Lasing in Bose-Fermi mixtures

Light amplification by stimulated emission of radiation, well-known for revolutionising photonic science, has been realised primarily in fermionic systems including widely applied diode lasers. The prerequisite for fermionic lasing is the inversion of electronic population, which governs the lasing threshold. More recently, bosonic lasers have also been developed based on Bose-Einstein condensates of exciton-polaritons in semiconductor microcavities. These electrically neutral bosons coexist with charged electrons and holes. In the presence of magnetic fields, the charged particles are bound to their cyclotron orbits, while the neutral exciton-polaritons move freely. We demonstrate how magnetic fields affect dramatically the phase diagram of mixed Bose-Fermi systems, switching between fermionic lasing, incoherent emission and bosonic lasing regimes in planar and pillar microcavities with optical and electrical pumping. We collected and analyzed the data taken on pillar and planar microcavity structures at continuous wave and pulsed optical excitation as well as injecting electrons and holes electronically. Our results evidence the transition from a Bose gas to a Fermi liquid mediated by magnetic fields and light-matter coupling.


Theoretical model
Low and high pumping regimes Let us, first, focus on the set of equations (1) for the exciton reservoir assuming the onset of polariton lasing at = ( ℎ) , where ( ℎ) is a critical exciton concentration corresponding to the population of polariton ground state equal to unity. In the following we assume that the electron and hole components are pumped with equal rates by the Gaussian-shaped laser beam, = ℎ ≡ − 2 2 ⁄ . The absence of the drift currents means equal concentrations for electrons and holes, n e = n h ≡ n, satisfying the following equation in the stationary limit: gives a good approximation preserving all the physical insight. Moreover, there are two limiting cases when Eq. (S1) allows for analytical solutions. The first one corresponds to a low pumping regime: in that case one deals with low concentrations and the nonlinear term in Eq. (S1) is small. The effective lifetime of a particle is then = 1 ( 0 ) ⁄ , and the spread of carrier density is = √ . Integration of Eq. (S1) over space domain gives: = ( √ ) ⁄ . This effective radius obviously should be greater than the radius of the pump spot, R, so that ≫ and ≪ * = 2 2 ( 4 ) ⁄ . The excitonic density at r = 0 is (0) = 0 2 (if one totally neglects the diffusion of excitons, i.e. sets D x ≡ 0) and it behaves quadratically with the pump intensity: In the opposite case of a high pump regime, we can neglect the diffusion term in Eq. (S1) resulting in a very short, linear in pump, expression for the exciton density: Since the transition between the discussed regimes is defined by J * which is inversely proportional to the fourth power of R, one should expect that the excitonic density distribution dependence on the pump power with and without a magnetic field significantly depends on the excitation spot radius. This effect is illustrated in Fig. S1a, where n x (0) is plotted for two spot radii different by one order of magnitude. One can see that for the sharper spot (red curve) J * significantly increases. Thus, for the fixed exciton density we switch between the high-pumping regime (black curve) and the low-pumping regime (red curve).
Application of a magnetic field alters parameters D and τ x . Magnetic field leads to suppression of the diffusion of charged carriers which can be described using the Einstein relation and the Hall expression for conductivity: Here 0 = ⁄ is the mobility of electron/hole gas at B = 0 and T is its temperature. The experimental data show that T might be significantly higher than the lattice temperature (e.g. T ≈ 50 K). Excitonic lifetime decreases with the magnetic field due to the decrease of the exciton Bohr radius ( ∝ 2 in 2D structures 6  parameter J * depends on the magnetic field via the diffusion coefficient, the magnetic field acts as a switch between the two regimes leading to a non-monotonous behavior of the excitons' density. This is illustrated in Fig. 3c in the main text of the article.

Results of calculations and parameter sets used
As it has been already mentioned, the onset of polariton lasing is described by a condition can be set manually. The photon lasing onset coincides with the Mott transition in the system and it is described by the following condition 8 : where κ is the coefficient of the order of unity. In the simulations we set κ = 1, while its exact value is not essential for the magnetic field dependence of the Mott transition threshold. Note that the absolute values of electron and hole concentration cannot be experimentally measured with a good accuracy. Using those two conditions, one can find the threshold pump intensity as a function of magnetic field. This function is plotted in Fig. 2c, Fig. 2d and Fig

Equations for the laser with current injection
The situation is quite different if we consider an electrically pumped laser. In this case an electron and a hole are excited in different spots of the sample, thus leading to small values of the parameter w. We will assume that in our sample the electrical injection is homogeneous along the quantum well planes meaning that the diffusion is neglected in the case of current injection. On the other hand, there are lots of possible mechanisms of electron and hole decays not leading to the exciton formation (including the 'fly-through' of the particles, non-radiative processes at the pillar surface, etc.), so that the non-radiative decay times τ e and τ h should be introduced. The kinetic equations similar to Eq. (1) therefore read: In the stationary limit, we obtain the algebraic system of equations, the solution of which reads: We will further make some simplifications, namely: = ℎ ≡ and = ℎ ≡ , so that = ℎ ≡ . In a realistic limit of small carrier lifetimes, 2 < 1, the carrier concentration is = , and for the exciton density one obtains: We will assume a usual dependence of τ x on magnetic field [see Eq. (S5)], which gives the following expressions for the polariton and photon thresholds: Here ℎ,0 = √ ℎ ( (0) 2 ) ⁄ is the polariton threshold at zero magnetic field. The calculated phase diagram is presented in Fig. 4a in the main text with ℎ,0 = 85 2 ⁄ , 1 = 7 T and ℎ,0 2 (0) = 0.65.
The temperature dependence of lasing thresholds is ruled by the strongly temperaturedependent exciton formation rate, w. We assume that the exciton formation rate in the lowtemperature region increases with temperature, following: ( ) = 0 exp (− 0 ⁄ ), where T0 is some effective temperature. It leads to a decrease of the thresholds which is illustrated in Fig. 4b of the main text. The value of T 0 is chosen T 0 = 5 K.  Figure S3 shows micropillar PL spectra of under pulsed excitation at the on-set and off-set of polariton lasing and at the maximum of polariton laser intensity. Figure S4 shows the PL spectra just below and just above the threshold to photon lasing. These spectra have been used to obtain the points in Figure 2(b) of the main text. We note that while the spectral range chosen for integration does have influence on the resulting value of the integrated intensity, of course, the positions of both polariton and photon laser thresholds extracted from the spectrally integrated data are essentially independent on the chosen spectral integration range. Figure S5 shows the spatial shape of polariton condensates in a pillar cavity in the absence and in the presence of the magnetic field. One can see that the field induces a supplementary localization of the condensate. Also, the spatial disorder effect on the shape of the condensate is quite pronounced. The disorder induced localization of polariton condensates has been studied e.g. in 11 . Figure  The additional data on the planar samples is summarized in Figs. S8 and S9. Fig. S8 shows PL spectra of the sample with a planar microcavity for varying pump intensities at a fixed magnetic field B = 5 T and negative detuning δ = −7 meV. One can see that the onset of a polariton lasing is clearly observed for both excitation spots. We conjecture that pronounced PL peaks observed at the high energy side of the lower polaritonic branch for a 10 μm spot and above threshold power originate from the cavity modes characterized by large k-vectors. Indeed, the lens with a larger NA (10 μm spot) integrates the PL signal in a wider k-space than that with a smaller NA (100 μm spot). Our additional study demonstrates that the phase diagrams (see Fig.   3d-f in the main text) are also sensitive to the detuning. Figure S9 shows the phase diagrams obtained for large 100 μm (upper panel) and small 10 μm (lower panel) spots measured at optimal negative detuning δ = −7 meV (the highest efficiency of lasing) and δ = −3.3 meV. One can see that deviation of the detuning from the optimal to less negative values significantly decreases the effect of the magnetic field on threshold for the large spot. In contrast the effect of detuning variation for the small spot is almost negligible. Figure S10 presents the Input-output characteristics and the emission linewidth of the electrically pumped microcavity sample. The polariton laser threshold is manifested by sharp increase in the intensity at 2 T and 4 T as shown in Fig. S8a, while the weak coupling laser threshold is characterized by a smooth S-curve at higher current values. The onset of polariton lasing is also accompanied by a sharp drop in the linewidth (see Fig. S8b).
Finally, figure S11 shows the data on the photoluminescence of the 8-µm diameter micropillar taken at the same sample as the data in Figures 1 and 2 of the main text. The threshold dependence on the magnetic field is shown in the lower panel to this figure. One can clearly see that lasing persists at cw pumping. The threshold dependence on the magnetic field is essentially the same at cw pumping ( Figure S11) and pulsed excitation ( Figure S7). Figure S2. The PL intensity (a), peak energy (b) and full width at half maximum (c) measured at a pillar microcavity of 20 µm diameter measured at the non-resonant cw excitation at the magnetic field of 5T. Figure S3 Characteristic PL spectra of a micropillar polariton laser taken at different pumping intensities at the onset of polariton lasing (blue curve), offset of polariton lasing (red curve) and at the maximum intensity of polariton lasing (black curve). The spectral integration in Figure 2(b) of the main text has been done within the area corresponding to the peak at the black curve. Figure S4. PL spectra of the pillar sample just before and after the threshold to photon lasing (red and black curves, respectively). The integration in Figure 2(b) of the main text has been performed in the spectral area of the black peak.     Figure S11. The photoluminescence spectra of a 8-µm diameter pillar microcavity from the same sample as one characterised in Figure 1 and 2 of the main text. The spectra have been taken at cw excitation. The upper left image is obtained at 0T, while the upper right plot corresponds to the magnetic field B=8 T. The lower panel shows the polariton laser threshold dependence as a function of magnetic field obtained from the data taken at cw pumping.