Exact diffractive three-particle scattering in one dimension for arbitrary nonintegrable contact couplings, its finite-range validity, and kinetic consequences

In one-dimensional quantum systems, integrability is synonymous with nondiffractive scattering: the set of outgoing momenta equals the set of incoming momenta, and the many-body wavefunction takes the Bethe-ansatz form. The consistency condition behind this is the Yang-Baxter equation for the two-particle S-matrix. Lamacraft [Phys. Rev. A 87, 012707 (2013)] showed that for three equal-mass particles with pairwise contact couplings $g_{ij}$ near the integrable manifold $g_{12}=g_{13}=g_{23}$, the leading diffracted three-body amplitude is a simple commutator-like combination of two-particle scattering matrices acting on the Bethe amplitudes, expressible entirely through two-body scattering data, with consequences ranging from bound-pair disintegration amplitudes to a three-body collision integral predicting relaxation rates scaling as $(n_1+n_2)n_1n_2$ in a two-component Bose gas. The task is to determine the exact diffractive amplitudes at arbitrary (not nearly equal) couplings, i.e. to solve the six Sommerfeld functional relations without the periodicity supplied by the Yang-Baxter equation; the rigorous finite-range validity of the near-integrable formula; nontrivial solutions of the three-particle Boltzmann equation; and the experimental observation of three-body diffractive relaxation in multicomponent one-dimensional gases.

Background

For a quantum system of particles moving in one dimension, scattering has a special structure: for two particles, energy and momentum conservation force the outgoing momenta to coincide with the incoming ones (possibly exchanged). The distinction between integrable and nonintegrable dynamics therefore appears first for three particles. A system is called nondiffractive when, in every scattering event, the set of outgoing single-particle momenta equals the set of incoming momenta; the corresponding wavefunction in the asymptotic region is a superposition of plane waves over the permutations of the incoming momenta, which is the coordinate Bethe ansatz that underlies all exactly solved one-dimensional many-body models such as the Lieb-Liniger gas (one-dimensional bosons with delta-function interactions).

Nonintegrable interactions instead produce diffraction: a continuum of outgoing momenta, restricted only by energy and momentum conservation, carried by a diffracted wave that decays as $e^{i(kr+\pi/4)}/\sqrt{2\pi kr}$ with angle-dependent amplitude $\mathcal{D}(\phi)$, in direct analogy with optical diffraction from a wedge.

The canonical family of models here consists of three equal-mass distinguishable particles with pairwise contact (delta-function) interactions of strengths $g_{12}, g_{13}, g_{23}$. When all three couplings coincide, the model is the three-body sector of the Lieb-McGuire system (the $N$-body delta-interacting gas) and is solved by the Bethe ansatz; the algebraic reason is that the two-particle reflection and transmission amplitudes satisfy the Yang-Baxter equation, the condition that three-particle scattering factorizes consistently into sequential two-particle collisions irrespective of ordering. For unequal couplings the Yang-Baxter equation fails and diffraction appears. Beyond the integrable case, only a few special cases have exact solutions: McGuire's equal-coupling Bethe solution, the impenetrable-particle limits (infinite coupling, where particles cannot pass one another) of McGuire and Hurst, the solvable three-body model with a bound state of Gaudin and Derrida, and Lipszyc's Sommerfeld-Malyuzhinets treatment of particular diffraction problems; the general formulation (McGuire and Hurst's algebraic approach, a system of coupled functional equations) has been regarded as forbiddingly complex since the 1970s-80s.

Lamacraft (2013) recast the problem using the Sommerfeld representation: after removing the center of mass, the three coincidence lines $x_i=x_j$ cut the plane into six sectors (the six orderings of the particles), and the wavefunction in each sector is an angular superposition with amplitude function $\mathcal{A}_Q(\alpha)$, where $Q$ labels the sector. Matching the contact boundary conditions yields six coupled functional relations (Eq. (15) of that paper) relating the $\mathcal{A}_Q(\alpha)$ through $6\times6$ matrices $\mathcal{S}_1, \mathcal{S}_2$ built solely from the two-particle reflection and transmission amplitudes $r_{ij}, t_{ij}$. The key structural result is that $2\pi$-periodicity of the $\mathcal{A}_Q(\alpha)$ is equivalent to the Yang-Baxter equation: periodicity forces the Sommerfeld integrals to collapse onto their pole residues (the Bethe-ansatz waves) and eliminates diffraction. When the couplings are close but not equal, the failure of periodicity is small, and Lamacraft obtained the leading-order diffracted amplitude $\mathcal{D} = [\mathcal{S}_1(\phi-2\pi/3)\mathcal{S}_2(\phi-\pi/3)\mathcal{S}_1(\phi) - \mathcal{S}_2(\phi)\mathcal{S}_1(\phi-\pi/3)\mathcal{S}_2(\phi-2\pi/3)]\mathcal{A}^{\mathrm{B}}$, the difference of the two orderings of two-particle scattering matrices acting on the Bethe amplitudes, which vanishes identically when the Yang-Baxter equation holds and is first order in the deviation of the couplings from the integrable point. Because only two-particle data enter, the result was argued (though not proven) to extend to finite-range interactions close to integrable. For attractive couplings, poles of the $\mathcal{A}_Q(\alpha)$ at complex angles describe two-particle bound states, and the same formula gives the leading amplitude for disintegration of a bound pair colliding with a third particle, a process forbidden in the integrable limit.

The physical stakes are kinetic. In a one-dimensional gas, binary collisions merely exchange or bounce momenta and so cannot change the single-particle momentum distribution at all; this is the mechanism behind the celebrated quantum Newton's cradle experiment (Kinoshita, Wenger and Weiss, 2006), in which a trapped one-dimensional Bose gas failed to thermalize. Relaxation therefore requires diffractive events, and at low density the leading mechanism is three-body diffraction. This is directly relevant to multicomponent one-dimensional gases, such as atoms in different hyperfine states, where the pairwise interaction strengths differ by a few percent: a natural, weakly nonintegrable system. Lamacraft used the diffracted amplitude to write the three-body collision integral of the Boltzmann equation for such gases, with the estimate that the relaxation rate of a two-component Bose gas scales as $(n_1+n_2)n_1n_2$ with the component densities $n_{1,2}$. Subsequent large programs on weak integrability breaking (diffusive hydrodynamics, and generalized hydrodynamics - the exact hydrodynamic theory of integrable one-dimensional systems - extended by collision integrals) have mostly addressed other perturbations, such as lattice discretization, atom loss, or external noise, and treat three-body diffraction in multicomponent contact gases only through the perturbative formula above. What is still missing is the exact theory away from the near-integrable regime, its rigorous domain of validity, the resulting kinetic solutions, and any experimental test.

Problem Statement

The problem is the completion of the quantitative theory of three-body diffractive scattering in one dimension beyond the near-integrable regime, as left open by Lamacraft (2013). Its components share one objective - the exact quantitative link between violation of the Yang-Baxter equation and diffractive three-particle scattering and its kinetic consequences - and are ordered by dependence as follows.

(1) Core: exact amplitudes at arbitrary couplings. For three equal-mass distinguishable particles in one dimension with pairwise contact (delta-function) interactions of real strengths $g_{12}, g_{13}, g_{23}$, determine the exact three-body scattering solution - in particular the diffracted amplitudes $\mathcal{D}_Q(\phi)=\mathcal{A}_Q(\phi-\pi)-\mathcal{A}_Q(\phi+\pi)$ giving the angular structure of the diffracted wave - for generic couplings at finite distance from the integrable manifold $g_{12}=g_{13}=g_{23}$. This means solving the six coupled functional relations obeyed by the Sommerfeld sector amplitudes $\mathcal{A}_Q(\alpha)$ (Eq. (15) of Lamacraft 2013, equivalent to the McGuire-Hurst algebraic formulation) without assuming the $2\pi$-periodicity in $\alpha$ that holds exactly when the two-particle S-matrix satisfies the Yang-Baxter equation. The answer must (i) go beyond first order in the Yang-Baxter violation, (ii) satisfy the contact boundary conditions in all six ordering sectors together with unitarity and flux conservation of the full scattering, (iii) reproduce the known limits: exact Bethe (nondiffractive) solution with $\mathcal{D}_Q\equiv 0$ at equal couplings, Lamacraft's leading-order formula $\mathcal{D}=[\mathcal{S}_1(\phi-2\pi/3)\mathcal{S}_2(\phi-\pi/3)\mathcal{S}_1(\phi)-\mathcal{S}_2(\phi)\mathcal{S}_1(\phi-\pi/3)\mathcal{S}_2(\phi-2\pi/3)]\mathcal{A}^{\mathrm{B}}$ near that manifold, the impenetrable (infinite-coupling) limits, and weak-coupling perturbation theory, and (iv) retain the correct pole structure describing two-particle bound states and their disintegration for attractive couplings.

(2) Finite-range validity. Establish rigorously, or refute with quantitative bounds, the claim that the leading diffractive amplitude is determined by two-particle scattering data alone and therefore extends to finite-range (non-contact) interactions close to integrable, including the size and structure of the corrections beyond the near-integrable regime.

(3) Kinetic theory. Using the diffractive collision integral, construct nontrivial solutions of the three-particle Boltzmann equation - for example self-similar solutions describing spatially constant but nonequilibrium distributions - and characterize the relaxation dynamics of multicomponent one-dimensional gases that follow from them, including the predicted $\propto(n_1+n_2)n_1n_2$ relaxation-rate scaling for a two-component Bose gas and its regime of validity.

(4) Experimental validation. Provide a quantitative measurement of three-body diffractive relaxation in a multicomponent one-dimensional atomic gas with independently characterized unequal interaction strengths, to be compared with the predicted density, coupling, and angular dependence.

An answer to the problem as a whole must make progress on the core (1); components (2)-(4) are the rigor, consequence, and observational test of the same theory and are judged against the amplitudes of (1). A complete solution of the core (1) for arbitrary real contact couplings is the deliverable that resolves the task; components (2)-(4) are optional follow-on consequences and are not required for completion. No finite-size, geometry, coupling-ratio, or answer-form restriction is intended: component (1) concerns arbitrary real contact couplings, and partial progress (for example an exact solution on a subfamily of nonintegrable couplings beyond the known special cases, or a controlled resummation of the violation) constitutes progress on the problem, but does not by itself complete it.

References

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