Traces of integrability in scattering of one-dimensional dimers on a barrier

We consider molecules made of two one-dimensional short-range-interacting bosonic atoms. We show that in the process of scattering of these molecules off a narrow barrier, odd incident waves produce \emph{no unbound atoms, even when the incident energy exceeds the dissociation threshold}. This effect is a consequence of a prohibition on chemical reactions acting in a generally unphysical Bethe Ansatz integrable system of a $C_{2}$-type, with which our system shares the spatially odd eigenstates. We suggest several experimental implementations of the effect. We also propose to use the monomer production as an alternative read-out channel in an atom interferometer: unlike in the standard interferometric schemes, no spatial separation of the output channels will be required.

The last two decades are marked by a revival of interest in Bethe Ansatz integrable particle systems [1][2][3][4][5][6][7], inspired by the emerged experimental relevance of the former in both many-body [8][9][10][11] and few-body [12,13] cold-atom systems (see [14] for a review). Integrability, besides providing a way to produce theoretical predictions, also induces new empirically observable integrals of motion. Traditionally, the conservation of the momentum distribution is emphasised [8,15]. However, another experimentally sound set of conserved quantities is derived from the conservation of the chemical composition [16], i.e. the decomposition of the system onto unbound atoms, dimers [12], trimers, all the way to the long polimers, the latter manifesting themselves as bosonic solitons [17,18].
Below we show that an integrability-related chemical stability also appears in a system that consists of two one-dimensional attractive short-range-interacting bosons and a narrow barrier. While generally not integrable, our system shares some of its eigenstates with a lesser studied integrable model based on a reflection group C 2 [19,20], a symmetry group of a square. As a result, a spatially odd incident wave of dimers shows a strictly vanishing rate of a monomer production, no matter how far above the dissociation threshold the incident energy is.
Model. In this work, we investigate the scattering states of a bosonic dimer on a potential barrier by modeling the system as two one-dimensional δ-interacting bosons in presence of a δ-potential barrier located at the center of the system. The Hamiltonian reads: with g ab > 0 being the strength of the potential barrier, g = − 2 /aµ < 0 the strength of the attractive interatomic interactions and where a is the one dimensional even-wave scattering length [21], µ = m/2 the reduced mass and m the mass of the atoms.
Prior to an encounter with the barrier, the energy of the dimer reads where k is the momentum of either particle in the dimer; the latter is related to the center-of-mass incident momentum, K, as k ≡ K/2. The dissociation condition, k > 1/a, can be readily inferred from the relationship (2): it ensures that the system has enough kinetic energy to invest towards dissociation [4]. The presence of the barrier in Eq. (1) breaks the integrability of the model. However, we show below that the spatially odd scattering states of the system described by Eq. (1) coincide with the ones of the integrable model described by the following Hamiltonian: The above Hamiltonian can be shown to be integrablein all symmetry sectors-using a Bethe Ansatz based on a symmetry group of a square, C 2 . This model, as well as all its multidimensional and affine generalizations, was first analyzed by Gaudin [4,19], albeit with a conjecture that there exists a single integrability-supporting value of the ratio between the coupling constants g ab and g. The works [22,23] show that integrability persists for any ratio between the constants, but restrict the treatment to the "identity" representation of the group. The paper [24] can be used to construct other representations of the corresponding reflection groups, in particular the one we are using in this Letter. In the affine case, it can be shown that the model supports three independent coupling constants [25]. Note that, the same rich choice arXiv:1806.01820v1 [cond-mat.quant-gas] 5 Jun 2018 of parameters is paralleled in the Calogero-Sutherland-Moser models [7].
Let us now show that the spatially odd sector of eigenstates of the Hamiltonians given in Eqs. (1) and (3) indeed coincide. We introduce the permutation, P ψ(x 1 , x 2 ) = ψ(x 2 , x 1 ), and the spatial reflection, P ψ(x 1 , x 2 ) = ψ(−x 1 , −x 2 ), transformations. They allow us to introduce a subspace of bosonic, spatially odd states, ψ +,− :P Note that [P ,Ĥ] = [P ,Ĥ] = 0 and [P ,P ] = 0. The reflection symmetry with respect to the x 2 = −x 1 lineone of the four symmetry axes of the C 2 model-can be written asR =P P ; it commutes with the transfor-mationsP andP and with the Hamiltonian (3). As a result, the ψ +,− eigenstates of the Hamilonian are, at the same time, odd eigenstates of the reflectionR, i.e. (3) acts. Hence, the spatially odd states ψ +,− are simultaneously eigenstates of Eqs. (1) and (3), allowing us to connect the eigenstates of an integrable system, with all the corresponding conservation quantities associated to them, with the ones of a generally non-integrable one. Among all the eigenstates of Eq. (1), we shall now focus on those corresponding to scattering states, i.e. non-normalizable states satisfying plane-wave incoming boundary conditions. Since our scattering potential conserves the spatial parity, we will be considering the even and the odd partial waves separately, along with the even and odd scattering solutions. We will show below that for the odd states, the hidden partial integrability revealed above leads to tangible consequences.
Preservation of chemical composition. Bethe Ansatz integrable systems are known to preserve the chemical composition [16]. In particular, in the C 2 -integrable model (3), any purely dimeric incident wave will not produce unbound monomers after a collision with the barrier, even at energies higher than the dimer dissociation threshold [27]. This preservation of the chemical composition can be demonstrated by considering the available rapidities produced by a dimeric incident wave, all of which being substantially complex and as such, supporting no monomers. Indeed, the underlying reflection group, induced by four mirrors with a 45 • angle between them can only permute and change sign of the incoming rapidities, but it is not capable of altering their imaginary parts. In particular, the above conclusion is valid for a spatially odd linear combination of the incident dimeric waves. The scattering solution induced by it will also be spatially odd. But as we have shown above, odd eigenstates of the Hamiltonian (3) are, at the same time, the eigenstates of the empirically relevant Hamiltonian (1). This brings us to the central result of this Letter: F odd (ϕ) is identically zero at all incident energies.
Hence, for both Hamiltonians, the corresponding odd scattering solution can be written as The scattering phase δ odd (K) can be obtained after a long but straightforward calculation that mirrors the one for the A 2 reflection group [3] (scattering of a dimer on a monomer for three distinguishable particles of the same mass, interacting with the same strength): δ odd (K) = 1 2 arctan 2a 2 k a aa ab k 2 − 1 + a ab (a(ak(a ab k − 1) − 1) + a ab )(a(ak(a ab k + 1) − 1) + a ab ) , where a ab ≡ − 2 mg ab is the scattering length associated with the interaction of a single particle with the barrier. This result also allows to define the dimer-barrier odd scattering length as .
Single incident dimeric wavepacket and its dissociation. By numerical solution of the time-dependent manybody Schrödinger equation associated to Eq. (1), we investigate the scattering of an incident (from the left) dimer, of the form where x = −x 0 is the position of the center of mass of the dimer at initial time. Figure 1 shows the particle density, |Ψ(x 1 , x 2 )| 2 , at long times after that the initial dimer has collided against the barrier. If the kinetic energy of the incoming dimer is smaller than the threshold of monomer formation, we observe that the dimer is partially reflected and partially transmitted. However, a "deflection" via formation of monomers is clearly visible above threshold.
For each region S of the (x 1 , x 2 ) plane, the probability P (S) = S dx 1 dx 2 |Ψ(x 1 , x 2 )| 2 with S being the four sectors defined as R ≡ {x 1 < 0, yields the transmission and reflection coefficients, P (T ) and P (R), respectively, along with the monomer formation probability, P (M ). Our results for the three coefficients as a function of the initial wavevector of the dimer are summarized in Fig. 2. We notice that the nonvanishing monomer production indeed requires k > 1/a. Two counterpropagating dimeric wavepackets. We consider next the solution of the two-body Schrödinger equation when the following initial condition is taken: In Fig. (3) we show the probability density when the input state is represented by a spatially even and a spatially odd linear combinations of dimeric wavepackets, thus corresponding to the choices φ = 0 and φ = π respectively. The figure corresponds to the case where ka = 4, i.e. the input kinetic energy of each dimeric wavepacket is above the threshold for monomer formation. The figure clearly shows that, while monomers are created in the spatially even configuration, a complete suppression of output monomers is achieved when choosing a spatially odd configuration, in full agreement with the predictions obtained by the spatial odd sector of the integrable C 2 model. The monomer formation probability as a function or the phase difference φ between the incident dimeric wavepackets is shown at Fig. (4). In the same figure we also show a measure proportional to the density-density correlation function ρ 2 (x, y), taken at zero distance and averaged over the sample, ie dxρ 2 (x, x). Notice that the local second-order correlation g 2 (0) has been already made experimentally accessible in a one-dimensional setting [28].
Potential experimental realizations. One-dimensional dimers appear in several areas of physics of ultracold atoms. One realization is offered by the waveguidetrapped spin-1 2 fermions [12]. While the two atoms constituting a dimer are formally distinguishable, the dimer state belongs to the bosonic sector of the model, and so will the scattering state with a moving dimer as the in- Total monomer production after the collision of a linear superposition of two motional states of the dimer against the barrier as a function of the relative phase between the two contrapropagating dimeric wavepackets. The integral shown on the right axis is proportional to the empirically relevant (see [28]) two-body correlation function g2. The input kinetic energy is set to two times the dissociation energy (ka = 4). The other parameters are the same as in Fig. (1). cident wave. In this case, the δ-interaction model is well justified in the regime where the size of the ground transverse vibrational state in the guide greatly exceeds the three-dimensional scattering length [21]. Apriori, the one-dimensional dimers described in this Letter can be constructed using any type of onedimensional bosonic particles provided that the corresponding interaction potential is sufficiently shallow; quantitatively, it will be required that the width of the interaction potential w exceeds its scattering length a [29]. A remarkable example is offered by the recently realized dimers of Rydberg polaritons [30], where the w is greater than a by at least an order of magnitude.
In both cases considered above, a narrowly focussed sheet of light can be used to generate a fixed one-particle barrier. A similar requirement, w ab |a ab |, must be applied to the sheet waist w ab .
Potential applications. One may regard the process of a collision between two dimeric wavepackets and the barrier as a recombination process in an atom interferometer. Indeed, the intensity of all three output channels of the scattering event-right moving dimers, let moving dimer, and the monomer production-are expected to depend periodically on the relative phase between the input packets (see Fig. (4) for the latter). Unlike the first two, the dimer production is a new possibility.
Recall that in a chip-based atom accelerometer [31], the interferometer arms need not be spatially separated: however the readout still requires the separation between the channels thus expanding the minimal size of the device. We suggest that in a dimer-based interferometer, the read-out stage of the process can also be made compact if the total dimer population, accessible through the two-body correlation function [28] is used as an output. Summary.
In this Letter we have shown that for short-range-attractive-interacting one-dimensional bosonic atoms, scattering of a spatially odd motional state of a dimer off a barrier produces-even above the dissociation threshold-no unbound atoms. This prohibition originates from a map-valid in the bosonic, spatially odd sector of the Hilbert space-between the Hamiltonian of the system and a known, generally unphysical, Bethe Ansatz integrable Hamiltonian associated with the symmetries of a square. Potential experimental realizations include the waveguide confined atomic dimers and bound states of two Ridberg polarons. We also suggest that in the context of chip-based atom accelerometers, using the monomer productionaccessible through the second-order local correlation function g 2 (0) right after recombination-as an output channel may allow to further miniaturize the readout.