Processing photons

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-formationsU_θ,0^π = ˆσ_θ,0 for the HWP and U_θ,0^π/2 = (i·11 + ˆσ_θ,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 U_0,φ^ω =icos(ω/2)11 + sin(ω/2)ˆσ_z. 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.

4See e.g. the experiments cited in context of state analysis in the overview of four-qubit states (sec-tion 3.2.3) and implementa(sec-tions of quantum gates in the introduc(sec-tion to theCP experiment (chap-ter 5)

5The upper index ofUθ,0^ω corresponds the phase shiftωintroduced by arbitrary waveplates. For definitions see section 2.1.1.

6Zero order, optically contacted, CeNing Optics/FOCtek

7FOKtek

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

T_H · a^0†_H +iR_H · b^0†_H

´

(4.10)

b^†_H → 1

√2

³

T_H · b^0†_H+iR_H· a^0†_H

´

(4.11)

a^†_V → 1

√2

³

T_V · a^0†_V +iR_V · b^0†_V

´

(4.12) b^†_V → 1

√2

³

T_V · b^0†_V +iR_V · b^0†_V

´

, (4.13)

where T_H (R_H) 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 |T_H|² =|T_V|² =|R_H|² =|R_V|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

8In the notation of the second quantization.

behave equally for both polarizations, but with different transmission and reflection am-plitudes. After aligning the beam splitters⁹for polarization independence they take values of|T_H|² =|T_V|² ≈0.6, |T_V|²=|R_V|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 |T_H|² = |R_V|² = 1, |T_V|² = |R_H|² = 0. Again the experimentally used beam splitters are not perfect. As we will see, we use PBSs for polarization analysis, were mainly |T_V|² =|R_H|² = 0 is important. Again by alignment, we reach with our PBSs¹⁰ a very good value of approximately|T_V|² =|R_H|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 |T_H|² = 1, |R_H|² = 0, |T_V|² = ¹₃, |R_V|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 sphere¹²and

’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

9LINOS

10Newport

11EXPLA, former EKSMA

12From the waveplate transformations it is easy to see that this is possible for any axis.