Microscopic theory of equilibrium polariton condensates

We present a microscopic theory of the equilibrium polariton condensate state of a semiconductor quantum well in a planar optical cavity. The theory accounts for the adjustment of matter excitations to the presence of a coherent photon field, predicts effective polariton-polariton interaction strengths that are weaker and condensate exciton fractions that are smaller than in the commonly employed exciton-photon model, and yields effective Rabi coupling strengths that depend on the detuning of the cavity-photon energy relative to the bare exciton energy. The dressed quasiparticle bands that appear naturally in the theory provide a mechanism for electrical manipulation of polariton condensates.

Figure 1

(a) Typical polariton condensate geometry. One or several quantum wells are placed between a pair of distributed Bragg reflectors (DBRs). The two-dimensional quantum-well conduction- and valence-band states are dressed by their interactions with condensed cavity photons and by electron-electron (e-e) interactions. (b) and (c) Polariton condensate chemical potential and photon fraction as a function of detuning $\delta$ and polariton density $n_{\mathrm{pol}}$.

Figure 2

(a)–(d) Detuning $\delta$ (red line) at which mutual equilibrium is established as a function of photon density $n_{\mathrm{ph}}$ for a series of chemical potentials $\mu$ which lie between the lower-polariton and the isolated exciton energies. The detuning value predicted by an exciton-photon model is plotted as a black line for comparison. The chemical-potential-dependent effective Rabi coupling $\mathrm{\Omega }$ values listed in the four panels were determined by fitting Eq. (18) in the main text to our numerical data.

Figure 3

(a)–(d) Polariton chemical potential $\mu$ as a function of polariton density $n_{\mathrm{pol}}$ at a series of fixed detuning values. The dashed line is a linear fit to the numerical data from which we determine the effective polariton-polariton interaction strength and lower-polariton energy as the slope and intercept. [See Eq. (21).] The value of $U_{\mathrm{pol}}$ predicted by the exciton-photon model [see Eq. (23)] is calculated using the Rabi coupling strength defined by the lower-polariton energy [Eq. (20)] and $U=6{\mathrm{Ry}}^{*}{a}_B^{*2}$ [1] and is given in the upper left of each panel. These values should be compared with the microscopic polariton-polariton interactions determined by the slopes of the $\mu$ vs $n_{\mathrm{pol}}$ plots.

Figure 4

Effective Rabi coupling $\mathrm{\Omega }$ determined by the smallest chemical potential value at which an equilibrium polariton condensate can be formed as a function of detuning $\delta .\mathrm{\Omega }$ is calculated from Eq. (20).

Figure 5

(a)–(d) Exciton fraction $x=n_{\mathrm{ex}}/n_{\mathrm{pol}}$ as a function of the density of polaritons $n_{\mathrm{pol}}$ at different fixed detuning values $\delta$. The red lines are obtained from microscopic mean-field-theory calculations, and the blue dashed lines are obtained from the analytic expressions [Eq. (15)] for $x$ in the simplified exciton-photon model. Note that red and blue points have different $y$ values which are shown by red and black marks, respectively.

Figure 6

(a)–(h) Quasiparticle bands at various detuning and polariton density values. The blue lines denote the dressed quasiparticle band structure, whereas the red dot-dashed lines illustrate the bands at the same value of $\mu$ when the self-energies responsible for interband coherence are neglected. The bare conduction- and valence-band extrema, marked by dashed horizontal lines, are located at $\vec{k}=0$ in all cases and have the values $\pm \mu /2$ in the undressed case. This figure is based on calculations with $m_e=m_h$.

Figure 7

Contour plots of the (a) e-e and (b) e-ph contributions to the maximum band-mixing self-energy as a function of polariton density and detuning. All energies are in ${\mathrm{Ry}}^{*}$ units, and the polariton density is in $a_B^{*-2}$ units. Panels (c) and (d) plot the total self-energies and the e-ph interaction fractional contribution to the total self-energies. Note that the e-e interaction self-energy is largest even when the polariton condensate is photon dominated and that the electron-electron interaction contribution is enhanced by the presence of the photon field.