Bound states of a localized magnetic impurity in a superfluid of paired ultracold fermions

We consider a localized impurity atom that interacts with a cloud of fermions in the paired state. We develop an effective scattering length description of the interaction between an impurity and a fermionic atom using their vacuum scattering length. Treating the pairing of fermions at the mean-field level, we show that the impurity atom acts like a magnetic impurity in the condensed matter context, and leads to the formation of a pair of Shiba bound states inside the superconducting gap. In addition, the impurity atom can lead to the formation of deeply bound states below the Fermi sea.

Figure 1

Diagrammatic representation of (a) Eq. (2) and (b) Eq. (3). Thin lines represent the clean (unperturbed) Green’s functions [$G^0(\mathbf{k},\omega )$], the thick lines the impurity-perturbed Green’s function ($G_{\mathbf{k}}$), and the dashed line the interaction of these fermions with the impurity ($V$).

Figure 2

Representation of the sign of the solutions given by the first and second subspaces as a function of $1/k_F{a}_{\uparrow }$ and $1/k_F{a}_{\downarrow }$. In the blue (grey) domain the solution given by the first subspace is negative, while in the white domain the one given by the second subspace is negative.

Figure 3

Energies of the two in-gap (Shiba) bound states as a function of $1/k_F{a}_{\uparrow }$ and $1/k_F{a}_{\downarrow }$. Here we use the approximation of Eq. (28), we took $\Delta =0.2{\epsilon }_F$, and $\omega$ is measured in units of ${\epsilon }_F$.

Figure 4

Comparison between the approximate analytical solution (blue dashed curve) and the exact numerical solution (black curve) for the frequency (in units of ${\epsilon }_F$) of the in-gap bound state (of the first Nambu subspace) as a function of $1/k_F{a}_{\uparrow }$, with $\Delta =0.2{\epsilon }_F$ and $1/k_F{a}_{\downarrow }=-0.5.$

Figure 5

Impurity induced correction to the spectral function $\Delta A(\mathbf{k},\omega )$ of the one-component Fermi gas for the case $a{k}_F=0.5$ (white – increase of spectral weight, blue – no change, red – decrease). (a) $\Delta A(\mathbf{k},\omega )$ as a function of momentum and frequency. The dashed white line indicates the position of the Fermi energy. $\Delta A(\mathbf{k},\omega )$ shows a depletion of spectral weight along the clean dispersion line $k^2/2m-{\epsilon }_F$ (indicated by the red line), an under-sea bound state at $\omega =-3{\epsilon }_F$, and excess spectral weight in the vicinity of the continuum band which corresponds to the impurity induced broadening. (b) $\Delta A(\mathbf{k},\omega )$ as a function of frequency only with momentum fixed at $k=0.5k_F$ [slice is indicated by the green line in (a)]. The spectral function can be decomposed into three (labeled) features: (1a) a $\delta$ function corresponding to the depletion of spectral weight along the clean dispersion line; (1b) part of the depleted weight is transferred into the vicinity of the clean dispersion line resulting in its broadening; (2) the remaining weight is transferred to a $\delta$ function corresponding to the under-sea bound state. We note that although the part labeled “Broadening” is divergent in the impurity density expansion, its frequency integral remains finite, and the spectral function fulfills the frequency sum rule.

Figure 6

Correction to the RF transition rate obtained for the one-component gas due to the presence of impurities as a function of the drive frequency $\omega$, with $k_{F}a=-0.5$ (top) and $k_{F}a=0.5$ (bottom). The RF spectrum for the clean case is sharply peaked at $\omega -{\mathrm{\omega ̃}}_3~0$ with the width set by either trap properties and temperature. The impurities have two main effects: (1) Since momentum is no longer a good quantum number, the impurities broaden the sharp absorption peak at $\omega -{\mathrm{\omega ̃}}_3~0$. This broadening is composed of the depletion of the $\delta$ function indicated by the blue arrow together with population of nearby-in-frequency states. (2) If there is a bound state, it induces an edge in the spectrum of transferred atoms followed by a broad feature indicated in pink (grey). The broadening correction cannot be accurately captured in an expansion in impurity density. In fact at first order in impurity density we find that the correction is divergent but integrable. Therefore, in the figure we cut it off with a wavy line. While feature (1) is present independently of the sign of the scattering length, feature (2) which corresponds to the coherent part of the transition rate correction (i.e., the bound state induced part) is present only for positive scattering length.

Figure 7

Impurity induced correction to the spectral function of (a) the $|\uparrow \rangle$ atoms and (b) $|\downarrow \rangle$ atoms as a function of momentum and frequency for the case $a_{\uparrow }{k}_F=0.5$, $a_{\downarrow }=-0.5$, and $\Delta /\epsilon =0.4$ (white – increase of spectral weight, gray – no change, gray over white – decrease). The dashed white line indicates the position of the Fermi energy. Both $A_{n_i,\uparrow }(\mathbf{k},\omega )$ and $A_{n_i,\downarrow }(\mathbf{k},\omega )$ show a depletion of spectral weight along the dispersion curve of the clean system indicated by the red line. $A_{n_i,\uparrow }(\mathbf{k},\omega )$ shows an under-sea bound state at $\omega \approx -3{\epsilon }_F$ as well as a Shiba state at $\omega \approx -0.13{\epsilon }_F$, while $A_{n_i,\downarrow }(\mathbf{k},\omega )$ shows only a Shiba state at $\omega \approx 0.13{\epsilon }_F$. In addition, there is spectral weight in the vicinity of the dispersion curve of the clean system which corresponds to impurity induced broadening.

Figure 8

(a) RF transition rate for BCS state as a function of the drive frequency $\omega$. Corrections to the transition rate for the $|\uparrow \rangle$ atoms (b) and $|\downarrow \rangle$ atoms (c). (d) Total transition rate (clean+corrections) for 10% concentration of impurities, with divergences smoothed out. Throughout we have used $a_{\uparrow }{k}_F=0.5$, $a_{\downarrow }=-0.5$, and $\Delta /\epsilon =0.4$. In (b) the coherent part of the transition rate correction, i.e., the part induced by the Shiba and the under-sea bound states, is indicated by pink (grey) shading, with the peak on the left corresponding to the Shiba state and the peak on the right to the under-sea state. Similar to the case of the single-component Fermi gas, the incoherent part of the transition rate correction is divergent at this order in impurity density (see Fig. 6). Therefore, the total transition rate correction plotted in (b) and (c) is also divergent, and we cut it off with wavy lines, as before.

Figure 9

(a) Effective mass $m^{*}$ of lithium and sodium atoms in a $532\hspace{0.5em}\mathrm{nm}$ lattice as a function of the lattice depth (measured in lithium recoil energies). Using a potential depth of $~4\hspace{0.5em}{E}_{R,\mathrm{Li}}$ it is possible to localize the sodium atoms (that serve as impurities) while lithium atoms remain itinerant. (b) Effective mass for lithium and rubidium atoms in a $820\hspace{0.5em}\mathrm{nm}$ lattice. Lattice depths between about $0.5$ and $4\hspace{0.5em}{E}_{R,\mathrm{Li}}$ can be used to localize the Rb atoms while Li remains itinerant.

Figure 10

Effective scattering length $a$ that describes the scattering of a free fermion of mass $m_{\alpha }$ on an impurity of mass $m_i$ localized in a harmonic potential of frequency ${\omega }_i$ as a function of the fermion-impurity atom interaction strength (scattering length in vacuum $a_{0,\alpha }$) computed in the frozen impurity approximation. For small $a_{0,\alpha }$, $a$ depends linearly on $a_{0,\alpha }$, Eq. (A4). However, for large $a$ we see a deviation from linear law. For large negative $a$ it is possible to form bound states of the fermion, which result in Feshbach resonances at $(4m_{\alpha }{a}_{0,\alpha }/\mu )\sqrt{m_i{\omega }_i/\pi }\approx \{-2.6,-17.8,\dots \}$.