bsc dissertation: 'quantum reflection of ultra-cold atoms

8/14/2019 BSc Dissertation: 'Quantum Reflection of Ultra-Cold Atoms'

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Quantum Reflection of

Ultra-Cold Atoms

F33SLT (10 Credits)

Theoretical Project S

Spring 2007/2008

School of Physics & Astronomy

University of Nottingham

Jonathan D. Algar (4036480)

Project Supervisor: Professor T.M. Fromhold

Project Partner: Aaron D. Crute

April 2008

I have read and understood the School and University guidelines on plagiarism.

I confirm that this work is my own, apart from the acknowledged references.

1



Abstract

We present a theoretical investigation into the reflection properties of

a free particle in one dimension acting under the Time Dependent

Schrodinger Equation (TDSE). We study the dynamics of the parti-

cle incident on both a constant potential step and the Casimir-Polder

potential to model an atom-surface interaction.

We develop this methodology further to study the dynamics of a Bose-

Einstein condensate (BEC) under the Gross-Pitaevskii (GP) Equation.

The form of the Casimir-Polder potential is found to have a profound

effect on the reflection properties of the BEC.

2



3.1.5 Casimir-Polder Potential (Surface Interaction) . . . . . . . 32

3.2 Bose-Einstein Condensate (BEC) . . . . . . . . . . . . . . . . . . 36

3.2.1 Time Evolution of the Wavefunction ψ(x, t) . . . . . . . . 36

3.2.2 Positive Potential Step . . . . . . . . . . . . . . . . . . . . 36

3.2.3 Casimir-Polder Potential (Surface Interaction) . . . . . . . 37

4 Conclusions 41

4.1 Limitations of Project . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.2 Surface interactions . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Directions for Further Research . . . . . . . . . . . . . . . . . . . 43

4.2.1 Higher Dimensional Analysis . . . . . . . . . . . . . . . . . 43

4.3 Project Management Skills . . . . . . . . . . . . . . . . . . . . . . 43

A MATLAB CODE 46

4



1 Introduction

Our investigation is motivated by developing a theoretical framework for under-

standing and expanding on key theoretical results obtained by Fromhold et al.

[1]. We limit our studies in this project to one dimension but include a discussion

of generalizing to higher dimensions in the Conclusion (4.2.1). Our investigation

is split into two major components.

1.1 A Single Non-Relativistic Particle

The first concerns the dynamics of a non-relativistic particle with zero spin and

mass mass m. The time evolution of the wavefunction ψ (r, t) of the particle is

described by the TDSE

Hψ (r, t) = i ∂ψ

∂t (r, t) (1.1)

where denotes Plank’s constant and i gives the imaginary number. The total

energy of the system, the Hamiltonian ( H ), is given by the sum of the kinetic

(T ) and potential (V ) energy terms

H = (T + V ) = − 2

2m∇2 + V (r) (1.2)

This can be expressed as a second order partial differential equation

− 2

2m

∂

2

ψ∂x2 + ∂

2

ψ∂y2 + ∂

2

ψ∂z 2

+ V (x)ψ = i ∂ψ∂t (1.3)

We are concerned with the time evolution in one dimensional space. The

equation can be reduced to

− 2

2m

∂ 2ψ

∂x2 + V ψ = i

∂ψ

∂t (1.4)

5



where now ψ is a function of x and t.

We solve this equation by using a numerical method called the Crank-

Nichelson scheme also used by Fromhold et al. [1]

Using the potential term V (x) we form a potential step at a well defined point

xs in the domain of the particle

V (x) =

V, xs < 0

0, xs ≥ 0

(1.5)

and investigate how the reflection probability (R) depends on the size of thepotential (V ) and the velocity of the incident particle (v). We move on to in-

vestigate the reflection dynamics for a free particle incident on a Casimir-Polder

potential. This requires the derivation of an appropriate form of V (x) to describe

the atom-surface interaction.

1.2 Bose-Einstein Condensate (BEC)

Once we have understood how a single particle interacts with a surface we take

the investigation on to a collection bosons forming a BEC. The quantum evo-

lution of a such system is described by the Time Dependent Gross-Pitaevskii

Equation (TDGPE) or Gross-Pitaevskii (GP) for short. This is a nonlinear equa-

tion which takes into account the interaction between the bose particles. As the

bose particles are in the same quantum state they can be described by the same

wavefunction [3]. Denote this ψ (r). The GP equation takes the form

i ∂ψ(r)

∂t =

−

2

2m∇2 + V (r) + g|ψ(r)|2

ψ(r) (1.6)

where

g = 4π 2a

m (1.7)

6



is the coupling constant of the interacting bosons and a is the so called s-

scattering length. ψ(r) is the time dependent condensate wavefunction normal-

ized such that |ψ(r)| is the number of atoms per unit volume. As before, our

investigation will be limited to the case of dimensional space. Then the GP

equation reduces to

i ∂ψ

∂t =

−

2

2m

∂ 2

∂x2 + V (x) + g|ψ|2

ψ (1.8)

where now ψ is a function of x and t.

Note that if the coupling constant g = 0 we recover the TDSE.Given the very similar form of the GP Equation to the TDSE we proceed by

modifying our existing Crank-Nichelson scheme.

This places us in a position to analyze the reflection dynamics of a Bose-

Einstein Condensate interacting with a surface by using the Casimir-Polder po-

tential.

1.3 Computational Tools

Throughout the project we have used MATLAB1, a proprietary package for nu-

merical analysis and algorithmic development. This was identified as the language

of choice as it has powerful routines available for inverting matrices. MATLAB

also has graphical functionality which will be useful for analysis of our results.

The alternative was the C programming language which does not have built func-

tions readily available for inverting large matrices. Although many libraries are

available for matrix computation in C, we could not justify learning how to im-

plement them given the time constrains. The University of Nottingham has a

site wide academic license for MATLAB. Cost was not an issue. An additional

1The MathWorks Ltd. Tel. +44 (0)1223 226 700 E-mail: [email protected]

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consideration may have been cost if we were working on a similar project in a

commercial environment.

2 Theoretical Method

2.1 Stationary Wavefunction ψ(x, t = 0)

Before proceeding to present a formal solution to the TDSE we briefly discuss

the stationary case (t = 0). A free particle with zero spin and mass mass m is

described by a Gaussian wavefunction of the form

ψ(x, t = 0) = C exp[−(x − x0)2/σ2]exp[ik0x] (2.1)

The center of the wavefunction is at x0 and has associated width σ. The

normalization constant C is obtained by ensuring the wavefunction is properly

normalizedF

p(−∞ < x < ∞) =

∞

−∞

| ψ |2 dx = 1 (2.2)

It is a standard derivation [4] that the wave has average velocity v0 = k0/m

where k0 denotes the wave vector. We now have an inital condition for the TDSE

at t = 0.

This was translated into MATLAB by initializing all the constants and then

setting up a space of values of x over which to evaluate the function. The nor-

malization constant C is obtained numerically by summing over all values in the

space

∞

−∞

| ψ |2 dx =xmaxxmin

| ψ |2 dx = 1 (2.3)

The complete code is listed in Appendix A under inital.m.

8



2.2 Time Evolution of the Wavefunction ψ(x, t)

We introduce time dependence to the Gaussian wavefunction [5] and obtain the

form

ψ(x, t) = C

α exp[−(x− x0 −

p0t

m )2/α2] exp[ik0(x−

p0t

m )] (2.4)

where

α = (σ2 + i t

m )

1

2 . (2.5)

and the other constants are as above. As a check let t = 0

ψ(x, t = 0) = C

σ exp[−(x − x0)2/σ2]exp[ik0x] (2.6)

as required. The evolution of the wavefunction is governed by the TDSE

− 2

2m

∂ 2ψ

∂x2 + V (x)ψ = i

∂ψ

∂t (2.7)

as discussed in the Introduction. This can be written in operator form as

Hψ = i ∂ψ

∂t (2.8)

where the Hamiltonian operator is defined as

H = − 2

2m

∂ 2

∂x2

+ V (x) (2.9)

Now (2.8) has the form of a first order differential equation in ψ(t). This has

formal solution [6]

ψ(x, t) = ψ(x, t = 0) exp−it H/ (2.10)

9



Take note that H is an operator and not just a number. Proceed none the

less and discretize space in steps of length ∆t. Now using (2.10) we obtain

ψ(x, t + ∆t) = ψ(x, t)exp−i∆t H/ (2.11)

Write the exponential factor in Cayley form

exp−i∆t H/ ≈ 1− i∆t H/2

1 + i∆t H/2 (2.12)

We use (2.11) to propagate the wave forward in time

ψ(x, t + ∆t) = 1 − i∆t H/2

1 + i∆t H/2 ψ(x, t) (2.13)

Then ψ(x, t + ∆t) satisfies the required stability condition given in [9]. Let us

check at this stage the wavefunction satisfies the normalization condition (2.2)

ψ(x, t + ∆t)∗ψ(x, t + ∆t)dx = 1 + i∆t H/2

1− i∆t H/2

ψ∗(x, t)1 − i∆t H/2

1 + i∆t H/2

ψ(x, t)dx

(2.14)

The quotient terms cancel and we obtain

ψ(x, t + ∆t)∗ψ(x, t + ∆t)dx =

ψ(x, t)∗ψ(x, t)dx = 1 (2.15)

by definition, as required. Now rearranging (2.12) yields

1 + i∆t H

2

ψ(x, t + ∆t) =

1 − i∆t H

2

ψ(x, t) (2.16)

Substituting back in the expression for H using (2.9)

1 +

i∆t

2

−

2

2m

∂ 2

∂x2 + V (x)

ψ(x, t+∆t) =

1−

i∆t

2

−

2

2m

∂ 2

∂x2 + V (x)

ψ(x, t)

(2.17)

10



ψ(m + 1, n + 1) = e(m, n)ψ(m, n + 1) + f (m, n) (2.22)

If we insert (2.22) into (2.20) we deduce that the e and f must take the form

e(m, n) = 2 + 2(∆x)2V (m) − 2iλ − 1

e(m − 1, n) (2.23)

and

f (m, n) = Ω(m, n) + f (m − 1, n)

e(m − 1, n) (2.24)

We confine our particle to a region of space such that the spatial index runs

from m = 0 to m = M and impose the boundary conditions ψ(0, n) = 0 and

ψ(M, n) = 0 ∀n ∈ Z+. That is to say the wavefunction vanishes at the boundary

of our space. Now equations (2.23) and (2.24) only apply in the interior of the our

system. We deduce from the boundary condition for the wavefunction at m = 0that the required expressions for e and f at this end of the system are

e(1, n) = 2 + 2(∆x)2V (1) − 2iλ (2.25)

and

f (1, n) = Ω(1, n) (2.26)

For the first time step, n = 0, the factors Ω(1, 0), e(1, 0) and f (1, 0) can all

be calculated explicitly from the initial wave function ψ(m, 0). Once we have

the values of e(1, 0) and f (1, 0) we use (2.23) and (2.24) to calculate e(m, 0) and

f (m, 0) for m = 0 to m = M .

12



Rearrange (2.22) to obtain

ψ(m, n + 1) = ψ(m + 1, n + 1) − f (m, n)

e(m, n) (2.27)

At the other end of the system where m = M the wavefunction also vanishes

ψ(M, n) = 0 ∀n and we obtain

ψ(M − 1, n + 1) = ψ(M, n + 1) − f (M − 1, n)

e(M − 1, n) = −

f (M − 1, n)

e(M − 1, n) (2.28)

We use (2.27) to obtain ψ(M − 1, 0) and then traverse the system backwards

from large to small m.

We now proceed to implement the algorithm in the MATLAB environment.

The methodology we derived above uses the convention that the time (m) and

space (n) indices start at 0. However in MATLAB the first element of a vector

is indexed by 1. To ensure consistency we define z = n + 1 and y = m + 1

throughout our MATLAB code

1. We begin by initializing a wavefunction ψ(m, 0). As we are considering the

dynamics of a free particle we use consider a Gaussian wavefunction:

z = 1; % corresponds to t = 0

alpha = sqrt(sigma^2 + i*hbar*t(z)/mass); psi = (1/alpha) *

exp((-(x-x_0-(p_0*t(z)/mass)).^2)/(alpha^2)) ...

.* exp((i*k_0*(x-(p_0*t(z))/2*mass)));

We then proceed to normalize using

% compute normalization constant

C = 1/sqrt(sum((psi .* conj(psi)) * dx));

psi = C * psi; % normalized wavefunction

13



2. The system is traversed from small m to large m and the functions e(m, 0)

and f (m, 0) are calculated using (2.23) and (2.24) initially

% calculate initial e value

e(2) = 2 + (2*dx^2*V(2)) - 2*i*lambda;

% calculate initial f value

f(2) = -psi(3) + (2*i*lambda + (2*dx^2*V(2)) + 2)*psi(2) - psi(1);

and (2.23) and (2.24) thereafter

for y = 3:(x_size-1)

e(y) = 2 + (2*dx^2*V(y)) - 2*i*lambda - 1/e(y - 1);

f(y) = -psi(y + 1) + (2*i*lambda + (2*dx^2*V(y)) + 2) ...

*psi(y) - psi(y-1) + f(y-1)/e(y-1);

end

3. The system is then traversed from large m to small m and ψ(m, 1) is cal-

culated using (2.28) initally

psi(x_size-1) = -f(x_size-1)/e(x_size-1);

and then (2.27) thereafter

y = x_size - 2;

while y > 1

psi(y) = (psi(y + 1) - f(y))/e(y) ;

y = y - 1;

end

14



This completes one iteration of the algorithm and yields the wavefunction

at t = ∆t.

4. We obtain the ψ as a function of time for n ≥ 1 by iterating over all m

(time) values

for z = 1:(t_size-1)

2.3 Notes on Animation in MATLAB

A crucial part in analyzing reflection behavior throughout the project has been

animating the dynamics of the particle in real time. MATLAB has powerful

animation built in for. To produce the animations in the project we used the

following general method based on [8]

1. Plot the inital wavefunction:

nframes = 1;

Frames = moviein(nframes); % plot inital wavefunction

plot(x,psi.*conj(psi),’k’) xlabel(’x’);

ylabel(’psi*psi*’);

axis([0 max(x) 0 max(psi*psi*)]);

k = 1 ;

Frames(:,k) = getframe;

2. Now plot the wavefunction for every 10th time step

for z = 1:(t_size-1) ...

% produce required graphical output (animation)

if mod(z,10) == 1

15



2.5 Potential Barriers

2.5.1 Free Particle

A free particle is one for which the potential is defined as zero everywhere, V (x) =

0 for all x in our space. This is implemented in MATLAB as

for n = 1:x_size

V(n) = 0;

end

2.5.2 Constant Potential Step

A constant potential step is defined as

V (x) =

0, x < xs

V s, x ≥ xs

(2.30)

where V s is the magnitude of the potential after some well defined point. This isimplemented in MATLAB as

for n = 1:x_size

if x(n) < x_s

V(n) = 0;

elseif x(n) >= x_s

V(n) = V_s;

end

end

2.5.3 Casimir-Polder Potential

The Casimir Polder potential is given in [1] as

17



i ∂ψ

∂t = Hψ (2.34)

where

H = − 2

2m

∂ 2

∂x2 + V (x) + g|ψ|2 (2.35)

Now using (2.16) we substitute in our expression for H

1 + i∆t

2 − 2

2m

∂ 2

∂x2 + V (x) + g|ψ|2ψ(x, t + ∆t)

=

1 −

i∆t

2

−

2

2m

∂ 2

∂x2 + V (x) + g|ψ|2

ψ(x, t) (2.36)

We proceed, as with the TDSE, to convert everything to finite difference form

ψ(m + 1, n + 1) + [2iλ − 2(∆x)2V (m) − 2(∆x)2g|ψ|2 − 2]ψ(m, n + 1) + ψ(m − 1, n + 1)

= −ψ(m + 1, n) + [2iλ + 2(∆x)2V (m) + 2(∆x)2g|ψ|2 + 2]ψ(m, n) − ψ(m − 1, n)

Now define

Ω(m, n) = −ψ(m+1, n)+[2iλ+2(∆x)2V (m)+2(∆x)2g|ψ|2+2]ψ(m, n)−ψ(m−1, n)

(2.37)

from which we obtain expressions for e and f of the form

e(m, n) = 2 + 2(∆x)2V (m) + 2(∆x)2g|ψ|2 − 2iλ − 1

e(m − 1, n) (2.38)

and

f (m, n) = Ω(m, n) + f (m − 1, n)

e(m − 1, n) (2.39)

19



Which yields

e(1, n) = 2 + 2(∆x)2V (1) + 2(∆x)2g|ψ|2 − 2iλ (2.40)

and

f (1, n) = Ω(1, n) (2.41)

A note here on the form of the potential. The total potential energy for each

atom of mass m in the Bose-Einstein Condensate is defined in [1] as

V (x) = V step + V trap (2.42)

where xs is defined as the region of rapid potential energy variation. V step

takes the form as defined in Section (2.5.1) (2.5.2) or (2.5.3). ωx determines the

potential energy of the harmonic trap

V trap = 1

2m[ω2

x(x− xs)2] (2.43)

Clearly an increase in ωx corresponds to an increase in the size of the harmonic

trap

for n = 1:x_size

... % define V_step(n)

V(n) = V_step(n) + (omega^2 * (x(n) - x_s)^2);end

We have simplifed this expression by ignoring V im [1] which is the term used

to model the absorbtion of the atoms by the surface.

One can then simply modify the MATLAB code by using the new expressions

for e and f to obtain an algorithm to give the time evolution of the wavefunction.

20



2.7 Variation of Quantities

The variational techniques we develop here are applicable for both a single particle

and a Bose-Einstein Condensate.

2.7.1 Variation of Velocity (v)

We developed a general method of understanding how R depends on the velocity

v of the incident wavefunction. This was achieved by looping over equally spaced

values of the velocity and recalculating the velocity dependent quantities within

the loop

for velocity = 2:16

v_0 = (1/8)*velocity*(dx/dt); % velocity

p_0 = mass * v_0; % momentum

k_0 = p_0/hbar; % wave vector

R = xs−∞

ψ(x, t)∗ψ(x, t)dx is then of calculated for each value value of v and

we plot each point

plot(velocity, R, ’x’); hold on;

The animation code was used to choose an appropriate range of values for v.

2.7.2 Variation of Constant Potential Step (V )

A similar method is used to measure R for an incident particle of constant v =

dx/dt incident on a Constant Potential Step. We now loop over equally spaced

values of potential

% loop over potentials

for potential_var = 1:30

21



% define potential

for n = 1:x_size

if x(n) < x_s

V(n) = 0;

else

V(n) = potential_var * V_s;

end

end

and calculate R = xs−∞

ψ(x, t)∗ψ(x, t)dx for each loop and plot each value

plot(potential_var, R, ’x’); hold on;

2.7.3 Variation of the Casimir-Polder Potential

Let us remind ourselves of the form of the Casimir-Polder potential

V (x) =

−C 4x3(x+3λa/2π2)

, x < xs − δ

−C 4(−δ)3(−δ+3λa/2π2)

, x ≥ xs − δ

(2.44)

where x = x − xs. The form of the potential depends directly on C 4 =

β 24 2/2m, δ and the effective atomic transition wavelength of the incoming particle

λeff . Clearly C 4 determines the amplitude of the potential over all points x. We

have already considered how the magnitude of the potential effects the reflection

probability R by varying the Constant Potential above. We now consider how

δ , which fundamentally changes the form of the potential, effects the reflection

probability R. We can then loop over a range of equally spaced values of δ

22



% loop over various delta

for delta_power=1:40

% constants used in Casimir-Polder potential

delta = delta_power*delta;

...

% define Casimir-Polder potential

for n = 1:x_size

if x(n) < x_s- delta

V(n) = -C_4/((x(n) - x_s)^3 * ((x(n) - x_s) + 3*lambda_eff/(2*pi^2)))

else

V(n) = -C_4/((-delta)^3 * ((-delta) + 3*lambda_eff/(2*pi^2)));

end

end

We calculate R = xs−∞

ψ(x, t)∗ψ(x, t)dx for each value of δ and plot each point

plot(delta_power, R, ’x’);

3 Results and Discussion

The caption of each figure contains a reference to which MATLAB file was used

to produce it. Complete code listings can be found in Appendix A. We present

the results and discussions in one chapter. As our project frequently uses pre-

vious results to produce new ones, this seems like the most appropriate format.

Otherwise we would be constantly be referring back to our results section within

the discussion. This, we feel, would be too distracting.

23



3.1 A Single Non-Relativistic Particle

3.1.1 Stationary Wavefunction ψ(x, t = 0)

Using (2.1) we plot the real R(ψ(x, t = 0)) and imaginary I(ψ(x, t = 0)) com-

poenents of the wavefunction ψ(x, t = 0) using the constants outlined in [6].

% constants

x_0 = 0.4; % input(’enter point of symmetry: ’);

sigma = sqrt(0.001); % input(’packet width: ’)

x_int = 10000; % input(’number of points in space: ’);

k_0 = 500; % input(’wave vector: ’);

t = 0; % inital time

24



Figure 1: inital.m OUTPUT

We then plot the probability density ψ(x, t = 0)ψ(x, t = 0)∗

25



Figure 2: inital.m OUTPUT

26



Figure 3: SCH step 0.m OUTPUT

This corresponds a travelling wave evolving in the correct temporal direction

with a constant velocity. We confirmed that the wavefunction was correctly

normalized at each step. The form of the probability density function follows two

trends with evolving time: The amplitude of the packet decreases and the width

of the packet increases. To ensure the wavefunction is correctly normalized this

is what we expect.

By the Heisenberg Uncertainty Principle

δxδp ≥ /2 (3.2)

where δx and δp correspond to the width of the packet in terms of space and

momentum respectfully. Now inserting the momentum expression p = k into

28



the algorithm required us to keep λ = 2(dx)2/(dt) invariant and meant the values

of t had to be reduced as well. This was not ideal as it corresponds to velocities

to the order of 109. Consequentially the results we present below do not strictly

correspond to those in the paper [1]. We present possible solutions to this problem

in the Conclusion (4.1.1).

3.1.4 Positive Potential Step

30



Figure 4: cntdse vary potential.m OUTPUT

We incident a single particle at speed v0 = dx/dt towards a positive postential

step and measure the probability of reflection. The average energy of the wave

packet is E packet = k20/2m = 2.00×1018. We see that as the size of the potential

increases past the average energy of the particle the proportion of the reflected

wavefunction R tends towards unity. We noted from the animated output that

for a particle which undergoes reflection, a stationary wavefunction forms, as

predicted.

Note that when E packet < V , classically the particle should undergo a to-

tal reflection. It is a purely quantum mechanical phenomena that the particle

undergoes partial transmittance and is called quantum tunnelling. [10].

31



Figure 5: cntdse vary velocity.m OUTPUT

We now analyze how the energy of the incident particle effects the probability

of reflection. This is achieved by modifying the incident velocity, v0, to give

E packet = k20/2m = ( p0/ )2/2m = 1

2mv2

0. The particle is then incident on a

constant potential of V = 2.3 × 1018. This makes sense. As v0 is increased the

kinetic energy of the particle can overcome the potential energy of the barrier.

The partial reflection is due to tunnelling, as described above.

3.1.5 Casimir-Polder Potential (Surface Interaction)

32



Figure 6: cntdse casimir.m OUTPUT

We use a value of δ = 0.05 × 10−6 to produce the form of the Casimir-

Polder Potential used in [1] to understand the atom-surface interaction. Note that

V (x) < 0∀x which corresponds to an attractive potential. Increasing the value of

δ increases the size of the slope and consequentially decreases the magnitude of

the potential step.

33



Figure 7: cntdse casimir vary velocity.m OUTPUT

We now investigate how the kinetic energy of the particle incident on a

Casimir-Polder Potential effects the reflection probability R. As the kinetic en-

ergy of the particle is increased the probability of reflection decreases. However

this time we see an exponential dependence of the speed v0 on the reflection prob-

ability R. This is an encouraging result as it corresponds to what was obtained

in [2]. We focus our investigation, as in [2], on values of 0 ≤ R ≤ 0.18. To obtain

values of R ≥ 0.18 would require us to take to use, relatively speaking, very small

velocities and thus vastly limit the range of values we can analyze in the graph.

34



Figure 8: cntdse casimir vary delta.m OUTPUT

We now vary the δ paramater and incident a particle at a fixed v = dx/dt.

In effect by modifying δ we are changing the form of the surface interaction.

Increasing δ corresponds to making the potential drop less steep and consequen-

tially reduces the probability of reflection. In the limit of δ → 0 we are effectively

modelling a negative potential step. We take note of some minor oscillatory be-

havior, which we put on hold until we investigate the surface interaction of a

Bose-Einstein condensate when we find these effects are exacerbated.

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3.2 Bose-Einstein Condensate (BEC)

3.2.1 Time Evolution of the Wavefunction ψ(x, t)

We move on to study how the wavefunction for the Bose-Einstein condensate

evolves with time. We used the animation code to study qualitatively how the

wavefunction evolution under the GP equation varies to that of the TDSE. We

saw that the amplitude of the packet decreases at a faster rate (corresponding

to the width of the packet increasing). Prof. Fromhold confirmed that it should

follow this general trend. If you wish to see the animations in action please visit

http://www.ug-home.nottingham.ac.uk/~ppyzjda/f33slt/animations/

3.2.2 Positive Potential Step

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Figure 9: cntdgp vary potential.m OUTPUT

We incident the Bose-Einstein Condensate at an average speed of v0 = dx/dt

onto a constant potential barrier. As for the free particle analysis, we varied the

size of the barrier to see what effect this would have on the reflection probability

R. This produces the pattern as was found with a single particle.

3.2.3 Casimir-Polder Potential (Surface Interaction)

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Figure 10: cntdgp casimar.m OUTPUT

To model to the interaction the Bose-Einstein Condensate has with the surface

we consider the Casimir-Polder Potential. The harmonic trap term ensures the

collection of atoms remains within a Condensate. This is a quadratic term and

corresponds to the quadratic slopes either side of the step. The form we display

above reproduced the form of potential used in [1].

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Figure 12: cntdgp varydelta.m

Finally and most importantly we want to understand how the form of the

atom-surface interaction effects the reflection probability for a Bose-Einstein

Condensate. As δ increases, the probability of reflection R appears to decay

exponentially. We obtain clear oscillation with peaks occurring at regular in-

tervals of δ = 8 × 1.25 × 10−4 = 10−3. We suspect it could be some form

of resonance. This cannot be explained by anything we have learnt in our

Elementry Quantum Physics courses. After research me and Aaron concluded

it may be explained by one of the following three phenomena:

1. Consider electrons incident on a rectangular potential barrier of length L.

“When particle-waves back-scattered from the step change in barrier poten-

tial at positions x = 0 and x = L interfere and exactly cancel each other,

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resulting in zero reflection from the potential barrier” Levi [10]. Perhaps

there exists values of δ such that the parts of the reflected wavefunction

cancel each other out.

2. We came across the concept of Feshbach Resonance [3]. “A Feshbach reso-

nance arises in cold atom scattering due to the complex interplay between

several coupled channels.” [11]. We thought it unlikely that our code had

manifested such a complex phenomena, but it is certainly something to

investigate further.

3. It could be a bug in the code! We would need experimental evidence to

confirm or dispel this.

4 Conclusions

As laid out in our objectives we have constructed a theoretical framework to

analyze the reflection effects of a Bose-Einstein Condensate incident on a surface.We have implemented the Crank-Nicelson algorithm in MATLAB to solve both

the TDSE and GP.

4.1 Limitations of Project

4.1.1 Units

We reached a stage in the project when we had working simulations for the

constants used in [6]. The algorithm that we developed used = m = 1 with the

other units in the book are presented in an apparently dimensionless form such as

v = 700. We used this approach as we wanted to verify that we had implemented

the Crank-Nichelson algorithm correctly.

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The next stage in the project required us to simulate experimental and re-

quired us to translate the units system to SI. We tried to rescale x, t and v0 to give

us appropriate units such that the velocities would be of order 10−

3. It became

apparent that the algorithm required us to retain the value of λ = 2(dx)2/(dt)

for it to work properly. This approach was therefore unworkable.

We tried to attack the problem by changing the values of and m to SI units

and then adapting the algorithm to include values of and m. We concluded

that this didn’t work due to different ratios of and m present in the algorithm,

/m and m/ 2 for example.

We tried to solve the problem using dimensional analysis but m = 1 corre-

sponds to the mass of a sodium atom which is not a fundamental unit.

We found no perfect solution to translate the units and concluded we must

come up with a compromise that still gave accurate results. This was an unfortu-

nate turn in the project, but a necessary decision had been made. We built our

existing code on strong foundations - using a scheme that that we could explicitly

verify was correct. Without completely scrapping the project and finding another

algorithm to solve the TDSE this is the best solution we could come up with.

If we were to have more time to undertake the project, we would find a suitable

algorithm that would allow us to specify SI values for and m from the outset.

4.1.2 Surface interactions

We have simplified the expression (2.42) by ignoring V im [1] which is the termused to model the absorbtion of atoms by the surface.

Further investigation could look at the prospect of including this imaginary

term.

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A MATLAB CODE

bsc dissertation: 'quantum reflection of ultra-cold atoms

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