Far-from-Equilibrium Quantum Magnetism with Ultracold Polar Molecules

Recent theory has indicated how to emulate tunable models of quantum magnetism with ultracold polar molecules. Here we show that present molecule optical lattice experiments can accomplish three crucial goals for quantum emulation, despite currently being well below unit filling and not quantum degenerate. The first is to verify and benchmark the models proposed to describe these systems. The second is to prepare correlated and possibly useful states in well-understood regimes. The third is to explore many-body physics inaccessible to existing theoretical techniques. Our proposal relies on a nonequilibrium protocol that can be viewed either as Ramsey spectroscopy or an interaction quench. The proposal uses only routine experimental tools available in any ultracold molecule experiment. To obtain a global understanding of the behavior, we treat short times pertubatively, develop analytic techniques to treat the Ising interaction limit, and apply a time-dependent density matrix renormalization group to disordered systems with long range interactions.

Figure 1

Dynamic protocol viewed as Ramsey spectroscopy, two microwave pulses of area $\theta$ and ${\theta }^{\prime }$ separated by time $t$. Inset: alternative view as an interaction quench, a $t=0$ sudden turn-off of an infinite strength field $h_{\theta }$ along $\theta$.

Figure 2

Left: dc electric field dependence of coupling constants $J_{\perp }$ (blue) and $J_z$ (purple) for $\{|0⟩,|1⟩\}$ (solid) and $\{|0⟩,|2⟩\}$ (dashed) rotational level choices to represent the spin-$1/2$. Top to bottom, at $E=0$, these are $J_{\perp }^1$, $J_z^1$, $J_z^2$, $J_{\perp }^2$ where superscript 1 and 2 denote the first and second choice of states, respectively. Units for KRb are in a 532 nm lattice. Right: rescaled short time coefficient $A/f^2$, defined by $⟨S_i^x(t)⟩=⟨S_i^x(0)⟩-A{\tau }^2+O(t^4)$, for a one-dimensional dipolar chain, for fillings $f=0,0.2,\dots ,1.0$, bottom to top, with $\tau \equiv (J_z-J_{\perp })t$. Results are from Eqs. (2).

Figure 3

Phase diagram illustrating crossovers of dynamics versus filling $f$, $J_{\perp }/J_z$, and $\theta$ using $t$-DMRG on chains. In each region, a plot shows dynamics for $\theta =0.1\pi$, $0.5\pi$ (top blue, bottom purple), labeled with a qualitative description. Shaded regions for $f=0.2$ indicate one standard deviation from disorder averaging 100 configurations. The partial curves at short times are obtained from our perturbative expansions, Eqs. (2). The label “KRb: 14 ms” indicates the time for KRb in a $5\mathrm{kV}/\mathrm{cm}$ dc $\mathbf{E}$ field, using $|0⟩$ and $|1⟩$ as the spin-$1/2$ (see Fig. 2). Dynamics in higher dimensions is faster due to the presence of more neighbors.

Figure 4

Correlation and squeezing dynamics. Top: $⟨S_j^x{S}_{j+i}^x⟩-⟨S_j^x⟩⟨S_{j+i}^x⟩$ (averaged over $j$) for $J_{\perp }=2J_z$ with $\theta =\pi /2$ as a function of time for $f=0.4$ (left) and $f=1$ (right), compared to the ground state (upper right bar). Other $f$, $\theta$, and $J_{\perp }$ are similar. Bottom: squeezing in decibels as a function of time, quantifying quantum-enhanced metrological utility, for $J_{\perp }=2J_z$, $f=1$; $J_{\perp }=0$, $f=1$; $J_{\perp }=2J_z$, $f=0.4$; and $J_{\perp }=0$, $f=0.4$ (top to bottom) and $\theta =\pi /2$.