Coherence loss and revivals in atomic interferometry: A quantum–recoil analysis
Abstract
The coherence effects induced by external photons coupled to matter waves inside a MachZehnder threegrating interferometer are analyzed. Alternatively to atomphoton entanglement scenarios, the model considered here only relies on the atomic wave function and the momentum shift induced in it by the photon scattering events. A functional dependence is thus found between the observables, namely the fringe visibility and the phase shift, and the transversal momentum transfer distribution. A good quantitative agreement is found when comparing the results obtained from our model with the experimental data.
pacs:
03.65.Ta, 03.75.Dg, 42.50.p, 42.50.Xa, 37.25.+k, 42.25.Hz1 Introduction
The remarkable refinement reached in matter wave interferometry in the last decades [1, 2] has made possible to explore experimentally fundamental key questions about wave particle duality and complementarity that have been studied since the very inception of quantum mechanics [3, 4]. In this regard, Chapman et al [5] carried out an outstanding experiment in 1995, where the influence of photonatom scattering events (inside an atomic MachZehnder interferometer) on the coherence properties of an atomic beam was investigated. This experiment was interpreted as a realization with atoms of Feynman’s “whichway” gedankenexperiment [6].
The most intriguing result from Chapman’s experiment was the revival of fringe contrast beyond the limits predicted by the complementary principle [5, 2, 7]. Furthermore, it was also observed [5] a regain of fringe contrast after postselecting atoms at the exit of the interferometer according to the momentum transferred in the photonatom scattering process. The regain of interference due to postselection in momentum space had been previously reported for optical [8] and neutron [9] experiments with presence of resonant spinflipper fields. In the case of the neutron experiments, a spectral modulation effect was observed by means of a proper postselection procedure, where the spatial shift of the wave trains greatly exceeds the coherence length of the neutron beams traversing the interferometer [1, 9].
By the time when the paper by Chapman et al[5] was published, a controversy on the origin of the disappearance of interference in “whichway” (actually, “whichslit”) doubleslit experiments was already in fashion: recoil vs decoherence. At a first glance, it seems that the primacy of recoil arguments [10] has been contested in favor of more general decoherence mechanisms, based on considering the entanglement between the observed system and its environment to be the source of the system loss of fringe contrast or visibility. Nevertheless, Storey et al[11] argued that, whenever interference is destroyed, transverse momentum has to be transferred according to the uncertainty principle.
Revivals observed beyond the limit of the complementarity principle enforced Chapman et al[5] and Cronin et al[2] to argue that “the momentum recoil by itself can not explain the loss of contrast (as it can in the diffraction experiments), but the path separation at the point of scattering and the phase shift imprinted by the entanglement in the scattering process must also be taken into account”. In addition, Cronin et al[2] argued that “focusing on the whichway information carried away by the scattered photons is not the only way decoherence may be understood. An alternative, but completely equivalent picture involves the phase shift between the two components of the atomic wave function”. These two views (whichway and dephasing) “correspond to two different ways to describe the scattered photon (position basis versus momentum basis). In these two cases, an observer in the environment can determine either which path the atom took or else the phase shift of its fringe pattern. The key point is that when the experimenter is completely ignorant of the state of the scattered photons, whether an apparatus has been set up to measure them or not, the whichpath and phase diffusion pictures are equally valid (Stern et al., 1990, [12]). Both predict decoherence, i.e., loss of contrast” [2].
It is important to note that the apparatus of Chapman et al[5] was set up to detect atoms, but not to measure the state of the scattered photons. Because of this, in the present work we study this experiment using a model [13, 14] that focuses on atomic states. It accounts for the effects caused on the atom timedependent wave function by the interferometer as well as the (environmental) photons scattered from the atoms when the latter are excited in a resonance fluorescence state by a laser beam. Due to the negligible timescales involved in the dynamics of the atomphoton scattering process (i.e., the absorption and then reemission of the photon by the atom) compared with the timescales involved in the experiment, the photon atom resonance scattering is described as a sudden change of the atom wave function accompanying the momentum transfer between the photon and the atom. Hence we assume each atom can be individually described by a pure state, and only when a collection of atoms is considered statistically, the decoherence effect arising from the photoninduced momentum displacements becomes apparent. More specifically, here we use the probability distribution of transverse momentum transfer to an atom in resonance fluorescence derived by Mandel [15, 16] from the angular distribution of spontaneously emitted photons.
According to such a model, here we present a functional dependence between the experimental observables, namely the fringe visibility and the phase shift, and the statistical distribution of photonatom transversally transferred momentum. From this relationship, a direct connection is established between the coherence losses and subsequent revivals undergone by the atoms, which arise as a consequence of the statistical distribution of the sudden momentum shifts induced in the atomic wave function by the photons (scatteringmediated momentum transfer processes). Furthermore, when some particular choices of momentum transfers are considered by selecting the outgoing atoms according to some prescribed momentum distributions, i.e., by postselecting the atoms, a regain of the coherence is observed. As it is shown, these results are in good agreement (both qualitatively and also quantitatively) when compared with the experimental data reported by Chapman et al[5]. Note therefore that this simple model thus provides a selfconsistent explanation of the experiment based on firstprinciplelike arguments rather than only a best fitting to some suitable function.
This work is organized as follows. In Section 2, to be selfcontained, we start by briefly introducing the experimental setup used by Chapman et al[5] as well as an also brief description of the two types of experiments they carried out. In Section 3, we introduce our theoretical description of this experiment together with the analytical tools that arise from it to later on evaluate the fringe visibility and phase shift, which are compared with the experimental data. As it will be seen, this entails the two features of a quantum particle within the same experiment: wave and corpuscle. In other words, with each individual atom that enters into and passes through the threegrating MachZehnder interferometer, and then arrives at the detector, there is a wave associated, which is described by a coherent wave function or pure state. In Section 4, results for different functional forms of the transversal momentum transfer distribution are analyzed and discussed. As it is shown, when these results are directly compared with the experimental data reported in [5], a good agreement is found even without using any bestfit method, but just introducing the experimental parameters into the functional forms derived from our theoretical model. Finally, the main conclusions arising from this work have been summarized in Section 5.
2 Description of the experiment
In the experimental setup utilized by Chapman et al[5] (a sketch is shown in figure 1a), a beam of atomic sodium with a narrow velocity distribution is produced, collimated and launched through an atomic MachZehnder interferometer. The interferometer consists of three 200 nm period nanofabricated Ronchi diffraction gratings (indicated by the vertical dotted lines in figure 1a) separated by cm. Each grating acts as a coherent beam splitter [17], with the zeroth and first order maxima being the relevant ones.
A polarized laser beam behind the first grating, , is switched on with the direction of the beam being parallel to this slit. This laser leads the atoms to a resonant excited state, from which they decay back to the ground state via spontaneous emission. The atomic flux collected behind the third grating, (see figure 1a), was then measured as a function of a shift produced in this grating along the axis, with the laser both off and on. This measurement was performed considering different values of the distance between and the laser beam. Then, next, the same set of measurements was repeated, but adding a selection slit behind , in front of the detector (see figure 1b). Each selection slit was associated with a particular range of values of the transferred transverse momentum.
The dependence of the measured values of the number of detected atoms on the shift , given by
(1) 
revealed interference [5]. In this expression, is the average atom count rate, is the period of the grating and is the relative contrast (or fringe visibility). When the laser was off, the contrast was typically about 20% and the phase was zero. When the laser was turned on, photon scattering events before and immediately after does not affect either the contrast or the phase. However, as increases, the contrast decreases, first linearly and then it sharply falls to zero. Afterward few revivals were observed. This behavior can be seen in figure 2 of [5], where the relative contrast (visibility) was represented as a function of , with being the photon wavelength and
(2) 
Chapman et al[5] interpreted the quantity as “the relative displacement of the two arms of the interferometer at the point of scattering”. However, Božić et al[14] pointed out that this quantity is equal to the separation between the two paths associated with the zeroth and first order interference maxima only in the far field, behind . On the contrary, in the near field, is equal to the distance between the prolongations of such paths. This distinction should be taken into account when interpreting the experimental data, since the photonatom scattering events in this experiment take place in the near field. In this work, this is explained in detail, taking into account the following fact:
(3) 
where is the socalled Talbot distance [18]. In the experiment, the ratio ranges between 0 and 2. From the values of the other experimental parameters, it follows that mm and the Talbot distance is mm.
The same set of measurements was repeated, but this time adding a selection slit behind , in front of the detector (see figure 1b). More specifically, this was done by arranging slits in three different positions, each selection slit being associated with a particular range of values of the transverse momentum transferred to the atom (i.e., with a particular momentum transfer distribution). This was possible because the deflection of the atom at the third grating, , is proportional to , the transverse momentum transferred to the atom. The curves shown in figure 3 of [5] show a substantial regain of contrast over the whole range of values for . In particular, a 60% of the contrast lost at was regained.
From these results, Chapman et al[5] concluded that the decrease of contrast to zero in the range confirms the complementarity in quantum mechanics, which suggests that fringe contrast must disappear when it is possible to acquire whichway information, i.e., for . Consequently, one should expect that beyond this value no coherence should be possible. On the contrary, the experiment revealed that the atomic coherence displayed revivals in the relative contrast beyond the first zero, thus allowing the atoms to also display some wavelike behavior beyond the limits of complementarity. Furthermore, in the second part of the experiment, it was also observed that the coherence could be regained; actually, no zero values were observed in the relative contrast.
In our opinion, analyzing this kind of experiments in terms of the idea of complementarity might result confusing, though very widespread. This was already pointed out by Englert [19] in 1996, who warned about the misunderstandings that may arise from the use of concepts like waveparticle duality unless they are clearly specified and disambiguated. As it is shown below, in the model here described, such concepts, namely wave and particle, are not mutually exclusive, but they both coexist in the experiment, giving a good account of the experimental data. In particular, the wave aspect of the atom is kept all the way through the interferometer, the photon only causing a deviation of its translational motion (due to the kick and subsequent momentum transfer during the scattering event).
Having in mind these ideas and the scheme displayed in figure 1a, in the derivations presented below, we assume the atomic beam incident onto the grating (at ) can be well approximated by a monochromatic or plane wave of finite transverse width with wavelength and wave vector . If the atomic beam crosssection is also assumed to be wide enough (in the experiment, this crosssection is about two orders of magnitude larger than the grating period [20]), not only it will cover a relatively large number of slits, but also an important extension along the direction. This causes a symmetry along the direction, which allows us to simplify the analysis by reducing it to the plane (for fixed , e.g., ).
3 Theoretical approach
3.1 Atom’s wave function evolution accompanying atom’s passage through the interferometer
Taking into account the description of the experiment made above, now we are going to analyze it here according to our model. Thus, consider the incident atomic wave function associated with atoms having a velocity is given by
(4) 
where , and describes the width of the initial wave function along the transverse direction. In the paraxial approximation, the outgoing wave evolving freely after the diffraction caused by is approximated by
(5) 
This function is a product of the plane wave along the longitudinal direction by the “transverse” wave function
(6) 
which describes the evolution along the direction. The function is the Fourier transform of the function which is determined by through the relation
(7) 
where is the given transmission function of the grating located at . It is also the transmission function of grating . More explicitly,
(8)  
(9) 
Evidently, is the timedependent transverse wave function in momentum representation.
Taking into account the length scales involved in the experiment, the paraxial approximation can be considered a good approximation. This implies, first, that the particle motion parallel to the direction can be treated as a quasiclassical (uniform) motion, i.e., satisfying the relation , with . Second, the wave function (7) behind the grating is such that is relevant only for (in other words, the spreading of the wave function is much slower than its propagation along the direction [21]). Accordingly, equation (6) can be parameterized in terms of the coordinate or, equivalently, the (propagation) time .
In the passage from to as well as beyond , a similar analysis can be conducted (see below). However, at a time and a distance after the grating the atom absorbs and reemits a photon. This process induces a sudden change in the atomic transverse momentum which is accompanied by the sudden change of the evolution of atom’s wave function. Arsenović et al[13] determined the evolution of atom’s wave function after photon atom scattering by assuming that atom’s wave function in momentum representation after photon atom scattering has to satisfy:
(10) 
The corresponding transverse wave function at time , in accordance to (6) is then given by
(11) 
It should satisfy
(12) 
As shown by Arsenović et al[13], from equations (9)(11) it follows that condition (12) will be fulfilled if
(13) 
Substituting (13) into (11) and then using (9), one finds that just after the photonatom scattering event, the atomic wave function becomes
(14) 
where
(15) 
Assuming (14) keeps the same form at any , we may write:
(16) 
By changing now the integration variable , (16) transforms into
(17) 
This wave function describes the evolution of (6) after the scattering event (i.e., for or, equivalently, ). After the scattering event the atom wave function evolves freely until it reaches the second grating . It is important to note that the wave function , associated with , describes also the evolution of the wave behind the first grating when laser is off.
It is useful to parameterize wave function (17) in terms of coordinate using the relation ,
(18)  
The integrals in (17) and (18) have no general analytic solution, except for large or values. In such a limit, when the dimensions of the diffracting object and the wavelength of the diffracted beam are relatively small compared with the typical propagation distances, the farfield or Fraunhofer condition, (with being a measure of the dimensions of the diffracting object), holds [22] and (18) can be approximated (see A) by
(19) 
when the laser is off, and
(20) 
for and the laser on. By comparing (19) and (20) we conclude that the overall form of the atom probability density is the same as for . However, the former will display a shift or displacement along the direction with respect to the latter given by
(21) 
The evolution of the wave function between and follows a similar description to the one prior to the scattering event. Thus, if the wave function incident onto is denoted as , which arises from evaluating (20) at , just before the second grating, then wave function evolution behind the second grating () is given by
(22)  
where the relation between the time and is now and the momentum probability density reads as
(23) 
From (22) and (23) one finds by numerical integration that the probability density incident onto for a given value of oscillates with period . This oscillatory pattern (figure 3 in [13]) is of finite width and its position along axis depends on . In other words, the oscillatory pattern corresponding to is shifted, with respect to the oscillatory pattern when laser is off, by the quantity
(24) 
which arises after considering the shift of the wave function at (according to (21)) and the influence of on the propagation direction of the wave function emerging from . This estimate of is consistent with the shifts determined through the numerical evaluation of the squared modulus of [13, 14].
3.2 Atomic flux behind the interferometer
In order to compare the results obtained from the theoretical model exposed above with the experimental data [5], we have first considered the number of atoms transmitted through that undergo a change of momentum during the scattering process. This number is proportional to
(25) 
where is a lateral shift of the third grating with respect to the alignment of and the integration limits extend over the region covered by the central maximum at . By numerical integration with the wave function determined as described in the previous section, it has been found [13, 14] that (25) the transmitted flux (25) is a simple periodic function:
(26) 
where is defined in (2), and and are constants independent of and . Far from the grating (i.e., large values of ), the distance is equal to the separation between the paths associated with the zeroth and first order interference maxima of the atomic wave diffracted by (see figure 1a). However, near the grating the emergent diffraction pattern is far more complex than a series of well defined paths, obeying a Talbotlike carpet structure [18]. This implies, as explained after (3) and in [13] that should not be interpreted as the distance between two atomic paths in the region covered by the laser light, for in this region there are, actually, many more paths than simply two, as it is generally assumed [2, 5].
The results reported in [5] essentially come from two types of measurements. The first type consists of simply counting all atoms that pass through ; in the second type, only a certain subset of the transmitted atoms are counted or postselected, in particular those with a certain momentum direction, which is done by positioning an additional slit beyond (see figure 1b). Therefore, the observable is not in general, but its integral over a set of transferred momenta ,
(27)  
where the weight denotes the transversal momentum transfer distribution of the detected atoms. More specifically, this quantity is the product of the atom momentum transfer distribution and the distribution function characterizing the way how the atoms are selected (postselected) by their momentum beyond the interferometer. That is, we have . In particular, when the postselection process will be included, we shall refer to the normalized function as the postselection momentum transfer distribution. Thus, if , with , is the corresponding normalized distribution, it is straightforward to verify that (27) reads as
(28) 
where the quantities and represent the fringe visibility or relative contrast and the phaseshift, respectively, and are determined through the relations
(29) 
with
(30) 
From a practical point of view, in order to evaluate and , it is useful to introduce the complex integral
(31) 
so that
(32) 
Taking this into account together with the standard definition of fringe contrast [16], from (28) we find
(33) 
When the laser is off, and hence and . The relative contrast then reads as
(34) 
which is a function of the ratio ( is the scattering photon wavelength), as it will be seen below.
4 Numerical results
In order to compare with the experiment, below we present some calculations, where we have considered the same parameter values used in the experiment [5]: ms, 10 m, nm (10 m), m, 10 m and 10 m. To evaluate the wave function, we have considered a total number of illuminated slits in , which is an acceptable range compared with experimental atomic beam crosssections (i.e., the coherence length of the atoms arriving in the grating) [20].
Apart from the Mandel distribution [15], which accounts for the bare transversal momentum transfer distribution, to compare with the experiment we have also considered the three postselection momentum transfer distributions used in the experiment, denoted by , and ). These distributions correspond to the combined effect of the momentum transfer process (described by Mandel’s distribution) and three different particular selections (postselections of atomic momenta (each one given by a different distribution), which are achieved by arranging a slit behind in three different positions (see figure 1b). The dependence of these four momentum transfer distributions as a function of the ratio between the transferred momentum and the incident photon wave number, , is displayed in figure 2a. Apart from these distributions, we have also considered several other theoretical forms for the momentum transfer distribution of the detected atoms, which are of interest to further analyze and better understand the dependence of coherence and visibility on the experimental distributions. In particular, a Dirac function distribution () and three constant distributions, , and , uniform over the intervals , and , respectively. These four distributions are displayed in figure 2b.
A straightforward evaluation according to the method indicated at the end of Section 3.2, leads us to the following expressions for the visibility and phase shift associated with these distributions:

As shown by Mandel [15], for photons incident with a momentum , the transversal momentum transfer distribution can be expressed as [16, 15]
(35) In this case, the visibility and phase shift read as
(36) (37) which are both functions of the ratio (black solid lines in figures 3a and 3b). As it can be seen, we find a good agreement between these theoretical expressions and the experimental data (black solid circles) without taking into account any fitting procedure. Both the coherence losses and subsequent regains are thus accounted for without abandoning the idea of pure state to describe the full evolution of the atom.

The case of is simulated by a halfGaussian,
(38) where determines the width of the Gaussian (here, we have chosen , so that ). In this case (see A),
(39) (40) where . As seen in figures 3c and 3d (black solid lines), there are no recurrences in (they are completely damped), while approaches a constat value of as increases. Again, as it can be seen, we find a fair agreement with the experiment (black solid circles).
If instead of , one would choose , i.e, the mirror image of with respect to , then
(41) (42) That is, the visibility is the same in both cases, but is an increasing linear function of (after reaches its maximum, steady value).

For we consider a displaced Gaussian,
(43) with its maximum at and , as before, so that . With this, we find
(44) (45) which are represented by blue dotted lines in figures 3c and 3d. In this case, since there relative contrast is very similar to that found for , no experimental data were reported. We only have experimental results for the phase shift (blue squares in figure 3d), where a good agreement is also found.

is described by means of an increasing exponential,
(46) where is the increase rate (see blue dasheddotted line in figure 2a). This distribution leads to
(47) (48) where . As seen in figures 3c and 3d (red dashed lines), now presents some damped recurrences and there is a significant phase shift. The same trend is also observed in the experimental data (red stars), which follow very closely the behavior of the theoretically predicted curves.
There are several simple cases of particular interest, because grosso modo they capture the essential features of the distributions used in the experiment, which are the finite, uniform momentum transfer distribution within the interval , being zero everywhere else,
(49) 
for . For this form we find
(50)  
(51) 
As can be noticed, the visibility is given in terms of the half distance between the limits of the interval, , while the phaseshift is proportional to their half sum, , which corresponds to the average momentum. This implies that the visibility will decay and oscillate faster as both and approach the limits of the interval, the phase behaving in a similar manner (i.e., increasing). On the contrary, if , we will be approaching the limit described by : will oscillate more and more slowly (behaving almost constant up to very large values of ), while its phase will approach . Now we will analyze each one of these cases separately:

In the case and , , which is a rough approximation to . Here, we find
(52) (53) 
If and , we have , which roughly describes and renders
(54) (55) 
And, and , we have , which can be an approximation to either , or , and gives rise to
(56) (57) Notice that in this case and the previous one, the visibility is the same, but not the phase shifts, which increases three times faster for than for .
As it can be noticed, the functional forms found with our model for the visibility and the phase shift associated with the different momentum transfer distributions are in good agreement with those reported in [5].
As it can be noticed, vanishes for , with being an integer, while and vanish when . This is related to the fact that, for these three distributions, the integrand in (30) is a periodic function of , with period . For the integration in (30) is carried out over the interval , which contains an integer number of periods when . For and the integration is performed over the intervals and , respectively, which contain an integer number of periods when . Nevertheless, it is worth going further and analyzing the physical reasons why the zeros of , and appear at these values of . To start with, let us remember that the phase that appears in arises as a consequence of the shift along the axis at displayed by the atom wave function after the change of atomic transverse momentum due to photonatom scattering. This shift, which is explicitly given by (24), contains the term . The latter is of the order of the grating constant , as can be noticed if we define , with for , for , and for . Thus, taking into account explicitly the value of , we find , which implies for , for , and for . Therefore, when , lies within the intervals , or depending on we have , or , respectively. This is why in the case of a uniform momentum transfer distribution along the interval the total number of detected atoms (27) does not depend on the lateral shift at and the contrast is zero. However, if the transferred momentum spans the interval , the displacement of the wave function spans half the grating constant and, therefore, the number of detected atoms will depend on the lateral shift at , then the contrast being greater than zero. On the other hand, when , lies within the intervals , and for , and , respectively. In the three cases the displacements thus span an integer number of grating periods. Therefore, in any of these cases, the total number of detected atoms will not depend on the lateral shift at and the contrast will vanish (see figures 4a and 4b).
It is insightful to analyze the experimental outcomes in the light of the constant distributions. One could therefore state that the contrast regain found in the experiment, compared with the Mandel distribution, arises from the change of the momentum transfer distribution of the detected atoms, which is an objective effect. Furthermore, the loss and revival of coherence in the case of the Mandel distribution are also objective effects, which are related to the properties of the atomic wave function incident onto .
5 Conclusions
In spite of the details involved in entanglementbased models aimed at describing complementarity in experiments like the one here analyzed, appealing to simpler models is also of interest in order to understand the underlying physics, even if they are not fully complete. In the case dealt with here, we have considered a description based on the recoil of the wave function describing the diffracted beam when a photon impinges on it within the interferometer. This model not only allows us to obtain a nice description of the evolution of the wave function throughout the matterwave MachZehnder interferometer, but also to explain the losses (e.g., the total loss at ), subsequent revivals (for ) and regains (for all values of of experimental interest) undergone by the (atom) fringe contrast in a very simple manner. In particular, here we have presented how such effects arise when the outgoing atomic probability density is sampled by a certain momentum distribution, either Mandel’s bare momentum transfer distribution or the corresponding postselection ones. In other words, these three effects can be attributed to the smearing out of the interference pattern induced by the distribution of transverse momentum that the photon or the postselection process cause on the atomic beam.
In order to obtain some extra information, other momentum transfer distributions of theoretical interest have also been considered. In this regard, it was shown that, if the atoms passing through