Excitation spectra of many-body systems by linear response: General theory and applications to trapped condensates

We derive a general linear-response many-body theory capable of computing excitation spectra of trapped interacting bosonic systems, e.g., depleted and fragmented Bose-Einstein condensates (BECs). To obtain the linear-response equations we linearize the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method, which provides a self-consistent description of many-boson systems in terms of orbitals and a state vector (configurations), and is in principle numerically exact. The derived linear-response many-body theory, which we term LR-MCTDHB, is applicable to systems with interaction potentials of general form. For the special case of a $\delta$ interaction potential we show explicitly that the response matrix has a very appealing bilinear form, composed of separate blocks of submatrices originating from contributions of the orbitals, the state vector (configurations), and off-diagonal mixing terms. We further give expressions for the response weights and density response. We introduce the notion of the type of excitations, useful in the study of the physical properties of the equations. From the numerical implementation of the LR-MCTDHB equations and solution of the underlying eigenvalue problem, we obtain excitations beyond available theories of excitation spectra, such as the Bogoliubov–de Gennes (BdG) equations. The derived theory is first applied to study BECs in a one-dimensional harmonic potential. The LR-MCTDHB method contains the BdG excitations and, also, predicts a plethora of additional many-body excitations which are out of the realm of standard linear response. In particular, our theory describes the exact energy of the higher harmonic of the first (dipole) excitation not contained in the BdG theory. We next study a BEC in a very shallow one-dimensional double-well potential. We find with LR-MCTDHB low-lying excitations which are not accounted for by BdG, even though the BEC has only little fragmentation and, hence, the BdG theory is expected to be valid. The convergence of the LR-MCTDHB theory is assessed by systematically comparing the excitation spectra computed at several different levels of theory.

Figure 1

Comparison of the standard LR-GP (BdG) and LR-MCTDHB(2) linear-response spectra computed for the systems of $N=100$ (lower two panels) and $N=10$ (upper two panels) bosons trapped in a one-dimensional harmonic potential and subject to linear and quadratic perturbations. The trap potential and ground-state density (red solid line) are shown in the insets. The lower part of each panel displays excitation frequencies on the $x$ axis and response weights, Eq. (70), on the $y$ axis for linear (odd) $f^{+}=f^{-}=x$ and quadratic (even) $f^{+}=f^{-}=x^2$ perturbations depicted by the solid (red triangle) and dashed (green squares) lines (symbols), respectively. The mean-field-based excitations, well described by both theories, are labeled 1,2,3. The many-body excitations, not describable by the BdG theory, are labeled as $2^{\prime }$ and $3^{\prime }$ (for more details, see the text and Fig. 2). The upper parts of the panels display the type of excitation $t_k$ computed as in Eq. (75): $t_k=0$ implies a purely orbital-like excitation, and $t_k=1$, a purely CI one. All quantities are dimensionless.

Figure 2

Density responses for $N=10$ bosons in a one-dimensional harmonic potential. Plotted are the real parts of the response densities $\Delta \rho \left(x\right)=\mathrm{Re}[\Delta {\rho }_o^k\left(x\right)+\Delta {\rho }_c^k\left(x\right)]$ corresponding to real-valued ${\gamma }_k$'s; see text for details. All excitations are numerated according to the underlying nodal structures (number of nodes) in the respective response density. The LR-GP (BdG) results for the first few excitations are depicted in the left column of the figure. The LR-MCTDHB results for these excitations are quite similar and, therefore, not shown. The many-body excitations, not describable by the BdG theory, are labeled by the numbers with a prime, and magnified for better visibility. All quantities are dimensionless.

Figure 3

Convergence of the LR-MCTDHB($M$) theory. Comparison of the linear-response spectra for $N=10$ bosons in a one-dimensional harmonic potential subject to linear and quadratic perturbations computed at the LR-MCTDHB(4) and LR-MCTDHB(5) levels of theory. The depicted response weights and state characterizations are done in the same way as in Fig. 1. The main observation is that the lowest-in-energy excitations are numerically converged already at a low level of the LR-MCTDHB($M$) theory. By increasing $M$ we improve the quality of description and converge the higher-energy excitations as well; see text, Table , and Fig. 4 for further details. All quantities are dimensionless.

Figure 4

Convergence of the center-of-mass excitations of $N=10$ bosons in the harmonic potential as a function of $M$ (data combined from Figs. 1 and 3). The horizontal lines are the exact analytical values. All quantities are dimensionless.

Figure 5

Comparison of the standard LR-GP (BdG) and LR-MCTDHB(2) linear-response spectra computed for the systems of $N=10$ (upper two panels) and $N=100$ (lower two panels) bosons trapped in a very shallow one-dimensional double-well potential, Eq. (77), and subject to linear and quadratic perturbations. The trap potential and ground-state density (red solid line) are shown in the insets. The depicted response weights and the scheme of the state enumeration and characterizations are the same as in the harmonic case; see Fig. 1 and text. The key observation here is that the number of low-lying excitations appearing on the many-body level and not described by the BdG theory is much larger in comparison with the harmonic-trap results. All quantities are dimensionless.

Figure 6

Density responses for $N=10$ atoms in a very shallow one-dimensional double-well potential, Eq. (77). Plotted are the real parts of the response densities $\Delta \rho \left(x\right)=\mathrm{Re}[\Delta {\rho }_o^k\left(x\right)+\Delta {\rho }_c^k\left(x\right)]$ corresponding to real-valued ${\gamma }_k$'s; see discussion in the text. All the excitations are numerated according to the underlying nodal structures (number of nodes) in the respective response density. The LR-GP (BdG) results for the first few excitations are depicted in the left column of the figure. The LR-MCTDHB results for these excitations are quite similar and, therefore, not shown. The many-body excitations, not describable by the BdG theory, are labeled by the numbers with a prime, and magnified for better visibility. All quantities are dimensionless.

Figure 7

Convergence of LR-MCTDHB($M$) theory. Comparison of the linear-response spectra for $N=10$ bosons in a very shallow one-dimensional double-well potential, Eq. (77), and subject to linear and quadratic perturbations. The depicted LR-MCTDHB(4) and LR-MCTDHB(5) response weights and state characterizations are obtained in the same way as in Fig. 1; see text for more details. The main observation is that the lowest-in-energy excitations are numerically converged already at a low level of the LR-MCTDHB($M$) theory. By increasing $M$ we improve the quality of description of the higher-energy excitations as well. All quantities are dimensionless.