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Numerical Simulation of the Double Slit Interference with Ultracold Atoms

We present a numerical simulation of the double slit interference experiment realized by F. Shimizu, K. Shimizu and H. Takuma with ultracold atoms. We show how the Feynman path integral method enables the calculation of the time-dependent wave function. Because the evolution of the probability density of the wave packet just after it exits the slits raises the issue of the interpreting the wave/particle dualism, we also simulate trajectories in the de Broglie-Bohm interpretation.


I. INTRODUCTION
In 1802, Thomas Young (1773-1829, after observing fringes inside the shadow of playing cards illuminated by the sun, proposed his well-known experiment that clearly shows the wave nature of light. 1 He used his new wave theory to explain the colours of thin films (such as soap bubbles), and, relating colour to wavelength, he calculated the approximate wavelengths of the seven colours recognized by Newton. Young's double slit experiment is frequently discussed in textbooks on quantum mechanics. 2 Two-slit interference experiments have since been realized with massive objects, such as electrons, 3,4,5,6 neutrons, 7,8 cold neutrons, 9 atoms, 10 and more recently, with coherent ensembles of ultra-cold atoms, 11,12 and even with mesoscopic single quantum objects such as C 60 and C 70 . 13,14 This paper discusses a numerical simulation of an experiment with ultracold atoms realized in 1992 by F. Shimizu, K. Shimizu, and H. Takuma. 11 The first step of this atomic interference experiment consisted in immobilizing and cooling a set of neon atoms, mass m = 3.349 × 10 −26 kg, inside a magneto-optic trap. This trap confines a set of atoms in a specific quantum state in a space of 1 mm, using cooling lasers and a non-homogeneous magnetic field. The initial velocity of the neon atoms, determined by the temperature of the magneto-optic trap (approximately T = 2.5 mK) obeys to a Gaussian law with an average value equal to zero and a standard deviation σ v = k B T m 1 m/s; k B is Boltzman's constant.
To free some atoms from the trap, they were excited with another laser with a waist of 30 µm. Then, an atomic source whose diameter is about 3 × 10 −5 m in and 10 −3 m in the z direction was extracted from the magneto-optic trap. A subset of these free neon atoms start to fall, pass through a double slit placed at l 1 = 76 mm below the trap, and strike a detection plate at l 2 = 113 mm. Each slit is b = 2 µm wide, and the distance between slits, center to center, is d = 6 µm. In what follows, we will call "before the slits" the space between the source and the slits, and "after the slits" the space on the other side of the slits. The sum of the atomic impacts on the detection plate creates the interference pattern shown in Fig. 1.
The first calculation of the wave function double slit experiment using electrons 4 was done using the Feynman path integral method. 15 However, this calculation has some limitations. It covered only phenomena after the exit from the slits, and did not consider realistic slits. The slits, which could be well represented by a function G(y) with G(y) = 1 for −β ≤ y ≤ β and G(y) = 0 for |y| > β, were modeled by a Gaussian function G(y) = e −y 2 /2β 2 . Interference was found, but the calculation could not account for diffraction at the edge of the slits. Another simulation with photons, with the same approximations, was done recently. 16 Recently, some interesting simulations of the experiments on single and double slit diffraction of neutrons 9 were done. 17 The simulations discussed here cover the entire experiment, beginning with a single source of atoms, and treat the slit realistically, also considering the initial dispersion of the velocity.
We will use the Feynman path integral method to calculate the time-dependent wave function. The calculation and the results of the simulation are presented in Sec. II. The evolution of the probability density of the wave packet just after the slits raises the question of the interpretation of the wave/particle dualism. For this reason, it is interesting to simulate the trajectories in the de Broglie 18 and Bohm 19 formalism, which give a natural explanation of particle impacts. These trajectories are discussed in Sec. III.

II. CALCULATION OF THE WAVE FUNCTION WITH FEYNMAN PATH IN-TEGRAL
In the simulation we assume that the wave function of each source atom is Gaussian in x and y (the horizontal variables perpendicular and parallel to the slits) with a standard deviation σ 0 = σ x = σ y = 10 µm. We also assume that the wave function is Gaussian in z (the vertical variable) with zero average and a standard deviation σ z 0.3 mm. The origin (x = 0, y = 0, z = 0) is at the center of the atomic source and the center of the Gaussian.
The small amount of vertical atomic dispersion compared to typical vertical distances, ∼ 100 and 200 mm, allows us to make a few approximations. Each source atom has an initial velocity v = (v 0x , v 0y , v 0z ) and wave vector k = (k 0x , k 0y , k 0z ) defined as k = mv/ .
We choose a wave number at random according to a Gaussian distribution with zero average and a standard deviation σ k = σ kx = σ ky = σ kz = mσ v / √ 3 2 × 10 8 m −1 , corresponding to the horizontal and vertical dispersion of the atoms' velocity inside the cloud (trap). For each atom with initial wave vector k, the wave function at time t = 0 is The calculation of the solutions to the Schrödinger equation were done with the Feynman path integral method, 20 which defines an amplitude called the kernel. The kernel characterizes the trajectory of a particle starting from the point α = (x α , y α , z α ) at time t α and arriving a at the point β = (x β , y β , z β ) at time t β . The kernel is a sum of all possible trajectories between these two points and the times t α and t β .
Using the classical form of the Lagrangian L(ẋ,ẏ,ż, z, t) = mẋ Feynman 20 defined the kernel by with +∞ −∞ +∞ −∞ K(β, t β ; α, t α ) dx α dy α dz α = 1. Hence, with For each atom with initial wave vector k, let us designate by ψ(α, t α ; k) the wave function at time t α . We call S the set of points α where this wave function does not vanish. It is then possible to calculate the wave function at a later time t α at points β such that there exits a straight line connecting α and β for any point α ∈ S. In this case, Feynman 20 has shown that: For the double slit experiment, two steps are then necessary for the calculation of the wave function: a first step before the slits and a second step after the slits.
If we substitute Eqs. (1) and (4) in Eq. (6), we see that Feynman's path integral allows a separation of variables, that is, References 11 and 21 treat the vertical variable z classically, which is shown in Appendix A to be a good approximation. Hence, we have z(t) = z 0 + v 0z t + gt 2 /2. The arrival time of the wave packet at the slits is For v 0z = 0 and z 0 = 0, we have t 1 = 2l 1 g = 124 ms and the atoms have been accelerated to v z1 = gt 1 = 1.22 m/s on average at the slit. Thus the de Broglie wavelength λ = /mv z1 = 1.8 × 10 −8 m is two orders of magnitude smaller than the slit width, 2µm.
Because the two slits are very long compared with their other dimensions, we will assume they are infinitely long, and there is no spatial constraint on y. Hence, we have for an initial fixed velocity v 0y : ψ y (y, t; k 0y ) = yα K y (y, t; y α , t α = 0)ψ 0 (y α ; k 0y ) dy α . Thus with s 0 (t) = σ 0 (1 + i t The wave packet is an infinite sum of wavepackets with fixed initial velocity. The probability density as a function of y is with Because we know the dependence of the probability density on y, in what follows we consider only the wave function ψ x (x, t; k 0x ).
A. The wave function before the slits Before the slits, we have It is interesting that the scattering of the wave packet in x is caused by the dispersion of the initial position σ 0 and by the dispersion τ of the initial velocity v 0x (see Fig. 2). Only 0.1% of the atoms will cross through one of the slits; the others will be stopped by the plate.

B. The wave function after the slits
The wave function after the slits with fixed z 0 and k 0z = mv 0z / for t ≥ t 1 (v 0z , z 0 ) is deduced from the values of the wave function at slits A and B (cf. Fig. 3) by using Eq. (6).
We obtain: with where ψ x (x a , t 1 (v 0z , z 0 ); k 0x ) and ψ x (x b , t 1 (v 0z , z 0 ); k 0x ) are given by Eq. (11) whereas The probability density is The arrival time t 2 of the center of the wave packet on the detecting plate depends on z 0 and v 0z . We have  Figure 4 shows the cross sections of the probability density (|ψ A +ψ B | 2 ) for z 0 = 0, v 0z = 0 (k 0z = 0) and for several distances (∆z = 1 2 gt 2 − 1 2 gt 2 1 ) after the double slit: 1 µm, 10 µ m, 0.1 mm, 0.5 mm, 1 mm and 113 mm.  Figure 4 shows the probability density (|ψ A | 2 +|ψ B | 2 ) for the same cases. Note that the difference between the two phenomena does not exist immediately at the exit of the two slits; differences appear only after some millimeters after the slits.

C. Comparison with the Shimizu experiment
In the Shimizu experiment, atoms arrive at the detection screen between t = t min and t max . To obtain the measured probability density in this time interval, we have to sum the probability density above the initial position z 0 and their initial velocity v 0z compatible with The positions at the detection screen can only be measured to about 80 µm, and thus to compare our results with the measured results. we perform the average

III. IMPACTS ON SCREEN AND TRAJECTORIES
In the Shimizu experiment the interference fringes are observed through the impacts of the neon atoms on a detection screen. It is interesting to simulate the neon atoms trajectories in the de Broglie-Bohm interpretation, 18,19 which accounts for atom impacts.
In this formulation of quantum mechanics, the particle is represented not only by its wave function, but also by the position of its center of mass. The atoms have trajectories which are defined by the speed v(x, y, z, t) of the center of mass, which at position (x, y, z) at time t is given by 24,25 v(x, y, z, t) = ∇S(x, y, z, t) m where ψ(x, y, z, t) = ρ(x, y, z, t) exp i S(x, y, z, t) and s is the spin of the particle. Let us see how this interpretation gives the same experimental results as the Copenhagen interpretation.
In both interpretations, ρ(x, y, z, t) = |ψ(x, y, z, t)| 2 is the probability density of the particles. But, in the Copenhagen interpretation, it is a postulate for each t (confirmed by (thanks to v = ∇S m + ∇ × ( ln ρ m )s), which is obviously the fluid mechanics equation of conservation of the density. The two interpretations therefore yield statistically identical results.
Moreover, the de Broglie-Bohm theory naturally explains the individual impacts.
In the initial de Broglie-Bohm interpretation, 18,19 which was not relativistic invariant, the velocity was not given by Eq. (18), but by v = ∇S m which does not involve the spin. In the Shimizu experiment, the spin of each neon atom in the magnetic trap was constant and vertical: s = (0, 0, 2 ). In our case the spin-dependent term ∇ log ρ m × s = 2mρ ∂ρ ∂y , − ∂ρ ∂x , 0 is negligible after the slit, but not before.
A. Trajectories before the slits Before the slits, Appendix B gives z(t) = z 0 . For a given wave vector k and an initial position (x 0 , y 0 , z 0 ) inside the wave packet, an atom of neon will arrive at a given position on the plate containing the slits. Notice that the term ∇ log ρ × s/m adds to the trajectory defined by ∇S/m a rotation of −π/2 around the spin axis (the z axis).
The source atoms do not all pass through the slits; most of them are stopped by the plate. Only atoms having a small horizontal velocity v 0x can go through the slits. Indeed an atom with an initial velocity v 0x and an initial position x 0 , y 0 arrives at the slits at t = t 1 at the horizontal position x(t 1 ) = v 0x t 1 + y 0 (σ 0 (t 1 )/σ 0 ). For this atom to go through one of the slits, it is necessary that |x(t 1 )| ≤ x, with x = (d + b)/2 = 4 × 10 −6 m and −2σ 0 ≤ y 0 ≤ 2σ 0 and t 1 0.124 s. Consequently, it is necessary that the initial velocities of the atoms satisfy |v 0x | ≤ (x + 2σ 0 (t 1 ))/t 1 = v 0x 3.9 × 10 −4 m/s. The double slit filters the initial horizontal velocities and transforms the source atoms after the slits into a quasi-monochromatic source. The system appears fully deterministic. If we know the position and the velocity of an atom inside the source, then we know if it can go through the slit or not. Figure 9 shows some trajectories of the source atoms as a function of their initial velocities. Only atoms with a velocity |v 0x | ≤ v 0x can go through the slits.

B. Velocities and trajectories after the slits
In what follows, we consider only atoms that have gone through one of the slits. After the slits, we still have z(t) = v 0z t + 1 2 gt 2 + z 0 (σ z (t)/σ z ), but now v x (t) and v y (t) and x(t) and y(t) have to be calculated numerically. The calculation of v x (x, t) is done by a numerical computation of an integral in x above the slits A and B (see Appendix B); x(t) is calculated acceleration. At the exit of the slit, ∆t is very small: ∆t 10 −8 s; it increases to ∆t 10 −4 s at the detection screen. Figure 10 shows the trajectories of the atoms just after the slits; x 0 and y 0 are drawn at random, z 0 = 0, with v 0x = v 0z = 0.

C. Impacts on the screen
We observe the impact of each particle on the detection screen as shown by the last image in Fig. 11. The classic explanation of these individual impacts on the screen is the reduction of the wave packet. An alternative interpretation is that the impacts are due to the decoherence caused by the interaction with the measurement apparatus. In the de Broglie-Bohm formulation of quantum mechanics, the impact on the screen is the position of the center of mass of the particle, just as in classical mechanics. Figure 11 shows our results for 100, 1000, and 5000 atoms whose initial position (x 0 , y 0 , z 0 ) are drawn at random. The last image corresponds to 6000 impacts of the Shimizu experiment. 11 The simulations show that it is possible to interpret the phenomena of interference fringes as a statistical consequence of a particle trajectories.

IV. SUMMARY
We have discussed a simulation of the double slit experiment from the source of emission, passing through a realistic double slit, and its arrival at the detector. This simulation is based on the solution of Schrödinger's equation using the Feynman path integral method. A simulation with the parameters of the 1992 Shimizu experiment produces results consistent with their observations. Moreover the simulation provides a detailed description of the phenomenon in the space just after the slits, and shows that interference begins only after 0.5 mm. We also show that it is possible to simulate the trajectories of particles by using the de Broglie-Bohm interpretation of quantum mechanics. from which we find Equation (B2) gives the classical trajectory if z 0 = 0 (the center of the wave packet).
The velocity (18) applied before the slit to Eqs. (9) and (11) gives the differential equations in the x and y directions: It then follows that with , and sin(ϕ 0 ) = y 0 √ where X A and X B are the centers of the two slits, and where f (x, u, t) = exp [(α + iβ t )u 2 + iγ x,t u] α = − 1 .

APPENDIX C: CONVERGENCE
In simulation, it is possible to make the Planck constant h tend towards zero. So, it is possible to show, about the example of the Young slits, the convergence of the quantum mechanics to the classical mechanics. In the following simulations, one take k 0x = k 0z = 0. The figure 12 shows, when h is divided by 100, how the interference fringes are getting strongly more and more narrow up to the distance between the slits.
The figure 13 shows, when h is divided by 1000, that the interference fringes disappear and the probability density converges on the classical probability density (h = 0).
The figure 14 shows how the trajectories become strongly narrower when h is divided by 100. After showing results of the evolution of the probability density and of the trajectories of atoms, it is difficult to throw down the hypothesis of trajectory.
Everything seems to happen also as if atoms have well a trajectory.

ACKNOWLEDGMENTS
We would like to thank the anonymous reviewers who provided valuable critiques and constructive suggestions for better presentation of the results.