Quantum theory of electron tunneling into intersubband cavity polariton states

Through a nonperturbative quantum theory, we investigate how the quasielectron excitations of a two-dimensional electron gas are modified by strong coupling to the vacuum field of a microcavity. We show that the electronic dressed states originate from a Fano-type coupling between the bare electron states and the continuum of intersubband cavity polariton excitations. In particular, we calculate the electron spectral function modified by light-matter interactions and its impact on the electronic injection of intersubband cavity polaritons. The domain of validity of the present theoretical results is critically discussed. We show that resonant electron tunneling from a narrow-band injector can selectively excite super-radiant states and produce efficient intersubband polariton electroluminescence.

Figure 1

Sketch of the dynamical coupling between quantum states in a microcavity-embedded QW containing a 2DEG. In the ground state, the first subband is doped with a dense 2DEG (bold lines at the bottom of the dispersions). Black dots represent bare electrons, while white dots denote holes in the 2DEG. The dashed cones depict the possible final states for an electron radiatively relaxing from the second to the first subband by emission of a cavity photon. The ground state with $N$ electrons is the standard Fermi sea $|F_N⟩$. The injection (e.g., through electron tunneling) of an additional electron in the second subband creates the state $|C⟩=c_{2,\mathbf{k}}^{†}|F_N⟩$, which, in the presence of light-matter interaction, is not an eigenstate. Spontaneous emission of a cavity photon couples the $|C⟩$ state to the states $|A,\mathbf{q}⟩=a_{\mathbf{q}}^{†}{c}_{1,\mathbf{k}-\mathbf{q}}^{†}|F_N⟩$. Reabsorption of the emitted cavity photon can couple back to the $|C⟩$ state or to the states $|B,\mathbf{q},{\mathbf{k}}^{\prime }⟩=c_{2,{\mathbf{k}}^{\prime }+\mathbf{q}}^{†}{c}_{1,{\mathbf{k}}^{\prime }}{c}_{1,\mathbf{k}-\mathbf{q}}^{†}|F_N⟩$. Spontaneous emission couples the $|B⟩$ states back to $|A⟩$ states or to states of the form $|D,\mathbf{q},{\mathbf{q}}^{\prime },{\mathbf{k}}^{\prime }⟩=a_{{\mathbf{q}}^{\prime }}^{†}{c}_{1,{\mathbf{k}}^{\prime }+\mathbf{q}-{\mathbf{q}}^{\prime }}^{†}{c}_{1,{\mathbf{k}}^{\prime }}{c}_{1,\mathbf{k}-\mathbf{q}}^{†}|F_N⟩$. Being the relevant cavity photon wave vectors very small compared to the Fermi wave vector, spontaneous emission can occur only on narrow emission cone in momentum space. Due to the small probability of photon absorption by electrons on the border of the Fermi sea, we can neglect $|D⟩$ states and assume that the system always jumps from $|B⟩$ states to $|A⟩$ states. Thus the relevant dynamics takes place only between the states in the shaded region. We can thus neglect the other marginal states while diagonalizing the light-matter Hamiltonian.

Figure 2

Electron spectral function $A_2^{+}\left(\mathbf{k},\omega \right)$ for the second subband for all $k>k_F$. The spectral function, defined in Eq. (3), is the density of quasielectron states, weighted by the overlap with the bare electron state. The integral of the spectral function over the frequency $\omega$ is by construction equal to one. Coupling parameter: ${\Omega }_R\left(q_{\text{res}}\right)=\mid \chi \left(q_{\text{res}}\right)\mid \sqrt{N}=0.1{\omega }_{12}$.

Figure 3

Extracavity electroluminescence spectra $N_{\text{ph}}\left(\mathbf{q},\omega \right)$. (a) The case of a broadband electrical injector (bandwidth equal to ${\omega }_{12}$, centered at $\omega ={\omega }_{12}$). The other panels show the results for a narrow-band injector (of width $0.05{\omega }_{12}$) centered, respectively, at (b) $\omega ={\omega }_{12}$, (c) $1.2{\omega }_{12}$, and (d) $0.8{\omega }_{12}$. The nonradiative relaxation rate $1/{\tau }_{\text{nr},\mathbf{k},\zeta }$ has been taken equal to $0.005{\omega }_{12}$. In all panels, the dashed-dotted lines are the frequency dispersions ${\omega }_{\pm },q$ of the two cavity polariton branches. In the first panel the solid line represents the edge of the light cone (Ref. 18).