Measuring correlations of cold-atom systems using multiple quantum probes

We present a nondestructive method to probe a complex quantum system using multiple-impurity atoms as quantum probes. Our protocol provides access to different equilibrium properties of the system by changing its coupling to the probes. In particular, we show that measurements with two probes reveal the system's nonlocal two-point density correlations, for probe-system contact interactions. We illustrate our findings with analytic and numerical calculations for the Bose-Hubbard model in the weakly and strongly interacting regimes, under conditions relevant to ongoing experiments in cold-atom systems.

Figure 1

Schematic of a Bose gas (blue shading) in an optical lattice (black line). Two ancillary two-level quantum systems (circles) are coupled to the Bose gas at distinct sites on the lattice, separated by a distance $\mathrm{\Delta }c$ in units of the lattice constant $a$.

Figure 2

Weak interactions. Normalized correlation function obtained with Eq. (11) simulating $N_{\exp }={10}^4$ experiments (crosses), compared with the analytic results, Eq. (13) (lines with symbols), for a system initially at equilibrium at inverse temperature $\beta J=10$ and $\beta J=100$. Other parameters used are $\eta =0.4J$ and $U/J=0.1$.

Figure 3

Real part of the coherence function with added Gaussian noise (symbols) and a parabolic fit (solid line). Inset: Real (solid blue line) and imaginary (dashed red line) parts of $\zeta \left(t\right)$, Eq. (16), for the case $\mathrm{\Delta }c=5$. Here we used a system with $M=1000$ lattice sites and $N=1000$ bosons and, therefore, an average density $\overline{\rho }=1$. The system is initially in a thermal state with $\beta J=10$; other parameters as in Fig. 2.

Figure 4

Correlation function $g^{\left(2\right)}\left(\mathrm{\Delta }c\right)$ for the ground state in the strongly interacting regime for different interactions strengths, $U/J=3$, 15, and 100, as indicated. These points represent the numerically exact expectation values of number operator pairs, $\left(\langle {\widehat{\rho }}_i{\widehat{\rho }}_j\rangle -\langle {\widehat{\rho }}_i\rangle \langle {\widehat{\rho }}_j\rangle \right)/{\langle {\widehat{\rho }}_i\rangle }^2$, from the TNT calculation.

Figure 5

Strong interactions. (a) Real (asterisks) and imaginary (crosses) parts of the coherence function $\zeta \left(t\right)$ for $U/J=3$. Other parameters are $\mathrm{\Delta }c=5, M=101, \overline{\rho }=1$, and $\eta =J$. (b) Normalized correlation function $g^{\left(2\right)}\left(\mathrm{\Delta }c\right)$ for the same parameters. The results of the measurement protocol with $N_{\exp }={10}^4$ (crosses) agree with the numerically exact values calculated with the TNT method (circles).

Figure 6

Correlation function $g^{\left(2\right)}\left(\mathrm{\Delta }c\right)$ for $\mathrm{\Delta }c\in \{0,1,2\}$ and different values of $U/J$. Bogoliubov theory was used for $U/J\le 0.4$ with a $\beta J=1000$ thermal state (shaded region), and TNT for larger values of $U/J$. For clarity, we do not include error bars for the $\mathrm{\Delta }c=2$ calculation; they are similar to those for $\mathrm{\Delta }c=1$.