Visualizing Bose-Einstein Condensates

VISUALIZATIONV ISUALIZATION C ORNER

VISUALIZING BOSE–EINSTEIN CONDENSATES

By Peter M. Ketcham and David L. Feder

OSE–EINSTEIN CONDENSATES ARE THE COLDEST SUB- ity to do so, and the result is the formation of tiny quantum vortices, each B STANCES EVER MADE. ALONG WITH THIS DISTINCTION, with a quantized amount of superflow around it. BECS HAVE OTHER INTRIGUING PROPERTIES: FOR EXAMPLE, THEY These quantum whirlpools are produced in BECs by rotating the mag- BEHAVE MUCH LIKE LIGHT DOES IN AN OPTICAL FIBER. IN FACT, netic trap in which the BEC is contained—much like trying to stir your many research groups have produced Quantum Vortices coffee by squeezing the cup and spin- “atom lasers” formed from traveling Laboratory-produced BECs aren’t ning yourself around. BECs range BECs. They also behave similarly to quite as simple as the ones Bose and from a millimeter to a centimeter in superfluids, much like liquid helium or Einstein originally predicted because size, so you’d think it should be easy to superconductors. By studying BECs, they are made from real alkali atoms, detect the vortices simply by shining we’ve gained important insights into not ideal gases. For instance, unlike an light on the cloud, taking a photo- much more complicated, but closely ideal gas, atoms can’t pass through one graph, and noting the tiny holes in the related, systems. another. They interact strongly, and at- picture where the atom density has BECs created from dilute alkali va- tract each other if they get too close to- gone to zero in the vortex cores. pors were first produced in the labora- gether. Thus, keeping the gas dilute Unfortunately, in spite of many ef- tory in 19951–4 using laser cooling and enough is very important; otherwise, forts, it wasn’t until several years after trapping techniques developed over you would end up with a lump of metal BECs were first formed that re- the past 20 years. Creating a BEC re- instead of a BEC. Conversely, the searchers saw quantum vortices in quires temperatures between one bil- atoms also repel each other if kept suf- gases made of a single species of atom.8 lionth and one millionth of a degree ficiently far apart, and it’s this repulsive (The first quantum vortices in BECs above absolute zero. interaction that gives rise to the most were produced in two-species gases, so Once the cooling technology became interesting properties of BECs. For that one species filled the core of the available, an explosion of research in more on the physical intricacies of vortex formed in the other species. the area followed; over 30 experimen- BECs, see the “Quantum Degeneracy This made the vortex large enough to tal groups around the world have made and BECs” sidebar. image, but the cloud as a whole had no BECs. However, BECs are not a new One of the most interesting phe- density depression in the vortex core.) idea. Satyendra Nath Bose and Albert nomena exhibited by superfluids is the The inability to see single-species gas Einstein predicted the Bose–Einstein formation of quantum vortices. When quantum vortices wasn’t properly un- condensation phenomenon in dilute an ordinary fluid is rotated—for exam- derstood until full three-dimensional ideal gases over 75 years ago.5–7 ple, when you stir milk into a cup of numerical simulations and visualiza- In this article, we present a method coffee—a large vortex, or depression, tions of rotating BECs began to for visualizing numerical simulations is formed in the center. But a super- emerge in the 1990s. These simula- of BECs, discuss the visualization fluid, by its very nature, wants to flow tions demonstrated that the rotation process, which includes selecting a without friction. This means that if frequencies needed to produce quan- color scheme and the application of a you tried to stir it with a spoon, it tum vortices were higher than previ- volume rendering model, and describe would attempt to flow around the ously expected. our visualization’s impact on associated spoon with no resistance. Of course, The visualizations also revealed that scientific research. there are some limitations on its abil- the vortices themselves were generally

A Bose–Einstein condensate forms when a gas of particles obeying Bose–Ein- stein statistics, bosons, becomes quantum degenerate. In his 1924 doctoral the- sis, Louis de Broglie proposed that all particles possess a quantum wave nature quite curved. Due to the small size of in addition to a quantum particle nature. When viewed as a wave, a particle the vortices, this curvature meant that has an associated quantum wavelength. This quantum wavelength is inversely any photograph of the entire BEC proportional to the particle’s mass and temperature, so the colder the temper- wouldn’t detect the holes because the ature and the lighter the mass, the longer the quantum wavelength. Particles position of the holes changes down the with long quantum wavelengths more readily manifest their wavy character. line of sight, thus the cloud itself would Quantum degeneracy occurs when the quantum wavelength becomes obscure the vortices. comparable to the average spacing between particles. In practice, this means Vortices were finally imaged experi- either very high particle densities or very cold temperatures. Metals are good mentally using two techniques. At conductors because the electrons carrying the charge are quantum degener- École Normale Supérieure, Jean Dal- ate, which is possible at room temperatures due to the high electron densities. ibard’s group found that if they re- Two kinds of particles exist in the known universe: bosons and fermions, the leased the atoms from their trap and al- latter of which obey Fermi–Dirac statistics. (In certain artificial two-dimensional lowed the BEC to expand, the vortex systems, such as semiconductor heterojunctions, particles can actually have frac- cores were magnified sufficiently to be tional statistics, intermediate between bosons and fermions. These instances give detectable.9,10 This approach works rise to the fractional quantum Hall effect at low temperatures and high magnetic well when the number of atoms isn’t fields.) Bosons, which have integer values of something called spin, include pho- too large, on the order of a few hun- tons (the quantum particles of light). Fermions, which have half-integer spin, in- dred thousand. Wolfgang Ketterle’s clude the electrons, protons, and neutrons making up ordinary atoms. BEC, however, contained several mil- Atoms themselves are either bosons or fermions, depending on the total lion atoms; his solution was to pump a number of their fermion constituents: an even number of fermions gives a thin section of the cloud into another bosonic atom. For example, hydrogen is a boson, with one proton and one quantum state using a laser and then electron. In fact, most efforts to produce a BEC over the past two decades fo- image only that slice.11 cused on hydrogen, and in 1998, a hydrogen BEC was finally produced in the laboratory by Dan Kleppner and Thomas Greytak’s group at Harvard Univer- Numerical Simulation sity.1 Hydrogen was long considered a good candidate because of its light We can numerically model the forma- mass. tion of quantum vortices in BEC su- In principle, the gas needed to be only a few thousandths of a degree above perfluids by solving differential equa- absolute zero to become a BEC. But achieving these temperatures in hydrogen tions very similar to those used to proved to be very difficult, and efforts turned instead to alkali atoms where laser model ocean or air currents. The cooling could be used. Alkali atoms are much heavier than hydrogen, however, main differences are the total lack of so the temperatures needed to reach BEC are perhaps 100 times lower. friction and the inclusion of terms that take the quantum nature of the atoms Reference into account. 1. D.G. Fried et al., “Bose–Einstein Condensation of Atomic Hydrogen,” Physical Rev. Letters, All the atoms in a BEC are in a sin- vol. 81, no. 18, 1998, pp. 3811–3814. gle 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 flow: the superfluid velocity ν is pro- The data values represent the wave φ ρ and a phase φ, so that ψρ= ei . portional to the gradient of the phase function ψ on a three-dimensional grid Here, i =−1 is the i m a g i n a r y ν ∝ ∇φ. with 200 grid points in each dimension. unit, so that exp (i × 0) = 1 and exp (i Visualization provides a way to examine ×π) = –1. The density gives the Visualization this data set at a high conceptual level. cloud’s overall profile, expressing the The product of the numerical simula- average number of atoms in a given tion of BEC vortex formation we used Process time. The phase is a quantum mechan- is a data set containing 8 million com- The visualization process includes ical variable that defines the superfluid plex numbers expressed in polar form. three main activities:

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of complex num- expressed linearly in the sense that it is bers expressed in not expressed in an angular or radial polar form. The fashion. This allows a mapping from HSV (hue, satura- the BEC wave function’s density to the tion, value) color value component of the HSV color model offers such model. A useful mapping is one that a scheme. maps low density to high brightness The HSV color and high density to low brightness. model expresses Because vortices correspond to local- hue (the degree of ized suppressions of the BEC wave red-, green-, or function, such a mapping exhibits vor- blueness) in an tex structures because it emphasizes angular fashion. low-density regions and subdues high- This angular ex- density ones. pression suggests a natural mapping Volume Rendering. Applying the from the phase of color scheme just described, we render the BEC wave the BEC wave function data set with a function to the dense emitter volume rendering hue component of model.12 With a dense emitter volume the HSV color rendering model, each point of the vol- Figure 1. A volume rendering of Bose–Einstein condensate model. In particu- ume emits (and possibly absorbs) light wave function data. The various colors reﬂect the changing lar, a phase of zero of a particular color. The color scheme phase of the wave function throughout the condensate. maps to red, a determines the color of light that each phase of 2π/3 point emits. maps to green, a An opacity scheme determines the • A review of the requirements and phase of 4π/3 maps to blue, and a phase amount of light that each point ab- conventions of physicists to ensure a of 2π maps once again to red, thus com- sorbs. We obtain the best results with meaningful presentation of the data pleting the circle. The remaining phase an opacity scheme in which every • The selection of an appropriate values are interpolated to the hue com- point absorbs no light—meaning, color scheme ponent in the expected manner. every point allows light from every • The volume rendering of the data to The HSV color model expresses sat- other point to pass through itself produce a ﬁnal image uration in a radial fashion. This suggests unimpeded. a mapping from the BEC wave func- Figure 1 displays a volume render- Color Scheme. As previously noted, tion’s density to the saturation compo- ing of BEC wave function data using the product of the numerical simula- nent of the HSV color model. However, the Open Visualization Data Explorer tion is a set of complex-valued data ex- such a mapping is not very useful: with (OpenDX) software application pack- pressed in polar form. Recall that a a dense emitter volume rendering mo- age. Twelve vortices appear as bright, complex number expressed in polar del (discussed next), high-density re- column-like, vertical bands. We take form has both an angular component gions of the BEC wave function obscure some care in rendering this data to and a radial component. For each com- low-density regions. This is unaccept- prevent uninteresting, low-density, plex number, the angular component is able because the low-density regions are outer regions from obscuring the in- the phase of the BEC wave function, of primary interest. teresting, low-density, inner vortex and the square of the radial component The HSV color model expresses structures. is the BEC wave function’s density. value (the degree of brightness) in a To visualize BEC wave function data, linear fashion. This is not meant to Research Impact we chose a color scheme that harmo- imply that the human eye perceives Visualization contributes not only to nizes effectively with the circular nature brightness in a linear manner. Value is the exploration of a data set, but to the

88 COMPUTING IN SCIENCE & ENGINEERING entire scientific research process as well. for their assistance: Charles W. Clark, che Klasse, vol. 1925, no. 1, 1925, pp. 3–14. Visualization assists with the pro- Judith E. Devaney, William L. 8. M.R. Matthews et al., “Vortices in a Bose–Ein- gramming and validation of the nu- George, Terence J. Griffin, Steven G. stein Condensate,” Physical Rev. Letters, vol. merical simulation code. In support of Satterfield (National Institute of 83, no. 13, 1999, pp. 2498–2501. NIST’s theoretical and experimental Standards and Technology); William 9. K.W. Madison et al., “Vortex Formation in a research on BECs, visualization ex- P. Reinhardt (University of Washing- Stirred Bose–Einstein Condensate,” Physical posed errors in early versions of the ton); and Barry I. Schneider (National Rev. Letters, vol. 84, no. 5, 2000, pp. 806–809. simulation code and lent confidence to Science Foundation). 10. F. Chevy, K.W. Madison, and J. Dalibard, the correctness of the final version of Certain commercial equipment and “Measurement of the Angular Momentum of the simulation. In general, a feedback software are identified to adequately a Rotating Bose–Einstein Condensate,” Physi- loop exists between data visualization specify or describe the subject matter cal Rev. Letters, vol. 85, no. 11, 2000, pp. and numerical simulation refinement. of this work. In no case does such 2223–2227. The visualization of the BEC wave identification imply recommendation 11. J.R. Abo-Shaeer et al., “Observation of Vortex function demonstrates at a high con- or endorsement by the National Insti- Lattices in Bose–Einstein Condensates,” Sci- ceptual level how the BEC behaves. In tute of Standards and Technology, nor ence, vol. 292, no. 5516, 2001, pp. 476–479. particular, the visualization displays the does it imply that the equipment or 12. B. Lucas, “A Scientific Visualization Ren- formation and evolution of vortex software is necessarily the best avail- derer,” Proc. Visualization ’92, IEEE CS Press, structures in a rotating BEC. It also able for the purpose. 1992, pp. 227–234. suggests why certain observational ap- 13. B.P. Anderson et al., “Watching Dark Soli- proaches (such as photography) could References tons Decay into Vortex Rings in a Bose–Ein- fail to detect the vortices in an actual, 1. M.H. Anderson et al., “Observation of stein Condensate,” Physical Rev. Letters, vol. physical experiment. Bose–Einstein Condensation in a Dilute 86, no. 14, 2001, pp. 2926–2929. The visualization revealed the theo- Atomic Vapor,” Science, vol. 269, no. 5221, retical existence of vortex structures in 1995, pp. 198–201. rotating BECs, which were confirmed 2. K.B. Davis et al., “Bose–Einstein Condensation by actual laboratory experiments. A in a Gas of Sodium Atoms,” Physical Rev. Let- Peter M. Ketcham is a computational scientist successful outcome stimulates further ters, vol. 75, no. 22, 1995, pp. 3969–3973. in the Information Technology Laboratory of numerical simulations, further simula- 3. C.C. Bradley et al., “Evidence of Bose–Ein- the National Institute of Standards and Tech- tion visualizations, and further labora- stein Condensation in an Atomic Gas with nology (NIST). His research interests include 13 tory experimentation. Attractive Interactions,” Physical Rev. Letters, scientific visualization and high-performance Visualization of BECs enhances and vol. 75, no. 9, 1995, pp. 1687–1690. computing. He received his MS in mathemat- accelerates the scientific investigation 4. C.C. Bradley, C.A. Sackett, and R.G. Hulet, ics from the University of Minnesota. Contact of this exotic state of matter. Visual- “Bose–Einstein Condensation of Lithium: Ob- him at [email protected]. ization is an integral part of the nu- servation of Limited Condensate Number,” merical simulation process and a key Physical Rev. Letters, vol. 78, no. 6, 1997, pp. David L. Feder is an assistant professor in the component of modern, collaborative 985–989. Department of Physics and Astronomy at the research efforts. In this work, visual- 5. S.N. Bose, “Plancks Gesetz und Lichtquan- University of Calgary in Alberta, Canada. He is ization enables scientific discoveries tenhypothese,” Zeitschrift für Physik, vol. 26, currently exploring the possibility of using that may be difficult to achieve by no. 3, 1924, pp. 178–181. Bose–Einstein condensates to operate rudi- other methods. We expect visualiza- 6. A. Einstein, “Quantentheorie des Einatomi- mentary quantum computers. Between 1997 tion to play an important role in many gen Idealen Gases,” Sitzungsberichte der and 2002, he was a guest researcher at NIST, of the emerging technologies using Preussischen Akademie der Wissenschaften, where he worked on the theory of Bose con- BECs, such as superfluid gyroscopes, Physikalisch-mathematische Klasse, vol. 1924, densation in dilute trapped gases, in close col- graviometers, and quantum informa- no. 22, 1924, pp. 261–267. laboration with experimental efforts there. He tion devices. 7. A. Einstein, “Quantentheorie des Einatomi- obtained his MSc and PhD degrees in theoret- gen Idealen Gases, Zweite Abhandlung,” ical condensed matter physics at McMaster Acknowledgments Sitzungsberichte der Preussischen Akademie University in Hamilton, Ontario, Canada. Con- We thank the following individuals der Wissenschaften, Physikalisch-mathematis- tact him at [email protected].

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