Observations of superfluid behaviour — flow without friction — of unusual character in a condensedmatter system pave the way to investigations of superfluidity in systems that are out of thermal equilibrium.
When in 1937 liquid helium was first observed to flow with negligible viscosity through a narrow gap, it was clear that, at low temperatures, helium was different from ordinary fluids. This prompted Pyotr Kapitza to name the phenomenon superfluidity^{1} by analogy with superconductivity. Since then, experiments on liquid helium and cold atoms have revealed other aspects of superfluidity (Table 1), including quantized vortices, undisturbed flow past an obstacle (for example, a structural defect), and (metastable) persistent flow in a doughnutshaped geometry.
Experiments by Amo et al.^{2}, reported on page 291 of this issue, reveal a new variety of dissipationless flow in semiconductor microcavity polaritons — entities comprising both matter and light. The properties of the polariton fluid demonstrated by Amo et al. have their origin in the way polariton–polariton interactions modify the propagation of these quasiparticles, preventing scattering from structural defects in the microcavity. The question of how the finite lifetime of polaritons distinguishes these systems from previous examples of superfluidity provokes questions about the relationships between different aspects of superfluidity^{3}.
The system studied by Amo et al. is a semiconductor microcavity, consisting of a pair of mirrors (built from layers of semiconductors with alternating refractive indices) and a semiconductor quantum well, placed between these mirrors. The quantum well confines excitons (electronic excitations in the semiconductor) and the mirrors confine light. Because excitons can recombine and emit light, and light can create new excitons, repeated interconversion leads to new quasiparticles: microcavity polaritons^{4,5}. These quasiparticles inherit properties from both of their constituents: from light comes their very small effective mass (about 0.0001 that of the electron); from the exciton come the polariton–polariton interactions.
Polaritons also have properties that neither of their constituents has alone. Crucially for Amo and colleagues' experiment, the polariton dispersion relation (the dependence of energy on momentum) does not take the simple quadratic form that a free particle takes. This means that a pair of polaritons injected at one energy is able to scatter to one highenergy polariton state and one lowenergy state, in a process known as parametric scattering^{6}. The rate of scattering depends on the type of quantum statistics particles obey. Because polaritons behave as bosons and thus obey Bose–Einstein statistics, the rate of scattering goes up as the populations of the final states increase. Amo et al. used this property to create a small seed population in the highenergy state. This, in turn, triggered parametric scattering of polaritons from the injected energy (pump state) to a lowerenergy and a higherenergy state, producing a pulse of lowenergy polaritons that travelled through the microcavity, continually fed by the parametric scattering.
Previous experiments on microcavity polaritons, in which polaritons were injected (incoherently) in a range of highenergy states instead of by parametric scattering, showed that they can form a Bose–Einstein condensate^{7} (BEC): a form of matter that emerges when particles obeying Bose–Einstein statistics are cooled to very low temperatures and collapse into the same lowestenergy state, behaving as a single, coherent whole. Recently, quantized vortices have also been seen^{8} in these systems, a feature associated with superfluidity.
The phenomenon of superfluidity is closely related, but not equivalent, to Bose–Einstein condensation^{3}. In an interacting BEC, the occupation of a single quantum state by a large fraction of bosons means that the system is described by a single quantum wavefunction that satisfies a nonlinear equation for classical waves. Ubiquitous in nonlinear physics, this equation is known as the nonlinear Schrödinger equation^{9}. This equation, written in terms of the density and velocity of the condensate, has a form almost equivalent to the usual Euler equation for the dynamics of a nonviscous fluid. Thus, one has the ingredients necessary to produce many of the aspects of superfluidity, such as frictionless flow below the Landau critical velocity (Table 1).
Frictionless flow has been observed in liquid helium using moving ions as probes^{10}, and in BECs of dilute atoms using laser beams^{11}. At subcritical velocities, the superfluid flow around such an obstacle is symmetrical fore and aft of the direction of motion, so there is no drag. This absence of drag on moving objects is known as d'Alembert's paradox. In a classical fluid, drag does arise, because viscosity breaks the fore–aft symmetry. In their experiments, Amo and coworkers observed that a cloud of polaritons passed an obstacle — a structural defect in the microcavity — without any detectable drag, realizing d'Alembert's paradox in a condensate out of thermal equilibrium. In superfluid helium and atomic BECs, the drag above the critical velocity has been attributed to the shedding of vortices around the obstacle^{10}. Although no such vortices have been directly observed in the polariton experiments of Amo et al., it remains plausible that drag at speeds between the critical and sound velocities could arise from vortex formation in polariton fluids.
Amo and colleagues' experiments differ from previous investigations of superfluids in several ways. Most obviously, the polaritons have a finite lifetime. But this alone need not preclude many of the regular signatures of superfluidity. Perhaps most remarkably, the linearization of the relevant part of the polariton's dispersion relation, and thus the quasiparticle's immunity to scattering from structural microcavity defects, results from the nonlinear interactions of the lowenergy polaritons with the pump polaritons as well as interactions among the lowenergy polaritons. By contrast, for liquid helium or cold atoms, such linearization arises entirely as a result of interactions between the lowenergy particles. The question of whether such polariton fluids can be called superfluid is, to us, less interesting than investigating the properties that these outofthermalequilibrium polariton fluids possess. While experiments continue apace to explore these properties, it seems better to go with the flow.
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Jonathan Keeling is in the Department of Physics, University of Cambridge, Cambridgeshire CB3 0HE, UK. jmjk2@cam.ac.uk
 Jonathan Keeling
Natalia G. Berloff is in the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridgeshire CB3 0WA, UK. n.g.berloff@damtp.cam.ac.uk
 Natalia G. Berloff
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