Coherent oscillations in small Fermi-polaron systems

We study the ground state and excitations of a one-dimensional trapped polarized Fermi gas interacting with a single impurity. First, we study the tunneling dynamics of the impurity through a potential barrier, such as one effectively created by a double-well trap. To this end, we perform an exact diagonalization of the full few-body Hamiltonian and analyze the results in a local-density approximation. Off-diagonal and one-particle correlation matrices are studied and are shown to be useful for discerning between different symmetries of the states. Second, we consider a radio-frequency spectroscopy of our system and the resulting spectral function. These calculations can motivate future experiments, which can provide further insight into the physics of a Fermi polaron.

Figure 1

Schematic representation of the system. A single impurity is oscillating in a double-well potential (1), but the oscillations are affected by the presence of a background gas of single-species fermions. The pairwise quasidegeneracy of the low-lying single-particle energy levels, where each pair contains states with an odd and an even wave function, is responsible for the shell structure of the resulting spectrum.

Figure 2

Polaron energy in a harmonic potential. Dashed lines show results obtained with McGuire formula adapted to an inhomogeneous geometry (4) and the crosses show the exact energies as obtained by exact diagonalization. For comparison, see Ref. [20].

Figure 3

Splitting of the first two energy levels in a double-well potential. Solid lines show a result obtained with McGuire formula adapted to an inhomogeneous geometry (4) and the crosses correspond to exact energies obtained by exact diagonalization. For the legend, see Fig. 2.

Figure 4

Spectra of the (a) $N_{\uparrow }=1$ and (b) $N_{\uparrow }=2$ problem in the HO potential and of the (c) $N_{\uparrow }=1$ and (d) $N_{\uparrow }=2$ problem in the double-well potential for $V_0=5\hslash {\omega }_0$. The lines are alternately colored with blue and red for clarity. We can see that the spectra for a double well are pairwise almost degenerate (i.e., blue and red lines overlap), which reflects the structure of the single-particle spectrum.

Figure 5

Example of densities of three fermions (thin blue line) interacting with an impurity (thick red line). It can be clearly seen that, for strong repulsion, initially noninteracting impurity starts behaving like the fourth fermion. The total density (shown with a green dashed line), i.e., $n_{\uparrow }+n_{\downarrow }$, coincides with that of $N_{\uparrow }=4$ noninteracting Fermi gas. For the harmonically trapped system in the strong attraction limit, the density of $N_{\uparrow }=2$ Fermi gas and that of a noninteracting molecule (of mass $m_M=2m$) are shown (dashed and dotted black lines, respectively). For definitions of the units, see Eq. (3).

Figure 6

Densities of the $N_{\uparrow }=3$ antisymmetric state in the degeneracy manifold at the $g\to +\infty$ limit for a harmonic trap (left) and a double well (right). The thick (red) line corresponds to the density of the impurity, the thin solid (blue) line to the density of the majority fermions, and the dashed (green) line to the total density $n_{\uparrow }+n_{\downarrow }$.

Figure 7

Off-diagonal correlation matrices of the polaron ${\rho }^{(\downarrow )}(x,x^{\prime })$ with $N_{\uparrow }=3$ background polarized fermions. Row (a) shows a harmonically trapped system and row (b) a double-well confinement. Rows (c) and (d) show the correlation matrices of the polarized gas ${\rho }^{(\uparrow )}(x,x^{\prime })$ for a harmonic trap and double-well potential, respectively. In the strong repulsion case we can see distinct peaks along a diagonal corresponding to the Friedel oscillations of the density. Higher intensities (brighter colors) correspond to higher values of the one-particle density matrix. In each panel, axis ranges for both $x$ and $x^{\prime }$ vary from $-4{\xi }_0$ to $4{\xi }_0$.

Figure 8

Total one-particle density matrices ${\rho }^{(\uparrow )}(x,x^{\prime })+{\rho }^{(\downarrow )}(x,x^{\prime })$ for (a) a harmonic trap and (b) a double-well potential. The correlation matrices of a noninteracting Fermi gas are shown for (c) a harmonic trap with $N_{\uparrow }=3$, (d) a harmonic trap with $N_{\uparrow }=4$, (e) a double well with $N_{\uparrow }=3$, and (f) a double well with $N_{\uparrow }=4$. The off-diagonal elements reveal the difference in behavior of the strongly repulsive impurity interacting with $N_{\uparrow }=3$ polarized fermions and $N_{\uparrow }=4$ noninteracting Fermi gas. Axis ranges the same as in Fig. 7.

Figure 9

Oscillations of $\langle x_{\downarrow }\left(t\right)\rangle$ in time. (a) Oscillations in a harmonic trap. The faint green line shows the sum of $\langle x_{\downarrow }\left(t\right)\rangle +{\sum }_i\langle x_i\left(t\right)\rangle$, i.e., the center-of-mass motion. As expected, it adds up to a perfect harmonic oscillation with a trap frequency ${\omega }_0$. (b) Oscillations in a double-well potential, with much lower frequency due to a barrier tunneling ($V_0=5\hslash {\omega }_0$). In both cases the solid line corresponds to the interaction $g_{1\text{D}}=0.4g_0$. In (b) the dashed line represents $g_{1\text{D}}=0$ for comparison.

Figure 10

Fourier transform of the time evolution of the state (9) of $N_{\uparrow }=2$ for both (a) the harmonic trap and (b) the double-well potential with $V_0=5\hslash {\omega }_0$. The peaks seem broader in (b) than in (a) because of the different scale of frequency range plotted. The vertical green dashed lines show the oscillation frequencies $\delta \epsilon$ calculated from the exact diagonalization according to (a) Eq. (10a) and (b) Eq. (10b). The red lines show the frequency estimate ${\mathrm{\Omega }}_{-1}$ [see Eq. (14)] for $\mathcal{F}=X_{\downarrow }$ (solid lines) and $X_{\downarrow }-X_{\uparrow }$ (dashed lines).

Figure 11

Spectral function $S_{\mathrm{\Omega }}$ [Eq. (18)] in a harmonic-oscillator potential for (a) $N_{\uparrow }=1$ and (b) $N_{\uparrow }=2$. Only odd peaks are nonzero due to the parity of the system and get exponentially suppressed for higher energies. The red lines show the corresponding polaron energies calculated with exact diagonalization (see Fig. 2).

Figure 12

Spectral function (18) in a double-well potential for (a) $N_{\uparrow }=1$ and (b) $N_{\uparrow }=2$. In the first case there are two peaks, where the first one arises because the two lowest levels are almost degenerate. In the latter case, the second peak is already farther apart and thus exponentially suppressed. The red lines show the corresponding polaron energies calculated with exact diagonalization (see Fig. 3).