We derive a phase-space description of squeezed fluctuations using the SU(1, 1) generalized coherent states. We introduce the coherent-diagonal representation of quantum states, based on squeezed coherent states. In this representation we derive generalized Fokker-Planck equations for a quasi-distribution in phase-space formed by SU(1, 1) states. We solve this equation for different representations of the SU(1, 1) group involving as examples parameteric amplifiers and two-mode upconvertors.
We derive a phase-space description of squeezed fluctuations using the SU(1, 1) generalized coherent states. We introduce the coherent-diagonal representation of quantum states, based on squeezed coherent states. In this representation we derive generalized Fokker-Planck equations for a quasi-distribution in phase-space formed by SU(1, 1) states. We solve this equation for different representations of the SU(1, 1) group involving as examples parameteric amplifiers and two-mode upconvertors.
The theory of generalized coherent states associated with the SU(2) and SU(1,1) Lie algebras is discussed and applied in an investigation of possible reductions of fluctuations. In the framework of a system of N two-level atoms the squeezing of angular-momentum [SU(2)] fluctuations is exhibited for optical coherent transients involving the photon echo. The SU(1,1) fluctuations are discussed and established for general two-photon processes involving dynamical variables different from the creation and annihilation operators.
The properties of correlated two-mode SU(1, 1) coherent states are studied. These states are a generalization of the two-mode squeezed vacuum states and are generated by applying the two-mode squeeze operator to a state in which there are initially q photons in one mode, |q, 0〉, the squeezed vacuum (q = 0) obviously being a special case. For these states the photon-number distribution and the nonclassical properties of photon antibunching, enhanced phase fluctuations, violations of the Cauchy–Schwarz inequality, two-mode squeezing, and sum squeezing are studied. A Q-function quasi-probability distribution for these states is also presented.
The semiclassical Dicke model and the ideal parametric amplifier are described by Hamiltonians linear in the SU(2) and SU( 1,1) group generators, respectively. We use the usual pseudospin vector as well as its SU(1,1) analog1 to give a unified description of the dynamics of the coherent states in these models. The pseudospin dynamics is governed by a precession (or pseudoprecession) equation, hence simple geometrical arguments can be used to infer the evolution of the coherent state parameters. To study fluctuations and squeezing we introduce a correlation tensor which is a higher-order generalization of the pseudospin vector. The correlation tensor obeys a generalized precession equation which can be factored into scalar, vector, and quadrupole parts and solved exactly by application of the Cayley-Hamilton theorem. Our approach to the calculation of squeezing and fluctuations provides an alternative to the more traditional approach employing disentangling theorems. It also provides an attractive and closely parallel geometrical description of coherent state evolution and squeezing in the two models.
We examine the non-classical properties of two-mode coherent states based on different unitary irreducible representations of SU(1, 1). Such states are generated by the action of the two-mode squeezing operator on initial states of the field containing arbitrary numbers of photons in each of the two modes. If the initial state of the field is a two-mode vacuum state, the final state is of course the two-mode squeezed vacuum. An initial occupation generalizes the idea of a squeezed vacuum to the SU(1, 1) coherent states. We show that fields in such states have remarkable quantum properties such as sub-Poissonian statistics, violations of the Cauchy-Schwarz inequality, strong correlations in the photon number fluctuations and squeezing. Using information theory formalism, we show that these coherent states are less correlated than the usual two-mode squeezed vacuum. Moreover, we show that an initial coherent amplitude contribution, in a large amplitude limit, can result in the reduction of correlations between modes.
A new operator representation, called squeezed coherent state representation, is introduced since Husimi operator has the form of squeezed coherent state. We fisrt introduce its specific integral expression. When κ = 1, this representation is reduced to the usual P function. As an example, we calculate the squeezed coherent state representation for thermal field to illustrate a difference between P function and the squeezed coherent state representation. Especially, in order to better apply this representation to quantum optics, we reveal the integral transformations between the squeezed coherent state representation, respectively, and the following three functions: Wigner function, Q function, and Husimi function.
The noise (variance squared) of a component of the electromagnetic field - considered as a quantum oscillator - in the vacuum is equal to one half, in appropriate units (taking Planck's constant and the mass and frequency of the oscillator all equal to 1). A practical definition of a squeezed state is one for which the noise is less than the vacuum value - and the amount of squeezing is determined by the appropriate ratio. Thus the usual coherent (Glauber) states are not squeezed, as they produce the same variance as the vacuum. However, it is not difficult to define states analogous to coherent states which do have this noise-reducing effect. In fact, they are coherent states in the more general group sense but with respect to groups other than the Heisenberg-Weyl Group which defines the Glauber states. The original, conventional squeezed state in quantum optics is that associated with the group SU(1,1). Just as the annihilation operator a of a single photon mode (and its hermitian conjugate a, the creation operator) generates the Heisenberg Weyl algebra, so the pair-photon operator a(sup 2) and its conjugate generates the algebra of the group SU(1,1). Another viewpoint, more productive from the calculational stance, is to note that the automorphism group of the Heisenberg-Weyl algebra is SU(1,1). Needless to say, each of these viewpoints generalizes differently to the quantum group context. Both are discussed. The following topics are addressed: conventional coherent and squeezed states; eigenstate definitions; exponential definitions; algebra (group) definitions; automorphism group definition; example: signal-to-noise ratio; q-coherent and q-squeezed states; M and P q-bosons; eigenstate definitions; exponential definitions; algebra (q-group) definitions; and automorphism q-group definition.