Kay Kirkpatrick

Bose-Einstein Condensation

Bose-Einstein condensates (BECs) are unusual states of matter near absolute zero that can be used to slow and briefly stop light, as well as convert light to matter and back. There are tantalizing applications, such as quantum information processing and increased accuracy in measurements by interferometry with atom lasers instead of traditional photon lasers. But the BEC is fragile and difficult to work with, so it is vital to work out the theory.

In a BEC, the particles are so supercooled (to a few billionths of a degree Kelvin) that they all fall into the ground state and exhibit quantum mechanical behavior macroscopically -- in effect they condense into a quantum super-particle. Around 1925 this phenomenon was predicted by Bose and Einstein, but only in 1995 was the BEC observed in a laboratory. Only two years ago was the BEC's macroscopic behavior explained mathematically in three dimensions: the microscopic repulsive interactions between quantum particles give rise, in the scaling limit, to quantum macro-behavior governed by the cubic nonlinear Schrödinger equation (CNLS):

      i ∂_t φ = - Δ φ + b |φ|² φ .

Two-dimensional cases are also of interest from the point of view of experimentalists studying important applications of Bose-Einstein condensation. Some superfluidity experiments involve constraining a BEC between two sheets of light, making the physical space effectively the plane. Other experiments for the sake of quantum computing involve a BEC in an optical lattice or constraining it to the surface of a torus.

These cases are addressed in my recent work with Benjamin Schlein and Gigliola Staffilani, which shows that the cubic NLS provides the macroscopic description of Bose-Einstein condensation in two dimensions, for both the plane and, more interestingly, the torus. The key novelty in our proof is for the uniqueness of the limiting object (typically the hardest part of these proofs). Where Fourier transforms appeared previously, we encounter Fourier series in the toroidal case; to analyze these sums, we resort to ingenious number-theoretical techniques.

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The rate of convergence has been studied for quantum many-body systems, and it's still not known exactly how fast these marginal densities converge. With Gérard Ben Arous, I am working on a large deviations principle (LDP) for the mean-field limit. We have defined empirical measures at time zero for factorized initial data with suitable regularity, and we can prove that the diagonal one-particle densities satisfy an LDP.

What happens after time zero is highly nontrivial, however, because quantum correlations immediately develop and destroy the factorization of the solution, though it is asymptotically factorized by the limit theorems. We have defined empirical measures for asymptotically factorized data, and we are working to use the contraction principle to prove an LDP for positive time. In addition, we are formulating a quantum probability framework with the full densities.

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It would also be fascinating to understand the phase transition between the BEC and a warmer boson gas (warmer than a microkelvin, anyway) -- and in particular, why the BEC is so likely to implode and then explode in what has been dubbed the bosenova.

I'm working with Sourav Chatterjee on a new probabilistic approach: we hope to say something interesting about a phase transition for the NLS's invariant measures. Besides the bosenova, this work might shed light on strange phenomena in other physical systems, such as Langmuir waves in plasmas, rogue waves on the ocean's surface, and self-focusing effects in nonlinear optics.

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(Figure: On the left, warmish bosons; on the right, supercold ones condense into a quantum super-particle in the lowest energy state. Courtesy of UC Boulder/NIST/JILA.)

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