Towards probing for hypercomplex quantum mechanics in a waveguide interferometer

In tests searching for higher-order interferences, any nonlinearity in the experiment can lead to a false-positive detection of higher-order interferences [17, 37]. Similarly, nonlinearities also have a systematic effect on the Peres test, because they bias the addition of the rates in (5)–(7). In our experiment, which uses low-power continuous-wave illumination, the photodetector and its subsequent electronics are the dominant sources of nonlinearity, which need to be accounted for. Therefore, we measured the nonlinearity of the detector in independent beam combination experiments, following the procedure outlined in [38]. Correcting the measured data of the Peres experiment for this detector nonlinearity results in a correction of F by

$$\begin{equation}{\Delta}{F}_{23{^\circ}\mathrm{C}}^{\mathrm{N}\mathrm{L}}=-4.7{\times}1{0}^{-5}\quad \text{and}\quad {\Delta}{F}_{30{^\circ}\mathrm{C}}^{\mathrm{N}\mathrm{L}}=-9.4{\times}1{0}^{-5}.\end{equation} \tag{ 9 }$$

These deviations are negligible compared to the observed deviation. Thus, we can rule out nonlinearity as a relevant source of error in our experiment.

The measurements shown in this work were taken over periods of 9 h and 13 h and temporal effects can occur on various timescales. The effect of slow drift in power or phase difference on a timescale much longer than a measurement cycle (∼1.75 min) is reduced by measuring all eight path combinations in random order. Power fluctuations occurring on timescales shorter than the measurement duration for one shutter combination (∼13 s) are averaged over multiple successive data points. Phase fluctuations on that timescale, however, reduce the interference contrast, which is considered in section 5.3. In this section, we concentrate on power and phase fluctuations on the intermediate timescale of one measurement cycle.

In a first scenario, we assumed fluctuations of the input power during a cycle and overall perfect phase stability. To simulate this, we expressed the power P_i of each single path combination i ∈ {A, B, C} as

$$\begin{equation}{P}_{i}={P}_{\text{in},i}\cdot {T}_{i}.\end{equation} \tag{ 10 }$$

The variable P_in,i is the average power coupled into the chip during the measurement of the combination i and T_i the fraction passing through the open shutter i and ending up at the path of the detector, effectively depending on the overall splitting factor of both beam splitters of the photonic circuit and the total transmission. For the two-path combinations i, j ∈ {A, B, C}, with i ≠ j, the power P_ij is given by

$$\begin{equation}{P}_{ij}={P}_{\text{in},ij}\left({T}_{i}+{T}_{j}+2\sqrt{{T}_{i}{T}_{j}}\enspace \mathrm{cos}({\Delta}{\phi }_{ij})\right).\end{equation} \tag{ 11 }$$

As the total transmission of the chip factors out in the analysis, it is sufficient to determine the normalized splitting factors t_i = T_i /(T_A + T_B + T_C), which were measured in a separate experiment as t_A = 0.26, t_B = 0.52 and t_C = 0.22. While the phase differences were calculated from the corrected normalized interference terms (see table 1). Each input power is treated as an independent normally distributed random variable with mean and standard deviation extracted from the measured data (for details see appendix A.2). Based on the measurement data we can determine the standard deviation of the fluctuations on either the timescale of a single shutter setting or the timescale of the whole measurement, but not directly on the scale of one cycle. The standard deviation on the timescale of the whole measurement is larger and thus leads to more deviations in the Peres parameter. We therefore use it as an upper-bound estimation for the effect of input-power fluctuations. We calculated the deviations of the Peres parameter from the expected value ΔF = F − 1 and its spread σ_F with a Monte Carlo simulation. The results in table 2 show that power fluctuations mainly lead to a random uncertainty of the measured Peres parameter and only a small systematic shift.

Table 2. Influence on the Peres parameter F caused by power and phase fluctuations, respectively. The mean of the distribution calculated with a Monte Carlo simulation is given as a deviation from the expected value ΔF = F − 1. The standard deviation is described by σ_F . The results are listed for the two datasets at different housing temperatures T_h.

	Power fluctuations		Phase fluctuations
Th	ΔFPow	${\sigma }_{{F}^{\mathrm{P}\mathrm{o}\mathrm{w}}}$	ΔFPhase	${\sigma }_{{F}^{\mathrm{P}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}}$
23 °C	5.1 × 10−4	1.8 × 10−2	0.8 × 10−4	1.0 × 10−2
30 °C	1.4 × 10−4	1.4 × 10−2	−8.1 × 10−4	1.4 × 10−2

To estimate the impact of phase fluctuations, we considered fluctuations on a timescale slower than one shutter combination. Any faster fluctuations would lead to a reduction of the interference contrast, which we discuss in section 5.3. To model the slower phase fluctuations we used equations (10) and (11) and assumed perfect stability of the input power for each combination P_in,ij = P_in,i = P_in. The phase differences of each two-path combination are modelled by independent Gaussian distributed numbers with mean values given by the corrected phase differences (see table 1) and the standard deviation given by the observed fluctuations on the timescale of the whole measurement (see appendix A.2 for details). We again performed a Monte Carlo simulation and obtained the values for ΔF and σ_F as given in table 2. As it turns out, the phase fluctuations have a similar impact as the power fluctuations, i.e. they mainly lead to a larger random uncertainty and only to a small systematic shift of the Peres parameter. The latter is for both types of fluctuations two orders of magnitude smaller than the observed ΔF and therefore negligible.

Phase fluctuations that are faster than the duration of one shutter setting lead to a reduction of the normalized interference terms χ ∈ {α, β, γ}. Our model of this effect describes the measured normalise interference terms $~{\chi }$ as

$$\begin{equation}\chi \rightarrow ~{\chi }={C}_{\chi }\cdot \chi ,\end{equation} \tag{ 12 }$$

where C_χ < 1 is the reduced interference contrast. This model predicts a strong dependence of the Peres parameter on the interference contrast C_χ of each pair of paths. Assuming the same interference contrast for all three interference terms (C_χ = 1 − ΔC), the deviation of the Peres parameter is calculated as follows:

$$\begin{equation}{\Delta}{F}^{\mathrm{C}}=2\left(\alpha \beta \gamma -1\right){\Delta}C-\left(4\alpha \beta \gamma -1\right){\Delta}{C}^{2}+2\alpha \beta \gamma \cdot {\Delta}{C}^{3}.\end{equation} \tag{ 13 }$$

To estimate the reduction of the interference contrast, we used the corrected phase fluctuations on the timescale between a shutter setting (∼13 s) and a single measurement (∼0.11 s). How these were extracted from the measured data is outlined in appendix A.2. We used these fluctuations to calculate the interference contrasts with a Monte Carlo simulation around the mean phase difference Δϕ = 0 for each interference term. Due to the maximum of the cosine function at this point, phase fluctuations lead to a unidirectional reduction of the power, such that the resulting interference contrast serves as an estimate at all possible mean phases. However this method does not consider the effect of fluctuations on a timescale below a single measurement (above ∼9 Hz). As worst a case estimate, we used the smallest of these interference contrasts (C = 0.999 998) for each normalized interference term $~{\chi }$ resulting in a deviation from the Peres parameter ΔF^C in the order 10⁻⁶ for both measurements. Compared to the measured deviations this effect is negligible.

We also separately measured the interference contrast of our setup for one path combination by thermally tuning the phase difference, which gives a contrast of C_α = 1.000(29). The measurement and its evaluation is described in appendix A.3. Note that the number of subsequent measurements per shutter setting is only 1/10 of the number in the main experiment. This leads to reduced duration of each shutter setting and therefore reduces the range of fluctuation frequencies affecting the contrast. Hence this interference contrast probably underestimates the degradation of contrast from fluctuations.

Ideally, the state of each shutter should only influence the path it is inserted to. However, experimentally one cannot exclude the possibility that closing one shutter also affects the light in the other two paths. The moving steel wires could for example lead to bending of the chip and with it to a change of the phase of the other paths. In theory, the shutter could also influence the transmitted power of other paths, however this seems less likely than an influence on the phase and is, therefore, not considered here.

In the following model of phase crosstalk, the closed shutter of path i changes the phase of another path j by ${\Delta}{\phi }_{j}^{i}$ as opposed to the open state in path i. This is shown for our triangular path arrangement in figure 3. We constrained these six parameters by symmetry considerations: Due to the slightly elliptical waveguide profiles, which are common for femtosecond laser writing, the symmetry axis of the interfero-meter runs vertically through path B (cf figure 3). Therefore, it seems very plausible to impose this mirror symmetry also on the crosstalk terms. That is, we assumed that the crosstalk originating from path A is equal to the one from path C and the crosstalk originating from B is the same for both of the other paths:

$$\begin{equation}\begin{aligned}{\Delta}{\phi }_{\mathrm{C}}^{\mathrm{A}}& ={\Delta}{\phi }_{\mathrm{A}}^{\mathrm{C}}\equiv {\Delta}{\phi }_{\mathrm{H}},\\ {\Delta}{\phi }_{\mathrm{B}}^{\mathrm{A}}& ={\Delta}{\phi }_{\mathrm{B}}^{\mathrm{C}}\equiv {\Delta}{\phi }_{\mathrm{D}}^{\text{CA}},\\ {\Delta}{\phi }_{\mathrm{A}}^{\mathrm{B}}& ={\Delta}{\phi }_{\mathrm{C}}^{\mathrm{B}}\equiv {\Delta}{\phi }_{\mathrm{D}}^{\mathrm{B}}.\end{aligned}\end{equation} \tag{ 14 }$$

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We further assume the two diagonal phase crosstalks ${\Delta}{\phi }_{\mathrm{D}}^{\mathrm{B}}$ and ${\Delta}{\phi }_{\mathrm{D}}^{\text{CA}}$ to be of same magnitude

$$\begin{equation}{\Delta}{\phi }_{\mathrm{D}}^{\mathrm{B}}={\Delta}{\phi }_{\mathrm{D}}^{\text{CA}}\equiv {\Delta}{\phi }_{\mathrm{D}}.\end{equation} \tag{ 15 }$$

These simplifications lead to the following equations for the normalized interference terms

$$\begin{equation}\begin{aligned}\alpha & =\mathrm{cos}({\Delta}{\phi }_{\text{BC}}+{\Delta}{\phi }_{\text{DH}}),\\ \beta & =\mathrm{cos}({\Delta}{\phi }_{\text{CA}}),\\ \gamma & =\mathrm{cos}({\Delta}{\phi }_{\text{AB}}-{\Delta}{\phi }_{\text{DH}}),\end{aligned}\end{equation} \tag{ 16 }$$

with Δϕ_DH = Δϕ_D − Δϕ_H. The effects on the normalized interference term β cancel each other out, leaving only the parameter Δϕ_DH to be quantified. For this purpose, we additionally performed the Sorkin test, which probes for higher-order interferences in quantum mechanics (see [12–20] and appendix B), on the same set of data used for the Peres test. The Sorkin test has the advantage that of all effects considered in this chapter only crosstalk and nonlinearity (which can here be excluded due to its miniscule effect on the measured data; cf section 5.1) can lead to a systematic shift in its result, [17]. Therefore, the deviation from the expected value = 0 can be used to estimate the magnitude of the crosstalk present in our setup. The measured (see appendix B) deviate significantly from = 0 for both temperatures. Using our crosstalk model one can then calculate the crosstalk strength Δϕ_DH required to produce the observed deviation. Based on these results we then calculated the expected deviation of the Peres parameter ΔF as listed together with Δϕ_DH in table 3. Note that we also tested an asymmetric model, where all phase crosstalk arises from a single path (e.g. from one wire moving under stronger friction than the others). This delivered consistently smaller $\left\vert {\Delta}F\right\vert$. The results of this section indicate that some crosstalk is present in our experiment. It could well lead to a non-negligible bias on the Peres test, yet the size of this effect seems to fall short of the observed deviations $\left\vert {\Delta}F\right\vert$ by one order of magnitude.

Unfortunately, our wire-based shutters exhibit finite transmissivity in their closed state. Therefore we assign to the shutter in each path i a closed-state transmissivity τ_i , due to residual light passing through the wire-blocked hole. This additional light can interfere with the light from open paths and influence the measured Peres parameter. To simplify things, we consider the same transmissivity for each shutter (τ_i = τ ∀i ∈ {A, B, C}) and the phase of the residual light to be shifted with respect to the phase in the open-state by ϕ_Sh. With this model the measured background P₀ (all shutters closed), the single path P_i and the two path combinations P_ij are described by

$$\begin{equation}{P}_{0}={~{P}}_{0}+{P}_{\text{in}}\tau \left(\sum\limits _{i}{\;T}_{i}+\sum\limits _{i,j}\;(1-{\delta }_{i,j})\sqrt{{T}_{i}{T}_{j}}\enspace \mathrm{cos}({\Delta}{\phi }_{ij})\right)\end{equation} \tag{ 17 }$$

$$\begin{equation}{P}_{i}={~{P}}_{0}+{P}_{\text{in}}\left({T}_{i}+\tau \sum\limits _{j\ne i}{\;T}_{j}+2\sqrt{\tau {T}_{i}}\enspace \sum\limits _{j\ne i}\sqrt{{T}_{j}}\enspace \mathrm{cos}({\Delta}{\phi }_{ij}+{\phi }_{\text{Sh}})\right.+\left.2\tau \sqrt{{T}_{k}{T}_{l}}\enspace \mathrm{cos}({\Delta}{\phi }_{kl})\right)\end{equation} \tag{ 18 }$$

$$\begin{equation}{P}_{ij}={~{P}}_{0}+{P}_{\text{in}}\left({T}_{i}+{T}_{j}+\tau {T}_{k}+2\sqrt{{T}_{i}{T}_{j}}\enspace \mathrm{cos}({\Delta}{\phi }_{ij})+\right.\left.2\sqrt{\tau {T}_{i}{T}_{k}}\enspace \mathrm{cos}({\Delta}{\phi }_{ik}+{\phi }_{\text{Sh}})+2\sqrt{\tau {T}_{j}{T}_{k}}\enspace \mathrm{cos}({\Delta}{\phi }_{jk}+{\phi }_{\text{Sh}})\right),\end{equation} \tag{ 19 }$$

with i, k, l ∈ {A, B, C} and i ≠ k ≠ l. The input power P_in already includes in- and out-coupling efficiencies. The transmissivities T_i are the percentage of light passing through path i and reaching the detector. The power ${~{P}}_{0}$ describes a generic shutter-independent incoherent background, e.g. from straylight or detector dark current. This term subsequently drops out, when calculating the normalized interference terms, as the measured background is subtracted from the single- and two-path combinations.

We estimated the shutter transmissivity based on the measured background signal P₀. For this calculation we assumed ${P}_{i}={~{P}}_{0}+{P}_{\text{in}}{T}_{i}$, i.e. neglecting the contribution of τ compared to T_i . Furthermore we assumed that the shutter-independent background is only given by the dark voltage of the detectors ${~{P}}_{0}={P}_{\text{dark}}$. Any additional shutter-independent background would further decrease the shutter transmissivity and, therefore, the result would act as an upper bound. Applying these assumptions to (17), results in

$$\begin{equation}\tau =\frac{{P}_{0}-{~{P}}_{0}}{({\overline{P}}_{\mathrm{A}}+{\overline{P}}_{\mathrm{B}}+{\overline{P}}_{\mathrm{C}}+2\sqrt{{\overline{P}}_{\mathrm{A}}{\overline{P}}_{\mathrm{B}}}\gamma +2\sqrt{{\overline{P}}_{\mathrm{A}}{\overline{P}}_{\mathrm{C}}}\beta +2\sqrt{{\overline{P}}_{\mathrm{B}}{\overline{P}}_{\mathrm{C}}}\alpha )},\end{equation} \tag{ 20 }$$

where ${\overline{P}}_{i}={P}_{i}-{~{P}}_{0}$, describing the experimentally measured single path terms P_i with the shutter-independent background ${~{P}}_{0}$ already subtracted. For the set of normalized interferences α, β and γ, we used the corrected points in phase space. Evaluation of this equation for each measured cycle gives an average shutter transmissivity of

$$\begin{equation}\langle {\tau }_{23{^\circ}\mathrm{C}}\rangle =2.20(2){\times}1{0}^{-4}\quad \text{and}\quad \langle {\tau }_{30{^\circ}\mathrm{C}}\rangle =2.707(4){\times}1{0}^{-4},\end{equation} \tag{ 21 }$$

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with the uncertainty being the standard error of mean resulting from variations of the corrected points in phase space, as well as variations of the measured single path and background signals during the experiment. Applying these shutter transmissivities to our model results in the data shown in figure 4 for each housing temperature. The deviation of the Peres parameter from its expected value can be negative as well as positive and strongly depends on the phase shift ϕ_Sh of the residual light. The fact that the bias is stronger for the 30 °C measurement is not due to the slightly larger shutter transmissivity, but due to its point in phase space causing an increased sensitivity to the effect. Due to the oscillations in ΔF with the unknown phase ϕ_Sh, we cannot reliably estimate a specific figure for the impact of the non-zero shutter transmissivity on our measured results, but can only provide a range given by its extrema. This range is wide enough to contain our experimental result (30 °C) or at least in the same order of magnitude as the observed deviation (23 °C).

The estimated effect of the various imperfections discussed and modelled in this work on the Peres parameter F is summarized in table 4. The results suggest that the only imperfection resulting in deviations of the same order of magnitude as experimentally measured is the residual light transmission through closed shutters. While in the case of the 30 °C measurement this effect could have caused the entire deviation, for the 23 °C measurement, it falls somewhat short of the observed deviation, even if all other considered effects are included. One has to keep in mind, though, that such a sum of all deviations does not take into consideration the interplay of multiple error sources and that each model rests upon certain assumptions and simplifications. For example, if one assumes that the residual light is only caused by shutter C (τ_A = τ_B = 0) instead of equally by all shutters (τ_A = τ_B = τ_C), the lower bound on the deviation of the Peres parameter is reduced by 1.1 × 10⁻² to −3.8 × 10⁻². This results in a total deviation of −4.4 × 10⁻², which would almost eliminate the observed difference. All considered potential systematic error sources, besides residual light, lead to effects at least one order of magnitude below the measured deviation. The different deviations for both housing temperatures show that the effect of each error source can be minimized by choosing an optimal set of phase differences.

In order to narrow down the range resulting from the residual light model, phase modulators in each of the paths would be necessary. This could be achieved by placing three heaters on the surface of the chip in between both splitters [39–41]. Arranging them in a similar triangle as the paths allows to create temperature gradients between each of the paths and with that tuning of the phase differences between them. Using phase modulators to sweep the phase of each path would allow determining ϕ_Sh and thus increase the accuracy of this analysis. Alternatively, one could cancel out the interference effect (relative magnitude $\sqrt{\tau }$) by scrambling the phase in all closed channels, which would reduce the effect of residual light transmission (to order τ). Control of each path's phase, would furthermore allow to use a specific feature of the three-amplitude Peres test: sweeping one phase leads to a modulation of the Peres parameter F in the case of QQM. This would permit to isolate the fingerprint of a quaternionic phase from experimental imperfections with different phase dependencies. Furthermore, this modulation technique would allow to identify cases, which otherwise would lead to negative results [26]. Even though the actual value of ϕ_Sh and, with it, the actual impact on F cannot be determined from our experiment, these results clearly show that shutters with a low residual transmission in the closed-state and, ideally, full phase control are critical for any interferometric test of hypercomplex quantum mechanics.

 The results of each systematic error model presented in this work depend on the point in phase space, where the Peres test is performed, i.e. the set of phase differences {Δϕ_BC, Δϕ_CA, Δϕ_AB}. For conventional interference of complex amplitudes, by definition, the sum of this set must be zero (see (1)). However experimental imperfections, like the ones analysed in this work (cf section 5), can lead to a deviation from zero and to seemingly non-physical phase differences. The goal here is to estimate the most probable original physical point in phase space, with phase differences summing to zero and therefore also F = 1. This provides a starting point to estimate the effect of each individual experimental influence on the Peres parameter. The following section shows how we determined this point in detail.

The phase space is three-dimensional, as shown in figure A1. The physically allowed phases Δϕ_BC + Δϕ_CA + Δϕ_AB = 2πn, with $n\in \mathbb{Z}$, form a periodic set of planes in phase space. From the measured set of phase differences we find the closest (using Euclidean distance) physical point in phase space by normal projection onto the planes. Due to the sign ambiguity in the cosine function (e.g. α = cos(Δϕ_BC)), the measured set {α, β, γ} corresponds to eight points. As a global sign flip of all phase differences has no effect on any of the models considered, this reduces to two sets of four inequivalent points, whose projections are shown in orange and blue, respectively. We therefore only consider one of the two sets e.g. the one depicted in blue. Within each set, two points project onto the n = 0 plane and two points onto the ±2π-planes. The latter ones are excluded, as they would require much larger shifts, most likely due to inconsistent sign assignments. From the two remaining points we chose the one closer to the n = 0 plane for each temperature. This results in the corrected phase differences listed in table 1. Note that choosing the second-closest point to the n = 0 plane gives results in the same order of magnitude. The only exception is the phase crosstalk at 30 °C, where the point farther from the n = 0 plane leads to no real solution for Δϕ_DH (see section 5.4).

To simulate the effect of power and phase fluctuations on the Peres parameter (see section 5.2) the magnitude of the respective fluctuations has to be determined. The signal of a single open path P_i is only influenced by fluctuations of the input power and we therefore used its standard deviation to describe this fluctuation. In the case of two open paths, the measured signal fluctuations are a superposition of input-power and phase fluctuations. To estimate the strength of the power fluctuations we calculated the standard deviation of the two-path signals based on the single-path power fluctuations using Gaussian error propagation. The signal of two open paths i and j is described by

$$\begin{equation}{P}_{ij}={P}_{i}+{P}_{j}+2\sqrt{{P}_{i}{P}_{j}}\enspace \mathrm{cos}({\Delta}{\phi }_{ij})\end{equation} \tag{ A.1 }$$

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${\sigma }_{{P}_{ij}^{\text{pow}}}$

$$\begin{equation}{\sigma }_{{P}_{ij}^{\text{pow}}}=\sqrt{{\left(\sqrt{\frac{{P}_{j}}{{P}_{i}}}\enspace \mathrm{cos}({\Delta}{\phi }_{ij})+1\right)}^{2}{\left({\sigma }_{{P}_{i}}\right)}^{2}+{\left(\sqrt{\frac{{P}_{i}}{{P}_{j}}}\enspace \mathrm{cos}({\Delta}{\phi }_{ij})+1\right)}^{2}{\left({\sigma }_{{P}_{j}}\right)}^{2}},\end{equation} \tag{ A.2 }$$

where ${\sigma }_{{P}_{i}}$ and ${\sigma }_{{P}_{j}}$ are the standard deviation of the single-path signal for path i and j respectively. We assumed phase and power fluctuations to be independent of each other and therefore the measured standard deviation of a two-path term is described by

$$\begin{equation}{\sigma }_{{P}_{ij}}=\sqrt{{({\sigma }_{{P}_{ij}^{\text{pow}}})}^{2}+{({\sigma }_{{P}_{ij}^{\text{ph}}})}^{2}}.\end{equation} \tag{ A.3 }$$

With this equation we calculated the phase fluctuation ${\sigma }_{{P}_{ij}^{\text{ph}}}$ using the experimentally measured fluctuation ${\sigma }_{{P}_{ij}}$ for the open paths i and j.

In order to measure the interference contrast independently from estimations of the magnitude of phase fluctuations, we reduced the interferometer to a two-path one by keeping one path closed. Sweeping the phase difference over a range including fully constructive or destructive interference allows to directly measure the interference contrast. By creating a vertical temperature gradient along the triangular path arrangement, only the phase differences between path B (tip of the triangular path arrangement) and path A or C (base of the triangle) can be tuned. We decided to closed path A and sweep the phase difference between path B and C. To tune this temperature gradient we heated the sample to about 34.9 °C and shut off the heater afterwards. We measured all four shutter combinations of path B and C in random order about 70 times, during the thermalization of the setup with its surrounding. To increase the time resolution we only took five successive measurements per combination. Each combination took about 5.71 s including 5.16 s for changing the shutter combination. Each cycle takes about 23.25 s with 0.41 s for logging additional parameters like input power and housing temperature.

The thermalization of the housing temperature T_h can be described by an exponential function

$$\begin{equation}{T}_{\mathrm{h}}={T}_{0}+{\Delta}T\enspace \mathrm{exp}(-\kappa k),\end{equation} \tag{ A.4 }$$

where k is the cycle index, T₀ the room temperature, ΔT the range of the temperature change and κ the thermalization coefficient. The recorded temperature is shown together with an exponential fit (T₀ = 22.04(3) C, ΔT = 13.269(28) C and κ = 0.020 55(10)/cycle) in figure A2. This description fits the data quite well and therefore the influence on the normalized interference can be modelled by

$$\begin{equation}\alpha ={C}_{\alpha }\enspace \mathrm{cos}({\Delta}{\phi }_{0}+\eta \cdot {\Delta}T\enspace \mathrm{exp}(-\kappa k)),\end{equation} \tag{ A.5 }$$

where C_α describes the interference contrast, Δϕ₀ the phase difference at room temperature and η the change of the phase difference per temperature change. This model, with the parameters κ = 0.0415(13)/cycle, Δϕ₀ = 3.44(7) rad, η = 0.203(9) rad °C⁻¹ and C_α = 1.000(29), fits the measured data very well as shown in figure A2. The fit parameter of C_α shows that the interference contrast of these two paths is, within the experimental precision of this measurement, quite high.

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Note that the parameters of the temperature fit were not used for fitting the normalized interference, due the position of the temperature sensor. It is mounted on the outside of the housing, between the insulation and the housing. Therefore the measured housing temperature does not perfectly describe the change of the temperature gradient and with it the change of the phase difference. Also the phase change per temperature η, resulting from the fit of the left plot in figure A2, can not be compared with the one given in section 3, which described a fully thermalized situation and here we have a dynamic thermal process.

The three-dimensional coupling between the waveguides in the fused silica chip leads to polarization rotations. These changes in polarization depend on the orientation of the adjacent waveguides in regards to the orientation of their elliptical waveguide profile. Therefore, the two waveguides at the base of the triangle are affected differently than the one at the tip. This effect leads to a superposition of light of different polarizations in the output ports of the chip. Without a polarizer in front of the detector the combined signal of both orthogonal polarizations is measured. For example if only one path i is open this can be described using the horizontal H and vertical V polarization basis as

$$\begin{equation}{P}_{i}={P}_{i}^{\mathrm{H}}+{P}_{i}^{\mathrm{V}}.\end{equation} \tag{ A.6 }$$

In case of a two open paths i and j the same principle can be applied leading to

$$\begin{equation}{P}_{ij}={P}_{i}^{\mathrm{H}}+{P}_{j}^{\mathrm{H}}+2\sqrt{{P}_{i}^{\mathrm{H}}{P}_{j}^{\mathrm{H}}}\enspace \mathrm{cos}({\Delta}{\phi }_{ij}^{\mathrm{H}})+{P}_{i}^{\mathrm{V}}+{P}_{j}^{\mathrm{V}}+2\sqrt{{P}_{i}^{\mathrm{V}}{P}_{j}^{\mathrm{V}}}\enspace \mathrm{cos}({\Delta}{\phi }_{ij}^{\mathrm{V}}).\end{equation} \tag{ A.7 }$$

A polarizer in front of the detector can be set to filter all H-components resulting in F = 1, which is trivial to show using (2) and (5)–(7).

Without a polarizer the resulting equation for F is quite long and does not reduce to 1. This can be shown for the case of equal powers for both polarizations ${P}_{i}^{\mathrm{H}}={P}_{i}^{\mathrm{V}}={P}_{i}/2$. This reduces P_ij to

$$\begin{equation}{P}_{ij}={P}_{i}+{P}_{j}+\sqrt{{P}_{i}{P}_{j}}(\mathrm{cos}({\Delta}{\phi }_{ij}^{\mathrm{H}})+\mathrm{cos}({\Delta}{\phi }_{ij}^{\mathrm{V}})).\end{equation} \tag{ A.8 }$$

Calculating F for this case results in

$$\begin{equation}F=\frac{1}{2}\left(1+{\alpha }^{\mathrm{H}}{\alpha }^{\mathrm{V}}+{\beta }^{\mathrm{H}}{\beta }^{\mathrm{V}}+{\gamma }^{\mathrm{H}}{\gamma }^{\mathrm{V}}-\frac{1}{2}\sum\limits _{i,j,k\in \left\{\mathrm{H},\mathrm{V}\right\}}{\alpha }^{i}{\beta }^{j}{\gamma }^{k}+{\alpha }^{\mathrm{H}}{\beta }^{\mathrm{H}}{\gamma }^{\mathrm{H}}+{\alpha }^{\mathrm{V}}{\beta }^{\mathrm{V}}{\gamma }^{\mathrm{V}}\right).\end{equation} \tag{ A.9 }$$

It is easy to find normalized interference terms αⁱ , βⁱ and γⁱ , which fulfill Fⁱ = 1 for each polarization i, but result in F ≠ 1. This shows that a measurement of a combination of orthogonal polarizations can lead to a deviation in the Peres parameter. In contrast, by projecting to a single polarization, this effect can be avoided.