse trains that
form the frequency combs in cooling of the internal degrees of freedom in
molecules and creating them in the fundamentally cold state—the ground
electronic state with zero vibrational and rotational quantum number.
The hallmark of a chirped pulse is a frequency which is time dependent.
So, as time evolves the frequency of the light of the laser pulse changes, thus
allowing for a range of frequencies to be generated over the time of lasing.
In a linear chirp, the instantaneous frequency is o(t) ¼ o0 þ at, which is linear
in t. The coefficient a is the chirp parameter and determines how quickly
the pulse’s frequency will change as time evolves. A pulse is said to be
up-chirped if a > 0 and said to be down-chirped if a < 0. The chirp parameter
can be controlled in an experiment by use of optical devices.13 The instantaneous frequency is the time derivative of the phase of the field
F(t) ¼ o0t þ at2/2 þ f, where f is an arbitrary phase constant. The electric
field at a point in space through which the pulse travels is given by
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Svetlana A. Malinovskaya et al.
EðtÞ ¼ E0 et
2
=t2
at2
þf :
cos o0 t þ
2
(1)
Here, the exponential term, if plotted on its own, forms the Gaussian pulse
envelope. The t is the pulse duration and the E0 is the peak amplitude of the
field. The value o0 is the carrier frequency of the pulse. In the laboratory, not
just one but many time-delayed pulses are utilized. Pulses are generated at a
set time interval T and may be described as
!
X
aðt nTÞ2
ðtnTÞ2 =t2
þf :
(2)
E0 e
cos o0 ðt nTÞ þ
EðtÞ ¼
2
n
Such pulse trains may be generated from mode-locked lasers. A modelocked laser is one in which we have a large number of oscillating modes.
These modes vibrate with different frequencies that are spaced by the same
amount, Dn. This even spacing of the frequencies of the various components
yields a final waveform that, even though it may appear chaotic, is periodic
in time, forming a pulse train. The relative phase between successive modes
is locked. This means that for two successive modes, l and l 1, the difference in phase obeys the relation fl fl 1 ¼ f.14
Now, let us consider what this pulse train looks like when we switch to the
frequency domain. Assume we have the case of 2n þ 1 oscillators, each
separated in the frequency domain by Dn and phases that obey the modelocking condition. If each oscillator contributing to the pulse has the same
maximum amplitude, plotting the pulse in the frequency domain will give
us a series of Dirac delta functions F(o) ¼ Sld[o (o0 2plDn)]. The function
F(o) is the frequency comb of the field E(t). Its spectral bandwidth is determined by t 1 and the spacing between the high intensity peaks is given by
Dn ¼ T 1. Modern mode-locked lasers typically produce pulse trains at the
repetition rate frep about 100 MHz. Then, the optical frequencies nn of the
comb lines can be written as nn ¼ nfrep þ f0, where n is a large integer number
of the order 106 and f0 is the offset frequency due to carrier–envelope phase
shift.15 The pulse train described by Eq. (2) will produce zero offset frequency
since the carrier–envelope phase difference is zero. An optical frequency
comb has been recognized as a new and unique tool for high-resolution
spectroscopic analysis of internal energy structure and dynamics as well as
for controlling ultrafast phenomena in atomic and molecular physics.16–19
Owing to its broadband spectrum, the frequency comb may efficiently
interact with the medium inducing one-photon, two-photon, and multiphoton resonances between finely structured energy levels. A unique ability of
the frequency comb is provided by the presence of about a million optical
modes in its spectrum with very narrow bandwidth and exact frequency
positions.20 During the past few years, the investigations have been carried
out on implementation of a femtosecond frequency comb to manipulate
ultracold gases. The pioneering works in quantum control in ultracold
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Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence
215
temperatures include the two-photon excitation of specific atomic levels
forming four-level diamond configuration in cold 87Rb using a phase-modulated, femtosecond optical frequency comb21 and a theory on piecewise
stimulated Raman adiabatic passage performed with two coherent pulse
trains possessing the pulse-to-pulse amplitude and chirped phase variation.22,23 In these papers, the authors reported on creation of ultracold KRb
molecules from Feshbach states, the highest excited vibrational states of the
ground electronic state, using the pump–dump stepwise technique that
coherently accumulates population in the ground vibrational state. Experimentally, a dense quantum gas of ultracold KRb polar molecules was produced from the Feshbach molecules using the STIRAP scheme with two
microsecond pulses.24 Coherent population transfer was demonstrated to
rovibrational ground state of the triplet and singlet electronic ground potential. We describe how to use a single optical frequency comb to induce twophoton Raman transitions from the Feshbach state to a deeply bound molecular state to create ultracold molecules.
Nowadays, chirped femtosecond pulses are widely used in investigations
of ultrafast phenomena in natural sciences. In the past decade, CARS microscopy has developed as a promising technique for imaging of various objects
of biological interest, for example, living cells, cancerous cells, and also for
combustion diagnostics, and monitoring of molecules. The recent advances
in shaping of ultrafast femtosecond laser pulses25 along with demonstration
of different experimental and theoretical techniques26–35,1,36,37 of driving a
system to the desired quantum yield have made CARS a major tool of
investigation of biological structures. Experimental configurations such as
Box-CARS,38 FM-CARS,39 EPI-CARS,40 heterodyne CARS,41 polarization
CARS,42 FT-CARS,43 and interferometric CARS44 are among the most promising ones. A major drawback of CARS technique is the nonresonant background signal. Implementation of the femtosecond pulses in combination
with the quantum control methods makes it possible to selectively drive a
predetermined Raman transition and effectively suppress the background
signal. It has been shown recently4 that an application of femtosecond,
chirped laser pulses induces adiabatic passage in a system and provides
an optimal CARS signal.
CARS is a spectroscopic method which relies on three photons in order to
probe a multilevel quantum system formed by Raman active vibrational
modes in a molecule. The process utilizes the two-photon transitions in the
following manner: a pump pulse of frequency op interacts with the system in
its initial, usually ground electronic and vibrational state exciting it to a
virtual level. Next or simultaneously, a Stokes pulse of frequency os interacts
with the system that is now in the virtual state and causes it to decay to one
of its excited vibrational states of the ground electronic state with creation of
a coherent superposition state. At this point, a probe pulse of frequency opr
interacts with the induced coherence in the system and excites the system
further to another virtual state, from which the system decays back to its
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Svetlana A. Malinovskaya et al.
ground electronic and vibrational state, emitting an anti-Stokes photon of
frequency oAS ¼ (op os) þ opr. If the resonance condition is satisfied, the
quantity (op os) is equal to the frequency between the ground and the
excited vibrational state of the system. The emitted anti-Stokes photons are
coherent, and thus this method of spectroscopy is more efficient than spontaneous Raman spectroscopy, in which only a single frequency source, the
pump pulse, excites the system to a virtual level and then allows spontaneous decay of the excited vibrational states down to the ground. The signal
from the spontaneous emission is orders of magnitude weaker than that
from the stimulated Raman scattering. Spontaneous Raman spectroscopy is
often regarded in combination with the surface enhancement techniques.
As with any spectroscopic method, the spectral resolution is of key importance. In order to identify the presence of certain chemical agents in a
sample, it is desirable to have as fine a ‘‘tunable’’ spectrum as possible so
that it could be set to resonance with the energy difference between certain
states of the agent to be detected, such as the vibrational state of the molecule.13,45 The use of chirped pulses allows for selective Raman excitations
which are necessary. Presently, using shaped femtosecond pulse techniques
a spectral resolution can be obtained within about 3 cm 1.45 Described in
this chapter, new control methods impose no limitation on the magnitude of
the frequency difference between Raman active vibrational modes.
The chapter is organized as follows. Section 2 is devoted to the description
of the quantum control methods aiming at creation of a maximum coherence
in a predetermined mode in CARS. Methods are distinguished depending
on the scheme of chirped pulse implementation and the nature of the
induced dynamics. In Section 3, the impact of the coupling between vibrational modes via external fields is investigated on controllability of selective
excitation of Raman transitions. In Section 4, we describe a method to create
a maximum coherence in the presence of fast decoherence by using chirped
pulse trains. The method utilizes a series of chirped pulses with locked
phase that form a frequency comb. Section 5 is dedicated to the application
of optical frequency combs to control the dynamics in ultracold temperatures, namely, to perform rovibrational cooling in molecules from the
Feshbach state to the ground electronic singlet state with zero rotational
and vibrational quantum number. Section 6 has conclusions.
2. CHIRPED PULSE CONTROL IN COHERENT ANTI-STOKES
RAMAN SCATTERING FOR IMAGING OF BIOLOGICAL
STRUCTURE AND DYNAMICS
Selective excitation of predetermined molecular vibrations in a complex
biological sample allows one to picture biological structure and ultrafast
dynamics using CARS. Recently, selective excitation of vibrational modes in
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Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence
217
vital samples such as live cells has been achieved by means of picosecond
laser pulses possessing a narrow spectral bandwidth.40 A series of articles,
both theoretical and experimental, demonstrate feasibility of modern laser
control methods for selective excitation of two-photon transitions.1,29–35
Coherent control methods may be used to execute vibrational excitations
in biological samples in order to achieve high chemical selectivity and
maximum intensity of the CARS signal. Implementation of laser control
methods to noninvasive biological imaging may bring investigations of
complex biological systems to a new level.
In this chapter, we present quantum control methods for selective excitation of vibrational modes among many, a large number of them having close
frequencies, by making use of ultrafast chirped laser pulses.1–4 The central
method, the chirped pulse adiabatic passage method, addresses vibrational
levels within a single electronic state via simultaneously applied chirped
pump and Stokes laser pulses; pulse parameters satisfy specific conditions
for the ac Stark shifts. These conditions provide fractional adiabatic passage
resulting in a maximum coherence between vibrational states of a predetermined mode and zero coherence between vibrational states of other Raman
active modes. This method is complementary to methods described in Refs.
46,47,26. In CARS, the amplitude of the anti-Stokes field, which is generated
at frequency 2op os, is determined by the induced coherence according to
the Maxwell–Bloch equations. Thus, the goal for a manipulation with
chirped pump and Stokes pulses is to create a maximum coherence between
the desired vibrational states to get a strong CARS signal from a predetermined molecular vibration and to suppress coherence in all other vibrational
states. We discuss three methods that make use of chirped pulses and provide selective excitation of vibrational modes in nonadiabatic and adiabatic
regime.
2.1. Theory
A semiclassical model of light–matter interaction is developed, in which
vibrational states are described by two-level systems (TLSs), and chirped
pump and Stokes pulses are described by classical electromagnetic wave
packets, having phase linearly depending on chirp. The schematic of the
model is shown in Figure 7.1. We consider two, TLSs having slightly different transition frequencies o21 and o43, such that o43 o21 ¼ d o21. Initially,
only ground states j1i and j3i of the TLSs are evenly populated.
One TLS represents a vibrational mode to be selectively excited and another
TLS represents an off-resonant vibrational mode or background to be suppressed. TLSs are uncoupled, meaning the probability for population
transfer from one TLS to another via interaction with electromagnetic fields
is zero.
The intense pump and Stokes pulses having central frequencies op and os
are defined as
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Svetlana A. Malinovskaya et al.
ws
wAS =2 w p −ws
wp
|4ñ
d
|2ñ
w43
w 21
|1ñ
|3ñ
Figure 7.1 Schematic of two-level systems representing Raman active vibrational modes
that interact with the intense pump, Stokes, and probe pulses, to generate an anti-Stokes
signal at frequency 2op os.
0
1
0
1
2
2
at
bt
Ep ðtÞ ¼ Ep0 ðtÞcos @op t A; Es ðtÞ ¼ Es0 ðtÞcos @os t A;
2
2
Ep0 ðtÞ ¼ 0
E0
02
11=4 e
t2 =2t2
Es0 ðtÞ ¼ 0
;
E0
02
1 et
2
=2t2
;
(3)
@1 þ b A
t40
@1 þ a A
t40
where Ep0(t) and Es0(t) are the time-dependent pump and Stokes field
envelope, a, a0 and b, b0 are the linear temporal and spectral chirps of the
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pump and Stokes pulses, respectively, and t ¼ t0 1 þ a0 2 =t40 is the chirpdependent pulse duration assumed to be the same for the pump and
Stokes pulses in our model. For zero chirp, pulse frequency difference
(op os) is in resonance with the o21 frequency and the pump and Stokes
pulse duration is t0. This specifies the value of the maximum temporal
linear chirp to be equal to 1/(2t20) and the related spectral linear chirp to
be equal to t20.26
The interaction Hamiltonian for a single TLS reads
0
1
1
O3 ðtÞ
ðd þ ða bÞt þ O1 ðtÞ O2 ðtÞÞ
B 2
C
B
C
H¼B
C; (4)
1
@
A
ðd þ ða bÞt þ O1 ðtÞ O2 ðtÞÞ
O3 ðtÞ
2
with d ¼ 0 for the j1i–j2i TLS and d ¼
6 0 for the j3i–j4i TLS. Here,
2
E20
02
1=2
=t40
et
2
=t2
2
E20
t =t
are the ac Stark
1=2 e
ð1þa Þ
ð1þb =t40 Þ
shifts originated from the two-photon transition, where m is the dipole
moment and D is the one-photon detuning from the excited state;
O1 ðtÞ ¼ 4ℏm2 D
; O2 ðtÞ ¼ 4ℏm2 D
02
2
2
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Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence
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O3 ðtÞ ¼ 4ℏm2 D "
!
E20
!#1=4 et
2
219
=t2
is the effective Rabi frequency. We
a
b
1þ
1þ
t40
t40
define O3 as the peak effective Rabi frequency.
The pump and Stokes pulse bandwidth at zero chirp is chosen to be about
the difference between transition frequencies of the TLSs and the effective
Rabi frequency to be larger than that.
Diagonalization of the Hamiltonian in Eq. (4) gives energy separation of
the dressed states, called the generalized Rabi frequency:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(5)
OðtÞ ¼ ðd þ ða bÞt þ Ot ðtÞ O2 ðtÞÞ2 þ 4O23 ðtÞ:
02
02
The generalized Rabi frequency depends on the chirps, the ac Stark shifts,
and the effective Rabi frequency.
The time-dependent probability amplitudes of the dressed states for two
uncoupled TLSs are described by the coupled differential equations
i
_
a_ 1ð3Þ ¼ OðtÞa1ð3Þ þ YðtÞa
2ð4Þ ;
2
i
_
a_ 2ð4Þ ¼ OðtÞa2ð4Þ þ YðtÞa
1ð3Þ :
2
(6)
_
Here, O(t) is the generalized Rabi frequency and YðtÞ
is the coupling
parameter, which determines the nonadiabatic coupling between the dressed
_
states. The coupling parameter YðtÞ
reads
_ 3 ðtÞ
LO_ 3 ðtÞ LO
_
;
YðtÞ
¼
2
OðtÞ
(7)
where L ¼ d þ (a b)t þ O1(t) O2(t). Note that the degree of adiabaticity of a
process may be evaluated by the Massey parameter M(t) defined as a ratio of
the energy splitting of the dressed states O(t) to the coupling parameter
_ .48 If M(t) 1, the light–matter interaction is essentially adiabatic, and
YðtÞ
if M(t) 1, nonadiabatic effects take place.
Notably, the adiabaticity of light–matter interaction is dependent on properties of the ac Stark shifts present in the interaction Hamiltonian described
by Eq. (4). In the following, we reveal the role of the ac Stark shifts in the
nature of two-photon Raman transitions and demonstrate how the choice of
the pump and Stokes pulse chirps determines the properties of the ac Stark
shifts and adiabaticity of the dynamics in TLSs.
2.2. On the role of the ac Stark shifts in the nature of two-photon
Raman transitions
Generally, it is convenient to analyze the adiabatic effects in the dressed state
picture. Assume that a transform-limited pump pulse (e.g., a ¼ 0) and a
linearly chirped Stokes pulse (e.g., b ¼ constant) are applied simultaneously
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Svetlana A. Malinovskaya et al.
in the CARS scheme shown in Figure 7.1. In this case, the ac Stark shifts
(O1(t) and O2(t)) are essentially different, resulting in the generalized Rabi
_
frequency O(t) and the coupling parameter YðtÞ
to be dependent on the ac
Stark shift difference (O1(t) O2(t)). In order to evaluate the adiabatic effects
via Massey parameter, we have calculated the energy of the dressed states
_
as a function of time. In the calcu 12 OðtÞ and the coupling parameter YðtÞ
lations, we used the following values of the system and the field parameters:
transition frequency o21 was chosen to be equal to one frequency unit and
the o43 to be equal to 1.1, giving d ¼ 0.1. This model is valid for description of,
for example, symmetric and asymmetric stretch modes in liquid methanol,49
or symmetric CH2 vibrational mode in lipids and the CH3 stretch mode in
the proteins, both constituting the myelin sheath.50 All the frequency param1
eters given in the section are in [o21] units, temporal parameters are in [o
21 ]
2
2
units, frequency chirps are in [o21 ] units, and temporal chirps are in [o21]
units. Duration of a transform-limited pulse was taken to be equal to 15,
giving 1/t0 ’ d.
In Figure 7.2, the dressed state energies and the coupling parameter are
shown as a function of time for the peak effective Rabi frequency equal to 0.7,
indicating a strong field regime, the spectral linear chirp equal to 270, which
is slightly above b0 jbmax ¼ 225, and d ¼ 0.1 (a) and d ¼ 0 (b). Black and red
curves show the energies of the upper and lower dressed states of a TLS,
and green curve shows the coupling parameter as a function of time. For
particular times, the energy separation of the dressed states O(t) decreases
_
with a significant increase in the coupling strength YðtÞ.
As a result, the
Massey parameter decreases to much less than unity, implying nonadiabatic
A
0.4
d = 0.1
0
-0.4
B
0.4
d=0
0
-0.4
0
50
100
150
tw
Figure 7.2 The dressed state energies and the coupling parameter as a function of time for
the peak effective Rabi frequency equal to 0.7, the spectral linear chirp equal to 270, and
d ¼ 0.1 (a), and d ¼ 0 (b) for the CARS scheme implementing a transform-limited pump pulse
and a linearly chirped Stokes pulse.
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Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence
221
nature of the light–matter interaction. The nonadiabatic coupling between
the dressed states takes place due to the ac Stark shifts. In the vicinity of
_
small energy separation of the dressed states, the coupling parameter YðtÞ
is
large enough to provide a population transfer between the dressed states. It
is known that the dressed states are a linear symmetric and antisymmetric
combination of the bare states j1i and j2i (or j3i and j4i). Population transfer
from the lower to the upper dressed state results in a final population of both
bare states. It may be only partial depending on the coupling strength. The
_
values of the O(t) and the coupling parameter YðtÞ,
that determine the
quantum yield, are dependent on the choice of the pulse chirp and the
field amplitude.
The nonadiabatic coupling gives rise to an oscillatory dependence of the
coherence in the TLSs on the spectral linear chirp and the peak effective Rabi
frequency, see Figure 7.3. Here, the coherence density plot of the resonant r12
(yellow) and the detuned r34 (black) TLSs is shown as a function of the peak
effective Rabi frequency O3 and the dimensionless chirp parameter b0 =t0 2 .
Regions where lines of different color coincide are the regions of selective
excitation of molecular vibrations. Projection of any point from such a region
onto abscise and ordinate axes gives values of the field parameters required
for selective excitation.
The plot shows that the ac Stark shifts, different for the lower and upper
levels of the resonant and detuned TLSs due to interaction with a transformlimited pump pulse and linearly chirped Stokes pulse, lead to nonadiabatic
4
W3
3
2
1
-20
-15
-10
-5
0
2
b¢/t0
Figure 7.3 The coherence density plot of the resonant (yellow) and the detuned (black)
TLSs as a function of the peak effective Rabi frequency and the dimensionless chirp
parameter b0 =t0 2 for the CARS scheme implementing a transform-limited pump pulse and
a linearly chirped Stokes pulse.
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Svetlana A. Malinovskaya et al.
coupling, whose value could be manipulated by variation of the pulse chirp,
giving a desired dynamics and the quantum yield (see also Ref. 1). Understanding of the role of the ac Stark shifts suggests an approach for selective
excitation based on the adiabatic passage: choosing totally overlapped
pulses may provide identical ac Stark shifts O1 and O2, resulting in their
cancellation in the interaction Hamiltonian, Eq. (4), and, thus, removing their
influence on the final quantum yield.
First, we consider equally chirped laser pulses in the CARS scheme, which
implies a ¼ b in Eqs. (3) giving the pump and Stokes pulse frequency difference op os being in resonance with the o21 frequency. Then, in the Eqs. (4)
and (5) the ac Stark shift difference (O1(t) O2(t)) is equal to zero. In this case,
the interaction Hamiltonian and the generalized Rabi frequency read
0
1
1
d O3 ðtÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B 2
C
B
C
(8)
H¼B
;
OðtÞ
¼
d2 þ 4O23 ðtÞ:
C
1 A
@
d
O3 ðtÞ
2
_
For d ¼ 0, the nonadiabatic coupling parameter YðtÞ
is exactly zero,
according to Eq. (7), two dressed states are uncoupled and the system
dynamics takes place within a single dressed state having lower energy.
In this case, the probability amplitude of the bare states, following the
ðt
Rabi oscillations, depends on the pulse area O3 ðt0 Þdt0 .
0
For d 6¼ 0, the nonadiabatic coupling between the dressed states is negligibly small, as shown by numerical calculations of Eq. (7), resulting in the
system dynamics to be also within a single dressed state. The solution for the
probability amplitudes of the bare states is the Rabi oscillations having
amplitude proportional to the ratio of the effective Rabi frequency O3(t)
and the generalized Rabi frequency O(t), resulting in low-amplitude oscillations vanishing with the increase in the chirp parameter. The chirp dependence of the effective Rabi frequency O3(t) establishes selectivity of
excitation of the resonant TLS in the framework of the Rabi solution.
In Figure 7.4a, the coherence density plot of the resonant TLS r12 is
depicted as a function of the peak effective Rabi frequency O3 and dimensionless frequency chirp parameter b0 =t0 2 . According to the Rabi solution,
the coherence oscillates as a function of O3, independent of b0 =t0 2 . The
coherence density plot of the off-resonant TLS r34 is shown in Figure 7.4b.
The coherence is zero in the large region of chirp parameters, b0 =t0 2 , for any
value of O3. It is the adiabatic dynamics in the TLS that gives zero population
transfer to the upper level. In the vicinity of zero chirp, the off-resonant Rabi
oscillations of the coherence are observed as a function of O3. Thus, control of
coherence in TLSs, having close frequencies, is achieved by means of the
_
chirp parameters that provide the conditions for adiabaticity, OðtÞ=YðtÞ
1,
and optimally chosen pulse area.
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1.6
1.1
1.1
W3
W3
1.6
0.6
0.1
-20
0.6
-10
0
10
0.1
-20
20
-10
0
10
20
b¢ / t02
2
b¢ / t0
Figure 7.4 The coherence density plot of the resonant TLS (a) and the detuned TLS (b) is
depicted as a function of the peak effective Rabi frequency and dimensionless frequency
chirp parameter b0 =t0 2 for the CARS scheme implementing equally chirped pump and
Stokes pulses. Maximum coherence is shown by blue color and minimum coherence, by
red color.
2.3. The ‘‘roof’’ method
Now, we consider the adiabatic passage control method implementing
chirped pulses for selective excitation of spectrally close Raman transitions
within the scheme in Figure 7.1. This method implies the chirp parameter of
the Stokes pulse b to be constant, giving monotonous change of the Stokes
pulse frequency, see Eq. (3), and chirp parameter of the pump pulse a to
have same magnitude and opposite sign before the central time t0, when the
pulse amplitude reaches maximum, and then to flip the sign of the pump
pulse chirp. At t0, the difference of the pump and Stokes pulse frequencies
comes into resonance with the o21 frequency and stays in resonance for the
rest of the time. We call this method the ‘‘roof’’ method in accordance with
the temporal profile of the pump pulse instantaneous frequency. The interaction Hamiltonian of a TLS and the generalized Rabi frequency reads
0
1
d
at O3 ðtÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B 2
C
B
C
(9)
; OðtÞ ¼ ðd þ 2atÞ2 þ 4O23 ðtÞ;
t < t0 ; H ¼ B
C
d
@
A
O3 ðtÞ
þ at
2
0
B
B
t t0 ; H ¼ B
@
d
2
O3 ðtÞ
O3 ðtÞ
1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C
C
; OðtÞ ¼ d2 þ 4O23 ðtÞ:
C
d A
2
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In accordance with the Eq. (9), the roof method induces qualitatively
different dynamics in the TLSs exposed to the pump and Stokes laser fields
with respect to previously considered methods. Because the ac Stark shifts
are the same and cancel each other in Eq. (4) giving the interaction Hamiltonian described in Eq. (9), the adiabatic passage takes place leading to the
maximum coherence in the resonant TLS, Figure 7.5a (blue), and minimum
coherence in the detuned TLS, Figure 7.5b (red), in a broad range of the peak
effective Rabi frequency O3 and dimensionless frequency chirp b0 =t0 2 . We
refer once again to the dressed state picture to understand the adiabatic
features of the system dynamics. The energies of the bare (dashed curves)
and the dressed (solid curves) states in the field-interaction representation
are shown as a function of time for the resonant TLS in Figure 7.6a. Bare
states come to resonance at central time t0 and further remain unchanged.
Since the nonadiabatic coupling is negligibly small, the systems time evolution takes place within a single dressed state. Initially, the lower dressed
state coincides with the populated bare ground state. As time approaches the
central time t0, both bare states acquire equal probability amplitudes and
remain unchanged giving maximum final coherence r12.
Coherence is more sensitive to the field amplitude in the detuned TLS and
depends on the sign of the chirp in weak fields. For the positive chirp,
Figure 7.6b, the crossing of the bare states takes place far before t0, when
the field amplitude is rather weak and therefore dressed states have small
energy separation. At this time, the nonadiabatic coupling (green curve) is
strong enough to couple them and to partly promote population to the upper
dressed state, leading to nonzero coherence r34. For the negative chirp, the
crossing does not occur and evolution is essentially adiabatic, giving zero
1.6
1.1
1.1
W3
W3
1.6
0.6
0.6
0.1
-20
-10
0
10
2
b¢ / t0
20
0.1
-20
-10
0
10
20
2
b¢ / t0
Figure 7.5 The coherence density plot of the resonant TLS (a) and the detuned TLS (b) as a
function of the peak effective Rabi frequency and the dimensionless chirp parameter b0 =t0 2
for the roof method. Maximum coherence is shown by blue color and minimum coherence,
by red color.
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0.1
0.05
0.05
0
0
-0.05
-0.05
-0.1
300
400
500
600
tc
700
800
900
300
400
500
600
tc
700
800
900
tw
Figure 7.6 The energies of the bare (dashed lines) and the dressed (solid lines) states in the
field-interaction representation are shown as a function of time for the resonant TLS, for
the peak effective Rabi frequency (before chirping) equal to 1.1 and b0 =t0 2 ¼ 10, (a) and the
detuned TLS, for the peak effective Rabi frequency (before chirping) equal to 0.2 and
b0 =t0 2 ¼ 10 (b). Blue curve shows the effective Rabi frequency as a function of time and
green curve shows the nonadiabatic coupling parameter _ðtÞ.
population transfer and zero coherence in the detuned system. In strong
fields, the dressed states have large energy separation all the time, resulting
in negligible nonadiabatic coupling and complete (zero) population transfer
to the upper level and zero coherence for a0 > 0 (a0 < 0) (see also Ref. 4).
Numerical calculations have shown that the conditions for the selectivity
of excitations 1/t d must be satisfied.
The roof method suggests the robust way to obtain a noninvasive image of
a biological structure. Various biological tissues contain molecular groups,
having CH vibrations which span from 2800 to 3100 cm 1 and may be
selectively excited to provide noninvasive image with high chemical sensitivity. For example, in order to get CARS imaging of axonal myelin in live
spinal tissues by obtaining a strong signal from the symmetric CH2 stretch
vibration that appears at 2840 cm 1, the chirped pulse adiabatic passage
method suggests to use femtosecond pulses generated from Ti:sapphire
oscillators, having pulse duration t0 ¼ 176 fs, linear spectral chirp
a ¼ 30 10 5 cm2, and field intensity in the range of 1012 W/cm2.
Similar to the multiplex CARS method,51–53 the adiabatic passage control
method may be applied by fixing the pump central frequency and scanning
the Stokes frequency to obtain the vibrational spectrum of unknown molecular species. However, in contrast to the multiplex CARS method, for each
instantaneous magnitude of the Stokes central frequency, the maximum
intensity of the CARS signal and the efficient suppression of the background
signal will be provided.
Recently, somewhat related idea of using chirped pulses in CARS spectroscopy has been demonstrated experimentally,13 where the two equally
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Svetlana A. Malinovskaya et al.
chirped pulses were used. Time delay between pulses has been utilized as
an effective way to tune two-photon detuning into resonance. That allowed
the authors to achieve high spectral resolution using broadband pulses.
One can see from the present analysis that imposing temporal delay gives
rise to ac Stark shifts. In turn, these time-dependent Stark shifts result in an
effective nonlinear avoided crossing and substantial nonadiabatic coupling.1 Optimization of the selective excitation of the vibrational coherence
under the condition of nonlinear avoided crossing could be a challenge
even in the adiabatic limit. Therefore, the method13 is sensitive to the pulse
parameters, and excitation selectivity is limited because of the nonadiabatic
coupling induced by the Stark shifts. In contrast, our methods employ full
advantage of adiabatic passage technique and provide a clear and simple
way of controlling predetermined vibrational coherence. The robustness of
the methods should facilitate experimental implementations. To reach adiabatic regime, one can use 100 fs transform-limited pulses at a repetition
rate 1 MHz with averaged beam power of 25 mW similar to that in Ref.
40. Assuming dipole moments to be equal to 1 D, we estimate Oef
R 200 THz
giving the Landau–Zener parameter to be well in the adiabatic range,
2
(Oef
R ) /a 40.
In summary, we have shown that selective excitation of spectrally close
vibrational modes may be achieved using chirped pump and Stokes pulses
in CARS spectroscopy and microscopy within the nonadiabatic and adiabatic regimes of light–matter interaction.
By simultaneously applying a transform-limited pump pulse and a linearly chirped Stokes pulse, selective excitation is achieved as a result of strong
nonadiabatic coupling of the dressed states originated from the induced ac
Stark shifts having substantially different values. A careful choice of a
spectral chirp of the Stokes pulse results in a desirable final quantum state
of a multimode system.
Two methods implementing chirped pump and Stokes pulses provide
selective excitation of predetermined Raman transitions based on adiabatic
regime of light–matter interaction. Choosing totally overlapped chirped
pulses, we removed the ac Stark shifts influence on the final quantum
yield. Implementation of two equal chirps in one scheme provides the Rabi
oscillation type of coherence control. The roof method makes use of a
constant chirp in the pump (Stokes) pulse and the flipping sign of the
chirp in the Stokes (pump) pulse. This method provides the adiabatic passage type of coherent control and gives a robust solution for a wide range of
the field amplitude and the chirp values. Both methods can be used for
noninvasive imaging of biological structure and femtosecond dynamics
studies, and as an efficient tool for suppression of the background signal in
the CARS spectroscopy and microscopy. All three methods demonstrate the
possibility of coherent control over spectrally close modes with frequencies
larger than the bandwidth of the pulse. Methods may be applied to control
an induced polarization, decoherence, and quantum gates.
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227
3. THE IMPACT OF PHASE AND COUPLING BETWEEN THE
VIBRATIONAL MODES ON SELECTIVE EXCITATION IN
CARS MICROSCOPY
In this chapter, we analyze the impact of the coupling between Raman active
vibrational modes on selectivity of their excitation in CARS microscopy
using method proposed in Ref. 4 and discussed in Section 2. We study two
theoretical models demonstrating control over the quantum yield and optimization of the CARS signal.6 Each model consists of two effective TLSs with
(a) degenerate ground states and (b) nondegenerate ground states, which
interact with two chirped femtosecond pulses within the Raman configuration in accordance with Ref. 4. The first model, the one with degenerate
lower states, may be used to describe the induced dipole moments coupled
via dipole–dipole interactions and subject to interaction with external electromagnetic fields,54 while the second one with nondegenerate lower states
is useful for the description of Raman modes present in a molecule and
interacting with light.1
The nonlinear nature of CARS is due to four-wave mixing. When the frequency difference between the pump and Stokes beams is in resonance with a
molecular vibration, it excites the molecule to a higher vibrational level creating a coherence on corresponding transition. On de-excitation by the probe
pulse, the anti-Stokes frequency light is emitted. It contains vibrational signature of the molecule which is unique in its nature. Thus, a large amount of
information can be extracted from the CARS spectrum. By applying femtosecond, chirped laser pulses as in Ref. 4 within two models, and performing
relative phase dependence studies, we analyze the role of coupling between
vibrational modes as well as relative phase dependence on optimizing the
coherence in the desired vibrational mode and suppressing the unwanted
one. We would like to point out that our results are related to the so-called
strong field control regime when perturbation theory with respect to the
external field amplitude is not valid and the exact solution of the Schrödinger
equation must be obtained to describe the excitation dynamics correctly.
3.1. Theory
We consider a semiclassical model of light–matter interaction, where strong
femtosecond laser pulses interact with two coupled TLSs representing two
Raman active vibrational modes in a molecule. A molecular medium of
interest may be considered as an ensemble of TLSs,4,26 with no relaxation
or collisional dephasing effects taken into account. We investigate two
models: one with degenerate ground states (Figure 7.7a) and the other
with nondegenerate ground states (Figure 7.7b). Here, the j1i–j2i TLS has
transition frequency o21 and the j3i–j4i TLS has transition frequency o43.
These two modes are coupled by an external field, meaning that all states are
effectively coupled. In addition, we take into account the phase relation
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Svetlana A. Malinovskaya et al.
|bñ
|bñ
D
D
ws
ws
wp
wp
|4ñ
|4ñ
d
|2ñ
|1ñ
|2ñ
|3ñ
(a)
3d /2
d /2
|1ñ
|3ñ
(b)
Figure 7.7 Schematic of the energy levels in the degenerate model (a) and nondegenerate
model (b).
between different modes, assuming that the relative phase between initially
populated states j1i and j3i is random in a large ensemble of molecules.
Therefore, most of the presented results are phase averaged. Note that the
bandwidth of the applied pulses is larger than the frequency mode difference. Our goal is to suppress coherence in the j3i–j4i TLS and to create a
maximum coherence in the j1i–j2i TLS. The frequency chirped pump and
Stokes pulses having central frequencies op and os are defined as
Ep;s ðtÞ
¼ Ep0;s0 ðtÞ cos½op;s ðt t0 Þ þ ap;s ðt t0 Þ2 =2;
E0
exp½ðt t0 Þ2 =2t2 ;
Ep0;s0 ðtÞ ¼
ð1 þ a0 2p;s =t40 Þ1=4
(10)
where Ep0(t) and Es0(t) are the time-dependent pump and Stokes field envelopes, ap,s and a0 p, s are the linear temporal and spectral chirps of the pump
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
and Stokes pulses, respectively, and tp;s ¼ t0 1 þ a0 2p;s =t40 is the chirp-dependent pulse duration. For zero chirp, the frequency difference op os is in
resonance with the frequency o21, and the pump and Stokes pulse duration
before chirping is t0.
Interaction of vibrational modes with ultrafast chirped laser pulses is
described in the rotating wave approximation by a semiclassical Hamiltonian
obtained using adiabatic elimination of the virtual state jbi. Within the fieldinteraction representation, the Hamiltonian for the degenerate model reads
0
1
O3 ðtÞ
O1 ðtÞ
O3 ðtÞ
dðtÞ Od ðtÞ
B O3 ðtÞ
C
dðtÞ þ Od ðtÞ
O3 ðtÞ
O2 ðtÞ
C:
(11)
H¼B
@ O1 ðtÞ
A
O3 ðtÞ
dðtÞ Od ðtÞ
O3 ðtÞ
O2 ðtÞ
O3 ðtÞ
d þ dðtÞ þ Od ðtÞ
O3 ðtÞ
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The interaction Hamiltonian for the nondegenerate model reads
0
1
dðtÞ Od ðtÞ
O3 ðtÞ
O1 ðtÞ
O3 ðtÞ
B O3 ðtÞ
C
dðtÞ þ Od ðtÞ
O3 ðtÞ
O2 ðtÞ
C:
H¼B
@ O1 ðtÞ
A
O3 ðtÞ
d=2 dðtÞ Od ðtÞ
O3 ðtÞ
O2 ðtÞ
O3 ðtÞ
3d=2 þ dðtÞ þ Od ðtÞ
O3 ðtÞ
(12)
Here, d(t) ¼ (as ap)(t t0)/2, Od(t) ¼ (O1(t) O2(t))/2, and O1, 2(t) ¼
m2E2p0, s0(t)/(4ℏ2D) are the ac Stark shifts originated from the two-photon
transitions, m mij is the dipole moment (for simplicity we consider all the
dipole moments to be equal to 1 D), D is the single photon detuning from the
excited state jbi (assumed equal for both pump and Stokes pulse frequencies), and O3(t) ¼ m2Ep0(t)Es0(t)/(4ℏ2D) is the effective Rabi frequency.
The diagonal elements of the Hamiltonian describe bare state energies in the
field-interaction representation, and they depend on the chirp parameters ap,s
and detuning d ¼ o43 o21. The off-diagonal elements represent coupling of
the bare states through the effective Rabi frequency and the ac Stark shifts.
Here, we consider the case when pump and Stokes chirp rates have equal
value, japj ¼ jasj ¼ a. The control in the TLSs is achieved by linearly chirped
pulses with chirp parameters such that the frequency difference of the pump
and Stokes pulses first reduces at 2a rate and comes to resonance with o21 at
the central time t0 without further change till the end of the pulse. This
method is known as the roof method (see Chapter 2 and Ref. 4). In the case
of two uncoupled TLSs, studied in Ref. 4, the proposed scheme resulted in a
creation of a maximum coherence in the resonant TLS and zero coherence in
the off-resonant TLS. Now, the model is modified by switching on the
coupling between two TLSs via external fields in order to analyze the impact
of the coupling on controllability of excitation and also the attendant relative
phase effects.
3.2. Numerical results and discussion
To analyze the dynamics of the population and coherence, we solve the timedependent Schrödinger equation with the Hamiltonians in Eqs. (11) and (12)
for the total wave function jC(t)i ¼ S4i ¼ 1 ai(t)jii, where ai(t) are the timedependent probability amplitudes. Calculations were performed using the
Runge–Kutta method55 under the initial conditions of equally populated
ground states, j1i and j3i, which is likely to be the case for molecules at
room temperature, particularly, for the nondegenerate model, because the
energies of the two nondegenerate ground states are very close compared to
the kT expected under all experimental conditions on biological samples.
The population of each state was chosen to be 0.5, giving the total population
in the system equal to unity and coherence value ranging from zero to the
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Svetlana A. Malinovskaya et al.
maximum value 0.5. Below we use a standard density matrix notation for the
population rii ¼ jai(t)j2 and coherence rij ¼ jai*(t)aj(t)j, i, j ¼ 1, 2, 3, 4. The
parameters of the fields and the systems used in numerical calculations
correlate with experimental conditions discussed in Refs. 33,49 and also
used in Ref. 35. They are also equally good fit to address different vibrational
modes of CHn molecular species in biological samples.40 We chose
o21 ¼ 84.9 THz (2840 cm 1), o43 ¼ 87.6 THz (2930 cm 1). The range of intensity of the laser fields is from 1011 to 1013 W/cm2, and the transform-limited
pulse duration is t0 ¼ 176 fs. The spectral chirps used in calculations are a0 p,s
¼ 31 10 5 cm 2, giving the chirped pulse duration t ¼ 1.8 ps.
Let us discuss phase averaging in detail to gain more understanding of its
importance in making a connection with experimentally measurable CARS
signal. Phase is embedded as the complex part in the probability amplitude
of the state. When states are coupled by external fields, the relative
phase between them just before the fields strike the medium is of key
importance since it determines the evolution of the population and coherence in the TLSs. Obviously, the quantum yield at the end of the pulse is
phase dependent. One can prepare a particular relative phase between
initially populated states by optical pumping56 into state j1i and creating a
j1i–j3i state coherence using a Raman scheme. Some values of the initial
relative phase are known to bring the system to an optimal quantum yield.57
In the bulk gas or liquid medium, molecules have all possible relative phases
between vibrational states at the instant when pulses strike the molecular
medium. The Raman signal measured from such a macroscopic ensemble of
molecules is phase averaged. To include this idea in our theoretical
approach, we take into account the effect of relative phase between initially
populated states by performing phase averaging. The procedure consists of
calculating state physical quantities at the end of the pulse for 100 initial
relative phases between initially populated states j1i and j3i ranging from
zero to 2p and averaging over the results obtained.
3.2.1. Degenerate model
The coherence density plots of the resonant and detuned Ðsystems are
depicted in Figure 7.8 as a function of effective pulse area A ¼ O3(t)dt and
dimensionless frequency chirp parameter a0 =t0 2 . The sign of the chirp
parameter, a0 , determines the direction of the pump chirp before the central
time when the sign changes. The Stokes chirp is the linear chirp and it has an
opposite sign to the pump chirp before the central time. The transformlimited pulse duration is t0 ¼ 15 [o 1], where o is the unit frequency equal
to o21. This value of t0 corresponds to 176 fs. The figure shows that there is a
broad parameter region providing maximum coherence r12 in the resonant
TLS (shown in blue color) for the positive chirp values. In the off-resonant
TLS, coherence is zero in the same region of field parameters. Negative
values of the chirp parameter do not provide selective excitation of coherence in the coupled TLSs. Note that the effective pulse area of 0.85p
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Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence
0.18
8
7
7
0
5
4
5
4
3
3
2
2
1
-20
-10
0
a ¢/t02
(a)
10
20
0
6
A [p]
A [p]
0.18
8
6
231
1
-20
-10
0
a ¢/t02
(b)
10
20
Figure 7.8 Degenerate model: Phase-averaged coherence density plot for the resonant
(a) and off-resonant (b) systems as a function of the effective pulse area, A, and chirp
parameter, a0 =t0 2 .
(corresponding to the chirped pulse peak intensity in the range of
1011 W/cm2) shows coherent control of the vibrational excitations in molecules. This is a useful addition to previously obtained results on selective
excitation in a strong field regime when the field intensity ranges from 1012
to 1013 W/cm2.4 Thus, chirped pulse adiabatic passage method—the roof
method—is feasible in achieving a maximum coherence in a predetermined
vibrational mode in the presence of strong coupling between many Raman
active vibrational modes via external fields.
The understanding of the mechanism of selective excitation under the
condition of the coupling between vibrational modes is gained through the
dressed state picture analysis. We refer in this case only to a particular value
of initial relative phase. To do so, we first consider the dependence of
coherence and population on the initial relative phase between the ground
states j1i and j3i at final time, which is shown in Figure 7.9. In the case of
the negative Stokes chirp, Figure 7.9b, for most values of the relative phase
between 0 and 2p, there is a strong coherence in the resonant TLS, while zero
coherence in the nonresonant TLS. The relative phase p can be inferred as the
one creating the dark state, as the coherences r12 and r34 both come to zero
and all population is collected in the ground states of TLSs, meaning that the
TLSs stay uncoupled from the external fields. In the case of the positive
Stokes chirp, Figure 7.9a, both coherences r12 and r34 have some value and
the relative phase dependence is almost symmetrical with respect to phase p,
so that averaging over the phase gives almost zero as shown in Figure 7.8
(see left side of the plots (a) and (b)). The asymmetry in the coherence value
at final time with respect to the chirping direction (positive or negative)
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Svetlana A. Malinovskaya et al.
0.8
r11
r44
Population, coherence
Population, coherence
0.8
0.6
r11
r33
0.4
r34
0.2
r12
r22
0
r33
0.6
r12
0.4
r22
0.2
r44
0
0
p/4
p/2
3p/4
p
5p/4 3p/2 7p/4
2p
0
p/4
p/2
3p/4
p
r34
5p/4 3p/2 7p/4
Relative phase
Relative phase
(a)
(b)
2p
Figure 7.9 Degenerate model: Relative phase dependence of population, rii, and coherence, rij, at final time for positive (a) and negative (b) chirp of the Stokes pulse; O3(t0) ¼ 0.35
[o21] (respective effective pulse area A ffi 2.95p), ja0 j/t02 ¼ 10.
might be clearly observed using the dressed states. As usual, the dressed
states can be obtained by diagonalizing the 4 by 4 Hamiltonians (Eq. (2) or
(3)) by solving the corresponding eigenvalue problem. In the present situation, the expressions for the dressed state energies and corresponding
dressed vectors are too complicated to be presented in a compact form in
the chapter. In general, each dressed state is a liner superposition of all four
bare states, jii, with coefficients depending on the Rabi frequency, detuning,
and pulse chirps. The energies of the dressed states are shown in Figure 7.10.
Within the adiabatic approximation, the system dynamics starts in
dressed states I and III that initially correlates with the bare state j1i and
j3i correspondingly, independent of the chirping direction. Consider first
the positive Stokes chirp (Figure 7.10a). As time evolves and the pulse
intensity increases, dressed state III reaches the region of avoiding crossing,
with dressed state IV following immediately after that by diabatic crossing
with state I. It is clear that in adiabatic approximation, the dressed state III
correlates with state j4i at final time and will provide population transfer to
this bare state. The dressed state I has several diabatic crossings with states
III and IV, and it correlates with state j1i at a later time. In fact, the dressed
states I, II, and IV coincide in energy at a later time; therefore (if follow this
route), the system will end up in coherent superposition of all three bare
states, j1i, j2i, and j3i, at final time. Population of the state j3i at final time
depends strongly on the adiabaticity of the first avoiding crossing between
the dressed states III and IV: the higher the adiabaticity parameter, the less
the population will be in state j3i at the end.
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0.2
0.1
|1ñ
|3ñ
Dressed states energy
Dressed states energy
Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence
|Iñ
|IIIñ
0.0
-0.1
-0.2
|4ñ
|IVñ
|2ñ
|IIñ
800
1000
1200
tw
(a)
1400
0.3
233
|IVñ
0.2
|4ñ
0.1
|2ñ
|IIñ
0.0
-0.1
-0.2 |1ñ
|3ñ |IIIñ
|Iñ
800
1000
1200
1400
tw
(b)
Figure 7.10 Degenerate model: Dressed states energies (solid lines) and bare states energies
(dashed lines) as a function of time for the positive (a) and negative (b) Stokes chirp; O3(t0) ¼
0.35 (respective effective pulse area A ’ 2.95p), ja0 j/t02 ¼ 10.
In the case of the negative Stokes chirp, Figure 7.10b, the dressed state
energies are much different. Now, the dressed state I effectively has only one
avoiding crossing at central time and it is located far below all other dressed
states. In adiabatic approximation, the wave function dynamics of states j1i
and j2i is very close to the case of TLS considered in Ref. 4. This is the case of
most robust solution which is supported by the results presented in
Figure 7.8 (see right side of the panel (a)), where maximum coherence in
the resonant mode can be prepared in a wide range of the chirp rates and
pulse areas. Of course, averaging over the relative phase reduces the maximum value of the coherence.
Figure 7.11 shows the dynamics of the state population and coherence as a
function of time for zero initial relative phase between states j1i and j3i. In
essence, this figure confirms the discussion of the dressed state analysis
presented above. For the positive Stokes chirp, Figure 7.11a, population is
mostly transferred to the state j4i because of adiabatic following in dressed
state III which correlates with the bare state j4i (Figure 7.9a). At final time,
the coherences r12 and r34 are of order 0.05. For the negative Stokes chirp,
Figure 7.11b, population dynamics follows the dressed state picture presented in Figure 7.9a. At final time, the state j4i is empty and whole population is almost equally distributed between states j1i, j2i, and j3i, which
gives maximum coherence for the resonant mode, r12 0.33. The result
additionally demonstrates a good correlation between the dressed state
picture and the exact solution, showing that by preparing a molecular
system in the initial state with a particular relative phase between vibrational modes we can achieve a high value of coherence for the resonant mode
and zero excitation in the off-resonant mode.
From the discussion above, it follows that there is a near adiabatic solution
for achieving a maximum coherence in a predetermined TLS in the presence
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0.5
r11
r44
0.6
Population, coherence
Population, coherence
0.8
W3(t) . 20
r11
r33
0.4
r34
0.2
r22
r12
r33
0.4
r22
0.3
r12
0.2
W3(t) . 10
r34
0.1
r44
0
0
0.8
1
1.2
1.4
10-3 tw
(a)
1.6
0.8
1
1.2
1.4
1.6
10-3 tw
(b)
Figure 7.11 Degenerate model: Coherences and state populations as a function of time for
the positive (a) and negative (b) Stokes chirp; O3(t0) ¼ 0.35 [o21] (respective effective pulse
area A ’ 2.95p), ja0 j/t02 ¼ 10.
of its coupling with another TLS via external fields. This solution may be
achieved in a relatively strong field regime. The results demonstrate that
coherence in both modes is sensitive to the initial relative phase between
originally populated states and to the field parameters such as intensity and
the chirp sign.
3.2.2. Nondegenerate model
In this section, we discuss the nondegenerate model in which the ground
states j1i and j3i are shifted by d/2 (Figure 7.7b). The dynamics of the system
is governed by the time-dependent Schrödinger equation with the Hamiltonian in Eq. (12). Figure 7.12 shows the phase-averaged coherence density
plots r12 and r34 as a function of the chirp parameter and effective pulse area.
The range of the effective pulse area corresponds to the peak intensity of a
transform-limited pulse in the range of 32 1011–32 1012 W/cm2. The chirp
parameter spans between a0 =t0 2 ¼ 20 corresponding to 62 10 5 cm 2.
Here, essentially for both coherences r12 and r34, the region of the positive
chirp (of the pump pulse before the central time) shows zero value in strong
fields, where one could expect adiabatic solution. However, there is a relatively large area of the moderate chirp rates, which provides r12 coherence of
order 0.15 (green area in Figure 7.12a, left side), while r34 is almost zero
(Figure 7.12b, left side). In the negative chirp region, the topology of the
coherence density plots is almost identical, meaning that there is no selectivity of the mode excitation.
It is clear that there is a dependence of the coherence on the initial relative
phase between states, and that a specific relative phase may provide the time
evolution leading to a maximum coherence in the j1i–j2i TLS and zero
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0.25
0.21
8
8
7
7
0
5
4
5
4
3
3
2
2
1
-20
-10
10
0
0
6
A [p]
A [p]
6
1
-20
20
-10
0
10
a ¢/t02
a ¢/t02
(a)
(b)
20
Figure 7.12 Nondegenerate model: Phase-averaged coherence density plot for the resonant (a) and off-resonant (b) systems as a function of the effective pulse area, A, and chirp
parameter, a0 =t0 2 .
r22
0.8
r33
r44
Population, coherence
Population, coherence
0.8
0.6
0.4
r34
0.2
0.6
r12
0.4
r11
0.2
r33
r11 r12 r22
0
r44
0
0
p/4
p/2
3p/4
p
5p/4 3p/2 7p/4
2p
0
p/4
p/2
3p/4
p
r34
5p/4 3p/2 7p/4
Relative phase
Relative phase
(a)
(b)
2p
Figure 7.13 Nondegenerate model: Relative phase dependence of population, rii, and
coherence, rij, at final time for positive (a) and negative (b) chirp of the Stokes pulse;
O3(t0) ¼ 0.7 [o21] (respective effective pulse area A ’ 5.9p), ja0 j/t02 ¼ 10.
coherence in the j3i–j4i TLS. Phase dependence of r12 and r34 at final time is
demonstrated in Figure 7.13. Notably, coherence r12 is nonzero and significant, while coherence r34 is negligible for any relative initial phase in the
case of the negative Stokes chirp (Figure 7.13b). The positive Stokes chirp
case shows relatively high values of the r34 coherence, while r12 is very small
(Figure 7.13a).
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Let us discuss the case of zero relative phase in more detail. In Figure 7.14,
the density plots of r12 and r34 are presented as a function of effective pulse
area and the spectral chirp parameter when the initial relative phase is zero.
Note that the field conditions are the same as those given in Figure 7.12. Blue
regions of a maximum coherence r12 and red regions of zero coherence r34
are observed for the both positive and negative chirps. It is interesting that
the selectivity of excitation of TLSs resulting in optimal values of coherence
is achieved for zero relative phase in a relatively weak field regime (0.85p
pulse area). This is an important observation that highlights the coupling as
an additional channel of controllability. In Ref. 4, where a model of two
uncoupled TLSs was investigated, it was demonstrated that no selectivity
can be achieved in the weak field regime. However, when coupling between
TLSs is present, it opens an additional channel for population transfer and,
depending on the initial phase conditions, leads to the desired selective
excitation with optimal value of coherence r12 and r34 in the moderate fields.
To see the adiabatic effects of light–matter interaction within the current
model, we analyzed the dressed state picture. We numerically diagonalized
the Hamiltonian in Eq. (12) and obtained the time-dependent energy of the
dressed states and eigenvectors. The diagonalization was carried out under
the conditions that (a) the ac Stark shifts are equal, O1(t) ¼ O2(t), due to
identical pulse envelopes of the pump and Stokes pulses, and (b) the chirp
parameter ap changes the sign at the central time, t0, while the absolute value
of the pump and Stokes pulse chirps is preserved. Figure 7.15 shows the
0.45
0.5
8
8
7
7
0
5
4
5
4
3
3
2
2
1
-20
-10
0
10
20
0
6
A [p]
A [p]
6
1
-20
-10
0
a ¢/t02
a ¢/t02
(a)
(b)
10
20
Figure 7.14 Nondegenerate model: Coherence density plot for the resonant (a) and offresonant (b) systems as a function of the effective pulse area, A, and chirp parameter, a0 =t0 2 ,
for zero initial relative phase between states j1i and j3i.
Author's personal copy
0.2
|3ñ |IIIñ
0.1
|1ñ
0
-0.1
-0.2
Dressed states energy
Dressed states energy
Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence
|Iñ
|IVñ
|4ñ
|2ñ
|IIñ
800
1000
1200
1400
237
0.3
0.2
|4ñ |IVñ
0.1
|2ñ
|IIñ
0.0
-0.1
-0.2
|3ñ
|IIIñ
|1ñ |Iñ
800
1000
1200
tw
tw
(a)
(b)
1400
Figure 7.15 Nondegenerate model: Dressed states energies (solid lines) and bare states
energies (dashed lines) as a function of time for the positive (a) and negative (b) Stokes chirp;
O3(t0) ¼ 0.7 (respective effective pulse area A ’ 5.9p), ja0 j/t02 ¼ 10.
dressed state energies (solid lines) and the bare state energies (dashed lines)
as a function of time.
For the case of the positive Stokes chirp, three dressed states are involved
in the time evolution of the system: dressed states I, III, and IV. Initially,
dressed states I and III are populated due to initial population of the bare
states j1i and j3i. As time evolves, dressed states I and IV approach avoiding
crossing, which is followed by the avoiding crossing between states I and III.
These two avoiding crossings are not really separated in time; most probably, they cannot be treated independently. This complex situation results in
essentially nonadiabatic population transfer between dressed states I, III,
and IV. In turn, populations and coherences in the bare state basis show no
sign of adiabatic control. However, there are areas of the parameters (blue
area in Figure 7.14b, left side) where r34 ¼ 0.5, which means all population is
now distributed between states j3i and j4i. In the dressed state, these regions
correspond to the case when only two dressed states I and III are populated,
and the first avoiding crossing between I and IV is adiabatic; at a later time,
the state I correlates with bare state j3i while the dressed state III correlates
with bare state j4i.
In the case of the negative Stokes chirp, Figure 7.15b, the dressed state
picture looks much better for realizing adiabatic control at least of the
resonant mode. Here, the dressed states I, II, and III are involved in the
system dynamics and the dressed state IV is well separated from all other
states. However, there is the avoiding crossing between states III and II,
which effectively involves the bare state j3i into evolution. In fact, the
oscillations in the coherence r12 at final time (Figure 7.14a, left side) as a
function of the effective pulse area at the fixed chirp rate demonstrate the
importance of the dynamical phase, meaning that several dressed states
provide a contribution to the bare state populations; in this case, they are
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Svetlana A. Malinovskaya et al.
the states I, II, and III. Note that at some values of the effective pulse area the
population is only in the states j1i and j2i which provide r12 ¼ 0.5 at the fixed
relative phase.
In summary, we investigate the impact of the coupling between Raman
active vibrational modes on the controllability of their excitation. We also
analyze a possibility for optimizing the CARS signal in the case of coupled
Raman active vibrational modes for enhanced imaging. Selective excitation
in a predetermined vibrational mode among many of them having close
transitional frequencies may be achieved by using the chirped pump and
Stokes laser pulses within CARS. A theory developed in this Section implements the roof method4 to a system of two coupled TLSs and gives us a
broader essence of the method implementation in laboratory. The use of
femtosecond, chirped laser pulses with chirp sign variation at central time
provides adiabatic or near adiabatic passage in two coupled TLSs leading to
a significant coherence in the resonant TLS and zero coherence in nonresonant TLS in the presence of the coupling between them via external electric
fields. The results show that by applying the roof method, one can stay in
low-intensity regime and gain coherent control over the system. The positive
chirp is desirable for the excitation in the resonant TLS for degenerate model.
Also, single phase calculations support this idea for optimizing coherence
among the states of interest. For the nondegenerate model, the phaseaveraged solution gives population transfer among all states in TLSs. Near
adiabatic passage resulting in substantial coherence in the resonant mode is
observed for a single, fixed phase between initially populated states in the
coupled TLSs. The dressed states analysis supports this conclusion by showing an optimal population transfer between the ground and excited states in
the resonant TLS, which is a desirable condition for having a high value of
coherence. Thus, the roof method can be used for noninvasive imaging of the
biological specimens in the presence of the coupling between vibrational
modes and be an efficient tool to suppress the contribution of the nonresonant background and, thus, improve the selectivity and chemical sensitivity
of the CARS signals.
4. SELECTIVE EXCITATION OF RAMAN TRANSITIONS BY
TWO CHIRPED PULSE TRAINS IN THE PRESENCE OF
DECOHERENCE
Maximum coherence between predetermined states in atoms and molecules
is an important aspect of many problems in quantum optics. One possible
example is the generation of optimal CARS signal in the CARS microscopy
aiming to reveal structures consisting of particular molecular groups. Presently, in experiments on biological imaging, the CARS signal is generated
with implementation of picosecond laser pulses,58 a narrow spectral bandwidth of which allows one to address a particular molecular group with a
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good accuracy. Aiming at improving chemical sensitivity and obtaining
an optimal CARS signal for a given pulse intensity, we propose to use the
quantum control methods for selective excitation of required vibrations in
complex molecular systems. We developed a novel method that allows one
to achieve a maximum coherence in a predetermined vibrational mode in a
multimode molecular system via the Raman transitions induced by the
pump and Stokes chirped pulses discussed in Section 2. According to
the Maxwell–Bloch equations, the maximum coherence is the condition for
the CARS signal to acquire the highest intensity upon propagation through a
molecular medium.5 The mechanism of creation of the maximum coherence
underlying the proposed method is based on the adiabatic passage.
An important aspect in the investigations associated with a condense
phase is decoherence. Two main sources of decoherence, the vibrational
energy relaxation and collisional dephasing, may significantly affect the
CARS signal intensity. The phase control of molecular processes in the
presence of collisions was demonstrated in Ref. 59. In this Section, we
show that the use of two chirped pulse trains, prepared in accordance with
the proposed method of chirped pulse adiabatic passage, allows one to
control coherence between selected molecular states in the presence of
decoherence caused by the spontaneous emission, which occurs on the
timescale close to an incident pulse duration.7–9
Femtosecond pulse trains have been recently utilized in the frequency
comb spectroscopy to control atomic and molecular systems.60–62 In Ref.
61, a coherent train of weak pump–dump pulse pairs is implemented to
perform the narrow-band Raman transitions. A piecewise adiabatic passage
is demonstrated in Ref. 61 on an example of the STIRAP scheme using a
series of femtosecond transform-limited pulses. The experiments on the
molecular trace presence, spectroscopic parameters evaluation, and realtime population dynamics implementing the frequency comb are described
in Ref. 62.
4.1. Theoretical model
In our model, we make use of a series of femtosecond, chirped pulses with
the phase changing in each pulse as f(t) ¼ o0t þ ap,st2/2. We describe molecular vibrational modes by TLSs with the lower and upper levels representing
the ground and excited molecular vibrational states of an individual mode
(Figure 7.16).
Transition frequencies of the TLSs o21 and o43 are chosen to be very close to
model a typically dense vibrational frequency spectrum, for example,
o43 o21 ¼ d o21. Initially, only ground states j1i and j3i are evenly populated. Let our goal be to generate and sustain high level coherence in the j1i–j2i
TLS and zero coherence in the j3i–j4i TLS in the presence of decoherence
caused by vibrational energy relaxation and collisions. Under these conditions,
a strong CARS signal will be generated from a molecular group represented by
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ws
was = 2wp - ws
wp
|4ñ
d
|2ñ
w21
|1ñ
w43
|3ñ
Figure 7.16 A scheme of the two, two-level systems representing Raman active vibrational
modes that interact with the intense chirped pump and Stokes pulses to generate maximum
coherence in the j1i–j2i TLS and zero coherence in the j3i–j4i TLS.
the j1i–j2i TLS, and there will be no signal from any other molecular group,
particularly, having close vibrational frequency or from the background.
To that end, two-photon Raman transitions are realized by a series of femtosecond pump and Stokes pulses having central frequencies op and os, which
(0)
2
are described as Ep, s(t) ¼ E(0)
p, s(t)cos(op, s(t t0) þ ap, s(t t0) /2). Here, Ep, s(t) ¼
2
2
E0 exp((t t0) /2t ) are the pump and Stokes pulse envelopes, and
t ¼ t0[1þ a0 p, s/t40]1/2 is the chirp-dependent pulse duration.26 We choose that
for ap,s ¼ 0, op os ¼ o21. The temporal (a) and spectral (a0 ) chirps are related as
4
4
02
ap, s ¼ a0 p, st
0 /(1 þ a p, s/t0).
The Hamiltonian that describes the interaction of the pump and Stokes
pulse pair with a single TLS with no decoherence taken into account, reads
as1
1
0
1
O3 ðtÞ
ðd þ ðap as Þt þ O1 ðtÞ O2 ðtÞÞ
C
B2
C
B
H
¼B
C;
1
A
@
ðd þ ðap as Þt þ O1 ðtÞ O2 ðtÞÞ
O3 ðtÞ
2
O1 ðtÞ ¼
m2
E20
m2
E20
2 2
t2 =t2
e
;
O
ðtÞ
¼
et =t ;
2
2 4 1=2
2 4 1=2
2
2
0
0
4ℏ D ð1 þ a p =t0 Þ
4ℏ D ð1 þ a s =t0 Þ
O3 ðtÞ ¼
m2
E20
2 2
et =t ;
2
2
4ℏ D ½ð1 þ a0 p =t40 Þð1 þ a0 2s =t40 Þ1=4
ð13Þ
with d ¼ 0 for the resonant, j1i–j2i TLS and d 6¼ 0 for the detuned, j3i–j4i TLS.
Here, O1(t) and O2(t) are the ac Stark shifts originated from two-photon
Raman transitions, and O3(t) is the effective Rabi frequency.
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Intensity
wp
t0
Time
wp
ws
t0
Time
Frequency
Frequency
Intensity
The proposed method suggests to take a constant value of as, to provide
monotonous change of the Stokes pulse frequency, and to choose ap to have
same magnitude and opposite sign before the central time t0, when the pump
and Stokes pulses reach maximum amplitude, and then to flip the sign.
Under these conditions, at t0, the difference of the pump and Stokes pulse
frequencies comes into resonance with the frequency o21 and does not
change further on. The adiabatic passage takes place in both TLSs and results
in zero coherence r34 in the detuned, j3i–j4i TLS and maximum coherence
r12 in the resonant, j1i–j2i TLS in a wide range of the field parameters
satisfying condition t ’ 1/d (see also Ref. 63). The pump and Stokes pulses
are described by the Wigner diagrams in Figure 7.17.
In the following, we analyze the vibrational energy relaxation and collisional dephasing as factors that cause decoherence in a selectively excited
Raman mode for characteristic times of these processes close to the pulse
duration. Then, we analyze a possibility of counteracting decoherence by
applying two chirped pulse trains, prepared in accordance with the method
stated above.
Using the Liouville von Neumann equation and including relaxation
terms, we derived a set of differential equations for the density matrix
evolution which reads as
ws
as
t0
Time
t0
Time
Figure 7.17 The Wigner presentation of the pump (left panel) and Stokes (right panel)
pulses. The positive and negative slopes of white dashed lines on the density plots
correspond to an upward and a downward frequency chirps.
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r_ 11 ¼ 2O3 ðtÞIm½r21 þ g2 r22 ;
r_ 22 ¼ 2O3 ðtÞIm½r21 g2 r22 ;
0
1
g
r_ 12 ¼ iðd þ ðas ap Þt þ O1 ðtÞ O2 ðtÞÞr12 þ iO3 ðtÞðr22 r11 Þ @ 2 þ GAr12 ;
2
0
1
g
r_ 21 ¼ iðd þ ðas ap Þt þ O1 ðtÞ O2 ðtÞÞr21 iO3 ðtÞðr22 r11 Þ @ 2 þ GAr21 :
2
(14)
Parameter g2 ¼ 1/T1 is the vibrational energy relaxation rate, and G ¼ 1/T2* is
the collisional dephasing rate. Parameters g2 and G can originate from the
inelastic and elastic collisions, respectively. In Eq. (14), we used indexes 1 and
2 that label states of the resonant, j1i–j2i TLS, in which case d ¼ 0. To get the
equations for the detuned, j3i–j4i TLS, the former indexes have to be
substituted by 3 and 4. This strategy has to be applied to all following equations.
4.2. Numerical results: The effects of the vibrational energy
relaxation and collisional dephasing on coherence loss
The values of parameters used in numerical calculations are as follows.
Raman transition frequencies are o21 ¼ 85.35 THz (2845 cm 1) and
o43 ¼ 88.68 THz (2956 cm 1). These are the frequencies of CH2 and CH
molecular groups, which are abundantly present in various biological tissues. Field parameters are O3 ¼ 28.4 1012 W/cm2 and t0 ¼ 177 fs,
a0 s ¼ 2.8 10 4 cm 2, giving t ¼ 1.58 ps. These parameters of the fields are
chosen in accordance with the data published in Ref. 63 to provide the
adiabatic passage in the TLSs resulting in a maximum value of r12 and
zero r34. The parameters that determine decoherence g2 and G vary in the
region from 0 to 0.085 THz, allowing the time of vibrational energy relaxation and collisional dephasing to change from infinity to 11.76 ps.
At first, we calculated r12 using Eq. (14) in the case when a single pair of
pump and Stokes laser pulses interacts with the TLS. In Figure 7.18, dotted
lines show the dependence of coherence r12 on g2 for various fixed values of
G: from G ¼ 0 (the upper curve) to G ¼ 10 3 (the lowest curve), solid lines
show the dependence of coherence r12 on G for analogous fixed values of g2.
The parameters g2 and G are given in the units of frequency o21. Given
o21 ¼ 85.35 THz, the value g2 ¼ G ¼ 10 3 corresponds to 0.085 THz. For
g2 ¼ 0, increasing G from 0 to 0.085 THz causes significant decrease in r12
from the maximum value (0.25) to 0.065. While for G ¼ 0, increasing g2 causes
twice less decrease in r12 in agreement with Eq. (14). This demonstrates that
the dynamics of induced coherence is very sensitive to collisional dephasing,
and to less extent to the vibrational energy relaxation. When decoherence
time is one order of magnitude larger than the pulse duration, coherence in
the resonant TLS is small, yielding a small amplitude of the generated
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0.25
r12
0.2
0.15
0.1
0.05
0
0
0.0002
0.0004
0.0006
0.0008
0.001
g 2/w, G /w
Figure 7.18 Dependence of coherence r12 on g2 and G. Dotted lines show the dependence
of coherence r12 on g2 for various fixed values of G: from G ¼ 0 (the upper curve) to G ¼ 10 3
(the lowest curve), solid lines show dependence of coherence r12 on G for analogous fixed
values of g2. The parameters g2 and G are given in the units of frequency o21.
Raman fields and a weak CARS signal. For the detuned TLS, numerical
calculations show minor changes in this picture, preserving characteristic
features of the behavior of the coherence as a function of the g2 and G
parameters.
4.3. Prevention of decoherence
In the framework of the dressed state picture in the adiabatic approximation,
valid for g2, G O3, the balance equations for the population of the dressed
states read as
r_ d11 ¼ ðg2 sin2 Y þ 1=2ðG g2 =2Þ sin2 2YÞrd11 þ ðg2 cos4 Y þ 1=2G sin2 2YÞrd22 ;
r_ d22 ¼ ðg2 sin4 Y þ 1=2G sin2 2YÞrd11 ðg2 cos2 Y þ 1=2ðG g2 =2Þ sin2 2YÞrd22 ;
(15)
where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sinY ¼ 1=2ð1 R0 =RÞ; cosY ¼ 1=2ð1 þ R0 =RÞ;
R0 ¼ d þ ðas ap Þt þ O1 ðtÞ O2 ðtÞ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R ¼ ðd þ ðas ap Þt þ O1 ðtÞ O2 ðtÞÞ2 þ 4O3 ðtÞ2 :
(16)
These equations demonstrate the time evolution of the population of the
dressed states assuming rd12 ¼ 0.
The correlation between the dressed states and the bare states in the fieldinteraction representation is as follows:
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rd11 ¼ cos2 Yr11 þ sin2 Yr22 2sinY cosYRe½r12 ;
rd22 ¼ sin2 Yr11 þ cos2 Yr22 þ 2sinY cosYRe½r12 :
(17)
According to Eq. (17), at t ! 1, for the detuned TLS, when d 6¼ 0, the coefficients are sin2 Y ¼ 0, and cos2 Y ¼ 1. It means that the dressed state population rd11 relaxes to the bare state population r11, and dressed state population
rd22 relaxes to the bare state population r22. In the resonant TLS, the asymptotic solution is qualitatively different. When d ¼ 0, it gives sin2 Y ¼ cos2
Y ¼ 1/2, and asymptotically the dressed states populations are
rd11 ¼ 1=2ðr11 þ r22 Þ Re½r12 ;
rd22 ¼ 1=2ðr11 þ r22 Þ þ Re½r12 ;
(18)
which means that as t ! 1, rd11 and rd22 approach r11/2. This behavior of the
dressed states populations may also be obtained from the numerical solution
of Eq. (15).
It follows from Eq. (18) that periodic restoration of the population of the
upper level r22 in the resonant TLS would provide the asymptotic solution
for the dressed state to be not r11/2 but a nonzero superposition of the lower
and upper bare states and coherence. This is the essence of the mechanism of
preventing decoherence. To do so in a selectively excited TLS, we propose to
use 2-fs pulse trains chirped in accordance with the chirped pulse adiabatic
passage. A series of pump and Stokes chirped pulses, when applied with the
period comparable to the vibrational energy relaxation time, create and
sustain high coherence in the selected vibrational mode. The values of g2
and G used in further calculations are equal to 0.085 THz. These values give
the time of vibrational energy relaxation and collisional dephasing (decoherence time) to be 11.76 ps.
Figure 7.19 shows results of interaction of the resonant TLS with two
sequential pump and Stokes pulse pairs having the time delay approximately four times longer than the decoherence time, namely, 44 ps. The
bold black curve shows the effective Rabi frequency O3, which is the same
for both the pump and Stokes pulses. The time dependence of the state
population and coherence, calculated in the presence of only vibrational
energy relaxation, is marked by r11 ðerÞ , r22 ðerÞ , and r12 ðerÞ ; the state population
and coherence, calculated in the presence of only collisional dephasing is
marked by r11 ðcdÞ , r22 ðcdÞ , and r12 ðcdÞ . Figure 7.19 shows that r12 ðerÞ achieves
the maximum value during the application of the first pulse pair and
undergoes decay up to the instant when the second pulse pair arrives and
restores population of the upper level and coherence. Notably, the second
pair of pulses does not affect coherence r12 ðcdÞ because it is not coupled to the
population distribution. The restoration of maximum coherence takes place
because of the time delay between the first and the second pulse pairs being
longer than the vibrational relaxation time. This results in almost complete
relaxation of the population to the ground state in the resonant TLS and
return of the system into its initial state. Figure 7.20 shows the time-
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Ultrafast Manipulation of Raman Transitions and Prevention of Decoherence
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0.5
r (er)
11
0.4
0.3
(cd) (cd)
r 22
r11
0.2
r (er)
r(er)
12
22
0.1
W3
r (cd)
12
0
0
1
2
3
5
4
tw 21 ⫻ 10
3
Figure 7.19 Dynamics in the resonant TLS in the field-interaction representation. Figureshows the results of application of two sequential pump and Stokes pulse pairs with the
time delay 44 ps. The parameters are g2 ¼ G ¼ 10 3 (0.085 THz).
(er)
0.3
(er)
r11, r22, r12
0.4
(er)
0.5
0.2
W3(t)/w 21
0.1
0.20
0.15
0.1
0.05
0
0
2
4
6
8
10
tw 21⫻103
Figure 7.20 Dynamics in the resonant TLS in the field-interaction representation. Figureshows the state population and coherence resulting from application of the pump and
Stokes chirped pulse trains. The pulse trains period is 11.76 ps, g2 ¼ 10 3 (0.085 THz).
dependent picture of the resonant TLS’s parameters resulting from its interaction with the sequence of pump and Stokes pulses. The pulse train period
is 11.76 ps, which is the same as the time of vibrational energy relaxation.
The upper panel shows the populations r11 (long-dashed line) and r22
(dotted line), and coherence r12 (solid line) in the TLS, and the lower panel
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Svetlana A. Malinovskaya et al.
shows the dimensionless Rabi frequency O3(t)/o21. Prepared by the first
pulse pair, the coherence r12 ðerÞ decreases from maximum (0.25) to 0.10 and
varies within a 0.10–0.15 region, subject to vibrational energy relaxation and
periodic interaction with the subsequent pulse pairs. The variation of coherence within a restricted region is determined by the population dynamics
which is different from the previous case. Here, according to Figure 7.20, the
population of the upper level in the TLS only partially relaxes to the lower
level during the time between pulses. This population distribution determines the ‘‘initial’’ condition for each subsequent pulse pair and provides a
partial adiabatic population transfer to the upper level resulting in coherence within indicated region. The choice of the period of the pulse train
depends on one’s need to either maintain uninterruptedly substantial coherence in a TLS or get maximum coherence periodically.
In the detuned TLS, the pump and Stokes chirped pulse trains suppress
generation of the coherence by preserving the population in the ground state.
Thus, the proposed method satisfies the goal of selective excitation and
creation of the coherence in a vibrational mode having known frequency.
In summary, we propose a novel method to control molecular vibrational
excitations and sustain high coherence in a predetermined vibrational mode
in the presence of decoherence by making use of the pump and Stokes
femtosecond, chirped pulse trains. The method employs a constant chirp
in the pump (Stokes) pulse train and the flipping sign of the chirp in the
Stokes (pump) pulse train. The pulse train period must be close to the time of
vibrational energy relaxation. High coherence is sustained in a selectively
excited vibrational mode (modeled by a TLS) in the presence of decoherence
because of a periodic, adiabatic pumping of the population to the upper level
by the pump and Stokes chirped pulse trains. The method can be used for
noninvasive imaging of biological structure and femtosecond dynamics
studies, and as an efficient tool for suppression of the background signal in
the CARS spectroscopy and microscopy. It demonstrates the possibility of
coherent control over spectrally close vibrational modes in the nonimpulsive
Raman scattering.
5. FESHBACH-TO-ULTRACOLD MOLECULAR STATE RAMAN
TRANSITIONS VIA A FEMTOSECOND OPTICAL
FREQUENCY COMB
Magnetically tunable Feshbach resonances are used to create loosely bound
Feshbach molecules by association of atom pairs in cold temperatures. The
methodology is based on magnetic field sweep across zero-energy resonance
between the diatomic vibrational bound state and the threshold for dissociation into an atom pair at rest.64 The technique was applied to produce
diatomic molecules consisting of identical atoms, for example, K2,65 Li2,66
and Cs2,67 or of different species of atoms, resulting in creation of polar
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molecules, for example, the KRb.20,68 The permanent dipole moment associated with polar molecules may be used for studies of long-range interactions
in many-body quantum systems. Optically induced transitions in Feshbach
molecules are an efficient route to reach the fundamentally lowest state in
molecules. In Ref. 24, using two narrow-band pulses applied to weakly
bound KRb molecules in accordance with the STIRAP scheme, the ultracold
gas of polar KRb molecules in the ground singlet or triplet electronic configuration and rovibrational state was created. Recently, a theory on piecewise
stimulated Raman adiabatic passage involving Feshbach states was performed using two coherent pulse trains.22,23 The ultracold KRb molecules
were created using the pump–dump stepwise technique that coherently
accumulated population in the ground electronic–vibrational state.
An optical frequency comb is a unique tool for high-resolution spectroscopic analysis of internal energy structure and dynamics as well as for
controlling ultrafast phenomena in atomic and molecular physics.16–19
Owing to its broadband spectrum, the frequency comb may efficiently
interact with the medium inducing one-photon, two-photon, and multiphoton resonances in atoms and molecules. A unique ability of the frequency
comb is provided by the presence of about a million optical modes in its
spectrum with very narrow bandwidth and exact frequency positions.69 In
this Section, aiming at creation of deeply bound ultracold molecules from
Feshbach states, the control of population dynamics in a molecular system is
investigated using a single optical frequency comb with zero
offset frequency.10–12 The frequency comb is generated by phase-locked femtosecond pulse train having unmodulated or sinusoidally modulated phase
across an individual pulse. We investigate the dynamics of rovibrational
cooling on an example of the KRb molecule, which involves the interaction
of loosely bound Feshbach KRb molecules with the femtosecond optical
frequency comb resulting in the molecular transfer to the ground electronic
and rovibrational state. The population dynamics takes place via two-photon
Raman transitions between three energy levels separated by terahertz region
frequencies. Coherent accumulation of population in the ultracold KRb state
with a negligible population (for the case of the sinusoidally modulated
optical frequency comb applied) of the excited state is accomplished by a
series of sequential pulses with zero carrier–envelope phase and within the
lifetime of the Feshbach KRb molecules, which is about 100 ms.68,70
5.1. Theoretical model
A semi-classical model of two-photon Raman transitions induced by a
femtosecond optical frequency comb in a l system, Figure 7.21, describes
the cooling process of Feshbach KRb molecules. A frequency comb known to
be characterized by two key parameters, the radio frequency ( fr), determined
by the pulse repetition rate and specifying the spacing between modes, and
the carrier–envelop offset frequency (f0 ¼ frDfce/(2p)), where Dfce is the
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|2Ò
d
wL
|1Ò
fr
|3Ò
Figure 7.21 The three-level l-system modeling the molecular energy levels, involved in
Raman transitions. State j1i is the Feshbach state, state j2i is the transitional, electronic
excited state, and state j3i is the cold molecular state.
carrier–envelope phase difference. Then, a mode in the comb has a frequency equal to f ¼ nfr þ f0. Both parameters, the f0 and the fr, were efficiently
used to manipulate dynamics in, for example, Ref. 21, to resonantly enhance
two-photon transitions in cold 87Rb atoms. Here, we make use of the radio
frequency (fr) only, and put f0 equal to zero by implying zero value of Dfce.
The pulse train that generates the optical frequency comb reads
2
2
EðtÞ ¼ SN1
k¼0 E0 expððt kTÞ =ð2t ÞÞ cosðoL ðt kTÞÞ:
(19)
Here, E0 is the peak field amplitude, T is the pulse train period, t is a single
pulse duration, N is the number of the pulses in the pulse train, and oL is the
carrier frequency. To get zero value of the Dfce, the temporal variation in the
Gaussian envelope and in the harmonic wave is taken in the form (t kT). It
guarantees the envelope maximum to coincide with the peak value of the
amplitude of the electric field. The fine structure of the optical broadband
comb is owing to the modes equally spaced by the radio frequency 1/T 1. In
our model, a single optical frequency comb induces Raman resonances in the
three-level l-system resulting in full population transfer from the initial state
j1i through intermediate state j2i to the final state j3i. We assume that state
j1i is the Feshbach state, state j2i is the transitional, electronic excited state,
or state manifold, and state j3i is the ultracold molecular state. There are a
large number of mode pairs in the frequency comb that differ by exactly the
transition frequency o31 and, thus, satisfy the condition of the Raman resonance. These lead to an efficient stepwise population accumulation in the
final state in the l-system.
The evolution of the density matrix of the l-system is investigated via a set
of coupled differential equations obtained from the Liouville von Neumann
equation:
r_ 11 ¼ 2Im½H12 r21 ;
r_ 22 ¼ 2Im½H21 r12 þ H23 r32 ;
r_ 33 ¼ 2Im½H32 r23 ;
r_ 12 ¼ iH12 ðr22 r11 Þ þ iH32 r13 ;
r_ 13 ¼ iH12 r23 þ iH23 r12 ;
r_ 23 ¼ iH23 ðr33 r22 Þ iH21 r13 :
(20)
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Two nonzero matrix elements of the interaction Hamiltonian are
Hij ¼ OR(t T) [exp{ i((oL þ oji)(t T))} þ exp{i((oL oji)(t T))}], where i
and j are the indexes of the basis set, i ¼ 1, 2, and j ¼ i þ 1, OR(t T) ¼ OR exp
(t T)2/(2t2) is the Rabi frequency, and OR is the peak value of the Rabi
frequency. Calculations are done beyond the rotating wave approximation.
5.2. Numerical results: Creation of ultracold molecules using a
standard optical frequency comb and a phase-modulated
optical frequency comb
For the l-system, we apply a set of parameters that corresponds to data on
molecular cooling of loosely bound KRb molecules from the Feshbach states
presented in Ref. 24. Experimental schemes used in Ref. 24 involve a transition to the ground electronic triplet or the ground electronic singlet state
with zero rovibrational quantum number. These states are achieved by
implementing the STIRAP control scheme with a pair of pulses having a
narrow bandwidth and being in resonance with the transitions between
three states involved in the process. These are the Feshbach state, the 23S
electronically excited state, and the triplet or singlet ground electronic state
with zero rotational and vibrational quantum number. In our model, we
implement parameters that correspond to the experiment involving the singlet ground electronic state, thus addressing a fundamentally cold molecule.
The parameters of the l-system are o21 ¼ 309.3 THz and o32 ¼ 434.8 THz,
making the frequency of two-photon transition o31 to be equal to
125.5 THz. The carrier frequency of the pulse train oL is chosen to be in
resonance with the one-photon transition frequency o32 in the l-system, and
the multiples of the radio frequency provide two-photon resonances when
the condition oL nfr ¼ o31 is satisfied, where n is integer number. The peak
Rabi frequency is OR ¼ 0.01o31 ¼ 1.255 THz, the pulse duration t is 3 fs, and
the pulse train period is T ¼ 6.4 105t (2 ns), giving the radio frequency
500 MHz. For a given peak Rabi frequency, the evaluated peak field amplitude is 106 –107 V/cm.
The results of population transfer are presented in the Figure 7.22. A stepwise adiabatic population transfer is observed from the initial j1i (black) to
the final j3i state (green) via the transitional state j2i (red), which gets
populated up to 45%. Each pulse brings a fraction of population to the
final state and contributes to the accumulative effect. Total population transfer occurs in 460 ns after 242 sequential pulses, which is within the lifetime of
the Feshbach KRb molecules. Next 242 pulses return population to the initial
state, the system returns to the initial conditions, and, then, the dynamics
repeats. Within the lifetime of the Feshbach states, it is possible to transform
the medium from highly vibrationally excited molecules to the ultracold
state and back. Notably, the detuning of the carrier frequency off-resonance
with the o32 gives very similar dynamics of population transfer to the
ultracold molecular state and back as in the resonance case with a difference
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1
Population
0.8
0.6
0.4
0.2
0
0
5000
10000
tw ⫻ 104
Figure 7.22 Population dynamics in the three-level l-system using an optical frequency
comb having fr ¼ 500 MHz, and zero offset frequency. Parameters of the pulse train are the
carrier frequency oL ¼ 434.8 THz, the pulse duration t0 ¼ 3 fs, and the peak Rabi frequency
OR ¼ 1.26 THz; the system parameters are o21 ¼ 309.3 THz, o32 ¼ 434.8 THz, and
o31 ¼ 125.5 THz. Black line shows population of the ground state, red line, population of
the transitional state, and green line, that of the final state. Time is given in the units of
[o 1], where o ¼ o31 ¼ 125.5 THz.
being some reduction in the number of pulses needed to accomplish this
transfer. The increase in the strength of the electric field decreases the
number of pulses and the overall time duration needed for the full population transfer, preserving the quality of the dynamics picture. Decreasing the
pulse repetition rate elongates the duration of the coherent accumulation
dynamics; however, it does not change the number of the required pulses
(given other field parameters are preserved). Since a single pulse duration
determines the bandwidth of the comb, the shorter it is the more efficient
excitation of Raman transitions takes place.
We performed the phase modulation across an individual pulse in the
pulse train in the form of a sinusoidal function. In previous works, the
sinusoidal modulation in the terahertz region was applied to the carrier
frequency, for example, to study absorption resonances in I2 with high
precision.71 A general form of a phase-modulated pulse train reads
Eðt; zÞ ¼
N
1
X
E0 expððt kTÞ2 =ð2t2 ÞÞ cosðoL ðt kTÞ þ F0 sinðOðt kTÞÞÞ:
k¼0
(21)
Here, O is the modulation frequency, and F0 is the modulation amplitude.
The carrier–envelope phase is zero in the pulse train resulting in zero offset
frequency of the generated optical frequency comb. The time-dependent
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phase across each pulse in the form of the sin function enriches the frequency
comb power spectrum with new peaks compared to a standard optical
frequency comb discussed above. More specifically, laser frequency oL
determines the center of the frequency comb, and the sinusoidal modulation
forms the sidebands at multiples of O with the amplitude dictated by F0.
Numerical analysis of the population dynamics in the l-system was done
using Eq. (21) with the Hamiltonian having two nonzero matrix elements
Hij ¼ OR(t T) [exp{ i((oL þ oji)(t T) þ M(t T))} þ exp{i((oL oji) (t T) þ
M(t T))}], where i ¼ 1, 2 and j ¼ i þ 1, and M(t T) ¼ F0 sin O(t T) is the
phase modulation in a single pulse. The carrier frequency of the pulse train
oL is chosen to be equal to the o32, and the modulation frequency O, to the
o21, so that the field frequencies are in resonance with the one-photon
transitions in the three-level l-system which automatically satisfies the
condition for the two-photon Raman resonance. Additionally, the modes
that are multiples of the radio frequency fr provide with pairs of frequencies
that differ by exactly the transition frequency o31. These lead to an efficient
stepwise population accumulation in the final state in the l-system. Parameters of the l-system used in these calculations were taken from Ref. 23. The
system levels addressed in Ref. 23 include the final state as the ground
electronic singlet state with vibrational quantum number v ¼ 22, which suggests the following parameters for the l-system: the transitional frequency
from the initial to the excited electronic state o21 ¼ 340.7 THz, and the transition frequency from the excited electronic state to the final state
o32 ¼ 410.7 THz. It makes the initial-to-final state frequency difference
o31 ¼ 70 THz. The carrier frequency and the modulation frequency of the
pulse train are in resonance with the o32 and o21, respectively; the peak Rabi
frequency is equal to o31, F0 is equal to 4 (the discussion of this choice will be
given below), a single pulse duration t is 3 fs, and the pulse train period T is
6400t (about 20 ps). The modulation of the carrier frequency having value
410.7 THz is to be done at frequency 340.7 THz. To achieve the needed
modulation, an approach, described in Ref. 72, to efficient generation of
a Raman-type optical frequency comb in an enhancement cavity may
be applied. The technique provides the whole comb bandwidth covering
300–900 THz.
The results of the numerical calculations, Figure 7.23, show a smooth
stepwise population transfer from the initial to the final state with a negligible population of the excited state, in analogy with the STIRAP stepwise
scheme proposed by Shapiro,23 however, performed with a single optical
frequency comb. In our opinion, making use of a single pulse train, in
contrast to the two pulse trains scheme in Ref. 23, significantly simplifies
the experimental conditions for cooling of rovibrational degrees of freedom
in molecules. Total population transfer occurs after 109 sequential pulses
and, thus, is accomplished in 2.5 ns.
Notably, there is a strong dependence of the efficiency of population
transfer on the value of the amplitude F0 of sinusoidal modulation of the
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1
Population
0.8
0.6
0.4
0.2
0
0
10
20
30
tw ⫻ 104
Figure 7.23 Population transfer in the three-level l-system, achieved via the resonant
Raman transitions using a phase-modulated optical frequency comb described by Eq. (21).
The values of the parameters are the carrier frequency oL ¼ 410.7 THz, the modulation
frequency O ¼ 340.7 THz, the modulation amplitude F0 ¼ 4, and the peak Rabi frequency
OR ¼ 70 THz. Stepwise, adiabatic accumulation of the population is observed in state j3i
(green), which is the ultracold KRb state. The population of the Feshbach state j1i (black),
comes gradually to zero, while the excited state manifold j2i (red), is slightly populated
during the transitional time. Full population transfer is accomplished in 109 pulses. Time is
given in the units of [o 1], where o ¼ o31 ¼ 70 THz.
phase across an individual pulse in the pulse train. For the resonant excitation, population dynamics was calculated using different values of the
parameter F0; however, for the values of F0 from 1 to 9, only F0 ¼ 4 gave
the desired population transfer to the cold state. To get insight into the
mechanism of the frequency comb–system interaction leading to a successful
cooling, we made a Fourier transform both analytically and numerically to
reveal its spectral properties linking to the molecular system resonances.
The Fourier transform reads
EðoÞ ¼ ðE0 tÞ=2Sn Jn ðF0 Þ expð1=2ðoL þ nO oÞ2 t2 ÞSk expðiokTÞ:
(22)
Here, Jn(F0) is the Bessel function of the order n and F0 is the modulation
index. When multiplied by exp ( 1/2(oL þ nO o)2t2), it determines the
shape of the power spectrum of the optical frequency comb.
Depending on the value of F0, the power spectrum has different number
of maxima as seen in Figure 7.24 showing the envelope of the power spectrum for F0 ¼ 3, 4, 5, 8. The increase in modulation index brings additional,
intense peaks of modes into spectrum and broadens it. These maxima
are located at different frequencies for different values of F0 affecting the
population dynamics in the l-system.
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0.8
0.4
E
2
WT
0.6
0.2
2
4
6
8
10
12
14
WT
2
E (wt)
Figure 7.24 The envelope of the power spectrum of the optical frequency comb described
by Eq. (21) (no dense radio-frequency comb lines are included). The pulse train parameters
are the effective Rabi frequency OR ¼ 70 THz, the carrier frequency oL ¼ 410.7 THz, the
modulation frequency O ¼ 340.7 THz, t ¼ 3 fs, and the modulation amplitudes F0 ¼ 3 (red),
F0 ¼ 4 (green), F0 ¼ 5 (blue), and F0 ¼ 8 (black).
0
wt
Figure 7.25 The fast Fourier transform of the phase-modulated optical frequency comb in
Eq. (21). The pulse train parameters are the effective Rabi frequency OR ¼ 70 THz, the carrier
frequency oL ¼ 410.7 THz, the modulation frequency O ¼ 340.7 THz, t ¼ 3 fs, and the modulation amplitude F0 ¼ 4.
The power spectrum of the pulse train with the modulation amplitude
F0 ¼ 4 has three maxima; the highest one is located at o ¼ 4.9 (which is in
resonance with the o21), making the pulse train with F0 ¼ 4 an optimal one
for coherent accumulation of the population in the final state of the l-system.
It provides full population transfer to the ground electronic–vibrational
state, thus cooling the KRb Feshbach molecule. The numerical result of the
Fast Fourier Transform of Eq. (21) with F0 ¼ 4, Figure 7.25, shows a power
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Svetlana A. Malinovskaya et al.
spectrum of the optical frequency comb with the fine structure of the radiofrequency spaced modes appearing as a solid color under the envelope. The
parameters of the phase-locked pulse train, needed to accomplish molecular
cooling from Feshbach states, have to be chosen based on the analysis of the
power spectrum of the sin-phase-modulated optical frequency comb and the
energy levels involved in dynamics of the molecular system. The offset
frequency not necessarily has to be zero. Its nonzero value does not affect
the two-photon resonance condition, while, for one-photon transitions, it
induces a detuning. The field carrier frequency detuned off-resonance with
the one-photon transition in the l-system leads to the same quantum yield
on a shorter timescale. Alternatively, the detuning may be compensated by
adjusting the radio frequency to keep the frequency comb modes in resonance with the l-system.
In summary, we have demonstrated a coherent population transfer from a
loosely bound Feshbach state to the ultracold molecular state using a single
femtosecond, optical frequency comb. We studied the phenomenon on an
example of the KRb molecule, modeled by a three-level l-system with the
energy levels taken from Refs. 23,24. Coherent accumulation of the population in the ground electronic and rovibrational state is achieved by applying
a standard optical frequency comb with zero offset frequency, or an optical
frequency comb generated by a pulse train with a phase modulation in the
form of the sinusoidal function across an individual pulse. The mechanism
of the accumulative effect leading to full population transfer is based on the
excitation of the two-photon Raman resonances by pairs of optical frequency
modes with the frequency difference matching Feshbach-to-ultracold molecular state transition. In the case of sinusoidally modulated optical frequency
comb, the Raman transitions are stimulated by the carrier and the modulation frequencies. A strong dependence of the population dynamics on the
amplitude of the sinusoidal modulation is demonstrated, suggesting to
choose this parameter based on the analysis of the envelope of the power
spectrum of the optical frequency comb.
6. CONCLUSIONS
Recent advancements in the application of ultrafast laser pulses to control
Raman transitions have led to the development of new quantum control
methods aiming at enhancement of CARS bioimaging techniques. The proposed methods utilize femtosecond, chirped pulses to induce two-photon
excitations in Raman active vibrational modes and provide adiabatic, via the
pulse area solution or by the roof method, fractional population transfer in a
predetermined vibrational mode, resulting in a maximum coherence. At the
same time, coherence in other Raman active vibrational modes is completely
suppressed. Methods provide high chemical selectivity between spectrally
close Raman modes, whose frequency difference d is on the order of t 1, the
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255
inverse chirped pulse duration. To control molecular vibrational excitations
and sustain high coherence in a predetermined vibrational mode in the
presence of decoherence, the method implementing two chirped pulse trains
was proposed with the chirp scheme as in the roof method. The pulse trains
period must be close to the time of vibrational energy relaxation. High
coherence is sustained because of periodic adiabatic pumping of the population to the upper level. A pair of chirped pulse trains periodically restores
population in the excited state and, therefore, coherence between the ground
and the excited vibrational states and, thus, prevents decoherence caused by
spontaneous emission. We have also reviewed our achievements in a theory
of ultracold gases manipulation aiming at creation of ultracold molecules
from Feshbach states. For this purpose, we investigated applicability and
robustness of an optical frequency comb, which may be a standard one,
produced by a phase-locked pulse train with no modulation, or a frequency
comb with a sinusoidal phase modulation. It was demonstrated that a single
optical frequency comb, with no modulation or modulated, may be used to
cool the internal degrees of freedom in molecules and create ultracold
molecules from Feshbach states via two-photon Raman transitions.
ACKNOWLEDGMENTS
Authors acknowledge Philip H. Bucksbaum, Jun Ye, Paul R. Berman, Vladimir
S. Malinovsky, John R. Sabin, N. Yngve Öhrn, Jeffrey L. Krause, Moshe
Shapiro, and Pierre Meystre for many fruitful discussions. This research is
partially supported by the National Science Foundation under Grant No.
PHY-0855391 and DARPA under Grant No. HR0011-09-1-0008.
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