Stabilizing Open Photon Condensates by Ghost-Attractor Dynamics. A. Abouelela, M. Turaev, R. Kramer, M. Janning, M. Kajan, S. Ray and J. Kroha in Phys. Rev. Lett. 135:053402 (2025). What the paper says!?
The authors present:
a systematic study of the coupled, temporal dynamics of continuously driven, dissipative photon BECs, their noncondensed fluctuations, and the dye-molecule excitations, based on the Lindblad formalism and treating all dynamical fields on the same footing by means of a second-order cumulant expansion.
They find that, unlike GPE treatments,
the photon BEC always decays to zero in the long-time limit.
Condensation is either sustained at high-driving but under a lasing regime (with inversion of population) or for finite times only, but as a result of
a previously undiscovered ghost attractor
A ghost attractor being one which is unphysical (outside of the physical space, here with a condensation fraction >100%) but which can still influence the dynamics inside the physical boundaries. As a result:
This leads to a constant, plateaulike photon BEC amplitude before it is repelled due to a positive Lyapunov exponent and then exponentially decays to zero.
Although condensation is of finite times only,
the plateau lifetime is macroscopically large, consistent with the experimental observation of a stationary photon BEC over the entire observation time
The lasing occurs when the attractor is not a ghost but becomes "real" (physical), which leads to a phase transition between those two regimes.
In their words:
at finite times the dynamics are governed by a previously undiscovered ghost attractor, a concept known from nonlinear systems [26–28] and realized here as a fixed point (FP) in configuration space which exists outside of the physically accessible realm with an unphysical condensate fraction, ν > 1.
and
The system dynamics evolve toward this FP but stall due to its inaccessibility. This leads to a constant, plateaulike photon BEC amplitude before it is repelled due to a positive Lyapunov exponent and then exponentially decays to zero.
Their model is based on the master equation for a single-mode cavity:
The rely on «second-order cumulant expansion» to derive nonlinear equations in terms of the following expectation values (although it is unclear where such correlators like $\langle aa\rangle$ later play a role):
and make several approximations, including decorrelation of the condensate with the rest of the system, which is against my mechanism:[1]
Polaritonic correlators between photonic and molecular degrees of freedom factorize, e.g., $\langle\ud{a}\sigma^-\rangle=\langle\ud{a}\rangle\langle\sigma^-\rangle$, since we work in a regime of weak photon-molecule coupling, where polaritonic states [31,32] are not formed, i.e., the photons are independent particles.
They say that they checked this approximation!
This is finally where they arrive to:
of which they conduct a stability analysis.
With no population inversion, the fixed point is unphysical (ghost attractor).
They seem to need a seed: «{{{1}}}». The dynamics of condensation is interesting:
- First, «the initial time period is dominated by an exponential growth of the noncondensed cavity-photon number n(t) − ».
- Then «a sudden population of the photon BEC».
- A plateau where the BEC is sustained metastably (what they call the open photon BEC), unable to reach its ghost attractor.
- An exponential decay of $\psi$ while population condensate (intensity) remains large.
The other regime is the «lasing state, i.e., a condensate with population inversion and infinite lifetime» and phase coherence.
Their conclusion is, however, wrong, as they conclude that:
Our findings can be tested experimentally by interference measurements of the condensate amplitude
thus assuming that there would be no fringes in interferences, which is certainly not the case; here they are mistaking quantum coherence with quantum optical coherence, existence of the phase as a convenient fiction, etc.
