Complementarity Relationship In The Asymmetric Mach-Zehnder Interferometer

Bohr's complementarity principle points that a single quantum system has mutually exclusive properties,and these characteristics can be converted to each other depending on the method of observation.The wave-particle duality is often used to reflect the Bohr's complementarity principle,so the wave-particle dual-ity has also attracted a lot of attention.Many research groups have conducted a number of theoretical and experimental studies to quantize the wave-particle duality,and many of these are based on the well-known Mach-Zehnder interferom-eter.The Mach-Zehnder interferometer has both symmetrical and asymmetrical types.The device studied in this paper consists of an asymmetric Mach-Zehnder interferometer?symmetry of the first beam splitter and asymmetry of the second beam splitter?and a detector placed on one of the paths of the interferometer We discuss the wave-like property and particle-like property of a single particle incident into the device.The fringe visibility is used to quantize the wave-like property.The distinguishability,the which-path information,and the correlation?including classical correlation and quantum correlation?are used to quantize the particle-like property.The specific content includes?1?For fringe visibility V in the asymmetric Mach-Zehnder interferometer,we find that the fringe visibility is affected by the input state of the particlecharacterized by the Bloch vector ???={S_x,S_y,S_z},the input state of the detector,the unitary operator performed on the detector and the second asymmetric beam splitter characterized by parameter ?.For the particle is initially in a pure state,it is found that the fringe visibility reaches the upper bound when S_x=-cos?.The fringe visibility obtain the maximum only if S_x=0 and ?=?/2 when the input particle is initially in a mixed state.?2?For the distinguishability D in the asymmetric Mach-Zehnder interfer-ometer,we find that the distinguishability is affected by the initial state of the particle and the detector,the unitary operator performed on the detector and the second asymmetric beam splitter.For the detector is initially in a pure state,it is found that the distinguishability reaches the lower bound when S_x=—cos?.The complementary relationship V²+D² ?1 is obtained,and the conditions for the equality are also presented.?3?We study the complementary relationship between the fringe visibility and the which-path information I_path in the asymmetric Mach-Zehnder interferometer.The minimum error measurement and the error-free measurement in the detec-tor axe used to extract the which-path information,respectively.The which-path information is affected by the initial state of the particle and the detector,the uni-tary operator performed on the detector and the second asymmetric beam splitter.The complementary relation between the fringe visibility and the which-path in-formation obtained by the minimum error measurement satisfies V²+I_path²?1,and the complementary relation between the fringe visibility and the which-path information obtained by the error-free measurement satisfies V+I_path?1.The conditions for the equality are also presented,respectively.?4?we study the complementary relationship between the fringe visibility and the correlation?including the classical correlation and the quantum correlation?in the asymmetric Mach-Zehnder interferometer.?1?When the final state of theparticle and the detector is pure,we calculate the classical correlation J(?₁^QD)and the quantum correlation D>(?₁^QD)between the particle and the detector,re-spectively.We find that the classical correlation and the quantum correlation are affected by the initial state of the particle and the detector,the unitary oper?ator performed on the detector and the second asymmetric beam splitter.The complementary relation between the fringe visibility and the classical correlation satisfies V²+J²(?₁^QD)?1,and the complementary relation between the fringe visibility and the quantum correlation satisfies V²+D²(?₁^QD):<1.?2?When the final state of the particle and the detector is mixed,we calculate the classical correlation J(?₂^QD)and the quantum correlation D(?₂^QD)between the particle and the detector,respectively.We find that the classical correlation and the quantum correlation are affected by the initial state of the particle and the detector,the unitary operator performed on the detector and the second asymmetric beam s-plitter.The complementary relation between the fringe visibility and the classical correlation satisfies V²+J²(?₂^QD)?1.The conditions for the equality are also presented,respectively.