Impurity coupled to an artificial magnetic field in a Fermi gas in a ring trap

The dynamics of a single impurity interacting with a many-particle background is one of the central problems of condensed-matter physics. Recent progress in ultracold-atom experiments makes it possible to control this dynamics by coupling an artificial gauge field specifically to the impurity. In this paper, we consider a narrow toroidal trap in which a Fermi gas is interacting with a single atom. We show that an external magnetic field coupled to the impurity is a versatile tool to probe the impurity dynamics. Using a Bethe ansatz, we calculate the eigenstates and corresponding energies exactly as a function of the flux through the trap. Adiabatic change of flux connects the ground state to excited states due to flux quantization. For repulsive interactions, the impurity disturbs the Fermi sea by dragging the fermions whose momentum matches the flux. This drag transfers momentum from the impurity to the background and increases the effective mass. The effective mass saturates to the total mass of the system for infinitely repulsive interactions. For attractive interactions, the drag again increases the effective mass which quickly saturates to twice the mass of a single particle as a dimer of the impurity and one fermion is formed. For excited states with momentum comparable to number of particles, effective mass shows a resonant behavior. We argue that standard tools in cold-atom experiments can be used to test these predictions.

Figure 1

A simple illustration of the system. $N-1$ uncharged fermions (light gray) and a single charged impurity (dark gray) are trapped on a ring. The impurity is interacting with the fermions via a $\delta$-function interaction. An artificial magnetic field couples exclusively to the impurity. The dynamics of the system depends on the interaction strength between particles and the total flux through the ring $\beta =\frac{qRA}{\hslash }$.

Figure 2

Energy of the lowest three states vs interaction strength for $N=2$ particles, for zero total angular momentum. Only scattering states are displayed. Energy is calculated by three different methods. Lines are from the Eq. (13) direct analytical solution without employing the BA, which is algebraically the same with the two-particle BA calculation [Eq. (12)]. Diamonds are from the general $N$-particle BA calculation [Eq. (21)].

Figure 3

Ground-state energy vs flux $\beta$ for $N=2$ particles with total angular momentum $n$. Eigenstates for flux $\beta$ with total angular momentum $n$ are also eigenstates for flux $\beta +1$ with total angular momentum $n+1$. The system can be analyzed by considering flux values between $-1/2<\beta <1/2$ for all $n$. As the flux is adiabatically increased by one, the system evolves to a higher excited state which has one more unit of total angular momentum. As can be observed, the crossings between different eigenstates are not a problem for adiabatic evolution since only states with different total angular momentum $n$ are degenerate.

Figure 4

Ground-state energy vs interaction strength for $N=1000$ particles and zero total angular momentum. The numerical solution of the BA equation (dots) given by Eq. (21) is virtually indistinguishable from the analytical solution (circles) $E=$ (Fermi energy of $N-1$ fermions) $+\mathrm{\Delta }E$. The error between the numerical and analytical solutions is too small to observe, even in the regimes where the assumptions for analytical calculation fail.

Figure 5

A representation of the graphical solution to the BA equation. The $cot\pi k$ term in Eq. (21) diverges at every integer $k$ and there is a root between every consecutive integer independently from $\lambda \left(\beta \right)$. By changing the value of $\lambda$, all roots can be adjusted so that the total angular momentum constraint is satisfied.

Figure 6

Angular momentum of the impurity vs interaction strength for $N=100$ particles for (a) attractive and (b) repulsive interactions from the analytic calculation; numerical solutions produce the same results. In the noninteracting limit, the impurity carries all the angular momentum $\left(n-\beta \right)$. $L_1$ saturates to almost zero for infinitely strong repulsive interactions as the total angular momentum is shared equally between all particles. The same behavior holds for excited states. For strongly attractive interactions, $L_1$ saturates to half the total value, signifying dimer formation with one background particle. The insets in both figures focus on the weak-interaction limit.

Figure 7

The angular momentum of the charged particle vs flux at varying interaction strength for $N=100$ particles. As expected, $L_1$ depends linearly on flux. The slope of the line decreases with increasing interaction strength, indicating higher values of effective mass of the impurity.

Figure 8

Effective mass of the impurity vs interaction strength for $N=50$ particles and zero total angular momentum $n=0$. (a) For attractive interactions, $m^{*}$ saturates to twice the mass of the impurity due to the formation of a tightly bound pair. Inset: The behavior around the zero interaction in more detail. For attractive interaction, the effective mass [given by Eq. (32)] is almost insensitive to flux change. (b) For repulsive interactions, $m^{*}$ converges to $N$. As flux $\beta$ increases, this saturation gets faster. The dependence on the flux is more prominent for small particle numbers.

Figure 9

Resonant behavior in $m^{*}$ for $\beta$ comparable to $N$. When the drag effect applied by the background particles overcomes the driving force of the magnetic field, $m^{*}$ can become negative. When the second derivative of the energy with respect to flux becomes zero, $m^{*}$ diverges. This divergence does not change the infinitely strong-interaction limit.

Figure 10

Effective momentum density in $k$ space for different interaction strengths. In the strongly interacting limit, the distance between adjacent BA roots (wave vectors) is one. For small $c$, the roots are closer to each other around $(n-\beta )$. The impurity carrying $n-\beta$ units of angular momentum in the noninteracting limit first disturbs the fermions which are momentum matched to that value.

Figure 11

Two-particle correlation function for $N=2$ particles. As expected, $g_{12}$ at zero separation decreases with increasing interaction strength. For weak interactions, the correlation function at zero decreases with increasing flux. However, for strong interactions, the correlation is almost insensitive to flux change due to the fermionization of the charged particle.

Figure 12

Two-particle correlation function for $N=50$ particles. (a) For weak interactions, Friedel oscillations occur as interference of two waves with wavelengths related to $k_F-\beta$ and $k_F+\beta$. (b) At strong interactions, the correlation becomes zero at zero separation since the impurity is effectively indistinguishable and $g_{12}$ and the frequency of Friedel oscillations are almost insensitive to flux change.

Figure 13

Kinetic energy of the particles vs interaction strength for $\beta =0.2$ and $\beta =10.2$ for $N=50$ particles. Inset: The interaction potential contribution to the total energy. The initial increase in interaction energy follows the increase in the interaction strength. However, beyond a certain strength, the tendency of the fermions to avoid the impurity is more dominant. These plots are obtained by taking the derivative of the total energy with respect to $c$. Alternatively, the interaction potential energy is also obtained by using the two-particle correlation function at zero separation. Both results are plotted in the inset, showing remarkable agreement.