Relation between the eigenfrequencies of Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian in a time-dependent variational approach

We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian stability matrix in a variational approach that maps the Gross-Pitaevskii equation to a system of equations of motion for the variational parameters. We do this for Bose-Einstein condensates with attractive contact interaction in an external trap and for a simple model of a self-trapped Bose-Einstein condensate with attractive $1/r$ interaction. The stationary solutions of the Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of the ground and excited state of the self-trapped monopolar condensate exhibit a Rydberg-like structure, which can be explained by means of a quantum-defect theory. On the variational side, we treat the problem using an ansatz of time-dependent coupled Gaussian functions combined with spherical harmonics. We first apply this ansatz to a condensate in an external trap without long-range interaction and calculate the excitation spectrum with the help of the time-dependent variational principle. Comparing with the full-numerical results, we find good agreement for the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational method is then applied to calculate the excitations of the self-trapped monopolar condensates and the eigenfrequencies of the excitation modes are compared.

Figure 1

Mean-field energy $E_{\mathit{MF}}$ and chemical potential $\mu$ of the ground and excited state of a self-trapped monopolar condensate as functions of the scattering length $a$. For a scattering length lower than the critical value of $a_{\mathrm{crit}}\approx -1.025$ no stationary solution exists. At $a=a_{\mathrm{crit}}$ the two solutions emerge in a tangent bifurcation. For the ground state, both $E_{\mathit{MF}}$ and $\mu$ stay negative in the range of the scattering length considered. These quantities diverge for the excited state in the limit $a\to 0$.

Figure 2

Frequencies of Bogoliubov excitations of the ground state in Fig. 1 for the angular momenta from $l=0$ to 3 as functions of the scattering length $a$. The seven lowest eigenvalues are shown for each angular momentum. The spectrum only contains real frequencies, i.e., the ground state is stable. The lowest mode for $l=0$ tends to zero as $a\to a_{\mathrm{crit}}$, which leads to the collapse of the condensate. The lowest $l=1$ mode corresponds to a displacement of the center of mass of the condensate, while its shape remains unaffected. The frequency of this mode is exactly the trapping frequency [24], in this case $\omega =0$. The frequencies of the other modes increase as the scattering length is decreased, finally merging with the modes of the excited state for $a\to a_{\mathrm{crit}}$ (see Fig. 3). Note that for fixed scattering length the distance between adjacent frequencies diminishes with growing radial quantum number, indicating the convergence of the frequencies to a (scattering-length-dependent) limit frequency.

Figure 3

Same as Fig. 2, but for the excited state. There is one imaginary frequency for $l=0$: The excited state is unstable with respect to small perturbations. As for the ground state, the lowest $l=1$ mode is $\omega =0$ and corresponds to a displacement of the center of mass of the condensate. Again, the frequencies of the stable modes apparently converge to a limit for fixed scattering length.

Figure 4

Bogoliubov functions $u_{nl}\left(r\right)$ and $v_{nl}\left(r\right)$ for the angular momentum $l=0$ and the radial quantum numbers $n=1,\cdots ,5$. The scattering length is $a=-0.4$. The mode with $n=1$ represents the gauge mode discussed in Sec. 2b and the functions $u_{10}\left(r\right)$ and $v_{10}\left(r\right)$ are equal to the stationary solution ${\psi }_0$, except for the sign. The functions $u_{n0}\left(r\right)$ have $n-1$ nodes, whereas all functions $v_{n0}\left(r\right)$ show qualitatively the same behavior and are nodeless for all $n$. It can be seen that with growing radial quantum number the functions $u_{n0}\left(r\right)$ extend to ever increasing values of $r$.

Figure 5

Frequencies of the Bogoliubov excitations of the ground state of a self-trapped monopolar BEC for a fixed scattering length $a=-0.4$, plotted for different values of the angular momentum. The dotted line gives the value of the chemical potential. Obviously, as shown in Figs. 2 and 3, the frequencies converge to a common limit, which is the chemical potential. The Bogoliubov spectrum can be described by a Rydberg formula with quantum defects.

Figure 6

(a) Wave function ${\psi }_0$ and (b) mean-field potential ${\phi }_0$ for the ground state of a self-trapped monopolar condensate at a scattering length of $a=-0.4$ as functions of the radial coordinate $r$. The wave function approaches zero exponentially, whereas the mean-field potential behaves like $-2/r$ for large values of $r$. In this region, the wave function can be neglected and the mean-field potential replaced by its asymptotic form in the BDG equations.

Figure 7

Calculated quantum defects ${\delta }_l$ for the Bogoliubov spectrum of the ground state (Fig. 2) dependent on the scattering length $a$ for different angular momenta $l$. The quantum defects for $l=0$ and 1 rise steeply and turn from negative to positive as the scattering length is decreased, while the quantum defect for $l=2$ shows only a weak dependence on the scattering length and drops close to the critical scattering length. As expected, for the higher angular momenta $l>2$ the quantum defects are close to zero.

Figure 8

Same as Fig. 7, but for the excited state. Since the Bogoliubov spectra of the ground and excited states merge at the critical scattering length, the same holds for the quantum defects. The quantum defects ${\delta }_0$ and ${\delta }_1$ grow as the scattering length is increased, whereas ${\delta }_2$ drops and tends to zero. As in the case of the ground state, the quantum defects for $l>2$ are close to zero.

Figure 9

Comparison of the full-numerical Bogoliubov spectrum of a BEC with attractive contact interaction with the spectrum obtained from the variational ansatz with coupled Gaussian functions and spherical harmonics (SH). The variational ansatz has been used with five coupled Gaussian functions and spherical harmonics up to an angular momentum of $l=3$. For the lowest modes we find excellent agreement. There are almost no deviations for frequencies $\omega <10$. Just slightly above the critical scattering length small differences can be seen in the figure. For the higher modes, differences become larger and the variational ansatz can only qualitatively describe the Bogoliubov modes correctly.

Figure 10

Comparison of both spectra as in Fig. 9, but here for a fixed scattering length of $a=-0.4$ and angular momenta up to $l=6$. For $l=0$ the variational ansatz reproduces the Bogoliubov frequencies very well for the four lowest modes and with only small deviations for the two lowest modes in the higher angular momentum bands.

Figure 11

Comparison of the full-numerical Bogoliubov spectrum of the ground state of a self-trapped monopolar BEC with the spectrum obtained from the variational ansatz with coupled Gaussian functions and SH. The variational ansatz has been used with six coupled Gaussian functions and spherical harmonics up to an angular momentum of $l=3$. For the lowest $l=0$ and 1 modes the results of both methods almost cannot be distinguished. The differences in the frequencies of the second lowest $l=0$ and 1 and the lowest $l=2$ modes are small in the range of the scattering length considered. The lowest $l=3$ mode is well approximated by the variational ansatz, but the differences in frequency are larger compared to the frequencies belonging to lower angular momenta. For the higher modes there is no quantitative agreement.

Figure 12

Comparison of both spectra as in Fig. 11 for a self-trapped monopolar BEC at the fixed scattering length of $a=-0.4$ and angular momenta up to $l=6$. For $l=0$ and 1 the two lowest modes and for $l=2$ and 3 only the lowest modes agree well. For higher angular momenta $l⩾5$ the lowest mode lies even above the limit of the numerical Bogoliubov spectrum (compare with Fig. 5).