Bose–Einstein condensation occurs when many particles enter into the same, coherent quantum state, and is now claimed to occur in various systems of 'quasiparticles' in solids. But is it the right term to use here?
The basic theory of the phenomenon known as Bose–Einstein condensation is well understood. When the temperature of a gas of integer-spin particles, or bosons, is low enough, thermo-dynamics causes a significant fraction of them to spontaneously enter a single quantum state. These particles form the condensate, and they act collectively as a coherent, classical wave.
In this issue, Kasprzak et al. (page 409)1 and Demokritov et al. (page 430)2 extend the concept of Bose–Einstein condensation to two different types of quasiparticles — discrete quanta of energy that can be treated as real particles — in solids. These papers, together with another published two years ago3, raise questions of terminology. First, two of the systems1,3 are two-dimensional. But wasn't it proved long ago that Bose–Einstein condensation is forbidden in two dimensions? Second, in one of the experiments1, the quasiparticles live for just a few picoseconds. Can we really talk of an equilibrium condensation in such a system? Third, in two of the systems1,2, a coherent field is driving the system, and the number of particles is not conserved. Under these circumstances, is the condensation really a thermodynamic phase transition? Or, with all these objections, is it legitimate to say that we now have another class of condensates? Should we put the Bose–Einstein condensation of quasiparticles in solids on the list along with condensates of atoms in optical traps and superfluid helium?
Rather than haggling over the exact meaning of the term Bose–Einstein condensation, we can adopt a more general concept that allows us to use the same language to talk about many systems. This concept is the spontaneous emergence of coherence. When helium condenses into a superfluid at low temperatures, it can be described as a strongly interacting, sponta-neously coherent system. Atoms in traps, which form the classic Bose–Einstein condensate, are an example of a weakly interacting, sponta-neously coherent system. Superconductivity stems from the spontaneous coherence of paired-up electrons known as Cooper pairs. The onset of coherent emission from a laser is also an example of spontaneous coherence, although in this case the coherence does not occur through a thermodynamic phase transition.
This spontaneous coherence can occur in two-dimensional systems just as well as in three dimensions. Experiments with liquid helium on surfaces, for example, have shown that two-dimensional helium can be superfluid, just like bulk helium4. But in two dimensions, the coherence length cannot be infinite5: over large distances, fluctuations will cause regions to go out of phase with one another. Such fluctuations can also occur in three-dimensional systems, but there is no upper bound to the coherence length6.
When the system is finite in two dimensions — in a trap, for instance — the distinc- tion between a two-dimensional and a three-dimensional condensate is moot, as both have an upper bound for the coherence length dictated by the system's size. Kasprzak et al.1, for example, created polaritons in a finite cloud in a two-dimensional structure using a focused laser. (Polaritons are quasiparticles formed from electronic excitations strongly coupled to photons.) The authors argue that the coherence length in this case is comparable to the size of the cloud of polaritons. An alternative experimental approach is to create a trap to confine polaritons to a finite region using an external force7, and initial results with trapped polaritons in gallium arsenide semiconductors also show spontaneous coherence.
Can quasiparticles with finite lifetime —whose numbers are not conserved — undergo Bose–Einstein condensation? The consensus on this issue, reached about a decade ago8,9, is that if the lifetime of the particles is much longer than the time they need to scatter with each other, condensation is possible, although there may not be time to measure the resulting superfluidity10.
In practice, atoms in optical traps do have a finite lifetime — they evaporate or form molecules — but this does not prevent condensation. For quasiparticles in solids, the situation is no different, but the timescales are shorter. Although the picosecond lifetime of Kasprzak and colleagues' polaritons1 might seem short, the particles can interact even faster. The evidence comes from the distribution function of the particles' energy, which fits a Maxwell–Boltzmann distribution (meaning that the curve tails off at high energy in a way that fits a straight line when plotted semi-logarithmically) when the particles are not condensed. This behaviour is clearly observed in the polariton experiments when the particles are near condensation1. When the particles condense, presumably they interact even faster. In the same way, in Demokritov and colleagues' study2, the number of magnons — quasiparticles representing collective spin excitations in a solid — can be approximately conserved on sub-microsecond timescales. If they interact on even shorter timescales, they can undergo condensation.
As spontaneous coherence is so essential to the phenomenon of Bose–Einstein condensation, to prove that one has a condensate, one must demonstrate two things: first, that there is coherence, and second, that this coherence is spontaneous. Experiments showing interference between two condensates11 are generally regarded as more convincing evidence of coherence, and therefore condensation, than observations of many particles seemingly in the ground state. Experimentally, with an energy distribution alone, it is difficult to tell whether particles really are in the same quantum state, or just close to it.
Direct evidence of spontaneous coherence of quasiparticles in solids has been hard to come by. Demokritov et al.2 demonstrate a build-up of a population of magnons near the ground state, but present no direct test of coherence. In earlier experiments3, coherence among excitons was deduced indirectly from measurements of their current. Kasprzak et al.1, however, do provide direct evidence of coherence among exciton polaritons, both in the interference and in the spontaneous linear polarization of light emitted by the system.
That the coherence should be spontaneous is essential. Many systems can be coherent if driven by another coherent source: any loudspeaker, for example, generates a coherent state of long-wavelength phonons (sound quanta). Calling this a 'driven condensate' would, however, seem to contradict the spirit of the term 'condensate' as a spontaneous thermodynamic phenomenon. Equally, the coherence of polaritons directly coupled to a coherent laser beam12 is not generally accepted as condensation, although the phenomenon has great promise for the development of nonlinear devices for optical switching13,14. In Kasprzak and colleagues' case1, as in earlier attempts15, the laser that creates the polaritons is not directly coupled to the condensate; numerous phonons are also emitted, which, one presumes, scramble any phase information from the laser. Thus, the coherence in the polariton condensate must be spontaneous, and indeed, there is a clear change from incoherent to coherent light emission at the transition point in these experiments.
The polariton condensate1 is a pumped system that emits coherent light. So is it fundamentally different from a laser? Some have given it the name 'polariton laser'15. Fundamentally, both are examples of spontaneous coherence. In the case of a laser, however, coherent emission occurs only when enough energy is pumped into a system so that an excited electronic state acquires a population far greater than that of the ground state. In the case of a polariton condensate, the thermodynamics of bosons drives the transition. This means that the coherent light emission occurs without a population inversion16. This may lead to very low-power coherent light emitters in the future.
After steady work for two decades, the field of the condensation of quasiparticles in solids is now blossoming. Besides the examples discussed1,2,3, experiments continue on trapping excitons in coupled quantum wells17, on excitons in bulk semiconductors18, and on tripletons in magnetic insulators19. The appeal of quasiparticles in solids as systems to observe condensation is, in part, their light mass. The smaller the mass, the higher the critical temperature at which condensation occurs, and there is no reason why condensation of one of these quasiparticles cannot occur at room temperature. In fact, Demokritov and colleagues' experiments with magnons2 supply evidence that this might already have been observed. Some of these schemes of quasiparticle condensation also, intriguingly, imply superconductivity20.
We now have several examples of systems with evidence of spontaneous coherence caused by a thermodynamic phase transition. Some people may not want to call certain cases Bose–Einstein condensation, but these systems collectively represent a new frontier in coherent phenomena.
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