Modeling coherence measurements on a spectrally diffusing single-photon emitter

We examine the possibility of measuring the emission linewidth $2\hslash {\Gamma }_2$ of a single-photon source by Michelson interferometry. Such an emitter is characterized by a limited fluorescence intensity and, when embedded in a solid matrix, a large spectral diffusion. We show that fast spectral diffusion renders standard Fourier spectroscopy irrelevant. We then calculate the correlations of the intensities detected at the two interferometer outputs, point out the existence of two-photon interferences (coalescence) even for a non-single-mode emission, and show that the correlations are not affected by spectral diffusion at correlation times shorter than spectral dynamics. This yields two ways to measure the decoherence rate ${\Gamma }_2$: the recently proposed photon-correlation Fourier spectroscopy, and the study of the coalescence dip. We examine the feasibility and spectral resolution of each method, depending on spectral-diffusion characteristics, and show that both methods could be applied to typical nanometer-sized emitters such as colloidal quantum dots.

Figure 1

(a) The model considered for the single-photon source: a two-level system pumped at a rate $r$, with an emission centered on $\hslash {\omega }_0$, an excited-state lifetime $1∕{\Gamma }_1$, a decoherence rate ${\Gamma }_2$, and spectral diffusion $\delta \omega \left(t\right)$. (b) The detection setup: a scanning Michelson interferometer with temporal path difference $d_t$ at time $t$, and two photon counters $a$ and $b$.

Figure 2

(Color online) Interferogram $I_a^{\delta t}\left(t\right)$ simulated following the procedure explained in Appendix , for a time resolution $\delta t=10\text{ms}$ and different SD times ${\tau }_c$, for a uniform translation $\left(d_t=\eta t\right)$, with $\mathcal{I}=2$, $\eta =3.3\times {10}^{15}$, $\eta {\omega }_0=10{\text{s}}^{-1}$, $\eta \sigma =0.1{\text{s}}^{-1}$, and $\eta {\Gamma }_2=0.05{\text{s}}^{-1}$.

Figure 3

Correlation functions $G_{aa}\left(t,\tau \right)=G_{bb}\left(t,\tau \right)$ (black lines) and $G_{ab}\left(t,\tau \right)=G_{ba}\left(t,\tau \right)$ (gray lines) as a function of $\tau$ at the $\mu s$ (left side) and s scales (right side), for different values of the path difference $d_t$ and the decoherence rate ${\Gamma }_2$: (a) $d_t=0.5\mu \text{s}$ and ${\Gamma }_2=3\mu {\text{s}}^{-1}$, (b) $d_t=2\mu \text{s}$ and ${\Gamma }_2=3\mu {\text{s}}^{-1}$, (c) $d_t=0.5\mu \text{s}$ and ${\Gamma }_2=0.5\mu {\text{s}}^{-1}$, (d) $d_t=2\mu \text{s}$ and ${\Gamma }_2=0.5\mu {\text{s}}^{-1}$. $\mathcal{I}=2$, $\eta {\omega }_0=10{\text{s}}^{-1}$, and ${\Gamma }_1+r=1\mu {\text{s}}^{-1}$. Black and gray arrows respectively indicate the antibunching and coalescence dips.

Figure 4

Correlation functions $G_{aa}\left(t,\tau \right)=G_{bb}\left(t,\tau \right)$ (black lines) and $G_{ab}\left(t,\tau \right)=G_{ba}\left(t,\tau \right)$ (gray lines) as a function of $\tau$ (in log scale) for different path differences $d_t$, with $\mathcal{I}=2$, ${\Gamma }_2=0.5\mu {\text{s}}^{-1}$, $\eta {\omega }_0=10{\text{s}}^{-1}$, and ${\Gamma }_1+r=1\mu {\text{s}}^{-1}$, in the absence of SD (dashed lines) and in the presence of a SD characterized by ${\tau }_c=100\mu \text{s}$ and $\sigma =2\mu {\text{s}}^{-1}$ (solid lines).

Figure 5

Correlation functions $G_{aa}\left(t,\tau \right)=G_{bb}\left(t,\tau \right)$ (black lines) and $G_{ab}\left(t,\tau \right)=G_{ba}\left(t,\tau \right)$ (gray lines) as a function of $d_t$ for different values of $\tau$, with $\mathcal{I}=2$, ${\Gamma }_2=2\mu {\text{s}}^{-1}$, $\eta {\omega }_0=10{\text{s}}^{-1}$, and ${\Gamma }_1+r=1\mu {\text{s}}^{-1}$. The dashed lines correspond to the absence of SD, and the solid lines correspond to the presence of a SD characterized by ${\tau }_c=100\mu \text{s}$ and $\sigma =3\mu {\text{s}}^{-1}$.

Figure 6

(Color online) SD effective linewidth ${\Gamma }_{\tau }$ at time scale $\tau ={10}^{-6}\text{s}$ as a function of the SD amplitude $\sigma$ and characteristic time ${\tau }_c$ (log scale).

Figure 7

(Color online) PCFS resolution $\delta {\Gamma }_{\text{PCFS}}$ as a function of the SD amplitude $\sigma$ and characteristic time ${\tau }_c$ (log. scale), with a $1∕{\Gamma }_1=1\mu \text{s}$ lifetime.

Figure 8

(Color online) Resolution $\delta {\Gamma }_{\text{coal}\text{.}}$ as a function of the SD amplitude $\sigma$ and characteristic time ${\tau }_c$ (log scale).