Hole burning: a discrete kinetic approach
Alexander Priill, Friedrich Hanser and Ferdinand Schiirrer
Institute for Theoretical Physics
Graz University of Technology, Graz, Austria
Abstract. Investigations in the field of quantum optics very often requires a molecular simulation of
transport phenomena. This paper presents a discrete kinetic model to deal with the quantum optical
problem of hole burning. Based on a full discretization of the velocity space, Boltzmann-like equations are
established to govern the temporal evolution of heavily disturbed Maxwell distributions of optically excited
particles diffusing in a buffer gas. These model equations involve elastic collisions and the interaction
of photons with three-level atoms by taking into account the Doppler effect. Important features of the
developed model are the realistic simulation of the relaxation process and the existence of a stable and
unique equilibrium solution.
INTRODUCTION
Manipulation of atomic motion by means of laser light is an interesting challenge in quantum optics and
saturated laser spectroscopy. Optical excitation of a small amount of particles diffusing in a buffer gas creates
heavily disturbed Maxwell distributions. This offers us the possibility to study their recovery under the
influence of collisions. This was first recognized by Berman [1], who investigated these phenomena by applying
the Fokker-Plank equation and recently by Bestgen et. al [2].
Our intention is to deal with strongly collisional regimes, where the elementary change of the velocity cannot
be considered small, and, consequently, the Fokker-Plank equation does not apply. This is the case when a
narrow-banded laser modifies the thermal distribution of sodium atomic vapor in the presence of argon at a
pressure of several Torr, by typically burning a hole in it. Because of velocity-changing collisions, the maximum
absorption is observed neither at the F=l hfs-sublevel nor at the F=2 hfs-sublevel of the sodium DI resonance
transition line but rather at a position halfway between. It is important to note that in the absence of a
buffer gas, atoms which have been excited from a given F-level return to a large fraction into another F-level
by keeping their original velocity. Consequently, these species are not in resonance with the pump laser, and
the fluorescence of the sodium atoms will stop after a few optical pumping cycles. The resulting absorption
spectrum is just a Doppler-broadened profile with maxima at the two hfs-sublevels. However, in the presence
of a buffer gas, the velocities of the sodium atoms are continuously changed, such that new fluorescence cycles
becomes possible. It has been observed that in this case the absorption spectrum is peaked halfway between
the two hyperfine transition resonance frequencies. The velocity-changing collisions lead to a strong increase of
the absorption through sodium atoms because of the higher number of fluorescence cycles [3], [4]. Renzoni et. al
[5] found experimentally that the hole in the velocity distribution is not destroyed by the velocity changing
collisions, but its shape and width are strongly affected by the presence of the buffer gas.
In this paper we present a discrete kinetic model [6] which allows one to investigate such scenarios for arbitrary
mass ratios of vapor and buffer gas particles. In particular, we consider diluted sodium vapor diffusing in a
buffer gas of argon atoms, where the laser frequency is tuned somewhere in between the two hyperfine lines of
the sodium DI transition.
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
59
THE DISCRETE KINETIC MODEL
Our aim is to establish a discrete kinetic model which tackles the interactions of sodium (Na) vapor with
laser light in the presence of a buffer gas of Argon (Ar). The approach is based on the assumption that the
configuration space is homogeneous and the number density of Ar is much greater than that of Na. This implies
that the Maxwellian of the Ar atoms is not affected by collisions with Na atoms. Because of the low density
of Na, we only take elastic collisions between Na and Ar atoms into account. The three-level Na atoms (two
ground states due to the hyperfine structure, and - a little bit idealized - one excited state) are realized in our
model by three different species A-\_, A2, and A% according to Fig. 1. Special attention is paid to the interaction
of the laser light with the Na atoms via absorption, spontaneous and stimulated emission. These events [7] are
modeled by means of the Einstein coefficients a and (3 as indicated in Fig. 1. The laser is tuned to a frequency
between the two hyperfine transition resonance frequencies of species A\ and A^. Due to the Doppler effect,
only atoms of species A\ and A% with the proper velocity interact with the laser light.
FIGURE 1. The internal energy levels F = 1 hfs-sublevel, F = 2 hfs-sublevel, and the excited state of the test particles
are treated as particles A\, A^, and AS, respectively. The quantities a1, /3l, and a2, {32 are the Einstein coefficients of
the transitions.
We implement the interactions on a microscopic level by introducing a two-dimensional equidistant velocity
grid for species AL, A^, and A% with (2q + I)2 different speeds:
vi := A-u • i = At?
-1
with
i\,ii G {— <?, —q + l,...,q — l,q}:= Ind,
(1)
where Av denotes the grid spacing. The velocity grid belonging to the Ar atoms will be discussed later.
The temporal evolution of the number densities N* corresponding to the species A8, s = 1,2,3, of the Na
atoms is governed by the the discrete Boltzmann-like equation
|^(*) = Jfw + JTw.
Two kinds of interactions are responsible for the change of N?. The first term, Jj
elastic collisions of buffer gas particles with particles of species As:
(2)
8
, represents the rate of
.£« = ]T Af [JV£(t)Bk - JTOBj] .
(3)
j,h,k
The fixed number densities of the buffer gas are denoted by B^ and Ayk indicates the transition rates. A more
detailed specification of the collision term can only then be given when the admitted collisions are defined.
The second expression in Eq. (3) describes the radiation interactions sketched in Fig. 1. Depending on the
kind of species, this term reads
=a1N?+f3ii{N?-pi}Nt,
(4)
=^N? + ^I?N?-^I?Nl
(5)
= -C"1 + *2)N? - (/31 1} + f32 1?) N? + f I} JV,1 + /32 1? N?.
60
(6)
Number densities N? multiplied by a1 and a2, respectively, are gain and loss terms due to spontaneous emission.
Terms containing (3l or /32 correspond either to photon absorption or stimulated emission events. It should be
noted that in three level atoms two different transitions appear, and therefore, two different discretized laser
intensities Ij, I £ {1, 2}, have to be taken into account.
In the continuous kinetic theory [7], the laser intensity 1l(v) referring to the l-th transition of a Na atom
moving with the velocity v is related to the spectral intensity /i/(r, £1, t) by
(r,n ) t)a'( I /).
(7)
The Lorentz profile
rules the absorption according to the l-th transition in the rest frame of Na atoms. The parameter d^^i denotes
the natural line width of the profile and Z/Q the transition frequency.
With reference to a moving atom, we observe a shift of the laser frequency Z/L to
(9)
due to the Doppler effect. The symbol v stands for the speed of the Na atoms in the fixed direction fio of the
laser beam and c for the speed of light. Usually, the width of the laser profile is some magnitudes smaller than
that of Na atoms. It is, therefore, justified to represent the spectral laser intensity by
Ji(r,n ) t) = j5 2 (n-n 0 )*(i/-^) )
(10)
where I measures the strength of the laser beam. Inserting Eqs. (8,9,10) into Eq. (7) results in
j<(v)=Jj
The relevant expression of the laser intensity within a volume element at a certain grid point vi defined by
Eq. (1), is obtained by averaging Tl(v) over the grid spacing in direction of the laser beam Q 0 (ti+l/2)At;
Tl
.
r^2£^_
,=(«)- *
f
__________ dv __________
(
>
Evaluating this integral, we get
Next, we discretize the two-dimensional velocity space of Ar atoms to specify the elastic collision term given
by Eq. (3). This cannot be done in a poorly arbitrary manner. Let us consider a collision between Na and
Ar atoms with masses mi and 7712, respectively, where the velocities of the particle pair change from (v, w) to
(v', w'). Due to energy and momentum conservation, the pre- and the post-collisional velocities are related by
V'(v, w,
0') = ^Lv ————^————• + -J^g 6',
mi + m2
mi + ^2
^H + ^2
mi
m2
mi
f\i\) = —————
w U(v, w, O
v +i ————— w — ————— ar\'
6 .
mi + m2
mi + m2
mi + m2
f\A\
(14)
With g we denote the magnitude of the relative velocity g = v — w and with O the unit vector in direction
of g. The primed symbols refer to post-collisional variables. A realistic discrete velocity model should allow
any change of velocities of Na atoms on their grid through collisions with Ar atoms. To find the corresponding
61
velocity grid for the Ar atoms, we need for each pair of pre- and post-collisional velocities v and v' of Na
atoms the corresponding pre- and post-collisional velocities w and w' of Ar atoms. In general, these relations
are nonlinear functions of the components of v and v' parametrized by Q'. However, in order to reduce the
set of possible collisions to a reasonable amount, we only choose those vectors for 0', which lead to simple
representations of w and w'. Based on the selection
,_ 1
9'
"
"3~#
9\9xJ
(15)
we obtain from Eq. (14) after some algebra three pairs of pre- and post-collisional velocities for the Ar atoms:
1
m 2 )v']
[(ra2 - mi)v
2m2
1
[(mi + m 2 )v + (rri2 - rai)v']
(16)
and
vfx- vfy)
mi(—v
^ ~x .-vy + vx-vry)+m2(vxf
2m2 \mi(~vx -vy + vx + v ) 1
m2(vx + vy + v'x - v'y)
w2 =
2m2 \mi(-vx-<"<)
w
2
=__
,
(17)
as well as
w3 =
Wo
~VX
-Vy+V'x
m2(vx
-vy~vx
2m2 V
4^
-v
2m2
(18)
The subscripts of w and w' indicate the type of collisions corresponding to G^, G 2 and ©3. This means that
we only allow collisions between Na and Ar atoms with a ?r/2, Tr/4 and 3?r/4 rotation of the relative velocity
to change from a certain pre- to a certain post-collisional velocity of Na atoms as shown in Fig. 2. It should
be noted that the Eqs. (16-18) reflect the symmetries
w'i(v,v') = wi(v',v),
w' 2 (v,v') = w 3 (v',v),
w' 3 (v,v') = w 2 (v',v),
(19)
which essentially simplify the determination of the necessary grid for Ar atoms and dramatically reduce the
computational expense.
FIGURE 2. Velocity diagrams representing the selected three types of elastic collisions between Na and Ar atoms.
The collision types are characterized by a Tr/2, ?r/4 and 3?r/4 (from left to right) rotation of the relative velocity.
The velocity grid points for Ar atoms are gained by substituting in Eqs. (16-18) the velocity grid points of
Na atoms vi and YJ, Eq. (1), for the velocities v, v' and their components, respectively. This yields
(20)
Wi
62
W
Av
(21)
U,2 = -y
Av
W
with /x =
(22)
U,3 =
as well as
w
ij,2 = WJU5,
w
(23)
ij,3
These results imply that the collision term, Eq. (3), transforms to
(24)
E
1=1j
The constant 1/3 in this formula refers to the assumption that each of the three types of collisions is equally
probable. The original transition rate Ayk separates into the product of the total cross section for elastic
scattering, 0"y,z, and the magnitude of the relative velocity, g-^i, which reads
h)2 =551,1
- Wy,i =
2
(25)
- jl - J2) 2
v
^d-
(26)
(27)
- jl - J2) 2
The three terms for / = 1,2 and 3 in the sum of Eq. (24) reflect the three different collision types and point out
the necessity to use for each collision type and subscript of N{ a special velocity grid for the Ar atoms. The
Eqs. (20-23) ensure that both the pre- and post-collisional Ar velocities lie on the corresponding grid. Their
grid spacings are given by
= | w iO'i+l,j2),l -
W
i0'i,j2),l
= ~5~(1
(28)
(29)
^(
(
' 2
(30)
As an example, we show in Fig. 3 the three different velocity grids for Ar atoms referring to the Na grid point
FIGURE 3. Two-dimensional velocity grids for Ar atoms interacting with Na atoms of velocity vi (i =
corresponding to a Tr/2, ?r/4 and 3?r/4 (from left to right) rotation of the relative velocity.
63
The fixed velocity distribution of the Ar atoms is assumed to be a Maxwellian:
2
This distribution transforms, with respect to the different Ar grids, to the following number densities:
(32)
(33)
-P
Based on all of these preliminaries, the system of kinetic equations describing the temporal evolution of the
three species of Na atoms given by Eq. (2), reads in matrix form
(34)
at
with N = (Afiq, . . . , Nl, . . . , JVq, AT^q, . . . , AT2, JV^q, . . . , N^)T. (This representation requires implicitly a one
to one map of the two-dimensional index set Ind2 onto the set {1, 2, 3, . . . , (2q + I) 2 })- It is interesting to note
that the coefficient matrix
'-(
MI
0
Gl3
0
M2
G23
G32
M3
:
(35)
consists of seven blocks reflecting the physics of hole burning. The matrices in the main diagonal of J are
given by
(36)
where the square matrix M = MI + M2 + MS with entries
(37)
represent the elastic collisions of Na atoms with the buffer gas. The diagonal matrices
= -/3 2 diag(..., /?,...),
(38)
contain loss terms caused by photon-atom interactions. The symbol 1 denotes the (2q + l)2-dimensional unit
matrix. Gain terms due to photon-atom interactions are represented by the matrices
= a 2 l-V 2 ,
G3i = -Vi,
G32 = -V2.
(39)
In our model, photon-atom interactions do not change the velocity of atoms. Consequently, the corresponding
reaction terms only appear as diagonal entries in the blocks.
64
Main properties of the kinetic model
Analyzing the mathematical structure of our model equations, we found that the sum over the components
of JN in Eq. (34) vanishes. This implies that the total number density of the Na atoms
= EE
(40)
s=l i
is conserved during the temporal evolution of the system. Inspection of the matrix J reveals that its kernel is
one-dimensional. This means that the equilibrium solution N resulting from JN = 0 is unique with respect
to the normalization condition Eq. (40). Moreover, we have proven by applying the theorem of Gersgorin [8]
that due to the particular structure of J all eigenvalues are either zero or have negative real parts. Thus, as
time t tends to infinity, the solution of equation (34) approaches the stable equilibrium N.
When the laser is turned off, the different quantum states of Na atoms are decoupled because of missing
inelastic particle-particle collisions. However, in this case it is possible to calculate the equilibrium state
analytically. A sufficient condition for stationarity is to demand that each term in the sum representing J^,
Eq. (24), vanishes separately. This yields e.g. in the case of / = 1
2kT
"*2WJ1,1
Aw2 = .
2kT
Aw2
(41)
With the help of Eq. (20), we obtain after some algebra
exp i
2kT
(42)
A7-1
iV
exp (
Thus, the equilibrium state of Na atoms is the usual discretized Maxwellian due to the temperature of the Ar
atoms.
NUMERICAL RESULTS
If the laser is turned on, the different quantum states are coupled. In this case, we solve the system of kinetic
equations (34) numerically by taking advantage of the group behavior of the matrix exponential function. A
typical result describing one-dimensionally a stationary hole-burning experiment is plotted in Fig. 4.
5el7
1000
-1000
-5el7
1000
-5el7
-3el7
FIGURE 4. Deviation of the velocity distribution AT/ from the corresponding Maxwellians N? for particles AI (left),
AZ (middle), and AS (right).
Due to the effect of velocity-changing collisions, the maximum absorption of laser photons appears neither
at the F = 1 hfs-sublevel nor at the F = 2 hfs-sublevel but at some frequency in between. In our example,
the pump laser is, therefore, tuned to a frequency halfway between the two hyperfine transition resonance
frequencies. This leads to a depletion of the velocity distribution for species AI (A 2 ) for particles with certain
positive (negative) values of speed and to a small peak on the opposite side of the origin of the speed axis.
65
The velocity distribution of the excited species of Na atoms displays the overpopulation caused by Na atoms
in ground states which are in resonance with the laser light due to the Doppler effect. It should be noted that
the three plots in Fig. 4 represent the deviation of the velocity distribution of species A\, A% and A% from the
corresponding Maxwellians.
CONCLUSION
Not only photon-atom interactions but also elastic and inelastic collisions in multicomponent gases with
species in different quantum states play an important role in quantum optics. The velocity distributions of
diluted gases are very often far away from equilibrium. Consequently, a kinetic modeling on a molecular level
has to be applied. We have shown in this paper that it is possible to treat such scenarios by taking advantage
of a full discretization of the velocity space. The obtained linear transport model fulfills all requirements on a
Boltzmann-like evolution equation. It involves elastic collisions between the test particle and the buffer gas as
well as the interaction of photons with three-level atoms by taking into account the Doppler effect. In addition,
any change of pro- to post-collisional velocities on the equidistant grid is possible. This behavior supports a
realistic relaxation towards equilibrium and avoids spurious collision invariants. It has been proven rigorously
that the model equations possess a stable and unique equilibrium solution. A tricky selection of admitted
collisions reduces essentially the computational amount. An interesting extension of the developed model by
taking inelastic particle-particle interactions into account is planned for future research.
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