Correlation functions with single-photon emitters under noisy resonant continuous excitation

To characterize the statistics and indistinguishability of a source, it is common to measure the correlation functions of the emitted field using various interferometers. Here, we present a theoretical framework for the computation of the correlation functions of a two-level system that is resonantly driven by a realistic noisy cw excitation laser. Analytic expressions of the first- and second-order autocorrelation functions are obtained where the various contributions of the noisy excitation source are correctly taken into account. We predict that, even in the low power regime, the noise source has a strong influence on the two-level system dynamics, which is not anticipated by simpler models. The characterization of photon indistinguishability in the pulsed excitation regime is usually done by measuring the value of the zero-delay intensity correlation obtained with a Hong-Ou-Mandel interferometer. We show that this figure is irrelevant in the cw excitation regime and we introduce the coalescence time window, a figure of merit based on a probabilistic interpretation of the notion of photon indistinguishability. We finally use the coalescence time window to quantify how noisy cw excitation influences photon indistinguishability.

Figure 1

Diagram of the different evolution regimes depending on the angular frequency fluctuation characteristics of the continuous driving: (i) $x$ axis: coherence time ${\tau }_C$ of the fluctuation; (ii) $y$ axis: square root of the fluctuation $\overline{\delta {\omega }^2}$. Both axes are implicitly in logarithmic scales. The diagonal dotted-dashed line delimitates the left bottom side where BPP approximation can be used either in the viscous or nonviscous regime. The vertical dashed line delimitates the region where the pseudoadiabatic approximation is valid. In the rest of the diagram, the correlation statistics can be reached through Monte Carlo (MC) simulations. The red dotted line represents the region where relaxation processes decoupling is valid (see Sec. 2a3). This line collapses with the BPP frontier if the field fluctuation amplitudes are smaller. Note that, in the region common to BPP and adiabatic regimes, BPP relaxation can be neglected. The red star spots refer to the experimental conditions investigated in Ref. [16].

Figure 2

$\overline{{\tilde{g}}^{\left(1\right)}}$ (a) and $\overline{g^{\left(2\right)}}$ (b) in the weak driving limit obtained by numerical integration of the Liouville equations taking into account pseudoadiabatic averaging. The simulations are represented by symbols and are computed for a resonant excitation and for ${\tau }_C=4\mathrm{ns},T_1=0.34\mathrm{ns},T_2=0.5\mathrm{ns},\overline{\mathrm{\Omega }}=0.1\mathrm{rad}/\mathrm{ns}$, and consequently a saturation parameter $s=1.7\times {10}^{-3}$ (weak driving). In both cases, $\sqrt{\overline{\delta {\mathrm{\Omega }}^2}}=\sqrt{\overline{\delta {\omega }^2}}=0.1\mathrm{rad}/\mathrm{ns}$ which corresponds to $Q=1$. The rotating frame blurring is not included in the calculation. Both $\overline{{\tilde{g}}^{\left(1\right)}}$ and $\overline{g^{\left(2\right)}}$ simulations have been done with a correlation factor between driving field amplitude and energy fluctuations $\epsilon =0$ (circles) and 0.8 (dots). The fluctuationless theories are represented as dashed lines and fail at describing the simulations. The weak driving pseudoadiabatic theories (15) and (16) are represented as thick lines and reproduce accurately the simulations.

Figure 3

(a) Reduced difference $\frac{\overline{g^{\left(2\right)}}\left(\tau \right)}{g^{\left(2\right)}\left(\tau \right)}-1$ between the pseudoadiabatic simulation $\overline{g^{\left(2\right)}}\left(\tau \right)$ and the fluctuation-free $g^{\left(2\right)}\left(\tau \right)$ for the case considered in Fig. 2 (thick black line). The solid red line indicates the fit result using the model proposed in Eq. (16) with ${\tau }_C=4\mathrm{ns},Q=1,A=1$, and $B=0.5$. (b) Coefficients A (black) and B (red) as a function of $1/Q$ obtained from fitting numerical integrations of a postaveraged Bloch-Liouville equation for $Q=[0.1;5]$. Thin plain lines are guides for the eye.

Figure 4

$\overline{{\tilde{g}}^{\left(1\right)}}$ (a) and $\overline{g^{\left(2\right)}}$ (b) in the strong driving limit obtained by numerical integration of the Liouville equations taking into account pseudoadiabatic averaging. The simulations are represented by symbols and are computed for a resonant excitation and for ${\tau }_C=4\mathrm{ns},T_1=0.34\mathrm{ns},T_2=0.5\mathrm{ns},\overline{\mathrm{\Omega }}=2\mathrm{rad}/\mathrm{ns}$, and consequently a saturation parameter $s=0.68$ (strong driving). In both cases, $\sqrt{\overline{\delta {\mathrm{\Omega }}^2}}=\sqrt{\overline{\delta {\omega }^2}}=2\mathrm{rad}/\mathrm{ns}$, which corresponds to $Q=1$. The rotating frame blurring is not included in the calculation. Both $\overline{{\tilde{g}}^{\left(1\right)}}$ and $\overline{g^{\left(2\right)}}$ simulations have been done with a correlation factor between driving field amplitude and energy fluctuations $\epsilon =0$ (circles) and 0.8 (dots). The fluctuationless theories are represented as dashed lines and the weak driving limit theory as thick lines. Both fail at describing the simulations when fluctuations are large.

Figure 5

Cross-correlation experiment to characterize the indistinguishability of photon emission of two identical sources. The HOM uses only one source at consecutive times using an extra beam splitter and an asymmetric delay on the two arms.

Figure 6

Scheme of the Hong-Ou-Mandel interferometer. Beam-splitter A separates the field scattered by the two-level system. A delay $\mathrm{\Delta }t$ is set between the two arms. Time-delayed fields beat on beam splitter B and the time correlation between the outcoming fields are characterized through the time correlation of two identical single-photon time-resolved detectors.

Figure 7

Comparison of computed HOM interferograms obtained without and with noisy driving. (a) Effect of the driving energy fluctuations alone on HOM interferograms $g_{\parallel }^{\left(2\text{X}\right)}$; the rotating frame blurring effect is captured by the laser coherence time $T_L={\mathrm{\Gamma }}_L^{-1}=0,20,\infty \mathrm{ns}$ (green, orange, and blue, respectively). Inset: zoom around zero delay. (b) Effect of the driving field amplitude fluctuations $Q=1,\infty$ (red and orange, respectively) for a fixed laser coherence time $T_L=20\mathrm{ns}$. The reference cross-polarized interferogram $g_{\perp }^{\left(2\text{X}\right)}$ for $Q=1$ (black) is represented as well. (c) Visibility computed from panel (b) as $V\left(\tau \right)=|g_{\parallel }^{\left(2\text{X}\right)}\left(\tau \right)-g_{\perp }^{\left(2\text{X}\right)}\left(\tau \right)|/g_{\perp }^{\left(2\text{X}\right)}\left(\tau \right)$. All simulations are done for a delay $\mathrm{\Delta }t=43\mathrm{ns}$, a field correlation time 4 ns, a field amplitude $\mathrm{\Omega }=0.1\mathrm{rad}/\mathrm{ns}$ corresponding to a saturation parameter $s=1.7\times {10}^{-3},T_1=0.34$ ns, $T_2=0.5$ ns, and driving field amplitude fluctuation parameter $Q=5.8$ ($Q^{-2}=3\%$).

Figure 8

CTW function of the laser coherence time $T_L$ for various driving field amplitude fluctuation strengths $Q^{-2}$ indicated on the figure. Simulation parameters are identical to Fig. 7, lower panel. For $T_L\le {\tau }_c$, the decoupling condition between rotating frame blurring and Bloch evolution is not fulfilled and the simulation result is not accurate. For $T_L\ge \mathrm{\Delta }t$, the CTW can be computed but it cannot be interpreted due to the single-photon self-interference terms as in the noiseless continuous excitation regime.

Figure 9

CTW/$T_L$ function of the driving amplitude expressed in terms of saturation parameter $s_0$ for various driving field amplitude fluctuation strengths (on the figure) and a fixed laser coherence time $T_L=20$ ns. Above saturation ($s_0\ge 1$), the linear approximation to the pseudoadiabatic regime used in this paper is not valid anymore (although a limit theory can be done for small $Q^{-2}$.) The simulation parameters are the same as in Fig. 7.