Detecting Entanglement Structure in Continuous Many-Body Quantum Systems

A prerequisite for the comprehensive understanding of many-body quantum systems is a characterization in terms of their entanglement structure. The experimental detection of entanglement in spatially extended many-body systems describable by quantum fields still presents a major challenge. We develop a general scheme for certifying entanglement and demonstrate it by revealing entanglement between distinct subsystems of a spinor Bose-Einstein condensate. Our scheme builds on the spatially resolved simultaneous detection of the quantum field in two conjugate observables which allows the experimental confirmation of quantum correlations between local as well as nonlocal partitions of the system. The detection of squeezing in Bogoliubov modes in a multimode setting illustrates its potential to boost the capabilities of quantum simulations to study entanglement in spatially extended many-body systems.

Figure 1

Sampled phase-space distributions of global and local observables. (a) Schematics of the level scheme for the physical preparation of the target state and of the longitudinally expanded atomic cloud. The initially populated $m_F=0$ level is tuned (shading) close to resonance with the third excited eigenmode in $m_F=\pm 1$ via off-resonant microwave dressing. In the right-hand part, the dashed lines indicate the evaluation regions with a length of $20\mu \mathrm{m}$ each. Panels (b) and (c) show the sampled phase-space distributions for the global observables and for the local subsystems, respectively. The corresponding partitioning functions are shown in the insets. The state has been prepared by 800 ms of spin-mixing dynamics which results in a non-Gaussian distribution in the local partitions. The red and yellow points highlight single experimental realizations illustrating the strong correlations between the individual subsystems. These correlations strongly suppress the fluctuations of the global observables leading to a Gaussian distribution as shown in (b). Panel (d) shows a quantitative analysis of the first-order coherence between different subsystems revealing strong anticorrelations between neighboring partitions.

Figure 2

Superposition of two squeezed vacuum states. (a) Illustration of simultaneous spin mixing in two spatial modes (blue and red). The initially populated $m_F=0$ level is tuned (shading) close to resonance with two energetically lowest eigenmodes in $m_F=\pm 1$ via off-resonant microwave dressing (upper part). This results in simultaneous squeezing in the two modes with relative squeezing angle $\mathrm{\Delta }{\mathrm{\Theta }}_S$ (lower part). (b) Phase-space distributions for different partitioning functions as indicated in the insets of each panel. We prepare two different relative squeezing angles of $\mathrm{\Delta }{\mathrm{\Theta }}_S\approx 0°,90°$ which are shown in the left-hand and right-hand part, respectively. The blue ellipses show the 2 s.d. interval of the distribution and the black dashed circles show the fluctuations expected for the initial vacuum state (including photon shot noise). The upper row reveals the squeezing in the individual modes, where the red lines indicate the axis of smallest fluctuations. In the case of $\mathrm{\Delta }{\mathrm{\Theta }}_S\approx 0°$ we find squeezing in the local partitions while for $\mathrm{\Delta }{\mathrm{\Theta }}_S\approx 90°$ the fluctuations are increased along all directions compared to the vacuum state (lower row).

Figure 3

Witnessing entanglement between subsystems. (a) Fluctuations in the observable $u$ quantifying correlations between the subsystems as a function of the local field projection angles, ${\theta }_L$ and ${\theta }_R$. Here, the blue regions signal fluctuations below the atomic shot noise limit. While for $\mathrm{\Delta }{\mathrm{\Theta }}_S=0°$ (left) we observe fluctuations for specific angle pairs, the continuous diagonal band of reduced fluctuations seen for $\mathrm{\Delta }{\mathrm{\Theta }}_S=90°$ (right) is what allows witnessing entanglement. (b) Entanglement witness as a function of local relative angles, where the blue regions signal entanglement. For each pair of local relative angles, we minimized the first line of Eq. (3). The point of minimal $\mathcal{W}$ is indicated by the black cross. The corresponding pair of local orientations is indicated in (a).

Figure 4

Simultaneous vacuum squeezing of multiple Bogoliubov modes. (a) Schematic of a boxlike trapping potential implemented by combining an elongated attractive harmonic dipole potential (red) with two repulsive barriers (green). The detected BEC density is shown in the lower part (yellow). (b) Sampled phase-space distributions for the partitionings indicated in the insets, which reveal vacuum squeezing in the four energetically lowest Bogoliubov modes. The colored solid lines indicate the 2 s.d. interval of the distribution and the dashed black lines show the fluctuations expected for the initial vacuum state including photon shot noise. (c) $C_{k,l}$ for the data shown in (b). This confirms the independence of the individual Bogoliubov modes as expected in the low depletion limit.