Theory of near-field matter wave interference beyond the eikonal approximation
A generalized description of Talbot-Lau interference with matter waves is presented, which accounts for arbitrary grating interactions and realistic beam characteristics. The dispersion interaction between the beam particles and the optical elements strongly influences the interference pattern in this near-field effect, and it is known to dominate the fringe visibility if increasingly massive and complex particles are used. We provide a general description of the grating interaction process by combining semiclassical scattering theory with a phase space formulation. It serves to systematically improve the eikonal approximation used so far, and to assess its regime of validity.
pacs:03.75.-b, 03.75.Dg, 34.35.+a
The ability of material particles to show wave-like interference is one of the central predictions of quantum mechanics. While the early experimental tests worked with elementary particles Davisson and Germer (1927); von Halban and Preiswerk (1936), interferometry of atoms is by now a matured field of physics Berman (1997); Miffre et al. (2006); Cronin et al. (2007). It is in particular the ability to cool and to control atoms using laser techniques that has propelled atom interferometry into a versatile tool for precision measurements, e.g. Weiss et al. (1993); Gustavson et al. (1997); Peters et al. (1999); Müller et al. (2007).
As far as more complex and more massive objects are concerned, we are currently witnessing this transition from the proof-of-principle demonstration of their wave nature to the use of interferometry for quantitative measurements. Specifically, the static Berninger et al. (2007) and the dynamic Hackermüller et al. (2007) bulk polarizability of fullerene molecules was measured recently with a molecule interferometer. Also the controlled observation of decoherence, due to collisions with gas particles col or due to the emission of thermal radiation the , has been used to characterize the interaction strength (or cross-section) of fullerenes with external degrees of freedom. Another motivation for studying interference with large molecules is to test quantum mechanics in unprecedented regimes by establishing the wave nature of ever more massive objects Arndt et al. (2005).
The above-mentioned interference experiments with large molecules are based on the near-field Talbot-Lau effect, where three gratings are used which serve, in turn, to produce coherence in the beam, to bring it to interference, and to resolve the fringe pattern. This is an established technique in atom and electron interferometry Cla ; Cahn et al. (1997); Deng et al. (1999); Cronin and McMorran (2006); Kohno et al. (2007), and it is the method of choice for massive and bulky molecules (such as fullerenes Brezger et al. (2002), meso-tetraphenylprorphyrins Hackermüller et al. (2003), or functionalized azobenzenes Gerlich et al. (2007)). The main reason is that a Talbot-Lau interferometer (TLI) tolerates beams which are relatively weakly collimated, thus alleviating the increasing difficulty in producing brilliant beams if the particles get more complex. Another important advantage compared to far-field setups, which is essential if one wants to increase the particle mass by several orders of magnitude, is the favorable scaling behavior of Talbot-Lau interferometers with respect to the de Broglie wavelength Berman (1997).
As a specific feature of Talbot-Lau interferometers, the forces between the particles in the beam and the diffraction grating influence the interference pattern much stronger than in a far field setup. This is due to the fact that the different diffraction orders do not get spatially separated in a TLI. Rather, all the orders interfere among each other, producing a resonant recurrence of the pattern whenever the so-called Talbot condition is met. As a consequence, even tiny distortions of the matter wave may lead to significant changes of the fringe visibility—requiring, for example, the modification of the van der Waals force due to retardation effects Casimir and Polder (1948) to be taken into account when describing the diffraction of fullerenes at gold gratings.
This effect of the dispersion force between the particle and the grating wall gets more important as the mass and structure of the molecule grows. In particular, it sets increasingly strict requirements on the monochromaticity, i.e, the velocity spread permissible in the molecular beam. A very recent development, undertaken to reduce this influence, is the Kapitza-Dirac Talbot-Lau Interferometer (KDTLI), where the second material grating is replaced by the pure phase grating produced by a standing light field Gerlich et al. (2007).
The available theoretical descriptions of molecular Talbot-Lau interference account for the grating forces in terms of a simple eikonal phase shift Patorski (1989); Brezger et al. (2003); Hornberger et al. (2004) (an expression originally derived by nuclear physicists as an asymptotic high-energy approximation to the Lippmann-Schwinger equation in scattering theory Molière (1947); Glauber (1959)). This approximation ceases to be valid with a growing influence of the particle-grating interaction, and, due to its non-perturbative nature, its range of validity is not easy to assess. Therefore, given the quest for testing quantum mechanics with ever larger particles and given the increased precision required in metrological applications, there is a clear need to extend the theoretical description of near-field matter wave interference beyond the eikonal approximation.
The main purpose of this paper is therefore to develop a generalized formulation of the coherent Talbot-Lau effect. By combining a scattering theory formulation with semiclassical approximations we incorporate the effect of the grating interaction systematically beyond the eikonal approximation. As a prerequisite for its implementation, the established theory of near field interference first needs to be extended to account for the effects of finite angular dispersion in the molecular beam. We show how this can be done transparently by using the phase space formulation of quantum mechanics. As a by-product, this formulation permits us to quantify the adjustment precision required in realistic experiments.
The structure of the article is as follows. In Section II we develop a generalized theory of Talbot-Lau interference, which is formulated independently of the particular choice of how to incorporate the grating interaction. We also establish the relation to the previous treatments by applying the eikonal approximation. In Section III, we review relevant realistic descriptions of the grating interaction, and numerically illustrate their effect in the eikonal approximation. Section IV is devoted to the development of a semiclassical formalism to go beyond the elementary eikonal approximation. Its predictions are numerically evaluated and compared in Section V, using the experimental parameters of the molecular interference experiments carried out in Vienna Brezger et al. (2002); Gerlich et al. (2007). Finally, we present our conclusions in Section VI.
Ii Phase space description of the Talbot-Lau effect
The effect of Talbot-Lau interference can be described in a particularly transparent and accessible fashion by using the phase space representation of quantum mechanics Wigner (1932); Ozorio de Almeida (1998); Schleich (2001); Zachos et al. (2005). This is demonstrated in Hornberger et al. (2004), where both the coherent effect and the consequences of environmental interactions are formulated in terms of the Wigner function of the matter wave beam. As also shown there, the analogy of the Wigner function with the classical phase space distribution allows one to evaluate the predictions of classical and quantum mechanics in the same framework, a necessary step if one wants to distinguish unambiguously quantum interference from a possible classical shadow effect.
However, the treatment in Hornberger et al. (2004) is based on a number of idealizations, which must be reconsidered in view of a more refined description of the particle-grating interaction. The least problematic approximation is to disregard the motion in the direction parallel to the grating slits, which is permissible due to the translational symmetry of the setup in this direction. It follows that a two-dimensional description involving the longitudinal motion (denoted by ) and the transverse motion (denoted by ) of the beam is required in principle. In front of the interferometer these two degrees of freedom are well approximated by a separable state involving transverse momenta . If the grating interaction is treated in eikonal approximation, as done in Hornberger et al. (2004), this implies that the transverse and the longitudinal motion remain separable throughout. One can then resort to an effectively one-dimensional description, characterized by a fixed longitudinal momentum . The coordinate then represents a time and the longitudinal propagation of the beam along effectively evolves the one-dimensional transverse beam state during the time . The finite distribution of the longitudinal momenta is then accounted for only in the end by averaging the results obtained with sharp values of .
A priori, such a treatment is no longer valid for a general grating interaction where different longitudinal momentum components of the beam get correlated. We will accordingly use a two-dimensional scattering formulation to describe the grating interaction in Sect. IV. However, we will see that for the parameters of typical experiments the main effect of taking the grating interaction beyond the eikonal approximation is on the transverse degrees of freedom. Since the effect on the longitudinal motion is much weaker we will retain the effectively one-dimensional description outlined above, postponing the physical discussion why this is permissible to Sect. IV.
Another idealization found in the basic treatments of the Talbot-Lau effect is to assume the transverse motion in front of the interferometer to be in a completely incoherent state. This would correspond to a constant distribution of the transverse momenta , and for general grating transformations, which depend on , it is no longer a valid approximation. We will therefore present a formulation that takes into account a realistic beam profile and that allows for a general state transformation at the diffraction grating. Going beyond the idealization of a perfectly incoherent state will also permit us to assess the adjustment precisions required in an experimental implementation.
ii.1 The Wigner function and its transformations
We start by briefly outlining how to describe matter wave interference by means of the phase space representation of quantum mechanics Hornberger et al. (2004); Hornberger (2006). As discussed above, one may restrict the dynamics to the transverse beam state, which is most generally specified by its density matrix . This state is equivalently described by the Wigner function
which is a function of the transverse phase space coordinates . Like it depends parametrically on the longitudinal momentum . Since we assume the beam to be collimated is non-zero only for .
The great advantage of the phase space formulation is that it permits a straightforward, yet realistic, description of the beam and its propagation through the interferometer. Most importantly, the free time evolution of a state during the time is given by the same shearing transformation as in the case of the classical phase space density,
with the particle mass. On the other hand, a quantum state transformation of the form , such as the effect of passing a grating, reads in phase space
Here the integral kernel is given by the the propagator
Specifically, an ideal grating is characterized by a grating transmission function , with , describing the multiplicative modification of an incoming plane wave. (The factor is non-zero only within the slit openings of the grating and it may there imprint a complex phase to account for the interaction potential between the grating walls and the beam particle, see Sect. III.) For such gratings the transformation reads , so that the grating propagator reduces to a convolution kernel
which is local in position Hornberger et al. (2004). This choice of will be required below to reduce the generalized formulation of Talbot-Lau interference to the eikonal approximation.
The convolution kernel provides a descriptive picture of the diffraction process. Suppose the incoming beam is perfectly coherent, i.e. a plane wave characterized by the longitudinal momentum and transverse momentum , corresponding to the (unnormalized) transverse Wigner function . The diffracted transverse state is then given by
In case of a periodic grating with period , the transmission function can be decomposed into a Fourier series,
so that, after a free propagation over the longitudinal distance , the transverse spatial density of the beam state is given by the marginal distribution
Here, we introduced the basic Talbot-Lau coefficients,
and the characteristic length scale is called the Talbot length.
Compare (9) to the Fourier coefficients of the transmission probability , which are given by the convolution . Equation (8) thus implies that the density distribution takes the form of the grating transmission profile whenever the distance is an integer multiple of the Talbot length, ,
This is the elementary Talbot effect Talbot (1836), and it is the backbone of the Talbot-Lau interferometer, which however does not require a coherent illumination.
ii.2 The general Talbot-Lau effect
The general setup of a Talbot-Lau interferometer is described in Fig. 1, along with the most relevant parameters. An incoherent particle beam represented by the shaded area passes a preliminary collimation slit on the left side of the figure and enters the Talbot-Lau interferometer through the first grating. The angular distribution of those particles in the beam which are finally detected is characterized by the spread , determined by the collimation slit and the finite area of the detector.
The first grating can be understood as an array of collimation slits of period preparing, after a distance , the transverse coherence required for diffraction at the second grating. In this region, the distortion of a matter wave front due to the grating interaction may significantly affect the interference pattern, so that the grating thickness enters the calculation. (It is replaced by the laser waist in case of a light grating.)
The different diffraction orders still overlap further downstream if the distance is on the order of the Talbot length , and they may thus interfere among each other producing a near field fringe pattern. This can be explained qualitatively from the elementary Talbot effect (10) by considering each of the slits in the first grating as independent point sources. The Talbot patterns due to the individual slits will add up constructively for appropriate choices of the grating periods , and of the distance factor , so that a distinct density pattern is created. It is verified without the need for a spatially resolving detector by superposing a third grating whose period equals that of the density pattern, and by measuring the total flux through it as a function of its transverse position .
We will now present a quantitative formulation of the successive propagation of the beam through the interferometer. As discussed above, one can take the longitudinal motion of the beam particles to remain unaffected by the gratings. We may therefore assume the longitudinal momentum to have a definite value for the time being, so that the longitudinal position plays the role of a time coordinate for the transverse motion.
ii.2.1 Sequential calculation
The transverse state of the beam entering the interferometer is far from pure. It is confined in position by the orifice of the first grating, whose size typically covers thousands of grating slits. The momenta are characterized by the angular distribution whose characteristic spread is denoted as .
In a typical experimental situation, rarely exceeds mrad corresponding to a fairly well-collimated beam. However, this still covers a range of transverse momenta that is by orders of magnitude larger than the grating momentum which separates the different diffraction orders. This explains why diffraction does not need to be taken into account at the first grating. The transverse Wigner function merely gets modulated by the grating profile, so that behind the first grating it reads
Here is the transmission function which is confined to the grating orifice, in principle. However, since the size is typically larger than the sensitive region of the final detector one may equally take it to be an unconstricted periodic function.
The free propagation of the beam over the longitudinal distance yields . The effect of passing the second grating is in general described by an operator to be specified below. The corresponding phase space transformation is given by (LABEL:eqn:generalprop) so that, after a second free propagation over the distance , the transverse beam state in front of the third grating reads
The corresponding spatial density distribution, given by , is now modulated by the third grating as a function of the grating shift . The detection signal is thus obtained as
where is the spatial transmission probability of the grating and the size of the detector. The latter typically covers hundreds of grating periods so that one may disregard the finiteness of when evaluating the basic interference effect. However, as shown below, it does play a role when the experimental adjustment precision needs to be evaluated.
The resulting interference pattern is most easily characterized by the contrast of the modulation signal, conventionally defined as the ratio between the amplitude of the signal variation and its mean value. However, since the experimental signal is usually very close to a sinusoidal curve, it is in practice more convenient and more robust to use the sinusoidal visibility defined in terms of the first to the zeroth Fourier expansion coefficient of the -periodic signal Brezger et al. (2003),
It can be easily obtained from noisy experimental data by fitting a sine curve and it coincides with the conventional definition in case of a sinusoidal signal.
ii.2.2 Decomposing the grating propagator
We proceed to evaluate the interference pattern (12) by noting a general property of the grating transformation . The periodicity of the grating implies that the position representation is -periodic with respect to the center position . This admits the series expansion
where the transformation within a single slit is now characterized by the corresponding Fourier coefficients
It is now convenient to specify these transformation functions in terms of their shifted Fourier transformation, which serves to generalize the grating coefficients introduced for the eikonal case in (7).
Indeed, for position-diagonal operators , these functions drop their momentum-dependence and reduce to the coefficients in (7).
The generalized grating coefficients (17) can now be used to construct the generalized Talbot-Lau coefficients, which are the central quantities for describing the interference effect.
We note that at integer values of the argument the phase factor in (18) reduces to a constant sign, while has the meaning of an incident momentum and that of a scale factor, as will be seen below.
By using the generalized Talbot-Lau coefficients the density distribution at the third grating takes the explicit form
This expression for the interference pattern is now further simplified by identifying the Talbot-Lau resonance condition.
ii.2.3 The resonance condition
Only a part of the double summation in (LABEL:eqn:interferencepattern1) contributes appreciably to the interference pattern. This follows from the fact that the ratio of grating distance to grating period is a large number, typically on the order of . As a result, the momentum dependence of the phase factor in (LABEL:eqn:interferencepattern1) occurs on a very different scale compared to the variation of the momentum distribution and of the Talbot-Lau coefficients . In fact, in the idealized case of both a completely incoherent illumination and an eikonal interaction the latter two functions are independent of , so that the momentum integral is finite only if the phase vanishes identically, i.e., for
It would imply that an interference pattern is observed only if is a rational number.
This strong resonance condition gets relaxed if we account for the weak momentum dependence of the remaining integrand in (LABEL:eqn:interferencepattern1). We assume that those index pairs of the double sum contribute appreciably where the phase variation in the momentum integration does not exceed about . Since the value of is bounded by the angular spread this leads to the relaxed condition
Here and are natural numbers without common divisor. They indicate the type of resonance and specify the set of index pairs contributing to the sum.
If is about mrad and the right hand side of (21) is on the order of , and in practice only a single resonance dominates for each set of parameters , , . The expression for the interference pattern thus simplifies to
Here, the period of the interference pattern is given by
Note that a large interference contrast is obtained for the low order resonances (with small values of the integers and ) because the Fourier and Talbot-Lau coefficients generically decrease with their order .
The standard choice in experiments is to take equal gratings, , in an equidistant configuration, . This corresponds to the resonance. One may obtain even larger contrasts with by using the basic resonance, at the expense of dealing with different gratings, .
Also magnifying or demagnifying interferometers can be realized Brezger et al. (2003). Consider, for example, the case where the first two gratings are equal, . The period of the interference signal is then given by and cannot thus be greater than . The condition with implies that must be strictly greater than . If one wishes to decrease , one must choose which requires , i.e., a compressed setup. However, the values of and grow with decreasing so that the visibility of the resulting interference pattern decreases significantly. A magnifying interferometer, on the other hand, can only be realized with different gratings .
If the aim is to create a high contrast interference pattern with a specific period a low-order resonance should be taken, such as or . The experimental setup parameters must then satisfy the equations
with desired resonance parameters . The solution is not unique, in general, so that one has a certain freedom to account for experimental limitations.
ii.3 The Talbot-Lau effect in eikonal approximation
The expression for the interference pattern can be further simplified if the grating interaction is treated in eikonal approximation. The grating coefficients (17) turn then into the momentum-independent Fourier coefficients of the transmission function (7). Similarly, the generalized Talbot-Lau coefficients (18) then reduce to the basic coefficients given in Eq. (9) so that the prediction for the density pattern (LABEL:eqn:interferencepattern1) simplifies to the series
It involves the Fourier transformation of the angular distribution,
It should be emphasized, though, that in order to establish that an observed fringe pattern is really due to a quantum effect one must compare the quantum prediction with the result of the corresponding classical calculation. This is necessary since a classical moiré-type shadow pattern might give rise to a similar observation, even tough the classical contrast is typically much smaller. It is shown in the appendix how the classical treatment can be formulated in the same framework as the quantum case using a phase-space description.
Finally, since Eq. (28) assumes the resonance condition to be exactly met, it cannot be used to assess the adjustment precision required in the experiment. As shown in the next section, one can evaluate the necessary precision in the eikonal approximation by explicitly taking into account the finite transverse momentum spread and the finite size of the signal detector.
ii.4 Adjustment requirements
In order to allow for deviations from the exact resonance condition we use Eq. (26) when evaluating the expression for the detection signal. According to Eq. (13) the detection signal is then characterized by the Fourier coefficients
Here, the are the Fourier expansion coefficients of the third grating profile with the period . Consider now small deviations from the setup parameters satisfying a particular Talbot-Lau resonance condition . The distances and can thus vary independently. Instead of approximating the characteristic function by a Kronecker-, we account for the small deviations by the refined approximation . It holds as long as the main argument is either zero or much greater in modulus than the width , and as long as .
One can thus split the argument of in (29) into the ideal resonance part and in the part containing the parameter deviations. The same procedure may be done with the sinc term, which is sharply peaked if the detector size exceeds the period by orders of magnitude. If the small parameter deviations are taken into account to first order one obtains the same resonance relation for the summation indices in (29) as in the ideal case, while the th Fourier expansion coefficient of the signal (with respect to the period ) is multiplied by the reduction factor
The absolute value of this factor is less than for so that the contrast of the detection signal is effectively reduced by the deviations. In particular, the sinusoidal visibility (14) of the signal is reduced by the factor . Typical experiments Brezger et al. (2002, 2003); Hackermüller et al. (2003); Gerlich et al. (2007) are characterized by , and so that already relative parameter deviations on the order of strongly affect the interference contrast.
To obtain a simple and conservative estimate we treat all imprecisions as independent, specializing to the standard case , where the grating distances and grating periods are equal. Bounds for the adjustment precisions are then obtained from Eq. (LABEL:eqn:reductionfactor) by requiring the arguments to be smaller than the widths of the momentum distribution and of the sinc function, respectively,
This quantifies to what degree a better collimation of the beam and a smaller detector size relax the required adjustment precision, albeit at the expense of a loss of signal.
Iii Effect of the interaction potential
The preceding section showed how the general coherent state transformation effected by a diffraction grating enters the Talbot-Lau calculation via the generalized grating coefficients (17). In general, this transformation is determined by the potential due to the long-range dispersion forces acting on the beam particles while they pass the grating structure.
Before we present a general way to account for the presence of this potential in Sect. IV it is helpful to discuss the most important grating-particle interactions in a simpler form, by using the eikonal approximation. It treats the grating as the combination of an absorption mask and a phase modification, and it can be characterized by a grating transmission function
whose amplitude describes the grating structure. Interaction-free gratings are modeled by a grating transmission function without phase, , while pure phase gratings, such as the standing laser field, wave are characterized by . The corresponding eikonal interference pattern is then obtained immediately as described in Sect. II.3.
We proceed with a short overview of the typical grating potentials, discussing how they affect the interference contrast in the eikonal approximation. By convention, the diffraction grating is located at the longitudinal position , as indicated in Fig. 1.
iii.1 Material gratings
A neutral particle located within the slit of a material grating experiences a potential determined mainly by the attractive dispersion forces due to the grating walls. Other forces, such as the exchange interaction or the electro-static attraction due to a permanent dipole, are much less important for interferometry because they either occur only at very close distances or because they are diminished by rotational averaging. In any case, for all positions within a slit the walls are well approximated by an infinite surface in the -plane Grisenti et al. (1999); Brühl et al. (2002). The grating potential is thus set to be solely -dependent and acting only within the time of passage throughout the grating of thickness , while it is set to be zero outside.
In general, the dispersion force between a polarizable particle and a material plane is described by the expression of Casimir and Polder Casimir and Polder (1948) and its generalizations, e.g. Wyl ; Buhmann et al. (2005). In the close distance limit it reduces to the van der Waals potential , with the distance to the surface, and at large distances, where retardation plays a role, one has the asymptotic form . The interaction constants are determined by the frequency dependent polarizability of the particle (as well as the dielectric function of the grating material), and the regime of validity of the limiting forms is delimited by the wavelengths corresponding to the strong electronic transitions Derevianko et al. (1999); Brühl et al. (2002); Madroñero and Friedrich (2007).
In any case, the dispersion force diverges on the grating walls, rendering the eikonal approximation invalid in close vicinity to the walls. Since the contributions of these regions do not alter the interference contrast appreciably, but lead to numerical noise, we will discuss a reasonable criterion to blind out the beam close to the grating walls in Section IV.
Figure 2 shows the interference visibility (14) for a typical Talbot-Lau experiment with C fullerene molecules of mass Brezger et al. (2002), plotted versus their de Broglie wavelength. All gratings are assumed to be separated by the distance and to be made of gold with a period of , a slit width of , and a thickness of . One can see two peaks corresponding to the first and second Talbot order, at and at , respectively. The solid line represents the eikonal approximation with a retarded asymptotic potential ( with the static polarizability Å obtained via the Clausius-Mossotti relation Compagnon et al. (2001)), while the dashed and the dotted lines correspond to the van der Waals potential () and to the absence of an intra-slit potential, respectively. One observes that the presence of the dispersion forces changes the interference characteristics significantly. The asymmetry in the double-peak structure compared to the interaction-free case is due to the fact that the particles with a larger velocity, i.e., with a smaller wavelength , receive a smaller eikonal phase than the slower particles. Moreover, note that the effect of retardation has a small but visible influence on the fringe contrast.
The dash-dotted line in Fig. 2 represents the moiré-like effect as expected from classical mechanics. It was calculated in a phase space formulation as described in the appendix, using the classical correspondence of the eikonal approximation with the -potential. As one expects, the dependence of the classical result on the fictitious “wavelength” , which is due to the velocity dependence of the classical deflection, differs strongly from the quantum results.
We emphasize that all numerical results presented in this paper are obtained for a fixed longitudinal velocity of the particles. A comparison with the experiment still requires the results to be averaged with respect to the velocity distribution in the beam. This may pose a severe restriction when using particles with larger polarizability because the increased dispersive interaction decreases the width of the double-peaks seen in Fig. 2 significantly Gerlich et al. (2007). One way to avoid this is to replace the material diffraction grating by a standing laser wave.
iii.2 Laser gratings
Recently, a Kapitza-Dirac-Talbot-Lau interferometer (KDTLI) was demonstrated, where the central grating is formed by a standing light wave Gerlich et al. (2007). In the eikonal approximation the phase of an incoming plane wave is modulated according to the potential created by the off-resonant interaction with the standing laser beam. It is determined by the energy of the induced electric dipole in the oscillating field, and is therefore proportional to the laser power and to the dynamic polarizability of the beam particles at the laser frequency,
Here, and are the waists of the Gaussian mode, and is chosen large compared to the detector size in order to guarantee a regular grating structure. We therefore disregard the -dependence by setting . Moreover, for sufficiently small an effective one-dimensional treatment of the potential is permissible, where one replaces the -dependence by a parametric time dependence , as discussed below in Sect. IV.3.
The -integration in (33) renders the eikonal phase independent of , which already indicates that the elementary eikonal approximation will cease to be valid if the laser waist is increased. Nonetheless, it yields an interference contrast that fits well to the measured data of the recent fullerene experiments Gerlich et al. (2007). Moreover, it admits a simple, closed expression for the Talbot-Lau coefficients (9),
where is the mass of the beam particles and stands for the th order Bessel function of the first kind Abramowitz and Stegun (1965).
The solid line in Figure 3 shows the visibility (14) for the Viennese KDTLI with fullerenes Gerlich et al. (2007) as obtained from (35). In contrast to Figure 2, the visibility drops to zero whenever the wavelength corresponds to an integer multiple of the Talbot condition. This is explained by the fact that in the elementary Talbot effect (10) a pure phase grating with no absorptive walls leads to a constant density at multiples of the Talbot length. At the same time, there are broad regions of high contrast. They render the KDTLI setup superior to a material grating interferometer for particles with a high polarizability and a substantial velocity spread. Moreover, the interaction strength and the grating period can be tuned in a KDTLI rendering the laser grating the preferable choice to explore the validity regime of the eikonal approximation, as done in Section V.
The dashed line in Figure 3 represents the classical version of the eikonal calculation as described in the appendix. Note that it differs significantly from the quantum result in the experimentally relevant wavelength regime, while the curves become indistinguishable for . Finally, it should be emphasized that a realistic description of the laser grating must also account for the possibility of photon absorption. The effect of the resulting transverse momentum kicks can be incorporated into the eikonal approximation by replacing the grating transformation kernel by a probabilistic sum of such kernels Gerlich et al. (2007).
Iv Semiclassical approach to the grating interaction
The de Broglie wavelength is by far the smallest length scale in the matter wave interference experiments considered in this paper. It seems therefore natural to use a semiclassical approach for calculating the grating transformation used in (16). We will show how the elementary eikonal approximation (33) can be derived from the appropriate semiclassical formulation if a high-energy limit is taken. Moreover, one can obtain a refined eikonal approximation, where an incoming plane wave is still merely multiplied by a factor. Since this factor depends on the transverse momentum of the incoming wave it renders the numerical implementation more elaborate.
However, the main result of this section is the expression (66)–(68) for the scattering factor which approximates the scattering transformation of a transverse plane wave in the semiclassical high-energy regime. As such it permits to evaluate the generalized grating coefficients (17) straightforwardly, see Eq. (39). As shown in Section V, this result outperforms the eikonal approximation and it may serve to characterize its regime of validity. The derivation is based on the semiclassical approximation of the two-dimensional time evolution operator determined by the grating interaction potential .
iv.1 The scattering factor
We proceed to incorporate the grating interaction by means of the formalism of scattering theory Taylor (1972). Its basic tool is the scattering operator defined as
with the free time evolution operator for the motion in the -plane and the complete time evolution operator, which includes the grating potential . It is pertinent to use instead of since the scattering operator transforms the state instantaneously leaving the asymptotic dynamics to be described by the free time evolution. Since the latter is easily incorporated using Wigner functions this fits to the phase space description of Sect. II.2, where the initial beam state entering a Talbot-Lau interferometer is propagated freely to the second grating before the scattering transformation is applied. In the expression (LABEL:eqn:generalprop) for the grating propagator, which serves to calculate the interference pattern (22), one may thus use the S-matrix (36) in place of the general unitary operator .
In the basis of the improper plane wave states the scattering operator (36) is conveniently described by the scattering factor
Notice that acts in the Hilbert space defined on the two-dimensional plane . However, as will be justified below, the longitudinal motion may be separated and treated classically so that the transformation is confined to the transverse dimension, while the -coordinate turns into an effective time coordinate for a given longitudinal momentum . The brings about the reduced scattering factor
which depends parametrically on and which is evaluated at the position of the center of the diffraction grating.
The reduced scattering factor describes the phase and amplitude modification of a transverse plane wave with momentum due to the grating interaction. It enters the Talbot-Lau calculation via the Fourier coefficients (16) after an expansion in the plane wave basis. It follows that the generalized grating coefficients (17) are directly related to by a Fourier transformation,
The calculation of the Talbot-Lau interference thus reduces to evaluating the scattering factor (38).
iv.2 Semiclassical calculation
We proceed to calculate the scattering factor (38) by means of the semiclassical asymptotic approximation. It assumes the action of those trajectories through the interaction region, which contribute to the path integral for its position representation, to be much larger than , and the particle wavelength to be much smaller than the scale where the interaction potential changes appreciably, ). These conditions are well satisfied in the typical experimental situation if we disregard the regions very close to the gratings where the potential exceeds the kinetic energy. We may thus approximate the time evolution operator in the position representation by the semiclassical van Vleck-Gutzwiller propagator Gutzwiller (1967)
Here, is the action of the classical trajectory travelling during time from the position to . In general, there might be more than one such trajectory, which would require taking special care of almost coalescing trajectories and of the associated Morse index. However, in the interferometric setup we are in the high-energy regime, where the interaction potential is much weaker than the energy of the incoming particles, . All relevant contributions are therefore characterized by a single, slightly deflected classical trajectory passing the interaction region.
It is now convenient to specify this trajectory in terms of the deviation from the undeflected straight line, as specified by the initial position and momentum . The momentum change after time is given by
so that the deflected trajectory reads
In the van Vleck-Gutzwiller propagator (40) the contributing trajectory is specified by the boundary values and . For the following calculation it is important to rewrite it as an initial value problem, specified by the initial phase space point of the trajectory Heller (1991).
The phase of the integrand is given by the action-valued function
while the amplitude is determined by the stability determinant of the associated trajectory.
The semiclassical van Vleck-Gutzwiller propagator (40) used here is a stationary phase approximation of the time evolution operator in path integral formulation Gutzwiller (1967). To remain at a consistent level of approximation it is therefore necessary to evaluate the phase space integral in (45) in the stationary phase approximation as well Bleistein and Handelsman (1975). The stationary point of the phase is determined by the condition
Noting the initial value derivatives of the classical action
and using the fact that the matrix is invertible for our trajectories, the stationary phase condition leads to the equations
They serve to determine the initial position of the trajectory implicitly. Using the general formula for a four-dimensional integral, e.g. (Hornberger and Smilansky, 2002, Eq. (A.30)), the 2d scattering factor (45) then takes the form
The amplitude modification
is obtained, in a tedious but straightforward calculation, by using the Poisson relation between the conjugate variables and of the trajectory Landau and Lifshitz (1960) when evaluating the product of two determinants. The matrix of derivatives in (53) can be computed by taking the initial value derivative of the equation of motion for the trajectory and solving the resulting ordinary differential equation.
Figure 4 shows how the stationary point can be understood from the point of view of a classical scattering trajectory. Given the phase space coordinates determining the matrix element of the scattered plane wave, the momentum fixes the initial momentum of the classical trajectory passing through the interaction region. The position determines the final position of the deflected trajectory, which is obtained after a free motion during time in a direction given by the final momentum of the trajectory. This free evolution ensures that the associated action (46) and stability amplitude (53) is independent of in the limit . Note that Fig. 4 overemphasizes the effect of the potential, since in all interferometrically relevant situations the deflection of the classical trajectories will be only a small correction to the free rectilinear path.
The limit in the semiclassical result (52) is obtained already at a finite time if the interaction potential may be considered to have a finite range. The relevant scale is the grating passage time, given by for material gratings and for laser gratings, respectively. This follows from the fact that outside of the grating potential the classical trajectories remain rectilinear, so that neither the stationary phase condition (50) nor the stationary phase and amplitude are affected by a further increase in .
iv.3 Iterative solution in the momentum deflection
We will now give explicit approximate solutions for the semiclassical scattering factor (52) which are valid in the high-energy limit, i.e., whenever the longitudinal kinetic energy of the incoming particles is large compared to the interaction potential . In general, one has to solve Eq. (50) for the initial value . Rewriting it in terms of the expressions (42) and (43) for the positions and momenta of the trajectory, one obtains an implicit equation for which may be solved iteratively.
iv.3.1 The Glauber eikonal approximation
In the semiclassical short interaction time and high-energy limit, which applies to typical interference experiments, the transverse momentum deflection is so weak that its contribution can be neglected. This corresponds to the zeroth order solution of Eq. (54),
In order to keep the approximation consistent one has to neglect the deflection in the expressions for the phase modification (46) and for the amplitude modification (53) as well, by setting the classical trajectory to be the free rectilinear path. Consequently, there is no amplitude modification and the scattering factor (52) reads
This is the Glauber eikonal approximation obtained by R. J. Glauber from the Lippmann-Schwinger equation in a quite different argumentation Glauber (1959). The reduced scattering factor (38) for the transverse dimension then reads
The elementary eikonal approximation (33) follows from this expression in the limit of vanishing transverse momentum, . It applies in the case of a well-collimated beam and a small grating thickness , i.e., .
iv.3.2 The deflection approximation
The Glauber eikonal approximation ceases to be valid as the interaction strength or interaction time increases, and one has to go to the first order in the momentum deflection when evaluating the stationary initial value (54),
The higher order terms, which are neglected here, involve derivatives of the momentum deflection .
The time-evolved trajectory starting from this initial value is approximated, again to first order in , by
This trajectory must be used when calculating the action (46) in order to ensure that all expressions are evaluated to the same order in the deflection . Since this holds also for the time integral a st order Taylor expansion of the potential is required. In total this yields
On the other hand, when evaluating the amplitude (53) the zeroth order solution of the initial value (55) must be used instead of (58) because the classical equation of motion for the matrix of initial value derivatives is governed by the derivative of the interaction force rather than by the force itself. Accounting for the modification (58) in calculating the stability determinant would therefore amount to a higher order correction in the momentum deflection. It follows that the amplitude factor is consistently approximated by
iv.3.3 Separation of the longitudinal motion
Based on the deflection approximation of the stationary scattering factor we may now derive a reduced scattering factor (38) that can be incorporated into the one-dimensional Talbot-Lau calculation and that significantly extends the validity regime of the eikonal approximation. It is necessary for this purpose that the longitudinal motion remains unaltered. To a very good approximation this is indeed the case for the small trajectory deflections required above in the deflection approximation. This is due to the energy conservation during the scattering process, ensuring where is the incoming momentum and the total deflection (41). It follows that a small transverse deflection yields in a well-collimated beam, , a total longitudinal deflection which is much smaller than . The longitudinal part of the classical trajectory (IV.3.2) may thus be treated as a free motion with constant momentum ,