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arXiv:2409.13595 · Physical Review Research

Geometric contribution to adiabatic amplification in non-Hermitian systems

Tomoki Ozawa, Henning Schomerus

Published Feb 18, 2025• 7 citations
arXivHTMLPDF

Abstract

Concepts from non-Hermitian quantum mechanics have proven useful in understanding and manipulating a variety of classical systems, such as those encountered in optics, classical mechanics, and metamaterial design. Recently, the non-Hermitian analog of the Berry phase for adiabatic processes was experimentally measured. In non-Hermitian systems, the Berry phase can have an imaginary part, which contributes to the amplification or decay of the total wave intensity. When the imaginary part of the Berry curvature is zero, this geometric amplification factor is determined solely by the initial and final points of the adiabatic path in parameter space, and it does not depend on how these points are connected by the path. We list classes of non-Hermitian Hamiltonians where this path independence is guaranteed by suitable symmetries, and we find that, for some of these classes, the amplification factor can be written only in terms of the Petermann factors of the initial and final points. Our result can, in turn, be used to experimentally obtain the Petermann factor by observing how the norm of the wave function changes under adiabatic processes. We validate our theory using a couple of concrete examples of physical relevance.

DOI
10.1103/physrevresearch.7.013173

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Concepts from non-Hermitian quantum mechanics have proven useful in understanding and manipulating a variety of classical systems, such as those encountered in optics, classical mechanics, and metamaterial design. Recently, the non-Hermitian analog of the Berry phase for adiabatic processes was experimentally measured. In non-Hermitian systems, the Berry phase can have an imaginary part, which contributes to the amplification or decay of the total wave intensity. When the imaginary part of the Berry curvature is zero, this geometric amplification factor is determined solely by the initial and final points of the adiabatic path in parameter space, and it does not depend on how these points are connected by the path. We list classes of non-Hermitian Hamiltonians where this path independence is guaranteed by suitable symmetries, and we find that, for some of these classes, the amplification factor can be written only in terms of the Petermann factors of the initial and final points. Our result can, in turn, be used to experimentally obtain the Petermann factor by observing how the norm of the wave function changes under adiabatic processes. We validate our theory using a couple of concrete examples of physical relevance.
The geometrical Berry phase is key to understanding the behaviour of quantum states under cyclic adiabatic evolution. When generalised to non-Hermitian systems with gain and loss, the Berry phase can become complex, and should modify not only the phase but also the amplitude of the state. Here, we perform the first experimental measurements of the adiabatic non-Hermitian Berry phase, exploring a minimal two-site PT\mathcal{PT}PT-symmetric Hamiltonian that is inspired by the Hatano-Nelson model. We realise this non-Hermitian model experimentally by mapping its dynamics to that of a pair of classical oscillators coupled by real-time measurement-based feedback. As we verify experimentally, the adiabatic non-Hermitian Berry phase is a purely geometrical effect that leads to significant amplification and damping of the amplitude also for non-cyclical paths within the parameter space even when all eigenenergies are real. We further observe a non-Hermitian analog of the Aharonov--Bohm solenoid effect, observing amplification and attenuation when encircling a region of broken PT\mathcal{PT}PT symmetry that serves as a source of imaginary flux. This experiment demonstrates the importance of geometrical effects that are unique to non-Hermitian systems and paves the way towards the further studies of non-Hermitian and topological physics in synthetic metamaterials.
The Berry phase is a geometric phase acquired during adiabatic evolution over a closed loop in parameter space. It plays an essential role in geometric quantum gates and other phase-based protocols. In non-Hermitian systems, the Berry phase is complex, introducing fundamentally new geometric effects, including state amplification. In this work, we report experimental measurement of both the real and imaginary components of a Berry phase in a fully quantum system using a superconducting transmon circuit with engineered dissipation. We also demonstrate the path-dependent effects of the imaginary part on the dissipation and its utility in the implementation of non-unitary quantum control. These findings establish a clear geometric distinction between the real and imaginary components of the Berry phase and experimentally confirm the unique adiabatic behavior of non-Hermitian quantum systems.
In a quantum system initially in the nnn-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. 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 inverted harmonic oscillator. Although a pseudo-inner product can be introduced to make resonance states formally normalizable, this procedure relies on a non-unitary metric operator and defines a modified Hilbert space; it does not restore normalizability nor self-adjointness in the standard L2(R)L^2(R)L2(R) framework.
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.
Unitary evolution in P\mathcal{P}PT\mathcal{T}T-symmetric quantum mechanics (QM) with a time-dependent metric is found to yield an interesting class of adiabatic processes. As an explicit example, a Berry-like phase associated with a P\mathcal{P}PT\mathcal{T}T-symmetric two-level system is derived and is interpreted as the flux of a fictitious monopole with a tunable charge plus a singular string component with nontrivial phase contributions. To gain more insight, the Hermitian analog of our non-Hermitian problem is also analyzed, which results in an intriguing class of geometric-phase problems in conventional QM as well, where the Hamiltonian includes a perturbative term that is proportional to the rate of change in adiabatic parameters.
Abstract This work revisit on the intrinsic connection between Dirac monopole theory and Berry geometric phases by extending Dirac's theory to the parameter space. Using the simplest two-mode Hamiltonian model, we explicitly visualize Dirac strings with endpoints in the parameter space, demonstrating that these endpoints correspond to accidental degeneracy points of energy eigenvalues in Hermitian systems. We show that non-integrable phase factors, induced by such Dirac strings, directly give rise to the well-known Berry connection and curvature, which can be re-derived by following the spirit of Dirac's monopole theory with the aid of the quantum adiabatic theorem. Our results indicate that the Berry geometric phase is essentially the non-integrable phase factor induced by Dirac strings with endpoints in the parameter space. These results further deepen our understanding on monopole and geometric phases, and would inspire more studies on them especially when Berry's framework is not applicable.