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Radiation effects and defects in solids

Unitary qubit lattice algorithms for spin-1 Bose Einstein condensates (BECs)

Linda L. Vahala, Min Soe, George Vahala, Jeffrey Yepez

Published Feb 1, 2019• 10 citations
Publisher

Abstract

A unitary qubit lattice gas algorithm is developed for spin-1 BEC and is used to examine vortex reconnections for singular quantum hyperfine linear vortices. It is shown that a simple diagonalization of the spin–orbit coupling term will break the unitarity of the qubit algorithm and can rapidly lead to code instability.
DOI
10.1080/10420150.2019.1577425

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Using the qubit unitary algorithm developed in the previous paper, quantum reconnection of non-Abelian quantum vortices is simulated and rung-like vortices are seen during the reconnection. Vortices in two of the four tetrahedral symmetry group are examined. Moreover, the algorithm performs exceptionally on classical supercomputers, as shown on timings on the IBM-Q supercomputer MIRA. It is expected that one should see the exponential speedup on a quantum computer since quantum entanglement is crucial for the qubit algorithm to recover the Gross-Pitaevskii equations for spin-2 systems.Keywords : spin-2 BECs, non-Abeliam vortices
Spinor Bose Einstein Condensates are intriguing because of their vast range of different topological vortices. These states occur when a BEC gas is trapped in an optical lattice rather than in a magnetic well (which would result in scalar BEC vortices). A spinor BEC states also occur in a quantum gas when several hyperfine states of the atom co-exist in the same trap. A unitary quantum lattice algorithm that is ideally parallelized to all available processors is used to solve the evolution of non-eigenstate Skyrmions in a coupled BEC system. The incompressible kinetic energy spectrum of the inner quantum vortex ring core rapidly deviates from the k-3 spectrum found in the evolution of scalar BECs.
Using a set of interleaved unitary collision-stream operators, a three-dimensional (3D) quantum lattice gas algorithm is devised which, on taking moments, recovers the Gross-Pitaevskii (GP) equation. If a zero-temperature Bose-Einstein condensate (BEC) is trapped in an a magnetic well, the evolution of the ground-state wave function satisfies the scalar GP equation, while if the BEC is trapped in an optical trap the ground-state wave function satisfies spin or GP equations. Quantum turbulence is studied in a scalar GP system on 5,7603 grid yielding not only the classical Kolmogorov k-5/3 cascade but also the quantum vortex k-3 spectrum. For a certain class of initial conditions, one finds an intermittent loss of tangled quantum vortices as the vortex cores attain minimal size, and thus prevent the Kelvin wave cascade (due to helical wave-wave coupling on the vortex). A coupled set of GP equations are solved for spin or BEC. Skrymions, which describe topologically-linked quantum vortices, are examined. One finds, for certain initial conditions that the incompressible kinetic energy spectrum for the condensate component of a vortex ring core rapidly departs from the k-3 linear quantum vortex spectrum.
A quantum vortex is a topological singularity with quantized circulation, unlike a classical vortex with its continuous circulation strength.Quantum turbulence, envisaged as strong tangle of quantum vortices, of a Bose-Einstein condensate is examined by developing a unitary qubit lattice algorithm for the solution of the Gross-Pitaevskii equation.Earlier, it was shown that a certain class of initial conditions had a very short Poincare recurrence time for this Hamiltonian system.Here it is shown quantitatively that increasing the internal energy of the initial state leads to a systematic degradation of this class of solutions.Coupled Bose-Einstein condensate systems are investigated for a Hopf link class of initial conditions in which a vortex ring core is threaded by a linear vortex core that then closes toroidal around the vortex ring.These states are known as skyrmions and play a role in particle physics, astrophysics and condensed matter physics.
The ground state wave function for a Bose Einstein condensate is well described by the Gross-Pitaevskii equation. A Type-II quantum algorithm is devised that is ideally parallelized even on a classical computer. Only 2 qubits are required per spatial node. With unitary local collisions, streaming of entangled states and a spatially inhomogeneous unitary gauge rotation one recovers the Gross-Pitaevskii equation. Quantum vortex reconnection is simulated - even without any viscosity or resistivity (which are needed in classical vortex reconnection).
The dynamics of vortex solitons is studied in a BEC superfluid. A quantum lattice-gas algorithm (measurementbased quantum computation) is employed to examine the dynamical behavior vortex soliton solutions of the Gross-Pitaevskii equation (ø4 interaction nonlinear Schroedinger equation). Quantum turbulence is studied in large grid numerical simulations: Kolmogorov spectrum associated with a Richardson energy cascade occurs on large flow scales. At intermediate scales, a new k-6 power law emerges, due to vortex filamentary reconnections associated with Kelvin wave instabilities (vortex twisting) coupling to sound modes and the exchange of intermediate vortex rings. Finally, at very small spatial scales a k-3power law emerges, characterizing fluid dynamics occurring within the scale size of the vortex cores themselves. Poincaré recurrence is studied: in the free non-interacting system, a fast Poincaré recurrence occurs for regular arrays of line vortices. The recurrence period is used to demarcate dynamics driving a nonlinear quantum fluid towards turbulence, since fast recurrence is an approximate symmetry of the nonlinear quantum fluid at early times. This class of quantum algorithms is useful for studying BEC superfluid dynamics and, without modification, should allow for higher resolution simulations (with many components) on future quantum computers.