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Acta Physica Polonica A

Enhancement of the Hong-Mandel Higher-Order Squeezing and Amplitude Odd-Power Squeezing in Even Coherent State by its Superposition with Vacuum State

Pankaj Kumar, R. Kumar, Hari Prakash

Published Jun 1, 2017• 3 citations
Publisher

Abstract

We study the Hong-Mandel 2n th-order squeezing of the Hermitian operator, X θ ≡ X1 cos θ + X2 sin θ and amplitude n th-power squeezing of the Hermitian operator,], of vacuum state and even coherent state defined by |α,, a is the annihilation operator, α, θ, r and ϕ are arbitrary and the only restriction on these is the normalization condition of the superposed state.We show that the Hong-Mandel 2n th-order squeezing and amplitude odd-power squeezing exhibited by even coherent state enhance in its superposition with vacuum state.Variations of these higher-orders squeezing with different parameters near its maxima have also been discussed.

DOI
10.12693/aphyspola.131.1485

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We study the Hong-Mandel 2n th-order squeezing of the Hermitian operator, X θ ≡ X1 cos θ + X2 sin θ and amplitude n th-power squeezing of the Hermitian operator,], of vacuum state and even coherent state defined by |α,, a is the annihilation operator, α, θ, r and ϕ are arbitrary and the only restriction on these is the normalization condition of the superposed state.We show that the Hong-Mandel 2n th-order squeezing and amplitude odd-power squeezing exhibited by even coherent state enhance in its superposition with vacuum state.Variations of these higher-orders squeezing with different parameters near its maxima have also been discussed.
The characteristics of the k th-power squeezing of a class of superposition states which involving two orthonormalized eigenstates of photon annihilation operator aN are studied.The result show that the characteristics of the k th-power squeezing of a class of superposition states which involving two orthonormalized eigenstates of photon annihilation operator aN are different from the characteristics of the k th-power squeezing of the orthonormalized eigenstates of photon annihilation operator aN clearly.No matter what N is an odd number or even number,the superposition states own amplitude k(k=Nt or Nt/2) th-power squeezing,but for the orthonormalized eigenstates of aN,only when N is even number there exist amplitude k(k=(2P+1)N/2) th-power squeezing.Owing to that,when phase difference δ=2mπ/t(m is integer),the superposition states don′t own amplitude k th-power squeezing;when δ=π,only N and t are odd number simultaneously,the superposition states may own amplitude k th-power squeezing;when δ=π/2,t≠4m,the superposition states own amplitude k th-power squeezing;thus the phase difference of parameter has key effect on amplitude k th-power squeezing.
In this paper, the multi-mode squeezed state theory is utilized to study the effects of even number-order unequal-power Njth power Y-squeezing on the two kind of multi-mode quantum superposition statesψ(2)± ■ that is made of the linear superposition of the multimode coherent stateR(a)j exp(-iφ■)■ and the multi-mode imaginary coherent states±iR(b)j exp(-iφ■)■.when the squeezing order number Nj of each mode is even number, viz. Nj=2pj(j=1,2,3,......q, pj=1,2,3,......), it is found that two kind ofψ(2)± ■can always display respectively the generalized nonlinear even-number-order unequal-power Njth power Y-squeezing effects which changes periodically while the some conditions are satisfied respectively by the initial phase φj of each mode,and by initial mixing-phase (θ[R]pq -θ(I)nq )■■R(a)j R(b)j which is composed of the initial phase difference θ(R)pq -θ(I)nq between the two components of the stateψ(2)± ■mentioned above and the sum ■R(a)j R(b)j of the products R(a)j R(b)j of amplitudes R(a)j and R(b)j .The squeezing-intensity is bigger when the pj is odd number.
The multi-mode squeezed state theory was utilized to study the effects of higher order unequal-order N_jth power Y-squeezing on the multi-mode quantum superposition state |Ψ_q that was made of the linear superpositions of the two nonsymmetry multi-mode coherent states |{z~((a)*)_j}_q and |{-z~((b)*)_j}_q.It was found that two quardrature phase components in state |Ψ_q could always display respectively the generalized nonliear higher-order unequal order N_jth power Y-squeezing effects which change periodically while some conditions are satisfied respectively by the product N_jφ_j of the initial phase φ_j of each mode, moreover the squeezing order number N_j of each mode(here j=1,2,3,...,q), and by the initial phase difference θ~((R))_(pq)-θ~((R))_(nq) between the two components of state |Ψ_q.The squeezing intensity was bigger when all of squeezing order number N_j of each mode were odd number.
The multi-mode squeezed state theory is utilized to study the effects of higher order unequal-order N_jth power H-squeezing on the multi-mode quantum superposition state |ψ〉_q that is made of the linear superpositinos of the two nonsymmetry multi-mode coherent states |{z~((a)~*)_j}〉_q and |-{z~((b)~*)_j}〉_q, it is found that two quardrature phase components in state |ψ〉_q can always display respectively the generalized nonliear higher-order unequal -order N_jth power Y-squeezing effects which changes periodically while some conditions are satisfied respectively by the sum ∑qj=1N_jφ_j of the multiplication N_jφ_j of the initial phase φ_j of each mode and squeezing order number N_j of each mode(here j=1,2,3,…,q), and by the initial phase difference θ~((R))_(pq)-θ~((R))_(nq) between the two components of state |ψ〉_q. The squeezing intensity is bigger when sum ∑qj=1N_j of squeezing order number N_j of each mode is odd number.
In this paper,a kind of nonsymmetry multi-mode quantum superposition state Ψ〉_q is constituted by the complex conjugate multi-mode coherent state z~((a)*)_j〉_q and the contrary state -iz~((b)*)_j〉_q of imaginary conjugate coherent state iz~((b)*)_j〉_q.It is studied for the equal-order even-number-order Y-squeezing effect in Ψ〉_q, by using multi-mode squeezed state theory.When squeezing order N is even number ,viz. N=2p(p=1,2,3,…),it is found that the two quardrature phase components on the Ψ〉_q present alternately equal-order even-number-orderY-squeezing effect which changes periodically whether p is odd number or even number,while some phase conditions are saisfied respectively by the initital phase φ_i of each mode in Ψ〉_q and by the initital mixing-phase Δθ+∑qj=1[R~((a))_jR~((b))_j] which is composed of the initital phase difference between the two components of the state Ψ〉_q and the sum ∑qj=1[R~((a))_jR~((b))_j] of the product R~((a))_jR~((b))_j of amplitude of each mode in z~((a)*)_j〉_q and amplitude R~((b))_j of each mode in -iz~((b)*)_j〉_q . The squeezing-intensity odd-number p is bigger than that under even-number p,which is not the result in symmetry multi-mode superposition state.
WT5”BZ]:The properties of the two order and higher order squeezing of phase in Kerr even and odd coherent states have been discussed via the measured phase operator introduced by S.M.Barnett and D.T.Pegg.The results exhibit that the measured phase operator cos M has not only periodic k order squeezing in cs type in even and odd coherent states but the k order squeezing in the type of cN when k is even.They also show that the degree of squeezing is relevant with ｜α｜ 2.When ｜α｜ 2 is comparatively big,the squeezing in Kerr even coherent states resembles that in Kerr odd coherent states.Nevertheless,when ｜α｜ 2 is small,the squeezing in Kerr odd coherent states appears shallow,even no squeezing effect occurs. [WT5”HZ]