Open Access
Add to BibSonomyAbstract
We examine the free time evolution of a rectangular one dimensional Schrödinger wave packet of constant phase during the early stage which in the paraxial wave approximation is identical to the diffraction of a scalar field from a single slit. Our analysis, based on numerics and the Cornu spiral reveals considerable intricate detail behavior in the density and phase of the wave. We also point out a concentration of the intensity that occurs on axis and propose a new measure of width that expresses this concentration.
© 2012 Optical Society of America
1. Introduction
The diffraction of a wave from a single-slit is a well-known phenomenon that has been verified experimentally for light as early as 1665 by Francesco Grimaldi [1] but in modern times it has been observed also with electrons [2,3], atoms [4], matter waves in the form of neutrons [5], and recently with molecules [6]. So far the analysis of this phenomenon has mainly concentrated on the near-field, or the far-field of the diffraction pattern. In the present article we illuminate the intermediate domain and point out three effects that to the best of our knowledge have not been noted before: (i) an intricate intensity and phase pattern in space and time, (ii) a self-similarity, and (iii) a narrowing of the wave packet interpreted as a focusing effect.
Needless to say, there has been an exhaustive literature on the diffraction of a single-slit. Most relevant to the present dicussion is the phenomenon [7] of diffraction in time, which arises in the time evolution of a plane wave restricted initially to the half-plane corresponding to the diffraction by only one edge. In contrast, we consider a slit defined by two edges. Moreover, [8] contains cuts of the reported intensity distribution at specific times, but does not analyze the continuous time evolution. Also, noteworthy is Ref. [9], which discusses the diffraction from the single-slit but does not address the phenomena reported in the present article.
The focusing necessary to achieve narrow structures in quantum lithography [10, 11] is an important application in the manipulation of matter waves. Examples of mechanisms through which wave packets are focused include the use of lenses [12], the realization of Talbot carpets with diffracting gratings [13, 14] and the creation of appropriate guiding potentials [15]. Particularly, noteworthy in this context is also the use of NOON-states [16] or even classical light fields [17]. The focusing reported in the present article is fundamentally different in the sense that it arises in the free evolution of a properly chosen wave packet due to its phase space correlations [18–21]. It should be noted that this effect is fundamentally different from the Poisson spot [1], recently found also in the context of molecular beams [22]. The effects discussed here are also different from the ones related to near-field microscopy [23, 24] since they do not deal with evanescent waves.
We emphasize that in the field of microscopy attempts to surmount the diffraction limit in experiments involving lenses have led to the investigation of an appropriate spatial shaping of beams [25] as well as the existence of beams which do not suffer diffraction due to a suitable preparation of packets [26]. Interestingly, our example shows that near-field focusing and diffraction are not mutually exclusive concepts.
Our article is organized as follows. Section 2 contains our main results. Here, we first present the density and phase patterns of the near-field by numerically evaluating the exact expression of the wave function in terms of Fresnel integrals. We then take advantage of the connection between the Fresnel integrals and the Cornu spiral to explain the main features of these patterns. Here we focus on three features: i) the formation of maxima along parabolas, ii) the concentration of intensity corresponding to a focusing of the wave on the optical axis at a distance z ≈ a2/3λ where a and λ are the width of the slit and the wavelength of the incident beam, respectively, and iii) a phase plateau around this focus. To quantitatively describe this focusing effect we introduce a new measure. We conclude in section 3 with a summary and an outlook.
2. Near-field pattern
Throughout the paper we concentrate on the one-dimensional motion of a particle of mass m along the x-axis, governed by the Schrödinger equation in the absence of a potential. However, we emphasize that this problem is equivalent to the diffraction of a scalar field from a single slit described in the paraxial approximation [27].
2.1. Probability density and phase
We consider the time evolution of an initial rectangular wave packet of length a according to the Schrödinger equation of a free particle. For this purpose it is convenient to introduce dimensionless variables such as position χ ≡ x/a and time τ ≡ ht/(ma2) where h is Planck’s constant. For the diffraction problem τ translates into the distance z from the plane of the slit by the substitution law τ = zλ/a2.
In terms of these dimensionless variables our initial normalized wave function
where Θ denotes the Heaviside step function, evolves according to the free-particle propagator [28] and the wavefunction at τ reads which can be expressed in terms of the standard Fresnel integrals [29] in the form giving rise to This expression can be evaluated numerically to give the near-field probability density |ψ|2 and the phase pattern of |ψ| in the τ − χ plane shown in Figs. 1(a) and 1(b), respectively. We observe a series of maxima culminating in a dominant one near τ = 1/3 and χ = 0 where the intensity is 1.8 times the one within the slit. In the neighborhood of this point, the phase is approximately constant in space. On the way towards the slit a structure of ever increasing complexity and fineness emerge.
Fig. 1 Near-field patterns (top) originating from the time evolution of a rectangular wave packet, and their explanation (bottom) with the help of the Cornu spiral. Here we depict the probability density (a) and the phase (b) of the wave function ψ given by Eqs. (4) and (5) in their dependence on the dimensionless time τ and coordinate χ. The white line located at τ = 1/3 indicates both the dominant peak of the probability density corresponding to the focusing of the wave packet and the main plateau of the phase of ψ. The Cornu spirals of (c) and (d) show the complex-valued function F(w) = FC(w) + iFS(w) where the arc length of the curve is parameterized by the argument w. Maxima of |F(w)| correspond to positions on the spiral where the separation from the origin assumes a local maximum. The blue arrows in (c) indicate the first two maxima with n = 0, 1. The corresponding maxima in terms of the space-time variables τ and χ are given by Eqs. (12) and (13). The red arrows indicate maxima of the phase of F(w), which coincide with maxima in the phase of ψ for χ = 0; again we only show the first two corresponding to k = 0, 1. The central arrow in (d) labeled 1 shows the approximate location of the intensity maximum at τ = 1/3 and χ = 0. Arrows 2 and 3 represent arguments of F for τ = 1/3 but with nonzero χ. Since the arguments of the two Fresnel integrals in Eq. (5) defining the wave function ψ are , we see that the two contributions are symmetrically placed around χ = 0 and thus ψ has a smaller amplitude but a similar phase compared to ψ at χ = 0.
2.2. Cornu spiral essentials
Insight into the structures shown in Figs. 1(a) and 1(b) springs from the well-known Cornu spiral [1,30] representing the Fresnel integral F. The complex value F(w) is given either by the real and imaginary parts FC(w) and FS(w) of the point on the spiral depicted in Fig. 1(c) and 1(d) with arc length w, or by the separation |F(w)| of this point from the origin together with the angle with respect to the real axis. In order to provide the background for our analysis of Figs. 1(a) and 1(b) we now briefly review the essential properties of the Cornu spiral.
From Fig. 1(c) we recognize that this separation from the origin first increases as we start from the origin and move along the spiral. It reaches its largest value when the angle with respect to the horizontal axis is approximately π/4 as indicated in Fig. 1(c) by the blue arrow marked n = 0. At this point we have the identity FC(w) = FS(w).
When we increase w further and thus continue on the spiral the separation |F(w)| decreases till we have reached again an angle of approximately π/4. Now we have reached a minimum of |F(w)|.
For even larger values of w the separation and thereby |F(w)| increases again and we obtain another maximum when the angle of the arrow assumes the value π/4. However, due to the spiral nature of the curve this maximum is now smaller than the first one. This second maximum is marked in Fig. 1(c) by n = 1.
This process of consecutive maxima and minima continues forever as we increase w. However, the values of |F| at these extremal points approach a common limit as we spiral into the center.
These geometrical considerations show that extrema, that is maxima and minima of |F(w)| occur for points on the spiral with FC = FS which yields with the help of the identity
and the definition Eq.(4) of FC and FS the condition Hence, |F| assumes extrema for argumentsIndeed, for l = 0 we find the largest maximum, for l = 1 the lowest minimum, for l = 2 again a maximum and so on. Thus the arguments wn corresponding to maxima of |F| read
where n = 0, 1, 2, 3.... The two largest maxima determined by n = 0 and n = 1 are depicted in Fig. 1(c).Likewise, the minima of |F| are given by
with m = 0 and m = 1 providing us with the smallest values.We conclude this brief summary of the Cornu spiral by discussing the phase of the Fresnel integral F given by the angle with respect to the horizontal axis. After the first maximum of |F| this angle and thus the phase oscillates around π/4 with a decaying amplitude. The two largest values of the phase are indicated in Fig. 1(c) by the two red arrows marked k = 0 and k = 1.
2.3. Application to the single-slit
We now apply the properties of the Cornu spiral reviewed in the preceding section to explain the patterns in the probability density and the phase shown in Figs. 1(b) and 1(c). In particular, we explain the origin of the focusing effect and the associated phase plateau.
According to Eq. (5) the wave function ψ is given by the sum of two Fresnel integrals at different arguments. If we neglect, for the moment, the interference between these contributions, the maxima of |ψ| will be determined by the maxima of the two Fresnel integrals with arguments
With the help of the expression Eq. (9) for the location of the maxima of |F(w)|, the individual contributions in Eq. (5) have a maximum at and respectively.So far we have neglected interference between the two contributions in Eq. (5). We now discuss its influence and start with the case χ = 0, that is the behavior along the optical axis. Here the two terms in Eq. (5) are identical and the interference is obviously constructive, generating a series of maxima at τ = 1/(3 + 8n), the largest of these occurring for τ = 1/3. This point corresponds to the focus, where the central peak is narrowest.
Next we turn to the explanation of the phase plateau in the neighborhood of the focus. For this purpose we note that when we move off the optical axis the arguments and of the two Fresnel integrals are represented by two vectors symmetrically located about the maximum at χ = 0, as shown in Fig 1(d). Thus we expect that their sum gives a reduced amplitude but with the same phase.
The maxima described by Eqs. (12) and (13) lie along parabolas. This is shown in Fig. 2, together with an intensity pattern similar to Fig. 1(a) but on a finer time scale. There we see that the curves in question seem to emerge from the original edges of the square packet, located at χ = ±1/2 and τ = 0. For τ > 0, the curves propagate in the region between the edges and intersect at many points. This gives a clear comparison of the intensity pattern with an emerging network of blue and red curves coming from left and right edges respectively. The points of intersection coincide with the bright regions of the pattern.
Fig. 2 Probability density |ψ|2 of a freely propagating rectangular wave packet as a function of the space-time variables τ and χ, similar to Fig. 1(a) but on a finer time scale close to the origin. Light and dark colors represent high and low densities, respectively. Superimposed are the parabolas of Eq. (12) and Eq. (13) indicated by blue and red lines, respectively. At the crossings of the parabolas maxima of the intensity pattern occur.
The crossings of the parabolas on the line χ = 0 are the maxima at τ = 1/(3 + 8n) already noted. Crossings of these lines for χ ≠ 0 are relative maxima of the terms for different n-values. But here we again refer to the graph of the Cornu spiral and note that the two terms in Eq. (5) have the same phase and thus the sum is also a maximum. The probability amplitude Eq. (4) is a maximum at these points.
Further consideration of the spiral reveals that minima occur between the maxima and, as shown in Fig. 1(c), the maxima of the phase of the field occurs midway between these maxima and minima of the field amplitude. This feature can clearly be seen by comparing the panels (a) and (b) of Fig. 1. A corollary of this property is the relatively stationary phase of the field at maxima and minima of the field intensity.
2.4. Focusing effect
One particularly striking feature of the near-field pattern of Fig. 1(a) is the strong concentration of the probability density on the optical axis in the region between 0.2 and 0.5. This concentration is interesting since it is well-known that for a real-valued symmetric initial wave function, the expectation value 〈x2〉 increases quadratically with time [20]. The reason for this behavior is that the second moment puts too much weight at large values of the coordinate. The family of Gaussian widths
avoids this misrepresentation. It is parameterized by the constant κ with dimensions of inverse length and the expectation value is performed with respect to the quantum mechanical probability density |ψ|2. In the limit of vanishing κ our measure reduces to the second moment. Moreover, it increases monotonically as a function of time when the initial wave packet is a Gaussian - a feature that any bona fide definition of a measure of width should have.In the case of our rectangular wave function, Eq. (1), we obtain the value of 𝒲 by substituting ψ(χ, τ) given by Eq. (5) into the definition Eq. (14) and performing the average numerically. The relevant length parameter is given by the dimensionless product κa. Here it is convenient to compare 𝒲 with its original value at τ = 0 in the form of a normalized width δ𝒲 ≡ 𝒲(κ, τ)/𝒲(κ, 0). The numerical results are shown in Fig. 3 with a plot of δ𝒲 as a function of κa and τ. The resulting surface clearly shows that for each cut of the form κa = constant there is a minimum at τ ≈ 1/3 indicating the focusing point. The plot in Fig. 3 also shows that there is a value of κa for which the measure yields a global minimum. Therefore, the best way to capture the shrinkage of the packet is by choosing the extremal point corresponding to κa ≈ 4.5. For τ > 1/3, i.e. after the focus, the measure δ𝒲 increases due to expansion in agreement with Fig. 1(a).
Fig. 3 Family of normalized Gaussian widths δ𝒲 ≡ 𝒲 (κ, τ)/𝒲(κ, 0) of the freely-propagating initial rectangular wave packet as a function of the dimensionless time τ ≡ ht/(ma2) and the parametrization κa of the measure. For the optimal parameter κa ≈ 4.5 and τ ≈ 0.3 a global minimum, corresponding to the focused probability peak occurs, in complete agreement with the numerical evaluation of the time-dependent probability density shown in Fig. 1(a) and the analytical considerations of section 2.3. This choice of the parameter κ indicates a maximal shrinkage of the width and therefore represents the best way of capturing focusing.
One might suspect that the unusual features in the probability density such as the focusing and the fine structure at early times shown in Figs. 1(a) and 2 crucially depend on the rectangular shape, and in particular, on the discontinuous nature [31] of the initial wave function. In Fig. 4 we address this question by studying numerically the time evolution of a constant distribution with soft edges. This initial wave function emerges when we divide a Gaussian of width Δχ at the center, separate the two parts and put a constant distribution between them. The four panels of Fig. 4 show the resulting spatio-temporal distributions for four increasing values of the width Δχ, that is for decreasing sharpness of the rectangular distribution. Indeed, we eventually lose the features characteristic for the rectangular distribution such as the focusing and the fine strucure but it is remarkable how robust they are for small values of Δχ. We can explain Fig. 4 with more detail by noting that the slope of the soft edges decreases faster than exponentially with an increasing width Δχ. The upper panels show that the focusing point in the red region is visible after increasing Δχ from 1/100 (left panel) to 1/10 (right panel), while the intensity peak decreases its value from 1.8 to 1.7. On the other hand, the lower row of Fig. 4 shows that increasing Δχ from 1 (left panel) to 10 (right panel) destroys the focusing effect as the maximum intensity decreases from 1.4 to 1.
Fig. 4 Influence of sharp edges on the spatio-temporal probability density patterns generated by flat distributions with Gaussian edges. In each frame a different slope is achieved by varying the width Δχ of the Gaussian: Δχ = 1/100 (top left), Δχ = 1/10 (top right), Δχ = 1 (lower left), and Δχ = 10 (lower right). For small values of Δχ, that is for large slopes at the edges we recover the focusing peak and the intricacies of the pattern, while for soft edges, that is for increasing values of Δχ the details near the slit disappear and the focusing effect is mitigated.
3. Conclusions and outlook
In summary, we have shown that a freely-evolving rectangular wave packet of constant phase exhibits a rich intensity and phase pattern in the near-field region. In particular, we have pointed out a rather counterintuitive effect: the wave packet first contracts before it undergoes the familiar spreading. We have explained this focusing effect using the Cornu spiral representation of the Fresnel integrals determining the wave function. However, we emphasize this phenomenon can also be traced back [32] in the framework of the Wigner function formulation of quantum mechanics to non-classical correlations between momentum and position in quantum phase space contained in the initial rectangular wave packet. In this sense, this focusing is not due to phase factors imprinted for example by a lens, but rather due to the internal structure of the initial state.
It is also remarkable that in the focal plane the phase is approximately constant around the peak. This feature would allow us to place a second slit of half the width of the original one at the focal plane and to achieve in this way an additional focusing. This process could be continued by positioning more slits whose widths are reduced by factors of two at distances decreasing by factors of four due to the a2-dependence of the focusing effect. Unfortunately this method of focusing without a lens but by selecting takes a toll on the integrated intensity.
It is also illuminating to think of experiments to verify this effect. Here, two realizations offer themselves.
With classical light of wavelength λ the distance z from the slit to the main intensity peak is determined by the relation
This condition, together with the one to observe diffraction a/λ ≫ 1, leads us to the estimate for the location of the focus. For the G-line of a Mercury vapor lamp with 435.8nm, this estimate gives the condition a > 138.7 nm. Preliminary results of an experiment along these lines have been obtained [33].A second possibility to create an experimental situation in which the effects discussed in this article can take place emerges in the context of Bose-Einstein condensates. The preparation of a square packet is possible [34] through a careful control of the confining potential within the Thomas-Fermi approximation, while the free Schrödinger evolution can be achieved by controlling the scattering length of the atoms in question with the help of a Feshbach resonance [35].
Acknowledgments
We are most grateful to M. Arndt, N. Gaaloul, M. Goncalves, G. Leuchs, E. M. Rasel, E. Wolf and M. S. Zubairy for stimulating discussions on this topic. As part of the QUANTUS collaboration, this project was supported by the German Space Agency DLR with funds provided by the Federal Ministry of Economics and Technology (BMWi) under grant number DLR 50 WM 0837.
References and links
1. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, London, 1964).
2. G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys. 155, 427–474 (1959). [CrossRef]
3. C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys. 42, 4–11 (1974). [CrossRef]
4. J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys. 37, 905–912 (1969). [CrossRef]
5. A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys. 60, 1067–1073 (1988). [CrossRef]
6. O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys. 71, 319–325 (2003). [CrossRef]
7. In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625–631 (1952). [CrossRef]
8. J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis. VI.3, 105–115 (1957).
9. M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch. 56a, 173–177 (2001).
10. M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter 15, R233–R255 (2003). [CrossRef]
11. W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett. 102, 070501 (2009). [CrossRef] [PubMed]
12. T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B. 54, 375–379 (1992). [CrossRef]
13. W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express 17(23), 20966–20974 (2009). [CrossRef] [PubMed]
14. A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A 71, 043612 (2005). [CrossRef]
15. See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett. 88, 083601 (2002). [CrossRef] [PubMed]
16. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000). [CrossRef] [PubMed]
17. Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett. 105, 183601 (2010). [CrossRef]
18. I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett. 89, 060404 (2002). [CrossRef] [PubMed]
19. M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen. 36, 7113–7128 (2003). [CrossRef]
20. K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics 375, 133–143 (2010). [CrossRef]
21. R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt. 57, 1437–1444 (2010). [CrossRef]
22. T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A 79, 053823 (2009). [CrossRef]
23. L. Novotny, “The history of near-field optics” in Progress in Optics vol. 50, E. Wolf, ed. (Elsevier, Amsterdam, 2007) pp. 137–184. [CrossRef]
24. D. Courjon, Near-field Microscopy and Near-Field Optics (World Scientific Publishing, Singapore, 2003). [CrossRef]
25. T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001). [CrossRef]
26. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]
27. The paraxial approximation is expected to hold as long as the wavelength is much less than the slit width.
28. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
30. R. W. Wood, Physical Optics (Optical Society of America, Washington, 1988).
31. M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen. 37, 9549–9556 (2004). [CrossRef]
32. E. Sadurní, W. B. Case, and W. P. Schleich, in preparation.
33. M. Gonçalves (personal communication, 2011).
34. A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A 82, 023613 (2010). [CrossRef]
35. T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B 79, 1013–1019 (2004). [CrossRef]
