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arXiv:quant-ph/9907023 · Physical Review A

Exact solutions ofn-level systems and gauge theories

Merced Montesinos, Abdel Pérez-Lorenzana

Published Sep 1, 1999• 5 citations
arXivHTMLPDF

Abstract

We find a relationship between unitary transformations of the dynamics of quantum systems with time-dependent Hamiltonians and gauge theories. In particular, we show that the nonrelativistic dynamics of spin-$\frac{1}{2}$ particles in a magnetic field ${B}^{i}(t)$ can be formulated in a natural way as an SU(2) gauge theory, with the magnetic field ${B}^{i}(t)$ playing the role of the gauge potential ${A}^{i}$. The present approach can also be applied to systems of n levels with time-dependent potentials, $U(n)$ being the gauge group. This geometric interpretation provides a powerful method to find exact solutions of the Schr\"odinger equation. The root of the present approach rests in the Hermiticity property of the Hamiltonian operators involved. In addition, the relationship with true gauge symmetries of n-level quantum systems is discussed.

DOI
10.1103/physreva.60.2554

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We find a relationship between unitary transformations of the dynamics of quantum systems with time-dependent Hamiltonians and gauge theories. In particular, we show that the nonrelativistic dynamics of spin-12\frac{1}{2}21​ particles in a magnetic field Bi(t){B}^{i}(t)Bi(t) can be formulated in a natural way as an SU(2) gauge theory, with the magnetic field Bi(t){B}^{i}(t)Bi(t) playing the role of the gauge potential Ai{A}^{i}Ai. The present approach can also be applied to systems of n levels with time-dependent potentials, U(n)U(n)U(n) being the gauge group. This geometric interpretation provides a powerful method to find exact solutions of the Schr"odinger equation. The root of the present approach rests in the Hermiticity property of the Hamiltonian operators involved. In addition, the relationship with true gauge symmetries of n-level quantum systems is discussed.
Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.
Given a time-independent Hamiltonian H~\widetilde HH , one can construct a time-dependent Hamiltonian HtH_tHt​ by means of the gauge transformation Ht=UtH~Ut†−iUt∂tUt†H_t=U_t\kern1pt \widetilde H \kern1pt U^\dagger_t-i\kern1pt U_t\kern1pt\partial_t U_t^\daggerHt​=Ut​HUt†​−iUt​∂t​Ut†​ . Here UtU_tUt​ is the unitary transformation that relates the solutions of the corresponding Schrödinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as physical. We formulate sufficient conditions on UtU_tUt​ ensuring that HtH_tHt​ is physical as long as H~\widetilde HH is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians HtH_tHt​ and H~\widetilde HH such that the driven many-body dynamics governed by HtH_tHt​ can be reduced to the quench dynamics due to the time-independent H~\widetilde HH . We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to a time-independent system without a magnetic field [F. Yan, L. Yang, and B. Li, Phys. Lett. A 251, 289–293; 259, 207–211 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where H~=H~t\widetilde H = \widetilde H_tH=Ht​ is time-dependent but dynamically integrable.
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 .
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.
This study shows that the usual time evolution operator can act as a U(1) gauge transformation on the initial wavefuntion. When considering the spin freedom in the system, the time evolution operator corresponds to a non-Abelian SU(2) gauge transformation for spin-1/2 particles and a SU(3) gauge transformation for spin-1 particles. If we adopt the adiabatic approximation in the magnetization dynamics of a ferromagnetic system driven by a spin-polarized current, a temperature-dependent non-Abelian thermal gauge potential will appear. We can employ a temperature-dependent Landau-Lifshitz-Gilbert (LLG) equation to describe the magnetization dynamics in the semi-classical limit, a linear approximated solution for this equation is given analytically.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.