Correlated Dynamics in a Synthetic Lattice of Momentum States

We study the influence of atomic interactions on quantum simulations in momentum-space lattices (MSLs), where driven transitions between discrete momentum states mimic transport between sites of a synthetic lattice. Low-energy atomic collisions, which are short ranged in real space, relate to nearly infinite-ranged interactions in momentum space. However, the added exchange energy between atoms in distinguishable momentum states leads to an effectively attractive, finite-ranged interaction between atoms in momentum space. In this Letter, we observe the onset of self-trapping driven by such interactions in a momentum-space double well, paving the way for more complex many-body studies in tailored MSLs. We consider the types of phenomena that may result from these interactions, including the formation of chiral solitons in zigzag flux lattices.

Figure 1

Interaction shifts of Bragg tunneling resonances. (a) Energy dispersion of noninteracting massive atoms $E_p^0$ (red solid line) and Bogoliubov dispersion $E_p$ of a homogeneous gas with weak repulsive interactions and a mean-field energy $U=4E_R$ (blue dashed line), for recoil energy $E_R$. Gray lines denote laser fields used to couple adjacent momentum states. (b) Effective momentum-space lattice site energies (with a common shift of $U$ removed and renormalized to $U$) experienced by weakly coupled excitations of a macroscopically populated $p=0$ condensate, shown for $U/E_R=0.1$, 1, and 4 (solid red, dash-dotted purple, and dashed blue lines, respectively). (c) Curves as in (b), but plotted on a semilog scale for a larger range of momenta, compared to the form $U-U^2/2E_p^0$ (dotted black lines) relevant in the free-particle limit ($E_p^0\gg 2U$). (d) Depiction of site energies shifted by interactions with a $p=0$ condensate for $U/E_R=0$ and $U/E_R=4$.

Figure 2

Interaction effects in a momentum-space double well. (a) Bragg spectroscopy of the $0\to 1$ transition showing an interaction-driven shift (dashed line) of 1.31(3) kHz from single particle resonance (solid line). The momentum distributions, relating to the integrated optical density after 18 ms time of flight, show population transferred to the state $p=2\hslash k$ by a $400\text{−}\mu \mathrm{s}$-long Bragg pulse vs detuning from single-particle resonance $\mathrm{\Delta }$. (b) Measured shifts for both $0\to \pm 1$ transitions overlaid onto the Bogoliubov dispersion (dashed blue, shifted by initial condensate momentum $-0.018\hslash k$) with single-particle dispersion for comparison (solid black). (c) Experimental protocol for double-well sweeps. Population begins in left well ($L$) and transfers to the right well ($R$) as the imbalance $\mathrm{\Delta }$ (detuning from single-particle resonance) is swept linearly across 0 (dashed line) in the positive (left, blue arrow) and negative (right, red arrow) directions over 1 ms. (d) Population in the right well $P_R$ plotted vs time $\tau$ (lower horizontal axis) and vs the ratio of the bias to the initial bias $\mathrm{\Delta }/{\mathrm{\Delta }}_i$ (upper horizontal axis). Positive (left, blue squares) and negative (right, red dots) sweeps are shown with single-particle predictions (dashed gray curves) and predictions taking into account the inhomogeneous density distribution with an average mean-field energy $U/\hslash \approx 2\pi \times 1.81\mathrm{kHz}$ [28] (solid black curves). (e) Adiabatic energy levels (I and II) of the noninteracting double well vs $\mathrm{\Delta }$. Cartoon insets depict the population distributions for large $|\mathrm{\Delta }/h|$. (f) Population projection of the adiabatic levels in (e) onto the right well vs $\mathrm{\Delta }$. (g),(h) Energy levels and population projections as in (e),(f), but with an added homogeneous mean-field energy of $U/\hslash \approx 2\pi \times 1.81\mathrm{kHz}$. Gray arrows $A$ and $B$ on the negative sweep denote forced tunneling pathways as the population transfer overshoots due to self-trapping. Error bars in (b) and (d) denote 1 standard error of the mean.

Figure 3

Interaction effects in a zigzag flux lattice. (a) Cartoon depiction of atoms initialized at the central site ($n=0$, gray) of a zigzag lattice with uniform magnetic flux $\phi$. Nearest- (solid black) and next-nearest (dashed red) neighbor tunneling links have uniform amplitude $t$. Shaded yellow region indicates two-site unit cell. (b) Site population distributions for evolution times $\tau =\{12.5,25,50,100\}\hslash /t$, shown for several combinations of interaction-to-tunneling ratios and flux values on a 401-site lattice: solid black denotes $(U/t,\phi )=(0,\pi /6)$; rightmost solid blue, $(7.2,\pi /6)$; dashed red, $(12,\pi /6)$; solid purple, (7.2,0); and leftmost solid orange, $(7.2,-\pi /6)$. (c) Average site position $⟨n⟩$ (red dashed line, left vertical axis with linear scale) and population in the most-populated site $P_n^{\max }$ (blue solid line, right vertical axis with logarithmic scale) versus $U/t$. Simulations are shown for $\phi =\pi /6$ after an evolution time $\tau =65\hslash /t$ on an 801-site lattice, with population never reaching the boundaries. Shaded gray region indicates chiral soliton stability.