Theoretical Study On The Symmetry And Coherence Of Quantum System

Symmetry is an important topic in the development of physics.As Noether's theorem stated,every sysmetry in classical physcis corresponds to a conserved quantity.Symmetry in physical system simplifies problems a lot.However,the description of symmetry and its broken still re-mains all or nothing and there is no research about the continuous quantification of symmetry property.In this paper,we propose the degree of symmetry(DoS)based on the Frobenius norm.The main idea of this DoS is that the distance between the transformed Hamiltonian or density matrix of one system and its original one carries its symmetry information.Four good properties of this DoS make it more reasonable in physics.With the DoS,we study the symmetry property of physical systems and obtain a quantitative relation between DoS and perturbation strength.The DoS of qubits is also studied.An important application of DoS is to describe the spontaneous symmetry breaking(SSB)of physical system.Spontaneous symmetry breaking happens with continuous phase transition where systems transfer from high-symmetry phase to low-symmetry one.Generally speaking,phase transition can be characterized with an order parameter.When the continuous phase transition occurs,its order parameter grows from zero to non-vanishing value continuously.Hence,in SSB systems,the decay of its DoS must correspond to the growth of its order parameter.Here,we take the Bose-Einstein Condensation and superconductor as examples and obtain the DoS formula with Order parameters.There exist strong connections between symnetry aad coherence.Generally speaking,coher-ence is a spectial kind of symmetry.Hence,as one part of this paper,we also study the decoherence theory of macroscopic objects.As we all know,systems in microscopic world such as electrons and atoms evolve under quantum mechanics while objects in macroscopic world moves under classical physics.The difference between quantum and classical physics is the quantum coher-ence and uncertainty induced by superposition principle.Therefore,one question arises naturally:Why is not there quantum coherence in macroscopic objects?One of the answers is the quantum decoherence theory.'In this theory,macroscopic systems interact with their environments whose degree of freedom is very large.After long time evolution,the off-diagonal element of the reduced density matrix tend to zero and the quantum coherence vanishes.In this letter,we study the de-coherence of one macroscopic system-an one-dimentional relativistic particle ring.Here we only consider the nearest-neighbour interaction and neglect the interaction with environment.We find that there exist interaction between the center of mass motion and the relative motion.What is more important,this interaction may induce the decoherence of the cemter of mass motion.In fact,the decoherence rate depends on the scale of the system,and the larger the onject is,the faster it decoheres.