Control of bound-pair transport by periodic driving

We investigate the effect of periodic driving by an external field on systems with attractive pairing interactions. These include spin systems (such as the ferromagnetic $XXZ$ model) as well as ultracold fermionic atoms described by the attractive Hubbard model. We show that a well-known phenomenon seen in periodically driven systems—the renormalization of the exchange coupling strength—acts selectively on bound pairs of spins (atoms), relative to magnon (bare atom) states. Thus one can control the direction and speed of transport of bound pair relative to magnon (unpaired) atom states, and thus coherently achieve spatial separation of these components. Applications to recent experiments on transport with fermionic atoms in optical lattices which consist of mixtures of bound pairs and bare atoms are discussed.

Figure 1

(Color online) CDT demonstrated in a $XXZ$ chain with $N=20$ and large $\Delta =8$. For $\Delta ⪢1$, initial spin states of adjacent spin flips $|\psi \left(t=0\right)⟩=|n,n+1⟩$ overlap only with bound-pair states; initial states with well-separated spin flips $|\psi \left(t=0\right)⟩=|n_1,n_2⟩=|5,15⟩$ overlap only with magnonlike scattering states. The fidelity $F\left(t\right)={\left|⟨\psi \left(t=0\right)\mid \psi \left(t\right)⟩\right|}^2$ is shown as a function of time. Insets plot $T_{90}$, the time taken for the fidelity to fall to 0.9, so show the CDT resonances. (a) $J^{\prime }=1/8$. CDT for magnon states. Spins freeze at zeros of ${\mathcal{J}}_0\left(B^{\prime }\right)\left(B^{\prime }\simeq 2.4,5.5\dots \right)$ but delocalize elsewhere. (b) $\frac{J^{\prime }}{2\Delta }=1/8$. CDT for bound pairs. Adjacent spins freeze at fields for which ${\mathcal{J}}_0\left(2B^{\prime }\right)=0\left(B^{\prime }\simeq 1.2,2.7\dots \right)$

Figure 2

(Color online) (a) Probability distribution ${\left|\psi \left(n,t\right)\right|}^2$ for an initially Gaussian wave packet containing a single excitation, near the DL regime, for $J^{\prime }=8$. Two zero-velocity cases are shown: $B^{\prime }=5.5$ corresponds to actual DL, with ${\mathcal{J}}_0\left(B^{\prime }\right)=0$ and suppression of hopping; while $B^{\prime }=6.3$ corresponds to $\text{sin}{B}^{\prime }=0$, so suppression of directed motion, but not of hopping: the excitation diffuses along the chain. (b) Average velocity of the wave packets as a function of $B^{\prime }$. Numerics were obtained from plots like those in (a). They are well fitted by the magnon velocity Eq. (11), while the predicted bound-pair behavior corresponds instead to Eq. (12) shown here for $\Delta =2$. The vertical lines $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ correspond to the multiply excited packets shown in Fig. 3.

Figure 3

(Color online) Dynamics of spin states which are superpositions of bound-pair and magnon states, for parameters shown in Fig. 2b. (a): $J^{\prime }=10$, $\Delta =2$, and $B^{\prime }=2.4$. The magnon states are stopped, while bound pairs travel to the end of the spin chain. (b) $J^{\prime }=10$, $\Delta =2$, and $B^{\prime }=2.53$. Bound-pair and magnon states move in opposite direction (bound pairs move upwards). (c) $J^{\prime }=10$, $\Delta =2$, and $B^{\prime }=5.34$. The faster magnon packets are slowed by the field so magnon and bound pair speeds become equalized and they now travel together. (d) $J^{\prime }=2$, $\Delta =5$, and $B^{\prime }=4.33$. Bound pairs remain static, magnons travel. (e) Spin-spin correlation, ${\left|⟨n_1,n_2\mid \psi \left(t\right)⟩\right|}^2$, at $t/T=10$ for the spin dynamics of (d). The bound-pair component remains stationary, in its initial position at the center, while the magnonlike components travel to the end of the chain.