Momentum-resolved and correlation spectroscopy using quantum probes

We address some key conditions under which many-body lattice models, intended mainly as simulated condensed-matter systems, can be investigated via immersed, fully controllable quantum objects, namely quantum probes. First, we present a protocol that, for a certain class of many-body systems, allows for full momentum-resolved spectroscopy using one single probe. Furthermore, we demonstrate how one can extract the two-point correlations using two entangled probes. We apply our theoretical proposal to two well-known exactly solvable lattice models, a one-dimensional (1D) Kitaev chain and 2D superfluid Bose-Hubbard model, and show its accuracy as well as its robustness against external noise.

Figure 1

(a) Sketch of a probe overlapping with either one or two lattice sites of a 1D chain. Measurements in both of the positions are required to perform momentum-resolved spectroscopy. (b) Transition probabilities (dimensionless) corresponding to the two probe positions, displayed in black and purple, respectively. (c) Reconstructed spectrum, where the momentum is expressed in units of $1/a$ and $a$ being the lattice parameter. The purple dots are the frequencies extracted from the transition probabilities data, while the black ones are the exact values. Here the source of error is given by the finite sampling in the probe frequency $\nu$. An additional random $2\%$ error has been added to the ratios of the peaks, and $g^{\prime }/g$ has been estimated by summing over them. The agreement is very good, although two points are missing in the dispersion relation. The corresponding frequency is clearly visible in the transition probability, but the errors hinder the estimation of the proper momentum. (d) Quantum correlation function (dimensionless) with a clear light-cone structure emerging while it spreads across the lattice. All the plots are drawn for $J/\mathrm{\Delta }=5,\alpha =0.3$, and $N_s=51$. Everywhere, $\mathrm{\Delta }$ is the energy and inverse time unit.

Figure 2

(a) Sketch of the probe positions in the three stages of the protocol. (b) Semilog plot of the low-energy transition probabilities ${\tilde{\mathrm{\Gamma }}}^{(\mathrm{I},\mathrm{II},\mathrm{III})}$ (dimensionless), rescaled by $|\frac{J_i}{J_0}{|}^2$, for the three cases shown in (a), drawn in black, yellow, and purple, respectively, for a lattice of $N_s=121$ sites. (c) Reconstructed dispersion relation (purple dots) in units of $J$, where the momentum (in $x$ and $y$ direction) is expressed in units of $1/a$ and $a$ being the lattice parameter. (d) The correlation function defined in Eq. (26) for a square lattice of $N_s={121}^2$ sites, displayed at three different times. In each plot, the value of the correlation function (dimensionless) has been scaled to its maximum and $J$ has been taken as energy and inverse time units, with $U=0.1J$.