Bethe Ansatz & Entanglement Amplification

Integrable Many-Body Systems, Platonic Solids, and Entanglement Amplification

Since the mid-70's, it has been believed that there exist only two types of reflection groups that can generate exact solutions to quantum N-body problems with short-range interactions: AN-1and CN. These, respectively, allow one to solve the problem of an ensemble of atoms of identical masses on a ring, and in a box.

We show[JO15], however, that the eigenenergies and eigenstates of a system consisting of four one-dimensional hard-core particles with masses 6m, 2m, m, and 3m in a hard-wall box can be found exactly using the Bethe ansatz based on the reflection group F4. The latter is associated with the symmetries and tiling properties of an octacube: a Platonic solid unique to four dimensions, with no three-dimensional analogues (pictured above). A particular section of the ground state wavefunction is shown on the left.

Recently, we extended the above method to non-crystallographic reflection groups, to the symmetries of the 3D and 4D icosahedra (H3 and H4) in particular[SSJO16].

We invoke reflection groups of regular polygons, I2(n), to predict and confirm numerically[HCO15] (see also [AGHJOA17]) the distinct peaks in the dependence of the relaxation time on the mass ratio in a one-dimensional mixture of atoms of two different masses, with hard-core interactions.

Harmonically trapped atoms lead to spherical billiards. Similarly to the relative motion of N particles on a line, integrability of N atoms in a harmonic potential is governed by the finite N-1-dimensional reflection groups; in particular, the integrable mass spectra are identical for these two models. We study explicitly the four-body integrability instances, given by the tetrahedral, cubic/octahedral, and dodecahedral/icosahedral symmetries, A3, C3, and H3 respectively, and in the case of H3, demonstrate numerically a transition from a Wigner-Dyson to Poisson statistics for an H3 mass ratio set[HODVJZ17].

Of a particular interest is the ability of integrable systems to stir the quantum evolution towards entangled states. We study this effect in the context of a so-called Galilean Cannon, in turn related to the symmetries of the N-dimensional tetrahedra, AN. In an interferometric scheme we suggest[OSDJ16], a light particle is beam-split and then, subsequently entangled with a heavy component of the system that never sees a beamsplitter. This scheme, if used as a sensor, shows an (Natoms)1/2 increase in sensitivity as compared to the conventional scheme, for the same total number of atoms Natoms. 

Other Bethe-Ansatz-based Projects

We study in detail a Calogero-Sutherland model with screening[PBOC17]. We find that an introduction of a radius r screening (cut-off of interactions between two atoms when the latter are separated by more than r-1 other atoms) does not affect the integrability of the system, and, as in the un-screened case, the spectrum is controlled by the AN-1 reflection group, for N atoms. 

We revisit correlations in the Lieb-Liniger model (δ-interacting bosons on a ring) and find a connection between the local three-body correlation function and the forth derivative of the one-body correlator[ODML17]. 

In the problem of scattering of a bosonic dimer on a barrier, we find that while the whole problem is not Bethe Ansatz integrable, it shares a part of its Hilbert space with an integrable problem associated with the C2 reflection group[GMO19]. As a result, the dimer dissociation becomes prohibited for spatially odd incident states. We suggest using this effect to effectuate a spatially compact readout off a chip-based atom interferometer. 

We realized that under a stereographic projection, finite four-dimensional reflection groups induce three-dimensional finite groups of sphere inversions. This allows one to generate 19 3-parametric families of solvable electrostatic problems[OSYDJ19].

We look at the high-momentum tails of the momentum distribution, for a finite temperature one-dimensinal Bose gas[RAOB23] . We study the effect of the so-called hole anomaly that manifests itself in fading of the conventional 1/|k|4 tails and an appearance of the 1/|k|3 ones.