Synthetic magnetic fields for cold erbium atoms

The implementation of the fractional quantum Hall effect in ultracold atomic quantum gases remains, despite substantial advances in the field, a major challenge. Since atoms are electrically neutral, a key ingredient is the generation of sufficiently strong artificial gauge fields. Here we theoretically investigate the synthetization of such fields for bosonic erbium atoms by phase imprinting with two counterpropagating optical Raman beams. Given the nonvanishing orbital angular momentum of the rare-earth atomic species erbium in the electronic ground state and the availability of narrow-line transitions, heating from photon scattering is expected to be lower than in atomic alkali-metal species. We give a parameter regime for which strong synthetic magnetic fields with good spatial homogeneity are predicted. We also estimate the size of the Laughlin gap expected from the $s$-wave contribution of the interactions for typical experimental parameters of a two-dimensional atomic erbium microcloud. Our analysis shows that cold rare-earth atomic ensembles are highly attractive candidate systems for experimental explorations of the fractional quantum Hall regime.

Figure 1

(a) Reduced level scheme of alkali-metal atoms, with an $S$-electronic ground state ($L=0$). (b) Reduced level scheme for a transition from a ground state with $L=1$ to an electronically excited state with $L^{\prime }=0$. This system gives an example for an electronic transition starting from a higher orbital angular momentum ground state, for which even with radiation far detuned from the electronically excited state Raman transitions between different ground-state spin projections become possible (the ground state $|g_0\rangle$ is shown in gray, because it is not relevant for the atom-light coupling here). For the shown case of $L=1$, the Raman transitions can be driven with a ${\sigma }^{+}-{\sigma }^{-}$ optical polarization configuration, inducing Raman transitions with $\mathrm{\Delta }{m}_{\mathrm{F}}=2$. (c) Schematic for synthetization of an artificial magnetic field for atoms using optical driving with two counterpropagating Raman beams and a transverse gradient of the (real) magnetic field.

Figure 2

Synthetization of magnetic fields in a three-level configuration. (a), (b) Energy quasimomentum dispersion relation for an undressed (${\mathrm{\Omega }}_{\mathrm{R}}=0$) and a dressed case (${\mathrm{\Omega }}_{\mathrm{R}}=16E_{\mathrm{L}}/\hslash$), both for a vanishing two-photon detuning $\delta$. (c), (d) Corresponding curves for the nonvanishing detuning values of $\delta =\pm 16E_{\mathrm{L}}/\hslash$, respectively, for ${\mathrm{\Omega }}_{\mathrm{R}}=16E_{\mathrm{L}}/\hslash$. The energy curves for the undressed case are plotted in (b)–(d) as gray dotted lines for comparison. (e) Generated vector potential $q^{*}{A}_x^{*}\left(\delta \right)/\hslash$ vs the two-photon detuning $\delta$ for ${\mathrm{\Omega }}_{\mathrm{R}}=16E_{\mathrm{L}}/\hslash$ (blue solid line) and the dependence obtained from a Taylor expansion up to lowest order in $\delta$ ($k_{x,\min }^{\mathrm{Taylor}}/k_{\mathrm{L}}=-\delta /{\mathrm{\Omega }}_{\mathrm{R}}$, see text), yielding a linear slope (orange dotted line). (f) The generated synthetic magnetic field, when applying a transverse detuning gradient ${\delta }^{\prime }/\left(2\pi \right)=2.66\mathrm{kHz}/\mu \mathrm{m}$ with a gradient of the real magnetic field vs position $y$. Here $q^{*}=e$ was assumed.

Figure 3

(a) Relevant atomic erbium levels for the $J=6\to J^{\prime }=7$ transition driven by Raman beams in a ${\sigma }^{+}-{\sigma }^{-}$ optical polarization configuration. The Raman beams are irradiated in a counterpropagating geometry. (b) Dispersion relation $E\left(k_x\right)$ of the seven undressed (with ${\mathrm{\Omega }}_{\mathrm{R}}=0$) states for $\delta =0$. (c) Dispersion of the dressed state system with moderate Raman coupling (${\mathrm{\Omega }}_{\mathrm{R}}=8E_{\mathrm{L}}/\hslash$, for which ${\mathrm{\Omega }}_{\mathrm{R}}<m_{F,\max }^2{E}_{\mathrm{L}}/\hslash$) and $\delta =0$. (d) Dispersion for a larger value of the Raman coupling (${\mathrm{\Omega }}_{\mathrm{R}}=96E_{\mathrm{L}}/\hslash$, for which ${\mathrm{\Omega }}_{\mathrm{R}}>m_{F,\max }^2{E}_{\mathrm{L}}/\hslash$, with $m_{F,\max }=6$), for which the lowest energetic dressed state level has a near parabolic shape. Here a nonvanishing two-photon detuning $\delta =4E_{\mathrm{L}}/\hslash$ was used, resulting in a minimum of the dispersion curve at $k_{x,\min }\ne 0$. (e), (f) Corresponding curves for the larger positive and negative detuning values $\delta =\pm 16E_{\mathrm{L}}/\hslash$, respectively.

Figure 4

(a) Variation of the synthetic vector potential $A_x^{*}$ vs the two-photon detuning $\delta$ for different values of the (unity Clebsch-Gordan coefficients) effective two-photon Rabi frequency ${\mathrm{\Omega }}_{\mathrm{R}}$. For too low values of ${\mathrm{\Omega }}_{\mathrm{R}}$ (roughly below $m_{F,\max }^2{E}_{\mathrm{L}}/\hslash$, with $m_{F,\max }=6$), no continuous variation is observed. (b) Corresponding synthetic magnetic field (blue) vs position along the $y$ axis for a transverse gradient of the real magnetic field of $\partial B_x/\partial y=70.3\mathrm{G}/\mathrm{cm}$. While the synthetic field is spiky for the case of small values of ${\mathrm{\Omega }}_{\mathrm{R}}$ (see inset for ${\mathrm{\Omega }}_{\mathrm{R}}=16E_{\mathrm{L}}/\hslash$), for ${\mathrm{\Omega }}_{\mathrm{R}}=96E_{\mathrm{L}}/\hslash$ over a relatively large spatial region good spatial homogeneity is reached (see main panel). For comparison, also the spatial variation of the synthetic magnetic field obtained for a pure three-level system as shown in Fig. 1 is shown (red dashed line), where a two-photon Rabi frequency ${\mathrm{\Omega }}_{\mathrm{R}}=16E_{\mathrm{L}}/\hslash$, $g=1$, and a magnetic-field gradient of $595\mathrm{G}/\mathrm{cm}$ was assumed, for which the desired value of $\hslash {\omega }_{\mathrm{c}}=E_{\mathrm{L}}$ in the center (at $y=0$) is achieved.

Figure 5

Comparison of results for the different narrow-line transition components $J=6\to J^{\prime }=5,6,$ and 7 (left, middle, and right panels, respectively) of the erbium transition, assuming ${\mathrm{\Omega }}_{\mathrm{R}}=96E_{\mathrm{L}}/\hslash$ in all cases. (a) Energy wave-vector dispersion $E\left(k_x\right)$ for a two-photon detuning $\delta =4E_{\mathrm{L}}/\hslash$. (b) Synthetic vector potential $A_x^{*}$ vs the two-photon detuning $\delta$ and (c) the synthetic magnetic field vs position $y$ for a transverse gradient of the (real) magnetic field of $73.9\mathrm{G}/\mathrm{cm}$ in the case of $J^{\prime }=5$ and of $70.3\mathrm{G}/\mathrm{cm}$ in the case of $J^{\prime }=7$, for which in both cases $\hslash {\omega }_{\mathrm{c}}=E_{\mathrm{L}}$ is reached in the center. For both the $J=6\to J^{\prime }=5$ and $J^{\prime }=7$ transitions for the used parameters the lowest energetic dispersion curve has a single minimum, allowing for the synthetization of a—within the central region—spatially relatively homogeneous synthetic magnetic field. For the case of the $J=6\to J^{\prime }=6$ transition the lowest-energy dispersion curve for the same value of the Raman coupling has two minima, with the absolute minimum alternating from $k_{x,\min }<0$ to $k_{x,\min }>0$ for $\delta >0$ and $\delta <0$, respectively, so that the synthetic vector potential exhibits a steplike behavior. The resulting expected synthetic magnetic field (shown here for a transverse gradient of the real magnetic field of $70.3\mathrm{G}/\mathrm{cm}$) exhibits a divergence at $y=0$, as understood from the here discontinuous variation of the vector potential vs $\delta$.