Engineering dynamical phase diagrams with driven lattices in spinor gases

We experimentally demonstrate that well-designed driven lattices are versatile tools to simultaneously tune multiple key parameters (spin-dependent interactions, spinor phase, and quadratic Zeeman energy) for manipulating phase diagrams of spinor gases with negligible heating and atom losses. This opens avenues for studying engineered Hamiltonians and dynamical phase transitions. Modulation-induced harmonics generate progressively narrower separatrices at driving-frequency-determined higher magnetic-field strengths. This technique enables exploration of multiple, previously inaccessible parameter regimes of spinor dynamics (notably high magnetic-field strengths, tunable spinor phase, and individually tunable spin-preserving and spin-changing collisions) and widens the range of cold-atom applications, e.g., in quantum sensing and studies of nonequilibrium dynamics.

Figure 1

(a) Solid and dotted lines display the timing of $U$-driven and $D$-driven lattice sequences, respectively. (b) TOF images taken via $U$-driven sequences at the minimum (left) and maximum (right) lattice depths, showing atoms occupying different momentum $p$ states. Here $k_L$ is the lattice wave vector. (c)–(e) Triangles display observed spin oscillations near $q^{*}/h\approx f/2=200\mathrm{Hz}$ via $U$-driven sequences when $q/h$ equals (c) $160\mathrm{Hz}$, (d) $195\mathrm{Hz}$, and (e) $235\mathrm{Hz}$. (f)–(h) Similar to (c)–(e) but taken in free space. Solid lines in (c)–(h) are sinusoidal or sigmoidal fits (see Appendix pp4).

Figure 2

(a) Triangles, circles, and diamonds display spin oscillation amplitudes observed via $U$-driven, $D$-driven, and free-space sequences, respectively, near $q/h=f/2=200\mathrm{Hz}$. Solid lines are DSMA predictions and the dashed line is an eye-guiding linear fit. (b) Drastically different spin dynamics observed via $D$-driven (upper graph) and $U$-driven (lower graph) sequences at $q/h=210\mathrm{Hz}$. Solid lines are sinusoidal or sigmoidal fits. (c) Phase diagram of $H_{\mathrm{mf},1}$ [see Eq. (1)] for our system demonstrating that the dynamical critical points (black solid lines) for driven lattice systems are determined by ${\mathcal{G}}_{1}cos\left[{\theta }_{\mathrm{eff},1}\left(0\right)\right]/{\mathcal{G}}_0$. The black and red dashed lines mark ${\mathcal{G}}_{1}cos\left[{\theta }_{\mathrm{eff},1}\left(0\right)\right]/{\mathcal{G}}_0=0.2$ and ${\mathcal{G}}_{1}cos\left[{\theta }_{\mathrm{eff},1}\left(0\right)\right]/{\mathcal{G}}_0=-0.2$ for the $D$-driven and $U$-driven data, respectively, studied in this paper.

Figure 3

(a) Triangles, circles, and diamonds display normalized spin oscillation periods observed via $U$-driven, $D$-driven, and free-space sequences, respectively. Some free-space data are adapted from our prior work [18]. Black and red solid lines and purple dashed lines are DSMA and SSMA predictions for our initial state, respectively, and the black dashed line is an eye-guiding piecewise fitting. The inset shows the observed dynamics via $D$-driven sequences at $q/h=160\mathrm{Hz}$. The solid line is a two-sine fitting with an overall slow oscillation of $T_{\mathrm{slow}}=14.5\left(5\right)\mathrm{ms}\approx h/2\left|q_{\mathrm{eff},1}\right|$ determined by $q_{\mathrm{eff},1}=q-hf/2$ plus a fast oscillation of $T_{\mathrm{fast}}=3.2\left(1\right)\mathrm{ms}\approx h/2\left|q_{\mathrm{eff},0}\right|$ determined by $q_{\mathrm{eff},0}=q$ (see the text for details). The eye-guiding dashed lines display the slow oscillation from the fitting at the extrema. (b) Similar to (a) but plotted against $q_{\mathrm{eff},1}/h=q/h-f/2$. (c) Similar to (b) but for the center of oscillations $\langle {\rho }_0\rangle$.

Figure 4

Time traces taken when $q/h$ equals (a) $410\mathrm{Hz}$ and (b) $398\mathrm{Hz}$, demonstrating a critical region at a higher harmonic with $U$-driven sequences. Solid lines are sinusoidal or sigmoidal fits. Blue (red) triangles display the (c) oscillation period and (d) amplitude observed via $U$-driven sequences near $q^{*}/h\approx 400\mathrm{Hz}$ ($200\mathrm{Hz}$), corresponding to $q_{\mathrm{eff},2}=0$ ($q_{\mathrm{eff},1}=0$). The top (bottom) axis is applied to both data sets (blue triangles). Red dashed and blue solid lines are DSMA predictions and eye-guiding Lorentzian fits, respectively (see Appendix pp4).

Figure 5

Analytic predictions for the (a) period, (b) amplitude, and (c) center of spin-mixing oscillations. We use parameters and initial conditions matching the results of Figs. 2 and 3: ${\mathcal{G}}_0/h=30\mathrm{Hz}$, ${\mathcal{G}}_1/h=5\mathrm{Hz}$, and $({\rho }_0\left(0\right),{\theta }_{\mathrm{eff},1}\left(0\right))=(1/2,{\varphi }_1)$ with phases ${\varphi }_1=0$ (black solid lines) and $\pi$ (red dotted lines).