E. C. G. Sudarshan

Completely positive dynamical semigroups of N‐level systems

We establish the general form of the generator of a completely positive dynamical semigroup of an N‐level quantum system, and we apply the result to derive explicit inequalities among the physical parameters characterizing the Markovian evolution of a 2‐level system.

Spin‐s Spherical Harmonics and ð

Recent work on the Bondi‐Metzner‐Sachs group introduced a class of functions sYlm(θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R3 and the properties of ð are derived from its relationship to an angular‐momentum raising operator. The relationship of the sTlm(θ, φ) to the spherical harmonics of R4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.

f-oscillators and nonlinear coherent states

The notion of f-oscillators generalizing q-oscillators is introduced. For classical and quantum cases, an interpretation of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of oscillation depends on the energy. The f-coherent states (nonlinear coherent states) generalizing q-coherent states are constructed. Applied to quantum optics, photon distribution function, photon number means, and dispersions are calculated for the f-coherent states as well as the Wigner function and Q-function. As an example, it is shown how this nonlinearity may affect the Planck distribution formula.

Properties of Quantum Markovian Master Equations

In this paper we give an essentially self-contained account of some general structural properties of the dynamics of quantum open Markovian systems. We review some recent results regarding the problem of the classification of quantum Markovian master equations and the limiting conditions under which the dynamical evolution of a quantum open system obeys an exact semigroup law (weak coupling limit and singular coupling limit). We discuss a general form of quantum detailed balance and its relation to thermal relaxation and to microreversibility.

INCONSISTENCY OF THE LOCAL FIELD THEORY OF CHARGED SPIN 3/2 PARTICLES

Abstract The relativistic quantum theory of Fermi Dirac fields of arbitrary spin is investigated and a general theorem is proved which aserts that for fields of half integral spin > 1 2 , the possibility of a consistent quantization requires that the equal-time anticommutators must be functions of the other fields to which the field in question is coupled. The case of spin 3 2 is studied in detail and the equivalence of various formulations of the theory is shown. The inconsistency of the relativistic local quantum theory of a charged spin 3 2 field in interaction with an external electromagnetic field is demonstrated by showing that the equal time commutation relations and relativistic covariance of the theory are not compatible. Finally, the mixed spin 3 2 - spin 1 2 (Bhabha) field is found to be characterized by the same inconsistency.

Partially coherent beams and a generalized ABCD-law

The beam width and the angular spread of a partially coherent beam, and their transformation as the beam passes through Sp(2, R) first order optical systems are studied using the method of generalized rays. A generalized abcd-law which governs this transformation is derived. Kogelniks abcd-law for coherent gaussian beams and its later generalization to gaussian Schell-model beams are shown to be special cases of this law.

Theory of photoelectric detection of light fluctuations

The basic formulae governing the fluctuations of counts registered by photoelectric detectors in an optical field are derived. The treatment, which has its origin in Purcells explanation of the Hanbury Brown-Twiss effect, is shown to apply to any quasi-monochromatic light, whether stationary or not, and whether of thermal origin or not. The representation of the classical wave amplitude, of the light by Gabors complex analytic signal appears naturally in this treatment. It is shown that the correlation of counts registered by N separate photodetectors at N points in space is determined by a 2Nth order correlation function of the complex classical field. The variance of the individual counts is shown to be expressible as the sum of terms representing the effects of classical particles and classical waves, in analogy to a well-known result of Einstein relating to black-body radiation. Since the theory applies to correlation effects obtained with any type of light it applies, in particular, to the output of an optical maser, although, for a maser operating on one mode, correlation effects are likely to be very small.

Completely positive maps and classical correlations

We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord [1]. We prove that initial states that have only classical correlations lead to completely positive reduced dynamics. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present.

Representations of parafermi rings

Abstract We consider the operator algebra corresponding to a system consisting of an arbitrary finite number v of parafermi oscillators with a view to obtaining all irrepresentations of the operator algebra defining it. It is shown that this algebra is isomorphic to the Lie algebra of the orthogonal group B v in 2 v + 1 dimensions. The known results of the representation theory of the orthogonal group are then used to find the representations of the ring of parafermi operators. In particular it is seen that the so-called Green ansatz for a parafermi ring arises in a natural way and that, in a certain sense, it furnishes the most general “solution” of the parafermi algebra. The case of a parafermi ring of order 2, which is the simplest non-trivial case, is considered in some detail.

Dynamics of initially entangled open quantum systems

Linear maps of matrices describing the evolution of density matrices for a quantum system initially entangled with another are identified and found to be not always completely positive. They can even map a positive matrix to a matrix that is not positive, unless we restrict the domain on which the map acts. Nevertheless, their form is similar to that of completely positive maps. Only some minus signs are inserted in the operator-sum representation. Each map is the difference of two completely positive maps. The maps are first obtained as maps of mean values and then as maps of basis matrices. These forms also prove to be useful. An example for two entangled qubits is worked out in detail. The relation to earlier work is discussed.