Numerical simulation of the double slit interference with ultracold atoms Michel Gondrana) 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 ensembles of ultracold atoms,11,12 and even with mesoscopic single quantum objects such as C60 and C70 . 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 nonhomogeneous magnetic field. The initial velocity 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k B T/m⯝1 m/s; k B is Boltzmann’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 subset of these free neon atoms start to fall, pass through a double slit placed at ᐉ 1 ⫽76 mm below the trap, and strike a detection plate at ᐉ 2 ⫽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 experiment using electrons4 was done using the Feynman path 1

Am. J. Phys. 73 共3兲, March 2005

http://aapt.org/ajp

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 ⫺ ␤ ⭐y⭐ ␤ and G(y)⫽0 for 兩 y 兩 ⬎ ␤ , were modeled 2 2 by a Gaussian function G(y)⫽e ⫺y /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 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 Broglie18 and Bohm19 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 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 standard 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 and wave vector k velocity v⫽( v 0x , v 0y , v 0z ) ⫽(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 © 2005 American Association of Physics Teachers

1

K z 共 z ␤ ,t ␤ ;z ␣ ,t ␣ 兲 ⫽ ⫻exp

冉

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

K 共 ␤ ,t ␤ ; ␣ ,t ␣ 兲 ⬃exp ⫽exp

冉

i S 共 ␤ ,t ␤ ; ␣ ,t ␣ 兲 ប cl

冉冕 i ប

t␤

t␣

冊

共1兲

共2兲

冊

L 共 x˙ ,y˙ ,z˙ ,z,t 兲 dt ,

共3兲

⫹⬁ ⫹⬁ 兰 ⫺⬁ K( ␤ ,t ␤ ; ␣ ,t ␣ )dx ␣ dy ␣ dz ␣ ⫽1. Hence with 兰 ⫺⬁

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 with

冉

m K x 共 x ␤ ,t ␤ ;x ␣ ,t ␣ 兲 ⫽ 2i ␲ ប 共 t ␤ ⫺t ␣ 兲

冉

冊

共4兲

冉

m K y 共 y ␤ ,t ␤ ;y ␣ ,t ␣ 兲 ⫽ 2i ␲ ប 共 t ␤ ⫺t ␣ 兲 ⫻exp 2

冉

冊

冊

(x ␣ ,y ␣ ,z ␣ )苸S

K 共 ␤ ,t; ␣ ,t ␣ 兲

共5a兲

冊

im 共 y ␤ ⫺y ␣ 兲 2 , ប 2 共 t ␤ ⫺t ␣ 兲

共5b兲

共6兲

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. 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 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 t 1 ( v 0z ,z 0 ) ⫽ 冑2(ᐉ 1 ⫺z 0 )/g ⫹( v 0z /g) 2 ⫺ v 0z /g. For v 0z ⫽0 and z 0 ⫽0, we have t 1 ⫽ 冑2ᐉ 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 ␭⫽ប/m v z1 ⫽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

冋

␺ y 共 y,t;k 0y 兲 ⫽ 共 2 ␲ s 20 共 t 兲兲 ⫺1/4 exp ⫺

1/2

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冕

Thus

1/2

im 共 x ␤ ⫺x ␣ 兲 2 ⫻exp , ប 2 共 t ␤ ⫺t ␣ 兲

共5c兲

⫻ ␺ 共 ␣ ,t ␣ ;k 0x ,k 0y ,k 0z 兲 dx ␣ ,dy ␣ ,dz ␣ .

The calculation of the solutions to the Schro¨dinger 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 ␣ ⫽(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

Feynman20 defined the kernel by

冊

g2 共 t ⫺t 兲 3 . 24 ␤ ␣

␺ 共 ␤ ,t;k 0x ,k 0y ,k 0z 兲 ⫽

2 2 ⫻ 共 2 ␲␴ 20 兲 ⫺1/4e ⫺y /4␴ 0 e ik 0y y

x˙ 2 y˙ 2 z˙ 2 L 共 x˙ ,y˙ ,z˙ ,z,t 兲 ⫽m ⫹m ⫹m ⫹mgz. 2 2 2

冉

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, Feynman20 has shown that:

2 2 ⫽ 共 2 ␲␴ 20 兲 ⫺1/4e ⫺x /4␴ 0 e ik 0x x

2 2 ⫻ 共 2 ␲␴ z2 兲 ⫺1/4e ⫺z /4␴ z e ik 0z z .

冊

冊

1/2

im 共 z ␤ ⫺z ␣ 兲 2 im g exp 共 z ⫹z 兲 ប 2 共 t ␤ ⫺t ␣ 兲 ប 2 ␤ ␣

⫻ 共 t ␤ ⫺t ␣ 兲 ⫺

Fig. 1. Schematic configuration of the experiment.

冉

m 2i ␲ ប 共 t ␤ ⫺t ␣ 兲

⫹ik 0y 共 y⫺ v 0y t 兲

册

共 y⫺ v 0y t 兲 2 4 ␴ 0s 0共 t 兲

共9兲

with s 0 (t)⫽ ␴ 0 (1⫹ iបt/2m ␴ 20 ). The wave packet is an infinite sum of wave packets with fixed initial velocity. The probability density as a function of y is Michel Gondran and Alexandre Gondran

2

Fig. 2. Density ␳ x (x,z) before the slits. The time evolution is obtained from the figure by using z⫽z 0 ⫹ v 0z t⫹ gt 2 /2. Fig. 3. Schematic representation of the experiment: calculation method of the wave function after the slits.

␳ y 共 y,t 兲 ⫽

冕

⫹⬁

⫺⬁

2

2

共 2 ␲ ␶ 2 兲 ⫺1/2e ⫺k y /2␶ 兩 ␺ y 共 y,t;k y 兲 兩 2 dk y

⫽ 共 2 ␲ ␧ 20 共 t 兲兲 ⫺1/2e ⫺y

2 /2␧ 2 (t) 0

共10兲

and ␴ 20 (t)⫽ ␴ 20 with ␧ 20 (t)⫽ ␴ 20 (t)⫹(បt ␴ k /m) 2 2 2 ⫹( t/2m ␴ 0 ) . Because we know the dependence of the probability 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

冋

共 x⫺ v 0x t 兲 4 ␴ 0s 0共 t 兲

2

⫹ik 0x 共 x⫺ v 0x t 兲 ,

冋

␳ x 共 x,t 兲 ⫽ 共 2 ␲ ␧ 20 共 t 兲兲 ⫺1/2 exp ⫺

共11兲 x2

2␧ 20 共 t 兲

册

.

共12兲

B. Wave function after the slits The wave function after the slits with fixed z 0 and k 0z ⫽m v 0z /ប for t⭓t 1 ( v 0z ,z 0 ) 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 ,z 0 兲 ⫽ ␺ A ⫹ ␺ B

␺ A⫽

冕

A

共13兲

K x 共 x,t;x a ,t 1 共 v 0z ,z 0 兲兲

⫻ ␺ x 共 x a ,t 1 共 v 0z ,z 0 兲 ;k 0x 兲 dx a , 3

B

K x 共 x,t;x b ,t 1 共 v 0z ,z 0 兲兲

⫻ ␺ x 共 x b ,t 1 共 v 0z ,z 0 兲 ;k 0x 兲 dx b ,

Am. J. Phys., Vol. 73, No. 3, March 2005

共14a兲

共14b兲

where ␺ x (x a ,t 1 ( v 0z ,z 0 );k 0x ) and ␺ x (x b ,t 1 ( v 0z ,z 0 );k 0x ) are given by Eq. 共11兲, whereas K x (x,t;x a ,t 1 ( v 0z ,z 0 )) and K x (x,t;x b ,t 1 ( v 0z ,z 0 )) are given by Eq. 共5a兲. The probability density is

冕

⫹⬁

⫺⬁

共 2 ␲␴ k 2 兲 ⫺1/2

冉

⫻exp ⫺

It is interesting that the scattering of the wave packet in x is caused by the dispersion of the initial position ␴ 0 and 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.

with

冕

␳ x 共 x,t;k 0z ,z 0 兲 ⫽

␺ x 共 x,t;k 0x 兲 ⫽ 共 2 ␲ s 20 共 t 兲兲 ⫺1/4 exp ⫺

册

␺ B⫽

2 k 0x

2 ␴ k2

冊

兩 ␺ x 共 x,t;k 0x ,k 0z ,z 0 兲 兩 2 dk 0x .

共15兲 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 t 2 ⫽ 冑2(ᐉ 1 ⫹ᐉ 2 ⫺z 0 )/g ⫹( v 0z /g) 2 ⫺ v 0z /g. For z 0 ⫽0 and v 0z ⫽0, t 2 ⫽ 冑2(ᐉ 1 ⫹ᐉ 2 )/g⫽196 ms and the atoms are accelerated to v z2 ⫽gt 2 ⫽1.93 m/s. The calculation of ␳ x (x,t;k 0z ,z 0 ) at any (x,t) with k 0z and z 0 given and t⭓t 1 is done by a double numerical integration: 共a兲 Eq. 共15兲 is integrated numerically using a discretization 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 sections of the probability density ( 兩 ␺ A ⫹ ␺ B 兩 2 ) for z 0 ⫽0, v 0z ⫽0 (k 0z ⫽0) and for several distances (⌬z⫽ 21 gt 2 ⫺ 21 gt 21 ) 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 evolution of the probability density when both slits are simultaneously 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 Michel Gondran and Alexandre Gondran

3

Fig. 6. 共Color online.兲 Evolution of the probability density ␳ x (x,t;k 0z ⫽0,z 0 ⫽0) for the first millimeter 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 t min⭐t2⭐tmax , that is Fig. 4. Comparison between 兩 ␺ A ⫹ ␺ B 兩 2 共plain line兲 and 兩 ␺ A 兩 2 ⫹ 兩 ␺ B 兩 2 共dotted line兲 with z 0 ⫽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.

␳ x 共 x,t min⭐t⭐t max兲 ⫽

冕

t min⭐t 2 ⭐t max 2

2

␳ x 共 x,t 2 ;k 0z ,z 0 兲 2

2

⫻e ⫺k 0z /2␶ e ⫺z 0 /2␴ z dk 0z dz 0 .

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 measured results, we perform the average

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.

Figure 8 compares those calculations to the results found in

Fig. 5. 共Color online.兲 Evolution of the probability density ␳ x (x,t;k 0z ⫽0,z 0 ⫽0) from the source to the detector screen.

Fig. 7. 共Color online.兲 Evolution of the probability density ␳ x (x,t;k 0z ⫽0,z 0 ⫽0) for the first 100 ␮m after the slits.

4

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␳ 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.

Michel Gondran and Alexandre Gondran

共17兲

4

冉 冊

⳵␳ ⵜS ⫽0, ⫹ⵜ• ␳ ⳵t m

共21兲

with initial conditions S(x,y,z,0)⫽S 0 (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 experience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 density of the presence of particles without any postulate because Eq. 共21兲 becomes the continuity equation

⳵␳ ⫹ⵜ• 共 ␳ v兲 ⫽0 ⳵t

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

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 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,18,19,22,23,27 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 by24,25 v共 x,y,z,t 兲 ⫽ ⫽

ⵜS 共 x,y,z,t 兲 ⵜ log ␳ 共 x,y,z,t 兲 ⫻s ⫹ m m

冋

册

ប s Im共 ␺ * ⵜ ␺ 兲 ⫹Re共 ␺ * ⵜ ␺ 兲 ⫻ , m␳ 兩 s兩

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 interpretation. If ␺ satisfies the Schro¨dinger equation iប

⳵␺ ប2 2 ⫽⫺ ⵜ ␺ ⫹V ␺ ⳵t 2m

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 1 ប 2 䉭 冑␳ ⫹ ⫽0, 共 ⵜS 兲 2 ⫹V⫺ ⳵ t 2m 2m 冑␳ 5

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共20兲

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 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. 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 ⵜ log ␳/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 (x 0 ,y 0 ,z 0 ) of the particle inside its wave packet 共normal distribution f ((0,0,0);( ␴ 0 , ␴ 0 , ␴ z ))). The trajectories are given by 1 ⳵S ប ⳵␳ dx ⫽ v x 共 x,t 兲 ⫽ ⫹ , dt m ⳵ x 2m ␳ ⳵ y

共23a兲

ប ⳵␳ 1 ⳵S dy ⫽ v y 共 x,t 兲 ⫽ ⫺ , dt m ⳵ y 2m ␳ ⳵ x

共23b兲

1 ⳵S dz ⫽ v z 共 x,t 兲 ⫽ , dt m ⳵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 ␺ y are given by Eqs. 共8兲–共13兲. A. Trajectories before the slits Before the slits, Appendix B gives z(t)⫽z 0 ␴ z (t)/ ␴ z ⫹ v 0z t⫹ 12 gt 2 , x(t)⫽ v 0x t⫹ 冑x 20 ⫹y 20 ␴ 0 (t)/ ␴ 0 cos ␸(t), and with ␸ (t)⫽ ␸ 0 y(t)⫽ v 0y t⫹ 冑x 20 ⫹y 20 ␴ 0 (t)/ ␴ 0 sin ␸(t), ⫹arctan(⫺បt/2m ␴ 20 ), cos ␸0⫽x0 /冑x 20 ⫹y 20 and sin ␸0 ⫽y0 /冑x 20 ⫹y 20 . 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 ⵜ log ␳⫻s/m adds to the trajectory defined by ⵜS/m a rotation of ⫺ ␲ /2 around the spin axis 共the z axis兲. Michel Gondran and Alexandre Gondran

5

Fig. 9. Trajectories of atoms before the slit. Note that if the initial velocities 兩 v 0x 兩 ⭓3.9⫻10⫺4 m/s, then no atoms can cross the slit.

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 ( ␴ 0 (t 1 )/ ␴ 0 ). For this atom to go through one of the slits, it is necessary that 兩 x(t 1 ) 兩 ⭐x ¯ , with ¯x ⫽(d⫹b)/2 ⫽4⫻10⫺6 m and ⫺2 ␴ 0 ⭐y 0 ⭐2 ␴ 0 and t 1 ⯝0.124 s. Consequently, it is necessary that the initial velocities of the atoms ¯ ⫹2 ␴ 0 (t 1 ))/t 1 ⫽¯v 0x ⯝3.9⫻10⫺4 m/s. The satisfy 兩 v 0x 兩 ⭐(x double slit filters the initial horizontal velocities and transforms the source atoms after the slits into a quasimonochromatic source. The horizontal velocity of an atom leads to a horizontal shift of the atom’s impacts on the detection 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⫽t 2 ⫺t 1 ⯝0.072 s); hence ⌬x⯝2.8⫻10⫺5 m. This shift does not produce a blurring of the interference fringes because 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 ᐉ 1 ⫽5 mm, then ⌬x⯝10 ⫻10⫺5 m), the slit would not filter enough horizontal velocities 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.

Fig. 10. Trajectories of atoms 共k 0x ⫽k 0z ⫽0兲.

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 mechanics, 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 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 particle trajectories. IV. SUMMARY We have discussed a simulation of the double slit experiment from the source of emission, passing through a realistic

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⫹ 12 gt 2 ⫹z 0 ( ␴ z (t)/ ␴ 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 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 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. 6

Am. J. Phys., Vol. 73, No. 3, March 2005

Fig. 11. Zoom of trajectories of atoms for the first millimeters after the slit (k 0x ⫽k 0z ⫽0). Michel Gondran and Alexandre Gondran

6

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.

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 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.

冉

␺ z 共 z,t;k 0z 兲 ⫽ 共 2 ␲ s z2 共 t 兲兲 ⫺1/4 exp ⫺

冋 冉 冊册

⫻exp ⫺

共 z⫺ v 0z t⫺gt 2 /2兲 2 4 ␴ zs z共 t 兲

im 共 v 0z ⫹gt 兲共 z⫺ v 0z t/2兲 ប

mg 2 t 3 6

共A2兲

,

where s z (t)⫽ ␴ z (1⫹ iបt/2m ␴ z2 ). Consequently we have 兩 ␺ z 共 z,t; v 0z 兲 兩 2 ⫽ 共 2 ␲␴ z2 共 t 兲兲 ⫺1/2

APPENDIX A: CALCULATION OF ␺ Z „Z,T;K 0z …

␺ 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 7

Am. J. Phys., Vol. 73, No. 3, March 2005

冋

⫻exp ⫺

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

冊

共 z⫺ v 0z t⫺gt 2 /2兲 2

2 ␴ z2 共 t 兲

册

,

共A3兲

with ␴ z (t)⫽ 兩 s z (t) 兩 ⫽ ␴ z (1⫹(បt/2m ␴ z2 ) 2 ) 1/2. Note that ␴ z (t) is negligible compared to ᐉ 1 ( ␴ z ⫽0.3 mm and t/2m ␴ z ⫽10⫺3 mm are negligible compared to ᐉ 1 ⫽76 mm for an average crossing time inside the interferometer of t⬃200 ms). Therefore (2 ␲␴ z2 (t)) ⫺1/2 exp关⫺ (z ⫺v0zt⫺gt2/2) 2 /2␴ z2 (t) 兴 ⯝ ␦ 0 (z⫺ v 0z t⫺ gt 2 /2), and if z 0 is the initial position of the particle, we have z⯝z 0 ⫹ v 0z t ⫹ gt 2 /2 at time t. Michel Gondran and Alexandre Gondran

7

APPENDIX B: CALCULATION OF THE ATOM’S TRAJECTORIES

1

␣ ⫽⫺ 4 ␴ 20

The velocity 共18兲 applied to Eq. 共A2兲 gives the differential equation for the vertical variable z 1 ⳵S dz 共 z⫺ v 0z t⫺gt 2 /2兲 ប 2 t ⫽ v z 共 z,t 兲 ⫽ ⫽ v 0z ⫹gt⫹ dt m ⳵z 4m 2 ␴ z2 ␴ z2 共 t 兲 共B1兲

␤ t⫽

m 2ប

冉

冉 冉 冊冊 1⫹

បt 1

2

共B9兲

,

2m ␴ 20

1 ⫹ t⫺t 1

冉 冉

t 1 1⫹

1 2m ␴ 20 បt 1

冊 冊冊 2

,

共B10兲

from which we find 1 ␴ z共 t 兲 . z 共 t 兲 ⫽ v 0z t⫹ gt 2 ⫹z 0 2 ␴z

Equation 共B2兲 gives the classical trajectory if z 0 ⫽0 共the center 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 ប ⳵␳ 1 ⳵S dx ⫽ v x 共 x,t 兲 ⫽ ⫹ dt m ⳵ x 2m ␳ ⳵ y 共 x⫺ v 0x t 兲 ប 2 t

⫽ v 0x ⫹

4m 2 ␴ 20 ␴ 20 共 t 兲

⫺

ប 共 y⫺ v 0y t 兲 2m ␴ 20 共 t 兲

共 y⫺ v 0y t 兲 ប 2 t

4m

2

␴ 20 ␴ 20 共 t 兲

⫹

ប 共 x⫺ v 0x t 兲 2m ␴ 20 共 t 兲

,

共B3a兲

.

共B3b兲

It then follows that x 共 t 兲 ⫽ v 0x t⫹ 冑x 20 ⫹y 20

␴ 0共 t 兲 cos ␸ 共 t 兲 , ␴0

共B4a兲

y 共 t 兲 ⫽ v 0y t⫹ 冑x 20 ⫹y 20

␴ 0共 t 兲 sin ␸ 共 t 兲 ␴0

共B4b兲

with ␸ (t)⫽ ␸ 0 ⫹arctan(⫺ បt/2m ␴ 20 ), cos(␸0)⫽ x0 /冑x 20 ⫹y 20 , and sin(␸0)⫽ y0 /冑x 20 ⫹y 20 . Equations 共B4a兲 and 共B4b兲 give the classical trajectory if x 0 ⫽y 0 ⫽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 calculated using Eqs. 共6兲, 共14a兲, and 共14b兲. We obtain v x 共 x,t 兲 ⫽

冋

冉 冉 冊 冊册

C 共 x,t 兲 1 ⫺1 x⫹ ␤ Im t⫺t 1 H 共 x,t 兲 2 共 ␣ 2 ⫹ ␤ 2t 兲 t ⫹ ␣ Re

冉

冊

C 共 x,t 兲 ⫺ ␤ t ␥ x,t H 共 x,t 兲

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

u⫽X ⫹b

f 共 x,u,t 兲 du,

共B6兲

u⫽X ⫹b

共B7兲

C 共 x,t 兲 ⫽ 关 f 共 x,u,t 兲兴 u⫽X A ⫺b ⫹ 关 f 共 x,u,t 兲兴 u⫽X B ⫺b , A

B

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 u 2 ⫹i ␥ x,t u 兴 , 8

mx . ប 共 t⫺t 1 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

dy ប ⳵␳ 1 ⳵S ⫽ v y 共 x,t 兲 ⫽ ⫺ dt m ⳵ y 2m ␳ ⳵ x ⫽ v 0y ⫺

␥ x,t ⫽⫺

共B2兲

Am. J. Phys., Vol. 73, No. 3, March 2005

共B8兲

Electronic mail: [email protected] T. Young, ‘‘On the theory of light and colors,’’ Philos. Trans. R. Soc. London 92, 12– 48 共1802兲. 2 R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics 共Addison–Wesley, Reading, MA, 1966兲, Vol. 3. 3 C. J. Davisson and L. H. Germer, ‘‘The scattering of electrons by a single crystal of nickel,’’ Nature 共London兲 119, 558 –560 共1927兲. 4 C. Jo¨nsson, ‘‘Elektroneninterferenzen an mehreren ku¨nstlich hergestellten Feinspalten,’’ Z. Phys. 161, 454 – 474 共1961兲, English translation ‘‘Electron diffraction at multiple slits,’’ Am. J. Phys. 42, 4 –11 共1974兲. 5 P. G. Merlin, C. F. Missiroli, and G. Pozzi, ‘‘On the statistical aspect of electron interference phenomena,’’ Am. J. Phys. 44, 306 –307 共1976兲. 6 A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, ‘‘Demonstration of single-electron buildup of an interference pattern,’’ Am. J. Phys. 57, 117–120 共1989兲. 7 H. V. Halbon, Jr. and P. Preiswerk, ‘‘Preuve expe´rimentale de la diffraction des neutrons,’’ C. R. Acad. Sci. Paris 203, 73–75 共1936兲. 8 H. Rauch and A. Werner, Neutron Interferometry: Lessons in Experimental Quantum Mechanics 共Oxford University Press, London, 2000兲. 9 A. Zeilinger, R. Ga¨hler, C. G. Shull, W. Treimer, and W. Mampe, ‘‘Single and double slit diffraction of neutrons,’’ Rev. Mod. Phys. 60, 1067–1073 共1988兲. 10 I. Estermann and O. Stern, ‘‘Beugung von Molekularstrahlen,’’ Z. Phys. 61, 95–125 共1930兲. 11 F. Shimizu, K. Shimizu, and H. Takuma, ‘‘Double-slit interference with ultracold metastable neon atoms,’’ Phys. Rev. A 46, R17–R20 共1992兲. 12 M. H. Anderson, J. R. Ensher, M. R. Mattheus, C. E. Wieman, and E. A. Cornell, ‘‘Observation of Bose-Einstein condensation in a dilute atomic vapor,’’ Science 269, 198 –201 共1995兲. 13 M. Arndt, O. Nairz, J. Voss-Andreae, C. Keller, G. van des Zouw, and A. Zeilinger, ‘‘Wave-particle duality of C60 molecules,’’ Nature 共London兲 401, 680– 682 共1999兲. 14 O. Nairz, M. Arndt, and A. Zeilinger, ‘‘Experimental challenges in fullerene interferometry,’’ J. Mod. Opt. 47, 2811–2821 共2000兲. 15 C. Philippidis, C. Dewdney, and B. J. Hiley, ‘‘Quantum interference and the quantum potential,’’ Il Nuovo Cimento 52B, 15–28 共1979兲. 16 P. Ghose, A. S. Majumdar, S. Guha, and J. Sau, ‘‘Bohmian trajectories for photons,’’ Phys. Lett. A 290, 205–213 共2001兲. 17 A. S. Sanz, F. Borondo, and S. Miret-Artes, ‘‘Particle diffraction studied using quantum trajectories,’’ J. Phys.: Condens. Matter 14, 6109– 6145 共2002兲. 18 L. de Broglie, ‘‘La me´canique ondulatoire et la structure atomique de la matie`re et du rayonnement,’’ J. de Phys. 8, 225–241 共1927兲; L. de Broglie, Une Tentative d’Interpretation Causale et Non Lineaire de la Mecanique Ondulatoire 共Gauthier-Villars, Paris, 1951兲. 19 D. Bohm, ‘‘A suggested interpretation of the quantum theory in terms of ‘hidden’ variables,’’ Phys. Rev. 85, 166 –193 共1952兲. 20 R. Feynman and A. Hibbs, Quantum Mechanics and Integrals 共McGraw– Hill, New York, 1965兲, pp. 41– 64. 1

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C. Cohen-Tannoudji, 具http://www.lkb.ens.fr/cours/college-de-france/199293/9-2-93/9-2-93.pdf典. 22 S. Goldstein, ‘‘Quantum theory without observers. Part one,’’ Phys. Today 51共3兲, 42– 46 共1998兲; ‘‘Quantum theory without observers. Part two,’’ 51共4兲, 38 – 42 共1998兲. 23 M. Gondran, ‘‘Processus complexe stochastique non standard en me´canique,’’ C. R. Acad. Sci. Paris 333, 592–598 共2001兲; M. Gondran, ‘‘Schro¨dinger proof in minplus complex analysis,’’ quant-ph/0304096. 24 P. R. Holland, ‘‘Uniqueness of paths in quantum mechanics,’’ Phys. Rev. A 60, 4326 – 4331 共1999兲. 21

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M. Gondran and A. Gondran, ‘‘Revisiting the Schro¨edinger probability current,’’ quant-ph/0304055. 26 J. D. Lambert and D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem 共Wiley, New York, 1991兲, Chap. 5. 27 For a presentation of the de Broglie–Bohm interpretation of quantum mechanics equations, see D. Bohm and B. J. Hiley, The Undivided Universe 共Routledge, London, 1993兲; P. R. Holland, The Quantum Theory of Motion 共Cambridge U. P., Cambridge, 1993兲.

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