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Berry phase in Heisenberg representation

V. A. Andreev, A. B. Klimov, P. B. Lerner

Published May 1, 1994• 0 citations
OpenAlex

Abstract

We define the Berry phase for the Heisenberg operators. This definition is motivated by the calculation of the phase shifts by different techniques. These techniques are: the solution of the Heisenberg equations of motion, the solution of the Schrodinger equation in coherent-state representation, and the direct computation of the evolution operator. Our definition of the Berry phase in the Heisenberg representation is consistent with the underlying supersymmetry of the model in the following sense. The structural blocks of the Hamiltonians of supersymmetrical quantum mechanics ('superpairs') are connected by transformations which conserve the similarity in structure of the energy levels of superpairs. These transformations include transformation of phase of the creation-annihilation operators, which are generated by adiabatic cyclic evolution of the parameters of the system.

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The evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval (0,T) is considered. It is shown that all the cyclic states of the system, mod Psi (T))=e i phi mod Psi (0)), are determined by the eigenvalues in (quasi-energies) and associated eigenvectors mod phi in (t)) of the Schrodinger operator S q (t)=H(t)-i( delta / delta t) acting on some Hilbert space. The set of linearly independent cyclic (quasi-energy) states possess some properties similar to the properties of the stationary states of a closed system. The Berry phases of the states associated with eigenvectors of the discrete spectrum of S q (t), which are single-valued functions of omega =2 pi /T, are supplied by the partial derivatives of the corresponding eigenvalues (quasi-energies) with respect to omega . The approach developed is illustrated by several applications to time-dependent systems: the system under an adiabatic perturbation, the forced harmonic oscillator, and the two-level system. Even in the case of a system with a time-independent Hamiltonian there exist nonstationary quasi-energy states that have nontrivial Berry phases ( beta /2 pi not an integer). The criterion of existence of such states is formulated in terms of the energies of the system, and the corresponding expression for the Berry phases is obtained. Some examples of nonstationary cyclic states of closed systems, including the coherent states of a force-free oscillator and the Wannier states of electrons in the parabolic band, are considered.

Journal of Physics A Mathematical and General•Feb 21, 1993•17 citations
We define the Berry phase for the Heisenberg operators. This definition is motivated by the calculation of the phase shifts by different techniques. These techniques are: the solution of the Heisenberg equations of motion, the solution of the Schrodinger equation in coherent-state representation, and the direct computation of the evolution operator. Our definition of the Berry phase in the Heisenberg representation is consistent with the underlying supersymmetry of the model in the following sense. The structural blocks of the Hamiltonians of supersymmetrical quantum mechanics ('superpairs') are connected by transformations which conserve the similarity in structure of the energy levels of superpairs. These transformations include transformation of phase of the creation-annihilation operators, which are generated by adiabatic cyclic evolution of the parameters of the system.
Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.
The sequences of supersymmetry for a cyclic adiabatic evolution governed by the supersymmetric quantum mechanical Hamiltonian are revised. The condition (so-called supersymmetric adiabatic evolution condition) under which the supersymmetric reductions of Berry (nondegenerated case) or Wilczek–Zee (degenerated case) phases of superpartners are taking place is pointed out. The analog of the Witten index (supersymmetric Berry index) is determined. The final expression for the new index has compact form of indBH=sDet U≡Det Uτ, where U is the cyclic evolution operator generated by supersymmetric Hamiltonian H and τ is a supersymmetric involution. As the examples of the suggested concept of the supersymmetric adiabatic evolution the holomorphic quantum mechanics on a complex plane and meromorphic quantum mechanics on Riemann surfaces are considered. The supersymmetric Berry indices for the models are calculated.
For a time-dependent $\ensuremath{\tau}$-periodic harmonic oscillator of two linearly independent homogeneous solutions of the classical equation of motion that are both bounded over time (stable), it is shown that there is a representation of states that are cyclic up to multiplicative constants under $\ensuremath{\tau}$ evolution or $2\ensuremath{\tau}$ evolution depending on the model. The set of the wave functions is complete. Berry's phase, which could depend on the choice of representation, can be defined under the $\ensuremath{\tau}$ or $2\ensuremath{\tau}$ evolution in this representation. If a homogeneous solution diverges as time goes to infinity, it is shown that such a Berry phase cannot be defined in any representation considered. Berry's phase for the driven harmonic oscillator is also considered. For the cases where Berry's phase can be defined, the phase is given in terms of solutions of the classical equation of motion.
We study the dynamics of a quantum system with a time-dependent Hamiltonian which is given by a linear combination of SU(1,1) generators and coupled with a heat bath. An effective Hamiltonian of this dissipative system is derived. With the help of a time-dependent gauge transformation, we obtain the exact solutions of the time-dependent Schrödinger equation, from which the time-evolution operator and the non-adiabatic and adiabatic Berry phases, which depend on time, are calculated explicitly. In the weak dissipation limit, an additional term besides the original Berry phase is found. The additional phase does not have a geometrical meaning due to the dissipation.
For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.
The evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval (0,T) is considered. It is shown that all the cyclic states of the system, mod Psi (T))=e i phi mod Psi (0)), are determined by the eigenvalues in (quasi-energies) and associated eigenvectors mod phi in (t)) of the Schrodinger operator S q (t)=H(t)-i( delta / delta t) acting on some Hilbert space. The set of linearly independent cyclic (quasi-energy) states possess some properties similar to the properties of the stationary states of a closed system. The Berry phases of the states associated with eigenvectors of the discrete spectrum of S q (t), which are single-valued functions of omega =2 pi /T, are supplied by the partial derivatives of the corresponding eigenvalues (quasi-energies) with respect to omega . The approach developed is illustrated by several applications to time-dependent systems: the system under an adiabatic perturbation, the forced harmonic oscillator, and the two-level system. Even in the case of a system with a time-independent Hamiltonian there exist nonstationary quasi-energy states that have nontrivial Berry phases ( beta /2 pi not an integer). The criterion of existence of such states is formulated in terms of the energies of the system, and the corresponding expression for the Berry phases is obtained. Some examples of nonstationary cyclic states of closed systems, including the coherent states of a force-free oscillator and the Wannier states of electrons in the parabolic band, are considered.