Geometrical Amplitude factors in the the adiabatic evolution

Abstract: In a quantum system initially in the n-th eigenstate, an adiabatic evolution of the Hamiltonian ensures that the system remains in the corresponding instantaneous eigenstate while acquiring a phase factor. This phase has two components: one resulting from standard time evolution and another associated with the dependence of the eigenstate on the varying Hamiltonian, known as the Berry phase. In this work, we explore the concept of geometric amplitudes in the context of a Hermitian Hamiltonian with imaginary eigenvalues. We introduce the notion of geometric amplitude and provide a novel derivation of this concept. Our study reveals that a system undergoing cyclic evolution under adiabatic conditions acquires an additional amplitude factor of purely geometric origin. To illustrate this idea, we apply it to a concrete case: a generalized harmonic oscillator.

• The paper introduces geometric amplitude factors as a key contribution to extend adiabatic evolution theory for Hermitian systems with imaginary eigenvalues.
• It develops a framework using a dynamic metric operator for normalizing non-unitary state evolution, demonstrated through a generalized harmonic oscillator model.
• The findings generalize the traditional Berry phase concept, offering new insights for quantum control and the study of non-standard quantum dynamics.

Geometrical Amplitude Factors in Adiabatic Evolution with Imaginary Eigenvalues

Introduction and Context

The study addresses the adiabatic evolution of quantum systems described by Hermitian Hamiltonians supporting imaginary eigenvalues, a scenario traditionally precluded in orthodox quantum mechanics due to implications for non-unitary evolution. Classical adiabatic theory, first formalized by Ehrenfest, Born, and Fock, ensures adiabatic following within the instantaneous eigenbasis of a slowly-varying, non-degenerate Hermitian Hamiltonian, yielding a state-dependent dynamical phase and a geometric phase—termed the Berry phase—manifesting in cyclically-evolved systems. While Berry’s phase and its generalizations are firmly embedded in the landscape of quantum theory and applications, especially in quantum control and topological phenomena, their extension to systems with non-orthogonal states, non-traditional Hilbert space structures, or non-Hermitian operators remains nontrivial.

This work pivots from the traditional restriction to purely real spectra and develops a theory of “geometrical amplitude” factors for Hermitian quantum systems exhibiting imaginary eigenvalues, motivated by edge cases such as certain operator polynomials (e.g., $q^3p + pq^3$) and by analogy to pseudo-Hermitian and PT-symmetric quantum mechanics.

Hermitian Hamiltonians with Imaginary Eigenvalues

Hermiticity, in the Dirac sense, is foundational to real spectra and unitary dynamics. Yet, as illustrated in the example of $O = q^3p + pq^3$, Hermiticity does not invariably guarantee real eigenvalues. In this framework, the eigenvalue problem $H |n\rangle = i E_n|n\rangle$, with $E_n \in \mathbb{R}$ for Hermitian $H$, fundamentally alters the norm evolution: $|\psi_n(t)\rangle = e^{E_n t}|n\rangle$ non-unitarily amplifies or damps the state. For time-dependent Hamiltonians $H(t)$, the need for a consistent quantum mechanical evolution leads naturally to the introduction of a dynamical metric operator ensuring positive-definite norms, following pseudo-Hermitian quantization strategies.

The recognition of Hermitian but non-diagonalizable, or otherwise structurally anomalous, Hamiltonians aligns with prior work on PT-symmetric and pseudo-Hermitian systems, yet the focus here remains strictly on Hermitian cases with complex spectra, rather than the broader class of non-Hermitian models.

Geometric Amplitude: Extension of Berry Phase

Cyclic adiabatic evolution under such Hamiltonians yields, in place of pure phase accumulation, both a geometric amplitude and a geometric phase. The wavefunction in this context is expressed as:

$$|\psi_n(t)\rangle = \exp\left[\int_0^t E_n(\tau) d\tau\right] \exp\left[i \gamma_n(t)\right] |n(t)\rangle$$

where $E_n$ may be real but appears in the exponent with a prefactor $i$, and $\gamma_n(t)$ is the geometric quantity, reducing to the Berry phase for traditional Hermitian, real spectrum cases. Notably, the geometric contribution manifests as an amplitude factor rather than a pure phase, reflecting the non-unitary evolution entailed by the imaginary spectrum.

A generalized scalar product is constructed by introducing a nontrivial, time-dependent metric $\eta(t)$, defined such that states remain normalized:

$$\langle m(R(t))| \eta(t) |n(R(t)) \rangle = \delta_{mn}$$

For evolved states, the metric operator acquires time-dependent dynamical and geometric contributions, ensuring consistency of the adiabatic projection and maintaining the orthonormal resolution despite the absence of conventional Hermitian unitarity.

Exemplification: The Generalized Harmonic Oscillator

These theoretical developments are concretized via the generalized, time-dependent harmonic oscillator:

$H(t) = Z(t) p^2 + Y(t)(pq + qp) + X(t)q^2$

With $XZ < Y^2$, the model describes an inverted oscillator with a purely imaginary effective frequency. After appropriate unitary (canonical) transformations, explicit eigenvalue and (generalized) eigenfunction solutions are obtained, revealing that their norm is ill-defined under the standard $L^2$ scalar product, thus necessitating the use of the aforementioned metric operator for consistent normalization.

The geometric amplitude is computed via:

$$\gamma_n(C) = -\left(n + \frac{1}{2}\right) \oint_C \frac{Z}{Y} d\omega$$

where the path $C$ is the closed loop traced by the parameters in the evolution, paralleling the geometric (Berry) phase in unitary settings but resulting here in an amplitude modulation. The metric operator used to regularize the norm is shown to be non-unique within a family parameterized by the dynamical variables.

Implications and Directions for Research

The introduction and formal derivation of geometric amplitude factors for Hermitian systems with imaginary spectra generalizes foundational aspects of adiabatic quantum theory and offers new perspectives on the geometric invariants in systems without unitary time evolution. The results suggest that amplitude-like geometric factors may emerge in other settings with anomalous spectrum operators, particularly those relevant to non-standard quantum dynamics, quantum measurement with gain/loss, and mathematical formulations of pseudo-Hermitian quantum mechanics.

The theoretical implications point toward reevaluating adiabatic theorems and geometric phase structures under generalized spectral conditions, potentially impacting quantum control, dissipative quantum systems, and extensions to quantum thermodynamics. Practically, realization in physical systems—such as those described by effective non-unitary Hamiltonians—may yield observable geometric amplitude effects. There is also scope for extending these results to open quantum systems or for exploring connections with geometric phases in infinite-dimensional representations and continuous spectra, wherein similar metric modifications are required.

Conclusion

This work systematically extends the adiabatic geometric phase to Hermitian quantum systems characterized by imaginary eigenvalues, establishing the notion of a geometric amplitude factor that arises in cyclic adiabatic evolution. Both the conditions for normalizability and the necessity of a nontrivial metric operator are identified, with the formalism substantiated through a detailed analysis of the generalized harmonic oscillator. These findings enlarge the conventional framework of quantum geometric phases, introducing new invariants and providing a foundation for further investigation in quantum systems exhibiting departures from unitary evolution due to non-standard spectral properties.