A. Vourdas

Quantum systems with finite Hilbert space

Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU(2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.

Analytic representations in quantum mechanics

Various Euclidean, hyperbolic and elliptic analytic representations are introduced and relations among them are discussed. The Bargmann analytic representation in the complex plane is considered and its relation to other phase-space methods for the harmonic oscillator is reviewed. The general theory that relates the growth of analytic functions with the density of their zeros is applied to Bargmann functions and it leads to theorems on the completeness of sequences of Glauber coherent states. Two hyperbolic analytic representations in the unit disc, based on SU(1, 1) coherent states and also on phase states are introduced. A third analytic representation in the complex plane based on Barut–Girardello states is also considered and transformations which relate it to the other ones are studied. In the case of systems with finite-dimensional Hilbert space, an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced. The Berezin formalism in the Euclidean, hyperbolic and elliptic cases is discussed. Contour analytic representations in these three cases are also presented.

Extraction of information from acoustic vibration signals using Gabor transform type devices

A technique for interpreting acoustic signals is presented based upon the use of Gabor type devices. This interpretation technique processes the information acquired from vibration sensors and presents the information as three time-varying parameters which represent the dominant signal frequency, the nominal breadth of the frequency content of the signal and its effective energy content. The technique is an extension of the chromatic analysis used successfully in the optical domain. The information produced by this analysis proves to be more convenient for defining a signal than that yielded by the conventional Fourier transform approach. By way of an example, the Gabor transform based technique is applied to the monitoring of acoustic signals generated by a high-voltage circuit breaker (as used on power transmission networks). Results are presented which indicate that the technique is capable of quantifying trends in complicated time-varying acoustical signals.

The Gabor transform basis of chromatic monitoring

Chromaticity based monitoring methodologies have evolved from polychromatic optical fibre based sensing systems, which was first implemented during the 1980s. The evolution has led to a recognition that such a methodology is applicable to monitoring systems more widely. It is shown that the method may be regarded as the manifestation of a particular form of signal representation by Gabor transforms. Examples of generically different practical implementations of the chromatic methodology to a range of different monitoring situations are given.

Time-frequency analysis and filtering of kinematic signals with impacts using the Wigner function: accurate estimation of the second derivative.

Biomechanical signals are represented in the time-frequency domain using the Wigner distribution function. Filtering of this representation for the case of a non-stationary displacement signal with impact is studied. Smoothed displacement data are then double differentiated and compared with references accelerometer data. It is shown that this technique is able to remove noise from these signals in a better way than conventional filtering techniques currently used in biomechanics.

Analytic representation of finite quantum systems

A transform between functions in R and functions in Zd is used to define the analogue of number and coherent states in the context of finite d-dimensional quantum systems. The coherent states are used to define an analytic representation in terms of theta functions. All states are represented by entire functions with growth of order 2, which have exactly d zeros in each cell. The analytic function of a state is constructed from its zeros. Results about the completeness of finite sets of coherent states within a cell are derived.

Symplectic transformations and quantum tomography in finite quantum systems

There is difference between coefficients of thermal expansion of the first member and the second member. A mismatch occurs between the walls of the first member and the second member when the temperature is changed significantly. Stress generated by this mismatch is distributed among and received by respective fitting sections of respective concave portions and respective convex portions. Because these fitting sections exist above and below the joining member, stress generated by the mismatch is distributed and received by the fitting sections before the stress reaches the joining member. Therefore, only a little stress acts on the joining member.

The angle-angular momentum quantum phase space

The angle-angular momentum quantum phase space is considered and the corresponding Heisenberg - Weyl group is studied. When (2j+1) is a power of a prime, Z(2j+1) is a Galois field and stronger results can be proved. SL(2,Z(2j+1)) transformations are explicitly constructed and various properties of the displacement operators are studied. Central extensions of the Abelian group by Z(2j+1) are studied and they provide all the ways of constructing the Heisenberg - Weyl group from G and Z(2j+1).

Coherent mixed states and a generalised P representation

The pure Glauber (harmonic oscillator) coherent states provide a very useful basis for many purposes. They are complete in the sense that an arbitrary state in the Hilbert space may be expanded in terms of them. Furthermore, the well known P representation provides a diagonal expansion of an arbitrary operator in the Hilbert space in terms of the projection operators onto the coherent states. The authors study the extensions of these results to the analogous mixed states which describe comparable harmonic oscillator systems in thermodynamic equilibrium at non-zero temperatures. Their results are given for the general density operator which describes the mixed squeezed coherent states of the displaced and squeezed harmonic oscillator. They show how these squeezed coherent mixed states similarly provide a very convenient complete description of a Hilbert space. In particular they show how the usual P and Q representations of operators in terms of pure states may be extended to finite temperatures with the corresponding mixed states, and various relations between them are demonstrated. The question of the existence of the generalised P representation for an arbitrary operator is further examined and some pertinent theorems are proven. They also show how their results relate to the Glauber-Lachs formalism in quantum optics for mixtures of coherent and incoherent radiation. Particular attention is focused both on the interplay between the quantum mechanical and thermodynamical uncertainties and on the entropy associated with such mixed states.

Quantum Statistics and Entanglement of Two Electromagnetic Field Modes Coupled via a Mesoscopic SQUID Ring

In this paper we investigate the behaviour of a fully quantum mechanical system consisting of a mesoscopic SQUID ring coupled to one or two electromagnetic field modes. We show that we can use a static magnetic flux threading the SQUID ring to control the transfer of energy, the entanglement and the statistical properties of the fields coupled to the ring. We also demonstrate that at, and around, certain values of static flux the effective coupling between the components of the system is large. The position of these regions in static flux is dependent on the energy level structure of the ring and the relative field mode frequencies, In these regions we find that the entanglement of states in the coupled system, and the energy transfer between its components, is strong.