Violation of classical inequalities by photon frequency filtering

The violation of the Cauchy-Schwarz and Bell inequalities ranks among the major evidence of the genuinely quantum nature of an emitter. The conventional theoretical approaches associate operators with spectral lines to study correlations between photons from real-state transitions. Instead, we use a formalism that studies directly correlations between the physical reality—the photons—with no prejudice as to their origin. This allows us to extend photon correlations to all frequencies in all the possible windows of detection and to reveal landscapes of two-photon correlations that delineate regions of quantum emission, i.e., where classical inequalities are violated. We show that quantum correlations are rooted in the joint emission of two photons involving virtual states of the emitter instead of, as previously assumed, cascaded transitions between real states. As a result, correlations can be optimized in a process akin to distillation by keeping only the emission which is quantum and filtering out that which is not.

Figure 1

Violation of the CSI and BIs by frequency-resolved correlations. (a) Spectrum of resonance fluorescence, where filtering is illustrated in the tails (T), sidebands (S), and central peak (C) of the Mollow triplet. (b) Two-photon de-excitation between rungs of the Mollow ladder involve an intermediate real state (blue, orange, and green arrows) or a virtual state (red arrows). The latter type conveys CSI and BI violation. It is found in the flanks or between the peaks, where the signal is, however, weaker. Parameters: $\Omega =10{\gamma }_{\sigma },\Gamma ={\gamma }_{\sigma },{\omega }_S\approx 2\Omega ,{\omega }_T=2.5\Omega$.

Figure 2

Test for the violation of Bell inequalities by frequency filtering. A source (S) emits photons in a broadband of frequencies. Frequency filters (F) select light at frequencies ${\omega }_1$ and ${\omega }_2$, described by the operators $a_1$ and $a_2$. Recombination at beam splitters (BS) with transmittivities given by $sin\theta$ and $sin\varphi$ gives a total of four output beams, which are collected at the photodetectors (PD) and correlated with coincidence counters (C). Alice (A) and Bob (B) test nonlocality by independently measuring the probability of detection at the output ports of the two beam splitters, $P_{\pm }^{A_{\theta }}$ and $P_{\pm }^{B_{\varphi }}$.

Figure 3

Landscapes of correlations in the frequency domain for three filter linewidths: (a) $g_{\Gamma }^{\left(2\right)}({\omega }_1,{\omega }_2)$, (b) $R_{\Gamma }({\omega }_1,{\omega }_2)$, and (c) $B_{\Gamma }({\omega }_1,{\omega }_2)$. In (b) [(c)], the color code is such that green [red] violates the CSI [BI] and thus corresponds to genuine quantum correlations between the detected photons in the corresponding energy windows, while black and white do not (with white maximizing the inequality). The violation originates from the emission that involves virtual states. Dashed lines I and II in (a) are the cuts in the frequency domain along which the curves in Fig. 4 are calculated. Spectra on the axes show which frequency windows are correlated. Parameters are the same as in Fig. 1. An animation of these landscapes as a function of the detector linewidth is provided in the Supplemental Material [40].

Figure 4

(a, b) Cuts of $R_{\Gamma }$ (a) and $B_{\Gamma }$ (b) along lines I $(\omega ,-\omega )$ and II $(\omega ,{\omega }_S-\omega )$ in Fig. 3. Parameters are the same as in Fig. 1, with $\Gamma ={\gamma }_{\sigma }$. (c, d) Photon correlation $g_{\Gamma }^{\left(2\right)}(\omega ,\pm \omega )$ computed exactly [solid (red) lines] or through the usual auxiliary operator approximation [dashed (blue) lines]. In (d), the absence of the latter curve in some domains corresponds to values which are, incorrectly, exactly 0 (the vertical axis is on log scale).

Figure 5

(a) Solid lines: $R_{\Gamma }({\omega }_i,-{\omega }_i)$ (top) and $B_{\Gamma }({\omega }_i,-{\omega }_i)$ (bottom) as a function of the detector linewidth for the three sets of frequencies $({\omega }_i,-{\omega }_i),i\in 1,2,3$, depicted in (b). Dashed lines: Amount of signal $\Gamma S_{\Gamma }\left(\omega \right)$ that can be collected for the corresponding filter linewidth. Filled (blue) circles illustrate how two configurations with the same amount of collected signal can yield different degrees of violation. (b) Resonance fluorescence spectrum, this time on a linear scale, displaying the characteristic Mollow triplet and three sensors with linewidth $\Gamma =2{\gamma }_{\sigma }$ centered at the frequencies used for (a): ${\omega }_1={\omega }_S,{\omega }_2=1.125{\omega }_S$, and ${\omega }_3=1.25{\omega }_S$. Parameters are the same as in Fig. 1.

Figure 6

Landscapes of correlations for the Jaynes-Cummings model featuring violation of both the Cauchy-Schwarz inequality (upper left) and the Bell's inequalities (bottom right). The full landscape for each case follows by symmetry. Parameters: ${\gamma }_{\sigma }={10}^{-3}g,{\gamma }_a=0.1g,P_{\sigma }=0.05g$, and $\Gamma =0.1g$.