$\eta$ Condensate of Fermionic Atom Pairs via Adiabatic State Preparation

We discuss how an $\eta$ condensate, corresponding to an exact excited eigenstate of the Fermi-Hubbard model, can be produced with cold atoms in an optical lattice. Using time-dependent density matrix renormalization group methods, we analyze a state preparation scheme beginning from a band insulator state in an optical superlattice. This state can act as an important test case, both for adiabatic preparation methods and the implementation of the many-body Hamiltonian, and measurements on the final state can be used to help detect associated errors.

Figure 1

(a) Preparation of an $\eta$ condensate: (1) Begin in an insulator state $|{\psi }_i⟩$ with attractive on-site interactions $U$ in an optical superlattice with depth $V_{\mathrm{SL}}$; (2) switch $U$ to a positive value larger than the band gap; (3) delocalize on-site pairs by adiabatic removal of the superlattice. (b) Energy spectrum of $H_{\mathrm{FH}}$ with $U>0$ in 1D for one atom of each spin species, plotted as a function of center-of-mass quasimomentum $k$, with $a$ the lattice spacing.

Figure 2

(a) Energy eigenstates of $H_{\mathrm{FH}}+V$ as a function of time for a small example system with $N_{\uparrow }=N_{\downarrow }=3$, $M=6$. Left: Lowest energy states for strong initial attraction $U/J=-30$, in the presence of a superlattice. The lowest energy state is the initial state in our preparation scheme, $|{\psi }_i⟩$. The shaded area denotes the excited manifold of states with one dissociated pair. Right: The highest energy levels of $H_{\mathrm{FH}}+V(t)$ as a function of time during the adiabatic ramp. The lowest energy eigenstate $|\psi (t)⟩$ in the upper manifold where all atoms exist in pairs is equal to $|{\eta }_N⟩$ at $t=T$. During the ramp, a gap of order $U$ always exists to the manifold (shaded area) where some atoms are unpaired, and a gap to higher levels is present when the superlattice is present. (b) Pair momentum distribution ${\mathcal{C}}_k^{|{\eta }_N⟩}$ of a perfect $\eta$ condensate (see text).

Figure 3

(a) Fidelities for the superlattice and parabolic trap ramp as a function of ramp time $T$ for different system sizes $M$, computed using $H_S$. The superlattice ramp shape is $V_{\mathrm{SL}}=2J(e^{-\nu t}-e^{-\nu T})/(1-e^{-\nu T})$, $\nu =J/8$. For the parabolic trap, we use the same shape with initial $V_P/J=0.1$, $\nu =J/12$, and values for $M=64$ are not shown as $\mathcal{F}\sim {10}^{-12}$. $U$ is decreased with the same shape as the potential in each case, with $U=30J$ at $t=0$, giving parameters for $U$ and the superlattice that are typical of current experiments. The inset shows results for longer ramp times with $M=16$. (b) On-site pair momentum distribution after $T=2400J^{-1}$ for the superlattice (solid lines) and parabolic trap (dotted lines) ramps for $M=32$ (left) and $M=64$ (right), computed using $H_S$. (c) Final state fidelity $\mathcal{F}$ as a function of correlation function distance $\mathcal{D}(T)$ from the perfect $\eta$ condensate, computed using $H_S$. (d) ${\mathcal{C}}_k(T)$ for superlattice ramps, computed using $H_{\mathrm{FH}}$, with a number of impurities $N_i=1$ (left) and $N_i=2$ (right), for $T=200J^{-1}$ (solid black lines), and $T=400J^{-1}$ (grey dashed lines). The number of states retained in decompositions, $\chi =200$ in (a–c), and $\chi =400$ for (d).

Figure 4

Stability of a state close to an $\eta$ condensate with imperfections. (a) Time dependence of ${\mathcal{C}}_{\pi /a}$ for the initial state with impurities as defined in the text on 32 sites, with 16 on-site pairs and varying impurity count $N_i$, for $U/J=4$ (dashed black lines), $U/J=10$ (solid grey lines). (b) Same as (a), but with additional trapping potential $V_P/J=1.25\times {10}^{-4}$. The number of states retained in state decompositions, $\chi =600$ in both parts.