Rydberg impurity in a Fermi gas: Quantum statistics and rotational blockade

We consider the quench of an atomic impurity via a single Rydberg excitation in a degenerate Fermi gas. The Rydberg interaction with the background gas particles induces an ultralong-range potential that binds particles to form dimers, trimers, tetramers, etc. Such oligomeric molecules were recently observed in atomic Bose-Einstein condensates. Understanding the effects of a correlated background on molecule formation, absent in bosonic baths, is crucial to explain ongoing experiments with Fermi gases. In this work we demonstrate with a functional determinant approach that quantum statistics and fluctuations have clear observable consequences. We show that the occupation of molecular states is predicated on the Fermi statistics, which suppresses molecular formation in an emergent molecular shell structure. At high gas densities this leads to spectral narrowing, which can serve as a probe of the quantum gas thermodynamic properties. Rydberg excitations in Fermi gases go beyond traditional impurity problems, creating an opportunity for studies of mesoscopic interactions in synthetic quantum matter.

Figure 1

Occupation of a shell structure induced by a Rydberg excitation in a Fermi sea. The Rydberg electron (brown sphere) of the impurity atom (red ion core) induces a molecular potential (dark brown) for the host atoms whose range can be tuned by the principal quantum number $n$. As illustrated in the close-up to the right (blue bubble), the potential supports bound molecular states in various angular momentum channels that are localized in the outermost potential well (not to scale). These bound states can be populated by the atoms initially occupying the single-particle states of the unperturbed Fermi sea up to the Fermi energy ${\epsilon }_{\mathrm{F}}$ (green shading). Since angular momentum is conserved in the scattering with the spherically symmetric Rydberg excitation, fermions must overcome the rotational barrier to occupy the $s$, $p$, and $d$ shells of the Rydberg atom leading to a shell structure whose occupation (shown in the inset) is determined by a Pauli-enforced Rydberg rotational blockade.

Figure 2

Rydberg molecular potential. The Rydberg potential $V_{\mathrm{Ryd}}\left(r\right)$ is shown for a ${}^{87}\mathrm{Rb}\left(71s\right)$ excitation as obtained from a quantum defect calculation in an $s$-wave scattering approximation. The potential is shown as a function of the distance $r$ between the Rydberg ion and an atom in the Fermi sea. The interacting single-particle bound radial wave functions $u_{k_{\alpha },l_{\alpha }}\left(r\right)$ are shown as colored lines. The inset shows the lowest three vibrational states for angular momenta $l=0$ (solid line) and $l=1$ (dashed line). The offset of the wave functions corresponds to their respective energies. This exemplary potential is used for all numerical results shown in this work.

Figure 3

Impact of quantum statistics at low density. The absorption spectrum $A\left(\omega \right)$ of the Rydberg impurity in the Fermi sea (solid line) and its $l=0$ component (dotted line) are compared to the spectrum obtained for a Rydberg excitation in a BEC (dashed line) at low density $\rho =5\times {10}^{11}{\mathrm{cm}}^{-3}$.

Figure 4

Rydberg rotational barrier. (a) Franck-Condon factors $|\langle 1_{\alpha },l|k_s,l\rangle {|}^2$ between the lowest molecular dimer states $|1_{\alpha },l\rangle$ and the noninteracting states $|k_s,l\rangle$ of nodal number $k=1$, 2, and 3 as a function of the rotational angular momentum $l$. (b) Comparison of the spatial structure of the interacting and noninteracting single-particle states that give rise to the superexponential decay with $l$ of the Franck-Condon factors shown in (a). In contrast to the noninteracting states (dashed lines), the molecular dimer states (solid lines) in the $s$, $p$, and $d$ orbitals of the shell structure are deeply localized in the outermost potential well at $R_0\approx 8800a_0$ and are thus hardly affected by the rotational barrier.

Figure 5

Spectral evolution with density. The absorption spectrum $A\left(\omega \right)$ of a Rydberg impurity is shown in a Fermi sea of density $\rho =5\times {10}^{11}$, $2.5\times {10}^{12}$, and $7.5\times {10}^{12}{\mathrm{cm}}^{-3}$. Here $A\left(\omega \right)$ exhibits an evolution with increasing densities to a broad distribution as characteristic of the Fermi superpolaron.

Figure 6

Effect of the rotational barrier on the initial Fermi sea. The height of the centrifugal barrier $E_l\left(R_0\right)=l(l+1)/2m{R}_0^2$ (blue line) is shown at the radius $R_0$ of the outermost potential well and the evolution is shown with energy of the Franck-Condon overlaps $|\langle 1_{\alpha },l|k_s,l\rangle {|}^2$ between the lowest interacting nodal state $|1_{\alpha },l\rangle$ and the noninteracting states $|k_s,l\rangle$ of nodal number $k$. The arrows indicate the Fermi energies chosen for the calculations of the spectra in Fig. 5.

Figure 7

Shell structure with increasing densities. A comparison between spectra of a Rydberg impurity in a Fermi sea and in a BEC is shown at density (a) $\rho =2.5\times {10}^{12}{\mathrm{cm}}^{-3}$ and (b) $\rho =7.5\times {10}^{12}{\mathrm{cm}}^{-3}$. In both cases, spectral suppression is visible for fermions at large detuning. At high densities an additional compression of the spectrum is found at small detunings that originates from suppressed density fluctuations in the fermionic medium. The response at large detuning [inset in (b)] reveals the formation of bound complexes of a large particle number.