Bulletin of the Korean Chemical Society
Limits of the adiabaticity assumption and conditions for
improving laser focusing of atomic matter wave
Journal: Bulletin of the Korean Chemical Society
Manuscript ID B-21-PC-370-A
Wiley - Manuscript type: Article
Fo
Date Submitted by the
25-Oct-2021
Author:
Keywords:
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Complete List of Authors: Lee, Chang Jae; SunMoon University, Division of Basic Tecnology
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The adiabaticity assumption (AA) and the resulting concept of optical
potential in the context of atom focusing with lasers were examined
numerically for various experimentally controllable laser parameters. The
Schrödinger equations for the atomic center-of-mass dynamics in a laser
field were calculated using with or without the AA, and the results were
Abstract:
compared. The validity of the AA was tested over a wide range of the
Rabi frequency to detuning ratios and the optical potential depths. From
the analysis several specific guidelines were provided on choosing laser
parameters that sustain the validity of the AA and improve atom
focusing outcomes.
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Page 1 of 20
Bulletin of the Korean Chemical Society
RESEARCH ARTICLE
Bulletin of the Korean Chemical Society
Limits of the adiabaticity assumption and conditions for
improving laser focusing of atomic matter wave
Chang Jae Lee
Division of Basic Technologies, Sunmoon University, Asan 31460, Korea
E-mail: coolcjl@sunmoon.ac.kr
a substrate rely crucially upon the atoms to remain in the lower
state throughout the light-atom interaction time and to evolve in
the relevant optical potential—a concept resulting from the
adiabaticity assumption (AA). Several types of optical potentials
have been used in many other applications,16-18 but without giving
much thought on their validity other than simply assuming that the
Rabi frequency to drive the transitions is much smaller than the
detuning. Successful demonstrations of the atom deposition
method were possible by setting out suitable experimental
conditions under which predictions based on the concept of
optical potential is applicable in manipulating atomic motion with
lasers. However, it is impractical to find a wide range of such
suitable conditions with expensive and time-consuming
experiments alone, or from particle optics approaches19 that
ignore wave nature of matter. To our knowledge such conditions
have not yet been systematically explored despite continued
interests in the problem.20-22 For such investigations, in silico
approaches based on accurate numerical simulations would be
much more efficient. Adiabaticity is a key assumption in atom
manipulation methods, so it is critical to examine if the assumption
is valid for a given set of experimental conditions. The adiabaticity
problem has been actively researched in many areas and in
widely different contexts23 and, if certain conditions are met, the
assumption has shown to work quite well.24 Thus, it should be
Of course, other imperfections such as the non-ideal atomic beam,
highly valuable to give an extensive assessment over a broad
mechanical vibrations of the experimental setups, migrations of atoms
range
of laser and
parameters
of etc
thewill
validity
ofmask
the AA
the atom
after focusing
deposition,
surely
the in
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with optimal laser parameters given below. However, technologies
provide
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finding
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developguidelines
fast, so these
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be substantially
atomic
motion.
suppressed
eventually. THus, the goal/merit of this work is to
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In this
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we perform
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accurately
solve
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et aleliminating
giave sims with
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withoutMcClelland
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but that’s
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expt cinditions,
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and compare
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results
withgenerally.
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there are twoones
typeswith
of simulations:
approximate
the AA. Unlike many others, atom focusing
one
with
imperfections
incorporated..to match expt w thory
is an application of light assisted manipulation of matter wave
one withultimate resolution that can be achieved. This paer, latter side.
where quantitative investigation on the validity of the AA can be
made. This is equivalent to examining how well the concept of
optical potential holds up against a wide rage of laser parameters.
This paper is organized as follows: In the section that follows next,
we describe the relevant time-dependent Schrödinger equations
and review how the concept of optical potential emerges from the
AA. The numerical method to be used for the subsequent section
is also described in this section. Following this section, we
describe simulation parameters and discuss basic assumptions
made in all calculations. Results for various laser parameters with
and without invoking the concept of optical potential are compared
and general atom focusing trends are elucidated, depending on
the parameters. Based on the analyses some recommended
parameters are presented. Conclusions and further remarks are
given in the final section.
Introduction
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Fo
Abstract: The adiabaticity assumption (AA) and the resulting concept
of optical potential in the context of atom focusing with lasers were
examined numerically for various experimentally controllable laser
parameters. The Schrödinger equations for the atomic center-of-mass
dynamics in a laser field were calculated using with or without the AA,
and the results were compared. The validity of the AA was tested over
a wide range of the Rabi frequency to detuning ratios and the optical
potential depths. From the analysis several specific guidelines were
provided on choosing laser parameters that sustain the validity of the
AA and improve atom focusing outcomes.
iew
ev
In regular optics and spectroscopy, it is customary to observe light,
resulting from the interaction with material. Considering the waveparticle duality of physical entities (light and matter), one may
expect analogous manifestations of wave nature of matter.
Diffraction of electrons was a pivotal step in the development of
quantum theory. There has been remarkable progress during the
past few decades in manipulating other matter waves such as
neutrons,1,2 atoms and molecules,3-5 and biological particles.6
Such progresses spawned a relatively new field named atom
optics, where the roles of light and matter have reversed: light
serves as mirrors, lenses, gratings, etc. to reflect, deflect, focus,
and diffract matter wave.3-5,7 They have deep implications of great
theoretical interests and, in addition, find numerous applications
in diverse fields in chemistry, physics, biology, and informational
technology.8-11
Key ingredients in these matter-wave controls are the forces
exerted by light on matter—the radiation pressure and the dipole
forces. The former is a dissipative one and is limited by
spontaneous emission rate and may be suppressed with large
detunings. The latter is conservative and may be considered
deriving from an optical potential.12 Dipole force occurs when light
field has spatial inhomogeneity, such as in standing waves. The
gradient of the optical potential is proportional to light intensity, so
the strength of the force can be readily controlled. The dipole force
has been extensively used for trapping, channeling, and focusing
atoms,7 and tweezers,10 realizing the formation of a Bose-Einstein
condensate optically.13 Incidentally, Timp et al. utilized it to
construct a nanostructure with a series of parallel atomic peaks.14
Since atoms and molecules have internal structures, the
interaction of light with them invariably causes transitions among
energy levels. Description of external motion accompanying these
transitions requires solving a complicated set of coupled timedependent Schrödinger equations. Such interaction of light with
multilevel atoms and molecules yields many novel phenomena.15
However, it is frequently the case that only two energy levels of
interest are selectively excited. To further simplify the problem,
often the upper state is adiabatically eliminated. Nanostructure
creation methods utilizing the dipole force to deposit atoms onto
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Bulletin of the Korean Chemical Society
Page 2 of 20
RESEARCH ARTICLE
Bulletin of the Korean Chemical Society
Theoretical Background
basis of the AA in this context. If we keep only two terms in Eq.
(6), the constant first terms do not affect dynamics and the second
terms are interpreted as respective optical potentials in which the
atom in states |𝑔⟩ and |𝑒⟩ moves. Then the evolution of the atomic
center of mass can be described by two uncoupled timedependent Schrödinger equations
Consider an atomic beam moving along the 𝑧 direction and
interacting with a single-mode laser standing wave applied along
the 𝑥 direction. Assume the laser is applied near-resonantly
between certain two energy levels in the entire atomic energy
level scheme, and in this manifold we may call these levels the
"ground" |𝑔⟩ and the "excited" |𝑒⟩ states. (Hereafter, we will use
the terms level and state interchangeably.) The transition
frequency between them is 𝜔𝐴, and the frequency and the
wavelength of the laser are 𝜔 and 𝜆, respectively. Because the
laser frequency is nearly resonant, 𝜔 ≈ 𝜔𝐴 ≫ 𝜔 ― 𝜔𝐴 ≡ 𝛥(the
detuning or the resonance offset), and the counter-rotating term
is negligible and the rotating wave approximation is usually made.
0
1
If we adopt the vector notation |𝑔⟩ = 1 and |𝑒⟩ = 0 , the total
()
𝑖ℏ
𝑖ℏ
𝑝2𝑥
ℏ𝛥
2
𝜎𝑧 +
ℏ𝛺(𝑡)cos 𝑘𝑥
2
()
𝜎𝑥,
(1)
(
)
Fo
1 0
where 𝑀 is the atomic mass, 𝑘 = 2𝜋/𝜆 and 𝜎𝑧 = 0 ―1 and
0 1
𝜎𝑥 = 1 0 are Pauli matrices. Because the atomic beam moves
along the 𝑧 direction, the 𝑧 coordinate is closely related to the
interaction time 𝑡. The time-dependent Rabi frequency 𝛺(𝑡) may
be written as a product of the maximum Rabi frequency 𝛺0 and a
function 𝑔(𝑡) that describes the laser beam profile along the 𝑧
direction: 𝛺(𝑡) = 𝛺0 𝑔(𝑡). We take time 𝑡 = 0 when the atom first
enters the laser beam. The kinetic energy of the atom is of the
2 2
order of the recoil energy, 𝐸𝑟 = ℏ 𝑘 /2𝑀. Typicaly 𝛥,𝛺0 ≫ 𝐸𝑟, so
the kinetic energy part may be treated separately from the rest of
the Hamiltonian to a first approximation. The total wave function
of the atom can be represented as a product of the center-of-mass
and the electronic parts, 𝛹(𝑥,𝑡) = 𝜓𝑔(𝑥,𝑡)|𝑔⟩ + 𝜓𝑒(𝑥,𝑡)|𝑒⟩.
The center-of-mass dynamics of the atom is governed by the
coupled time-dependent Schrödinger equation that may be
written as
(
)
∂𝑡
∂𝜓𝑔(𝑥,𝑡)
∂𝑡
𝑝𝑥
2
= 2𝑀𝜓𝑒(𝑥,𝑡) ―
=
𝑝𝑥
ℏ𝛺2cos2 𝑘𝑥
2
2𝑀𝜓𝑔(𝑥,𝑡)
𝜓𝑒(𝑥,𝑡),
4𝛥
ℏ𝛺2cos2 𝑘𝑥
+
4𝛥
(7a)
𝜓𝑔(𝑥,𝑡).
(7b)
Spontaneous emission may be incorporated by including a
non-Hermitian term ―𝑖ℏ𝛤|𝑒⟩⟨𝑒|/2, where 𝛤 is the decay rate of the
upper state, in the Hamiltonian, Eq. (1) and carrying out MonteCarlo wave function simulations.27-31 Spontaneous emission
degrades atom focusing somewhat and large detuning and Rabi
frequency values are used to suppress spontaneous emission.
Again, in this regime we are dealing with the dipole force rather
than the dissipative force. In this paper, we assume such
experimental conditions are met and ignore spontaneous
emission from now on.
If initially the atom is in the ground state, 𝛹(𝑥,0) = 𝜓𝑔(𝑥,0)|𝑔⟩,
and if the AA is made and thus the upper state is adiabatically
eliminated, the atom evolves in the optical potential given by Eq.
(7b) alone, which can be solved numerically rather
straightforwardly. On the other hand, for more accurate
calculations the full coupled equations in Eq. (2) still need to be
solved. There are several numerical methods for solving these
equations and we chose the pseudospectral method,32 which is
particularly useful for periodical systems such as in Eq. (1).
The interaction time 𝑇int is divided into 𝑁 ≫ 1 segments,
each interval being 𝛿𝑡 = 𝑇int/𝑁. The time-evolution operator is
also divided into the same number of segments, and the evolution
operator at time 𝑡𝑛 = 𝑛𝛿𝑡 (𝑛 = 0,1,…,𝑁) for the Hamiltonian in Eq.
(1) may be approximately split as
Hamiltonian for the two-level atom moving in the laser standing
wave, in a suitable rotating frame, may be written as
𝐻 = 2𝑀 ―
∂𝜓𝑒(𝑥,𝑡)
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(
𝑖ℏ ∂𝑡
)
𝜓𝑒(𝑥,𝑡)
𝜓𝑔(𝑥,𝑡) =
(
𝑝2𝑥
2𝑀
ℏ𝛺
2
―
ℏ𝛥
ℏ𝛺
2
2
cos 𝑘𝑥
𝑝2𝑥
cos 𝑘𝑥
2𝑀
+
ℏ𝛥
2
)
(𝜓𝜓 (𝑥,𝑡)
(𝑥,𝑡)).
𝑒
𝑔
ℏ
ℏ
𝑈𝑉(𝑡) = cos
(3)
𝛺eff𝛿𝑡
𝛺eff 𝛿𝑡
2 2
2 2
( )𝕝 ― 𝑖sin (
, cos θ =
(𝛺eff ― 𝛥)2 + 𝛺2
(𝛺eff ― Δ)2 + 𝛺2
.
)(―
𝛥
𝛺eff𝜎𝑧
𝛺
)
+ 𝛺eff𝜎𝑥 , (9)
(10)
―1
where ℱ is a fast Fourier transform to 𝑘-space and ℱ
is an
inverse Fourier transform back to 𝑥-space. Equation (7b) can be
solved likewise, the only difference being that the potential is onedimensional:
(4)
(
sin 𝜃 =
)
𝛿𝑡
≡ 𝑈𝑉𝑈𝑇𝑈𝑉,
𝛺2cos2 𝑘𝑥 𝛿𝑡
𝑈𝑉(𝑡) = exp ―𝑖
𝛺eff ― 𝛥
𝑖
𝛹(𝑥,𝑡 + 𝛿𝑡) = 𝑈𝑉 ℱ ―1{𝑈𝑇 ℱ[𝑈𝑉𝛹(𝑥,𝑡)]},
where
𝛺
) (
where 𝕝 is the 2 × 2 unit matrix. As is well known, the evolution
operator is applied to the wave function in the sequence
and the eigenvectors are
|1𝑥⟩ = sin 𝜃 𝜓𝑔(𝑥,𝑡)|𝑔⟩ + cos 𝜃 𝜓𝑒(𝑥,𝑡)|𝑒⟩,
|2𝑥⟩ = cos 𝜃 𝜓𝑔(𝑥,𝑡)|𝑔⟩ ― sin 𝜃 𝜓𝑒(𝑥,𝑡)|𝑒⟩;
𝑖
where 𝑇 is the the kinetic energy operator and 𝑉 denotes the rest
("potential'') of the Hamiltonian. The potential part contains
noncommuting operators and can be expressed as33
Let us attempt to solve the equation approximately by ignoring the
small kinetic energy part first, similarly to the Born-Oppenheimer
approximation. The rest of the Hamiltonian can be diagonalized
in the dressed-state basis25 and the eigenvalues are
ℏ
𝛿𝑡
(8)
(2)
𝐸1 = + 2 𝛥2 + (𝛺co𝑠 𝑘𝑥)2 ≡ + 2𝛺eff, 𝐸2 = ― 2𝛺eff;
) (
𝑖
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On
∂
(
𝑈(𝑡𝑛) ≈ exp ― ℏ𝑉(𝑡𝑛) 2 exp ― ℏ𝑇𝛿𝑡 exp ― ℏ𝑉(𝑡𝑛) 2
(5)
4𝛥
2
).
(11)
Equations (1) and (7b) along with the pseudospectral method are
the primary tools to compute the atomic center-of-mass dynamics.
For large detunings, 𝛥 ≫ 𝛺0, these eigenvalues can be expanded
as
𝐸1 ≈ +
2
1 𝛺cos 𝑘𝑥 2
[1 + (
ℏ𝛥
2
𝛥
)
Simulation details with various laser
parameters
]
+ ⋯ , 𝐸2 = ― 𝐸1 (6)
Simulation parameters. In actual implementation of the
equations in the previous section we express frequency and time
and we note that |1𝑥⟩ and |2𝑥⟩ mostly correspond to 𝜓𝑔|𝑔⟩ and 𝜓𝑒
|𝑒⟩, respectively,26 throughout the interaction time. This is the
2
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Bulletin of the Korean Chemical Society
RESEARCH ARTICLE
Bulletin of the Korean Chemical Society
𝜔𝑟―1,
2
in units of recoil frequency 𝜔𝑟 = ℏ𝑘 /2𝑀 and recoil time 𝑡𝑟 =
and the lengths in units of the optical wavelength λ. This makes
the results of simulation universally applicable and nonspecific to
a particular atom or laser. In the recoil units the parameters
reported in the literature for the chromium atom deposition with a
4
laser standing wave are close to 10 for both Δ and Ω0, and 0.1 for
the atom-laser interaction time 𝑇int.31,34 In this paper we designate
4
these parameters 𝛥 = 𝛺0 = 10 , 𝑇int = 0.1 as the reference
values. Parameters for other atom deposition experiments should
be different, but we will consider a wide range of parameters (in
recoil units) below. Thus, we stress again that results obtained
are not specific to Cr atom deposition but hold true for other atoms.
Usually, the atomic beam from a hot oven is velocity-selected, so
we assume that the beam is monoenergetic with the speed along
the 𝑧 direction large and undamped as it crosses and interacts
with the laser beams. Thus, the 𝑧 coordinate has a simple linear
relationship to the interaction time 𝑡. Cooling techniques are also
applied to make the atomic beam have nearly zero momentum
along the laser beam axis. Lasers with a Gaussian rather than a
top-hat beam profile were employed in the experiments, so we
represent the profile function 𝑔(𝑡) introduced in the previous
section as a Gaussian
Fo
[ ( ―1 + 2 ) ],𝑑 = 2.14597.
2
𝑔(𝑡) = exp ― 𝑑
𝑡
𝑇int
2
Figure 1. Change of populations of the atomic states as a function of interaction
time for 𝛥 = 𝛺0 = 10,𝑇int = 0.1. All parameter values are in recoil units. Solid line:
lower state population, Dashed line: upper state population, Dotted line: total
population.
Results for reference parameters. We first compute the atomic
center-of-mass dynamics with the reference values 𝛥 = 𝛺0 = 10,
𝑇int = 0.1. These values correspond to ROR = 1 and DF = 10.
Without making the AA, Eq. (2) is numerically integrated with the
pseudospectral method discussed in the previous section. Figure
1 shows how the populations of the atomic electronic states 𝑃𝑠(𝑡)
(12)
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In the above, the value of 𝑑 is chosen so that the function has 1%
of the peak value at 𝑡 = 0 and 𝑡 = 𝑇int, the beginning and the end
of the atom-laser interaction. The initial atomic beam (with atoms
prepared in the ground electronic state) is assumed to be given
by a Gaussian wave packet
ev
𝜓𝑔(𝑥,0) = 𝑁exp ( ― 𝑥2/2𝑑0 + 𝑖𝑝0𝑥)
𝐿
= 𝑁2∫ ―𝐿|𝜓𝑠(𝑥,𝑡)|2𝑑𝑥,(𝑠 = 𝑔,𝑒) change as the atom moves across
the laser. Similar results have been reported elsewhere.31 It is
rather surprising to see that the maximum population of the upper
state is only about 5% (with the average value 1.59%),
considering the ROR = 1 that certainly jeopardizes the validity of
the expansion in Eq. (6). For some reason, the assumption that
the atom remains adiabatically in the ground state during the
interaction with the the laser seems to hold very well.
(13)
𝐿
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with no momentum 𝑝0 = 0 and the width of the wave packet large,
𝑑0 ≫ 1, to represent a uniform atomic beam. The normalization
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On
2
2
constant 𝑁 is determined from 𝑁 ∫ ―𝐿|𝜓𝑔(𝑥,0)| 𝑑𝑥 = 1, where 2𝐿
= 2.56 is the space we chose that is large enough to avoid
boundary effects in simulations.
Two parameters 𝛥 and 𝛺0 may be independently varied and
the resulting focusing performance be analyzed. Of course, there
is an infinite set of these parameters to consider and blindly
probing all these sets would be futile. A much more intuitive
approach is discovering, instead, patterns of focusing
performances in terms of either the Rabi frequency to resonance
offset ratio (ROR), 𝛺0/𝛥 or the depth of the potential—quantities
related to the strength of the atom-laser interaction. From the
expression for the optical potential given in Eq. (7) the potential
2
depth is proportional to 𝛺0/𝛥 and this ratio will be called in this
paper the Depth Factor (DF). Thus, In the following analyses we
keep the atomic beam speed the same 𝑇int = 0.1 while varying the
two parameters ROR and DF. In subsequent discussions we will
3
express 𝛥, 𝛺0, and DF values in multiples of 10 for simplicity, but
full values are used in all actual calculations.
3
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RESEARCH ARTICLE
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Bulletin of the Korean Chemical Society
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Fo
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Figure 3. (a) Variation of densities at 𝑥 = 0.25 as a function of interaction time.
The focal planes calculated with Eqs. (2) and (7b) are at 𝑡 = 0.053 and 0.052,
respectively. (b) Densities about 𝑥 = 0.25 at the respective focal planes shown
in (a). Solid line: Eq. (2). Dotted line: Eq. (7b).
Figure 2. (a) Variation of the total atomic density with the same parameters as
in Fig. 1. Darker shade denotes higher density. Only the range ―1 ≤ 𝑥 ≤ 1 is
shown. (b) Variation of the upper state density. Note that the highest upper
state density is less than 1% that of the total density.
For a more quantitative comparison, we show in Fig. 3(a)
the densities calculated with and without the AA at one of the
nodes of the optical potential, 𝑥 = 0.25, as a function of interaction
time; and in Fig. 3(b) the densities around the node at the
respective focal planes. The small peaks after the major one in
Fig. 3(a) is due to diffraction of which pattern can be clearly seen
in Fig. 2(a) of the matter wave and will not be regarded as focused
positions. An obvious measure of focusing performance is the
maximum density height. The highest densities calculated with
Eqs. (2) and (7b) are 0.019057 and 0.020879 at the respective
focal planes at 𝑡 = 0.053 and 0.052. The errors are roughly 10%
in the maximum density height and 2% in position of the focal
plane. A second measure is the full width at half maximum along
the 𝑥 axis (FWHM-x). Here, the maximum height is defined as the
difference between the density at a node and that at an antinode
of the optical potential. FWHM-x values are 0.0123 (with the AA)
and 0.0122 (without the AA)—a 1% error. Thus, for this 𝛺0 and 𝛥
pair the calculations with the adiabaticity assumption deviate only
slightly from the ones done without the assumption, and
consequently, the assumption works quite well for these
parameters. We already hinted in Fig. 1 that even with this strong
coupling conditions the contribution of the upper state would be
small. It is intriguing to see the robustness of the AA despite that
the reference ROR we used violates the condition Ω0/Δ ≪ 1 for
the expansion Eq. (6). We take up this point in the next subsection.
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The same parameters are used to calculate without the AA
the variation of the atomic density as the atomic beam traverses
the standing-wave laser beam. The resulting total and lower state
densities are shown in Fig. 2. We can see that the uniform atomic
beam tends to be focused by the laser at the lattice points 𝑥 = ±
0.25 × (2𝑛 + 1), (𝑛 = 0,1, 2, …) at a time (or position along the 𝑧
axis) slightly past 𝑡 = 0.05. Here, the concept of optical potential
comes in handy in explaining this focusing behavior. These
positions along the 𝑥 axis where the atomic density is the highest
coincide with the nodes (the lowest part) of the optical potential in
2
Eq. (7b), which varies as cos 𝑘𝑥. The potential is essentially a
Mathieu potential, and the wavefunctions and the energy bands
can be calculated numerically.35 It can be shown that the
amplitude of the wavefunction for each "crystal momentum" within
the first Brillouin zone (in the parlance of solid-state theories) of
the optical lattice tends to be maximum at about the bottom of the
potential, so it comes as no surprise that the atomic density is the
highest at these optical lattice minima. On the other hand, the
focusing for the atom in the upper electronic state gravitates
towards the antinodes of the optical potential, 𝑥 =± 0.25 × 2𝑛,
(𝑛 = 0,1,2,…). But the focusing is very weak and the highest upper
state density contributes only about 1% to the total density. The
ground state atomic density at the nodes (not shown in Fig. 2)
resulting from the use of the AA with the optical potential given in
Eq. (7b) is somewhat higher but very close to the total density
because the density "lost" to the upper state is insignificant.
Effects of laser beam profile. In order to trace the origin of the
robustness of the AA, let us digress on the focusing performance
of a laser with a top-hat profile, which is given by 𝑔(𝑡) = 1,
(0 ≤ 𝑡 ≤ 𝑇int) and 𝑔(𝑡) = 0, otherwise. We did simulations with and
without making the AA for the top-hat laser profile with the same
reference parameters and compared the results in Fig. 4. It is
4
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Bulletin of the Korean Chemical Society
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Bulletin of the Korean Chemical Society
obvious from these figures that for the top-hat laser profile the
results with the AA substantially differ from those with the coupled
Schrödinger equation, Eq. (2). Also, upon comparing these
figures with those with the Gaussian profile shown in Fig. 3, we
find that results with the AA are very similar to each other (peak
densities: 0.020879 vs 0.020685) and are not sensitive to the
laser profile. On the contrary, results without making the AA reveal
the sensitivity to the laser profile and show that the density at the
focal plane, 𝑡 = 0.052, predicted by the AA gives rise to a
significantly smaller central peak accompanied by many
undesirable side peaks. Thus, for the top-hat laser profile, should
the parameters obtained with the AA be used, the outcome of
lithography experiments could turn out to be quite suboptimal. The
failure of the AA for the top-hat profile can be attributed to the fact
that the atom-laser interaction with this profile amounts to a
sudden perturbation. For this profile, ROR = 1 is simply too large
a value for the expansion in Eq. (6) to be truncated to two terms.
Note, however, that it is actualy the time-dependent Rabi
frequency that is used in the expansion. For the case of the
Gaussian profile, Ω(t) varies smoothly over time and that 𝛺(𝑡)/𝛥
< 1 all the time except for 𝑡 = 𝑇int/2. Consequently, the success
of the AA for the Gaussian profile can be attributed to that it leads
to a slowly varying perturbation rather than an abrupt one.
Fo
𝛥 values of 1, 10 (the reference), and 500. The corresponding
Rabi frequencies are 3.1623, 10, and 70.711; and the RORs are
3.1623, 1, and 0.14142. The variation of atomic densities with
respect to the interaction time for these parameters is shown in
Fig. 5. The curves with various line types are the results of
calculations without the AA. The circles denote data calculated
with the AA for the detuning value 𝛥 = 500. Note that for a given
value of the DF the optical potentials are identical regardless of
the detuning values. Hence, for a fixed value of the DF,
calculations with the AA must give an identical result regardless
of ROR values. In the figure the curve with solid line is for the
ROR = 3.1623, which is larger than the reference value of 1, and
differs greatly from the data with the AA (circles). The dash-dot
curve is for the reference ROR, and this curve and the circles are
equivalent to the plots shown in Fig. 3(a), just displayed on
different time ranges. We see much improvement in the reliability
of the AA as compared to the result with the ROR larger than 1.
Finally, for the small ROR = 0.14142 (𝛥 = 500) the AA delivers
results virtually indistinguishable from those without the AA
(dotted line). Specifically, the peak density heights at the focal
planes, FWHM-x, and the respective errors (in parentheses) are:
0.013229 (57.8% error) at 0.055 (5.5% error) and 0.016 (23.1%
error) for 𝛥 =1, 0.020813 (0.3% error) at 0.052 (0% error) and
0.0123 (0% error) for 𝛥 =500. It is noted that, since the ROR =
iew
ev
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DF/𝛥, for a given depth of the potential the AA and predictions
based on the concept of optical potential become more accurate
as larger detunings are used.
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On
Figure 5. Comparison of focusing as the detuning is varied with a fixed potential
depth, DF = 10. The circles denote results obtaned with the AA, and lines are
3
results without the AA and with the detuning values (Δ/10 ) shown on the legend.
Dependence on potential depth. In this subsection we isolate
the effects of the potential depth by fixing the ROR (= 1) and
varying the DF values over a wide range: DF = 1, 2, 5, 10, 20, 50,
100, 200, and 500. The results of calculations with these
parameters are compared in Fig. 6. The peaks again denote
evolution of densities at 𝑥 = 0.25 in the laser field. Here, we
observe that as the DF increases, both the number of peaks (the
focal planes) and the height of the tallest peak for each DF
increase, while position of the first focal plane and the separation
between subsequent focal planes steadily decrease across the
plots (a) and (g). The height of the most prominent density peak
(calculated without the AA) in each plot is 0.0057646 (not fully
focused), 0.010102, 0.015469, 0.019057, 0.021952, 0.023142,
0.025407, 0.029707, and 0.026357, respectively. The most
prominent peaks with the AA are always higher than those without
the AA. As with the case DF = 10, the difference between the
results with and without the AA is not significant for all DF values,
especially the locations of focal planes, because of the smooth
perturbation due to the Gaussian profile as discussed earlier. So,
in the rest of this subsection we set aside issues related to the
accuracy of the AA and concentrate on some general trends due
to the potential depths using the results without the AA.
Figure 4. Comparison of focusing performances of the top-hat laser profile with
(dotted line) and without (solid line) the AA. (a) Variation of densities at 𝑥 = 0.25
as a function of interaction time. (b) Densities along the 𝑥 axis at 𝑡 = 0.052 (the
focal plane predicted by the AA).
Dependence on ROR. The use of the concept of optical potential
has been a pivotal step in developing matter-wave manipulation
techniques. As stated earlier, validity of the concept is closely tied
to the accuracy of the AA. Certainly, RORs smaller than 1 would
render the expansion in Eq. (6) converge more rapidly and one
can expect results with the AA to be more accurate. Thus, let us
vary the ROR while keeping the DF the same at DF = 10, the
reference value. The Rabi frequency 𝛺0 is related to the DF and
the detuning as 𝛺0 = DF ∙ 𝛥. We consider three widely separated
5
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Bulletin of the Korean Chemical Society
(2𝑛 + 1) 𝜋 /2,(𝑛 = 1, 2, 3, …) and 1/(𝐿0 𝜋) at 𝜔𝑡𝑛 = 𝑛𝜋 ,
(𝑛 = 1, 2, 3, …). With a view to our interest in focusing an initially
broad atomic wave packet, it is appropriate to assume 𝐿0 ≫ 1.
Then the maximum density height corresponds to 𝑚𝜔𝐿0/(ℏ 𝜋),
not Delta but DF
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ev
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Fo
and the FWHM-x at each period is 2 𝑙𝑛 2 ℏ/(𝑚𝜔𝐿0). Note that the
maximum height is proportional, and the FWHM-x and the period
inversely proportional to the slope of the harmonic potential. For
a given range of 𝑥 values a steeper slope amounts to a deeper
potential. Consequently, the wavepacket dynamics at least
qualitatively explains why potentials having increased depths lead
to taller focused peaks along with shorter refocusing periods.
A second notable point from these figures is that, while the
peak heights get taller as the potential gets deeper in general, the
tallest peak does not appear in (g) corresponding to the deepest
potential, but in (f) instead. This somewhat erratic behavior can
be understood by looking at the densities at the respective focal
planes of the tallest peaks, as shown in Fig. 7. All densities in the
figure are results of calculations without the AA. One can
immediately recognize in the figure that at the node of the
potential, 𝑥 = 0.25, peak heights for bigger detuning values are
larger than those with smaller ones, but they accompany many
noise-like smaller peaks. Especially, for ∆ = 500, the two
pronounced side peaks about 0.18 and 0.33 seem to take away
the density at the center, lowering it below the central peak for ∆
= 200. The noisy small peaks will act to reduce the contrast of
the feature created with laser focusing. Usually, the first focal
plane for each DF gives the cleanest focusing. Consequently, for
producing neat feature it is advisable to avoid selecting those
focal planes occurring at later times.
-
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On
Figure 6. Variation of the atomic density (vertical axis) as a function of the atomlaser interaction time (horizontal axis) for the fixed reference value ROR = 1.
The DF parameters used for calculations are 1, 2, 5, 10, 20, 50, 100, 200, and
500, respectively. In subplots (b) to (g) solid lines are results without the AA and
dotted lines with the AA, and axis scales are the same as in (a).
The DF params in (b)-(g) are 10,20,…500, repectively .
To explain this DF effects on the peak height and the distance
2
2
between peaks we reexpress the DF as DF = 𝛺0/𝛥 = (𝛺0/𝛥) 𝛥
2
= (ROR) 𝛥. So, the DF increases as either the ROR, the
detuning, or both increase. Bacause the ROR is fixed at 1, the
detuning values and the DF values are the same. Although the
optical lattice potential is not a harmonic oscillator potential, it may
be regarded as approximately parabolic near each node. Thus,
some insights can be gained from the wave packet dynamics in a
harmonic oscillator potenial. Such dynamics is well known.36
Suppose initially the wave packet is given by a Gaussian function
1
located at the center of a harmonic oscillator potential 𝑉(𝑥) = 2𝑚
𝜔2𝑥2:
𝜓(𝑥,𝑡0) =
1
𝐿0 π
(
𝑥2
)
exp ― 2𝐿 2 ,
0
Figure 7. Atomic densities about the potential node, x = 0.25, at the respective
focal planes that correspond to the tallest peak for each potential depth. The
densities are results of calculations done without the AA, and parameters are
the same as in Fig. 6.
A third trait to note regarding the potential depth is that for
deeper potentials the density variation with respect to the atomlaser interaction time becomes more rapid. This pattern may be
accounted for using again the pre-exponential factor in Eq. (15).
2
The half maximum of the wave packet density |𝜓(0,𝑡)| is 𝑚𝜔𝐿0/
(2ℏ π) and, after some algebra, one can obtain the FWHM of the
density with respect to time (FWHM-t) as
(14)
where, 𝐿0 is an arbitray number related to the width of the initial
wave packet. At time 𝑡 the probability density is given by
1
1
|𝜓(𝑥,𝑡)|2 = 𝐿(𝑡)
𝜋
(
𝑥2
)
exp ― 𝐿(𝑡)2 ,
[
(
―1
FWHM-t = 𝜔 cos
(15)
― 3ℏ
) ― cos (
𝐿04𝑚2𝜔2 ― ℏ2
―1
3ℏ
)]. (16)
𝐿04𝑚2𝜔2 ― ℏ2
It can be easily verified that the FWHM-t given above is smaller
for a deeper harmonic potential, rationalizing the simulation
results. The inverse of the FWHM-t may be loosely considered as
a measure of how sensitively focusing is influenced by the atomic
beam speed and/or the substrate position. This sensitivity
consideration provides a useful guideline about the laser power.
Lasers with high powers are good for tight focusing. However, a
2
2
2
2
where 𝐿(𝑡) = 𝐿0 cos 𝜔𝑡 + (ℏ/𝑚𝜔𝐿0) sin 𝜔𝑡 . The width and the
height of the probability density oscillate with time. We are
interested in the height of the density at the center of the potential
(𝑥 = 0), so we are concerned only with the time-varying preexponential factor in Eq. (15). It is easy to show that the factor has
two extreme values: 𝑚𝜔𝐿0/(ℏ 𝜋) occurring periodically at 𝜔𝑡𝑛 =
6
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2
DF = (ROR) 𝛥, we may vary both the ROR and the detuning
values. Raising the ROR beyond the reference value surely
worsens the validity of the AA, so one must choose ROR values
less than 1. But this will make the potential shallower, so the
detuning needs to be boosted to compensate for the reduction of
the potential depth. Thus, detunings less than the reference value
10 are precluded from consideration. Let us take a coarsegraining approach and consider potential depths having some
integer multiples of the reference: 20, 30, 50, 100. Since the
optical potential is not strictly harmonic, there is no precise
relation of the positions of the focal planes to the depth of the
optical potential, unlike calculations done with classical particle
optics.37,38 Thus, we attempt to find the depth that would give the
desired focal plane by trial and error. But the parameter space to
search is not infinite. From the discussion given above, we
immediately impose the conditions that must be satisfied: ROR
< 1 (the reference ROR) and DF > 10 (the reference DF). As an
2
example, consider Δ = 20. Since ROR = Ω0/20 < 1 and DF = Ω0
/20 > 10, we can limit the range that the Rabi frequency can have,
200 < Ω0 < 20. Then the Rabi frequency is incremented from the
lowest bound each time by one until the focal plane coincides with
the laser beam center. The Rabi frequency Ω0 = 16 turns out to
give the best fit, for which the ROR = 0.8 and the desired DF
= 12.8. As remarked earlier, for a given DF calculations done with
the AA give an identical result regardless of the magnitude of the
detuning. However, the deviation from the results without the AA
does depend on detuning values. Once the desired DF value is
determined, it is a simple matter to find the matchinging Rabi
frequencies and hence the ROR values for other detunings Δ =
30, 50, and 100. These are Ω0 = 19.596, 25.298, and 35.777; and
ROR = 0.653, 0.506, and 0.358, respectively. Since a smaller
ROR reduces the errors caused by the AA, agreements between
the results with the AA and those without the AA will get better as
the ROR decreases. Figure 9 shows % errors in peak density
height, sensitivity, and FWHM-x of the results with the AA
compared to those without the AA. As expected, it shows that the
difference of the results between the two methods diminishes as
the ROR gets smaller. More specifically, as far as focusing atoms
on the Gaussian center is concerned, predictions for atom
focusing based on the AA and hence the concept of optical
potential are accurate about 95% or more if the ROR less than
about 0.65 is used.
slight misalignment along the 𝑧 axis between the substrate
position and an intended focal plane would mar deposition
outcome. Conversely, if the laser power is not high enough, only
mediocre focusing would be achieved. Therefore, laser powers
should be chosen to balance the focusing power against the
sensitivity to achieve satisfactory deposition results.
Fo
rR
Figure 8. Maximum heights, FWHM along 𝑥, and sensitivities to the interaction
time of the densities for the DF cases 2 – 500, from left to right) relative to the
height and the sensitivity for DF = 10. Because of the wide DF range considered,
the DF values are given on a logarithmic scale.
iew
ev
Since the density profile along the 𝑥 axis exhibit undesirable
side peaks at focal planes other than the first one, let us consider
the density peaks at the first focal plane for the respective DF
values. The first density peak heights shown in Fig. 6 calculated
without the AA are 0.0057646, 0.010102, 0.019057, 0.021952,
0.023142, 0.024479, 0.026526, and 0.025337, respectively. With
the first DF (= 1) atoms are not fully focused, so the corresponding
focusing data will not be included in the discussion. As expected,
the peak height grows, but only 2.5 times when the depth changes
250 fold from DF = 2 to DF = 500. The respective FWHM-x values
―3
are: 28.5, 16.7, 12.2, 10.9, 8.29, 9.34, 6.79, and 8.48 ( × 10 ).
The FWHM-x is reduced 3.36 times during the same DF variation.
Regarding the FWHM-t, we choose to define the half maximum of
a density as the difference between the height at the beginning of
the atom-laser interaction 𝑡 = 0 and that at the first focal plane,
divided by two. The FWHM-t values along the time (or the 𝑧) axis
―3
are 26.8, 8.70, 5.13, 3.50, 2.65, 2.48, 2.17, and 1.57 ( × 10 ),
respectively. Thus, sensitivity (the inverse of the FWHM-t) values
are 37.3, 115, 195, 286, 377, 403, 461, and 637—an increase by
17 times across the DF values. Figure 8 shows the peak heights
and FWHM-x, and sensitivities for the above eight DF cases
relative to those for the DF = 10. One can notice that the variation
in sensitivity is much more rapid than those in height and width
anlong the 𝑥 axis. Thus, we have an additional guideline on
parameter choice: the gain in peak height and the reduction of
FWHM-x by employing a high-powered laser and a large detuning
are far outweighed by the risk of sensitivity to the interaction time.
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Optical potential for focusing on the laser beam center. As
remarked earlier, the optical lattice potential is a key concept in
designing and interpreting atom focusing methods with lasers.
The results presented in the previous subsections can be utilized
to find parameters needed to focus atoms at any position along
the time axis using the optical potential concept. Here, let us
assume the goal is to focus atoms precisely at the center of the
Gaussian laser beam profile (𝑡 = 𝑇int/2). It was shown earlier that
with the reference ROR the first focal planes appear slightly past
the beam center: 0.052 and 0.053 ( × 𝑇int) with and without the
AA, respectively. Thus, we need a deeper potential to bring the
focal plane to the Gaussian beam center. Incidentally, this will
improve other measures as well, which we now discuss. Because
Figure 9. Percent errors vs. ROR in peak height of the density, sensitivity, and
FWHM along the 𝑥 axis obtained with the AA as compared with the results
without the AA.
Conclusions and Discussion
The adiabatic approximation (AA) gives rise to the extensively
utilized concept of optical potential. In this paper we explored the
focusing performance of an atomic matter wave beam with a laser
standing wave over wide-ranging laser parameters. Extensive
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disribution less severe. Thus, it gives another reason to avoid
using too deep optical potentials (at the cost of less tight focusing)
in addition to the sensitivity matter we pointed out earlier in this
paper. To conclude, setting right laser parameters is highly critical
for satisfactory atom focusing and also for numerous other matter
wave manipulations with optical potentials that are currently of
intense research interests.39 The results reported in this paper can
be utilized in easing the task of securing those optimal parameters.
numerical simulations with and without the AA were carried out
and analyzed the accuracy of the approximation. It was found
that the AA worked well even for strong coupling cases (ie. the
Rabi frequency and the detuning are similar in magnitude)—a
condition that normally goes beyond the realm of validity of the
approximation. It was shown that the source of this robustness of
the approximation was the Gaussian beam profile of the laser,
which led to a slowly varying perturbation to the time evolution of
the atomic matter wave. The examination of the effects of the
laser parameters were facilitated with two quantities, the Rabi
frequency to the resonance offset ratio (ROR) and the depth
factor (DF) that is related to the depth of the optical potential. The
focusing performance variation due to either the ROR and the DF
were evaluated using the three measures: the peak density height,
feature width along the 𝑥 axis (FWHM-x), and sensitivity of
focusing along the 𝑧 (or time) axis. For a given DF the accuracy
of the AA was shown to improve as the detuning increased (or
equivalently, the ROR reduced). Thus, to accurately describe
atom focusing in terms of the optical potential, it is necessary to
sufficiently detune the laser frequency from the resonance
frequency, but not detune so much as to make the rotating wave
approximation that led to Eq. (1) fail. Next, we studied effects of
the DF with the ROR locked. As the DF increased, the following
tendencies were identified: Both the number of focal planes and
the height of the tallest density for each DF increased, position of
the first focal plane and the separation between subsequent focal
planes steadily decreased, and the variation of density with
respect to time was more sensitive. These general tendencies
were accounted for, albeit qualitatively, using the quantum
dynamics of a wave packet in a harmonic oscillator potential.
Close scrutiny of the measures—the height, the FWHM-x, and the
sensitivity to the time atoms move in the laser field— of the peak
density showed that the risk of sensitivity effect was much greater
than the benefits in other measures gained by using deeper
optical potentials. Therefore, it was recommended against using
too strong laser powers and large detunings to generate deeper
potentials. We recommended using parameters that would let the
focusing occur just once in the laser field. Specifically, these
analyses allowed us to find the parameters for focusing atoms at
the center of the Gaussian profile of the laser beam. It was shown
that with ROR = 0.65 and DF = 12.8 the calculation with the
adiabaticity assumption was 95% accurate. In other words, with
these parameters one can predict atom focusing outcomes with
95% confidence when using the concept of optical potential. The
same approach can be equally extended to obtain laser
parameters for which the optical potential can be used safely to
predict other locations to focus atoms in the laser field.
The calculations in this paper were carried out with some
assumptions: the omission of the counter-rotating term, a uniform
atomic beam with perfect cooling along the laser beam axis (the
𝑥 axis), atoms having a same speed along the 𝑧 direction initially
and while moving in the the laser field, and ignoring atoms initially
in the upper state. A non-uniform atomic beam and motion along
the 𝑥 axis would surely degrade feature widths. These cannot be
remedied by manipulating laser parameters, so suppression of
momentum along this direction by cooling should be done to
ensure better focusing results. Atoms emitted from a hot oven
have distributions in both the electronic energy and the speed. As
shown in Fig.2(b), atoms at the upper state are focused on the
antinodes of the potential. Reducing the ROR does not help,
because even then atoms in either state will evolve adiabatically
in respective potentials, still producing peaks at the antinodes.
Thus, if this is not what is intended, the initial upper state
population
poputation in the atomic beam needs to be minimized. Atoms with
slower speed along the 𝑧 direction have more time to evolve in
the laser field and have higher probability to be focused more than
once, resulting in more noisy features. Conversely, atoms with
faster speed will not have enought evolution time to be tightly
focused. So, the speed distribution degrades optimal focusing
and it must be carefully contained. Laser parameter choice can
help in alleviating this problem as well. Evolution under a
shallower optical potential is more gradual than under a deeper
one, so it will make the difference in focusing due to the speed
Acknowledgements
This work was supported by a 2019 Research Grant from Sun
Moon University.
Keywords: adiabatic approximation, optical potential, matter
wave focusing, atom lithography
Fo
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Change of populations of the atomic states as a function of interaction time for Δ = Ω0=10,Tint=0.1. All
parameter values are in recoil units. Solid line: lower state population, Dashed line: upper state population,
Dotted line: total population.
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(a) Variation of the total atomic density with the same parameters as in Fig. 1. Darker shade denotes higher
density. Only the range -1≦ x ≦ 1 is shown.
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Variation of the upper state density. Note that the highest upper state density is less than 1% that of the
total density.
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(a) Variation of densities at x=0.25 as a function of interaction time. The focal planes calculated with Eqs.
(2) and (7b) are at t = 0.053 and 0.052, respectively. (b) Densities about x=0.25 at the respective focal
planes shown in (a). Solid line: Eq. (2). Dotted line: Eq. (7b).
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Comparison of focusing performances of the top-hat laser profile with (dotted line) and without (solid line)
the AA. (a) Variation of densities at x=0.25 as a function of interaction time. (b) Densities along the x axis
at t=0.052 (the focal plane predicted by the AA).
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Comparison of focusing as the detuning is varied with a fixed potential depth, DF = 10. The circles denote
results obtaned with the AA, and lines are results without the AA and with the detuning values (Δ/103)
shown on the legend.
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Atomic densities about the potential node, x\ =\ 0.25, at the respective focal planes that correspond to the
tallest peak for each potential depth. The densities are results of calculations done without the AA, and
parameters are the same as in Fig. 6.
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Maximum heights, FWHM along x, and sensitivities to the interaction time of the densities for the DF cases 2
– 500, from left to right) relative to the height and the sensitivity for DF = 10. Because of the wide DF range
considered, the DF values are given on a logarithmic scale.
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1333x999mm (72 x 72 DPI)
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Percent errors vs. ROR in peak height of the density, sensitivity, and FWHM along the x axis obtained with
the AA as compared with the results without the AA.
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Variation of the atomic density (vertical axis) as a function of the atom-laser interaction time (horizontal
axis) for the fixed reference value ROR = 1. The DF parameters used for calculations are 1, 2, 5, 10, 20, 50,
100, 200, and 500, respectively. In subplots (b) to (g) solid lines are results without the AA and dotted lines
with the AA, and axis scales are the same as in (a).
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