Coordinated self-interference of wave packets: a new route towards classicality for structurally stable systems

This is a study of proton transmission through planar channels of tungsten, where a proton beam is treated as an ensemble of noninteracting wave packets. For this system, the structural stability manifests in an appearance of caustic lines, and as an equivalence of self-interference produced waveforms with canonical diffraction patterns. We will show that coordination between particle self-interference is an additional manifestation of the structural stability existing only in ensembles. The main focus of the analysis was on the ability of the coordination to produce classical structures. We have found that the structures produced by the self-interference are organized in a very different manner. The coordination can enhance or suppress the quantum aspects of the dynamics. This behavior is explained by distributions of inflection, undulation, and singular points of the ensemble phase function, and their bifurcations. We have shown that the coordination has a topological origin which allows classical and quantum levels of reality to exist simultaneously. The classical behavior of the ensemble emerges out of the quantum dynamics without a need for reduction of the quantum to the classical laws of motion.

Let a L , and N a denote lattice constant and a number of atoms contained in the BCC unit cell of the tungsten crystal. Spacing between [111] planes is d 111 = √ 3a L /6, while corresponding surface density is σ 111 = N a d 111 /a 3 L . We assume that y0z plane of the Cartesian coordinate system is parallel to [111] planes of the tungsten crystal with x axis orthogonal to them. Quasi-parallel proton beam of kinetic energy E k = 2MeV was assumed to be aligned with the z axis of the coordinate system.
Let r = (x, y, z) denotes a proton-tungsten separation vector. To construct protoncrystal interaction potential we start from the Molieré's approximation of the Thomas-Fermi's proton-tungsten interaction potential [1,2] where Z 1 = 1, and Z 2 = 74 are proton and tungsten atomic numbers, e is elementary charge, with a standard deviation given by the following relation here A r = 183.85 is tungsten atomic weight, m u = 1.6605 · 10 −27 kg is universal atomic mass unit, Θ D = 310 K is the tungsten Debye temperature [2], k B = 1.3806 · 10 −23 J/K is Boltzmann's constant, T is sample's absolute temperature, and D f is the Debye's function.
Since scattering angles of channeled particles are small, the continuous approximation could be used. The resulting continuous planar-averaged potential is given by multiple integrals For the Molière's potential integrals (4) can be evaluated analytically giving here erfc is complementary error function [3]. In the final step potential of the planar channel was expressed as a sum of contributions of individual atomic planes. In the coordinate system attached to the midpoint between two nearest planes, the arrangement of planes can be viewed as a layered structure with each layer comprised of two plains. Planes of the m-th layer are to be found at distances d m = (m + 1 2 )d 111 from the coordinate origin. They enclose all planes of the layer order less than m. The channel potential is a periodic function with period d 111 given by the following expression Each term of the sum gives a contribution of one layer which is in its own right a sum of contributions coming from the left and the right plane, respectively. Constant V 0 = 2 m V th 111 (d m ) was introduced in order to have V (0) = 0. In principle number of terms in the sum should be infinite, however since potential (5) is a rapidly decreasing function, sum (6) can be truncated after N -th term.
In channeling it is usually E k V (d 111 /2). As a consequence maximal deflection angle, a classical particle could have while still being channeled, is given by the expression which is called a critical channeling angle. For thin crystals, considered here, energy loss and fluctuation of the scattering angle can be neglected. Newton's equations governing proton dynamics are where m p is proton mass, while t denotes the time. This means that motion in y and z directions is given by equations Since dynamics in the direction parallel to crystal planes is trivial it will be disregarded, and the problem treated as essentially one-dimensional.
In the transverse direction, proton dynamics is governed by Hamilton's equations of In order to represent parallel beam initial conditions should be in form θ x (t = 0) = 0, and x(t = 0) = b, with uniform distribution of the proton impact parameter b. Trajectories will be parameterized by variable Λ = 1 2π ωt, called reduced time [4] or reduced crystal thickness [5]. Here variable represents an angular frequency of proton trajectories in the center of the potential well.
For fixed value of Λ trajectory family define maps of the particle starting position b to its current position denoted X(b) ≡ x(Λ; b), and to its current scattering angle denoted . Their critical sets consist of all points satisfying equations ∂ b X(b) = 0, and ∂ b Θ x (b) = 0, which are called spatial and angular rainbow points, respectively. Note that when rainbow points exist inverse maps X → b and Θ x → b are multivalued, therefore singular. Considered together maps X(b) and Θ x (b) define a curve in the phase space , called a rainbow diagram, whose critical points are spatial and angular rainbows [6]. In the case of regular dynamics, the rainbow diagram is also known as a whorl [7].
Trajectories of critical points ∂ b x(Λ; b) = 0, and ∂ b θ x (Λ; b) = 0 form lines called caustics [8]. They are identical to envelope lines associated with the trajectory family [9]. The density of trajectories is infinite at caustic lines, therefore a probability for finding a classical particle on caustic lines is very large. The caustic pattern is structurally stable [10][11][12]. This means that it can be modeled locally by the appropriate catastrophic prototype of codimension one In quantum mechanics, it is meaningless to speak about actual values of position and scattering angle particle might have. Instead, a state of the system is specified by a wave function Ψ(r, t) which in its own right determines only probability distributions of physical quantities. In analogy with the classical case, the energy loss of channeled particles will be neglected. Evolution of the quantum state is then given by the following Schrödinger equation with initial condition of the form here k z represents proton's initial wave vector (E k = k 2 z /2m p ), while ψ 0 (x)ζ 0 (y) is the transverse part of the initial wave function. The dependence of the potential (6) only on the coordinate x makes motions in the y, and z directions free. Therefore, a general wave function can always be represented in the form and K(y,ȳ) is the free space propagator As a result, an evolution of the transverse wave function ψ(x, t) is governed by the reduced Schrödinger equation Since motion in y and z direction is trivial it will be disregarded from the subsequent analysis.
The evolution of the quantum dynamics will be also parameterized by the variable Λ.
In quantum mechanics, the closest analog of a classical particle is a wave packet. We have assumed that initial wave function is a Gaussian function of mean value b 0 , and standard deviation σ x . The wave function in the angular representation The initial wave function in the angular representation is therefore also Gaussian function Standard deviations σ x and σ θ are linked through Heisenberg's uncertainty relation k z σ θ σ x = 1/2.
Hamilton's principal functions in the spatial and angular representations are defined by Let us introduce reduced principal functionsS x andS θ by relations S x (x) = k zSx (x), and S θ (θ x ) = − k zSθ (θ x ), respectively. Semiclassical wave functions in the initial value representation [13,14] are then given by integrals where dS . Note that phase functions are defined on the interval of the length d 111 . So that integrands can be considered as rapidly oscillating functions, and semiclassical representation (22) valid, it is necessary that k z d 111

1.
The proton beam is represented as an ensemble of noninteracting wave packets where states ψ b (x, Λ) forming the ensemble are parameterized by the impact parameter b. According to the rules of the quantum mechanics state of the ensemble is specified by the density matrix operatorρ, whose spatial and angular representations, ρ x , and ρ θ , are given by relations respectively. Expansion coefficient p b represents relative frequency of the state ψ b in the ensemble ( b p b = 1). Note that the assumption of an absolutely parallel beam is incompatible with the representation of particles as wave packets. Therefore, we assume that the incoming beam has a Gaussian profile of very small angular divergence Ω. Unknown parameters σ θ σ x , the distribution of the impact parameters, and their statistical weights p b , should be determined in such way that ρ x (x, 0) is uniform distribution in the interval of Using Eqs. (20) and (23) it is easy to show that σ θ = Ω, and σ x = 1/(2k z Ω). An 1D grid of M impact parameters was taken to uniformly cover the interval −d 111 /2 ≤ x ≤ d 111 /2, with p b = 1/M . The number M is minimal for which the difference between ρ x (x, 0) and the uniform distribution is smaller than some predetermined quantity.
In the (x, θ x ) phase space Wigner function associated with the state ψ b is given by the following expression [15] or equivalently by here † stands for Hermitian conjugation. A Wigner function of the ensemble is given by the expression It is the closest quantum analog of the classical probability density in the phase space. The Wigner function has many properties out of which the following are important for the subsequent analysis. It should be stressed that in the classical limit Wigner function is given by the relation [16,17] W which value is 0 except for the points belonging to the rainbow diagram R = (Q(b), P (b)).
Multiplicities of the maps X → b and Θ x → b are responsible for the enhancement of the wave packet self-interference in the vicinity of the classical caustic lines, and rainbow diagram in the phase space [4,11,18]. Structurally stable structures also appear in the quantum domain. They are locally isomorphic to the canonical diffraction patterns defined by the relation [11,12] [19,20], giving average energy loss of δE k = ν∆E k = 3.5354 keV, which is utterly negligible. For considered crystal thickness, the average variance of the proton scattering angle fluctuations given by the relation is sufficiently small that the dechanneling process can be neglected.
Our decision to neglect both effects can be justified on the following grounds. First, the electron density is not uniform in the region of the channel. Its maximal value at the atomic plane is approximately 10 times larger than its minimal value at the center of the channel.
Energy loss and dispersion of the scattering angle of the well-channeled protons are grossly overestimated since particles spend only a fraction of time in regions of high electron density.
Fluctuations of scattering angle decrease the sharpness of the observed structures and limit the resolution of measured distributions (see section C). This is a serious limitation of any distribution recording system based on image detection. However, modern particle counting detectors based on the microchannel plate technology offer a very large dynamical range. In the pulsed operation, they can provide reliable count rates up to 1GHz [21], meaning that dynamical range is limited only by available time to accumulate sufficient statistics.
Second, in this study, we are primarily investigating to which extent coordinated selfinterference of wave packets is responsible for the formation of patterns on a global scale.
It will be shown when coordinated self-interference amplifies the classical aspects and when it amplifies wave-like aspects of the ensemble dynamics. The mentioned processes are unaffected by the slight inaccuracies of the obtained results on the smallest scale.
which can be evaluated analytically giving

D. Catastrophic modeling of caustics
We wish to model a caustic line from the first rainbow cycle by a bifurcation set of the A 5 catastrophe which is defined by the relation Here, state variable η is taken to be proportional to the impact parameter b, while values of parameters c 1 , . . . , c 4 are to be determined. The critical points and degenerate critical points of polynomial A 5 are solutions of equations The first of Eqs. (35) defines the function which we assume to be proportional to the spatial deflection function X(b). Because of symmetry c 3 = 0. In that case the second of Eq. (35) can be solved giving Note that only for c 4 < 0 catastrophe A 5 has the correct multiplicity of real critical points. Bifurcation set of the obtained model is shown in Fig. 1(a). For c 4 < 0 unfolding of the catastrophe A 5 proceeds as follows. For Λ < 0.16 there are no degenerate critical points. For Λ = 0.16 two double degenerate critical points appear forming apexes of the cusps. For Λ > 0.16 degeneracy is lifted and apex points split into two pairs of degenerate cryptical points. This behavior is typical of the cusp catastrophe A 3 , while the evolution of the single degenerate cryptical points is equivalent to the behavior of the catastrophe A 2 . Therefore, it could be said that catastrophe A 3 brakes into two A 2 catastrophes. Outer branches emanating out of cusp points move towards the channel axis where they meet for Λ = 0.25. Two A 2 catastrophes merge to form a new A 3 catastrophe and then disappear.
Inner branches cross the channel axis and become outer branches. They are only degenerate critical points that exist for Λ > 0.25. The described unfolding is schematically represented in Fig. 1(b where ζ = 5 16k 4 z p 1 /5s 6 θ x is an auxiliary function, while χ 4 is the swallowtail canonical diffraction pattern.