The depiction of Hamiltonian PT-symmetry

Abstract

The theory of PT-symmetry describes the non-hermitian Hamiltonian with real energy levels, which means that the Hamiltonian <i>H</i> is invariant neither under parity operator <i>P</i>, nor under time reversal operator <i>T</i>, <inline-formula><tex-math id="M2">\begin{document}$ PTH = H $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M2.png"/></alternatives></inline-formula>. It originated from the Hamiltonian is real and symmetric is not a necessary condition for guarantee the fundamental axioms of quantum mechanics: real energy levels and unitary of the time evolution. The theory of PT-symmetry plays a significant role in the quantum physics and quantum information science. Where the problem that how to describe PT-symmetry of Hamiltonian was focused on deeply. In the paper, we propose the definition of operator <i>F</i> based on the PT-symmetry theory and the normalized eigenfunction of Hamiltonian. Then we give the first description of the PT-symmetry of Hamiltonian in dimensionless cases after finding the features of commutator and anti-commutator of operator <inline-formula><tex-math id="M3">\begin{document}$ CPT $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M3.png"/></alternatives></inline-formula> and operator <i>F</i>. Furthermore, we discover this method also can quantify the PT-symmetry of Hamiltonian in dimensionless cases. <inline-formula><tex-math id="M4">\begin{document}$ I(CPT,F)=\|[CPT,F]\|^{CPT} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M4.png"/></alternatives></inline-formula> represents the part of PT-symmetry broken, and <inline-formula><tex-math id="M5">\begin{document}$ J(CPT,F)=\|\{CPT,F\}\|^{CPT} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M5.png"/></alternatives></inline-formula> represents the part of PT-symmetry. If <inline-formula><tex-math id="M6">\begin{document}$ I(CPT,F)=\|[CPT,F]\|^{CPT} = 0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M6.png"/></alternatives></inline-formula>, Hamiltonian <i>H</i> is global PT-symmetric. Once <inline-formula><tex-math id="M7">\begin{document}$ I(CPT,F)= $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M7.png"/></alternatives></inline-formula><inline-formula><tex-math id="M7-1">\begin{document}$ \|[CPT,F]\|^{CPT}\neq0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M7-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M7-1.png"/></alternatives></inline-formula>, it shown that Hamiltonian H is PT-symmetric broken. In addition, we propose another method to describe PT-symmetry of Hamiltonian based on real and imaginary parts of eigenvalues of Hamiltonian, which only used to judge whether the Hamiltonian is PT symmetric. <inline-formula><tex-math id="M8">\begin{document}$ ReF=\dfrac{1}{4}\|(CPTF + F)\|^{CPT} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M8.png"/></alternatives></inline-formula> represents the sum of squares of real part of the eigenvalue <inline-formula><tex-math id="M9">\begin{document}$ En $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M9.png"/></alternatives></inline-formula> of Hamiltonian <i>H</i>, <inline-formula><tex-math id="M10">\begin{document}$ ImF=\dfrac{1}{4}\|(CPTF - F)\|^{CPT} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M10.png"/></alternatives></inline-formula> is the sum of imaginary part of the eigenvalue <inline-formula><tex-math id="M11">\begin{document}$ En $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M11.png"/></alternatives></inline-formula> of a Hamiltonian <i>H</i>. If <inline-formula><tex-math id="M12">\begin{document}$ ImF= 0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M12.png"/></alternatives></inline-formula>, Hamiltonian <i>H</i> is global PT-symmetric. Once <inline-formula><tex-math id="M13">\begin{document}$ ImF\neq0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M13.png"/></alternatives></inline-formula>, Hamiltonian <i>H</i> is PT-symmetric broken. If <inline-formula><tex-math id="M14">\begin{document}$ ReF = 0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M14.png"/></alternatives></inline-formula> it implies that Hamiltonian <i>H</i> is PT-asymmetry, but it is a sufficient condition, not necessary. The later is easier to operate in the experiment, but the study conditions are tighter, and it further requires that <inline-formula><tex-math id="M15">\begin{document}$ CPT\phi_n(x)=\phi_n(x) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_M15.png"/></alternatives></inline-formula>. If we only pay attention to whether PT-symmetry is broken, use the latter method is simpler. The former method is perhaps better to quantify the PT-symmetry broken part and the part of local PT-symmetry.