Numerically exact quantum dynamics of bosons with time-dependent interactions of harmonic type

Axel U. J. Lode, Kaspar Sakmann, Ofir E. Alon, Lorenz S. Cederbaum, and Alexej I. Streltsov

Phys. Rev. A 86, 063606 – Published 6 December 2012

Abstract

The exactly solvable quantum many-particle model with harmonic one- and two-particle interaction terms is extended to include time dependency. We show that when the external trap potential and interparticle interaction have a time dependency, the numerically exact solutions of the corresponding time-dependent many-boson Schrödinger equation are still available. We use these exact solutions to benchmark the recently developed multiconfigurational time-dependent Hartree method for bosons (MCTDHB) [Phys. Rev. Lett. 99, 030402 (2007); Phys. Rev. A 77, 033613 (2008)]. In particular, we benchmark the MCTDHB method for (i) the ground state; (ii) the breathing many-body dynamics activated by a quench scenario where the interparticle interaction strength is suddenly turned on to a finite value; (iii) the nonequilibrium dynamic for driven scenarios where both the trap- and interparticle-interaction potentials are time-dependent. Excellent convergence of the ground state and dynamics is demonstrated. The great relevance of the self-consistency and time adaptivity, which are the intrinsic features of the MCTDHB method, is demonstrated by contrasting the MCTDHB predictions and those obtained within the standard full configuration interaction method spanning a Fock space of the same size, but utilizing as one-particle basis set the fixed-shape eigenstates of the one-particle potential. Connections of the model's results to ultracold Bose-Einstein condensed systems are addressed.

Received 20 July 2012

DOI:https://doi.org/10.1103/PhysRevA.86.063606

Authors & Affiliations

Axel U. J. Lode¹, Kaspar Sakmann¹, Ofir E. Alon², Lorenz S. Cederbaum¹, and Alexej I. Streltsov^1,*

¹Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany
²Department of Physics, University of Haifa at Oranim, Tivon 36006, Israel

^*Alexej.Streltsov@pci.uni-heidelberg.de

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 86, Iss. 6 — December 2012

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Numerical convergence of the self-consistent MCTDHB and fixed-orbital FCI (ED) methods for the ground-state energy of the HIM. Systems with $N = 2, 10, 50, 100$ , and 1000 bosons are considered; the strengths of the interparticle interactions $K_{0}$ have been chosen to keep $Λ = K_{0} (N - 1) = 0.5$ constant. We plot the relative differences between the total energies computed using the MCTDHB (filled symbols) and FCI (ED) (open symbols) many-body methods and respective exact energies in percents, $100 (E_{MB} - E_{exact}) / E_{exact}$ , for different orbital number $M$ . For a given $M$ both many-body methods span the same Fock space, i.e., the respective secular matrices to be diagonalized are of the same size. The advantage of the appropriate, i.e., self-consistent, choice of the one-particle basis functions is evident—the self-consistent MCTDHB method converges much faster than the fixed-orbital FCI (ED) one. Note the logarithmic scale and number of decades spanned. All quantities shown are dimensionless.Reuse & Permissions
Figure 2
A sudden change (quench) of the interparticle interaction leads to breathing dynamics of the system. We study the HIM system with $N = 2$ bosons where the interparticle interaction strength is quenched from zero to $K_{0} = 0.5$ . The evolution of the one-particle density at the origin $ρ (x = x^{'} \equiv 0)$ is plotted as a function of time. The exact dynamics reveal oscillations namely, with the main breathing frequency $ω_{breath} = 2 \sqrt{ω^{2} + 4 K_{0}}$ augmented by overtones $2 ω_{breath}$ , $3 ω_{breath}, ...$ (see text for more details). The solid (black) line depicts the guiding $\sim \cos (w_{breath} t)$ function. The numerical MCTDHB and FCI (ED) results are contrasted and compared with the exact ones, plotted by a bold (red) line. The left panel depicts the density oscillations at short times. At the FCI (ED) level an accurate description of the dynamics is achieved by using at least eight fixed-shape orbitals. To gain similar accuracy within the MCTDHB method one needs only three time-adaptive orbitals. Four time-adaptive orbitals [MCTDHB( $M = 4$ )] provide a numerically exact description. The right panel shows the density oscillations at longer times. To describe the dynamics in this case a larger Fock space (more orbitals) is (are) required. A numerically exact description is obtained by using six time-adaptive [MCTDHB(6)] or twelve fixed-shape orbitals [FCI(12)]. See text for further discussion. All quantities shown are dimensionless.Reuse & Permissions
Figure 3
Breathing dynamics of the HIM system with $N = 10$ for the same interaction quench scenario as in Fig. 2. The evolution of the one-particle density at the origin $ρ (x = x^{'} \equiv 0)$ is plotted; notice the different scales for the short and long times. The density oscillation is formed by the main breathing frequency $ω_{breath} = 2 \sqrt{ω^{2} + 2 N K_{0}}$ with strong contributions of the overtones $2 ω_{breath}, 3 ω_{breath}, ...$ (see text for more details). The solid (black) line depicts the guiding $\sim \cos (w_{breath} t)$ function. The MCTDHB( $M = 8$ ) method with eight time-adaptive orbitals provides a very accurate description of the breathing dynamics for the short and long times. The MCTDHB results for $M = 9$ and $M = 10$ are identical, indicating that the exact description has been numerically reached. The FCI( $M = 16$ ) results plotted by triangles start to deviate from the exact solution already for short times. The FCI (ED) method with 16 fixed-shape orbitals provides a reasonable description of the dynamics for a very short time only, i.e., it is incapable of describing more than a half of the first breathing cycle. The exact results could not be obtained in this model from the analytical solution, Eq. (7), because it is more difficult to perform the needed ten-dimensional integrations than to solve the problem numerically exactly by MCTDHB. All quantities shown are dimensionless.Reuse & Permissions
Figure 4
The HIM model with time-dependent trap $V (x) = ω_{0} [1 + f (t)] x^{2}$ and time-dependent interparticle interaction $W (x_{i} - x_{j}) = K_{0} [1 - \frac{ω_{0}^{2}}{2 N K_{0}} f (t)] {(x_{i} - x_{j})}^{2}$ permits exact solution. The exact expectation value of the total Hamiltonian of the system, Eq. (13), reads $〈 Ψ (t) | \hat{H} (t) | Ψ (t) 〉 = ε (t) + const .$ , with the time-independent constant equal to $\frac{D}{2} (N - 1) δ_{N}$ and $D = 1$ . The driving function $f (t) = f_{1} (t)$ and the time-dependent part of the energy $ε (t) = ε_{1} (t)$ for $N = 10$ bosons with $K_{0} = 0.5$ are plotted. The convergence of $ε_{1} (t)$ when increasing the number of the time-adaptive orbitals $M$ is depicted. The Gross-Pitaevskii results [GP $\equiv$ MCTDHB(1)], plotted by a bold solid line, are inaccurate even for very short time. The MCTDHB(3) provides excellent description up to $t \approx 5$ , the MCTDHB(5) works well till $t \approx 15$ , the MCTDHB(6) till $t \approx 30$ , and the MCTDHB(7) results coincide with the exact solution at all the times depicted. See text for discussion. All quantities shown are dimensionless.Reuse & Permissions
Figure 5
The modified HIM model with time-dependent trap and interparticle interaction driven by a complicated function $f_{2} (t)$ [Eq. (14)]. The function is depicted in the upper panel. The time-dependent contribution $ε_{2} (t)$ to the total energies is computed at several different levels of the MCTDHB( $M$ ) theory for $N = 10$ [ $M = 6, 7$ ] and $N = 50$ [ $M = 5, 6$ ] bosons. The strength of the interparticle interaction is $K_{0} = 0.5$ . The considered time dependency of the one- and two-body interaction potentials guarantees that the exact $ε_{2} (t)$ , plotted by open red circles, is the same for both systems. The MCTDHB(5) for $N = 50$ and MCTDHB(6) for $N = 10$ provide a converged description of the dynamics till $t \approx 25$ ; for longer times more orbitals are needed for numerical exactness. The corresponding Gross-Pitaevskii results, marked by arrows, are semiqualitative for very short initial times only, till $t \approx 1$ . See text for discussion. All quantities shown are dimensionless.Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information