Quantum dynamics of two bosons in an anharmonic trap: Collective versus internal excitations

We deal with the effects of an anharmonic trap on an interacting two-boson system in one dimension. Our primary focus is on the role of the induced coupling between the center of mass and the relative motion as both anharmonicity and the (repulsive) interaction strength are varied. The ground state reveals a strong localization in the relative coordinate, counteracting the tendency to fragment for stronger repulsion. To explore the quantum dynamics, we study the system’s response upon (i) exciting the harmonic ground state by continuously switching on an additional anharmonicity and (ii) displacing the center of mass, this way triggering collective oscillations. The interplay between collective and internal dynamics materializes in the collapse of oscillations, which are explained in terms of few-mode models.

Figure 1

(Color online) Ground state of the two-boson system in an anharmonic trap with different anharmonicity factors $\lambda$. Shown are the one-particle densities for the relative coordinate ${\rho }_1\left(x\right)$ for interaction strengths of (a) $g=0.0$, (b) $g=1.0$, (c) $g=6.0$, and (d) $g=20.0$ (in harmonic-oscillator units).

Figure 2

(Color online) (a)–(c) Two-particle densities ${\rho }_2(R,x)$ of the ground state of the two-boson system in an anharmonic trap with anharmonicity factor $\lambda =6.0$ and (a) $g=0.0$, (b) $g=1.0$, and (c) $g=20.0$. (d)–(f) same, but without the $Rx$-coupling term in the Hamiltonian.

Figure 3

(Color online) (a) Mean relative distance of the two particles and (b) energy of the groundstate of the two-boson-system both as a function of the anharmonicity factor $\lambda$ and the interaction strength $g$.

Figure 4

(Color online) Dynamics of the two-boson system after switching on a quartic anharmonicity. Square of the mean relative distance of the two particles $⟨x^2⟩$ for (a) $g=1.0$, $\lambda =0.1$, (b) $g=1.0$, $\lambda =0.5$, (c) $g=20.0$, $\lambda =0.1$, (d) $g=20.0$, $\lambda =0.5$, and (e) $g=20.0$, $\lambda =0.1$ (without the $Rx$-coupling term)

Figure 5

(Color online) Excitation spectra for (a) $g=20.0$, $\lambda =0.1$ and (b) $g=20.0$, $\lambda =1.0$. The numbers indicate the excited states, and 0 denotes the ground state.

Figure 6

(Color online) Fits of $⟨x^2⟩$ for a system with parameters $g=20.0$ and $\lambda =0.1$. The open pentagons represent the exact simulation. (a) Two-mode fit for time interval $t∊[10,40]$ (b) and (c) More detailed two-mode- and four-mode fits, (b) for time interval $t∊[20,26]$ and (c) for time interval $t∊[33,39]$.

Figure 7

(Color online) Dynamics in the anharmonic trap after elongating the center of mass. The open pentagons represent the exact simulation. (a) Three-mode fit of the center-of-mass position $⟨R⟩$ for $g=20.0$, $\lambda =0.1$, $\kappa =1.0$. (b) and (c) Fits of the mean-square distance of the two particles $⟨x^2⟩$ for $g=20.0$, $\lambda =0.1$, and $\kappa =1.0$ in the time interval $t∊[15,35]$ (b) three-mode fit and (c) seven-mode fit.