Dipole dynamics of an interacting bosonic mixture

We unravel the coupled dipole dynamics of a two-species Bose-Einstein condensate with tunable interspecies interaction. We produce a degenerate mixture of $^{41}$K-$^{87}$Rb in an optical trap and we study the dipole oscillations of both atomic species in the linear response regime. Varying the interspecies interaction from the weakly to the strongly attractive side, we measure the frequencies and the composition of the two dipole eigenmodes. For enough strong interactions, even beyond the mean-field collapse, we find that the frequency of the low-energy eigenmode is determined only by the bare trap frequencies and the species population imbalance. The experimental results are well reproduced by numerical simulations based on two coupled Gross-Pitaevskii equations. Our findings provide a detailed picture of the dipole excitations in asymmetric bosonic mixtures.


I. INTRODUCTION
Multicomponent quantum systems emerge in a variety of physical contexts ranging from condensed matter to cosmology.In multicomponent superfluids, the interactions among constituents give rise to a number of fascinating and still not fully understood phenomena, which may be relevant in superconductors [1][2][3][4], liquid helium [5,6] and neutron stars [7][8][9].Quantum degenerate atomic mixtures have proven to be ideal platforms to study multicomponent superfluidity [10][11][12], thanks to the high control and tunability offered by ultracold gases experiments.
The dipole mode is the most basic example of collective oscillations: in a single-component superfluid trapped in a harmonic potential, it corresponds to rigid oscillations of the density, whose frequency, equal the corresponding harmonic value, is independent of the interatomic interactions [23].For two interacting superfluids, this simple picture is no longer valid since the mutual interaction mixes the dipole modes of the two components and shifts their frequencies.Thus, in this case, the study of dipole dynamics allows to access the intercomponent interaction [12,20] and may provide new insights into the nature of superfluidity [11,24].The experimental realization of binary mixtures of Fermi-Bose superfluids [11,12,25] has renewed interest in the study of collec-tive oscillations [26][27][28][29][30][31].Conversely, the excitation spectrum of Bose-Bose mixtures has remained relatively unexplored and the experiments have mainly focused on either symmetric spin mixtures [18,19,21,32,33], strongly imbalanced [22], or immiscible two-species Bose-Einstein condensates (BECs) [10].Further, in the aforementioned works, no systematic analysis has been done varying the intercomponent interaction.
In this work, we fill the gap by studying the dipole modes of a tunable 41 K-87 Rb bosonic mixture, confined in a harmonic trap.We induce the center-of-mass (CM) oscillations of both atomic species and, focusing on the linear response regime, we investigate their coupled dynamics.We measure the frequencies and the composition of the mixed dipole modes as a function of the interspecies interaction, from the weakly interacting to the strongly attractive regime, even beyond the meanfield collapse.For enough strong interactions, we observe that the two condensates almost move as one.In this regime, we directly explore the role of the population imbalance and find that the dipole dynamics is governed by the in-phase mode whose frequency depends on the bare trap frequencies and the population imbalance.The experimental results are in good agreement with numerical simulations performed by solving two coupled Gross-Pitaevskii (GP) equations.We also discuss a simple model, based on the Ehrenfest theorem, which captures the main features of the dipole dynamics.

II. SYSTEM PREPARATION
We prepare binary condensates of 41 K and 87 Rb, following the methods described in Refs.[34,35].Briefly, we start by loading both atomic species in a hybrid potential, consisting of a magnetic quadrupole plus a dimple laser beam.Here, 87 Rb is evaporatively cooled, while 41 K is cooled by thermal contact with 87 Rb.The cooling sequence ends with the production of a degenerate mixture, with both species in the |F = 1, m F = 1 hyperfine ground-state, in a purely optical potential.The final trap is formed by the dimple beam and an auxiliary beam, crossing at 45 • in the horizontal plane.The typical condensate atom number N 1 + N 2 (hereafter we use the notation: 1 → 41 K and 2 → 87 Rb) is 3 × 10 5 and the species population imbalance, α = N 1 /N 2 , is adjusted by controlling the number of 41 K and 87 Rb atoms that are initially loaded in the hybrid trap [36].
The interspecies scattering length a 12 is magnetically tuned by the Feshbach field, an homogeneous magnetic field B z , directed along the vertical ẑ axis (aligned antiparallel to gravity).In the range where B z is varied, the intraspecies scattering lengths are both positive and almost constant: a 11 = 62a 0 [37] and a 22 = 100.4a0 [38].We start the experiment with a non-interacting mixture by setting B z close to 72 G, in correspondence of the zero-crossing point (a 12 = 0) between two interspecies Feshbach resonances [39].
The trapping potential minima of 41 K, z 0 1 , and 87 Rb, z 0 2 , are vertically shifted due to the different harmonic trap frequencies.In order to overlap the two minima, we compensate the differential sag with a vertical magnetic field gradient b z , taking advantage of the different magnetic moments of 41 K and 87 Rb in the B z range [35].The gradient is adiabatically ramped to −16.6 G/cm, ensuring z 0 1 z 0 2 .Then, B z is linearly varied in 20 ms from the zero-crossing to the target value of a 12 and, finally, the dipole dynamics is started by increasing b z up to −14.6 G/cm in about 1 ms.This changes the positions of the trapping minima along ẑ by ∆ 1 2 µm and ∆ 2 1 µm, triggering the dipole dynamics.
The trap shifts are chosen as a compromise between the contrasting requirements of small perturbations and an acceptable signal-to-noise ratio of the recorded oscillations.It is worth noticing that for the typical in-trap amplitude oscillations, used in this work, the maximum relative velocity between the two superfluids remains below the sum of their sound velocities.This represents a necessary condition for preventing the emergence of dissipation [14,40,41].
In the non-interacting case, the measured trapping frequencies are (ω x,1 , ω y,1 , ω z,1 ) 2π × (65, 190, 185) Hz for 41 K, and (ω x,2 , ω y,2 , ω z,2 ) 2π × (45, 140, 135) Hz for 87 Rb.Since the trap frequencies are very sensitive to the alignment of the trap beams, they are recorded daily and used as a normalization factor for the dipole mode frequencies observed for a 12 = 0. Once the dipole dynamics is excited, the two condensates are held for a variable time in the optical dipole potential, then they are released from the trap and let expand.After a timeof-flight (TOF) t T OF =35 (38) ms, we set B z = 0 and record the 41 K ( 87 Rb) density distribution in the yz plane by absorption imaging [42].The effect of the interspecies interactions during the expansion is minimized by extinguishing b z 1 ms later than the optical potential.The corresponding species-selective magnetic force contributes to separate the two falling condensates.From the density distribution we extract the vertical displacements of the CM positions of 41 K and 87 Rb, ∆z CM,1 and ∆z CM,2 .
From the size and the optical depth of the atomic clouds we determine the nominal atom numbers, that are subsequently multiplied by the calibration factors.The latter are obtained by matching the TOF expansion of single-species BECs with the expected theoretical behaviour [43].We estimate an uncertainty in the atomic number correction of 20% and 10% for 41 K and for 87 Rb, respectively.

III. COUPLED DIPOLE OSCILLATIONS
As common to many physical situations, two coupled oscillators exhibit two oscillation modes whose frequencies differ from the natural frequencies of the uncoupled systems and whose amplitudes generally depend on the initial conditions.In our case, the two oscillating BECs are coupled by the coupling constant g 12 which is proportional to a 12 : A simple theoretical description of the present system can be obtained by means of the Ehrenfest theorem, or, equivalently, of the sum-rules approach, as discussed in Appendix A. In this case the frequencies ω ± of the two modes have the analytical expression given by Eq. (A7), and their shifts from the bare trap frequencies (ω z,1 and ω z,2 ), for a given mixture, are fixed by g 12 and by the equilibrium density distributions.In particular, in the attractive (repulsive) regime the eigenmode frequencies are raised (lowered) by the interspecies interactions.Analyzing the eigenvectors of the matrix (A6), one can gain information on the relative phase of the two modes [44].More specifically, in the attractive regime, the low-energy mode corresponds to the in-phase oscillations of the two condensates, while the high-energy mode to the out-ofphase ones.The situation is reversed in the repulsive regime, where the in-phase dynamics occurs for the highenergy mode.
However, it is worth noticing that since this model relies on the knowledge of the equilibrium configuration only, its predictions may become inaccurate if during the dynamics the shape of density distributions deviates significantly from the initial one, (i.e., when the evolution of each condensate cannot be described by a rigid translation).Therefore, in order to have a more accurate description of the observed oscillations, we perform full dynamical GP simulations (see Appendix.A).
In Fig. 1 we show the experimental CM shifts, ∆z CM,1 and ∆z CM,2 , as a function of the in-trap time for a weakly attractive mixture with a 12 = −(30.7 ± 1.2)a 0 , N 1 7 × 10 4 and α ∈ [0.4,0.6].The variation range of α is due to fluctuations in the 41 K and 87 Rb atom numbers during the measurement.We observe that, in this regime, the 87 Rb dipole dynamics is dominated by a single mode with frequency ω − , while the 41 K one clearly shows two distinct modes with frequencies ω − and ω + .The measurements are well reproduced by the corresponding GP simulations (continuous lines) where, by assuming that the interaction effects during the TOF are negligible, the expansion is obtained via the mapping ∆z CM (t + t T OF ) = ∆z CM (t) + t T OF v CM (t) (v CM being the vertical CM velocity at time t).In Fig. 1 we also report, as a reference, the GP simulations in the noninteracting regime (dashed lines).The comparison with the latter shows that the frequencies of both modes are up-shifted by the mean-field attraction, as expected (see Appendix A).

IV. GENERAL RESULTS AND DISCUSSION
We have systematically investigated the dipole dynamics of the 41 K-87 Rb mixture, varying the interaction coupling, from the weakly to the strongly attractive side.We restrict our analysis to this interaction range in order to remain in the linear response regime and to avoid the counter-flow instability [41].Indeed, repulsive interactions increase the relative velocity between the two condensates by pushing them apart, and, at the same time, they are expected to lower the value of the critical velocity associated with the relative motion [14].
We measure both the frequencies (ω − , ω + ) and the amplitudes, (A − , A + ) [45], of the dipole eigenmodes as a function of a 12 .These quantities are extracted by fitting FIG. 2. Normalized eigenmode frequencies of a 41 K-87 Rb mixture as a function of a12: the frequency of the high-energy eigenmode is obtained by fitting the 41 K oscillation (green circles, upper panel); the frequency of the low-energy eigenmode is obtained by fitting the 87 Rb oscillation or, whenever possible (see text), it is given by the mean of the values obtained by fitting the motion of both atomic species (yellow circles, lower panel).The colored areas indicate the GP numerical predictions corresponding to the maximum experimental variation of α: dashed colored lines for α = 0.84 and continuous lines for α = 0.32.For comparison, the corresponding results from Eq. (A7) are also reported as gray lines: dashed for α = 0.84 and continuous for α = 0.32.
the CM positions of the two condensates with ∆z CM = κ=± A κ e −t/τκ sin(ω κ t + φ κ ). ( The decay times (τ + , τ − ) are introduced to take into account damping effects which may emerge due to reasons both external and intrinsic to the dipole dynamics, as discussed in the following.We initially consider the case of slightly imbalanced mixtures, with α ∈ [0.32, 0.84], N 1 7 × 10 4 for −80a 0 < a 12 ≤ 9a 0 , and N 1 5 × 10 4 for a 12 ≤ −80a 0 .In Fig. 2, we show the frequencies of the high and lowenergy eigenmodes, respectively normalized to ω z,1 and ω z,2 , versus a 12 .On the attractive (repulsive) side, both ω + and ω − are up (down) shifted and this shift increases with the strength of the interspecies interaction.The experimental data are compared to the results of the GP simulations (colored regions) and to the predictions of the Ehrenfest theorem in Eq. (A7) (gray regions).In addition, in Fig. 3 we show the amplitudes A + and A − extracted from the CM oscillations of both 41 K (upper panel) and 87 Rb (lower panel).The amplitude of each condensate is normalized to the corresponding amplitude in the non-interacting case (that is, A 1 for 41 K and A 2 for 87 Rb).In the weakly interacting regime, the 41 K oscillations are characterized only by the high-energy eigenmode, with relative amplitude A + /A 1 (data points).As the attractive interaction increases, also the low-energy eigenmode, with relative amplitude A − /A 1 (data points), starts to appear and becomes dominant in the strongly attractive regime.On the contrary, the 87 Rb oscillation is always dominated by the low-energy eigenmode and only the relative amplitude A − /A 2 can be measured.
In order to reproduce the experimental results we perform corresponding GP simulations.The colored areas in Figs.2-3 show the numerical predictions obtained by considering the maximum variation of α during the measurements: the continuous line corresponds to α = 0.32 and the dashed one to α = 0.84.The agreement between GP theory and experiment is good in the full range of explored interactions.For completeness, we also show in Fig. 2 the eigenmode frequencies calculated with Eq. (A7).Approaching the strongly attractive side, we see that only the lower frequency mode, corresponding to the in-phase motion of the two condensates, is well reproduced by the Ehrenfest model.
Even for moderate couplings (a 12 < −10a 0 ), we measure a non-negligible damping of the high-energy dipole mode with a decay time τ + of about 20 ms.This effect, observable also in the GP simulations, is the result of non-linear coupling to other collective modes for large enough excitations.We observe a decay of the dipole oscillations, depending on their amplitude, also for noninteracting mixtures due to the anharmonicity of the trap potential away from its minimum.However, in this case the decay time is of about 200 ms, for trap shifts of the order of 2 µm, substantially longer than τ + .
For strong enough interactions, namely for a 12 < −70a 0 , we observe that the two condensates oscillate in The systematic uncertainty due to the atom number calibration mentioned in text corresponds to multiplying all α values for a scaling factor of (1.0 ± 0.22).The continuous line is the result of GP simulations and the gray dashed line is the solution of Eq. 2.
unison, with the same frequency and amplitude.In this limit, the eigenmode frequency ω − tends to the value It is worth noticing that this expression depends only on the bare trap frequencies and on the total masses N i m i of the two components.We verify the accuracy of Eq. ( 2) in the strongly interacting regime for a 12 = −(83.4± 1.2)a 0 , where the mixture is stabilized against the mean-field collapse by quantum fluctuations [46], i.e., for g 12 < − √ g 11 g 22 (for our mixture the critical point corresponds to a c 12 = −73.6a0 ).
To this end, we measure the low-energy eigenmode frequency once the species population imbalance spans a wider range, for α ∈ [0.07, 1.34].In Fig. 4, we show the corresponding values of ω − as a function of α.The experimental data points are compared with the results of both the GP simulations including the Lee-Huang-Yang correction (continuous line, see Appendix A), and Eq.(2) (dashed line).The frequency shift from the bare value ω z,2 decreases by decreasing α and eventually tends to ω z,2 when α 1.Because the dipole dynamics is insensitive to the presence of liquid droplets, we probe their possible existence using TOF detection since, differently from the unbound gas component, droplets do not expand after the trap removal.Following the system evolution up to t T OF = 25 ms [47], we observe that, in the same conditions of Fig. 4, i.e., for a 12 = −83.4a0 , both atomic species expand, even if the atom number exceeds the critical value for droplet formation [35].This effect can be ascribed to the fact that the system is prepared in a configuration far from the equilibrium one in free space.We verified that stable droplets form by further increasing the interspecies attraction.However, such increase implies a substantial reduction of the mixture lifetime, due to the enhancement of three-body losses.This hinders the study of the in-trap dynamics in this range of parameters.We conclude that the production of long-lived droplet states requires a trap geometry matching the droplet size in free space, which is out the scope of the present work [48,49].

V. CONCLUSIONS
We have studied the coupled dipole dynamics of a binary condensate by exploiting an asymmetric bosonic mixture, where the two components have different masses and experience different trapping potentials.In particular, we have investigated the dipole excitations of the two condensates as a function of their mutual coupling, exploring different regions of the mixture phase diagram.In the weakly interacting regime, where the mixture is miscible, we have excited both the in-phase and out-phase dipole modes, both observable in the CM oscillations of component 1, and measured their dependence on the intercomponent interaction.We find that the frequencies of both eigenmodes are shifted by the coupling, differently from the case of symmetric spin mixtures, where the in-phase dipole frequency always matches the trap frequency.Further, we have extended our analysis to the strongly attractive regime, where the mixture is stabilized by quantum fluctuations.Approaching this region, the most significant features are the dominance of the in-phase dynamics, and the unison motion of the two condensates.In this limit, the dipole frequency is determined only by the bare trap frequencies and the total masses of the two components.Our findings provide a solid ground for future experimental studies aimed at exploring the coupled dynamics of bosonic mixtures in different regimes, even beyond the linear limit, such as in the presence of dissipation or in proximity of a phase transition.Further extensions of our work could also include the exploration of higher-order excitations, which may be useful for probing quantum fluctuations [50], exotic matter states [51] and topological structures [52] in multicomponent superfluids.and The frequencies ω ± of the two dipole modes are obtained from the eigenvalues of M , namely From the above equation one can easily see that in the attractive regime (g 12 < 0) the two modes are up-shifted with respect to the bare trapping frequencies ω 1 and ω 2 of the two condensates, while in the repulsive (g 12 > 0) regime they are down-shifted.It is also worth noticing that these frequencies are fixed by the equilibrium configuration of the system (i.e., the ground state in the same harmonic potential in which the evolution takes place) through the integral I = ∂ z n 01 ∂ z n 02 [55], as in the case of the sum-rules approach (see, e.g., Refs.[26,56]).Indeed, it can be show that the two approaches are equivalent [44].
For a general description, beyond the linear regime of small oscillations and of the rigid displacement ansatz [see below Eq. (A4)], we perform dynamical simulations of the full GP equation (A1), to take into account the deformations of the density profiles during the evolution of the system.In addition, since at the mean-field level the mixture becomes unstable against collapse at g 12 = − √ g 11 g 22 , in the strongly attractive regime we also include the Lee-Huang-Yang (LHY) correction [57], that accounts for the stabilizing effect of quantum fluctuations [46].In particular, in Eq. (A1) we include a term V LHY,i ≡ ∂E LHY /∂n i , where the LHY energy density E LHY can be written in the form [58,59]

FIG. 1 .
FIG. 1. Oscillations of ∆zCM,1 (blue circle, upper panel) and ∆zCM,2 (red squares, lower panel) in the weakly attractive regime, for a12 = −(30.7 ± 1.2)a0.The vertical CM positions, extracted from the density profiles, are measured after a TOF expansion of 35 ms (upper panel) and 38 ms (lower panel) for 41 K and 87 Rb, respectively.Error bars denote standard deviation of mean over five independent measurements.The continuous lines are the corresponding GP simulations where all parameters are fixed to the experimental ones (α = 0.4), with the exception of the initial phases, which are fitting parameters.As a reference, GP simulations in the non-interacting regime for the respective atomic species are also displayed as the grey dashed curves.

FIG. 3 .
FIG. 3. Normalized eigenmode amplitudes of a 41 K-87 Rb mixture as a function of a12.Upper panel: experimental amplitudes of the low-energy (yellow squares) and high-energy (green squares) eigenmodes of 41 K. Lower panel: experimental amplitude of the low-energy (yellow squares) eigenmode of 87 Rb.The colored areas indicate the GP numerical predictions corresponding to the maximum variation of α during the measurements: the dashed line is for α = 0.84, while the continuous one is for α = 0.32.

2 FIG. 4 .
FIG. 4. Upper panel: absorption images of a balanced and imbalanced dual-species BEC, in the strongly attractive regime, for a12 = −(83.4± 1.2)a0.Images are taken after 35 ms of TOF for 41 K and 38 ms for 87 Rb.N1 = (4.8± 1.0) × 10 4 and N2 = (5.5 ± 0.6) × 10 4 , in the balanced case, and N1 = (4.7 ± 1.0) × 10 4 and N2 = (3.4± 0.3) × 10 5 , in the imbalanced case.Lower panel: normalized frequency of the low-energy eigenmode as a function of the species population imbalance α, for a12 = −(83.4± 1.2)a0.The horizontal error bars correspond to the binning intervals of α values.The systematic uncertainty due to the atom number calibration mentioned in text corresponds to multiplying all α values for a scaling factor of (1.0 ± 0.22).The continuous line is the result of GP simulations and the gray dashed line is the solution of Eq. 2.