Frank-Tamm formula for the frequency spectrum of Vavilov-Cerenkov radiation in a Bose-Einstein condensate

Sergio Rica

A Frank-Tamm formula for the frequency spectrum of Vavilov-Cerenkov radiation caused by a moving particle with velocity v in a Bose-Einstein condensate is derived in the frame of the Gross-Pitaevski{\v\i} equation for the condensate wave function. The radiated energy per traveled distance, $x$, and per frequency $\ omega$ is: $$\frac{d^2 E(\omega) }{dx d\omega} = - , \frac{ \alpha \rho }{\pi} \frac{ \omega }{ \kappa_1 ( c_s^4+4 \alpha ^2 \omega ^2) } . $$ where, $\alpha = \frac{\hbar}{2 m} $, $\rho$ in the condensate number density, $c_s$ is the sound speed and $ \kappa_1(\omega) = \left[ \frac{1}{2} \left(\frac{\sqrt{ c_s^4+4 \alpha ^2 \omega ^2}}{\alpha^2}-\frac{c_s^2}{\alpha ^2}-\frac{2 \omega ^2}{v^2}\right)\right]^{1/2} .$ This formula tells us that the radiation is highest for large frequencies. A similar treatment may be done for the case of a propagating electromagnetic wave over inclined cylinder.