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Modern Physics Letters B

ENTANGLEMENT OF AMPLITUDE AND OPTICAL PHASE INVOLVED IN THE EPR EIGENSTATE

Hong-Yi Fan, Yue Fan

Published Dec 10, 2000• 5 citations
Publisher

Abstract

By noticing that although single-mode Susskind–Glogower phase operator does not commute with photon number operator, two-mode nonlinear phase operator commutes with number-difference operator. We reveal that the common eigenstate <η| of two-particles' relative coordinate and total momentum involves the entanglement between phase and amplitude. We also reveal entanglement between number difference and amplitude inherent in the state |D,|η|>, where [Formula: see text] is a deductive state of |η>.

DOI
10.1142/s0217984900001208

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After a penetrating analysis of the known number–phase commutative relations we find that the common eigenvector |η⟩ of two particles' relative coordinate and total momentum also embodies the entanglement with respect to the correlative amplitude–operational phase. Based on the fact that the number-difference operator is the canonical conjugate to the Noh–Fougères–Mandel operational phase operator, the corresponding entanglement is also briefly discussed.

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Comment on Correlative amplitude-operational phase entanglement embodied by the EPR-pair eigenstate

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In a recent paper in this journal Fan (Fan H 2002 J. Phys. A: Math. Gen. 35 1007 ) discards the possibility of using a genuine phase-difference operator to investigate number-phase entanglement because of the lack of unitarity of the Susskind–Glogower phase operators. However, Fan overlooked the existence of a bona fide unitary operator exponential of the phase difference. Here we find the amplitude–phase maximally entangled states as the simultaneous eigenstates of the total number and the phase-difference operators.

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Physica Scripta•Oct 1, 2012•0 citations
By noticing that although single-mode Susskind–Glogower phase operator does not commute with photon number operator, two-mode nonlinear phase operator commutes with number-difference operator. We reveal that the common eigenstate <η| of two-particles' relative coordinate and total momentum involves the entanglement between phase and amplitude. We also reveal entanglement between number difference and amplitude inherent in the state |D,|η|>, where [Formula: see text] is a deductive state of |η>.
In a recent paper in this journal Fan (Fan H 2002 J. Phys. A: Math. Gen. 35 1007 ) discards the possibility of using a genuine phase-difference operator to investigate number-phase entanglement because of the lack of unitarity of the Susskind–Glogower phase operators. However, Fan overlooked the existence of a bona fide unitary operator exponential of the phase difference. Here we find the amplitude–phase maximally entangled states as the simultaneous eigenstates of the total number and the phase-difference operators.
We define a unitary phase operator for photons in a single momentum mode. The operator acts on a Hilbert space with basis consisting of all number states in both polarizations. The Susskind Glogower operator, E^=eiϕ^{\hat{E}} = e^{i \hatϕ}E^=eiϕ^​, thus defined is unitary and acts as a true ladder operator over all the states. It satisfies the commutation relation [E^,n^]=E^[{\hat{E}}, {\hat{n}}] = {\hat{E}}[E^,n^]=E^; consequently, the phase ϕ^\hatϕϕ^​ satisfies the canonical commutation relation [n^,ϕ^]=i[{\hat{n}}, {\hatϕ}] = i[n^,ϕ^​]=i. The eigenstates of the Susskind Glogower operator are found. Our model is able to directly account for phase measurements when polarization changes cannot be neglected and at the same time gives a well defined phase operator.
We demonstrate that a common eigenvector |η> (Einstein, Podolsky and Rosen, EPR states) of two particles' relatively coordinate and total momentum possesses a property that the projection (a Bell measurement) of two pairs of EPR states |η>1,2⊗| η'>3,4 onto the subspace spanned by 1,4<η″|, yields a EPR state |η″-η*-η′* >2,3 for particles 2 and 3. Entanglement swapping operator is also derived by virtue of the technique of integration within an ordered product of operators.
Since the commutation relation [x1-x2,a1+a2]=0,their common eigenvector |β,x〉 is constructed by using the technique of integration within an ordered product(IWOP) of operators.This state exhibits the properties of the coherent and entangled states.Its entangled properties and the relations with the EPR entangled state can be confirmed by Schmidt decomposition and the unilateral integral for|β,x〉.The superposition of the states |β=β1+iβ2,x〉 along β1 axis yields the ideal two-mode entangled state.
We define a Susskind–Glogower operator that is unitary and acts as a true ladder operator for photons of a given frequency and any polarization. The operator acts on a Hilbert space consisting of all number states in both polarizations. The phase operator derived from this ladder operator satisfies the canonical commutation relations with the number operator. The eigenstates of this ladder operator are also found. Our method based on all the observable states enables us to write the operators for objects such as beam splitters, phase shifters or polarizers, as constant matrices; those used to study classical polarized beams. The Pegg–Barnett operator and the two-mode operator of Ban are special cases of our operator. We compared the phase state defined here with that obtained from the original Susskind–Glogower case, as well as that obtained by the method of Pegg and Barnett in the limit of a very small number of photons being present, and found that there are differences which may enable a direct experimental comparison of these operators.