Complex dynamics and coherent states

Eva-Maria Graefe, Roman Schubert

The complex geometry underlying the Schrödinger dynamics of coherent states for non-Hermitian Hamiltonians is investigated. In particular two seemingly contradictory approaches are compared: (i) a complex WKB formalism, for which the centres of coherent states naturally evolve along complex trajectories, which leads to a class of complexified coherent states; (ii) the investigation of the dynamical equations for the real expectation values of position and momentum, for which an Ehrenfest theorem has been derived in a previous paper, yielding real but non-Hamiltonian classical dynamics on phase space for the real centres of coherent states. Both approaches become exact for quadratic Hamiltonians. The apparent contradiction is resolved building on an observation by Huber, Heller and Littlejohn, that complexified coherent states are equivalent if their centres lie on a specific complex Lagrangian manifold. A rich underlying complex symplectic geometry is unravelled. In particular a natural complex structure is identified that defines a projection from complex to real phase space, mapping complexified coherent states to their real equivalents.

Yousef Yousefi, Х. Х. Муминов

It was studied coherent states in complex variables in SU(2), SU(3), SU(4) groups and in general in SU(n) group. Using the completeness relation of the coherent state, we obtain a path integral expression for transition amplitude which connects a pair of SU(n) coherent states. In the classical limit, a canonical equation of motion is obtained.

Blagovest A. Nikolov, D. A. Trifonov

V. M. Akulin

Alexander V. Gorokhov

Applications of Kähler manifolds to the temporal dynamics of multilevel quantum systems in external fields are considered. It is shown by using the representation of coherent states (CS), the temporal evolution of the state vector is reduced to the “classical” dynamics of the complex parameters of the CS taking values in the coset space of the dynamical group of the Hamiltonian.