Optical Signature of Quantum Coherence in Fully Dark Exciton Condensates.

We predict that the collision of two fully dark exciton condensates produces bright interference fringes. So, quite surprisingly, the collision of coherent dark states makes light. This remarkable effect, which is many body in essence, comes from the composite boson nature of excitons, through the fermion exchanges they can have which transform dark states into bright states. The possibility of optically detecting quantum coherence in a regime where the system is hidden by its total darkness was up to now considered as hopeless.

Bose-Einstein condensation and indirect excitons: a review.

We review recent progress on Bose-Einstein condensation (BEC) of semiconductor excitons. The first part deals with theory, the second part with experiments. This Review is written at a time where the problem of exciton Bose-Einstein condensation has just been revived by the understanding that the exciton condensate must be dark because the exciton ground state is not coupled to light. Here, we theoretically discuss this missed understanding before providing its experimental support through experiments that scrutinize indirect excitons made of spatially separated electrons and holes. The theoretical part first discusses condensation of elementary bosons. In particular, the necessary inhibition of condensate fragmentation by exchange interaction is stressed, before extending the discussion to interacting bosons with spin degrees of freedom. The theoretical part then considers composite bosons made of two fermions like semiconductor excitons. The spin structure of the excitons is detailed, with emphasis on the crucial fact that ground-state excitons are dark: indeed, this imposes the exciton Bose-Einstein condensate to be not coupled to light in the dilute regime. Condensate fragmentations are then reconsidered. In particular, it is shown that while at low density, the exciton condensate is fully dark, it acquires a bright component, coherent with the dark one, beyond a density threshold: in this regime, the exciton condensate is 'gray'. The experimental part first discusses optical creation of indirect excitons in quantum wells, and the detection of their photoluminescence. Exciton thermalisation is also addressed, as well as available approaches to estimate the exciton density. We then switch to specific experiments where indirect excitons form a macroscopic fragmented ring. We show that such ring provides efficient electrostatic trapping in the region of the fragments where an essentially-dark exciton Bose-Einstein condensate is formed at sub-Kelvin bath temperatures. The macroscopic spatial coherence of the photoluminescence observed in this essentially dark region confirms this conclusion.

Effects of fermion exchange on the polarization of exciton condensates.

Exchange interaction is responsible for the stability of elementary boson condensates with respect to momentum fragmentation. This remains true for composite bosons when single fermion exchanges are included but spin degrees of freedom are ignored. Here, we show that their inclusion can produce a spin fragmentation of the dark exciton condensate, i.e., an unpolarized condensate with an equal amount of spin (+2) and (-2) excitons not coupled to light. The composite boson many-body formalism allows us to predict that, for spatially indirect excitons, the condensate polarization switches from unpolarized to fully polarized when the distance between the layers confining electrons and holes increases. Importantly, the threshold distance for this switch lies in a regime fully accessible to experiments.

"Gray" BCS condensate of excitons and internal Josephson effect.

It has been recently suggested that the Bose-Einstein condensate formed by excitons in the dilute limit must be dark, i.e., not coupled to photons. Here, we show that, under a density increase, the dark exciton condensate must acquire a bright component due to carrier exchange in which dark excitons turn bright. This, however, requires a density larger than a threshold which seems to fall in the forbidden region of the phase separation between a dilute exciton gas and a dense electron-hole plasma. The BCS-like condensation which is likely to take place on the dense side, must then have a dark and a bright component--which makes it "gray." It should be possible to induce an internal Josephson effect between these two coherent components, with oscillations of the photoluminescence as a strong proof of the existence for this "gray" BCS-like exciton condensate.

From spherical to periodic symmetry: the analog of orbital angular momentum for semiconductor crystals.

The angular momentum formalism provides a powerful way to classify atomic states. Yet, requiring a spherical symmetry from the very first line, this formalism cannot be used for periodic systems, even though cubic semiconductor states are commonly classified according to atomic notations. Although never noted, it is possible to define the analog of the orbital angular momentum, by only using the potential felt by the electrons. The spin-orbit interaction for crystals then takes theL^⋅S^form, withL^reducing toL^=r^×p^for spherical symmetry. This provides the long-missed support for using the eigenvalues ofL^andJ^=L^+S^, as quantum indices to label cubic semiconductor states. Importantly, these quantum indices also control the phase factor that relates valence electron to hole operators, in the same way as particle to antiparticle, in spite of the fact that the hole is definitely not the valence-electron antiparticle. Being associated with a broader definition, the(L^,J^)analogs of the(L^,J^)angular momenta, must be distinguished by names: we suggest 'spatial momentum' forL^that acts in the real space, and 'hybrid momentum' forJ^that also acts on spin, the potential symmetry being specified as 'cubic spatial momentum'. This would castJ^as a 'spherical hybrid momentum', a bit awkward for the concept is novel.

Energy of N Cooper pairs by analytically solving the Richardson-Gaudin equations for conventional superconductors.

This Letter provides the solution to a yet unsolved basic problem of solid state physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 yr old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these N pairs via N parameters coupled through N nonlinear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the N-pair energy depends on N as N(N-1) only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.

Finite temperature formalism for composite quantum particles.

This Letter provides the missing part of the newly constructed many-body formalism for composite quantum particles: the introduction of a finite temperature. The finite T formalism we propose deeply relies on the existence of a compact closure relation for the (overcomplete) set of N-composite-particle states. As a first application, we here calculate the energy mean value of the exciton gas outside the condensation regime. We show that carrier exchanges increase its temperature dependence compared to elementary bosons, a signature of the degree-of-freedom increase resulting from the particle composite nature.

Optical traps for dark excitons.

We propose a mechanism for optical trapping of dark excitons by linearly polarized unabsorbed standing waves, with a potential depth of the order of a few meV. Since this trapping, based on carrier exchanges with virtual excitons coupled to unabsorbed photons, equally acts on bright and dark states, Bose-Einstein condensation of excitons--which occurs in dark states--must appear as dark spots in a cloud of bright excitons, at the trap potential minima, when the temperature decreases.

Quantum control of a trapped electron spin in a quantum dot using photon polarization.

We present an original scheme to rotate at will one electron spin trapped in a quantum dot by just acting on pump-laser polarization: The quantum control is based on the virtual excitation of electron light-hole pairs with pi symmetry, as possibly done by using a single laser beam with a propagation axis slightly tilted with respect to a weak magnetic field. This allows us to fully control the effective axis of the electron spin rotation through the pump polarization. Our analysis shows that quantum dots with inverted valence states are ideal candidates for ultrafast, high-fidelity, all optical control.

Bose-Einstein condensation in semiconductors: the key role of dark excitons.

Bose-Einstein condensation in semiconductors is controlled by the nonelementary-boson nature of excitons. Pauli exclusion between the fermionic components of composite excitons produces dramatic exchange couplings between bright and dark states. In microcavities, where bright excitons and photons form polaritons, they force the condensate to be linearly polarized, as observed. In bulk, they also force linear polarization, but of dark states, due to interband Coulomb scatterings. To evidence this dark condensate, indirect processes are thus needed.