Chapter 19 addresses bosonic condensation in composite bosons. It first considers the coboson ground-state energy, which is analytically known by now in the case of N Cooper pairs interacting via the reduced BCS potential but not yet known for N Wannier or Frenkel excitons, owing to the complexity of the Coulomb potential. Still, the chapter provides hints on how to derive the density linear term of the N-exciton ground-state energy in the case of Wannier excitons. It also discusses the role of the exciton composite nature in condensate momentum fragmentation, spin fragmentation, and condensate brightness. Indeed, the excitons that have the lowest energy and thus are the ones expected to condense are not coupled to light; so, the exciton condensate must be dark. However, a bright exciton condensate can appear, coherent with the dark exciton condensate, above a density threshold, owing to coupling induced by carrier exchange between dark excitons and bright excitons.
Chapter 19 addresses bosonic condensation in composite bosons. It first considers the coboson ground-state energy, which is analytically known by now in the case of N Cooper pairs interacting via the reduced BCS potential but not yet known for N Wannier or Frenkel excitons, owing to the complexity of the Coulomb potential. Still, the chapter provides hints on how to derive the density linear term of the N-exciton ground-state energy in the case of Wannier excitons. It also discusses the role of the exciton composite nature in condensate momentum fragmentation, spin fragmentation, and condensate brightness. Indeed, the excitons that have the lowest energy and thus are the ones expected to condense are not coupled to light; so, the exciton condensate must be dark. However, a bright exciton condensate can appear, coherent with the dark exciton condensate, above a density threshold, owing to coupling induced by carrier exchange between dark excitons and bright excitons.
Chapter 16 introduces the last part that addresses the bosonic condensation of composite bosons. To bridge the gap between elementary bosons and composite bosons, elementary fermion systems like semiconductor electrons and holes are also studied. Whereas the previous chapters provided a fairly precise knowledge of bosonic particles relevant for condensed matter physics, namely, Wannier excitons, Frenkel excitons, Cooper pairs, biexcitons, and polaritons, this chapter addresses an important question of current interest: what is the ground state of N composite bosons from the perspective of Bose-Einstein condensation? Or rather, how does the genuine Bose-Einstein condensation for elementary bosons persist in composite boson systems? This chapter lists important results with respect to condensate momentum fragmentation or spin fragmentation, correlation energy, and the dark nature of exciton condensate, to emphasize the connection between elementary bosons, elementary fermions, elementary fermion pairs, and composite bosons, from the composite boson perspective of their condensation.
Abstract This book bridges the gap between two major fields in condensed matter physics: semiconductors and superconductors. Through an original perspective that their key particles, excitons and Cooper pairs, are composite bosons (cobosons), the book raises fundamental questions: how does the Pauli exclusion principle affect the fermionic components of bosonic particles at the microscopic level, and how does this appear in the coboson many-body physics at the macroscopic level? What insights would the study of Wannier excitons, Frenkel excitons, and Cooper pairs provide about the “bosonic condensation” of composite bosons and how it differs from the Bose-Einstein condensation of elementary bosons? The book starts from a solid mathematical and physical foundation to derive excitons and Cooper pairs and introduces Shiva diagrams as a graphic support to grasp the many-body physics induced by fermion exchange in the absence of fermion-fermion interaction–a mechanism not visualized in standard Feynman diagrams. The book also covers composite bosons related to excitons: trions, biexcitons, and polaritons. The last part of the book is devoted to composite boson condensation, from the prototypical Bose-Einstein condensation of free elementary bosons and the consequence of interaction, to interacting electrons in the dense regime, to electron-hole pairs in the dense and dilute regimes. Advanced undergraduate and graduate students in physics, with no prior background, will benefit from this book, as numerous appendices provide the materials required to follow the various chapters. The concepts and formalism presented should also prove useful in research on ultracold atomic gases, polariton condensation, and quantum information.
Chapter 17 first considers noninteracting elementary bosons for zero and finite temperatures and then discusses the effects of temperature and space dimension on Bose-Einstein condensation. Condensation of massive bosons in a potential trap is also discussed. Next, the chapter discusses condensate fragmentation, showing that the condensate stability against momentum fragmentation requires the introduction of a boson-boson repulsive interaction: exchange processes between the two parts of the fragmented condensate provide a positive extensive contribution to the N-boson energy, which stabilizes the unfragmented condensate. By contrast, spin fragmentation can occur: the condensate is “spin fragmented,” that is, unpolarized, for same-spin bosons undergoing repulsive interactions. The effect of interaction on the elementary boson ground-state energy is also presented, with emphasis on the physical origin of the singularity of the boson correlation energy, and its similarity with the singularity of the electron correlation energy.
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.
This book bridges a gap between two major communities of Condensed Matter Physics, Semiconductors and Superconductors, that have thrived independently. Through an original perspective that their key particles, excitons and Cooper pairs, are composite bosons, the authors raise fundamental questions of current interest: how does the Pauli exclusion principle wield its power on the fermionic components of bosonic particles at a microscopic level and how this affects the macroscopic physics? What can we learn from Wannier and Frenkel excitons and from Cooper pairs that helps us understand "bosonic condensation" of composite bosons and its difference from Bose-Einstein condensation of elementary bosons? The authors start from solid mathematical and physical foundation to derive excitons and Cooper pairs. They further introduce Shiva diagrams as a graphic support to grasp the many-body physics induced by fermion exchange - a novel mechanism not visualized by standard Feynman diagrams. Advanced undergraduate or graduate students in physics with no prior background will benefit from this book. The developed concepts and methodology should also be useful to present researches on ultracold atomic gases, exciton-polaritons, and quantum information.
This paper shows the necessity of introducing a quantum object, the ``coboson,'' to properly describe, through a fermion scheme, any composite particle, such as the exciton, which is made of two fermions. Although commonly dealt with as elementary bosons, these composite bosons---cobosons in short---differ from them due to their composite nature which makes the handling of their many-body effects quite different from the existing treatments valid for elementary bosons. As a direct consequence of this composite nature, there is no correct way to describe the interaction between cobosons as a potential V. This is rather dramatic because, with the Hamiltonian not written as H=H0+V, all the usual approaches to many-body effects fail. In particular, the standard form of the Fermi golden rule, written in terms of V, cannot be used to obtain the transition rates of two cobosons. To get them, we have had to construct an unconventional expression for this Fermi golden rule in which H only appears. Making use of this expression, we give here a detailed calculation of the time evolution of two excitons. We compare the results of this exact approach with the ones obtained by using an effective bosonic Hamiltonian in which the excitons are considered as elementary bosons with effective scatterings between them, these scatterings resulting from an elaborate mapping between the two-fermion space and the ideal boson space. We show that the relation between the inverse lifetime and the sum of the transition rates for elementary bosons differs from the one of the composite bosons by a factor of 1∕2, so that it is impossible to find effective scatterings between bosonic excitons giving these two physical quantities correctly, whatever the mapping from composite bosons to elementary bosons is. The present paper thus constitutes a strong mathematical proof that, in spite of a widely spread belief, we cannot forget the composite nature of these cobosons, even in the extremely low-density limit of just two excitons. This paper also shows the (unexpected) cancellation in the Born approximation of the two-exciton transition rate for a finite value of the momentum transfer.