Physics Today, volume 55, Number 3, page 18, March 2002
From Superfluid to Insulator: Bose-Einstein Condensate Undergoes a Quantum Phase Transition
The atoms in a BEC assemble gregariously into a coherent whole, but in a periodic potential that's sufficiently strong, they can separate into an array of isolated atoms.
Bose-Einstein condensates (BECs) have opened yet another promising avenue of
experimental research. This time, the road leads to an opportunity to study
quantum phase transitions in a very clean and controlled manner. Specifically,
researchers from the Max Planck Institute for Quantum Optics in
Matthew Fisher, a condensed matter theorist
Classical phase transitions are well known, the most obvious example being the melting of ice. At the melting point, thermal fluctuations drive the system from the liquid to the solid phase, or vice versa. A quantum phase transition is one that occurs at absolute zero: Thermal fluctuations are absent and the system is instead governed by quantum fluctuations.2 Only in the past decade or two have theorists begun to study in earnest such quantum phase transitions, and it's been difficult to find experimental systems that bear close resemblance to the idealized models.
One example of a quantum phase transition is that between a superfluid and a Mott insulator. In a superfluid, the atoms move in phase with one another, all part of a single macroscopic wavefunction. In a Mott insulator, each atom occupies its own separate quantum well, unaffected by any of its neighbors. As different as these two phases are, they are described by the same Hamiltonian and are characterized by the competition between two interactions: the tendency of the particles to hop into adjacent wells, and the interparticle forces that keep them in separate wells. Depending on the relative strengths of these two interactions, the system can go from a superfluid to an insulator and back again, much as ice melts and refreezes as the air gets warmer or cooler.
A transition to a Mott insulator becomes possible when a superfluid like a BEC is placed in a periodic potential. In 1989, theorists used a Bose-Hubbard model, which describes interacting bosons in a periodic potential, to study transitions in a system like superfluid helium-4 absorbed in a porous medium.3 Their predictions could not be unambiguously validated because of the imprecisely known interactions and the presence of disorder, or imperfections, in the confining lattices.
Three years ago, a team of theorists from
At the time of the Innsbruck-Victoria
paper, experimenters were able to make optical lattices in one, two, and three
dimensions, but they had not been successful in getting atoms to occupy more
than a small percentage of the lattice sites. (To picture an optical lattice,
think of the two-dimensional case, which is simply an egg-carton potential.) In
1998, Mark Kasevich and his group at
Bloch and company had to find room for the three pairs of lasers needed to create a 3D lattice, so they magnetically steered the ultracold atoms from a magneto-optic trap, which already has six lasers for cooling, to a separate trap (see the cover of this issue), where they formed a BEC and imposed the optical lattice.
With an average occupancy of one to three
atoms per lattice site, the
Interference patterns in absorption images (gauged by scale on right) result when a gas of cold atoms in a three-dimensional optical lattice is in its superfluid phase; no interference is seen in a Mott insulating phase. The depth of the potential wells in the lattice is systematically increased from 0 at (a) to 20 Er at (h), where Er is a reference energy. The phase transition occurs somewhere between (f) and (g). (Adapted from ref. 1.)
To find out what phase is present, the
Amazingly, experimenters can take the
atoms back and forth between these two phases. The phase coherence that's lost
when the atoms enter the insulating state is promptly restored when the system
re-enters the superfluid. The transition is rather
sharp as a function of well depth and comes at a value that agrees with the
predictions of the Innsbruck-Victoria group. Peter Zoller,
Although the quantum phase transition
technically occurs only at absolute zero, the atoms in the
A year ago, Kasevich,
together with coworkers from Yale and the
In both the Yale-Tokyo and the
(a) Two atoms (dark blue balls) occupy neighboring potential wells. U is the energy cost for them to be in the same well (pale blue balls). (b) Lowering one well relative to the other allows atoms originally in separate wells (light blue) to occupy the same well (dark blue). (Adapted from ref. 1.)
One prediction of the theory is that the
formation of a Mott insulator should be accompanied by the opening of an energy
gap in the excitation spectrum; as shown in the top panel of Figure 2, it costs an energy U to move an
atom from the left-hand to the right-hand well. Bloch and his colleagues came
up with a clever way to measure this energy gap. With the system in its
insulating phase, they applied an energy gradient to the potential wells, which
in 2D would be like tilting the egg carton. The effect is shown in the bottom
panel of the figure: Once the energy gradient has raised the relative energy of
the left-hand well by an amount U, the left-hand atom can hop, and both
atoms end up on the same site. The tilt threshold that results in such
tunneling tells experimenters the value of the energy
As for applications, Zoller said that the Mott insulator should allow interesting chemistry to happen. For example, "One might load exactly two atoms per lattice site and engineer the formation of molecules by way of a photoassociation process." Zoller and Ignacio Cirac (Max Planck Institute for Quantum Optics) have also proposed a scheme to entangle atoms for quantum computation using cold, controlled collisions.6,7 Zoller views the Mott insulator as an ideal starting point for their scheme.
Barbara Goss Levi
1. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, I. Bloch, Nature 415, 39 (2002).
2. S. Sachdev, Quantum Phase
3. M. P. A. Fisher, P. B. Weichman, G. Grinstein, D. S. Fisher, Phys. Rev. B 40, 546 (1989).
4. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, P. Zoller, Phys. Rev. Lett. 81, 3108 (1998).
5. C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, M. A. Kasevich, Science 291, 2386 (2001).
6. D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, P. Zoller, Phys. Rev. Lett. 82, 1975 (1999).
7. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000).