The property of generalized nonlinear equal-order N-TH power Y-squeezing in a new type of multi-mode schrödinger-cat state light field with the superpositions of macroscopically distinct two quantum states

According to the linear superposition principle in quantum mechanics in this paper, it is constructed a new type of multi-mode Schrödinger-Cat state light field ψ⁽²⁾>_q, which is made of the linear superposition of macroscopically distinct two quantum states named the contrary state | { - Z_j}>_q, of multi - mode ( or q - mode ) coherent state | { Z_j } >_q, and the multi - mode imaginary coherent state I {i Z_j } >_q, respectively . By using the theory of multi - mode squeezed states established recently by Yang Zhiyong and Hou Xun in references 25 - 28, the property of generalized nonlinear equal-order N-th power Y-squeezing of the state ψ ⁽²⁾>_q mentioned above is studied. It is found that l) if the squeezed order N = 2p and p = 2m(m = 1, 2, 3, ⋯, ⋯), the state l ψ⁽²⁾>_qis always stayed in the N-Y minimum uncertainty state; 2) if the squeezed order N = 2p and p = 2m +1 (m = 0, 1, 2, 3, ⋯⋯), while the quantization conditions are satisfied by the initial phase φ of each mode by the initial phase difference (θ_pq⁽¹⁾ - θ_nq^(r)) between the two components in the state |ψ⁽²⁾>_q, and by the sum σ_j=1^qR_j² of average photon mumbers of all the single-mode coherent state light field |Z _q>(here, j= 1, 2, 3, ⋯, q), the multi-mode Schrodinger-Cat state | ψ⁽²⁾>_q, can presents any order N-th power Y-squeezing effect which changes periodically;and 3), if the squeezed order N = 2p' +1, whether p' =2m (m = 0, 1, 2, 3, ⋯, ⋯) or p' - 2m + 1 (m = 0, 1, 2, 3, ⋯, ⋯), meanwhile if some certain conditions of quantization are satisfied by the initial phase φ_j of each mode, one while the probability amplitudes satisfy r_pq⁽¹⁾ = r_nq^(r), the state ψ⁽²⁾>_q, is only stayed in the N-Y uncertainty state rather than the n-Y minimum uncertainty state, and two, while r_pq⁽¹⁾ ≠r_nq ^(r), the N-Y uncertainty state, the N-Y minimum uncertainty state, and the N-th power Y-squeezing can never present.