Quench dynamics of Rydberg-dressed bosons on two-dimensional square lattices

We study the dynamics of bosonic atoms on a two-dimensional square lattice, where atomic interactions are long ranged with either a box or soft-core shape. The latter can be realized through laser dressing ground-state atoms to electronically excited Rydberg states. When the range of interactions is equal or larger than the lattice constant, the system is governed by an extended Bose-Hubbard model. We propose a quench process by varying the atomic hopping linearly across phase boundaries of the Mott insulator-supersolid and supersolid-superfluid phases. Starting from a Mott insulating state, the dynamical evolution of the superfluid order parameter exhibits a universal behavior at the early stage, largely independent of interactions. The dynamical evolution is significantly altered by strong, long-range interactions at later times. Particularly, we demonstrate that density wave excitation is important when the quench rate is small. Moreover, we show that the quench dynamics can be analyzed through time-of-flight images, i.e., measuring the momentum distribution and noise correlations.

Figure 1

Long-range interaction and quench scheme. (a) Long-range soft-core (red dashed) and box (black solid) interaction. The range $r_c$ of interactions can be larger than the optical lattice (dotted) constant $d$. (b) Fourier transform of the interaction potential. The line style is the same as shown in panel (a). (c) Quench protocol. At $t<0$, we prepare the ground state in a Mott insulator with $V_{\mathrm{latt}}\ne 0$ and $J=0$. The tunneling $J\left(t\right)/U=a_{Q}t$ (orange) is increased linearly during $t\in (0,t_f)$. This is done by reducing lattice depth (gray). At the same time, long-range interactions (green dotted) are turned on. When $t>t_f$, atoms are released from the optical lattice to initialize the time-of-flight experiment.

Figure 2

Phase diagram and roton instability. Phase diagram for the box-type interaction with $r_c=d$ (a) and for the soft-core interaction with $r_c=2.5d$ (b) at unit filling. When the long-range interaction is weak, a MI-SF transition is found by increasing $J$. Starting from the DW phase (at stronger $V_0$), the ground state undergoes a DW-SS and then a SS-SF transition when increasing $J$. The dashed line is calculated from the roton instability analysis, which is close to the SS-SF phase boundary. (c) $\mathrm{\Sigma }\left(\mathbf{k}\right)$ as defined in Eq. (5). The momentum ${\mathbf{k}}_{\mathrm{rot}}$ is found at the minimal $\mathrm{\Sigma }\left(\mathbf{k}\right)$ when the spectra become complex. (d) Bogoliubov spectra for $J=0.5U$. The white areas indicate roton instability. In (c) and (d) we consider the soft-core interaction with $r_c=2d$ and $V_0=0.6U$. (e) Critical tunneling $J_l$ for box-type (black solid) and soft-core (red dashed) interactions. The interaction lengths are $r_c=\{d,2d,3d\}$, respectively. (f) Momentum $|{\mathbf{k}}_{\text{rot}}|$ at the roton minimum, in terms of its inverse, for box (square) and soft-core (dot) interactions. Here the interaction strength is $V_0=2U$.

Figure 3

Superfluid fraction ${\rho }_s$ (a),(c) and density variance ${\sigma }_n$ (b),(d). In weak interacting cases (a),(b), we consider $V_0=0$ (dashed black) and $V_0/U=0.1$ (colored). The lower panel shows the difference of ${\rho }_s$ and ${\sigma }_n$ between interacting and noninteracting cases. In strong interacting cases with $V_0/U=0.5$ (c),(d), the evolution apparently depends on the strength and radius of the long-range interaction. In the simulation, $a_Q/U=0.01$ and $r_c/d=\{1$ (blue), 2 (green), 3 (orange)$\}$. Dotted lines indicate probe times for time-of-flight (TOF) interference. Arrows indicate the Kibble-Zurek time. See text for details.

Figure 4

Dynamics of the vortex nucleation. (a) Evolution of superfluid vortex density for box interactions with $V_0/U=0.5$, $r_c=d$, and $J\left(t_f\right)/U=0.5$. The quench rate $a_Q/U$ is ${10}^{-3}\left(\mathrm{black}\right),{10}^{-2}\left(\mathrm{red}\right)$. The local phase of SF order parameter for quench labeled as magenta triangle and blue circle is shown for (b) $J\left(t\right)/U=0.25$ and (c) $J\left(t\right)/U=0.5$.

Figure 5

Kibble-Zurek time scale. Dynamics of global variables for box and soft-core interactions. (a) Kibble-Zurek time, in terms of frozen parameter $\widehat{\epsilon }$, fitted from the turning point of superfluid fraction. The interaction configurations include noninteracting (black), box interaction with $V_0/U=0.6$ and $r_c=d$ (blue), $r_c=2d$ (green), soft-core interaction with $V_0/U=0.6$ and $r_c=d$ (yellow), $r_c=2d$ (orange). (b) Density of vortices counted at Kibble-Zurek time. Density variance ${\sigma }_n$ with respect to hopping $J\left(t\right)$ with box (c) and soft-core (d) interaction for both $r_c=d$ and $V_0/U=0.6$. The hopping parameter at KZM time $J\left(\widehat{t}\right)$ (solid line), roton instability $J_l$ (dashed line), and MI-SF transition of BHM $J_s$ (dotted line) are plotted together. (e) Density distributions of specific quench rate and time for box interactions shown in (c), where ① $a_Q/U=0.004$, $J\left(t\right)/U=1.5$, ② $a_Q/U=0.06$, $J\left(t\right)/U=1.5$, ③ $a_Q/U=0.004$, $J\left(t\right)/U=0.3$, and ④ $a_Q/U=0.06$, $J\left(t\right)/U=0.3$.

Figure 6

TOF snapshots at time $Ut=17.5$. Before the atoms are released from the optical lattice, the interaction between them is a box type. The quench rate $a_Q/U=0.01$. TOF density $n\left(\mathbf{k}\right)$ (b1),(b2) and covariance $C\left(\mathbf{q}\right)$ (d1),(d2) are compared with the Fourier transform of order parameter ${\tilde{\varphi }}_{\mathbf{k}}$ (a1),(a2) and density ${\tilde{n}}_{\mathbf{q}}$ (c1),(c2). Panels (a1)–(d1) show results of the BHM and (a2)–(d2) results for box interaction with $r_c=d$ and $V_0/U=0.66$.

Figure 7

Radial distribution of the momentum density and covariance. By integrating the angular direction, we obtain radial distribution of the momentum density $n\left(k\right)$ (a1)–(a4) and covariance $C\left(q\right)$ (b1)–(b4) at different interaction strengths. The probe time is $Ut=28$ with quench rate $a_Q/U=0.01$. We consider box interactions with radius $r_c/d=\{1,2,3,4\}$ from left to right, respectively. Dotted lines indicate the momentum where the covariance has maximal values. See Fig. 8 for snapshots of momentum distributions.

Figure 8

TOF snapshots at time $Ut=28$. The quench rate is $a_Q/U=0.01$. TOF density $n\left(\mathbf{k}\right)$ (b1)–(b3) and covariance $C\left(\mathbf{q}\right)$ (d1)–(d3) are compared with the Fourier transform of order parameter ${\tilde{\varphi }}_{\mathbf{k}}$ (a1)–(a3) and density ${\tilde{n}}_{\mathbf{q}}$ (c1)–(c3). Box interactions are considered with $r_c/d=\{0$ (noninteracting)$,1,3\}$ and strengths $V_0/U=\{0,0.66,0.36\}$ (top to bottom).

Figure 9

Superfluid fraction (a), density variance (b), and vortex nucleation (c),(d) with soft-core interactions. (a),(b) The soft-core radius is $r_c/d=2.5$ and quench rate $a_Q/U=0.01$. The interaction strengths are $V_0/U=0.1$ (blue), 0.5 (green), and 0.8 (yellow). (c) Superfluid vortex density under different quench rate $a_Q/U={10}^{-2}\left(\mathrm{red}\right),{10}^{-3}\left(\mathrm{black}\right)$. The interaction strength is $V_0/U=0.5$ and radius $r_c/d=2.5$. (d) The phase of local SF order parameter $arg\left(\varphi \right)$ at $J\left(t_f\right)=0.5U$ with $a_Q/U={10}^{-3}$.

Figure 10

Momentum and covariance for atoms with soft-core interactions. Radial distribution (a) and two-dimensional distribution (b) of the momentum density $n\left(\mathbf{k}\right)$. Radial distribution (c) and two-dimensional distribution (d) of the covariance $C\left(q\right)$. The soft-core radius $r_c=3d$, quench rate $a_Q/U=0.01$, probe time $Ut=28$, and interaction strength $V_0/U=0.36$. The dotted vertical line marks the momentum corresponding to the roton minimum.