A Possible Generalization of Dirac’s Phase Operator and its Relation to Paul’s Phase Operator

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• 作者：Jun Zhou ; Hong-yi Fan
• 关键词：Dirac’s phase operator ; Phase operator in doubled Fock space ; Entangled state representation ; Paul’s phase operator
• 刊名：International Journal of Theoretical Physics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：56
• 期：2
• 页码：514-520
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics, general; Quantum Physics; Elementary Particles, Quantum Field Theory; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer US
• ISSN：1572-9575
• 卷排序：56

文摘

In this paper we generalize Dirac’s phase operator \(\frac {1}{\sqrt {N}}a\) by defining a new phase operator \(\sqrt {\frac {a+\tilde {a}^{\dagger }}{a^{\dagger }+\tilde {a}}}\) in doubled Fock space, where \(\tilde {a}\) is a fictitious mode which annihilates the fictitious vacuum state \(\left \vert \tilde {0}\right \rangle \). It turns out that \(\sqrt {\frac {a+\tilde {a}^{\dagger }}{a^{\dagger }+\tilde {a}}}\) corresponds to a classical phase in the entangled state representation and is unitary. Remarkably, \(\left \langle \tilde {0}\right \vert \sqrt {\frac {a+\tilde {a}^{\dagger }}{a^{\dagger }+\tilde {a}}}\left \vert \tilde {0}\right \rangle \) is just the Paul’s phase operator whose antinormally ordered form is \({\vdots } \frac {1}{\sqrt {N}}a\vdots \). We also employ the method of integration within ordered product of operators to obtain the Fock representation of Paul’s phase operator, from which one can see how it deffers from Susskind-Glogower’s phase operator.