Fermionization of a strongly interacting Bose-Fermi mixture in a one-dimensional harmonic trap

We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{-4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient $C$ is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions “fermionize” under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.

Figure 1

(Color online) Bosonic (red solid line) and fermionic ($\times 20$, green dashed line) density profiles (in units of $1/{\ell }_0$ and normalized to the total number $N$ of particles) as a function of the distance $x$ from the trap center (in units of ${\ell }_0$) as predicted by a two-fluid mean-field model [13] for a BF mixture confined in a one-dimensional (1D) harmonic trap. The length scale unit ${\ell }_0$ is the harmonic-oscillator length as defined in Sec. 2. The boson-boson and boson-fermion interaction strengths have been chosen such that ${\gamma }_{BB}=1.5\times {10}^{-4}$ and ${\gamma }_{BF}=0.63$ (see the text for the definitions of ${\gamma }_{BB}$ and ${\gamma }_{BF}$).

Figure 2

(Color online) Plot of the off-diagonal behavior of the fermionic and bosonic one-body density correlators $g_F^{\left(1\right)}\left(x,-x\right)$ and $g_B^{\left(1\right)}\left(x,-x\right)$ [Eq. (12)] as a function of the spatial coordinate $x$ (in units of ${\ell }_0$). Panel (a) shows $g_F^{\left(1\right)}$ for the BF mixture (in log-log scale) with $N=20$ particles and three different bosonic concentrations $N_B/N=0.75$ $(+)$, 0.5 $(\ast )$, and 0.25 (◼). The comparison to the fitting function $\sim {\left|x\right|}^{-\alpha }$ (continuous, short-dashed, and dotted curves) allows extraction of the slope. Panel (b) shows the results for the power-law exponent $\alpha$ obtained at large distances $x$ for the mixture with $N=20$ particles when the boson concentration $N_B/N$ is varied. Panel (c) shows the log-log plots of $g_B^{\left(1\right)}$ for a pure TG gas with $N=N_B=20$ particles $(\ast )$ and of $g_F^{\left(1\right)}$ for the pure Fermi gas with $N=N_F=20$ particles $(+)$. For the pure TG gas, the comparison to the fitting function $\sim {\left|x\right|}^{-\alpha }$ (double-dotted line) gives $\alpha =0.44$, a value close to the one found for the homogeneous case where $\alpha =0.5$. For the pure Fermi gas, the comparison to the fitting function $\sim \text{sin}\left(2k_{F}x\right)/{\left|x\right|}^{\alpha }$ (dot-dashed line) gives $\alpha =0.93$, a value close to the one found for the homogeneous case where $\alpha =1$. Panel (d) is the same as panel (c) but is in linear scale.

Figure 3

(Color online) Fermionic density profiles $n_F\left(x\right)$ (in units of $1/{\ell }_0$) versus $x$ (in units of ${\ell }_0$), as obtained for a BF mixture with a fixed total number of $N=7$ particles when the number of fermions is increased from $N_F=1$ (solid line) up to $N_F=7$ (double-dashed line). Normalizing each curve by $N_F/N$ would collapse them on the same single curve.

Figure 4

(Color online) Fermionic momentum distribution $n_F\left(p\right)$ (in units of $1/p_0$) versus $p$ (in units of $p_0$), as obtained for a BF mixture with a fixed total number of $N=7$ particles when the number of fermions is raised from $N_F=1$ (solid line) up to $N_F=7$ (double-dashed line). As fermions are added, $N_F$ “fermionic” oscillations develop in the momentum distribution.

Figure 5

(Color online) Fermionic momentum distribution $n_F\left(p\right)$ (in units of $1/p_0$) versus $p$ (in units of $p_0$), at fixed number of fermions $N_F=4$ when bosons are gradually added. The curves range from $N_B=0$ (solid line) down to $N_B=3$ (double-dashed line). As one can see, adding more and more bosons smoothens the fermionic oscillations.

Figure 6

(Color online) Momentum distribution $n_F\left(p\right)$ (in units of $1/p_0$) as a function of $p$ (in units of $p_0$) for the case of a BF mixture made of $N_F=2$ fermions and $N_B=5$ bosons. The plot shows the comparison between the analytical asymptotic prediction $C{p}^{-4}$, with $C$ being given by Eq. (23) (dashed line), and the exact result computed from Eq. (14) (squares).

Figure 7

(Color online) Scaled momentum distribution $n\left(p\right)$ (in units of $1/p_0$) versus $p$ (in units of $p_0$) of an expanding mixture at different times (in units of ${\omega }_0^{-1}$) as indicated in the panels. The different lines correspond to $N_B=7$ bosons (continuous red lines), $N_F=2$ fermions mixed with $N_B=5$ bosons (dashed green lines), and $N_F=7$ fermions (double-dashed black lines). The distributions have been rescaled by the total number $N$ of particles in order to be normalized to unity.