By a quantum mechanical analysis of the additive rule Fa [Fβ[f]] = Fα＋β[f], which the fractional Fourier transformation （FrFT） Fα [f] should satisfy, we reveal that the position-momentum mutual- transformation ...By a quantum mechanical analysis of the additive rule Fa [Fβ[f]] = Fα＋β[f], which the fractional Fourier transformation （FrFT） Fα [f] should satisfy, we reveal that the position-momentum mutual- transformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation （CFrFT）, which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.展开更多
We propose an entangled fractional squeezing transformation （EFrST） generated by using two mu- tually conjugate entangled state representations with the following operator： e-iα（a1a2＋a1a2）eiπa2a2; this transfo...We propose an entangled fractional squeezing transformation （EFrST） generated by using two mu- tually conjugate entangled state representations with the following operator： e-iα（a1a2＋a1a2）eiπa2a2; this transformation sharply contrasts the complex fractional Fourier transformation produced by e-ia（a1a1＋a2a2）eiπa2a2 （see Front. Phys. DOI 10.1007/s11467-014-0445-x）. The EFrST is obtained by converting the triangular functions in the integration kernel of the usual fractional Fourier transformation into hyperbolic functions, i.e., tan α→ tanh α and sin α→ sinh α. The fractional property of the EFrST can be well described by virtue of the properties of the entangled state representations.展开更多
基金The work was supported by the National Natural Science Foundation of China （Grant Nos. 11105133 and 11175113） and the National Basic Research Program of China （973 Program） （Grant No. 2012CB922001）.
文摘By a quantum mechanical analysis of the additive rule Fa [Fβ[f]] = Fα＋β[f], which the fractional Fourier transformation （FrFT） Fα [f] should satisfy, we reveal that the position-momentum mutual- transformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation （CFrFT）, which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.
文摘We propose an entangled fractional squeezing transformation （EFrST） generated by using two mu- tually conjugate entangled state representations with the following operator： e-iα（a1a2＋a1a2）eiπa2a2; this transformation sharply contrasts the complex fractional Fourier transformation produced by e-ia（a1a1＋a2a2）eiπa2a2 （see Front. Phys. DOI 10.1007/s11467-014-0445-x）. The EFrST is obtained by converting the triangular functions in the integration kernel of the usual fractional Fourier transformation into hyperbolic functions, i.e., tan α→ tanh α and sin α→ sinh α. The fractional property of the EFrST can be well described by virtue of the properties of the entangled state representations.
