Observation of phase memory and dynamical phase transitions in spinor gases

TL;DR

The paper addresses nonequilibrium dynamics and dynamical phase transitions in $F=1$ spinor Bose-Einstein condensates, showing that spinor phases provide an order parameter complementary to spin populations. They use a single-mode approximation with Hamiltonian $H/h = \frac{c_2}{2N}\mathbf{S}^2 + p_B \hat S_z + q\sum_i(\hat s_{i,z})^2 + p(t)\hat S_y$ to model dynamics and demonstrate that the relative phase $\theta$ can be extracted from $\rho_0(t)$; they define an order parameter $\beta = 2 - A_{\mathrm{pp}}$, where $A_{\mathrm{pp}} = \max[\cos(\theta/2)] - \min[\cos(\theta/2)]$, which sharply distinguishes the interaction-dominated and Zeeman-dominated dynamical regimes. They show that one can infer the spin-dependent interaction $c_2$ from a single time trace and examine phase memory phenomena in driven lattices, including non-ergodic relaxation where long-time values of $\rho_0$ depend on the initial phase $\eta$ among nonzero-spin components, in contrast to ETH predictions. The findings advance quantum simulation, sensing, and state preparation by enabling phase-resolved control of nonequilibrium spinor dynamics and revealing new routes to dynamical phase diagrams and non-thermal memory effects.

Abstract

Utilizing ultracold spinor gases as large-scale, many-body quantum simulation platforms, we establish a toolbox for the precise control, characterization, and detection of nonequilibrium dynamics via internal spinor phases. We develop a method to extract the phase evolution from the observed spin population dynamics, allowing us to define an order parameter that sharply identifies dynamical phase transitions over a wide range of conditions. This work also demonstrates a technique for inferring spin-dependent interactions from a single experimental time trace, in contrast to the standard approach that requires mapping a cross section of the phase diagram, with immediate applications to systems experiencing complex time-dependent interactions. Additionally, we demonstrate experimental access to and control over non-ergodic relaxation dynamics, where states in the (nominally) thermal region of the energy spectrum retain memory of the initial state, via the manipulation of spinor phases, enabling the study of non-ergodic thermalization dynamics connected to quantum scarring.

Figures (3)

• Figure 1: (a) Equal energy contours, derived from Eq. \ref{['FS_Ham']} for the S-state, demonstrate the phase diagram consisting of the interaction regime (where $|q/c_2|<1$) and the Zeeman regime (where $|q/c_2|>1$). The red (blue) solid contour marks $q/c_2=0.60$ ($q/c_2=1.33$). (b) Triangles (circles) display $\rho_0$ dynamics observed after a quench in $q$ from $41~\mathrm{Hz}$ to $31~\mathrm{Hz}$ ($69~\mathrm{Hz}$) at time $t=0$ from the S-state in free space. (c) $c_2$ (triangles) extracted from the $q=31~\mathrm{Hz}$ spin dynamics shown in panel (b); consistent with values (dotted line) inferred from the observed separatrix shown in panel (e) and results (circles) derived from the observed atom number and trapping frequencies. (d) $\theta$ extracted from the $\rho_0$ time traces shown in panel (b). (e) Squares (diamonds) display the order parameter $\beta$ ($T$) mapping the experimental phase diagram after a sudden quench in $q$ from $41~\mathrm{Hz}$ to a given value for the initial S-state in free space. Purple data points are adapted from Ref. Lichao2014. In panels (b)-(e) solid (dashed) lines are Eq. \ref{['FS_Ham']} predictions (eye-guiding fits).
• Figure 2: (a) Markers display $\Delta\beta$, the change in $\beta$ after a sudden quench in $q$, for the data shown in Fig. \ref{['FreeSpace']}(e), versus $q/c_2$. No DPT occurs when $\Delta\beta\sim 0$. (b) Triangles (circles) display $\Delta\beta$ after the application of a 1D lattice whose depth is driven sinusoidally at a frequency $f=400~\mathrm{Hz}$ between 0 and $5E_R$ starting from the initial state of $\rho_0(0)\approx0.5$, $M=0$, and $\theta(0)=0$ ($\theta(0)=\pi$). Solid lines are SMA predictions.
• Figure 3: (a) Expectation values of $\rho_0$ in eigenstates versus energy. Pink vertical lines show the energy range of U-states, a group of spin-coherent initial states with $\rho_0(0)=0.6$, $M(0)=0$, and $\theta(0)=\pi$ with varying $\eta(0)$. (b) Time-traces of $\rho_0$ for U-states with $\eta(0) = 0$, $\pi/2$, $\pi$. The shaded region marks the range of ETH expectation values at the corresponding energy region (see the pink shaded region in panel (a)). (c) Deviation of long-time relaxed values at $t\approx1~\mathrm{s}$ from ETH predictions, $\rho_{0,\mathrm{av}}-\rho_{0,\mathrm{th}}$, for spin-coherent states with $\rho_0(0)=0.6$ and $M(0)=0$ versus $\theta(0)$ and $\eta(0)$. The white dashed line marks U-states. (d) Triangles display the equilibrated $\rho_0$, observed at a long holding time of $t=300~\mathrm{ms}\sim 10/c_2$, as a strong function of the initial phase $\eta(0)$ for U-states at $q=40~\mathrm{Hz}$ and $p\approx 8~\mathrm{Hz}$SM. The solid (dashed) line is the Eq. \ref{['SMA_Ham']} (ETH) prediction. All theory calculations in this figure were performed for $q=40~\mathrm{Hz}$ and $p\approx 8~\mathrm{Hz}$.