Tuning Topological Charge and Gauge Field Anisotropy in a Spin-1 Synthetic Monopole

Abstract

Higher-dimensional Hilbert spaces in quantum simulation, as in all quantum science, expand the range of accessible phenomena. In this work, we experimentally realize a synthetic monopole using an ultracold spin-1 ensemble, where the monopole charge is quantified by the topologically invariant first Chern number and sources a synthetic magnetic field quantified by the Berry curvature. By using a three-level system with tunable spin-tensor coupling, we introduce anisotropy to the field, directly measure the Chern number, and observe a topological phase transition. We verify the robustness of the monopole's topological charge under deformation, and observe signatures of the topological phases using spin-texture and Majorana-star measurements. This work demonstrates spin-tensor coupling as a tuning parameter for engineering both geometric anisotropy and a rich topological phase space.

Figures (9)

• Figure 1: Berry curvature of the anisotropic monopole.A-C. The Berry curvature $\tilde{B}^r$ is plotted in the $m_x$-$m_z$ plane through the origin for hyperparameter value $\beta = 0$ and $\alpha = 0$ (A), $\alpha = 0.5$ (B), and $\alpha = 1.5$ (C), showing the growth of a "string" of positive Berry curvature (blue) from the origin to the north pole as $\alpha$ increases, and the change of sign (red color) as the string inverts direction above the phase transition at $\alpha = 1$. The colorbar indicates $\tilde{B}^r$ values in this and all figures; note that colors are intentionally saturated at $\tilde{B}^r = \pm2$, though curvature values diverge well beyond this. D. Mapping Berry curvature over the sphere $r = 1$, with a cut showing internal values, for a variety of hyperparameter values $(\alpha,\beta)$. All cases retain a monopole divergence of $\tilde{B}^r$ at the origin, but the hyperparameter-driven anisotropy results in different structures across the phase diagram. Regions of different topological charge are shaded, with $\mathcal{C}_1$ taking on values of 2, 1, 0, and -1, as indicated.
• Figure 2: Characterizing the synthetic monopole. In the case of the uniform monopole $(\alpha, \beta) = (0,0)$, the method of not-quite-adiabatic evolution is used both along the path $r =1$(A-C) and from a deformed path from $m_z = 2$ to $-2$ via $m_x = 1$(D-F). Spin projections $\left\langle \hat{F}_y \right\rangle$ (A, D) are measured vs. final polar angle $m_\theta$, from which the curvature vector $\tilde{B}^r$ (B, E) and tensor $\tilde{B}_{\theta\phi}$ (C, F) are determined. Calculations of the evolution using all experimental parameters are shown via grey lines in all subfigures; uncertainties shown reflect variation over repeated measurements. The insets in (B, E) illustrate the path used for measurement and the calculated $\tilde{B}^r$ on the parameter-space sphere. (G) Spherical path through parameter space for (A-C) measurements, superimposed on sphere representing $\tilde{B}_r(\mathbf{m})$ (colorscale as in Fig. \ref{['fig:phase_diagram']}). (H) Deformed path through parameter space for (D-F) measurements together with $\tilde{B}_r(\mathbf{m})$; (I) Using the same measurement scheme as in (A-C), $\tilde{B}^r$ is measured at different distances from the monopole, indicating the divergence of this field at the origin. Solid line is a calculation using experimental parameters. Inset shows three representative paths in parameter space.
• Figure 3: Topological phase transition of anisotropic monopole. By measuring the Berry curvature vector $\tilde{B}^r$(A,C) and tensor $\tilde{B}_{\theta\phi}$(B,D) values over the polar angle $m_\theta$ for $(\alpha,\beta$) = (0.2,0) (A-B) and (2.0,0) (C-D), the first Chern number is obtained via numerical integration of $\tilde{B}_{\theta\phi}$. Insets to (A,C) show $\tilde{B}^r$ over the $m_r = 1$ parameter space sphere using the colorscale from Fig. \ref{['fig:phase_diagram']}. Uncertainties shown reflect variation over repeated measurements. (E) Chern number $\mathcal{C}_1$ measured vs. $\alpha$ (round points), indicating the phase transition at $\alpha = 1$. Solid line shows the calculation including experimental parameters, while dashed line indicates the idealized expectation of infinite-time measurement.
• Figure 4: Complementary signatures of the anisotropic monopole.(A,B) Spin-projection measurements of $\mathbf{F}$ for systems adiabatically prepared across m-parameter space for $\alpha = 0$ (A) and $\alpha = 2$ (B) and $\beta = 0$. Each arrow represents the orientation of $\mathbf{F}$ in a set of spin axes centred at the arrow tail (orientation as shown, right). Outset from (B) is the projection in the $m_x$-$m_y$-plane at $m_\theta = 0.436$ rad $(=25^\circ)$. (C) The $\left\langle F_z \right\rangle$ projection measurement at the north (${\bf m} = (0,0,m_z = 1)$, black) and south (${\bf m} = (0,0,m_z =-1)$, green) poles as a function of $\alpha$ for $\beta = 0$. Error bars are smaller than data points. (D) The $\left\langle F_z \right\rangle$ projection measurement at the north [${\bf m} = (0,0,m_z =1/\sqrt{2})$ or $(m_\theta,m_\phi) = (\pi/4,0)$] pole as a function of $\beta$ for $\alpha = 0$. Error bars represent statistical variations. (E, F) Majorana star projections on the Bloch sphere for systems adiabatically prepared along the path [Eq. \ref{['Eq:trajectory']}] through m-parameter space. for $\alpha = 0$ (E) and $\alpha = 2$ (F) and $\beta = 0$. The two stars are represented as blue six-point and orange four-point stars; statistical experimental uncertainties are represented by shaded ellipses. Theoretical star locations represented as the colored curves, with the trajectory position (parameterized by $\xi$) indicated as per the colorbar. In both cases, two trajectories are present: in (E), they lie on top of each other, but in (F) lie in opposite hemispheres.
• Figure 5: Experimental diagram. (A) Level diagram of three levels we select for spin-1 system. (B) Experimental diagram simplified: the ultracold atomic cloud is prepared in an ODT. Permanent magnets create a Zeeman splitting. An open-ended waveguide irradiates the atoms with microwaves. (C) Time-ordering of experimental sequence (not to scale): MOT is magneto-optical trap; MT is magnetic trap; rf evap is radiofrequency evaporation, ODT is optical dipole trap; QST is quantum state tomography; SG is Stern Gerlach separation; TOF is time of flight.