Raman processes and effective gauge potentials

A technique is described by which light-induced gauge potentials allow systems of ultracold neutral atoms to behave like charged particles in a magnetic field. Here, atoms move in a uniform laser field with a spatially varying Zeeman shift and experience an effective magnetic field. This technique is applicable for atoms with two or more internal ground states. Finally, an explicit model of the system using a single-mode two-dimensional Gross-Pitaevskii equation yields the expected vortex lattice.

Figure 1

Geometry and level diagram for realization of light-induced gauge potentials. (a) The panel depicts the two counterpropagating laser beams with momenta ${\mathbf{k}}_{\mathbf{1}}$ and ${\mathbf{k}}_{\mathbf{2}}$, aligned along $+\widehat{x}$ and $-\widehat{x}$, respectively, and the physical magnetic field along $\widehat{y}$. (b) Level diagram for two electronic ground states. (c) Level diagram for three electronic ground states; a small quadratic Zeeman shift $ϵ$ of state $|2⟩$ is not depicted. Additional transitions, relevant for physical atoms, contribute to the state-independent light shift in this configuration.

Figure 2

(Color online) Each panel pictures the dressed-state dispersion relations for a two-level dressed-state atom. The horizontal axis is quasimomentum ${\tilde{k}}_x$, and the vertical axis is the dressed-state energy $E_{\pm }\left({\tilde{k}}_x\right)$. The black lines are the exact eigenvalues of Eq. (2), and the red dashed lines are the analytic approximation. (a) Bare potentials (undressed) with Raman beams on resonance; (b) dressed potentials $\left(\Omega =16E_r,\delta =0E_r\right)$; (c) bare potentials (undressed) with Raman beams off-resonance $\left(\delta =5E_r\right)$; and (d) dressed potentials with Raman beams off-resonance $\left(\Omega =16E_r,\delta =5E_r\right)$.

Figure 3

Effective vector potential $e{A}_x/\hslash k_r$ versus detuning $\delta$ for $\Omega =16E_r$. The solid line is the exact result, and the dashed line is the lowest-order expansion in $\delta$. The insets depict predicted quantities relevant to experiment, computed for ${}^{87}\text{R}\text{b}$ with a detuning gradient ${\delta }^{\prime }\left(y\right)=900\text{Hz}/\mu \text{m}$ and $\lambda =800\text{nm}$ Raman lasers. The top inset shows the effective magnetic field $B$, indicating the degree of field inhomogeneity. The bottom inset shows the filling fraction $\nu$ for a $R_{TF}=37\mu \text{m}$ BEC chosen so $\nu$ is nearly constant near the system’s center.

Figure 4

(Color online) Each panel denotes the dressed-state dispersion relations $E\left({\tilde{k}}_x\right)$ for atoms dressed by counterpropagating Raman beams. The horizontal axis is quasimomentum ${\tilde{k}}_x$, and the vertical axis is energy in the RWA. (a) Bare potentials (undressed) with Raman beams on resonance; (b) dressed potentials $\left(\Omega =32E_r,\delta =0E_r\right)$; (c) bare potentials (undressed) with Raman beams off-resonance $\left(\delta =5E_r\right)$; and (d) dressed potentials with Raman beams off-resonance $\left(\Omega =32E_r,\delta =10E_r\right)$.

Figure 5

The left panel shows the optimized effective fields $\beta$, and the right panel shows the optimized filling fraction $\nu$. In each plot, $\Omega =16E_r$ and the solid curve shows the case optimized for uniform field $\left(ϵ=-1.657E_r\right)$ and the dashed curve shows that for uniform $\nu$ $\left(ϵ=-0.064\right)$.

Figure 6

Inverse vortex spacing versus detuning gradient ${\delta }^{\prime }\left(y\right)$ in a three-level system. The displayed symbols are obtained by solving the GPE for 3D BEC with $\left(3–5\right)\times {10}^5$ in the presence of an effective magnetic field with a detuning $\delta =0$ at the system’s center and a coupling $\Omega =16E_r$. The simulation assumes a $10\mu \text{m}$ Thomas-Fermi radius along $\widehat{z}$, and solves a 2D GPE along the remaining two directions. The typical vortex spacing is obtained from the Fourier transform of the density distribution ${\left|\Psi \left(\mathbf{r}\right)\right|}^2$. The uncertainties, reflecting the vortex-spacing distribution, are obtained from the half-width of the first peak in the same Fourier transform. The dashed line is the approximation [Eq. (8)], and the solid line results from numerical diagonalization of Eq. (7). Inset: calculated in situ density distribution for a detuning gradient ${\delta }^{\prime }\left(y\right)=0.023E_r{k}_r$ showing the expected vortex lattice structure in a nonsymmetric and nonrotating system.