Novel Instability in Periodic Structured Bose-Einstein Condensate with Long Range Self-Interaction

Ui-Han Zhang^1*, Tak-Pong Woo^1,2, Tzihong Chiueh^1,2,3

¹Institute of Astrophysics, National Taiwan University, Taipei, Taiwan
²Department of Physics, National Taiwan University, Taipei, Taiwan
³Center for Theoretical Physics, National Taiwan University, Taipei, Taiwan

* Presenter:Ui-Han Zhang, email:zhanguihan@gmail.com

Extended Bloch theorem has been applied to the periodic Bose-Einstein Condensate (BEC) nonlinear background wave function for analyzing the small amplitude perturbation, where the dispersion relation is derived. Inspired by the interference fringes of wave dark matter, we have adopted long-range gravitational interactions as a working example. A 2D (coupling strength-wavenumber) phase diagram is obtained, where 3 types of oscillations are identified: (1) real 𝜔, (2) imaginary 𝜔 and (3) complex 𝜔, where 𝜔 is the oscillation frequency. Among them, (3) is the most peculiar unstable mode for a quantum system, as the quantum mechanical problem is often described by the Sturm-Liouville theory, and moreover the Brillouin zone center is largely populated with these peculiar modes. We find the peculiarity is rooted in the fact that the periodic background state is not a ground state of the Hamiltonian operator, for which ⟨δφ│H ̂│δφ⟩ can be negative and 𝛿𝜑 is the perturbed wave function. Contrary to the conventional wisdom, the Madelung transformation of the Schrӧdinger equation fails to hold for this perturbed system. We further find that when the eigenfunction 𝛿𝜑 is real, ω^2 is real and the system is Sturm-Liouville; however, when ω^2 is complex, 𝛿𝜑 is also complex but must additionally satisfy ⟨δφ│H ̂│δφ⟩=0, the Sturm-Liouville theory fails. The theoretical framework developed in this work can be extended to local interactions, such as Gross-Pitaevskii equation.

Keywords: Bloch theorem, Bose-Einstein Condensate, Sturm-Liouville Theory, Brillouin zone, Madelung transformation