Floquet-induced superfluidity with periodically modulated interactions of two-species hardcore bosons in a one-dimensional optical lattice

We consider two species of hard-core bosons with density-dependent hopping in a one-dimensional optical lattice, for which we propose experimental realizations using time-periodic driving. The quantum phase diagram for half-integer filling is determined by combining different advanced numerical simulations with analytic calculations. We find that a reduction of the density-dependent hopping induces a Mott-insulator to superfluid transition. For negative hopping, a gauge-dressed superfluid state is found where one species induces a gauge phase of the other species, which leads to a superfluid phase of gauge-paired particles. The corresponding experimental signatures are discussed.

Figure 1

(a) The realization of time-periodically modulated interspecies interaction by imposing a periodically modulated magnetic field (see inset) near the Feshbach resonance. (b) Schematic picture of the standard simulated Raman transition. An atom jumps from the hyperfine state $a$ to the intermediate state $m$ by absorbing a photon from a pump laser (red line, frequency ${\omega }_P$, fast rotating linearly polarized, coupling strength ${\mathrm{\Omega }}_P$). Similarly, an atom jumps from $m$ to $b$ by emitting a photon from a Stokes laser (blue line, frequency ${\omega }_S$, linearly polarized, coupling strength ${\mathrm{\Omega }}_S$) [49]. One realization of the pump laser is the output of a circularly polarized laser passing a $1/4$ wave plate which is connected to a fast rotating mechanical motor (frequency $\omega$ sets several kHz).

Figure 2

(a) Hopping processes of one species $a$ (red filled circle) in the effective model in Eq. (21). The other species $b$ is denoted by blue filled circles. Hopping between two neighboring single occupied or double/empty sites is suppressed by ${\mathcal{J}}_0\left[K\right]$. (b) Quantum phase diagram of the effective model in Eq. (21) at half filling.

Figure 3

Different observables at $J/\overline{U}=0.4$. (a) Fidelity susceptibility ${\chi }_F$ and entanglement entropy $\mathcal{S}$ from iDMRG ($L=\infty$); charge gap ${\mathrm{\Delta }}_c$ and superfluid density ${\rho }_s$ from DMRG ($L=100$); compressibility $\kappa$ from QMC ($L=100$). (b) Single-particle correlation $G_c\left(r\right)=\langle {\widehat{a}}_0^{†}{\widehat{a}}_r\rangle$ ($○$) and density-hole-pair correlation $G_s\left(r\right)=\langle {\widehat{a}}_0^{†}{\widehat{b}}_0{\widehat{a}}_r{\widehat{b}}_r^{†}\rangle$ ($\square$) as a function of distance $r$ relative to $L/4$ for ${\mathcal{J}}_0\left[K\right]=0.4$ (solid line) and 1 (dashed line) obtained by DMRG ($L=100$).

Figure 4

Superfluid density ${\rho }_s$ per site calculated by DMRG with $L=100$ at $J=\overline{U}$. Inset: Momentum distribution $n_k^b$ at ${\mathcal{J}}_0\left[K\right]=\pm 0.35$, which is normalized by its maximal value.

Figure 5

Finite-size scaling of the (a) charge gap and (b) compressibility for the case of $J=1$ and $\overline{U}/J=0.4$. In DMRG, we choose the maximal truncation dimension $m=4096$ for various system size $L=20$ (black $○$), 40 (red $\square$), 80 (blue $\diamond$), 100 (magenta $\mathrm{\Delta }$), 160 (cyan $⬠$), 320 (orange $⎔$), and 640 (green $\nabla$) with open boundary condition. In QMC, the inverse temperature $\beta /\overline{U}=2L$ with $L=50$ (black $○$), 100 (red $\square$), 150 (blue $\diamond$), 200 (magenta $\mathrm{\Delta }$), 250 (cyan $⬠$), 300 (orange $⎔$), and 400 (green $\nabla$).

Figure 6

Determinant on BKT-type transition points from the Mott-insulator to superfluid phase. (1a),(2a) The peaks of fidelity susceptibility measured from iDMRG indicate the transition points (1a) ${\mathcal{J}}_0{\left[K\right]}_c=0.285\left(3\right)$ for $J/\overline{U}=0.28$, and (2a) ${\mathcal{J}}_0{\left[K\right]}_c=0.624\left(6\right)$ for $J/\overline{U}=0.4$. (1b),(2b) We adopt the level-spectroscopic technique to achieve finite-size scaling. In the first step, we calculate excitation gaps ${\mathrm{\Delta }}_c^{+}=E_p-E_0+\overline{U}/2$ (red hexagon) and ${\mathrm{\Delta }}_s^0=E_1-E_0$ (black pentagon), where $E_0$ and $E_1$ are the ground state and the first-excited state in the Hilbert space $N^a=N^b=L/2$, respectively, and $E_p$ is the lowest energy of the system where we add in one more particle relative to the ground state. Obviously, we find a level crossing between them called quasicritical points such as (1b) ${\mathcal{J}}_0{\left[K\right]}_{qc}=0.415$ for $L=16$ and $J/\overline{U}=0.28$, and (2b) ${\mathcal{J}}_0{\left[K\right]}_{qc}=0.924$ for $L=24$ and $J/\overline{U}=0.4$. In the second step, we plot the quasicritical points as a function of $1/L$ in the inset and find that they can be linearly extrapolated to the thermodynamical limit very well. We get the best extrapolation value ${\mathcal{J}}_0{\left[K\right]}_c=0.287$ for $J/\overline{U}=0.28$ and ${\mathcal{J}}_0{\left[K\right]}_c=0.630$ for $J/\overline{U}=0.4$, which remains consistent with the results from the iDMRG calculations in (1a) and (2a).

Figure 7

Quasienergy spectrum of the Floquet Hamiltonian (E1) as a function of $K_U, K_{\mathrm{\Omega }}$, and $\overline{U}$, respectively. Here, $L=6, \omega =20$ and $N^a+N^b=6$. (a),(b) $N^a=N^b=3$. (a) $K_{\mathrm{\Omega }}=0$ and $\overline{U}/J=1$, (b) $K_{\mathrm{\Omega }}=0$ and $K_U=1$, (c) $K_U=0$ and $\overline{U}/J=1$, and (d) $K_U=0$ and $K_{\mathrm{\Omega }}=1$.

Figure 8

Real-time dynamics for the model (7). Here we choose $L=6, \hslash \omega /J=20, K_{\mathrm{\Omega }}=0$, and $N^a=N^b=3$. Initially, we prepare the system in the ground state of the nondriven model with ${\overline{U}}^i=1$. $K_U$ linearly grows until $t_1$ and persists as a working Rabi oscillation. At the moment $t_1$, we start to linearly increase $\overline{U}$ to a desired value ${\overline{U}}^f$ by $t_2$ and persist until the measurement. We consider two cases: (1)–(4) $t_1=100T$ and $t_2=200T$ for the fast switching-on, (5)–(8) $t_1=300T$ and $t_2=600T$ for the slow switching-on. For each case, we show the time-evolving behavior related to four different sets of parameters: (1) and (5) for $K_U^f=1$ and ${\overline{U}}^f=2$, (2) and (6) for $K_U^f=4$ and ${\overline{U}}^f=2$, (3) and (7) for $K_U^f=1$ and ${\overline{U}}^f=6$, and (4) and (8) for $K_U^f=4$ and ${\overline{U}}^f=6$. For each scheme, we show (a) modulated $K_U$ (black lines) and $\overline{U}$ (red lines) as a function of time $t/T$, (b) time-evolving overlap $P$ and energy $e_0$, and (c),(d) structure factors of ${\tilde{n}}_k^b$ and ${\tilde{n}}_k^{\beta }$ at the moment $t=0$ (blue circles), $t=t_1$ (magenta squares), and $t=t_2$ (green diamonds), respectively.

Figure 9

Real-time dynamics for the model (11). Here we choose $L=6, \hslash \omega /J=20, K_U=0$, and integer-1 filling $N^a+N^b=6$. Initially, we prepare the system in the ground state of the nondriven model with ${\overline{U}}^i=1$. $K_{\mathrm{\Omega }}$ linearly grows until $t_1$ and persists as a working Rabi oscillation. At the moment $t_1$, we start to linearly increase $\overline{U}$ to a desired value ${\overline{U}}^f$ by $t_2$ and persist until the measurement. We consider two cases: (1)–(4) $t_1=100T$ and $t_2=200T$ for the fast switching-on, (5)–(8) $t_1=300T$ and $t_2=600T$ for the slow switching-on. For each case, we show the time-evolving behavior related to four different sets of parameters: (1) and (5) for $K_{\mathrm{\Omega }}^f=0.5$ and ${\overline{U}}^f=2$, (2) and (6) for $K_{\mathrm{\Omega }}^f=2$ and ${\overline{U}}^f=2$, (3) and (7) for $K_{\mathrm{\Omega }}^f=0.5$ and ${\overline{U}}^f=6$, and (4) and (8) for $K_{\mathrm{\Omega }}^f=2$ and ${\overline{U}}^f=6$. For each scheme, we show (a) modulated $K_{\mathrm{\Omega }}$ (black lines) and $\overline{U}$ (red lines) as a function of time $t/T$, (b) time-evolving overlap $P$ and energy $e_0$, and (c),(d) structure factors of ${\tilde{n}}_k^b$ and ${\tilde{n}}_k^{\beta }$ at the moment $t=0$ (blue circles), $t_1$ (magenta squares), and $t_2$ (green diamonds), respectively.