Detuning-controlled internal oscillations in an exciton-polariton condensate. N. S. Voronova, A. A. Elistratov and Y. E. Lozovik in Phys. Rev. Lett. 115:186402 (2015). What the paper says!?
The Authors discuss the regimes of Rabi dynamics in presence of detuning. They neglect nonlinearities (interactions) that they find are negligible (cf. their discussion after Eq. (9)), so they really focus on the pure Rabi dynamics. This is not uninteresting, especially in the light that:
one may say that Rabi oscillations are density oscillations between the
photon and exciton condensates.
However they constantly mention blueshifts anyway and refer profusely to the Josephson dynamics as well. For instance, they say that the
regime of oscillations is a kind of interplay between the modified Rabi dynamics and an analog of the internal Josephson effect: for small-amplitude oscillations, one may say that the shift of natural frequency corresponds to Josephson “plasma frequency” $\omega_\mathrm{JP}=\delta\Omega_R$.
This Josephson plasma frequency looks an overkill here, since there is no Josephson effect and what they see is merely renormalization of the oscillations due to detuning. Also the expression is unclear, it does not even seem dimensionally correct.
Their criterion for Josephson dynamics in such conditions is a running phase:
a dramatic change from Rabi-like to Josephson-like dynamics: while the density imbalance oscillates around its new equilibrium value [...], the relative phase between the photon and
exciton condensates S(t) becomes monotonically increasing.
We showed however that this is not the case.[1] As a consequence, most of their discussion is in error. They give much importance to the phase, which is not a reliable observable, if observed clumsily (in fact the phase of the photons alone as excitons remain hidden).
we demonstrate that at any nonzero detuning different types of oscillations are possible, from harmonic and anharmonic modifications of Rabi oscillations up to the transition to a so-called Josephson regime analogous to the
internal Josephson effect in a two-state BEC of $^{87}$Rb atoms.
We showed on the opposite that the dynamics is trivial when described on its canonical space (the Bloch sphere). All the distortions they see, e.g., in their Fig. 3 with dramatic interplay of these phases (the relative phase is more meaningful, although also very distorted), arise from projection of a circle on a tilted axis.[1]
The critical parameters (compare to our effective counterparts in Ref. [1]):
Their analytical solutions in Eqs. (8-9) are written in a bit obscure form ($H$ depends on $\cos S$, which in turns is expressed as a function of $H$ and Eq. (9) is directly an integral, for an apparent typo $t$ instead of $\rho(t)$)
Instead of quantum state, they speak in terms of "energy" to describe the various solutions, which I find confusing, although this is equivalent:
Let us address the preparation of initial states, namely
ρ(0) and S(0), which define the energy h.
This entices them to make a phase-diagram of Rabi vs Josephson regimes depending on the quantum state (on the "energy" in their approach), while we have shown this does not depend on the quantum state itself, but on the interaction.[1] Precisely, we show that Josephson-looking dynamics is obtained in the Rabi case. It is instructive to compare their Fig. 2 to our Fig. 5a.
Closed trajectories representing finite motion at low energies
belong to the regime of Rabi-like oscillations
Overall the work looks much more sophisticated than it really is. They refer to Rabi dynamics as "internal oscillations", which is fine, especially as this is how Bosonic Josephson Junctions came to be from atomic internal oscillations (their Refs. [44-45]), although also a terminology not needed here. More irritatingly, they speak of "Rabi-like" and "Josephson-like", with no clear reason what the "-like" attends to. It seems that "pure Rabi" for them are exact sine oscillations, but we have also shown that such "distortions" of the dynamics are artifacts of projection on the observable-basis (photon)[1]. As for Josephson, it seems the "-like" is to distinguish from Cooper-pair oscillations or spatially coupled BEC.
The description starts in terms of general Gross-Pitaevskii equations which are however later reduced to single-modes. No dissipation (unlike our treatment[1]).
They have a related paper.[2]
