Cavity-polaritons

Coulomb problem, and thus are Coulomb excitons. The Coulomb excitons can be found as solutions of Maxwell s equations in the limit c — oo, i.e. neglecting the retardation. 

Besides Coulomb excitons, it is convenient to use also the so-called mechanical excitons. These are the excited states of the crystal found in the approximation that the contribution of the macroscopic electric field to the intermolecular interaction is omitted, and only the contribution of the short-range intermolecular interaction is taken into account. The energies of the mechanical excitons determine the poles of the dielectric tensor  

Let us return to the structure of the dielectric tensor of a crystal of anthracene, tetracene type (i.e. with two molecules in the unit cell). Based on the above arguments and supposing that we know the energies tub and fru 2 of the mechanical excitons, which for simplicity we take to be dispersionless, we can write 

We suppose that the microcavity was grown in such a way that the dipole moments Pi and P2 are parallel to the microcavity plane, and the 2-axis denotes 

We shall describe the electric field by its in-plane longitudinal (E q), in-plane transversal (EtTq), and -components (Ez). Let p denote the angle between the in-plane wavevector q and the. x-axis. From eqn (10.3 a) we find that Ez = = (iq/K2c)dEi/dz, where nc = Juj 2 c/c2 — q2. The fields Ei and Et are related to each other by the system of equations obtained from eqn (10.3 a)  

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Alexey Kavokin and Guillaume, Cavity Polaritons, Volume 32, 2003. 

In Subsection 10.2.2, the Maxwell equations in an anisotropic microcavity are discussed and some important facts concerning Coulomb and mechanical excitons are summarized. In Subsection 10.2.3 the dispersion equation of cavity polaritons is derived. Its solutions for the cases of crystals with one and two molecules in the unit cell are discussed in Subsections 10.2.4 and 10.2.5, respectively. The main results are summarized in the conclusions. 

In this subsection we introduce some notations for the bare excitations whose interaction leads to the formation of cavity polariton states. These bare excitations are cavity photons and Coulomb excitons. 

The determinant of this system of equations yields the dispersion law of the cavity polaritons. 

We introduce Q2 = q2 + k2 and write the dispersion equation of the cavity polaritons in the form ... 

Thus the cavity polariton dispersion has a simple interpretation. Let us calculate the electric fields from eqn (10.7) for the modes (10.16). Neglecting small terms of the order of q2 /n2, the fields Ei and Et are related in these modes by Ei = —Et cot (p. Then the y-component of the fields Eu,l is equal to zero. In other words, with accuracy up to small terms (of the order of q2 /k2) the total in-plane electric field in the polaritonic modes is parallel to the dipole moment Pi for any direction of the wavevector q, and the value of the Rabi splitting energy thus does not depend on the wavevector direction. 

When the unit cell of an organic crystal contains two or more molecules, the spectrum of the cavity polaritons strongly depends on the relation between (i) the detuning u = coc — u>ly (ii) the energy of the Rabi splittings W1 and W2, and... 

Now let us consider also states with energies below EiJ(q< i>n), where the imaginary part of the wavevector is no longer small in comparison with its real part. These states are of interest, since, first, they determine the distribution of photoluminescence for the directions near to the normal to the surface of the microcavity, and second, because just these low-energy states can be important in the discussion of condensation of cavity polaritons. 

Now let us consider the wavevector broadening of the upper polariton states for large q. At large wavevectors the upper polariton dispersion curve tends to that of the cavity photon, and 5q 7o(A2e63/2/cj2h c3) Rabi splitting, for large q the upper cavity polariton branch contains the coherent states only. 

In this section we develop a microscopic quantum model which accounts both for positional and orientational disorder in such a system. Using the bosonic Hamiltonian for the system, we find the structure of the eigenstates (i.e. the weights of the electronic excitations on different molecules of the disordered medium) in the intervals where the wavevector of the cavity polaritons is a good quantum number. These weights will be used in Ch. 13 in consideration of the upper polariton nonradiative decay and also for estimations of the rate transition from incoherent states to the lowest energy polariton states. 

Substituting /% from the second of eqn (10.36) into the first equation, we find that the relation which determines the energies and amplitudes of cavity polaritons is ... 

In this section we completely ignored the damping of the molecular and the cavity photon states. In other words, the cavity polariton wavevector was treated as a good quantum number. Therefore, based on the results of the previous Section, we conclude that the relations we have obtained are only applicable for the wavevectors q n < q < qmax for the lower branch, and q > q n for the upper branch. 

One of the unsolved problems arising in the framework of the macroscopic approach is the nature of the lowest energy local states in the microcavity with disordered organics. The nature of states which formally appear in the region of molecular resonance are clear they do not interact with the cavity photon and for this reason are similar to states in non-cavity organic material. However, the nature of cavity polaritons near the lowest end-points deserves careful... 

Just such a situation takes place for microcavity dispersion at the bottom of the lower and upper polariton branches in a microcavity with a = h/2M where M is the effective mass of the cavity polariton. Of course, specific features of the low-energy wavepackets stem from the fact that the polariton dispersion near the... 

We note that by changing the physical parameters of the microcavity and organic material, as well as conditions for polariton excitation, one can influence the dynamics described above. One should also be aware that the evolution times are limited by the actual lifetimes r of small wavevector cavity polaritons. Long lifetimes r on the order of 10 ps can be achieved only in microcavities with high quality factors Q = cut. 

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