Excitation spectra of fragmented condensates by linear response: General theory and application to a condensate in a double-well potential

Linear response of simple (i.e., condensed) Bose-Einstein condensates is known to lead to the Bogoliubov–de Gennes equations. Here, we derive linear response for fragmented Bose-Einstein condensates, i.e., for the case where the many-body wave function is not a product of one, but of several single-particle states (orbitals). This gives one access to excitation spectra and response amplitudes of systems beyond the Gross-Pitaevskii description. Our approach is based on the number-conserving variational time-dependent mean-field theory [Alon, Streltsov, and Cederbaum, Phys. Lett. A 362, 453 (2007)], which describes the time evolution of best-mean-field states. Correspondingly, we call our linear-response theory for fragmented states LR-BMF. In the derivation it follows naturally that excitations are orthogonal to the ground-state orbitals. As applications excitation spectra of Bose-Einstein condensates in double-well potentials are calculated. Both symmetric and asymmetric double wells are studied for several interaction strengths and barrier heights. The cases of condensed and twofold fragmented ground states are compared. Interestingly, even in such situations where the response frequencies of the two cases are computed to be close to each other, which is the situation for the excitations well below the barrier, striking differences in the density response in momentum space are found. For excitations with an energy of the order of the barrier height, both the energies and the density response of condensed and fragmented systems are very different. In fragmented systems there is a class of “swapped” excitations where an atom is transferred to the neighboring well. The mechanism of its origin is discussed. In asymmetric wells, the response of a fragmented system is purely local (i.e., finite in either one or the other well) with different frequencies for the left and right fragments. This finding is in stark contrast to that for condensed systems.

Figure 1

Schematic comparison between condensed, fragmented, and stable excited fragmented states in a double-well potential. The single orbital of a simple BEC [${\varphi }^0\left(x\right)$] is delocalized (left chart). For a fragmented BEC the orbitals corresponding to different fragments [${\varphi }_L^0\left(x\right)$ and ${\varphi }_R^0\left(x\right)$] are localized (right charts). For a given barrier height, the stable fragmented state (lower right chart) is lower in energy than the condensed one up to some (possibly very high) atom number. Then, the condensed state becomes the lowest in energy. However, an energetically close stable excited fragmented state (upper right chart) typically exists. It is separated by an energy barrier from the condensed one.

Figure 2

Excitation spectra of a BEC in a symmetric double-well potential ($a=0$) versus interaction strength with high barrier height ($b=20$). The total atom number is $N=100$, and the overall harmonic confinement is given by ${\omega }_{ho}=\sqrt{2}$. We compare the linear response of LR-GP, shown by the blue solid lines, and LR-BMF, shown by the red dashed lines for direct, and orange dotted lines for swapped excitations. The excitations are grouped into pairs of lines with gerade-ungerade symmetry and marked with numbers. The swapped excitations of LR-BMF are marked with primed numbers. Inset: We plot the potential by the black solid line. The corresponding ground-state orbital of GP for ${\lambda }_{0}N=10$ is shown by the blue solid, and the left orbital of BMF by the red dashed line. All quantities are dimensionless.

Figure 3

Sketch of the two types of excitations occurring for fragmented BECs in a double-well potential. Direct (swapped) excitations can be interpreted as the promotion of a boson from one orbital to an excited state which dominates in the same (other) well, e.g., from ${\varphi }_L$ to $u_L^1$ ($u_L^{1^{\prime }}$). The transfer of atoms to direct (red solid line) costs less energy than to swapped (orange dashed lines) excitations. This is because a depletion, related to $v_L^1$ (green dashed-dotted line), reduces the energy of direct excitations. For the swapped excitations, $v_L^{1^{\prime }}\approx 0$. Response weights of direct and swapped excitations of LR-BMF are proportional to the overlap between an orbital and the corresponding response amplitudes. Hopping excitations of LR-BH are proportional to the much smaller overlap of the two ground-state orbitals.

Figure 4

Example of response amplitudes ${\mathbf{u}}^k$ and ${\mathbf{v}}^k$ for the same double-well system as in Fig. 2 and for ${\lambda }_{0}N=10$. (a) The amplitudes $u$ and $v$ of LR-GP, shown by the blue solid and short-dashed lines, respectively, are symmetric functions. For LR-BMF, instead, they are localized functions. We show $u_L^{1g}$ by the red dashed line, and $v_L^{1g}$ by the red dotted line. (b) Amplitudes of the swapped excitations $u_L^{1g^{\prime }}$ and $v_L^{1g^{\prime }}$ of LR-BMF, shown by the orange dashed and dotted lines, respectively. See text for more details. All quantities are dimensionless.

Figure 5

Position-space densities and density oscillations for the same double-well system as in Fig. 2. The interaction strength is ${\lambda }_{0}N=10$ in the left panels and ${\lambda }_{0}N=20$ in the right panels. The ground-state density of GP is shown in (a) and (b) by the broad gray solid line and is scaled for better comparison to the density of BMF plotted by the broad black dashed line. The GP and BMF results are seen to coincide. Gerade density oscillations of the indicated excited states are shown in (a)–(f). The LR-GP results are shown by the blue solid lines. The dashed red lines (orange dashed-dotted lines) show results of LR-BMF for direct (swapped) excitations. See text for more details. All quantities are dimensionless.

Figure 6

Momentum space densities and density oscillations for the same double-well system as in Fig. 2. The interaction strength is ${\lambda }_{0}N=10$. The ground-state density of GP is shown in (a) by the broad gray solid line and is scaled for better comparison to the ground-state density of BMF, which is shown by the broad black dashed line. In momentum space they are completely different. Momentum space density oscillations of gerade type of the indicated excited states (see Fig. 5) are shown in (a), (d), and (g), and those of ungerade type in (b), (e), and (h). Blue solid lines correspond to density oscillations of LR-GP and red dashed lines to direct excitations of LR-BMF. Swapped excitations of LR-BMF are shown in (c), (f), and (i). Orange solid (dashed) lines correspond to gerade-type (ungerade-type) density oscillations. See text for more details. All quantities are dimensionless.

Figure 7

Example of position-space density oscillations for the same double-well system as in Fig. 2 and for ${\lambda }_{0}N=20$. Gerade [see Fig. 5f] as well as ungerade density oscillations of LR-GP are shown by the blue solid and dashed-dotted lines, respectively. Direct gerade (swapped ungerade) density oscillations of LR-BMF are shown by the red dashed (orange dashed-double dotted) lines, and are multiplied by a factor of $\sqrt{2}$ for easier comparison. See text for more details. All quantities are dimensionless.

Figure 8

Excitation spectra of a BEC in a symmetric double-well potential ($a=0$) versus barrier height for a fixed interaction strength ${\lambda }_{0}N=20$. The total atom number is $N=100$, and the overall harmonic confinement given by ${\omega }_{ho}=\sqrt{2}$. We compare the linear response of LR-GP, shown by the blue solid lines, and LR-BMF, shown by the red dashed lines for direct, and orange dotted lines for swapped excitations. The excitations are grouped into pairs of lines with gerade-ungerade symmetry and marked with numbers. The swapped excitations of LR-BMF are marked with primed numbers. Inset: We plot the potential by the black solid line. The corresponding ground-state orbital of GP for $b=14$ is shown by the blue solid, and the left orbital of BMF by the red dashed line. All quantities are dimensionless.

Figure 9

Position-space densities and density oscillations for the same double-well system as in Fig. 8 for a low barrier height $b=14$. The ground-state density of GP is shown in (a) by the broad gray solid line and is scaled for better comparison to the BMF density shown by the broad black-dashed line. The GP and BMF results are seen to almost coincide. The blue solid lines in (a), (c), and (d) show density oscillations of LR-GP. The red dashed lines in (c) and (d) show density oscillations of direct excitations of LR-BMF. (a) Low-lying excitation 0 of LR-GP, which is absent for LR-BMF. (b) The gerade and ungerade swapped excitations of LR-BMF, marked as $1^{\prime }$ and shown by the orange solid and dashed lines, respectively. (c) The gerade and (d) the ungerade excitations of LR-GP and LR-BMF, marked as 1. See text for more details. All quantities are dimensionless.

Figure 10

Momentum space densities and density oscillations for the same double-well system as in Fig. 8 for a low barrier height $b=14$. The ground-state density of GP is shown in (a) by the broad gray solid line and is scaled for better comparison to the one of BMF, which is shown by the broad black dashed line. We show the same indicated excited states as in Fig. 9. The blue solid lines in (a), (c), and (d) show density oscillations of LR-GP. The red dashed lines in (c) and (d) show results for direct excitations of LR-BMF. (a) Low-lying excitation 0 of LR-GP, which is absent for LR-BMF. (b) The gerade-type and ungerade-type swapped excitations of LR-BMF, marked as $1^{\prime }$ and shown by the orange solid and dashed lines, respectively. (c) The gerade-type and (d) the ungerade-type excitations of LR-GP and LR-BMF, marked as 1. See text for more details. All quantities are dimensionless.

Figure 11

Excitation spectra of a BEC in an asymmetric double-well potential ($a=0.1$) versus occupation of the left orbital. We choose a high barrier height $b=20$ and interaction strength ${\lambda }_{0}N=10$. The total atom number is $N=100$, and the overall harmonic confinement given by ${\omega }_{ho}=\sqrt{2}$. The top of the filled gray area shows the BMF ground-state energies as a function of occupation of the left orbital. The vertical dashed line marks the location of the optimal occupation $n_L/N\approx 0.55$. We compare the linear response of LR-GP, shown by the blue solid lines, and LR-BMF, shown by the red dashed lines for direct, and orange dotted lines for swapped excitations. The excitations are grouped into pairs of lines with gerade-ungerade symmetry and marked with numbers. The swapped excitations of LR-BMF are marked with primed numbers. Inset: We plot the potential by the black solid line. The corresponding ground-state orbital of GP at the optimal occupation is shown by the blue solid line. The left (right) orbital of BMF is shown by the red dashed (short-dashed) line. All quantities are dimensionless.

Figure 12

Position-space densities and density oscillations for the same double-well system as in Fig. 11 at the optimal occupation $n_L/N\approx 0.55$. The ground-state density of GP is shown in (a) by the broad gray solid line and is scaled for better comparison to the one of BMF, which is shown by the broad black line in (b). The density oscillations for an indicated excited state show up two lines for LR-GP (see Fig. 11). The lower in energy density oscillations are shown by the blue solid, and those higher in energy by the blue dotted lines in panels (a), (c), and (e). A similar statement holds for the direct excitations of LR-BMF, where the respective lower excited states are shown by the red solid, and the respective upper ones by the red dotted lines in panels (b), (d), and (f). The lower (upper) swapped excitation are shown by the orange dashed (dashed-dotted) lines in (b), (d), and (f). See text for more details. All quantities are dimensionless.

Figure 13

Momentum space densities and density oscillations for the same double-well system as in Fig. 11 at the optimal occupation $n_L/N\approx 0.55$. The ground-state density of GP is shown in (a) by the broad gray solid line and is scaled for better comparison to the one of BMF, which is shown by the broad black dashed line. We show the same indicated excitations as in Fig. 12. The density oscillations for a given number show two lines for LR-GP (see Fig. 11). The lower in energy density oscillations are shown by the blue solid lines in (a), (d), and (g), and those higher in energy by the blue dotted lines (b), (e), and (h). A similar statement holds for the direct excitations of LR-BMF, where the respective lower excited states are shown by the red dashed lines in (a), (d), and (g), and the respective upper ones in (b), (e), and (h). The lower (upper) swapped excitations of LR-BMF are shown by the orange solid (dashed) lines in (c), (f), and (i). See text for more details. All quantities are dimensionless.