Interacting multiple zero mode formulation and its application to a system consisting of a dark soliton in a condensate

To formulate the zero modes in a finite-size system with spontaneous breakdown of symmetries in quantum field theory is not trivial, for in the naive Bogoliubov theory, one encounters difficulties such as phase diffusion, the absence of a definite criterion for determining the ground state, and infrared divergences. An interacting zero mode formulation that has been proposed for systems with a single zero mode to avoid these difficulties is extended to general systems with multiple zero modes. It naturally and definitely gives the interactions among the quantized zero modes, the consequences of which can be observed experimentally. In this paper, as a typical example, we consider an atomic Bose-Einstein condensed system with a dark soliton that contains two zero modes corresponding to the spontaneous breakdown of the U(1) gauge and translational symmetries. Then we evaluate the standard deviations of the zero mode operators and see how the mutual interaction between the two zero modes affects them.