Pairing correlations of cold fermionic gases at overflow from a narrow to a wide harmonic trap

Within the context of Hartree-Fock-Bogoliubov theory, we study the behavior of superfluid Fermi systems when they pass from a small to a large container. Such systems can be now realized thanks to recent progress in experimental techniques. It will allow researchers to better understand pairing properties at overflow and in general in rapidly varying external potentials.

Figure 1

The different trapping potentials $U\left(R\right)$ used in this work, as defined in Eq. (2). The ratio among the two harmonic potentials is $r={\omega }_1/{\omega }_2=0.1$, 0.2, 0.4, 1. $U$ and $R$ are in trap units, i.e., in units of $\hslash {\omega }_1$ and $\sqrt{\hslash /m{\omega }_1}$, respectively [cf. text below Eq. (2)].

Figure 2

Evolution of the pairing field at the center of the trap $\Delta (R=0)$ as a function of the chemical potential $\mu$ in the trap configuration $r={\omega }_2/{\omega }_1=1$ for different cutoff values $E_C$ for $g=-1.56$. All quantities are expressed in trap units [cf. text below Eq. (2)].

Figure 3

Number of particles as a function of the chemical potential (in units of $\hslash {\omega }_1)$ for two trap configurations.

Figure 4

Level density as given in Eq. (5) for the two extreme traps as shown in Fig. 1. In the left panels the calculations are done at $\mu =24$, while in the right panels $\mu =26$. In panels (a) and (b) we show the level densities from HFB calculations and in (c) and (d) the ones without pairing. The discrete level densities are folded with Gaussians of width $\sigma =0.05$ for the case with pairing [(a) and (b)] and $\sigma =0.1$ without pairing [(c) and (d)]. All quantities are expressed in trap units.

Figure 5

Evolution of the lowest quasiparticle energies as a function of the chemical potential (both in units of $\hslash {\omega }_1)$ for various potential configurations.

Figure 6

Level density as given in Eq. (5) for the trap configuration $r={\omega }_2/{\omega }_1=0.1$ at $\mu =26$ and four different values of coupling constant. The dashed lines for each color code show the results obtained using the LDA approximation. All quantities are expressed in trap units.

Figure 7

Diagonal matrix elements of the pairing field, ${\Delta }_{nlnl}$, as a function of the energy of the basis state $n$ at $\mu =25$ for two different configurations: $r=1$ and $r=0.1$ $({\Delta }_{nlnl},e_{nl}$, and $\mu$ in units of $\hslash {\omega }_1)$.

Figure 8

Radial wave function $v_{nl}\left(R\right)$ as defined in Eq. (1) corresponding to the lowest quasiparticle as shown in Fig. 5 for the trap configuration $r=0.1$ and the cases $\mu =24$ and $\mu =26$. All quantities are expressed in trap units.

Figure 9

Pairing field $\Delta \left(R\right)$ calculated at $\mu =26$ for different trap configurations. The filled circles indicate the result obtained using the LDA for $r=0.1$. All quantities are expressed in trap units.

Figure 10

Comparing the quantal and LDA calculations of pairing field $\Delta (R=0)$ as a function of the chemical potential $\mu$ (both in units of $\hslash {\omega }_1)$ for the trap configuration $r=0.1$.

Figure 11

Same as Fig. 5, but for overflow at $\mu =12.25$ and $R_0=4.95$ (in trap units). See text for more explanation.

Figure 12

Pairing field at the center of the trap $\Delta (R=0)$ as a function of the chemical potential $\mu$ (both in units of $\hslash {\omega }_1)$ for the same trap configuration as in Fig. 11 and different ratios $r$.