Sintonización del TMD

However, this situation changes significantly when densities become so large that interactions between polaritons do matter. Being bosons, their occupution numberNiof stateiwith energy

E_i follows the Bose-Einstein distribution,159 N_i= 1 eEi −µ kB T −1 , (2.28)

where E₀ denotes the energetic ground state of the system,µ <E₀ the chemical potential, k_B the Boltzmann constant and T the temperature. µis a function of temperature and density. At high temperatures, where the chemical potential lies well belowE₀, a Maxwell-Boltzmann distribution with a negligible occupation of the ground state is observed. It can be seen that in the limitµ_→E₀(or T <T_c),N₀ diverges, whereasN_i remains finite fori>0.29,160When this phase transition occurs, the occupation of the ground state is maximised and becomes macroscopic. The new phase is called the Bose-Einstein condensate (BEC).54The critical tem- perature depends on the massmand densityρof the particles as159

Tc≈3.31 ħh2 k_Bmρ

2/3 _(2.29)

and describes the point where the wave functions of the particles start to overlap. Two points of importance follow: (1) the low mass of the polariton favours a highTc and (2) the system can be driven to condensate at constant temperature by increasing the density of the ground state.

It should be noted that BEC is originally only defined for 3D systems in equilibrium, whereas polaritons in microcavities are non-equilibrium particles in 2D. As a consequence, large systems show a Berezinskii-Kosterlitz-Thouless transition that is unique to 2D systems and that posesses a phase of higher order with bound vortex–antivortex pairs.88However, for systems of finite size—which is defined by the pump spot—, the phase of higher order is more similar to a BEC and thus termed quasi-BEC. Here, the steady state of the polaritons (including pumping and decay) can form a quasi-equilibrium outwith the bath.∗25,29It has been argued, though, that the polariton condensate is distinct from a BEC, with distinct behaviour, and should thus not be called BEC.161

As microcavities containing polariton condensates emit coherent light, they are also re- ferred to as polariton lasers.† This might be confusing since the underlying physical prin-

∗_{Recently, microcavities of sufficiently high Q-factors have been fabricated to fully thermalise the polaritons before}

they leak from the cavity.93

†_{Sometimes, the distinction between polariton condensates and polariton lasers is made, depending on the extent} of equilibrium,152,162_{but in this thesis, the two terms will be used interchangeably.}

ciple differs fundamentally: photon lasing relies on the population inversion of a radiative transition—is thus inherently far from thermal equilibrium—and the bosonic stimulation acts on photons. By contrast, polariton condensation does not require a population inversion and the particles undergoing bosonic scattering/cooling are of mixed light–matter nature. The threshold for polariton condensation is thus expected to be lower than that for photon las- ing.54Comparing the emission characteristics, however, the two processes look similar as both thresholds exhibit a non-linear increase of photon emission, linewidth narrowing (both in real and in momentum space) and the onset of temporal and spatial coherence.29,85A uniform po- larisation across the spot is likewise expected in BECs and in most lasers.29,163Even a thermal- like distribution of the background and a spontaneous build-up of polarisation—previously thought to be the smoking gun for condensates25,162—have also been demonstrated in photon lasers.164,165

In order to unambiguously identify a condensate, two indicators need to be verified in addition to the features mentioned above. (1) Does the non-linear emission (still) arise from the coupled LP branch as opposed to the uncoupled branch of the cavity photon? (2) Does the emission show a small but continuous blue-shift with excitation power? The latter can be attributed to Coulomb repulsion of the polaritons, originating from their matter part. The attentive reader may notice an interdependence between (1) and (2), as the blue-shift will move the condensate in energy above the LP and towards the photon branch. This blue- shift should be small compared toħhΩ, especially since lasing has also aready been shown to originate slightly red-shifted from the photon branch.164 In organic microcavities, the blue- shift will be comparably small with respect to inorganic cavities due to the smaller exciton size and thus smaller nonlinearities (see Section 2.4.3). Nonetheless, ambiguities sometimes persist, especially in microcavities where LPs have a large photonic fraction and are thus very similar in energy to the energy of the uncoupled cavity photon.63 Another proof to clearly distinguish polariton condensation from photon lasing is the presence of a second threshold at higher excitation powers. Here, the lower threshold corresponds to polariton lasing and the higher one to photon lasing. However, a demonstration of the second threshold is not always possible—in particular for organic materials. Low stability of the material can render the second threshold elusive, especially since the system can only be optimised for one of the phenomena.27,62

ferent bosonic effects, great care has to be taken to distinguish them experimentally.