Spin-Induced Fractal Time-Crystal-Like Dynamics and Non-Markovian Memory in the Bateman Dual Oscillator

Abstract: Can a closed quantum system generate time-crystal-like nonequilibrium behavior, self-similar scaling structures, and non-Markovian memory without external driving or coupling to a macroscopic environment? We address this question within the quantum Bateman oscillator formulated in a nonrelativistic (2 + 1)-dimensional phase-space noncommutative framework generated by spin-induced spatial deformation. The resulting doubled quantum dynamics is governed by a time-independent Hermitian Hamiltonian and exhibits an underlying SU(1, 1) structure with amplified and damped collective modes. We show that these modes satisfy an exact discrete scaling covariance, leading to self-similar temporal evolution without external driving. Upon tracing over one oscillator sector, the reduced dynamics becomes intrinsically non-Markovian and is governed by a history-dependent memory kernel. The same scaling structure admits a geometric representation in terms of logarithmic-spiral trajectories associated with the amplified and damped branches of the Bateman system. Because the mechanism relies on nonequilibrium reduced dynamics rather than equilibrium expectation values of local observables, it lies outside the assumptions underlying conventional no-go theorems for equilibrium time crystals. Our results identify spin as the common physical origin of the amplified and damped Bateman dynamics, self-similar scaling periodicity, logarithmic-spiral structures, and non-Markovian memory, also suggesting a natural extension of the mechanism to relativistic anyonic systems.

• The paper demonstrates that spin-induced deformation creates a noncommutative phase-space in Bateman dual oscillators, leading to fractal self-similarity and time-crystal-like behavior.
• It employs canonical quantization and Bopp’s shift to reveal a doubled SU(1,1) structure that generates amplified and damped mode dynamics with exact scaling relations.
• The study uncovers non-Markovian memory effects linking spin-induced dissipation to fractal geometry, offering insights for designing quantum systems with engineered gain-loss dynamics.

Spin-Induced Fractal Scaling and Non-Markovian Dynamics in the Bateman Dual Oscillator

Introduction

This paper presents a rigorous noncommutative phase-space analysis of the quantum Bateman dual oscillator, emphasizing spin-induced deformation as a key driver for generating fractal self-similar dynamics and non-Markovian memory effects within a globally unitary framework. The analysis unfolds in $(2+1)$ dimensions, leveraging the second central extension of the planar Galilei group to demonstrate how nonrelativistic spin alters the symplectic structure and seed noncommutativity, dissipation, and mode amplification/damping in closed systems. The quantum Bateman system realizes a doubled $SU(1,1)$ structure, with explicit connections to fractal scaling, logarithmic-spiral trajectories, and kernel-based non-Markovian subsystem evolution.

Symplectic Deformation and Spin-Induced Noncommutativity

The symplectic structure is augmented by the spin-induced term $s\epsilon_{ij}d\pi_i\wedge d\pi_j$, leading to a noncommutative phase-space in which spatial coordinates possess nontrivial Dirac brackets. The Bateman oscillator emerges naturally via the induced coupling between damped and amplified sectors, with the dissipation parameter $\gamma$ linearly proportional to the spin deformation: $\gamma = \Omega^2 s / m$. This establishes a direct correspondence between fractional spin and dissipative quantum dynamics, as parameterized in $(2+1)$ dimensions.

Canonical Quantization and Doubled Dynamics

Upon canonical quantization, the Dirac bracket structure is embedded using Bopp’s shift, enabling the representation of noncommutative variables $(\hat{Y}_i, \hat{\pi}_i)$ in terms of auxiliary canonical pairs $(\hat{X}_i, \hat{P}_i)$. The resulting Hamiltonian, recast in light-cone coordinates, acquires an explicit coupling characterized by $\Gamma = m\Omega^2\theta/2\hbar$, with $\theta \sim \hbar s/m^2$. Ladder operators for both sectors yield a two-mode Hamiltonian featuring a Bogoliubov-type $SU(1,1)$0 structure. The Heisenberg equations of motion generate amplified and damped branches, leading to time-dependent $SU(1,1)$1 squeezed coherent states.

Self-Similar Scaling and Logarithmic Spirals

The evolution of collective amplified/damped modes $SU(1,1)$2 yields exact self-similar scaling relations of the form $SU(1,1)$3, with characteristic scaling period $SU(1,1)$4. This scaling, fundamentally different from ordinary temporal periodicity, is geometrically realized as logarithmic-spiral trajectories in the complex plane (Figure 1):

Figure 1: Trajectory in the complex plane illustrating the logarithmic spiral structure arising from the combined oscillatory and scaling dynamics.

Each spiral trajectory combines exponential scaling and rotation, parameterized by fractional spin, with both expanding and contracting branches encoding the amplified/damped sectors. The scaling exponent $SU(1,1)$5 is directly related to the underlying spin: $SU(1,1)$6, confirming the physical origin of the fractal self-similarity.

Non-Markovian Memory and Subsystem Dynamics

Tracing over one oscillator sector results in reduced dynamics governed by a non-Markovian evolution equation containing a history-dependent kernel, $SU(1,1)$7. Although the explicit scaling law $SU(1,1)$8 is inaccessible at the subsystem level (no observable periodicity), the full density matrix retains memory of cross-sector correlations via the kernel, which depends on past states $SU(1,1)$9 and operator combinations $s\epsilon_{ij}d\pi_i\wedge d\pi_j$0 and $s\epsilon_{ij}d\pi_i\wedge d\pi_j$1. The non-Markovian kernel transmits influence from the doubled structure, enabling indirect imprint of the self-similar scaling behavior in reduced evolution. This mechanism lies entirely outside equilibrium assumptions and evades standard no-go theorems for time crystals.

Fractal Geometry and Koch Structures

The geometric interpretation is extended to fractal constructions such as the Koch curve. The iterative self-similarity of the Koch fractal, parameterized as $s\epsilon_{ij}d\pi_i\wedge d\pi_j$2 with $s\epsilon_{ij}d\pi_i\wedge d\pi_j$3, mirrors the doubling and scaling properties of the Bateman modes. The discrete scaling structure and time-lattice emerge, and the scaling exponent $s\epsilon_{ij}d\pi_i\wedge d\pi_j$4 is again controlled by spin-induced deformation. The mapping between fractal geometry and quantum mode doubling consolidates the conceptual framework (Figure 2):

Figure 2: Successive stages in the construction of the Koch curve, illustrating the emergence of self-similar fractal structure.

Energy Exchange and Time-Crystal-Like Oscillations

Even as the total Hamiltonian remains time-independent and globally unitary, coherent energy exchange between coupled sectors yields persistent oscillations in local observables. The expectation values of sector occupation numbers oscillate with opposite phases, governed by canonical Bogoliubov transformations, and serve as direct signatures of dissipative gain-loss mechanisms. The amplified/damped branches and associated $s\epsilon_{ij}d\pi_i\wedge d\pi_j$5 symmetry ensure strict global conservation but nontrivial subsystem evolution, characteristic of time-crystal-like ordering without symmetry breaking.

Implications and Outlook

The precise mechanism links spin-induced spatial noncommutativity, fractal scaling dynamics, and non-Markovian memory. Practical implications include the design of quantum systems or synthetic platforms (photonic resonators, optomechanics, circuit QED, trapped ions) supporting engineered gain-loss dynamics and exhibiting rich memory effects and scaling behavior. The theoretical structure—globally unitary, time-independent Hamiltonian with intrinsic dissipative dynamics—provides a template for open quantum system modeling, quantum thermodynamics, and dynamical pattern formation. The framework is primed for extension to relativistic anyonic systems, with the potential for Lorentz-covariant scaling and novel dephasing statistics.

Conclusion

This paper systematically establishes the centrality of nonrelativistic spin in inducing dissipation, noncommutative geometry, and fractal self-similar scaling in quantum Bateman dual oscillators. The doubled $s\epsilon_{ij}d\pi_i\wedge d\pi_j$6 dynamics, exact scaling periodicity, and non-Markovian kernel-based memory effects are shown to originate from internal mechanisms, not external driving or symmetry breaking. The geometric realization in logarithmic spirals and fractal Koch structures unifies physical and mathematical aspects. Experimentally, the model suggests observable non-Markovianity and information backflow, with prospects for extension to relativistic and anyonic sectors. Overall, the study delineates a robust path towards understanding self-similar scaling, dissipative quantum dynamics, and memory in closed noncommutative systems.