2.1.1.

Cavity photons

The confined cavity photons with the quantized wave vector $q_{zl}$ (where $l$ is the quantization number) have a discrete set of energy branches: $\varepsilon (k)=(\frac{\mathrm{\hbar }c}{n_0})\sqrt{q_{zl}^2+k^2}$, where $c$ is the speed of light in free space, $n_0$ is the refractive index, and $\mathbf{k}$ is the in-plane component of wave vector $\mathbf{q}$. Within a quantized branch, while $q_{zl}$ is a constant, the 2D in-plane part $k$ is a continuous variable. Thus, the low-energy dispersion ($k\ll q_{zl}$) is given by an effective 2D dispersion relation:

In quantum theory, the microcavity photons are represented by a bosonic field ${\mathrm{\Psi }}_{C\sigma }(\mathbf{r})$ with the spin polarization projection $\sigma$ along the $z$ axis (growth axis of the microcavity), which is associated with the left and right circular polarizations of light. With a polarization-dependent effective mass $m_{\sigma }$, the cavity photons can be described by a Hamiltonian:

This is a common form of a Hamiltonian describing particles with a nonzero mass^[45]. Using the reciprocal space representation of the field ${\mathrm{\Psi }}_{C\sigma }(\mathbf{r})$ as given by Eq. (2), the photonic Hamiltonian $H_{\mathrm{ph}}$ can be rewritten in a simpler form:

While for simplicity the dispersion relation ${\varepsilon }_{\sigma }(\mathbf{k})$ in Eq. (6) is considered diagonal for $\sigma =\pm 1$, in general, dispersion relations might have off-diagonal elements. Thus, a general form of the Hamiltonian for cavity photons is given by

Parameters ${\varepsilon }_{\sigma {\sigma }^{\prime }}(\mathbf{k})$ represent elements of an energy matrix written in the basis of polarization projections ($\sigma ,{\sigma }^{\prime }=\pm 1$) on the growth axis ($z$ axis). While the diagonal elements represent the energy of a photon for a given polarization, the off-diagonal elements represent the coupling between different polarization modes. For instance, birefringence may cause a different effective mass along the crystal axis ($x$ axis) and the perpendicular axis ($y$ axis). Furthermore, the dispersion relations can be polarization dependent due to crystal anisotropy^[47], such that the effective dispersion for the $X$ and $Y$ polarization components is given by

2.1.2.

Excitons

In semiconductor microcavities, thin layers of semiconductors are placed inside the cavities to realize strong interaction between light and matter. The low-energy optical excitation corresponds to the creation of electron–hole pairs (so-called excitons) in the semiconductor layer. Excitons emerge from the Coulomb interaction between electrons in the conduction band and the holes in the valence band. A sufficiently thin semiconductor layer acts as a 2D QW, where the electrons and holes are confined. Since electrons and holes are oppositely charged, they form bound states like hydrogen atoms in 2D with a characteristic length called the Bohr radius. Depending on the relative size of the Bohr radius with respect to the crystal lattice constant, excitons are typically classified as Frenkel excitons (with Bohr radii similar to lattice constants) and Wannier–Mott excitons (with Bohr radii extending several lattice constants). It is interesting to note that while the Bohr radius is inversely related to the exciton binding energy, it is directly related to the interaction between excitons^[48,49]. Since the nonlinear interaction strength scales linearly with the exciton binding energy but quadratically with the Bohr radius, an intermediate value of the Bohr radius is thus preferable to obtain a sizable interaction strength as well as a decent binding energy.

To obtain room temperature polaritons, excitons must withstand corresponding thermal excitation energy $k_{B}T\approx 26\mathrm{meV}$, where $k_B$ is the Boltzmann constant, and $T$ is the temperature. This is possible for exciton binding energies $E_b\gg k_{B}T$. In this regime, low-energy physics (where the typical distance between excitons is much larger than the exciton Bohr radius) can be understood by considering excitons as bosonic quasiparticles (composite bosons)^[50]. The corresponding Hamiltonian describing the excitons is given by

2.1.3.

Light–matter interaction

To obtain strong coupling with the cavity mode, typically QWs are placed at the antinodes of the optical field, and the energy of the cavity mode is placed near the exciton energy. The interaction between excitons and cavity photons can be described by an interaction Hamiltonian $H_{\mathrm{int}}$^[20,41,52]:

2.1.4.

Exciton polaritons

The Hamiltonian describing a system of cavity photons, excitons, and their interaction is given by $H_{\mathrm{ep}}=H_{\mathrm{ph}}+H_{\mathrm{ex}}+H_{\mathrm{int}}$^[20]. We find that the total Hamiltonian $H_{\mathrm{ep}}$ is quadratic in field operators. We can thus diagonalize the Hamiltonian with a linear transformation:

Operators $a_{\mathrm{LP}\sigma }(\mathbf{k})$ and $a_{\mathrm{UP}\sigma }(\mathbf{k})$ represent the annihilation operators of exciton polaritons, which are composites of excitons and photons. Prefactors ${\eta }_{X\sigma }(\mathbf{k})$ and ${\eta }_{C\sigma }(\mathbf{k})$ are known as Hopfield coefficients. Subscripts LP and UP represent the lower and upper polariton branches corresponding to the eigenvalues $E_{\mathrm{LP}\sigma }(\mathbf{k})$ and $E_{\mathrm{UP}\sigma }(\mathbf{k})$, respectively, given by

In Figs. 1(e) and 1(f), we show the lower polariton (LP) and upper polariton (UP) energy branches as functions of the in-plane wave vector k for different bare exciton energies ${\varepsilon }_{\mathrm{ex}}$. While the bare cavity photon ${\varepsilon }_{\sigma }(\mathbf{k})$ and exciton ${\varepsilon }_{\mathrm{ex}}(\mathbf{k})$ branches (dashed lines) can cross each other [Fig. 1(e)] at ${\varepsilon }_{\sigma }(\mathbf{k})={\varepsilon }_{\mathrm{ex}}(\mathbf{k})$, the polariton branches show anticrossing with a minimum gap $2\mathrm{\hbar }{\mathrm{\Omega }}_R$, which is considered a hallmark of the strong coupling regime. Since the composite of two bosonic fields is also bosonic, the polariton field operators satisfy the Bose commutation relations shown by Eq. (1). The Bose commutation relations yield ${|{\eta }_{X\sigma }(\mathbf{k})|}^2+{|{\eta }_{C\sigma }(\mathbf{k})|}^2=1$. Typical variations of Hopfield coefficients for different parameters are shown in Figs. 1(g) and 1(h). Hopfield coefficients represent the amount of excitonic and photonic content in the exciton polaritons. Notably, near the crossing points between the bare cavity photon energy and the exciton energy, the Hopfield coefficients are equal. In most experiments, the relevant phenomenon can be described only by the LP branch, such that the effective Hamiltonian is given by^[42]

Henceforth, we will consider only the LP branch for our discussion. From an experimental point of view, there are some prominent methods used to show polariton dispersion in an energy-momentum diagram. For example, goniometer-like angle scanning, pinhole translation, and Fourier-space projection methods can be used^[53]. However, the most commonly used method is Fourier-space spectroscopy, i.e., angle-resolved optical spectroscopy, where the angle information in the far-field projection is used to provide direct access to the energy-momentum dispersion of exciton polaritons. Typically, a system of lenses is used to perform the imaging of the back focus of a microscope objective (objective lens), which encodes information about the angle of emitted, reflected, or transmitted light from a sample [Fig. 2(a)]. The back focal plane of the objective lens, which shows a radial–symmetric intensity profile, is then collected and focused through the entrance slit of a spectrometer and finally onto the sensor of a 2D charge-coupled device (CCD). The center of the collected far-field circle represents 0°, while the edge represents the maximum collected angle, determined by the numerical aperture of the objective. The narrow slits at the entrance of the spectrometer serve as a spatial filter that selects only the signal with $k_y=0{\mathrm{\mu m}}^{-1}$, or alternatively any other narrow slice of momentum space. The center of the ring $(k_x,k_y)=0{\mathrm{\mu m}}^{-1}$ is related to the ground state of the polaritonic dispersion. The signal is then dispersed by a grating inside the spectrometer and imaged onto the CCD. The energy-momentum dispersion relation is obtained from the CCD detector in which each pixel line includes spectral information for a corresponding angle (in-plane momentum). Furthermore, the spatial and temporal coherence properties of exciton polaritons are typically investigated by optical interferometry where the most common configuration is a Michelson interferometer [Fig. 2(b)]. For this purpose, the emission from the sample is, in general, studied in real space. For details, the real space emission from the microcavity is sent to a beam splitter that splits the signal into two new paths. Along one path, a mirror is used to reflect the signal back, while along the second path, a backreflector is in general preferred. In this way, the reflected signal from the second arm is flipped in a centrosymmetric way, thus allowing to study the coherence properties along different directions in space. To change the time delay between the two arms, thus obtaining information about the coherence time, the mirror is mounted on a motorized delay line. Two reflected signals are then combined onto the sensor of a CCD and generate an interference pattern whose amplitude and visibility are used to study the first-order spatial and temporal coherence properties of polaritons.

Fig. 2

Commonly used experimental methods. (a) Schematic of the angle-resolved photoluminescence setup with Fourier image configuration^[35]. (b) Top, schematic of the Michelson interferometer setup. Bottom, illustration of centro-symmetrically interfered images.

2.1.5.

Polariton–polariton interaction

Polaritons interact with each other due to their excitonic components. In the mean field regime, the interaction between polaritons manifests as nonlinearity, which plays an important role for many practical applications of polaritons. Although the actual form of the interaction between polaritons is due to Coulomb interaction between electrons and holes, the low-energy physics in the LP branch can be well described by an effective Hamiltonian^[49,54]:

The form of the interaction Hamiltonian ${\mathit{H}}_{\mathrm{P}-\mathrm{P}}$ is known as the contact interaction. Here, two polaritons experience interaction only when they contact each other (same position). This local nature of the interaction motivates us to express the interaction Hamiltonian ${\mathit{H}}_{\mathrm{P}-\mathrm{P}}$ in real space. However, ${\mathit{H}}_{\mathrm{P}-\mathrm{P}}$ can be equivalently written in momentum space^[54].

2.1.6.

External potential

Exciton polaritons are excellent candidates for simulating various model Hamiltonians. For this, designing various types of potentials for in-plane motion is required. Several strategies are considered for implementing such potentials in semiconductor microcavities. For instance, external metallic meshes^[55–57], optically induced high-energy excitons^[58], and chemical etching techniques^[59,60] are used for generating external potentials (see Schneider et al. for a review^[8]). In the domain of room temperature exciton polaritons, one of the most used methods is to modulate the local thickness of the top DBR mirror using an etched layer of a polymer on top of the semiconductor QW^[47]. This method proved to be effective in implementing a range of lattice structures. It has been shown that it can implement a strong lattice where the bandgap between the lowest-energy s-band and p-band is much larger than the linewidth of the LP mode. In such structures, the potential is induced by modulating the local thickness of the cavity, such that the cavity photons experience a change in the local energy along the in-plane motion. This can be represented by a potential function $V_{\mathrm{ph}}(\mathbf{r})$ for the cavity mode. Since the polariton is partly light, the potential is also experienced by the polaritons through its photonic component. The effective potential for the LP branch is approximately given by

2.1.7.

Plasmonic cavity

Although the majority of exciton polariton studies are based on a planar microcavity structure, alternatively, plasmonic nanostructures can concentrate electromagnetic (EM) fields to deep sub-diffraction volumes ($V$) with high field enhancement, which serves as another type of cavity for exciton polariton studies. Hence, despite the high loss from the metallic part^[61], and low $Q$-factor compared to microcavities, plasmonic nanostructures can still achieve high figures of merit ($Q/\sqrt{V}$) for strong coupling at room temperature^[62]. Plasmonic cavities have been carefully designed to realize strong couplings for quantum dots (QDs), molecules, and 2D materials^[63]. Plasmonic cavities can be realized in different forms. For example, individual metal particles, such as nanoprisms^[64,65], nanorods^[66–68], or nanocubes^[69], that support localized surface plasmons can be the simplest plasmonic cavities. These kinds of plasmonic cavities are widely used because they are not only able to realize strong coupling with different materials, but are also a good platform to study the behaviors and characteristics of exciton polaritons as well as tunability^[70]. Metal particle arrays that support plasmonic resonance are also utilized to achieve strong coupling^[71,72].

Other widely used plasmonic cavities usually consist of paired plasmonic structures, such as bowties^[73,74] or nanoparticle-on-mirror geometry^[75]. When the gap between the paired metal nanoparticles is in sub-nanometer range, an ultra-low mode volume and ultra-strong field enhancement can be obtained, which is more likely to overcome the limit of high emitter scattering at room temperature to realize strong coupling^[75–77]. These kinds of ultra-compact plasmonic cavities are widely used to study strong coupling with different materials^[62,78]. Their ultra-compact and small mode volumes can not only improve the vacuum Rabi splitting but also further reduce the effective exciton number contributing to coupling down to the single-digit level^[62], which is promising for quantum optics applications.

Despite the high metallic loss, high-$Q$ plasmonic cavities were also achieved via several mechanisms^[79], such as surface lattice resonance, multipolar resonance, plasmonic Fano resonance, and Tamm plasmon resonance, and strong coupling and exciton polaritons were also realized at room temperature^[80,81]. Otherwise, by combining plasmonic nanostructures with conventional microcavities, the plasmon microcavity can be obtained to study ultra-strong coupling in plasmon-microcavity exciton polaritons^[82]. Plasmonic cavities with their diverse structures and versatile properties will still play important roles for research of strong coupling and exciton polaritons at room temperature.

2.2.1.

Dissipation

The DBR mirrors in the microcavity are not perfect, allowing photons to leave the system after a certain number of coherent oscillations. Similarly, excitons are quasiparticles with finite lifetimes. These effects can be incorporated as an effective lifetime for exciton polaritons ${\tau }_{\mathrm{LP}}$, which can be represented by a loss term in the GP equation^[86]

2.2.2.

Coherent excitation

Polaritons can be injected coherently by a near resonant laser, such that the injected particles can reach the polariton mode without losing the coherence of the applied field. Coherent excitations can be introduced into the driven-dissipative GP equation by a source term^[41,52]:

2.2.3.

Incoherent excitation

Although there are different methods to implement incoherent excitation, theoretically, it can be described by an effective non-Hermitian potential $i{P}_{\mathrm{in}}(\mathbf{r})/2$. The equation of motion for polaritons under incoherent excitation is given by^[86]

In addition, here, a generalized complex nonlinearity coefficient ${\tilde{\alpha }}_{\mathrm{LP}}={\alpha }_{\mathrm{LP}}-i{\alpha }_{\mathrm{LP}}^{\prime }$ is introduced^[87]. The imaginary part ${\alpha }_{\mathrm{LP}}^{\prime }$ represents nonlinear loss, which activates an effective decay channel at high polariton density. This term is effectively responsible for saturation in the polariton intensity. The microscopic origin of such a term can be found in kinetic theories of the polariton condensates, which essentially indicates that saturations in polariton density arise due to saturations in high-energy exciton reservoirs^[86,88]. As an alternative to Eq. (21), one can introduce incoherent excitation through a polariton reservoir with a local density $n_R(\mathbf{r},t)$:

One of the widely used methods of implementing incoherent excitation is to apply a far-off-resonant laser, such that the injected polaritons go through scattering processes with several scattering channels before reaching the low-energy polariton mode. While microscopic processes involved in different systems could be different, typically incoherent injections result in an accumulation in the so-called bottleneck region of LP dispersion (this effect can be described with a kinetic Monte Carlo approach^[89]). Further relaxation to the lowest energy state can involve impurity scattering, phonon–polariton^[90], polariton–polariton, and polariton–exciton channels. In the phonon–polariton channel, exciton polaritons lose energy through coupling with the lattice vibration. The polariton–polariton channel gets activated at a large polariton density when the incoherent excitation strength becomes large enough. In this process, a polariton can relax to a lower-energy state by colliding with another one, which scatters into a higher-energy state^[42]. In the polariton–exciton channel, polaritons lose energy by scattering with excitons^[91]. During these processes, polaritons lose their initial coherence of the applied laser field. Similarly, incoherent excitation can be induced by electrical injection. These methods of providing gain into the system allow to demonstrate strong evidence for spontaneous coherence building in polariton condensates.

Although quantum theory can provide a more accurate and general description of exciton polaritons, the mean-field approach is highly successful in describing the condensate regime, where a large polariton population together with weak interaction supresses quantum fluctuations. In the next few sections, we will use the mean-field theory to describe various phenomena, such as polariton condensation, superfluidity, and related effects.

2.3.1.

Global phase coherence

In quantum theory, a particle is described by a wave function $\psi (\mathbf{r},t)$. The same particle can be equivalently described by $e^{i\theta }\psi (\mathbf{r},t)$, where $\theta$ is a global phase (which is a constant for the whole space). This arbitrariness of the global phase is due to so-called gauge symmetry. In a gauge symmetric system, each particle is independently associated with an arbitrary global phase $\theta$. Remarkably, all particles in a BEC can be described by a single wave function $\psi (\mathbf{r},t)$. Here, there is only a single global phase associated with the whole condensate rather than each particle having an independent one. Furthermore, since $\psi (\mathbf{r},t)$ represents a macroscopic number of particles, the global phase of a condensate can be determined from interference experiments^[94]. This technique was used in exciton polariton systems to show long-range coherence^[28].

2.3.2.

Exciton polariton BEC

Exciton polaritons are bosons and thus can condense into the lowest-energy state $k=0$ in the LP branch^[95]. Although microcavity exciton polaritons are confined in 2D semiconductor planes and thus do not exhibit a strict BEC (Hohenberg–Mermin–Wagener theorem), local condensates with finite size can occur in 2D, where macroscopically large occupations and long-range coherence can be observed^[4]. Moreover, the driven-dissipative nature allows to induce various kinds of condensate properties, which do not spontaneously arise in traditional bosonic systems. For instance, polariton BEC can be obtained with nonzero in-plane wave vectors or stable quantized vortices. In a driven-dissipative system, BEC occurs in a quantum state with the largest gain. The control on excitation profiles allows to induce maximum gain in a chosen spatial profile of the polariton BEC. In most experimental systems, polaritons have lifetimes shorter than energy relaxation time, which does not permit thermal equilibrium. Thus, polariton BECs might not follow the exact equilibrium Bose distribution. However, polaritons with longer lifetimes can show equilibrium BEC in high-quality microcavities^[96]. Since polaritons escape the system due to finite lifetimes, they are widely studied in the steady state condition where the rate of loss is compensated for with an equal amount of external injection through coherent and incoherent excitations. The experimental signatures of the room temperature polariton BEC are discussed in Section 3. A polariton BEC is described by the driven-dissipative GP equation given by Eq. (18). Under incoherent excitation, the steady state solution can be obtained from the equation

For simplicity, considering a uniform incoherent excitation $P_{\mathrm{in}}(\mathbf{r})=P_{\mathrm{in}}$, below the threshold $P_{\mathrm{in}}<{\gamma }_{\mathrm{LP}}$, the only steady state solution is given by ${\psi }_{\mathrm{LP}}=0$. However, above the threshold $P_{\mathrm{in}}>{\gamma }_{\mathrm{LP}}$, one obtains nonzero solutions ${\psi }_{\mathrm{LP}}=\sqrt{(P_{\text{in}}-{\gamma }_{\mathrm{LP}})/(2{\alpha }_{\mathrm{LP}}^{\prime })}{e}^{i\theta }$ for the condensate wave function, where $\theta$ is an arbitrary global phase. This suggests that below the excitation threshold, polariton BECs cannot occur due to excessive loss. Above threshold, the system spontaneously chooses a solution with a particular value of the global phase $\theta$ (note that the global phase can be time dependent).

Under coherent excitation $F(\mathbf{r},t)=F_0{e}^{i{\mathbf{k}}_p\mathbf{r}-i{\omega }_{p}t}$, the steady state solution can be found from Eq. (20). For simplicity, we consider a uniform planar cavity with a constant coupling function ${\eta }_{\mathrm{LP}}(\mathbf{r})={\eta }_{\mathrm{LP}}$ and define an effective coherent excitation amplitude $P_{\mathrm{co}}=i{\eta }_{\mathrm{LP}}{F}_0$. The steady state solution can be obtained with an ansatz ${\psi }_{\mathrm{LP}}(\mathbf{r},t)={\varphi }_{\mathrm{LP}}{e}^{i{\mathbf{k}}_p\mathbf{r}-i{\omega }_{p}t}$, where ${\varphi }_{\mathrm{LP}}$ satisfies

These turning points can be obtained from the condition $\mathrm{d}{N}_{\mathrm{LP}}/\mathrm{d}{I}_{\mathrm{co}}=0$. Bistability is a signature of nonlinear behavior of polariton condensates, where the polariton density abruptly jumps to a higher value from a lower one for a relatively small change in excitation intensity. Bistability is a feature of a single nonlinear polariton mode, but the same nonlinearity can also induce intriguing effects when multiple modes are coupled, such as parametric scattering^[99] and oscillation^[100].

2.4.1.

Quantized vortices

A fluid can form a vortex due to flow of mass with a finite rotational velocity around an axis. A superfluid with a uniform global phase does not allow such a rotational flow of mass. However, a sufficiently large perturbation can change the scenario with a spatially inhomogeneous phase $\theta (\mathbf{r})$. The wave function can then be written as

Let us consider that the phase of a vortex is given by $\theta (r)=Q_v\phi$, where $\phi$ is the angle in polar coordinates. Here, the phase winds $Q_v$ times around the point $r=0$. Considering loop $C$ as a circle with radius $r$, one finds

2.4.2.

Berezinskii–Kosterlitz–Thouless transition

In a 2D Bose liquid, vortices are the elementary topological excitations, which may appear with positive and negative winding numbers. Excitations with positive and negative winding numbers are known as vortices and antivortices, respectively. At a finite temperature, vortex–antivortex pairs can be spontaneously unpaired by thermal excitations. In a thermodynamically large system, there is a characteristic temperature $T_{\mathrm{BKT}}$ above which the thermal energy is sufficient to activate enough numbers of freely moving vortices and antivortices to destroy the global superfluidity. Below $T_{\mathrm{BKT}}$, the thermal energy is insufficient to unpair the tightly bound vortex–antivortex pairs, and thus, they can only weakly perturb the local superfluidity. Thus, if the temperature is tuned around $T_{\mathrm{BKT}}$, the system goes from a normal fluid to superfluid phase transition. This is known as the Berezinskii–Kosterlitz–Thouless transition^[102,103].

2.5.1.

Collective quantum effects

In this regime, systems of many coupled quantum modes are considered. While each quantum mode may appear simple to understand, all modes together can represent a complex interacting many-body quantum system. Here, collective phenomena such as phase transition, critical effects, and long-range correlations are studied. A system of driven coupled quantum modes can be described by the Hamiltonian $H_{\mathrm{Q}\mathrm{C}}$:

2.5.2.

Single-mode quantum effects

So far, we have presented polariton systems with many coupled quantum modes, each of which is associated with a real (e.g., $a_j$) or reciprocal [e.g., $a_{\mathrm{LP}\sigma }(\mathbf{k})$] space index. This is indeed suitable for simulating collective many-body physics. However, there is another quantum regime where one controls the polariton modes individually. This regime is relevant to the rapidly emerging quantum technologies^[104–106], such as quantum computing and quantum information processing. Here, an isolated single quantum mode is considered and manipulated individually. An isolated quantum mode can be associated with an annihilation (creation) operator $a$ ($a^{†}$) and can be described by a Hamiltonian given by

For a strong polariton blockade, $g_2(t,0)\to 0$, and noninteracting modes with $g_2(t,0)\to 1$ show the absence of blockades. An alternative mechanism to achieve a polariton blockade is the so-called unconventional blockade, which utilizes an interference induced effect^[108–110]. An unconventional blockade occurs even in weakly interacting polaritons, which are more accessible in the current experimental conditions. However, in an unconventional blockade, the correlation function $g_2(t,\tau )$ shows rapid oscillations as a function of $\tau$, which makes it hard to observe for polaritons. More recently, dynamical mechanisms have been proposed, where a polariton blockade is induced by a time-dependent parameter in the Hamiltonian^[111,112]. For instance, a combination of pulsed and continuous wave excitation can be used to induce periodic (in time $t$) windows of a strong blockade. In this dynamical blockade, there is no rapid oscillation in $g_2(t,\tau )$, and thus, it is suitable for observation in polariton systems. Another advantage of a dynamical blockade is that it can operate in regimes of both weak and strong interactions^[111].

Beyond the blockade effect, a strongly interacting single-mode polariton condensate can be used as an elementary quantum bit (qubit) for quantum computation^[113]. A qubit is a two-level quantum system, where a single quantum of excitation is controlled and manipulated. Although a condensate itself can be occupied by a large number of polaritons, strong interaction can induce significant quantum fluctuation. Quantum fluctuation on top of a single-mode condensate can be described by considering the number operator $a^{†}a=N_c+\widehat{n}$, where $N_c$ is the average number of polaritons in the condensate, and $\widehat{n}$ is the number fluctuation on top of the condensate. Using the Hamiltonian $H_{\mathrm{iso}}$, one can show that the fluctuation is described by an effective Hamiltonian $H_{\mathit{f}}$^[113]:

2.6.1.

Polariton laser

Bose–Einstein condensation is a natural source of a large population in a single energy mode with phase coherence, which can be used for the same function as lasers do for photons. Indeed, atomic BEC was proposed as a laser providing coherent atoms^[114,115]. In this context, polaritons in semiconductor microcavities are attractive. They are hybrid particles combining characteristics of matter and light. While polaritons can condense into a single energy state with a large population like an atomic BEC, they can also emit photons retaining the phase coherence and monochromaticity of the condensate. Polariton lasing was thus envisioned with intriguing characteristics and closely related to BEC^[116]. An attractive feature of polariton lasers is that they can operate with significantly lower thresholds compared to their photonic counterparts^{[35,117–120]}.

Even though the output actions of both types of lasers are the same, the underlying mechanism of polariton lasing is very different from that of standard photon lasers^[11]. In a photon laser, phase coherence is established by stimulated emission in a system with a massive number of photons with the help of so-called electron population inversion. Note that photon and polariton lasing can both occur in the same system, but in different regimes of parameters. In semiconductor microcavities, photon lasing may occur in the so-called Mott density limit, where the exciton binding energy diminishes because of Coulomb screening and the interaction between excitons. In this limit, electron–hole pairs cannot be approximated as bosonic quasiparticles. Here, the formation of an electron–hole plasma, the stimulated photon emission into the cavity mode, and electron population inversion are all allowed. With these conditions fulfilled, photon laser action begins^[121].

In polariton lasers, electron population inversion is not required. Here, the coherence originates from the formation of BECs [see Fig. 3(a) as reported by Kasprzak et al.^[122]], which occurs at an exciton density much lower than the Mott density. Since excitons are injected via external excitation, achieving the Mott limit for photon lasing would require more energy than reaching the BEC regime (a lower threshold). In this regime, the bosonic nature of excitons is conserved, and strong coupling between excitons and cavity photons allows the formation of polaritons. These composite quasiparticles then can obey Bose statistics, which is at the heart of polariton lasing^[116]. Here, injected high-energy polaritons go through several scattering processes (e.g., polariton–polariton, polariton–exciton, and polariton–phonon scatterings) and relax to a low-energy state in the LP branch where it forms a polariton BEC. The photon emission from the polariton BEC due to decay (finite lifetime) then acts as lasing action.

Fig. 3

Applications of microcavity exciton polaritons. (a) Angle resolved photoluminescence below threshold (left), at threshold (middle), and above threshold (right). The polariton condensate above threshold acts as a polariton laser^[122]. (b) Lattice simulator for realizing topological insulator using exciton polaritons under a strong magnetic field^[134]. (c) Left: schematic of a room temperature lattice simulator using intrinsic anisotropy of perovskite microcavities. Right: eigenvalues in trivial (W = 0) and topological (W = 1) phases of the simulated Su–Schrieffer–Heeger model^[135]. (d) Top: scheme of an all-optical polariton transistor. Bottom: real space emission image of address beams A and B and control beam C^[14]. (e) Top: schematic representation of a polariton neuromorphic computer for handwritten digit recognition. Bottom: experimental scheme of modulating a laser beam with spatial light modulator (SLM) and a CCD camera to get the output^[161]. (f) Left: proposal for realizing qubits using quantum fluctuations in polariton condensates. Right: fidelities of polariton quantum gates (SWAP and sSWAP) as functions of lifetime compared to the interaction strength^[168].

2.6.2.

Lattice simulation

BEC represents a macroscopic number of particles in the ground state of a system. It is thus suitable for simulating materials at low temperatures, where the ground state properties are the dominating features. Typically, the Hamiltonians representing materials are complex, and thus exact analyses of them are difficult. While approximate theories can be used to understand many important features, they are not reliable in all parameter regimes. This problem can be solved by simulating Hamiltonians using BEC in tunable physical systems. While tunability provides the reach to different parameter regimes, BEC provides accurate ground state features. Note that measurements on macroscopic numbers of particles provide accurate results with small statistical errors.

Typically, BEC is used to simulate lattice systems, which describe various condensed matter systems. In cold atomic systems, counterpropagating lasers are used to design optical lattices^[123]. Polariton BEC is also suitable for simulating intriguing lattice systems^[124], e.g., topological^[125–127], nonreciprocal^[128,129], flatband^[130,131], and honeycomb lattices^[132,133]. For instance, see Fig. 3(b) for a scheme of a simulated topological insulator using an exciton polariton lattice as was shown by Klembt et al.^[134]. One of the major advantages of polariton BEC is that it can operate at room temperature, which is much easier to achieve than reaching an extreme cryogenic temperature for atomic BEC lattices. For this reason, polariton simulators are scalable and readily implementable for room temperature operations [see Fig. 3(c), which is adopted from Su et al.^[135]]. Moreover, exotic features of materials such as halide perovskites and TMDs provide additional controls on the lattice design.

In polariton BEC, lattices can be implemented by using periodic external potentials. A single site in a lattice is created by a local potential well $V_j(\mathbf{r})$, which is centered around position ${\mathbf{r}}_j$. For instance, the potential well can be a circular region around point ${\mathbf{r}}_j$ such that $V_j(\mathbf{r})=-V_{\mathrm{Lat}}\mathrm{\Theta }(|\mathbf{r}-{\mathbf{r}}_j|-r_{\mathrm{Lat}})$, where $V_{\mathrm{Lat}}$ is the depth of the potential, and $r_{\mathrm{Lat}}$ is the radius of the circular region ($\mathrm{\Theta }$ is the Heaviside step function). This isolated site represented by $V_j(\mathbf{r})$ is a finite 2D potential box, which can have many bound eigenstates for a sufficiently large value of $V_{\mathrm{Lat}}$. Let us consider that these bound eigenstates are $|{\phi }_n^j\rangle$ with eigenvalues $E_n^j$, such that $[H_{\mathrm{KE}}+V_j(\mathbf{r})]|{\phi }_n^j\rangle =E_n^j|{\phi }_n^j\rangle$, where $n=0,1,2,\dots$, and $H_{\mathrm{KE}}$ is the kinetic energy part of the Hamiltonian. A lattice can be generated by introducing many such sites with $j=1,2,3,\dots$, such that the total Hamiltonian is given by $H_{\mathrm{KE}}+\sum _j{V}_j(\mathbf{r})$. The low-energy subspace of the total Hamiltonian can then be spanned by eigenvectors $|{\phi }_0^j\rangle$:

Lattice design can be greatly enhanced with extra features such as anisotropy and birefringence in perovskite microcavities. Here, the effective masses of polaritons along $x$ and $y$ axes are different and polarization dependent. Depending on the linear polarization projections ($X$ and $Y$) and crystal axes ($x$ and $y$), the effective LP mass can take four different values $m_{\mathrm{LP},iS}$, where the directional index $i=x,y$, and the linear polarization index $S=X,Y$. Thus, for a given polarization projection $S$, one obtains different hopping amplitudes by aligning the lattice sites in different directions. For instance, if two sites $j$ and $k$ are aligned along the $x$ axis,

Another element in lattice simulation is the so-called artificial gauge fields^[136]. Exciton polaritons are electrically neutral particles, and thus, they do not couple to the EM field in the same way as charged particles do. However, exciton polaritons do respond to the applied EM field due to their nonzero dipole moment. Using this effect, one can induce artificially created gauge fields for polaritons to simulate charged particles in an actual EM field^[137]. Such artificial gauge fields can be used to simulate various physical effects in polariton systems, e.g., quantum Hall effect, topologically protected propagation, and Hofstadter’s butterfly^[126,138]. Furthermore, other exotic effects can be simulated by utilizing polariton nonlinearity^[44,139,140].

2.6.3.

Optical computing

Optical computing uses light for propagation and processing information^[141]. Optical computers promise ultra-low power consumption, high processing speed, and extremely low losses. However, realizing an all-optical computer is a complex task. Present-day computers process data in the form of binary digits, which can have a value of zero or one. It is then processed with operations made with elementary gates, such as AND, OR, and NOT gates. One of the major challenges in the domain of optical computing is to find strong nonlinearity, such that these elementary gates can be implemented simultaneously in the same system. Exciton polaritons with strong nonlinear properties can be an excellent platform for optical computing. However, if these computers must compete with electronic computers, their room temperature operation is necessary. In this direction, an exciton-polariton-based optical transistor was shown to operate at room temperature^[15], and previously at lower temperatures as shown by Fig. 3(d)^[14]. In fact, transistors are used in digital computing as electronic binary switches, which can be put into “on” and “off” states by controlling flows of electrical currents with gate voltages. An optical version of this binary switch can be obtained, where propagation of an exciton polariton condensate can be controlled by optical fields. Recently, room temperature operation of an all-optical binary switch was indeed demonstrated using self-assembled perovskite microwires^[142]. Other required optical components such as memory devices^[143,144], diodes^[145,146], routers^[147–149], and optical amplifiers^[150] are required for room temperature optical computing. A universal logic gate set and a method of cascading them are also required^[151].

Another promising computing architecture is the optical spin simulator. Here, certain computational tasks are transformed to problems of finding the ground states of spin models. For instance, $XY$ spin simulators were shown to find the global ground state configuration using interconnected graphs of local exciton polariton condensates^[152]. In this method, the global minimum of the $XY$ Hamiltonian is reached by the so-called bottom-up approach, such that the system naturally condensates at the minimum energy state^[153]. This approach is very different from commonly used methods in classical and quantum annealers. Similarly, the classical Ising model can be realized with exciton polariton condensates, which can be used for global optimization of non-deterministic polynomial-time-hard (NP-hard) problems^[154,155]. Alternatively cellular automata based on exciton polaritons can be used for computing^[156].

2.6.4.

Neuromorphic computing

Artificial neural networks (ANNs) are information processing systems inspired by biological brains. ANNs are collections of interconnected nonlinear nodes that mimic biological neurons, and their connections are like synapses in brains. Typically, neurons are arranged in layers, where the connections can be allowed within and between layers. Information travels from the first (input) layer to the last (output) layer, possibly after going through multiple (hidden) layers in between. ANNs can be implemented as software in digital computers based on the so-called von Neumann architecture, where the processing unit is physically separated from the memory unit. The communication between these two physical units causes a time delay, which puts a limit on computing speed no matter how fast a given processor can work. This is known as the von Neumann bottleneck, which is considered as a problem that can be overcome only by changing the computing architecture. Since ANNs do not require physical separations between memory and processors, they can be physically implemented as independent computers. The physically implemented ANNs are known as neuromorphic computers, which benefit from native parallelization and are free of the von Neumann bottleneck. Among various possibilities, exciton polaritons can be excellent platforms for neuromorphic computing due to their unprecedented photonic nonlinearity. However, ANN works on the principle of learning, where network connections are slowly adopted to a set of parameters such that it can produce a desired output. For this, individual connections between nodes must be controlled precisely^[157,158]. This is a challenging task for networks with massive numbers of nodes. This problem was solved using a relatively lesser-known neural network architecture called reservoir computing^[159]. In reservoir computers, the main physical part of the neural network is kept fixed with a random set of connections and only a single layer of output connections is controlled^[160]. Here, the random network can be formed with polariton BECs in lattices with random hopping amplitudes [see, for instance, Fig. 3(e) where a polariton neural network is used for classifying handwritten digits]^[161,162]. The output layer is controlled externally after a readout in the form of photoluminescence from the polariton lattice. It was also shown that the driven-dissipative nature of exciton polaritons can be taken as an advantage for better performance^[159], and that that small-scale polariton reservoir computers can be used as building blocks for digital computing with natural robustness to noise and error^[163]. So far, these networks work in the domain of cryogenic temperature. The recent advancement in room temperature exciton polaritons inspires the possibility of polariton neural networks working in ambient conditions.

2.6.5.

Quantum simulation

Quantum simulators are devices that use quantum effects (e.g., entanglement and superposition) to understand physical systems through studying model systems^[125,164]. They can be used for a wide range of applications, including drug design and material manufacturing. It was suggested by Feynman that quantum devices would be better for modeling certain systems that cannot be efficiently done by classical devices^[165]. For instance, typically quantum systems are simulated using numerical tools such as mean-field theory, dynamical mean-field theory, density functional theory, and Monte Carlo algorithms. All of them have their limitations and cannot be applied for all systems in all regimes. Some can be used as complementary methods in different regimes of parameters. However, there are many examples where no numerical method can escape the slowdown in computational time due to an exponential rise of the Hilbert space dimension with respect to the number of particles. A typical example of such a system is a quantum spin model with arbitrary couplings between spins. For this reason, quantum devices are considered crucial for modeling quantum systems in large scale.

Exciton polaritons can be used as a platform to implement quantum simulators. In the strongly interacting regime, where interaction between exciton polaritons exceeds the linewidth, quantum correlations between different particles become important. In this regime, exciton polaritons can show quantum effects beyond mean-field condensate behavior. These effects can be used for quantum simulations. For instance, let us consider a Bose–Hubbard model, which is a prototypical system to study a Mott insulator to superfluid transition. The corresponding Hamiltonian is given by

Depending on the ratio between $U$ and $J$, one can get a superfluid phase with $J/U\gg 1$ or Mott insulator phase with $J/U\ll 1$. This Hamiltonian can be obtained in an exciton polariton lattice with strong polariton–polariton interaction. Similarly, the Jaynes–Cumming model^[166] and quantum spin models^[167] can be implemented, in principle, in semiconductor microcavities. Here, the driven-dissipative nature could be a limitation for simulating strictly equilibrium properties. However, driven-dissipative versions of these quantum models are equally interesting and can be used for simulating open quantum systems.

2.6.6.

Quantum computing

We have seen that an effective two-level system can be obtained using a strongly interacting single-mode polariton condensate. This two-level system can contain at most one quantum of excitation provided by an applied laser as schematically shown in Fig. 3(f), which is adopted from Ghosh et al.^[168]. Thus, this system can indeed serve as an elementary quantum bit or qubit for quantum computation. Let us now discuss how this system can be effectively controlled by the applied laser field. From Eq. (36), we find that the low-energy Hamiltonian describing the qubit is given by

Even though finding a suitable system for a qubit is a significant step, suitable quantum gates must be designed to obtain a functional quantum computer. It is shown that any quantum operation can be approximated as a combination of single- and two-qubit quantum gates. Furthermore, these gates can be chosen from a small set of universal gates. For instance, single-qubit gates and controlled-NOT (cNOT) are a universal gate set. Similarly, the square root of a SWAP (sSWAP) gate and single-qubit gates are also a universal gate set.

As can be seen in Eq. (40), the single-qubit Hamiltonian $H_Q$ is fully tunable by laser parameters. Thus, by sending a laser pulse, one can induce arbitrary unitary gate $U_Q(P,\varphi ,{\omega }_p)=\exp (-i\tau H_Q/\mathrm{\hbar })$, where $\tau$ is the pulse duration, and the pulse is a constant for a time window $0\le t<\tau$. Achieving two-qubit quantum gates is more subtle^[113]. Here, two qubits need to interact for a brief time to induce a gate. Indeed, polaritons in two different condensates can interact. For instance, polariton condensates can be achieved in the two different polarization components, and they can interact via the Hamiltonian ${\mathit{H}}_{\mathrm{P}-\mathrm{P}}$ given by Eq. (15). This interaction Hamiltonian in qubit space can induce a cNOT gate. Furthermore, polariton condensates can be coupled simply by placing them close to each other. If the distance is small enough, quantum tunneling of polaritons induces the so-called Josephson coupling between condensates. Such a coupling can induce sSWAP and SWAP gates between two qubits. Note that the Josephson coupling or the interaction between couplings can be effectively switched off by introducing a large energy detuning between qubits. Similarly, quantum gates can be realized using polariton propagation in nonlinear waveguides, where polariton interaction plays a central role^[170]. There are also schemes of quantum computing based on continuous variable states^[171,172].

3.1.1.

Polariton condensation in GaN

In 2002, GaN-based microcavities with a large exciton binding energy ($\sim 30\mathrm{meV}$ for bulk layers and $>40\mathrm{meV}$ for QWs), large Rabi splitting, and an efficient polariton relaxation rate were theoretically proposed as candidates for the realization of room temperature polariton lasers^[201]. Earlier works failed to reach the strong coupling regime in a GaN microcavity with monolithically grown GaN/AlGaN DBRs. This was caused by insufficient exciton–photon coupling in low-quality cavities due to a large lattice mismatch and thermal expansion coefficient. Afterwards, to reach the strong coupling regime in GaN, many efforts were devoted towards the improvement of microcavity quality via epitaxy approaches^{[200,202–204]}. Later, several works experimentally confirmed that Rabi splitting can reach up to 60 meV in GaN-based microcavities, which is about one order of magnitude larger than that of GaAs-based microcavities^[202–204]. In 2007, the first demonstration of room temperature polariton lasing using a GaN microcavity was reported^[31]. To get rid of the broad linewidth and quantum-confined Stark effect in nitride-based QWs, bulk GaN microcavities with a hybrid design were used, where the bulk $3/2\lambda$ GaN central spacer was sandwiched between the top and bottom DBRs [inset of Fig. 4(a), top left]. The sample structure and experimental reflectivity spectrum showing an LP mode at 3.4 eV are reported in Fig. 4(a), top left. In Fig. 4(a), bottom left, the theoretical angular dispersion of absorption without and with the resonant excitonic component was found at 3.42 eV, which suggests the realization of the strong coupling regime for this bulk GaN microcavity. Under pulsed non-resonant excitations, the occurrence of polariton condensation at 300 K was evidenced by showing typical experimental signatures, such as rapid thermalization, linewidth narrowing, superlinear increase of photoluminescence intensity, and blueshifts of emission energy, as shown in Fig. 4(a), right top and bottom panels. A low threshold density of $29\mathrm{\mu J}{\mathrm{cm}}^{-2}$ was achieved, which was one order of magnitude smaller than what was obtained in vertical surface-emitting lasers based on (In,Ga)N^[205]. Subsequently, room temperature polariton lasers were realized in microcavities based on bulk GaN, hybrid AlInN/AlGaN QW, GaN nanowires, and GaN membranes with low thresholds^{[173,206–208]}.

Fig. 4

Exciton polaritons in GaN microcavities and ZnO microcavities. (a) GaN polariton lasing at room temperature^[31]. Top left panel: bulk GaN microcavity (inset) and reflectivity at 300 K with lower polariton mode marked by the arrow. Bottom left panel: theoretical simulation of polariton dispersion. Right top and bottom panels: photoluminescence spectra as a function of pump power (20 μW to 2 mW). (b) Electrical injection GaN-based microcavity diode polariton lasing at room temperature^[210]. Top left panel: schematic image of GaN-based microcavity diode. Top right panel: measured data (black dot) of polariton dispersion simulated by a coupled harmonic oscillator model. Bottom left panel: angle-resolved electroluminescence showing the polariton condensation at ground state at high current injection densities. Bottom right panel: transition from polariton lasing to photon lasing. (c) Polariton parametric scattering (PPS) in ZnO microcavity at room temperature^[222]. Top left panel: scanning electron microscope (SEM) image of ZnO microwire with whispering gallery modes. Top right panel: multiple lower polariton modes along TE polarization (left image) and TM polarization (right image). Bottom left panel: polariton lasing in the ZnO microcavity at pumping power of 0.33 μW. Bottom middle panel: polariton parametric scattering in the ZnO microcavity at pumping power of 1.3 μW. Bottom right panel: schematic diagram of the PPS between the adjacent polariton branches. (d) Weak lasing in one-dimensional ZnO-Si polariton superlattices at room temperature^[29]. Top left panel: SEM image of ZnO microwire placed on a periodic Si grating. Top right panel: folded polariton dispersion caused by a polariton superlattice. Bottom left panel: polariton lasing at the excited polariton states (edges of the Brillouin zone). Bottom right panel: spatially resolved photoluminescence showing the emission spots settled in the periodic potential.

3.1.2.

Electrical injection in GaN

An important milestone from a practical point of view was achieving an electrically injected polariton laser. The early experimental demonstrations of electrically pumped polariton lasers were presented in GaAs-based QWs by Bhattacharya et al. and Schneider et al. in 2013, which operated at cryogenic temperatures and required a magnetic field^[12,209]. In 2014, Bhattacharya’s team achieved room temperature electrically pumped polariton lasing in a bulk GaN-based microcavity diode, where current injection and optical feedback in the microcavity were in the orthogonal directions^[210]. The bulk GaN diode consists of a thick GaN p-n junction, while the microcavity consists of DBR mirrors deposited on opposite sides along the cavity length, as schematically shown in Fig. 4(b), top left. Under a low forward bias current at room temperature, the two experimental emission peaks simulated by a coupled harmonic oscillator model are determined to be the LP and UP transitions in the strong coupling regime, as shown in Fig. 4(b), top right. Accompanied by a collapse of the emission linewidth and small blueshift of the emission peak, polariton condensation occurs towards the ground state above the threshold of current injection density at $169\mathrm{A}/{\mathrm{cm}}^2$ in Fig. 4(b), bottom left. With a further increase in current injection density, photon lasing at the second threshold of $44\mathrm{kA}/{\mathrm{cm}}^2$ was observed, as reported in Fig. 4(b), bottom right. Such a demonstration of a room temperature electrically injected polariton laser paves the way to exploit polaritons towards practical on-chip device applications with low power consumption.

3.1.3.

Polariton condensation and parametric scattering in ZnO

Owing to the large exciton binding energy ($E_b\sim 60\mathrm{meV}$) and strong oscillator strength^[211], ZnO emerges as another type of wide-bandgap semiconductors, which can sustain stable microcavity exciton polaritons at room temperature. Zamfirescu et al. predicted a large Rabi splitting around 50 meV in ZnO-based microcavities and proposed polariton lasing with an ultra-low threshold^[211]. Experimentally, the existence of UP and LP branches, and thus, the achievement of strong light–matter coupling, were first observed through polarized photoluminescence and reflectance measurements in ZnO epitaxial films^[212]. The large longitudinal-transverse splitting ($\sim 5\mathrm{meV}$) was found in the reflectivity, which is two orders of magnitude larger than that of GaAs-based microcavities^[213]. Later, the value of Rabi splitting was experimentally measured in ZnO systems with different designs. Standing-wave exciton polaritons were observed in ZnO nanowire cavities with a Rabi splitting larger than 100 meV^[214], while a Rabi splitting of about 50 meV was observed in bulk ZnO-based hybrid microcavities^[215,216]. Subsequently, realizations of room temperature polariton lasers were reported with hybrid bulk ZnO microcavities and ZnO whispering gallery mode resonators^[30,217,218]. The polariton condensate (tuning the excitonic fraction ranging from 17% to 96%) was achieved in a high-quality ($Q\sim 4000$) hybrid bulk ZnO microcavity, showing a large Rabi splitting of $\sim 240\mathrm{meV}$^[219].

All these works also stimulated the study of nonlinear effects in ZnO-based systems. In particular, parametric scattering processes, which are intrinsically linked to the excitonic component of exciton polaritons, have been investigated. The polariton parametric effect was first achieved in planar microcavities embedding InGaAs QWs^[23] and later observed in a CdTe microcavity up to 220 K^[220], Similar results were also obtained in GaAs microcavities in both 2D and 1D geometries at cryogenic temperature^[221] and further extended to room temperature operation by Xie et al. in 2014 with a ZnO microwire cavity^[222]. Uniaxial ZnO microwires with hexagonal planes [Fig. 4(c), top left] intrinsically form whispering gallery resonators, providing multiple polariton branches with TE and TM polarizations as observed in Fig. 4(c), top right. Under non-resonant excitation, polaritons condense at the ground state of a polariton branch, when pump power reaches a certain threshold value ($\mathit{P}=0.33\mathrm{\mu W}$); see Fig. 4(c), bottom left. With a further increase in excitation power ($\mathit{P}=1.3\mathrm{\mu W}$), driven by the nonlinear parametric process, the polariton condensate at the ground state scatters to an adjacent branch, forming idler and signal states with the same energy but opposite momenta at high-$k$ states, as reported in Fig. 4(c), bottom middle. Unlike previous studies of resonant degenerate parametric scattering in CdTe- or GaAs-based microcavities at low temperatures, where injection is performed at the inflection point, the non-resonant interbranch parametric scattering in a ZnO microwire at room temperature provides a novel way for generating balanced polariton pairs and developing polariton nonlinearity.

3.1.4.

Polariton condensation in ZnO superlattices

Initially, the low-temperature condensate in polariton superlattices was demonstrated in GaAs-based systems. In 2015, room temperature weak lasing was achieved in ZnO systems by Zhang et al.^[29]. A ZnO microrod with a hexagonal cross section was placed on a Si grating that introduced periodic potentials to trap polaritons. Limited by the small trapping potential, a small gap was seen in the ZnO polariton dispersion; see Fig. 4(d), top right. Above the threshold, room temperature polariton lasing takes place at the excited states ($\pi$-states), namely, the edges of the mini-Brillouin zone in reciprocal space near the gap, rather than at the ground state (zero state), as can be seen in Fig. 4(d), top right. The corresponding spatially resolved photoluminescence [Fig. 4(d), bottom left] shows that the emission spots settle in the periodic potential, where the period of emission in real space is the same as the period of the superlattice [Fig. 4(d), bottom right]. However, with a further increase in excitation power, the period of the condensate was found double the period of the superlattice, suggesting that the system is operating in a weak lasing regime with spontaneously broken symmetry. Interestingly, ZnO systems are also used for realizing Bosonic cascade lasers^[223].

However, there are certain limitations in GaN- and ZnO-based polaritonic devices. For instance, these systems have relatively weak nonlinearity, which is a major limitation for realizing polaritonic applications. While the large bandgaps of these systems enable device stability, they constrain the systems to ultraviolet emissions and low conductivities for electric injection. The quality of microcavity fabrication also has not been improved to the level of GaAs or CdTe systems. Note that a limited quality factor is a major hindrance for practical polaritonic devices such as polaritonic switches and transistors. However, horizontal polariton lasers and amplifiers in ZnO-based waveguides are recently reported, suggesting an alternative geometry for high-quality microcavities for future applications^[224].

3.2.1.

Strong coupling and BEC in OSCs

The first demonstration of room temperature exciton polariton formation in an OSC-based microcavity was reported by Lidzey et al. in 1998 by using a polystyrene-doped Zn-porphyrin (4TBPPZn) film embedded in a hybrid DBR–metallic planar microcavity, achieving a Rabi splitting of 110–160 meV^[32]. Since then, drop-casted low-molecular-weight emitters and J-aggregated cyanine dyes, either in pure film or hosted in matrices, have become the most popular organic materials for polaritonic studies^[233–239]. This choice has been dictated by the presence of a sharp and narrow absorption peak with an extremely small Stokes shift as well as large oscillator strength. Indeed, although OSCs have in general large oscillator strength, it was thought that the large Stokes shift as well as the inhomogeneous broadening could prevent the formation of exciton polaritons, and these materials possess a favorable physical structure suitable to achieve the strong coupling regime. For instance, the film used by Lidzey et al. shows a relatively narrow Soret band (FWHM of 90 meV at 2.88 eV), while the cyanine dye in the J-aggregate form is characterized by a strong and narrow absorption shifted to lower energy with respect to the monomer (FWHM of 50 meV at 1.78 eV). However, this situation is not universally true, and materials strongly affected by inhomogeneous broadening and a large Stokes shift, e.g., conjugated polymers, have been successfully integrated in planar microcavities achieving Rabi splitting up to 400 meV^[240]. Follow-up works demonstrated that the Stokes shift could affect the relaxation dynamics but does not hinder the formation of polaritons^[241]. Moreover, it is important to note that under a strong coupling regime, the modification of the energy levels of the molecule leads, in some cases, to increased quantum yields and reduced inhomogeneous broadening, thus making molecules with low quantum yield and broad linewidths still attractive for polaritonic applications^[242,243].

Nevertheless, the formation of a polariton condensate remained elusive, and these early works identified the large static disorder present in OSC films as first and foremost responsible for hindering BEC formation. In disordered films, not all molecules have the same transition dipole moment orientation, and this misalignment could induce Förster transfer between them, thus suppressing the population^[244,245]. Consequently, the attention drifted on to the study of more ordered thermally evaporated polycrystalline thin films and single organic crystals, which are of particular interest to control vibronic states, decrease optical losses (thus increasing the quality factor), increase the damage threshold, as well as exploit nonlinear polariton processes^[246–249].

All these efforts eventually culminated in the first realization of polariton lasing in a planar microcavity reported by Kéna–Cohen and Forrest in 2010^[33]. Anthracene single crystals were formed in nanometer-thick lithographically defined channels between the two DBRs, as shown in Fig. 5(a), left. The strong birefringence of the crystalline structure leads to the formation of two LP branches, which depend on the sample orientation^[249]. The absorption spectrum shows clear vibronic transitions, namely, 0–0, 0–1, and 0–2, that are strongly coupled with the cavity mode, thus forming two additional middle branches (MP1b, MP2b), as reported in Fig. 5(a), right. Under femtosecond pulsed excitation, a spectral narrowing, spatial modulation of the pump spot, as well as shortening of the emission lifetime and changes in polariton distribution are observed above threshold. Moreover, polariton lasing is achieved only when the energy separation between the ground state of the LP branch and the excitonic reservoir is equal to the energy of the first vibronic sublevel, which confirms a vibronic-assisted radiative pumping of the LP branch as the main relaxation pathway. However, even though compelling evidence of polariton lasing has been reported, the nonlinear polariton character, i.e., the blueshift of the polariton mode, was not observed, and the study of coherence properties was not reported.

Fig. 5

Exciton polariton condensate and superfluidity in OSC microcavities. (a) Left: schematic of the microcavity and anthracene crystal structure. Right: calculated dispersion along the a-crystal axis showing the formation of LP branch, UP branch, and two middle branches arising from strong coupling of intramolecular vibronic modes, as indicated in the absorption spectrum^[33]. (b) Left top and bottom: DBR microcavity and chemical structure of the MeLPPP polymer. Top middle and right: momentum resolved map for excitation power below (middle) and above threshold (right). Bottom middle and right: same as in the top panel but for real space emission. The insets show the g⁽¹⁾ interferograms. Scale bars are 20 µm^[176]. (c) Top left and right: g⁽¹⁾ maps for increasing pumping fluence. Scale bars are 5 µm. Bottom left and right: vertical cuts of the g⁽¹⁾ maps reported in the top panel^[250]. (d) Left: sketch of the optical setup and microcavity embedding TDAF. Top middle and right: real space emission for low (middle) and high polariton density (right). Bottom middle and right: same as in the top panel but for the momentum space emission^[6].

In 2014, two works showed the presence of such interactions in OSC-based polariton microcavities by using two different molecular structures: a small organic molecule and a polymer^[175,176]. Daskalakis et al. used a thermally evaporated 2,7-bis[9,9-di(4-methylphenyl)-fluoren-2-yl]-9,9-di(4-methylphenyl) fluorene (TDAF) film pumped with a femtosecond pulsed laser^[175]. TDAF is employed mainly as light-emitting diode material and shows strong absorption due to a preferential orientation of the molecules along the in-plane direction. Plumhof et al. used a spin-coated methyl-poly(para-phenylene) (MeLPPP) film pumped with a picosecond pulsed laser^[176]. A scheme of the microcavity along with the chemical structure is reported in Fig. 5(b), left top and bottom. The polymeric film, although intrinsically amorphous, has a limited inhomogeneous broadening, thus showing absorption linewidths narrower than that of typical conjugated polymers. In both materials, the presence of vibronic replicas is visible in the absorption (MeLPPP) or emission (TDAF) spectrum. When the pumping power exceeds the power threshold, the presence of polariton interactions is confirmed through the blueshift of the LP branch, as reported in Fig. 5(b), top, for the polymeric cavity. Moreover, the build-up of spatial coherence [Fig. 5(b), bottom] and linear polarization as well as the observation of shot-to-shot spatial fluctuations were also reported in both systems. Despite the intrinsically disordered nature of the active material, a follow-up work demonstrated the increase in spatial correlation of a TDAF-based microcavity as a function of the above-threshold pumping power. Specifically, the coherence length of the condensate (${\lambda }_c$) increases from ${\lambda }_c=12\mathrm{\mu m}$ (at $1.2{\mathit{P}}_{\text{th}}$) up to ${\lambda }_c=38\mathrm{\mu m}$ (at $1.9{\mathit{P}}_{\text{th}}$), as shown in Fig. 5(c), left and right, respectively^[250].

Recently, great attention has been paid in reducing the condensation threshold and achieving tunable polariton lasing. This led to the observation of polariton lasing in a great variety of small molecules, self-assembled crystals, and polymers. To date, polariton condensation has been achieved in planar microcavities embedding oligomer pentafluorene, boron-dipyrromethene (BODIPY), H-aggregate 4,4-bis[4-(di-p-tolylamino)styryl]-biphenyl (DPAVBi) microbelt, oligo(p-phenylene), and poly-(9,9-dioctylfluorene) (PFO)^{[119,251–256]}.

At the state of the art, polariton lasing has been achieved in both amorphous and crystalline OSCs. Although a high quantum yield and fine tuning of the vibronic coupling are fundamental to achieve condensation, the role of static disorder is debated. On one side, the disorder leads to detrimental effects such as wrong dipole orientation and increased Förster transfer, while on the other side, the presence of disorder potentials maximizes the local polariton population, thus helping condensation^[41].

3.2.2.

Superfluidity in organics

The presence of polariton interactions in OSCs led to the observation of a fundamental condensate-related phenomenon, i.e., polariton superfluidity, which profoundly depends on the nonlinear character of polaritons. The same TDAF-based microcavity employed by Daskalakis et al. was used by Lerario et al. in 2017 to investigate superfluidity at room temperature^[6]. The adopted excitation scheme closely follows the one used in GaAs-based microcavities, thus probing the frictionless flow around obstacles through the formation and suppression (for sufficient polariton density) of the Rayleigh scattering ring^[5,257]. The system is pumped by using a femtosecond pulsed excitation resonant with the LP branch, which coherently injects polaritons with finite in-plane momentum (enabling in-plane propagation) in a transmission geometry, as reported in Fig. 5(d), left. At low polariton densities, the artificially created obstacle induces isoenergetic Rayleigh scattering, thus generating a counterpropagating flow that interferes with the incoming polariton population. Consequently, an interference pattern and a shadow cone are formed in real space, while a bright scattering ring appears in momentum space, as reported in Fig. 5(d), top and bottom middle panels, for an upward polariton flow. At high polariton density, the shadow cone, interference pattern, and scattering ring disappear, thus showing the frictionless character of the polariton condensate, as shown in Fig. 5(d), top and bottom right panels. In good agreement with the theoretical description provided in Section 2.4, the experimental results can be interpreted considering the perturbation theory for elementary excitations developed by Bogoliubov. Therefore, for a suitable polariton density, the scattering suppression is the result of the LP branch linearization around the pumping point, which is renormalized such that no isoenergetic states are available for scattering. Moreover, although the phase is initially imprinted by the pump, the authors employed a homodyne interferometric technique to study the phase of the polariton fluid, thus revealing the proliferation of vortex pairs after excitation. The formation of a vortex–antivortex pair is observed for low particle density and is fully suppressed in the superfluid regime, further corroborating the observation of room temperature polariton superfluidity. The pulsed excitation employed by the authors allows a free evolution of the system after the initial phase imprinting, and it is an important advantage over previous demonstrations of polariton superfluidity in GaAs-based microcavities, which have been carried out under continuous wave (CW) resonant excitation^[5]. Indeed, a follow-up theoretical paper interpreted the scattering suppression under CW pumping as a result of the external fixing of the phase rather than superfluidity, suggesting that vortices and persistent currents cannot be formed when the phase is locked by the external CW pump^[258].

3.2.3.

Strong coupling and BEC in plasmonic structures

Although the planar microcavity remains the preferred design for polaritonic studies, great attention has been paid in the investigations of different optical platforms capable of reducing the volume of the optical mode. In this respect, plasmonic systems, despite offering quality factors lower than dielectric microcavities, are able to confine and enhance the EM field at a subwavelength level, thus bringing light–matter interaction at the nanoscale. Like early works on OSC-based microcavities, the strong coupling between surface plasmons sustained on thin metallic layers and organics was first observed in J-aggregate cyanine films, reporting Rabi splitting exceeding 200 meV^[259–261]. Thereafter, TDBC J-aggregates, rodhamine 6G, fluorescent molecule (DiD), modified rylene dyes, β-carotene, and tetracene crystals have been strongly coupled in nanoparticle arrays, individual dimers, and nanorods^{[65,72,78,262–271]}. However, individual plasmonic nanostructures, although confining the EM field at the nanoscale, cannot provide the necessary small volume to couple a single or few molecules, therefore hindering access to quantum effects at room temperature^[272]. To create such a small nanocavity, Chikkaraddy et al. used the so-called nanoparticle-on-mirror geometry in which an extremely small volume is achieved in the gap between a flat gold mirror and a plasmonic nanosphere. The bottom-up nanoassembly approach employed by the authors allowed them to carefully align the molecule into the plasmonic gap, i.e., the molecule transition dipole is aligned along the plasmonic field within the gap. Under these conditions, a single methylene blue molecule has been successfully coupled as shown in Fig. 6(a), left^[78]. The strong coupling is confirmed by studying two films with a different molecule orientation. A Rabi splitting of 90 meV is visible when the transition moment is aligned with the TM polarization of the gap plasmon (vertical molecule orientation), as shown in Fig. 6(a), right. The possibility to bring polariton physics at the single-molecule level could enable quantum effects at room temperature.

Fig. 6

Plasmonic systems coupled with OSCs. (a) Left: schematic of a methylene-blue molecule in the gap of a plasmonic nanoparticle-on-mirror geometry. Right: scattering spectra of isolated nanoparticles with the transition dipole moment parallel (top) and perpendicular to the mirror (bottom)^[78]. (b) Left top and bottom: schematic of the nanoparticle array covered with dye molecules and corresponding absorption and emission spectra of the rylene dye in PMMA. Right top and bottom: experimental g⁽¹⁾ interferogram (top) and map of g⁽¹⁾ correlation function (bottom)^[72,275].

Moreover, early attempts to achieve polariton lasing in plasmonic systems remained unsuccessful due to the inefficient relaxation and saturation of strong coupling^[273]. The first work reporting polariton lasing employed an array of silver nanoparticles strongly coupled with a thin layer of a modified rylene dye^[72]. The sample structure along with absorption and emission spectra is shown in Fig. 6(b), left top and bottom. This kind of plasmonic structure produces diffracted waves that couple the localized plasmon of individual nanoparticles together, leading to narrower resonances and higher quality factors^[266,274] Polariton lasing is obtained from a dark mode and shows a pronounced threshold in emission intensity accompanied by spectral narrowing. More importantly, a follow-up work confirmed the buildup of long-range spatial coherence and the presence of nonlinear interactions, as shown in Fig. 6(b), right top and bottom^[275].

3.2.4.

BEC under quasi-CW excitation

The above-discussed demonstrations of polariton lasing in organics have in common one important aspect, i.e., ultrashort pulsed excitation (100 fs–10 ps), which is required to minimize exciton–exciton annihilation. However, from a practical viewpoint, achieving CW or quasi-CW polariton lasing would certainly lead the way to further development towards electrical injection and the study of hydrodynamics of polaritons. In this respect, polariton condensation in novel organic molecules has been recently reported by using nanosecond excitation^[276–278]. Dietrich et al. used a biologically produced enhanced green fluorescent protein (eGFP) in which a $\mathrm{\beta }$-sheets structure surrounds the chromophore^[276]. The $\mathrm{\beta }$-sheets structure acts as a barrier and helps in reducing the exciton−exciton annihilation, suppressing biomolecular quenching, and ensuring a 3 to 4 nm separation between chromophores, as reported in Fig. 7(a), left top and bottom. The microcavity, fabricated by using a simple lamination process, consists of two DBRs sandwiching a 500 nm thick eGFP film and is pumped by using nanosecond pulses (7 ns). When the threshold is crossed, a clear polariton condensate is formed in the ground state and exhibits a blueshift by further increasing the pumping power, a sign of nonlinear interactions, as shown in Fig. 7(a), right top. Importantly, for excitation power one order of magnitude higher than the polariton lasing threshold, a transition to weak coupling, and thus to photonic lasing, is observed, as shown in Fig. 7(a), right bottom. In this power range, no interaction-induced blueshift is visible. At the state of the art, this is the only organic system showing the presence of two thresholds. Recently, similar results were obtained in a high-quality-factor 0D microcavity filled with a red fluorescent protein (mCherry) and pumped with a nanosecond laser (7 ns)^[277]. The condensate formation is underlined by a nonlinear increase in emission, linewidth narrowing, and build-up of linear polarization. Remarkably, the coherence time exceeds 150 ps. Moreover, a nanosecond pumped polariton condensate has also been recently achieved in a BODIPY film (4 ns)^[278].

Fig. 7

BEC under quasi-CW pumping, polariton devices, and topological polariton lasing in OSCs. (a) Left top and bottom: schematic of the eGFP molecular structure (top) and corresponding molecular arrangement in solid state (bottom). Top left, middle, and right: momentum space map for excitation power below (left), at threshold (middle), and above threshold (right). Bottom: two thresholds behavior and transition from polariton lasing to photonic lasing^[276]. (b) Left: schematic of the MeLPPP microcavity and pumping scheme. The pump is tuned one vibron above the control beam (ground state). Top: schematic of the polariton transistor. Bottom left and right: schematic of the first amplification stage (left) and second amplification stage (right) demonstrating the cascadability^[15]. (c) Left: ground state stimulated polariton population by decreasing the seed power (from top to bottom). Right: total population triggered by 2.7 (top) and one photon per pulse (bottom)^[279]. (d) Top: schematic of the Su–Schrieffer–Heeger chain; red dots indicate the boundary and edge defects. Middle panel: below threshold energy-resolved momentum space collected by exciting the bulk of the chain (left) and the boundary defect (middle). The corresponding real space emission including the boundary defect is reported in the right map. Bottom: above threshold energy-resolved momentum space collected by exciting the boundary defect (left). The corresponding real space interferogram and g⁽¹⁾ correlation function including the boundary defect are reported in the right-top and right-bottom images, respectively^[293].

3.2.5.

OSC polariton devices

The use of OSCs for the realization of polaritonic devices could remove the constraint imposed by cryogenic temperature operation of GaAs- or CdTe-based microcavities^[2]. In 2019, Zasedatelev et al. successfully realized an all-optical polariton transistor in an MeLPPP microcavity by making use of a vibron-mediated bosonic stimulation process^[15]. The stimulation process was achieved by employing two beams: a weak control resonant with the ground state of the LP branch and a strong off-resonance pump tuned one vibron above the ground state, as shown in Fig. 7(b), left. The control beam can trigger ultrafast vibronic-assisted bosonic stimulation, and the whole process can be seen as a switch where the pump forms the address state gated by the weak control beam, as shown in Fig. 7(b), right top. The authors also demonstrated the cascadability of the device by implementing two-stage amplification in which the amplified emission from the first stage is redirected onto the chip and further amplified by a second pump in a double-pump–control scheme, as shown in Fig. 7(b), right bottom. Moreover, OR and AND logic gate operations were also realized by coupling three transistors in a single-pump–double-control scheme. A follow-up work performed in the same microcavity investigated the control over the condensate occupancy^[279]. The microcavity detuning was further optimized leading to a reduced threshold for spontaneous condensate formation (no seed in the ground state) of ${\mathit{P}}_{\text{th}}\approx 8\mathrm{\mu J}/{\mathrm{cm}}^2$. The same two-beam excitation scheme was adopted, and the power of the control beam was decreased at the few photons level. Even though by decreasing the control power the emission of the triggered condensate decreases, a clear stimulation was visible by injecting less than three photons per pulse, as shown in Fig. 7(c), left. Moreover, by switching on and off the seed (averaged over 300 single-pulse realizations), a clear difference was achieved at the single-photon level, as shown in Fig. 7(c), right, for 2.7 (top) and one (bottom) photon per pulse. The single-photon switch of the condensate population opens new horizons for practical implementations at the fundamental quantum limit. However, although these results are important steps towards the realization of OSC-based polariton devices, realistic applications in cascadable-on-chip technologies require suitable in-plane propagation, a property that has been exploited in high-quality III–V semiconductor microcavities. At the state of the art, OSC-based microcavities present short lifetimes and the rigid microcavity design limits the group velocity, thus preventing suitable in-plane propagation. In this respect, high group velocity lossless modes, such as the Bloch surface wave (BSW), have been proposed to maximize the propagation properties of organic polaritons^[280–284]. Remarkably, by strongly coupling a thermally evaporated layer of Lumogen F Red with the BSW, it has been demonstrated that polaritons are able to propagate up to 100 µm ($\mathit{X}\approx 15\%$). The presence of nonlinear interactions was also evidenced. The main drawback of the BSW is the excitation/detection scheme, which requires the use of an oil immersion objective, prism, or grating to extract the radiation confined in total internal reflection. In this respect, Tamm plasmons, i.e., a hybrid metallic–dielectric mode located within the light cone and sustained at the interface between a metal and a DBR, have been proposed as a powerful platform to enhance propagation. This platform has been recently used to couple BODIPY dye dispersed in a transparent matrix^[285]. Moreover, cavity-free systems, such as 1D wire and microribbons, have been employed to observe polariton propagation^[286,287]. In particular, the N,N’-bis(2,6-diisopropyl phenol)-3,4,9,10-perylenetetracarboxylic diimide (PDI-O) microribbons, which naturally sustain Fabry–Pérot modes, are extremely interesting due to their capability to undergo polariton condensation. Indeed, above threshold, the condensate can be tuned such that it acquires a finite wave vector, and thus propagation into the microribbons. Finally, it is important to note that other open systems, such as metallic and dielectric metasurfaces, have been suggested as suitable platforms to overcome the limitations imposed by the large lateral footprint of planar microcavities, and are thus suitable to realize miniaturized and compact devices^[288,289].

3.2.6.

Potential landscape engineering and topology

The possibility to achieve strongly interacting BECs at room temperature in OSC-based microcavities makes organic polaritons extremely interesting for quantum simulation applications and topological lasers. In III–V semiconductor microcavities, the desired spatial potential can be easily introduced by local modifications of the photonic mode via common lithographic techniques. However, organics are less tolerant to solvent exposure, and the spatial potential is in general introduced by using solvent-free techniques. For instance, the pattern can be imprinted into the substrate via focused ion beam milling and subsequently transferred into the DBR grown on top. The as-fabricated mirror can be used either in open cavity systems or as a master for the deposition of organic film. This fabrication approach has been used to integrate 0D, 1D, and 2D potentials in microcavities embedding MeLPPP and mCherry^{[131,277,290–292]}. In early studies, the feasibility of the method was demonstrated through the realization of single- and double-Gaussian defects capable of emulating photonic molecules and sustaining the formation of a localized condensate in MeLPPP microcavities^[290,291]. More importantly, in a follow-up work, the authors used a 2D Lieb lattice, achieving condensation into the s-, p-, and d-lattice bands by tuning the microcavity detuning, a key step toward the investigation of topologically protected modes^[131]. However, the high disorder of the polymeric film limited the quality factor ($Q_{f }\approx 600$), thus hindering the observation of the full Bloch band structure.