Solving the Quantum Many-Body Problem via Correlations Measured with a Momentum Microscope

In quantum many-body theory, all physical observables are described in terms of correlation functions between particle creation or annihilation operators. Measurement of such correlation functions can therefore be regarded as an operational solution to the quantum many-body problem. Here, we demonstrate this paradigm by measuring multiparticle momentum correlations up to third order between ultracold helium atoms in an $s$-wave scattering halo of colliding Bose-Einstein condensates, using a quantum many-body momentum microscope. Our measurements allow us to extract a key building block of all higher-order correlations in this system—the pairing field amplitude. In addition, we demonstrate a record violation of the classical Cauchy-Schwarz inequality for correlated atom pairs and triples. Measuring multiparticle momentum correlations could provide new insights into effects such as unconventional superconductivity and many-body localization.

Figure 1

Atomic momenta as measured by the quantum many-body momentum microscope. Individual momenta of detected atoms are reconstructed in 3D momentum space, with the main image showing the collision halo, with dense (yellow) patches on the north and south poles showing unscattered atoms from the pair of colliding condensates. The highlighted balls and arrows are an illustration of the underlying microscopic interactions—the binary $s$-wave collisions. The 2D histograms below show an equatorial slice through the experimental data, where the red arrows ${\mathbf{k}}_1$, ${\mathbf{k}}_2$, and ${\mathbf{k}}_3$ indicate three arbitrarily chosen momenta for which, e.g., three-atom correlations can be analyzed via coincidence counts. Experimental data from ten runs is shown, which approximates the density present in a single halo, given our detection efficiency of $\sim 10\%$. Individual atoms can be seen in the magnified inset, represented as 2D Gaussians with a width equal to the detector resolution. The size of the balls representing the individual atoms on the main 3D image are not to scale.

Figure 2

Three-body momentum correlation functions. The various back-to-back (a)–(c) and collinear (d)–(f) correlation functions between three atoms in the scattering halo, with the maximum expected value of each correlation function indicated. (g) Surface plot showing the correlation function between two collinear and one back-to-back atom ${\overline{g}}_{BB}^{(3)}(\mathrm{\Delta }{k}_1,\mathrm{\Delta }{k}_2)$ is shown for the $s$-wave halo that has a mean occupation of $n=0.010(5)$ atoms per mode. The red and blue solid lines show cases (a) and (b), respectively [21], while the green solid line is the 1D Gaussian fit used to extract ${\overline{g}}_{BB}^{(3)}(0,0)$. (h) The correlation function between three collinear atoms ${\overline{g}}_{CL}^{(3)}(\mathrm{\Delta }{k}_1,\mathrm{\Delta }{k}_2)$ is shown for the $s$-wave halo with $n=0.44(2)$. The red and blue solid lines show cases (d) and (e), respectively, while the green solid line is the 1D Gaussian fit used to extract ${\overline{g}}_{CL}^{(3)}(0,0)$ [21]. Insets show visual representations of the relevant cases at four selected points on each plot.

Figure 3

Peak two- and three-atom correlation amplitudes and anomalous occupancy $|m|$ vs halo mode occupancy $n$. (a) The measured peak two-atom back-to-back and collinear correlation amplitudes ${\overline{g}}_{BB}^{(2)}(0)$ (blue circles) and ${\overline{g}}_{CL}^{(2)}(0)$ (grey squares), respectively, plotted against the average halo mode occupancy ($n$) for different halos. The dashed (blue) line shows the analytic prediction of Eq. (1). We expect theoretically that ${\overline{g}}_{CL}^{(2)}(0)=2$ for all $n$, but due to the finite resolution of the detector and the bins used to calculate the correlation function, this is reduced slightly. The dotted line shows the mean value of ${\overline{g}}_{CL}^{(2)}(0)$. (b) The quantity $(2C_2-1{)}^{1/2}\simeq |m|/n$ [where $C_2\equiv {\overline{g}}_{BB}^{(2)}(0)/{\overline{g}}_{CL}^{(2)}(0)$], with $|m|$ the anomalous occupancy, is plotted against $n$ along with the theoretical prediction (dashed red line) of $|m|/n=(1+1/n{)}^{1/2}$. The horizontal dotted line at unity is drawn for reference, showing that $|m|>n$ for all our data points. (c) ${\overline{g}}_{BB}^{(3)}(0,0)$ vs $n$, with green diamonds showing experimental data [extracted from fits to ${\overline{g}}_{BB}^{(3)}(\mathrm{\Delta }k,\mathrm{\Delta }k)$, as shown by the green line in Fig. 2] and the dashed line showing the theoretical prediction of Eq. (2). Red circles are reconstructed using experimental data for $C_2$. Error bars for all three plots show the combined statistical and fit uncertainties [21].