Manipulating atom-number distributions and detecting spatial distributions in lattice-confined spinor gases

We present an experimental study demonstrating the manipulation of atom-number distributions of spinor gases after nonequilibrium quantum quenches across superfluid to Mott-insulator phase transitions in cubic optical lattices. Our data indicate that atom distributions in individual Mott lobes can be tuned by properly designing quantum quench sequences, which suggests methods of maximizing the fraction of atoms in Mott lobes of even occupation numbers and has applications in attaining different quantum magnetic phases including massively entangled states. Spatial distributions of gases in three-dimensional lattices are derived from the observed number distributions, which reveal complex spatial dynamics during the quantum quenches. Qualitative agreements are also found between our experimental data and numerical simulations based on time-dependent Gutzwiller approximations in two-dimensional systems.

Figure 1

(a) Observed dynamics of spin-0 atoms at $v_{\mathrm{ramp}}=14\left(1\right)E_R/\mathrm{ms}$ and $u_{L0}=0E_R$. Markers represent the average of approximately 15 repeated shots at the same conditions and error bars are one standard error. The solid line is a fit based on our empirical model to guide the eye [8]. (b) Markers show FFTs over the first 40 ms of $t_{\mathrm{hold}}$ on the data shown in (a). Vertical lines mark the predicted energy signature $f_n$ for each $n$, while the solid line represents Gaussian fits to each observed peak [26]. (c) Number distributions ${\chi }_n$ extracted from the FFT spectrum shown in (b) (see text).

Figure 2

(a) Observed number distributions ${\chi }_n$ at various $v_{\mathrm{ramp}}$ and $u_{L0}=0E_R$. Shades of blue (red) represent even (odd) occupation numbers $n$ with the shades getting darker as $n$ increases from 2 to 8 (from 3 to 9). The height of each shaded box represents ${\chi }_n$ for a given $n$, while the combined height of the blue (red) boxes corresponds to the total number distribution in even (odd) Mott lobes. (b) Diamonds represent the experimentally found ${\chi }_{\mathrm{even}}$ for each quench sequence shown in (a). The solid line is a linear (an exponential) fit to the data when $v_{\mathrm{ramp}}$ is faster (slower) than $39\left(1\right)E_R/\mathrm{ms}$. (c) Similar to (a) but varies $u_{L0}$ while holding $v_{\mathrm{ramp}}$ at $28\left(1\right)E_R/\mathrm{ms}$ during the lattice ramp from $u_{L0}$ to $35E_R$ (see text).

Figure 3

(a)–(c) Spatial distributions of atoms corresponding to the measured ${\chi }_n$ shown in Fig. 2 at $v_{\mathrm{ramp}}=54\left(1\right)$, 28(1), and $14\left(1\right)E_R/\mathrm{ms}$, respectively (see text). The $z$ axis and color scale correspond to the observed ${\chi }_n$ while the radial distance from the center denotes the predicted radii of each Mott plateau and is normalized so that the outer radius of the $n=1$ Mott lobe is 1. (d) Markers represent the observed ${\chi }_{2\&3}$ from the distributions presented in Fig. 2. The solid line is an exponential fit.

Figure 4

(a) Predicted number distributions ${\chi }_n$ derived from our numerical simulations for 2D lattice-confined sodium spinor gases after ramp sequences with $u_{L0}=0.1E_R$ at various $v_{\mathrm{ramp}}$ and for IS, the initial superfluid ground state. Shades of blue (red) represent even (odd) $n$ with the shades getting darker as $n$ increases from 2 to 8 (3 to 9). The height of each shaded box represents the predicted ${\chi }_n$ for a given $n$, while the combined height of the blue (red) boxes represents the total number distributions in even (odd) Mott lobes. (b) Markers represent fractions of even Mott lobes extracted from simulated lattice quenches similar to those shown in (a) but at various $v_{\mathrm{ramp}}$ and two $u_{L0}$. (c) Similar to (b) but markers represent ${\chi }_{2\&3}$. Solid lines in (b) and (c) are fitting curves to guide the eye. (d)–(g) Similar to Figs. 3–3 but after the simulated quenches at $v_{\mathrm{ramp}}=700$, 300, 100, and $35E_R/\mathrm{ms}$, respectively.