An $Sp(4, R)$ geometrization on the quantum dynamics of two-component Bose-Einstein condensate

Chang-Yan Wang, Yan He

Published Sep 7, 2021• 0 citations

Abstract

The method of geometrization arises as an important tool in understanding the entanglement of quantum fields and the behavior of the many-body system. The symplectic structure of the boson operators provide a natural way to geometrize the quantum dynamics of the bosonic systems of quadratic Hamiltonians, by recognizing that the time evolution operator corresponds to a real symplectic matrix in $Sp(4,R)$ group. We apply this geometrization scheme to study the quantum dynamics of the spinor Bose-Einstein condensate systems, demonstrating that the quantum dynamics of this system can be represented by trajectories in a six dimensional manifold. It is found that the trajectory is quasi-periodic for coupled bosons. The expectation value of the observables can also be naturally calculated through this approach.

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An $Sp(4, R)$ geometrization on the quantum dynamics of two-component Bose-Einstein condensateAn $Sp(4, R)$ geometrization on the quantum dynamics of two-component Bose-Einstein condensate

Chang-Yan Wang, Yan He

Published Sep 7, 2021• 0 citations

OpenAlex

Abstract

Sp(4,R)

group. We apply this geometrization scheme to study the
quantum dynamics of the spinor Bose-Einstein condensate systems, demonstrating
that the quantum dynamics of this system can be represented by trajectories in
a six dimensional manifold. It is found that the trajectory is quasi-periodic
for coupled bosons. The expectation value of the observables can also be
naturally calculated through this approach.

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The quantum dynamics of two-component Bose–Einstein condensate: an S p ( 4 , R ) symmetry approach

Chang-Yan Wang, Yan He

) groups have been heavily studied and applied in the study of quantum many body systems. However, the non-compact groups such as the real symplectic groups are less touched. In this paper, it is revealed that the quantum dynamics of two-component Bose-Einstein condensate can be described by a non-compact real symplectic groupSp(4,R). With this group, an explicit form of the wavefunction in any time of the evolution can be given, meanwhile, this whole time evolution can be shown to correspond to a trajectory in a six-dimensional manifold. By introducing a polar coordinate, we can visualize this six-dimensional manifold in 2d unit disk and reveal the relation between the behavior of the trajectory in this manifold and the eigenenergies of the Hamiltonian. Furthermore, the time evolution of expectation value of a physical observable such as number operator is proven closely related to the behavior of the trajectory in this manifold.) groups have been heavily studied and applied in the study of quantum many body systems. However, the non-compact groups such as the real symplectic groups are less touched. In this paper, it is revealed that the quantum dynamics of two-component Bose-Einstein condensate can be described by a non-compact real symplectic groupSp(4,R). With this group, an explicit form of the wavefunction in any time of the evolution can be given, meanwhile, this whole time evolution can be shown to correspond to a trajectory in a six-dimensional manifold. By introducing a polar coordinate, we can visualize this six-dimensional manifold in 2d unit disk and reveal the relation between the behavior of the trajectory in this manifold and the eigenenergies of the Hamiltonian. Furthermore, the time evolution of expectation value of a physical observable such as number operator is proven closely related to the behavior of the trajectory in this manifold.

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David Dahlbom, Hao Zhang, Cole Miles +3 more

The Landau-Lifshitz equation describes the time evolution of magnetic dipoles and can be derived by taking the classical limit of a quantum mechanical spin Hamiltonian. To take this limit, one constrains the many-body quantum state to a tensor product of coherent states, thereby neglecting entanglement between sites. Expectation values of the quantum spin operators produce the usual classical spin dipoles. One may also consider expectation values of polynomials of the spin operators, leading to quadrupole and higher-order spin moments, which satisfy a dynamical equation of motion that generalizes the Landau-Lifshitz dynamics [Zhang and Batista, Phys. Rev. B 104, 104409 (2021)]. Here we reformulate the dynamics of these ${N}^{2}\ensuremath{-}1$ generalized spin components as a mean-field Schr\"odinger equation on the $N$ -dimensional coherent state. This viewpoint suggests efficient integration methods that respect the local symplectic structure of the classical spin dynamics.The Landau-Lifshitz equation describes the time evolution of magnetic dipoles and can be derived by taking the classical limit of a quantum mechanical spin Hamiltonian. To take this limit, one constrains the many-body quantum state to a tensor product of coherent states, thereby neglecting entanglement between sites. Expectation values of the quantum spin operators produce the usual classical spin dipoles. One may also consider expectation values of polynomials of the spin operators, leading to quadrupole and higher-order spin moments, which satisfy a dynamical equation of motion that generalizes the Landau-Lifshitz dynamics [Zhang and Batista, Phys. Rev. B 104, 104409 (2021)]. Here we reformulate the dynamics of these ${N}^{2}\ensuremath{-}1$ generalized spin components as a mean-field Schr\"odinger equation on the $N$-dimensional coherent state. This viewpoint suggests efficient integration methods that respect the local symplectic structure of the classical spin dynamics.

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Sp(4, R)

geometrization on the quantum dynamics of two-component Bose-Einstein condensate

Sp(4,R)

This work reformulates quantum mechanics on the real plane

(\mathbb{R}^2)

, establishing an algebraic isomorphism to the complex field

(\mathbb{C})

and demonstrating that the complex structure inherent to the Schrödinger equation can be fully captured by coupled real-valued dynamics. By constructing a field

(\mathbb{R}^2, +, \cdot)

isomorphic to

\mathbb{C}

, the real counterpart of the Schrödinger equation is derived, yielding a system of coupled real partial differential equations. This system reveals a symplectic structure where the symplectic matrix

\mathcal{J}

geometrically represents the imaginary unit

i

, acting as a phase-space rotation operator. Time-independent solutions decouple into identical real Helmholtz equations, preserving spectral equivalence. The reformulation provides geometric insights into foundational aspects: the geometric representation of

i

, the emergence of probability conservation from real-component interactions, and the encoding of quantum phase relationships through the multiplicative structure of the

\mathbb{R}^2

field. This perspective reinforces connections to classical Hamiltonian mechanics and geometric quantization, suggesting that complex numbers are structural, emerging from field operations on the real plane, rather than fundamental. Furthermore, quantum computation elements like qubits and gates are shown to have equivalent representations as real symplectic and special orthogonal matrices, maintaining unitary equivalence. We introduce a framework for quantum computation within a real vector space and provide a generalized protocol for generating entangled Bell states. These results offer a new perspective on the necessity of complex numbers in quantum theory, providing a framework with potential computational advantages for hybrid quantum-classical systems and enhanced geometric interpretability.

N

-dimensional coherent state. This viewpoint suggests efficient integration methods that respect the local symplectic structure of the classical spin dynamics.