Quantum systems with finite Hilbert space where position x and momentum p take values in Z(d) (integers modulo d) are considered. Symplectic tranformations S(2¿,Z(p)) in ¿-partite finite quantum systems are studied and constructed explicitly. Examples of applying such simple method is given for the case of bi-partite and tri-partite systems. The quantum correlations between the sub-systems after applying these transformations are discussed and quantified using various methods. An extended phase-space x¿p¿X¿P where X, P ¿ Z(d) are position increment and momentum increment, is introduced. In this phase space the extended Wigner and Weyl functions are defined and their marginal properties are studied. The fourth order interference in the extended phase space is studied and verified using the extended Wigner function. It is seen that for both pure and mixed states the fourth order interference can be obtained. / Ministry of Higher Education, Sultanate of Oman

No / Vortices circulating in a ring made from a Josephson array in the insulating phase are studied. The ring contains a `dual Josephson junction' through which the vortices tunnel. External non-classical microwaves are coupled to the device. The time evolution of this two-mode fully quantum mechanical system is studied, taking into account the dissipation in the system. The effect of the quantum statistics of the photons on the quantum statistics of the vortices is discussed. Entropic calculations quantify the entanglement between the two systems. Quantum phenomena in the system are also studied through Wigner functions. After a certain time (which depends on the dissipation parameters) these quantum phenomena are destroyed due to dissipation.

A covariant Wigner-Weyl quantization formalism on the manifold that uses pseudo-differential operators is proposed. The asymptotic product formula that leads to the symbol calculus in the presence of gauge and gravitational fields is presented. The new definition is used to get covariant differential operators from momentum polynomial symbols. A covariant Wigner function is defined and shown to give gauge-invariant results for the Landau problem. An example of the covariant Wigner function on the 2-sphere is also included.

Made available in DSpace on 2014-07-29T15:07:07Z (GMT). No. of bitstreams: 1 simone ferreira.pdf: 699351 bytes, checksum: 41d071a2fd7b0fc8c68e3c0a7dcc415e (MD5) Previous issue date: 2006-03-24 / We present a feasible proposal for the construction of the conditional displacement operator using a Kerr medium between two beam splitters fed from coherent states highly excited. The device allows the implementation of the generation of a new class of states the quantized electromagnetic field (arbitrary superpositions of states with states moved) and the measurement of the Wigner function for arbitrary states. The application special case of the number of states and study their nonclassical properties were also considered. / Apresentamos uma proposta factível para a construção do operador deslocamento condicional usando um meio Kerr entre dois divisores de feixes alimentados por estados coerentes altamente excitados. O dispositivo permite implementar a geração de uma nova classe de estados do campo eletromagnético quantizado (superposições de estados arbitrários com estados deslocados) bem como a medição da função de Wigner para estados arbitrários. A aplicação especial ao caso de estados de número e o estudo de suas propriedades não-clássicas, foram também consideradas.

The displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed.

The displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed.