Stabilized by quantum fluctuations,dipolar Bose-Einstein condensates can form self-bound liquid-like droplets.However in the Bogoliubov theory,there are imaginary phonon energies in the long-wavelength limit,implying dynamical instability of this system.A similar instability appears in the Bogoliubov theory of a binary quantum droplet,and is removed due to higher-order quantum fluctuations as shown recently[Gu Q and Yin L 2020 Phys.Rev.B102 220503(R)].We study the excitation energy of a dipolar quantum droplet in the Beliaev formalism,and find that quantum fluctuations significantly enhance the phonon stability.We adopt a self-consistent approach without the problem of complex excitation energy in the Bogoliubov theory,and obtain a stable anisotropic sound velocity which is consistent with the superfluid hydrodynamic theory,but slightly different from the result of the extended Gross-Pitaevskii equation due to quantum depletion.A modified Gross-Pitaevskii equation in agreement with the Beliaev theory is proposed,which takes the effect of quantum fluctuations into account more completely.

Fan Zhang;Lan Yin

School of Physics,Peking University,Beijing 100871,China

中国物理快报（英文版）

[1]Fan Zhang;Lan Yin-.Phonon Stability of Quantum Droplets in Dipolar Bose Gases)[J].中国物理快报（英文版）,2022(06):1-6

Dipolar,B102,Beliaev

Phonon,Stability,Quantum,Droplets,Bose,Gases,Stabilized,by,quantum,fluctuations,dipolar,Einstein,condensates,self,bound,liquid,like,droplets,However,Bogoliubov,theory,there,are,imaginary,phonon,energies,long,wavelength,limit,implying,dynamical,instability,this,system,similar,appears,binary,removed,due,higher,order,shown,recently,Gu,Yin,Phys,Rev,We,study,excitation,energy,formalism,find,that,significantly,enhance,adopt,consistent,approach,without,problem,complex,obtain,stable,anisotropic,sound,velocity,which,superfluid,hydrodynamic,but,slightly,different,from,result,extended,Gross,Pitaevskii,equation,depletion,modified,agreement,proposed,takes,effect,into,account,more,completely

0.599209