Directed motion of doublons and holes in periodically driven Mott insulators

Periodically driven systems can lead to a directed motion of particles. We investigate this ratchet effect for a bosonic Mott insulator where both a staggered hopping and a staggered local potential vary periodically in time. If driving frequencies are smaller than the interaction strength and the density of excitations is small, one obtains effectively a one-particle quantum ratchet describing the motion of doubly occupied sites (doublons) and empty sites (holes). Such a simple quantum machine can be used to manipulate the excitations of the Mott insulator. For suitably chosen parameters, for example, holes and doublons move in opposite directions. To investigate whether the periodic driving can be used to move particles “uphill,” i.e., against an external force, we study the influence of a linear potential $-gx$. For long times, transport is only possible when the driving frequency $\omega$ and the external force $g$ are commensurate, $n_{0}g=m_0\omega$, with $\frac{n_0}{2},m_0\in \mathbb{Z}$.

Figure 1

Illustration of the lattice model at a fixed time $t$. Both the local hopping parameter and potential are staggered. $J_1,J_2,V_1,V_2$ are defined according to Eqs. (2) and (3).

Figure 2

Doublon (solid red line) and hole (dashed blue line) asymptotic velocity for (a) the sudden switching and (b) the adiabatic case. Velocities are given in units of $a(J_{s,o}+J_{s,e})$, i.e., lattice constant times average static doublon hopping rate. Parameters: $\phi =-\pi /2$, $J_{s,e}=0.67$, $J_{s,o}=0.33$, $J_e=0.67$, $J_o=-0.17$, $V_s=0.4$, $V_e=-0.4$, and $V_o=0.4$, measured in units of $(J_{s,o}+J_{s,e})$. Vertical lines indicate the first 10 doublon resonances, i.e., $\omega =\Delta E/n$, $n\in \{1,...,10\}$.

Figure 3

Avoided crossing at the main resonance $(\omega =\Delta E/1)$ of the doublon bands (parameters as in Fig. 2) at the lower edge of the first Floquet zone (gray shaded area). Dashed lines correspond to eigenenergies of $H^F$ in the absence of time-dependent terms in the original Hamiltonian. Points illustrate population of the respective band for (a) the sudden switching and (b) the adiabatic case. In (a) the system changes the band as function of $\omega$, while in (b) it remains in the Floquet band.

Figure 4

Illustration of states in a mixed real-space–Floquet-space representation. The vector from $(0,0)$ to $(n_0,m_0)$ fulfills the commensurability condition (16). The practical truncation of the Floquet matrix is done around this vector and only states lying in the blue shaded region are used in calculations. In most cases, it is sufficient to take only a few Floquet states per real-space index into account.

Figure 5

Asymptotic (a) doublon and (b) hole velocity as function of driving frequency $\omega$ and external force $g$ for an adiabatically switched on ratchet [velocities are measured in units of $a(J_{s,o}+J_{s,e})$ and parameters are as in Fig. 2]. Combined results are shown for real-space unit-cell sizes $n_0=14,16,18$, and 60 and arbitrary values of $m_0$.

Figure 6

Average doublon velocity (solid lines) at $t=10,25$, and 50 after a sudden switching on of the ratchet potential as function of the external force $g$ for a fixed driving frequency $\omega =4.4$ ($n_0=120$). The parameters are as in Fig. 2 and velocities are measured in units of $a(J_{s,o}+J_{s,e})$. The discrete points at $g_c=\frac{m_0}{60}\omega$ show the average velocity in the limit $t\to \infty$. The inset in (c) reveals the characteristic pattern of minima and maxima caused by effective Bloch oscillations (see text) near the $\frac{m_{0r}}{n_{0r}}=\frac{1}{4}$ resonance.

Figure 7

Absolute values of doublon velocities for the main resonance for different values of $\epsilon ={\epsilon }_1={\epsilon }_2$ (other parameters are as in Fig. 2). Colored lines represent exact numerical data and corresponding dashed lines are plots of (A18). Sharp spike appearing on curve for $\epsilon =0.05$ corresponds to second resonance, which is not captured by (A18).

Figure 8

Doublon velocities around the main resonance for different values of $\epsilon ={\epsilon }_1={\epsilon }_2$ (other parameters as in Fig. 2). Colored lines represent exact numerical data and corresponding dashed lines are plots of (B12).

Figure 9

Comparison of doublon velocities of exact numerical data (red, solid line) and plots of Eq. (C10) (gray, dashed line) for large frequencies $\omega >J_o,J_e,V_o,V_e$ (parameters as in Fig. 2).