Scattering amplitudes for dark and bright excitons

Abstract

Using the composite boson many-body formalism that takes single-exciton states rather than free carrier states as a basis, we derive the integral equation fulfilled by the exciton-exciton effective scattering from which the role of fermion exchanges can be unraveled. For excitons made of (±1/2)-spin electrons and (±3/2)-spin holes, as in GaAs heterostructures, one major result is that most spin configurations lead to brightness-conserving scatterings with equal amplitude ∆, in spite of the fact that they involve different carrier exchanges. A brightness-changing channel also exists when two opposite-spin excitons scatter: dark excitons (2, −2) can end either in the same dark states with an amplitude ∆e, or in opposite-spin bright states (1, −1), with a different amplitude ∆o, the number of carrier exchanges being even or odd respectively. Another major result is that these amplitudes are linked by a striking relation, ∆e + ∆o = ∆, which has decisive consequence for exciton Bose-Einstein condensation. Indeed, this relation leads to the conclusion that the exciton condensate can be optically observed through a bright part only when excitons have a large dipole, that is, when the electrons and holes are well separated in two adjacent layers. In contrast to the structureless 4 He, a number of bosonic condensates have more than one component inherited from the internal spin and orbital degrees of freedom of their constituents, the superfluid then being multi-component. Dipolar Bose gases [1] and superfluid phases of 3 He [2] are prime examples. This also occurs to excitons, which are composite bosons (cobosons for short) made of one conduction electron and one valence hole. Since electrons and holes carry spins, so do excitons, their condensate depending on these internal degrees of freedom. Recently, it has been shown that signatures of exciton condensates were long held back by the missed fact that the lowest-energy states are dark [3, 4], that is, not coupled to light. This fact precludes a direct photoluminescence observation of exciton condensate in a very dilute regime. Unambiguous optical evidences for Bose-Einstein condensation are bound to the density regime where dark and bright components coexist coherently [5]. The expected darkening[6-10] of the exciton gas upon cooling has been recently seen. Macroscopic spatial coherence of the bright component has also been revealed [7,10], in this way providing the most unambiguous evidence for the coexistence of dark and bright exciton condensates.As a rule, the energy of a condensate depends weakly on its internal degrees of freedom, all possible "spin" phases being essentially degenerate. These degeneracies are lifted once interactions are taken into account. One then has to determine which combination of the competing phases produces the lowest energy that rules the macroscopic properties of the condensate. In this Letter, we show that the relation between brightness-conserving and brightness-changing scattering amplitudes constitutes a crucial element to charact...

Temporal coherence of spatially indirect excitons across Bose–Einstein condensation: the role of free carriers

“…Bright excitons lying at a slightly higher energy than dark ones [5,6], Bose-Einstein condensation leads to a macroscopic occupation of lowest energy dark states so that the condensate is not detectable easily. In fact, it is only in double quantum wells (DQWs) that a condensate can be imaged directly [7], when a small fraction (∼20%) of bright excitons is possibly introduced coherently in a dark condensate by exciton-exciton interactions at sufficiently high densities (∼10 10 cm −2 ) [7][8][9]. The condensate then becomes gray and is signaled by the weak photoluminescence it radiates.…”