Dynamical Disentangling and Cooling of Atoms in Bilayer Optical Lattices

We show how experimentally available bilayer lattice systems can be used to prepare quantum many-body states with exceptionally low entropy in one layer, by dynamically disentangling the two layers. This disentangling operation moves one layer—subsystem $A$—into a regime where excitations in $A$ develop a single-particle gap. As a result, this operation maps directly to cooling for subsystem $A$, with entropy being shuttled to the other layer. For both bosonic and fermionic atoms, we study the corresponding dynamics showing that disentangling can be realized cleanly in ongoing experiments. The corresponding entanglement entropies are directly measurable with quantum gas microscopes, and, as a tool for producing lower-entropy states, this technique opens a range of applications beginning with simplifying production of magnetically ordered states of bosons and fermions.

Figure 1

Dynamical disentangling in bilayer systems: the example of single-component bosons. (a) Two tunnel-coupled layers (with interlayer tunneling $J_p$) are prepared at the same chemical potential (identical trap depths in the vertical direction). Particles are delocalized between the layers, which are entangled at zero temperature. By manipulating the relative trap depth of the layers and then removing the tunnel coupling, layer $A$ can be prepared in a Mott-insulating state. Having a gapped state in layer $A$ strongly suppresses entanglement of the two layers at zero temperature. At nonzero temperatures, the entropy per particle is much higher in layer $B$, where atoms are free to move.

Figure 2

Equilibrium entropies in the limit $J_p$, $J_p^{\sigma }\to 0$. (a) Entropy per particle $S_A/N_A$ in layer $A$ of a 1D bilayer system of length 6 with eight spinless bosons, as a function of total entropy per particle, $S/N$. Black stars and a fitted dotted line show the case where $U/J=5$ and $\mathrm{\Delta }\mu =0$, demonstrating nonzero entanglement of the two layers at zero temperature. Remaining points show target parameters for dynamical disentangling, with $\mathrm{\Delta }\mu =U/2$. Blue crosses denote $U/J=8$ ($\beta J\in [0.1,1.5]$), red squares $U/J=20$ ($\beta J\in [0.1,2]$), and green circles $U/J=50$ ($\beta J\in [0.1,1]$). For large $U/J$, $S_A/N_A$ is strongly suppressed for low $S/N$. Inset: The zero-temperature mean-field phase diagram for the Bose-Hubbard model in the local density approximation, showing two superfluid layers with less than unit filling (central diamond) being separated in chemical potential so that one becomes Mott insulating and the other remains superfluid (upper and lower diamonds). (b) $S_A/N_A$ of a 1D bilayer system of length 5 with four spin-up and four spin-down lattice bosons with $U_{\uparrow ,\downarrow }\to \infty$, $V=10J_{\downarrow }$, as a function of $S/N$ ($\beta J_{\downarrow }\in [1,20]$; purple stars). As $J_{\uparrow }=3J_{\downarrow }$, $\mathrm{\Delta }{\mu }_{\uparrow }=8.5J_{\downarrow }$, and $\mathrm{\Delta }{\mu }_{\downarrow }=5J_{\downarrow }$, layer $A$ realizes gapped AFM order with exactly five bosons with a large charge and a small spin gap. Again, $S_A/N_A$ is strongly suppressed at low $S/N$ (see the text). For comparison, we show the results from (a) with $U/J=20$ and plot $S_A/N_A=S/N$ in black. We include maximally four bosons per site in our calculations.

Figure 3

Time-dependent disentangling. (a) At zero temperature, the fidelity of the final state of the ramp to the ground state of the system with $\mathrm{\Delta }\mu =U/2$, and then to $J_p=0$, against total ramp time $T$ (measured in units of $J^{-1}$), computed using time-dependent density matrix renormalization group techniques for up to $M=16\times 2$ lattice sites, always taking $N=3M/4$ bosons at $U=8J$. We begin in the ground state with $\mathrm{\Delta }\mu =0$ and $J_p=J$, ramp linearly in time to $\mathrm{\Delta }\mu =U/2$, and then to $J_p=0$. (b) At a finite temperature, $S_A/N_A$ at the end of the ramp in a 1D bilayer system with $M=5\times 2$ for spinless bosons with $N=7$ and $T=96{\mathrm{J}}^{-1}$ (blue crosses and magenta diamonds), with $\mathrm{\Delta }\mu :0\to 10J$ in $T(\mathrm{\Delta }\mu )=92{\mathrm{J}}^{-1}$, followed by $J_p:J\to 0$ within $T(J_p)=4{\mathrm{J}}^{-1}$. The blue crosses show a ramp with $U$ initially kept at a low constant value $U=J$ for a time $30{\mathrm{J}}^{-1}$ and subsequently $U:J\to 20J$ within $T(U)=62{\mathrm{J}}^{-1}$. Magenta diamonds show the same protocol, but with $U=20J$ kept constant throughout. Green squares show the first type of ramp again, but for two-component bosons, $N_{\uparrow ,\downarrow }=4$ and $T=100{\mathrm{J}}^{-1}$. This ramp into an antiferromagnetic state with gaps to both charge and spin excitations uses $J_{\uparrow }=3J_{\downarrow }$, $U_{\uparrow ,\downarrow }=\infty$, with $\mathrm{\Delta }{\mu }_{\uparrow }:0\to 8.5J_{\downarrow }$, $\mathrm{\Delta }{\mu }_{\downarrow }:0\to 5J_{\downarrow }$ with $T(\mathrm{\Delta }{\mu }_{\uparrow ,\downarrow })=50{\mathrm{J}}_{\downarrow }^{-1}$. In parallel, $V$ is kept initially at $\text{value}=J_{\downarrow }$ over time span $15{\mathrm{J}}_{\downarrow }^{-1}$, after which $V:J_{\downarrow }\to 10J_{\downarrow }$ with $T(V)=35{\mathrm{J}}_{\downarrow }^{-1}$. Afterwards, $J_p^{\uparrow }:3J_{\downarrow }\to 0$, $J_p^{\downarrow }:J_{\downarrow }\to 0$, with $T(J_p^{\uparrow ,\downarrow })=50{\mathrm{J}}_{\downarrow }^{-1}$, thereby accounting for the small gap to spin excitations in layer $A$. For all shown calculations, maximally four bosons per site have been allowed.

Figure 4

(a) Bilayer disentangling for a dimerized lattice: Layers $A$ and $B$ are connected with tunneling $J_p$. In each layer, we have alternating tunneling amplitudes $J$ and $J_{ID}$. (b) Characterization of spin singlets produced from fermions on sites with tunneling $J$ in from (a), with initial parameters $J_p=J$, $J_{ID}=0$, and $\mathrm{\Delta }{\mu }_{\uparrow ,\downarrow }=0$, ramping to $\mathrm{\Delta }{\mu }_{\uparrow ,\downarrow }=U/2$ in time $T$ and then to $J_p=0$ in time $T$. We show the difference of the average correlation over all dimers in $A$ $\overline{⟨S_i^{+}{S}_{i+1}^{-}⟩}$ at the end of the ramp to the value on a single dimer with one spin-up and spin-down fermion, for system sizes $L=8$ (blue circles), $L=16$ (red squares), and $L=32$ (green diamonds). Here, $U/J=8$ and $N_{\uparrow }=N_{\downarrow }=3L/4$. (c) Schematic overview of antiferromagnetic state preparation starting from the final state of (b) (isolated pairs of singlets) and adiabatically increasing tunneling between dimers. (d) Plot of $1-F(T)$ for fidelities $F(T)$ at the end of a ramp of time scale $T$, where $J_{1D}$ is increased from $J_{1D}=0$ to $J_{1D}=J$, with ramp function $1-(e^{-\nu t}-e^{-\nu T})/(1-e^{-\nu T})$, $\nu ≔T/10$. We show results for $U/J=8$, for system sizes $L=8$, $L=16$, and $L=32$ [symbols as for (b)].