Complementary Relationship Between Fringe Visibility And Path Information In An Asymmetric Mach-Zehnder Interferometer

The particle-like property and the wave-like property are inherent properties of microscopic particles.They exist at the same time,but they cannot observed.In the same experiment,once interference fringes are observed,the particle-like will property disappear.Conversely,when the path of the particle is accurately obained,the wave-like property will disappear.This is Bohr’s principle of comple-mentarity which is related to wave-particle duality.Many studies on the quantifi-cation of wave-particle duality and their complementarity are based on symmetric Mach-Zehnder interferometers.But Mach-Zehnder interferometers are not neces-sarily symmetric.And the study of asymmetric Mach-Zehnder interferometers is more universal.So we studied the complementarity of particle and volatility in an asymmetric Mach-Zehnder interferometer.The difference from the symmetric Mach-Zehnder interference experimental device is the asymmetry of the second beam splitter.We use the fringe visibility to quantify the wave-like property,use the path information to quantify the particle properties,and explore the comple-mentary relationship between the fringe visibility and path information.The main research contents are as follows:1.The visibility of fringes of asymmetric Mach-zehnder interferometer was studied.It is found that the fringe visibility of asymmetric Mach-Zehnder inter-ferometer is not only affected by the initial state of particle and detector,but also by the second asymmetric beam splitter.There is an upper limit on the fringe visibility,that is,V≤C,and the maximum value C is obtained only when the initial state of incident particles is pure,and the parameter satisfies cosβ=-S_x.2.We obtain the path information of asymmetric Mach-Zehnder interferom-eter by the minimum error discrimination method.The path information is found to be zero in the following cases:（1）The second beam splitter is in the case of total transmission or total reflection,that isβ=0 orβ=π;（2）S_x=±1,that is single photon only propagates in path a or b;（3）C=1,the detector state does not change.In addition,the size of the I_pathpeak is determined by C,the position of the peak is determined by S_xandβ.When cosβ=-S_x,the path information reaches the upper limit.3.The the fringe visibility V and the which-path information I_pathin an asymmetric Mach-Zehnder interferometer are studied,complementary relation-ship:V²+I²_path≤1.the equation is established to satisfy the following three conditions:（1）the particle is a pure state,（2）cosβ=-S_x,（3）C=0 or C=1.