Minimizing nonadiabaticities in optical-lattice loading

In the quest to reach lower temperatures of ultracold gases in optical-lattice experiments, nonadiabaticities during lattice loading represent one of the limiting factors that prevent the same low temperatures being reached as in experiments without lattices. Simulating the loading of a bosonic quantum gas into a one-dimensional optical lattice with and without a trap, we find that the redistribution of atomic density inside a global confining potential is by far the dominant source of heating. Based on these results we propose adjusting the trapping potential during loading to minimize changes to the density distribution. Our simulations confirm that a very simple linear interpolation of the trapping potential during loading already significantly decreases the heating of a quantum gas, and we discuss how loading protocols minimizing density redistributions can be designed.

Figure 1

Sketch of optical-lattice loading. We show the potential at various times corresponding to lattice strengths $V_0$ ranging from 0 to $8E_R$, illustrating how the lattice is ramped up (a) in a homogeneous system; (b) in a constant trapping potential; and (c) while changing the trap to minimize density redistribution.

Figure 2

Dependence on the ramp time $t_R$ of (a) the fidelity of the final state after ramping, (b) the same data plotted as the probability of being in an excited state $1-|\langle {\psi }_0\left(V_f\right)|\psi \left(t_R\right)\rangle {|}^2$ on a logarithmic scale, and (c) the excess energy per particle $q$ in units of the recoil energy $E_r$ and of the effective final hopping amplitude $J_{\mathrm{eff}}$. The optical lattice is ramped up to a final strength of $V_f=8E_r$ at fixed interactions $g=2E_r\lambda /2$. Results are shown for three different ramp profiles, displayed in the inset of (a), and described in Eqs. (4, 5, 6). Full symbols refer to homogeneous system with $N=24$ particles in $L=24$ sites with hard-wall boundary conditions. Open symbols refer to an inhomogeneous system of the same size but with with $N=12$ particles in a confining harmonic potential with $\omega =0.3{(\hslash /E_r)}^{-1}$.

Figure 3

Dependence of the final excess energy per particle $q\left(t_R\right)$ in units of the recoil energy $E_r$ and of the effective final hopping amplitude $J_{\mathrm{eff}}$. The optical lattice is ramped up to a final strength of $V_f=8E_r$ at fixed interactions $g=2E_r\lambda /2$ for various system sizes $L=16,20,24,28,32$ at unit filling. Results in the subpanels show three different ramp profiles (a) linear, (b) exponential, and (c) sigmoid, as plotted in Fig. 2 and described in Eqs. (4, 5, 6).

Figure 4

(a) Local density profile for the initial state with the same trapping frequency ${\omega }_i={\omega }_f=0.3{(\hslash /E_r)}^{-1}$ as the final target state (bold solid line in the upper panel), and for the optimal initial state with ${\omega }_i\approx 0.16{(\hslash /E_r)}^{-1}$ (thin solid line in the upper panel). The final state, a unit-filling Mott insulator, is shown in the lower panel. (b) Excess energy per particle $q=[E\left(t_R\right)-E_0]/N$ at the end of the ramp up as a function of the initial trap frequency ${\omega }_i$ for various ramp times. (c) Fidelity of the final wave function $\left|\psi \right(t_R)\rangle$ against the true ground state $|{\psi }_0\left(V_f\right)\rangle =|{\psi }_0(V_0=V_0\left(t_R\right))\rangle$ as a function of the initial trap frequency ${\omega }_i$ for various ramp times. All calculations are performed on a system of $L=24$ optical lattice sites with $N=12$ particles and targeting a final state with trapping frequency ${\omega }_f=0.3{(\hslash /E_r)}^{-1}$.

Figure 5

(a) Local density profile for the initial state with the same trapping frequency ${\omega }_i={\omega }_f=0.4{(\hslash /E_r)}^{-1}$ as the final target state (bold solid line in the upper panel), and for optimal initial state with ${\omega }_i\approx 0.25{(\hslash /E_r)}^{-1}$ (thin solid line in the upper panel). The final state, a Mott insulator with a superfluid core, is shown in the lower panel. (b) Excess energy per particle $q=[E\left(t_R\right)-E_0]/N$ at the end of the ramp up as a function of the initial trap frequency ${\omega }_i$ for various ramp times. (c) Fidelity of the final wave function $\left|\psi \right(t_R)\rangle$ against the true ground state $|{\psi }_0\left(V_f\right)\rangle =|{\psi }_0(V_0=V_0\left(t_R\right))\rangle$ as a function of the initial trap frequency ${\omega }_i$ for various ramp times. All calculations are performed on a system of $L=24$ optical lattice sites with $N=16$ particles and targeting a final state with trapping frequency ${\omega }_f=0.4{(\hslash /E_r)}^{-1}$.

Figure 6

Dependence on the ramp time $t_R$ of the excess energy per particle. Black solid lines refer to the target state of Fig. 4 with fixed trap $(T_1$, fixed) and optimal initial trap $(T_1$, optimal). Blue dashed lines refer to the target state of Fig. 5 with fixed trap $(T_2$, fixed) and optimal initial trap $(T_2$, optimal).

Figure 7

Comparison of linear (solid blue line) and exponential (dotted red line) lattice loading in a trapped system of length $L=24$ and $N=12$ particles. (a) Dependence on the ramp time $t_R$ of the excess energy per particle $q=[E\left(t_R\right)-E_0]/N$ in units of the recoil energy $E_r$ for an initial trap frequency ${\omega }_i=0.12{(\hslash /E_r)}^{-1}$ which provides the best results for the exponential loading [see (b)]. (b) Excess energy per particle $q$ at the end of the ramp up as a function of the initial trap frequency ${\omega }_i$ for a ramp time $t_R=128\hslash /E_r$.