Light scattering and dissipative dynamics of many fermionic atoms in an optical lattice

We investigate the many-body dissipative dynamics of fermionic atoms in an optical lattice in the presence of incoherent light scattering. Deriving and solving a master equation to describe this process microscopically for many particles, we observe contrasting behavior in terms of the robustness against this type of heating for different many-body states. In particular, we find that the magnetic correlations exhibited by a two-component gas in the Mott insulating phase should be particularly robust against decoherence from light scattering, because the decoherence in the lowest band is suppressed by a larger factor than the time scales for effective superexchange interactions that drive coherent dynamics. Furthermore, the derived formalism naturally generalizes to analogous states with $\text{SU}\left(N\right)$ symmetry. In contrast, for typical atomic and laser parameters, two-particle correlation functions describing bound dimers for strong attractive interactions exhibit superradiant effects due to the indistinguishability of off-resonant photons scattered by atoms in different internal states. This leads to rapid decay of correlations describing off-diagonal long-range order for these states. Our predictions should be directly measurable in ongoing experiments, providing a basis for characterizing and controlling heating processes in quantum simulation with fermions.

Figure 1

(a) Atomic structure of ${}^{171}\text{Yb}$ (nuclear spin $I=1/2$). We use the spectroscopic notation for the sublevels and show hyperfine-structure splittings of the lowest singlet levels (energies are not drawn to scale). The ground states have total electron spin of zero and states in this manifold essentially only differ in the nuclear spin component $m_I$. We write the two ground states as spin-down and spin-up states for $m_F=-1,1$, respectively. (b) Reduction of this hyperfine structure to an effective four-level system where, for very large detuning ($\Delta \gg {\delta }_{\text{hfs}},|{\omega }_{\uparrow }-{\omega }_{\downarrow }|$), we can neglect the possibility of a spin flip and can take the photons scattered from each spin system to be identical.

Figure 2

Diagram of the atomic structure of ${}^6\text{Li}$ (nuclear spin $I=1$) showing the lowest hyperfine manifolds (energies not drawn to scale). The ground states in the $S$ sublevel are labeled by $A,B,\cdots ,F$ and the excited states in $P$ sublevels are labeled by $1,2,\cdots ,18$. These names will be used in the text in discussing transitions between different levels.

Figure 3

Decoherence of fermions in a double well. In the limit of strong interactions for $U>0$, the ground state of the Fermi-Hubbard Hamiltonian with strong repulsive interactions is primarily a spin singlet (therefore symmetric spatially across the double well). The population of doubly occupied sites is small $\left[O\right(J^2/U^2\left)\right]$. Now, for $U<0$, the initial ground state is a coherent superposition of states with doubly occupied sites with $O(J^2/U^2)$ population in the spin singlet state. Spontaneous emission events over a significant period of time lead to decoherence of virtual double occupations, and populate states in which the final steady-state population is evenly distributed in the state with single occupancy and those with doubly occupied sites.

Figure 4

Decoherent dynamics starting from the ground state of one particle of each spin species interacting strongly in a double well $(M=2)$ for $\gamma =0.1J$, computed in both our perturbative approach (red dots) and exact diagonalization (solid line): (a) decay of the rescaled spin correlations between the sites [Eq. (26)] for $U=8J$. The dashed line indicates the steady-state expectation value. (b) Time evolution of rescaled doublon correlation [Eq. (27)] for $U=-8J$ which vanishes in the final steady state.

Figure 5

Comparison of the decay of spin correlations [see Eq. (25)] averaged over the chain obtained from exact diagonalization using the expokit package [90] for $M=4,8$ and tDMRG with $D=128$ for $M=32$. (a) Decay of the on-site contribution $S(\Delta x=0,t)$ for different system sizes $M=4,8,32$ at $U=8J$ and $\gamma =0.1J$ using 500 trajectories for $M=4,8$, and 250 trajectories for $M=32$. (b) Example fit to the data shown in panel (a) for $M=32$. (c) The decay rates $\beta$ extracted from numerical fits as shown in panel (b) as a function of $N_D\gamma$. The dashed lines in (b) and (c) are linear fits to the data for different system sizes $[M=4$ (black squares) and $M=32$ (red triangles)], which exhibit the scaling $\beta \sim N_D\gamma$ predicted by perturbation theory. Panel (d) shows the rescaled $S_r(\Delta x,t)$ [Eq. (26)] for $M=32$, $U=8J$, and different $\Delta x$ which shows only a weak distance dependence, especially at larger times.

Figure 6

Comparison of doublon correlation obtained numerically and in perturbation theory (dashed line) in the strong attractive interaction limit, averaged over chain and rescaled by the initial value [Eq. (27)]. (a) Time dependence of the doublon correlation at $\Delta x=2,5$ for system size $M=8$ at $U=-8J$ and $\gamma =0.05J$. We see that the quantum trajectory results (diamonds) from tDMRG with $D=64$ are in good agreement with the result obtained by doing exact diagonalization (squares) using the expokit package [90] and that perturbation theory does not take into account the rebuilding of correlations destroyed by spontaneous emissions and hence underestimates the decay at short distances, but overestimates the decay at large distances. Using same line symbols in panel (b) we show the quantum trajectory results for spatial dependence for $M=32$ at $U=-8J$ and $\gamma =0.05J$ qualitatively similar to $M=8$. (c) Effects of different decay rates for $M=32$ at $U=-8J$. (d) Dependence on interaction strength for $M=32$, $\gamma =0.05J$. The time for which our perturbation theory is reliable scales with $U$. Panels (b) to (d) show tDMRG data using a bond dimension $D=128$ and the number of trajectories used in all of the calculations here is 528.

Figure 7

Zeeman diagram of the different hyperfine levels shows the lifting of degeneracy obtained by numerical diagonalization for qualitative values of magnetic field. The electron and nuclear spins get decoupled in high enough $B$ field. (a) $P_1$ sublevel in ${}^{171}\text{Yb}$ (the $S$ sublevel is already decoupled). (b) Splitting of the $S_{1/2}$ sublevel in ${}^6\text{Li}$. (c) $P_{1/2}$ sublevel in ${}^6\text{Li}$ needs weaker $B$ field to get decoupled than the previous case as the hyperfine coupling is weaker. (d) Even smaller $B$ field is needed for the even more weakly hyperfine-coupled $P_{3/2}$ sublevel in ${}^6\text{Li}$.