The observation of Bose–Einstein condensation in an atomic gas was a seminal result. Two-dimensional gases are more complex, and an intriguing interference experiment has exposed a different superfluid transition.

Atomic quantum gases are unique systems for investigating elementary concepts in many-particle physics. The most prominent example is Bose–Einstein condensation — the formation of a gaseous superfluid through the collapse of an ensemble of atoms into a single quantum state — which was first observed in an ultracold gas of rubidium atoms^{1}. Atomic gases are useful because their ingredients are familiar: the atoms interact through well-defined two-body collisions, and the trapping potentials confining the gas are known. That simplicity allows experimental insights to be linked directly to theoretical fundamentals.

But can physics in two dimensions — which is intrinsically different from that in three — also be investigated in atomic quantum gases? Hadzibabic and colleagues set out to do just this, and their results are published on page 1118 of this issue^{2}. In an intriguing experimental set-up, the authors used laser-induced optical potentials to squeeze an ultracold ‘Bose’ gas — consisting of atoms with integer spin, in this case rubidium-87 — into flat traps so that the atoms could move only within a plane. To find out what the gas was doing, they cunningly placed two traps on top of one another and released the atoms from both. Images of the resulting interference patterns yielded information on crucial aspects of the two-dimensional gas^{3}.

The history of physics in two dimensions is littered with puzzles and surprises. In 1966, a theorem proclaiming the absence of a class of phase transitions in one and two dimensions was proved^{4,5}. Essentially, minute thermal fluctuations prevent order from emerging over large distances in such systems. A Bose–Einstein condensate consists of atoms whose quantum-mechanical wavefunctions are ‘coherent’ — perfectly in phase — and so exhibit a form of long-range order. Thus, an immediate consequence of this theorem was that, except purely theoretically at a temperature of absolute zero, a Bose–Einstein condensate cannot exist in one or two dimensions.

In two dimensions, however, the destruction of order by thermal fluctuations is only marginal: order can develop over short distances, decaying slowly with increasing distance so that quasi-long-range order survives. This reduced order is sufficient to generate another property of condensed matter: that of stiffness, such as occurs when a solid resists the shear force applied to two opposite surfaces of the body. This shear stiffness is lost when the solid melts and becomes a liquid. Analogously to the behaviour of a solid, below a certain temperature a Bose gas resists a ‘twist’ in the phase of its atoms applied to its boundaries. This phenomenon is the prerequisite for establishing a superfluid that flows without friction.

Usually, such stiffness is established by a phase transition into an ordered phase. But in two dimensions, stiffness can be present without true order. The Soviet physicist Vadim L'vovich Berezinskii was the first to understand the curious nature of this superfluid yet non-ordered, low-temperature phase. In 1971, he predicted the existence of a phase transition^{6} that he attributed to the uncoupling of bound pairs of vortices with superfluid currents circulating in opposite directions^{7}. Some months later, Michael Kosterlitz and David Thouless^{8,9} presented an elegant thermo-dynamic argument that established the correct expression for the temperature, *T*_{BKT}, at which this Berezinskii–Kosterlitz–Thouless (BKT) transition occurs. Going above this temperature, quasi-long-range order suddenly disappears, as does superfluidity — the superfluid density collapses to zero in a characteristic jump that depends only on *T*_{BKT} and the mass of the atoms involved^{10}.

The particular case relevant to the work of Hadzibabic *et al*.^{2} — a two-dimensional Bose gas whose atoms interact only weakly — has been studied in detail^{11}. Recent quantitative numerical work^{12} has established the precise location of *T*_{BKT} for the superfluid transition in such a gas as a function of the strength of interaction, *U*, between its atoms (the *U*–*T* plane in Fig. 1). As *U* diminishes, the superfluid density decreases and *T*_{BKT} approaches zero. This projected trend agrees with the observation that a non-interacting gas (*U*=0) does not support superfluidity.

The question of whether the finite room for movement that is induced by a trapping potential such as that of Hadzibabic *et al*.^{2} changes the physics of a non-interacting Bose gas has also been analysed^{13}. The dramatic conclusion was that such a confinement modifies the density of allowed states such that a crossover to a Bose–Einstein condensed (and hence ordered) phase is also possible in one- and two-dimensional traps. By considering trapping potentials with the algebraic form *R*^{η} (where *R* is the radial distance from the centre of the trap and *η* is a parameter indicating the steepness of the trap's walls), the authors established a continuous connection between a ‘harmonic’ trap (with *η*=2) and a trap with a flat bottom and extremely steep walls (*η*→∝). This latter limit mimics the behaviour of a homogeneous gas of finite density and, as expected, the transition temperature to a Bose–Einstein condensate, *T*_{BEC}, goes to zero as *η* increases (the *η*–*T* plane in Fig. 1).

A full description of Hadzibabic and colleagues' experimental situation further requires the inclusion of a non-zero collisional interaction (*U*>0), as well as the harmonic trapping potential. In such a set-up, below a crossover temperature, T̃_{BEC}, the decrease of phase order with distance can be so slow that phase coherence is preserved over all atoms of the sample, and a Bose-Einstein condensate forms^{14} Above T̃_{BEC}, larger phase fluctuations destroy the condensate, but should still leave the system in a superfluid state with quasi-long-range order. Hence, a second crossover at which superfluidity disappears when vortex pairs unbind must be expected at a higher temperature, T̃_{BKT}. The consistency of this picture (Fig. 1) is supported by a vanishing jump found in the superfluid stiffness when approaching the ideal (non-interacting) gas limit^{15}, and by numerical simulations^{16}.

Hadzibabic and colleagues^{2} carried out their experiments along the black line in the phase diagram of Figure 1. They analysed the contrast in the interference patterns of atoms emerging from their two traps as a function of the spatial range^{2}. With increasing temperature, they identified a sudden decrease in the degree of phase order, indicative of a jump in the superfluid density at T̃_{BKT.} Simultaneously, the interference patterns were flooded with dislocation lines (Fig. 4 on page 1120), signalling the presence of vortices^{16}.

The jump in density and flooding with vortices appeared over an appreciable range in temperature, as would be expected for a transition phenomenon in an inhomogeneous sample. Caveats do remain, however. The temperature range over which the jump was observed has yet to be related to the trap configuration, and the second smooth crossover into the Bose–Einstein condensed phase, which should be fully coherent over the entire sample size, has not been observed. Nevertheless, Hadzibabic and colleagues' direct observation^{2} of the Berezinskii–Kosterlitz–Thouless transition shows once again how fundamental concepts of physics can be extracted from experiments in a quantum gas.

## References

- 1
Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E. & Cornell, E. A.

*Science***269**, 198–201 (1995). - 2
Hadzibabic, Z., Krüger, P., Cheneau, M., Battelier, B. & Dalibard, J.

*Nature***441**, 1118–1121 (2006). - 3
Polkovnikov, A., Altman, E. & Demler, E.

*Proc. Natl Acad. Sci. USA***103**, 6125–6129 (2006). - 4
Hohenberg, P. C.

*Phys. Rev.***158**, 383–386 (1967). - 5
Mermin, N. D. & Wagner, H.

*Phys. Rev. Lett.***17**, 1133–1136 (1966). - 6
Berezinskii, V. L.

*Sov. Phys. JETP***32**, 493–500 (1971). - 7
Berezinskii, V. L.

*Sov. Phys. JETP***34**, 610–616 (1972). - 8
Kosterlitz, J. M. & Thouless, D. J.

*J. Phys. C***5**, L124–L127 (1972). - 9
Kosterlitz, J. M. & Thouless, D. J.

*J. Phys. C***6**, 1181–1204 (1973). - 10
Nelson, D. R. & Kosterlitz, J. M.

*Phys. Rev. Lett.***39**, 1201–1205 (1977). - 11
Popov, V. N.

*Functional Integrals in Quantum Field Theory and Statistical Physics*(Reidel, Dordrecht, 1983). - 12
Prokof'ev, N., Ruebenacker, O. & Svistunov, B.

*Phys. Rev. Lett.***87**, 270402 (2001). - 13
Bagnato, V. & Kleppner, D.

*Phys. Rev. A***44**, 7439–7441 (1991). - 14
Petrov, D. S., Holzmann, M. & Shlyapnikov, G. V.

*Phys. Rev. Lett.***84**, 2551–2555 (2000). - 15
Holzmann, M., Baym, G., Blaizot, J. -P. & Laloe, F. preprint available at http://www.arxiv.org/cond-mat/0508131 (2005).

- 16
Simula, T. P. & Blakie, P. B.

*Phys. Rev. Lett.***96**, 020404 (2006).

## Author information

### Affiliations

#### Department of Physics, ETH-Zurich, Zurich, CH-8093, Switzerland

- Tilman Esslinger
- & Gianni Blatter

### Authors

### Search for Tilman Esslinger in:

### Search for Gianni Blatter in:

## Rights and permissions

To obtain permission to re-use content from this article visit RightsLink.

## About this article

## Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.