Emerging bosons with three-body interactions from spin-1 atoms in optical lattices

We study two many-body systems of bosons interacting via an infinite three-body contact repulsion in a lattice: a pairs quasicondensate induced by correlated hopping and the discrete version of the Pfaffian wave function. We propose to experimentally realize systems characterized by such interaction by means of a proper spin-1 lattice Hamiltonian: spin degrees of freedom are locally mapped into occupation numbers of emerging bosons, in a fashion similar to spin-1/2 and hardcore bosons. Such a system can be realized with ultracold spin-1 atoms in a Mott insulator with a filling factor of 1. The high versatility of these setups allows us to engineer spin-hopping operators breaking the SU(2) symmetry, as needed to approximate interesting bosonic Hamiltonians with three-body hardcore constraint. For this purpose we combine bichromatic spin-independent superlattices and Raman transitions to induce a different hopping rate for each spin orientation. Finally, we illustrate how our setup could be used to experimentally realize the first setup, that is, the transition to a pairs quasicondensed phase of the emerging bosons. We also report on a route toward the realization of a discrete bosonic Pfaffian wave function and list some open problems for reaching this goal.

Figure 1

Sketch of the proposed mapping. A Mott insulator with a filling factor of $1$ whose atoms have three relevant degrees of freedom can simulate a system of bosons with an infinite contact repulsion via the mapping $W$. The three relevant degrees of freedom can be identified with a $F=1$ hyperfine manifold.

Figure 2

Plot of the superlattice potential in Eq. (10) with $V_0=20E_r$ and $\lambda =1.0$, where $E_r=\hslash {}^2{k}^2/(2m)$. We show the exact Wannier wave functions of the first and third band of the lattice, corresponding to the two bound states we use in our proposal.

Figure 3

Sketch of the scheme we propose to induce hopping between the levels of the $F=1$ manifold, that is, the adiabatic elimination of one $F=2$ state trapped in the intermediate minimum (red solid arrows). Because of orthogonality properties of Wannier functions, the coupling cannot be realized with microwave fields. Optical Raman transitions through an excited state carry non-negligible momentum and can therefore be a solution. In Appendix we also discuss the effect of spurious couplings as those depicted with orange dashed arrows.

Figure 4

Splittings of the levels of the $F=1$ and $F=2$ hyperfine manifolds in ${}^{87}\mathrm{Rb}$ due to an external magnetic field. The splitting between the two manifolds is not to scale. Red arrows describe the effective couplings we want to engineer via Raman transitions; $\delta$’s and $\Omega$’s are the effective parameters describing these transitions.

Figure 5

Phase diagram of the 1D model in Eq. (13) calculated with a DMRG algorithm. Two phases appear, characterized respectively by quasi-long-range order of the one-body density matrix (AQC) and by exponential decay of the one-body-density matrix and quasi-long-range order of the two-body one (PQC). The red dashed line is the mean-field result obtained via Gutzwiller ansatz.

Figure 6

Phase diagram of the system described by Eq. (14) in which the three-body interactions have been substituted by two-body ones. Instead of a PQC a phase appears which is unstable toward collapse. At incommensurate filling and $J=0$ the ground-state manifold is largely degenerate and is spanned by Fock states with less than double local occupancies (Mott Glass).

Figure 7

Phase diagrams of the Hamiltonian $H_{3\mathrm{hbc}}$ realized with our proposal which approximates the model in Eq. (13). The two plots are drawn for the realistic values of $U_2/U_0=-0.005$ (${}^{87}\mathrm{Rb}$, left) and $U_2/U_0=-0.04$ (right); the case $U_2/U_0=0.04$ (${}^{23}\mathrm{Na}$) is not shown since it is qualitatively equivalent to the case of ${}^{87}\mathrm{Rb}$. On the left, even if a PQC phase appears, there are no signatures of AQC. Instead, at density $n=1.0$ we find a MI, whereas at $n\ne 1.0$ an inhomogeneous phase appears (phase separation). On the right, the phase diagram shows that even a small tuning of $U_2/U_0$ from the atomic values let the AQC phase arise.

Figure 8

Plot of the density profile $\langle n(x)\rangle$ of the ground state of the system for $U_2=-0.04U_0$, $n=1.125$, and $J/K=1.54$. The two insets show the algebraic decay of the particle-particle (left) and pair-pair (right) correlators (log-log plots). These data allow us to identify the phase as an AQC.

Figure 9

Plot of the density profile $\langle n(x)\rangle$ of the ground state of the system for $U_2=-0.04U_0$, $n=1.125$, and $J/K=1.43$. The two insets show the exponential decay of the particle-particle correlator (left, log plot) and the algebraic decay of the pair-pair correlator (right, log-log plot). The presence of this last quasi-long-range order allows us to identify the phase as a PQC.

Figure 10

(Top) Plot of the first $20$ energy levels of the Hamiltonian (20) studied on a $4\times 4$ torus with the parameters listed in Table . A threefold quasidegenerate ground state can be recognized. (Bottom) Cut of the first 20 energy bands in the $({\theta }_x,{\theta }_y)$ space. The lowest (red) line, threefold degenerate, does not mix with the higher bands; that is, the ground-state multiplet is always well defined and separated from higher energy levels. This is a crucial ingredient for applying the Hatsugai method for the computation of CNs.

Figure 11

Plot of the auxiliary field $\Omega ({\theta }_x,{\theta }_y)$ for the system with Hamiltonian (20). The parameters of the system are those in Table . The three highlighted vortices mean that the CN of the system is equal to 3. The definition of the field $\Omega$ and the way it can be computed are discussed extensively in Refs. [72, 73, 75], to which we refer the interested reader.

Figure 12

(Top) Plot of the first $20$ energy levels of the spin-1-based model with the parameters listed in Table , studied on the same $4\times 4$ torus as before. No threefold quasidegenerate ground state can be recognized. Moreover, the gap here is approximately one order of magnitude smaller than in the Pfaffian case (here we take $J=-0.0016$ $U_0$). (Bottom) Cut of the first 20 energy bands in the $({\theta }_x,{\theta }_y)$ space. No definite threefold ground state multiplet can be recognized. This result strongly tells us that the system is far from featuring a ground state sharing the topological properties of the Pfaffian wave function.

Figure 13

The six-level model used to study the coupling of different hyperfine levels with one Raman transition. Levels are labeled by two quantum numbers, $F$ and $k$. Energies are not to scale; the orders of magnitude of the parameters are as follows: $d~10-100$ kHz, $\delta ~100-500$ kHz, and $\Delta ~1-10$ GHz. We propose to adiabatically eliminate the upper manifold and to study the dynamics of the lowest one with an effective Hamiltonian $H_{\mathrm{pert}}$.

Figure 14

Exact time evolution of the populations with $F_z=0$ of the $6\times 3$-level model describing the dynamics of the hyperfine levels of the ground state under the action of three Raman couplings. The initial state is $|1,1\rangle$. (Inset) The maximum population reached in each level. The parameters used are listed in Table .

Figure 15

Time evolution of the $|1,1\rangle$ levels with different magnetic numbers. As explained in the text, with the parameters listed in Table  it is possible to tune the different hopping rates to very different values.