Dynamics and condensation of polaritons in an optical nanocavity coupled to two-dimensional materials

We present a comprehensive investigation of the light-matter interaction dynamics in two-dimensional materials coupled with a spectrally isolated cavity mode in the strong coupling regime. The interaction between light and matter breaks the translational symmetry of excitons in the two-dimensional lattice and results in the emergence of a localized polariton state. Employing an approach involving transformation to exciton reaction coordinates, we derive a Markovian master equation to describe the formation of a macroscopic population in the localized polariton state. Our study shows that the construction of a large-scale polariton population is affected by correction terms addressing the breakdown of translational symmetry. Increasing the spatial width of the cavity mode increases the Coulomb scattering rates while the correction terms saturate and affect the system's dynamics progressively less. Tuning the lattice temperature can induce bistability and hysteresis with different origins than those recognized for quantum wells in larger microcavities. We identify a limit temperature $T_{\mathrm{l}}$ as a key factor for stimulated emissions and forming a macroscopic population, enriching our understanding of strong coupling dynamics in systems with extreme confinement.

Figure 1

Ground-state polariton occupation as a function of pump power for different in-plane confinement lengths, $L$, of the optical cavity. The curves are computed by adapting the model from Ref. [10] to the case of a system with a single optical mode. For each confinement length, the star indicates the Mott transition.

Figure 2

Schematic representation of a sheet of 2D material coupled to an optical nanocavity, with a single optical mode localized at the center of the bowtie. Strong light-matter coupling between the resonant electromagnetic field and the excitons in the 2D sheet leads to the formation of a localized polariton state.

Figure 3

Schematic representation of the construction of the exciton reaction coordinate formulation. (a) The localized resonant field, with $a$ the operator describing the quantum properties of the resonant field, is coupled to a continuum of exciton modes with momentum $\mathbf{k}$, described by the annihilation operators $b_{\mathbf{k}}$. (b) The strength of the light-matter interaction is described through the coupling strength $G_0$ between the resonant field and a single collective exciton reaction coordinate with annihilation operator $B_0$, which, in turn, is coupled to an environment of residual exciton modes described by the operators $B_i$. (c) In the strong coupling regime, the Hopfield transformation [20, 21] enables a description of the system through polariton quasiparticles with annihilation operators $p_{\mathrm{lp}/\mathrm{up}}$, forming the upper and lower polariton branch. For the present applications, the former is energetically decoupled from the dynamics and is therefore neglected.

Figure 4

Sketch of the scattering between two excitons in the reservoir. After the scattering, a particle is captured into the lower polariton state, and an exciton is scattered to a high-energy state in the reservoir. The subscript $i$ labels the bosonic states in the ERC and replaces the in-plane momentum $\mathbf{k}$ in the new basis.

Figure 5

Numerically computed in-scattering rate for $G_0=3.8\mathrm{meV}$ and different optical cavity in-plane confinements. The black dotted line is the result expected for a quantum-well model where translation invariance is preserved, in which case it is possible to derive an analytic expression for $W_{\mathrm{in}}$ [10]. The inset shows the behavior of the scattering-in rate for $L=20\mathrm{nm}$, for three different coupling strengths.

Figure 6

Sketch of the scattering between a lower polariton and a higher-energy exciton. After the scattering, two lower exciton states are filled. The lower polaritons can scatter with any exciton in the reservoir as long as energy is conserved in the process.

Figure 7

Numerically computed $W_{{\mathrm{in}}_{\mathrm{c}}}$ scattering rate for $G_0=3.8\mathrm{meV}$ and different optical cavity in-plane confinement lengths. Increasing $L, W_{{\mathrm{in}}_{\mathrm{c}}}$ initially grows faster, but increasing the in-plane cavity length even more, its value remains stable.

Figure 8

Top plot, polariton occupation as a function of pump power for different in-plane confinement lengths of the optical cavity, $G_0=3.8\mathrm{meV}$ and $T_{\mathrm{L}}=8\mathrm{K}$. The small dots in each curve label the point where the corresponding occupation number of the excitons $N^x\sim 1$ and Maxwell-Boltzmann distribution breaks down in this limit. The black star refers to the value corresponding to the Mott transition of $n_x$. The black dotted line represents the system's dynamics without correction terms, assuming analytic expressions for in-scattering and out-scattering rates [10] and considering a single polariton state. The bottom plot shows the variation in the exciton reservoir temperature.

Figure 9

Top plot, the evolution of the scattering rates as a function of the pump power $p_x$ for $G_0=3.8\mathrm{meV}$ and $L=20\mathrm{nm}$. Middle plot, evolution of the total scattering-out rate ${\mathrm{\Gamma }}_{\mathrm{out}}\left(p_x\right)=n_x^2{W}_{{\mathrm{out}}_1}+n_x^2{W}_{{\mathrm{out}}_2}+n_x^2{W}_{{\mathrm{out}}_{\mathrm{c}}}$ compared to $n_x{W}_{{\mathrm{out}}_1}+n_x^2{W}_{{\mathrm{out}}_2}$. Bottom plot, corresponding evolution of the exciton density $n_x$. The dotted black line is the $p_x$ value corresponding to the $N_0$ threshold. The black stars indicate the onset of the Mott transition.

Figure 10

Polariton occupation number $N_0$ as a function of the pump power $p_x$ for different coupling strengths $G_0$ and lattice temperatures. In the top plot $T_{\mathrm{L}}=1\mathrm{K}$, in the middle $T_{\mathrm{L}}=8\mathrm{K}$ and in the bottom $T_{\mathrm{L}}=12\mathrm{K}$. The small dots in each curve label the point where the corresponding occupation number of the excitons $N_x\sim 1$ and Maxwell-Boltzmann distribution breaks down. The black star indicates the density at which the Mott transition for the exciton reservoir occurs. Dashed regions of the curves indicate an exciton density higher than the limit imposed by Maxwell-Boltzmann when the exciton energy is still defined by the initial lattice temperature $T_{\mathrm{L}}$.

Figure 11

Color map of the steady-state value of the ground-state population, $N_0^{\mathrm{ss}}$ versus the exciton reservoir density $n_x$ and temperature $T_x$. The black line shows the threshold exciton density $n_{x,\mathrm{th}}\left(T_x\right)$, where $N_0^{\mathrm{ss}}$ diverges. The red line, $T_{\mathrm{l}}\simeq 12\mathrm{K}$, defines the limit temperature, behind which the system cannot reach a threshold anymore.

Figure 12

Three-dimensional plot of the steady-state values of $N_0$ versus $p_x$ and $T_{\mathrm{L}}$, for fixed $G_0=3.8\mathrm{meV}$ and $L=20\mathrm{nm}$. The color map represents the variation in the polariton occupation number, lighter colors correspond to higher values of $N_0$. There are three phases corresponding to bistability, threshold, and not-threshold. These are reached by increasing the value of the lattice temperature $T_{\mathrm{L}}$ and letting $N_0$ evolve as a function of the pumping $p_x$.

Figure 13

Numerically computed out-scattering rate, $W_{{\mathrm{out}}_1}$, for $G_0=3.8\mathrm{meV}$ and different optical cavity in-plane confinement lengths. The black dotted line is the result expected for a quantum-well model with translational invariance, in which case it is possible to derive an analytic expression for $W_{{\mathrm{out}}_1}$ [10].

Figure 14

Numerically computed out-scattering rate, $W_{{\mathrm{out}}_2}$, for $G_0=3.8\mathrm{meV}$ and different optical cavity in-plane confinements.

Figure 15

Numerically computed out-scattering rate, $W_{{\mathrm{out}}_{\mathrm{c}}}$, for $G_0=3.8\mathrm{meV}$ and different optical cavity in-plane confinements.