Rigorous solutions, Berry phases and quantum-classical correspondence for time-dependent non-Hermitian SU(2)-SU(1, 1) Hamiltonians
Abstract
<p indent="0mm">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 <inline-formula id="INLINE77"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" mathsize="9.0pt"><mml:mrow><mml:msub other="0"><mml:mi mathsize="9.0pt" other="0">G</mml:mi><mml:mi mathsize="6.3pt" other="1">c</mml:mi></mml:msub><mml:mrow><mml:mo mathsize="9.0pt">(</mml:mo><mml:mi mathsize="9.0pt">ω</mml:mi><mml:mo mathsize="9.0pt">)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> 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 <inline-formula id="INLINE78"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" mathsize="9.0pt"><mml:mrow><mml:msub other="0"><mml:mi mathsize="9.0pt" other="0">G</mml:mi><mml:mi mathsize="6.3pt" other="1">c</mml:mi></mml:msub><mml:mrow><mml:mo mathsize="9.0pt">(</mml:mo><mml:mi mathsize="9.0pt">ω</mml:mi><mml:mo mathsize="9.0pt">)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>. 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, <inline-formula id="INLINE79"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" mathsize="9.0pt"><mml:mrow><mml:msub other="0"><mml:mi mathsize="9.0pt" other="0">γ</mml:mi><mml:mi mathsize="6.3pt" other="1">n</mml:mi></mml:msub><mml:mrow><mml:mo mathsize="9.0pt">(</mml:mo><mml:mi mathsize="9.0pt">T</mml:mi><mml:mo mathsize="9.0pt">)</mml:mo></mml:mrow><mml:mo mathsize="9.0pt">=</mml:mo><mml:mo mathsize="9.0pt">−</mml:mo><mml:mrow><mml:mo mathsize="9.0pt" minsize="2.5">(</mml:mo><mml:mrow><mml:mi mathsize="9.0pt">n</mml:mi><mml:mo mathsize="9.0pt">+</mml:mo><mml:mfrac><mml:mn mathsize="9.0pt">1</mml:mn><mml:mn mathsize="9.0pt">2</mml:mn></mml:mfrac></mml:mrow><mml:mo mathsize="9.0pt" minsize="2.5">)</mml:mo></mml:mrow><mml:mtext mathsize="9.0pt">∆Θ</mml:mtext></mml:mrow></mml:math></inline-formula>.</p>
