Polaronic polariton quasiparticles in a dark excitonic medium

The concept of quasiparticles has revolutionized our understanding of complex quantum systems. A famous example is Landau’s quasiparticles within Fermi liquid theory Abrikosov and Khalatnikov (1959); Pines and Nozières (1966) which describe the collective behavior of interacting fermions, and which have provided a remarkably successful description of Fermi systems ranging from liquid ³He to the semiconductors that underpin modern electronics. Another important example is the so-called polaron quasiparticle, which forms when a mobile impurity particle becomes “dressed” by excitations of a background quantum medium. This latter scenario was initially confined to the case of electrons interacting with phonons in a crystal lattice Devreese and Alexandrov (2009) but has since emerged in a variety of systems, including ultracold atomic gases Massignan et al. (2014); Scazza et al. (2022) and, most recently, doped semiconductors Sidler et al. (2017); Efimkin and MacDonald (2017).

In this work, we extend the quasiparticle paradigm, as embodied by polarons and Landau’s quasiparticles, and show that it provides a new framework for understanding hybrid light-matter systems in semiconductor microcavities, as depicted in Fig. 1. Here cavity photons are strongly coupled to excitons—bound electron-hole pairs—in a two-dimensional (2D) semiconductor, leading to the formation of exciton polaritons (or polaritons), emergent hybrid particles that are superpositions of light and matter Weisbuch et al. (1992); Kavokin and Malpuech (2003); Deng et al. (2010); Carusotto and Ciuti (2013). Most notably, polaritons possess a small mass inherited from their photonic component, allowing them to Bose condense at high temperatures Byrnes et al. (2014), while their excitonic component can potentially generate strong optical nonlinearities Ferrier et al. (2011); Delteil et al. (2019); Muñoz-Matutano et al. (2019).

The polaritons’ hybrid nature makes this platform attractive for the investigation of a rich variety of phenomena ranging from superfluidity Amo et al. (2009); Lagoudakis et al. (2008); Sanvitto et al. (2010); Lerario et al. (2017) and topological effects St-Jean et al. (2017); Klembt et al. (2018); Gianfrate et al. (2020); Solnyshkov et al. (2021) to the emulation of the classical XY and Kardar-Parisi-Zhang models Berloff et al. (2017); Fontaine et al. (2022). Polaritons also have potential applications in optoelectronics Ballarini et al. (2013); Nguyen et al. (2013); Marsault et al. (2015); Zasedatelev et al. (2019); Sanvitto and Kéna-Cohen (2016), while the recent advent of atomically thin materials with robust exciton bound states further widens the capabilities of light-matter systems Wang et al. (2018); Schneider et al. (2018). On the other hand, it is well known that such microcavity polaritons are often accompanied by a “dark reservoir” Deng et al. (2010); Carusotto and Ciuti (2013) consisting of excitons which are not coupled to light (e.g., momentum forbidden) yet can interact with polaritons via their excitonic part and thus impact the behavior of polaritons. Furthermore, the density of dark excitons can be substantial Madéo et al. (2020); Pieczarka et al. (2022), even when polaritons are directly excited with a resonant laser Gavrilov et al. (2012); Vishnevsky et al. (2012); Takemura et al. (2016); Walker et al. (2017); Stepanov et al. (2019).

We show here that the presence of a dark excitonic medium can fundamentally modify the polaritons themselves by dressing them to form polaron-like quasiparticles, as illustrated in Fig. 1(a). In addition to shifting the energy of the polaritons, the polaronic dressing cloud can reduce the polariton’s light-matter coupling and lifetime, as well as generate a new quasiparticle—the biexciton polariton—a polaron that is adiabatically connected to the biexciton (two bound excitons) at vanishing exciton density [Fig. 1(b)]. Reservoir induced energy shifts have already been observed experimentally and have been exploited to create on-demand potentials for polaritons via a spatially varying laser pump Wertz et al. (2010); Tosi et al. (2012); Askitopoulos et al. (2013); Cristofolini et al. (2013); Gao et al. (2015); Pieczarka et al. (2021); Estrecho et al. (2019); Pieczarka et al. (2022). Our theory of polaronic polaritons goes further and suggests that the very character of the polariton quasiparticle can be controlled by varying the reservoir density.

To describe the many-body polariton-reservoir system, we formulate a novel Green’s function approach involving one and two-body correlators that allows us to incorporate the exact low-energy scattering between polaritons and reservoir excitons for arbitrary spin. In particular, it captures the coherent scattering of excitons out of the “bright” (light-coupled) state, leading to a polaron quasiparticle that is a superposition of photon, bright exciton and scattered dark excitons (dressing cloud). This provides a natural mechanism for reducing the light-matter coupling with increasing exciton density. Moreover, unlike conventional theories Schmitt-Rink et al. (1985); Rochat et al. (2000); Glazov et al. (2009), it does not invoke any Pauli blocking between the electrons and holes within the excitons. Thus, we expect it to dominate in typical experiments which are conducted below the Mott density.

Figure 1(b) summarizes the key implications of our polaronic polariton quasiparticles for the optical spectrum. We find that the upper and lower polaritons (UP and LP) are affected differently by the interactions with the medium, such that the UP polaron generally broadens with increasing density while the LP polaron shifts upwards in energy. This results in an effective reduction in the Rabi splitting between polariton branches, which mirrors the diminished light-matter coupling in the polarons and which is consistent with observations in experiment Houdré et al. (1995); Rhee et al. (1996). Crucially, this differs from the standard picture of saturation since the loss of the Rabi splitting is primarily accomplished by the disappearance of the UP quasiparticle rather than the merger of two quasiparticle branches. We show that this result can explain recent experimental measurements of polariton interactions in transition metal dichalcogenides (TMDs) Stepanov et al. (2021).

We furthermore see the appearance of the biexciton polariton (BP) at lower energies, which exhibits a pronounced Zeeman splitting since it relies on the presence of the biexciton bound state and is thus sensitive to the spin composition of both the reservoir and the polariton. This is associated with resonantly enhanced opposite-spin interactions that emerge from the dark-exciton-analog of a polariton Feshbach resonance Wouters (2007); Vladimirova et al. (2010); Takemura et al. (2014); Bleu et al. (2020). Remarkably, we find that coherent BP polarons exist even though the excitonic medium is incoherent, and that such quasiparticles are stabilized by the strong coupling to light.

The paper is organized as follows. In Section II, we outline the model and we present the Green’s function formalism used in our work. We use this to solve the two-body problem exactly. In Section III, we generalize our result to the many-body case of an incoherent excitonic reservoir, and discuss the key properties of the resulting quasiparticle branches, the BP, LP, and UP polarons. We consider two particular experimental configurations in Section IV, both of which have recently been explored in microcavities containing a MoSe₂ monolayer as the active medium Stepanov et al. (2021); Tan et al. (2023). While those experiments were interpreted fully in terms of polaritons, we argue that an incoherent exciton reservoir could have played a key role in both cases. We conclude in Section V.

We consider a 2D semiconductor embedded in an optical microcavity, as depicted in Fig. 1(a). To describe an exciton polariton immersed in an incoherent dark excitonic medium, we employ the Hamiltonian $\hat{H}=\hat{H}_{0}+\hat{V}$, which consists of the following single-particle and interaction parts (we set the system area and $\hbar$ to 1):

Here the bosonic operator $\hat{x}^{\dagger}_{{\bf k}\sigma}$ creates an exciton with in-plane momentum ${\bf k}$ and pseudospin $\sigma=\uparrow,\downarrow$, corresponding to the spin of the electrons involved in the optical excitation of the exciton. We treat the excitons as featureless bosons, which is reasonable as long as the exciton binding energy is large compared to the energy scales of interest, as is typically the case in 2D semiconductors such as TMD monolayers and single GaAs quantum wells. We also assume that we are close enough to the band edge that the excitonic dispersion is given by $\epsilon_{{\bf k}}=|{\bf k}|^{2}/2m_{X}\equiv k^{2}/2m_{X}$ with exciton mass $m_{X}$. All energies are with respect to the exciton energy at ${\bf k}=0$.

Equation (1b) describes the interactions between excitions via the interaction potential $v_{\sigma\sigma^{\prime}}({\bf q})$. Importantly, this depends on the exciton spin, where we obviously require $v_{\uparrow\downarrow}=v_{\downarrow\uparrow}$ and we also assume that $v_{\uparrow\uparrow}=v_{\downarrow\downarrow}$. A complete description of the exciton-exciton interactions is presented in Sec. II.2.

For simplicity, we consider a single cavity mode of the microcavity, which is described by the photon creation operator $\hat{c}^{\dagger}_{\sigma}$ with spin $\sigma$ and detuning $\Delta_{0}$ with respect to the energy of the exciton at ${\bf k}=0$. The cavity photon is assumed to be at normal incidence to the semiconductor and thus it only couples to the exciton at ${\bf k}=0$ via the Rabi coupling $\Omega_{0}$ in Eq. (1a). The excitons with ${\bf k}\neq 0$ remain uncoupled to light and can thus naturally form a reservoir of optically dark excitons. Note that this scenario also applies to planar microcavities where the cavity photons have in-plane momenta, since the light-coupled excitons have $k$ much smaller than the typical momentum scales in the semiconductor. Hence, we can simply incorporate the effect of finite photon momentum by varying the light-matter detuning.

To connect our theory with the optical spectrum that is observed in experiments, we use the retarded photon Green’s function which can be defined as

where $\theta(t)$ is the Heaviside function and the average $\expectationvalue{\cdots}$ is taken over the state of the microcavity system assuming a small or negligible photon number, e.g., an incoherent medium of dark excitons. Here we work within the Heisenberg picture such that we have time-dependent operators $\hat{c}_{\sigma}(t)=e^{i\hat{H}t}\hat{c}_{\sigma}e^{-i\hat{H}t}$. The Fourier transform, $G_{C\sigma}(\omega)$, of the photon Green’s function then gives us access to the spectrum for each photon spin/polarization $\sigma$. In particular, the absorption is related to the spectral function, given by

while the transmission is proportional to $|G_{C\sigma}(\omega)|^{2}$ Ćwik et al. (2016). The frequency is shifted by an infinitesimal positive imaginary part due to the Heaviside function in Eq. (2).

Importantly, while the Green’s functions are diagonal in the spin basis, experiments are free to consider any polarization angle, which will in general probe the spectrum associated with both $G_{C\uparrow}(\omega)$ and $G_{C\downarrow}(\omega)$. For example, when the photon is of the form $\hat{c}_{\theta}=\cos(\theta)\hat{c}_{\uparrow}+\sin(\theta)\hat{c}_{\downarrow}$, the spectral function becomes a weighted sum

where $A_{\uparrow}$ and $A_{\downarrow}$ are the spin-resolved spectral functions defined in Eq. (3).

We now turn to the many-body problem and discuss how the interaction with a dark excitonic medium can strongly influence the behavior of exciton polaritons due to the formation of polaron quasiparticles. Our predictions include non-trivial shifts of the lower and upper polariton energies, modifications of their exciton fraction and associated light-matter coupling strength, saturation of their Rabi splitting, and even the emergence of new light-matter-coupled quasiparticles in the spectrum.

Similarly to the exact two-body formulation in Eq. (19), the effects of the dark medium are all encoded in the self-energy $\Sigma_{\sigma}(\omega)$, such that the exciton Green’s function satisfies the Dyson equation [Fig. 3(a)] Fetter and Walecka (2003)

This modifies the Green’s matrix in Eq. (5) to the spin-dependent interacting form

where the interaction with the medium only affects the excitonic component. Thus we obtain the spin-dependent photon Green’s function

Before proceeding to consider the specific form of the self-energy that arises from interactions with a dark excitonic reservoir, let us first make some general observations that are valid for any medium, dark or otherwise. Provided the imaginary part of the self-energy is sufficiently small such that the upper and lower polaritons remain well-defined quasiparticles, we can treat the effect of the medium by considering their modified quasiparticle properties. The new medium-dressed and light-matter-coupled quasiparticles may be found by considering the poles of the Green’s functions, i.e., we require $\real[G^{-1}_{C,X\sigma}(\omega_{p})]=0$ ¹¹1Note that, in general, the poles of the two Green’s functions can be slightly different due to the imaginary part of the self-energy. However, the difference is typically very small, for instance close to zero detuning it is $\simeq\frac{1}{2}\Gamma_{\sigma}^{2}/\Omega_{\sigma}$.. For frequencies close to such a pole, we then have

and

Here we have introduced the quasiparticle residue $Z_{\sigma}$, a key quasiparticle parameter that is defined as Fetter and Walecka (2003)

and satisfies $0\leq Z_{\sigma}\leq 1$. Comparing Eqs. (25) and (26) with the non-interacting Green’s functions in Eqs. (6) and (7), we find that $Z_{\sigma}$ changes the weight of the exciton Green’s function and leads to the effectively reduced light-matter coupling

Physically, $Z_{\sigma}$ corresponds to the fraction of the matter component of the polaron quasiparticle that remains in the bright ${\bf k}=0$ exciton, while the remaining fraction ($1-Z_{\sigma}$) is in the polaronic dressing cloud composed of dark ${\bf k}\neq 0$ excitons and scattered medium particles. This physics relies on the frequency dependence of the self-energy, since a constant self-energy simply gives $Z_{\sigma}=1$ according to Eq. (27). The interactions with the medium also cause the exciton resonance to shift and broaden according to

respectively. In particular, we see that whether the exciton strongly shifts or broadens depends sensitively on whether the self-energy is predominantly real or imaginary. The former typically happens when the quasiparticle is detuned from a continuum of states, which is the case for the lower polariton. Conversely, the latter occurs when the quasiparticle sits in a continuum of states, as is typically the case for the upper polariton due to its upwards shift from the bare exciton line. Hence, the medium typically shifts the lower polariton while broadening the upper polariton, leading to a modified Rabi splitting, as illustrated in Fig. 1(b). For most parameter regimes, the Rabi splitting is reduced by the interaction with the medium; however, we show below that it is even possible to enhance the splitting between lower and upper polaritons in certain cases, even though the underlying light-matter coupling is reduced as per Eq. (28).

Let us contrast our ideas with the conventional treatment of polaritons in a (dark) excitonic reservoir Ferrier et al. (2011); Walker et al. (2017); Estrecho et al. (2019); Fontaine et al. (2022). Here it is assumed that the shifts of the lower and upper polaritons have the simple mean-field form

where $n$ is the reservoir density and $g$ is a constant exciton-exciton interaction strength, such as might be obtained in the Born approximation. This corresponds to taking the constant self-energy $\Sigma_{\sigma}(\omega)=gn$ and assuming $gn/\Omega_{0}\ll 1$. Expressions like in Eq. (31) are often used when modeling effective reservoir-induced potentials for an exciton-polariton superfluid. However, this approach does not properly describe the full energy-dependent scattering of excitons and thus misses the polaronic physics we describe here. In particular, it gives $Z_{\sigma}=1$ and it does not capture the different interactions for the upper and lower polaritons. For instance, close to zero detuning where $|X_{0}|^{2}=|C_{0}|^{2}=1/2$, Eq. (31) predicts that both branches shift equally and thus the Rabi splitting remains constant. Therefore, in order to explain any reduction of the Rabi splitting between upper and lower polariton branches (i.e., saturation), one must invoke “phase-space filling” due to the fermionic nature of the electronic constituents of the excitons Schmitt-Rink et al. (1985). By contrast, our polaron theory naturally yields a saturation of the Rabi splitting, without the need to appeal to Pauli exclusion.

To make our discussion more concrete, we now consider a specific excitonic self-energy based on the exact solution of the two-body problem presented in Sec. II.2 above. Specifically, we perform a weighted sum over the possible momenta and spins of the excitons in the reservoir, such that we obtain the self-energy depicted in Fig. 3(b):

where $n_{{\bf Q}\sigma}$ is the occupation of a dark exciton with momentum ${\bf Q}$ and spin $\sigma$. Here, we have assumed that the relevant energy scales are such that we can take the universal expression for the $T$ matrix in Eq. (21). This self-energy corresponds to a so-called ladder approximation Rath and Schmidt (2013). As such, it neglects higher order correlations between the impurity and multiple excitations of the dark excitonic medium, as is reasonable for an incoherent reservoir where $n_{{\bf Q}\sigma}$ is approximately a Boltzmann distribution.

A further important simplification arises from the strong light-matter coupling combined with the fact that the dark medium particles are much heavier than the polaritons themselves. This implies that the collision energy in Eq. (32) is naturally dominated by the Rabi coupling rather than the exciton kinetic energy. Neglecting $\epsilon_{\bf Q}$ in Eq. (32), the self-energy takes the particularly simple analytic form

where we have defined $\varepsilon_{n_{\sigma}}=4\pi n_{\sigma}/m_{X}$, with $n_{\sigma}$ the density of dark excitons for a given spin $\sigma$. The factor 2 in the first term originates from the indistinguishability of same-spin excitons, while the second term involves opposite spin interactions with $\bar{\sigma}\neq\sigma$. Provided $\omega$ is not resonant with the biexciton, we find that Eq. (33) is an excellent approximation when $\sum_{{\bf Q}\sigma}n_{{\bf Q}\sigma}\Theta(\epsilon_{\bf Q}-\Omega_{0})\ll n$, with $n$ the total density of the dark medium.

Using our analytic expression for the self-energy, Eq. (33), we can immediately plot the evolution of the $\uparrow$ and $\downarrow$ spectral functions with increasing total reservoir density, as parameterized by $\varepsilon_{n}\equiv 4\pi n/m_{X}$. The result is shown in Fig. 4 for a reservoir with spin proportions $n_{\uparrow}=0.75n$ and $n_{\downarrow}=0.25n$. We see that each spectrum features three lines, rather than two, and that the two lower branches can either blueshift or redshift with increasing density. Furthermore, the corresponding lines in $A_{\uparrow}$ and $A_{\downarrow}$ can be Zeeman split with respect to each other. We now discuss each of these features in detail.

We now consider two experimentally relevant scenarios of particular interest. In the first, we assume that the exciton reservoir is polarized along the same angle as the polariton and determine the spectral function as a function of polarization angle. This allows us to clearly identify the contributions to interactions due to same and opposite spin interactions, including the effect of the biexciton resonance, as well as the polarization-dependent reduction of the effective light-matter coupling. Second, we consider a $\downarrow$ polariton in a fully polarized spin-$\uparrow$ dark reservoir and we investigate signatures of the biexciton polariton as a function of the detuning.

Such scenarios have recently been experimentally realized in MoSe₂ monolayers embedded in microcavities—see Refs. Stepanov et al. (2021) and Tan et al. (2023). In both cases, the medium was assumed to consist of bright polaritons, rather than dark excitons. However, we find that we can capture the main features of these experiments using our polaron theory with a dark excitonic reservoir, thus suggesting that dark excitons could have played a role. Therefore, in the following we will discuss these two experimental scenarios using parameters from the experiments.

In conclusion, we have developed a theory of exciton polaritons in an incoherent excitonic reservoir which exposes the polaronic nature of the light-matter coupled quasipartcles. By incorporating the exact low-energy interactions with the medium into a many-body Green’s function formalism, we have demonstrated that interactions with the reservoir lead to a wealth of intriguing effects, many of which are missed by standard mean-field approaches. Our predictions include a medium-induced reduction of the light-matter coupling and Rabi splitting, without needing to invoke Pauli blocking, and the appearance of a new quasiparticle, the biexciton polariton polaron. We expect the physics that we have uncovered here to apply to polariton systems more generally, thus providing a new understanding of the physical mechanism responsible for the saturation of the Rabi splitting.

The presence of an excitonic reservoir is already likely to play a non-trivial role in current experiments, particularly those with a non-resonant pump where the reservoir density may be significant, or those where the lower polariton is close to the biexciton resonance such as is generally the case in monolayer TMD microcavities. Our simple analytic expressions for the modifications of energies, exciton and photon fractions, and lifetimes are likely to aid future experiments in ascertaining the impact of the reservoir on the observed physics.

We have furthermore analyzed two particular experimentally relevant scenarios, related to recent experiments carried out in MoSe₂ monolayer microcavities Stepanov et al. (2021); Tan et al. (2023). For the case of a polariton co-polarized with the reservoir, we have demonstrated that the blueshift of the lower polariton can be largest for linear polarization due to the presence of the biexciton resonance. Our results are in quantitative agreement with those of Ref. Stepanov et al. (2021) with only a single fitting parameter, the reservoir density, which is particularly remarkable given that the polarization-dependent behavior of the blueshift could not even be qualitatively explained by conventional mean-field theories Stepanov et al. (2021). In the case of a counter-polarized reservoir, we have shown that the incoherent nature of the reservoir can lead to a significant broadening of the biexciton feature, in agreement with the observations of Ref. Tan et al. (2023) as well as previous measurements Saba et al. (2000); Takemura et al. (2014). Thus, the significant biexciton linewidth observed in experiment may arise from the tail of the exciton momentum distribution, rather than due to an intrinsic property of the biexciton.

Our results open up new possibilities to engineer and manipulate exciton-polaritons as well as their interactions. For example, they have immediate implications for optical trapping of polariton condensates, a key experimental technique due to its great flexibility Sanvitto et al. (2011); Tosi et al. (2012). Recently, there have been experimental studies on ‘reservoir optics’, i.e., non-resonantly pumping the system across a spatial area, and using the resulting reservoir to focus and trap polariton condensates Schmutzler et al. (2015); Wang et al. (2021); Aristov et al. (2023). Our work suggests that since the polariton-reservoir interaction can be dramatically enhanced and even its sign reversed, these reservoir optics devices can be engineered to provide strong responses through the manipulation of the laser spin and photon-exciton detuning, while at the same time maintaining the flexibility and predictability of systems in which the laser alone provides a well-understood trapping potential. In particular, this potentially provides a versatile platform for achieving strongly correlated photons via reservoir engineering of the polariton polarons’ mutual interactions Vishnevsky and Laussy (2014).