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Acta Sinica Quantum Optica

量子光学中的双模幺正相位算符及其经典对应

De-Hui Zhan, 范洪义 Hong-yi FAN

Published Jan 1, 2021• 0 citations
Publisher

Abstract

众所周知,传统定义的单模相位算符不是幺正算符。我们选择另一种定义相算符的途径,鉴于相位的测量总是相对的,就像势能的测量是相对于零势能位置一样,所以我们在双模空间中引入相位算符。具体做法是在纠缠态表象的基础上提出幺正的双模相位算符,它的本征值就是经典位相。在此理论框架内,相角对应的正则共轭算符是光子数差算符。并且给出了数差-相测不准关系以及此相位算符的Weyl-Wigner经典对应,发现它是双变数相空间(q1,p1;q2,p2)中的相exp［iargtanp1+p1q1-q2］。

DOI
10.3788/jqo20212704.0101

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众所周知,传统定义的单模相位算符不是幺正算符。我们选择另一种定义相算符的途径,鉴于相位的测量总是相对的,就像势能的测量是相对于零势能位置一样,所以我们在双模空间中引入相位算符。具体做法是在纠缠态表象的基础上提出幺正的双模相位算符,它的本征值就是经典位相。在此理论框架内,相角对应的正则共轭算符是光子数差算符。并且给出了数差-相测不准关系以及此相位算符的Weyl-Wigner经典对应,发现它是双变数相空间(q1,p1;q2,p2)中的相exp［iargtanp1+p1q1-q2］。
We give the explicit form of the common eigenstate \ensuremath{\Vert}\ensuremath{\xi}〉 of the center-of-mass coordinate ${\mathrm{\ensuremath{\mu}}}_{1}$Q1{\mathit{Q}}_{1}Q1​+${\mathrm{\ensuremath{\mu}}}_{2}$Q2{\mathit{Q}}_{2}Q2​ and the mass-weighted relative momentum ${\mathrm{\ensuremath{\mu}}}_{2}$P1{\mathit{P}}_{1}P1​-${\mathrm{\ensuremath{\mu}}}_{1}$P2{\mathit{P}}_{2}P2​ of two particles, which is more complicated than the common eigenstate of the other pair of commutative operators Q2{\mathit{Q}}_{2}Q2​-Q1{\mathit{Q}}_{1}Q1​ and P1{\mathit{P}}_{1}P1​+P2{\mathit{P}}_{2}P2​. The orthonormal and completeness relation of \ensuremath{\Vert}\ensuremath{\xi}〉 are investigated easily by virtue of the technique of integration within an ordered product of operators. The normally ordered squeezing operator for ${\mathrm{\ensuremath{\mu}}}_{1}$Q1{\mathit{Q}}_{1}Q1​+${\mathrm{\ensuremath{\mu}}}_{2}$Q2{\mathit{Q}}_{2}Q2​ and ${\mathrm{\ensuremath{\mu}}}_{2}$P1{\mathit{P}}_{1}P1​-${\mathrm{\ensuremath{\mu}}}_{1}$P2{\mathit{P}}_{2}P2​ is also derived by using the \ensuremath{\Vert}\ensuremath{\xi}〉 representation.
本文把量子场论中算符的正规乘积应用于相干态. 利用正规乘积下的积分方法可以简捷地导出一系列新的量子算符公式、某些著名的么正变换和泛函积分的转换矩阵元、给出了Wigner算符的显示形式和相干态形式,并由此分别就玻色、费米两种情况证明了Weyl对应的实质是相干态对应,其中费米子Weyl对应和Wigner算符是根据费米子相干态的理论给出的,由它可以进一步导出无边界项的费米子泛函积分、母泛函、费米子Wigner定理和Moyal括号,从而建立了费米系统量子力学及其赝经典对应的理论.
Para Hamiltonianas analiticas, e possivel estender equacoes de movimento reais para coordenadas complexas q = q1 + ip2 e p = p1 + iq2 . Dessa forma obtemos H(q,p) = H1 (q1 ,p1 ,q2 ,p2 ) + iH2 (q1 ,p1 ,q2 ,p2 ), onde H1 e H2 sao funcoes reais das variaveis reais q1 ,p1 ,q2 ,p2 e cujas equacoes de movimento sao geradas apenas por H1. Neste trabalho, utilizamos tais coordenadas para calcular as trajetorias complexas do oscilador harmonico, para o qual obtivemos trajetorias periodicas analogas as do caso real para dois graus de liberdade. Para a Hamiltoniana do oscilador harmonico ao quadrado obtivemos um comportamento mais rico, com trajetorias periodicas em apenas em alguns casos especiais. Essas trajetorias tem papel importante no calculo do limite semiclassico na representacao de estados coerentes.
在实验上重构了从准周期到相干塌陷混沌激光的相空间Wigner准概率分布函数，为混沌保密通信中熵源的精确表征提供了重要依据。通过调控偏置电流和光反馈强度制备了准周期（带宽为3.2 GHz）、中等强度（带宽为7.3 GHz）、相干塌陷状态（带宽为11.5 GHz）的混沌激光，进而利用平衡零拍量子层析测量及最大似然法重构了获得的三种不同状态的混沌激光的密度矩阵及相空间Wigner准概率分布，测得的混沌激光相空间Wigner准概率相较于真空散粒噪声极限增大了1.5~3.0倍。同时，在重构测量过程中，去除背景噪声并对现有实验测量系统的损耗进行补偿，三种状态混沌激光相空间Wigner准概率分布的测量保真度分别从74.9%、81.2%、84.8%显著提升到95.5%、97.0%、97.6%。
Statistical reformulation of quantum mechanics in terms of phase-space distribution functions as given by Moyal using Weyl's correspondence rule between classical functions and operators has been extended to various different correspondence rules. The dynamical bracket in the Weyl correspondence (the ``Moyal'' or the ``sine'' bracket) is shown to be a Lie bracket. It is further shown that if the theory is restricted to Lie brackets of the form [u(p, q),υ(p, q)]={f(∂2∂q1 ∂p2−∂2∂q2 ∂p1)u(p1, q1)υ(p2, q2)},evaluated for p1 = p2 = p; q1 = q2 = q after differentiation, then the only admissible functional form of f is f(x) = β[(sin αx)/α], where α and β are constants. A law of multiplication which is associative and distributive with respect to addition is also introduced in each case. It gives a correct correspondence between operator multiplication and the multiplication of classical functions. The dynamical brackets obtained in each case are also found to be Lie brackets. Conditions on the phase-space distribution functions to describe pure states are also given.