November 2011
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Day November 9, 2011

Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians

Eva-Maria Graefe, Roman Schubert

The complex geometry underlying the Schr\”odinger 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.
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Analytic Results for a PT-symmetric Optical Structure

H. F. Jones

Propagation of light through media with a complex refractive index in which gain and loss are engineered to be PT symmetric has many remarkable features. In particular the usual unitarity relations are not satisfied, so that the reflection coefficients can be greater than one, and in general are not the same for left or right incidence. Within the class of optical potentials of the form \(v(x)=v_1\cos(2\beta x)+iv_2\sin(2\beta x)\) the case \(v_2=v_1\) is of particular interest, as it lies on the boundary of PT-symmetry breaking. It has been shown in a recent paper by Lin et al. that in this case one has the property of “unidirectional invisibility”, while for propagation in the other direction there is a greatly enhanced reflection coefficient proportional to \(L^2\), where \(L\) is the length of the medium in the direction of propagation.

For this potential we show how analytic expressions can be obtained for the various transmission and reflection coefficients, which are expressed in a very succinct form in terms of modified Bessel functions. While our numerical results agree very well with those of Lin et al. we find that the invisibility is not quite exact, in amplitude or phase. As a test of our formulas we show that they identically satisfy a modified version of unitarity appropriate for PT-symmetric potentials. We also examine how the enhanced transmission comes about for a wave-packet, as opposed to a plane wave, finding that the enhancement now arises through an increase, of \(O(L)\), in the pulse length, rather than the amplitude.
Optics (physics.optics); Quantum Physics (quant-ph)