On the influence of resonance photon scattering on atom interference

Abstract

Here, the influence of resonance photon-atom scattering on the atom interference pattern at the exit of a three-grating Mach-Zehnder interferometer is studied. It is assumed that the scattering process does not destroy the atomic wave function describing the state of the atom before the scattering process takes place, but only induces a certain shift and change of its phase. We find that the visibility of the interference strongly depends on the statistical distribution of transferred momenta to the atom during the photon-atom scattering event. This also explains the experimentally observed (Chapman et al 1995 Phys. Rev. Lett. 75 2783) dependence of the visibility on the ratio d_p/\lambda_i = y'_{12} (2\pi/kd\lambda_i), where y'_{12} is distance between the place where the scattering event occurs and the first grating, k is the wave number of the atomic center-of-mass motion, $d$ is the grating constant and \lambda_i is the photon wavelength. Furthermore, it is remarkable that photon-atom scattering events happen experimentally within the Fresnel region, i.e. the near field region, associated with the first grating, which should be taken into account when drawing conclusions about the relevance of "which-way" information for the interference visibility. Comment: 9 pages, 1 figure

Full text

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arXiv:0910.2979v1 [quant-ph] 15 Oct 2009
On the influence of resonance photon scattering on
atom interference
M Boˇ zi´ c1, D Arsenovi´ c1, A S Sanz2and M Davidovi´ c3
1Institute of Physics, University of Belgrade, 11080 Belgrade, Serbia
2Instituto de F´ ısica Fundamental - CSIC, Serrano 123, 28006 - Madrid, Spain
3Faculty of Civil Engineering, University of Belgrade, 11000 Belgrade, Serbia
E-mail: arsenovic@phy.bg.ac.yu,bozic@phy.bg.ac.yu,
asanz@imaff.cfmac.csic.es,milena@grf.bg.ac.yu
Abstract.
interference pattern at the exit of a three-grating Mach-Zehnder interferometer
is studied. It is assumed that the scattering process does not destroy the atomic
wave function describing the state of the atom before the scattering process takes
place, but only induces a certain shift and change of its phase. We find that
the visibility of the interference strongly depends on the statistical distribution
of transferred momenta to the atom during the photon-atom scattering event.
This also explains the experimentally observed (Chapman et al 1995 Phys. Rev.
Lett. 75 2783) dependence of the visibility on the ratio dp/λi = y′
where y′
12is distance between the place where the scattering event occurs and the
first grating, k is the wave number of the atomic center-of-mass motion, d is the
grating constant and λiis the photon wavelength. Furthermore, it is remarkable
that photon-atom scattering events happen experimentally within the Fresnel
region, i.e. the near field region, associated with the first grating, which should be
taken into account when drawing conclusions about the relevance of “which-way”
information for the interference visibility.
Here, the influence of resonance photon-atom scattering on the atom
12(2π/kdλi),
PACS numbers: 03.65.Ta, 42.50.Xa, 03.75.Dg, 37.25.+K
1. Introduction
With the rise and advancement of neutron [1] and atom interferometry [2, 3], it
has become feasible the realization of the well-known gedanken experiments devised
by Einstein during his famous discussions with Bohr [4], later also considered by
Feynman [5]. These discussions were focused on the understanding and interpretation
of the wave-particle duality and, therefore, the completeness of Quantum Mechanics.
In particular, Bohr [4] and Feynman [5] argued that wave and particle properties were
complementary, i.e. they could not be simultaneously observed experimentally. On
the other hand, aimed to disprove the concept of complementarity, Einstein devised
double-slit type experiments [4] where it should be possible to obtain “which-way”
information without influencing the interference pattern. Einstein’s viewpoint based
on the compatibility of the wave and particle properties, i.e. that both are present
simultaneously and, therefore, can be observed in quantum interference experiments,
was supported by De Broglie [6] and Bohm [7]. For these authors, the quantum system
comprises both a wave and a particle, the former guiding the motion (evolution) of
the latter, which leads to a hydrodynamic-like view of Quantum Mechanics [8,9].

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On the influence of resonance photon scattering on atom interference
2
More recently, Rauch and Vigier have pointed out [10] that de Broglie’s and
Bohm’s argumentation was based on so-called einweg experiments, which should
explicitly show that individual particles go along one trajectory, but without
evidencing which particular trajectory. Of course, the “which-way” argumentation
implies the einweg one, but not vice versa. The difference between these two types
of argumentation arises from the different signatures of wave and particle properties
invoked in them. More specifically, for Bohr and Feynman the particle signature is the
“which-way” information, while the wave signature is the visibility or relative contrast
of the interference pattern. On the contrary, for de Broglie [6], Bohm [7], Philippidis et
al [11] or Sanz et al [12–14], the particle signature is the arrival of individual quantum
particles to a screen (array of detectors) and the time evolution of the distribution
of these arrivals [15]; the wave signature associated with each quantum particle is
the visibility of the interference pattern together with the fact that it comes from the
accumulation of arrivals of a large number (theoretically, an infinite number) of atoms,
photons, electrons, etc.
To perform experimentally Feynman’s gedanken experiment, Chapman et al [16]
scattered single photons from Na atoms within a three-grating Mach-Zehnder atom
interferometer. By measuring the transmission of atoms through the third grating,
the influence of photon scattering processes (which take place at a distance y′
the first grating) on the visibility of the atom interference pattern was investigated.
These results have intensified a controversial discussion on the wave-particle duality
issue. At the time when the experiment was carried out, this controversy evolved
towards a discussion around the question: Is complementarity more fundamental than
the uncertainty principle? [17–20] This discussion continued [21–25] with the aim to
determine the cause of the visibility decrease: Does the visibility decrease arise (a)
from a random momentum transfer between the atom and the photon or (b) from the
correlations between the “which-way” detector and the atomic motion? More recently,
the statement (b) has been reformulated as [3]: Is the visibility decrease (decoherence)
the result of entanglement between a quantum system (the atom) and an environment
(the emitted photon, which carries information about the atom’s path).
Previously, we explained [26] the experiment carried out by Chapman et al using
the solution of the time-dependent Schr¨ odinger equation for an atom interacting with
a photon and the gratings in a three-grating Mach-Zehnder interferometer. In our
explanation, wave and particle properties were compatible, since in our opinion both
are present and play a role. We derived an analytic expression for the visibility
dependence on the ratio dp/λi = y′
the atomic center-of-mass motion, d is the grating constant and λi is the photon
wavelength. This theoretical result was in fairly good agreement with the visibility
measured in the experiment [16]. The distribution of transferred momentum during
the photon scattering process leads to the visibility decrease as dp/λi approaches
0.5 as well as several subsequent revivals with decreasing maxima. Here, we provide
additional arguments which support our conclusions of reference [26]. In particular, we
study the visibility dependence on the features of the atom selection at the exit of the
interferometer, before the detection takes place. We also show that in the experiment
the photon-atom interaction takes place within the Fresnel region, i.e. the near field
region, associated with the first grating, something that has to be taken into account
within any dynamical (i.e. time-dependent) description of the experiment.
12from
12(2π/kdλi), where k is the wave number of

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On the influence of resonance photon scattering on atom interference
3
2. Wave function of an atom after interacting with a grating and a photon
Consider an initial stationary atomic monochromatic wave, which spreads along the
y-axis and is incident to a one-dimensional grating parallel to the x-axis at y = 0,
Ψ(x,y,t) = e−iωtψi(x,y) = Bie−iωteiky, y < 0, (1)
with Bibeing a constant.
diffracted), this incident wave transforms into
After interacting with the grating (i.e. after getting
Ψ(x,y,t) = e−iωtψ(x,y), (2)
where
ψ(x,y) =eiky
√2π
?∞
−∞
dkxc(kx)eikxxe−ik2
xy/2k, (3)
satisfies the Helmholtz equation [27]. If the grating is completely transparent inside
the slits (the union of slit areas is denoted by A) and completely absorbing outside
them, c(kx) can be expressed [27] as
?∞
=
√2π
A
where ψi(x′,0−) and ψ(x′,0+) denote the wave function just before and just after the
first grating, respectively. The solution of the Helmholtz equation, ψ(x,y), given by
(3), is equivalent to the Fresnel-Kirchhoff solution
c(kx) =
1
√2π
1
−∞
?
dx′ψ(x′,0+)e−ikxx′
dxψi(x′,0−)e−ikxx′, (4)
ψ(x,y) =
?
k
2πye−iπ/4eiky
?∞
−∞
dx′ψ(x′,0+)eik(x−x′)2/2y. (5)
The photon-atom scattering event induced by the laser light at a distance y′
from the first grating leads to a change of the atomic transverse momentum, ∆kx,
and, therefore, to a shift of the wave function in the momentum representation. Hence,
after an atom absorbs and re-emits again a photon somewhere along the x axis at a
time t′
12and a distance y′
wave function takes the form [26]
12
12= vt′
12= (?k/m)t′
12from the first grating, the atomic
ψ∆kx(x,y) =eiky
√2πei∆kx(x+∆x0)−i∆k2
?∞
xy/k
×
−∞
dk′
xc(k′
x)e−ik′2
xy/2keik′
x(x+∆x0−∆kxy/k), (6)
where
∆x0=∆kx?t′
12
m
=∆kxy′
12
k
. (7)
The wave function (6), evaluated at the distance y12, which separates the second from
the first grating, was used [26] as a wave incident onto the second grating. Then, the
wave function propagating towards the third grating was determined using (5), where
ψ(x′,0+) consists of the parts of the incident wave which are transmitted through the
slits of the second grating.
This is the way how the evolution of the initial plane wave (1) was determined.
After interacting with the first grating, a photon at the distance y′
12from this slit,

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On the influence of resonance photon scattering on atom interference
4
and a second grating, the resulting wave function evolves freely again up to the
third grating. In order to illustrate this evolution, in figure 1 we have plotted the
atom probability density within the interferometer at several distances from the first
grating. Two values for the transferred momentum are considered: ∆kx= 0 (blue)
and ∆kx = 1.5ki (red). Far from a grating, the straight lines represent the paths
along which the maxima of the probability density move; near the grating, these
straight lines would just be the prolongation of the paths. Note that in the near field
associated with a grating, the wave function has a very complex form and the lines
do not exactly represent the paths of the atoms. Within this region, according to a
Bohmian picture [13,14], there are many paths which, as the distance from a grating
increases, converge towards three main paths (only two are plotted in figure 1). The
extension of the near field is of the order of 10LT, where Lt= 2d2/λ is the so-called
Talbot distance [14].
3. Dependence of visibility on the distribution of transferred momentum
In the experiment [16], the number of Na atoms transmitted through the third grating
is counted for the laser off and laser on varying the distance y′
different values of the shift ∆x3of the third grating along the x-axis. In a first round
of measurements, all transmitted atoms were collected (counted) without carrying
out any selection. Thus, the resulting interference curve can be directly associated
with the distribution of transferred momentum, which is given by the Mandel-Wolf
expression [28,29]
12as well as considering
PMW(∆kx) =
3
8ki
?
1 +
?
1 −∆kx
ki
?2?
. (8)
Then, in subsequent measurements, specific subsets of transmitted atoms were
counted after being selected using certain slits positioned after the third grating. As
explained in [16], each selection is equivalent to a particular distribution of transferred
momentum during the photon-atom scattering process. The experimental data show
that the interference pattern visibility strongly depends on the ratio dp/λias well as
on the probability distribution for the transferred momentum, P(∆kx).
In order to explain and interpret these experimental data within our approach[26],
it is necessary to study the function
T(y′
12,∆x3) =
?2ki
0
d(∆kx)P(∆kx)˜T(y′
12,∆kx,∆x3), (9)
where
˜T(y′
12,∆kx,∆x3) =
?
slits
|ψ∆kx(x,y = y12+ y23)|2dx. (10)
The latter function is proportional to the number of atoms transmitted through the
third grating that undergone a change of momentum ∆kx during the photon-atom
scattering process behind the first grating. By numerical integration, it was shown [26]
that (10) has the general form
˜T(y′
12,∆kx,∆x3) = a + bcos(2π∆x3/d + dp∆kx),(11)
where a and b are constants, and
dp= (2π/kd)y′
12.(12)

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On the influence of resonance photon scattering on atom interference
5
Figure 1.
interferometer at several distances from the first grating for two values of the
transferred momentum: ∆kx = 0 (blue) and ∆kx = 1.5ki (red). The values of
the parameters used are: vNa= 1400 m/s, k = mNavNa/? = 5.09 × 1011m−1,
ki= 2π/(589 nm) = 1.07 × 107m−1, y12= y23= 65 cm, d = 200 nm, slit width
δ = 100 nm and number of slits illuminated by the incident atomic plane wave
n = 24. The distance y′
12= 2.863 mm corresponds to dp/λi = 0.3 mm, which
lies within the near field region, this being evident from the fact that the Talbot
distance is Lt= 2d2/λ = 6.484 mm.
Atom probability density within a three-grating Mach-Zehnder

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On the influence of resonance photon scattering on atom interference
6
For y′
with the zeroth and first order interference maxima; near the grating, it refers to the
distance between the prolongations of these paths. Based on these results, we may
assume that for certain classes of distribution functions P(∆kx), T(y′
general form
12far from a grating, dp denotes the separation between two paths associated
12,∆x3) has the
T(y′
12,∆x3) = a + bV cos(2π∆x3/d + ϕ),(13)
where the visibility V and phase-shift ϕ are both functions of the ratio dp/λi, and
their particular form is determined by P(∆kx), which is assumed to be normalized on
the interval [0,2ki].
In order to understand the features of V (dp/λi) and ϕ(dp/λi), consider, for
instance, the uniform distribution P(∆kx) = 1/2ki. The evaluation of the integral
of the first term in T(y′
12,∆x3) is trivial, and for the second term we have
?2ki
=
kidp
From this result, we reach
b
kidp
and, therefore,
1
kidp
As can be seen, V vanishes for dpki= nπ, which is easy to explain as follows. The
integrand in (14) is a periodic function of ∆kx, with period 2π/dp. Hence, for
2ki
2π/dp
λi
the integration is performed over an integer number of periods of a simple periodic
function, the result being zero.
If we now consider the distribution PMW(∆kx), described by (8), we obtain the
transmission function [26]
0
1
2kid(∆kx)bcos(2π∆x3/d + dp∆kx)
b
sin(dpki)cos(2π∆x3/d + dpki). (14)
T(y′
12,∆x3) = a +
sin(dpki)cos(2π∆x3/d + dpki). (15)
V =
sin(dpki) andϕ = dpki.(16)
=2dp
= n,i.e. for
dp
λi
=n
2,(17)
T(y′
12,∆x3) = a + bV cos(2π∆x3/d + dpki),(18)
where the visibility reads as
V =
3
4π
λi
dp
??
1 −
1
(2π)2
λ2
d2
i
p
?
sin
?2πdp
λi
?
+
1
2π
λi
dp
cos
?2πdp
λi
??
.(19)
The zeros of (19) are very close to those of the visibility obtained from the constant
distribution above (see figure 2 in reference [26]).
Finally, if we consider the distribution
PI(∆kx) = γe−(∆kx/Nki)2,
where γ = 2/Nki√π, we obtain
?
(20)
V =
?
?
??e−(Ndpki/2)2?2+
?
Ndpki
√π
∞
?
k=1
(−1)k−1
(2k − 1)!!
?Ndpki
√2
?2(k−1)?2
,
(21)

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On the influence of resonance photon scattering on atom interference
7
tanϕ =
Ndpki
√π
∞
?
k=1
(−1)k−1
(2k − 1)!!
e−(Ndpki/2)2
?Ndpki
√2
?2(k−1)
.(22)
The visibility given by (21) does not have zeros; for any dp/λi, this function is always
above the values of the visibility arising from the previous two distributions. This is
in agreement with the experimental data displayed in figure 2 of reference [16]. From
this fact, Chapmann et al concluded [16] that coherence was not really destroyed, but
only entangled with the final state of the reservoir (photons).
As can be seen, our results agree with the first part of the conclusions of
Chapmann at al [16]. In evaluating the visibility, we have assumed that the scattering
process does not destroy the atomic wave function (i.e. coherence), which describes
the state of the atom before the scattering process, but only induces a certain shift
and change of its phase. The visibility dependence on dp/λiis thus a consequence of
the statistical distribution of transferred momenta to the atom during photon-atom
scattering process. Through the atom selection, which is equivalent to varying the
distribution of transferred momenta, the visibility can change substantially. Within
our approach, we have not invoked entanglement with the reservoir states; rather, we
have determined and used the evolution of an atomic wave function before and after
photon-atom scattering events.
Finally, we would like to stress that experimentally photon-atom scattering events
take place at distances from a grating within the Fresnel region. This can be easily
seen by means of the relation
2πdp=dp
λi
ki
λi
In the experiment, the ratio dp/λi goes from 0 to 2. By the values of the other
parameters given in the caption of figure 1, it follows that y′
LT= 6.48 mm.
y′
12=kd
kd
=dp
LT
2
λi
d.
(23)
12∈ [0,19.09] mm, with
4. Conclusions
To explain atom interference experiments with presence of photon-atom scattering
processes, in opinion it is necessary to use the atom wave function as well as to take
into account its particle properties (i.e. the change of momentum during photon-atom
scattering events). The experimentally established visibility dependence on dp/λiwas
previously explained [26] by considering a random change of the atomic transverse
momentum induced by the scattering with photons.
The experimental regain of visibility induced by selecting a subset of atoms from
the set of all those transmitted through the third grating is explained by studying the
visibility dependence on the probability distribution of transferred momenta. This
atom selection does not provide any information about the place along the x-axis where
the photon scatters from the atom. Consequently, it is not necessary to attribute
the decrease and disappearance of visibility as dp/λi increases to an increase of an
observer’s (potential) knowledge about the atomic path behind the first grating.
From our description, we also find that photon-atom scattering processes happen
within the Fresnel region, where the atomic wave function has a very complex form
(see figure 1). But, as has been shown within the context of the Talbot effect [14], the
topology of the trajectories also becomes very complex within this region. Therefore,
not two but many atomic paths exist near the grating, where the photon hits the

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On the influence of resonance photon scattering on atom interference
8
atom. As one moves further away from the grating (towards the Fraunhofer region),
those numerous trajectories basically group along three main paths. Here, we have
considered two of them for a chosen value of the transferred momentum to explain the
experiment.
The agreement between our theoretical expressions for the visibility and the
experimental curves thus supports, in our opinion, the views of Einstein, de Broglie,
Bohm and others, i.e. that individual micro-objects can be described by a wave and a
particle simultaneously.
Acknowledgments
M. Boˇ zi´ c, D. Arsenovi´ c and M. Davidovi´ c acknowledge support from the Ministry of
Science of Serbia under Project “Quantum and Optical Interferometry”, N 141003.
A. S. Sanz acknowledges support from the Ministerio de Ciencia e Innovaci´ on (Spain)
under Project FIS2007-62006 and the Consejo Superior de Investigaciones Cient´ ıficas
for a JAE-Doc Contract.
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