Two-component few-fermion mixtures in a one-dimensional trap: Numerical versus analytical approach

We explore a few-fermion mixture consisting of two components that are repulsively interacting and confined in a one-dimensional harmonic trap. Different scenarios of population imbalance ranging from the completely imbalanced case where the physics of a single impurity in the Fermi sea is discussed to the partially imbalanced and equal population configurations are investigated. For the numerical calculations the multiconfigurational time-dependent Hartree method is employed, extending its application to few-fermion systems. Apart from numerical calculations we generalize our ansatz for a correlated pair wave function proposed recently [I. Brouzos and P. Schmelcher, Phys. Rev. Lett. 108, 045301 (2012)] for bosons to mixtures of fermions. From weak to strong coupling between the components the energies, the densities and the correlation properties of one-dimensional systems change vastly with an upper limit set by fermionization where for infinite repulsion all fermions can be mapped to identical ones. The numerical and analytical treatments are in good agreement with respect to the description of this crossover. We show that for equal populations each pair of different component atoms splits into two single peaks in the density while for partial imbalance additional peaks and plateaus arise for very strong interaction strengths. The case of a single-impurity atom shows rich behavior of the energy and density as we approach fermionization and is directly connected to recent experiments [G. Zürn et al., Phys. Rev. Lett. 108, 075303 (2012)].

Figure 1

(a) Energy as a function of the inverse interaction strength $1/g$ for all cases examined in this work (impurity 2 : 1, 3 : 1, and 4 : 1; balanced 2 : 2; and imbalanced 3 : 2). For each case the lower line corresponds to the numerical results employing the MCTDH method and the line slightly above corresponds to the energy computed from the ansatz CPWF (2). Contour plots of the density distribution $|\Psi (x_1,x_2,x_3){|}^2=0.01$ for the configuration 2 : 1 using the CPWF for a (b) weak $g=0.5$ and (c) strong $g=5.0$ interaction strength. Lengths and energies are scaled by $a_{\parallel }$ and $\hslash {\omega }_{\parallel }$, respectively, and $g=g_{1D}/\hslash {\omega }_{\parallel }{a}_{\parallel }$.

Figure 2

One-body densities of the (a)–(d) impurity and (e)–(h) majority fermions for the 2 : 1 case with varying interaction strength $g$ using the numerical (MCTDH) approach and the analytical (CPWF) and variational ansatz (var). Lengths are scaled by $a_{\parallel }$.

Figure 3

One-body densities of the (a)–(d) impurity and (e)–(h) majority fermions for the 3 : 1 case with varying interaction strength $g$ using the numerical (MCTDH) approach and the analytical (CPWF) ansatz. Lengths are scaled by $a_{\parallel }$.

Figure 4

One-body densities for (a) the case 2 : 2 (the densities for the two components are identical) and the case 3 : 2 for the (b) minority and (c) majority fermions. Lengths are scaled by $a_{\parallel }$.