1,427 research outputs found
Upper quantum Lyapunov Exponent and Anosov relations for quantum systems driven by a classical flow
We generalize the definition of quantum Anosov properties and the related
Lyapunov exponents to the case of quantum systems driven by a classical flow,
i.e. skew-product systems. We show that the skew Anosov properties can be
interpreted as regular Anosov properties in an enlarged Hilbert space, in the
framework of a generalized Floquet theory. This extension allows us to describe
the hyperbolicity properties of almost-periodic quantum parametric oscillators
and we show that their upper Lyapunov exponents are positive and equal to the
Lyapunov exponent of the corresponding classical parametric oscillators. As
second example, we show that the configurational quantum cat system satisfies
quantum Anosov properties.Comment: 17 pages, no figur
An Algorithmic Test for Diagonalizability of Finite-Dimensional PT-Invariant Systems
A non-Hermitean operator does not necessarily have a complete set of
eigenstates, contrary to a Hermitean one. An algorithm is presented which
allows one to decide whether the eigenstates of a given PT-invariant operator
on a finite-dimensional space are complete or not. In other words, the
algorithm checks whether a given PT-symmetric matrix is diagonalizable. The
procedure neither requires to calculate any single eigenvalue nor any numerical
approximation.Comment: 13 pages, 1 figur
On the computation of quantum characteristic exponents
A quantum characteristic exponent may be defined, with the same operational
meaning as the classical Lyapunov exponent when the latter is expressed as a
functional of densities. Existence conditions and supporting measure properties
are discussed as well as the problems encountered in the numerical computation
of the quantum exponents. Although an example of true quantum chaos may be
exhibited, the taming effect of quantum mechanics on chaos is quite apparent in
the computation of the quantum exponents. However, even when the exponents
vanish, the functionals used for their definition may still provide a
characterization of distinct complexity classes for quantum behavior.Comment: 11 pages Latex, 4 ps-figures. Phys. Lett. A, to appea
Coherent states and the reconstruction of pure spin states
Coherent states provide an appealing method to reconstruct efficiently the pure state of a quantum mechanical spin s. A Stern-Gerlach apparatus is used to measure (4s + 1) expectations of projection operators on appropriate coherent states in the unknown state. These measurements are compatible with a finite number of states which can be distinguished, in the generic case, by measuring one more probability. In addition, the present technique shows that the zeros of a Husimi distribution do have an operational meaning: they can be identified directly by measurements with a Stem-Gerlach apparatus. This result comes down to saying that it is possible to resolve experimentally structures in quantum phase space which are smaller than (h) over bar
A quantum search for zeros of polynomials
A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical measurements of the observable output directly the numerical values of the zeros. Performing the measurements is the only computational resource involved
Reconstruction of the spin state
System of 1/2 spin particles is observed repeatedly using Stern-Gerlach
apparatuses with rotated orientations. Synthesis of such non-commuting
observables is analyzed using maximum likelihood estimation as an example of
quantum state reconstruction. Repeated incompatible observations represent a
new generalized measurement. This idealized scheme will serve for analysis of
future experiments in neutron and quantum optics.Comment: 4 pages, 1 figur
Quantum diagonalization of Hermitean matrices
To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource: Hermitean (N Ă—N) matrices can be diagonalized, in principle, by performing appropriate quantum mechanical measurements. To do so, one considers the given matrix as an observable of a single spin with appropriate length s which can be measured using a generalized Stern-Gerlach apparatus. Then, each run provides one eigenvalue of the observable. As the underlying working principle is the `collapse of the wavefunction' associated with a measurement, the procedure is neither a digital nor an analogue calculation - it defines thus a new example of a quantum mechanical method of computation
Adiabatic motion of a neutral spinning particle in an inhomogeneous magnetic field
The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck’s constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution
Small denominators, frequency operators, and Lie transforms for nearly integrable quantum spin systems
Based on the previously proposed notions of action operators and of quantum integrability, frequency operators are introduced in a fully quantum-mechanical setting. They are conceptually useful because another formulation can be given to unitary perturbation theory. When worked out for quantum spin systems, this variant is found to be formally equivalent to canonical perturbation theory applied to nearly integrable systems consisting of classical spins. In particular, it becomes possible to locate the quantum-mechanical operator-valued equivalent of the frequency denominators that may cause divergence of the classical perturbation series. The results that are established here link the concept of quantum-mechanical integrability to a technical question, namely, the behavior of specific perturbation series
Signatures of quantum integrability and nonintegrability in the spectral properties of finite Hamiltonian matrices
For a two-spin model which is (classically) integrable on a five-dimensional
hypersurface in six-dimensional parameter space and for which level
degeneracies occur exclusively (with one known exception) on four-dimensional
manifolds embedded in the integrability hypersurface, we investigate the
relations between symmetry, integrability, and the assignment of quantum
numbers to eigenstates. We calculate quantum invariants in the form of
expectation values for selected operators and monitor their dependence on the
Hamiltonian parameters along loops within, without, and across the
integrability hypersurface in parameter space. We find clear-cut signatures of
integrability and nonintegrability in the observed traces of quantum invariants
evaluated in finite-dimensional invariant Hilbert subspaces, The results
support the notion that quantum integrability depends on the existence of
action operators as constituent elements of the Hamiltonian.Comment: 11 page
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