Adiabatic cooling of bosons in lattices to magnetically ordered quantum states

We suggest and analyze a scheme to adiabatically cool bosonic atoms to picokelvin temperatures which should allow the observation of magnetic ordering via superexchange in optical lattices. The starting point is a gapped phase called the spin Mott phase, where each site is occupied by one spin-up and one spin-down atom. An adiabatic ramp leads to an $xy$-ferromagnetic phase. We show that the combination of time-dependent density matrix renormalization group methods with quantum trajectories can be used to fully address possible experimental limitations due to decoherence, and demonstrate that the magnetic correlations are robust for experimentally realizable ramp speeds. Using a microscopic master equation treatment of light scattering in the many-particle system, we test the robustness of adiabatic state preparation against decoherence. Due to different ground-state symmetries, we also find a metastable state with $xy$-ferromagnetic order if the ramp crosses to regimes where the ground state is a $z$ ferromagnet. The bosonic spin Mott phase as the initial gapped state for adiabatic cooling has many features in common with a fermionic band insulator, but the use of bosons should enable experiments with substantially lower initial entropies.

Figure 1

Setup for adiabatic preparation of magnetic states. (a) Two-component bosons on a single lattice site with occupation number two and strong interactions can be represented as three different spin-1 states. (b) When the intercomponent interaction $U_{AB}$ is negligible compared to the intracomponent interaction $U$, the ground state of the system corresponds to a spin Mott state, for $U_{AB}\lesssim U$ to a planar $xy$-ferromagnetic state, shown here as a mean-field depiction. (c) Spin-dependent lattices can be used to adiabatically tune the system from a spin Mott state to an $xy$-ferromagnetic regime.

Figure 2

Magnetic phase diagram and correlations. (a) Phase diagram for two-component bosons in a 1D optical lattice. The color coding shows the gap in a small system with 12 particles on six lattice sites. The black solid line indicates the mean-field phase transition from a spin Mott to a $xy$-ferromagnetic phase, and the dashed line shows the transition line predicted from the 1D spin model. Our adiabatic ramp is along the orange arrow. Along this path a phase transition to the $xy$ ferromagnet occurs at $U_{AB}/U=0.956\pm 0.001$ [36]. (b) The $xy$-ferromagnetic ground state is characterized by the onset of algebraically decaying $〈S_i^{+}{S}_{i+j}^{-}〉$ correlations (DMRG calculations for 100 bosons on 50 lattice sites, $U=10J,n_{\max }=4)$. (c) The same type of correlations as in (b) but now obtained with a time-dependent ramp with a final ramp speed of $d{U}_{AB}/dt=0.01J^2$ (t-DMRG calculation, $n_{\max }=4)$. (d) The same plots as in (b) and (c) on a double-logarithmic scale, clearly demonstrating the onset of algebraically decaying correlations. The solid black line is a reference for a DMRG calculation in the effective spin-1 model with 200 sites and for an effective $U_{AB}/U=0.98$.

Figure 3

Many-body state fidelity $\mathcal{F}$ during adiabatic ramps. (a) The fidelity of the adiabatically evolved state for different ramp times in a system with 100 particles on 50 sites. The fidelity reduces when crossing the phase-transition point at $U_{AB}/U\sim 0.96$. For slower ramps, a larger fidelity can be achieved. (b) The fidelity with which the $xy$-ferromagnetic state at $U_{AB}/U=0.98$ can be prepared as a function of the preparation time and for different system sizes $\mathcal{N}$. With increasing $\mathcal{N}$, a larger preparation time is required to reach high state fidelities. (c), (d) State fidelities for the $U_{AB}/U=0.97$ state in a system of 40 particles with competing processes. (c) $\mathcal{F}$ for different magnetic field gradients $\mathrm{\Delta }$. (d) $\mathcal{F}$ in the presence of spontaneous emissions with rates $\gamma$. The quantum noise dramatically reduces the achievable state fidelities and there is an optimum speed for the ramp [(a)–(c) $n_{\max }=4$, (d) $n_{\max }=3]$. The labeled points refer to the parameter values taken for panels in Fig. 4.

Figure 4

Comparison of correlations and state fidelity in the presence of spontaneous emission. $〈S_i^{+}{S}_{i+j}^{-}〉$ correlations for the adiabatically prepared state with $U_{AB}/U=0.97$ in a system with 40 particles on 20 sites, compared to the ground-state correlation (dashed lines). The parameter values here are indicated as circles in Fig. 3. (a), (b) High spontaneous emission rate, $\gamma ={10}^{-3}J$. (c), (d) Low spontaneous emission rate $\gamma ={10}^{-4}J$. (a) and (c) are for a faster ramp with $t_{a}J=38.9$, and (b) and (d) for the slower ramp with $t_{a}J=117.5$. The state fidelities are given in the plots.