1. Ra
H = 0 =RbH
The reflectivity of the overlap beam splitter P DBSO for horizontal polarization should be zero.
2. RaVRbV
Ta
VTVb = 2
This ratio of transmission and reflection amplitudes ofP DBSOis the substitute for the fixed values ofRV =√2TV =p2/3 in the ideal case.
3. Ta
HAH =TVaAV
This is the only condition on the attenuation plate in mode a. The transmission amplitude for both polarizations needs to be such that the polarization of a single photon transmitted through the whole setup from mode a is conserved, i.e. the effect of P DBSO is compensated.
4. Tb
HBH =TVbBV
The same condition as 3.) for the attenuation plate in mode b.
The parameters of the originally presented ideal phase gate follow from these condi- tions under the assumption of a loss free, symmetric overlap beam splitter. The whole calculation was performed for indistinguishable photons to discuss the distinct imperfec- tions separately. The considerations of section 5.1.2 and section 5.1.3 are combined by expressing the matrices ˆMtt and ˆMrr (section 5.1.2) in terms of the parameters derived here.
These four conditions establish requirements on the three components independently and therefore turn out to be extremely useful for the alignment of the experimental setup, which will be presented in detail in the following section.
5.2 Experimental setup
In this section, the actual experimental implementation of the phase gate will be described. This includes the quantification of the distinguishability of the photons used to test the phase gate via an Hong-Ou-Mandel type (HOM) interference experiment. Further, the components used in the setup are characterized with respect to the parameters discussed in the last section. For a detailed description of how the setup was built, the interested reader may refer to the Diploma thesis by Ulrich Weber [156].
For the experimental investigation of the gate we use photon pairs emitted from a non- collinear SPDC source, as described in section 4.1.1. As perfect entanglement cannot be guaranteed, we prefer to use separable states as input for the phase gate. Therefore, a horizontally and a vertically aligned polarizer are placed in the two emission modes, respectively, before they are coupled into single mode fibers. These fibers guide the photons to the phase gate. Via stress induced birefringence in the fiber, the polarizations of the photons are rotated at will to arbitrary separable input states. Thus, the preparation of the input states is completely independent of the phase gate setup itself. The spectral filtering of the photons (filter bandwidth ≈ 2 nm FWHM) is achieved by interference filters mounted in the phase gate setup together with the polarization analysis. While the actual position of the filters is irrelevant, this choice has the advantage that by filtering in the output ports of P DBSO possible differences between the filter functions have less
Figure 5.2: Photo of the linear optics controlled phase gate setup. On the left hand side, the two couplers where the photons enter the gate are located. The actual gate operation takes place at the three beam splitters in the center of the picture. The photons are overlapped at the cube and attenuated at the roundP DBSplates. On the right hand side, the polarization analysis is mounted consisting of two motorized rotation mounts for the wave plates, an interference filter, a PBS and multimode fibers that guide the photons to the APDs (section 4.2).
influence, because the filter function does not ”label” the photon by its mode. Thus, better indistinguishability is guaranteed.
A picture of the experimentally realized phase gate is shown in figure 5.2. The photons enter the setup in modes a and b via two different fiber coupling systems. In mode a, the fiber coupler is mounted on a translation stage moving parallel to the photons spatial mode. Thereby, the optical path length of the photon from the SPDC crystal toP DBSO can be set. This allows to compensate arbitrary time delays of the two photons at the overlap beam splitter. The other coupler allows alignment of the transverse position and direction of the spatial mode b. In addition, the position of the focus of the emitted gaussian beam can be set (the corresponding divergence is equal for both modes due to usage of identical optics components). Thus, the spatial overlap of mode b with mode a can in principle be achieved. For details on the alignment procedure, refer to [156].
The three beam splitters P DBSO, P DBSa and P DBSb are custom made and were coated for perfect transmission of p- and 2/3 transmission for s- polarization for photons with a wavelength of 780 nm. The overlap beam splitterP DBSO is a cube mounted such that |Hiis perfectly transmitted. For attenuation, we use beam splitting plates (P DBSa and P DBSb) with the same coating as P DBSO, however the mount is rotated by 90◦
such that the roles of horizontal and vertical polarizations are exchanged. Real beam splitters do not exactly reproduce the desired values and are, in addition, dependent on the angle of incidence (AOI) of the photons. Therefore, it is very useful to apply the four conditions derived in the previous section (see equation 5.1.3, page 66) for their alignment. Conditions 1.) and 2.) are requirements on P DBSO only and cannot be compensated by additional components. Thus, we first seek to align the AOI ofP DBSO
5.2 Experimental setup
Figure 5.3: Hong-Ou-Mandel interference at the polarization dependent overlap beam splitterP DBSO in the phase gate for a |V V iinput. In case of perfect interference the
count rate should drop down to 20 % leading to a theoretically achievable dip visibility of 80 %.
such that these conditions are fulfilled as good as possible. Because we observed that condition 1.) does not dependent on the AOI ofP DBSO we optimize 2.) reaching a value of 2.018±0.003 for the respective ratio3. The corresponding value for condition 1.) is then|Ta H|, ¯ ¯Tb H ¯
¯≈0.993. Finally, following conditions 3.) and 4.), the attenuation plates P DBSaand P DBSb are tilted such that the polarization of a single photon transmitted through the whole setup (mode a or b) is preserved as good as possible.
As argued before, a perfect overlap of the two gaussian modesaandb can be achieved by alignment of the coupler in mode b. Furthermore, we need to ensure equal optical path length for both photons between the crystal and P DBSO. To do so, we use a HOM-type measurement [158, 179] providing information about the position of zero delay and about the quality factor Q0 that describes the indistinguishability of the photons
for each delay. By scanning the delay between the two photons, using the coupler in mode a, it is possible to sample the |V V i-coincidence count rates for the transition from completely distinguishable to maximally indistinguishable photons. The outcome of a typical measurement is shown in figure 5.3. When the photons become less distinguishable, the count rate drops. This is due to the fact that, as argued in section 5.1.2, the probability of a coincident detection of |V V iat the output of the phase gate is lower when interference occurs (in the ideal case by a factor of five).
The gaussian shape of the resulting curve is mainly determined by the interference filters used for spectral filtering of the photons. Thus, we can use a gaussian function to fit this curve:
c(x) =A µ 1− Ve−(x−x0) 2 2∆2 ¶ , (5.21)
where Ais the count rate for big delay,V is the visibility of the HOM-dip, ∆ is its width and x0 is the position of the minimum, where zero delay between the two path lengths is reached. All measurements for the characterization of the gate made in the following are
3The ratioR
V/TV is equal for bothP DBSO input modes. For a loss free beam splitter this corresponds to a transmittivity (reflectivity) of the vertical polarization ofT2
taken with the translation stage in mode a atx0.
The value of interest for determination of the indistinguishability of two photons is V. It can be expressed as V = (c∞−c0)/c∞ where c∞ is the count rate at positions with
big delay, i.e. perfect distinguishability, andc0 the rate at zero delay. In a typical HOM- experiment at a symmetric beam splitter, the ideal value is 1, which means vanishing count rates for perfect indistinguishability. Here, we have to consider the special splitting ratio of the beam splitter. For large delay, the success probability p∞
V V = 5/9, which reduces in the ideal case to p0V V = 1/9 for perfect indistinguishability. This results in an ideal visibility ofVideal = 80%. Experimentally we findVexp = 72.8%±0.7%. From this we can calculate the fraction of indistinguishable photons asQ=Vexp/Vth= 91.0%±0.9%. This also means that the probability of observing a coincidence from two vertically polarized input photons is increased by an admixture of the state |V V ihV V |, which is obtained for any input state that includes |V V i-terms. This admixture plays an important role in the further analysis and will henceforth be refered to asV V-noise. It is directly visible in figure 5.3 as the difference between the minimum count rate and the 20%-level.