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Journal of the Optical Society of America B

Pseudospin approach to the dynamics and squeezing of SU(2) and SU(1, 1) coherent states

P. K. Aravind

Published Jul 1, 1988• 37 citations
Publisher

Abstract

The semi-classical 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) analog to give a unified description of the dynamics of the coherent states in these models. The pseudospin vector obeys a precession (or pseudoprecession) equation; hence simple geometrical arguments can be used to relate its evolution to that of the coherent-state parameters. To study fluctuations and squeezing we introduce the pseudospin tensor, which is a higher-order generalization of the pseudospin vector. The pseudospin tensor obeys a generalized precession equation that can be factored into scalar, vector, and quadrupole parts and solved exactly by application of the Cayley–Hamilton theorem. Squeezing is related directly to the behavior of the. pseudospin tensor and can therefore be calculated without invoking the group-theoretical disentangling formulas. Our approach is thus elementary in character. It also provides an attractive and closely parallel geometrical account of coherent-state evolution and squeezing in the two models.

DOI
10.1364/josab.5.001545

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The semi-classical 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) analog to give a unified description of the dynamics of the coherent states in these models. The pseudospin vector obeys a precession (or pseudoprecession) equation; hence simple geometrical arguments can be used to relate its evolution to that of the coherent-state parameters. To study fluctuations and squeezing we introduce the pseudospin tensor, which is a higher-order generalization of the pseudospin vector. The pseudospin tensor obeys a generalized precession equation that can be factored into scalar, vector, and quadrupole parts and solved exactly by application of the Cayley–Hamilton theorem. Squeezing is related directly to the behavior of the. pseudospin tensor and can therefore be calculated without invoking the group-theoretical disentangling formulas. Our approach is thus elementary in character. It also provides an attractive and closely parallel geometrical account of coherent-state evolution and squeezing in the two models.
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.
The atomic pseudospin operators describing a two-level atom, and obeying the SU(2) Lie algebra, can exhibit reduced quantum fluctuations (“squeezing”) in certain states of the atom. These states include both pure and mixed states of the atom. Parameterizing the atom by the spherical polar components S, θ, and ϕ of its pseudospin vector S, we determine the parameter ranges in which squeezing can occur and the magnitude of this squeezing. We illustrate our considerations by examining two models, namely, the coherent-state Jaynes–Cummings model and the optical Bloch equations, in which mixed-state SU(2) squeezing arises dynamically.
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.
In this thesis, we explore the Dicke model, well-known for its explanation of the superradiant phase transition and signatures of Classical and Quantum Chaos. We then propose a variational approach to determine the semi-classical (SC) equations of motion, using coherent states. The SC dynamics exhibits signs of classical chaos both with increasing coupling and as the energy of the system is gradually increased, eventually leading to it completely engulfing the 2 dimensional phase space of the pseudospin, which we examine through Poincare sections. We then use numerical diagonalization to study the Quantum Dicke model - it is observed that at and after exactly the critical value of the coupling, the system shows a net macroscopic photon expectation value, and the pseudospin component in the z direction starts to steadily decline - implying a net fraction of the atoms are shifting to the excited state. We then study the dynamics of the Quantum Dicke Hamiltonian using coherent states and compare the expectation values of the dimensional with the classical trajectories from the semi-classical dynamics for stable and chaotic orbits. We then introduce a linear quench which drives the Dicke model across the Superradiant phase transition by changing the coupling strength. The quenched dynamics is simulated using both semi-classical and quantum dynamics - it is seen that dynamically varying quantities are periodically oscillating around their instantaneous steady-state coupling values.
We analyze the dynamics of spin-mixing interactions generated by coupling spin-1 atoms to the mode of a high-finesse optical cavity. We show that the dynamics can be understood in terms of generators of the noncompact Lie group SU(1, 1) and introduce a set of SU(1, 1) coherent states which are preserved under Hamiltonian evolution. In terms of these coherent states the resulting dynamics may be interpreted as classical motion on the unit disk. We explicitly compute the trajectories of this classical motion and show that the motion is equivalent to spin-nematic squeezing in the atomic ensemble. Non-uniform coupling between the atomic ensemble and the cavity mode leads to departures from this simple behavior; we introduce a toy model that captures this non-uniformity and solve it exactly.