Breaking the resilience of a two-dimensional Bose-Einstein condensate to fragmentation

A two-dimensional Bose-Einstein condensate (BEC) split by a radial potential barrier is investigated. We determine on an accurate many-body level the system's ground-state phase diagram as well as a time-dependent phase diagram of the splitting process. Whereas the ground state is condensed for a wide range of parameters, the time-dependent splitting process leads to substantial fragmentation. We demonstrate the dynamical fragmentation of a BEC despite its ground state being condensed. The results are analyzed using a mean-field model and suggest that a large manifold of low-lying fragmented excited states can significantly impact the dynamics of trapped two-dimensional BECs.

Figure 1

(a) The two-dimensional circular trap split by a radial barrier of radius $R$. (b) The inner-disk and (c) outer-annulus model potentials used to interpret the ground-state phase diagram.

Figure 2

Ground-state many-body phase diagram of a BEC in a circular trap. The three panels correspond, from top to bottom, to the interaction strengths ${\lambda }_0=0.002,0.02$, and $0.2$. The number of bosons is $N=100$. The splitting changes the coherence properties of the BEC. The BEC is mostly condensed, except for a narrow window (magenta shaded area) of radii $R$, which depends on ${\lambda }_0$. The weaker the interaction is, the narrower the window in which the BEC is fragmented is. A static model based on the GP theory is shown. The energies ${\varepsilon }_{\mathrm{disk}}^{\mathrm{GP}}$ (solid blue line) and ${\varepsilon }_{\mathrm{annulus}}^{\mathrm{GP}}$ (dashed orange line) as a function of $R$ are depicted. The maximal fragmentation on the many-body level is encountered when ${\varepsilon }_{\mathrm{disk}}^{\mathrm{GP}}={\varepsilon }_{\mathrm{annulus}}^{\mathrm{GP}}$. All quantities are dimensionless.

Figure 3

The densities of the ground state and of the wave function after the splitting process for ${\lambda }_0=0.02$. (top) For different values of $R$ the ground state is located either inside the inner-disk part, outside in the annular part, or in a combination of both. (middle) Without the radial barrier, the BEC is spread out in the circular trap $V_{\mathrm{trap}}\left(\mathbf{r}\right)$. (bottom) The density after the splitting process is located both in the disk and annulus for a wide range of radii $R$.

Figure 4

Two cuts through the time-dependent phase diagram for $N=100$ bosons and interaction strength ${\lambda }_0=0.02$. The fragmentation depicted refers to the end of the splitting process. (a) Remarkably, the dynamical splitting process leads to fragmentation over the entire examined range of radii $R$; for comparison, see Fig. 2. The splitting rate is $\beta =1$. (b) The dynamical splitting process leads to fragmentation over two orders of magnitude of the splitting rate. The radius is $R=3$. In the splitting process the system has a high affinity to fragment. All quantities are dimensionless.

Figure 5

Same as Fig. 4 at barrier radius $R=3$ but for different particle numbers $N$. The interaction parameter ${\lambda }_0(N-1)=2$ is kept fixed. All quantities are dimensionless.

Figure 6

Accuracy of the MCTDHB computations. Time evolution of the natural occupation numbers $n_j$ (divided by the number of bosons $N)$ for the split dynamics of Fig. 4 at barrier radius $R=3$. No more than two orbitals are macroscopically occupied. The inset shows the occupation of the third and fourth natural orbitals. All quantities are dimensionless.