BCS-BEC crossover of ultracold ions driven by density-dependent short-range interactions in a quantum plasma

We study theoretically a Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) crossover of two-species ions in a three-dimensional quantum plasma at zero temperature. Central to this crossover is an effective short-ranged, attractive interaction potential between the ions shielded by the surrounding degenerate electrons. The interaction range and magnitude can be tuned nonmonotonically by varying the carrier density of the quantum plasma. Low-energy collisions between two ions are characterized by the $s$-wave scattering length when the interaction range and the inter-ion spacing are comparable. We show that the $s$-wave scattering length can be changed from $-\infty$ to $\infty$, leading to a BCS-BEC crossover driven purely by the plasma density. Through numerical and analytical calculations, we find that the quantum acoustic waves in the plasma exhibit distinct dispersion relations in the different regimes, providing a route to probe the crossover. Our paper shows that the quantum plasma may offer a different platform to quantum simulate the BEC-BCS crossover and exotic phases with added tunability that might be difficult to achieve in conventional solid-state systems and ultracold atom gases.

Figure 1

Schematic illustration the BCS-unitary-BEC regimes. Binding potentials between two ions are shown in the upper row. The depth of the potential depends on the plasma density, which also changes $1/k_{F}a$ with $k_F$ and $a$ to be the Fermi momentum and $s$-wave scattering length. BCS-BEC crossover of ion pairs is induced by this short-range attractive potential. The orange (blue) spheres represent ${}^4\text{He}^{+}$ ions in the spin-up (-down) state and the gray dots are electrons. The Thomas-Fermi screening length of the ion sphere is $1/q_s\approx 0.1/k_F\text{–}0.5/k_F$. See text for details.

Figure 2

Short-range attractive potential and $s$-wave scattering length. (a) The screened potential $\varphi \left(r\right)$ is tuned by varying $\alpha$ (through the density $n_0={10}^{20},2\times {10}^{22},\text{and}{10}^{24}{\mathrm{cm}}^{-3}$, respectively, to $\alpha \approx 70,12,\text{and}3$). Decreasing $\alpha$ (increasing $n_0$), the potential supports the BCS-unitary-BEC crossover. (b) The $s$-wave scattering length as a function of the binding potential depth, $\left|\varphi \right(d\left)\right|$, with the equilibrium distance $d$ at which the potential takes its minimum value. (c) The dependence of $\alpha$ on the scattering length $a$ ($1/k_{F}a$). (d) The normalized order parameter $\mathrm{\Delta }/E_F$ as a function of $1/k_{F}a$.

Figure 3

Tuning the screened potential via the plasma density. We show (a) the minimum potential $\varphi \left(d\right)$, (b) the corresponding $d$ compared to the $s$-wave scattering length $a$, (c) the quantum recoil parameter $\alpha$, and (d) the normalized $s$-wave scattering length $k_{F}a$ as a function of the plasma density $n_0$. The tunability provided by the screened potential allows us to realize the BCS-BEC crossover.

Figure 4

Phonon dispersion in different regimes. Phonon frequency $\omega /{\omega }_{ip}$ and ${\omega }_0/{\omega }_{ip}$ (normalized by the ion plasma frequency ${\omega }_{ip}=\sqrt{4\pi e^2{n}_0/m}$) when varying (a) $1/k_{F}a$ and (b) $n_0$. It is found that the frequency is higher in the paired plasma. Phonon dispersion in the (c) BCS, (d) unitary, and (e) BEC regime. Other parameters are (c) $1/k_{F}a=-1$ ($n_0=2\times {10}^{19}{\mathrm{cm}}^{-3}$), (d) $1/k_{F}a=0$ ($n_0=2\times {10}^{22}{\mathrm{cm}}^{-3}$), and (e) $1/k_{F}a=1$ ($n_0=4\times {10}^{23}{\mathrm{cm}}^{-3}$).

Figure 5

The screened potential $\varphi \left(r\right)$, varying with the plasma density $n_0=2\times {10}^{27}(\alpha =0.25),n_0=3\times {10}^{24}(\alpha =2.2),n_0={10}^{23}(\alpha =6.9),n_0=2\times {10}^{22}(\alpha =12),n_0={10}^{21}(\alpha =32),n_0={10}^{20}(\alpha \approx 70){\mathrm{cm}}^{-3}$.

Figure 6

(a) The distance $d$ defined by $\frac{\partial \varphi \left(r\right)}{\partial r}=0$, (b) the potential $\varphi \left(d\right)$, (c) its second derivative ${\varphi }^{\prime \prime }\left(d\right)$ at $r=d$, and (d) the acoustic wave period $2\pi /k_{F}d$ under the potential, as a function of $1/k_{F}a$, over the BCS-BEC crossover.

Figure 7

The energy per particle ${\epsilon }_s/E_{Fs}$ (solid line), chemical potential ${\mu }_s/E_{Fs}$ (dashed line), and sound velocity $c_{s0}/v_{Fs}$ (dash-dotted line) as a function of $1/k_{F}a$ (a) and $n_0$ (b). Phonon frequency $\omega /{\omega }_{ip}$ for $q=0.2k_F$ (solid line), $q=0.4k_F$ (dashed line), $q=0.8k_F$ (dash-dotted line), $q=1.2k_F$ (dotted line) as a function of $1/k_{F}a$ (c) and $n_0$ (d).

Figure 8

Phonon frequency $\omega /{\omega }_{ip}$ as a function of $1/k_{F}a$ for different $q$.

Figure 9

Phonon frequency $\omega /{\omega }_{ip}$ as a function of $q$ for different $1/k_{F}a$ over the BCS-BEC crossover (a–c), and the contour of $\omega /{\omega }_{ip}$ as functions of $1/k_{F}a$ and $q$ (d).