Imaginary gauge potentials in a non-Hermitian spin-orbit coupled quantum gas

Abstract

In 1996, Hatano and Nelson proposed a non-Hermitian lattice model containing an imaginary Peierls phase [Phys. Rev. Lett. 77 570-573 (1996)], which subsequent analyses revealed to be an instance of a new class of topological systems. Here, we experimentally realize a continuum analog to this model containing an imaginary gauge potential using a homogeneous spin-orbit coupled Bose-Einstein condensate (BEC). Non-Hermiticity is introduced by adding tunable spin-dependent loss via microwave coupling to a subspace with spontaneous emission. We demonstrate that the resulting Heisenberg equations of motion for position and momentum depend explicitly on the system's phase-space distribution. First, we observe collective nonreciprocal transport in real space, with a "self-acceleration" that decreases with the BEC's spatial extent, consistent with non-Hermitian Gross-Pitaevskii simulations. We then examine localized edge states: the relatively strong interactions in our BEC suppress the formation of topological edge states, yielding instead highly excited states localized by an interplay between self-acceleration and wavefunction spreading. Finally, we confirm that our non-Hermitian description remains valid at all times by comparing to a multi-level master-equation treatment.

Figures (5)

• Figure 1: Concept. a Level diagram for creating non-Hermitian SOC in a system subspace linked to a reservoir subspace with a microwave field of strength $\Omega_{\mu}$. Dissipation is induced when the excited state $\ket{e}$ decays with rate $\Gamma$. b Geometry. A homogeneous BEC was confined in a box-shaped trap in the $\mathbf{e}_x$-$\mathbf{e}_y$ plane, and Raman-dressed by a pair of laser beams projected through a high-resolution microscope objective. Dissipation is induced by a resonant laser. c Imaginary gauge potential. Left: Spin up population and imaginary gauge field as a function of momentum (solid) along with their linear approximations (dashed). Right: corresponding momentum dependent loss. Both computed for $\Omega / \omega_0 = 5.7$. Bottom: Gaussian momentum distribution before (black) and accelerated after (red) evolving with $\gamma = 0.19 \omega_0$ for $10\ {\rm ms}$. The solid red curve plots the final distribution including the overall loss term in Eq. \ref{['eq:imaginary_gauge_potential']}, while the dashed curve includes only the imaginary gauge potential.
• Figure 2: Nonreciprocal transport. a Time-dependence of 1D atomic density at $\mathcal{B}/p_0=0.16$. Top: density profiles at $t=1$ and $t=9~\mathrm{ms}$ (red) along with the $t=0$ profile (black-dashed) and their difference $\Delta n$ (purple). Bottom: $\Delta n$ as function of evolution time along with the measured CoM position (markers). In b, c and d, markers are experimental data and solid curves are GPE simulations. b Center of mass time evolution for a range of $\mathcal{B}/p_0$. Each point reflects the average of about 10 measurements and the uncertainties are the standard error of the mean. c Initial acceleration as a function of $\mathcal{B}/p_0$. Markers result from fits to the experimental data in a; solid curves are simulated acceleration with (red) and without (blue, magnified by 20$\times$) atomic interactions. Dashed curves plot the corresponding EoM-predicted acceleration averaged for the first 6 ms. d Reversed gauge field. Single spin transport $\langle x \rangle_{\downarrow}$ at $|\mathcal{B}/p_0|=0.16$.
• Figure 3: Dynamical edge modes.a CoM dynamics of an initially canted state for a range of ${\mathcal{B}}$ (markers) along with quadratic fits (dashed curves), with $\mathcal{B}/p_0$ = 0.19, 0.11, 0.05, 0.01 (top to bottom). b Optimal ${\mathcal{B}}/p_0$ from data as in a versus CoM position $x_0$ (markers, single error bar is representative of all data). Curves show calculations with (red) and without (blue) interactions.
• Figure 4: Transition from coherent to quantum Zeno regime. a Data with $\Omega_{\rm RF} = 0$ showing fractional population in $\ket{\uparrow}$ versus time for four values of $\Omega_{\rm L}$, all with $\Omega_\mu = 2\pi\times 3.08(4)\ \rm {kHz}$. Solid curves are fits as described in the text. b Lifetime and oscillation frequency extracted from the fits as in a, plotted as a function of $\Omega_{\rm L}/\Omega_{\mu}$. Solid curves result from time-domain master equation simulations (no free parameters), with $\omega_{\rm eff}$ and $\gamma$ obtained from fits as in a. c Fractional population in $\ket{\uparrow}$ versus time, now with rf coupling $\Omega_{\rm RF} = 2\pi\times 1.20(5) \rm\ kHz$. Solid curves show the predictions of the non-Hermitian Hamiltonian (no free parameters).
• Figure 5: Example data for $\mathcal{B}/p_0=0.11$ and $t=4~\mathrm{ms}$. a, b Optical depth of atoms initially in $\ket{\uparrow}$ measured via partial transfer absorption imaging. The image in b reflects the average of 10 measurements. c Integrated optical depth ${\rm OD}_{\uparrow}$ (red), ${\rm OD}_{\downarrow}$ (blue) and ${\rm OD}_{\rm tot}$ (black). Two dashed vertical lines are $x=0$ (grey) and CoM position (purple).