Nonequilibrium Floquet Steady States of Time-Periodic Driven Luttinger Liquids

Time-periodic driving facilitates a wealth of novel quantum states and quantum engineering. The interplay of Floquet states and strong interactions is particularly intriguing, which we study using time-periodic fields in a one-dimensional quantum gas, modeled by a Luttinger liquid with periodically changing interactions. By developing a time-periodic operator algebra, we are able to solve and analyze the complete set of nonequilibrium steady states in terms of a Floquet-Bogoliubov ansatz and known analytic functions. Complex valued Floquet eigenenergies occur when integer multiples of the driving frequency approximately match twice the dispersion energy, which correspond to resonant states. In experimental systems of Lieb-Liniger bosons we predict a change from power-law correlations to dominant collective density wave excitations at the corresponding wave numbers as the frequency is lowered below a characteristic cutoff.

Figure 1

Coupling constants $g_2=g_4=(1/K^2-1)/2$ for the Lieb-Liniger model as a function of $mg/n$ [63, 64, 65], which can be determined for any value of $a_0$ and $a_{\perp }$.

Figure 2

Top: value of $\nu =2\mathrm{\Delta }/\omega$ as a function of $q{v}_F/\omega$ using $\overline{\rho }=-0.6$ and amplitude $\rho =0.25$. Shaded regions indicate complex values of $\mathrm{\Delta }$. Bottom: stability chart of the Mathieu equation with $\overline{\rho }=-0.6$. Grey areas are the instability regions around the resonance points $q_{\ell }=\ell \omega /2\overline{v}$.

Figure 3

Characteristic exponent $\nu$ for $g_4=-0.4$, while only $g_2(t)$ is driven with $\overline{\rho }=-0.6$ and $\rho =0.25$ in Eq. (4).

Figure 4

Time average of ${\eta }_q(t)$, plotted as a function of $q{v}_F/\omega$ for $\overline{\rho }=-0.6$ using different amplitudes $\rho$. For $q\to 0$ the static limit $|{\overline{\gamma }}_2{|}^2=(1/\overline{K}+\overline{K}-2)/4$ is recovered (red dot). Inset: with finite lifetime ${\tau }_0={10}^4/v_{F}q$ the divergent regions are turned into overwhelmingly large maxima.