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Effects of polarization-dependent loss and fiber birefringence on photon-pair entanglement in fiber-optic channels Milja Medic and Prem Kumar Center for Photonic Communication and Computing, EECS Department, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3118, USA
[email protected]
Abstract: Quantum communication requires that photon-pair entanglement be preserved as the photons are distributed to remote locations. We model the effects of loss and birefringence on polarization-entangled photon pairs propagating in optical fibers. c 2006 Optical Society of America ° OCIS codes: (270.5290) Photon statistics; (060.2330) Fiber optics communications.
In some applications involving entangled photon pairs, the photons need to be propagated to long distances through fibers. Here we explore the effects of loss and birefringence during that propagation on the launched two-photon state. While loss and birefringence have been studied in the context of optical communications [1], we address photon-pair distribution relevant for quantum communications. We develop a model which gives the output state of the photon pair for an arbitrary state at the input of the fiber. We quantify entanglement by tangle [2], which depends on the density matrix of the two-photon state. To get simple and intuitive results on how pair propagation changes entanglement, we then use a maximally-entangled Bell state as the initial state of the photon pair. We first describe our model and then present the obtained dependence of the tangle for the Bell-state photon pairs on polarization-dependent loss and birefringence in the fiber segment of an optical-communication channel. We use the method of antinormally-ordered quantum characteristic function [3] by which it is convenient to find the output state of the photon pair when their input state and the transformation between the annihilation operators of the input and output modes are known. Photon-pair propagation in fibers is in general a four-mode problem, since the entangled photons may have different frequencies (signal and idler) and we need to keep track of the polarizations at each frequency (horizontal-vertical bases are used). The output state ρˆ of the photons can be found from the antinormally-ordered output characteristic function χA . The Heisenberg picture is used, so the initial state and the output-mode operators determine the output characteristic function. To find the output characteristic function, we need relations between the input and output radiation modes a’s and b’s, respectively. Loss is represented with unitary transformations applied to radiation-mode annihilation operators of all the four modes. To preserve the commutators, appropriate vacuum modes are mixed in: q q bhs = ηsh ahs + 1 − ηsh vsh , (1) where ahs (bhs ) denotes the annihilation operator for the input (output) horizontally-polarized signal mode, vsh denotes the annihilation operator for the corresponding vacuum mode, and ηsh is the loss coefficient specific to that mode with values ranging from 0 to 1. The transformations for the remaining three modes are analogous. Birefringence transformations are applied at the output of the lossy channel. This is justified by the fact that polarization-dependent loss, which is negligible in modern fibers, and polarization-mode dispersion can be viewed consecutively in short fiber segments (on the order of a few 10’s of km) [4]. We model fiber birefringence by applying general rotations on the Poincar´e spheres representing state transformations taking place during propagation of the signal and idler modes. We apply the following transformations: |Hi → cos θ|Hi + eıφ sin θ|V i |V i → − sin θ|Hi + e−ıφ cos θ|V i.
(2)
The general result of the model for an arbitrary input state is a cumbersome expression which provides little intuition. We, therefore, apply the model to the input Bell-state density matrix: 1 0 0 1 1 0 0 0 0 . (3) ρˆ0 = 2 0 0 0 0 1 0 0 1
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The corresponding antinormally-ordered output quantum characteristic function we get is: χA (ξsh , ξsv , ξih , ξiv ) = I exp(−|ξsh |2 − |ξsv |2 − |ξih |2 − |ξiv |2 ), with
(4)
q 1 [(1 − |ξsh |2 ηsh )(1 − |ξih |2 ηih ) + (1 − |ξsv |2 ηsv )(1 − |ξiv |2 ηiv ) + (ξsh∗ ξsv ξih∗ ξiv + ξsh ξsv∗ ξih ξiv∗ ) ηsh ηsv ηih ηiv ], (5) 2 where ξ’s are complex arguments of the characteristic function. This characteristic function gives the following output two-photon-state density matrix: p ηsh ηih 0 0 ηsh ηsv ηih ηiv 1 0 0 0 0 . (6) ρˆ = h h v 0 0 0 ηs ηi + ηsv ηi p 0 v v h v ηsh ηsv ηi ηi 0 0 ηs ηi I=
We see that in propagating through a lossy channel, the Bell-state density matrix acquires dependence on η’s, the loss coefficients in the four modes. Since tangle, the measure of entanglement we use, is calculated from the density matrix [2], the tangle of the output state depends on the loss coefficients. Exploring this dependence, we find the following interesting results: If the loss in all modes is the same, the output state is maximally entangled, just as the input Bell state (for 0 < η < 1). Even if the signal and idler frequencies suffer different losses, so long as their losses are polarization independent (ηsh = ηsv 6= ηih = ηiv ), the output state is still maximally entangled. However, if the horizontal and vertical polarizations experience different losses, i.e., polarization-dependent loss (PDL), the tangle is degraded. If we define the ratio of loss in horizontallyh and vertically-polarized modes as: x ≡ ηηv , where ηh ≡ ηsh = ηih and ηv ≡ ηsv = ηiv , then the output-state tangle is equal to T = 4x2 /(1 + x2 )2 [shown in Fig. 1(a)]. To include the effect of birefringence, for simplicity, we apply the transformations from Eq. (2) to the signalmode basis vectors only. In this case, the output-state density matrix for the lossy channel [Eq. (6)] becomes dependent on the birefringence angles. When the loss coefficients η’s are polarization-independent, the tangle is 1 (the output state is maximally entangled) for all angles θ and φ. For a fixed value of the PDL parameter x, the dependence of the tangle on birefringence angles θ and φ is as shown in Fig. 1(b) for x = 1.1. The dependence is periodic with twice the frequency in θ than in φ; the minimum values of the tangle are zero and the maximum values are determined by x via the equation for T shown above. The profile of any tangle peak fits well to a Gaussian. If the birefringence averages close to zero, and the standard deviations of the angle distributions are small enough, the resulting tangle can be on the plateau in Fig. 1(b) with an almost undiminished value after propagation.
1
η η
0.75 T 0.5 0.25 0
1
0 -1 0 -1 1
Fig. 1. a) Dependence of the tangle on PDL. b) Dependence of the tangle on birefringence angles θ and φ for PDL parameter x = 1.1. Note the plateau around (0,0).
These results indicate that polarization entanglement can be well preserved in photon-pair propagation through optical fibers, since the polarization-dependent loss in fibers is negligible, and if the distribution of birefringence angles along the fiber has a small standard deviation and the average values are close to zero. This work is supported in part by the NSF under Grant No. EMT-0523975 and by DARPA and the AFOSR through an HP subcontract under AFOSR contract No. FA9550-05-C-0017. References 1. 2. 3. 4.
B. Huttner, C. Geiser, and N. Gisin, IEEE J. Sel. Top. Quant. Electron. 6, 317 (2000). D. F. V. James et al., Phys. Rev. A 64, 052312 (2001). L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1995). M. Shtaif and O. Rosenberg, J. Lightwave Tech. 23, 923 (2005).
