Nonexponential Tunneling due to Mean-Field-Induced Swallowtails

Typically, energy levels change without bifurcating in response to a change of a control parameter. Bifurcations can lead to loops or swallowtails in the energy spectrum. The simplest quantum Hamiltonian that supports swallowtails is a nonlinear $2\times 2$ Hamiltonian with nonzero off-diagonal elements and diagonal elements that depend on the population difference of the two states. This work implements such a Hamiltonian experimentally using ultracold atoms in a moving one-dimensional optical lattice. Self-trapping and nonexponential tunneling probabilities, a hallmark signature of band structures that support swallowtails, are observed. The good agreement between theory and experiment validates the optical lattice system as a powerful platform to study, e.g., Josephson junction physics and superfluidity in ring-shaped geometries.

Figure 1

Scaled energy levels of ${\widehat{H}}_{\mathrm{TS}}$ for $C/(\alpha \tau )=0.268$ as a function of $t/\tau$ for (a) $C/\mathrm{\Omega }=2.14$, (b) 1.07, (c) 0.428, and (d) 0.306. Black and red solid lines show the adiabatic energy levels [in (c) and (d), the black lines are covered by the symbols]. Blue circles and green squares show the dynamic energy levels for $\hslash /(\alpha {\tau }^2)=7.68\times {10}^{-2}$ and $\hslash /(\alpha {\tau }^2)=7.68\times {10}^{-3}$, respectively [in (d), the blue circles are covered by the green squares]. The tunneling probability is appreciable in (a) and (b) and essentially zero in (c) and (d). The inset in (a) shows an enlargement of the swallowtail; ${\gamma }_{c,1}/(\alpha \tau )$ and ${\gamma }_{c,2}/(\alpha \tau )$ correspond to the boundaries of the swallowtail. Purple dotted lines in (c) and (d) show the adiabatic energy levels, labeled by their eigenstates, for $\mathrm{\Omega }=0$. The parameters used to make the solid lines and blue circles are the same as those used in Fig. 2.

Figure 2

Experiment-theory comparison of population imbalance $\mathrm{\Delta }b(t)$ for a linear ramp with $\alpha =h\times 9\mathrm{kHz}/\mathrm{ms}$ [$\hslash /(\alpha {\tau }^2)=7.68\times {10}^{-2}$] as a function of $t/\tau$ for various $\tilde{C}(0)/\mathrm{\Omega }$. The experimental results (red error bars show the standard deviation from two independent runs) are for $N=3.1\times {10}^5$, ${\omega }_{x,y,z}=2\pi \times (147,160,29.8)\mathrm{Hz}$, and $E_L=h\times 1.08\mathrm{kHz}$. Blue squares and green dashed lines are obtained from ${\widehat{H}}_{\mathrm{GP}}$ and ${\widehat{H}}_{\mathrm{TS},t}$, respectively (analyzing the dynamic states). In both cases, the initial state is prepared in an axially symmetric trap with ${\omega }_{\rho }=({\omega }_x+{\omega }_y)/2$. The mean-field energy $\tilde{C}(t)$ is equal to 1.27 $E_L$ and 1.07 $E_L$ for $t=-\tau$ and $t=0$, respectively. Black solid lines show $\mathrm{\Delta }b(t)$ for the lower adiabatic eigenstate of ${\widehat{H}}_{\mathrm{TS},t}$.

Figure 3

Experiment-theory comparison of tunneling probability $|b_0(t){|}^2$ at $t=\tau$ for linear ramps with varying $\alpha$ for $\tilde{C}(-\tau )/\mathrm{\Omega }=2.75$ as a function of $\pi {\mathrm{\Omega }}^2/(2\hslash \alpha )$. The experimental results (red circles; error bars show the standard deviation from three independent runs) are for $N=2.3\times {10}^5$, ${\omega }_{x,y,z}=2\pi \times (193,218,29.8)\mathrm{Hz}$, $E_L=h\times 1.08\mathrm{kHz}$, and $\mathrm{\Omega }=0.52$ $E_L$. Blue squares and green dashed lines are obtained for ${\widehat{H}}_{\mathrm{GP}}$ and ${\widehat{H}}_{\mathrm{TS},t}$, respectively [analyzing the dynamic states; the initial state is prepared in an axially symmetric trap with ${\omega }_{\rho }=({\omega }_x+{\omega }_y)/2$]. Both datasets follow the nonexponential trend of the gray-shaded region, which shows Eq. (S11) for $C/\mathrm{\Omega }$ values ranging from $\tilde{C}(-\tau )/\mathrm{\Omega }$ to $\tilde{C}(0)/\mathrm{\Omega }$ [34]. The black solid line, which oscillates around the linear Landau-Zener formula [gray dash-dotted line; Eq. (1)], shows the tunneling probability for ${\widehat{H}}_{\mathrm{TS}}$ with $C=0$. The experimental data are better described by the nonexponential gray-shaded family of curves than by the linear Landau-Zener formula.

Figure 4

Theoretical GP and experimental densities for the ramp ending at $t=-\tau +0.6\mathrm{ms}$ [$t/\tau =0.25$ in Fig. 2]. (a) and (b) Density cuts before time-of-flight expansion for $\rho =0$ and $y=z=0$, respectively. Blue solid and black dashed lines are for states ${\psi }_0(\overrightarrow{r},t)$ and ${\psi }_2(\overrightarrow{r},t)$, respectively. Panels (c) and (d) show, respectively, the theoretical and experimental integrated densities $n(x,z,t)$, $n(x,z,t)={\int }_{-\infty }^{\infty }|\mathrm{\Psi }(\overrightarrow{r},t){|}^{2}dy$, after 16.5 ms time-of-flight expansion.