We use the interference scheme in Fig. 7.1 to study spatial entanglement. A pump beam creates twin photons in a non-linear crystal by the process of SPDC. The orientation of the crystal with respect to the propagation direction of the pump beam and the polarisation of the pump beam are such, that the twin photons that propagate collinearly have orthogonal polarisation. The walk-offs that result from the birefringence of the crystal, are compensated by compensating crystals. For simplicity they are not shown in the picture. A pinhole is used to select collinear twins. The crystal birefringence together with the direction of propagation defines a unique basis for the polarisation, consisting of the unit vectors~εH and~εV. A trans- lation∆~sV in the transverse direction is imposed onV-polarised photons. This is done by propagation of the beam through a tilted birefringent crystal. Then also a time difference be- tweenH- andV-polarised photons results, introducing polarisation information in the arrival time of the photons at the detectors. This polarisation information can be erased by propaga- tion through a compensating crystal with the appropriate thickness. The collinear twins fall on a 50% : 50% beam splitter. In one of the output channels a translation∆~sHis imposed in the transverse direction onH-polarised photons. Coincidences are detected by the detectorsa
andbin the output channels. In front of both detectors are narrow-band filters and linear po- larisers set to transmit when the polarisation is at an angle of 45◦with respect to both~εHand ~εV. The narrow-band filters have a centre frequency of half the frequency of the pump beam. The detectors are bucket detectors that integrate both over the time and the transverse space. In front of detectorbis also a circular aperture with radiusd. For this setup the coincidence detection rate is considered as a function of the translation∆~sH for fixed value of∆~sV. Now two amplitudes are relevant. The first is the amplitude that theH-polarised photon is detected by detectoraand theV-polarised photon is detected by detectorb. For the second amplitude it is the other way around. It is necessary that the polarisers are both at 45◦, because then the information about the polarisation is completely erased, and these amplitudes can interfere.
7.3.2
The coincidence detection rate
In section 7.2 we argued that the transfer functions that describe the propagation from the crystal to the detector, factorise in transverse and temporal parts. In the interferometer in Fig. 7.1 there are no optical elements that introduce any polarisation dependence in the temporal properties. As a consequence, the temporal part of the problem drops out, and we can write
7. Interference between entangled photon states in space and time
Figure 7.1:Scheme of the spatial interferometer. The pump beam creates twin photons in a non-linear crystal. Collinear twins with orthogonal polarisations are selected with a pinhole. On V -polarised photons a transverse translation∆~sV is imposed. Then the
beam falls on a beam splitter. In one of the output channels a transverse translation
∆~sHis imposed on H-polarised photons. The detectors a and b detect coincidences in
the two output channels. In front of both detectors is a narrow-band filter and a linear polariser at45◦. There is a pinhole in front of detector b.
the two-photon state in Eq. (7.2) as
|Ψi∝
Z
d~ρG(~ρ)aˆ†H(~ρ)aˆV†(~ρ)|0i, (7.9) where ˆa†H(~ρ)and ˆa†V(~ρ)create a photon at the transverse position~ρin the crystal plane, with
HandV polarisation, respectively. For simplicity the translation∆~sVis imposed at the output plane of the crystal, and the translation∆~sHis imposed immediately behind the output plane of the beam splitter. The detectors are both at a distancezfrom the crystal along the optical lines. The transfer function for free space propagation over a distancezis given by
hf(~ρ,~ρ0;z) = k 2πizexp ik 2z ~ρ−~ρ 02 , (7.10)
wherekis the wavenumber of the light. We consider as an example the transfer function for the propagation from the output plane of the crystal to the detection plane of detectorafor a beam withHpolarisation. We split up the optical line in two parts with lengthsz1andz2,
where the first part stretches from the crystal plane to the output plane of the beam splitter, and the second from the latter plane to the detector plane. The vectors~ρ,~ρ0, and~ρarefer to a point in the plane of the crystal, beam splitter, and detector, respectively. Then the transfer function for the propagation from the crystal to detectorafor anH-polarised beam is given
by Z
where z=z1+z2. We have ignored that the beam splitter changes the handedness of the
basis of the transverse space. We see that for the transfer function it is not relevant in which plane the translation ∆~sH is imposed. Now we can express the positive-frequency parts of the electric-field operators in the detector planes in terms of the annihilation operators ˆaH(~ρ) and ˆaV(~ρ), that annihilate a photon at the location~ρ in the detector plane withH andV polarisation, respectively. Because the polarisers in front of the detectors are at 45◦, only photons with polarisation vector(~εH+~εV)/
√
2 are detected. Therefore we only consider the component of the electric-field operator in the 45◦direction. We have
ˆ Ea+(~ρa) = Z d~ρhf(~ρa,~ρ+∆~sH;z)aˆH(~ρ) +hf(~ρa,~ρ+∆~sV;z)aˆV(~ρ) , ˆ Eb+(~ρb) = Z d~ρhf(~ρb,~ρ;z)aˆH(~ρ) +hf(~ρb,~ρ+∆~sV;z)aˆV(~ρ) . (7.12)
By using Eq. (7.9) and the commutation rules in Eq. (7.3) we find that the coincidence detection amplitude in Eq. (7.8) is given by
A(~ρa,~ρb) =h0|Eˆa+(~ρa)Eˆb+(~ρb)|Ψi = Z d~ρG(~ρ)hf(~ρa,~ρ+∆~sH;z)hf(~ρb,~ρ+∆~sV;z) +hf(~ρa,~ρ+∆~sV;z)hf(~ρb,~ρ;z) , (7.13)
where only the cross products∝aˆHaˆV from ˆEa+Eˆb+contribute. The coincidence detection rate is given by R= Z d~ρa Z II d~ρb|A(~ρa,~ρb)|2, (7.14) where II indicates that the integration domain is over the opening of pinhole II. We perform the integration over~ρaand use that the transfer function for free space propagation is unitary:
Z
d~ρhf(~ρ2,~ρ;z)h∗f(~ρ,~ρ1;z) =δ(~ρ2−~ρ1). (7.15)
We then find that
R=k 2d2 2πz2 Z d~ρ|G(~ρ)|2+2 Re Z d~ρG(~ρ)G∗(~ρ+∆~sV−∆~sH) × Z II d~ρbhf(~ρb,~ρ;z)h∗f(~ρb,~ρ+2∆~sV−∆~sH;z). (7.16)
For the integration over~ρbwe use the property Z II d~ρ0hf(~ρ0,~ρ+∆~s/2;z)h∗f(~ρ0,~ρ−∆~s/2;z) = kd 2πzk∆~skJ1(kdk∆~sk/z)exp[ik~ρ·∆~s/z], (7.17) where Jn(x) = 1 2πin Z 2π 0
7. Interference between entangled photon states in space and time
Figure 7.2:The coincidence detection rate R as a function of the x component of∆~sH−
2∆~sV for a Gaussian pump beam tilted at0.6◦. The intensity of the pump beam drops
off to1/e at a distance of3 mmfrom the beam axis. The wavelength of the pump beam is400 nm. The distance from the crystal to the detectors is2 m. For the radius of the pinhole in front of detector b we have d=3 mm. The value of the x component of∆~sV
is fixed at0.1 mm.
are the Bessel functions of the first kind. By using this property we find that
R∝1 2 Z d~ρ|G(~ρ)|2+ z kdk∆~skJ1(kdk∆~sk/z) ×Re Z d~ρexp(ik~ρ·∆~s/z)G(~ρ+∆~s/2)G∗(~ρ−∆~sH/2), (7.19)
where∆~s=∆~sH−2∆~sV. To see interference fringes, the exponential factor inside the integral must oscillate at least once over the range of the pump spot, which puts a lower bound on
k∆~sk. The visibility of the fringes is determined by the factor in front of the integral, which puts an upper bound on the radiusd of pinhole II. As a consequence, to see interference fringes the spot size must in general not be smaller than the size of pinhole II.
As an example we use a Gaussian function for the pump beam profileG(~ρ). In order to obtain interference fringes, we assume that the pump beam is slightly tilted, such that the propagation direction is in thexzplane. Then the profile of the pump beam has a phase pat- tern. The translations∆~sHand∆~sVare imposed in thexdirection. In Fig. 7.2 the coincidence detection rate is given for this case as a function of thexcomponent of∆~sHfor fixed value of
∆~sV. We see in Fig. 7.2 that for∆~sH=2∆~sV the visibility is 100%. The reason for this can be understood by considering again the two relevant amplitudes that we discussed at the end of Section 7.3.1. We first consider the amplitudeA(H→a;V →b)where theH-polarised photon is detected by detectora, and theV-polarised photon by detectorb. These photons are translated by∆~sH−∆~sV with respect to each other. For the amplitudeA(H→b;V →a) where theH- and theV-polarised photon are detected by detectorbanda, respectively, the photons are translated by∆~sVwith respect to each other. Since in front of the detector there is a linear polariser at 45◦, the information about the polarisation is erased. As a consequence, these two amplitudes are indistinguishable when the polarisation is concerned. For interfer- ence with maximum visibility, the two amplitudes must also be indistinguishable concerning
Figure 7.3:The two relevant amplitudes for the case that∆~sH=2∆~sV, where the loca-
tion of birth of the twin photons for the amplitude A(H→a;V→b)in (a) differs by∆~sV
with respect to the location of birth for the amplitude A(H→b;V→a)in (b). The thick lines are the optical axes of the system and the paths of the photons are dashed. We see that for both amplitudes the location at detector a at which a photon arrives, is the same, but that the polarisation of the photon is different. The same holds for the photon arriving at detector b. Nevertheless, the two amplitudes are indistinguishable since the polarisers in front of both detectors erase the information about the polarisation.
their spatial properties. This is the case when, for the two amplitudes, the vectors over which theH- andV-polarised photon are translated with respect to each other, are identical. That is, when∆~sH=2∆~sV, indeed. Under this condition there might still be another spatial prop- erty that distinguishes the two amplitudes. For the amplitudeA(H→a;V →b)one photon is translated by∆~sV, and the other by 2∆~sV, while for the amplitudeA(H→b;V →a)the translations are 0 and ∆~sV. The relative translation is the same for both, but the absolute translations are different. The reason that the visibility is 100% anyway, is because of the spatial entanglement: the photons of the twins are created at the same location in the crystal, but this location itself is undetermined within the spot size of the pump beam on the crystal. Therefore, when∆~sVis small with respect to the spot size, the two amplitudes cannot be dis- tinguished by a difference in the absolute translation mentioned above. This can be seen in Fig. 7.3, where the location of birth of the twin photons for the amplitudeA(H→a;V →b)
differs by∆~sV with respect to the location of birth for the amplitudeA(H→b;V→a). Then the two amplitudes cannot be distinguished. The width of the envelope in Fig. 7.2 depends on the radius of the pinhole in front of detectorb. A smaller pinhole radius decreases the spa- tial distinguishability of the detector. Because the detector is then less able to obtain spatial information, the width of the envelope becomes larger.
By imposing the translation∆~sH onH-polarised photons the spatial information intro- duced by the translation∆~sV on theV polarisation can be erased. In Fig. 7.1 we see that the erasing is done in one of the output channels of the beam splitter. The condition for erasing is then that there must be spatial entanglement. The information can also be erased by im- posing a translation∆~sH=∆~sVonH-polarised photons before the beam splitter. Then spatial entanglement is not necessary, because before the beam splitter the collinear twins are not polarisation entangled. Erasing behind the beam splitter without using spatial entanglement
7. Interference between entangled photon states in space and time
Figure 7.4:Scheme of the temporal interferometer. The pump beam creates twin pho- tons in a non-linear crystal. A pinhole selects collinear twins with orthogonal polarisa- tion. On V -polarised photons a delay∆τV is imposed. Then the beam falls on a beam
splitter. In one of the output channels a delay∆τHis imposed on H-polarised photons.
Coincidences are detected by the detectors a and b in the two output channels. In front of both detectors is a narrow-band filter and a linear polariser at45◦.
can be done by imposing the translation∆~sH=∆~sV onH-polarised photons in both output channels of the beam splitter.