Wave Packet Dynamics in Synthetic Non-Abelian Gauge Fields

It is generally admitted that in quantum mechanics, the electromagnetic potentials have physical interpretations otherwise absent in classical physics as illustrated by the Aharonov-Bohm effect. In 1984, Berry interpreted this effect as a geometrical phase factor. The same year, Wilczek and Zee generalized the concept of Berry phases to degenerate levels and showed that a non-Abelian gauge field arises in these systems. In sharp contrast with the Abelian case, spatially uniform non-Abelian gauge fields can induce particle noninertial motion. We explore this intriguing phenomenon with a degenerated Fermionic atomic gas subject to a two-dimensional synthetic SU(2) non-Abelian gauge field. We reveal the spin Hall nature of the noninertial dynamic as well as its anisotropy in amplitude and frequency due to the spin texture of the system. We finally draw the similarities and differences of the observed wave packet dynamic and the celebrated Zitterbewegung effect of the relativistic Dirac equation.

Figure 1

(a) Real-space configuration of the tripod laser beams. The three co-planar beams (red arrows) define the plane $(Ox,Oy)$ orthogonal to the gravity pull. A 67 G bias magnetic field, applied along $Ox$, allows us to isolate the tripod system among the $({}^1{S}_0,F_g=9/2)\to ({}^3{P}_1,F_e=9/2)$ Zeeman manifold of the intercombination line. Counterpropagating beams 1 and 3 along $Ox$ have opposite circular polarizations and address ${\sigma }_{+}$ and ${\sigma }_{-}$ transitions, respectively. The orthogonal beam 2 is linearly polarized along $Ox$ and addresses a $\pi$ transition. (b) The tripod laser beam $a$ drives the transition $|a⟩\leftrightarrow |e⟩$ with a detuning ${\delta }_a$ ($a=1$, 2, 3). The resonant Rabi frequencies are all equal to $\mathrm{\Omega }/2\pi =210\mathrm{kHz}$. (c) The internal dressed-state basis of the system features two degenerate dark states (in blue) uncoupled to the laser beams and two bright states (in red) shifted from the dark states by $\pm \sqrt{3}\hslash \mathrm{\Omega }/2$.

Figure 2

(a) Time-of-flight fluorescence image recorded at $v_0=0$ after the system has been initialized in the dark state $|D_2⟩$. The ballistic time is 9 ms. As expected from Eq. (4), the measured velocity distribution shows three peaks centered at $\mathbf{v}=0$, $\mathbf{v}=-v_r({\widehat{\mathbf{e}}}_x+{\widehat{\mathbf{e}}}_y)$, and $\mathbf{v}=-2v_r{\widehat{\mathbf{e}}}_x$ ($v_r=\hslash k/m$ is the recoil velocity). These peaks correspond to states $|3⟩$, $|2⟩$, and $|1⟩$, respectively. (b) Temporal oscillations of the Cartesian coordinates of the averaged velocity obtained for a boost velocity $(v_0=4\sqrt{2}{v}_r,{\theta }_0=0.6\pi )$. The solid lines are obtained by numerically integrating the time evolution of the system in the gauge field, initialized in $|3⟩$. We include the $10\mu \mathrm{s}$ ramping stage of the tripod beams and finite momentum distribution of our fermionic gas at $T=30(3)\mathrm{nK}=0.21(4)T_F$. The dashed lines correspond to the plane-wave model given by Eq. (7) at $\mathbf{q}=m{\mathbf{v}}_0$. Conveniently the time origin is shifted to match the phase oscillations with experimental signal. This time shift is justified inasmuch as Eq. (7) does not incorporate the effect of the laser ramping stage. Its value of about $5\mu \mathrm{s}$ is essentially half the ramping duration. (c) Time-of-flight fluorescence images at times $t_1=18$, $t_2=30$, $t_3=42$, and $t_4=62\mu \mathrm{s}$. These times are indicated by vertical dotted lines in (b).

Figure 3

Direction and amplitude of the wave packet oscillation and spin texture. (a) The oscillatory velocity direction $\eta$ as function of ${\theta }_0$. The blue points are the experimental data whereas the grey plain curve represents the plane-wave model prediction. The inset shows the direction locking of the oscillation at $\eta ={\theta }_0\pm \pi /2$. (b) Velocity oscillation amplitude as a function of the boost velocity polar angle ${\theta }_0$ for $v_0=4\sqrt{2}{v}_r$ (points). The solid gray line is the plane-wave prediction $|f({\theta }_0)|$ of Eq. (8). (c) Velocity oscillation amplitude as a function of $v_0$ at angles ${\theta }_0=0.6\pi$ (green points) and ${\theta }_0=0.75\pi$ (magenta points). The green and magenta solid lines are the theoretical predictions from the plane-wave model. The data points in (a), (b), and (c) are obtained with a damped-sinusoidal fit function representing the time evolution of the mean velocity of the atoms [see example in Fig. 2]. All error bars represent 1 standard deviation of uncertainty. (d) Spin texture $\mathbf{S}$ of the lower energy branch (arrows), Eq. (10), along a circle in the $(Ox,Oy)$ momentum plane centered at the Dirac point. The spin orientation lies in the $(Ox,Oz)$ plane, and the color code corresponds to the amplitude of the $S_x$ component. The central inset shows the evolution of the angle $\zeta$ of the spin texture in a Bloch sphere representation as a function of ${\theta }_0$. The angle $\zeta$ is defined with respect to the initial spin orientation $⟨D_2|\widehat{\mathit{\sigma }}|D_2⟩$; see example on the top-left inset, where ${\theta }_0=3\pi /2$ and $\zeta =\pi /3$.

Figure 4

Oscillation frequency. (a) Variation of the oscillation frequency as a function of the magnitude $v_0$ of the boost velocity at fixed angles (${\theta }_0=0.60\pi$ for the green data points and ${\theta }_0=0.75\pi$ for the magenta data points) (b) Angular variations of the frequency observed at $v_0=4\sqrt{2}{v}_r$. The solid lines in (b) and (c) are the theoretical predictions inferred from the plane-wave model given by Eq. (9). The error bars are 1 standard deviation of uncertainty.