Boundary bound diffraction: A combined spectral and Bohmian analysis
BBoundary Bound Diffraction: A combined Spectraland Bohmian Analysis
Jalal Tounli, Aitor Alvarado and ´Angel S. Sanz
Department of Optics, Faculty of Physical Sciences,Universidad Complutense de Madrid,Pza. Ciencias 1, Ciudad Universitaria E-28040 Madrid, SpainE-mail: [email protected]
Abstract.
The diffraction-like process displayed by a spatially localized matterwave is here analyzed in a case where the free evolution is frustrated by thepresence of hard-wall-type boundaries (beyond the initial localization region).The phenomenon is investigated in the context of a nonrelativistic, spinlessparticle with mass m confined in a one-dimensional box, combining the spectraldecomposition of the initially localized wave function (treated as a coherentsuperposition of energy eigenfunctions) with a dynamical analysis based on thehydrodynamic or Bohmian formulation of quantum mechanics. Actually, such adecomposition has been used to devise a simple and efficient analytical algorithmthat simplifies the computation of velocity fields (flows) and trajectories. Asit is shown, the development of space-time patters inside the cavity dependson three key elements: the shape of the initial wave function, the mass of theparticle considered, and the relative extension of the initial state with respect tothe total length spanned by the cavity. From the spectral decomposition it ispossible to identify how each one of these elements contribute to the localizedmatter wave and its evolution; the Bohmian analysis, on the other hand, revealsaspects connected to the diffraction dynamics and the subsequent appearance ofinterference traits, particularly recurrences and full revivals of the initial state,which constitute the source of the characteristic symmetries displayed by thesepatterns. It is also found that, because of the presence of confining boundaries,even in cases of increasingly large box lengths, no Fraunhofer-like diffractionfeatures can be observed, as happens when the same wave evolves in free space.Although the analysis here is applied to matter waves, its methodology andconclusions are also applicable to confined modes of electromagnetic radiation(e.g., light propagating through optical fibers).PACS numbers: 03.65.Ta, 03.75.Dg, 42.50.-p, 42.50.Xa, 37.25.+k, 42.25.Hz
1. Introduction
Localization, confinement and diffusivity settle a rather broad context, which includesmyriads of physical problems both fundamental and applied. Pattern formation insidecavities constitutes, in this regard, a well-known topic in the literature. Withinthis context, the present work is intended to render some light on the issue of howthe presence of boundaries affects the evolution (diffusion) of an initially localizedmatter wave, i.e., how boundaries influence general diffraction (in time) processes.Specifically, it is motivated by a need to elucidate whether there are universal keyelements that determine the evolution of a quantum carpet associated with a confined a r X i v : . [ qu a n t - ph ] D ec oundary Bound Diffraction t = 0 is a spatially localized state between two impenetrable walls, i.e., a wave functionwith an everywhere vanishing amplitude except on a specific region between the boxboundaries, where it has finite values. This state describes a nonrelativistic, spinlessparticle of mass m in one dimension, with the cavity length being L and the localizationregion having a width w . This can be the case, for example, of a neutron matterwave entering a waveguide [1] with rectangular cross section at a relatively high speed(compared to other transversal dynamical scales) through a shutter characterized by anaperture smaller than the size of the incoming wave and with a particular transmissionfunction (not necessarily homogeneous along the opening).Thus, to better appreciate the scope of the work, let us consider the case oflocalized matter waves. As is well known, if such waves are spatially localized at agiven time, they eventually undergo delocalization once they are released and allowedto freely evolve. This behavior is exhibited, for example, by trapped neutral atoms orcondensates that are suddenly released [2,3]; once the atomic cloud is released, it startsloosing localization as it spreads in time. An analogous behavior is also displayed bymatter wave crossing openings [4] or gratings [5–8]. In this case, the passage throughthe opening or openings produces a number of transmitted (spatially localized) beamsthat, with time, undergo delocalization by diffraction, giving rise to the appearancein the long-time regime to a spatial redistribution of the probability — the so-calledFrauhofer diffraction pattern. The latter scenario is actually directly related to so-called diffraction in time phenomenon and the Moshinsky shutter problem, introducedby Moshinsky in 1952 [9,10] to explain the appearance of transient terms in dynamicaldescriptions of resonance scattering [11]. In this phenomenon, diffraction-like featuresarise when a rather nonlocalized wave (e.g., a plane wave) is suddenly truncated(by the action of a straight-edged shutter), thus producing a localized wave. Thesubsequent transverse evolution of this wave is analogous to the evolution of a wavediffracted by a real opening. Diffraction in time has been a subject of interest in theliterature ever since [12–19], being first confirmed with the time-domain analogous ofsingle- and two-slit diffraction [20]. It is worth noting that, actually, this phenomenonis analogous to considering the evolution of a wave under paraxial conditions, whichallows to separate the problem into the longitudinal (fast) propagation, characterizedby a classical-like motion, and the transverse propagation, describable in terms of aSchr¨odinger equation of reduced dimensionality [21, 22].The delocalization displayed by an initially localized matter wave can be, however,spatially limited by adding some extra boundaries, which gives rise to additionalphenomena. Think, for instance, of such a matter wave as in the Moshinsky problem,i.e., and extended wave entering a cavity. The initial localization of the ingoing state,produced by the size of the input shutter, evolves into a rather symmetric patter inspace and time displaying, at some positions and times, recurrences and even fullrevivals of the initially localized state [23, 24]. Due to the similarity between thesepatterns and usual carpets, they are called quantum carpets [25], which may showfractal features under certain conditions [26,27]. The emergence of such a pattern canbe explained in terms of a complex interference process involving a number of energyeigenstates of the confining cavity. Now, although this is a bound effect, it is worthnoting that an analogous situation also takes place in the continuum when considering oundary Bound Diffraction ‡ since both are parabolic partial differential equations [37].That is, the evolution of matter waves and the propagation of heat both follow anequation of the kind ∂u∂t − D ∇ u = 0 , (1)where the distribution or field variable u ( r , t ) specifies the state of the system (i.e., thevalue of the probability amplitude or the temperature, respectively) at any (allowed)position r at a given time t . In this equation, the diffusivity constant D plays afundamental role, because it determines the diffusion rate of u . Now, while in the heatequation D is a real number without a specific predetermined value, when dealing withquantum systems, D is a pure imaginary constant with value i (cid:126) / m [38, 39] (otherauthors [40] have considered alternative approaches where D is real, although itsmodulus remains the same). On the one hand, by virtue of the complex-valuednessof D we can observe interference traits when dealing with quantum systems (whichis not the case in heat transfer problems). On the other hand, we find that, because D depends on an external parameter, namely the mass associated with the matterwave (in many-body problems there can be several of these constants, each one witha different associated mass), the smaller the mass, the larger its diffusivity, whichtranslates into a faster spreading or delocalization rate.In direct connection with the problem dealt with here, particularly themethodology that is considered to tackle it, let us recall that is is a typical boundaryvalue problem described by Eq. (1), where a given (initial) field function u ( x, t = 0, is constrained to vanish at the boundaries at any time, i.e., u (0 , t ) = u ( L, t ) = 0. With these boundary conditions, the energy eigenfunctionsor eigenmodes of the cavity form a very convenient set of solutions for Eq. (1). As itis well-known from any elementary course on quantum mechanics, general solutions ‡ Of course, there is an important limitation in this analogy: while the diffusion equation arises froma conservation law, Schr¨odinger’s equation does not describe the evolution of any conserved quantity— unlike the wave function, though, the associated probability density obeys the continuity equation,which is a true transport equation, because it describes a conservation law. oundary Bound Diffraction D . The subsequent evolution of these states can then be understood as acomplex interference process among all the involved eigenfunctions, but also analyzedin terms of the flux associated with the wave function as a whole. This means thatthe analysis is more efficient when combining the use of the energy spectrum of thecavity, to analytically determine the evolution of the system quantum state, withthe numerically determined Bohmian trajectories, used to follow the evolution ofthe system at a local level inside the box. More specifically, in the problem hereinvestigated it will be seen that the former provides us with an efficient method tocompute the governing velocity (flux) fields (apart from other quantities of interest,such as the probability density), while the latter offers an insightful picture on howand why the probability evolves in the way it does, explaining the pattern formationcharacteristic of these systems. Besides its inherently fundamental interest to betterunderstand the processes of interference and recurrences in this kind of systems, asit will be seen the analysis here conducted has an also intrinsic applied interest as aground for the development of efficient Bohmian-based numerical methodologies.Following the above prescriptions, it is shown that the three aspects ruling thedynamical behavior of the system, which we are looking for, are: the shape of theinitial localized state, the particle mass, and the relative extension of the cavitywith respect to the size of the localization region of the state. The first factor isrelated to the way how a shutter may transmit a matter wave incident on it. Fromoptics, we are used to homogeneous functions, although this may not be the caseif there are short-range interactions between the particles described by the matterwave (e.g., electrons, neutrons, atoms, molecules, etc.) and the constituents of thematerial support where the shutter is, which are often neglected, although they mayhave an important influence [6, 41, 42]. The second factor, the mass of the particle,is important regarding the visualization of wave effects, since larger masses shoulddisplay classical-like features. This introduces the question of the classical limit in amore natural way than the standard one typically based on analyzing the behavior ofenergy eigenfunctions under some particular limit. Finally, the third factor, the ratiobetween the size of the cavity and the size of the region where the state is localizedwill render some light on the behavior exhibited by the system when it gets graduallyfreer (by free it has to be understood the condition when the confining boundaries goto infinity).The work has been organized as follows. A general analysis of quantum diffractionin terms of eigenfunctions of the infinite square well potential is presented in Section 2,as well as the method to compute the corresponding Bohmian trajectories in termsof such eigenfunctions. It is precisely by virtue of this analysis, where we readilynotice that the nonlinearity of the transport relation (Bohmian equation of motion orguidance condition), that the mathematical superposition principle does not have adirect physical counterpart. Accordingly, the evolution of the quantum system cannotbe naively described by appealing to independent waves (eigenfunctions), since thedynamics is governed by the collective effect of all of them as a whole. This non-separability is a fundamental quantum trait coming from the quantum phase, whichtranslates into a non-crossing in the streamlines or trajectories obtained in Bohmianmechanics. In Section 3 this analysis is applied to the study of the three differentelements that influence the evolution of the quantum system here considered: (1) theparticular shape of the system initial state, (2) its relative extension with respect to oundary Bound Diffraction
2. Theory
In the boundary value problem we are dealing with here, Eq. (1) takes the form of thetime-dependent Schr¨odinger equation, i (cid:126) ∂ψ ( x, t ) ∂t = − (cid:126) m ∂ ∂x ψ ( x, t ) , (2)where ψ ( x, t ) is constrained to the boundary condition ψ ( − L/ , t ) = ψ ( L/ , t ) = 0 atany time t and L is the total length of the box where the wave function is confined.The initial condition is specified by the localized state ψ ( x ) = (cid:26) f ( x ) , | x | ≤ w/ , w/ < | x | ≤ L/ , (3)with w being the effective size of the input shutter that allows the matter wave toenter the cavity. At t = 0, any general solution ψ to (2) can be recast in terms of a coherentsuperposition of energy eigenfunctions, as ψ ( x ) = (cid:88) α c α ϕ α ( x ) . (4)Since in one dimension the ϕ α are real functions, the coefficients are determined fromthe overlapping integral c α = (cid:90) ϕ α ( x ) ψ ( x ) dx, (5)although the real-valuedness of ϕ α does not ensure the real-valuedness of c α , whichalso comes from the value of ψ — for instance, if ψ ( x ) is a a traveling wave, e.g.,a constant amplitude multiplied by a phase factor e ikx , then the c α are complex-valued quantities. In the particular case of the infinite square well here considered,the time-independent eigenfunctions read as ϕ eα ( x ) = (cid:114) L cos( k α x ) , (6) ϕ oα ( x ) = (cid:114) L sin( k α x ) , (7)with k α = παL . (8)These solutions display, respectively, even ( e ) and odd ( o ) symmetry with respect to x = 0, i.e., φ eα ( − x ) = φ eα ( − x ) for α = 2 n − φ oα ( − x ) = − φ oα ( x ) for α = 2 n ,with n = 1 , , , . . . Physically, these solutions indicate that only an integer number ofhalf wavelengths can be accommodated between the box boundaries, with the largest oundary Bound Diffraction L , between them. The confiningwalls thus act in a way analogous to a space frequency (wavelength) filter, removingany component that does not match such condition.Following (4), any general initial condition can then be recast as ψ ( x ) = (cid:88) n c e n − ϕ e n − ( x ) + (cid:88) n c o n ϕ o n ( x ) . (9)At any subsequent time, the wave function reads as ψ ( x, t ) = (cid:88) α c α ϕ α ( x ) e − iE α t/ (cid:126) = (cid:88) n c e n − ϕ e n − ( x ) e − iE n − t/ (cid:126) + (cid:88) n c o n ϕ o n ( x ) e − iE n t/ (cid:126) , (10)since the time-evolution for ϕ α is given by ϕ α ( x, t ) = ϕ α ( x ) e − iE α t/ (cid:126) , (11)where E α = p α m = π (cid:126) α mL (12)is the corresponding energy eigenvalue (with p α = (cid:126) k α ). Accordingly, if thetransmitted wave function (initial condition) is described by (3), we find threepossibilities:i) If f ( x ) is an even function, only the cosine series contributes to (10).ii) If f ( x ) is an odd function, only the sine series contributes to (10).iii) If f ( x ) has no definite parity (asymmetric function), a general combination ofcosine and sine functions contributes to (10).In cases (i) and (ii) the parity or symmetry of the wave function at any subsequenttime is fully preserved. The time-dependent phases (11) only affect the amplitude ofthe real and imaginary parts of the corresponding eigenfunctions, but not their parity.Hence, when the collective effect of all the contributing eigenfunctions is taken intoaccount, the parity of their total linear combination is also preserved. The same holdsfor f ( x ) ∈ C . In this case, the function can be split up into its real and imaginarycomponents, which are then recast in terms of the corresponding eigenfunctiondecompositions. Contrary to directly operating over the full complex function, thisprocedure allows us to take advantage of the symmetry of each component separatelyto perform the analysis.Without loss of generality, from now on we shall consider the case of even-symmetric wave functions with respect to x = 0 (mirror symmetry), like (3). Thecorresponding time-dependent wave function reads as ψ ( x, t ) = (cid:114) L (cid:88) n c n − cos( k n − x ) e − iE n − t/ (cid:126) = (cid:114) L e − iE t/ (cid:126) (cid:88) n c n − cos( k n − x ) e − iω n − , t , (13)with ω n − , ≡ E n − − E (cid:126) = 2 π (cid:126) mL ( n − n (14) oundary Bound Diffraction n ≥ n = 1, ω , = 0). From a dynamical viewpoint, the preceding globaltime-dependent phase factor in (13) can be neglected, as it is seen bellow with theaid of Bohmian mechanics. The behavior exhibited with time is ruled by the set ofcharacteristic frequencies ω n − , , which introduce a series of related time-scales, τ n − , = 2 πω n − , = mL π (cid:126) n − n . (15)Whenever the evolution time equals an integer multiple of the largest of these periods,that is, the one for which n = 2, τ , = mL π (cid:126) , (16)we observe a full recurrence of the wave function (leaving aside the aforementionedglobal phase factor), since ω n − , τ , = ( n − nπ (17)is always an even integer of π . From now on we shall refer to τ , as the system recurrence time , which will be denoted by τ r . This is a universal quantity that doesnot depend on the initial shape of the wave function or its width w , but only on thetotal length L spanned by the cavity and the system mass m . Apart from τ r , there areother sub-multiples of this quantity for which fractional recurrences can be observed,as will be seen in Sec. 3. In the particular case of initial wave functions characterizedby non-differentiable boundaries, the evolution is characterized by a series of alternateregular and fractal-like (at irrational fractions of τ r ) replicas [26, 27]. These systemspresent an additional symmetry known as selfsimilarity.Apart from the time symmetry implicit in the fractional (or even fractal)recurrences mentioned above, the time-evolving wave function also displays (spatial)mirror symmetry (the symmetry of the initial state is preserved at any subsequenttime) and time-reversal symmetry with respect to τ r . This can easily be seen throughthe probability density arising from (13), ρ ( x, t ) = 2 L (cid:88) n,n (cid:48) c n − c n (cid:48) − cos( k n − x ) cos( k n (cid:48) − x ) cos( ω n − , n (cid:48) − t )= 2 L (cid:88) n c n cos ( k n − x )+ 2 L (cid:88) n,n (cid:48) n (cid:54) = n (cid:48) c n − c n (cid:48) − cos( k n − x ) cos( k n (cid:48) − x ) cos( ω n − , n (cid:48) − t ) , (18)where the bare sum of separate densities plus the sum of the coherence terms hasbeen made more apparent in the second equality. Although the symmetries canbe determined as well from (13), the particular functional form displayed by (18)makes this expression more convenient to better understand the role of the coherenceterms in the appearance of revivals. Actually, it is easy to see that, regardless of thecomplexity displayed by quantum carpets, if all coherences are removed at once, weimmediately recover | ψ ( x ) | , since the resulting ρ ( x, t ) becomes independent of time.This is of particular interest when introducing decoherence in this type of systems [43],which can happen, for instance, by reproducing the experiment with entangled pairsof photon [44, 45]. oundary Bound Diffraction ω n − , n (cid:48) − = E n − − E n (cid:48) − (cid:126) = 2 π (cid:126) mL (cid:2) (2 n − n − (2 n (cid:48) − n (cid:48) (cid:3) . (19)This expression generalizes the previous expression (14) to any pair of n and n (cid:48) components [in (14), we had n (cid:48) = 1]. If we exchange x by − x in Eq. (18), we readilyfind that ρ ( − x, t ) = ρ ( x, t ) , (20)which is satisfied at any time t . According to this symmetry, all what happens in onehalf of the space inside the box has a mirror replica in the other half.Regarding the time symmetry, consider two times symmetrically picked up aroundthe recurrence time, i.e., t = τ r / − t and t = τ r / t . Evaluating (18) at t andthen at t , we find ρ ( x, t ) = ρ ( x, t ) , (21)with τ r / t = τ r / involution , passing through all the previous stages until iteventually recollapses , reaching a state equal to the departure state (with respect tothe probability density, since the wave function, as seen above, accumulates a globalphase factor that makes it to be exactly the same we had at the beginning). To someextent the situation is analogous to a closed universe (in terms of density rather thanshape), where after reaching maximum expansion, it collapses again. This behavioris independent of the shape of the initial state, the system mass, or the extension ofthe confining box; in all cases an expansion (diffraction) and recollapse of the systemis expected (unless dissipation and/or decoherence are somehow present). The above spectral analysis allows us to understand the evolution of the matter waveinside the cavity by means of a complex interference process among different energyeigenfunctions. Instead of appealing to the energy representation, the same process canalso be understood in the configuration representation, where the spatial interferenceobserved (see Sec. 3) is usually explained in terms of semi-classical argumentationsbased on the computation of classical orbits [46]. In this regard, Bohmian mechanicsprovides us with an alternative and complementary analysis tool based on locallymonitoring the quantum flux with the aid of trajectories. This is possible by meansof the nonlinear (polar) transformation ψ ( r , t ) = ρ / ( r , t ) e i S ( r ,t ) / (cid:126) , (22)which recast a complex-valued field ( ψ ) in terms of two real-valued fields, namelythe probability density, ρ , and the (wave function) phase, S . After substitution of(22) into the time-dependent Schr¨odinger equation, two (real-valued) coupled partialdifferential equations arise, ∂ρ∂t + ∇ · J = 0 , (23) ∂S∂t + ( ∇ S ) m + V + Q = 0 , (24) oundary Bound Diffraction J = ρ ∇ S/m being the usual probability current or quantum flux [47], J = D ( ψ ∇ ψ ∗ − ψ ∗ ∇ ψ ) . (25)Equation (23) is the well-known continuity equation for the conservation of theprobability, while (24), more interesting from a dynamical viewpoint, is the quantumHamilton-Jacobi equation governing the particle motion under the action of a totaleffective potential: V eff = V + Q . The last term in the left-hand side of (24), Q = − (cid:126) m ∇ ρ / ρ / , (26)is the so-called quantum potential , which depends on the system quantum statethrough the density field.In the classical Hamilton-Jacobi theory, S represents the mechanical action of thesystem at a time t , with the trajectories describing the system evolution correspondingto the paths perpendicular to the constant-action surfaces at each time. Given thatquantum mechanics is just a wave theory (regardless of the physical meaning that onemay wish to assign to the wave function), one can proceed similarly according to theabove polar transformation, which allows us to operate with S in analogy to its classicalcounterpart. Accordingly, the classical-like concept of trajectory emerges in quantummechanics in a natural fashion: particle trajectories are defined as the solutions of anequation of motion that admits different functional (convenient) functional forms,˙ r = ∇ Sm = J ρ = (cid:126) m Im (cid:8) ψ − ∇ ψ (cid:9) = 1 m Re (cid:26) ˆ pψψ (cid:27) . (27)Here, ˆ p = − i (cid:126) ∇ is the usual momentum operator in the configuration representation.Notice that v = ˙ r specifies a velocity field predetermined at each time by the value ofthe system wave function ψ (through its phase S or, equivalently, the flux J ). Thisis particularly interesting at t = 0, since the initial momentum is predetermined bythe initial wave function, ψ . This means that trajectories (or, equivalently, quantumfluxes) must evolve in time following a given prescription, this being a dynamicalmanifestation of the so-called quantum coherence . Notice here the difference withrespect to point-like classical mechanical systems, with their initial momenta beingindependent of their initial positions. In this sense, although both quantum andclassical systems evolve under a similar equation, namely a Hamilton-Jacobi equation,they cannot be directly compared because their dynamics are very different. In thequantum (Bohmian) case, dynamics take place in configuration space, thus only beingdependent on coordinates [momenta are fixed at each point by the phase field, as itcan be inferred from Eq. (27)], while classical dynamics develop in phase space, wherecoordinates and momenta are both independent variables.Taking into account the explicit functional form of the wave function (13), theequation of motion (27) for a general superposition of energy eigenfunctions takes theform ˙ x = 1 m (cid:80) α,α (cid:48) | c α || c α (cid:48) | sin( k α x ) cos( k α (cid:48) x ) sin( ω α,α (cid:48) t + δ α,α (cid:48) ) (cid:80) α,α (cid:48) | c α || c α (cid:48) | cos( k α x ) cos( k α (cid:48) x ) cos( ω α,α (cid:48) t + δ α,α (cid:48) ) . (28)In this equation, the coefficients preceding each eigenfunction have been recast in polarform, c α = | c α | e iδ α , (29)assuming that they may also introduce a complex phase factor. This explains thephase shifts δ α,α (cid:48) = δ α − δ α (cid:48) that appear in both the numerator and the denominator oundary Bound Diffraction c α coefficients are real, for which δ α = 0, and therefore Eq. (28) acquires the simpler functional form˙ x = 1 m (cid:80) α,α (cid:48) c α c α (cid:48) sin( k α x ) cos( k α (cid:48) x ) sin( ω α,α (cid:48) t ) (cid:80) α,α (cid:48) c α c α (cid:48) cos( k α x ) cos( k α (cid:48) x ) cos( ω α,α (cid:48) t ) . (30)Because the velocity field (30) satisfies exactly the same symmetry conditionsdisplayed by ψ , the trajectories will also manifest this kind of overall feature[48]. However, it is also possible to go the other way around and extract valuableinformation about the topology displayed by the trajectories and, from it, about thedynamical behavior of the system. For example, the fact that the solution trajectoriesobtained from (30) cannot cross the same space point at the same time [36,49] impliesthat the dynamical behavior of the system can be split up into different domains.Specifically, in the cases considered here the mirror symmetry with respect to x = 0translates into two separate dynamical regions, with the trajectories from one domainnever penetrating the other one, and vice versa. This can easily be inferred from thefact that v ( x = 0) = 0 at any time, which means that the quantum flux splits up intotwo separate fluxes, each one confined in one half of the box [36, 49].
3. Numerical simulations
Diffraction is typically associated with functions or states characterized by well-definededges, even if this implies their non-differentiability on some particular space points.This is an interesting aspect to be analyzed, for such edges strongly determine notonly the speed or rate of the system diffusion inside the box, but also the type ofrecurrences that can be expected, which determines in the last instance the transportproperties of the box if it simulates, for instance, an optical fiber or the depth of a slitto diffract matter waves (e.g., electrons, neutrons or atoms). Physically, this initialshape can be related to the transmission function associated with the shutter, whichdoes not necessarily has to be a flat function all over its extension. On the contrary,it can be given in terms of a modulation function, as happens when we insert opticalfilters for light or, in the case of matter waves, specified by the effect of the potentialmediating the interaction between the diffracted particle and the constituents of theaperture. Analogously, as can be noticed through the functional form displayed bythe time-dependent wave function (13) and its recurrence time (15) [or its frequency(14)], the system mass m as well as the box length L (in relation to the dimensionof the shutter or, equivalently, the space region where the initial wave function isnonzero) are also going to play an important role in the subsequent evolution of thewave and the type of interference features that it will develop with time. Below,the effects of these three aspects on the system diffusion are going to be discussedwith the aid of a series of numerical simulations based on the analytical forms (18)for the probability densities, and (30) for the velocity field and, by integration, thecorresponding Bohmian trajectories.To better understand the effects of these influential aspects and particulary toacquire a more quantitative idea of them, in the analysis we are going to consider somequantities of interest. One of them is the overlapping probability , here defined as theoverlap between the exact wave function, Ψ( x, t ), and its associated series truncated oundary Bound Diffraction ψ f ( x ) c α ( α = 2 n − √ w (cid:114) wL sinc ( k α w/ (cid:114) w (cid:18) − | x | w (cid:19) (cid:114) w L sinc ( k α w/ (cid:114) w (cid:34) − (cid:18) xw (cid:19) (cid:35) w (cid:114) wL k α (cid:2) sinc ( k α w/ − cos ( k α w/ (cid:3) Half-cosine (cid:114) w cos ( k x ) , k = π/w √ Lw (cid:18) k k − k α (cid:19) cos ( k α w/ k α (cid:54) = k (cid:114) wL , for k α = k Half-cosine squared (cid:114) w cos ( k x ) , k = π/w (cid:114) w L (cid:20) (2 k ) (2 k ) − k α (cid:21) sinc ( k α w/ k α (cid:54) = k (cid:114) Lw k , for k α = k Gaussian (cid:18) πσ (cid:19) / e − x / σ (cid:114) L (cid:0) πσ (cid:1) / e − σ k α Table 1.
Different functional shapes considered in this work for the (simulated)diffracted wave function ψ and their corresponding Fourier components. Thewidth of the Gaussian function, σ = w/ π , has been chosen so that, infirst approximation, it equals the squared half-cosine function (moreover, withthis value, the corresponding probability density has decreased to about 4% at | x | = w/ at the N th term, Ψ N ( x, t ), i.e., P N = (cid:90) Ψ ∗ N ( x, t )Ψ( x, t ) dx = N (cid:88) n =1 c n − . (31)This quantity provides us with a direct measure of the convergence of the series and,therefore, how the above parameters influence the superposition and the subsequentinterference traits developed along time. Another two quantities of interest are the relative weight c n − of each component of the superposition as a function of n , justto get an ide on the relevance of each contributing eigenfunction, and the expectationvalue of the energy for Ψ N ( x, t ), (cid:104) H (cid:105) N = (cid:80) Nn =1 | c n − | E n − P N , (32)which is also a measure of convergence in terms of the energy added by each componentto the superposition. First we are going to analyze the spreading or diffusion and subsequent interferenceand recurrences in the position (configuration) space of a series of diffracted waves ψ ,an analysis that emphasizes the relationship between such traits or phenomena andthe relative curvature of the diffracted function. We have chosen a series of functional oundary Bound Diffraction Figure 1. (a) Reconstruction of the six wave functions described in Table 1 witha total of N = 500 eigenfunctions each: square (black), triangle (red), parabola(green), half cosine (blue), half-cosine squared (cyan), and Gaussian (magenta).For a better comparison, all wave functions have been normalized to their valueat x = 0 (Ψ (0)). In all cases: L = 50, w = 10, and (cid:126) = m = 1. (b) Probability P N (31) as a function of the number N of contributing eigenfunctions. For visualclarity, an enlargement of P N for low N is shown in the inset. (c) Expectationvalue of the Hamiltonian (32) as a function of the number N of eigenfunctions. forms f ( x ) for the initially localized wave packet (see Table 1) in a range that coversvarious intermediate functions, from the square function to the Gaussian wave packet[see Fig. 1(a)]. The square function constitutes a paradigm of transmission functionin both optics and quantum mechanics, although more realistic in the first case thanin the latter due to the faster propagation of light with respect to usual matter waves.The Gaussian function, in many cases of physical interest, has a more convenientcomputationally functional form, apart from being more realistic when short-rangeinteractions with the opening boundaries are non-negligible. As intermediate caseswe have chosen a triangle, a parabola, a half-cosine and a half-cosine squared, whichpresent different degrees of differentiability and curvature. All these intermediatecases have been chosen in such a way that they vanish at x = ± w/
2. As for theGaussian function, its width has been chosen in a way that, in first approximation,its functional form equals that of the half-cosine squared, as can be seen in Fig. 1(a)by means of the overlap of both functions for | x | (cid:46) /
3. As can be seen in the figure,the triangular function is quite close to these two functions, while the parabola andhalf-cosine functions are closer between themselves. The decomposition of all thesefunctions in terms of energy eigenfunctions of the infinite square well potential canbe seen in Table 1 in terms of the generic α th coefficient, with α = 2 n −
1. All theinitial ans¨atze considered in Table 1 share a general common feature worth mentioning,as can be noticed in their eigenfunction decomposition: Ψ does not depend on thesystem mass ( m ), but on the ratio w/L . However, despite this fact, the dynamicsdisplayed by Ψ is strongly dependent on the mass, since it appears in a key dynamicalelement, namely the frequencies (14) and (19). According to the expressions for thesefrequencies, the larger the mass, the lower the frequency (energy). These facts arerelated to the time-reverse and mirror symmetries above discussed. Large massesand/or box lengths will imply longer recurrence times, i.e., slower dynamics, as willbe seen in more detail in Sec. 3.2.Let us now discuss in more detail some properties of the six wave functions. InFig. 1(a) we observe a reconstruction of all the functions considered. A total of 500eigenfunctions has been considered in each case. As it can be seen (and is well known),the shape of each function is well converged, except the square wave function due to the oundary Bound Diffraction x = ± w/
2. This mismatch is produced by the well-known Wilbraham-Gibbs phenomenon [50–52], which, in the context of Fourier analysis, states that aFourier series will display a finite increase or decrease of the value of the sampledfunction at those points where the function has a discontinuity, independently of howmany Fourier components are considered in the series. Strictly speaking, althoughwe are not performing Fourier analysis, the decomposition of the function in terms ofa basis set associated with a certain potential function (an infinite square potential)is analogous. As a consequence, although the sum of components approaches veryslowly the normalization to unity, as seen in Fig. 1(b), the expectation value of theHamiltonian is unbound, as shown in Fig. 1(c). This is connected to fractal likefeatures in the evolution of the square function [26] each time that one looks at atime that is an irrational submultiple of τ r . This behavior quickly disappears as thediscontinuity at ± w/ P N approaches the unity witha very few eigenfunctions for all functions; in Fig. 1(c) it is shown that, in spite ofthe nondifferentiability at ± w/ x = 0, the convergence of the energy for the trianglefunction is relatively slower than the other cases, since the number of contributingeigenfunctions is larger.So far we have commented on properties related to the construction of wavefunctions with different initial shapes, which physically describe ways in which ashutter operates on an incoming wave larger than its opening (e.g., a plane wave actedby a collimating slit). Let us now focus on the subsequent time-evolution of such waves.To that end a series of contourplots with the corresponding Bohmian trajectories havebeen represented in Fig. 2 for each case considered in Table 1. The six top panelsrepresent the time-evolution (vertical axis; normalized to the recurrence time τ r ) ofthe corresponding probability densities, while the six bottom panels (labeled with aprime) describe the evolution of the associated velocity fields. In both cases, andfor each function, a set of 20 Bohmian trajectories (white solid lines) is also shown,with the initial conditions evenly distributed along the opening (or a bit further awayfor the Gaussian wave function, just to also sample the dynamics of its “wings”).Perhaps such a distribution can be considered as misleading, since it is not a bona fiderepresentation or mapping of the evolution of the probability density ρ ( x, t ). However,the purpose here is not to illustrate this behavior, which can be found elsewhere (see,for instance, [53] for diffraction and interference in the open), but to get a glimpse onthe features characterizing diffraction under confinement conditions, which are moreprominent for marginal trajectories than from those associated with large values ofthe probability density.By inspecting the behavior of the probability density, the first we notice is thepresence of the two kind of symmetries mentioned earlier. The space mirror symmetrydisplayed by the probability is very apparent for the whole evolution of the wavefunction, from t = 0 to t = τ r = 397 .
9, although the set of Bohmian trajectories revealsthe specificities of the dynamics, that is, there is a fast motion from some maximato others, while avoiding those regions where ρ is negligible. This is particularlyrelevant near the boundaries of the box: although initially the trajectories spreadvery fast towards the boundaries, after reaching them they start undergoing a seriesof bounces in order to avoid staying close. Nonetheless, except in the case of thesquare function, where trajectories display fractal features [27] and close to the borders oundary Bound Diffraction Figure 2.
Contour-plots showing the quantum carpets displayed by the six wavefunctions described in Table 1 along their evolution: (a) square, (b) triangle, (c)parabola, (d) cosine, (e) cosine square, and (f) Gaussian. The six upper panelsrepresent the probability density, while the six lower panels (labeled with primes)refer to the corresponding velocity fields. Sets of quantum trajectories have beensuperimposed in order to illustrate the dynamical evolution of the flux in eachcase. In all cases: N = 500, L = 50, w = 10, and (cid:126) = m = 1. For visual clarity, theprobability density contours have been taken from zero to a half of the maximumvalue of the probability density; in the case of the velocity field, contours aretaken from v = − v = 1. The initial conditions for the trajectories have beentaken following a constant distribution along the aperture for all cases to betterappreciate the border effect on the trajectory dynamics. oundary Bound Diffraction Figure 3. (a) Weights | c α | associated with each one of the components ( n ,with α = 2 n −
1) used in the reconstruction of wave functions with square (redcircles) and half-cosine squared (black squares) wave functions (see Table 1). Inall cases: L = 50, w = 10 and (cid:126) = 1. For a better visualization, log-scale hasbeen used in box axes; the linear-scale plot is displayed in the inset of the figure.(b) Probability P N as a function of the number N of eigenfunctions for the twocases considered in panel (a). (c) Expectation value of the Hamiltonian, (cid:104) H (cid:105) N as a function of the number N of eigenfunctions for different values of the mass: m = 1 (black), m = 10 (red), m = 100 (green), and m = 1000 (blue). Solid linesrepresent results for the cosine-squared wave function, while dashed lines refer toa square wave function. they undergo very fast oscillations, in the other five cases the border trajectories arerelatively well-behaved, particularly in the Gaussian case. Besides, the trajectoriesalso make apparent that the system, for practical purposes, behaves as composed oftwo independent halves, since the flux dynamics for x < x >
0, and vice versa. This dynamical behavior is well understood if we look atthe carpets corresponding to the velocity field (bottom panels), characterized by theproperty of mirror antisymmetry, i.e., v ( − x, t ) = − v ( x, t ). The pronounced regionswhere the velocity field has large values are characterized by sudden and also largevalues of the modulus of its first derivative, which provokes a fast dispersion of thetrajectories. On the other hand, the trajectories tend to accumulate in the regionswhere the first derivative of the velocity field is relatively small and smooth.When we examine the probability and velocity carpets along time, the secondsymmetry, namely the time-reversal symmetry, immediately becomes apparent. Aftera very fast initial boost, the wave function starts undergoing different recurrencesby interference after having interacted with the box boundaries, which generate thespecific pattern of the carpet. Now, interestingly, at t = τ r /
2, there is a neat recollapseof the wave function, which gathers two features: the probability density is split upin the form of a coherent superposition of two identical images of the initial density,each one centered just at the center of each half of the box. If we look at the velocitycarpets, what happens is that the flux is eventually confined within these two localizedregions at t = τ r /
2, that is, the trajectories are constrained to these two regions,like if there where two openings precisely at such positions. From this time on, thebehavior of the system reverts until we observe a full recollapse of the wave to its initialstate (neglecting the global phase factor accumulated with time, which is dynamicallyirrelevant, as it was pointed out in Sec. 2). oundary Bound Diffraction As can be seen in Table 1, the system mass has no influence on the superpositionitself. In Figs. 3(a) and (b) the weights | c α | and the probability P N are shownfor the square and half-cosine squared wave functions. We have chosen here thesetwo particular functions, because they both are nonzero only within the interval | x | ≤ x/
2, with the particularities that the former is an example of non-differentiablefunction and the latter is close in behavior to the Gaussian. Since the initial spectraldecomposition of the wave function does not depend on the system mass, the plots inthese figures (for each function) are the same for the four masses considered in thissection: m = 1, m = 10, m = 100 and m = 1000. Figure 3(a) allows us to observethe oscillatory behavior of the weighting coefficients in both cases, which explains thealso oscillatory behavior of P N or the stepped structure of (cid:104) ˆ H (cid:105) N that we already sawin the previous section. Interestingly here, when comparing the square and the half-cosine squared functions, we notice that while the contribution of the eigenfunctionsto the superposition (measured through | c α | ) decreases slowly with n for the former,the decrease is very fast for the latter (the same has also been observed for the otherwave functions). For example, while about 10 eigenfunctions have a weight above10 − for the half-cosine squared function, there are about 100 eigenfunctions in thecase of the square function, which explains why (cid:104) ˆ H (cid:105) N displays very clear steps in thelatter case, while the same cannot be seen for the former [see Fig. 1(c)]. The linearscale in the inset makes more apparent how, while the | c α | coefficients are negligiblebeyond n = 10 for the half-cosine squared function, the same does not happen forthe square function. The manifestation of this fact can be readily seen in Fig. 3(b): P N is already about 1 for n ≈
10 for the half-cosine squared function, while for thesquare function it converges very slowly to 1. Furthermore, from a simple least squarefitting, we have observed that the | c α | decay as n − for the square function and as n − for the half-cosine squared, which has interesting implications and an explanationfor the unbound increase of the expectation value of the energy in the case of thesquare function. As seen in Sec. 2, the eigenenergies increase with n approximatelylike n . So, if we compute the expectation value of the energy, we will have somethinglike (cid:104) ˆ H (cid:105) ∝ (cid:88) n n β n , (33)with β being the exponents obtained from the fittings. Accordingly, for the squarefunction, we have (cid:104) ˆ H (cid:105) ∝ (cid:88) n → ∞ , (34)which is unbound, while for the half-cosine squared function we obtain a convergentseries, (cid:104) ˆ H (cid:105) ∝ (cid:88) n n = ζ (4) = π . (35)These are precisely the behaviors observed in Fig. 3(c) for each mass.From a dynamical perspective, though, mass plays an important role in thetime-evolution of the system, as can easily be seen by inspecting the behavior of theexpectation value of the energy. This influence arises through the kinetic operator,thus here going like m − , as seen in Fig. 3(c) for the four masses referred to above. oundary Bound Diffraction (cid:104) ˆ H (cid:105) N are alwaysparallel, decreasing in the same proportion in which m increases. In other words, alarger inertia implies a slower diffraction. This effect has an interesting manifestationin the time-evolution of the wave or, equivalently, the corresponding quantum carpet.According to (16), the recurrence time τ r increases proportionally to m , which meansthat the diffraction and subsequent diffusion of the wave slows down. The massesconsidered here increase gradually in one order of magnitude, which means that thecorresponding recurrence times are also going to increase in the same way. Thus, if weconsider as a reference the recurrence time for m = 1, i.e, τ r = 397 .
9, we already noticea remarkable suppression of the system diffusion when the mass has been increasedby just one order of magnitude, as seen Fig. 4(b) when compared with Fig. 4(a).In Fig. 4(b) we notice that all the structure of the quantum carpet associated withinterference is completely absent; we only observe the effect of the initial diffractionundergone by the wave function and the bounces at the boundaries of the box, apartfrom some marginal interference, which becomes relevant almost at the end of theevolution. If the mass is increased by another order of magnitude, as seen in Fig. 4(c),there is still some flux associated with the edges of wave function that can reach theboundaries of the box, but essentially all the flux remain confined within a regionaround the wave function, which is slightly diffracted. Finally, when the mass isincreased by three orders of magnitude above the reference mass, the wave functiondoes not show much diffraction, as it is shown in Fig. 4(d). In this latter case, noticethat the trajectories remain nearly parallel one another.It is worth noting that this latter case is the quantum analog to the geometricoptics limit, where diffraction effects are neglected behind an opening. Consider theshutter is illuminated by monochromatic light with a negligible wavelength comparedto the shutter width ( λ (cid:28) w ), and that a screen is allocated a certain distance L = τ r /c . In a first approximation, the imaging problem at L can be describedby means of the geometric optics. Accordingly, there will be a spot of light just infront of the shutter, with nearly its same width (if the incident radiation is a planewave), and shadow everywhere else. However, if the distance to the screen increases,a series of diffraction traits start appearing because light start displaying its wavebehavior. Furthermore, if a constraint is imposed on its spatial diffusion (reflectingwalls, e.g., mirrors) and L becomes larger and larger, interference traits will manifestand eventually we will observe the same behavior as in our case for m = 1 (or aperiodic representation of the same if the length of the box or, equivalently, thepropagation time is further increased). With the matter wave we have exactly thesame, as seen in Fig. 4, if the wave entering the cavity is highly coherent. For veryshort times or very large masses, the system can be described in a first approximationwith classical mechanics, since Bohmian trajectories are going to closely behave asNewtonian ones. Actually, this situation is what can be denoted as the Ehrenfest-Huygens regime [54]. However, as time increases or smaller masses are considered,wave-like features, like diffraction or interference, become dominant in the evolutiondisplayed by the trajectories and classical mechanics is no longer a good description ofthe system dynamics — notice in Fig. 4(d) that, if the system is left to evolve up to thecorresponding recurrence time (a thousand times larger), we shall observe a pictureexactly the same as in Fig. 4(a). Typically, according to the standard view, waveand particle behaviors are incompatible. This simple example here shows that thisstatement is not true, but that all depends on the scale of time (or mass) considered to oundary Bound Diffraction Figure 4.
Contour-plots showing the quantum carpets displayed by a half-cosinesquared wave function (see Table 1) along its evolution and for different valuesof the mass: (a) m = 1, (b) m = 10, (c) m = 100, and (d) m = 1000. A set ofBohmian trajectories with equidistant initial positions has also been superimposedin order to illustrate the dynamical evolution of the flux, particularly at theborders of the lattice. In all cases: N = 200, L = 50, w = 10 and (cid:126) = 1. Inthe first panel, for a better visualization, the contours have been taken from zeroto half the maximum value of the probability density; in all cases, the transitionfrom darker (dark blue) to lighter (red) colors indicates increasing density values. analyze the system. Within this context, classical mechanics (or geometric optics, ifwe are dealing with light) is just a first-order approximation to the behavior displayedby the system in the very short term, regardless of its mass — of course, anothermatter beyond the scope of this discussion, but also very important at a fundamentallevel, is the whether by more mass one means more complex, i.e., a many-body object. To complete the analysis, the effects of the size of the box on the system have alsobeen studied, since the influence of this parameter not only comes from the recurrencetime (16) (as L ), but also through the momenta k α , according to (8) (as L − ), andthe relative weight | c α | in the superposition (as L − ). As in the previous sections,before going into detail with the dynamics, let us get some conclusions from thecorresponding superpositions. To that end, using again a half-cosine squared wavefunction, four different box sizes have been considered, including as a reference theprevious one L = 50. These sizes range from L = w , so that the shutter openingcovers the whole box, to L = 20 w , which is large enough compare to the shutteropening. As can readily be seen in Fig. 5(a), as L increases the superposition becomesmore and more structured, including a larger number of eigenfunctions with the samerelative weight. For example, if we consider a threshold of contributions around 0.1%or above, for L = w we have one major contribution, which is nearly a 100% of the oundary Bound Diffraction Figure 5. (a) Weights | c α | associated with each one of the components ( n ,with α = 2 n −
1) used in the reconstruction of a half-cosine squared inside a boxwith different lengths: L = 10 (black squares) L = 50 (red circles), L = 100 (bluetriangles), and L = 200 (red diamonds). For a better visualization, log-log scalehas been used in both axes. (b) Probability P N as a function of the number N ofeigenfunctions for the cases cases considered in panel (a). (c) Expectation valueof the Hamiltonian, (cid:104) H (cid:105) N , as a function of the number N of eigenfunctions. Inall cases, the shutter width is w = 10 and the system mass m = 1 (with (cid:126) = 1). superposition, the next falls below 10%, and the third drops to about 0.1%. For L = 5 w , there are 9 contributions above 0.1%, with 3-4 of them close in importance,above 10%. These numbers remarkably increase with L = 10 w and L = 20 w , as seenin the figure. Actually, the trend is that the number of eigenfunctions contributingwith nearly the same weight increases with L . If we consider those contributions whichdiffer from the first one up to 25%, i.e.,∆ ,α = (cid:18) − | c α | | c | (cid:19) × , (36)we find that for L = w there is only 1, for L = 5 w there are 2, for L = 10 w there are4, and for L = 20 w there are 17, which follows a nearly quadratic dependence on L .A similar trend can also be seen in case of P N and (cid:104) ˆ H (cid:105) N , as it is shown in panels (b)and (c) of Fig. 5, where an increasing number of L means that a remarkable number ofeigenfunctions is needed to better represent the original wave function and thereforean slower convergence to it and its energy.If we now go to the corresponding quantum carpets, displayed in Fig. 6 for L = w , L = 5 w and L = 10 w , we notice an increasing degree of complexity and structuringwith increasing L , which is expected as the number of eigenfunctions involved, andhence the number of frequencies ω α,α (cid:48) , also increases. This gives rise to a highly noisydynamics, as seen through the corresponding Bohmian trajectories, which comes fromthe fact that interference traits become more prominent due to the appearance of moreprofiled dips and ridges [compare panels (b) and (c)], which forces the trajectories tojump relatively fast from some regions to others, since the velocity field is too largein between. Nonetheless, for short times, of the order of 1/25 of the τ r correspondingto the case of L = 5 w and 1/100 of the one corresponding to L = 10 w , we find avery similar early-time evolution, as seen in panels (b’) and (c’), respectively, whichis in correspondence with the fact that at these stages there is not time enough yet tonotice the fine structuring effect coming from all the main contributions (many morein the latter case than in the former, as seen in the upper panels). oundary Bound Diffraction Figure 6.
Contour-plots showing the quantum carpets displayed by a cosine-squared wave function (see Table 1) along its evolution and for different values ofthe box size: (a) L = w , (b) L = 5 w and (c) L = 10 w . Panels (b’) and (c’) areenlargements of the regions of (b) and (c), respectively, for the same time displayedin panel (a). A set of Bohmian trajectories with equidistant initial positions hasalso been superimposed in order to illustrate the dynamical evolution of the flux,particularly at the borders of the lattice. In all simulations here: N = 200, w = 10and m = 1 (with (cid:126) = 1). In the first panel, for a better visualization, the contourshave been taken from zero to half the maximum value of the probability density;in all cases, the transition from darker (dark blue) to lighter (red) colors indicatesincreasing density values.
4. Concluding remarks
From a dynamical viewpoint, we have that the delocalization of a released matterwave is analogous to the diffraction it undergoes after crossing an opening — in thislatter regard, the opening would act as the localizing element and its subsequentcrossing would play the role of the release. On the other hand, regardless of the initialphysical context considered (whether a trapped atomic cloud or a diffracted atomicor molecular beam), if some extra boundaries are added, the new confining conditionswill produce the appearance with time of a series or recurrences. The pattern thatdevelops with time is commonly known as a quantum carpet, which displays somesymmetries in both space and time according to the interference of the wave withthe new confining boundaries. Actually, at some time, a full revival of the initialstate (except for a global phase factor) is observed, which is repeated in time onceand again unless some dissipative or decohering mechanisms act on the system. Thisis particularly remarkable in the case of the well-known problem of the particle ina one-dimensional box, assuming such a particle is nonrelativistic, spinless and withmass m .In this work we have focused on this classical problem with the purpose todetermine which are the main elements that affect the evolution of the bounddiffraction process, and more specifically how such elements influence the symmetry oundary Bound Diffraction oundary Bound Diffraction Acknowledgments
Financial support from the Spanish MINECO (Grant No. FIS2016-76110-P) isacknowledged.
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