Classical and Quantum Coherent States

Paavo Tikka

The topic of this Bachelor's thesis is the quantum-to-classical crossover using generalized coherent states (GCS). In other words, the thesis consists of a mathematically rigorous consideration of the transition between quantum theory and classical theory, and how this can be carried out using the theory of GCS. The necessary mathematical background for this is also reviewed. For the theory of generalized coherent states, Lie algebras and Lie groups are first reviewed. Lie groups are groups with a structure of a differentiable manifold. For the Lie groups we consider, there exist coresponding Lie algebras, whose elements generate the Lie groups using the exponential map. Generalized coherent states are a generalization of Glauber states, which are continuous quantum states used in describing the harmonic oscillator. The generalization is carried through using some Lie group other than the Heisenberg-Weyl group used in defining the Glauber states. A special property of GCS is their semiclassical structure: a structure resembling the classical phase space may be defined in the differentiable manifold used in generating the GCS. An example of GCS are the spin coherent states, generated using the Lie group SU(2). For the coherent states to be applicable in the quantum-to-classical crossover, one has to also find some symmetry so that the system under consideration may be described using a single GCS. In the reference literature it is assumed that a global symmetry enables the description of a system of N elements by a coherent state corresponding to a coherent state of a single element. In the spin chain example however, the symmetry can be explicitly applied on the system and it turns out that this assumption is valid: using suitable symmetries, a spin chain may be represented by a single spin coherent state. The quantum-to-classical crossover can be carried out by defining some classical operators whose expectation value with respect to the GCS corresponds to the functions of classical mechanics. In addition to this, the coherent states in question have to fulfill some requirements. In the case of spin coherent states these requirements are fulfilled and the quantum-to-classical crossover may be carried through.

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

Gérard G. Emch, Gerhard C. Hegerfeldt

A noncommutative version of the Cramer theorem is used to show that if two quantum systems are prepared independently, and if their center of mass is found to be in a coherent state, then each of the component systems is also in a coherent state, centered around the position in phase space predicted by the classical theory. Thermal coherent states are also shown to possess properties similar to classical ones.

Constantin Brif, A. Mann, M. Revzen

We review classical properties of harmonic-oscillator coherent states. Then we discuss which of these classical properties are preserved under the group-theoretic generalization of coherent states. We prove that the generalized coherent states of quantum systems with Lie-group symmetries are the unique Bell states, i.e., the pure quantum states preserving the fundamental classical property of satisfying Bell’s inequality upon splitting.

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau