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Annalen der Physik

Generalized Gauge Transformation with PT‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System (Ann. Phys. 6/2022)

Yan Gu, Xiao‐Lei Hao, Jiu‐Qing Liang

Published Jun 1, 2022• 15 citations
Publisher

Abstract

Gauge Transformation The front cover design is inspired by an ancient Chinese folk tale, “the Cowherd and the Vega”. The Vega was a carefree goddess, and the Cowherd was a hard-working cattle herder on the earth. They lived in two different worlds, but accidentally met and fell in love. According to the tale, they meet every year on the bridge of magpies. In the cover design, depicting the research reported by Jiu-Qing Liang and co-workers in article number 2200069, the Vega represents the Berry phase, which comes from the quantum world. The Cowherd represents the Hannay angle, which comes from classical world. The bridge of magpies is the PT-symmetric non-Hermitian. A certain connection between the Berry phase and Hannay angle is called the quantum–classical correspondence.

DOI
10.1002/andp.202270012

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Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

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Abstract This paper demonstrates that the parity‐time ()‐symmetric non‐Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time‐dependent transformation operator has to be symmetric and non‐unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of the non‐Hermitian Hamiltonian. The wave functions and associated non‐adiabatic Berry phase for the nth eigenstate are obtained analytically. The classical version of the non‐Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with symmetric canonical‐variable transfer in the classical gauge transformation. Moreover, with the change of position‐momentum to angle‐action variables it is revealed that the non‐adiabatic Hannay's angle and Berry phase satisfy precisely the quantum‐classical correspondence, .

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Rigorous solutions, Berry phases and quantum-classical correspondence for time-dependent non-Hermitian SU(2)-SU(1, 1) Hamiltonians

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Based on the generalized gauge transformation and invariant methods, we study in this work the rigorous solution for non-Hermitian Hamiltonians consisting of SU(1, 1) and SU(2) generators, which describe the harmonic oscillator and a single spin of arbitrary spin value, respectively, in periodically time-dependent fields. A non-Hermitian Hamiltonian with real eigenvalues is called pseudo-Hermitian. Such a Hamiltonian can be converted to a Hermitian one by a similarity transformation. We extend the concept of pseudo-Hermitian Hamiltonians to the time-dependent case. Under the generalized gauge transformation with a time-dependent Hermitian operator, the Hamiltonian becomes Hermitian and time-independent in the new gauge. The rigorous solution, as well as the Berry phase, is obtained by applying the inverse transformation to return to the original gauge. The time-dependent Schrödinger equation is also solved in terms of non-Hermitian invariant operators, which possess conserved real eigenvalues. We derive the time-dependent solutions and corresponding geometric phases by eigenstate expansion of the invariant operators, combined with the time-evolution operator. It is observed that the non-Hermitian coupling constant indeed has an exceptional point Gc(ω) for the SU(2) non-Hermitian Hamiltonian, where all eigenstates are degenerate with zero eigenvalue and the Berry phase diverges. The eigenvalues become complex beyond Gc(ω) . However, the eigenvalues are real within the entire region of coupling constant for the SU(1, 1) system, different from the common belief that the exceptional point must exist in the non-Hermitian Hamiltonian. The classical counterpart of the SU(1, 1) non-Hermitian Hamiltonian is, in fact, a complex function of position-momentum variables. It can be converted to a real function via the generalized gauge transformation, accompanied by a canonical-variable transformation. This process corresponds exactly to the Hermitization procedure of the pseudo-Hermitian Hamiltonian. We obtain the classical Hannay angle for the periodically time-dependent SU(1, 1) pseudo-Hermitian Hamiltonian. The non-adiabatic Hannay angle and the Berry phase satisfy precisely the quantum-classical correspondence, γn(T)=−(n+12)∆Θ .

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Correspondences between quantum and classical orbits Berry phases and Hannay angles for harmonic oscillator system

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On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantum-classical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantum-classical orbits and geometric phases. Here, the two one-dimensional oscillators refer to a common harmonic oscillator and a rotation and translation harmonic oscillator. In terms of the generalized gauge transformation, we obtain the stationary Lissajous orbits and Hannay's angle. On the other hand, the eigenfunctions and Berry phases are derived analytically with the help of time-dependent gauge transformation. We may draw the conclusion that the nonadiabatic Berry phase in the original gauge is-n times the classical Hannay's angle, here n is the eigenfunction index. As a matter of fact, the quantum geometric phase and the classical Hannay's angle have the same nature according to Berry. Finally, by using the SU(2) coherent superposition of degenerate two-dimensional eigenfunctions for a fixed energy value, we construct the stationary wave functions and show that the spatial distribution of wave-function probability clouds is in excellent accordance with the classical orbits, indicating the exact quantum-classical correspondence. We also demonstrate the quantum-classical correspondences for the geometric phase-angle and the quantum-classical orbits in a unified form.

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Generalized gauge transformation and the corresponding Hermitian counterparts of SU(1, 1), SU(2) pseudo-Hermitian Hamiltonians

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Abstract We study in this paper both the stationary and time-dependent pseudo-Hermitian Hamiltonians consisting respectively of SU(1, 1), SU(2) generators. The pseudo-Hermitian Hamiltonians can be generated from kernel Hermitian-Hamiltonians by a generalized gauge transformation with a non-unitary but Hermitian operator. The metric operator of the biorthogonal sets of eigenstates is simply the square of the transformation operator, which is formulated explicitly. The exact solutions of pseudo-Hermitian Hamiltonians are obtained in terms of the eigenststates of the Hermitian counterparts. We observe a critical point G c of coupling constant, where all eigenstates of the stationary Hamiltonians are degenerate with a vanishing eigenvalue. This critical point is modified as G c (ω) in the time-dependent case including the frequency of external field. Returning to the original gauge we obtain analytically the wave functions and associated non-adiabatic Berry phase, which diverges at the critical point for the SU(2)Hamiltonian. Beyond the critical point Berry phase becomes a complex domain.

Physica Scripta•Feb 14, 2023•6 citations
Gauge Transformation The front cover design is inspired by an ancient Chinese folk tale, “the Cowherd and the Vega”. The Vega was a carefree goddess, and the Cowherd was a hard-working cattle herder on the earth. They lived in two different worlds, but accidentally met and fell in love. According to the tale, they meet every year on the bridge of magpies. In the cover design, depicting the research reported by Jiu-Qing Liang and co-workers in article number 2200069, the Vega represents the Berry phase, which comes from the quantum world. The Cowherd represents the Hannay angle, which comes from classical world. The bridge of magpies is the PT-symmetric non-Hermitian. A certain connection between the Berry phase and Hannay angle is called the quantum–classical correspondence.
Generalized Gauge Transformation with PTPTPTPT‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System
Abstract This paper demonstrates that the parity‐time ()‐symmetric non‐Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time‐dependent transformation operator has to be symmetric and non‐unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of the non‐Hermitian Hamiltonian. The wave functions and associated non‐adiabatic Berry phase for the nth eigenstate are obtained analytically. The classical version of the non‐Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with symmetric canonical‐variable transfer in the classical gauge transformation. Moreover, with the change of position‐momentum to angle‐action variables it is revealed that the non‐adiabatic Hannay's angle and Berry phase satisfy precisely the quantum‐classical correspondence, .
Based on the generalized gauge transformation and invariant methods, we study in this work the rigorous solution for non-Hermitian Hamiltonians consisting of SU(1, 1) and SU(2) generators, which describe the harmonic oscillator and a single spin of arbitrary spin value, respectively, in periodically time-dependent fields. A non-Hermitian Hamiltonian with real eigenvalues is called pseudo-Hermitian. Such a Hamiltonian can be converted to a Hermitian one by a similarity transformation. We extend the concept of pseudo-Hermitian Hamiltonians to the time-dependent case. Under the generalized gauge transformation with a time-dependent Hermitian operator, the Hamiltonian becomes Hermitian and time-independent in the new gauge. The rigorous solution, as well as the Berry phase, is obtained by applying the inverse transformation to return to the original gauge. The time-dependent Schrödinger equation is also solved in terms of non-Hermitian invariant operators, which possess conserved real eigenvalues. We derive the time-dependent solutions and corresponding geometric phases by eigenstate expansion of the invariant operators, combined with the time-evolution operator. It is observed that the non-Hermitian coupling constant indeed has an exceptional point Gc(ω) for the SU(2) non-Hermitian Hamiltonian, where all eigenstates are degenerate with zero eigenvalue and the Berry phase diverges. The eigenvalues become complex beyond Gc(ω) . However, the eigenvalues are real within the entire region of coupling constant for the SU(1, 1) system, different from the common belief that the exceptional point must exist in the non-Hermitian Hamiltonian. The classical counterpart of the SU(1, 1) non-Hermitian Hamiltonian is, in fact, a complex function of position-momentum variables. It can be converted to a real function via the generalized gauge transformation, accompanied by a canonical-variable transformation. This process corresponds exactly to the Hermitization procedure of the pseudo-Hermitian Hamiltonian. We obtain the classical Hannay angle for the periodically time-dependent SU(1, 1) pseudo-Hermitian Hamiltonian. The non-adiabatic Hannay angle and the Berry phase satisfy precisely the quantum-classical correspondence, γn(T)=−(n+12)∆Θ .
On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantum-classical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantum-classical orbits and geometric phases. Here, the two one-dimensional oscillators refer to a common harmonic oscillator and a rotation and translation harmonic oscillator. In terms of the generalized gauge transformation, we obtain the stationary Lissajous orbits and Hannay's angle. On the other hand, the eigenfunctions and Berry phases are derived analytically with the help of time-dependent gauge transformation. We may draw the conclusion that the nonadiabatic Berry phase in the original gauge is-n times the classical Hannay's angle, here n is the eigenfunction index. As a matter of fact, the quantum geometric phase and the classical Hannay's angle have the same nature according to Berry. Finally, by using the SU(2) coherent superposition of degenerate two-dimensional eigenfunctions for a fixed energy value, we construct the stationary wave functions and show that the spatial distribution of wave-function probability clouds is in excellent accordance with the classical orbits, indicating the exact quantum-classical correspondence. We also demonstrate the quantum-classical correspondences for the geometric phase-angle and the quantum-classical orbits in a unified form.
Abstract The decomplexification procedure allows one to show mathematically (stricto sensu) the equivalence (isomorphism) between the quantum dynamics of a system with a finite number of basis states and a classical dynamics system. This unique way of connecting different dynamics was used in the past to analyze the relationship between the well-known geometric phase present in the quantum evolution discovered by Berry and its generalizations, with their analogs, the Hannay phases, in the classical domain. In here, this analysis is carried out for several quantum hermitian and non-hermitian PT-symmetric Hamiltonians and compared with the Hannay phase analysis in their classical isomorphic equivalent systems. As the equivalence ends in the classical domain with oscillator dynamics, we exploit the analogy to propose resonant electric circuits coupled with a gyrator, to reproduce the geometric phase coming from the theoretical solutions, in simulated laboratory experiments.
The well-known geometric phase present in the quantum adiabatic evolution discovered by Berry many years ago has its analogue, the Hannay phase, in the classical domain.We calculate the Berry phase with examples for quantum hermitian and non-hermitian PTPTPT-symmetric Hamiltonians and compare with the Hannay phase in their classical equivalents.We use the analogy to propose resonant electric circuits which reproduce the theoretical solutions in simulated laboratory experiments.
Abstract We study in this paper both the stationary and time-dependent pseudo-Hermitian Hamiltonians consisting respectively of SU(1, 1), SU(2) generators. The pseudo-Hermitian Hamiltonians can be generated from kernel Hermitian-Hamiltonians by a generalized gauge transformation with a non-unitary but Hermitian operator. The metric operator of the biorthogonal sets of eigenstates is simply the square of the transformation operator, which is formulated explicitly. The exact solutions of pseudo-Hermitian Hamiltonians are obtained in terms of the eigenststates of the Hermitian counterparts. We observe a critical point G c of coupling constant, where all eigenstates of the stationary Hamiltonians are degenerate with a vanishing eigenvalue. This critical point is modified as G c (ω) in the time-dependent case including the frequency of external field. Returning to the original gauge we obtain analytically the wave functions and associated non-adiabatic Berry phase, which diverges at the critical point for the SU(2)Hamiltonian. Beyond the critical point Berry phase becomes a complex domain.