Recovering Quantum Correlations in Optical Lattices from Interaction Quenches

Quantum simulations with ultracold atoms in optical lattices open up an exciting path toward understanding strongly interacting quantum systems. Atom gas microscopes are crucial for this as they offer single-site density resolution, unparalleled in other quantum many-body systems. However, currently a direct measurement of local coherent currents is out of reach. In this Letter, we show how to achieve that by measuring densities that are altered in response to quenches to noninteracting dynamics, e.g., after tilting the optical lattice. For this, we establish a data analysis method solving the closed set of equations relating tunneling currents and atom number dynamics, allowing us to reliably recover the full covariance matrix, including off-diagonal terms representing coherent currents. The signal processing builds upon semidefinite optimization, providing bona fide covariance matrices optimally matching the observed data. We demonstrate how the obtained information about noncommuting observables allows one to quantify entanglement at finite temperature, which opens up the possibility to study quantum correlations in quantum simulations going beyond classical capabilities.

Figure 1

Tomographic reconstruction. (a) Input data for the reconstruction based on out-of-equilibrium data of local particle numbers $N_x(t_i)$ measured at $K=10$ equidistant times after the quench to nearest-neighbor hopping in the tilted lattice. We model statistical fluctuations by sampling occupation numbers ${\mathcal{N}}_{\text{shots}}={10}^3$ times for each $t_i$ and estimating $N_x(t_i)$ via the empirical mean. (b) The input data have been obtained by evolving a thermal covariance of ${\widehat{H}}_{\mathrm{NN}}$ with inverse temperature $\beta =3$ such that there are relatively large currents to be recovered. (c) Results of the reconstruction ${\mathrm{\Gamma }}^{(\text{R}\mathrm{ec})}$. The extent of deviations shown in the inset is $\max |{\mathrm{\Gamma }}_{x,y}-{\mathrm{\Gamma }}_{x,y}^{(\text{R}\mathrm{ec})}|\approx 0.1$ and is explained by the fact that among the data some $N_x(t_i)$ can fluctuate statistically by 2 or 3 standard deviations.

Figure 2

Entanglement cost at finite temperature. (a) The covariance matrix ${\mathrm{\Gamma }}^{(D)}=U_D{\mathrm{\Gamma }}^{(\beta )}{U}_D^{†}$ for the inverse temperature $\beta =3$ after a suitable local transformation $U_D=U_A\oplus U_B$ preserving the entanglement. The inset shows the submatrix of the covariance matrix reflecting one mode in subsystem $A$ and one in $B$. (b) This figure shows how the lower bound for the entanglement depends on whether it is applied on one or two modes in systems $A$ and $B$ each. Selecting merely one mode each provides a positive entanglement cost for relatively large temperatures ($E_G^{(1+1)}$), while two modes are better suited to detect entanglement at low temperatures [$E_G^{(2+2)}(\beta )$].