Two-dimensional boson-fermion mixtures

Using mean-field theory, we study the equilibrium properties of boson-fermion mixtures confined in a harmonic pancake-shaped trap at zero temperature. When the modulus of the $s$-wave scattering lengths are comparable to the mixture thickness, two-dimensional scattering events introduce a logarithmic dependence on density in the coupling constants, greatly modifying the density profiles themselves. We show that, for the case of a negative boson-fermion three-dimensional $s$-wave scattering length, the dimensional crossover stabilizes the mixture against collapse and drives it toward spatial demixing.

Figure 1

(Color online) Boson and fermion density profiles for ${}^6\text{L}\text{i-}{}^7\text{L}\text{i}$ mixture with $N_B={10}^6$ and $N_F=5\times {10}^5$; radial trapping frequencies ${\omega }_B/2\pi =4000\text{Hz}$, ${\omega }_F/2\pi =3520\text{Hz}$; and scattering lengths $a_{BB}=5.1a_0$, $a_{BF}=38a_0$, where $a_0$ is the Bohr radius. The length unit is the radial harmonic-oscillator length for bosons $a_{B\perp }=\sqrt{\hslash /m_B{\omega }_B}$. The density given is in units of ${10}^{-4}{a}_{B\perp }^{-2}$ and is normalized to unity. The three regimes $a_{BF}/a_z=0.1,1,10$ correspond to values of the asymmetry parameter $\lambda \approx {10}^3$, ${10}^5$, and ${10}^7$, respectively.

Figure 2

(Color online) Same as in Fig. 1 for a ${}^{40}\text{K}\text{−}{}^{87}\text{R}\text{b}$ mixture with $N_B={10}^6,N_F=5\times {10}^5$. The values of $\left|a_{BF}\right|/a_z=0.3,10$ correspond to $\lambda \approx 2\times {10}^2$ and $2\times {10}^5$.

Figure 3

(Color online) Density profiles for the ${}^{40}\text{K}\text{−}{}^{87}\text{R}\text{b}$ mixture calculated with the Q2D model for various values of $\lambda$ (same units as in Fig. 1).

Figure 4

(Color online) Effective BF interaction strength (in dimensionless units) for the ${}^{40}\text{K}\text{−}{}^{87}\text{R}\text{b}$ mixture within the Q2D model as a function of the anisotropy parameter $\lambda$. Dots refer to the numerical calculation performed in the TF approximation, namely, neglecting the Laplacian terms in Eqs. (3, 4), while triangles refer to the full solution of the same equations (GPE).