Two faces of quantum sound

Fluctuations around a Bose-Einstein condensate can be described by means of Bogolubov theory leading to the notion of quasiparticle and antiquasiparticle familiar to nonrelativistic condensed-matter practitioners. On the other hand, we already know that these perturbations evolve according to a relativistic Klein-Gordon equation in the long-wavelength approximation. For shorter wavelengths, we show that this equation acquires nontrivial corrections which modify the Klein-Gordon product. In this approach, quasiparticles can also be defined (up to the standard ambiguities due to observer dependence). We demonstrate that—in the low-energy as well as in the high-energy regimes—both concepts of quasiparticle are actually the same, regardless of the formalism (Bogolubov or Klein-Gordon) used to describe them. These results also apply to any barotropic, inviscid, irrotational fluid, with or without quantum potential. Finally, we illustrate how the quantization of these systems of quasiparticles proceeds by analyzing a stationary configuration containing an acoustic horizon. We show that there are several possible choices of a regular vacuum state, including a regular generalization of the Boulware vacuum. Issues such us Hawking radiation crucially depend on this vacuum choice.

Figure 1

Dispersion relation (47) for various values of $\omega$ and of $|v|$, scaled at $c=1$. The intersection points between the line $\omega -vk$ (in blue) and the various branches of the dispersion relation (in black) mark the real (normalizable) mode solutions for a given $\omega$. The critical frequency ${\omega }_c$ represents the frequency at which additional, “extraordinary” roots appear for the maximal $|v|$ attained in the configuration. Then, for $\omega >{\omega }_c$, there are always only two normalizable solutions. For $\omega <{\omega }_c$, there can be either two or four normalizable solutions, depending on the value of $|v|$.

Figure 2

The in basis. $\omega >{\omega }_c$: (1) represents a scattering process in which there is an incoming wave from the left; (2) represents a scattering process in which there is an incoming wave from the right. $0<\omega <{\omega }_c$: $(1^{\prime })$ represents a scattering process in which there is an incoming wave from the left; $(2^{\prime })$ represents a scattering process in which there is an incoming wave from the right; $(3^{\prime })$ represents a scattering process in which there is an extraordinary incoming wave from the left. Solid arrows represent ordinary modes, while dashed arrows represent extraordinary modes.

Figure 3

The out basis. The interpretation is as in (time-reversed) Fig. 2.

Figure 4

Preliminary stationary basis. $\omega >{\omega }_c$: (1) represents a scattering process in which there is an outgoing wave to the right; (2) represents a scattering process in which there is an incoming wave from the right. $0<\omega <{\omega }_c$: $(1^{\prime })$ represents a scattering process in which there is an outgoing wave to the right; $(2^{\prime })$ represents a scattering process in which there is an incoming wave from the right; $(3^{\prime })$ represents a scattering process in which there are neither incoming nor outgoing waves on the right. These modes are not orthogonal. The stationary orthonormal basis is obtained by linear combinations of these modes. Solid arrows represent ordinary modes, while dashed arrows represent extraordinary modes.