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# Theory

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## A. Phase transitions with exciton polaritons

Interacting Bose-Fermi systems are a promising platform for investigating novel many-body physics in a highly controllable setting. However, virtually all implementations of Bose-Fermi systems up to date have used ultracold bosonic and fermionic atomic gases. In these systems, Feshbach resonances have been used to tune the strength of interactions. This allowed the investigation of very interesting many-body physics [1]^{}. such as the competition between various phase transitions like the supersolid formation and superconductivity. However, the parameter space that can be covered by ultra cold atoms systems is often very limited. For example the Fermi energies that can be achieved are very small and therefore ultracold atomic gases actually correspond to very hot solid-state systems. Furthermore, contact interactions are relevant for ultracold atoms, restricting the possibilities for exploring phases that appear in the presence of finite range interactions.

### 1. Two-dimensional Bose-Fermi mixtures

One research direction in our group consists in analysing a novel solid-state, Bose-Fermi system. In this system the fermions consist of a two-dimensional electron system (2DES) while the bosons are the elementary excitations of an exciton-polariton Bose-Einstein condensate (BEC) in two dimensions. The boson-fermion interaction is structurally similar to the electron-phonon interaction. Unlike the familiar electron-phonon coupling this interaction strength depends on the number of polaritons in the BEC which can be controlled optically. Furthermore, the electron-polariton coupling has a finite range and the interaction peaking at a finite tunable wavevector. This highly tunable solid-state system can vercome many of the limitations of the ultracold gases and allows us to investigate a richer phase space. Experimentally, this system can be implemented either in semiconductor heterostructures or in the newly-emerging transition metal dichalcogenide monolayers. We are interested in both the effect of the 2DES on the BEC as well as the effect of the BEC on the 2DES. It turns out that, due to the interaction with the 2DES, the BEC becomes unstable towards the formation of a supersolid, a spatially ordered material with superfluid properties. On the other hand the 2DES also becomes unstable and should exhibit a transition towards a superconducting phase. Since the boson-fermion interaction is strongly peaked in momentum space we can also expect a charge density wave instability. The interplay and competition between these different phases in two dimensions is likely to reveal rich physics. At the same time, the possibility to interface quantum Hall physics, superconductivity and nonequilibrium polariton condensates is likely to provide fertile ground for investigation of completely new physical phenomena. In addition to providing a platform for investigating exotic phases of matter the all optical control of superconductivity in semiconductor heterostructures could enable the realization of new device concepts compatible with semiconductor nanotechnology [2].

### 2. Number conserving p-wave pairing in wire geometry

An interesting application of the polariton induced attractive electron-electron interactions is the realisation of unconventional pairing in a one-dimensional system. The experimental setup for this project would involve a nanowire or a gate defined one-dimensional electron system. Recent work has shown a one-dimensional spin-polarized superconductor hosts exotic quasiparticles at its edges. The non-local characteristics of these quasiparticles (termed Majorana zero modes), make them an attractive candidate for applications in fault-tolerant quantum computation.

One of the big challenges of incorporating Majorana fermions as building blocks of quantum computation architectures is that there has not yet been a systems where the Majorana fermions can be easily manipulated. In a joint work with S. Huber, we explore the possibilities for manipulating Majorana modes using a laser that controls the polariton density. The high tunability of our setup allows us to explore many other interesting phenomena emerging in one-dimensional physics. Tuning the filling and the periodicity of the system by electromagnetic means, spin and charge density instabilities as well as elusive superconducting phases such as the FFLO states can be explored. Interestingly, finite range interactions in one dimensions is not treatable using the usual bosonization method, which is suitable for low energy physics in the vicinity of the Fermi energy. Therefore, the retardation effects that arise from the momentum dependence of the polariton induced electron-electron interactions cannot be treated satisfactorily. We explore the possibility of using a combination of mean field and exact numerical methods to determine the existence of unconventional pairing in our one-dimensional setup.

## B. Trion-polaritons

Another research direction we pursue focuses on trion-polaritons, a bound 4-body complex consisting of a hole, two electrons and a photon. These particles are the optical excitations of a low density 2DES strongly coupled to a cavity mode. Until now, trions have been observed in the optical spectra of 2DES, carbon nanotubes and more recently in transition metal dichalcogenides. Variational calculus suggests that the 2D trion can be approximated by two electrons orbiting a fixed hole, with one of the electrons having a larger Bohr radius. Recent theoretical and experimental investigations suggests that by coupling the trions to a two-dimensional cavity mode, trion-polaritons can be formed. Furthermore, the competition between the Coulomb interaction and light matter coupling determines the spin properties of the trion-polaritons. In a joint work with C. Ciuti, we address is the strength of polariton-polariton interactions and their enhancement by coupling to a 2DES.

## References

- [1] Many-body physics with ultracold gases Immanuel Bloch, Jean Dalibard, Wilhelm Zwerger,Reviews of Modern Physics, 80(3):885, 2008.
- [2] Superconductivity and other phase transitions in a hybrid Bose-Fermi mixture formed by a polariton condensate and an electron system in two dimensions Ovidiu Cotleţ, Sina Zeytinoǧlu, Manfred Sigrist, Eugene Demler, Ataç Imamoǧlu, arXiv preprint arXiv:1510.02001, 2015