Violation of Bell's inequality with quantum-dot single-photon sources

We investigate the possibility of realizing a loophole-free violation of Bell's inequality using deterministic single-photon sources. We provide a detailed analysis of a scheme to achieve such violations over long distances with immediate extensions to device-independent quantum key distribution. We investigate the effect of key experimental imperfections that are unavoidable in real-world single-photon sources including the finite degree of photon indistinguishability, single-photon purity, and the overall source efficiency. We benchmark the performance requirements to state-of-the-art deterministic single-photon sources based on quantum dots in photonic nanostructures and find that experimental realizations appear to be within reach. We also evaluate the requirements for a postselected version of the protocol, which relaxes the demanding requirements with respect to the source efficiency.

Figure 1

(a) Setup proposed in Ref. [6], consisting of two local stations (Alice and Bob) and a central heralding station (CHS). The different optical components involved are indicated: beam splitters with transmittances $T$ (at Alice and Bob's stations) and 50% (at the CHS); half- and quarter-wave plates (HWP and QWP), switchable mirrors (SM), polarizing beam splitters (PBS), and single-photon sources (SPS). The photodetectors are labeled as $D_1$, $D_2$, $D_3$, and $D_4$ at the CHS and $A_H$ ($B_H$) and $A_V$ ($B_V$) for Alice's (Bob's) detectors, with the subindex indicating the polarization of the corresponding incoming photons. The operators that model the creation of photons at each section of the setup are indicated as well. (b) Example of quantum-dot single-photon source and its energy-level schemes. The light-matter interaction between a quantum-dot and a waveguide mode is enhanced by the nanostructure realizing a deterministic single-photon source. The quantum dot is driven by a resonant excitation pulse $\left({\mathrm{\Omega }}_i\right)$ and subsequently emits a single photon into the waveguide by spontaneous emission. Inset: The optical transitions are subject to two different types of decoherence: (1) rapidly fluctuating phonon interactions induce pure dephasing with a rate ${\gamma }_d$ and (2) slow drifts of the levels induce a slowly varying detuning ${\mathrm{\Delta }}_i$.

Figure 2

Degree of indistinguishability of photons from different sources, i.e., the HOM visibility $V_{\beta }^{\left(0\right)}$. The indistinguishability depends on the width $\sigma$ of the probability distribution for the frequency fluctuations $\mathrm{\Delta }$ as well as the rate of pure dephasing ${\gamma }_d$ relative to the spontaneous emission rate $\gamma$.

Figure 3

Sketch of the efficiencies ${\eta }_i$ and definitions of the creation operators of the lost photons ${\widehat{c}}^{†}$.

Figure 4

(a) Violation of the CHSH inequality $S\ge 2$, as a function of the HOM visibility of the local sources $V_{\alpha }^{\left(0\right)}$. The curves represent both the case in which photons from Alice's and Bob's stations and equally indistinguishable ($V_{\alpha }^{\left(0\right)}=V_{\beta }^{\left(0\right)}$) and when there is inhomogeneous broadening with a width $\sigma \ne 0$ so that $V_{\alpha }^{\left(0\right)}>V_{\beta }^{\left(0\right)}$. The threshold for CHSH violation $S=2$ corresponds to the visibilities $(V_{\alpha }^{\left(0\right)},V_{\beta }^{\left(0\right)})=\{(0.79,0.79),(0.86,0.67),(0.93,0.51),(0.96,0.40)\}$. (b) Contour plot of the CHSH value $S$ as a function of the visibilities $V_{\alpha }$ and $V_{\beta }$. The dashed isoline $S=2$ delimits the regions with and without violation of Bell's inequality.

Figure 5

Evaluation of CHSH threshold for a postselected protocol and with finite multiphoton contributions. In both figures the CHSH parameter $S$ is shown as a function of the second-order correlation function for different values of the HOM visibility, in the limit in which the transmission and local efficiencies are low (${\eta }_t={\eta }_{2,i}=0.1$) and for postselected events. As postselection is performed, we set $T=0.5$, which will increase the heralding rate without affecting the performance of the protocol. (a) The crossed and local HOM visibilities $V_{\alpha }$ and $V_{\beta }$ are identical. (b) The local HOM visibility is fixed to $V_{\alpha }=0.95$ and $V_{\beta }$ varies between 0.6 and $V_{\alpha }$.

Figure 6

CHSH parameter $S$ as a function of the local efficiency ${\eta }_l={\eta }_{1,i}{\eta }_{2,i}(1-T)$ of Alice's and Bob's stations for different HOM visibilities. To focus on the effect of the local efficiency we consider a situation with a very limited transmission to the CHS ($T={10}^{-3}$ and ${\eta }_t=0.1$) and vary the efficiency of the final arm ${\eta }_2$ with ${\eta }_{1,i}=1$.

Figure 7

Threshold for the violation of the CHSH Bell's inequality $(S\le 2)$. We vary ${\eta }_{2,i}$ for different values of the second-order correlation function and determine the threshold visibility $V_{\beta }^{\left(0\right)}$ required to violate the CHSH inequality. The local HOM visibility has been set to $V_{\alpha }^{\left(0\right)}=1$ and we assume a low transmission efficiency ($T={10}^{-3}$, ${\eta }_t=0.1$).

Figure 8

Effect of finite transmittance $T$. (a) CHSH parameter $S$ as a function of the transmittance $T$ of the beam splitters at the end stations for several sets of single-source error parameters: visibility ($V_{\alpha }^{\left(0\right)}=V_{\beta }^{\left(0\right)}$), purity ($g^{\left(2\right)}$), and the local efficiency (${\eta }_2$ with ${\eta }_{1,i}=1$). A large transmission probability decreases the performance of the protocol since there will be more events where no photons are detected at end stations. (b) Number of standard deviations $Z^{\prime }$ with which a violation of Bell's inequality is achieved (normalized to the number of experimental runs) as a function of the transmittance $T$. In an experiment with $n$ experimental runs the CHSH inequality is violated by $Z^{\prime }\sqrt{n}$ standard deviations. An optimal $T$ value that maximizes the Bell violation can be found for each parameter set. Note that the legend is divided into two parts inserted in (a) and (b) but applies to both subfigures (a) and (b), i.e., half the lines are defined by the legend in (a), the others in (b).