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Quantum Energy-Loop Gain: Geometric Accumulation of Energy in Closed Parameter Spaces

Jincheng Zhang

Published Aug 11, 2026• 0 citations
Publisher

Abstract

The exploration of geometric phases in quantum mechanics has fundamentally transformed our understanding of cyclic adiabatic evolutions, revealing how state vectors acquire non-trivial phase factors dictated by the underlying parameter manifolds. While traditional geometric phases govern quantum state wavefunctions and superposition coefficients, the energetic consequences of cyclic parameter manipulation remain bound to instantaneous expectation values within conventional frameworks. In this paper, we introduce and formalize the concept of Quantum Energy-Loop Gain (QELG), demonstrating that quantum systems subjected to closed parameter loops can exhibit an irreducible, topologically protected net energy shift. We define this phenomenon through an energy vector potential acting directly on the parameter space, presenting its mathematical foundations, gauge transformation properties, adiabatic and non-adiabatic dynamics, and thermodynamic consistency. Furthermore, we outline potential experimental platforms capable of observing this geometric accumulation of energy, opening a novel pathway for non-dissipative energy manipulation and advanced quantum control.

DOI
10.5281/zenodo.21888663

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The exploration of geometric phases in quantum mechanics has fundamentally transformed our understanding of cyclic adiabatic evolutions, revealing how state vectors acquire non-trivial phase factors dictated by the underlying parameter manifolds. While traditional geometric phases govern quantum state wavefunctions and superposition coefficients, the energetic consequences of cyclic parameter manipulation remain bound to instantaneous expectation values within conventional frameworks. In this paper, we introduce and formalize the concept of Quantum Energy-Loop Gain (QELG), demonstrating that quantum systems subjected to closed parameter loops can exhibit an irreducible, topologically protected net energy shift. We define this phenomenon through an energy vector potential acting directly on the parameter space, presenting its mathematical foundations, gauge transformation properties, adiabatic and non-adiabatic dynamics, and thermodynamic consistency. Furthermore, we outline potential experimental platforms capable of observing this geometric accumulation of energy, opening a novel pathway for non-dissipative energy manipulation and advanced quantum control.
This thesis consists of several studies performed over different few-dof quantum systems exposed to the effect of an uncontrolled environment. The primary focus of the work is to explore the relation between decoherence and environmentally-induced dissipative effects, and the concept known as geometric phases. The first mention of such an object in the context of quantum mechanics goes back to the seminal work by Berry. He demonstrated that the phase acquired by an eigenstate of a time-dependent Hamiltonian in an adiabatic cycle consists of two distinct contributions: one termed 'geometric' and the other known as the dynamical phase. Since Berry's work, the notion of geometric phase has been extended far beyond the original context, encompassing definitions applicable to arbitrary unitary evolutions. These geometric phases naturally arise in the geometric description of Hilbert space, where they manifest as holonomies and possess significance in the fundamental understanding of quantum mechanics and its mathematical framework, and in explaining various physical phenomena, including the Fractional Hall Effect. Moreover, from a modern perspective, geometric phases hold promise for practical applications, such as constructing geometric gates for quantum information processing and storage. However, in practice, a pure state of a quantum system is an idealized concept, and every experimental or real-world implementation must account for the presence of an environment that interacts with the observed system. This interaction necessitates a description in terms of mixed states and non-unitary evolutions. The definition of a geometric phase applicable in such scenarios remains an open problem, giving rise to multiple proposed solutions. Consequently, characterizing these geometric phases encompase motivations from fundamental aspects of quantum mechanics to technological applications.
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The quantum geometric potential is a gauge invariant carrying novel geometric features between any two energy levels or bands in quantum systems. In generic time-dependent systems it gives a vital physical modification for the instantaneous energy gaps, laying down more appropriate quantum adiabatic conditions for both non-degenerate and degenerate systems. Remarkably, for generic parameterized quantum systems, the integration of the quantum geometric potential on a closed loop leads to a novel type of quantized winding number, which is a quantum counterpart of the Gauss-Bonnet theorem. The effects of the quantum geometric potential had been indirectly supported by the experiments on the quantum adiabatic evolution, however, a direct experimental observation so far is lacking. In this paper we propose an interference measurement to directly probe the quantum geometric potential, where the relevant parameters are easily accessible by current experimental apparatus. A direct confirmation of this new physical quantity could motive further theoretical and experimental investigations as well as its potential real applications.
In [1], a new way to generate an observable geometric phase on a quantum system by means of a completely incoherent phenomenon has been proposed. The basic idea is to force the ground state of the system to evolve cyclically by “adiabatically” manipulating the environment with which it interacts. The specific scheme analyzed in [1], consisting of a multilevel atom interacting with a broadband squeezed vacuum bosonic bath whose squeezing parameters are smoothly changed in time along a closed loop, is here solved in a more direct way. This new solution emphasizes how the geometric phase at the ground state of the system is indeed due to a purely incoherent dynamics.
Traditional quantum mechanics and thermodynamics predominantly treat energy as a state function, focusing on initial and final states while treating the intermediate trajectory as a secondary detail dictated solely by the equations of motion. In this paper, we challenge this paradigm by introducing the framework of Quantum Energy-Path Curvature (QEPC). We propose that between the exact same initial state and final state, different quantum evolution paths can incur fundamentally distinct effective energy costs. By defining a rigorous path energy functional and a corresponding path curvature metric over the control and state manifold, QEPC reveals a deeper geometric structure underlying quantum transitions. We formulate the mathematical foundations of path energy and curvature, derive key properties governing quantum evolution under geometric constraints, and present testable predictions where two distinct paths connecting identical initial and final states yield differing energy expenditures (E_Gamma_1 != E_Gamma_2). This framework broadens the conceptual boundaries of energy state functions and offers profound implications for quantum control, quantum thermodynamics, and fundamental quantum mechanics.