An effective density matrix approach for intersubband plasmons coupled to a cavity field: electrical extraction/injection of intersubband polaritons

TL;DR

The work develops a density-matrix framework with bosonized α (ISB plasmon) and β (extractor) transitions inside a cavity to model electrical extraction and injection of intersubband polaritons under strong coupling. It derives selection rules for tunnel coupling that forbid bright–dark transitions between α and β, while incorporating a cavity mode and decoherence/dissipation via a quantum master equation to separate intra- from intersubband dynamics. The model quantitatively fits photocurrent spectra from midinfrared QCDs near the onset of strong coupling, reproducing peak amplitudes and their bias dependence and showing that extraction proceeds coherently via bright-state tunneling. It further argues that electrically pumped polariton emitters cannot rely on simple dark-to-bright injection, requiring mechanisms to establish coherence, and outlines implications for the design of polaritonic LEDs and the role of dark states in injection pathways.

Abstract

The main technological obstacle hampering the dissemination of modern optoelectronic devices operating with large light-matter coupling strength $Ω$ is an in-depth comprehension of the carrier current extraction and injection from and into strongly coupled light-matter states, the so-called polaritonic states. The main challenge lies in modeling the interaction between excitations of different nature, namely bosonic excitations (the plasmonic ISB excitations) with fermionic excitations (the electrons within the extraction or injection subband). In this work, we introduce a comprehensive quantum framework that encompasses both the ISB plasmonic mode and the extractor/injector mode, with a specific emphasis on accurately describing the coherent nature of transport. This reveals inherent selection rules dictating the interaction between the ISB plasmon and the extraction/injection subband. To incorporate the dynamics of the system, this framework is combined to a density matrix model and a quantum master equation which have the key property to distinguish intra and intersubband mechanisms. These theoretical developments are confronted to experimental photocurrent measurements from midinfrared quantum cascade detectors ($λ$ = 10 $μ$m) embedded in metal-semiconductor-metal microcavities, operating at the onset of the strong light-matter coupling regime (2$Ω$ = 9.3 meV). We are able to reproduce quantitatively the different features of the photocurrent spectra, notably the relative amplitude evolution of the polaritonic peaks with respect to the voltage bias applied to the structure. These results on extraction allow us to elucidate the possibility to effectively inject electronic excitations into ISB plasmonic states, and thus polaritonic states.

Figures (8)

• Figure 1: (center) Schematic representation of the system in momentum space. Bright and dark states are represented for both $\alpha$ (0$\rightarrow$1) and $\beta$ (0$\rightarrow$2) transitions. Note that the $\beta$-bright state is degenerated with the $\beta$-dark states. The different important operators and their effect on the transport of the excitations are represented. The blue path represents a detection process, whereas the red path represents an injection process. (Top) Typical bandstructure for a quantum cascade detector. The main extraction pathway is represented in blue. (Bottom) Typical bandstructure for a quantum cascade emitter. The main injection pathway is represente in red. The cavity electric field $E_z$ is also schematically superimposed on the figure.
• Figure 2: Normalized photocurrent measurements (continuous lines) and quantum master equation global fit (dashed lines), for two cavity geometries [a] $s=1.50$ µ m, $p=7$ µ m and [b] $s=1.55$ µ m, $p=7$ µ m. Offsets are added for clarity. Filled areas represent the errors of the fit parameters propagated onto the spectra. The extractor frequency $\omega_\beta(F)$, dependant of the electric field $F$, and the plasma-shifted ISB transition $\tilde{\omega}_\alpha$ are both superimposed on the spectra. Additional results can be found in Appendix \ref{['section:photocurrent_additional']}.
• Figure 3: Left-side scale: extraction rate $\gamma_\beta$ as a function of the applied electric field $F$. Red cross: predicted values computed using a standard sequential transport model. Blue plus sign: values returned by the global fit using a quantum master equation model. Right-side scale: experimental photocurrent integrated amplitude, for two different ($s$, $p$) couples of cavity parameters.
• Figure 4: Light-matter coupling $\Omega$ parametric study for the different quantities of interest: [a] Total absorption of the system (ISB absorption, extractor dissipation and cavity absorption) [b] Internal QCD absorption (ISB absorption and extractor dissipation) [c] Photocurrent (extractor dissipation) [d] Transfer function between the extractor dissipation ($\mathcal{A}_\beta := \mathcal{J}_\beta/|s_+|^2$) and the internal QCD absorption ($\mathcal{A}_\alpha + \mathcal{A}_\beta$). The blue dashed line represents an equivalent situation weak coupling situation where the cavity is not included in the model, and the ISB $\alpha$ transition is directly pumped (every other parameters are exactly the same as the strong coupling situations).
• Figure 5: Sharpness of the transfer function $\mathcal{T}$ as a function of the light-matter coupling strength ratio $\Omega/\omega_\alpha$ and the intrasubband scattering predominance $\gamma^\text{intra}$ in both $\alpha$ and $\beta$ transitions with respect to the total scattering $\gamma_{\alpha\beta}$. This map gives intuitive validity domains for all the different physical models considered up to now, but the limits between the domains are arbitrary: a flat transfer function ($r< 0.3$) suggests a sequential transport model, whereas a sharp transfer function ($r> 0.7$) suggests a delocalized transport model, described by the Coupled Mode Theory. The quantum master equation model presented in this article covers the whole domain.