many research groups have produced

“atom lasers” formed from traveling

BECs. They also behave similarly to

superfluids, much like liquid helium or

superconductors. By studying BECs,

we’ve gained important insights into

much more complicated, but closely

related, systems.

BECs created from dilute alkali vapors were first produced in the laboratory in 19951–4 using laser cooling and

trapping techniques developed over

the past 20 years. Creating a BEC requires temperatures between one billionth and one millionth of a degree

above absolute zero.

Once the cooling technology became

available, an explosion of research in

the area followed; over 30 experimental groups around the world have made

BECs. However, BECs are not a new

idea. Satyendra Nath Bose and Albert

Einstein predicted the Bose–Einstein

condensation phenomenon in dilute

ideal gases over 75 years ago.5–7

In this article, we present a method

for visualizing numerical simulations

of BECs, discuss the visualization

process, which includes selecting a

color scheme and the application of a

volume rendering model, and describe

our visualization’s impact on associated

scientific research. 

Quantum Vortices

Laboratory-produced BECs aren’t

quite as simple as the ones Bose and

Einstein originally predicted because

they are made from real alkali atoms,

not ideal gases. For instance, unlike an

ideal gas, atoms can’t pass through one

another. They interact strongly, and attract each other if they get too close together. Thus, keeping the gas dilute

enough is very important; otherwise,

you would end up with a lump of metal

instead of a BEC. Conversely, the

atoms also repel each other if kept sufficiently far apart, and it’s this repulsive

interaction that gives rise to the most

interesting properties of BECs. For

more on the physical intricacies of

BECs, see the “Quantum Degeneracy

and BECs” sidebar. 

One of the most interesting phenomena exhibited by superfluids is the

formation of quantum vortices. When

an ordinary fluid is rotated—for example, when you stir milk into a cup of

coffee—a large vortex, or depression,

is formed in the center. But a superfluid, by its very nature, wants to flow

without friction. This means that if

you tried to stir it with a spoon, it

would attempt to flow around the

spoon with no resistance. Of course,

there are some limitations on its ability to do so, and the result is the formation of tiny quantum vortices, each

with a quantized amount of superflow

around it.

These quantum whirlpools are produced in BECs by rotating the magnetic trap in which the BEC is contained—much like trying to stir your

coffee by squeezing the cup and spinning yourself around. BECs range

from a millimeter to a centimeter in

size, so you’d think it should be easy to

detect the vortices simply by shining

light on the cloud, taking a photograph, and noting the tiny holes in the

picture where the atom density has

gone to zero in the vortex cores.

Unfortunately, in spite of many efforts, it wasn’t until several years after

BECs were first formed that researchers saw quantum vortices in

gases made of a single species of atom.8

(The first quantum vortices in BECs

were produced in two-species gases, so

that one species filled the core of the

vortex formed in the other species.

This made the vortex large enough to

image, but the cloud as a whole had no

density depression in the vortex core.)

The inability to see single-species gas

quantum vortices wasn’t properly understood until full three-dimensional

numerical simulations and visualizations of rotating BECs began to

emerge in the 1990s. These simulations demonstrated that the rotation

frequencies needed to produce quantum vortices were higher than previously expected.

The visualizations also revealed that

the vortices themselves were generally

86 Copublished by the IEEE CS and the AIP 1521-9615/03/$19.00 © 2003 US Gov’t Work Not Protected by US Copyright COMPUTING IN SCIENCE & ENGINEERING

VISUALIZING BOSE–EINSTEIN

CONDENSATES

By Peter M. Ketcham and David L. Feder

B OSE–EINSTEIN CONDENSATES ARE THE COLDEST SUBSTANCES EVER MADE. ALONG WITH THIS DISTINCTION,

BECS HAVE OTHER INTRIGUING PROPERTIES: FOR EXAMPLE, THEY

BEHAVE MUCH LIKE LIGHT DOES IN AN OPTICAL FIBER. IN FACT,

Editor: Jim X. Chen, jchen@cs.gmu.edu

VISUALIZATION V ISUALIZATION C ORNER
JANUARY/FEBRUARY 2003 87

quite curved. Due to the small size of

the vortices, this curvature meant that

any photograph of the entire BEC

wouldn’t detect the holes because the

position of the holes changes down the

line of sight, thus the cloud itself would

obscure the vortices.

Vortices were finally imaged experimentally using two techniques. At

École Normale Supérieure, Jean Dalibard’s group found that if they released the atoms from their trap and allowed the BEC to expand, the vortex

cores were magnified sufficiently to be

detectable.9,10 This approach works

well when the number of atoms isn’t

too large, on the order of a few hundred thousand. Wolfgang Ketterle’s

BEC, however, contained several million atoms; his solution was to pump a

thin section of the cloud into another

quantum state using a laser and then

image only that slice.11

Numerical Simulation

We can numerically model the formation of quantum vortices in BEC superfluids by solving differential equations very similar to those used to

model ocean or air currents. The

main differences are the total lack of

friction and the inclusion of terms that

take the quantum nature of the atoms

into account.

All the atoms in a BEC are in a single quantum state (which is why it’s

called quantum degeneracy). This

state is represented by a wave function ψ, which is a complex number

characterized by two values: a density

ρ and a phase φ, so that .

Here, is the i m a g i n a r y

unit, so that exp (i × 0) = 1 and exp (i

× π) = –1. The density gives the

cloud’s overall profile, expressing the

average number of atoms in a given 

flow: the superfluid velocity ν is proportional to the gradient of the phase

ν ∝ ∇φ.

Visualization

The product of the numerical simulation of BEC vortex formation we used

is a data set containing 8 million complex numbers expressed in polar form.

The data values represent the wave

function ψ on a three-dimensional grid

with 200 grid points in each dimension.

Visualization provides a way to examine

this data set at a high conceptual level.

Process

The visualization process includes

three main activities:

i = −1

ψ ρ φ = e

i

Quantum Degeneracy and BECs

A Bose–Einstein condensate forms when a gas of particles obeying Bose–Einstein statistics, bosons, becomes quantum degenerate. In his 1924 doctoral thesis, Louis de Broglie proposed that all particles possess a quantum wave nature

in addition to a quantum particle nature. When viewed as a wave, a particle

has an associated quantum wavelength. This quantum wavelength is inversely

proportional to the particle’s mass and temperature, so the colder the temperature and the lighter the mass, the longer the quantum wavelength. Particles

with long quantum wavelengths more readily manifest their wavy character.

Quantum degeneracy occurs when the quantum wavelength becomes

comparable to the average spacing between particles. In practice, this means

either very high particle densities or very cold temperatures. Metals are good

conductors because the electrons carrying the charge are quantum degenerate, which is possible at room temperatures due to the high electron densities.

Two kinds of particles exist in the known universe: bosons and fermions, the

latter of which obey Fermi–Dirac statistics. (In certain artificial two-dimensional

systems, such as semiconductor heterojunctions, particles can actually have fractional statistics, intermediate between bosons and fermions. These instances give

rise to the fractional quantum Hall effect at low temperatures and high magnetic

fields.) Bosons, which have integer values of something called spin, include photons (the quantum particles of light). Fermions, which have half-integer spin, include the electrons, protons, and neutrons making up ordinary atoms.

Atoms themselves are either bosons or fermions, depending on the total

number of their fermion constituents: an even number of fermions gives a

bosonic atom. For example, hydrogen is a boson, with one proton and one

electron. In fact, most efforts to produce a BEC over the past two decades focused on hydrogen, and in 1998, a hydrogen BEC was finally produced in the

laboratory by Dan Kleppner and Thomas Greytak’s group at Harvard University.1 Hydrogen was long considered a good candidate because of its light

mass.

In principle, the gas needed to be only a few thousandths of a degree above

absolute zero to become a BEC. But achieving these temperatures in hydrogen

proved to be very difficult, and efforts turned instead to alkali atoms where laser

cooling could be used. Alkali atoms are much heavier than hydrogen, however,

so the temperatures needed to reach BEC are perhaps 100 times lower.

Reference

1. D.G. Fried et al., “Bose–Einstein Condensation of Atomic Hydrogen,” Physical Rev. Letters,

vol. 81, no. 18, 1998, pp. 3811–3814.

time. The phase is a quantum mechanical variable that defines the superfluid 
88 COMPUTING IN SCIENCE & ENGINEERING

• A review of the requirements and

conventions of physicists to ensure a

meaningful presentation of the data

• The selection of an appropriate

color scheme

• The volume rendering of the data to

produce a final image

Color Scheme. As previously noted,

the product of the numerical simulation is a set of complex-valued data expressed in polar form. Recall that a

complex number expressed in polar

form has both an angular component

and a radial component. For each complex number, the angular component is

the phase of the BEC wave function,

and the square of the radial component

is the BEC wave function’s density.

To visualize BEC wave function data,

we chose a color scheme that harmonizes effectively with the circular nature

of complex numbers expressed in

polar form. The

HSV (hue, saturation, value) color

model offers such

a scheme.

The HSV color

model expresses

hue (the degree of

red-, green-, or

blueness) in an 

angular fashion.

This angular expression suggests

a natural mapping

from the phase of

the BEC wave

function to the

hue component of

the HSV color

model. In particular, a phase of zero

maps to red, a

phase of 2π/3

maps to green, a

phase of 4π/3 maps to blue, and a phase

of 2π maps once again to red, thus completing the circle. The remaining phase

values are interpolated to the hue component in the expected manner.

The HSV color model expresses saturation in a radial fashion. This suggests

a mapping from the BEC wave function’s density to the saturation component of the HSV color model. However,

such a mapping is not very useful: with

a dense emitter volume rendering model (discussed next), high-density regions of the BEC wave function obscure

low-density regions. This is unacceptable because the low-density regions are

of primary interest.

The HSV color model expresses

value (the degree of brightness) in a

linear fashion. This is not meant to

imply that the human eye perceives

brightness in a linear manner. Value is

expressed linearly in the sense that it is

not expressed in an angular or radial

fashion. This allows a mapping from

the BEC wave function’s density to the

value component of the HSV color

model. A useful mapping is one that

maps low density to high brightness

and high density to low brightness.

Because vortices correspond to localized suppressions of the BEC wave

function, such a mapping exhibits vortex structures because it emphasizes

low-density regions and subdues highdensity ones.

Volume Rendering. Applying the

color scheme just described, we render

the BEC wave function data set with a

dense emitter volume rendering

model.12 With a dense emitter volume

rendering model, each point of the volume emits (and possibly absorbs) light

of a particular color. The color scheme

determines the color of light that each

point emits.

An opacity scheme determines the

amount of light that each point absorbs. We obtain the best results with

an opacity scheme in which every

point absorbs no light—meaning,

every point allows light from every

other point to pass through itself

unimpeded.

Figure 1 displays a volume rendering of BEC wave function data using

the Open Visualization Data Explorer

(OpenDX) software application package. Twelve vortices appear as bright,

column-like, vertical bands. We take

some care in rendering this data to

prevent uninteresting, low-density,

outer regions from obscuring the interesting, low-density, inner vortex

structures.

Research Impact

Visualization contributes not only to

the exploration of a data set, but to the

V ISUALIZATION C ORNER

Figure 1. A volume rendering of Bose–Einstein condensate

wave function data. The various colors reflect the changing

phase of the wave function throughout the condensate.
JANUARY/FEBRUARY 2003 89

entire scientific research process as well.

Visualization assists with the programming and validation of the numerical simulation code. In support of

NIST’s theoretical and experimental

research on BECs, visualization exposed errors in early versions of the

simulation code and lent confidence to

the correctness of the final version of

the simulation. In general, a feedback

loop exists between data visualization

and numerical simulation refinement.

The visualization of the BEC wave

function demonstrates at a high conceptual level how the BEC behaves. In

particular, the visualization displays the

formation and evolution of vortex

structures in a rotating BEC. It also

suggests why certain observational approaches (such as photography) could

fail to detect the vortices in an actual,

physical experiment.

The visualization revealed the theoretical existence of vortex structures in

rotating BECs, which were confirmed

by actual laboratory experiments. A

successful outcome stimulates further

numerical simulations, further simulation visualizations, and further laboratory experimentation.13

Visualization of BECs enhances and

accelerates the scientific investigation

of this exotic state of matter. Visualization is an integral part of the numerical simulation process and a key

component of modern, collaborative

research efforts. In this work, visualization enables scientific discoveries

that may be difficult to achieve by

other methods. We expect visualization to play an important role in many

of the emerging technologies using

BECs, such as superfluid gyroscopes,

graviometers, and quantum information devices.

Acknowledgments

We thank the following individuals

for their assistance: Charles W. Clark,

Judith E. Devaney, William L.

George, Terence J. Griffin, Steven G.

Satterfield (National Institute of

Standards and Technology); William

P. Reinhardt (University of Washington); and Barry I. Schneider (National

Science Foundation).

Certain commercial equipment and

software are identified to adequately

specify or describe the subject matter

of this work. In no case does such

identification imply recommendation

or endorsement by the National Institute of Standards and Technology, nor

does it imply that the equipment or

software is necessarily the best available for the purpose.

References

1. M.H. Anderson et al., “Observation of

Bose–Einstein Condensation in a Dilute

Atomic Vapor,” Science, vol. 269, no. 5221,

1995, pp. 198–201.

2. K.B. Davis et al., “Bose–Einstein Condensation

in a Gas of Sodium Atoms,” Physical Rev. Letters, vol. 75, no. 22, 1995, pp. 3969–3973.

3. C.C. Bradley et al., “Evidence of Bose–Einstein Condensation in an Atomic Gas with

Attractive Interactions,” Physical Rev. Letters,

vol. 75, no. 9, 1995, pp. 1687–1690.

4. C.C. Bradley, C.A. Sackett, and R.G. Hulet,

“Bose–Einstein Condensation of Lithium: Observation of Limited Condensate Number,”

Physical Rev. Letters, vol. 78, no. 6, 1997, pp.

985–989.

5. S.N. Bose, “Plancks Gesetz und Lichtquantenhypothese,” Zeitschrift für Physik, vol. 26,

no. 3, 1924, pp. 178–181.

6. A. Einstein, “Quantentheorie des Einatomigen Idealen Gases,” Sitzungsberichte der

Preussischen Akademie der Wissenschaften,

Physikalisch-mathematische Klasse, vol. 1924,

no. 22, 1924, pp. 261–267.

7. A. Einstein, “Quantentheorie des Einatomigen Idealen Gases, Zweite Abhandlung,”

Sitzungsberichte der Preussischen Akademie

der Wissenschaften, Physikalisch-mathematische Klasse, vol. 1925, no. 1, 1925, pp. 3–14.

8. M.R. Matthews et al., “Vortices in a Bose–Einstein Condensate,” Physical Rev. Letters, vol.

83, no. 13, 1999, pp. 2498–2501.

9. K.W. Madison et al., “Vortex Formation in a

Stirred Bose–Einstein Condensate,” Physical

Rev. Letters, vol. 84, no. 5, 2000, pp. 806–809.

10. F. Chevy, K.W. Madison, and J. Dalibard,

“Measurement of the Angular Momentum of

a Rotating Bose–Einstein Condensate,” Physical Rev. Letters, vol. 85, no. 11, 2000, pp.

2223–2227.

11. J.R. Abo-Shaeer et al., “Observation of Vortex

Lattices in Bose–Einstein Condensates,” Science, vol. 292, no. 5516, 2001, pp. 476–479.

12. B. Lucas, “A Scientific Visualization Renderer,” Proc. Visualization ’92, IEEE CS Press,

1992, pp. 227–234.

13. B.P. Anderson et al., “Watching Dark Solitons Decay into Vortex Rings in a Bose–Einstein Condensate,” Physical Rev. Letters, vol.

86, no. 14, 2001, pp. 2926–2929.

Peter M. Ketcham is a computational scientist

in the Information Technology Laboratory of

the National Institute of Standards and Technology (NIST). His research interests include

scientific visualization and high-performance

computing. He received his MS in mathematics from the University of Minnesota. Contact

him at peter.ketcham@nist.gov.

David L. Feder is an assistant professor in the

Department of Physics and Astronomy at the

University of Calgary in Alberta, Canada. He is

currently exploring the possibility of using

Bose–Einstein condensates to operate rudimentary quantum computers. Between 1997

and 2002, he was a guest researcher at NIST,

where he worked on the theory of Bose condensation in dilute trapped gases, in close collaboration with experimental efforts there. He

obtained his MSc and PhD degrees in theoretical condensed matter physics at McMaster

University in Hamilton, Ontario, Canada. Contact him at feder@phas.ucalgary.ca.