Superconductivity and other collective phenomena in a hybrid Bose-Fermi mixture formed by a polariton condensate and an electron system in two dimensions The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Cotleţ, Ovidiu, Sina Zeytino#lu, Manfred Sigrist, Eugene Demler, and Ataç Imamo#lu. 2016. “Superconductivity and Other Collective Phenomena in a Hybrid Bose-Fermi Mixture Formed by a Polariton Condensate and an Electron System in Two Dimensions.” Physical Review B 93 (5). https://doi.org/10.1103/physrevb.93.054510. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:41412179 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP 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,1, ∗ Sina Zeytinoˇglu,1, 2 Manfred Sigrist,2 Eugene Demler,3 and Ataç Imamoˇglu1 1Institute of Quantum Electronics, ETH Zürich, CH-8093, Zürich, Switzerland 2Institute for Theoretical Physics, ETH Zürich, CH-8093, Zürich, Switzerland 3Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA Interacting Bose-Fermi systems play a central role in condensed matter physics. Here, we ana- lyze a novel Bose-Fermi mixture formed by a cavity exciton-polariton condensate interacting with a two-dimensional electron system. We show that that previous predictions of superconductivity [F.P. Laussy, Phys. Rev. Lett. 10, 104 (2010)] and excitonic supersolid formation [I.A. Shelykh, Phys. Rev. Lett. 14, 105 (2010)] in this system are closely intertwined– resembling the predictions for strongly correlated electron systems such as high temperature superconductors. In stark contrast to a large majority of Bose-Fermi systems analyzed in solids and ultracold atomic gases, the renor- malized interaction between the polaritons and electrons in our system is long-ranged and strongly peaked at a tunable wavevector, which can be rendered incommensurate with the Fermi momen- tum. We analyze the prospects for experimental observation of superconductivity and find that critical temperatures on the order of a few Kelvins can be achieved in heterostructures consisting of transition metal dichalcogenide monolayers that are embedded in an open cavity structure. All optical control of superconductivity in semiconductor heterostructures could enable the realization of new device concepts compatible with semiconductor nanotechnology. In addition 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. I. INTRODUCTION predicted that the 2DES can undergo a superconducting phase transition [3–5]. Our work unifies and extends the Interacting Bose-Fermi systems are regarded as a above mentioned prior work and shows that the predicted promising platform for investigating novel many-body phase transitions are closely intertwined. physics. Recent advances demonstrating mixtures of ul- The central finding of our work is that when screening tracold bosonic and fermionic atomic gases have inten- effects are taken into account, the long-range polariton- sified research efforts in this class of systems. Feshbach electron interaction is peaked at a wavevector q0 that is resonances in two-body atomic collisions can be used to determined by the distance between the 2DES and the tune the strength of interactions into the strong coupling quantum well (QW) hosting the polaritons. Remarkably, regime, allowing for the investigation of competition be- increasing the polariton condensate occupancy by in- tween various phase transitions such as supersolid for- creasing the resonant laser intensity leads to a substantial mation and superconductivity. Motivated by two recent softening of the polariton dispersion at qr (near q0) which proposals, we analyze a solid-state Bose-Fermi mixture in turn enhances the strength of the polariton-electron in- formed by an exciton-polariton Bose-Einstein condensate teraction, making it even more strongly peaked. Leaving (BEC) interacting with a two dimensional electron sys- a detailed analysis of competition between superconduc- tem (2DES). Unlike most solid-state systems, the inter- tivity and potential charge density wave (CDW) state as- action strength between the polaritons and electrons can sociated with polariton mode softening as an open prob- be controlled by adjusting the intensity of the laser that lem, we focus primarily on the superconducting phase drives the polariton system. We find that this system can transition. be used to reach the strong-coupling regime, evidenced After introducing the system composed of a bosonic by dramatic softening of the polariton dispersion at a polariton condensate interacting with a 2DES in Section tunable wavevector. II A, we investigate its strong coupling limit in Section Before proceeding, we remark that the interactions be- II B. Here, we summarize the effects of many body inter- tween a 2DES and an indirect-exciton BEC has been actions, leaving the more detailed calculations to the Ap- theoretically shown to lead to the formation of an exci- arXiv:1510.02001v1 [cond-mat.quant-gas] 7 Oct 2015 pendix A. In Section III we use the theoretical framework tonic supersolid [1, 2]. This prior work however, did not developed earlier and analyze the interactions between a take into account the effect of the exciton BEC on the polariton condensate and a 2DES self-consistently. We 2DES. Concurrently, the effect of a polariton condensate notice that the strong interactions can lead to instabili- on a 2DES has been investigated without considering the ties both in the condensate and in the 2DES. We inves- back-action of electrons on the polaritons and it has been tigate quantitatively the instability of the 2DES towards superconductivity [3–5] while also taking into account the effect of the 2DES on the BEC. We also comment briefly on the instability of the 2DES towards the forma- ∗ [email protected] tion of an unconventional CDW ordered state as a con- 2 sequence of the renormalized electron-polariton interac- tion becoming strongly peaked at wavevector qr. In Sec- tion IV we investigate how to reach the strong coupling regime experimentally in order to observe these phase transitions. We find that superconductors with tempera- tures of a few Kelvins can be obtained in transition metal dichalcogenide (TMD) monolayers. We briefly summa- rize our results and provide a short description of new physics and applications enabled by our analysis in Sec- tion V. II. THEORETICAL INVESTIGATION A. Description of the coupled electron-polariton system The system that we investigate is similar to the sys- tem in Ginzburg’s proposal for high temperature exciton- mediated superconductors [6]. It consists of a 2DES in close proximity to a quantum well (QW) in which exci- tons can be created by shining a laser resonantly without influencing the 2DES. The whole system is embedded in a cavity formed by a pair of distributed Bragg reflec- tors (DBRs), which confine light and and allow for a strong interaction between excitons and photons. Due to the non-perturbative light-matter coupling the new eigenstates are composite particles called polaritons. The polaritons can form a BEC either under non-resonant or under direct resonant excitation by a laser [7]. The in- teraction between neutral polaritons and the electrons in the 2DES is due to the excitonic content of the polari- tons and can be enhanced by enhancing the size of the k dipole of the exciton using a DC electric field. In this kph/10 ph scenario, an attractive interaction between electrons can q/kF be mediated by the polariton excitations of the BEC. As we will show below, the strength of the interaction is pro- Figure 1. Upper panel: The schematic of the semiconduc- portional to the number of polaritons in the condensate tor heterostructure that is analyzed. Lower panel: bare po- which can be tuned experimentally. The schematic de- lariton (blue) and electron (red) dispersion in logarithmic sign of the experimental setup is presented in the upper scale. The vertical dashed lines are at the photon wavevector panel of Figure 1. kph = 3:3Ec(0)=(¯hc) (corresponding to the maximal momenta In the following we will assume for simplicity that the that we can investigate optically; the 3:3 factor comes from polariton condensate is generated by a resonant laser field the GaAs index of refraction) and at kph=10 (roughly corre- for the k = 0 state of the lower polariton branch. The sponding to the momentum where the polariton dispersion switches from photonic to excitonic). The parameters used Hamiltonian describing the coupled exciton and cavity- are typical GaAs parameters: g0 = 2meV, me = 0:063m0, mode dynamics can be diagonalized through a canonical 11 −2 mh = 0:046m0, ne = 2×10 cm , Ec(0) = Ex(0) = 1:518eV. transformation and the resulting lower and upper polari- ton eigenstates are superpositions of exciton and photon states. Since the whole system is translationally invari- by: ant, in-plane momentum k is a good quantum number ! for polaritons. The annihilation operator for a lower po- 1 δE(k) X(k) 2 = 1 ; (2) lariton of momentum k is: p 2 2 j j 2 − δE (k) + 4g0 (x) p 2 (c) b = X(k)a + 1 X(k) a ; (1) where g0 is the light-matter coupling strength and k k − k δE(k) = Ex(k) Ec(k) is the energy difference be- where a(x) and a(c) are the exciton and cavity-photon tween the exciton− and the cavity mode. Denoting the annihilation operators. X(k) is the exciton fraction of k = 0 detuning between the photon and exciton disper- the lower polariton mode with wavevector k and is given sion by ∆ we express the energy difference: δE(k) = 3 2 2 −1 −1 ∆+¯h k (m m )=2 where m , m , mx = m +m , inducing a dipole in the excitons.