Transmission of entangled photons studied by quantum tomography: do we need plasmonic resonances?

The transmission (T) spectra of the gold films were measured at normal incidence with the UV–Vis Cary spectroscope on circular areas of 0.5 cm diameter. The spectrum of the noncorrugated gold film (see figure 3, upper) exhibits the well-known spectrum of a semi-transparent film, as it is rather opaque in the visible range [31]. The optical spectrum of the plasmonic nanostructures is also presented (see figure 3, lower). The optical properties of metallic particles are influenced by the particle size and orientation. The optical spectrum of plasmonic nanostructures is significantly structured. Due to the anisotropy and chirality of the array, a small difference in transmission (on different parts of the array) can be observed using different polarizations of the incident light (results not shown). In our experimental results depicted in figure 3, transmission and absorption of both the continuous and structured metallic films were measured using nonpolarized light and through a large spot, so that it was not possible to observe the small difference in transmission related to the anisotropy and chirality of the array. Thanks to the plasmonic resonances, the overall transmission of the nanostructured film increases, and becomes larger than the continuous planar film (see figure 3), even though the nanostructured film is thicker than the continuous film (25 versus 10 nm). It exhibits resonances with an asymmetric peak, which resembles very much the one previously reported for extraordinary transmission of different metallic subwavelength nanostructures [2, 31, 32]. In the case of the structured film, the peak in the absorption probably arises from the excitation of dipolar mode of the plasmonic array.

Zoom In Zoom Out Reset image size

We start by evaluating the effect of the plasmonic nanostructures on polarisation-entanglement, in transmission. We realise a quantum tomography to reconstruct the density matrix of the system of the initial biphoton state $\hat{\rho }$ obtained without the sample and the state $\hat{\sigma }$ obtained after transmission through the sample with the plasmonic nanostructures and at normal incidence. We realise quantum tomography of a system of two qubits with two detectors per qubit. In this case, 9 different measurement bases are sufficient to realise the quantum tomography [33]. Although using 36 different analysis settings can improve the measurement precision, in order to minimise the total time of tomography and increase the stability during the complete course of measurements, quantum tomography is performed with 9 different bases. Each basis measurement without and with the nanostructured sample is realised within a time total of 60 s and with a time window of 16 ns for coincidence counting. All results are treated with the code developed by Paul Kwiat's quantum information group to determine the maximum likelihood estimation of each state, the metrics describing the features of the quantum state and its uncertainties [33, 34]. The fidelity $F\left({\hat{\rho }}_{1},{\hat{\rho }}_{2}\right)={\left(Tr\left[\sqrt{\sqrt{{\hat{\rho }}_{1}}{\hat{\rho }}_{2}\sqrt{{\hat{\rho }}_{1}}}\right]\right)}^{2}$ is a measurement of similarity between the two states ${\hat{\rho }}_{1}$ and ${\hat{\rho }}_{2}$ and can take value in the interval [0,1]. In our experiment, $F\left(\hat{\rho },\hat{\sigma }\right)=0.9966\pm 0.0012$ is very close to one. Furthermore the computed fidelities with the Bell state $\left|{\phi }_{+}\right.\rangle ,$ $F\left(\left|{\phi }_{+}\right.\rangle ,\hat{\rho }\right)=0.8630\pm 0.0018$ and $F\left(\left|{\phi }_{+}\right.\rangle ,\hat{\sigma }\right)=0.8627\pm 0.0012$ are nearly identical. These show that the initial and final states are nearly identical. We conclude that, at normal incidence, the nanostructured sample does not change significantly the polarisation, the polarisation correlations, and the entanglement of the transmitted photon pairs. This is confirmed by the similarity between the graphical representations of these states (see figure 4). These results are in line with previous observations, which indicate that plasmonic nanostructures are able to preserve entanglement [1, 2].

Zoom In Zoom Out Reset image size

In order to assess the effect of plasmonic resonances on preserving quantum correlations, we performed a detailed quantum tomography study on the planar structure without plasmonic resonances. For classical waves, our continuous, planar film does not affect the polarisation at normal incidence. Transmittance is the same for all polarisations as it is isotropic in the surface plane. For non-normal incidence, the transmittance becomes a function of the polarisation. We investigate the effect of introducing the sample with different angles of incidence on the entangled polarisation state produced by the quantum source, as described below.

Firstly we measure both the initial biphoton state $\hat{\rho }$ obtained without the sample and the state $\hat{\sigma }$ obtained after transmission through the continuous metallic film at normal incidence. We realise a quantum tomography to reconstruct the density matrix of the system. Polarisation measurements are made in 36 different bases. Each measurement without (with) the planar sample is realised within a time total of 30 s (60 s) and with a time window of 16 ns (16 ns) for coincidence counting. The graphical representations of these states are also very similar (see figure 5).

Zoom In Zoom Out Reset image size

The fidelity between the two states $F\left(\hat{\rho },\hat{\sigma }\right)=0.9989\pm 0.0003$ is very close to one. In other words, the two-photon states before and after the sample at normal incidence are closely similar. Furthermore, the computed fidelities with the Bell state $\left|{\phi }_{+}\right.\rangle ,$ $F\left(\left|{\phi }_{+}\right.\rangle ,\hat{\rho }\right)=0.8683\pm 0.0008$ and $F\left(\left|{\phi }_{+}\right.\rangle ,\hat{\sigma }\right)=0.8673\pm 0.0006$ are nearly identical and lead to the same conclusion. At normal incidence the thin, continuous, metallic film does not change significantly the polarisation of the transmitted photon pairs. Consequently, polarisation entanglement is completely preserved within the experimental uncertainties.

Secondly, we compare the initial, unperturbed state ${\hat{{\rho }}}_{0}$ and the states obtained after the sample with 4 different orientations: at normal incidence $({\hat{{\sigma }}}_{0}),$ 20° $({\hat{{\sigma }}}_{1}),$ 35° $({\hat{{\sigma }}}_{2})$ and 50° $({\hat{{\sigma }}}_{3})$ with respect to the surface normal. In order to maximise the stability of the experiment throughout the measurements, quantum tomography is performed with 9 different bases with a total time of 60 s for each basis measurement and a coincidence time window of 16 ns. To compare the measured quantum state to the expected state produced by each sample rotation (0°, 20°, 35° and 50°), we calculate the theoretical renormalized state of the system after the sample:

$$\begin{eqnarray*}&&{\hat{\sigma }}_{t}=\frac{(\hat{M}{\hat{\rho }}_{0}{\hat{M}}^{\dagger })}{{Tr}\,(\hat{M}{\hat{\rho }}_{0}{\hat{M}}^{\dagger })}\,\text{with}\,\hat{M}=\hat{I}\otimes ({t}_{H}{\hat{P}}_{H}+{t}_{V}\,{\hat{P}}_{V}),\end{eqnarray*}$$

which we evaluate using the theoretical complex transmission coefficients through the metallic film sample of p(t_H) and s(t_V) polarisations and the projectors ${\hat{P}}_{H}$ and ${\hat{P}}_{V}$ on the horizontal and vertical polarisation states, respectively. As the transmission coefficients are complex numbers, the impact of phase differences between the transmitted s and p polarisations is taken into account in our evaluation of the transmitted two-photon state and in our calculations of the fidelities to the initial state. The transmission coefficients are obtained numerically with the matrix transfer method applied to flat multilayers of Au/Cr/glass, with thickness of 7 nm/3 nm/1 mm respectively. We use the complex refractive index at wavelength $\lambda =818\,{\rm{nm}}$ for the gold ${n}_{Au}=0,15905+5,0572\,i$ [35], chromium ${n}_{Cr}=3,1945+3,4772\,i$ [36] and glass N-BK7 (SHOTT) ${n}_{glass}=\,1,5104+9,{3260.10}^{-9}\,i$ obtained from [37]. Rapid oscillations of small amplitude of the wavelength-dependent transmission coefficients were averaged out. These arise from multiple reflections between the optical interfaces. They are not observed experimentally due to the finite coherence time and beam size.

The entanglement of formation ${E}_{f}\left(\hat{\sigma }\right)$ is a measurement of entanglement for a state $\hat{\sigma }.$ For a system of two qubits, ${E}_{f}\left(\hat{\sigma }\right)=-x{\mathrm{log}}_{2}\left(x\right)-\left(1-x\right){\mathrm{log}}_{2}(1-x)$ with $x=\tfrac{1+\sqrt{1-{C}^{2}}}{2}$ and the concurrence $C\,=\,\max ({\lambda }_{4}-{\lambda }_{3}-{\lambda }_{2}-{\lambda }_{1},0)$ where ${\lambda }_{i}$ with i = 1,2,3,4 are the square roots of the eigenvalues of $\hat{\rho }\left({\hat{\sigma }}_{y}\otimes {\hat{\sigma }}_{y}\right){\hat{\rho }}^{* }\left({\hat{\sigma }}_{y}\otimes {\hat{\sigma }}_{y}\right)$ in increasing order. We measure the entanglement of formation ${E}_{f}\left({\hat{\sigma }}_{i}\right)$ and the experimental state fidelity with respect to the state without the sample $F({\hat{\rho }}_{0},{\hat{\sigma }}_{i}),$ and with respect to the maximally entangled Bell state theoretically produced F($\left|{\phi }_{+}\right.\rangle ,{\hat{\sigma }}_{i}$). These properties quantify the influence of the planar film on the entanglement and polarisation state of the transmitted photon pairs. As our aim is to quantify the preservation of quantum entanglement in transmission through the sample, we first measure and compute the entanglement of formation. However, the degree of entanglement can be left unchanged by transmission while the state is transformed to another (equally) entangled state. For this reason, it is also important to quantify the closeness between the states before and after transmission through the sample, which we do by calculating the fidelity. In theory, the state of the photon pairs transmitted through the planar sample at normal incidence ${\hat{\sigma }}_{0}$ should be the same as in the configuration without the sample ${\hat{\rho }}_{0}.$ This is confirmed experimentally with the measure of the fidelity $F\left({\hat{\rho }}_{0},\,{\hat{\sigma }}_{0}\right),$ which is nearly equal to one. This is an immediate consequence of the equality between the transmission coefficients for s and p polarisations. For this reason, we use both ${\hat{\rho }}_{0}$ and ${\hat{\sigma }}_{0}$ as a reference to calculate the theoretical fidelity with respect to the initial state as a function of the incidence angle (see figure 6). We use these two references to provide a bound on experimental uncertainties due to the global stability of the system over long acquisition times. The theoretical curves associated with these two references are really close to each other, as shown by the Y-axis scales. The experimental entanglement of formation ${E}_{f}\left({\hat{\sigma }}_{i}\right)$ and the fidelities $F\left(\left|{\phi }_{+}\right.\rangle ,{\hat{\sigma }}_{i}\right)$ and $F({\hat{\rho }}_{0},{\hat{\sigma }}_{i})$ decrease slowly with the angle of incidence (see figure 6) and are in agreement with the theoretical predictions.

Zoom In Zoom Out Reset image size

For each state ${\hat{\sigma }}_{i}$ with $i\in \{0,1,2,3\},$ quantum tomography reveals that the probability of measuring a final polarisation $\left|HH\right.\rangle$ ($\left|VV\right.\rangle$) after the planar, continuous, metallic film increases (decreases) with the angle of rotation of the sample (see figure 7). The asymmetry between the coloured diagonal rods (blue or red) increases with the angle. This is explained by the fact that, out of normal incidence, the transmission coefficient of p polarisation is larger than the transmission coefficient of s polarisation. We quantify this behaviour experimentally and we compare it with theory (see figure 8). We represent the probabilities of measuring a final polarisation $\left|HH\right.\rangle$ and $\left|VV\right.\rangle$ of a system in an initially entangled state like the $\left|{\phi }_{+}\right.\rangle$ Bell state. Each point represents the mean value of 30 measurements of 5 s.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We conclude that the introduction of the flat, metallic film sample at normal incidence decreases the coincidence rate (a loss of $\sim 55 \% \,$theoretically and $\sim 57 \%$ experimentally) but it does not affect the entangled quantum state. Without perturbation of the polarisation of the system, the density matrix of the polarisation degree of freedom remains the same. This occurs at normal incidence, since the sample does not have any polarising property. For non-normal incidence, the polarisation entanglement decreases. It is related to the different transmission coefficients of s and p polarisation through the sample. This fact indicates that quantum correlations will be affected only if polarisation is affected. Although the nanostructured sample provides a transmission $\sim 50 \%$ better than the planar metallic film, the fidelities and trace distance between the initial and transmitted states at normal incidence indicate that the initial state preservation is nevertheless (marginally) better in the case of the planar metallic film. This effect could be the result of the anisotropy of the nanostructured sample that induces a small polarising effect upon transmission at normal incidence, contrary to the planar, isotropic film.