Spin dipole oscillation and relaxation of coherently coupled Bose-Einstein condensates

We study the static and the dynamic response of coherently coupled two component Bose-Einstein condensates due to a spin-dipole perturbation. The static dipole susceptibility is determined and it is shown to be a key quantity to identify the second order ferromagnetic transition occurring at large inter-species interaction. The dynamics, which is obtained by quenching the spin-dipole perturbation, is very much affected by the system being paramagnetic or ferromagnetic and by the correlation between the motional and the internal degrees of freedom. In the paramagnetic phase the gas exhibits well defined out-of-phase dipole oscillations, whose frequency can be related to the susceptibility of the system using a sum rule approach. In particular in the interaction SU (2) symmetric case, i.e., all the two-body interactions are the same, the external dipole oscillation coincides with the internal Rabi flipping frequency. In the ferromagnetic case, where linear response theory in not applicable, the system show highly non linear dynamics. In particular we observe phenomena related to ground state selection: the gas, initially trapped in a domain wall configuration, reaches a final state corresponding to the magnetic ground state plus small density ripples. Interestingly the time during which the gas is unable to escape from its initial configuration is found to be proportional to the square root of the wall surface tension.


I. INTRODUCTION
Two-component Bose-Einstein condensates (BECs) with an interconversion or Rabi term between the two species -hereafter also called spinor condensates -represent one of the easiest example of a superfluid with a vector order parameter. Due to the internal degrees of freedom and the lack of relative particle number conservation their behaviour is very different and richer with respect to the case of a single-component BEC. In particular spinor condensates show a zero-temperature ferromagnetic-like transition (see, e.g., the experiment [1] and reference therein), they have a gapless density mode and a gapped spin mode in the homogeneous case (see, e.g., [2,3]) and show dimerised vortices known as meron pairs [4][5][6]. Coherently driven BECs are moreover an extension of quantum optics concepts to condensates [7,8], a candidate to realise Schrödinger-cat-like states [9], as well as to study the effects of quenching in Hopf-like bifurcations [10][11][12]. All these effects are governed by the interplay between intra-and inter-particle interactions and the exchange coupling.
In this paper we study the spin-dipole properties of spinor condensates, both at the static and at the dynamical level. The paper is organised as it follows. In Sec. II we introduce the system modeled by two Gross-Pitaevskii (GP) equations coupled through the interspecies interaction and the Rabi term and we revisit the emergence of a paramagnetic/ferromagnetic like transition in the presence of an external harmonic trapping potential. In Sec. III we study the effect of a spin-dependent potential and the role of the spin-dipole susceptibility. The latter is shown to bear a clear signature of the phase transition. Then, we address the problem of the dynamics of the spin-dipole mode both in the para-and in the ferromagnetic phase. In the former case (Sec. IV A) linear response theory combined with a sum-rule approach * Electronic address: recati@science.unitn.it provides an accurate estimate of the spin-dipole mode frequency, which well compares with the numerical solution of the GP equation. In the ferromagnetic case (Sec. IV B) we find the system exhibits ground state selection and an interesting dynamics due to the presence of a magnetic-like domain wall at the center of the trap. Specifically, we identify a waiting time in which the system appears to be unable to leave the local minimum configuration, consisting of a domain wall in the polarisation, before relaxing and targeting the ground state. We find that this characteristic time is proportional to the square root of the domain wall energy.

II. GROSS-PITAEVSKII EQUATION FOR COHERENTLY COUPLED BECS
We consider an atomic Bose gas at zero temperature, where each atom of mass m has two internal levels a and b. The atoms interact via both their density through s-wave contact interactions and their phase due to an interconversion term. In the weakly interacting limit the system is well described by a spinor order parameter (ψ a (r, t), ψ b (r, t)), whose dynamics is determined by coupled Gross-Pitaevskii equations [7] i The couplings g i , with i = a, b, ab are the intra-and interspecies atomic interaction strengths and (where not differently specified) we consider g a = g b ≡ g, V a and V b are the external trapping potentials and Ω is the Rabi coupling between the two internal atomic levels. Due to the presence of the Rabi coupling only the total number of atoms N = N a +N b is conserved, while its polarisation The ground state of the system can be described -at least for values of the polarisation |P | not too close to unity -using local density approximation (LDA), i.e., neglecting the quantum pressure. Within this approach the ground state densities obey the relations (see, e.g. the review [13] and reference therein) where without any loss of generality we assumed Ω to be real and positive and, consequently, the relative phase of the order parameter components to be π. In the following, for the sake of simplicity, we consider a mean-field one-dimensional situation, where the transverse degrees are frozen [20]. Experimentally such configuration is realised by a strong transverse confinement. The trapping potentials are taken to be harmonic with a state-independent trapping frequency ω 0 . When V a = V b the system exhibits two different regions: an unpolarised one with n a = n b and a polarised region with n a = n b . In particular the position X P of the polarised phase is fixed by the condition g ab > g + 2Ω/n(X P ) with n(x) = n a (x) + n b (x) the total local density. Clearly, if such condition is not satisfied at the center of the trap, where the total density is maximum, then the whole system is unpolarised. This allows us to introduce a critical value of Rabi coupling defined by For values Ω ≥ Ω cr the system is unpolarised everywhere and the density profile n a = n b is easily obtained from Eq.(4): where R TF 2 = 2(µ + Ω)/(mω ho 2 ). In the case Ω < Ω cr a typical configuration within LDA is shown in Fig. 1.
Let us here remind that for a Bose-Bose mixture in the absence of Rabi coupling (Ω = 0), where the relative particle number can be chosen at will, the situation is very different. In that case there exists a first order phase transition to a phase separated state once g ab > g. The ground state structure in a trap depends on the trapping potential and on the value of P , but within LDA there are no coexistence regions [14][15][16].

III. STATIC DIPOLE POLARIZABILITY
In this Section we calculate the static response of a trapped spinor gas to a spin-dipole perturbation. A spin-dipole perturbation corresponds to a shift of the harmonic traps for the two components by a quantity d a ho , The GP ground state solution for the spinor gas in the displaced potentials is reported in Fig. 2 (see also [7]), where for the sake of concreteness we assume g ab > g to show the difference between a mixture and a coherently driven spinor gas. For d = 0 (first line) we see the features of the Ω-induced phase transition: below the critical value the linear coupling prevents the phase separation by creating a global polarisation in the system ( Fig. 2 plots (a2) and (a3)), while a mixture without any Rabi coupling Ω = 0 would be in a phase separated state ( Fig. 2 plots (a1)). Above the critical value the gas is unpolarised ( Fig. 2 plot (a4)). Applying a potential shift (second line) is equivalent to apply a local magnetic field and thus the ferromgnetic part of the gas is strongly affected: as a result a magnetic domain wall is created at the center of the trap.
In order to calculate the spin-dipole susceptibility we first determine the spin-dipole moment D, defined as The spin-dipole susceptibility is then defined by the limit where λ = dmω 2 ho is the perturbation associated with the spin-dependent component of the potential (7).
In the global paramagnetic phase (Ω > Ω cr ) it is easy to obtain an analytical expression for χ sd within LDA. For comparison we report also the ground state for a Bose-Bose mixture, i.e., Ω = 0. In the latter case, being the number of particle in each component fixed, no polarisation appears, and the ground state (a1) and (b1) are essentially equal. The effect of quantum pressure can be clearly noticed in the plots (a2) and (a3) (analogues to Fig. 1). The bifurcation points are not sharp as instead predicted by the Thomas-Fermi approximation.
In this case one can employ the energy functional relative to the spin degrees of freedom of the problem, where the spin (magnetic) susceptibility for an homogeneous system of density 2n 0 (see, e.g., [13]). Variation with respect to the spin density (n a − n b ) yields the result and the spin dipole polarizability finally reads After integratiion we obtain the result for the dimensionless ratio D/d where we have introduced the dimensional function f (t) = 3t 1 − √ 1 + t arccoth( √ 1 + t) [21] and used the notation n 0 = n 0 (0).
We notice that for Ω → 0, i.e., a Bose-Bose mixture, f (t → 0) → 0 and therefore the spin-dipole susceptibility diverges at the phase separation point g ab = g, while in our case instead the susceptibility remains finite, since f (t → ∞) → −1 + 2/(5t). Above the critical point the response of the system is no longer linear. The system is partially ferromagnetic and has the tendency to form a magnetic domain wall at the centre of the trap (see Appendix). A detailed analysis of this behaviour is shown in Fig. 3 where we calculate numerically the spin-dipole of the gas as a function of the trap separation d with the choice g ab = g. The spin-dipole moment allows for a clear identification of the phase transition point, above which the induced dipole moment D changes its behavior.

IV. SPIN DIPOLE DYNAMICS
The spin-dipole polarizability plays an important role in characterising the behavior of the spin-dipole frequency [17,18]. The spin-dipole oscillation of the system corresponds to an oscillation of the average value of the operatorŜ d = xσ z that acts on the spinorial wave function (ψ a , ψ b ). In particular for a mixture (Ω = 0) with g ab = 0 the spin-dipole frequency would simply coincides with the trap frequency ω ho .
A. Paramagnetic phase: sum rule approach As already mentioned, in the paramagnetic phase a small trap displacement corresponds to a small deviation with respect to the ground state and thus linear response theory can be applied. In order to estimate the spindipole frequency we use a sum rule approach [19] that provides an upper bound to the frequency of the lowest energy state excited by the operatorŜ d . In particular we use the relation where are the moments of the strength distribution function relative to S d = i (x ai − x bi ). The energy weighted moment (m 1 ) can be easily rewritten in terms of a double commutator as |0 . The only terms in H that do not commute with S d are the kinetic energy and the Rabi coupling H R = −Ωσ x . The former gives the usual N 2 /(2m) contribution, while the latter is straightforwardly evaluated as −4Ωx 2σ x . Averaging on the ground state, we obtain the result The inverse energy weighted sum rule (m −1 ) is directly related to the susceptibility of the ground state through the relation and using the definition Eq. (9) together with the result Eq. (14) we obtain the following upper bound to the spin-dipole frequency ω SD 2 = ω ho 2 g − g ab g + g ab 1 + 8Ωn 0 (g + g ab )/(5 ω 2 0 ) 1 + f (Ω/((g − g ab )n 0 )) . : Spin-dipole frequency as a function of Ω for different values of interactions; g ab /g = 0.9 (purple circles), g ab /g = 1 (red triangles), g ab /g = 1.01 (blue squares), g ab /g = 1.02 (black diamonds). Lines are analytical results from Eq. (19) and points are numerical data. In order to have a fully paramagnetic phase for g ab > g one needs Ω ≥ Ωcr (see text and Eq. (5)). In Fig. 4 the sum-rule result is compared with the predictions of the solutions of a time dependent Gross-Pitaevskii calculation. From the numerical (or experimental) point of view, the excitation of the spin-dipole mode is achieved starting with an equilibrium configuration in the presence of slightly displaced trapping potentials, as described by Eq. (7), and suddenly setting d = 0.
Notice that at the transition point the frequency does not go to zero. This has to be compared with the mixture case, which is recovered sending Ω → 0. In this case the spin-dipole frequency vanishes close to the critical point following the law and the sum-rule approach give the exact result as shown in Fig. 5.
For Ω = 0 and g ab = g, the frequency given by sumrules is exact (see lower panel of Fig. 4). The magnetic energy of the spinor gas in this regime depends on the relative density only through the Rabi coupling which breaks the SU (2) symmetry of the system. The spindipole frequency behaves in this case as ω SD (g ab = g) = 2Ω 1 + 5 16 which is essentially twice the Rabi frequency and almost independent of the tapping frequency.
In the more general case, when both Ω and (g ab −g) are different from zero, the frequency is given by the full eq. (19) in which both the coherent and the interspecies swave couplings play a role. In this more general case one observes that that the sum rule approach provides only an upper bound to the numerical solution, due to the appearance of more frequencies in the numerical signal resulting in beating effects.

B. Ferromagnetic phase: ground state relaxation
In the previous section we have studied the dynamics for a completely paramagnetic gas, i.e., Ω > Ω cr . The behaviour is very different when the system presents a ferromagnetic behavior. In this case the ground state of the system with equal trapping potentials is polarized as shown in Fig. 2 plots (a2) and (a3). When the traps are shifted, the ground state is instead globally unpolarized (N a = N b ) but with a large spin-dipole moment (depending on the values of Ω and d) as one can see in Fig. 2 plots (b2) and (b3). Therefore the initial state and the ground state are very far from each other. This circumstance results in a non-trivial non-linear dynamics as shown by the dynamics of the spin-dipole and of the polarisation reported in Fig. 6. At the beginning, the spinor gas oscillates around the initial configuration, trapped in the unpolarized state. After a certain time, τ wait , the domain wall starts moving and a finite polarisation appears. The system then bounces back and forth between the initial magnetic state and its magnetic ground state to eventually relax to the latter one. [22] If the global polarisation of the ground state is large, the effects of non linearity and the number of bounces are large. When the system is slightly in the ferromagnetic regime no bounces are observed and the system after τ wait soon reaches its ground state (see right lower panel in Fig. 6). Notice that even if the system is isolated, it can approach in the long time limit an asymptotic steady state, as a result of destructive interference of several time oscillating factors, present in the evolution of expectation values of observables. Specifically,in the case of a large and dense collection of frequencies, the interference phenomenon results in a dephasing mechanism similar to inhomogeneous dephasing.
As we already mentioned in Sec. II the initial configuration in the ferromagnetic case contains a domain wall at the centre of the trap. We have identified a close relation between the observed waiting time and the square root of the domain wall energy (see Appendix) σ ∝ |(g − g ab )n + 2Ω| 3/2 Ω .
As a matter of fact the Ω dependence of the waiting time follows the law τ wait ∝ √ σ with surprising accuracy, as we show in Fig. 6. The fact that for Ω → 0 the waiting time diverges can be easily understood noticing that the initial state and the ground state are very far from each other (see, e.g., panels (a2) and (b2) of Fig. 2). Eventually, in the strict Ω = 0 case, the system cannot reach the totally polarised ground state and it remains in the phase separated state (see panels (a1) and (b1) of Fig. 2).

V. CONCLUSIONS
In the present work we analyse in details the static and dynamic response of a trapped coherently driven 2component condensate to spin-dipole probe. We show that the spin-dipole susceptibility is a good quantity able to identify the appearance of a ferromagnetic-like region in the cloud.
For the dynamics we study the spin-dipole mode frequency by starting in a configuration with displaced harmonic potentials, which are suddenly brought to the same value. When the system is paramagnetic such a frequency is well reproduced by a sum-rule approach. In particular the f-sum rule is strongly modified by the Rabi coupling in the symmetric interaction case (g = g ab ) and the inverse energy sum rule is proportional to the spindipole susceptibility and coincides with the second spatial momentum of the local magnetic susceptibility (see Eq. (13)).
When the system has a ferromagnetic domain a linear response cannot be applied anymore and the dynamics is highly non-linear. The initial configuration within displayed potentials is unpolarised and contains a magnetic kink centred at the origin. The dynamics is trapped for a time, τ wait , in the initial configuration, after which the system is able to relax to its polarised ground state. We find that τ wait is proportional to the square root of the kink surface tension.
Our study improves the characterisation of coherently driven BECs, enlightening their differences with respect to Bose-Bose mixtures. Moreover measuring the spindipole dynamics opens new perspective to experimentally access important magnetic properties of the system, as, e.g., its susceptibility or the domain wall surface tension.
accounts for the density-density interaction and the Rabi terms. For a homogeneous magnetization minimisation δE/δM = 0 leads to the usual equation for the paraand ferro-magnetic like states. From Eq. (A1) one sees that close to the phase transition, i.e., M 1 a standard Ginzburg-Landau theory for the order parameter M , is valid, where the kinetic energy is just the gradient of M and the effective potential takes the usual quadratic plus quartic form As usual the Z 2 symmetry broken ground state is obtained for r < 0. A kink in M is the field solution interpolating between the two degenerate minima. Its surface tension, σ, which coincides with its energy in a one-dimensional situation, can be easily computed yielding the result σ ∝ 2 n 2 m |r| 3/2 u ∝ 2 n 2 m |δgn + 2Ω| 3/2 Ω (A4)