# PhD position – Strong coupling in moiré TMD cavities: Blockade and Topology

# PhD position - Strong coupling in moiré TMD cavities: Blockade and Topology

*This thesis will take in the framework of an ANR project between Institut Pascal (CNRS/UCA, Clermont-Ferrand), MajuLab CNRS and NTU (Singapore) and CRHEA, CNRS (Nice). The project title is “Engineering light-matter coupling in TMD-based optical microcavities: from free excitons to Moiré exciton-polaritons”. The project involves fabrication of microcavities with embedded monolayer and Moiré arranged bilayers of Transitional Metal Dichalcogenides (TMD), their experimental and theoritical study. The present thesis represents the theoritical part of this project. The supervisors are G. Malpuech and D. Solnyshkov in Institut Pascal and T. Liew in MajuLab. The PhD student will mostly be based in Clermont-Ferrand but should also work for some time in Singapore. *

**Introduction**

The study of 2D materials started with graphene, a single sheet of carbon atoms forming a honeycomb lattice. Graphene is a gapless material exhibiting Dirac points (mass-less electrons) at the origin of its unique electronic transport properties. A new field of research emerged a decade ago, when people started to consider the so-called bilayer graphene, with a small difference in the orientation of the two layers. This leads to the presence of an extra period, much larger than the one of the original lattice. Electronic bands are folded and become much smaller, and it has been shown that for “magic” twist angles[1], the folded bands could become flat and topologically non-trivial. Then, experimental works have demonstrated quite unique transport properties, such as superconductivity[2], which could be associated with the properties of these topological flat bands.

Transitional metal dichalcogenide (TMD) monolayers represent another type of 2D materials. They also form a honeycomb lattice with two different types of atoms, which means that a gap is opened at the Dirac points. The corresponding material is a direct band gap semiconductor, suitable for optics. They are very actively studied for a decade[3] because of their exceptional properties, such as exciton and trion stability[4], as well as optical selection rules[5] (spin-valley locking). Indeed, the split Dirac points can be described by a massive Dirac Hamiltonian, with the non-trivial geometry (topology) of the corresponding bands being at the origin of the spin-valley locking. It is then possible to make Moiré TMDs. Two monolayers combined at a non-zero angle give a moiré superlattice[6] with an arbitrary period controlled by the angle of orientation. The folded band structure again shows new properties which deeply affect the optical response of the bilayer system, but another effect is that this superlattice creates a potential for excitons, allowing to localize them in a regular lattice of quantum dots[7], opening access to a broad range of phenomena, such as the photonic blockade.

Embedding TMD structures into microcavities allows achieving the strong light-matter coupling regime of excitons or trions with cavity photons[8] as we demonstrated a few years ago. Strong coupling affects transport properties of the excitons and trions, and allows to efficiently create and detect them with the techniques of quantum optics[9].

This theoretical thesis, performed in collaboration with experimental groups, will comprise two main research directions.

**Research to be done**

**1- Quantum Blockade**

The first main goal is to determine theoretically the most favorable configuration for observing the polariton quantum blockade regime, and to experimentally assess Kerr nonlinearities and quantum correlations. In TMDs, the Berry curvature concentration near the valley extrema leads to the spin-valley locking. This property is preserved in TMD heterostructures and Moiré superlattices, and determines the spin/polarization eigenstates of excitons. Thus, initially excitons will be theoretically modelled by a Bose-Hubbard Hamiltonian including spin/valley degree of freedom. In Moiré superlattices, the models’ parameters are the spin-dependent tunnelings and the on-site spin-dependent interactions. Experimentally these parameters can be controlled by an external electric field. Depending on system parameters and on the average filling of sites (experimentally controlled by the pumping power), different collective modes could occur, including Mott-insulator and superfluid-like phases. Once this model is implemented, we will include the coupling to the optical mode following two sequential stages:

1st) First we will describe the coupling of inter- and intra-layer excitons with photonic modes. This will be combined with the simultaneous description of the interactions of these two types of excitons.

2nd) Second, if excitonic blockade is achieved and if the direct exciton tunneling between sites is negligible with respect to the light induced tunneling, it will be possible to simplify the model to the Tavis-Cummings one, in which each Moiré site is modelled as an isolated two-level system.

In both cases (**Bose-Hubbard & Tavis-Cummings**) it will be crucial to consider the open nature of our system, including dissipation using the Lindblad formalism. This will require numerical simulations employing the wavefunction Monte Carlo technique.

Our goal is to establish the limit on the number Moiré sites coupled to an optical mode **for blockade to operate**, which is equivalent to **establish the twist angles of the Moiré lattices displaying polariton blockade**. Importantly, this formalism will model the performance of blockade geometries in terms of second order correlation function and average occupation, which will be determined experimentally and compared with the theoretical predictions.

**2- Topological properties of Moiré Polaritons**

The second objective of the thesis is to analyze how the topology of TMD-polaritons is dictated by the topology of each of their constituents and how it can be affected by interactions and non-hermiticity. Photonic modes in planar microcavities form a series of 2D modes with quantized vertical wavevectors (*k**z*). Each of these modes forms a polarization doublet (TE and TM) degenerate at *k**x,y*=0 in isotropic microcavities. TE-TM splitting acts as a spin-orbit coupling similar to the Rashba spin-orbit coupling but possessing a winding number 2. This makes photonic bands topologically non-trivial, displaying a Berry monopole at *k**x,y*=0. Linear birefringence splits the dispersion at *k**x,y*=0, leading to the formation of a pair of tilted Dirac cones, each carrying a topological charge ½[10]. Rashba-Dresselhaus spin orbit coupling can also take place at the crossing of modes of different parity[11]. Further, as optical microcavities are open systems, non-Hermiticity is inherently present and leads to the emergence of exceptional points in *k*-space[12]. In TMDs, besides the spin-valley locking, the interplay between the excitonic and photonic topologies has been hindered because the exciton valley extension in *k*-space is much larger than the photonic one. However, in excitonic Moiré superlattices the band folding effect compresses the *k*-space extension of the excitonic Berry curvature and quantum metric distribution. Thus, the first goal of this task will be to study theoretically the **interplay between the excitonic band topology and the photonic one, **focusing on the potential advantages of lattices of excitons (as in Moiré superlattices). Further, we will address theoretically **how exciton-exciton interactions**, especially in strongly-interacting configurations (i.e. interlayer excitons), **affect the band topology and the way in which strongly-correlated topological phases could emerge **from the interplay between the spin dependent exciton-exciton interaction in Moiré superlattices and the polariton band topology. Mean field interactions and strong correlations are expected to modify the topology of the underlying lattice, switching it from trivial to non-trivial, both at the level of excitonic Moiré lattice and at the level of resulting polariton modes. To compare with experiments, the subjacent goal will be to highlight phenomena that can be verified explicitly (e.g. through the measurement of the quantum metric), providing a clear proof of nonlinear effects in topological photonics.

**Application Deadline**

Until the position is filled.

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**References**

[1] R. Bistritzer and A. H. MacDonald, Proceedings of the National Academy of Sciences 108, 12233 (2011).

[2] Y. Cao et al., Nature, 556, 43 (2018).

[3] S. Manzeli et al, Nature Reviews Materials 2, 17033 (2017).

[4] K. He et al, Phys. Rev. Lett. 113, 026803 (2014).

[5] K. F. Mak et al, Nat. Nanotechnol. 7, 494–498 (2012).

[6] K. F. Mak, J. Shan, Nat. Nanotechnol. 17, 686 (2022).

[7] K. Tran et al, Nature 567, 71 (2019).

[8] S. Dufferwiel et al, Nat. Comm. 6, 8579 (2015).

[9] G. Munoz et al, Nat. Mater. 18, 213 (2019).

[10] A. Gianfrate, O. Bleu, et al., Nature 578, 381, (2020).

[11] M. Rechcinska et al, Science, 366, 727, (2019).

[12] M. Krol, I. Septembre et al., Nature Communications 13, 5340 (2022).