Dynamic freezing and defect suppression in the tilted one-dimensional Bose-Hubbard model

We study the dynamics of a tilted one-dimensional Bose-Hubbard model for two distinct protocols using numerical diagonalization for a finite sized system ($N\le 18$). The first protocol involves periodic variation of the effective electric field $E$ seen by the bosons which takes the system twice (per drive cycle) through the intermediate quantum critical point. We show that such a drive leads to nonmonotonic variation of the excitation density $D$ and the wave function overlap $F$ at the end of a drive cycle as a function of the drive frequency ${\omega }_1$, relate this effect to a generalized version of Stückelberg interference phenomenon, and identify special frequencies for which $D$ and $1-F$ approach zero leading to near-perfect dynamic freezing phenomenon. The second protocol involves a simultaneous linear ramp of both the electric field $E$ (with a rate ${\omega }_1$) and the boson hopping parameter $J$ (with a rate ${\omega }_2$) starting from the ground state for a low effective electric field up to the quantum critical point. We find that both $D$ and the residual energy $Q$ decrease with increasing ${\omega }_2$; our results thus demonstrate a method of achieving near-adiabatic protocol in an experimentally realizable quantum critical system. We suggest experiments to test our theory.

Figure 1

A schematic representation of the quenching protocols studied in this paper. The black dotted line is the equation of the critical line $U-E=-1.85J$. The red dashed line corresponds to the periodic variation of $E$, and thus $U-E$, for fixed $J=1$. The quenching starts from the paramagnetic phase, crosses the quantum critical point (intersection of the path with the black dotted line), and reaches the AFM phase after which it returns back to the PM phase after one period. The blue solid line is the two-rate protocol with ${\omega }_1=0.05,{\omega }_2=0.1,U-E_i=100,\Delta E=200,\epsilon =0$, and $\Delta J=1$. The arrow shows the direction of the quenching which terminates at the quantum critical point. The slope of the blue line for an arbitrary set of parameters corresponds to $-\Delta E{\omega }_1/\left(\Delta J{\omega }_2\right)$.

Figure 2

(a) and (b) Plot of variation of the dipole excitation density $D\left(t\right)$ as a function of ${\omega }_{1}t$ for different scaled frequencies ${\omega }_1/(E_c-U)$ (shown in the figure) up to one (a) and seven (b) drive periods. (c) and (d) Plots of the defect density $1-F\left(t\right)$ as a function of ${\omega }_{1}t$ for the same parameters as (a) and (b), respectively.

Figure 3

Plot of the position of the dip in $D$, $(E^{*}-E_c)/E_c$, as a function of scaled frequency ${\omega }_1/(E_c-U)$. The dip is closer to the critical point for smaller frequencies.

Figure 4

Plot of $ln\left(D\right)$ and $ln(1-F)$ after one complete cycle as a function of ${\omega }_1$. (a) Small frequency region where for certain frequencies $F$ is close to unity (or $1-F\sim 0$) which signals near perfect freezing of the wave function after one complete cycle. As the frequency increases, the freezing phenomena becomes less effective and ultimately disappear for $ln[{\omega }_1/(E_c-U)]>2$ as shown in (b).

Figure 5

Plot of the wave function overlap coefficients $|c_{\alpha }{\left(t\right)|}^2$ for a selected set of $\alpha$ satisfying the condition $|c_{\alpha }{\left(t\right)|}^2\ge 1E-5$ as a function of ${\omega }_{1}t$ during the drive cycle. The top figure corresponds to ${\omega }_1/(E_c-U))=0.051$ corresponding to a minima of $D$, while that at the bottom has ${\omega }_1/(E_C-U)=0.074$ corresponding to a maxima of $D$.

Figure 6

Plots of the wave function overlap $F$ (top) and the dipole excitation density $D$ (bottom) as a function of ${\omega }_1/(E_c-U)$ showing near perfect match between analytical [Eq. (13)] and exact numerical calculations. The points correspond to numerical results and the lines to Eq. (13).

Figure 7

Plot of the variation of $|c_{\alpha }{|}^2$ at $t=T$ as a function of ${\omega }_1$ for $\alpha =0,14$, and 15.

Figure 8

Plot of the variation of the dipole excitation density $D$ with ${\omega }_1$ for different $r$ with fixed ${\omega }_2={\omega }_1^r$. The plot shows a clear crossover in behavior of $D$ from increasing to decreasing function of ${\omega }_1$ as $r$ passes through $3/2$. Here we have set ${\nu }_0=0.01$.

Figure 9

Plot of $ln\left(D\right)$ with ${\omega }_1$ for different system sizes and fixed ${\omega }_2/(U-E_i)=0.0005$. The ramp of $E$ and $J$ starts from $U-E_i=100$ to $U-E_f$, where $E_f$ is the value of the effective electric field at the critical point as discussed in the text. The scaling behavior of $D$ predicted in Eq. (16) is over a finite frequency range $-8.4\ge ln({\omega }_1/(U-E_i))\ge -9.0$; see text for details. Inset: Plot of $ln\left(D\right)$ as a function of ${\omega }_1$ for ${\omega }_2/(U-E_i)=0.0005$ (red solid line), $0.001$ (green dashed line), and $0.005$ (blue dash-dotted line) with system size set to $N=16$. All other parameters are the same as in Fig. 8.

Figure 10

Plot of $ln\{Q/\left[N(U-E_i)\right]\}$ with ${\omega }_1$ for different system sizes and fixed ${\omega }_2/(U-E_i)=0.0005$ demonstrating the agreement between the theoretically predicted $Q\sim {\omega }_1^3$ and the numerical simulations over a range of ${\omega }_1$ (see text for details). All other parameters are the same as in Fig. 8.

Figure 11

Plot of $ln\left(D\right)$ as a function of rate ${\omega }_2$ for different system sizes and fixed ${\omega }_1/(U-E_i)=0.001$ showing a decrease of $D$ with ${\omega }_2$. The theoretically predicted slope obtained from the scaling theory is also shown for comparison. The figure also demonstrates the role of finite size effects; the $N=18$ data follow the power law till larger values of ${\omega }_2$. Inset: Plot of $ln\left(D\right)$ as a function of ${\omega }_2$ for fixed $N=18$ and ${\omega }_1/(U-E_i)=0.0005$ (red solid line) and $0.001$ (blue dotted line). All other parameters are the same as in Fig. 8.