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PHYSICAL REVIEW X 9, 041019 (2019)

Transport of Neutral Optical Excitations Using Electric Fields

Ovidiu Cotleţ ,1,* Falko Pientka,2,* Richard Schmidt,2,3 Gergely Zarand,4 Eugene Demler,2 and Atac Imamoglu1 1Institute of Quantum Electronics, ETH Zürich, CH-8093, Zürich, Switzerland 2Department of , Harvard University, Cambridge, Massachusetts 02138, USA 3Max Planck Institute of Quantum Optics, 85748 Garching, Germany 4Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, H-1521, Hungary

(Received 23 March 2018; revised manuscript received 12 June 2019; published 25 October 2019)

Mobile quantum impurities interacting with a fermionic bath form known as Fermi . We demonstrate that a force applied to the bath can generate a drag force of similar magnitude acting on the impurities, realizing a novel, nonperturbative Coulomb drag effect. To prove this, we calculate the fully self-consistent, frequency-dependent transconductivity at zero temperature in the Baym-Kadanoff conserving approximation. We apply our theory to and interacting with a bath of charge carriers in a doped embedded in a microcavity. In external electric and magnetic fields, the drag effect enables electrical control of excitons and may pave the way for the implementation of gauge fields for excitons and polaritons. Moreover, a reciprocal effect may facilitate optical manipulation of transport. Our findings establish transport measurements as a novel, powerful tool for probing the many-body physics of mobile quantum impurities.

DOI: 10.1103/PhysRevX.9.041019 Subject Areas: Atomic and Molecular Physics, Condensed Physics, Quantum Physics

I. INTRODUCTION Exciton polaritons are charge-neutral particles, and hence their center-of-mass motion does not couple directly to dc Polaritons are composite bosonic particles formed by electric or magnetic fields. Prior work has exploited a hybridization of propagating and quanta of polari- combination of -orbit coupling of light and magnetic zation waves in a . A particular realization that has field response of a polarization degree of freedom recently been extensively studied involves two-dimensional in lattice structures. Alternatively, it might be possible to (2D) cavity exciton polaritons, implemented in monolithic exploit the interaction of polaritons with charge carriers to III–V semiconductor heterostructures [1],aswellasintran- induce effective photonic gauge fields that are linearly sition metal dichalcogenide (TMD) monolayers embedded proportional to the external fields. In this regard, semi- in open dielectric cavities [2–5].Remarkably,thesecavity- conducting TMD monolayers are a particularly promising polariton excitations combine an ultralight effective mass platform featuring exceptionally large exciton binding dictated by their photonic content with a sizable interparticle energies and strongly attractive interactions between exci- interaction strength stemming from their excitonic character. tons and charge carriers. This unique combination allows for the realization of a In a simple picture, excitons in the presence of degen- driven-dissipative interacting bosonic system that has been erate can be considered as mobile impurities shown to exhibit a myriad of many-body phenomena such as – interacting with a Fermi sea, which constitutes a funda- nonequilibrium condensation [6 10], [11],and mental problem of many-body physics [16–25].Tolower the Josephson effect [12]. its energy, an exciton can bind an additional charge carrier Recent progress in the realization of topological states – forming a charged trion. Alternatively, the exciton can, of polaritons [13 15] demonstrates the importance of however, also create a polarization cloud in its environ- implementing effective gauge fields in photonic systems. ment, forming an attractive exciton , shortly referred to as a polaron. In this case, the exciton remains *These authors have contributed equally to this work. a neutral that is dressed by fluctuations of the Fermi sea, which renormalize its energy and effective mass. Published by the American Physical Society under the terms of The competition of trion formation and polaronic dressing the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to was previously observed in cold atomic systems near a the author(s) and the published article’s title, journal citation, Feshbach resonance in two and three dimensions [26–29]. and DOI. Recently, it was demonstrated both theoretically [4,30] and

2160-3308=19=9(4)=041019(33) 041019-1 Published by the American Physical Society OVIDIU COTLEŢ et al. PHYS. REV. X 9, 041019 (2019) experimentally [4,31] that polaron physics also plays a key indirect excitons. Because of the pairing process, any force role in semiconductor photonic materials hosting a two- applied on one of the particles is felt by the as a dimensional electron system (2DES). In particular, absorp- whole, leading to the of phenomena such as tion experiments in TMD monolayers show one of the key perfect Coulomb drag. The direct analog in our system is signatures of Fermi polaron formation: a redshifted optical provided by trions—a bound molecular state of an exciton resonance with an oscillator strength that increases with and a single electron; we expect the trion to respond as a increasing electron density ne, which is accompanied by a charged particle to any applied electric field and exhibit strong blueshift and broadening of the bare-exciton reso- perfect Coulomb drag. nance [31]. These observations demonstrate that the optical The drag of polarons, on the other hand, which is the excitation spectrum should be described in terms of focus of the present work, is more subtle and was not attractive and repulsive exciton polarons. In contrast, trion analyzed previously. We find that the response of polarons, states have negligible weight in absorption spectra due to which are many-body optical excitations, is very different their vanishing oscillator strength. Moreover, we remark from the response of trions, which are two-body bound that the polaron states introduced in Ref. [4] correspond to molecular excitations. Indeed, in equilibrium setups with the mixed exciton-trion states previously introduced by disorder and static electric fields, polarons do not respond Suris et al. in Ref. [32] to explain the optical spectrum of at all to a force on the bath particles (at zero temperature). doped quantum wells. Remarkably, however, in the case of dynamical fields (more In this work, we propose to use exciton polarons as a new specifically, when the frequency of the field is much larger means to control photons by dc electric or magnetic fields. than the inverse disorder lifetime of the bath particles) or in In contrast to trions, polarons are amenable to polariton nonequilibrium scenarios where the impurities have a finite formation, and recent TMD-cavity experiments have dem- lifetime, the response of the polarons can be as large as the onstrated strong coupling of exciton polarons and light [4]. response of the . The (zero-temperature) polaron We find that, although polarons are charge-neutral optical drag effect that we investigate can be understood as arising excitations, their interaction with the Fermi sea forces them from a shift in the polaron dispersion proportional to the to follow the motion of charge carriers in an electric or velocity of the majority particles and therefore can be magnetic field. This phenomenon bears similarity to understood as an effective gauge potential for the impurity Coulomb drag between two semiconductor layers [33–36], particles. motivating us to coin the term polaron drag. A key Furthermore, our work establishes transport measure- difference to previous work on drag [36–38] ments as powerful tools to probe the nature of the many- originates from the nonperturbative nature of the polaronic body ground state in mobile impurity experiments, and it coupling, which gives rise to a remarkably efficient drag should be of central interest to a large subset of low energy mechanism. Indeed, we find that the zero-temperature physics, ranging from ultracold through semicon- (T ¼ 0) drag conductivity of polarons can ideally be of ductor quantum optics to strongly correlated materials. the same order as the electron conductivity, implying the Indeed, the fundamentally different transport response of realization of sizable photonic gauge fields. polarons compared to molecules provides a way of exper- From a more general perspective, our work addresses imentally identifying the polaron-to-molecule transition in the problem of nonperturbative Coulomb drag, and it is two dimensions, as well as also providing a way to measure unique in that it analyzes the drag effect in the regime the mass of the polaron or molecule quasiparticles. of strong interactions that can lead to the formation of Arguably, the most important aspect that distinguishes interspecies molecular bound states (trions in our setup) as our work is the extension to nonequilibrium drag. As an well as many-body optical excitations (exciton polarons). experimental realization, we assume that exciton polarons Although in this work we treat the regime of large density are injected resonantly by laser fields at a given point in the imbalance between the different species, which allows us to sample. The exciton-polaron drag leads to a spatial dis- model the problem using the quantum impurity framework, placement of the emission generated as the excitons decay our results will likely be relevant even in the balanced into photons propagating out of the plane (Fig. 1). Hence, density regime [39]. As mentioned previously, there are spatially resolved excitation and detection of emission from two types of emerging quasiparticles in this system, which the sample constitute the equivalent of source and drain have very different drag properties. contacts to the exciton system, enabling us to measure the The drag of the molecular states is closely related to drag-induced displacement and thereby giving a physical previous work on indirect excitons in bilayer systems in the meaning to the transconductivity. quantum Hall regime, where the phenomenon of perfect We start our analysis by a heuristic derivation of the Coulomb drag has been observed [36,40]. In these systems, polaron drag force based on intuitive arguments in Sec. II. one of the layers hosts holes, while the other contains Moreover, we solve the semiclassical equations of motion electrons. The strong interlayer attractive interaction leads of a polaron in an electric field. In Sec. III, we corroborate to the formation of bound molecular states known as these results by a microscopic calculation of the polaron

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the bare exciton with an effective mass mx. The accel- eration of this in the electron rest frame is thus F=mx ¼ −aemx=mx. Going back to the lab frame, we have to add the acceleration of the reference frame, and we thus arrive at an exciton acceleration a 1 − mx a x ¼ e: ð1Þ mx

This equation relates any force acting on the electron system to a somewhat smaller force on the dressed exciton. This force can be qualitatively understood as friction between electrons and excitons, which originates from the ability of the excitons to minimize their energy by FIG. 1. Schematic of the experimental setup. A photon is following the dressing cloud surrounding them. absorbed by the TMD monolayer, forming a polariton that is then dragged by the drifting electrons and subsequently emitted at a The above discussion is completely general and does not different position. make any assumptions regarding the nature of the quasi- particles, and therefore it is equally valid for both molecules and polarons. This is because we assumed that all electrons drag conductivity in an electric field using diagrammatic accelerate with ae. However, in the presence of disorder, perturbation theory within the Kadanoff-Baym conserving this is no longer the case, leading to the emergence of a approximation. To make this paper self-contained, we start distinction between the trion and polaron response. For this section by reviewing the Fermi polaron problem as well trions, the situation is relatively simple: Because the trion is as the conserving approximation. In Sec. IV, we extend our a two-particle , formed by an exciton that binds results to polaritons and comment on nonequilibrium to a single electron from the Fermi sea, it follows the effects in Sec. V. After discussing the extension of our motion of this particular electron. Therefore, the trion analysis to the case of magnetic field response in Sec. VI, motion is still described by Eq. (1) as long as we let ae we conclude by presenting an outlook on possible exten- denote the instantaneous acceleration of one electron. This sions of our work. We remark that the theoretical calcu- means that trions will respond to applied fields like charged – lations in Secs. II IV and VI assume an equilibrium particles, and they will exhibit drag properties similar to the scenario where the impurity does not decay radiatively, ones investigated in the work of Ref. [40]. In contrast, we and therefore these results do not apply directly to the recall that exciton polarons are many-body excitations experimental scenario that we proposed. Our purpose for where an exciton is dressed with a polarization wave of taking this approach was twofold. First, this keeps the the electron system. Since all electrons in the Fermi sea discussion completely general and allows our results to be contribute to this polarization wave, we can describe the directly applied to the usual bilayer systems or to cold- polaron motion using Eq. (1) if and only if we let ae denote experiments. Second, it allows us to identify the physical the average acceleration of the electron system. This process responsible for Coulomb drag and then analyze interpretation is justified a posteriori by the rigorous the polaron-polariton drag using a Boltzmann equation in diagrammatic calculation in the following section. In the Sec. V, without performing an even more cumbersome following, we focus only on the response of polarons. nonequilibrium diagrammatic calculation. We remark that the electrons should also experience a drag force when the excitons are accelerated by an external force. II. HEURISTIC DERIVATION Equation (1) corresponds to the combined force all electrons OF POLARON DRAG exert on a single polaron. Hence, the average inverse drag We consider a mobile exciton with mass m interacting force on an electron in the Fermi sea in the presence of an x exciton force Fx should be ðnx=neÞð1 − mx=mxÞFx,where with a Fermi sea of electrons with mass me. For simplicity, we neglect the finite radiative lifetime of the exciton in this ne;x is the density of electrons (excitons). section. Applying a force on the electrons causes an We can write down general semiclassical equations of motion for the coupled electron and polaron motion: acceleration ae of the Fermi sea. We now go to the rest frame of the electrons. Because this is a noninertial d n F ðtÞ reference frame, a fictitious force acts on the exciton m v ðtÞ¼F ðtÞþ x ðm − m Þ x ; ð2Þ e dt e e n x x m F ¼ −mxae accelerating it. Crucially, the force is propor- e x tional to the bare mass of the exciton. The interaction with d F t the electrons, however, impairs the motion of the exciton, eð Þ mx vxðtÞ¼FxðtÞþðmx − mxÞ : ð3Þ and hence, the dressed exciton quasiparticle is heavier than dt me

041019-3 OVIDIU COTLEŢ et al. PHYS. REV. X 9, 041019 (2019) − Ωτ We can evaluate the polaron drag conductivity in an ac E Ω nx 1 − mx i x f Ω E e ð Þ¼ xð Þ: ð9Þ electric field ðtÞ in the presence of disorder for both ne mx 1 − iΩτx electrons and excitons by choosing The polaron drag force in Eq. (1) is fully determined by v v F − E − me eðtÞ F − mx xðtÞ the force acting on the electrons and the mass renormal- eðtÞ¼ e ðtÞ ; xðtÞ¼ ; ð4Þ τe τx ization, and specifically, it does not depend on the exciton velocity. This enables us to reinterpret the polaron drag where τeðτxÞ is the transport lifetime of electrons (exciton effect as a consequence of an effective gauge field. This polarons). interpretation is further supported by a microscopic inves- To leading order in the polaron density, we can ignore the tigation of the drag mechanism. As we show in the drag force on electrons. The solution in Fourier space reads Appendixes B and C, the exciton-polaron dispersion ζk shifts by an amount proportional to the electron drift −eEðΩÞ τ velocity (implying that k ¼ 0 excitons experience a finite v ðΩÞ¼ e ; ð5Þ A e 1 − Ωτ group velocity). We can introduce a gauge potential xðtÞ me i e ˜ to write the new dispersion as ζk ¼ ζk− A . In the case of a e x m −iΩτ time-dependent drive, this gauge potential is given by v Ω 1 − x x v Ω : 6 xð Þ¼ 1 − Ωτ eð Þ ð Þ mx i x − − τ E −iΩt mx mx e e e eAxðtÞ¼ðmx − mxÞveðtÞ¼ ; ð10Þ me 1 − iΩτe Notice that in the absence of excitonic disorder, τx → ∞, the drag is the only force acting on the excitons, and their E which corresponds to an effective electric field x given by drift velocity is simply vx ¼ð1 − mx=mxÞve. This case is particularly relevant for exciton polaritons, which are − Ω m − m iΩτ eEe i t expected to be largely immune to exciton disorder (see E − A_ x x e e xðtÞ¼ e ðtÞ¼ 1 − Ωτ : ð11Þ Sec. IV). We emphasize that this result only holds at small me i e polaron densities as we neglect electron drag forces Although the field strength vanishes in the dirty regime generated by polarons. From the exciton drift velocity, Ωτ ≪ 1, the finite (constant) gauge potential can still have we obtain the transconductivity to first order in the polaron e measurable effects. Indeed, in Sec. V, we discuss at length density n : x the role of this gauge potential on photon transport and show, for a conservative choice of material, that photons en m iΩτ τ σ Ω x 1 − x e x : 7 could be dragged for distances of a few hundred xeð Þ¼ 1 − Ωτ 1 − Ωτ ð Þ me mx ð i eÞð i xÞ nanometers. We now investigate the magnitude of the polaron drag As mentioned above, the reverse effect also exists. An effect and the role played by the trion binding energy, f electric current should flow in response to an ac force xðtÞ which determines the energy scale of the exciton-electron applied to the excitons. Such a force can be affected by interaction in the limit of vanishing electron density. applying an ac field perpendicular to the 2D plane that Figure 2 shows the dependence of the (normalized) polaron modifies the exciton energy through a quantum-confined mass mx=me on the ratio of the Fermi energy (ϵF) to the Stark effect with a spatial and/or time dependence deter- trion binding energy (ϵT): Since the magnitude of the mined through the applied laser field [41]. We can find the effective electric field seen by the polaron is proportional to drag conductivity of electrons from Eqs. (2) and (3) with ½ðmx − mxÞ=me, we conclude that for a given electron density, a stronger trion binding leads to a larger mass v v F − me eðtÞ F f − mx xðtÞ renormalization for the polaron and, consequently, to a eðtÞ¼ ; xðtÞ¼ xðtÞ : ð8Þ τe τx more efficient drag. Remarkably, the polaron drag can ideally be even more efficient than the drag of trions. In this case, we need to include the drag force on electrons It is important to note that our analysis so far has in Eq. (2). We can, however, neglect the drag term in the neglected incoherent scattering of electrons and polarons. polaron equation of motion (3) since Fe ∝ ve ∝ nx=ne only In addition to coherent scattering of electrons and excitons contributes at higher order in polaron density. With this that lead to polaron formation, there may also be incoherent method, we again arrive at the transconductivity given by scattering events that lead to a finite lifetime of polarons in Eq. (7), as guaranteed by Onsager’s reciprocity principle. an excited state. We investigate this effect in more detail in An experimentally more relevant quantity is the electric Secs. III E and VB, and discuss in what parameter regimes voltage that builds up as a response to an exciton force it can be neglected. when no current can flow. The electric field can be found by To conclude this section, we emphasize that in the setting ve ¼ 0 in the equations of motion, which yields exciton-polaron transconductivity problem we are

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proportional to Ω, in the opposite (clean) limit (Ωτe, Ωτx ≫ 1), the transconductivity is proportional to 1=Ω. This clean limit result shows that second-order processes do not vanish as Ω → 0 at T ¼ 0,in contrast to previous claims that in these limits the only nonzero contributions to drag (in both the clean and dirty regime) must be higher order in the interspecies interaction [35,36]. It would be inter- esting to check whether our result holds also in the regime of balanced interspecies densities analyzed in Ref. [35]. (iii) The electrons should also experience a drag force, when the excitons are accelerated by an external FIG. 2. Polaron mass as a function of Fermi energy. For this force, in accordance with Onsager’s reciprocity plot, we used a contact interaction model for exciton-electron principle. interaction and solved the polaron problem using the non-self- (iv) The drag effect that we investigate emerges from an consistent T-matrix approach developed in Ref. [23]. We took the exciton mass to be twice as large as the electron mass m ¼ 2m . effective gauge potential that the exciton polaron x e experiences due to the motion of the electrons. Even in the Ωτe ≪ 1 limit, when the effective electric field vanishes, the finite gauge potential still has analyzing, the drag process takes place in an interacting important consequences for the exciton-polaron system consisting of a fixed number of excitons and transport (Sec. V). electrons. Even in the case of a finite exciton or polariton (v) The semiclassical analysis presented here can be lifetime, an underlying assumption is that laser excitation extended to include an external magnetic field (see leads to creation of real particles with a lifetime much also Refs. [44,45]). The drag force could then give longer than the characteristic timescales associated with rise to a Hall effect and a cyclotron resonance of their interactions with electrons. This case is in stark exciton polarons, a phenomenon that we discuss at contrast to polarons in an interacting electron- length in Sec. VI. system where dynamic screening of the electron by the lattice can be described in terms of virtual phonon emission and absorption processes. Consequently, in the limit of III. DIAGRAMMATIC CALCULATION OF linear response at vanishing temperature, there is no natural TRANSCONDUCTIVITY USING THE CONSERVING APPROXIMATION setting to talk about drag conductivity of . The opposite limit of large applied electric fields, on the other We now evaluate polaron drag within a microscopic hand, constitutes a very interesting problem where electron theory in order to verify the heuristic results discussed in transport can be associated with multiple real phonon the previous section. To this end, we use diagrammatic emission and absorption processes and the associated perturbation theory, taking into account the effect of phonon drag could have signatures in current noise [42,43]. electron-exciton interactions as well as disorder for electrons Before proceeding, we summarize the consequences of and excitons. We are interested in the nonperturbative effect our results: of the interaction and must therefore proceed with care. (i) The polaron drag can be extremely efficient, prom- Simply evaluating a certain class of diagrams might lead to ising a drag mobility of the same order as the erroneous results, as an incomplete set of diagrams does not electron mobility. This result is a consequence of necessarily satisfy the conservation laws of the physical the nonperturbative interaction that is responsible for systems. A powerful technique to generate diagrams obeying the polaron formation, which leads to a drag trans- conservation laws is the conserving approximation (see conductivity proportional to the mass renormaliza- Ref. [46] for a pedagogical introduction). tion of the polaron. We start by introducing the Hamiltonian of the exciton (ii) When the interspecies interaction is weak, the mass electron system and by discussing the polaron problem in renormalization of the polaron is proportional to V2 the absence of fields. We then review the basic principles (where V denotes the interspecies interaction of the conserving approximation and use it to find an strength; the contribution linear in V can only lead approximation to the drag conductivity within linear to a shift of the polaron energy and will not affect the response. As a crucial simplification, we focus on the limit polaron mass). According to Eq. (7), this conse- of small polaron density. This limit is also implicitly quence will result in a zero-temperature drag assumed in the semiclassical analysis in Sec. II,aswe transconductivity proportional to V2. While in the neglect interactions between polarons. Moreover, the (dirty) limit (Ωτe, Ωτx ≪ 1) the transconductivity is quasiparticle picture of polarons eventually breaks down

041019-5 OVIDIU COTLEŢ et al. PHYS. REV. X 9, 041019 (2019) at sufficiently high densities. A small polaron density also (a) justifies the standard diagrammatic description of polarons in the ladder approximation presented in Sec. III A. Hence, our calculation will only include contributions to leading order in the polaron density. (b)

A. Exciton-polaron problem We consider noninteracting excitons coupled via contact interactions of strength V to a Fermi sea of noninteracting FIG. 3. The Green’s functions and T matrix. electrons. To be specific, we focus on the two-dimensional case; however, our results for the transconductivity apply to −1 G ðpÞ¼ω − ξp − Σ ðpÞ; ð15Þ three dimensions as well. We focus on the limit of very e dis;e small exciton density, where the statistics of excitons −1 ω − ω − Σ − Σ becomes irrelevant. We exploit this fact by treating the Gx ðpÞ¼ p intðpÞ dis;xðpÞ; ð16Þ low-density excitons as an effective Fermi , which allows us to simplify calculations. Since we are moreover where we introduced the dispersions of electrons ξp ¼ 2 2 interested in the regime where no exciton condensation p =ð2meÞ − μe and excitons ωp ¼ p =ð2mxÞ − μx and the Σ takes place, treating excitons effectively as allows exciton self-energy from interactions with electrons int,as Σ Σ us to take the corresponding limit of T → 0 without well as the disorder self-energies e;dis and x;dis. Assuming technical complications from Bose condensation. We a small density of polarons in the system, we can neglect emphasize that all results are independent of the statistics. the effect of excitons on the electron system. Our system is described by the Hamiltonian The calculation of the self-energies requires some approximations. We treat the effect of disorder in the Z 2 † ∇ self-consistent Born approximation and therefore ignore H ¼ dr ΨeðrÞ − − μe ΨeðrÞ 2me any quantum interference effects such as weak localization. Σ ∇2 For the evaluation of the interaction self-energy int, we use Ψ† r − − μ Ψ r þ xð Þ 2 x xð Þ the self-consistent T-matrix approach in the ladder approxi- mx mation displayed in Fig. 3(b). This choice gives the leading † † † ðeÞ þ VΨxðrÞΨxðrÞΨeðrÞΨeðrÞþΨeðrÞΨeðrÞU ðrÞ contribution in the limit of small exciton density [47] and has been successfully used to describe the physics † ðxÞ þ ΨxðrÞΨxðrÞU ðrÞ ; ð12Þ of Feshbach resonances in cold atomic systems [22,48]. The T matrix does not have any vertex corrections due to disorder because of our choice of Gaussian-correlated where we introduced the creation operators of the electrons Ψ† Ψ† μ white-noise disorder. We will see in Sec. III B how these and excitons e and x and the chemical potentials e;x. self-energies can be obtained within a conserving approxi- ðe;xÞ We model the disorder potentials U by Gaussian ensem- mation. We point out that the non-self-consistent T-matrix bles with zero means and variances hUðe;xÞðrÞUðe;xÞðr0Þi ¼ approximation is equivalent to the Chevy ansatz for the γ δ r − r0 γ e;x ð Þ,where parametrizes the strength of disorder. variational polaron wave function [49,50]. We introduce the exciton and electron Green’s functions The disorder self-energies can be evaluated as Z r0 0; r − Ψ† r0 0 Ψ r k − Ge;xð t ;tÞ¼ ihT e;xð ;tÞ e;xð ;tÞi: ð13Þ Σ d γ i ω dis;e=xðpÞ¼ 2 e=xGe=xðkÞjϵ¼ω ¼ sgnð Þ; ð2πÞ 2τe=x After disorder averaging, the system is translationally invariant, which allows us to work in Fourier space, ð17Þ Z where the lifetimes are defined as ipr−iωt Ge;xðr;tÞ ≡ dpGe;xðpÞe ; ð14Þ 1 τe;x ¼ ; ð18Þ where we have introduced p ≡ ðp; ωÞ for notational sim- 2πρe;xγe;x plicity [we will later also use k ≡ ðk; ϵÞ]. Moreover, we incorporate factors of 1=2π into the definition of the with ρe;x the densities of states per unit volume at the Fermi integral measure such that dp ¼ dpdω=ð2πÞ3. The effect surface. Importantly, in the self-consistent theory, the of disorder and interactions is taken into account by exciton Green’s function in Eq. (17) is dressed by disorder introducing self-energy corrections to the Green’s functions and interactions, and hence ρx refers to the renormalized shown in Fig. 3(a), exciton dispersion to be determined below.

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The exciton self-energy due to interactions can be evaluated in the self-consistent T-matrix approximation Z Σ − intðpÞ¼ i dkGeðkÞTðk þ pÞ; ð19Þ where we introduced the self-consistent T matrix shown in Fig. 3(b), Z

TðpÞ¼V þ iV dkGeðkÞGxðp − kÞTðpÞ: ð20Þ

FIG. 4. Self-consistent spectral function of excitons at zero The contact interaction allows for a simple solution, exciton density, μe ¼ ϵT =2, mx ¼ 2me, and disorder broad- Z ening 1=2τx ¼ ϵT =100. −1 −1 T ðpÞ¼V − i dkGeðkÞGxðp − kÞ: ð21Þ cold atomic quantum close to Feshbach resonances In two dimensions, a bound state of electrons and excitons [24,27,29,51,52]. (i.e., a trion) exists for arbitrarily weak interactions in the ϵ limit of a single exciton, and its energy T is determined by 1. Attractive polaron quasiparticles the pole of the T matrix. The ground state of a single exciton described by At vanishing exciton density (i.e., nx ¼ 0), we can → R the model in Eq. (12) depends on the dimensionless evaluate the self-energy by replacing Gx Gx in the ϵ ϵ above equations (for more details about this step, see interaction strength given by the ratio F= T of Fermi Sec. III C). We can evaluate all frequency integrals by energy of electrons and trion energy. While the attractive closing the contour in the upper half-plane, where the polaron is stable at higher electron densities, diagrammatic retarded exciton Green’s function is analytical, and obtain Monte Carlo simulations predict a trion ground state for Fermi energies below 0.1ϵ for contact interaction models Z T 0 dk [53,54]. We henceforth assume a sufficiently large electron Σð Þ ξ ð0Þ ω ξ k p int ðpÞ¼ 2 nFð kÞT ð þ k; þ Þ; ð22Þ density so that the physics at low exciton densities is ð2πÞ dominated by the formation of attractive polarons. In this Z k regime, it is instructive to introduce an effective (or ð0Þ −1 −1 d T ðpÞ ¼ V þ ½1 − n ðξkÞ ’ ¯ ð2πÞ2 F projected) Green s function Gx, describing the propagation of attractive polaron quasiparticles (see Appendix A): R × Gx ðω − ξk; p − kÞ; ð23Þ ¯ 1 0 G p : 25 where the superscript ð Þ denotes quantities at n 0 and xð Þ¼ ð Þ x ¼ ω − ζp þ i=2τxðpÞsgnðωÞ nFðxÞ is the Fermi-Dirac distribution function. In order to regularize the contact interaction V, we introduce a UV Here, we have introduced the polaron dispersion momentum cutoff Λ. The interaction strength is then related to the experimentally accessible trion binding p2 ζp ¼ − μ ; ð26Þ energy ϵT at zero electron density by [23] x 2mx Z −1 dk 1 − with the polaron chemical potential μx measured from the V ¼ 2 k2 k2 : ð24Þ k Λ ð2πÞ ϵ j j< T þ 2m þ 2m bottom of the polaron band. Moreover, we have defined e x the effective polaron mass mx, the polaron lifetime τxðpÞ, We solve Eq. (22) self-consistently, by discretizing momen- and the quasiparticle weight Z as tum and energy and using an iterative method. The self- −1 −1 −1 2 −π R m Zm Z∂pReΣ p; 0 ; 27 consistent exciton spectral function AðpÞ¼ ImGx ðpÞ ð xÞ ¼ x þ intð Þjp¼pF ð Þ for nx ¼ 0 and μe ¼ ϵT=2 is plotted in Fig. 4. 1 2τ p 2τ − Σ p ζ The plot shows two spectral features: At negative = xð Þ¼Z= x ZIm intð ; pÞ; ð28Þ frequencies, the attractive polaron is a well-defined quasi- −1 1 − ∂ Σ p ω particle excitation at sufficiently low momenta, while a Z ¼ ω intð F; Þjω¼0; ð29Þ second well-defined, metastable repulsive polaron quasi- particle exists at positive energies. Both excitations have where we introduced the exciton Fermi momentum pF. The been observed in transition metal dichalcogenides [4] and polaron density of states in Eq. (18) is hence ρx ¼ mx=2π.

041019-7 OVIDIU COTLEŢ et al. PHYS. REV. X 9, 041019 (2019) Z The polaron lifetime has a constant part from disorder d2G−1ð1; 2ÞGð2; 3Þ¼δð1; 3Þ; ð31Þ scattering, as well as a momentum-dependent part due to incoherent electron-polaron scattering [see Appendix D for Σ an estimate of Im intðpÞ, as well as the discussion in where we have defined the operator Ref. [48] ]. We emphasize that it is crucial to evaluate −1 the polaron self-energy self-consistently to obtain the G ð1; 2Þ¼½i∂τ − hð1Þδð1; 2Þ − Σð1; 2Þ; ð32Þ momentum-dependent lifetime. In Sec. III E, we make use of the fact that the full exciton where h is the single-particle Hamiltonian and τ a time on ’ Green s function can be approximated by the effective the Keldysh contour. We now consider the variation δG of ≃ ¯ ω ≃ ζ expression, GxðpÞ ZGxðpÞ near the resonance p the Green’s function with respect to some perturbation p ≪ and j j kF. The interaction vertex between electrons and of the single-particle Hamiltonian δh. From Eq. (31),we polarons, however, is determined by virtual transitions to obtain excited states, and knowledge of the full exciton Green’s Z function is required to accurately determine the vertex −1 −1 functions. d2½δG ð1; 2ÞGð2; 3ÞþG ð1; 2ÞδGð2; 3Þ ¼ 0; ð33Þ

B. Conserving approximation and thus We now turn to finding an expression for the trans- Z conductivity within the conserving approximation [46,55]. −1 δGð1; 3Þ¼ d2d4Gð1; 2ÞδG ð2; 4ÞGð4; 3Þð34Þ We first review the basic principles of the conserving approximation and then apply this formalism to the polaron Z problem. In this section, we temporarily adopt the Keldysh − 2 4 1 2 δ 2 4 δ 4 δΣ 2 4 notation, which is most convenient for this purpose. ¼ d d Gð ; Þ½ ð ; Þ hð Þþ ð ; Þ Intuitively, an approximation that satisfies conservation × G 4; 3 ; 35 laws can be derived from a quantity that is invariant under ð Þ ð Þ symmetry operations. In the diagrammatic language, such with δΣ ¼ Σ½G þ δG − Σ½G. Equation (35) defines a quantities are represented by vacuum diagrams, i.e., dia- Rrecursive relation for δG as we can write δΣð1; 2Þ¼ grams that appear in the expansion of the thermodynamic 3 4 1 3; 2 4 δ 3 4 δ2Φ δ 2 Φ d d Kð ; ; Þ Gð ; Þ, where K ¼ = G is the potential. Our starting point is a functional ½G of the Φ Green’s function defined as the sum of certain two-particle irreducible two-particle vertex, which is obtained from irreducible connected vacuum diagrams. The choice of by cutting two lines. We can reorganize Eq. (35) to find diagrams determines the accuracy of the approximation. Z We can obtain a self-energy from the generating func- δGð1; 3Þ¼− d2d4δð2; 4Þδhð4ÞLð1; 2; 3; 4Þ; ð36Þ tional Φ½G by a functional derivative

where Lð1; 2; 3; 4Þ is a reducible vertex function, which is δΦ½G Σð1; 2Þ¼ ; ð30Þ related to the irreducible vertex K via the Bethe-Salpeter δGð2; 1Þ equation

Lð1; 2; 3; 4Þ¼Gð1; 4ÞGð2; 3Þ where the arguments are space-time coordinates on the Z Keldysh contour. In a diagrammatic language, this pro- 5 6 7 8 1 5 6 3 cedure amounts to simply cutting a Green’s function line in þ d d d d Gð ; ÞGð ; Þ the vacuum diagrams leading to the desired self-energy, 5 8; 6 7 7 2; 8 4 which is called Φ derivable. Crucially, the Green’s function × Kð ; ; ÞLð ; ; Þð37Þ obtained from a Dyson equation with this self-energy turns out to be conserving; i.e., physical quantities constructed shown in Fig. 5(c). with this Green’s function obey conservation laws such as The two-particle function L relates the change of the the continuity equation [56]. Importantly, Σ needs to be Green’s function to a perturbation in the single-particle evaluated self-consistently, which means that all internal Hamiltonian and can be used to calculate arbitrary response Green’s function lines in Σ represent full lines dressed by functions. For instance, the current response to a vector the self-energy. potential reads Response functions can also be derived within the Z ρ conserving approximation (for a detailed discussion, see δJμð1Þ¼ d2χμρð1; 2ÞδA ð2Þ; ð38Þ Ref. [46]). We start from the equation of motion

041019-8 TRANSPORT OF NEUTRAL OPTICAL EXCITATIONS USING … PHYS. REV. X 9, 041019 (2019)

(a) all possible pairs of lines in Φ, we obtain a set of diagrams that form the irreducible vertex K shown in Fig. 5(b), which is the kernel of the Bethe-Salpeter equation (37). Note that K includes exciton-electron as well as exciton-exciton vertices. Here, we have neglected any self-energy or vertex ’ (b) corrections of the electronic Green s function due to interactions, anticipating that these do not contribute to the transconductivity to leading order in the polaron density. For the same reason, we restrict the response (c) function to diagrams with a single polaron loop. Moreover, we have used the fact that vertex corrections due to disorder are absent for Gaussian white noise. The solution of the Bethe-Salpeter equation (37) shown (d) in Fig. 5(c) is the reducible two-particle vertex L, which is directly related to the response function by Eq. (39).Asa result, we obtain two types of diagrams, displayed in Fig. 5(d). We emphasize that these are the only diagrams contributing to the linear order in the polaron density nx within the conserving approximation, as any diagram with an additional internal exciton loop would yield a result 2 proportional to nx. These diagrams are closely related to the so-called Maki-Thompson and Aslamazov-Larkin dia- FIG. 5. Linear response theory for polaron drag within the grams that describe superconducting fluctuations (similar conserving approximation to lowest order in polaron density. diagrams appear in Ref. [57], which considers transport in a (a) The functional Φ. Dashed (wavy) lines represent disorder two-component ). We point out that these dia- (interactions). Blue (dark gray) lines indicate dressed electron grams recover the perturbative Coulomb drag results if propagators, and red (light gray) lines indicate dressed exciton expanded to lowest order in the interaction [34,35]. propagators as defined in Fig. 3. (b) The irreducible two-particle Crucially, however, we need to include vertex corrections 2 2 vertex K ¼ δ Φ=δG . Only diagrams to leading order in exciton for the polarons, which arise from the third diagram in density are retained. (c) The Bethe-Salpeter equation (37) for the Fig. 5(b). Moreover, we emphasize that the T matrices are reducible two-particle vertex L. (d) The transconductivity dia- to be evaluated self-consistently in order to remain within gram obtained from Eq. (39). The second and third lines show the the conserving approximation. Indeed, one can readily solution for L based on the irreducible vertex K in panel (b). verify that the self-consistent self-energy and the vertex corrections at zero external frequency satisfy the Ward where the Einstein sum convention is implied and identity (cf. Sec. III D). Finally, we point out that, while there are no vertex corrections from disorder, the effect of 1 μ − 10 μ 2 ρ − 20 ρ p ; p ; p ; p ; 0 0 impurities is included as a broadening of the electron and χμρð1;2Þ¼−i Lð1;2;1 ;2 Þ : ’ 2m1 2m2 10→1 exciton Green s functions [see Fig. 3(a)]. 20→2 We now switch to Fourier space to make use of the ð39Þ translational symmetry restored after disorder averaging. The transconductivity is related to the response function Diagrammatically, this amounts to connecting pairs of χ in Fig. 5(d) by outer legs of the two-point function L to current vertices as shown in Fig. 5(d). e σαβðΩÞ¼ χαβðΩÞ; ð40Þ We now derive the transconductivity diagrams for the iΩ electron-exciton system. Our starting point is the functional Φ depicted in Fig. 5(a). The prefactors of the individual and the response function reads terms are determined by the number of symmetry oper- Z ations of each diagram. The choice of the diagram is χ Ω − π ; Ω Ω Γ ; Ω motivated by the ladder approximation for the T matrix αβð Þ¼ i dp αðp ÞGxðpÞGxðp þ Þ βðp Þ; discussed above. Summing over all possible ways to cut a single line in this diagram, we obtain the disorder self- ð41Þ energies for excitons and electrons as well as the polaron self-energy in the ladder approximation. The resulting where p þ Ω ≡ ðp; ω þ ΩÞ. Here, we have defined the Green’s functions are displayed in Fig. 3(a). By cutting exciton current vertex

041019-9 OVIDIU COTLEŢ et al. PHYS. REV. X 9, 041019 (2019) Z Z p k δΣ Γ ; Ω 0 0 − Ω intðpÞ ðp Þ¼ þ dkdk GeðqÞGeðp þ k k þ Þ πðp; ΩÞ¼−i dk GeðkÞGeðk þ ΩÞ ð46Þ mx me δGeðkÞ × Tðp þ k0ÞTðp þ k0 þ ΩÞG ðkÞ Z x k Ω Γ ; Ω ¼ −i dk G ðkÞG ðk þ ΩÞ Tðp þ kÞ × Gxðk þ Þ ðk Þ: ð42Þ m e e Z e 0 0 0 2 0 The vertex Γ describes the renormalization of the exciton þ i dk Geðk ÞTðp þ k Þ Gxðp þ k − kÞ : velocity due to many-body interactions [see Fig. 5(d)]. The vertex function π describes the coupling of excitons to an ð47Þ electric current and is given by We can alternatively apply the vertex corrections to the Z k vertex π and rewrite Eq. (41) as πðp; ΩÞ¼−i dk G ðkÞG ðk þ ΩÞ Tðp þ k þ ΩÞ m e e Z Z e pβ χαβðΩÞ¼−i dpΠαðp; ΩÞGxðpÞGxðp þ ΩÞ ; ð48Þ 0 0 0 0 m þ i dk Geðk ÞTðp þ k ÞTðp þ k þ ΩÞ x where we introduced the dressed vertex Π defined by 0 × Gxðp þ k − kÞ : ð43Þ Πðp; ΩÞ Z

¼ πðp; ΩÞþ dkΠðk; ΩÞGxðkÞGxðk þ ΩÞwðk; pÞ: C. Low exciton density expansion Before embarking on the evaluation of the response ð49Þ function, we clarify the regime of validity of our calcu- We now outline our strategy for expanding Eqs. (41) lation. As mentioned above, we are interested in the result and (44) to linear order in n . While we have restricted to linear order in the exciton density n ∝ μ. Nevertheless, x x x the calculation to diagrams with only a single exciton we assume both electron and exciton Fermi levels to exceed loop, thereby neglecting certain higher-order contributions, the frequency and disorder scattering rate, which allows us single-loop diagrams may still contain terms nonlinear in to linearize the dispersions around the Fermi levels and n that should be eliminated. Our starting point is the neglect localization effects. To be precise, we consider the x following decomposition of the exciton Green’s function limiting case Ω, 1=τx ≪ μx and Ω, 1=τe ≪ μe. At the same time, we assume μ ≪ μ , which ensures that the Fermi- x e G p GR p 2i GA p θ −ω : polaron picture remains valid. xð Þ¼ x ð Þþ Im x ð Þ ð Þ ð50Þ Within this approximation, we can simply set Tðp þ In the diagrams, each Green’s function G appears inside ΩÞ ≃ TðpÞ to lowest order in Ω. The external frequency x a frequency integration, and we can write thus only enters in products of Green’s functions, Z Z GxðpÞGxðp þ ΩÞ and GeðkÞGeðk þ ΩÞ, where Ω separates ’ R the branch cuts of the two Green s functions. We can now dpfðpÞGxðpÞ¼ dpfðpÞGx ðpÞ write the exciton current vertex as Z Z þ dpfðpÞ2iImGAðpÞθð−ωÞ; ð51Þ p x Γðp; ΩÞ¼ þ dkwðp; kÞGxðkÞGxðk þ ΩÞΓðk; ΩÞ; mx where the second term is proportional to nx for any function ð44Þ fðpÞ that does not have poles in the lower half-plane ω < 0. Hence, we can use Eq. (51) to expand a loop where we introduced the kernel diagram in powers of nx. The zeroth order is given by replacing all exciton Green’s functions in a loop by retarded δΣ ðpÞ functions. This contribution to the transconductivity trivi- wðp;kÞ¼ int ally vanishes. The decomposition (51) thus suggests a δGxðkÞ Z simple recipe to generate the diagrams at first order in nx: ¼ dqG ðqÞG ðpþq−kÞT2ðpþqÞ; ð45Þ Consider all diagrams, where a single exciton line is e e A replaced by ImGx ðpÞθð−ωÞ and all others are assumed to be retarded functions. which corresponds to the last term in Fig. 5(b). The vertex Indeed, this recipe works for the functions wðp; kÞ and function π reads πðp; ΩÞ in Eqs. (45) and (47) as all exciton Green’s

041019-10 TRANSPORT OF NEUTRAL OPTICAL EXCITATIONS USING … PHYS. REV. X 9, 041019 (2019) functions in these expressions, including the internal expanding the Green’s functions in terms of delta functions Green’s functions in the definition of the T matrix, are around the quasiparticle resonance ϵ ¼ ξk. In Appendix F, of the form of Eq. (51). The only exception to this simple we show that the approximation expansion rule are the expressions in Eqs. (41) and (44), where products of two exciton Green’s functions within the τ Ω k ∂ ∂ k ði e m ϵ þ kÞGeðkÞ same frequency integral appear. The expansion of such G ðkÞG ðk þ ΩÞ ≃ e ð56Þ m e e 1 − iτ Ω terms will be derived in Sec. III E below. e e To simplify bookkeeping, we consider π and w to be is valid to leading order in Ω; 1=τ ≪ μ . We cannot π ≃ πð0Þ πð1Þ e e functionals of Gx. The expansion þ can then immediately use this relation for the exciton Green’s be expressed to first order in nx as function because its nonperturbative interaction self-energy correction precludes a description in terms of on-shell πð0Þ π R ¼ ½Gx ; ð52Þ properties only. We discuss this issue in more detail in Z Sec. III E below. δπ½GR πð1Þ ¼ dp x 2iImGAðpÞθð−ωÞ; ð53Þ δGRðpÞ x x D. Evaluation of the vertex functions R to zeroth order in nx where π½Gx means that all exciton Green’s functions inside the vertex have been replaced by retarded functions. We begin with the evaluation of the vertex functions Similarly, the function w can be expressed as w ≃ Γðp; ΩÞ, πðp; ΩÞ, and Πðp; ΩÞ to zeroth order in the wð0Þ þ wð1Þ, with polaron density by simply replacing all exciton Green’s functions by retarded functions. Equation (44) reads, at ð0Þ R w ¼ w½Gx ; ð54Þ zeroth order in nx, Z Z R δw½G 0 p 0 0 2 ð1Þ x 2 A θ −ω Γð Þ p ≃ dkwð Þ p; k Γð Þ k GR k ; 57 w ¼ dp R iImGx ðpÞ ð Þ: ð55Þ ð Þ þ ð Þ ð Þ x ð Þ ð Þ δGx ðpÞ mx

R R R 2 Naively, it may seem cumbersome to expand all exciton where we have approximated Gx ðkÞGx ðk þ ΩÞ → Gx ðkÞ Green’s functions as described above for both diagrams to lowest order in Ω. Using the chain rule shown in Fig. 5(d); however, we can considerably simplify Z the solution by symmetry arguments. Importantly, both δΣð0Þ ð0Þ int ðpÞ R diagrams have been derived by functional derivatives of ∂pΣ p dq ∂qG q ; 58 int ð Þ¼ δ R x ð Þ ð Þ the Φ functional, and therefore, the representation of the Gx ðqÞ n expansion in x in terms of functional derivatives is Σ particularly suitable for our problem. Most terms in the which immediately follows from the definition of int and T δΣð0Þ δ R expansion simply correspond to some higher-order deriv- in Eqs. (19) and (21), as well as int ðpÞ= Gx ðqÞ¼ atives of the Φ functional or the self-energy. The fact that wð0Þðp; qÞ, we can readily verify that the solution of the order of differentiation does not matter leads to Eq. (57) is given by the Ward identity important symmetry properties, for instance, wðp; p0Þ¼ δΣ δ 0 0 p 0 ðpÞ= Gxðp Þ¼wðp ;pÞ, which can be readily verified Γð0Þ ∂ Σð Þ ðpÞ¼ þ p int ðpÞð59Þ from Eq. (45). Moreover, this relation is immediately mx obvious from the diagrammatic representation of w, shown as the last term in Fig. 5(b). R −2 R ¼½Gx ðpÞ ∂pGx ðpÞ: ð60Þ We make use of symmetry properties as well as the Ward identity in the evaluation of the transconductivity below. In order to evaluate πð0Þðp; ΩÞ, defined by Eq. (47),we Indeed, we find at the end of Sec. III E that the various employ Eq. (56) and write contributions from the expansion of the self-energy and vertex corrections eventually combine into a simple final Z k iτeΩ ∂ϵ þ ∂k expression that can be readily computed. As mentioned πð0Þðp; ΩÞ¼i dkG ðkÞ me Tð0Þðp þ kÞ above, however, the product G ðpÞG ðp þ ΩÞ cannot be e 1 − iτ Ω x x Z e simply expanded using functional derivatives. This term 0 0 ð0Þ 0 2 R 0 accounts for the mobility of excitons and depends sensi- þ i dk Geðk ÞT ðp þ k Þ Gx ðp þ k − kÞ ; tively on the parameter Ωτx. A similar expression, ∼GeðkÞGeðk þ ΩÞ, occurs in the definition of πðp; ΩÞ in ð61Þ Eq. (46) and contains information about the electron mobility. The latter expression can be simplified by where we have performed a partial integration. Using

041019-11 OVIDIU COTLEŢ et al. PHYS. REV. X 9, 041019 (2019) Z Z ð0Þ ∂ 2 ∂ − δπα ðp; ΩÞ pTðpÞ¼i dkT ðpÞGeðkÞ pGxðp kÞ; ð62Þ χi Ω 2 0 A 0 θ −ω0 α;βð Þ¼ dpdp R 0 ImGx ðp Þ ð Þ δGx ðp Þ × ∂ GR p : 70 which follows directly from the definition of the T matrix in pβ x ð Þ ð Þ Eq. (21), we find that the momentum derivative in Eq. (61) drops out, and we obtain Writing the first-order term in nx as a functional derivatives turns out to be very useful. We first use Eq. (46) to express Z π Σ − τ Ω k − p the vertex αðpÞ in terms of the self-energy int, and we ð0Þ i e R π ðp; ΩÞ¼ dk wðk; pÞ∂ϵG ðkÞ: ð63Þ subsequently have to evaluate δΣ ðpÞ=δG ðp0Þ¼ 1 − iτ Ω m x int x e e Wðp; p0Þ. From its definition in Eq. (45), however, we immediately observe that the function Wðp; p0Þ is sym- ð0Þ The dressed vertex Π ðp; ΩÞ at zero polaron density can metric under the exchange of momentum arguments, which be determined by plugging this expression into Eq. (49) and implies the simple relation R R R 2 approximating Gx ðkÞGx ðk þ ΩÞ ≃ ½Gx ðkÞ . Using the 0 chain rule δπαðp; ΩÞ δπαðp ; ΩÞ 0 ¼ : ð71Þ Z δGxðp Þ δGxðpÞ ð0Þ ð0Þ R 2 R −1 ∂ωΣ ðpÞ¼ dkw ðp; kÞ½G ðkÞ ∂ϵ½G ðkÞ ð64Þ int x x Making use of this expression together with the chain rule Z 0 δπ ; Ω ∂ Σð Þ R −1 0 αðp Þ 0 and an analogous identity for p ðpÞ as well as ½Gx ¼ dp ∂p0 G ðp Þ¼∂pπαðp; ΩÞ; ð72Þ int δ 0 x ω−ω 2τ −Σð0Þ Gxðp Þ p þði= xÞ int ðpÞ, one can readily verify that Eq. (49) is satisfied by the expression we find Z 0 iτeΩ mx p ð0Þ ð0Þ Πð Þ ; Ω ∂ ∂ Σ χi Ω 2 A θ −ω ∂ πα Ω ðp Þ¼ p þ ω int ðpÞ: ð65Þ α;βð Þ¼ dpImGx ðpÞ ð Þ pβ ðp; Þ: ð73Þ 1 − iτeΩ me me The structure of the second term given by Eq. (68) is similar to the first contribution. Using the Ward identity as n E. Evaluation of transconductivity to first order in x well as the expansion of w in Eq. (55), we obtain With the results above, we are prepared to evaluate the Z 0 response function χαβ in Eq. (41) to linear order in nx.As ii 0 ð Þ R 2 0 χα βðΩÞ¼2 dpdkdk Πα ðp; ΩÞG ðpÞ θð−ϵ Þ outlined previously, we can use the representation of the ; x ’ 0 exciton Green s function in Eq. (51). By expanding the δwð Þ p; k A 0 ð Þ ∂ R different terms of Eq. (41) separately, we obtain three ×ImGx ðk Þ kβ Gx ðkÞ: ð74Þ δGRðk0Þ contributions x We now use the identities χ Ω χi Ω χii Ω χiii Ω O 2 α;βð Þ¼ α;βð Þþ α;βð Þþ α;βð Þþ ðnxÞ; ð66Þ δwðp; kÞ δ2Σ ðpÞ δwðp; k0Þ Z int ; 75 δ 0 ¼ δ δ 0 ¼ δ ð Þ ð1Þ 2 ð0Þ Gxðk Þ GxðkÞ Gxðk Þ GxðkÞ χi Ω −i dpπα p; Ω GR p Γ p ; 67 α;βð Þ¼ ð Þ x ð Þ β ð Þ ð Þ Z δw p; k ∂ ∂ 0 ð Þ ∂ 0 Z ð p þ kÞwðp; kÞ¼ dk 0 k0 Gxðk Þ; ð76Þ δGxðk Þ χii Ω − Πð0Þ ; Ω R 2 ð1Þ α;βð Þ¼ i dpdk α ðp ÞGx ðpÞ W ðp; kÞ where the second line immediately follows from the R 2 ð0Þ definition of T and w in Eqs. (21) and (45). We arrive at × Gx ðkÞ Γβ ðkÞ; ð68Þ Z Z ii ð0Þ R 2 χα βðΩÞ¼2 dpdkΠα ðp; ΩÞGx ðpÞ θð−ϵÞ χiii Ω − Πð0Þ ; Ω Γð0Þ Ω ; α;βð Þ¼ i dp α ðp Þ β ðpÞGxðpÞGxðp þ Þ: A ∂ ∂ ð0Þ ×ImGx ðkÞð pβ þ kβ Þw ðp; kÞ: ð77Þ ð69Þ The third contribution χiii in Eq. (66) contains a term To evaluate the first contribution, we use Eq. (53) as well as GxðpÞGxðp þ ΩÞ, which cannot be expanded by the simple the Ward identity for Γð0Þ in Eq. (60) and obtain recipe in Eq. (51) because two exciton Green’s functions

041019-12 TRANSPORT OF NEUTRAL OPTICAL EXCITATIONS USING … PHYS. REV. X 9, 041019 (2019) Z are evaluated at nearby frequencies. A similar expression ð0Þ ð0Þ 2 ¯ ¯ χiii ðΩÞ¼−i dpΠα ðp; ΩÞΓ ðpÞ½Z G ðpÞG ðp þ ΩÞ for the electronic Green’s function has been evaluated in α;β β x x Eq. (56); however, we cannot use the same formula for 3 þ 4i½GRðpÞ ImΣ ðpÞθð−ωÞ: ð82Þ excitons because their self-energy has an energy-dependent x int imaginary part. In particular, when evaluated away from the Σ ω ζ Σ To make a connection with Eq. (77), we express Im ðpÞ polaron resonance at ¼ p,Im intðpÞ is not necessarily ’ in terms of ImGxðpÞ using straightforward manipulations small. Hence, the imaginary part of the exciton Green s (see Appendix E), and employing the Ward identity, we function cannot be simply replaced by a delta function at arrive at the polaron resonance, and the energy integral has both on- Z shell and off-shell contributions. χiii Ω Πð0Þ ; Ω − Γð0Þ 2 ¯ ¯ Ω The evaluation of the on-shell contribution, where α;βð Þ¼ dp α ðp Þf i β ðpÞZ GxðpÞGxðp þ Þ ¯ Gx ≃ Gx given by Eq. (25), is further complicated by the ð0Þ R 2 þ 2ImG ðkÞθð−ϵÞw ðp; kÞ∂p½G ðpÞ g: ð83Þ momentum-dependent lifetime broadening 1=2τx þ x x Σ Im intðpÞ of the polarons. For simplicity, we neglect this momentum dependence in the following, writing Adding all three contributions, we obtain Z ¯ 1 2 ð0Þ ð0Þ ¯ ¯ G p ≃ 78 χα βðΩÞ¼−iZ dpΠα ðp; ΩÞΓβ ðpÞG ðpÞG ðp þ ΩÞ xð Þ ω − ζ 2τ ω ð Þ ; x x p þ i= xsgnð Þ Z 2 ¯ θ −ω ∂ Πð0Þ ; Ω þ Z dpImGxðpÞ ð Þ pβ α ðp Þ with 1=τx ¼ Z=τx, which is formally justified in the limit Σ 0 −μ ≪ 1 2τ Z Im intð ; xÞ = x. This approximation ignores the ð0Þ effect of electron-polaron scattering on transport, which is þ 2 dpdkΠα ðk; ΩÞθð−ωÞ expected to be suppressed at small external frequencies due A ∂ ð0Þ R 2 to the small available space. We discuss this issue in ×ImGx ðpÞ kβ ½w ðp; kÞGx ðkÞ : ð84Þ more detail in Sec. III F below. The distinction between on- and off-shell contributions We canR perform a partial integration in the last term, and ð0Þ R 2 can now be made explicit by writing using dkw ðp; kÞGx ðkÞ ≃−Z∂ωΣðωÞ ≃ 1 − Z, we can write the response function in the form ¯ R 2 Σ ω ImGxðpÞ¼ZImGx þjG ðpÞj Im intðp; Þ: ð79Þ Z χ Ω − 2 Πð0Þ ; Ω Γð0Þ ¯ ¯ Ω For the relevant energies ω < 0, the first term is entirely α;βð Þ¼ iZ dp α ðp Þ β ðpÞGxðpÞGxðp þ Þ determined by on-shell contributions, whereas the second Z ¯ ∂ Πð0Þ ; Ω term vanishes near the polaron resonance. With the help of þ Z dpGxðpÞ pβ α ðp Þ; ð85Þ Eq. (50), we find R Ω whereR we have used Eq. (51) and dpGxðpÞ¼ GxðpÞGxðp þ Þ ¯ dpGxðpÞ. ≃ R 2 ¯ θ −ω ½Gx ðpÞþ iZImGxðpÞ ð Þ We have arrived at an expression that depends exclu- R ¯ ’ ¯ × ½Gx ðp þ ΩÞþ2iZImGxðp þ ΩÞθð−ωÞ sively on the on-shell Green s function GxðpÞ. Using the explicit expression in Eq. (78), we can rewrite Eq. (85) 4 R R 2 Σ ω θ −ω þ iGx ðpÞjG ðpÞj Im intðp; Þ ð Þ; ð80Þ using Eq. (56) as Z where we have approximated ω þ Ω → ω in the second ¯ ð0Þ Σ 2 ∝ 2 χα β Ω − ∂p Πα ; Ω term and we have neglected terms of order Im intðpÞ nx. ; ð Þ¼ iZ dpGxðpÞ β ðp Þ Using this expression, we can rewrite Eq. (69) as Z 2 ð0Þ ð0Þ Z − iZ dpΠα ðp; ΩÞΓβ ðpÞ iii ð0Þ ð0Þ 2 ¯ ¯ χ ðΩÞ¼−i dpΠα ðp; ΩÞΓ ðpÞ½Z G ðpÞG ðp þ ΩÞ p α;β β x x Ω β ∂ ∂ τ ¯ ð ω þ pβ i= xÞGxðpÞ mx : 4 R 2 Σ θ −ω × ð86Þ þ iReGxðpÞjGx ðpÞj Im intðpÞ ð Þ; ð81Þ Ω þ i=τx ¯ where we have replaced Gx → ZGx in the first term as this Moreover, the vertex functions in Eqs. (60) and (65) can be integral is dominated by on-shell contributions. The second evaluated explicitly, term has only off-shell contributions, and we can therefore R 2 R 3 0 p write, to leading order, ReGxðpÞjGx ðpÞj ≃ Gx ðpÞ .We Γð Þ p ζ ð ; pÞ¼ ; ð87Þ obtain Zmx

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− τ Ω p the same arguments as in Sec. III B, one can readily ð0Þ i e mx Π p; ζp; Ω 1 − : 88 Π ð Þ¼1 − τ Ω ð Þ convince oneself that this vertex is given by Z ¼ i e Zme mx ∂ ∂A Σ A ½ =ð Þ int½Geð ÞjA¼0. The evaluation of the vertex R 0 − ¯ Πð Þ is straightforward and has been performed in Using theseR expressions and the identities i dpGxðpÞ¼ ¯ Sec. III D. n and dω∂ωG ðωÞ¼0 as well as integration by parts, x x The attractive polaron current resulting from the we readily obtain Hamiltonian H0 is 2 δαβn τ τΩ χ − x 1 − mx e x X p E −iΩt αβ ¼ : ð89Þ ˆ e e ð0Þ † 1 − τ Ω 1 − τΩ J Z∂pΠ p; ζp; Ω apap me mx ð i e Þð i x Þ x ¼ þ Ω ð Þ p mx i σ χ Ω The longitudinal transconductivity xe ¼ e αα=i thus E iΩt ˆ e e ð0Þ reads ¼ j þ Z∂pΠ ðp; ζp; ΩÞnˆ ; ð93Þ x iΩ x Ωτ τ P enx mx i e x jˆ ≡ p † σ Ω 1 − ; 90 where we introduced x pð =mxÞapap, and we used xeð Þ¼ 1 − Ωτ 1 − Ωτ ð Þ me mx ð i eÞð i xÞ ð0Þ the fact that ∂pΠ ðp; ζp; ΩÞ does not depend on momen- which is identical to the expression in Eq. (7). tum. The first term corresponds to the paramagnetic contribution that can be evaluated using Kubo’s formula. The second term is the diamagnetic contribution that F. Transconductivity in terms of originates from the change in polaron velocity due to the low-energy excitations shift of the dispersion implied by Eq. (92). These two terms We can rederive the result in Eq. (85) starting from an precisely recover Eq. (85). effective theory based on attractive polarons as the only We can hence interpret Eq. (85) as the paramagnetic low-energy excitations of the system. The effective low- and diamagnetic contributions to the conductivity in terms energy Hamiltonian in terms of the attractive polaron ¯ of effective polaron quasiparticles with propagator Gx. operators ap in the absence of electric fields reads In the derivation of this equation in Sec. III E,wehave X X neglected the momentum dependence of the lifetime, † qr † ζ i j H0 ¼ papap þ Ue apþqap; ð91Þ thereby ignoring incoherent electron-polaron scattering. p p;q;j Here, we have not made such an assumption, which suggests that Eq. (85) holds even for a more general where the first term denotes the dispersion of polarons, momentum-dependent lifetime. while the second term corresponds to disorder scattering of Nevertheless, the result for the transconductivity remains ’ polarons. The effectiveR polaron Green s function can be unchanged. Obviously, only quasiparticles within a thin ¯ ω −iωt 0 † 0 0 written as GxðpÞ¼ d e h jTapðtÞapð Þj i. shell with width of about Ω around the Fermi energy − Ω In the presence of an electric field EðtÞ¼Ee i t, the contribute to the conductivity. In close analogy to Landau polaron quasiparticles are no longer eigenstates of the Fermi theory, we find that the electron scattering rate system as the field induces a drift in the Fermi sea. Solving of an attractive polaron of energy Ω is proportional to Ω2 v the polaron problem with average electron velocity e,we (see Appendix D). In accordance with our expansion to Ω Σ find that the polaron dispersion is shifted in momentum lowest order in , the electron-scattering lifetime Im intðpÞ p → p v − space þ eðmx mxÞ (see Appendix B for a detailed of the quasiparticles relevant for transport can therefore calculation). This shift can be interpreted as an effective be neglected. vector potential for polarons induced by the electric field. Assuming an electronic drift velocity v ¼ −eτ Ee−iΩt= e e IV. TRANSCONDUCTIVITY OF með1 − iΩτeÞ, we can account for the shifted dispersion by introducing an additional term in the effective polaron POLARON POLARITONS Hamiltonian The above calculation can be readily generalized to the X case of exciton-polaron polaritons. We simply add a term 0 − † Πð0Þ p ζ ; Ω A H ¼ apapZ ð ; p Þ · ðtÞ; ð92Þ Φxν describing the coupling of excitons to the cavity mode p to the functional Φ discussed in Sec. III B. The term is depicted in Fig. 6 and reads explicitly Πð0Þ where is the vertex evaluated at vanishing exciton Z −iΩt density given by Eq. (88) and AðtÞ¼Ee =iΩ is the 2 Φ ν g d1d2G 1; 2 Gν 2; 1 ; electric vector potential. x ¼ xð Þ ð Þ ð94Þ Alternatively, Eq. (92) can be derived by evaluating the effective electron current vertex of polarons. Following where the photon propagator is defined as

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p2 γp ¼ γ0 þ : ð100Þ 2mγ

Hence, near the lower polariton resonance, G takes the T x approximate form

T Zγ GγðpÞ¼ω − γ − Σ Σ : ð101Þ FIG. 6. Functional describing the exciton-photon interaction p iZγIm½ intðpÞþ dis;xðpÞ Φxν½Gx;Gν and the vertex correction Γ for polaritons with photon propagators represented by green (dark gray) lines. When the cavity photon is tuned into resonance with the attractive polaron, Δ ¼ ζ0, we find −1 ffiffiffiffi GνðkÞ ¼ ω − νk þ i0sgnðωÞð95Þ p γ0 ¼ Δ − g Z; ð102Þ and we introduced the dispersion of cavity photons, 1 1 1 1 2 −5 νk ¼ k =2mν þ Δ, where mν ≃ 10 m and we assume ¼ þ ; ð103Þ e mγ 2 m mν for simplicity that photons have an infinite lifetime. The x Φ functional xν leads to an additional self-energy for the Z exciton Zγ ¼ 2 : ð104Þ

2 We have arrived at an effective polariton Green’s Σ 2 g xνðkÞ¼g GνðkÞ¼ ; ð96Þ function that is formally identical to the attractive polaron ω − νk þ i0sgnðωÞ Green’s function in Eq. (25), albeit with renormalized para- Σ meters. We emphasize that the broadening Im½ intðpÞþ and the exciton propagator is changed accordingly, Σ dis;xðpÞ has to be calculated self-consistently, and thus 1 disorder scattering as well as electron scattering will be GxðpÞ¼ : ð97Þ strongly suppressed due to the small polariton density of ω − ωp − Σ ðpÞ − Σ ðpÞ − Σ νðpÞ int dis;x x states ∼mν ≪ mx. The calculation of the transconductivity between elec- To make connection with our previous result, we assume trons and polaron polaritons is closely related to the one for that we can describe the exciton as an attractive polaron polarons. The relation for the exciton current vertex in neglecting other excitations such as the trion. In our Eq. (44) acquires an additional term and reads approach, this is formally justified if g is much smaller p 2 p than the energy difference between the attractive polaron Γðp; ΩÞ¼ þ g GνðpÞGνðp þ ΩÞ and trion. Nevertheless, we expect our results to be valid mxZ mν also at somewhat stronger couplings because, in reality, the Ω Γ ; Ω coupling strength between cavity photons and trions is þ dkWðp; kÞGxðkÞGxðk þ Þ ðk Þ: vanishingly small, even though this fact is not captured by ð105Þ our simple model. Hence, approximating the bare exciton Green’s function This equation is also displayed in Fig. 6. Following the by the attractive polaron Green’s function in Eq. (25), same steps as in Sec. III D, we can verify that the vertex at we find zeroth order in the polariton density satisfies the Ward identity Z GxðpÞ ≃ : ω − ζ − Σ Σ − Σ ð0Þ R −2 R p iZIm½ intðpÞþ dis;xðpÞ Z xνðpÞ Γ ðpÞ¼½Gγ ðpÞ ∂pGγ ðpÞ: ð106Þ ð98Þ pffiffiffiffiffiffiffiffi For small wave vectors jpj ≪ mγg, where GxðpÞ ≃ GγðpÞ,wehave The resonances of Gx are determined by the equation

0 p ω − ζ ω − ν 2 Γð ÞðpÞ¼ : ð107Þ ð pÞð kÞ¼Zg : ð99Þ Zγmγ

Near zero momentum, the lower-energy branch can be In contrast, the derivation of the vertex function Πð0Þðp; ΩÞ approximated by a quadratic dispersion in Eq. (65) remains unchanged. Moreover, the real part of

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Σ the self-energy Re intðpÞ is largely independent of the scattering times can be shorter than the radiative cavity coupling, and hence the on-shell expression for lifetime. In this case, the resonantly generated polar- Πð0Þðp; ΩÞ in Eq. (88) is retained. ons will be scattered to momentum states outside In Sec. III E, most expressions involving Gx remain the light cone, where they can decay nonradiatively unchanged as they involve an integral over a large area in or by phonon-assisted radiative decay. The nonzero momentum space. The only exceptions occur after Eq. (79), drag velocity may be detected in photolumi- where the on-shell Green’s function G¯ of occupied states is nescence. used. In these expressions, we can thus simply substitute (ii) Alternatively, our results may be relevant to hetero- ¯ ¯ bilayers at very large electron density. In this limit, ZGx with ZγGγ. These changes result in an additional factor screening of the interaction between valence band Zγ=Z in the final result, Eq. (90). In addition, the lifetime τx holes and conduction band electrons has to be taken is replaced by the polariton lifetime τγ ¼ τ m =mγ ≫ τ . x x x into account, which invalidates our assumption that Based on these considerations, we finally arrive at the excitons can be regarded as rigid quasiparticles. transconductivity between electrons and polaritons Instead, we may consider the valence band hole as a Ωτ τ quantum impurity interacting with a Fermi sea of Zγ enx mx i e γ σγ Ω 1 − : 108 conduction band electrons. For sufficiently high eð Þ¼ 1 − Ωτ 1 − Ωτ ð Þ Z me mx ð i eÞð i γÞ electron densities, it may become favorable for the hole to form a polaron rather than an exciton. The additional factor Zγ=Z takes into account the fact that This approach bears some similarity with Ref. [58], the drag requires a finite excitonic quasiparticle weight. At and it can be used to analyze the Fermi-edge resonance, this factor reduces the polaron velocity to half its singularity problem within the framework of Fermi value. Moreover, we emphasize that polariton drag is much polarons. Our analysis of the transport problem less affected by excitonic disorder because the small density carries over to the case of hole polarons with the of states of polaritons suppresses disorder scattering. trion binding energy replaced by the screened exciton binding energy. To ensure that Ωτ > 1 is V. NONEQUILIBRIUM EFFECTS satisfied, it may be possible to use microwave or terahertz irradiation and monitor the polaron re- We have seen that our intuitive picture of polaron drag sponse as sidebands in optical response. developed in Sec. II correctly reproduces the results of the (iii) Arguably, the most promising platform for the fully microscopic model presented in Sec. III. This is observation of electric-field-induced displacement encouraging as the semiclassical theory can be extended to of neutral optical excitations is provided by exciton- include other effects that could not be captured within the polaron polaritons observed when a monolayer with linear-response calculation but that are potentially relevant a 2DES is embedded inside a 2D microcavity [2–4] for experiments. Most notably, optically excited excitons or when a monolayer is embedded in a dielectric have a finite lifetime due to recombination processes, structure that supports in-plane propagating pho- which requires a nonequilibrium calculation. tonic modes [59]. Small-momentum excitations in For exciton polarons in monolayer TMDs, an ultrashort the lower-energy polaron-polariton branch have two radiative lifetime of about 1 ps for low-momentum excitons striking features: First, because of their extremely implies that the assumption of an equilibrium exciton gas is small effective mass, polaritons are, to a large extent, not justified. Moreover, disorder scattering in state-of-the- protected from disorder scattering. This effect was art samples is comparable to the radiative decay rate, observed in exciton polaritons in the 1990s in GaAs rendering it unlikely that a spatial displacement in exciton heterostructures [60,61]. Second, low-momentum photoluminescence induced by an applied low-frequency polaritons can only decay radiatively through cavity electric field can be observed. mirror loss: It is therefore possible to ensure that the In general, we envision three different experimental polariton lifetime is much longer than that of excitons scenarios, where our findings are potentially observable: by using high-quality-factor cavities. Nevertheless, (i) In the case of interlayer excitons in TMD hetero- interactions between polaritons are also weak, and we bilayers, where electrons and holes occupy conduc- cannot expect a low density of polaritons to thermal- tion and valence band states in different monolayers, ize, rendering it essential to develop a nonequilibrium the exciton lifetime can be tuned electrically and can description of transport. well exceed 100 ns. Since timescales for disorder scattering are considerably shorter, we expect the interlayer excitons to be in equilibrium. Provided A. Boltzmann equation that the spatially indirect trion state remains bound, Our aim is to develop a kinetic theory for the distribu- our results could also be used to describe drag of tion function gkðr;tÞ of exciton polarons, including the indirect exciton polarons, where disorder or electron effects of pumping, recombination, and disorder as well as

041019-16 TRANSPORT OF NEUTRAL OPTICAL EXCITATIONS USING … PHYS. REV. X 9, 041019 (2019) a nonzero electron drift velocity veðtÞ due to an electric correspond to collision processes, either due to exciton field EðtÞ. In the most general case, we can write the disorder or incoherent scattering off electrons. These terms Boltzmann equation as will be discussed in more detail below. Integrating Eq. (112) over momentum space, we obtain dgkðr;tÞ ∂gkðr;tÞ ∂gkðr;tÞ ∂gkðr;tÞ ≡ k_ r_ Rthe time evolution of the exciton density nðtÞ nxðtÞ¼ ¼ þ þ k : ð109Þ 2 dt ∂t ∂k ∂r ðdk=4π ÞgkðtÞ as

Importantly, the electric field does not exert a direct force nðtÞ k _ − on polarons, whose canonical momentum is conserved. nðtÞ¼R τ ; ð113Þ Instead, it shifts the polaron dispersion by changing the r electron velocity. A straightforward solution of the polaron where we have used that the collision integrals conserve problem discussed in Sec. III A in the presence of an the number of polarons. Moreover, we are interested in electron drift (see Appendix B) yields the dispersion evaluating the exciton current density defined as ˜ Z ζ ζ kðtÞ¼ kþðmx−mxÞveðtÞ; ð110Þ dk nðtÞv¯ ðtÞ¼ gkðtÞv ðk;tÞ: ð114Þ x ð2πÞ2 x which is shifted from the equilibrium dispersion ζk ¼ k2 2 k 0 = mx such that the polaron state at ¼ has a velocity Differentiating with respect to time, we obtain 1 − v ð mx=mxÞ eðtÞ. Even though an electric field does not Z k_ 0 v¯ k v k affect the conjugate momentum of the polaron, ¼ ,it d xðtÞ _ v¯ d d xð ;tÞ v k nðtÞ þ nðtÞ xðtÞ¼ 2 gkðtÞ changes its kinetic momentum mx xð ;tÞ with the polaron dt ð2πÞ dt velocity dgkðtÞ þ vxðk;tÞ : ð115Þ ˜ dt ∂ζkðtÞ k m v k;t 1 − x v t : 111 xð Þ¼ ∂k ¼ þ eð Þ ð Þ mx mx The first term on the right-hand side can be readily k_ 0 For simplicity, we assume a spatially homogeneous dis- evaluated using Eq. (111) and ¼ as ∂ r 0 Z tribution, rgkð ;tÞ¼ , and we suppress the spatial dk dvxðk;tÞ mx − mx dependence in the following. Hence, the distribution gkðtÞ ¼ nðtÞ F ðtÞ; ð116Þ 2π 2 e function does not have an implicit time dependence, and ð Þ dt mxme Eq. (109) can be written as with F ¼ m v_ ðtÞ. The second integral in Eq. (115) can be e e e ∂ ∂ evaluated using the Boltzmann equation (112) and, with the dgkðtÞ δ k − gkðtÞ gkðtÞ gkðtÞ ¼ R ð Þ τ þ ∂ þ ∂ : help of Eq. (113), we obtain dt r t dis t int ð112Þ dv¯ ðtÞ R m x ¼ ½ðm − m Þv ðtÞ − mv¯ ðtÞ x dt nðtÞ x x e x x The first term is due to pumping of polarons at a rate R in − k 0 mx mx F F F the polaron state at ¼ by resonant laser absorption. The þ eðtÞþ disðtÞþ intðtÞ; ð117Þ incidence angle of a collimated laser field determines the me in-plane momentum of the exciton polarons, which is, in In this expression, the first and second terms account turn, much smaller than the other characteristic momentum for polarons in the k 0 state with velocity v 0;t scales in the problem, such as k and m v . By tuning the ¼ xð Þ¼ F e e 1 − m =m v t that are generated by the laser or lost by frequency of a normal-incidence single-mode laser field, it ð x xÞ eð Þ the recombination of excitons, respectively. The third term is possible to ensure that only k ¼ 0 attractive polarons corresponds to the drag force acting on the exciton system can be created. The second term in Eq. (112) corresponds to due to the polaronic coupling to electrons, which was the the loss of attractive polarons due to the recombination main focus of the equilibrium calculation in Sec. III. The processes at a rate 1=τ . In general, τ is expected to depend r r last two terms correspond to the friction due to exciton on momentum since the recombination rate should be disorder and incoherent scattering with electrons, strongest for small momenta k that lie inside the light cone, Z whereas the decay from states outside the light cone m dk ∂gk requires the generation of additional excitations such as F t x v k;t ; disð Þ¼ 2π 2 ∂ xð Þ ð118Þ phonons. Here, we neglect the momentum dependence of nðtÞ ð Þ t dis τ Z r, for simplicity, although the generalization of our results m dk ∂gk to include this effect is straightforward. The last two terms F t x v k;t : intð Þ¼ 2π 2 ∂ xð Þ ð119Þ in Eq. (112) conserve the number of polarons and nðtÞ ð Þ t int

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These forces depend sensitively on the polaron distribution −vemx=mx and thus are in an excited state. Excited polaron and can lead to nonlinear effects in the time evolution. We states, however, have a finite lifetime due to the interaction discuss them in more detail in the next section. with electrons and will decay into lower-energy states, which also have a smaller absolute velocity. Hence, B. Estimation of the friction forces from disorder incoherent electron-polaron scattering will lead to a relax- and incoherent scattering with electrons ation of the polaron velocity to zero in the comoving frame. Friction from excitonic disorder can be described by the In the lab frame, this case corresponds to an acceleration of v collision integral polarons until they reach a velocity e, which justifies the Z following ansatz for the electron friction force on the 0 ∂gk dk ˜ polarons, ¼ ½gk0 ðtÞ − gkðtÞMkk0 ; ð120Þ ∂t ð2πÞ2 dis v¯ ðtÞ − v F ¼ −m x e ; ð124Þ ˜ int x τ t where Mkk0 denotes the matrix element corresponding to intð Þ k k0 the scattering of a polaron from state to state . In the τ simplest case, we can assume Gaussian white noise as in with an effective timescale int that depends on details of the τ Sec. III A, and we obtain polaron distribution at time t. A rough estimate for int is given by the interaction lifetime of an excited polaron state − v ˜ 1 ˜ ˜ with velocity mx e=mx. After this time, all the polarons at Mkk0 δ ζk t − ζk0 t : 121 ¼ ντ ½ ð Þ ð Þ ð Þ zero momentum have scattered at least once. In the comov- x x ing frame of electrons, the scattering probability does not Note that the matrix elements depend on the shifted have a very strong dependence on the direction of the final dispersion in the presence of the electric field. With this polaron momentum. Hence, the polaron reaches an average v approximation, the collision integral simplifies to velocity similar or equal to e already after a few scattering Z events, even though the time for each individual polaron to ∂ 1 k0 reach that velocity is expected to be much longer. gk d δ ζ˜ − ζ˜ − gkðtÞ ¼ 2 gk0 ðtÞ ½ kðtÞ k0 ðtÞ ; The expression for the friction force in Eq. (124) ∂t νxτx ð2πÞ τx dis captures the conventional Coulomb drag effect, e.g., in ð122Þ electron bilayer systems [33–36]. Indeed, for perturbative F interactions, int is the only contribution to drag. As we and the friction force reads have shown in Sec. III, nonperturbative interactions result in an additional polaron drag effect that dominates at low mv¯ ðtÞ F t − x x : 123 temperatures and frequencies, whereas the drag force in disð Þ¼ τ ð Þ x Eq. (124) is a subleading correction that we have neglected in the linear response calculation in Sec. III. We emphasize Hence, for Gaussian white noise correlated disorder, the that in nonequilibrium systems or at finite temperatures, polaron disorder scattering time is also the relaxation time F int cannot necessarily be ignored. for the drift velocity of the exciton system. We emphasize An explicit expression for the collision integral reads that this result holds, even though we cannot treat the Z collision integral in Eq. (120) in the relaxation time 0 ∂gk dk ˜ ˜ approximation. ¼ ½gk0 ð1 − gkÞQk0k − gkð1 − gk0 ÞQkk0 ; ∂t 2π 2 The force introduced in Eq. (119) corresponds to an int ð Þ additional drag force originating from the residual inter- ð125Þ action between electrons and polarons. While coherent ˜ scattering events of electrons and excitons result in polaron where the transition probability Qkk0 denotes the scattering formation and the polaron drag phenomenon described in rate of an attractive polaron of momentum k to a momen- Secs. II and III, incoherent collisions lead to a lifetime tum k0 due to interactions with an electron Fermi sea broadening of the polarons. This broadening appears as a drifting at velocity ve. In order to derive an expression for Σ ’ ˜ term Im intðpÞ in the Green s function description of Qpp0 , we first consider the polaron scattering rate Qpp0 polarons discussed in Sec. III A. when the electron drift velocity is zero. The electron-polaron The qualitative effect of incoherent broadening can be scattering amplitude is simply given by the T matrix, and, estimated from a simple argument by temporarily disre- hence, Fermi’s golden rule yields garding polaronic disorder. For concreteness, we imagine a Z narrow distribution of polarons around zero momentum, 2 dk 2 Qpp0 ¼ Z π jTðp þ k; ζp þ ϵkÞj fk½1 − fk p−p0 which corresponds to the distribution shortly after the ð2πÞ2 þ laser has been switched on. In the comoving frame of δ ζ − ζ − ε ε the electrons, the zero-momentum polarons have a velocity × ð p p0 kþp−p0 þ kÞ; ð126Þ

041019-18 TRANSPORT OF NEUTRAL OPTICAL EXCITATIONS USING … PHYS. REV. X 9, 041019 (2019) where f denotes the equilibrium electron distribution. ZTðkF; 0Þ, which is valid in the relevant limit jpj ≪ kF One can readily verify that this scattering rate reproduces and jkj ≃ kF. The scattering time at zero temperature the lifetime broadening of the on-shell polaron Green’s can then be obtained from a straightforward calculation function (see Appendix D3for the case of an empty polaron (assuming m ≡ mx ≃ me, see Appendix D)as band), 3 Z 1 2 ðmveÞ 0 ∼ ρ U : 132 dk ˜ τ e ð Þ ImΣ ðp; ζpÞ¼ ð1 − gk0 ÞQkk0 : ð127Þ intðtÞ kF int ð2πÞ2 At frequencies close to the trion energy, the T matrix is In the presence of a nonzero electron drift velocity, dominated by the trion pole. The interaction between Eq. (126) has to be modified by shifting both the polaron attractive polarons and electrons originates from virtual ˜ dispersion ζp → ζp and the electron distribution fk → scattering events into a higher energy state comprising a fk−meveðtÞ. A straightforward calculation along the lines of trion and a hole, which results in an effective attractive Appendix B establishes a relation between the scattering interaction. This yields the estimate amplitude with and without an electric field, ZϵT ˜ U ≃− ; ð133Þ 0 0 ρ Δ Qpp ¼ Qp−mxveðtÞ;p −mxveðtÞ: ð128Þ e

Indeed, this result confirms the intuition that the friction where Δ is the energy separation between the trion-hole between polarons and electrons results in the relaxation of continuum and thep attractiveffiffiffiffiffiffiffiffiffiffiffiffi polaron. Assuming a quasi- the polaron velocity in the comoving frame of electrons. To particle weight Z ≃ μe=ϵT, the friction force on polarons see this result, we observe that the lowest energy state has an at zero momentum is of the order of infinite lifetime, i.e., Q0;k ¼ 0. In the presence of an electric p v 2 field, this means a polaron state with momentum ¼ mx e m k ϵ F ≃ F T v3v : is stable with respect to scattering off electrons. According to int Δ2 e e ð134Þ Eq. (111), this state has a velocity ve; i.e., it is moving at the same speed as the electron Fermi sea. The friction force therefore scales with the electric field as In order to obtain an estimate for the friction force F ∝ 4 int Fint E , which illustrates why this effect has not been in Eq. (119) on a narrow distribution centered at zero captured in the linear response calculation in Sec. III. momentum, gk ≃ nδðkÞ, we can make the same approxi- We emphasize that this result is valid at zero temperature. mation as above that the average velocity after a single At finite temperatures, the friction force is still expected to v scattering event is e. We can hence write the force from the have the form in Eq. (124), but the scattering rate acquires first term in Eq. (125) as an additional temperature-dependent contribution due to Z the enhanced phase space available for electron-polaron k k0 v mx d d v k 1 − ˜ ≃ mx e scattering. 2 2 xð ;tÞgk0 ð gkÞQk0k τ ; nðtÞ ð2πÞ ð2πÞ intðtÞ ð129Þ C. Equations of motion for polarons and polaritons After the pump has been switched on for a time of about and for the second term, we obtain τ , the polaron density saturates to the value n ¼ Rτ .Even Z r r k k0 v¯ though the density has reached a steady-state value, the mx d d 0 ˜ mx x v ðk;tÞgkð1 − gkÞQkk0 ≃ : solution of the nonlinear Boltzmann equation (112) nðtÞ 2π 2 2π 2 x τ ðtÞ ð Þ ð Þ int depends sensitively on the ratio of the various timescales ð130Þ and, in general, requires numerical calculations. To ensure a strong hybridization of polarons and photons, it is desirable Here, we have defined the scattering time to work in a limit where the radiative lifetime is the shortest timescale, τ ≪ τ, τ . In this case, the polarons remain 1 X r x int ˜ mostly near k ¼ 0 and therefore within the light cone, as τ ¼ Q0k0 ; ð131Þ intðtÞ k0 disorder and electron scattering, which change the polaron momentum, are suppressed. which is identical to the interaction lifetime of a polaron at This limit guarantees that the friction caused by electrons momentum −mxve. With these results, we indeed re- can be described by Eq. (124), and we can make the cover Eq. (124). connection to the semiclassical equations in Sec. II more τ τ To obtain a rough estimate of int, we can approximate explicit. Assuming a density nðtÞ¼R r, Eq. (117) can be the T matrix by a constant ZTðp þ k; ζp þ εkÞ ≃ U≡ written as

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dv¯ t m − m v t mv¯ t mv¯ t resonant laser field, upon which they travel distances xð Þ ð x xÞ eð Þ − x xð Þ − x xð Þ mx ¼ 10 μ dt τr τr τx exceeding m while decaying due to cavity losses: By interfering, the polariton emission with the same laser m − m m(v¯ ðtÞ − v ) x x F − x x e field, drag-induced polariton displacement of the order of þ eðtÞ τ : ð135Þ me intðtÞ 30 nm can be easily measured [64]. The photonic dis- placement induced by the applied electric field can easily A comparison with the corresponding Eq. (3) in Sec. II be increased by using higher quality cavities leading to reveals three extra contributions in the nonequilibrium longer polariton lifetimes [65]. case. The first term originates from pumping polarons into the state k ¼ 0 with velocity v ðk ¼ 0;tÞ¼ x VI. MAGNETIC FIELD RESPONSE ð1 − m =mÞv ðtÞ. The second term describes recombina- x x e OF EXCITON POLARONS tion of polarons. Finally, the last term originates from incoherent scattering of electrons, which has been neg- We can include the effect of a dc magnetic field by lected in Sec. II. This assumption is justified in the limit of adding a Lorentz force to Eq. (2), weak electric fields. Equation (135) can be readily generalized to polaritons. d eEðtÞ eB veðtÞ veðtÞ¼aeðtÞ¼− − veðtÞ × − : ð137Þ As has been argued in Sec. IV, polaritons are much less dt me me τe affected by scattering processes because of their ultra-low mass and the correspondingly small scattering phase space. We can readily solve this equation and find Resonant coupling between polarons and the cavity mode E Ω B − E Ω τ−1 − Ω creates a local minimum near zero momentum in the v Ω e ð Þ × e =me ð Þð e i Þ eð Þ¼ −1 2 2 ; ð138Þ polariton dispersion. For a sufficiently strong coupling me ðτe − iΩÞ þ ωc 2 [g>ðmxveÞ =2mx in the notation of Sec. IV], disorder or interaction scattering of low-energy polaritons to large where ωc ¼ eB=me is the cyclotron frequency. As the momenta of about mxjvej is energetically forbidden. In magnetic field does not directly couple to excitons, Eq. (6) 1 τ 1 τ ω ≫ Ω 1 τ Ωτ ≫ 1 this case, the polariton scattering rates = x and = int are remains valid. Hence, when c ; = e and x , reduced by a factor ðmγ=mxÞ compared to exciton polarons. the electrons and polarons drift in the direction perpen- −5 dicular to the electric and magnetic fields realizing a Hall As typical values are ðmγ=mxÞ ∼ 10 , we conclude that such scattering processes can be neglected. Following effect of neutral excitons. similar steps as above, we arrive at the equation of motion More generally, Eq. (6) predicts that excitons will follow for polaritons, the trajectory of electrons (scaled by a factor) on timescales shorter than the exciton impurity scattering time. In the dv¯ ðtÞ m v ðtÞ v¯ ðtÞ m F ðtÞ absence of an electric field, excitons should therefore move x ¼ 1 − x e − x þ 1 − x e : dt m 2τ τ m 2m in cyclotron orbits, which suggests that polarons could x r r x e experience a phenomenon similar to Landau quantization. ð136Þ Equivalently, one can argue that excitons should be affected by the quantizing magnetic field as they are dressed by The additional factor of 1=2 in the first and third terms particle-hole excitations with a discrete energy spectrum due reflects the ratio of the polariton and polaron quasiparticle to electronic Landau levels. A signature of this case would be weights. This equation can be readily solved. Assuming a a polaron spectral function with a series of peaks on top of an F 0 static electric field, such that eðtÞ¼ , the polaritons incoherent background present at higher energies roughly ¯ 1 − v 2 move at a velocity vx ¼ð mx=mxÞ eðtÞ= during their spaced by the cyclotron frequency. A related phenomenon has entire lifetime, which corresponds to an approximate been discussed in the context of Bose polarons, in particular, Δ 1 − v τ 2 distance ¼ð mx=mxÞ eðtÞ r= . for electrons strongly coupled to dispersionless phonons. In We also perform a quick estimate of the magnitude of Δ. this case, phonon shake-off processes lead to a series of broad The ratio mx=mx depends on the Fermi energy but is of peaks in the polaron spectral function separated in energy by the order 1, and therefore we can conservatively take multiples of the phonon frequency [66]. 1 2 mx=mx ¼ = . The electron drift velocity can be calculated Observation of Landau levels in the absorption spectrum μ from ve ¼ E . In TMDs, the applied source-drain electric where the energy separation is given by the electron field E can be of the order of 1000 V=cm, while the cyclotron frequency would represent yet another manifes- 4 2 electron mobilities can reach μ ¼ 10 cm =ðVsÞ [62,63], tation of the central role played by polaron physics in which results in drift velocities of the order of ve ¼ optical excitations out of a 2DES. If bound trions were 5 10 m=s. Assuming a polariton lifetime of τr ≈ 10 ps, observable in absorption, the observed Landau-level spacing we obtain a drag-induced polariton displacement of would be determined by the trion mass, which is a factor roughly Δ ≈ 250 nm. We envision an experiment where of 3 larger than that of the electron in TMD monolayers. polaritons are injected with a finite group velocity using a In the opposite limit of very high electron densities, we

041019-20 TRANSPORT OF NEUTRAL OPTICAL EXCITATIONS USING … PHYS. REV. X 9, 041019 (2019) expect screening to lead to ionization of excitons: In this Another exciting extension of our work is the analysis of regime, the level separation will be determined by the the regime of a strong ac drive of the interacting electron- reduced mass of electron-hole pairs. polariton system where electrons occupy Floquet bands. Conversely, the motion of electrons in a magnetic field We expect a particularly strong modification of polaron can be influenced by the presence of excitons. Excitons at formation if the external field resonantly drives an appreciable density can lead to a polaronic dressing of resonance of degenerate electrons. Introduction of spatial electrons, thereby increasing their effective mass, which modulation of the electron density using Moire patters or could be detected in Shubnikov–de Haas oscillations or surface acoustic waves could be used to engineer nontrivial cyclotron resonance measurements. With increasing den- band structure for electrons: It is thereby possible to realize sity of excitons, the electronic resonance frequency is either stationary or Floquet topological bands for electrons, expected to shift as a result of a polaronic dressing. In which will, in turn, modify exciton-polaron transport. An contrast, bound trions would appear as a new resonance in alternative strategy to investigate the interplay between addition to the bare electron resonance, and no shift is topological order and polaron formation is to study optical expected as a function of trion density. excitations from fractional quantum Hall states. As we highlighted earlier, application of our formalism VII. OUTLOOK AND CONCLUSION to 2D materials is particularly exciting: On the one hand, these materials exhibit a valley pseudospin degree of Our work opens up new frontiers in nonequilibrium freedom and a nontrivial band geometry with a sizable many-body physics by showing that external electric and Berry curvature. On the other hand, they allow for creating magnetic fields could be used to control and manipulate hybrid materials combining different functionalities; an elementary optical excitations such as excitons or polar- exciting recent example is a heterostructure based on itons. The requisite element leading to this intriguing exchange coupled semiconducting and a ferromagnetic functionality is the presence of nonperturbative interactions monolayers. Optical excitations in such a system will be between excitons and electrons, leading to the formation of Bose polarons where the magnetic moments of valley exciton polarons. In addition to the potential applications in excitons are screened by excitations out of the realizing effective gauge fields for photonic excitations ferromagnet. that we already highlighted, we envision several extensions of our work that, by themselves, constitute open theory problems. ACKNOWLEDGMENTS Arguably, the most interesting extension of our work is We thank Wilhelm Zwerger for inspiring discussions and the investigation of the degenerate Bose-Fermi mixture important input in the initial stages of this work. We further regime, which can be accessed by increasing the optical acknowledge stimulating discussions with M. Fleischhauer, pump strength. In the limit of perturbative electron-exciton B. I. Halperin, L. Jauregui, A. Kamenev, P. Kim, A. interactions, this problem could be formulated as a cou- Levchenko, M. Lukin, Y. Oreg, H. Park, A. Rosch and S. pling between Bogoliubov excitations out of a polariton Zeytinoglu. A. I. and O. C. were supported by a European condensate and electrons close to the Fermi surface [67]. Research Council (ERC) Advanced Investigator Grant In the opposite limit of nonperturbative exciton-electron (POLTDES). F. P. acknowledges financial support by the interactions and polariton density np lower than that of STC Center for Integrated Quantum Materials, NSF electrons ne, the ground state could be described as a Grant No. DMR-1231319. E. D. was supported by the polaron-polariton condensate. We expect the confluence of Harvard-MIT CUA, NSF Grant No. DMR-1308435 and these two approaches to give rise to new physics where the by the AFOSR-MURI: Photonic Quantum Matter, Grant nature of polariton screening by electrons could be dra- No. FA95501610323. matically modified due to degeneracy favoring a high quasiparticle weight. Moreover, it is precisely in this regime Note added.—Recently, an experimental work was posted that the modification of the electronic transport properties [70] reporting a response of polaritons to external electric due to degenerate-polaron formation would become signifi- fields. At this point, it is not clear to us if the observed cant. It is possible that the previous proposals for polariton- response is related to the polaron drag we predict. mediated may have to be revisited in light of new features that emerge from a rigorous analysis of APPENDIX A: EFFECTIVE ATTRACTIVE polaron-polariton condensation [67–69]. Last but not least, a POLARON PROPAGATOR gradual increase (decrease) of polariton (electron) density from the np ≪ ne to np ≫ ne regime takes us from a Fermi- In this section, we show, starting from the full Green’s polaron problem to a Bose-polaron problem: The exciton- function Gx, how one can introduce an effective (or polariton system allows for such tuning by changing the projected) propagator, which describes the propagation optical pump strength together with applied gate voltages of the low-energy excitations (i.e., the attractive polarons). that control the electron density. Starting from the general Green’s function

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1 where we introduced the polaron mass mx and also GxðpÞ¼ω − ω − Σ 2τ ω ; ðA1Þ μ p intðpÞþi= xsgn introduced a new chemical potential x, measured from the bottom of the attractive polaron dispersion. The above we introduce the dispersion of attractive polarons as the equation is correct for small momenta p ≪ kF, while for lowest-energy pole of the above, i.e., larger momenta, it will start to deviate from a quadratic dispersion. p2 Σ p; ω ω ζ ≡ ω Σ p ζ ≃ − μ Expanding the self-energy ð Þ to linear order in , p p þ Re intð ; pÞ x; ðA2Þ 2mx we can write

1 ≃ GxðpÞ ω − ω − Σ p ζ − Σ p ζ ω − ζ ∂ Σ p ω 2τ ω p Re intð ; pÞ iIm intð ; pÞþð pÞ ω intð ; Þjω¼ζp þ i= xsgnð Þ 1 ¼ ω − ζ ω − ζ ∂ Σ p ω 2τ ω − Σ p ζ p þð pÞ ω intð ; Þjω¼ζp þ i= xsgnð Þ iIm intð ; pÞ

Zp ¼ ω − ζ 2τ ω − Σ p ζ ; ðA3Þ p þ iZp= xsgnð Þ iZpIm intð ; pÞ

−1≡ ∂ Σ p ω ≃ 1 − −1 where we introduced the renormalization factor Zp ω intð ; Þjω ζ Z : ðA8Þ 1 − ∂ Σ p ω ¼ p ω intð ; Þjω¼ζp . It is clear from this definition that, in general, the renormalization factor Zp is complex (jZpj To evaluate the momentum derivative, we solve Eq. (A2) to denotes the quasiparticle weight) since the self-energy is obtain complex. However, as long as

dζp p ∂ωImΣ ðp; ωÞjω ζ ≪ ImΣ ðp; ζpÞ; ðA4Þ int ¼ p int ¼ dp mx ω Σ p ζ we can neglect the complex part of Zp. We prove that the d p dðRe intð ; pÞÞ ¼ þ above condition is satisfied for small momenta p ≪ k in p p j j F d d Appendix D, where we explicitly evaluate the imaginary p dζp part of the self-energy. In this limit, we can approximate the ¼ − ∂pReΣ ðp; ζpÞþ ∂ωReΣ ðp; ωÞ : m int dp int quasiparticle weight by a constant: x ω¼ζp ðA9Þ −1 ≃ 1 − ∂ Σ p ω Zp ωRe intð ; Þjω¼ζp ≃ 1 − ∂ Σ p ω ≡ −1 From the above, we immediately obtain ωRe intð F; Þjω¼0 Z : ðA5Þ p p We can also introduce the lifetime of the attractive ∂ Σ p ζ −1 − pRe intð ; pÞ¼Z : ðA10Þ polaron as mx mx

1 2τ p 2τ Σ p ζ It is useful to obtain an explicit expression for the = xð Þ¼iZ= x þ iZjIm intð ; pÞj; ðA6Þ polaron dispersion ζp with respect to the value (and where we introduced the absolute value of the imaginary derivatives) of the self-energy at the Fermi surface p p part of the self-energy, which changes sign at ω ¼ 0. Using ¼ F, by expanding the self-energy in a Taylor series the above, we can rewrite the exciton propagator for low with respect to these points: momenta as Σ p ω ≃ Σ p 0 p∂ Σ p 0 Re intð ; Þ Re intð F; Þþ pRe intð ; Þjp¼p Z F ≃ ≡ ¯ p2 GxðpÞ ZGxðpÞ; ðA7Þ 2 ω − ζp þ iτxðpÞsgnðωÞ þ ∂pReΣ ðp; 0Þjp p 2 int ¼ F

¯ þ ω∂ωReΣ ðp ; 0Þjω 0: ðA11Þ which defines the projected operator GxðpÞ. int F ¼ In the main text, we sometimes need to evaluate the partial derivatives of ΣðpÞ on shell, i.e., at p ¼ðp; ζpÞ. The pole ω of Eq. (A1), which yields ζp, therefore satisfies From Eq. (A5), we immediately have the equation

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ω − ωp − ReΣ p ; 0 − p∂pReΣ p; 0 APPENDIX B: POLARON DISPERSION FOR A intð F Þ intð Þjp¼pF 2 DRIFTING FERMI SEA p 2 − ∂pReΣ ðp; 0Þjp p − ω∂ωReΣ ðp ; 0Þjω 0 ¼ 0; 2 int ¼ F int F ¼ Here, we determine the polaron dispersion, when the electrons are drifting at velocity v . For simplicity, we ðA12Þ e restrict ourselves to the case of zero polaron density, μx ¼ 0. Moreover, we focus on sufficiently low energies which can be solved to obtain (i.e., below the trion energy) such that we can represent the exciton Green’s function by the effective polaron Green’s function p2 ζp ¼ Z ωp þ ReΣ ðp ; 0Þþ − p∂pReΣ ðp; 0Þjp p 1 int F 2m int ¼ F G¯ RðpÞ¼ ; ðB1Þ x x ω − ζ 2τ − Σ 2 p þ iZ= x iZIm intðpÞ p 2 − ∂pReΣ ðp; 0Þjp p ; ðA13Þ 2 int ¼ F 2 where ζp ¼ p =2mx is the polaron dispersion at zero density, when the electron Fermi sea is at rest. According to Eq. (22), we can write the self-energy in ζ where we introduced 0. From the above, it is clear that the the absence of a drift as renormalized mass of the polaron can be written as Z k ð0Þ d ð0Þ Σ ðω; pÞ¼ n ðξkÞT ðω þ ξk; k þ pÞ: ðB2Þ int ð2πÞ2 F 1 1 2 ¼ Z þ ∂pReΣ ðp; 0Þjp p : ðA14Þ int ¼ F mx mx The T matrix in Eq. (23) can be expressed as

Z q n ξq − 1 −1 ω p −1 d Fð Þ T ð ; Þ¼V þ 2 ω − ξ − ζ 2τ − Σ ω − ξ p − q ; ðB3Þ ð2πÞ q p−q þ iZ= x iZIm intð q; Þ where we have substituted the exciton Green’s function by Eq. (B1). This method is justified because, in the domain of integration, the energy argument of Gx always remains smaller than ω, and hence Eq. (B1) is a good approximation. We can now simply introduce a drift velocity of the Fermi surface of electrons by shifting the distribution function ˜ nFðξkÞ → nFðξk−AÞ with A ¼ veme. We denote the self-energy with a shifted Fermi surface as Σ and find Z k ˜ ð0Þ d ˜ Σ ðω; p; AÞ¼ n ðξk−AÞTðω þ ξk; k þ p; AÞ: ðB4Þ int ð2πÞ2 F

The self-consistent T matrix needs to be changed accordingly, Z dq n ðξq−AÞ − 1 T˜ −1ðω; p; AÞ¼V−1 þ F ; ðB5Þ 2π 2 ω − ξ − ζ˜ 2τ − Σ˜ ω − ξ p − q A ð Þ q p−q þ iZ= x iZIm intð q; ; Þ ˜ where we have defined the polaron dispersion in the presence of a Fermi sea ζp that we seek to obtain. Shifting the variable of integration, the self-energy reads Z k ˜ ð0Þ d ˜ Σ ðω; p; AÞ¼ n ðξkÞTðω þ ξk A; k þ A þ p; AÞ; ðB6Þ int ð2πÞ2 F þ with

˜ −1 T ðω þ ξk A; k þ A þ p; AÞ þZ dq n ðξqÞ − 1 ¼ V−1 þ F : ðB7Þ 2π 2 ω ξ − ξ − ζ˜ 2τ − Σ˜ ω ξ − ξ k p − q A ð Þ þ kþA qþA kþp−q þ iZ= x iZIm intð þ kþA qþA; þ ; Þ

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We make the following general ansatz for the new polaron δp ¼ veðmx − mxÞ; ðB16Þ dispersion: 2 veðmx − mxÞ 2 δE ¼ − ; ðB17Þ ˜ ðp þ δpÞ 2 ζp ¼ þ δE; ðB8Þ 2m x and hence the polaron dispersion when the electrons are where δp and δE are constants to be determined. It is drifting at a constant velocity reads straightforward to rewrite the real part of the denominator (p v − )2 2 − of Eq. (B7) as ζ˜ þ eðmx mxÞ − veðmx mxÞ p ¼ : ðB18Þ 2mx 2 ω ξ − ξ − ζ˜ ω0 ξ − ξ − ζ þ kþA qþA kþp−q ¼ þ k q kþp0−q; ðB9Þ APPENDIX C: POLARON TRANSPORT USING where we have introduced A VARIATIONAL APPROACH A complementary way to show the emergence of this 0 p ¼ p þ δp − Amx=me; ðB10Þ force is using a variational approach. We start from the following Hamiltonian, which incorporates the interaction p δp A A2m between excitons and electrons in the presence of an ω0 ω − ð þ Þ x − δ ¼ þ 2 E: ðB11Þ electric field. We use the Coulomb gauge, such that the m 2m e e effect of the electric field is to shift the electron dispersion and therefore preserve translational invariance. We obtain Notice that the right-hand side of Eq. (B9) involves the bare X X polaron dispersion at zero electron drift velocity. Using this † † HðtÞ¼ ξk v ckck þ ωkxkxk relation, we can express the T matrix in the presence of an þme eðtÞ k k electron drift by the bare T matrix at shifted energy and ˜ X V momentum arguments, Tðω þ ξk A; k þ A þ p; AÞ¼ † † þ þ xk qc 0 ck0 xk; ðC1Þ ω0 ξ k p0 A þ k −q Tð þ k; þ Þ. Similarly, the self-energy can be ex- k;k0;q pressed as where c† is the electron creation operator, while x† denotes Σ˜ ð0Þ ω p A Σð0Þ ω0 p0 ξ int ð ; ; Þ¼ int ð ; Þ: ðB12Þ the creation of an excitonic impurity. Furthermore, k ¼ 2 ½k =ð2meÞ − μe is the electron dispersion, while ωk ¼ 2 Here, we have used the relation ½k =ð2mxÞ − μx denotes the impurity dispersion. Notice that we investigate only the case of vanishing exciton Σ˜ ω ξ − ξ k p − q A μ 0 Im intð þ kþA qþA; þ ; Þ density, i.e., x

Substituting A ¼ meve, we find, after some straightforward To obtain the ground-state energy, we have to minimize manipulations, hΨpjH − EjΨpi, where the energy E is the Lagrange

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(p − v )2 multiplier ensuring the normalization of the wave function. þðmx mxÞ eðtÞ ζp t − μ : C6 The minimization will yield the dispersion of the polarons ð Þ¼ x ð Þ 2mx 2 ζpðtÞ¼ζ0 þ p =ð2mxÞ (see Refs. [4,49] for details regard- ing the minimization procedure; the mass mx might not be Assuming that the electric field is small enough, the the same as the mx obtained self-consistently since this is a evolution is adiabatic, and we can focus only on the non-self-consistent derivation). attractive-polaron states and ignore all the other higher Having solved the problem in the absence of the electric lying states; therefore, we can write down an effective field, we now find a mapping from the instantaneous Hamiltonian in terms of attractive polarons: eigenstates of the Hamiltonian in the presence of an electric X p − v 2 field to the states in the absence of any field. To show this ½ þðmx mxÞ eðtÞ † H t − μ apap; C7 ð Þ¼ 2 x ð Þ mapping we first go to a frame that is comoving with the p mx electrons with the unitaryR UðtÞ¼eiSðtÞ, where SðtÞ¼ r pˆ r t 0v 0 † eðtÞ T, with eðtÞ¼ 0 dt eðt Þ. Since UðtÞxkU ðtÞ¼ which is similar to the Hamiltonian introduced in Sec. III F. −ikr ðtÞ xke e , the conservation of total conjugate momentum † implies that UðtÞHU ðtÞ¼H, so the only contribution to APPENDIX D: POLARON LIFETIME the Hamiltonian comes from the time dependence of the transformation −iUðtÞf½∂U†ðtÞ=ð∂tÞg ¼ f½∂SðtÞ=ð∂tÞg ¼ In this Appendix, we investigate the residual interac- ˆ tions between polarons and the Fermi sea and calculate ðe=meÞAðtÞpT. The new Hamiltonian becomes X X the corresponding scattering rates associated with these † † processes. We first calculate the scattering lifetime using HðtÞ¼ ξkckck þ ωk− v xkxk mx eðtÞ ’ k k Fermi s golden rule since it is more transparent, but then we X show that the same result can be obtained by explicitly V † † Σ p ξ þ x e 0 ek0 xk; ðC3Þ evaluating Im ð ; pÞ. In the following, we are interested A kþq k −q k;k0;q in the case of only one polaron in the system with momentum jpj ≪ kF, and therefore we choose μx < 0. where now the exciton dispersion is shifted by an − v amount mx eðtÞ. Assuming that the electric field is small 1. Interaction between polarons and electrons enough, the system will remain in the many-body ground The interaction between this polaron and an electron of state, according to the adiabatic theorem. To determine k the instantaneous ground state of HðtÞ, we have to momentum is given by minimize hΨpðtÞjHðtÞ − EjΨpðtÞi, where the Chevy ansatz Uðp; kÞ¼ZTðp þ k; ζp þ ξkÞ; ðD1Þ is given by

† where Z is the quasiparticle weight of the polaron. Since, jΨpðtÞi ¼ apj0i by assumption, the polaron has a small energy ζp ≪ e , the X F † † † polaron will only be able to scatter electrons in a thin shell ¼ ϕpðtÞxp þ ϕp k qðtÞx c cq j0i: ; ; pþq−k k around the Fermi surface. Therefore, for our purposes, we k;q have k ≃ kF, and thus, ðC4Þ

U ≃ ZTðkF; 0Þ: ðD2Þ One can check that, up to some irrelevant constants, Ψ Ψ Ψ Ψ h pðtÞjHðtÞj pðtÞi¼h p−mxveðtÞjHj p−mxveðtÞi (one way In the above, T denotes the self-consistent T matrix. We to see this is to explicitly expand these terms and compare know that the T matrix has a a simple pole at the trion them), which illustrates the mapping to the states in the resonance ωTðkÞ. Therefore, for energies in the vicinity of absence of an electric field. From the above, we immedi- the trion resonance, we can approximate the T matrix by ately see that k ω ≃ C p − v )2 Tð ; Þ ; ðD3Þ ˜ ð mx eðtÞ ω − ωTðkÞþiγT ζp t ζp− v − μ ; C5 ð Þ¼ mx eðtÞ ¼ x ð Þ 2mx where ωTðkÞ is the trion energy measured from the exciton where we used a tilde to denote the dispersion in the frame chemical potential μx, and γT denotes the lifetime of the co-moving with the electrons. We remark that polarons trion state. acquire a backward velocity −mx=mxveðtÞ, which is Since it is not immediately obvious how to calculate C smaller than the electron velocity due to the mass renorm- for the self-consistent T matrix, we proceed to estimate it. alization. Moving back to the lab frame, the dispersion Since the self-consistent T matrix cannot be too much becomes different than the non-self-consistent T matrix T0 (which

041019-25 OVIDIU COTLEŢ et al. PHYS. REV. X 9, 041019 (2019) Z Z we know how to calculate exactly), we use the analytical p 2π 2pq cos ϕ − q2 Γ ≃ 4ρ U2 dqq dϕIm2ρ expression of T0 to obtain an order-of-magnitude estima- e e 0 0 2mxμe tion for T. Considering, for simplicity, the case mx ¼ me, k we know that [23] × F θð2pq cos ϕ − q2ÞðD10Þ jqj 2 3 Z T0ð0; ωÞ¼ ϵ p k 1 ρ ðlnð T ÞþiπÞ 4ρ 2 F e ωþμx ¼ eU dq 2mμ 0 2 Z x e 2ϵT 1 ω þ ϵT þ μx 2π ¼ þ O ; 2 2 ρ ω þ ϵ þ μ ϵ × dϕð2q cos ϕ − q Þθð2q cos ϕ − q Þ: ðD11Þ e T x T 0 ðD4Þ The integral is easypffiffiffiffiffiffiffiffiffiffiffiffi to evaluate, and it is similar to or equal ≃ 2ϵ ρ which shows that C T= e. Denoting the energy differ- to 1. Since Z ≃ μe=ϵT, we finally obtain ence between the trion of momentum kF and the attractive polaron of momentum p with Δ, we can approximate the ϵ 3 Γ ∼ T p kF interaction U as 2 : ðD12Þ Δ memx 2Zϵ U ≃− T : ðD5Þ ρeΔ 3. Polaron lifetime from self-energy The same expression can be derived by explicitly evalu- ’ ating the imaginary part of the self-energy. To evaluate the 2. Polaron lifetime using Fermi s golden rule imaginary part, one can use the optical theorem. In our case, According to Fermi’s golden rule, the scattering rate it is just as simple to directly evaluate the imaginary part of from a state of momentum p is given by the self-energy. The polaron broadening due to interaction X with the Fermi sea is related to the self-energy by Γ 2π 2 δ ζ − ζ − ξ ξ ¼ U ð p p0 kþp−p0 þ kÞ p0 p k j j

041019-26 TRANSPORT OF NEUTRAL OPTICAL EXCITATIONS USING … PHYS. REV. X 9, 041019 (2019) Z 2k 2k0 energy −ω and momentum −p, which can scatter into Σ p ω d d p k ω ξ 2 Im ð ; Þ¼ 2 2 Tð þ ; þ kÞ p0 0 ≥ ζ ≥ ω ð2πÞ ð2πÞ states with momenta obeying the condition p0 0 by creating electron-hole pairs. × (1 − n ξ )n ξk Fð kÞ Fð Þ Using similar arguments as in Appendix D2, the A 0 ’ ×ImGx ðp þ k − k ; ω þ ξk − ξk0 ÞðD18Þ scattering rate can be calculated using Fermi s golden Z rule: 2k 2k0 d d 2 0 X T p k; ω ξk (1 − n ξ ) ¼ 2 2 ð þ þ Þ Fð kÞ 2 0 ð2πÞ ð2πÞ Γðp; ωÞ¼2U Imχðp − p ; ω − ζp0 Þθðζp0 − ωÞ p0 p0 p ξ π δ ω − ζ ξ − ξ ;j j

2 X ζ − ζ 4U ρekF p p−q Γðp; ζpÞ ≃ θðζp−q − ζpÞðD24Þ μe jqj q;jp−qj

0 In the above, one of the theta functions ensures that ζp0 ≥ ω, while the other theta function ensures that jp j

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The derivative of Γ can be evaluated similarly using Eq. (D9): 4 2ρ X ω − ζ ∂ Γ p ω U ekF ∂ p−q θ ζ − ω ω ð ; Þω¼ζp ¼ ω ð p−q Þ ðD28Þ μ q;jp−qj

Similarly to the evaluation of Γ, we can evaluate the above by turning the sum into integrals. We immediately see that the last term in the brackets vanishes, and we are left with 2ρ ∂ Γ p ω U ekF ω ð ; Þω¼ζp ¼ 2 pFI1ðp=pFÞ; ðD30Þ μeπ where we introduced the integral of order 1: Z Z 2 2π 2 2 2 I1ðuÞ ≡ dq dϕθðq − 2qu cos ϕÞθð1 þ 2uq cos ϕ − u − q Þ; ðD31Þ 0 0

APPENDIX E: RELATION BETWEEN ImΣint AND ImGx In Sec. III E of the main text, we encounter an expression of the type Z R Σ θ −ω dpA½Gx ðpÞIm intðpÞ ð Þ; ðE1Þ

R where A is a functional of Gx . In this Appendix, we show that this expression can alternatively be written as Z Z I ≡ R Σ θ −ω R A ð0Þ θ −ϵ dpA½Gx ðpÞIm intðpÞ ð Þ¼ dpdkA½Gx ðpÞImGx ðkÞw ðk; pÞ ð Þ Z 0 R A 0 0 R 0 2 ¼ dpdkdk A½Gx ðpÞImGx ðkÞGeðk ÞGeðp þ k − kÞ½T ðp þ k Þ θð−ϵÞ; ðE2Þ

R where A is a functional of Gx . We first derive an expression for the imaginary part of the self-energy. Using GeðkÞ¼ R Ge ðkÞþ2iImGeðkÞθð−ϵÞ and a similar relation for the T matrix, we can write the self-energy as Z Σ θ −ω − θ −ω R R R 2 A θ −ω − ϵ intðpÞ ð Þ¼ i ð Þ dk½Ge ðkÞT ðk þ pÞþGe ðkÞ iImT ðk þ pÞ ð Þ

A R A A þ 2iImGe ðkÞθð−ϵÞT ðk þ pÞ − 4ImGe ðkÞImT ðk þ pÞθð−ω − ϵÞθð−ϵÞ ðE3Þ Z A A ¼ 2θð−ωÞ dkfReGeðkÞImT ðk þ pÞθð−ω − ϵÞþImGe ðkÞReTðk þ pÞθð−ϵÞ

A A þ iImGe ðkÞImT ðk þ pÞ½θð−ω − ϵÞ − θð−ϵÞg: ðE4Þ

The imaginary part of the T matrix is Z 2 0 A 0 A 0 0 0 ImTðk þ pÞ¼2jTðk þ pÞj dk ImGe ðk ÞImGx ðk þ p − k Þθð−ϵ Þθðϵ − ω − ϵÞ: ðE5Þ

With this result, we can express the imaginary part of the self-energy as Z Σ θ −ω 4θ −ω 0 2 A A 0 A − 0 Im intðpÞ ð Þ¼ ð Þ dkdk jTðk þ pÞj ImGe ðkÞImGe ðk ÞImGx ðp þ k k ÞðE6Þ

× θð−ϵ0Þ½θð−ω − ϵÞ − θð−ϵÞθðϵ0 − ϵ − ωÞ: ðE7Þ

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0 R 0 A 0 0 To prove Eq (E2), we make the replacement Geðpþk −kÞ¼Ge ðpþk −kÞþ2iImGe ðpþk −kÞθðϵ−ω−ϵ Þ. The first term is canceled by the integral over ω since all poles are located in the same complex half-plane; hence, after shifting k0 → k0 − p, we find Z 0 R A 0 A 0 R 0 2 0 I ¼ 2i dpdkdk A½Gx ðpÞImGx ðkÞGeðk − pÞImGe ðk − kÞ½T ðk Þ θð−ϵÞθðϵ − ϵ Þ: ðE8Þ

0 A 0 The shift of variables conveniently allows us to repeat the same trick writing Geðk − pÞ¼Ge ðk − pÞ− A 0 0 2iImGe ðk − pÞθðϵ − ωÞ. The first term is again canceled by the ω integration, and we find, after returning to the original integration variables, Z 0 R A A 0 A 0 R 0 2 0 0 I ¼ 4 dpdkdk A½Gx ðpÞImGx ðkÞImGe ðk ÞImGe ðp þ k − kÞ½T ðk þ pÞ θð−ϵÞθðϵ Þθðϵ − ϵ − ωÞ: ðE9Þ

By comparison with Eq. (E7) and using the simple identity

θð−ϵÞθðϵ0Þθðϵ − ϵ0 − ωÞ¼θð−ϵ0Þ½θð−ω − ϵÞ − θð−ϵÞθðϵ0 − ϵ − ωÞθð−ωÞ; ðE10Þ we arrive at Z Z R Σð0Þ θ −ω R A ð0Þ θ −ϵ dpA½Gx ðpÞIm int ðpÞ ð Þ¼ dpdkA½Gx ðpÞImGx ðkÞw ðk; pÞ ð Þ: ðE11Þ

1. Exciton density R We now show that the integral −i dpGxðpÞ over the exciton Green’s function indeed yields the correct polaron density Z X X X X dω ≡ † p † 0þ − p ω iω0þ nx h xpðtÞxpðtÞi ¼ nxð Þ¼ hT½xpðtÞxpðt þ Þi ¼ i 2π Gxð ; Þe ; ðE12Þ p p p p where the infinitesimal ω0þ originates from time ordering and ensures that the contour is closed in the upper-half plane. Using the decomposition GðpÞ¼GRðpÞþ2iImGAðpÞθð−ωÞ, we find Z Z Z Z 1 − 2 A θ −ω R 2θ −ω 2 Σð0Þ θ −ω R 2 i dpGxðpÞ¼ dpImGx ðpÞ ð Þ¼ dp jGx ðpÞj ð Þþ dpIm int ðpÞ ð Þ½Gx ðpÞ ; ðE13Þ τx

where we have expanded the second term to lowest order in nx.As1=τx is a small parameter, the first term is dominated by on-shell contributions ω ≃ ζp, and we can write

1 1=2τ GR p 2θ −ω ≃ 2πZ x δ ω − ζ θ −ω ≃ 2πZθ −ω δ ω − ζ ; E14 τ j x ð Þj ð Þ 1 2τ Σ ð pÞ ð Þ ð Þ ð pÞ ð Þ x = x þ Im intðpÞ

Σ ≪ 1 τ ζ ≃ ω ≤ 0 where the last equality follows from Im intðpÞ = x for on-shell energies p consistent with the assumptions made in Sec. III E. The second term can be replaced by Eq. (E11), which yields Z Z A ð0Þ R 2 −i dpGxðpÞ¼Znx þ 2 dpImGx ðkÞθð−ϵÞW ðk; pÞ½Gx ðpÞ ; ðE15Þ R ð0Þ R 2 where the density is nx ¼ μxνx.Wehave dpW ðk; pÞ½Gx ðpÞ ¼ −Z∂ωΣðpÞ¼1 − Z, and hence, Z

−i dpGxðpÞ¼nx: ðE16Þ

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APPENDIX F: PROOF OF EQ. (56) IN MAIN TEXT We now demonstrate Eq. (56) of the main text, which reads

k k ðiτ Ω ∂ϵ þ ∂kÞG ðkÞ e me e GeðkÞGeðk þ ΩÞ ≃ ðF1Þ me 1 − iτeΩ

−1 R to leading order in Ω and 1=τe, where Ge ¼½ϵ − ξk þ ið1=2τeÞsgnϵ . We start by writing GeðkÞ¼Ge ðkÞþ A 2iImGe ðkÞθð−ϵÞ, and we obtain

R R R A GeðkÞGeðk þ ΩÞ¼Ge ðkÞGe ðk þ ΩÞþReGe ðkÞ2iImGe ðk þ ΩÞθð−ϵ − ΩÞ R A A A þ ReGe ðk þ ΩÞ2iImGe ðkÞθð−ϵÞþ2ImGe ðkÞImGe ðk þ ΩÞ½θð−ϵÞ − θð−ϵ − ΩÞ: ðF2Þ

We are interested in a result to zeroth order in Ω and 1=τe. At this order, all but the first term vanish everywhere except when ϵ ¼ ξk. Our strategy is to expand these expressions in terms of δ functions around the resonance; for instance, we write

R A 0 00 ReGe ðkÞImGe ðk þ ΩÞ¼α0δðϵ − ξkÞþα1δ ðϵ − ξkÞþα2δ ðϵ − ξkÞþ… ðF3Þ

We can obtain the coefficients from the following integrals: Z −πΩ α ϵ R A Ω 0 ¼ d ReGe ðkÞImGe ðk þ Þ¼ −2 2 ; ðF4Þ τe þ Ω Z π 2 2τ −2 Ω2 α − ϵ ϵ − ξ R A Ω − ½ ð eÞ þ 1 ¼ d ð kÞReGe ðkÞImGe ðk þ Þ¼ −2 2 ; ðF5Þ τe þ Ω Z πΩ 3 2τ −2 Ω2 α ϵ ϵ − ξ 2 R A Ω − ½ ð eÞ þ 2 ¼ d ð kÞ ReGe ðkÞImGe ðk þ Þ¼ −2 2 : ðF6Þ τe þ Ω

The second order is already linear in the small parameters and can hence be ignored. The expansion then reads π 1 R A Ω ≃ −Ωδ ϵ − ξ − Ω2 δ0 ϵ − ξ ReGe ðkÞImGe ðk þ Þ −2 2 ð kÞ 2 þ ð kÞ : ðF7Þ τe þ Ω 2τe

Similar considerations yield π 1 R Ω A ≃ Ωδ ϵ − ξ − δ0 ϵ − ξ ReGe ðk þ ÞImGe ðkÞ −2 2 ð kÞ 2 ð kÞ ; ðF8Þ τe þ Ω 2τ π 1 Ω A A Ω ≃ δ ϵ − ξ δ0 ϵ − ξ ImGe ðkÞImGe ðk þ Þ −2 2 τ ð kÞþ2τ ð kÞ : ðF9Þ τe þ Ω e e

Substituting these relations in Eq. (F2), we find 2πΩτ R 2 0 e GeðkÞGeðk þ ΩÞ ≃ Ge ðkÞ − 2iπδ ðϵ − ξkÞθð−ϵÞþ δðϵ − ξkÞδðϵÞ; ðF10Þ 1 − iΩτe where we have expanded the step function θð−ϵ − ΩÞ ≃ θð−ϵÞ − ΩδðϵÞ. The right-hand side of Eq. (F1) can be evaluated R explicitly using GeðkÞ¼Ge ðkÞþ2πiδðϵ − ξkÞθð−ϵÞ. We obtain k k 2πτ Ω R 2 0 e iτeΩ ∂ϵ þ ∂k GeðkÞ¼ð1 − iτeΩÞ Ge ðkÞ − 2πiδ ðϵ − ξkÞθð−ϵÞþ δðϵ − ξkÞδðϵÞ ; ðF11Þ me me 1 − iΩτe which concludes the proof.

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