Heating dynamics of bosonic atoms in a noisy optical lattice

We analyze the heating of interacting bosonic atoms in an optical lattice due to intensity fluctuations of the lasers forming the lattice. We focus in particular on fluctuations at low frequencies below the band-gap frequency, such that the dynamics is restricted to the lowest band. We derive stochastic equations of motion, and analyze the effects on different many-body states, characterizing heating processes in both strongly and weakly interacting regimes. In the limit where the noise spectrum is flat at low frequencies, we can derive an effective master equation describing the dynamics. We compute heating rates and changes to characteristic correlation functions both in the perturbation theory limit and using a full time-dependent calculation of the stochastic many-body dynamics in one dimension based on time-dependent density-matrix–renormalization-group methods.

Figure 1

(a) We consider bosons in an optical lattice where noise in the lattice depth leads to noise in the hopping amplitude and in the interaction energy. (b) Relative change of the hopping and the interaction parameter with lattice depth in an isotropic three-dimensional cubic optical lattice, generated in the standard way by three counterpropagating laser beams. Note that the change in the hopping and interaction parameter is anticorrelated in this setup.

Figure 2

Short-time heating rates of superfluid ($U=2J$) and Mott insulator states ($U=6J$) in one dimension as a function of the relative magnitude of noise on $J$ and $U$. We parametrize the correlations between the noise on $J$ and $U$ by $\theta$ and $\lambda$ as $\sqrt{S}(dJ/dV)/J=-\lambda {\cos }^2\left(\theta \right)$ for $0\le \theta <\pi /2$; $\sqrt{S}(dU/dV)/U=\lambda {\sin }^2\left(\theta \right)$ and $\sqrt{S}(dJ/dV)/J=\lambda {\cos }^2\left(\theta \right)$ for $\pi /2\le \theta <\pi$. The usual anticorrelated case corresponds to $\theta <\pi /2$, and the sweet spot $\xi =0$ of Eq. (3.2) to $\theta =3\pi /4$. The heating rates are calculated from linear regression over 500 t-DMRG trajectories in a system with 30 particles on 30 sites (open boundary conditions). In both cases heating is strongly suppressed in the vicinity of the sweet spot. Thin lines show results in the presence of a harmonic trap with ${\varepsilon }_i/J=0.0356i^2$, $\sqrt{S_0}\frac{d{\varepsilon }_i}{dV}\frac{1}{{\varepsilon }_i}=5\times {10}^{-3}{J}^{-1/2}$; in both cases we used $\lambda =0.02J^{-1/2}$, time step $\Delta t={10}^{-2}/J$.

Figure 3

Elementary heating processes in the limiting cases of weak and strong interaction. (a) For weak interactions ($U\ll J$) lattice fluctuations create pairs of Bogoliubov excitations with opposite quasimomenta on top of the Bose-Einstein condensate at $q=0$. (b) In the strongly interacting case ($U\gg J$) lattice fluctuations create pairs of excess particles and holes of opposite quasimomenta $\pm q$ on top of the Mott insulator.

Figure 4

Comparison of the short-time heating rates from the Gutzwiller ansatz and t-DMRG simulations to the analytical results for weak and strong interactions. The Gutzwiller calculation is for a homogeneous infinite system, the DMRG simulation for a 1D system of 30 particles on 30 sites. We consider a small anticorrelated noise ($\sqrt{S}\frac{1}{J}\frac{dJ}{dV}=-0.01J^{-1/2}$, $\sqrt{S}\frac{1}{U}\frac{dU}{dV}=0.01J^{-1/2}$). We average over $n_t=1000$ ($n_t=500$ for t-DMRG) noise trajectories and estimate statistical errors of the mean. Convergence has been checked with time steps $1\times {10}^{-3}\le \Delta tJ\le 1\times {10}^{-2}$ and Hilbert space truncations $8\le d_l\le 16$. The t-DMRG results are converged with a bond dimension of $D=200$. The inset shows the averaged mean energy as a function of time (solid lines) together with the linear increase (dashed lines) with heating rate given by Eq. (3.1). This shows that the energy increase is well captured by a constant heating rate for several hopping times. The parameters and methods used to calculate the inset are the same as in Fig. 5b.

Figure 5

(a) The time evolution of the condensate fraction for short times in a large system with $M=N=48$ (averaged over 60 trajectories, bond dimension $D=256$, local dimension truncation $d_l=6$), (b) for long times in a small system with $N=M=10$ (500 trajectories), and (c) for a homogeneous infinite Bose-Hubbard model in mean-field theory (Gutzwiller ansatz, $d_l=8$, 1000 trajectories, solid grey lines obtained by a small noise approximation [33]). The fraction in general decreases except for the extreme Mott insulating case of $U/J=30$ and in the long-time limit in mean-field theory. In (a) the noise is anticorrelated with $\sqrt{S}\frac{1}{J}\frac{dJ}{dV}=-0.025J^{-1/2}$, $\sqrt{S}\frac{1}{U}\frac{dU}{dV}=0.025J^{-1/2}$, in (c) with $\sqrt{S}\frac{1}{J}\frac{dJ}{dV}=-0.01J^{-1/2}$, $\sqrt{S}\frac{1}{U}\frac{dU}{dV}=0.01J^{-1/2}$. In all three plots, the straight dashed green lines correspond to the result obtained in perturbation theory (see text).

Figure 6

The evolution of the population in different quasimomentum modes. Shown are averages over 500 noise realizations for a superfluid state ($U/J=2$) for a 1D system with $N=10$ particles on $M=10$ sites. The lattice fluctuations lead to a decrease of the population in the condensate mode at $q=0$ in agreement with the prediction from perturbation theory given by the dashed line. The noise is anticorrelated with $\sqrt{S}\frac{1}{J}\frac{dJ}{dV}=-0.025J^{-1/2}$, $\sqrt{S}\frac{1}{U}\frac{dU}{dV}=0.025J^{-1/2}$.

Figure 7

(a) Parity correlations in the ground state of a small system with $N=M=10$. The next-neighbor and longer-ranged correlations assume a maximum at $U/J\approx 6$ (solid lines are for open, dashed lines are for periodic boundary conditions). (b) Long-time evolution of these correlations calculated with exact diagonalization in the small system, averaged over 500 noise trajectories (periodic boundary conditions). The transient state develops long-range correlations. (c) The evolution of the parity-parity correlation function of a single noise trajectory in a Mott insulating state with $U/J=6$ in a 1D system with $N=M=48$ (bond dimension $D=265$, local dimension truncation $d_l=6$). The amplitude noise excites single particle-hole pairs which spread out through the system as a light cone. In all simulations, the noise is anticorrelated with $\sqrt{S}\frac{1}{J}\frac{dJ}{dV}=-0.025J^{-1/2}$, $\sqrt{S}\frac{1}{U}\frac{dU}{dV}=0.025J^{-1/2}$.