Dispersion cancellation and quantum eraser experiments analyzed in the Wigner function formalism

where we have represented the pumping laser beam b plane wave of amplitude V andg is a dimensionless coupling constant. A similar expression holds for the idler beam exchanging the indices ‘‘ s’’ and ‘‘ i .’’ F0s (1) is the vacuum field entering the crystal in the direction of the signal bea and F0i (1) is the vacuum field in the direction of the idle beam.vs and v i are the average frequencies of the bea with wave vectorsks , k i , respectively, fulfilling the matching conditionsvs1v i5v0, ks1k i'k0, with v0 andk0 being the frequency and wave vector of the pumping la beam.G and J are linear operators expressing the intera tion, within the crystal, of the laser with the zero-point fiel The correlation properties of these fields are as follows: (a) Autocorrelations . Taking the signal field at a point r and timest and t8, we have


I. INTRODUCTION
Parametric down-conversion experiments in the Wigner representation have been recently studied ͓1,2͔. In this section we present a brief summary of the most importants results in order to apply them to new experiments. In the Wigner representation the electric field corresponding to a narrow light beam may be written ͑without taking the polarization into account͒ where we assume that the light beam contains frequencies k in an interval ( min , max ), and wave vectors with a limited transverse component ͉k tr ͉Ӷ min /c. It is convenient to work with slowly varying amplitudes defined by F ͑ ϩ ͒ ͑ r,t ͒ϭe i a t E ͑ ϩ ͒ ͑ r,t ͒, ͑2͒ a being an average frequency more or less midway between min and max . If the complex amplitude ␣ k (t) has a free evolution of the form then the amplitude F (ϩ) (r B ,t) in terms of the amplitude F (ϩ) (r A ,t) at another point of the light beam is where r AB ϭr B Ϫr A , r AB ϭ͉r AB ͉, and it is assumed that On the other hand, to second order in perturbation theory, the signal beam leaving the crystal can be expressed as where we have represented the pumping laser beam by a plane wave of amplitude V and g is a dimensionless coupling constant. A similar expression holds for the idler beam by exchanging the indices ''s' ' and ''i.'' F 0s (ϩ) is the vacuum field entering the crystal in the direction of the signal beam, and F 0i (ϩ) is the vacuum field in the direction of the idler beam. s and i are the average frequencies of the beams with wave vectors k s , k i , respectively, fulfilling the matching conditions s ϩ i ϭ 0 , k s ϩk i Ϸk 0 , with 0 and k 0 being the frequency and wave vector of the pumping laser beam. G and J are linear operators expressing the interaction, within the crystal, of the laser with the zero-point field. The correlation properties of these fields are as follows: (a) Autocorrelations. Taking the signal field at a point r and times t and tЈ, we have Here (tЈϪt) is a function that vanishes when ͉tЈϪt͉ is greater than the coherence time between signal and idler. From Eq. ͑8͒ it is possible to derive all cross correlations at different points r rЈ by using Eq. ͑4͒. Finally, the quantum theory of detection in the Wigner representation gives us the following results for single and joint detection probabilities: (a) Single probability. The following result is a general expression for calculating single probabilities in the Wigner representation: where I(r 1 ,t)ϭ͉E (ϩ) (r 1 ,t)͉ 2 , and I 0 (r 1 ) is the intensity of the vacuum field at the position of the detector.

II. DISPERSION CANCELLATION
In this section we present a study of dispersion cancellation in a fourth-order interferometer ͓4͔, using the Wigner formalism. This kind of process has been considered as an example of nonlocality in quantum mechanics, due to the fact that there is no dispersion cancellation in ''classical'' optics ͓5͔. However, the Wigner formalism suggests a fully local interpretation of this and many other phenomena, in the sense that a description in terms of fields propagating in space time is possible without ever surpassing the velocity of light. This possibility rests upon the fact that, in parametric down-conversion, the Wigner function is positive definite ͓1,2͔ and it may be interpreted as a probability distribution. Consequently, the Wigner representation of the experiments offers a counterexample to the claim that no local realist model may account for the said experiments.
The experimental setup is shown in Fig. 1. This is similar to the Hong-Ou-Mandel interferometer ͓3͔, but with a dispersive medium inserted in one arm. In order to calculate the joint probability we are going to express the fields at the detectors D 1 and D 2 , by propagating the slowly varying functions F (ϩ) from the crystal to the beam splitter BS ͑the phase factor from BS to the detectors is dropped because it is not important͒. The main difference with the other experiments that we have explained in the Wigner formalism lies in the fact that we have to propagate the field F s (ϩ) through the dispersive medium. Let us start by writing the fields at the detectors D 1 and D 2 at different times t and tϩ. Assuming, for simplicity, that TϭRϭ1/ͱ2 for the beam splitter, we have F ͑ϩ͒ ͑r 2 ,tϩ͒ϭ 1 ͱ2 ͓F s ͑ϩ͒ ͑r 2 ,tϩ͒ϩiF i ͑ϩ͒ ͑r 2 ,tϩ͔͒,

͑12͒
where and similarly for F i (ϩ) (r 2 ,tϩ). ␦l is the optical path length in the lower arm of the interferometer. In order to obtain F s (ϩ) (r 1 ,t) we shall use the expressions ͑1͒ and ͑2͒. We have where we have replaced the sum by an integral and extended the range of the integral to Ϯϱ because the function ␣( k s ) is peaked at k s Ϸ s , and we have introduced a constant K, which includes some other constants that are irrelevant for our purposes. We may expand the wave number k( k s ) to second order in a Taylor series about s as follows: with ␣ and ␤ being constants appropriate for the dispersive medium.
In order to express F s (ϩ) at rϭr 1 in terms of F s (ϩ) at rϭ0, we take into account that ␣( k s ) is the inverse Fourier transform of E s (ϩ) (0,t), that is, By taking into account Eqs. ͑16͒, ͑14͒, ͑12͒, and ͑13͒ we finally have

͑17͒
with a similar expression for F (ϩ) (r 2 ,tϩ). The coincidence detection probability is given by the correlation ͗F ͑ ϩ ͒ ͑ r 1 ,t ͒F ͑ ϩ ͒ ͑ r 2 ,tϩ ͒͘ϭ where we have used Eqs. ͑7͒, ͑8͒, and defined the Fourier transform of Multiplying Eq. ͑18͒ by its complex conjugate and using Eq. ͑11͒ we can calculate the joint detection probability. After some easy algebra, making use of the relation ͵ 0 ϱ dx sinax sinbxϭ 2 ͓␦͑aϪb͒Ϫ␦͑aϩb͔͒, and assuming that () is symmetric in , we have C being a constant. Finally, by substituing Eq. ͑15͒ into Eq. ͑20͒ and defining ϭ k s Ϫ s we obtain the final result for P 12 : This result is similar to the one obtained in Eq. ͑12͒ of ͓4͔.

III. THE QUANTUM ERASER
In 1992 Kwiat and co-workers ͓6͔ performed an experiment to show how the information may be erased from the state vector. This effect is known as the quantum eraser and shows the relation between quantum coherence and distinguishability. An outline of the experimental setup is shown in Fig. 2. A half wave plate at an angle (/2) to the horizontal is placed in one arm of a Hong-Ou-Mandel interferometer giving rise to a change in the polarization state of the light in this arm. Two polarizers P 1 and P 2 at angles 1 and 2 to the horizontal are inserted in front of detectors D 1 and D 2 , respectively.
We now present an analysis of this experiment in the Wigner formalism. This time we have to take into account the polarization of both the light beam and the vacuum field. The field is now represented by a vector ⑀ k, being orthonormal polarization vectors (ϭ1 denotes horizontal polarization and ϭ2 vertical polarization͒. It can easily be proved that the expressions for the detection probabilities ͑9͒, ͑10͒ remain valid. Moreover, the final expression for the joint detection probability when we deal with parametric down-conversion experiments involving polarization is P 12 ͑r 1 ,t;r 2 ,tϩ͒ϰ ͚ ͚ Ј ͉͗F ͑ϩ͒ ͑r 1 ,t͒F Ј ͑ϩ͒ ͑r 2 ,tϩ͉͒͘ 2 .

͑23͒
In order to apply Eq. ͑23͒ we must calculate the fields at the detectors D 1 and D 2 . For the sake of simplicity we shall consider that the half wave plate is placed at rϭ0. The signal beam coming out the half wave plate and the idler beam outgoing the crystal are F s ͑ ϩ ͒ ͑ 0,t ͒ϭF s ͑ ϩ ͒ ͑ 0,t ͒͑ cos,sin ͒, The field corresponding to the output port of the 50:50 beam splitter ͑placed at rϭR) in the direction of detector D 1 is ,tϪ 1 ͒e i s 1 cos,iF s ͑ϩ͒ ͑0,tϪ 1 ͒e i s 1 sin.

͑28͒
with ␦ϭ 1 Ϫ 2 , and C being a constant. When ␦ϭ0 we have P 12 ϭCЈsin 2 sin 2 ͑ 2 Ϫ 1 ͒, ͑29͒ CЈ being another constant. This expression is similar to the one obtained in the Appendix of ͓6͔.