Chaos and quantum-nondemolition measurements

The problem of chaotic behavior in quantum mechanics is investigated against the background of the theory of quantum-nondemolition (QND) measurements. The analysis is based on two relevant features: The outcomes of a sequence of QND measurements are unambiguously predictable, and these measurements actually can be performed on one single system without perturbing its time evolution. Consequently, QND measurements represent an appropriate framework to analyze the conditions for the occurrence of ‘‘deterministic randomness’’ in quantum systems. The general arguments are illustrated by a discussion of a quantum system with a time evolution that possesses nonvanishing algorithmic complexity

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

Structure of nonlinear gauge transformations

Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations (NGT) defined in terms of a wave function $\psi(x)$ do not form a group. To get a group property one has to consider transformations that act differently on different branches of the complex argument function and the knowledge of the value of $\psi(x)$ is not sufficient for a well defined NGT. NGT that are well defined in terms of $\psi(x)$ form a semigroup parametrized by a real number $\gamma$ and a nonzero $\lambda$ which is either an integer or $-1\leq \lambda\leq 1$. An extension of NGT to projectors and general density matrices leads to NGT with complex $\gamma$. Both linearity of evolution and Hermiticity of density matrices are gauge dependent properties.Comment: Final version, to be published in Phys.Rev.A (Rapid Communication), April 199

Chaotic Evolution in Quantum Mechanics

A quantum system is described, whose wave function has a complexity which increases exponentially with time. Namely, for any fixed orthonormal basis, the number of components required for an accurate representation of the wave function increases exponentially.Comment: 8 pages (LaTeX 16 kB, followed by PostScript 2 kB for figure

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

Topological quenching of the tunnel splitting for a particle in a double-well potential on a planar loop

The motion of a particle along a one-dimensional closed curve in a plane is considered. The only restriction on the shape of the loop is that it must be invariant under a twofold rotation about an axis perpendicular to the plane of motion. Along the curve a symmetric double-well potential is present leading to a twofold degeneracy of the classical ground state. In quantum mechanics, this degeneracy is lifted: the energies of the ground state and the first excited state are separated from each other by a slight difference ¿E, the tunnel splitting. Although a magnetic field perpendicular to the plane of the loop does not influence the classical motion of the charged particle, the quantum-mechanical separation of levels turns out to be a function of its strength B. The dependence of ¿E on the field B is oscillatory: for specific discrete values Bn the splitting drops to zero, indicating a twofold degeneracy of the ground state. This result is obtained within the path-integral formulation of quantum mechanics; in particular, the semiclassical instanton method is used. The origin of the quenched splitting is intuitively obvious: it is due to the fact that the configuration space of the system is not simply connected, thus allowing for destructive interference of quantum-mechanical amplitudes. From an abstract point of view this phenomenon can be traced back to the existence of a topological term in the Lagrangian and a nonsimply connected configuration space. In principle, it should be possible to observe the splitting in appropriately fabricated mesoscopic rings consisting of normally conducting metal

Constructing Mutually Unbiased Bases in Dimension Six

The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex 6x6 Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.Comment: As published version. Added discussion of the impact of numerical approximations and corrected the number of triples existing for non-affine families (cf Table 3

Separation of variables in the quantum integrable models related to the Yangian Y[sl(3)]

There being no precise definition of the quantum integrability, the separability of variables can serve as its practical substitute. For any quantum integrable model generated by the Yangian Y[sl(3)] the canonical coordinates and the conjugated operators are constructed which satisfy the ``quantum characteristic equation'' (quantum counterpart of the spectral algebraic curve for the L operator). The coordinates constructed provide a local separation of variables. The conditions are enlisted which are necessary for the global separation of variables to take place.Comment: 15 page

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

Integrability and action operators in quantum Hamiltonian systems

For a (classically) integrable quantum mechanical system with two degrees of freedom, the functional dependence $\hat{H}=H_Q(\hat{J}_1,\hat{J}_2)$ of the Hamiltonian operator on the action operators is analyzed and compared with the corresponding functional relationship $H(p_1,q_1;p_2,q_2) = H_C(J_1,J_2)$ in the classical limit of that system. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence is, in general, non-uniform. The existence of the function $\hat{H}=H_Q(\hat{J}_1,\hat{J}_2)$ in the integrable regime of a parametric quantum system explains empirical results for the dimensionality of manifolds in parameter space on which at least two levels are degenerate. The comparative analysis is carried out for an integrable one-parameter two-spin model. Additional results presented for the (integrable) circular billiard model illuminate the same conclusions from a different angle.Comment: 9 page