This letter provides a theory for pure energy relaxation in the Gross-Pitaevski equation description of polaritons. It is done by adding a gradient in the quantum hydrodynamics approach to the phase part of the Madelung decomposition of the wavefunction.
This is regarded as different from coupling to the exciton reservoir:
incoherent excitons are usually described in the framework of the Wouters-Carusotto model, which was first formulated for scalar polaritons [11] and then generalized for the spinor
case [12].
which is also neglected here:
we neglect all other dissipa- tive processes, such as finite lifetimes, external pumping, and coupling with an incoherent excitonic reservoir, for which
well-established theoretical approaches exist already
So they consider, instead, phonon-relaxation.
Beyond gain and loss mechanisms, another crucial as- pect of polariton condensate dynamics is energy relaxation, which plays an important role in the formation of steady
states and the redistribution of momentum.
The problem was tackled by Solnyshkov et al.[1] but
a closer inspection shows that such energy-dependent damping alters both the am- plitude and the phase of the macroscopic wave function, leading to a nonconservation of the total number of parti- cles. Therefore, this method cannot be interpreted as a pure energy relaxation mechanism, as it simultaneously induces
particle loss.
The GPE is written in the Madelung form separating density and phase (their Eq. (1)). From that they get an equation for the density (10):
Pure energy relaxation should not affect particle number con-
servation, so we can do nothing but leave this equation as is.
To describe energy relaxation, they add the term −δH/δθ to the phase equation (Eq. (9)). This leads to the sought lowering of the total energy while keeping the total number of particles constant. At this point, they can come back to the original equation for the wavefunction:
This model leads to qualitatively different behaviours:
models of dis- sipative polariton fluids without pure energy relaxation give a qualitatively different dispersion, with flat regions in energy
bands [11,19], and no clean superfluid behavior [20].
The linear analysis providing the spectrum of elementary excitations is complemented with numerical analysis of propagating a potential in the fluid. They find that in the supersonic regime, the emitted waves decay an localize the field [Fig. 2(c)], as the amplitudes of the scattered waves with high momenta get suppressed. In the subsonic regime, superfluidity is maintained.
They also study propagating polariton packets (that they call "droplets"), finding a slow down of their speed [Fig. 4].