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arXiv:1507.00372 · International Journal of Modern Physics A

Lie algebras and generalized thermal coherent states

Sérgio Floquet, Marco A. S. Trindade, J. David M. Vianna

Published Jan 25, 2017• 4 citations
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Abstract

In this paper, we developed an algebraic formulation for the generalized thermal coherent states with a thermofield dynamics approach for multi-modes, based on coset space of Lie groups. In particular, we applied our construction on SU(2) and SU(1,[Formula: see text]1) symmetries and we obtain their thermal coherent states and density operator. We also calculate their thermal quantum Fidelity and thermal Wigner function.

DOI
10.1142/s0217751x17500154

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In this paper, we developed an algebraic formulation for the generalized thermal coherent states with a thermofield dynamics approach for multi-modes, based on coset space of Lie groups. In particular, we applied our construction on SU(2) and SU(1,[Formula: see text]1) symmetries and we obtain their thermal coherent states and density operator. We also calculate their thermal quantum Fidelity and thermal Wigner function.
A generalized Wigner function W T(α, ) including the thermal degrees of freedom is presented to calculate the ‘reduced’ Wigner functions for all kinds of coherent and squeezed states with thermal noise by the formalism of thermofield dynamics. The results show that the different definitions of coherent and squeezed state with the same corresponding density matrix give rise to the same reduced Wigner function. By our method, the reduced Wigner function can be calculated for an arbitrary vector in the thermal Fock space, of which the density matrix is too complicated to use.
This paper constructs a new type of finite-dimensional thermal coherent states (FDTCS), which differs from the proceeding thermal coherent state in construction, as realisations of SU(2) Lie algebra. Using the technique of integration within an ordered product of operator, it investigates the orthonormality and completeness relation of the FDTCS. Based on the thermal Wigner operator in the thermal entangled state representation, the Wigner function of the FDTCS is obtained. The nonclassical properties of the FDTCS are discussed in terms of the negativity of its Wigner function.
Within the framework of thermofield dynamics, we construct a thermalized coherent thermal state, which is a general type of the coherent state with the thermal effects and can be presumably produced experimentally. The wavefunction and the density matrix element in the coordinate repersentation are calculated, and furthermore we give the probability densities, average values and variances of the position, momentum and particle number, which in special cases are consistent with those in the literature. All calculations are performed in the coordinate representation.
A quick review of some Lie algebras related to well-known groups is given. We start with the Heisenberg-Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1,1) algebras and their coherent states. From there we go on to describe the binomial states and their extensions as the finite dimensional pair coherent states and their nonlinear versions as realizations of the SU(2) group. This is followed by considering the negative binomial states, the single mode and two-mode squeezed states and their variants as realizations of the SU(1,1) group. Generation schemes based on physical systems are considered for some of these states.
Usual quantum statistics is written in Fock space but it is not an algebraic theory. We show that at a deeper level it can be algebraically formalized defining the different statistics as (multi-mode) coherent states of the appropriate (but different from the usual ones) Lie-Hopf groups. The traditional connection between groups and statistics, established in vacuum, is indeed subverted by the interaction with the thermal bath. We show indeed that h(1), related in quantum field theory to bosons, must be used to define in presence of a bath the Boltzmann statistics while, to build the Bose statistics, we have to take into account su(1,1). Astonishing to describe fermions we are forced to use not the superalgebra h(1|1) but su(2) in the fundamental representation. Higher representations of su(2) allow also to give a possible definition of anyon statistics with generalized Pauli principle. Physical implications are discussed; the results is more general then the usual on the discrete spectrum, but everything collapses to standard theory when the continuum limit is performed.