Quantum vacuum properties of the intersubband cavity polariton field

We present a quantum description of a planar microcavity photon mode strongly coupled to a semiconductor intersubband transition in presence of a two-dimensional electron gas. We show that, in this kind of system, the vacuum Rabi frequency ${\Omega }_R$ can be a significant fraction of the intersubband transition frequency ${\omega }_{12}$. This regime of ultrastrong light-matter coupling is enhanced for long-wavelength transitions, because for a given doping density, effective mass and number of quantum wells, the ratio ${\Omega }_R∕{\omega }_{12}$ increases as the square root of the intersubband emission wavelength. We characterize the quantum properties of the ground state (a two-mode squeezed vacuum), which can be tuned in situ by changing the value of ${\Omega }_R$, e.g., through an electrostatic gate. We finally point out how the tunability of the polariton quantum vacuum can be exploited to generate correlated photon pairs out of the vacuum via quantum electrodynamics phenomena reminiscent of the dynamical Casimir effect.

Figure 1

(a) Sketch of the considered planar cavity geometry, whose growth direction is called $z$. The cavity spacer of thickness $L_{\mathrm{cav}}$ embeds a sequence of $n_{\mathit{QW}}$ identical quantum wells. The energy of the cavity mode depends on the cavity photon propagation angle $\theta$. (b) Each quantum well contains a two-dimensional electron gas in the lowest subband (obtained through doping or electrical injection). The transition energy between the first two subbands is $\hslash {\omega }_{12}$. Only the TM-polarized photon mode is coupled to the intersubband transition and a finite angle $\theta$ is mandatory to have a finite dipole coupling. (c) Sketch of the energy dispersion $E_1\left(q\right)$ and $E_2\left(q\right)=E_1\left(q\right)+\hslash {\omega }_{12}$ of the first two subbands as a function of the in-plane wave vector $q$. The dispersion of the intersubband transition is negligible as compared to the one of the cavity mode. For a typical value of the cavity photon in-plane wave vector $\mathit{k}$, one has in fact $E_2\left(\mid \mathit{k}+\mathit{q}\mid \right)-E_1\left(q\right)\simeq \hslash {\omega }_{12}$.

Figure 2

Coupling ratio ${\Omega }_{R,k}∕{\omega }_{12}$ as a function of the intersubband emission wavelength ${\lambda }_{12}\left(\mu \mathrm{m}\right)$. Parameters: $f_{12}=12.9$ (GaAs quantum well), cavity spacer refraction index $\sqrt{{ϵ}_{\infty }}=3$, $n_{\mathit{QW}}^{\mathrm{eff}}=50$, $N_{2\mathrm{DEG}}=5\times {10}^{11}{\mathrm{cm}}^{-2}$ and $\theta =60°$. The Fabry-Perot resonator is a $\lambda ∕2$ microcavity. Results obtained from the analytical expressions in Eqs. (14, 15).

Figure 3

Normalized polariton frequencies ${\omega }_{\mathrm{LP},k}∕{\omega }_{12}$ and ${\omega }_{\mathrm{UP},k}∕{\omega }_{12}$ as a function of ${\Omega }_{R,k}∕{\omega }_{12}$ for $D_k={\Omega }_{R,k}^2∕{\omega }_{12}$. The calculation has been performed with ${\omega }_{\mathrm{cav},k}={\omega }_{12}$. Note that for a given microcavity system, ${\Omega }_{R,k}∕{\omega }_{12}$ can be tuned in situ by an electrostatic bias, which is able to change the density of the two-dimensional electron gas.

Figure 4

Mixing fractions for the lower polariton (LP) mode as a function of ${\Omega }_{R,k}∕{\omega }_{12}$ [see Eq. (18) in the text]. The calculation has been performed for the resonant case ${\omega }_{\mathrm{cav},k}={\omega }_{12}$, as in the previous figure. Panel (a): ${\mid w_{\mathrm{LP},k}\mid }^2$ (thin solid line), ${\mid x_{\mathrm{LP},k}\mid }^2$ (thick solid line). Note that for ${\Omega }_{R,k}∕{\omega }_{12}⪡1$, ${\mid w_{\mathrm{LP},k}\mid }^2\simeq {\mid x_{\mathrm{LP},k}\mid }^2\simeq 1∕2$. Panel (b): ${\mid y_{\mathrm{LP},k}\mid }^2$ (thin dashed line), ${\mid z_{\mathrm{LP},k}\mid }^2$ (thick dashed line). For ${\Omega }_{R,k}∕{\omega }_{12}⪡1$, ${\mid y_{\mathrm{LP},k}\mid }^2\simeq {\mid z_{\mathrm{LP},k}\mid }^2\simeq 0$. The upper polariton (UP) fractions (not shown) are simply ${\mid w_{\mathrm{UP},k}\mid }^2={\mid x_{\mathrm{LP},k}\mid }^2$, ${\mid x_{\mathrm{UP},k}\mid }^2={\mid w_{\mathrm{LP},k}\mid }^2$, ${\mid y_{\mathrm{UP},k}\mid }^2={\mid z_{\mathrm{LP},k}\mid }^2$, ${\mid z_{\mathrm{UP},k}\mid }^2={\mid y_{\mathrm{LP},k}\mid }^2$.

Figure 5

Solid line: normalized differential zero-point (ZP) energy (per mode) $\Delta {\omega }_{\mathrm{ZP}}\left(k\right)∕{\omega }_{12}$ as a function of ${\Omega }_{R,k}∕{\omega }_{12}$. Dashed line: $D_k∕{\omega }_{12}$. Dotted line: normalized correlation contribution $\Delta {\omega }_{\mathrm{ZP}}^{\mathit{corr}}\left(k\right)∕{\omega }_{12}$. The calculation has been performed with ${\omega }_{\mathrm{cav},k}={\omega }_{12}$.

Figure 6

Sketch of a possible setup for the generation of correlated photon pairs in the intersubband cavity system. The vacuum Rabi frequency of the intersubband cavity system can be modulated through an electric gate, which changes the density of the two-dimensional electron gas or, alternatively, the dipole moment of the intersubband transition. A modulation of the bias is expected to induce the emission of correlated photon pairs with opposite in-plane wave vectors. This kind of radiation can be optimally guided out of the cavity through wedged lateral facets, with inclination equal to the resonance angle ${\theta }_{\mathrm{res}}$.