Figure 1

Splitting a trapped condensate of $N=200$ bosons (${}^{87}\mathrm{Rb}$ atoms) with interaction strength ${\lambda }_0=0.1$ during a ramp-up time of $T_{\mathrm{ramp}}=1000$. (a) Natural occupation numbers ${\rho }_1(t)$ and ${\rho }_2(t)$ as a function of time on a logarithmic scale [initial conditions: ${\rho }_1(0)=99.62\%$, ${\rho }_2(0)=0.38\%$]. During the splitting process the many-body wave function evolves from a condensed towards a twofold fragmented state. Inset shows oscillatory behavior of ${\rho }_2(t)$ around $0.14T_{\mathrm{ramp}}$. (b) Energy difference $\Delta E$ between the many-body ground state (○; taken as reference energy) and first excited state (●) as a function of barrier height of the time-dependent trap potential $V(x,t)$; see text for more details. Time is expressed in units of $\frac{m{L}^2}{\hslash }=1.37\mathrm{msec}$ and energy in units of $\frac{{\hslash }^2}{m{L}^2}=116\mathrm{Hz}$.Reuse & Permissions

Figure 2

Quantifying how difficult it is to (fully) fragment a condensate. Plotted are the natural occupation numbers ${\rho }_1(t)$ and ${\rho }_2(t)$ as a function of time for three ramping-up times $T_{\mathrm{ramp}}$. The parameters used are the same as in Fig. 1. Increasing $T_{\mathrm{ramp}}$ the time-dependent many-body state approaches the ground state of the double-well potential which is a (fully) fragmented condensate. For $T_{\mathrm{ramp}}=10000=13.7\sec$ which is of the order of a condensate lifetime the amplitude of oscillations is still about 10%.Reuse & Permissions

Figure 3

Same as in Fig. 1 except for the stronger interaction strength ${\lambda }_0=0.3$. (a) Natural occupation numbers ${\rho }_1(t)$ and ${\rho }_2(t)$ as a function of time [initial conditions: ${\rho }_1(0)=99.32\%$, ${\rho }_2(0)=0.68\%$]. For $T_{\mathrm{ramp}}=75$ the many-body wave function evolves from a condensed towards the twofold fragmented ground state. The “counterintuitive” regime is uncovered by employing a longer ramp-up time, e.g., $T_{\mathrm{ramp}}=500$, in which the evolution of the condensate is not to the fragmented ground state, but to a low-lying coherent excited state. (b) Energy differences $\Delta E$ between the lowest-in-energy coherent state (○; taken as reference energy) and fragmented states (● curves) as a function of barrier height. Around $V_{\mathrm{cr}}\approx 0.41V_{\max }$ the coherent ground state and lowest-excited fragmented state come very close to each other and interchange their order; see text for more details.Reuse & Permissions