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Jiguang zazhi

Multimode entangled state with three states superposition on the (N+1)-dimensional Hilbert space and their N-th power difference squeezing properties

Wenhui Chen, Sun Zhong-yu

Published Jan 1, 2007• 0 citations
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Abstract

According to the principle of linear suporposition of quantum mechanics,the multimode entangled states ｜｛Ψ~((3))_(2q) is formed by linearly superposing three quantum states: the multimode complex conjugate coherent state ｜｛iZ~_j｝_(2q),multimode complex conjugate imaginary coherent state ｜｛iZ~_j｝_(2q) and multimode vacuum state ｜｛0_j｝_(2q) on the finite-dimensional(N+l) Hilbert space.and their difference squeezing properties of generalized nonlinear equal-power N-th power is studied by utilizing the general theory of multimode squeezed states.The results show that on the finite-dimensional Hilbert space,while some conditions are satisfied,the two quadratures of the state ｜Ψ~((3))_(2q) present the equal-power N-th power difference squeezing properties,but under some other conditions,the difference squeezing effects of two quadratures can be displayed at the same time,and the squeezed depth or squeezed degree in this space is different from that on the full-dimensional Hilbert space.It is called two-sided difference squeezing.

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According to the principle of linear suporposition of quantum mechanics,the multimode entangled states ｜｛Ψ~((3))(2q) is formed by linearly superposing three quantum states: the multimode complex conjugate coherent state ｜｛iZ_j｝_(2q),multimode complex conjugate imaginary coherent state ｜｛iZ_j｝(2q) and multimode vacuum state ｜｛0_j｝(2q) on the finite-dimensional(N+l) Hilbert space.and their difference squeezing properties of generalized nonlinear equal-power N-th power is studied by utilizing the general theory of multimode squeezed states.The results show that on the finite-dimensional Hilbert space,while some conditions are satisfied,the two quadratures of the state ｜Ψ~((3))(2q) present the equal-power N-th power difference squeezing properties,but under some other conditions,the difference squeezing effects of two quadratures can be displayed at the same time,and the squeezed depth or squeezed degree in this space is different from that on the full-dimensional Hilbert space.It is called two-sided difference squeezing.
According to the principle of linear superposition of quantum mechanics, the multimode entangled states (as stated below) is formed on the finite-dimensional(N+1) Hilbert space by linearly superposing three quantum states: the multimode complex conjugate coherent state (as stated below), multimode complex conjugate imaginary coherent state (as stated below) and multimode vacuum state (as stated below), and their difference squeezing properties of generalized nonlinear equal-power N-th power is studied by utilizing the general theory of multimode squeezed states. The results show that on the finite -dimensional Hilbert space, while some conditions are satisfied, the two quadratures of the multimode entangled states (as stated below) present the equal-power N-th power difference squeezing properties, but under some other conditions, the difference squeezing effects of two quadratures can be displayed at the same time. The squeezed depth or squeezed degree in this space is different from that on the full-dimensional Hilbert space. The later result is not in conformity with the uncertainty principle. It is called "two -sided〞difference squeezing.
On the bases of the linear superposition principle of quantum state in quantum mechanics, |ψ (3) 3q -the third kind of the three state superposition multimode superposition state light field is constructed that is composed of linear superposition of multimode complex conjugation coherent state |{Z * j} q ?multimode complex conjugation imaginary coherent state |{iZ * j} q and multimode vaccum state |{0 j} q . By utilizing the theory of multimode squeezed states, the properties of generalized nonlinear unequal-power N j-th power H-squeezing in the state |ψ (3) 3q is studied in detail. It is found that when the sum of the squeezing power of each mode is even number only and certain conditions are satisfied, the first perpendicular phase component and the second one of the state |ψ (3) 3q can always display any power generalized nonlinear unequal-power N j-th power H-squeezing effects that change periodically, respectively.
By using the general theory of multimode squeezed states, the difference of generalized nonlinear unequal - power higher - power in the multimode Schrodinger -cat state entangled light field is studied in detail. This light field is formed by linear superposition from three quantum states, i.e. multimode complex conjugation coherent state, multimode complex conjugation imaginary coherent state and multimode vacuum state. The conclusion shows that 1) the difference of the cat state entangled light field is independent of the vacuum state; 2) when certain conditions are satisfied, the two quadrature phases of this cat state entangled light field present unequal - power higher - power difference properties respectively, and under some other conditions, the difference effects of two quadrature phases can be displayed at the same time. The former is in conformity with the uncertainty principle. But the later is not, and is called two-sides difference squeezing considering the application of squeezed light in the light communication.
The third of the three state superpositions multimode superposition state light field |ψ(3)3q was constructed by linear superposition of multimode complex conjugation coherent state,|{Zj}q multimode complex conjugation imaginary coherent state |{iZj}q and multimode vaecum state |{0j}q.By utilizing the theory of multimode squeezed states,the properties of generalized nonlinear unequalpower Njth power Ysqueezing (or amplitude squeezing) in the state |ψ(3)3q was discussed.It is found that when the squeezing power of each mode Nj is even number only,and some certain conditions are satisfied,the first or second perpendicular phase component in the state |ψ(3)3q can always display any even number power generialized nonlinear unequalpower Njth power Ysqueezing effeets respectively,which is comlementary,also changes periodically.
By utilizing the general theory of multimode squeezed states, the effects of generalized nonlinear unequal-power higher-power difference in the multimode Schrodinger-cat state entangled light field is studied, that is formed by the linear superposition of three macroscopical distinguishable quantum states named the multimode complex conjugate coherent state, multimode complex conjugate imaginary coherent state and multimode vacuum state. It is found that 1) the difference of the cat state entangled light field is independent of its vacuum state component; 2) in some cases, the two quadrature phase components of this cat state entangled light field present unequal-power higher-power difference properties respectively ; 3) under some other conditions, the difference effects of two quadrature phase components of the state mentioned above can exist at the same time. The two preceding results stated above are in conformity with the uncertainty principle, but the last is not. It is called two-sided difference squeezing phenomenon and could be very useful in the application of squeezed light on light quanta communication.
The properties of equal-higher-power difference squeezing of multimode entangled state light - field composed of multimode complex conjugate coherent state, multimode complex conjugate imaginary coherent state and multimode vacuum state are studied by utilizing the multimode squeezing theory in decoherence. The results show that when some fixed conditions are satisfied the mixed entangled state light - field mentioned above not only has the equal - higher - power difference squeezing in one of the quadratures but also displays the similar effects in the quadratures simultaneously, that is called two - sided difference squeezing. This is a newly physical phenomenon which contradicts the principle of uncertainty.