Rayleigh-Taylor instability in a phase-separated three-component Bose-Einstein condensate

Three-component Bose-Einstein condensates (BECs) offer a rich platform for exploring inquisitive quantum phenomena. We investigate the Rayleigh-Taylor instability at the two interfaces in a phase-separated three-component BEC in the mean-field framework. The instability in the system can be instigated in various ways owing to the versatility of the three-component system and the subsequent dynamics for certain cases has been studied in detail. We also study the impact of immiscibility of the intermediate component on the onset of instability in the system. The rotational symmetry of the system breaks when the atom-atom interaction is tuned in such a way that the interface between the components becomes unstable giving rise to nonlinear patterns of mushroom shapes which grow exponentially with time. We identify these nonlinear patterns as the solutions of the angular Mathieu equation, representing the normal modes.

Figure 1

Schematic diagram of a phase separated three-component Bose-Einstein condensate.

Figure 2

Ground-state density profile (a1) $\mathrm{BEC}\text{−}1$, (b1) $\mathrm{BEC}\text{−}2$, and (c1) $\mathrm{BEC}\text{−}3$ of a phase-separated three-component Bose-Einstein condensate in an axisymmetric trap with $(\omega ,{\omega }_z)=2\pi \times (50,2500)\mathrm{Hz}$ and number of atoms in each component, $N_j=60000$ ($j=1,2,3$).

Figure 3

(a1)–(a8), (b1)–(b8), and (c1)–(c8) shows the density profiles $n_1$, $n_2$, and $n_3$ of components $\mathrm{BEC}\text{−}1$, $\mathrm{BEC}\text{−}2$, and $\mathrm{BEC}\text{−}3$, respectively, at different time instances shown in columns (1) $t={\tau }^{\prime }=105.8\mathrm{ms}$, (2) $t=\tau =117.7\mathrm{ms}$, (3) $t=120.9\mathrm{ms}$, (4) $t=129.7\mathrm{ms}$, (5) $t=135.2\mathrm{ms}$, (6) $t=140.8\mathrm{ms}$, (7) $t=157.5\mathrm{ms}$, and (8) $t=169.5\mathrm{ms}$. The scattering length $a_{33}$ is linearly decreased from $100.4a_0$ to $50a_0$ between $t=0\mathrm{ms}$ to $t=200\mathrm{ms}$, after which it is fixed to $50a_0$. The three-component BEC is initialized in an axisymmetric trap with $(\omega ,{\omega }_z)=2\pi \times (50,2500)\mathrm{Hz}$ with number of atoms in each component, $N_j=60000$ ($j=1,2,3$).

Figure 4

Variation of contact interaction energies $E_{11}^I$ [green (upper line)], $E_{22}^I$ [red (middle line)], $E_{33}^I$ [blue (lower line)] with time during the dynamics of the phase-separated three-component BEC for the case of Rayleigh-Taylor instability seeded by the adiabatic decrease in $s$-wave intraspecies scattering length $a_{33}$ from $100.4a_0$ to $50a_0$.

Figure 5

(a1)–(a4), (b1)–(b4), and (c1)–(c4) shows the density profiles $n_1$, $n_2$, and $n_3$ of components $\mathrm{BEC}\text{−}1$, $\mathrm{BEC}\text{−}2$, and $\mathrm{BEC}\text{−}3$, respectively, at different time instances shown in columns (1) $t=\tau =141.6\mathrm{ms}$, (2) $t=148.1\mathrm{ms}$, (3) $t={\tau }^{\prime }=153.5\mathrm{ms}$, (4) $t=159.1\mathrm{ms}$. The scattering length $a_{11}$ is linearly increased from $92.4a_0$ to $240a_0$ between $t=0\mathrm{ms}$ to $t=200\mathrm{ms}$, after which it is fixed to $240a_0$. The three-component BEC is initialized in an axisymmetric trap with $(\omega ,{\omega }_z)=2\pi \times (50,2500)\mathrm{Hz}$ with number of atoms in each component, $N_j=60000$ ($j=1,2,3$).

Figure 6

Variation of contact interaction energies $E_{11}^I$ [green (upper line)], $E_{22}^I$ [red (middle line)], $E_{33}^I$ [blue (lower line)] with time during the dynamics of the phase-separated three-component BEC for the case of Rayleigh-Taylor instability seeded by the adiabatic increase in $s$-wave intraspecies scattering length $a_{11}$ from $92.4a_0$ to $240a_0$.

Figure 7

(a1)–(a4), (b1)–(b4), and (c1)–(c4) shows the density profiles $n_1$, $n_2$, and $n_3$ of components $\mathrm{BEC}\text{−}1$, $\mathrm{BEC}\text{−}2$, and $\mathrm{BEC}\text{−}3$, respectively, at different time instances shown in columns (1) $t={\tau }^{\prime }=67.6\mathrm{ms}$, (2) $t=76.3\mathrm{ms}$, (3) $t=\tau =78.7\mathrm{ms}$, (4) $t=79.5\mathrm{ms}$. The scattering length $a_{22}$ is linearly increased from $94.5a_0$ to $240a_0$ between $t=0$ ms to $t=200\mathrm{ms}$, after which it is fixed to $240a_0$. The three-component BEC is initialized in an axisymmetric trap with $(\omega ,{\omega }_z)=2\pi \times (50,2500)\mathrm{Hz}$ with the number of atoms in each component, $N_j=60000$ ($j=1,2,3$).

Figure 8

Variation of contact interaction energies $E_{11}^I$ [green (upper line)], $E_{22}^I$ [red (middle line)], $E_{33}^I$ [blue (lower line)] with time during the dynamics of the phase-separated three-component BEC for the case of Rayleigh-Taylor instability seeded by the adiabatic increase in $s$-wave intraspecies scattering length $a_{22}$ from $94.5a_0$ to $240a_0$.

Figure 9

(a1)–(a4), (b1)–(b4), and (c1)–(c4) shows the density profiles $n_1$, $n_2$, and $n_3$ of components $\mathrm{BEC}\text{−}1$, $\mathrm{BEC}\text{−}2$, and $\mathrm{BEC}\text{−}3$, respectively, at different time instances shown in columns (1) $t=\tau =241.1\mathrm{ms}$, (2) $t={\tau }^{\prime }=256.2\mathrm{ms}$, (3) $t=261\mathrm{ms}$, (4) $t=264.1\mathrm{ms}$. The scattering length $a_{22}$ is linearly decreased from $94.5a_0$ to $50a_0$ between $t=0\mathrm{ms}$ to $t=100\mathrm{ms}$, after which it is fixed to $50a_0$. The three-component BEC is initialized in an axisymmetric trap with $(\omega ,{\omega }_z)=2\pi \times (50,2500)\mathrm{Hz}$ with number of atoms in each component, $N_j=60000$ ($j=1,2,3$)

Figure 10

(a1)–(a4), (b1)–(b4), and (c1)–(c4) shows the density profiles $n_1$, $n_2$, and $n_3$ of components $\mathrm{BEC}\text{−}1$, $\mathrm{BEC}\text{−}2$, and $\mathrm{BEC}\text{−}3$, respectively, at different time instances shown in columns (1) $t=\tau =118.5\mathrm{ms}$, (2) $t={\tau }^{\prime }=129.7\mathrm{ms}$, (3) $t=137.6\mathrm{ms}$, (4) $t=142.4\mathrm{ms}$. The scattering length $a_{22}$ is suddenly decreased from $94.5a_0$ to $50a_0$, after which it is fixed to $50a_0$. The three-component BEC is initialized in an axisymmetric trap with $(\omega ,{\omega }_z)=2\pi \times (50,2500)\mathrm{Hz}$ with number of atoms in each component, $N_j=60000$ ($j=1,2,3$).

Figure 11

Ground-state density profile (a1) $\mathrm{BEC}\text{−}1$, (b1) $\mathrm{BEC}\text{−}2$, and (c1) $\mathrm{BEC}\text{−}3$ of a three-component Bose-Einstein condensate with partially immiscible intermediate component. The system is confined in an axisymmetric trap with $(\omega ,{\omega }_z)=2\pi \times (50,2500)\mathrm{Hz}$ and number of atoms in each component, $N_j=60000$ ($j=1,2,3$).

Figure 12

(a1)–(a4), (b1)–(b4), and (c1)–(c4) shows the density profiles $n_1$, $n_2$, and $n_3$ of components $\mathrm{BEC}\text{−}1$, $\mathrm{BEC}\text{−}2$, and $\mathrm{BEC}\text{−}3$, respectively, at different time instances shown in columns (1) $t=89.9\mathrm{ms}$, (2) $t=93.9\mathrm{ms}$, (3) $t=99.4\mathrm{ms}$, (4) $t=101.8\mathrm{ms}$ for the RTI in the system as depicted in Fig. 11. The scattering length $a_{11}$ is linearly increased from $92.4a_0$ to $240a_0$, after which it is fixed to $240a_0$. The three-component BEC is initialized in an axisymmetric trap with $(\omega ,{\omega }_z)=2\pi \times (50,2500)\mathrm{Hz}$ with number of atoms in each component, $N_j=60000$ ($j=1,2,3$).

Figure 13

Variation of contact interaction energies $E_{jj}^I$ (rows 1 to 3) with the time propagation of phase-separated three-component BEC for all the cases (columns 1 to 3) of Rayleigh-Taylor instability discussed in the main text for different number of particles in each component, $N_j=(4,6,8,10)\times {10}^4$.