Coherent Control of Photon Correlations in Trapped Ion Crystals

Abstract

While the spontaneous emission from independent emitters provides spatially uncorrelated photons - a typical manifestation of quantum randomness, the interference of the coherent scattering leads to a well-defined intensity pattern - a feature described by linear optics. We here demonstrate experimentally how the interplay between the two mechanisms in large systems of quantum emitters leads to spatial variations of photon correlations. The implementation with trapped ion crystals in free space allows us to observe the anti-correlation between photon rates and variance of the photon number distributions in chains of up to 18 ions. For smaller crystals of four ions, the transition from a sub-Poissonian to a super-Poissonian variance of the photon number in the scattered light is reported. For higher numbers of scatterers, the photon statistics still display a strong deviation from the fully incoherent scattering case. Our results illustrate how the interference of coherent scattering, combined with spontaneous emission, provides a control mechanism for the light statistics.

Figures (7)

• Figure 1: Schematic description of the experiment, where the light scattered by a long chain ($N=18$) of non-interacting $^{40}\text{Ca}^+$ ions is monitored through a Hanbury-Brown Twiss (HBT) setup. The weakly driven ($s=0.6$) two-level emitters present an interference pattern of the intensity $I$ from the elastically scattered light, which is anti-correlated to the two-photon correlations signal $g^{(2)}(0)$.
• Figure 2: (a-b) Experimental data on the temporal two-photon correlations for $N=4$ ions and $s=0.62$, measured in constructive and destructive interference directions. In panel (a), the dip at zero time delay, $g^{(2)}(0)<g^{(2)}(\tau)$, corresponds to antibunching. (c) Two-photon equal-time correlations $g^{(2)}(0)$, as the distance between ions is tuned to switch from constructive to destructive interference through the trap frequency $\omega_z$. The red and blue circles are the experimental points for photon counts and $g^{(2)}(0)$, respectively, while the solid lines show the corresponding theoretical predictions. For each axial trapping frequency $\omega_z$, the values were averaged over $10^3$ numerical realizations. This accounts for the jitter in the positions of the ions and for the occupation of electronic levels which do not scatter in the monitored light mode (see main text), and results in the visible fluctuations. The discrepancy between the experimental and numerical curve may be due to the ion motion, which induces timescales not accounted for by the theory. The horizontal lines stand for Poissonian light (dashed line) and the phase-randomized prediction \ref{['eq:g20SE']} (solid dark-blue line).
• Figure 3: (a) Second-order photon autocorrelation function $g^{(2)}(0)$ in the constructive interference direction, for an increasing number of ions, $N=2...18$. The blue circles and red triangles correspond to the experimental data, for $s\approx 0.6$ and $1.2$, respectively, while the lines stand for the theoretical prediction \ref{['eq:g20cons']}, for the same parameters. Values above $1$ (shaded area) correspond to sub-Poissonian statistics, $g^{(2)}(0)<1$, while the black dash-dotted line refers to the spontaneous emission prediction $2(1-1/N)$. The theoretical prediction for $g^{(2)}(0)$, based on the hypothesis of a speckle pattern and which aims at capturing the large-$N$ behaviour, does not converge to $0$ for $N=1$. (b) Mandel $Q$ parameter, normalized by the square saturation parameter $s^2$ for clarity, as a function of the particle number $N$, and for saturation parameters $s=0.6$ and $1.2$. The dashed lines correspond to linear fits over the points for $N<10$.
• Figure S1: Experimental scheme for observation of photon correlations on light from long ion crystals in a coherent scattering regime and relevant energy level scheme of $^{40}$Ca${^+}$ ion.
• Figure S2: Measured temporal profiles of second-order correlation function $g^{(2)}(\tau)$ for different axial frequencies of the linear crystal with $N=4$ ions. The temporal dependence close to a zero time delay reveals the transition from anti-bunched [i.e., $g^{(2)}(0) < g^{(2)}(|\tau|\approx 50\,{\rm ns})$] to bunched photon statistics for close to constructive to destructive interference of the scattered photons at $\approx 700$ kHz and $\approx 750$ kHz, respectively. The higher noise in the measurements corresponding to the close-to-destructive interference results from the lower detectable photon rate.