Numerical simulation of the double slit interference with ultracold atoms

Numerical simulation of the double slit interference with ultracold atoms

Michel Gondran^a)

EDF, Research and Development, 1 av. du General de Gaulle, 92140 Clamart, France Alexandre Gondran

Paris VI University, 60 av. Jean Jaures, 92190 Meudon, France 共Received 10 October 2003; accepted 17 December 2004兲

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 interpreting the wave/particle dualism, we also simulate trajectories in the de Broglie–Bohm interpretation. © 2005 American Association of Physics Teachers.

关DOI: 10.1119/1.1858484兴

I. INTRODUCTION

In 1802, Thomas Young 共1773–1829兲, 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 colors of thin films共such as soap bubbles兲, and, relating color to wavelength, he calculated the approximate wavelengths of the seven colors 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– 6}neutrons,^7,8 cold neutrons,⁹ atoms,¹⁰ and more recently, with coherent en- sembles of ultracold atoms,^11,12 and even with mesoscopic single quantum objects such as C₆₀and C₇₀.^13,14

This paper discusses a numerical simulation of an experi- ment with ultracold atoms realized in 1992 by F. Shimizu, K.

Shimizu, and H. Takuma.¹¹The first step of this atomic in- terference 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 nonhomogeneous magnetic field. The initial ve- locity of the neon atoms, determined by the temperature of the magneto-optic trap共approximately T⫽2.5 mK) obeys a Gaussian distribution with an average value equal to zero and a standard deviation␴v⫽

冑

^kBT/m⯝1 m/s; k_B is Boltz- mann’s constant.

To free some atoms from the trap, they were excited with another laser with a waist of 30␮m. Then, an atomic source whose diameter is about 3⫻10^⫺5 m and 10^⫺3 m in the z direction was extracted from the magneto-optic trap. A sub- set of these free neon atoms start to fall, pass through a double slit placed atᐉ1⫽76 mm below the trap, and strike a detection plate at ᐉ2⫽113 mm. Each slit is b⫽2␮^{m wide,} and the distance between slits, center to center, is d

⫽6␮m. 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 ex- periment using electrons⁴was done using the Feynman path

integral method.¹⁵ However, this calculation has some limi- tations. 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⫺␤⭐y⭐␤ ^{and G( y )}⫽0 for兩y兩⬎␤, were modeled by a Gaussian function G( y )⫽e^⫺y2/2␤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 Some interesting simulations of the experiments on single and double slit diffraction of neutrons⁹ were done.¹⁷

The simulations discussed here cover the entire experi- ment, 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 Broglie¹⁸and Bohm¹⁹ for- malism, which give a natural explanation of particle impacts.

These trajectories are discussed in Sec. III.

II. CALCULATION OF THE WAVE FUNCTION WITH FEYNMAN PATH INTEGRAL

In the simulation we assume that the wave function of each source atom is Gaussian in x and y 共the horizontal variables perpendicular and parallel to the slits兲with a stan- dard deviation ␴0⫽␴x⫽␴y⫽10␮m. We also assume that the wave function is Gaussian in z 共the vertical variable兲 with zero average and a standard deviation␴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/ប. We choose a wave number at random according to a Gaussian distribution with zero average and a standard deviation ␴k⫽␴k_x⫽␴k_y⫽␴k_z

⫽m␴v/)ប⯝2⫻10⁸ m^⫺1, corresponding to the horizontal

and vertical dispersion of the atoms’ velocity inside the cloud 共trap兲. For each atom with initial wave vector k, the wave function at time t⫽0 is

␺0共x, y ,z;k_0x,k_0y,k_0z兲⫽␺0_x共x;k_0x兲␺0_y共z;k_0y兲␺0_z共z;k_0z兲

⫽共2␲␴0

2兲^⫺1/4e^⫺x2/4␴02e^ik0xx

⫻共2␲␴0

2兲^⫺1/4e^⫺y2/4␴02e^ik0yy

⫻共2␲␴z

2兲^⫺1/4e^⫺z2/4␴z2e^ik0zz. 共1兲 The calculation of the solutions to the Schro¨dinger equa- tion 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␣⫽(x_␣, y_␣,z_␣) at time t_␣ and arriving at the point␤

⫽(x_␤, y_␤,z_␤) at time t_␤. The kernel is a sum of all possible trajectories between these two points and the times t_␣ and t_␤.

By using the classical form of the Lagrangian L共x˙, y˙ ,z˙,z,t兲⫽mx˙²

2 ⫹my˙² 2 ⫹mz˙²

2 ⫹mgz. 共2兲

Feynman²⁰ defined the kernel by

K共␤^,t␤;␣^,t␣兲⬃exp

冉

^{បi Scl共␤,t␤;␣,t␣兲}

冊

⫽exp

冉

^បi

^冕

^t␣t␤L共x˙, y˙ ,z˙,z,t兲dt

冊

^{, 共3兲}

with兰_⫺⬁^⫹⬁兰_⫺⬁^⫹⬁K(␤^,t_␤^;␣^,t_␣^)dx_␣^{d y}_␣^dz_␣⫽1. Hence K共␤^,t␤;␣^,t␣兲⫽K_x共x_␤,t_␤;x_␣,t_␣兲K_y共y_␤,t_␤; y_␣,t_␣兲

⫻K_z共z_␤,t_␤;z_␣,t_␣兲 共4兲 with

K_x共x_␤,t_␤;x_␣,t_␣兲⫽

冉

^{2i␲ប共mt}␤⫺t_␣兲

冊

^1/2

⫻expim

ប

冉

^{共2x共␤t⫺}␤⫺^xt^␣_␣^兲兲²

冊

^{, 共5a兲}

K_y共y_␤,t_␤;y_␣,t_␣兲⫽

冉

^{2i␲ប共mt}␤⫺t_␣兲

冊

^1/2

⫻expim

ប

冉

^{共2y共␤t⫺}␤⫺^yt^␣_␣^兲兲²

冊

^{, 共5b兲}

K_z共z_␤,t_␤;z_␣,t_␣兲⫽

冉

²ⁱ␲ប共^mt_␤⫺t_␣兲

冊

^1/2

⫻expim

ប

冉

^{共2z共␤t⫺}␤⫺^z␣t_␣^兲2兲

冊

^expimប

冉

^{g2共z␤⫹z␣兲}

⫻共t_␤⫺t_␣兲⫺g²

24共t_␤⫺t_␣兲³

冊

^{. 共5c兲}

For each atom with initial wave vector k, let us designate by␺⁽␣^,t␣;k) the wave function at time t_␣. We call S the set of points ␣ where this wave function does not vanish. It is then possible to calculate the wave function at a later time t_␣ at points ␤such that there exits a straight line connecting␣ and␤for any point␣苸S. In this case, Feynman²⁰has shown that:

␺共␤^,t;k0x,k_0y,k_0z兲

⫽

冕

(x_␣,y_␣,z_␣)苸S

K共␤^,t;␣^,t␣兲

⫻␺共␣^,t_␣^;k0x,k_0y,k_0z兲dx_␣,d y_␣,dz_␣. 共6兲 For the double slit experiment, two steps are then neces- sary for the calculation of the wave function: a first step before the slits and a second step after the slits.

If we substitute Eqs. 共1兲 and共4兲 in Eq. 共6兲, we see that Feynman’s path integral allows a separation of variables, that is

␺共x,y ,z,t;k_0x,k_0y,k_0z兲

⫽␺x共x,t;k_0x兲␺y共y ,t;k_0y兲␺z共z,t;k_0z兲. 共7兲 References 11 and 21 treat the vertical variable z classi- cally, which is shown in Appendix A to be a good approxi- mation. Hence, we have z(t)⫽z₀⫹v0zt⫹gt²/2. The arrival time of the wave packet at the slits is t₁(v0z,z₀)

⫽

冑

²⁽ᐉ1⫺z₀)/g⫹(v0z/g)²⫺v0z/g. For v0z⫽0 and z₀

⫽0, we have t₁⫽

冑

²ᐉ1/g⫽124 ms and the atoms have been accelerated to vz1⫽gt₁⫽1.22 m/s on average at the slit.

Thus the de Broglie wavelength ␭⫽ប/mvz1⫽1.8⫻10^⫺8 m is two orders of magnitude smaller than the slit width, 2␮^m.

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

␺y共y ,t;k_0y兲⫽

冕

y_␣

K_y共y ,t; y_␣,t_␣⫽0兲␺0共y_␣;k_0y兲dy_␣. 共8兲 Thus

␺y共y ,t;k_0y兲⫽共2␲^s0

2共t兲兲^⫺1/4exp

冋

^{⫺共4y␴⫺v0s0y0共tt兲兲2}

⫹ik_0y共y⫺v_0yt兲

册

^共9兲

with s₀(t)⫽␴0(1⫹iបt/2m␴0 2).

The wave packet is an infinite sum of wave packets with fixed initial velocity. The probability density as a function of y is

Fig. 1. Schematic configuration of the experiment.

␳y共y ,t兲⫽

冕

⫺⬁

⫹⬁共2␲␶²兲^⫺1/2e^⫺ky2/2␶2兩␺y共y ,t;k_y兲兩²dk_y

⫽共2␲␧0

2共t兲兲^⫺1/2e^{⫺y2/2␧02(t)} 共10兲 with ␧0

2(t)⫽␴0

2(t)⫹(បt␴k/m)² and ␴0 2(t)⫽␴0

2

⫹( t/2m␴0

2)². Because we know the dependence of the prob- ability density on y , in what follows we consider only the wave function ␺x(x,t;k_0x).

A. Wave function before the slits

Before the slits, we have

␺x共x,t;k_0x兲⫽共2␲^s0

2共t兲兲^⫺1/4exp

冋

^{⫺共4x⫺␴0vs0x0共tt兲兲2}

⫹ik_0x共x⫺v_0xt兲

册

^{, 共11兲}

␳x共x,t兲⫽共2␲␧0

2共t兲兲^⫺1/2exp

冋

^{⫺2␧x022共t兲}

册

^{. 共12兲}

It is interesting that the scattering of the wave packet in x is caused by the dispersion of the initial position␴0and by the dispersion ␴k of the initial velocity v_0x 共see Fig. 2兲. Only 0.1% of the atoms will cross through one of the slits; the others will be stopped by the plate.

B. Wave function after the slits

The wave function after the slits with fixed z₀ and k_0z

⫽mv0z/ប for t⭓t₁(v0z,z₀) is deduced from the values of the wave function at slits A and B 共see Fig. 3兲by using Eq.

共6兲. We obtain

␺x共x,t;k_0x,k_0z,z₀兲⫽␺A⫹␺B 共13兲 with

␺A⫽

冕

A

K_x共x,t;x_a,t₁共v_0z,z₀兲兲

⫻␺x共x_a,t₁共v0z,z₀兲;k_0x兲dx_a, 共14a兲

␺B⫽

冕

B

K_x共x,t;x_b,t₁共v_0z,z₀兲兲

⫻␺x共x_b,t₁共v_0z,z₀兲;k_0x兲dx_b, 共14b兲 where␺x(x_a,t₁(v_0z,z₀);k_0x) and␺x(x_b,t₁(v_0z,z₀);k_0x) are given by Eq. 共11兲, whereas K_x(x,t;x_a,t₁(v_0z,z₀)) and K_x(x,t;x_b,t₁(v_0z,z₀)) are given by Eq.共5a兲.

The probability density is

␳x共x,t;k_0z,z₀兲⫽

冕

⫺⬁

⫹⬁共2␲␴k2兲^⫺1/2

⫻exp

冉

^{⫺2k␴0x2k2}

冊

^{兩␺x共x,t;k0x,k0z,z0兲兩2dk0x.}

共15兲 The arrival time t₂ of the center of the wave packet on the detecting plate depends on z₀ and v0z. We have t₂

⫽

冑

²⁽ᐉ1⫹ᐉ2⫺z₀)/g⫹(v0z/g)²⫺v0z/g. For z₀⫽0 and v0z⫽0, t₂⫽

冑

²⁽ᐉ1⫹ᐉ2)/g⫽196 ms and the atoms are ac- celerated tovz2⫽gt₂⫽1.93 m/s.

The calculation of ␳x(x,t;k_0z,z₀) at any (x,t) with k_0z and z₀ given and t⭓t₁ is done by a double numerical inte- gration: 共a兲 Eq. 共15兲 is integrated numerically using a dis- cretization of k_0x into 20 values; 共b兲 the integration of Eq.

共13兲using Eqs.共14兲is done by a discretization of the slits A and B into 2000 values each. Figure 4 shows the cross sec- tions of the probability density (兩␺A⫹␺B兩²) for z₀⫽0, v_0z

⫽0 (k_0z⫽0) and for several distances (⌬z⫽¹2gt²⫺¹2gt₁²) after the double slit: 1 and 10␮m, and 0.1, 0.5, 1, and 113 mm.

The calculation method enables us to compare the evolu- tion of the probability density when both slits are simulta- neously open共interference:兩␺A⫹␺B兩²) with the sum of the evolutions of the probability density when the two slits are successively opened 共sum of two diffraction phenomena:

(兩␺A兩²⫹兩␺B兩²). Figure 4 shows the probability density (兩␺A兩²⫹兩␺B兩²) for the same cases. Note that the difference

Fig. 2. Density␳x(x,z) before the slits. The time evolution is obtained from the figure by using z⫽z0⫹v_0zt⫹gt²/2.

Fig. 3. Schematic representation of the experiment: calculation method of the wave function after the slits.

between the two phenomena does not exist immediately at the exit of the two slits; differences appear only after some millimeters after the slits.

Figures 5–7 show the evolution of the probability density.

At 0.1 mm after the slits, we know through which slit each atom has passed, and thus the interference phenomenon does not yet exist共see Figs. 4 and 7兲. Only at 1 mm after the slits do the interference fringes become visible, just as we would expect by the Fraunhoffer approximation共see Figs. 4 and 6兲.

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 z₀and their ini- tial velocity v0z compatible with t_min⭐t₂⭐t_max, that is

␳x共x,t_min⭐t⭐t_max兲

⫽

冕

t_min⭐t₂⭐t_max␳x共x,t₂;k_0z,z₀兲

⫻e^{⫺k0z2/2␶2}e^{⫺z02/2␴z2}dk_0zdz₀. 共16兲 The positions at the detection screen can only be measured to about 80␮m, and thus to compare our results with the mea- sured results, we perform the average

␳measured共x,t_min⭐t⭐t_max兲

⫽ 1

80 ␮^m

冕

x⫺40 ␮m x⫹40 ␮m

␳共u,t_min⭐t⭐t_max兲du. 共17兲 Figure 8 compares those calculations to the results found in

Fig. 4. Comparison between兩␺A⫹␺B兩²共plain line兲and兩␺A兩²⫹兩␺B兩²共dot- ted line兲with z₀⫽0 and k_0z⫽0 at共a兲1␮^m,共b兲10␮^m,共c兲0.1 mm,共d兲0.5 mm,共e兲1 mm, and共f兲113 mm after the slits.

Fig. 5. 共Color online.兲 Evolution of the probability density ␳x(x,t;k_0z

⫽0,z0⫽0) from the source to the detector screen.

Fig. 6. 共Color online.兲 Evolution of the probability density ␳x(x,t;k0z

⫽0,z0⫽0) for the first millimeter after the slits.

Fig. 7. 共Color online.兲 Evolution of the probability density ␳x(x,t;k_0z

⫽0,z0⫽0) for the first 100␮m after the slits.

Ref. 11. The experimental fringe separation is narrower than in our calculation, see Figs. 8 and 11. This difference is explained by a technical problem in the Shimizu experiment.¹¹

III. IMPACTS ON SCREEN AND TRAJECTORIES In the Shimizu experiment the interference fringes are ob- served through the impacts of the neon atoms on a detection screen. It is interesting to simulate the neon atoms’ trajecto- ries in the de Broglie–Bohm interpretation,18,19,22,23,27which 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共x,y ,z,t兲⫽ⵜS共x,y ,z,t兲

m ⫹ⵜlog␳共x,y ,z,t兲⫻s m

⫽ ប

m␳

冋

^{Im共␺*ⵜ␺兲⫹Re共␺*ⵜ␺兲⫻兩ss兩}

册

^{, 共18兲}

where ␺(x, y ,z,t)⫽

冑

␳(x,y ,z,t) exp((i/ប)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 inter- pretation.

If␺satisfies the Schro¨dinger equation iប⳵␺

⳵^{t ⫽⫺}

ប²

2mⵜ²␺⫹V␺ 共19兲

with the initial condition ␺^{(x,y ,z,0)}⫽␺0(x,y ,z)

⫽

冑

␳0(x,y ,z) exp((i/ប)S₀(x,y,z)), then␳and S satisfy

⳵^S

⳵^{t ⫹} 1

2m共ⵜS兲²⫹V⫺ ប² 2m

䉭

冑

␳

冑

␳ ^{⫽0, 共20兲}

⳵␳

⳵^{t ⫹ⵜ•}

冉

^␳ⵜmS

冊

^{⫽0, 共21兲}

with initial conditions S(x, y ,z,0)⫽S₀(x, y ,z) and

␳^{(x,y ,z,0)}⫽␳0(x, y ,z).

In both interpretations, ␳(x, y ,z,t)⫽兩␺(x, y ,z,t)兩² is the probability density of the particles. But, in the Copenhagen interpretation, it is a postulate for each t 共confirmed by ex- perience兲. In the de Broglie–Bohm interpretation, if

␳0(x,y ,z) is the probability density of presence of particles for t⫽0 only, then ␳(x,y ,z,t) must be the probability den- sity of the presence of particles without any postulate be- cause Eq.共21兲becomes the continuity equation

⳵␳

⳵^{t ⫹ⵜ•共}␳^v兲⫽0 共22兲

共thanks to v⫽ⵜS/m⫹ⵜ⫻(ln␳/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 ex- plains the individual impacts.

In the initial de Broglie–Bohm interpretation,^18,19 which was not relativistic invariant, the velocity was not given by Eq. 共18兲, but by v⫽ⵜS/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,ប/2).

In our case the spin-dependent term ⵜlog␳^/m⫻s

⫽ប/2m␳⁽⳵␳^/⳵^{y ,}⫺⳵␳^/⳵x,0) is negligible after the slit, but not before.

For the simulation, we choose at random 共from a normal distribution f (0,0,0;␴k,␴k,␴k) the wave vector k

⫽(k_0x,k_0y,k_0z) to define the initial wave function共1兲of the atom prepared inside the magneto-optic trap. For the de Broglie–Bohm interpretation, we also choose at random the initial position (x₀, y₀,z₀) of the particle inside its wave packet 共normal distribution f ((0,0,0);(␴0,␴0,␴z))). The trajectories are given by

dx

dt⫽vx共x,t兲⫽ 1 m

⳵^S

⳵^x⫹ ប 2m␳

⳵␳

⳵^{y, 共23a兲} dy

dt⫽vy共x,t兲⫽1 m

⳵^S

⳵^y⫺ ប 2m␳

⳵␳

⳵^{x, 共23b兲} dz

dt⫽vz共x,t兲⫽1 m

⳵^S

⳵^{z, 共23c兲}

where ␳^{(x,y ,t;k}0x,k_0y)⫽兩␺x(x,t;k_0x)␺y( y ,t;k_0y)兩² and ␺x

and␺yare given by Eqs.共8兲–共13兲. A. Trajectories before the slits

Before the slits, Appendix B gives z(t)⫽z₀␴z(t)/␴z

⫹v_0zt⫹¹2gt², x(t)⫽v_0xt⫹

冑

^x0

2⫹y₀²␴0(t)/␴0cos␸^{(t), and} y (t)⫽v0yt⫹

冑

^x0

2⫹y₀²␴0(t)/␴0sin␸^{(t), with} ␸^(t)⫽␸0

⫹arctan(⫺បt/2m␴0

2), cos␸0⫽x₀/

冑

^x0

2⫹y₀² and sin␸0

⫽y₀/

冑

^x0

2⫹y₀². For a given wave vector k and an initial po- sition (x₀, y₀,z₀) inside the wave packet, an atom of neon will arrive at a given position on the plate containing the slits. Notice that the termⵜlog␳⫻s/m adds to the trajectory defined by ⵜS/m a rotation of ⫺␲/2 around the spin axis 共the z axis兲.

Fig. 8. Comparison of the probability density measured experimentally by Shimizu共see Ref. 11兲 共left兲and the probability density calculated numeri- cally with our model共right兲.

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 velocityv_0xand an initial position x₀, y₀ arrives at the slits at t⫽t₁ at the horizontal position x(t₁)

⫽v_0xt₁⫹y₀(␴0(t₁)/␴0). For this atom to go through one of the slits, it is necessary that 兩x(t₁)兩⭐¯ , with xx ¯⫽(d⫹b)/2

⫽4⫻10^⫺6 m and⫺2␴0⭐y₀⭐2␴0and t₁⯝0.124 s. Conse- quently, it is necessary that the initial velocities of the atoms satisfy 兩v0x兩⭐(x¯⫹2␴0(t₁))/t₁⫽v¯_0x⯝3.9⫻10^⫺4 m/s. The double slit filters the initial horizontal velocities and trans- forms the source atoms after the slits into a quasi- monochromatic source. The horizontal velocity of an atom leads to a horizontal shift of the atom’s impacts on the de- tection screen. The maximum shift is ⌬x⫽v¯_0x⌬t, where⌬t is the time for the atom to go from the slits to the screen (⌬t⫽t₂⫺t₁⯝0.072 s); hence⌬x⯝2.8⫻10^⫺5 m. This shift does not produce a blurring of the interference fringes be- cause the interference fringes are separated from one another by 25⫻10^⫺5 mⰇ⌬x. Note that if the source was nearer to the double slit 共for example if ᐉ1⫽5 mm, then ⌬x⯝10

⫻10^⫺5 m), the slit would not filter enough horizontal ve- locities and consequently the interference fringes would not be visible.

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兩v_0x兩⭐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)

⫽v0zt⫹¹2gt²⫹z₀(␴z(t)/␴z), but now vx(t) and vy(t) and x(t) and y (t) have to be calculated numerically. The calcu- lation of vx(x,t) is done by a numerical computation of an integral in x above the slits A and B 共see Appendix B兲; x(t) is calculated with a Runge–Kutta method.²⁶ We use a time step ⌬t which is inversely proportional to the accel- eration. 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; x₀ and y₀ are drawn at random, z₀⫽0, with v0x⫽v0z

⫽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. 12. 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 mechan- ics, the impact on the screen is the position of the center of mass of the particle, just as in classical mechanics. Figure 12 shows our results for 100, 1000, and 5000 atoms whose ini- tial position (x₀, y₀,z₀) 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 phe- nomena of interference fringes as a statistical consequence of particle trajectories.

IV. SUMMARY

We have discussed a simulation of the double slit experi- ment from the source of emission, passing through a realistic

Fig. 9. Trajectories of atoms before the slit. Note that if the initial velocities

兩v0x兩⭓3.9⫻10^⫺4m/s, then no atoms can cross the slit.

Fig. 10. Trajectories of atoms共k0x⫽k0z⫽0兲.

Fig. 11. Zoom of trajectories of atoms for the first millimeters after the slit (k0x⫽k0z⫽0).

double slit, and its arrival at the detector. This simulation is based on the solution of Schro¨dinger’s equation using the Feynman path integral method. A simulation with the param- eters of the 1992 Shimizu experiment produces results con- sistent with their observations. Moreover, the simulation pro- vides 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 inter- pretation of quantum mechanics.

APPENDIX A: CALCULATION OF␺Z„Z,T;K_0z… Because there are no constraints on the vertical variable z, we find using Eqs.共5c兲and共6兲for all t⬎0 共before and after the double slit兲that

␺z共z,t;k_0z兲⫽

冕

S

K_z共z,t;z_␣,t⫽0兲⫻␺0_z共z_␣;k_0z兲dz_␣, 共A1兲 where the integration is done over the set S of the points z_␣, where the initial wave packet ␺0_z(z_␣;k_0z) does not vanish.

We obtain

␺z共z,t;k_0z兲⫽共2␲^sz

2共t兲兲^⫺1/4exp

冉

^{⫺共z⫺v40z␴tz⫺sz共gtt兲2/2兲2}

冊

⫻exp

冋

^imប

冉

^{共v0z⫹gt兲共z⫺v0zt/2兲}

⫺mg²t³

6

冊册

^{, 共A2兲}

where s_z(t)⫽␴z(1⫹iបt/2m␴z

2). Consequently we have

兩␺z共z,t;v0z兲兩²⫽共2␲␴z 2共t兲兲^⫺1/2

⫻exp

冋

^{⫺共z⫺v20z␴t⫺z2共tgt兲2/2兲2}

册

^{, 共A3兲}

with␴z(t)⫽兩s_z(t)兩⫽␴z(1⫹(បt/2m␴z 2)²)^1/2.

Note that ␴z(t) is negligible compared to ᐉ1 (␴z

⫽0.3 mm and t/2m␴z⫽10^⫺3 mm are negligible compared toᐉ1⫽76 mm for an average crossing time inside the inter- ferometer of t⬃200 ms). Therefore (2␲␴z

2(t))^⫺1/2exp关⫺(z

⫺v0zt⫺gt²/2)²/2␴z

2(t)兴⯝␦0(z⫺v0zt⫺gt²/2), and if z₀ is the initial position of the particle, we have z⯝z₀⫹v0zt

⫹gt²/2 at time t.

Fig. 12. Atomic impacts on the screen of detection. Results for 共a兲100, 共b兲 1000, and共c兲5000 atoms;共d兲the last image corresponds to 6000 impacts of the Shimizu experiment.

APPENDIX B: CALCULATION OF THE ATOM’S TRAJECTORIES

The velocity共18兲applied to Eq.共A2兲gives the differential equation for the vertical variable z

dz

dt⫽v_z共z,t兲⫽1 m

⳵^S

⳵^{z⫽v0z⫹gt⫹}

共z⫺v0zt⫺gt²/2兲ប²t 4m²␴z

2␴z 2共t兲

共B1兲 from which we find

z共t兲⫽v0zt⫹1

2gt²⫹z₀␴z共t兲

␴z

. 共B2兲

Equation共B2兲gives the classical trajectory if z₀⫽0 共the cen- ter of the wave packet兲.

The velocity 共18兲 applied before the slit to Eqs. 共9兲 and 共11兲gives the differential equations in the x and y directions

dx

dt⫽vx共x,t兲⫽ 1 m

⳵^S

⳵^x⫹ ប 2m␳

⳵␳

⳵^y

⫽v0x⫹共x⫺v_0xt兲ប²t 4m²␴0

2␴0

2共t兲⫺ប共y⫺v_0yt兲 2m␴0

2共t兲 , 共B3a兲

d y

dt⫽vy共x,t兲⫽1 m

⳵^S

⳵^y⫺ ប 2m␳

⳵␳

⳵^x

⫽v_0y⫺共y⫺v0yt兲ប²t 4m²␴0

2␴0

2共t兲⫹ប共x⫺v_0xt兲 2m␴0

2共t兲 . 共B3b兲

It then follows that

x共t兲⫽v0xt⫹

冑

^x0

2⫹y₀²␴0共t兲

␴0

cos␸共t兲, 共B4a兲

y共t兲⫽v0yt⫹

冑

^x0

2⫹y₀²␴0共t兲

␴0

sin␸共t兲 共B4b兲 with ␸^(t)⫽␸0⫹arctan(⫺បt/2m␴0

2), cos(␸0)⫽x₀/

冑

^x0 2⫹y₀², and sin(␸0)⫽y₀/

冑

^x0

2⫹y₀². Equations 共B4a兲 and 共B4b兲 give the classical trajectory if x₀⫽y₀⫽0 共the center of the wave packet兲.

After the slits, the velocity v_x(x,t)

⫽ប/m Im(⳵␺^/⳵^x␺*)/␺␺* given by Eq. 共18兲 can be calcu- lated using Eqs.共6兲,共14a兲, and共14b兲. We obtain

v_x共x,t兲⫽ 1

t⫺t₁

冋

^{x⫹2共␣⫺2⫹1␤t2兲}

冉

^␤tIm

冉

^{HC共共x,tx,t兲兲}

冊

⫹␣^Re

冉

^{HC共共x,tx,t兲兲}

冊

^{⫺␤t␥x,t}

冊 册

^共B5兲

with

H共x,t兲⫽

冕

X_A⫺b X_A⫹b

f共x,u,t兲du⫹

冕

X_B⫺b X_B⫹b

f共x,u,t兲du, 共B6兲

C共x,t兲⫽关f共x,u,t兲兴_u⫽X

A⫺b u⫽X_A⫹b

⫹关f共x,u,t兲兴_u⫽X

B⫺b u⫽X_B⫹b

, 共B7兲 where X_A and X_B are the centers of the two slits, and where f共x,u,t兲⫽exp关共␣⫹i␤t兲u²⫹i␥x,tu兴, 共B8兲

␣⫽⫺ 1

4␴0

2

冉

^1⫹

冉

^2mបt␴102

冊

²

冊

^{, 共B9兲}

␤t⫽m

2ប

冉

^t⫺1t1⫹t1

冉

^1⫹

冉

^{12mបt␴102}

冊

²

冊 冊

^{, 共B10兲}

␥x,t⫽⫺ mx

ប共t⫺t₁兲. 共B11兲

ACKNOWLEDGMENTS

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

a兲Electronic mail: michel.gondran@chello.fr

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