Transition from classical to quantum loss of light coherence

Light is a precious tool to probe matter, as it captures microscopic and macroscopic information on the system. However, the measurement will be limited by the coherence of the light, both spatial and temporal, which itself reveals certain properties of the emitters. We here report on the transition from a thermal (classical) to a spontaneous emission (SE) (quantum) mechanism for the loss of light temporal coherence from a macroscopic atomic cloud. The coherence is probed by intensity-intensity correlation measurements realized on the light scattered by the atomic sample, and the transition is explored by tuning the balance between thermal coherence loss and SE via the pump strength. The transition occurs only at low temperatures, which illustrates the potential of cold atom setups to investigate the classical-to-quantum transition in macroscopic systems.

Figure 1

Coherence time of the light radiated by a macroscopic cloud, as a function of the saturation parameter (dashed: theory; dots: experiment with error bars, see main text). Left inset: For a weak drive on the atoms, the elastically scattered light acquires a frequency shift due to the Doppler effect. The finite temperature of the cloud broadens the light spectrum, setting the coherence time. Right inset: Each of the strongly driven atoms emits a broadened spectrum (Mollow triplet).

Figure 2

(a) Schematic setup of the experiment (see main text for details). (b) Temporal evolution of the second-order coherence $g^{\left(2\right)}\left(\tau \right)$, in a low-saturation temperature-dominated regime $[s=(4.0\pm 0.8)\times {10}^{-3}]$ and in a high-saturation SE-dominated regime ($s=24\pm 5$). Dashed lines: fits of the decay capturing the coherence time; the solid thicker line is a fit containing the (coherent) Rabi oscillation of the saturated regime. (c) Observation of the Siegert relation, which writes ${\tilde{g}}^{\left(2\right)}\left(\omega \right)=\delta \left(\omega \right)+{\tilde{g}}^{\left(1\right)}\left(\omega \right)⊛{\tilde{g}}^{\left(1\right)*}\left(\omega \right)$ in the frequency space [$⊛$ the convolution and $\delta \left(\omega \right)$ the Dirac function], for the large-$s$ regime, $s\approx 60$. The elastic component is broadened by the temperature; the spectra are normalized to one.

Figure 3

Theoretical coherence time of the light scattered by the atomic cloud, at an angle ${90}^{\circ }$ from the probe beam axis, as a function of the cloud temperature $T$ and the saturation parameter $s$ of the probe. The small oscillations close to the transition originate from the fitting procedure when the fitting model for the Doppler broadening and for the SE are competing with comparable weights to fit the $g^{\left(2\right)}$ curves.

Figure 4

(a) Evolution of the emitted intensity from a cloud of moving classical dipoles with positions ${\mathbf{r}}_j\left(t\right)$, where the Doppler effect combines with interference to provide temporal fluctuations: $I\propto |{\sum }_{j}exp[({\mathbf{k}}_L-k\widehat{n})·{\mathbf{r}}_j\left(t\right)]{|}^2$, with $\widehat{n}$ the direction of observation, and $\langle {\dot{r}}_j^2\rangle =k_{B}T/M$. (b) Stochastic evolution of the excited population of an ensemble of $N$ atoms driven by a strong pump, in the steady state, obtained from exact simulations of strongly driven two-level atoms [32] (with $N=10$ and $s=8$). The quantum jumps toward a lower population state correspond to the emission of a photon (photodetection events in red).