# Dipole modes of a trapped bosonic mixture: Fate of the sum-rule approach

Chiara Fort  Dipartimento di Fisica e Astronomia and LENS, Università di
Firenze, 50019 Sesto Fiorentino, Italy Istituto Nazionale di Ottica, CNR-INO,
50019 Sesto Fiorentino, Italy  Michele Modugno  Department of Physics,
University of the Basque Country UPV/EHU, 48080 Bilbao, Spain  IKERBASQUE,
Basque Foundation for Science, 48013 Bilbao, Spain  EHU Quantum Center,
University of the Basque Country UPV/EHU, 48940 Leioa, Biscay, Spain

###### Abstract

We present a general discussion of the dipole modes of a heteronuclear bosonic
mixture in a harmonic trap, by comparing the prediction of the sum-rule
approach with full Gross-Pitaevskii (GP) calculations. Yet in the range of
interaction in which the mixture is miscible and stable at the mean-field
level, $g_{12}^{2}<g_{11}g_{22}$, we find that there are regimes where the sum
rules display significant deviations from the GP solution. We provide an
interpretation for this behavior in terms of the Ehrenfest approach, which we
show to be equivalent to the sum-rule approach for dipole excitations.

##  I Introduction

Sum rules are a powerful and widely employed tool for investigating the
collective modes in different many-body systems, including nuclear collective
states [1, 2] and giant resonances in nuclei [3], Fermi liquids at low
temperatures [4], free electrons in metal spheres, and cavities and shells
[5], among others. Since the first experimental realization of Bose-Einstein
condensation in trapped dilute gases (see, e.g., [6]), and the subsequent
explosion of ultracold atom physics, the use of the sum-rule approach has
become customary in this field [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21]. In particular, it has been broadly used for describing collective
excitations of Bose-Fermi [9, 12, 13, 14, 15, 17] and Bose-Bose [16, 18, 19]
mixtures. The advantage of this approach is that the frequency of the
collective modes can be estimated by means of algebraic techniques, without
having to solve the equation of motion, based only on the knowledge of the
ground state solution [22]. At the same time, this also constitutes a limit of
this approximate method, since it cannot account for dynamical effects that
may break the self-similarity of the density distribution.

In order to shed some light on the above point, as a proof of concept here we
consider the case of the dipole mode of a mixture of Bose-Einstein condensates
(BECs) that was recently considered in a number of experiments [23, 19, 24,
25]. It is worth noticing that the collective excitation spectrum of a two-
species BEC has been originally addressed within the framework of the
Bogoliubov theory [26], but a general discussion of the sum-rule approach for
this case is still lacking, contrary to the case of Bose-Fermi mixtures. The
latter was first considered in Ref. [9] for the case of symmetric mixtures
(same mass and trapping frequency) with tunable interactions, and then
extended to the heteronuclear case (different masses) in Ref. [15].
Remarkably, in Refs. [12, 14] it was also shown that the sum-rule approach may
not always provide an accurate prediction for the dipole oscillation
frequencies as obtained from the full dynamical equations. This deviation is
particularly noticeable in the repulsive case, especially for the fermionic
component. Indeed, the different statistics of the Bose-Fermi mixture and the
large spatial extension of the Fermi clouds resulting in different regions
where the two clouds are superimposed or not, further complicates the coupled
dipole dynamics as the Fermi component may be characterized by several
oscillations modes [14].

In light of the above considerations, it is interesting to address to what
extent the sum-rule approach can be considered a suitable approximate method
for describing the collective dipole oscillations of a bosonic mixture. To
this end, we first extend the sum-rule formalism of Refs. [9, 15] to the
general case of a Bose-Bose degenerate gas mixture with different masses,
frequencies, and atom numbers. Then, we show that the sum-rule predictions for
the dipole frequencies can also be obtained from an effective approach based
on the Ehrenfest theorem [27], which also provides the amplitudes of the modes
and turns out to be helpful also for understanding the limits of validity of
these approximations. These results are compared with the frequency and
amplitudes of the dipole modes extracted from the full Gross-Pitaevskii
dynamics as a function of the interspecies interaction, in the whole regime in
which the mixture is miscible and stable against mean-field collapse.

Motivated by a recent experiment by Cavicchioli et al. [25], we consider in
particular the case of a 41K-87Rb mixture, discussing also the effect of the
atom number imbalance. We find that, for a balanced mixture, the sum-rule and
Ehrenfest approaches provide an accurate prediction for the low-lying dipole
mode, from the strongly attractive regime up to weakly repulsive interactions.
Instead, the high-lying mode is well reproduced in the opposite regime, from
weak attraction up to the repulsive regime, provided the system does not enter
(local) phase separation. This sort of specularity stems from the fact that
the two oscillation modes exchange their character from in-phase to out-of-
phase behavior when crossing the noninteracting point at $a_{12}=0$. For
imbalanced mixtures, we find that the accuracy of the sum-rule and Ehrenfest
approaches gets reduced depending on the degree of superposition of the two
clouds, similar to the case of a Bose-Fermi mixture [14].

This paper is organized as follows. In Sec. II we present the general
formalism, namely we review the sum-rule approach (Sec. II.1), we introduce
the GP formalism, and we discuss the effective description of the dipole
dynamics based on the Ehrenfest approach (Sec. II.2). In Sec. III we work out
a thorough comparison between the exact GP results and the predictions of the
sum-rule and Ehrenfest approaches, for the case of a heteronuclear Rb-K
mixture. Final remarks are given in the conclusions.

##  II General formalism

Let us consider a binary mixture of degenerate gases, each consisting of
$N_{i}$ atoms with masses $m_{i}$ ($i=1,2$), confined in harmonic potentials
$V_{i}^{ho}$ with frequencies $\omega_{ki}$ ($k=x,y,z$). The total Hamiltonian
of the system can be written as

$H=\sum_{i=1}^{2}H_{i0}+H_{int},$ (1)

with

$H_{i0}=\sum_{\ell=1}^{N_{i}}\left[\frac{\bm{p}_{i\ell}^{2}}{2m_{i}}+V_{i}^{ho}(\bm{r}_{i\ell})+g_{ii}\sum_{\ell^{\prime}<\ell}\delta(\bm{r}_{i\ell}-\bm{r}_{i\ell^{\prime}})\right],$
(2)
$H_{int}=g_{12}\sum_{\ell=1}^{N_{1}}\sum_{\ell^{\prime}=1}^{N_{2}}\delta(\bm{r}_{1\ell}-\bm{r}_{2\ell^{\prime}}).$
(3)

The coupling constants are $g_{ii}=4\pi\hbar^{2}a_{ii}/m_{i}$ and
$g_{12}=2\pi\hbar^{2}a_{12}/\mu$, with $\mu=m_{1}m_{2}/(m_{1}+m_{2})$ being
the reduced mass of the system and $a_{12}$ being the interparticle scattering
lengths.

The dipole operator along a certain direction, let us say $x$, can be defined
as

$D[\alpha_{1},\alpha_{2}]\equiv\sum_{i=1}^{2}\alpha_{i}\sum_{\ell=1}^{N_{i}}x_{i\ell}\equiv\sum_{i=1}^{2}\alpha_{i}D_{i},$
(4)

where $\alpha_{i}$ parametrize the mixing between the two species.

###  II.1 Sum-rule approach

An estimate of the oscillation frequencies corresponding to the above dipole
operator can be obtained by means of the sum-rule approach, in terms of the
energy weighted moments $\mu_{1}$ and $\mu_{3}$ [2, 4, 3, 9],

$\omega^{2}=\frac{1}{\hbar^{2}}\frac{\mu_{3}}{\mu_{1}},$ (5)

which can be evaluated in terms of the commutators

$\displaystyle\mu_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle\left[D,[H,D]\right]\rangle,$ (6)
$\displaystyle\mu_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle\left[[D,H,],[H,[H,D]]\right]\rangle,$ (7)

yielding

$\omega^{2}=\frac{\displaystyle\sum_{i=1}^{2}\frac{\alpha^{2}_{i}N_{i}}{m_{i}}\left(\omega^{2}_{i}-\frac{g_{12}I}{m_{i}N_{i}}\right)+2\alpha_{1}\alpha_{2}\frac{g_{12}I}{m_{1}m_{2}}}{\sum_{i=1}^{2}{\alpha^{2}_{i}N_{i}}/{m_{i}}},$
(8)

where $\omega_{i}\equiv\omega_{xi}$ (the result is independent of the trapping
frequencies along the directions orthogonal to that of the dipole operator
$D$) and the volume integral
$I\equiv\int\partial_{x}n_{01}\partial_{x}n_{02}d^{3}r$ is determined by the
ground-state configuration of the system characterized by the densities
$n_{0i}$. This clearly highlights the pros and cons of the sum-rule approach:
On the one hand, it provides an estimate of the dipole mode frequencies on the
basis of only static properties; on the other hand, it may fail if the system
configuration substantially changes during the dynamical evolution (see
below).

Following Ref. [9] the weights $\alpha_{i}$ can be conveniently parametrized
as $\alpha_{1}=\left(\cos\theta+\sin\theta\right)/\sqrt{2}$,
$\quad\alpha_{2}=\left(\cos\theta-\sin\theta\right)\sqrt{2}$, so that the
total dipole operator $D$ can be written as a linear combination of in-phase
and out-of-phase oscillations of the two species, with $\theta$ playing the
role of the mixing angle

$D=\frac{1}{\sqrt{2}}(D_{1}+D_{2})\cos\theta+\frac{1}{\sqrt{2}}(D_{1}-D_{2})\sin\theta.$
(9)

Then, Eq. (8) becomes

$\omega^{2}(\theta)=\frac{B_{+}+B_{-}\sin 2\theta+2\gamma_{12}\cos
2\theta}{A_{+}+A_{-}\sin 2\theta},$ (10)

with $\gamma_{12}={g_{12}I}/{m_{1}m_{2}}$ and

$\displaystyle A_{\pm}$
$\displaystyle=\frac{N_{1}}{m_{1}}\pm\frac{N_{2}}{m_{2}},$ (11) $\displaystyle
B_{\pm}$
$\displaystyle=\left(\frac{\omega_{1}^{2}N_{1}}{m_{1}}\pm\frac{\omega_{2}^{2}N_{2}}{m_{2}}\right)-g_{12}I\left(\frac{1}{m_{1}^{2}}\pm\frac{1}{m_{2}^{2}}\right).$
(12)

The above set of equations (10)-(12) constitutes the analog of the Bose-Fermi
results of Refs. [9, 15], here for the case of a bosonic mixture trapped in a
generic harmonic potential (with the two species having different masses,
frequencies, and atom numbers) 111One can easily check that in the special
case of spherically symmetric potentials, with the same frequencies
($\omega_{1}=\omega_{2}\equiv\omega_{0}$) and masses ($m_{1}=m_{2}\equiv m$)
for the two species, Eq. (10) reproduces the result in Eq. (6) of Ref. [9],
namely
$\omega^{2}(\theta)=\omega_{0}^{2}-({4g_{12}I}/{m})\sin^{2}\theta/[N_{1}+N_{2}+(N_{1}-N_{2})\sin
2\theta]$, with $I=\Omega/3$.

Finally, one has to choose a criterion for determining the possible mixing
angles and the corresponding frequencies. First of all, since we have two
components, two different modes are expected, one “in phase” and the other one
“out of phase” (see also the discussion in the next section). Then,
considering that sum rules provide an upper bound for the mode frequencies
(see, e.g., Ref. [22]), it is natural to identify the mixing angle of the low-
lying mode with the one that minimizes $\omega^{2}(\theta)$, for which
$\partial\omega^{2}/\partial\theta=0$. In Ref. [9] the other mode was taken to
be orthogonal to the former, but this is not necessarily so, because in
general they are eigenmodes of a non-symmetric, but diagonalizable, matrix.
This will be clear from the discussion in Sec. II.2. Therefore, here we
conjecture that both frequencies of the dipole modes can be obtained with the
stationary condition

$\displaystyle\frac{\partial\omega^{2}}{\partial\theta}=$
$\displaystyle\left(A_{+}B_{-}-A_{-}B_{+}\right)\cos 2\theta$
$\displaystyle-2\gamma_{12}\left(A_{-}+A_{+}\sin 2\theta\right)=0,$ (13)

whose solution will be denoted as $\omega_{\pm}$ (with mixing angles
$\theta_{\pm}$). Notice that in the non-interacting limit ($g_{12}=0$) we get
$\omega_{-}=\omega_{1}$ ($\theta={\pi}/{4}$) and $\omega_{+}=\omega_{2}$
($\theta={3\pi}/{4}$), which fixes the hierarchy of the two frequencies.
Therefore, it is convenient to set $\omega_{1}\leq\omega_{2}$ in order to
identify $\omega_{-}$ and $\omega_{+}$ with the frequencies of low-lying and
high-lying modes, respectively.

###  II.2 Gross-Pitaevskii equation and the Ehrenfest theorem

Let us now contextualize the above results within the framework of the Gross-
Pitaevskii (GP) theory, namely the case in which the binary mixture is
composed of two Bose-Einstein condensates. Their trapped evolution is
described by a set of (three-dimensional) GP equations of the form

$i\hbar\partial_{t}\psi_{i}(\bm{r},t)=\left[-\frac{\hbar^{2}}{2m_{i}}\nabla^{2}+V^{tot}_{i}(\bm{r},t)\right]\psi_{i}(\bm{r},t)$
(14)

where $V^{tot}_{i}$ represents the total mean-field potential acting on the
species $i$ ($\neq j$),

$V^{tot}_{i}(\bm{r},t)=V_{i}^{ho}(\bm{r})+g_{ii}n_{i}(\bm{r},t)+g_{ij}n_{j}(\bm{r},t),$
(15)

with $n_{i}(\bm{r},t)\equiv|\psi_{i}(\bm{r},t)|^{2}$ and $\int
n_{i}d^{3}r=N_{i}$. For simplicity here we consider the harmonic potentials
acting on the two species to be concentric.

It is interesting to note that the sum-rule approach discussed in the previous
section is equivalent to the prediction provided by the Ehrenfest theorem,
namely,

$\frac{d^{2}}{dt^{2}}\langle
x\rangle_{i}=-\frac{1}{m_{i}}\langle\partial_{x}V^{tot}_{i}\rangle.$ (16)

This goes as follows. By defining $\delta x_{i}(t)\equiv\langle
x\rangle_{i}(t)$ it is straightforward to get 222We have used the fact that
$\int n_{i}\partial_{x}n_{i}d^{3}r=(1/2)\int d(n_{i}^{2})d^{3}r\equiv 0$
because of the vanishing boundary conditions.

$\ddot{\delta x_{i}}(t)=-\omega^{2}_{i}\delta
x_{i}(t)-\frac{g_{ij}}{m_{i}N_{i}}\int n_{i}\partial_{x}n_{j}d^{3}r.$ (17)

The integral in the last term can be evaluated explicitly by assuming that the
densities translate rigidly, $n_{i}(x,t)=n_{0i}[x-\delta x_{i}(t)]$ (for ease
of notation we omit the dependence on the “transverse” coordinates $y$ and
$z$). This assumption is consistent with the sum-rule approach in the sense
that it relies on the information encoded only in the ground state. Then, in
the limit of small oscillations, we have

$n_{i}(x,t)\simeq n_{0i}(x)-\partial_{x}n_{0i}(x)\delta x_{i}(t).$ (18)

Within these approximations, by defining $\eta_{i}\equiv
g_{12}I/(m_{i}N_{i})$, Eq. (17) can be recast in the following form 333A
related approach, based on the Boltzmann equation for the distribution
functions of a Bose-Fermi mixture in the normal phase, can be found in Ref.
[39].

$\ddot{\delta x_{i}}=-\left(\omega^{2}_{i}-\eta_{i}\right)\delta
x_{i}-\eta_{i}\delta x_{j},$ (19)

or, equivalently, in matrix form as

$\ddot{\delta x}=-M\delta x,\quad\delta x\equiv\begin{pmatrix}\delta x_{1}\\\
\delta x_{2}\end{pmatrix}$ (20)

and

$M=\begin{pmatrix}\omega_{1}^{2}-\eta_{1}&\eta_{1}\\\
\eta_{2}&\omega_{2}^{2}-\eta_{2}\end{pmatrix}.$ (21)

The frequencies $\omega_{\pm}$ of the two dipole modes are obtained as
solutions of

$\det\left(M-\omega^{2}I_{2}\right)=0,$ (22)

namely

$\omega^{2}_{\pm}=\frac{1}{2}\sum_{i}(\omega^{2}_{i}-\eta_{i})\pm\frac{1}{2}\sqrt{\left[\sum_{i}(\omega^{2}_{i}-\eta_{i})\right]^{2}+4\left(\omega_{1}^{2}\eta_{2}+\omega_{2}^{2}\eta_{1}\right)-4\omega_{1}^{2}\omega_{2}^{2}}.$
(23)

At this point we are able to prove the equivalence of the sum-rule approach in
Sec. II.1 and the Ehrenfest approach discussed above. Indeed, by plugging Eq.
(8) into Eq. (22), it is not difficult to verify that the resulting equation
corresponds to Eq. (13) of the sum-rule approach. Namely, the solutions of the
latter equation are also solutions of Eq. (22), as well as those in Eq. (23),
and therefore the two must coincide. Owing to this, all the results that we
shall discuss in the following apply to both the Ehrenfest and sum-rule
approaches. As a matter of fact, in the case of the dipole mode, the two
approaches are completely equivalent.

Figure 1: Top panel: Ground-state linear density profile $n_{0i}(x,0,0)$ (in
arbitrary units) for the two components ($|1\rangle$: red dashed line,
$|2\rangle$: blue solid line) as a function of the intra-species interaction
$a_{12}$. As a reference we plot the dotted line corresponding to the maximum
density of $|1\rangle$ for $a_{12}=0$. Middle and bottom: Center-of-mass (CM)
evolution of the two condensates performing dipole oscillations, for the same
values of $a_{12}$ as indicated in the top panels. The dotted horizontal lines
correspond to the initial shift of each component (see text), which we
consider either in the same direction (middle panel) or in opposite directions
(bottom panel).

In order to analyze the amplitude of the two modes contributing to the dipole
oscillation of the mixture, let us now consider the solutions of Eq. (20).
First of all, we recall that the components of the eigenvectors of the matrix
$M$, which we shall denote as $v_{1}=(v_{11},v_{12})$ and
$v_{2}=(v_{21},v_{22})$, define the transformation matrix $P$ that
diagonalizes $M$, $P^{-1}MP=\textrm{diag}(\omega_{-}^{2},\omega_{+}^{2})$, as

$P=\left[\begin{array}[]{cc}v_{11}&v_{21}\\\ v_{12}&v_{22}\end{array}\right].$
(24)

In the following, we shall take the vectors $v_{1}$ and $v_{2}$ both with the
norm equal to $1$. Notice that in general the matrix $M$ is not symmetric, so
that the eigenvectors are not necessarily orthogonal. Then, by left
multiplying Eq. (20) by $P^{-1}$, and introducing the “diagonal” coordinates
$\xi=P^{-1}\delta x$, we can write the corresponding solutions as [we restrict
the analysis to the case of initial vanishing velocity, $\dot{\xi}(0)=0$]

$\begin{cases}\xi_{1}(t)=A\cos(\omega_{-}t),\\\
\xi_{2}(t)=B\cos(\omega_{+}t),\end{cases}$ (25)

yielding

$\begin{cases}x_{1}(t)=v_{11}A\cos(\omega_{-}t)+v_{21}B\cos(\omega_{+}t)\\\
x_{2}(t)=v_{12}A\cos(\omega_{-}t)+v_{22}B\cos(\omega_{+}t).\end{cases}$ (26)

From the above equations, we can express the relative weights of each of the
two eigenmodes in the center-of-mass motion of the two condensates in terms of
the ratios $r_{-}=v_{12}/v_{11}$ and $r_{+}=v_{21}/v_{22}$, respectively. By
construction, these quantities are vanishing in the noninteracting limit,
$r_{\pm}=0/1$ for $g_{12}=0$. It is also worth noticing that they do not
depend on the amplitude of the oscillations (namely, they are independent of
$A$, and $B$). By explicitly calculating the eigenvectors of the matrix $M$,
we can express the two ratios as

$\displaystyle r_{\pm}^{\pm
1}=\frac{1}{2}\left[\left(1-\beta\right)+\frac{\omega_{1}^{2}-\omega_{2}^{2}}{\eta_{2}}\right]$
(27)
$\displaystyle\quad\pm\sqrt{\frac{1}{4}\left(1+\beta\right)^{2}+\left(1-\beta\right)\frac{\omega_{1}^{2}-\omega_{2}^{2}}{2\eta_{2}}+\left(\frac{\omega_{1}^{2}-\omega_{2}^{2}}{2\eta_{2}}\right)^{2}},$

where $\beta\equiv m_{2}N_{2}/(m_{1}N_{1})$ represents the ratio between the
total masses of the two components.

##  III A tunable bosonic mixture

In order to compare these analytical predictions with GP simulations, we
consider an experimentally feasible system. In particular, we choose a mixture
of 87Rb (species $|1\rangle$) and 41K (species $|2\rangle$) both in the
$F=1,m_{F}=1$ state, where Feshbach resonances can be used to tune the
interspecies interactions [31, 32]. Then, we have $m_{1}=m_{Rb}$,
$m_{2}=m_{K}$, $a_{11}=100.4a_{0}$, and $a_{22}=62a_{0}$, with $a_{0}$ being
the Bohr radius. As for the values of interspecies scattering length $a_{12}$,
here we restrict the analysis to the regime in which the mixture is miscible
and stable against mean-field collapse, such that $g_{12}^{2}<g_{11}g_{22}$
[33]. The corresponding critical value of the scattering length is
$|a_{12}^{c}|\simeq 73.6a_{0}$. The mixture is trapped in a spherical optical
potential where the frequencies are $\omega_{1}=2\pi\times 100$ Hz,
$\omega_{2}=2\pi\times 136$ Hz $\simeq\omega_{1}\sqrt{\gamma m_{1}/m_{2}}$ and
$\gamma\simeq 0.87$ correspond to the relative polarizability of the two
components for a trapping laser wavelength $\lambda=1.064$ $\mu$m.

###  III.1 Balanced case

Let us start by considering the case of a balanced mixture, with
$N_{1}=N_{2}=5\times 10^{4}$ (the effect of an imbalance in the population of
the two components will be discussed in the following section). The
equilibrium (ground-state) density profiles of the two components are shown in
the top panel of Fig. 1 for different intraspecies interactions, ranging from
$a_{12}=-60a_{0}$ to $a_{12}=+60a_{0}$ 444The ground state of the system is
obtained by minimizing the GP energy functional, see e.g. Ref. [6], by means
of a steepest-descent algorithm [40]. The typical size of the numerical box is
$18$ $\mu$m per side, each discretized in 64 points.. The corresponding
center-of-mass oscillations, obtained by solving the time-dependent GP
equations (14) 555The time-dependent GP equation is solved by using a fast-
Fourier-transform split-step algorithm; see, e.g., Ref. [41]. are shown in the
middle and bottom panels. In particular, dipole oscillations are induced by
shifting the centers of mass of the two components by the same amount either
in the same direction (middle panel) or in opposite directions (bottom panel).
In all cases, the dynamics takes place in the configuration in which the two
traps are concentric (to which the ground state calculations refer), and the
initial shift is small enough to be in the linear regime of small amplitude
oscillations.

Then, by fitting the center-of-mass dynamics with a double sinusoidal
function, $f(t)=a\cos(\omega_{-}t)+b\cos(\omega_{+}t)$, we extract the
frequencies $\omega_{\pm}$ of the two modes and their relative amplitudes
$r_{\pm}$ defined in the previous section. These quantities, obtained from the
GP simulations, are plotted in Fig. 2 (solid lines) as a function of the
interspecies scattering length $a_{12}$ 666Notice that the behavior of the two
dipole frequencies (as a function of $a_{12}$) presents the same general
features as the case discussed in Ref. [26] for a Rb-Na mixture (see their
Fig. 1)., along with the predictions of the sum-rule and Ehrenfest approach
(dashed lines).

Figure 2:  Behavior of the high-lying and low-lying modes [in green (dark
gray) and yellow (light gray), respectively] as a function of the interspecies
scattering length $a_{12}$, for $|a_{12}|\leq|a_{12}^{c}|$. (a) The
frequencies $\omega_{\pm}$ extracted from the GP simulations (solid lines) are
compared to the prediction of the sum-rule and Ehrenfest approach in Eq. (23)
(dashed lines). Their relative difference
$\Delta\omega\equiv\left(\omega_{\pm}^{Ehr}-\omega_{\pm}^{GP}\right)/\omega_{\pm}^{GP}$
is shown in the inset. In the main panel we also show as a reference the trap
frequencies $\omega_{1}$ (lower dotted red line) and $\omega_{2}$ (upper
dotted blue line). Notice that all the frequencies are expressed in units of
$\omega_{1}$. (b) Relative weights of the two modes, $r_{-}$ [yellow (light
gray)] and $r_{+}$ [green (dark gray)]. In both panels, the thin dashed lines
represents the analytical limits of the Ehrenfest approach (see text).

As one can see from the inset in Fig. 2a, the sum-rule and Ehrenfest approach
provides an excellent result for the low-frequency mode, in the whole
attractive regime, up to weakly repulsive interactions ($\simeq 30a_{0}$). It
is worth noticing that in the limit of large attractive interactions [for
$g_{12}<0$ and $|\eta_{i}|\gg\omega_{i}^{2}$, see Eq. (23)], $\omega_{-}$
tends to the value

$\omega_{b}\equiv\displaystyle\sqrt{\frac{N_{1}m_{1}\omega_{1}^{2}+N_{2}m_{2}\omega_{2}^{2}}{N_{1}m_{1}+N_{2}m_{2}}},$
(28)

which corresponds to the center of mass frequency of the two condensates
oscillating as a whole [thin dashed yellow (light gray) line in Fig. 2(a)]. As
for the high-frequency mode, the sum-rule and Ehrenfest approach well
reproduces the GP results for $-30a_{0}\lesssim a_{12}\lesssim 50a_{0}$.

Instead, outside those regimes significant deviations are clearly visible.
This has the following explanation. The sum-rule and Ehrenfest approach is
based on only the knowledge of the ground-state density distribution, and
therefore it cannot account for effects stemming from modifications of the
condensate shape during the dynamics. This is particularly clear in the
derivation based on the Ehrenfest theorem, which follows from the assumption
that the density profiles of the two condensates perform only a rigid
translation during their motion. Clearly, this condition is violated for out-
of-phase oscillations [corresponding to negative values of $r$; see Fig. 2(b)]
in the regime where the two components interacts strongly (for both attractive
and repulsive interactions). In addition, one should also consider that when
the two components are trapped by different frequencies and interact with each
other, as in the present case, the Kohn theorem [37] does not hold. This
implies not only that even the frequency of the in-phase mode becomes
interaction-dependent but also that when the two components approach the
phase-separation regime (for repulsive interactions), the sum-rule and
Ehrenfest approach may fail again for the same reason as before. Indeed,
significant modification of the density may be expected for sufficiently
strong interactions, whenever the two components do not move as a whole.

Let us now turn to the discussion of the amplitudes of the two modes. In
particular, we consider the relative weights $r_{\pm}$ defined in the previous
section, which are shown in Fig. 2(b) as a function of $a_{12}$. Several
comments are in order. First of all, it is evident that the character of the
two modes is fixed by the sign of $a_{12}$. Namely, for attractive
interactions the low-lying mode is associated with in-phase oscillations
($r_{-}>0$), whereas the high-lying mode determines an out-of-phase dynamics
($r_{+}<0$). On the repulsive side, the situation is reversed. Then, we notice
that the Ehrenfest prediction for $r_{\pm}$ well reproduces the GP results in
a wide range of interactions, for $a_{12}^{c}\lesssim a_{12}\lesssim 20a_{0}$.
In addition, we can also derive an analytic estimate for the asymptotic values
in the strongly attractive regime (for $a_{12}\to a_{12}^{c}$). Indeed, in the
limit $\eta_{2}^{-1}(\omega_{1}^{2}-\omega_{2}^{2})\ll 1-\beta$, from Eq. (27)
we obtain:

$r_{+}=-\beta,\quad r_{-}=1,$ (29)

which are shown as thin dashed lines in Fig. 2. Remarkably, the fact that the
low-lying mode contributes with equal weight to the center-of-mass oscillation
of the two condensates ($r_{-}=1$) is consistent with the result in Eq. (28),
namely, the fact that the two condensates oscillate as a whole.

###  III.2 Effect of the population imbalance

We now turn to a discussion of the role of the population imbalance of the two
components. To this end, as illustrative cases, we consider a weakly
attractive mixture at $a_{12}=-30a_{0}$, and the corresponding repulsive case
at $a_{12}=+30a_{0}$ (as in Fig. 1 for the balanced mixture). We keep the
total number fixed to $N_{1}+N_{2}=10^{5}$, as in Sec. III.1, and we vary the
ratio $N_{1}/N_{2}$ in the range $[10^{-4},10^{4}]$. In analogy with the
previous discussion, in Fig. 3 we compare the frequencies $\omega_{\pm}$ of
the two modes computed from the GP simulations with the predictions of the
sum-rule and Ehrenfest approach, this time as a function of $N_{1}/N_{2}$.

Figure 3:  Behavior of the dipole frequencies $\omega_{\pm}$ [in green (dark
gray) and yellow (light gray), respectively] obtained from the GP simulations
(solid lines) and from the sum-rule and Ehrenfest approach (dashed lines), as
a function of $N_{1}/N_{2}$ (in log scale, with $N_{1}+N_{2}=10^{5}$), for (a)
$a_{12}=-30a_{0}$ and (b) $a_{12}=+30a_{0}$. For comparison we also show the
two trapping frequencies $\omega_{i}$ as dotted horizontal lines, and the two
limiting values $\tilde{\omega}_{i}$ for the minority component in Eq. (30) as
horizontal lines ($|1\rangle$: red dashed line, $|2\rangle$: blue solid). All
the frequencies are expressed in units of $\omega_{1}$.

Let us first discuss the attractive case at $a_{12}=-30a_{0}$, in Fig. 3(a).
The plot clearly shows that when the mixture becomes strongly imbalanced,
$N_{1}/N_{2}\ll 1$ or $N_{1}/N_{2}\gg 1$, the frequencies of the two
eigenmodes match the expected values for the impurity limit discussed in Refs.
[38, 24]: The majority component oscillates at its own (bare) trap frequency,
whereas the frequency of the minority component $i$ is shifted to the
following value:

$\tilde{\omega}_{i}=\omega_{i}\sqrt{1-\frac{g_{12}m_{j}\omega_{j}^{2}}{g_{jj}m_{i}\omega_{i}^{2}}}\qquad(i\neq
j).$ (30)

The latter expression, which is valid when the majority component $j$ can be
accurately described within the Thomas-Fermi (TF) approximation, can be
obtained either from the GP equation – by considering the effective harmonic
potential acting on the minority component due to the mean-field interaction
with the other species (neglecting the back action of the former on the
latter) [38, 24] – or, equivalently, from the sum-rule and Ehrenfest approach.
Indeed, in the TF limit for the majority species, the integral $I$ entering
Eq. (8) and the definition of $\eta_{i}$ can be expressed as

$I=-\frac{m_{j}\omega_{j}^{2}}{g_{jj}}\int
x\partial_{x}n_{0i}d^{3}r=\frac{m_{j}\omega_{j}^{2}N_{i}}{g_{jj}},$ (31)

so that one can easily recover the two asymptotic values from Eq. (23), by
taking the limit $N_{i}/N_{j}\ll 1$. In this regard, it is worth noticing (see
Fig. 3) that the sum-rule and Ehrenfest prediction reaches those values much
later than the GP case. Indeed, a close inspection reveals that the assumption
of neglecting the effect of the interspecies interactions on the density
profile of the majority component is justified only for $N_{i}/N_{j}\lesssim
10^{-3}$, where the integral $I$ gets close to the noninteracting value
[notice that the expression in Eq. (31) does not depend on $g_{12}$]. As a
matter of fact, the minority component is still able to produce local density
modifications on the majority component (which, in turn, produce a back-action
on the former) in the regime of intermediate imbalances ($N_{i}/N_{j}\sim
10^{-2}$), where its effect on the dipole motion of the majority component can
already be safely neglected. This suggests that the GP results converge
earlier to the asymptotic limit of Eq. (30) as a consequence of additional
contributions to the mean-field renormalization of the harmonic potential,
beyond the simple ansatz discussed above.

Figure 3(a) also shows that, for attractive interactions, the low-lying mode
is mostly dominated by species $|1\rangle$ (whose characteristic frequency is
represented by the horizontal red dashed line) and the high-lying one by
$|2\rangle$ (horizontal blue solid line). For the low-lying mode, significant
deviations between the GP result and the sum-rule and Ehrenfest prediction (up
to $\sim 10\%$) are visible in the regime with a slight minority of the
dominant species ($|1\rangle$), whereas for the high-lying mode they are
present in the whole intermediate regime (although at different degrees). The
reason for this behavior possibly lies in the fact that when the two
components have different spatial extensions the system is characterized by
different regions where the two clouds are superimposed or not, further
complicating their coupled dipole dynamics, as suggested for the case of Bose-
Fermi mixtures [14]. As discussed in the previous section, this effect is
amplified in the case of out-of-phase oscillations, namely, for the high-lying
mode (at $a_{12}=-30a_{0}$).

As for the repulsive case at $a_{12}=+30a_{0}$, in Fig. 3(b), the most
relevant difference from the previous case is the fact that the high-lying
mode is always associated with the majority species, and the low-lying mode is
associated with the minority one. This is signaled by the two asymptotic
values of each frequency: The limits of $\omega_{+}$ are the bare trap
frequencies of the two species, whereas $\omega_{-}$ tends to the values fixed
by Eq. (30) (for $i=1,2$). This behavior also provides an explanation of the
different degrees of accuracy of the sum-rule and Ehrenfest approach, in this
interaction regime. Indeed, the high-lying mode, being associated with the
majority component (less affected by the presence of the other) and also
having an in-phase character (see Fig. 2), is very well reproduced.
Conversely, significant deviations are appreciable for the low-lying
frequency, for the same reasons mentioned above for the corresponding case in
Fig. 3(a).

##  IV Conclusions

We have presented a systematic discussion of the sum-rule approach for the
dipole collective modes of a harmonically trapped bosonic degenerate mixture.
We have shown that the sum-rule prediction for the dipole frequency can be
alternatively obtained from a mean-field effective approach based of the
Gross-Pitaevskii (GP) equation and the Ehrenfest theorem. Remarkably, the
latter also provides an estimate for the amplitude of the collective modes.
Then, by direct comparison with full GP calculations, we have analyzed the
accuracy of the sum-rule and Ehrenfest predictions as a function of the
interspecies interaction, in the whole regime in which the mixture is miscible
and stable against mean-field collapse.

For a balanced mixture, namely with the same population in each species, we
have found that the sum-rule and Ehrenfest approach is quite accurate for not
too strong interspecies interactions, whereas some deviation from the exact GP
result becomes significant for the low-lying dipole mode in the repulsive
regime, especially for the high-lying mode for strongly attractive
interspecies interactions. This behavior is associated with the out-of-phase
character of the two oscillation modes, and it can be explained in terms of a
breaking of the self-similarity assumption in the dynamical evolution of the
density distributions, on which the effective Ehrenfest approach is based.

We have also analyzed the effect of a population imbalance, for both
moderately attractive and repulsive mixtures, finding that the sum-rule and
Ehrenfest approach well reproduces the frequency of the modes dominated by the
majority component, and it is especially accurate in reproducing the high-
lying frequency in the repulsive case. Explanations for the discrepancies
observed in other regimes have been provided, along with a comparison with the
analytic prediction in the limit of strongly imbalance mixtures.

Our analysis provides a framework for further theoretical studies aimed at
benchmarking the accuracy of sum-rule or related approaches in multi-component
systems, and it may prove valuable for future experimental studies of dipole
excitations in bosonic mixtures.

###### Acknowledgements.

We thank A. Burchianti, L. Cavicchioli, and F. Minardi for useful comments and
suggestions. MM acknowledges support from the Grant PGC2018-101355-B-I00
funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”,
and from the Basque Government through Grant No. IT1470-22.

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