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Entanglement generation by collisions of quantum solitons in the Born approximation

We present analytical expressions describing the generation of entanglement in collisions of initially uncorrelated quantum solitons. The results, obtained by means of the Born approximation (for fast solitons), are valid for both integrable and non-integrable quasi-one-dimensional systems supporting soliton states.


Introduction
Entanglement is a fundamental property of bipartite quantum systems [1]. Apart from being a major resource for quantum-information techniques, entanglement exhibits itself in perhaps its most spectacular form in the breakdown of Bell's inequalities and in the Einstein-Podolsky-Rosen 'paradox'. In experiments, strongly entangled states, and in particular those exhibiting 'nonlocality', are typically created with microscopic particles produced by the same source, or interacting prior to the detection, such as pairs of photons [2,3] or ions [4]. In their original paper [5], Einstein, Podolsky and Rosen considered two particles whose ordinary degrees of freedom, center-of-mass positions and relative momenta, were correlated. This situation is realized with the entanglement of Gaussian states of light [6] or atomic ensembles [7].
Particularly interesting is the possibility of generating entanglement between macroscopic (or mesoscopic) objects that may be transmitted over long distances. In this connection, solitons, i.e. stable solitary waves that propagate without distortion [9], may be considered as possible robust quantum-information carriers. In particular, the solitons as collective excitations in nonlinear media (unlike ions or other material objects) may be created with a desirable shape and effective mass, and allow a much greater degree of control by means of various 'management' techniques [11].
The objective of the present work is to analyze the possibility of creating entanglement between solitons by means of collisions between them. To the best of our knowledge, this problem has not been considered before (except for a preliminary version of the present work [12]). However, a very recent paper [13] has presented a rigorous proof of the generation of entanglement between constituent solitons in oscillating two-and three-soliton bound states, in the integrable model based on the nonlinear Schrödinger (NLS) equation with cubic nonlinearity. It is relevant to note that multisoliton solutions of the integrable NLS equations are actually unstable against slow separation, and therefore the entanglement, generated by the interactions between the solitons while they remain bound, may be kept after the separation.
Natural candidates for the study of collision-induced entanglement are matter-wave solitons, that can be built of ultracold bosonic atoms, which form the Bose-Einstein condensate (BEC) [8], and photonic solitons in nonlinear optical waveguides [9,10]. The studies of BECs have led to the creation of dark [14,15] and bright [16,17,19] solitons in trapped BECs (with repulsive and attractive interactions, respectively). Recently, the observation of stable bright-dark soliton pairs has been reported, too [21]. In addition, similarities to nonlinear optics [9] have triggered interest in theoretical studies of discrete (lattice) BEC solitons [22,23], and have also led to the seminal observation of gap solitons, i.e. robust localized matter-wave packets supported by the interplay of repulsive interactions and an effective negative mass of collective excitations in the condensate, induced by the periodic optical-lattice potential [20]. The analysis reported below suggests that the matter-wave solitons in a BEC with attractive inter-atomic interactions have the greatest potential for the generation of entanglement through collisions; in particular, the velocity of the moving solitons, which is the crucial parameter in determining collision-induced entanglement (see below), can be easily controlled for BEC solitons.
While most of the previous studies of matter-wave solitons have concentrated on their classical and mean-field aspects, more recently considerable interest has been devoted to the role of thermal noise [23,24] and quantum fluctuations. The latter may cause filling of 3 the dark-soliton's core through the quantum depletion process, as was predicted using the Bogoliubov-de Gennes (BdG) equations [25]. Making use of the discrete NLS equation and the time-evolving block-decimation algorithm [26] in the framework of the Bose-Hubbard model, it was confirmed that quantum effects lead to the filling in of dark BEC solitons, and it has been demonstrated that collisions between them become inelastic [27]. The BdG approach and its generalizations were also employed to study the solitons' stability [28]- [30] and excitations caused by opening of the trap [31]. In the latter work, exact Lieb-Liniger solutions were also used to calculate internal correlation functions of positions of the particles. A noisy version of standing bright solitons was studied by means of the exact diagonalization and quantum Monte Carlo method [32].
Being typically associated with integrable one-dimensional (1D) classical models, stable solitons are known too in non-integrable systems, including multidimensional ones. One of the remarkable properties of solitons is their stability to perturbations. This stability can be extended to quantum settings, in which the mean-field description admits quantum solitons in the semiclassical form. As mentioned above, the robustness of mean-field solitons against quantum fluctuations (including finite-temperature effects) was studied, using the time-dependent Hartree-Fock-Bogoliubov equations, for both ordinary matter-wave solitons, supported by attractive interactions between atoms [29], and for gap solitons [31]. Although under extreme conditions the quantum fluctuations may split a mean-field soliton [29], it has been concluded that, in a broad range of parameters relevant to the experiments, the matter-wave solitary waves predicted by the mean-field description (i.e. found as stable localized solutions to the Gross-Pitaevskii equation (GPE) [8]) are completely robust objects-in fact, in perfect agreement with the experimental observations of these solitons [17]- [20]. As concerns the relation between the mean-field and quantum descriptions of localized objects in BEC, it is relevant to mention that an alternative derivation of the GPE from a consistent many-body quantum theory, based on the variational approach in the multiconfigurational space, was recently presented in [33].
In this work, we analyze the collision-induced generation of entanglement in pairs of fast solitons for a class of equations of the NLS type by means of the Born approximation, which is valid when the kinetic energy of the moving objects (solitons) is much larger than the potential of the interaction between them, hence the interaction may be treated as a perturbation [34]. This is the opposite case to that studied in [13], where entanglement was considered between bound solitons created with zero relative velocity. We aim to present simple analytical expressions for the collision-generated entanglement of two quantum solitons, valid for generic quasi-1D systems-integrable or not-that admit soliton solutions. Another approach is possible for nearly quiescent solitons, where the entanglement-generating perturbation is the weak interaction between them, assuming that the distance between the solitons is large enough. Recently, this approach was developed in [35] for kinks in the sine-Gordon equation. In that model, the kinks were maintained in the quiescent state by boundary conditions, as the model was defined in a domain of a finite size.
The rest of the paper is organized as follows. In section 2, we introduce the model, starting with the known system of classical equations of motion for a pair of well-separated solitons [10], and then proceed to the respective quantum system. We confine the analysis to the basic case of a pair of symmetric solitons with equal amplitudes. Mismatched amplitudes make the interaction effectively incoherent [36], thus suppressing the entanglement generation (we estimate the size of the mismatch up to which it may be neglected). The initial quantum state, corresponding 4 to that for well separated solitons, is taken as a product of two independent wave packets. In section 3, the calculation of the correction to this factorized state, generated by the collision between the fast solitons, is performed in an analytical form, by means of the above-mentioned Born approximation. The collision-induced correction to the wave function features explicit entanglement in terms of two relevant degrees of freedom, viz the distance between the solitons and their relative phase, r and χ. The results are reported for the initial condition of two types: a more sophisticated one, with the Gaussian localization of χ around a definite value, and also for a simple phase-independent initial distribution. Both types of the initial quantum state may be realized in the experiment, under different specific conditions. The paper is concluded by section 4, where, in particular, we discuss the robustness of the predicted entanglement against external noise, and then possible extensions of this work.

The classical soliton pair
For the condensate with attractive interactions between atoms, the scaled form of the GPE in the free 1D space, which describes the BEC in the mean-field approximation, is tantamount to the integrable NLS equation [8], i.e.
The commonly known soliton solution to equation (1) is where η is the soliton's amplitude, dξ/dt = c its velocity, and the intrinsic frequency. More realistic forms of the GPE in 1D include various terms that break the integrability of the NLS equation. In particular, the full equation must include the axial trapping potential, but, as concerns interactions between solitons, the axial potential is not a crucially important factor, according to the available experimental results [17] and theoretical analysis [37,39]. Another physically relevant feature that may affect soliton-soliton collisions is a quintic nonlinear term. It may account for three-body collisions in the condensate, provided that they are lossless [40], but a more general (in fact, universal) source of the quintic term is the deviation of the condensate loaded into a cigar-shaped trap from one-dimensionality. This term can be derived by means of a perturbative analysis [41], or by the expansion of a more general equation that takes into regard the underlying three-dimensionality via the non-polynomial nonlinearity [42]. The universal quintic term always corresponds to self-attraction (irrespective of the sign of the binary interactions between atoms), with the coefficient in front of it proportional to the transverse-confinement frequency and square of the collisional scattering length. By means of straightforward rescalings, the equation with the combination of attractive cubic and quintic terms can be cast in the form that contains no free parameters, Although the cubic-quintic (CQ) NLS equation (4) is not integrable, its exact soliton solutions are well known [43], the entire family being stable [44]: where φ(t) and ξ(t) have the same meaning as in equation (2). The CQ model based on equation (4) finds other physical realizations in nonlinear optics [45]. The analysis that treats the interaction between well-separated quasi-classical (mean-field) solitons (2) with common amplitude η as a perturbation leads to the conclusion that they may be considered as a pair of quasi-particles with an effective Hamiltonian [10], where the overdot stands for the differentiation with respect to rescaled time τ ≡ 2 √ 2η 2 t, the normalized distance between the solitons and center-of-mass coordinate are defined as while the relative and overall phases of the soliton pair are A noteworthy fact, which is obvious in expression (6), is that the effective mass corresponding to the phase degrees of freedom in the two-soliton set is negative [46]. Actually, Hamiltonian (6) is universal, applying to any model which supports exponentially localized solitons with an intrinsic phase degree of freedom [38]. In particular, this Hamiltonian governs the interaction between solitons (5) of the NLS equation with the CQ nonlinearity, equation (4), up to a proper rescaling of coefficients [39].

Quantization
Proceeding to the quantum version of the model considered above, we treat, as usual [47], each soliton as a quantum particle with two degrees of freedom, namely position and phase (in the experimentally relevant situation, the effects of quantum fluctuations around the quasiclassical shape of the solitons may be negligible for matter-wave solitons, as discussed above). Thus, the quantum counterpart of the classical Hamiltonian (6) gives rise to the following linear Schrödinger equation for the wave function of the soliton pair, (tot) , which depends on the total set of four degrees of freedom describing the pair: where T ≡hτ and ε ≡ 1/h 2 , withh the renormalized Planck's constant (measured in scaled units in which Hamiltonian (6) was written). The reduced form of equation (9) for the wave function, which depends only on the relative variables, r and χ, is Our objective is to analyze collisions between rapidly moving quantum solitons, with large relative momentum K 0 . The initial state is taken as a naturally expected non-entangled 6 factorized one, centered around definite initial values of the dynamical variables, ±ξ 0 and ±χ 0 : This state assumes equal widths of the wave packets for the two solitons, and , with the pair's center of mass set at x = 0, and initial separation 2ξ 0 ≡ ξ 1 (T = 0) − ξ 2 (T = 0). The mean value of the initial overall phase is also fixed to be zero, while the initial phase difference between the solitons is 2χ 0 ≡ φ 1 (T = 0) − φ 2 (T = 0). Finally, initial state (11) can be written in terms of the variables defined in equations (7) and (8), where and r 0 ≡ 2ηξ 0 .
To conclude the formulation of the model, it is relevant to consider the possibility of the excitation of an intrinsic mode (IM) in the colliding solitons (if the IM exists). A known principle is that solitons of integrable equations do not support IMs, but a non-integrable model may feature an IM. In particular, exactly one IM exists in the excited state of solitons (5) of the CQ NLS equation, with the self-focusing sign of both nonlinear terms [44]. The possibility of using solitons' IMs as carriers of quantum information was proposed in [35]. However, in the case of collisions between fast solitons, which is considered below, the excitation of the IM, as well as generation of nonsoliton modes ('radiation'), may be disregarded in the lowest approximation, simply because the intensity of these effects is inversely proportional to the square of the collision velocity [10].

The Born approximation
Reduced equation (10) scattering solutions are generated by the incident wave, (r → ∞) = exp[iK r + iκχ − i(K 2 − κ 2 )T /2], where κ is an integer. The full solution is sought as with the stationary part of the wave functions obeying the following equation: As said above, our basic assumption is that we consider the collision between fast solitons, i.e. K 2 is a large parameter in comparison with ε (κ 2 may be large too). In other words, the kinetic energy of the relative motion is much larger than the potential of the soliton-soliton interaction.
In fact, the two-particle Hamiltonian (6) can be used for the description of collisions between solitons only in this case; otherwise, one cannot neglect deformation of the solitons in the course of the collision, as well as excitation of the IM, if it exists in the solitons, and the generation of the radiation modes. 7 We apply the Born approximation [34], looking for a solution to equation (14) as where perturbation ϕ, which is assumed to be a slowly varying function of r in comparison with exp(iK r ), obeys the simplified equation: −iK ϕ r + (1/2)ϕ χ χ + iκϕ χ = εe −|r | cosχ. Solutions to this equation satisfying the necessary boundary condition, ϕ(r = −∞) = 0, can be readily found in an analytical form. The outcome of the collision is determined by the asymptotic form of the solution, which is found to be ϕ(r → +∞, χ ) = (iε/K ) e iχ e i(κ−1/2)r/K + e −iχ e −i(κ+1/2)r/K .

Analysis of the collision-induced entanglement
In fast collisions, the overall variables, ρ and θ , may be treated as 'frozen' ones in the wave packet generated by initial state (12). Then, it is natural to decompose the initial state over the set of plane waves with respect to variables r and χ: Note that the Gaussian distribution of angular wavenumber κ, which must be an integer, is valid for |κ| 1, i.e. 2π (recall that is the width of the initial distribution of the phase variables introduced in equations (11) and (12)).
Next, recombining wave packet (17) with the collision-induced perturbation of the wave function, as per equations (15) and (16), and again making use of the fact that K 0 is large, we arrive at an expression for the net change of the wave function, which is generated by the fast collision between the two solitons, in the first order of the perturbation theory: with phase shifts A nontrivial feature of expression (18), which represents the entanglement proper, is the combination of two harmonics, exp(±iχ), multiplied by the Gaussian factors, whose maxima are located along two spirals in the plane of (r, χ): r max = ∓K 0 (χ − χ 0 ). These maxima also determine the correlation between variables r and χ, which is an inherent part of the entanglement. The width of the maxima gradually spreads out with the growth of time (T ), proportionally to √ 4 4 + T 2 , as does any coherent state evolving in the free space. It is relevant here to estimate the dephasing effect of a possible mismatch between the amplitudes of the colliding solitons, η. It may be estimated through the respective change of the relative phase, χ ∼ η η t coll ∼ η/c, generated by the mismatch during the collision time, t coll ∼ 1/(ηc) (see equations (1)- (3)). The dephasing is negligible if its size is small in comparison with the perturbation amplitude in expression (18), i.e.
The sophisticated experimental methods currently available for the creation of solitons in BEC [18], as well as various theoretically elaborated schemes of matter-wave soliton lasers [50], make it possible to generate nearly identical solitons, with a sufficiently small difference between their amplitudes.

The initial phase-uniform state
Instead of the initial wave packet (11), we can take one uniformly spread over phases φ 1 and φ 2 , which corresponds to an experimental situation in which the initial phases of solitons are not controlled. Then, modifying the expression in equation (12) accordingly, decomposition (17) is replaced by its simplified counterpart: Recombining this with the result of the Born approximation, as per equation (16), gives rise to the following expression for the collision-induced perturbation of the wave function: Although this result is much simpler than the one given by equation (18), in cases where the solitons' phases were initially allocated certain values, it is nontrivial too, demonstrating the dependence on relative phase χ of the perturbation generated, in the course of the collision, by the phase-independent initial wave function. Actually, this feature may be regarded as the simplest manifestation of the collision-induced entanglement. The correlation properties of the entanglement in this approximation are obvious, being, as a matter of fact, determined by factor cos χ.

Conclusion
In this work, our aim was to present a proof of the principle that two initially uncorrelated quantum (actually, semi-classical) solitons may become entangled through colliding with one another. To this end, we have restricted the analysis to macroscopic coordinates of the solitons, namely position and phase, which may be justified for matter-wave solitons in BEC. In this sense, they are regarded as Einstein-Podolsky-Rosen particles with an additional internal degree of freedom (the phase). We have presented simple analytical expressions characterizing the resulting entangled states, represented by the collision-induced corrections to the initial factorized wave functions of the soliton pair. The results were obtained by means of the Born approximation, which is valid for collisions between fast solitons. The entanglement and quantum correlations are described in terms of the relative position and phase of the solitons, and the (conserved) relative momentum and 'angular momentum' (the latter is the canonical momentum conjugate to the relative phase). A simple but noteworthy effect is that the collision of solitons with a completely uncertain (delocalized) relative phase leads to a partial phase 9 localization. The predicted results are expected to be directly relevant to matter-wave solitons in BEC formed by atoms with attractive interactions, loaded into a nearly 1D trapping potential. A realization in terms of optical solitons may be possible too, in principle. The entanglement predicted by the above analysis is weak, as the consideration was limited to the Born approximation, i.e. the first approximation of the perturbation theory. However, under more general conditions, the entanglement may reach significant levels (cf [13] where the entanglement was analyzed by means of an exact quantum solution for NLS solitons interacting at zero velocity). What then are the possible advantages of using matterwave solitons instead of individual particles, like photons or atoms? The obvious answer is that the solitons constitute macroscopic, or at least mesoscopic, objects, and thus exhibit completely different properties in measurements. To some extent, they are similar to atomic ensembles [7], but, in contrast to the latter, the solitons represent coherent moving objects. In principle, Bell's inequalities can be tested with entangled solitons, and even though such tests would not be flawless, they would evidently exhibit a different scale and level of error than previously studied systems.
As said above, one may attain stronger correlations by going beyond the Born approximation limit. However, in that regime the macroscopic (mean-field) description may cease to be quantitatively correct-in particular, because the deformation of the slowly colliding solitons makes it irrelevant to use the description solely in terms of their collective degrees of freedom. It may then be appropriate to consider quantum solitons using the time-dependent GPE (or similar equation(s)) to describe the coherent soliton states proper, and BdG equations for quantum fluctuations around them, cf the analysis of the stability of matter-wave solitons against quantum fluctuations performed in [29,30]. In particular, care should be taken over the collisional excitation of an IM, if the solitons support it. However, such a consideration is beyond the scope of this work.
Similar comments concern another important aspect of the study of entanglement, namely decoherence. In the present case, part of the decoherence is due to the excitation of manybody degrees of freedom (such as BdG modes), rather than just quantum fluctuations of the solitons' positions and phases. Nevertheless, in the case of collision between fast solitons we may restrict attention to the collective degrees of freedom of the solitons, and look at the respective covariance matrix, as with Gaussian states. Applying the entanglement criteria for Gaussian states [51]- [53] makes it possible to estimate the level of the entanglement of the present (ideal) state, as well as the maximum allowance of the contribution to the decoherence from the neglected many-body modes.
The estimate of the robustness of the entanglement is particularly simple for the solution corresponding to equation (21), because the degrees of freedom corresponding to the distance and phase shift between the solitons separate, hence the wave function of the two-soliton state can be written, in terms of dynamical variables (7) and (8), as (cf equations (18) and (21)) where the states of the relative phase (χ ), |0 , | ± 1 , correspond to harmonics with the respective numbers, and ζ ≡ ε/(2 √ 2K 0 ). The phase part of state (22) is obviously entangled, since its partial transpose has three negative eigenvalues, namely, −ζ, −ζ, −ζ 2 [54]. Therefore, the state remains entangled after adding arbitrary decoherence (noise) whose density matrix has an operator norm smaller than ζ . This actually means that one can add 'white noise' with norm ζ < ζ to state (22), and retain its non-positive partial transpose property. As concerns the dephasing due to a possible difference in the amplitudes of the colliding solitons, the estimate performed above demonstrates that it is negligible under condition (20).
A straightforward extension of the present analysis, still within the framework of the description based on the mean-field collective degrees of freedom and the Born approximation, may be elaborated for the description of the generation of entanglement by collisions between 2D solitons supported by a quasi-1D potential in the self-attractive BEC, as proposed (in the mean-field approximation) in [48]. Another extension, also possible in the framework of the quasi-particle approach, is to consider entanglement induced by collisions between gap solitons in a self-repulsive BEC loaded into an optical-lattice potential. It is known that such solitons may be mobile in 1D and 2D geometries, with a negative effective mass [49]. The interaction potential in the corresponding Hamiltonian is expected to be different from the above expression (6), namely, being ∼exp (−|r |)cos (qr )cos χ, with some wavenumber q, due to the oscillatory shape of the tails of the gap solitons.