International Journal of Modern Physics B

We study the correspondence between classical and quantum measurements on a harmonic oscillator that describes a one-mode bosonic field with annihilation and creation operators a and a^† with commutation [a, a^†]=1. We connect the quantum measurement of an observable Ô=Ô (a, a^†) of the field with the possibility of amplifying the observable Ô ideally through a quantum amplifier which achieves the Heisenberg-picture evolution Ô→ gÔ, where g is the gain of the amplifier. The "classical" measurement of Ô corresponds to the joint measurement of the position and momentum of the harmonic oscillator, with following evaluation of a function of the outcome α=q+ip. For the electromagnetic field the joint measurement is achieved by a heterodyne detector. The quantum measurement of Ô is obtained by preamplifying the heterodyne detector through an ideal amplifier of Ô and rescaling the outcome by the gain g. We give a general criterion which states when this preamplified heterodyne detection scheme approaches the ideal quantum measurement of Ô in the limit of infinite gain. We show that this criterion is satisfied and the ideal measurement is achieved for the case of the photon number operator a^† a and for the quadrature , where one measures the functions and of the field, respectively. For the photon number operator a^† a the amplification scheme also achieves the transition from the continuous spectrum |α|²∈ℝ to the discrete one n∈ℕ of the operator a^† a. Moreover, for both operators a^† a and the method is robust to nonunit quantum efficiency of the heterodyne detector. On the other hand, we show that the preamplified heterodyne detection scheme does not work for arbitrary observable of the field. As a counterexample, we prove that the simple quadratic function of the field has no corresponding polynomial function — including the obvious choice f=Im (α²) — that allows the measurement of through the preamplified heterodyne measurement scheme.

Dynamical symmetries of Hamiltonians quantized models of discrete nonlinear Schrödinger chain (DNLS) and of Ablowitz–Ladik chain (AL) are studied. It is shown that for n-sites the dynamical algebra of DNLS Hamilton operator is given by the su(n) algebra, while the respective symmetry for the AL case is the quantum algebra su_q(n). The q-deformation of the dynamical symmetry in the AL model is due to the non-canonical oscillator-like structure of the raising and lowering operators at each site.

Invariants of motions are found in terms of Casimir central elements of su(n) and su_q(n) algebra generators, for the DNLS and QAL cases respectively. Utilizing the representation theory of the symmetry algebras we specialize to the n=2 quantum dimer case and formulate the eigenvalue problem of each dimer as a nonlinear (q)-spin model. Analytic investigations of the ensuing three-term nonlinear recurrence relations are carried out and the respective orthonormal and complete eigenvector bases are determined.

The quantum manifestation of the classical self-trapping in the QDNLS-dimer and its absence in the QAL-dimer, is analysed by studying the asymptotic attraction and repulsion respectively, of the energy levels versus the strength of nonlinearity. Our treatment predicts for the QDNLS-dimer, a phase-transition like behaviour in the rate of change of the logarithm of eigenenergy differences, for values of the nonlinearity parameter near the classical bifurcation point.

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The critical behaviour of a dilute spin-S Bose gas in a magnetic field is investigated with the help of the renormalization group method. A complete set of critical exponents is given both at constant pressure and constant density. The magnetic properties are also discussed.

The magnetic susceptibilities of the degenerate and nondegenerate relativistic electron gases, and of the nondegenerate electron–positron gas are presented.

By investigating numerically a circle map with two cubic inflection points, we find that the fractal dimension D of the set of quasiperiodic windings at the onset of chaos has a variety of values, instead of a unique value like 0.87. This fact strongly suggests that a family of universality classes of D appears as the map has two various inflection points. On the other hand, at the quasiperiodic transition with the golden mean winding number, the ratios δ_n of the width of the mode lockings when going from one Fibonacci level to the next do not converge to a fixed value or a limit cycle in most cases. In this sense, local scaling is broken due to the interaction of the two inflection points of the map. Based on the above observations, it seems that the global scaling is more robust than the local one, at least for the maps we considered.

The relaxation dynamics in the case of a discretized atom-field interaction model, being of great importance in quantum optics, is investigated numerically by an algorithm based on Seke's self-consistent projection operator method. The transition from reversible to irreversible dynamics in the continuous limit is simulated numerically. The dependence of the time evolution of atomic nondecay probability on the number of discrete-field modes, coupled to the atom, is discussed.