Pruebas de MNFO (Minifalloff) y DataFRAC

where Cab and Ca denote the probability of detecting photons in modes a and b simul- taneously and a detection in mode a, respectively. As the photons are always created in pairs we know that each single photon detection indicates a lost partner qubit, which is why η corresponds to the total detection efficiency, i.e. the probability that a created photon is really detected. As argued before, emission stems also from pump photons that are slightly off the central wavelength, therefore it is important to remember that spectral filtering often filters one photon of a pair that is emitted from a photon slightly off the central wavelength, while the other is transmitted. This decreases the coincidence to sin- gle ratio strongly. An additional effect is that the bandwidth of the emissions is slightly different for the two polarizations, therefore also η is polarization dependent.

We can also estimate the value of c in equation 4.1.1 and equation 4.7 from the single and coincidence detection. The probability for a coincidence is the probability for creation of a photon pair ≈2c2 _{(for Z ≈1) times the probability that both photons are detected} ηa andηb:

Kab= 2c2ηaηb⇒2c2 = _SKab

aSb (4.9)

The probability for the pair creation and the efficiency of detecting the photons are basic parameters in the description of our photon sources and valuable for estimating, for ex- ample, the admixed noise due to higher order emissions. Thus, I conclude this section by summarizing typical values of these parameters and the corresponding coincidence count rate estimated from the detection rates with both configurations using 3 nm bandwidth filtering. Note that without further explanation of details the additional factors enter- ing due to splitting of the photons from the collinear configuration have been taken into account.

non collinear collinear Creation Probability 2c2 _{0.008 0.034}

Total Efficiency η 0.09 0.1

Coincidences [1/s] 5500 28000

Table 4.1: Approximate values of the parameters describing the two configurations of SPDC used in our experiments.

4.2 Processing photons

In order to observe a certain quantum state, the photons need to be processed further, as the source of photons does in general not directly create the desired state. Due to the fact that interactions of photons rely on extremely inefficient nonlinear processes, other more efficient methods to control the states need to be found. Such methods rely on linear optics and interferometry.

In a celebrated work, Knill, Laflamme and Milburn showed that universal linear optics quantum computation is indeed possible [151], i.e. any quantum operation can be realized with linear optics networks. Thus, one can also create an arbitrary quantum state with

single photon sources and linear optics. At the current stage, even small networks require practically an unrealistic amount of resources. Even though combinations of their ideas with the one-way quantum computer approach require much less resources [34, 152, 153], some further research is necessary until it is possible to create and control the large amount of photons needed, even for small experiments.

Here, like in many other experiments4, we rely on the technique of conditional detec- tion. A linear optics setup and interferometric methods are used to observe a quantum state under the condition of detecting one photon in each of four specified output modes. Basically, the nonlinearity is introduced by a projection of the complete photonic state onto the part of the state, where one photon is in each spatial mode – in the experiment this projection is realized by selecting the four-fold coincidences. The disadvantage is that the observed entangled state cannot be used for further processing, if the projection onto this subspace cannot be guaranteed. On the other hand, multiparty communication tasks rely on the distribution of the quantum state onto several parties, thus it is ensured that the photons do not need to interact anymore.

4.2.1 Wave plates

Unitary transformations of single qubits can easily (and without conditional detection) be implemented with birefringent crystals. The most important ones are half and quarter wave plates that introduce a phase shift of π and π/2 respectively, between the linear polarizations parallel to the fast and slow axis of the crystal. This introduces the trans- formationsUπ

θ,0 = ˆσθ,0 for the HWP and U

π/2

θ,0 = (i·11 + ˆσθ,0)/ √

2 for the QWP5_{. Here,θ} is the angle between the rotation axis and the horizontal axis. In our experiments we use zero order HWP and QWP6 in order to have little error on the introduced phase shift.

Another important tool are birefringent elements that shift the phase between the horizontal and vertical polarizations, represented by the operation Uω

0,φ =icos(ω/2)11 + sin(ω/2)ˆσz. Here, the fast and slow axes are aligned along horizontal and vertical polariza- tions, and the thickness of the plate, which determines the phaseω, is changed by rotation around the vertical axis. Thus, no zero order plates can be used. In our experiments we use for each compensation a pair of custom made Yttrium-Vanadate crystals (Y V O4) of 200µm thickness7_{. They were chosen due to their strong birefringence at 780 nm (for} more detail see [154]). The pair configuration allows to compensate dispersion effects that depolarize the photons strongly due to their comparably broad band width of about 3 nm. Here, we use the Y V O4 crystals to compensate phase shifts that are usually introduced by imperfect beam splitters.

4_{See e.g. the experiments cited in context of state analysis in the overview of four-qubit states (sec-}

tion 3.2.3) and implementations of quantum gates in the introduction to theCP experiment (chap- ter 5)

5_{The upper index ofU}ω

θ,0corresponds the phase shiftωintroduced by arbitrary waveplates. For definitions

see section 2.1.1.

6_{Zero order, optically contacted, CeNing Optics/FOCtek} 7_FOKtek

4.2 Processing photons

Figure 4.2: The notation of the in- and output modes of a general beam splitter are shown in a). In b) the polarization analysis as used in all of the presented experiments is depicted. A half and a quarter wave plate (HWP and QWP) serve to choose the basis of analysis. A polarizing beam splitter (PBS) separates photons with two corresponding eigenstates for detection in Silicon avalanche photo diodes (APD).

4.2.2 Beam splitters

The basic elements for making joint operations on photon pairs are beam splitters (they are, however, also useful for some single qubit operation). An ideal general beam split- ter8 _{introduces the following transformations, where the creation operators are named} according to the modes in figure 4.2 a):

a†_H → √1 2 ³ TH · a0†_H +iRH · b0†_H ´ (4.10) b†_H → √1 2 ³ TH · b0†_H+iRH· a0†_H ´ (4.11) a†_V → √1 2 ³ TV · a0†V +iRV · b0†V ´ (4.12) b†_V → √1 2 ³ T_V · b0†_V +iR_V · b0†_V ´ , (4.13)

where TH (RH) is the transmission (reflection) amplitude of the horizontal polarization and analogously for vertical polarizations. The phase i for the reflected modes is re- sponsible for a negative phase between the cases where photons from both input modes are transmitted or reflected. Without this, the transformation is not unitary (see article [155]). One imperfection of real beam splitters is that they introduce an additional phase shift between horizontal and vertical polarization in each mode. We use the phase shifters introduced before to compensate this effect (see setup for Dicke state figure 7.1 on page 96).

Beam splitters can be designed with basically any splitting ratio, there are however some special cases. Symmetric or 50/50 beam splitters (SBS) exhibit equal values for the parameters |TH|2 =|TV|2 =|RH|2 =|RV|2 = 1/2.

Usually, the commercially available beam splitters are not perfect and exhibit some asymmetry between the polarizations or the spatial modes (or both). They can, how- ever, usually be aligned to behave as polarization independent beam splitters (BS) that

behave equally for both polarizations, but with different transmission and reflection am- plitudes. After aligning the beam splitters9_{for polarization independence they take values} of|T_H|2 _=|T

V|2 ≈0.6, |TV|2=|RV|2 ≈0.4. In our experiment this influences, as we will see, only the efficiency, not the quality of the state.

Polarizing beam splitters (PBS) are specified by zero reflectivity of horizontal and zero transmittivity of vertical polarization |TH|2 = |RV|2 = 1, |TV|2 = |RH|2 = 0. Again the experimentally used beam splitters are not perfect. As we will see, we use PBSs for polarization analysis, were mainly |TV|2 =|RH|2 = 0 is important. Again by alignment, we reach with our PBSs10 _{a very good value of approximately|T}

V|2 =|RH|2≈0.002. The only other kind of beam splitter that is relevant here is used in the controlled phase gate (chapter 5) and will simply be called polarization dependent beam splitter (PDBS). It’s splitting ratios are in the ideal case |TH|2 = 1, |RH|2 = 0, |TV|2 = 1₃, |RV|2 = 2₃. A detailed characterization of our custom made11 _{beam splitters can be found in the} Diploma thesis by Ulrich Weber [156].

If one relies on conditional detection, there are several tasks beams splitters can be used for:

ˆ Attenuation of one polarization. Polarization dependent beam splitters can be used to attenuate one polarization under the condition that the transmitted photon is detected. This is a non-unitary transformation and corresponds, together with the conditional detection, to a POVM measurement [157].

ˆ Splitting of photons from one into two modes. Two photons that are incident on a polarization independent BS split up probabilistically. As conditional detection discards the other cases, the BS effectively works as a lossy photon splitter, where the smallest losses are achieved for a symmetric BS.

ˆ Via second order interference (HOM-type, see [158]) the overlap of photons from different input modes allows the implementation of quantum gates and Bell state analysis (see e.g. [159–161]). The implementation of a controlled phase gate with this technique is demonstrated in detail in chapter 5.

ˆ Polarization dependent beam splitters can in combination with half- and quarter wave plates, be used for polarization analysis, as demonstrated in figure 4.2 b). The measurement basis to be analyzed is turned to the Z axes of the Bloch sphere12_and ’clicks’ in the output modes of the PBS indicate the detection of one or the other eigenstate.

In all of the experiments presented here the configuration shown in figure 4.2 b) is used for the analysis of each photon. In order to optimize the analysis of a state the wave plates for the choice of the basis are motorized and driven according to a computer program. This allows to frequently change the setting automatically (which is usually done in intervals of 10 minutes). Thus all measurement settings are a time average of

9_LINOS 10_Newport

11_{EXPLA, former EKSMA}