Accurate multi-boson long-time dynamics in triple-well periodic traps

In order to solve the many-boson Schrödinger equation we utilize the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). To be able to attack larger systems and/or to propagate the solution for longer times, we implement a parallel version of the MCTDHB method, thereby realizing the recently proposed [Streltsov et al., Phys. Rev. A 81, 022124 (2010)] idea on how to construct efficiently the result of the action of the Hamiltonian on a bosonic state vector. As an illustrative example of its own interest, we study the real-space dynamics of repulsive bosonic systems made of $N=12$, 51, and 3003 bosons in triple-well periodic potentials. The ground state of this system is threefold fragmented. By suddenly strongly distorting the trap potential, the system performs complex many-body quantum dynamics. At long times it reveals a tendency to an oscillatory behavior around a threefold fragmented state. These oscillations are strongly suppressed and damped by quantum depletions. In spite of the richness of the observed dynamics, the three time-adaptive orbitals of MCTDHB($M=3$) are capable of describing the many-boson quantum dynamics of the system for short and intermediate times. For longer times, however, more self-consistent time-adaptive orbitals are needed to correctly describe the nonequilibrium many-body physics. The convergence of the MCTDHB($M$) method with the number $M$ of self-consistent time-dependent orbitals used is demonstrated.

Figure 1

Scenario of the many-body dynamics in a triple-well periodic trap. Schematic plots of the initial trap [dashed (black) sinusoidal line], normalized initial wave-packet density ${\rho }_{\mathrm{MB}}(x)$ [dotted (magenta) line], and the final trap potential [bold solid (black) sinusoidal line] in which the dynamics evolves. The ground state of $N$ bosons in the initial triple-well periodic potential [see Eq. (23) for $t<0$] is fully threefold fragmented and, therefore, can be viewed as a perfect Mott-insulating state with $N/3$ bosons per site. The final potential, obtained at $t=0$ by suddenly shifting and ramping down the initial trap, supports six bands within the wells, plotted as thin dashed (black) horizontal lines. In this scenario the initial state is strongly pushed out of equilibrium and its energy per particle [depicted as a bold double-dashed (red) horizontal line] is situated between the fourth and fifth bands. We consider systems with $N=12$, 51, and 3003 bosons interacting via repulsive interparticle contact potential of strength ${\lambda }_0=0.15/(N-1)$. For more details, see the text. The quantities shown are dimensionless.

Figure 2

Time-dependent densities for $N=12$ bosons. The time-dependent Schrödinger equation is solved for the scenario of Fig. 1 for $N=12$ bosons by using different levels $M$ of the MCTDHB($M$) theory. The plotted one-particle densities ${\rho }_{\mathrm{MB}}(x,t)$ obtained for $M=3,6,9,12$ [double-dashed (red), dashed-dotted (green), dashed (blue), and solid (magenta) lines] demonstrate the convergence of the MCTDHB($M$) results with $M$ for short ($t=10\approx 3.7\tau$), intermediate ($t=100\approx 37\tau$), and long ($t=300\approx 111\tau$) propagation times. Whenever one sees only the solid (magenta) line, the curves are all on top of each other. The trap potential in which the bosons evolve is shown schematically [bold solid (black) sinusoidal line]. The energy difference between the second and first bands of the final trap defines the time scale of the problem $\tau \approx 2.7$. The MCTDHB($M=3$) approach utilizing three time-adaptive orbitals provides an excellent description of the evolving one-body density. Small discrepancies appearing at longer propagation times can be attributed to quantum depletion, i.e., excitations to higher orbitals. The quantities shown are dimensionless.

Figure 3

Properties of the time-dependent many-body wave packets for $N=12$ bosons. The reduced one-particle density is diagonalized for the system of Fig. 2. The obtained eigenvalues, i.e., natural occupation numbers, are plotted as a function of time. The reference MCTDHB($M=12$) results for evolutions of all 12 natural occupation numbers are shown in the upper left-hand panel. The evolutions of the three most-occupied natural occupation numbers computed at different levels $M=3,6,9,12$ [double-dashed (red), dashed-dotted (green), dashed (blue), and solid (magenta) lines] of the MCTDHB($M$) theory reveal convergence with $M$ as depicted in the upper right-hand panel, and in the lower left-hand and right-hand panels for $n_1$, $n_2$, and $n_3$, respectively. The long-time oscillations around the pure threefold fragmented state are damped by the quantum depletions, i.e., excitations to higher orbitals. The quantum dynamics for the short and intermediate propagation times are adequately described at the MCTDHB($M=3$) level. At longer propagation times contributions from higher excited states become more relevant as time proceeds, and more self-consistent orbitals are needed to provide the correct description of these excitations. Numerical convergence up to $~$65 time units is achieved with $M=3$ and up to 150 time units with $M=9$. The quantities shown are dimensionless.

Figure 4

Many-body dynamics of $N=51$ bosons and the scenario of Fig. 1. Shown are the one-particle densities ${\rho }_{\mathrm{MB}}(x,t)$ computed at the $M=3$ and $M=6$ levels [double-dashed (red) and dashed-dotted (green) lines] of the MCTDHB($M$) theory for short ($t=10\approx 3.7\tau$), intermediate ($t=100\approx 37\tau$), and long ($t=300\approx 111\tau$) propagation times. Whenever one sees only the double-dashed (green) line, the curves are on top of each other. The trap potential in which the bosons evolve is shown schematically [bold solid (black) sinusoidal line]. Very small differences between the three- and the six-orbital computations start to appear at long propagation times ($t=300$). The quantities shown are dimensionless.

Figure 5

Properties of the evolving many-body wave packet for $N=51$ bosons. The evolutions of the natural occupation numbers $n_i$ obtained by using three- and six-orbital [double-dashed (red) and dashed-dotted (green) lines] levels of the MCTDHB($M$) theory are plotted as a function of time. For the short and intermediate propagation times the three-orbital ansatz is quantitatively correct. At longer propagation times the quantum depletions become more and more relevant, and more time-adaptive orbitals are needed. Note that the one-particle densities computed with $M=3$ and $M=6$ do not differ from each other even for the longer propagation time (see Fig. 4). The quantities shown are dimensionless.

Figure 6

Many-body dynamics of $N=3003$ bosons and the scenario of Fig. 1. Shown are the MCTDHB($M=3$) one-particle densities ${\rho }_{\mathrm{MB}}(x,t)$ obtained for the short ($t=10\approx 3.7\tau$), intermediate ($t=100\approx 37\tau$), and long ($t=300\approx 111\tau$) propagation times [double-dashed (red) line]. The trap potential in which the bosons evolve is shown schematically [bold solid (black) sinusoidal line]. In this work we keep the interparticle interaction energy, i.e., nonlinearity parameter $\lambda ={\lambda }_0(N-1)$, the same for all the considered systems and choose ${\lambda }_0=0.15/(N-1)$. The time-dependent densities presented in this figure for $N=3003$ bosons are quite similar to these depicted in Figs. 2 and 4 for $N=12$ and $N=51$ bosons, respectively. The systems made of different number of particles but characterized by the same $\lambda$ reveal quite similar density dynamics. The quantities shown are dimensionless.

Figure 7

Properties of the time-dependent many-body wave packets for $N=3003$ bosons. Evolutions of the three natural occupation numbers $n_{i=1,2,3}$ obtained within the MCTDHB($M=3$) theory are plotted as a function of time [dashed (blue) lines]. For comparison, also the results for $N=12$ [double-dashed (red) lines] and $N=51$ [dashed-dotted (green) lines] bosons are shown. The evolution of the systems made of $N=51$ and $N=3003$ bosons are much closer to each other than to the $N=12$ ones. The many-body properties seem to saturate to some average values already at $N=51$. The quantities shown are dimensionless.