Ultracold Bose gases deviate from the textbook picture

Statistical-physics textbooks typically present Bose–Einstein condensation in a way that’s similar to Albert Einstein’s original description. In a gas of identical bosons, they say, the statistical weighting of each state is such that the total occupation of the excited states is capped at a critical value N_c, which depends on temperature. If the temperature is decreased, or the particle number is increased, so that the number of identical particles exceeds N_c, the thermally excited portion of the system becomes saturated, and all additional particles must occupy the ground state.

The textbook picture doesn’t include interparticle interactions, but it does assume thermal equilibrium, which cannot exist without interactions of some form. It’s no surprise, therefore, that the picture doesn’t describe real systems exactly. But how far wrong is it? Physics is full of simplified descriptions that neglect some of the complexities of real systems but are still close enough to be applicable in many typical cases. Is the saturation picture of Bose–Einstein condensation one of them?

Zoran Hadzibabic and colleagues at Cambridge University have now shown that it’s not. ¹ Even when the interparticle interactions are moderate, the thermal component of the gas continues to grow far beyond N_c, and the condensate’s population is only about half what the saturation picture says it should be. By tuning the interaction strength, the researchers showed that as the system converged to the noninteracting limit, the saturation result was recovered. Since a noninteracting Bose–Einstein condensate (BEC) is itself not physically realizable, the demonstration of convergence constitutes the first experimental verification of Einstein’s description.

Inspiration from flatland

Studies of interparticle interactions in BECs are not new. Ever since the first experimental demonstration of Bose–Einstein condensation in 1995, some researchers have been exploring the properties of the condensed component, which are entirely controlled by interactions. Others have shown that, thanks to interactions, the critical number N_c (or the critical temperature T_c for constant particle number) isn’t quite what the textbook picture would predict. ² But the question of whether the thermal component actually becomes saturated for particle numbers above N_c has received much less attention.

Hadzibabic became interested in the saturation of Bose gases while, as a postdoc with Jean Dalibard at the École Normale Supérieure in Paris, he was studying two-dimensional systems of ultracold atoms (see PHYSICS TODAY, August 2006, page 17 ). Einstein’s statistical argument for saturation doesn’t work in two dimensions, but there is still a phase transition (called a Berezinskii-Kosterlitz-Thouless transition) to a superfluid state. The Paris researchers’ plan was to look side by side at the 2D system (which, as expected, did not show saturation) and a 3D gas (which they expected would saturate). But to their surprise, the 3D system was not saturated either.

“At the time, we could not make sense of it theoretically,” recalls Hadzibabic. “And we also could not prove anything experimentally because we were working with rubidium-87, so we had no access to a Feshbach resonance,” which would have allowed them to tune the interatomic interaction strength with an applied magnetic field. (For more on Feshbach resonances, see the Reference Frame by Daniel Kleppner in PHYSICS TODAY, August 2004, page 12 .) Although ⁸⁷Rb does have Feshbach resonances, they’re all either too narrow or at too high a magnetic field to allow practical control over the interactions.

Another alkali atom isotope, potassium-39, does have a convenient Feshbach resonance. But ³⁹K is much harder to cool to BEC temperatures, in part because of its tiny scattering length a, a quantity related to the square root of the scattering cross section extrapolated to zero collision energy. Applying a field near the Feshbach resonance can increase a, but that works only if the atoms are confined to an optical trap, for which they must already be quite cold. But the Feshbach resonance was so appealing that Hadzibabic, who by that time had joined the faculty at Cambridge, worked with his group to create one of the only ³⁹K-condensing machines in the world. (The first such apparatus was created by Massimo Inguscio and colleagues to study Anderson localization; see the article by Alain Aspect and Inguscio in PHYSICS TODAY, August 2009, page 30 .) They achieved the initial cooling by using ⁸⁷Rb to sympathetically cool the ³⁹K atoms.

Unsaturated gas

To vary the number of atoms in their ³⁹K gas, Hadzibabic and colleagues hold the gas in an optical trap for up to two minutes. During that time, some atoms escape (through collisions with the background gas and other processes) and the rest re-equilibrate. The researchers then release the gas from the trap and allow it to freely expand; the cloud of thermal atoms expands faster than the condensate, so the researchers can segregate the two components and determine their respective populations. Repeating that process some 100 times, varying the holding time while keeping the temperature and magnetic field constant, they could obtain a plot like the one in figure 1a. When the temperature is kept stable enough (within a few nanokelvin) over the course of the experiment, the data clearly show that the thermal population increases beyond the expected point of saturation.

Why would an ultracold Bose gas in an optical trap have more thermal atoms than the saturation picture predicts? Qualitatively, the researchers attribute the effect to repulsion between the thermal and condensed components, which turns the effective potential for the thermal atoms from a harmonic well into a Mexican hat. (The Mexican hat shape stems from the statistics of identical bosons, which cause the condensed atoms to interact twice as strongly with the thermal atoms as they do with one another. Were that not the case, the effective potential in the region of the condensate would be flat.) As a result of the repulsion, the thermal atoms spread out over a larger volume, and additional excited states become thermally accessible. The larger the population of the condensate, the more the thermal atoms spread out.

Figure 1b shows similar data series for 18 sets of experimental conditions, with bluer colors representing lower temperatures and shorter scattering lengths. As either the temperature or the scattering length decreases, the population of the condensate appears to approach the saturated-gas prediction shown by the solid black line, but that qualitative observation doesn’t prove that the system actually converges to the saturation picture at T = 0 or a = 0. For temperature, though, the convergence is trivial: At zero temperature, by definition, there is no thermal component, so N_c = 0 and the population of the condensate equals the total number of atoms in the gas.

To the limit

To figure out which numbers they should look at to quantify the convergence to the noninteracting limit, the researchers simulated the gas using Hartree–Fock theory. Hartree–Fock is a mean-field theory in that it treats each particle as moving in the potential energy landscape created by all the others. It’s inexact because it neglects interparticle correlations, but it provides a pretty good approximation of many quantities for many systems (including the electrons in molecules; see the article by Martin Head-Gordon and Emilio Artacho in PHYSICS TODAY, April 2008, page 58 ).

According to Hartree–Fock theory, for atom numbers greater than N_c, the thermal population N_therm should be linearly related to the condensed population N_cond raised to the 2⁄5 power. Furthermore, the slope of that line (which the researchers called the nonsaturation slope, since the slope for a saturated gas is zero) should be directly proportional to x = ξT²a^2/5, where ξ is a combination of system parameters that makes x dimensionless. As a goes to zero, therefore, the nonsaturation slope should also go to zero, and the saturation picture should be recovered.

For small condensates (with atom numbers just above N_c), the data matched the Hartree–Fock predictions well. For larger condensates, the plot of N_therm versus N_cond^2/5 strayed from the Hartree–Fock linear relationship, but the researchers were able to redefine the nonsaturation slope, and they found, as shown in figure 2, that it was still proportional to x. Even two data points taken with ⁸⁷Rb condensates at different temperatures (remember, the scattering length for ⁸⁷Rb can’t be changed) lie on the line.

Although the researchers observed that the scaling behavior is universal with respect to temperature and atomic species, it’s not universal with respect to trap geometry: The amount by which the thermal atoms can spread out—and whether they can spread out at all—is expected to depend critically on the size, shape, and harmonic nature of the optical trap. One of the next steps the researchers hope to take is to look at the effect of different trapping potentials, especially disordered potentials that might change the effective dimensionality of the system.