Characterization of Bose-Hubbard models with quantum nondemolition measurements

We propose a scheme for the detection of quantum phase transitions in the one-dimensional (1D) Bose-Hubbard (BH) and 1D Extended Bose-Hubbard (EBH) models, using the nondemolition measurement technique of quantum polarization spectroscopy. We use collective measurements of the effective total angular momentum of a particular spatial mode to characterize the Mott insulator to superfluid phase transition in the BH model and the transition to a density wave state in the EBH model. We extend the application of collective measurements to the ground states at various deformations of a superlattice potential.

Figure 1

Schematic diagram of the quantum polarization spectroscopy setup: atoms are trapped in a deep optical lattice potential (red) and illuminated by a probe beam forming a standing wave over a sample of trapped atoms (yellow). The beam is split by a beam splitter (BS) and measured by homodyne detection (HD).

Figure 2

(a) The dependence of the variance of $\widehat{J}$ on the on-site repulsion, with integer values $U/t=\{1,...,10\}$ and lattice size $L=160$. (b) The normalized quantity ${\left(\Delta \widehat{J}\right)}_N^2$ obtained from ${\left(\Delta \widehat{J}\right)}^2$ by dividing by its value at $k=0.1\pi$ for $U/t=1$, 5, and 10. All quantities plotted in this figure and subsequent figures are dimensionless.

Figure 3

The variance of $\widehat{J}$, taken at $k=\pi /2$, against the on-site repulsion, for the lattice sizes $L=40$, 80, and 160.

Figure 4

For illustrative purposes, the tail of the rescaled ${\left(\Delta \widehat{J}\right)}^2$ is shown for $U/t=5$ (MI phase) and $L=160$, with 100 data points.

Figure 5

Phase transition for the BH model: the quantity $d$, defined in the text, is plotted against $U/t$ for different lengths. The vertical dashed line represents the critical point $U_c/t=3.3$ for the MI-SF phase transition, according to Ref. [26]. All lattice lengths $L$ were calculated with DMRG block sizes of $m=160$, except for $L=240$ with $m=200$.

Figure 6

(Blue solid) Phase transition of the BH model for $L=160$ and $k=\{0.0396\pi ,0.0604\pi ,0.1\pi \}$. (Red dashed) Phase transition for $L=160$ using 100 $k$ values, for comparison and as shown in Fig. 5. The shaded region indicates the error around the calculated value, determined by the dependence of the results on moving the three $k$ values. The vertical line is the estimated value $U_c/t=3.3$.

Figure 7

(a) Schematic phase diagram of the extended BH model with its different phases: Mott-insulator (MI), superfluid (SF), Haldane-insulator (HI), and density-wave (DW). The arrow indicates the range across the HI-DW transition that we investigate here. (b) Phase transition for the EBH model. All lattice lengths $L$ were calculated with DMRG block sizes of $m=100$. The dashed line indicates the HI to DW phase transition [19].

Figure 8

(a) Superlattice potential (blue, solid) formed by two counterpropagating waves, one as a larger trapping potential (red, dashed line), the other as a perturbation (green, solid line). (b) Sublattice fillings for a $k_1=4/5k_2$ superlattice potential. The intersection point $P$ occurs at $\varphi =\frac{1}{2}arctan\left(\frac{1}{3}\right),U/A=\sqrt{10}/10$. The red vertical and green horizontal arrows show the two pairs of points for which we calculate the signal in Fig. 9.

Figure 9

The $\widehat{J}$ variance for different sublattice fillings from QPS measurements on a superlattice, for $t/U={10}^{-2}$. (a) Variance distribution of two fillings resulting from a change in the relative phase $\Delta \varphi =\pi /8$. A clear difference can be seen between the two fillings at $k=\pi /4$. (b) Variance distribution of two fillings corresponding to a change in the on-site repulsion $\Delta U/A=2/3$ (scaled by the relative amplitude). The two fillings are differentiated by the peak at $k=\pi /3$ for $|1012\rangle$. DMRG parameters: lattice length, $L=40$; block size, $m=40$.