We describe a general method to solve a time-dependent Hamiltonian system exhibiting dynamical symmetry under the Lie algebra SU(1,1). The instantaneous eigenstates can be obtained by applying a displacement operator which is constructed explicitly. The construction proves to be equivalent to the solution of the classical equations of motion. One can also obtain the exact solution of the quantum Hamiltonian using this displacement operator. We define and discuss the differences between the cyclic evolution in the space of the physical states and in the parameter space. This leads to a method to define and find the T-cyclic and periodic states and their phases for a system with dynamical symmetry SU(1,1) just by using its time evolution operator. For SU(1,1) one can show that T-cyclic states only exist for some values of T which can be determined through an inequality which is found explicitly and discussed. The example used to check these results has been the degenerate optical parametric oscillator with dynamical symmetry SU(1,1). The system has no periodic states but it does possess T-cyclic states. The characteristic interaction times are distributed in forbidden and allowed regions as in the band theory of particles in a periodic potential. We also define exact phases and adiabatic phases along the lines of the above discussion on T-cyclic and periodic states. For sufficiently low frequencies the initial eigenstates of the system acquire non-trivial Berry phases in each cycle of the evolution of the functional parameters on which H depends. Such a phase can be measured experimentally and from the discussion of its possible values under various circumstances emerges an interesting relationship with the golden mean.
We describe a general method to solve a time-dependent Hamiltonian system exhibiting dynamical symmetry under the Lie algebra SU(1,1). The instantaneous eigenstates can be obtained by applying a displacement operator which is constructed explicitly. The construction proves to be equivalent to the solution of the classical equations of motion. One can also obtain the exact solution of the quantum Hamiltonian using this displacement operator. We define and discuss the differences between the cyclic evolution in the space of the physical states and in the parameter space. This leads to a method to define and find the T-cyclic and periodic states and their phases for a system with dynamical symmetry SU(1,1) just by using its time evolution operator. For SU(1,1) one can show that T-cyclic states only exist for some values of T which can be determined through an inequality which is found explicitly and discussed. The example used to check these results has been the degenerate optical parametric oscillator with dynamical symmetry SU(1,1). The system has no periodic states but it does possess T-cyclic states. The characteristic interaction times are distributed in forbidden and allowed regions as in the band theory of particles in a periodic potential. We also define exact phases and adiabatic phases along the lines of the above discussion on T-cyclic and periodic states. For sufficiently low frequencies the initial eigenstates of the system acquire non-trivial Berry phases in each cycle of the evolution of the functional parameters on which H depends. Such a phase can be measured experimentally and from the discussion of its possible values under various circumstances emerges an interesting relationship with the golden mean.
We build the exact solution and the instantaneous diagonalization of a quantum system exhibiting a SU (2) or dynamical symmetry. A generalized displacement operator determines in both cases the temporal evolution and the diagonalization operators of the system. These operators can be used to characterize the exact and the adiabatical evolution of both periodical and cyclic states whose exact and adiabatical phases can be explicitly found. We consider two examples: a charged particle in the presence of a rotating magnetic field and the degenerate optical parametric oscillator. In particular we present and calculate nontrivial geometrical and dynamical phases for the optical parametric oscillator depending upon the coupling parameter that could indeed be measured using an adequate optical experiment.
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 .
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schrödinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions are cyclic. Criteria for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, cannot be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can change from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schr\\"odinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions, are cyclic. Criterions for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, can not be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can changes from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.
Using the time-dependent unitary transformation instead of the invariant operator, the solutions of SU(1, 1) and SU(2) time-dependent quantum systems are obtained. It is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the Hamiltonian and the other correspond to the Floquet operator which gives the cyclic states and their associated phases over one period of the evolution. The non-adiabatic Berry's (or Aharonov-Anandan) phases are determined totally by such a unitary transformation.
We introduce the notion of the geometrically equivalent quantum systems (GEQS) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQS. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is T-periodic, the corresponding class of GEQS includes a system with a T-periodic Hamiltonian. We apply our general results to study the classes of GEQS that include a system with a cranked Hamiltonian H(t) = e −iKt H0e iKt. We show that the cranking operator K also belongs to this class. Hence, in spite of the fact that it is time-independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator. PACS numbers: 03.65.Bz 1