Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics B

Time-dependent gauge transformation for non-hermitian periodically driven system with a controllable non-hermitian strength

Yajiang Hao, Shuchang Hao

Published May 22, 2026• 0 citations
Publisher

Abstract

In this paper we introduce a general periodically driven non-Hermitian SU(2) system. In this model, the parameter [Formula: see text] is introduced to define the non-Hermitian strength. By constructing a generalized invariant operator using the time-dependent gauge transformation we obtain the exact solution to this model and the dependence of bi-orthogonal basis, metric operator, Berry phase and time-evolution operator on the non-Hermitian parameter.

DOI
10.1142/s0217979226501833

Related papers

PT -symmetric non-Hermitian Hamiltonian and invariant operator in periodically driven SU(1,1) system

Yan Gu, Xue-Min Bai, Xiao‐Lei Hao +1 more

We study in this paper the time evolution of PT-symmetric non-Hermitian Hamiltonian consisting of periodically driven SU(1,1) generators. A non-Hermitian invariant operator is adopted to solve the Schrödinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a PT-symmetric but non-unitary transformation operator in the construction of the non-Hermitian invariant. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adiabatic one in the slow time-variation limit. A non-unitary time- evolution operator is found analytically. As a consequence of the non-unitarity the ket (|ψ(t)〉) and bra (〈ψ(t)|) states are not normalized each other. While the inner product of two states can be evaluated with the help of a metric operator. It is shown explicitly that the model can be realized by a periodically driven oscillator.

arXiv:2201.00158•Results in Physics•May 10, 2022•11 citations

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

JunLi XIN, Jiu‐Qing Liang, JunXia SHEN

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 .

Zhongguo kexue. Wulixue Lixue Tianwenxue•Jul 28, 2025•0 citations

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

Yan Gu, Xiao‐Lei Hao, Jiu‐Qing Liang

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, .

arXiv:2209.01393•Annalen der Physik•Apr 17, 2022•7 citations

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

Ni Liu, Yan Gu, J.-Q. Liang

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

The pseudo Hermitian invariant operator and time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry

Walid Koussa, Naima Mana, Oum Kaltoum Djeghiour +1 more

We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.

Journal of Mathematical Physics•Jul 1, 2018•18 citations

Time evolution of quantum systems with time-dependent non-Hermitian Hamiltonian and the pseudo Hermitian invariant operator

Mustapha Maamache, Oum Kaltoum Djeghiour, Walid Koussa +1 more

We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators.With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator.

arXiv:1705.08298•May 21, 2017•2 citations
In this paper we introduce a general periodically driven non-Hermitian SU(2) system. In this model, the parameter [Formula: see text] is introduced to define the non-Hermitian strength. By constructing a generalized invariant operator using the time-dependent gauge transformation we obtain the exact solution to this model and the dependence of bi-orthogonal basis, metric operator, Berry phase and time-evolution operator on the non-Hermitian parameter.
We study in this paper the time evolution of PT-symmetric non-Hermitian Hamiltonian consisting of periodically driven SU(1,1) generators. A non-Hermitian invariant operator is adopted to solve the Schrödinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a PT-symmetric but non-unitary transformation operator in the construction of the non-Hermitian invariant. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adiabatic one in the slow time-variation limit. A non-unitary time- evolution operator is found analytically. As a consequence of the non-unitarity the ket (|ψ(t)〉) and bra (〈ψ(t)|) states are not normalized each other. While the inner product of two states can be evaluated with the help of a metric operator. It is shown explicitly that the model can be realized by a periodically driven oscillator.
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 .
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, .
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.
We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.