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Theory Project B: F34YY6
MATHEMATICAL PHYSICS
29 APRIL 2005
Mixed stable-chaotic trajectories of an isolated cold sodium atom in a one-dimensional
optical lattice subject to a two-dimensional magnetically perturbing potential.
Mark Saunders and Andrew S. Chisholm
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
(submitted 29 April 2005)
We study the motion of a low energy point particle in an optical lattice, and
discuss the effects of adding an elliptical magnetic trap. The orientation of
these confining wells gives an interesting mixture of stable and chaotic
results. We develop the one-dimensional system to that of a quantum
particle, using Schrödinger’s equation to time-evolve its wavefunction.
SAUNDERS et al.
Page 1
a force, given by F  V . We can view the
effect of this force as causing the energy band to
tilt, as seen in Fig. 1(a). Now consider this atom
to be moving through the system. Its total energy
must remain constant, and so as x increases, the
particle’s wavefunction eigenenergy must
increase. On reaching the top of the ground state
energy band, the only possible motion of the
particle is for it to change direction, decreasing
the eigenvalues of the wavefunction. This is
known as Bragg reflection [4].
The sodium atom also Bragg reflects when it
reaches the bottom of the energy band, as no
lower energy eigenvalues are possible. This
phenomenon is known as Bloch oscillation [5]. It
is observed in figure 1(b), where we plot the
position of a point particle as a function of time.
(a)
Bandwidth
=22.1 peV
Bragg
Reflection
E
x
1x10
Ground state
energy band
Trajectory
of particle
-6
(b)
0
position (metres)
There have been many recent studies of
dynamical systems that yield a mixture of stable
and chaotic results [1-3]. In this letter, we
investigate one such system: that of a twodimensional magnetic trap superposed onto a onedimensional optical lattice (OL) [1,2].
We begin by considering the motion of a
particle in a 1D OL. An OL is a periodic potential
formed from the interaction between an alkalimetal atom and the stationary wave produced by
two
counter-propagating
laser
beams.
Superposing the two electromagnetic waves along
the
x-axis
yields
an
electric
field, E  E0 sin( kx) sin( t ) xˆ . Here, k is the
wavevector for the laser, with frequency   500
THz.
Adding an atom to this electric field induces an
electric dipole moment proportional to E. The
atom experiences a potential,
V  E 02 sin 2 (kx) sin 2 (t ) .
(1)
This leads to nodes and antinodes along the xaxis.
The state of a particle can be described by a
wavefunction  ( x, t ) .
As the potential is
spatially periodic, the energy eigenstates lie very
close together. This forms a continuum of energy
states, setting up a band-like structure of allowed
eigenvalues, as shown in figure 1(a). In this
work, we restrict the theory to considering the
motion of an atom within the ground state energy
band. We therefore assume that the band gap
energy between the two bands to be much larger
than the total energy of the system.
For a sodium atom within such a potential,
detuning one of the lasers linearly with time
causes the potential minima (or optical lattice
sites) to move. This motion of the potential
minima causes the gradient of the potential to
change. Therefore, the sodium atom experiences
-1 x 1 0
-2 x 1 0
-3 x 1 0
-6
-6
-6
0
0 .0 0 2
0 .0 0 4
0 .0 0 6
0 .0 0 8
0 .0 1 0
tim e (s e c o n d s )
Figure 1: (a) The trajectory of a particle within the energy
band structure. (b) The position for a sodium atom in an
OL. Bloch oscillations are observed in the semi-classical
(dashed line) and quantum regimes (solid line).
© 2005 The University of Nottingham
Theory Project B: F34YY6
MATHEMATICAL PHYSICS
xt
The repetition of Bragg reflection, shown by the
dashed line, causes the particle to oscillate with
Bloch period TB  1.67 ms and Bloch amplitude
of 2.6 μm. This means that the Bloch frequency is
of the order of 103 Hz, many orders of magnitude
smaller than the frequency of the laser light.
Therefore, we can neglect the time-dependence of
the OL potential in equation (1). We establish a
time-averaged
system
described
by
x
V ( x)  V0 sin 2 ( ) where d    294.5 nm,
2
d
and V0  562.52 peV.
(a)
(2)
In the equations above, kx is the wave vector of
the particle. These equations effectively model
Bloch oscillations found in the ground-state
energy band.
This one-dimensional theoretical study of
optical lattices can be experimentally realized as
three-dimensional. This is due to the fact that a
laser beam has finite width, and so in higher
dimensional research, we consider the nodes and
antinodes of the OL to correspond to the values
lying on the x-axis.
By superposing a magnetic trap onto a threedimensional system, we create a confining
potential in which the particle is forced towards
the centre. This set-up can be viewed as a
potential well [6]. However, the boundary of this
system is not a hard wall, but a much softer
constraining force, extending beyond the limits of
the particle’s motion.
The symmetry of the magnetic trap is slightly
broken, to yield interesting results: we use a trap
for which ellipse shaped contours of constant
potential are obtained. Furthermore, this trap is
rotated by an angle, θ with respect to the OL, as
shown in figure 2(a).
z(μm)
(a)
zt
θ
θ
Optical Lattice
z
Magnetic Trap
x
E
(b)
E  E2  E
z
x
k max
E tot  E 2
E tot  E1
x
k max
E
For a sodium atom of mass m  3.82 10 26 kg
within such a regime, the dynamics are governed
by the semi-classical equations of motion:
dx 1 d
 (
)  sin( k x  d )

dt 2
dk x
ma

dt

29 APRIL 2005
px
Figure 2: (a) The OL (dashed line) is along the x-axis. The
shaded ellipse represents the magnetic trap. The (xt,zt) plane is
shown to be rotated w.r.t. (x,z) by angle θ. The axes share a
common origin, marked by the white arrow.
(b) The energy band structure of the OL, with two possible initial
particle energies shown as E1 and E2. Kinetic energy (bold
curve) of a particle with a given momentum in the x-direction
In order to compare a large collection of results,
we combine multiple trajectories. In order to do
this, we ensure that the initial conditions for each
of these trajectories are comparable. We start
each particle off on a contour of constant total
energy.
Calculating the values for the particle’s initial
conditions is complicated because there are
restrictions on how much kinetic energy the
particle can posses along the x-axis. This is due to
the energy band motion already discussed. A
view of the energy band can be seen in figure
2(b). If this initial energy is greater than Emax, the
difference must be assigned as kinetic energy in
the z-direction.
A new set of axes (xt ,zt) shown in Fig. 2(a)
conveniently allows the potential of the magnetic
trap to be given by
V ( xt , z t ) 


1
m  12 xt2   22 y 2   32 z t2 (3)
2
where angular frequencies ω1=1479 rad s-1,
ω2=4702 rad s-1 and ω3=2561 rad s-1 are taken
(b)
(c)
(d)
(e)
(g)
(h)
(i)
(j)
x (μm)
pz (10-28 kgms-1) (f)
z(μm)
Figure 3: Examples of the results obtained from the motion of particle in the system described in the text. Axes inset.
(a) - (e): Many particles are considered, starting on the ellipse shaped equipotential. These initial starting points are
represented by the small circles. Examples are given of typical trajectories: Figures (a) to (c) correspond to an untilted
magnetic trap; Similarly for figures (d) and (e) with θ=45o. The trajectories are plotted for the
the first
fist 60 ms after the simulation
began. (f)-(j): The Poincaré sections correspond to that of the trajectory (a)-(e) respectively.
SAUNDERS et al.
Page 2
© 2005 The University of Nottingham
Theory Project B: F34YY6
MATHEMATICAL PHYSICS
from experiments on sodium atoms [2]. Motion
in the y-direction decouples from the (x,z) plane,
corresponding purely to simple harmonic motion.
We may suppress the y-axis without loss in
generality, and consider the two-dimensional (x,z)
system.
Within this potential, we model the motion of a
particle using the fourth-order Runge-Kutta
method [1,7] to evolve the point particle’s
position in time using the semi-classical equations
of motion (2), as in the one-dimensional approach.
We calculated the trajectory of a particle starting
with a constant total energy, E. We considered
various magnetic trap orientations and examined
how the particle’s trajectory was dependent upon
its starting point within the system. The results of
this computational experiment are shown in figure
3. The ellipse shaped curve is an equipotential of
the magnetic trap, with value E=22.1 peV. A
point on this curve shows where a particle begins
its trajectory.
Its initial velocity can be
determined by computing how much energy it
was allowed to have at the top of the band, as
shown in Fig 2(b).
To quantify how the particle motion changes
with the trap orientation, we plotted points in the
(z,pz) plane through phase space at each turning
point along the x-direction for which p x  0 and
dp x / dt  0 [8].
Examples of these Poincaré
sections are shown below their corresponding
trajectory in figure 3.
For a range of values of θ, the multiple constant
energy trajectories are superposed, as is shown in
figure 4. For   0  and 90  [left-hand figures],
these points lie on a series of concentric ellipses
corresponding to simple harmonic motion along
the z-axis. For intermediate angles, the system
reveals a phase space characterised by a mixture
of crescent shaped stable islands, surrounded by
chaotic seas.
For the figures on the right, we again present
Poincaré sections, this time corresponding to the
Poincaré plane given by points (x,px) in the phase
space plane for which p z  0 and dp z / dt  0
[8]. We have demonstrated the convenience of
Poincaré sections for allowing both stable and
chaotic results to be visualised on the same
diagram.
Adapting this model from a point particle,
whose dynamics incorporate the semi-classical
equations of motion; we now consider a quantum
mechanical wavefunction. The time evolution of
a non-interacting atom is governed by the timedependent Schrödinger equation:

2 2
i  ( x , t )  
 ( x, t )  V ( x) ( x, t ) (4)
t
2m x 2
SAUNDERS et al.
Page 3
29 APRIL 2005
θ = 0o
θ = 15o
θ = 30o
θ = 45o
θ = 60o
θ = 75o
px(10-28kgms-1)
pz(10-28kgms-1)
θ = 90o
z(μm)
x(μm)
Figure 4: Poincaré sections for a magnetic trap tilted at
angle θ with respect to the optical lattice. Left: (z,pz)
Poincaré sections, as described in the text. The figure
corresponding to θ=0 is made up, in part, from Figs. 3a),
3b) and 3c). The figure corresponding to θ=45 degrees is
made up from Figs. 3d) and 3e). Right: (x,px) Poincaré
sections corresponding to their adjacent figure. Axes inset.
Consider the case of the one-dimensional OL,
which contributes a potential energy term as given
by equation (1). In order to demonstrate that the
observable results are consistent with the semiclassical calculations (2), we show that Bloch
oscillations take place.
The wavefunction of the isolated sodium atom
was initially described as an entirely real
Gaussian. This is because it is localised in space,
has a well-defined width, and its mean position
does not move unless a potential is applied. A
Gaussian is smooth and therefore has well defined
derivatives.
Having decided to work with a Gaussian
wavepacket, we chose its standard deviation to be
between two and five lattice periods. This allows
the atom to feel the true periodic form of the
potential, whilst keeping its form localised in
space.
The Crank-Nicholson method [1,7] was used to
time-evolve the Schrödinger equation (4) by one
time-step, leading to a new wavefunction
© 2005 The University of Nottingham
MATHEMATICAL PHYSICS
Theory Project B: F34YY6
7.5x10
6
6.0x10
6
4.5x10
6
3x10
3.0x10
6
2x10
1.5x10
6
1x10
5x10
(a)
0
-10
6x10
6
4x10
6
4x10
-5
5
Probability
Density
6x10
-5
0
5
6
6
2x10
10
-5
0
5
10
-5
0
5
10
6
6
(f)
6
4x10
6
0
-10
5
6
(e)
4x10
0
(d)
0
-10
6x10
-5
6
3x10
10
6
6
6
1x10
0
-10
(b)
6
4x10
2x10
6
6
0
-10
10
5x10
(c)
2x10
0
2x10
-5
0
5
10
6
6
0
-10
Figure 5: The probability density distribution for the quantum
mechanical wavefunction of the 1D OL, plotted at various times.
a) Shortly after Gaussian wavepacket evolution. (t = 0.001 ms)
b) Half way to the first Bragg reflection. (t = 0.418 ms)
c) At first Bragg reflection. (t = 0.835 ms)
d) ¾ the way through the Bloch oscillation. (t = 1.253 ms)
e) After 1 Bloch oscillation. (t = 1.67 ms)
f) After 5 Bloch oscillations. (t = 8.35 ms)
describing the system. Repeating this process
yielded the snapshots in figure 5, which shows
this wavefunction to be Bloch oscillating.
The expectation value of the atom’s position
can be evaluated at each time using
x  dx * x, t x x, t  .
The expectation

values for the simulated wavepacket are plotted
against time by the solid curve in figure 1(b). As
predicted, the particle Bloch oscillates
with TB  1.67 ms.
Its amplitude, however,
[1] M. Saunders, A.S. Chisholm and T.M.
Fromhold,
Low
Temperature
Atom
Dynamics, (April 2005).
[2] R.G. Scott, S. Bujkiewicz, T.M. Fromhold,
P.B. Wilkinson and F.W. Sheard,
Phys. Rev. A 66, 023407(2002).
[3] T.M. Fromhold et al., Nature 428, 726 (April
2004).
[4] J.R. Hook and H.E. Hall, Solid State Physics,
Second Edition, Wiley, 1990.
[5] M. Ben Dahan, E.Peik, J.Reichel, Y.Castin,
and C. Salomon,
Phys. Rev. Lett. 76, 4508(1996).
[6] For similar examples for a variety of different
shaped billiards see M.C. Gutzwiller, Chaos
in Classical and Quantum Mechanics,
(Springer, New York, 1990).]
[7] W.H. Press, S.A. Teukolsky, W.T. Vetterling
and B.P. Flannery, Numerical Recipes in C,
Cambridge University Press, (1992).
SAUNDERS et al.
29 APRIL 2005
6
Page 4
differs from what we initially expected based on
the semi-classical calculations. This is due to the
method used to calculate the width of the energy
band in which the atom is moving.
This
difference is a consequence of the form of the
band structure in the quantum simulation.
Bose-Einstein condensates have recently
received considerable attention [9], the dynamics
of which can be incorporated into this research via
the Gross-Pitaevskii equation [10].
To summarise, we have explored the dynamics
of a low energy sodium atom in a variety of
systems and models.
Starting with the one-dimensional optical lattice,
we modelled the system semi-classically. Plotting
the trajectory in one dimension allowed Bloch
oscillations to be modelled for a point particle.
Modifying to take the wavefunction of a sodium
atom into account allowed the full quantum
mechanical nature to be realised. The resulting
wavepacket exhibited Bloch oscillations.
The semi-classical results are very interesting
for the two-dimensional system. For various
magnetic trap orientations, the mixture of both
chaotic and stable results proves to give a unique
insight into semi-classical mechanics. The use of
Poincaré sections to express these results was a
very powerful tool to convey this nature of the
system. The Poincaré sections enabled stable
islands and chaotic seas to be observed.
We wish to thank Professor Mark Fromhold for
his advice and support throughout this project.
This letter is dedicated to the life of George C.
Grieveson.
[8] A given Poincaré plane produces two possible
sections, one the mirror image of the other.
To avoid this overlap, we consider only the
intersections in which the sign of the
momentum changed from positive to
negative.
[9] R.G. Scott, A.M. Martin, T.M. Fromhold,
S. Bujkiewicz, F.W. Sheard, and
M. Leadbeater,
Phys. Rev. Lett. 90, (11) 110404(2003);
R.G. Scott, A.M. Martin, S. Bujkiewicz,
T.M. Fromhold, N. Malossi, O. Morsch,
M. Cristiani, and E. Arimondo,
Phys. Rev. A, 69, 033605(2004);
R.G. Scott, A.M. Martin, and T.M. Fromhold,
ibid. 69, 063607(2004).
[10] S.A. Gardiner, D. Jaksch, R. Dum, J.I. Cirac
and P. Zoller,
Phys. Rev. A 62, 023612(2000);
R. Bach, K. Burnett, M.B. d’Arcy and S.A.
Gardiner, arXiv:physics/0310143v4(2004).
© 2005 The University of Nottingham