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Note that this is not the case in the fully degenerate parametric oscillator considered, e.g., in Ref.: the signal and the idler are in this case emitted into the same discrete mode and no continuous U (1) symmetry is spontaneously broken.
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The damping of sound in Bose gases in the collisional regime has a Im [ω (k)] k2 dependence on a wave vector, which is, however, to be contrasted to the Im [ω (k)] |k| dependence of a Bose-Einstein condensate in the collisionless regime.
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The eigenvalues of a linear operator L (k) have a Taylor expansion around every value of k, except for some special points where fractional powers 1p can occur (see Ref., p. 65). If the Taylor expansion exists, stability requires Im [ω (k)] <0 for all k, which implies that only even powers of k are allowed in the Taylor expansion of Im [ω (k)] around k=0. If accidentally k=0 is a special point of order p, there are p eigenvalues which have a Puiseux series instead of a Taylor series λh (k) =λ+ α1 e2πihp k1p + α2 e4πihp k2p + for h=0,1,...,p-1. If p=2, stability requires that Im (αm) =0 for any odd m, so that no half-integer powers are allowed in the expansion of Im [λ (ω)]. The case with p>2 is ruled out just because it would be forced to give an eigenvalue with positive imaginary part.
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Rigorously speaking, a simultaneous and opposite rotation of the signal-idler phases without affecting the pump phase requires a boundary to be defined separating them. This can be done without breaking the wave function's continuity in k space only if the polariton population is negligible in between. This is a reasonable assumption if the spot boundaries are smooth and the spot size is σp | ks - kp | -1, i.e., much wider than the wavelength of the interference pattern formed by the signal, pump, and idler fields. In this case, the interference pattern can freely shift through the spot as a consequence of the signal-idler phase rotation and the symmetry breaking restoring force can be generally neglected. The physics is very different in the case of Bénard cells in a box geometry with sharp boundaries, as pointed out in.
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Rigorously speaking, a simultaneous and opposite rotation of the signal-idler phases without affecting the pump phase requires a boundary to be defined separating them. This can be done without breaking the wave function's continuity in k space only if the polariton population is negligible in between. This is a reasonable assumption if the spot boundaries are smooth and the spot size is σp | ks - kp | -1, i.e., much wider than the wavelength of the interference pattern formed by the signal, pump, and idler fields. In this case, the interference pattern can freely shift through the spot as a consequence of the signal-idler phase rotation and the symmetry breaking restoring force can be generally neglected. The physics is very different in the case of Bénard cells in a box geometry with sharp boundaries, as pointed out in. In this case, the displacement of the roll pattern must be accompanied by a deformation of the cells close to the boundaries, which introduces a significant restoring force and consequently opens a gap in the Bogoliubov spectrum.
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