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Atomic gas in flatland

Naturevolume 441pages10531054 (2006) | Download Citation

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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 atoms1. 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 issue2. 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 gas3.

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 proved4,5. 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 transition6 that he attributed to the uncoupling of bound pairs of vortices with superfluid currents circulating in opposite directions7. Some months later, Michael Kosterlitz and David Thouless8,9 presented an elegant thermo-dynamic argument that established the correct expression for the temperature, TBKT, 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 TBKT and the mass of the atoms involved10.

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 detail11. Recent quantitative numerical work12 has established the precise location of TBKT for the superfluid transition in such a gas as a function of the strength of interaction, U, between its atoms (the UT plane in Fig. 1). As U diminishes, the superfluid density decreases and TBKT approaches zero. This projected trend agrees with the observation that a non-interacting gas (U=0) does not support superfluidity.

Figure 1: Qualitative phase diagram for a two-dimensional trapped atomic Bose gas.
Figure 1

T is temperature, U collisional interaction strength. The trapping parameter, η, characterizes confinement: an infinite value of η (so at 0 on the 1/η axis) corresponds to the flat system with steep walls, and η=2 (marked at the point 1/η=1/2) describes a harmonically curved trap. On the U-T plane at 1/η = 0, a homogeneous gas will undergo a Berezinskii–Kosterlitz–Thouless (BKT) transition at TBKT into a superfluid phase, but a Bose–Einstein condensate (BEC) cannot form. The thickness of the blue line indicates the size of the jump in superfluid density at the transition point. The η-T plane at U=0 describes a trapped ideal (non-interacting) gas with a Bose–Einstein crossover, at TBEC, into a condensed phase that is not superfluid, with no BKT transition. Away from these planes, two crossovers are expected: at BKT, a sharp crossover takes the system into a quasi-long-range ordered phase with finite superfluid stiffness (blue region). The broad crossover at the lower temperature BEC marks the onset of order throughout the entire trap, establishing a condensate with a finite superfluid response. Hadzibabic and colleagues' experiments2 were carried out on the black line. On increasing the temperature, they observed both the disappearance of quasi-long-range order and the appearance of vortices (black dot).

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 analysed13. 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 Rη (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, TBEC, goes to zero as η increases (the ηT 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, T̃BEC, 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 forms14 Above T̃BEC, 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, T̃BKT. 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 limit15, and by numerical simulations16.

Hadzibabic and colleagues2 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 range2. With increasing temperature, they identified a sudden decrease in the degree of phase order, indicative of a jump in the superfluid density at T̃BKT. Simultaneously, the interference patterns were flooded with dislocation lines (Fig. 4 on page 1120), signalling the presence of vortices16.

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 observation2 of the Berezinskii–Kosterlitz–Thouless transition shows once again how fundamental concepts of physics can be extracted from experiments in a quantum gas.

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  1. Department of Physics, ETH-Zurich, Zurich, CH-8093, Switzerland

    • Tilman Esslinger
    •  & Gianni Blatter

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