Synchronized and desynchronized phases of coupled nonequilibrium exciton-polariton condensates. M. Wouters in Phys. Rev. B 77:121302(R) (2008). What the paper says!?
In this work, Wouters studies the Polaritonic Josephson Junction (in space) under the angle of synchronization (or lack of it) between two polariton condensates (multiple condensates numerically). He finds that
For small detuning between the different wells, the Joseph- son currents and densities reach a steady state. For too large detuning on the other hand, a steady state no longer exists and the Josephson currents cause density oscillations in the different condensates. Both interactions of the condensate with itself and with the reservoir are shown to enhance the
synchronization.
This describes the experimental observation from Ref. [1] that « depending on the configuration of the external potential (position on the sample), both the case of a fully coherent condensate with a single frequency (synchronized) and the case of incoherent condensates with different frequencies (desynchronized) can be realized.»
He links the phenomenon to mode locking in lasers:
The existence of a synchronized state up to a critical de- tuning is very similar to the phenomenon of mode locking in
lasers that is most simply described by the Adler equation
Claim on what's the simplest case study:
A realistic description of the actually realized polariton condensates requires to take into account the interactions between polaritons, pumping,
losses, and the disordered potential landscape.
In the presence of such a potential, the question naturally arises whether a single condensate is formed that is modulated by the disorder potential yet phase coherent over its size, or rather several independent condensates with different fre-
quencies are formed.
The dynamics is described by a Gross-Pitaevskii equation (also with an equation for the reservoir $n_j$ (Eq. (2)):
This allows him to study both pure condensate dynamics or effects of the reservoirs:
whether rather the condensate-condensate or condensate-reservoir interactions are the dominant mechanism for the synchronization
He finds that:
the expected window for the relative phase when condensate-condensate
interactions prevail is $0<—\theta, <\pi/4$, where $\pi/2<
He gives «the phase difference $\theta$ as a function of the effective condensate detuning $\Delta\omega$» analytically (his Eqs. (8) and (9)). The way I understand those equations, is that if they have no solution, then synchronization is not possible. It would be clearer to think of the phase (that the condensates adapt to) as a function of the externally imposed detuning, but since the result is as a function of the phase, he keeps the discussion in this way. This means, however, that for a give detuning, many relative phase solutions are possible. The choice of which is made is not clearly discussed.
the stationary syn- chronized state does not coincide with a linear eigenstate of the two-well system, for which $\theta\in\{0,\pi\}$, but that the con- densation occurs in a new state that is formed above the
threshold for condensation.
He makes the prediction of small amount of coherence in the desynchronized case, due to each condensate having a small overlap with the other, as seen in the right-column spectrum.
Thanks to dissipation, he has a neat separation between his two cases: phase damping or oscillating:
The time dependent phase difference induces an oscillating Josephson current that bears a striking analogy with the ac Josephson effect, where the application of a constant voltage (chemical
potential difference) also leads to an alternating current.
This appears to be a specificity of polaritons:
A crucial difference with the ac Josephson effect in atomic Bose–Einstein condensates concerns the relaxation: where the ac Josephson oscillations of an atomic condensate are damped at any finite temperature and its steady state is at thermodynamical equilibrium with a single chemical poten- tial, no relaxation to a single frequency state is possible for a
desynchronized nonequilibrium polariton condensate
This comment is strange for the stationary case, which must be unique, probably meaning initial conditions either go to that or to different oscillating solutions:
both the solutions with stationary and oscillating densities can be reached, de-
pending on the initial conditions.