Macroscopic Self-Trapping and Dynamical Phase Transition in Momentum Space Bose-Einstein Condensates

TL;DR

The paper demonstrates macroscopic quantum self-trapping ( MQST ) and a dynamical phase transition ( DPT ) in a momentum-space Bose-Einstein condensate realized with Raman-induced spin-orbit coupling and a matching optical lattice. It derives a two-mode Josephson-like model with Hamiltonian $H_\text{eff}= -\tfrac{1}{2}gn\chi^2 s_z^2 - \hbar\Omega_L\chi\sqrt{1-s_z^2}\cos\theta - \hbar\delta s_z$ and validates it against Gross-Pitaevskii simulations, showing how the lattice strength $\hbar\Omega_L$ tunes between self-trapped and delocalized regimes. MQST is observed via linear ramps of detuning $\delta(t)$, while a quench in $\delta$ reveals a DPT characterized by a non-analytic, time-averaged spin polarization $\bar{S}_z$ and a diverging oscillation period near a critical lattice strength. Finite-size effects and the off-resonant $|1,1\rangle$ state shift the phase boundary, and supplementary three-state analyses provide a more complete picture of the dynamics. Together, these results establish momentum-space BECs with SOC + ML as a versatile platform for exploring non-equilibrium critical phenomena and precision control in nonlinear quantum systems.

Abstract

Self-trapping is a hallmark phenomenon of nonlinear dynamics. It has significant applications in modern physics, including band structure engineering, phase transition dynamics, quantum metrology, and more. Dilute-gas Bose-Einstein condensates (BECs), in which self-trapping can arise from interatomic interactions, are a prime testbed for probing nonlinear dynamics. In this Letter, we report the observation of self-trapping in a spin-orbit coupled BEC subjected to a stationary optical lattice. We employ Raman-induced spin-orbit coupling, complemented by a matching optical lattice that facilitates coupling between momentum eigenstates of the spin-orbit coupled system. By ramping the Raman detuning, we probe atomic current flow between these eigenstates and identify a clear distinction between a delocalized mixed state and a self-trapped regime. Following a quench of the Raman detuning, the time-averaged atomic current exhibits non-analytic behavior across the transition between these two regimes in certain parameter ranges, signaling a dynamical phase transition in the system.

Figures (8)

• Figure 1: Experimental setup for the SOC + ML system. (a) Diagram of laser configuration. BEC (blue oval) is held by a cross-dipole trap (red line and x). Raman and optical lattice beams are shown by the green and yellow beams intersecting with the BEC. (b) Schematic state diagram for Raman coupling between different $m_{F}$ states within the F=1 hyperfine manifold of the $^{87}$Rb $5^{2}S_{1/2}$ ground state. (c) Example image of SOC + ML BEC taken using time of flight and Stern-Gerlach imaging. (d) The lowest band in the single-particle dispersion relation of the SOC system for $\hbar \Omega_{R}=\qty{2.7}{E_{R}}$ and $\delta=2\pi\times\qty{0}{Hz}$. Coupling between the two band minima is enhanced by a weak optical lattice with Rabi coupling $\hbar\Omega_L$.
• Figure 2: Self-trapping dynamics. (a)-(c) Self-trapping spin dynamics on the ground state of the system following an adiabatic Raman detuning ramp. Panels (a), (b), and (c) correspond to $\hbar\Omega_{L}=$ 1.0, 0.3, and $\qty{0.2}{E_{R}}$ respectively. The blue dots (solid curves) show spin polarization of experimentally (numerically) acquired results at $\delta$ given a linear Raman detuning ramp from $\delta_i=2\pi\times\qty{5}{kHz}\rightarrow\delta_f=-2\pi\times\qty{1.5}{kHz}$ in $t_r = \qty{50}{ms}$. The red diamonds (dashed lines) signify experimental (numerical) results given a ramp from $\delta_i=-2\pi\times\qty{5}{kHz}\rightarrow\delta_f=2\pi\times\qty{1.5}{kHz}$. The blue solid (red dashed) arrow in panel (a) indicates the direction of the detuning ramp in the negative (positive) $\delta$ direction for all panels (a)-(c). (d) Ground state phase diagram of the SOC + ML system. Orange dashed lines show cuts across the spin-polarization phase transition at the three values of $\hbar\Omega_{L}$ seen in (a)-(c). (e) Same experimental data as in panel (c), for $t_r=\qty{50}{ms}$ and $\hbar\Omega_{L}=\qty{0.2}{E_{R}}$. Numerical solutions for different ramp times at $\hbar\Omega_{L}=\qty{0.2}{E_{R}}$ for $t_{r}$ ranging from $\qty{10}{ms}$ to $\qty{100}{ms}$ showing a convergence of dynamics at longer ramp times. Blue dots (red diamonds) are experimental data while solid curves (dashed curves) are GP simulations for negative (positive) detuning ramps. The black dotted lines in (a)-(c) and (e) represent solutions to the two-mode model.
• Figure 3: Dynamical phase transition. (a)-(c) Individual spin oscillations over a substantial evolution time with $\hbar\Omega_{L} = 0.1$, $0.3$, and $\qty{0.6}{E_{R}}$ for (a), (b), and (c) respectively. Light blue square, triangle, and diamond dots represent experimentally collected spin-polarization averaged over three images with the error bar representing the standard deviation between them. If unseen, the standard deviation is less than the height of the point. The green curves represent the GP derived numerical time trace curves. The colored dashed lines are the total spin-polarization average of all experimental points for separate values of $\hbar\Omega_{L}$. Inset (d) Analytical phase diagram of order parameter $\bar{S}_z$ averaged over $\qty{10}{ms}$ for different values of $\hbar\Omega_{L}$ and final quench Raman detuning $\delta$. Experimental data was taken along the red dashed line which represents a cut of the phase diagram at $\delta=2\pi\times\qty{280}{Hz}$. The dashed white line indicates the theoretical phase boundary in the homogeneous limit obtained from the effective potential model sup_mat. (e) Cut of $\bar{S}_{z}$ along red dashed line in (d). Each point represents an experimental time average of the spin-polarization for different $\hbar\Omega_{L}$. The time domain over which we average is such that it is sufficient to resolve several oscillations between spin states sup_mat. The green curve represents GP numerics while the red shaded region denotes the analytic solution to the system bounded by a detuning range of $\qty{260}{Hz} \leq \delta/(2\pi) \leq \qty{300}{Hz}$. The green square, orange triangle, and purple diamond correspond to experimentally determined spin-polarization time averages of (a), (b), and (c) respectively with their error bars representing standard error between each of the three run's time-averaged spin-polarization.
• Figure 4: Dynamical phase transition beyond the two-state model. (a) Dynamical phase diagram from GP simulations showing time-averaged polarization for $25$ ms runs, with the two-mode separatrix (dashed line) overlaid. Unlike the 10 ms case, the delocalized phase exhibits uniformly negative polarization. (b)-(c) Polarization density $j(z) = \int(|\psi_\uparrow|^2-|\psi_\downarrow|^2)\text{d}x\text{d}y/N$ in the delocalized phase (b) and near the boundary (c). Spin domains form after $\sim\qty{15}{ms}$, signaling loss of integrability. (d)-(e) Normalized transverse density for the spin-up component $\eta_\uparrow(x)=\int |\psi_\uparrow|^2 \text{d}y\text{d}z/N_\uparrow$ at the same points. No transverse motion is seen in the delocalized phase, while near the boundary a resonance with the transverse mode induces oscillations, consistent with the new phase boundary in (a).
• Figure S1: Effective potential across the dynamical phase transition. (a) The dotted line is the time-averaged polarization $\bar{s}_z$ obtained from the model simulation at $\delta = 2\pi\times\qty{0}{Hz}$ as a function of $\hbar\Omega_L$ for a time interval $T=\qty{40}{ms}$. A dynamical phase transition is observed, with the critical point located around $\hbar\Omega_L \sim \qty{0.20}{E_R}$. The critical point in the experiments and GP simulations is higher ($\sim0.3E_R$) due to the detuning shift from the $|1,\ 1\rangle$ state. The diamond, square, and triangle correspond to the the particular values of $\hbar\Omega_L$ shown in panels (b–d). (b–d) Effective potential $W(s_z)$ for (b) $\hbar\Omega_L = \qty{0.16}{E_R}$, (c) $\hbar\Omega_L = \qty{0.19}{E_R}$, and (d) $\hbar\Omega_L = \qty{0.22}{E_R}$.