Atomic Three-Body Loss as a Dynamical Three-Body Interaction

We discuss how large three-body loss of atoms in an optical lattice can give rise to effective hard-core three-body interactions. For bosons, in addition to the usual atomic superfluid, a dimer superfluid can then be observed for attractive two-body interactions. The nonequilibrium dynamics of preparation and stability of these phases are studied in 1D by combining time-dependent density matrix renormalization group techniques with a quantum trajectories method.

Figure 1

(a) Bosons in an optical lattice in the presence of three-body loss at a rate ${\gamma }_3$. (b) Example model parameters estimated for Cs at a magnetic field of 15 Gauss (where the scattering length is $20a_0$ and the recombination length $\sim 500a_0$ [10] where $a_0$ is the Bohr radius) as a function of lattice depth $V_0$, showing ${\gamma }_3$ (solid line), $U$ (dashed), and $J$ (dotted). Values of ${\gamma }_3$ are obtained by integrating the measured three-body recombination rates in free space over a state with three particles in a single Wannier function. (c) The probability that at least one loss event has occurred at time $tJ=2$, beginning with a single particle on each of 10 sites, and $U/J=3$ (solid line), 5 (dashed), and 10 (dotted), computed using $t$-DMRG methods (see text for details).

Figure 2

Equilibrium analysis of the projected Bose-Hubbard model $PHP$. (a) Mean-field phase diagram as a function of $U/(Jz)$ and density, $n$. (b,c) Magnitude of off-diagonal elements of (b) the single particle density matrix $|S(i,j)|=|⟨b_i^{†}{b}_j⟩|$ and (c) the dimer density matrix, $|D(i,j)|=|⟨b_i^{†}{b}_i^{†}{b}_j{b}_j⟩|$, as a function of $|i-j|$, for $U/J=10$ (thin solid line), 5 (dotted), 0 (dot-dash), $-5$ (dashed), and $-10$ (thick solid line). These results are computed for 20 particles on 20 lattice sites in one dimension, with box boundary conditions and $i+j=21$, using imaginary time evolution in $t$-DMRG, and plotted on a logarithmic scale.

Figure 3

Dynamics of adiabatic ramps into a dimer superfluid regime. (a) We begin with (i) a Mott-insulator state (ramping $U/J$), and (ii) a state with preprepared dimers in a superlattice (removing the superlattice). (b)-(c) The sum of kinetic ($E_K$) and interaction ($E_I$) energy and (inset) particle number as a function of time for two example trajectories, one with no loss events (dashed lines) and one with several loss events (solid lines). Here, (b) shows a ramp from $U/J=30$ to $U/J=-8$, with $U(t)=\alpha J/(100+3tJ)+\gamma$, with $\alpha$ and $\gamma$ ramp parameters, and (c) shows a ramp with a superlattice potential, ${\epsilon }_l=V_0\cos (2\pi l/3)$, where $V_0\approx 30J\exp (-0.1tJ)$, adjusted so that $V_0(tJ=100)=0$, with fixed $U/J=-8$. In each case, ${\gamma }_3=250J$. For (b), we use 20 atoms on 20 lattice sites, for (c), 14 atoms on 23 lattice sites. (d) Plot showing the probability that no loss event has occurred after time $t$ for the ramps in (b) (dashed line) and (c) (solid).

Figure 4

Comparison of (a) lossless and (b) lossy trajectories from Fig. 3c. We show the mean density $⟨n_i⟩$ as a function of position and time, and magnitude of the dimer correlation function $|D(i,j)|$ ($i\ne j$) at the end of the ramp.