Theory of polariton-electron interactions in semiconductor microcavities

Let us first discuss how the formation of an exciton is described within the contact interaction model (II). To this end, we consider the most general state of an electron-hole pair with zero center-of-mass momentum:

with the normalization condition $\bra{\Phi}\ket{\Phi}=\sum_{\mathbf{k}}\absolutevalue{\phi_{\mathbf{k}}}^{2}=1.$ Here, $\ket{0}$ is the electron-hole vacuum. In the absence of coupling to light, i.e., at $g=0$, the wave function $\phi_{\mathbf{k}}$ satisfies the Schrödinger equation

where we define the electron-hole kinetic energy $\bar{\epsilon}_{\mathbf{k}}\equiv\epsilon^{e}_{\mathbf{k}}+\epsilon^{h}_{\mathbf{k}}=k^{2}/2m_{r}$ with $m_{r}=(1/m_{e}+1/m_{h})^{-1}$ the electron-hole reduced mass.

Equation (3) admits a single bound state with binding energy $\varepsilon_{B}$, corresponding to the $1s$ exciton state (see, e.g., Ref. [28]), which has the associated effective Bohr radius $a_{X}\equiv 1/\sqrt{2m_{r}\varepsilon_{B}}$. The corresponding wave function $\phi_{X\mathbf{k}}$ satisfies the equation

Importantly, the right hand side of Eq. (4) does not depend on momentum, and therefore we immediately find that the $1s$ exciton has the wave function

where $Z_{X}$ comes from the normalization condition

Equation (4) also allows us to relate the bare coupling constant, $V_{0}$, to $\varepsilon_{B}$: Acting with the operator $\sum_{\mathbf{k}}\frac{1}{\varepsilon_{B}+\bar{\epsilon}_{\mathbf{k}}}(\cdot)$ on Eq. (4), we find

Here, the sum on $\mathbf{k}$ is logarithmically divergent, and we have therefore explicitly introduced the ultraviolet momentum cutoff $\Lambda$. Once the coupling constant and $\Lambda$ are related to the exciton binding energy via Eq. (7), all dependence on these bare parameters is eliminated from the problem [28].

It is instructive to compare the operator formalism described above for the exciton problem with the electron-hole $T$ matrix [28]:

This is discussed further below in Sec. II.3. In this formalism, the bound state emerges as a pole at the negative energy $E=-\varepsilon_{B}$, where the presence of the infinitesimal positive imaginary part $+i0$ shifts the pole slightly into the lower half of the complex plane. In the vicinity of the pole, we expand the $T$ matrix to find $\mathcal{T}_{0}(E)\simeq Z_{X}/(E+\varepsilon_{B}+i0)$. Thus, the normalization $Z_{X}$ naturally emerges in both approaches.

Let us now discuss how to obtain the polariton spectrum within the model (II), and how to relate this to experimentally measurable quantities. As we will show, the Hamiltonian (II) is analytically solvable in the case of the single-polariton problem. In the following, we provide a derivation of the exact wave function for a polariton at normal incidence (i.e., zero momentum), but our results can be straightforwardly extended to describe polaritons at finite momentum — see Appendix A.

To proceed, we write the most general electron-hole-photon wave function as:

with the normalization condition:

The Schrödinger equation can be obtained by projecting $(E-\hat{H})\ket{\Psi}=0$ onto the electron-hole and photon parts of Eq. (9), which gives:

In the following, we assume that the exciton binding energy is larger than other relevant energy scales, such as the photon-exciton detuning and the light-matter Rabi coupling. This is a good approximation in the TMDs and in a single GaAs quantum well. As we show, this condition need not be strictly satisfied, and therefore our results also apply to structures containing multiple GaAs quantum wells.

Before delving into the technical details, we will briefly summarize our main conclusions. We emphasize that the results presented here act as a starting point for both few-body physics, as discussed in this paper, and many-body physics within the model (II). As we show in the following, the exact polariton energies $E_{\pm}$ satisfy the transcendental equation

where

is the effective exciton-photon Rabi coupling. The subscripts $-$ and $+$ indicate the lower and upper polaritons, respectively, and we assume that $E_{\pm}<0$ ¹¹1If $\Omega\gtrsim\varepsilon_{B}$ and/or $\delta\gtrsim\varepsilon_{B}$, the upper polariton enters the continuum of unbound electron-hole states. In that case, one must analytically continue the energy slightly into the complex plane, $E\to E+i0$. Like in the exciton problem, the right hand side of Eq. (11a) is independent of momentum, and hence the electron-hole wave function has the same functional form as the exciton wave function $\phi_{X\mathbf{k}}$ in Eq. (5) ²²2The variational approach developed by Khurgin [76] approximated the functional form of the electron-hole wave function as unchanged in the presence of light-matter coupling. While this is an approximation in the Coulomb case studied in that work [24], we see that it is exact for strongly screened electron-hole interactions.:

Here the numerator follows from the normalization condition in Eq. (10), and

is the generalization of the exciton normalization $Z_{X}$ in Eq. (6). The corresponding photon and exciton Hopfield coefficients $C_{\pm}$ and $X_{\pm}$ take the form

Finally, we note that the photon frequency relative to the exciton energy is shifted by a finite amount in the presence of the active medium, such that the effective photon-exciton detuning $\delta$ is

Figure 1 shows the polariton energies and the corresponding photon and exciton fractions according to Eqs. (12) and (16). We find that they agree extremely well with the results of treating the excitons and photons as two coupled oscillators [31], even when the Rabi coupling is comparable to the exciton binding energy as in structures containing multiple GaAs quantum wells. In particular, our scheme provides an improved agreement between the two models than that obtained in Ref. [23], which considered a similar electron-hole-photon model. Moreover, our scheme is technically simpler to implement since Eqs. (12)-(17) are fully analytic and do not require the introduction of an additional infrared cutoff. We now proceed to derive these results.

In this section, we provide an alternative description of exciton-polaritons using a diagrammatic formulation. While this is equivalent to the operator approach described above, the advantage of the diagrammatic formulation is that it allows a straightforward characterization of repeated scattering processes. As we shall see, this enables us to obtain the polariton propagator in closed form, and it provides a convenient starting point for further few-body studies such as the polariton-electron scattering discussed in Sections III and IV.

We start by considering the properties of a photon interacting with an undoped 2D semiconductor in a microcavity. The microcavity photon will be modified by repeated interactions with electron-hole pairs, which includes both bound excitons and unbound electron-hole pairs in the continuum. Referring to Fig. 2, the resulting “dressed” photon is characterized by the self-energy $\Sigma$ [33]. Following Ref. [24], we obtain the dressed photon propagator $D(\mathbf{Q},E)$ from the Dyson equation in Fig. 2(a):

where $E$ and $\mathbf{Q}$ are, respectively, the energy and momentum of the photon. In the absence of the active medium, we have the bare photon propagator

where the imaginary infinitesimal $+i0$ shifts the poles into the lower half of the complex plane and ensures that the photon propagates forward in time. Note that we can also incorporate the decay of photons from the microcavity by setting the imaginary part equal to the decay rate. However, we neglect this decay rate since it is much smaller than the Rabi coupling in the regime of strong light-matter coupling, and it therefore only has a negligible effect on the scattering. In other words, we consider a closed quantum system. In the following, we will always implicitly assume that the energy carries an infinitesimal positive imaginary part, that is, we will be working with retarded Green’s functions.

As shown in Fig. 2(b), the photon self-energy consists of two terms: $\Sigma(\mathbf{Q},E)=\Sigma^{(1)}(E-\epsilon^{X}_{\mathbf{Q}})+\Sigma^{(2)}(E-\epsilon^{X}_{\mathbf{Q}})$, with $\epsilon^{X}_{\mathbf{Q}}=Q^{2}/2m_{X}$ the exciton kinetic energy and $m_{X}=m_{e}+m_{h}$ the exciton mass. These contain all possible processes that involve the excitation of an electron-hole pair, and they thus only depend on the energy in the exciton center-of-mass frame. Hence, the $\mathbf{Q}$ dependence simply appears as a shift in the energy.

Within the model (II), we have

where $\Pi(E)$ is the polarization bubble diagram corresponding to an electron-hole pair. Since $\Pi(E)\sim\ln\Lambda$ and the bare coupling $g\sim 1/\ln\Lambda$, the term $\Sigma^{(1)}(E)$ vanishes when the momentum cutoff $\Lambda\to\infty$. On the other hand, we have $g^{2}\Pi^{2}(E)=\Omega^{2}/Z_{X}$ from Eq. (20) and the fact that $V_{0}\Pi(E)\to-1$ as $\Lambda\to\infty$.

The electron-hole $T$ matrix $\mathcal{T}_{0}(E)$ (or exciton propagator) is shown in Fig. 2(c), and corresponds to

We can renormalize the bare $V_{0}$ using Eq. (7) to finally obtain the standard expression in Eq. (8), see Appendix B.

Combining all of these expressions yields the explicit form for the dressed photon propagator:

where $\omega$ can be related to the physical photon-exciton detuning $\delta$ via Eq. (17). To simplify the notation, we have not explicitly written the imaginary infinitesimal on the energy. The poles of the photon propagator contain information about the entire polariton spectrum. Specifically, the denominator of Eq. (33) is zero when we have

Indeed, for $\mathbf{Q}=0$, this exactly corresponds to Eq. (25) from the operator approach. For general $\mathbf{Q}$, we denote the polariton energy solutions of Eq. (34) as $E_{\pm}(\mathbf{Q})$.

In addition to the dressed photon propagator, we have what we call the “polariton propagator” $\mathcal{T}(\mathbf{Q},E)$ shown in Fig. 3, which instead focuses on the behavior of the matter component of the polariton. Like the dressed photon, it contains all the possible scattering processes involving an electron-hole pair or photon. In the absence of light-matter coupling, it reduces to $\mathcal{T}_{0}(E-\epsilon^{X}_{\mathbf{Q}})$, the electron-hole $T$ matrix at center-of-mass momentum $\mathbf{Q}$. Suppressing the dependence on energy and momentum, Fig. 3 reads:

Here, we again use $g\sim 1/\ln\Lambda$ and $\Pi\sim\ln\Lambda$ to remove terms that vanish in the limit $\Lambda\to\infty$, while in the last line, we use the relation $D=(D_{0}^{-1}-g^{2}\Pi^{2}\mathcal{T}_{0})^{-1}$. Thus, the polariton propagator finally reads:

It is also straightforward to show that the polariton and photon propagators are related via

The polariton propagator has precisely the same poles as the dressed photon propagator in Eq. (33). However, the poles of the photon propagator have residues corresponding to the photon fractions of the polaritons:

while the residues for the poles of the polariton propagator instead correspond to the matter component,

with the electron-hole normalization factors

At $\mathbf{Q}=0$, the Hopfield coefficients and the normalization factors reduce to the corresponding expressions in Eqs. (16) and (15).

The analytic expression for the polariton wave function obtained in the previous section (see also the finite-momentum generalization in Appendix A) allows us to straightforwardly evaluate the Born approximation for polariton-electron scattering. For simplicity, we consider the frame where the center-of-mass momentum is zero, since in that case the different partial wave components of the scattering separate. Since the electron has a relatively flat dispersion, our results are likely to be insensitive to the actual electron momentum and will thus be dominated by the polariton momentum.

The Born approximation consists in approximating the scattering $T$ matrix by the first term in the Born series, which corresponds to evaluating the matrix element of the polariton-electron interaction potential. Taking the incoming (outgoing) polariton and electron to have momenta $\pm\mathbf{p}_{1}$ ($\pm\mathbf{p}_{2}$), this matrix element takes the form

where the operator $P^{\dagger}_{\mathbf{p}}$ creates a polariton with momentum $\mathbf{p}$, see Eq. (58) in Appendix A. We assume energy and momentum conservation such that the scattering is elastic, i.e., we take $|\mathbf{p}_{1}|=|\mathbf{p}_{2}|=p$. Here, the subtraction of the energy of the non-interacting particles from the Hamiltonian, Eq. (II), ensures that we only evaluate the polariton-electron interaction potential. Using the exact polariton wave function from Appendix A, we find

where the sign in the middle line arises from the exchange of identical fermions. As anticipated, we see that the exciton fraction $|X_{-}(\mathbf{p}_{1})|^{2}$ naturally emerges from the normalization of the polariton wave function.

In Fig. 4 we illustrate the Born approximation for elastic polariton-electron scattering. The exciton fraction quickly increases at low momentum, which dominates the behavior at momenta up to the polariton dispersion inflection point above which the exciton fraction is very close to unity. Apart from the Hopfield coefficient $|X_{-}(\mathbf{p}_{1})|^{2}$, there is also a momentum and angular dependence through the kinetic energies and the wave function normalization; however we find that this dependence is generally quite weak. Above the inflection point, it leads to a slow decline for increasing momentum.

Specializing to zero momentum, we may define a low-energy polariton-electron interaction constant $g_{eP}^{B}$, which takes the form

This is precisely the exciton fraction multiplied by the exciton-electron interaction, $g_{eX}^{B}=4\pi\varepsilon_{B}a_{X}^{2}=2\pi/m_{r}$. Note that this is quite close to the corresponding result obtained in the case of Coulomb electronic interactions [37]: Using the formalism developed in Ref. [24], we find $g_{eX}^{B,\mathrm{Coulomb}}=(2\pi/m_{r})(2-3\pi/8)\simeq 0.8g_{eX}^{B}$. We emphasize that the polariton-electron Born approximation in Eq. (44) only depends on the light-matter coupling through the exciton fraction, while it is independent of the absolute strength of the light-matter coupling, $\Omega/\varepsilon_{B}$.

The polariton-electron scattering process can also be calculated diagrammatically, as shown in Fig. 5. In this case, the Born approximation corresponds to the first diagram on the right hand side, which involves the exchange of a hole with momentum $\mathbf{p}_{1}+\mathbf{p}_{2}$ [38]. This gives

where the minus sign is a consequence of fermionic statistics and

is the free hole Green’s function. Each external polariton line in the diagram contributes the square root of the residue of the propagator $\cal{T}$ at the pole, and the normalization of the incoming and outgoing lines is the same since $|\mathbf{p}_{1}|=|\mathbf{p}_{2}|$. The residue is precisely $\operatorname{Res}\left[\mathcal{T}(\mathbf{p},E_{-}(\mathbf{p})\right]=Z_{-}(\mathbf{p})|X_{-}(\mathbf{p})|^{2}$ — see Eq. (39). Using the definition of the hole propagator, and comparing Eq. (III.1) with Eq. (III.2) we thus see that the Born approximation within the two approaches coincides, as it should.

We now argue that the Born approximation is conceptually important since it provides an upper bound on the polariton-electron interaction constant $g_{eP}$. We start by noting that the ground-state energy for an electron and a lower polariton in an area $A$ can be written as

where $\ket{\Psi_{eP}}$ is the exact interacting polariton-electron state. Here we have assumed that there are no lower-energy polariton-electron bound states so that $E_{eP}$ consists of the non-interacting kinetic energy plus a two-body interaction term. We momentarily keep the area $A$ explicit so that we can keep track of powers of $a_{X}^{2}/A$. In particular, note that $\innerproduct{\Psi_{eP}}{\Psi_{eP}}=1+O(a_{X}^{2}/A)$, where the last term arises from the composite nature of the polariton.

Rearranging Eq. (47) and keeping only terms up to order $a_{X}^{2}/A$ then gives

where the second line follows from the fact that the non-interacting state $P_{\mathbf{0}}^{\dagger}e_{-\mathbf{0}}^{\dagger}\ket{0}$ can be viewed as a variational approximation to the exact interacting state $\ket{\Psi_{eP}}$. Thus, we finally arrive at

Hence, the Born approximation serves as a strict upper bound on the interaction energy shift. This observation is quite general and is independent of the details of the underlying interactions, e.g., it also applies to polariton-polariton scattering and to more realistic interactions between charged particles. The only requirement is that there are no lower energy bound states (such as trions) and that the Hamiltonian faithfully reproduces the low-energy physics. Therefore, any diverging interaction strength obtained in the absence of a bound state must be an artefact of the approximation [35].

It is of particular interest to consider the scattering of polaritons at vanishing momentum due to the realization of polariton Bose-Einstein condensation [26]. First, it is important to clarify precisely what we mean by this limit. It is well-known [25, 28] that the scattering $T$ matrix of 2D quantum particles with short range interactions (such as the exciton-electron interaction between a charge and an induced dipole) must approach zero as the collision energy vanishes. However, in the case of polariton-electron scattering, this only becomes relevant for exceedingly small momenta $p$ below those where the typical scale of exciton-electron scattering $\sim 1/m_{e}$ is comparable to $1/[m_{c}\ln(\varepsilon_{B}/\epsilon_{\mathbf{p}}^{c})]$ [35]. Because of the very small photon mass, this momentum scale corresponds to a length scale much larger than the size of the universe, and this effect can therefore be discarded. Thus, we can safely take the limit $p\to 0$ in our numerical solutions, since we will never encounter such an extreme scale. We emphasize that this is a dramatic consequence of the lack of Galilean invariance in the polariton system due to the strong coupling of excitons and photons.

For scattering at small momenta, we set $\mathbf{p}_{2}=0$ and $\mathbf{p}_{1}=\mathbf{p}$ in Eq. (50) and perform the integration over the angle between $\mathbf{p}$ and $\mathbf{q}$ assuming $E<0$. The remaining integral equation is then solved using Gauss-Legendre quadrature; for more details, see Appendix C. As a result, we obtain the exact polariton-electron interaction constant

where we use the same normalization as in Eq. (51). Note the similarities between this expression and that in Eq. (44) within the Born approximation.

Figure 6 shows a comparison of the polariton-electron interaction constant obtained from different methods for parameters relevant to both TMDs and GaAs quantum wells. The exact diagrammatic calculation, $g_{eP}$, can be used as a standard to gauge the correctness of other calculations. We see that generally the interaction strength increases as the polariton goes from being photonic at negative detuning to more excitonic at positive detuning, in agreement with the expectation from the Born approximation, Eq. (44). However, we find that the exact calculation features a peak at small positive detuning when the ratio $\Omega/\varepsilon_{B}$ is not too large, which is a qualitative feature that is missed by the Born approximation. Furthermore, the Born approximation generically overestimates the interaction strength, and this effect is most dramatic for relatively small values of the ratio $\Omega/\varepsilon_{B}$ such as in TMDs. For larger $\Omega/\varepsilon_{B}$, the exact result approaches $g_{eP}^{B}$ and thus we expect the Born approximation to be more accurate for GaAs quantum wells, which is consistent with recent measurements of the polariton-polariton interaction strength [57]. Note that $\Omega/\varepsilon_{B}\gtrsim 0.5$ applies to the case of multiple GaAs quantum wells and thus a full description requires us to take account of the non-trivial effects of multiple layers [35].

We can understand these results by noting that at momenta above the inflection point, the polariton propagator in Eq. (36) quickly approaches the exciton propagator, ${\cal T}_{0}$, in Eq. (8). Taking the limit ${\cal T}\to{\cal T}_{0}$, we observe that Eq. (50) has the exact same functional form as in exciton-electron scattering [47], the only difference being the change in collision energy due to the light-matter coupling. This motivates the interpretation of polariton-electron scattering as an off-shell exciton-electron scattering process, similarly to recent results on polariton-polariton interactions [35]. To make this explicit, we consider the universal form of the low-energy exciton-electron scattering $T$ matrix

where we define $m_{eX}=m_{e}m_{X}/(m_{e}+m_{X})$ as the reduced exciton-electron mass. Unlike the light-matter coupled system, the exciton-electron scattering only depends on the collision energy, since the exciton-electron system is Galilean invariant. Equation (53) is valid at collision energies $|E+\varepsilon_{B}|\ll\varepsilon_{eX}$, where the energy scale $\varepsilon_{eX}\sim\varepsilon_{B}$ depends on the precise details of the electronic interactions and on the electron-hole mass ratio ⁴⁴4In general, there are finite range corrections to Eq. (53), which add a term $\sim m_{eX}r_{e}^{2}(E+\varepsilon_{B})$ in the denominator [28]. In the absence of other scales, the effective range $r_{e}\sim a_{X}$ and therefore such terms are not important when $|E+\varepsilon_{B}|\ll\varepsilon_{eX}$.. We thus estimate the polariton-electron scattering in the limit of low momentum by the off-shell exciton-electron expression

which is valid when $|E_{-}+\varepsilon_{B}|\ll\varepsilon_{B}$.

Equation (54) yields the yellow dashed line in Fig. 6, which is seen to closely follow the exact polariton-electron scattering result for sufficiently low Rabi coupling, reproducing all qualitative features. In particular, it works exceptionally well in the case of TMD monolayers, which supports the approximation for polariton-polariton interactions introduced in Ref. [35]. Note that Eq. (54) predicts a resonance when the (negative) collision energy is comparable to $\varepsilon_{eX}$, such as in GaAs with a very large Rabi coupling in panel (b). However, this resonance is spurious and can be cured by including higher order terms in the phase shift, and it is also absent in our exact calculation.

While Eq. (54) is conceptually similar to the Born approximation, Eq. (44), we see that the fact that the exciton-electron scattering should be considered off-shell leads to a qualitative difference in the behavior as a function of detuning. We stress that there is no a priori reason why this conclusion should not generalize beyond contact electronic interactions to Coulomb or Keldysh [59] type potentials in, e.g., GaAs or TMDs: At large electron-exciton separation $r$, both of these electronic potentials will result in an effective exciton-electron potential that decays like $-1/r^{4}$ corresponding to the interaction between a charge and an induced dipole. Such a potential qualifies as short ranged [27], and therefore the low-energy exciton-electron scattering necessarily follows the form in Eq. (53). Furthermore, this approximation relies only on a single quantity, the low-energy scale $\varepsilon_{eX}$, which can be fitted in experiment or calculated in exact solutions of the exciton-electron three-body problem such as in the recent work by Fey et al. [60]. Finally, we note that since $\varepsilon_{eX}$ only appears under a logarithm, our results are not very sensitive to realistic variations in this parameter, and they may therefore be expected to be quite accurate even at a quantitative level.

We now turn to the interaction of an electron and a polariton at finite momentum which, as we shall show, can be strongly enhanced compared with their exciton-electron counterpart. As previously noted, we will restrict our attention to the case of zero center-of-mass momentum, which allows us to perform a partial wave decomposition of the $T$ matrix as follows [25]:

where $l$ is the angular momentum quantum number and $\phi_{12}$ is the angle between $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$. Obviously, the scattering in the limit of zero momentum discussed in the previous subsection corresponds to $s$-wave, $l=0$. This decomposition allows us to solve the integral equation as a function of a single parameter, the magnitude of the incident relative momentum. The main challenge then is that the electron-hole $T$ matrix in Eq. (50) has a simple pole when $q=p_{2}$ [44], and we discuss how to deal with this in Appendix C. Once we have solved for the partial wave amplitudes $t_{l}$, we obtain the normalized scattering $T$ matrix in the $l$’th partial wave via

similarly to Eq. (51). In this work, we focus on $s$-wave scattering, which we denote by $T_{s}(p)$. However, we note that higher partial waves (in particular $p$-wave) can become important in the scattering of polaritons with heavy holes [45, 47] due to the presence of hole-hole-electron trion bound states which have previously been discussed in the context of quantum wells [61] and TMD monolayers [62, 63] or strongly screened electronic interactions [64, 51].

Figures 7 and 8 show our results for the polariton-electron scattering at finite relative momentum for the case of a TMD monolayer and for a GaAs quantum well, respectively. A striking aspect of our results is the presence of a resonance-like peak at momenta close to the polariton inflection point. The peak is most pronounced for zero and negative detuning and it is enhanced with increased Rabi coupling. By contrast, we find that the peak is completely absent in the case of exciton-electron scattering (also shown in the figures), where we obtain the corresponding $T$ matrix using Eq. (53) at collision energy $E+\varepsilon_{B}=p^{2}/2m_{eX}$. At a qualitative level, the results for GaAs and a TMD monolayer embedded in a microcavity are similar, although the generally larger ratios of Rabi coupling to exciton binding energy in GaAs means that the relative enhancement of interactions in GaAs microcavity is larger. For relative momenta above the polariton inflection point, we see that the polariton-electron $T$ matrix quickly reduces to the exciton-electron $T$ matrix, as we would expect since in that limit the exciton fraction approaches unity.

As in the case of scattering at low momenta, we can understand the resonance-like feature in the polariton-electron scattering at finite momentum in terms of off-shell exciton-electron scattering. Using a similar reasoning to that which led to Eq. (54), we approximate

We illustrate this in Fig. 9 for the case of the TMDs with a smaller ratio of Rabi coupling to exciton binding energy (MoSe₂, WSe₂, or MoS₂). We see that this approximation works extremely well for almost all momenta, the only difference being a narrow region around the momentum where the collision energy $E+\varepsilon_{B}$ vanishes. By analyzing the two contributions in Eq. (57), we conclude that the resonance-like feature is due to a competition between the exciton fraction — which quickly approaches unity when the momentum approaches the polariton inflection point — and the fact that scattering is suppressed when the collision energy approaches zero. Figure 9 also shows that the polariton-electron $T$ matrix is nearly purely real at the peak, which we can also understand from Eq. (57) since exciton-electron scattering only features an imaginary part for positive collision energy.

Crucially, our results in Figures 7-9 imply that the polariton-electron $T$ matrix is essentially never smaller than that of an electron and an exciton at a given momentum, the only exception being at a very large negative detuning. This is in spite of the suppression due to the reduced exciton fraction in the polariton, and it demonstrates that the strong energy shift of the scattering process due to the light-matter coupling is the dominant effect. Our result can thus be understood as an effective photon-mediated enhancement of exciton-electron scattering, which is a major qualitative difference compared with the commonly applied assumption in the Born approximation, Eq. (41). As we have derived this effect in a fully microscopic model, and since it agrees with the universal low-energy behavior, Eq. (57), we expect that this conclusion carries over to more realistic interactions between charged particles.

Given the general nature of our arguments, the enhancement of interactions between polaritons and electrons is also expected to carry over to other geometries, such as systems where a 2D electron gas is in proximity to a 2D exciton-polariton condensate. Here, it would be interesting to investigate whether the enhanced interactions could lead to stronger drag effects [65], or to more robust superconductivity [66, 67].