Solitons And Majorana Fermions In A Topological Superfluid

In this dissertation,we mainly focus on the Majorana fermions existing on the edges of the topological superfluid and the situation when a dark soliton coexists in this system.In the presence of the spin-orbit coupling,superfluid will be driven into a topological state,which gives rise to the Majorana fermions on the edges of the boundary of the topological phase.Furthermore,if we introduce a dark soliton into the topological superfluid,it can accommodate extra Majorana fermions in its core and the filling status of the core can indicate the state of the system.In the first chapter,we will briefly introduce some basic knowledge concerning this dissertation.Firstly,we will go over two important condensed systems,BEC(Bose-Einstein condensate)and BCS(Bardeen-Copper-Schrieffer)superfluid.Theo-ries and experiments will be addressed for both systems,and then we mention a useful and powerful experimental technique——Feshbach resonance,which can change the atomic scattering length,i.e.,the interactions between atoms.With the help of this technique,we can transform a BEC state with a repulsive interaction into a BCS state with an attractive interaction.This process is the so-called B EC-BCS crossover,which was of much interest to researchers during the last decade.Secondly,we will go to nonlinear physics and familiarize ourselves with an important notion——solitary wave.This notion was proposed by a Scottish engineer J.S.Russell in 1834.He discovered a wave which could propagate a long range without any distortion in a canal.However,the theoretical description of this exotic object was not established until more than 50 years of efforts were made.In 1895,D.J.Kortweg and G.de Vries proposed an equa-tion,which is called KdV equation,to describe this unusual wave.KdV equation is a nonlinear equation,which means the solitary wave is in fact a nonlinear phenomenon.The attribute of the solitary wave that it can carry energy without any loss during the long-range propagation renders it the importance in optical research and communica-tion engineering.Since the condensed matter is governed by a nonlinear equation when we take the interactions between atoms into account,it is not surprising that solitary wave can play an important role in this field in the form of a matter wave.We can even properly define the momentum and effective mass for the solitary wave,which makes it behave like a particle.In this case,this matter wave is also called soliton.Soliton has the form of a bulge or a notch corresponding to the bright soliton and dark soliton,respectively.With the development of the refrigeration technique,soliton was observed in ultracold atomic gases soon after the realization of the BEC and became a hotspot in the field of condensed matter physics.In the second chapter,we turn our attention to a well-studied topic in recent years——Majorana fermions.Proposed by Ettore Majorana in 1937,Maj orana fermions are such kind of particles that they are their own antiparticles.In the beginning,the search for the Majorana fermions was mainly carried out in the field of particle physics.Re-cently,people realized that Majorana fermions can exist in condensed matter physics in the form of the quasiparticle.For example,in Kitaev chain model,when the sys-tem is in a topological non-trivial phase,two Majorana fermions will emerge at each end of the chain.This type of fermions is of peculiar interests because of its poten-tial value of applications in the quantum computation.The keys to the quantum bits in quantum computation are the coherence,fault-tolerance and stability under exter-nal disturbance.Protected by its topology,a Majorana fermion possesses those great attributes mentioned above,which makes it a promising candidate as a quantum bit.In this case,it is understandable that much effort has been devoted to the search for Majorana fermions in condensed matter physics these years.In chapter three,we will introduce a special ultracold atomic system——topological superfluid.In the presence of the spin-orbit coupling,atomic superfluid may be driven into a topological phase.For those regions where the magnetic field exceeds a critical filed of the system,a topological phase emerges and Majorana fermions can exist on its edges.It is noteworthy that in such a spin-imbalanced situation,the ground state of the system is supposed to be FFLO(Fulde-Ferrell-Larkin-Ovchinnikov)state.How-ever,we find that spin-orbit coupling can actually suppress the FFLO state and drive the system into a topological one.We also introduce a dark soliton to the topological superfluid and find that in the presence of a strong spin-orbit coupling,the dark soli-ton can accommodate extra Majorana fermions in its core.What is more interesting is that the filling status of the core of the dark soliton is associated with the state of the system,which means we can acquire the information of the system by measuring the polarization of the core of the soliton.In the last chapter,we give a systematic summary for this dissertation and propose some possible outlooks for future researches.