Superfluidity in Two Dimensions
Author: Christof Weitenberg
Affiliation: Laboratoire Kastler Brossel, Ecole Normale Supérieure, UPMC, CNRS, Paris, France.
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In our daily life experience, flow is always accompanied by friction. But in quantum physics, some materials can form a superfluid, in which the friction vanishes completely. This flow without friction was first observed in liquid helium in 1938 [1,2], when cooled below the lambda temperature. Also the electron gas in a metal can form a superfluid, which leads to the phenomenon of superconductivity, i.e. electric current without resistance.
Superfluid flow with respect to an external body is a metastable state. The ground state would be the fluid at rest, but the decay of the superflow is protected by an energy barrier. The superflow can decay via collective excitations such as phonons or vortices, which are only activated if the flow surpasses a critical velocity. Equivalently, an obstacle moving in a superfluid at rest can only create excitations if it surpasses the critical velocity.
The physical origin of superfluidity is intimately related to Bose-Einstein condensation, which is the macroscopic occupation of the lowest energy state and which occurs below a critical temperature. The system can then be described by a macroscopic wavefunction, which implies irrotational flow and a long-ranged phase coherence between different parts of the system. A three-dimensional (3D) Bose gas at low temperatures is both a Bose-Einstein condensate (BEC) and a superfluid.
Things get more involved in lower dimensions. Here the increased role of thermal fluctuations prohibits a conventional phase transition to a state with long-ranged order. In particular, the uniform two-dimensional (2D) Bose gas does not form a BEC. Now the question arises whether superfluidity does survive the reduction to lower dimensions despite the absence of BEC. There is indeed a phase transition to a superfluid state with quasi-long-ranged order via the Berezinskii-Kosterlitz-Thouless (BKT) mechanism.
The original system to study superfluidity was liquid helium, and it was studied also in 2D films . With the achievement of BEC in dilute atomic gases , there is a new highly-controllable superfluid system at hand. It has weak isotropic interactions, greatly facilitating the comparison with theory. Cold gases confined to two dimensions have been used to study the phase coherence and the microscopic origin of the BKT transition [5-7], but the direct observation of the superfluidity of the low temperature phase was so far missing. In our recent results published in Nature Physics , we present the first direct observation of superfluidity in a 2D atomic Bose gas.
To create the 2D gas, we tightly confine the atoms in the vertical direction. When the energy of the first excited state in this direction is larger than the thermal energy and the chemical potential, then the motion in this direction is essentially frozen and the system can be described as 2D. In the horizontal plane, the atoms are also trapped, but with a much weaker confinement, such that the overall shape of the atom cloud resembles a pancake.
Figure 1: Schematics of the experiment. The pancake-shaped atom cloud is stirred by a tightly focused laser beam, which acts as a repulsive obstacle. By stirring in circles, one can probe at a fixed density.
Following the method developed by the MIT group in 3D systems , we stir the cloud with an obstacle formed by a tightly focused laser beam (Figure 1). We observe the resulting heating as a function of the stirring velocity and find no dissipation below a critical velocity (see Figure 2). Only above this velocity can excitations occur, which lead to a heating of the cloud. This threshold behavior is the signature of superfluidity.
Figure 2: Measuring the critical velocity. The curve shows the temperature T of the cloud for varying stirring velocities v of the laser beam. There is no heating below a critical velocity vc, indicating the superfluid response of the atom cloud.
The phase transition to the superfluid state occurs above a critical phase space density, i.e. when the temperature is sufficiently low and the density sufficiently high. Because the atomic gas is in equilibrium, the temperature is constant over the cloud. The density, however, varies across the cloud, being largest in the center. Therefore we expect the gas to be in the superfluid state in the center of the cloud and in the normal state in the wings. The position of the boundary between the two states depends on the total atom number and temperature of the cloud.
In the experiments, we stir in circles centered on the cloud. In this way, we can probe at a fixed density. By varying the stirring radius as well as the atom number and temperature, we can map out the density and temperature dependence of the transition between the superfluid and the normal state (see Figure 3).
Figure 3: Mapping out the BKT phase transition. The non-zero critical velocities vc indicate the superfluid state, which occurs above a critical ratio of the local chemical potential µloc at the stirring radius and the temperature T of the cloud.
Our results complete the thermodynamic picture of the atomic 2D Bose gas after the measurement of the equation of state [10, 11]. The 2D superfluidity is also interesting in the context of the not-yet-understood cuprate high Tc superconductors, which are based on superfluidity whithin 2D layers of the cuprate oxide material .
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