From Quantum Mechanics to Classical Mechanics and Back, via Coherent States

Yiannis N. Kaznessis

Augusto César Lobo, Clyffe de Assis Ribeiro

The theory of Non-Relativistic Quantum Mechanics was created (or discovered) back in the 1920's mainly by Schrödinger and Heisenberg, but it is fair enough to say that a more modern and unified approach to the subject was introduced by Dirac and Jordan with their (intrinsic) Transformation Theory. In his famous text book on quantum mechanics [1], Dirac introduced his well-known bra and ket notation and a view that even Einstein (who was, as well known, very critical towards the general quantum physical world-view) considered the most elegant presentation of the theory at that time[2]. One characteristic of this formulation is that the observables of position and momentum are truly treated equally so that an intrinsic phase-space approach seems a natural course to be taken. In fact, we may distinguish at least two different quantum mechanical approaches to the structure of the quantum phase space: The Weyl-Wigner (WW) formalism and the advent of the theory of Coherent States (CS). The Weyl-Wigner formalism has had many applications ranging from the discussion of the Classical/Quantum Mechanical transition and quantum chaos to signal analysis[3,4]. The Coherent State formalism had a profound impact on Quantum Optics and during the course of time has found applications in diverse areas such as geometric quantization, wavelet and harmonic analysis [5]. In this chapter we present a compact review of these formalisms (with also a more intrinsic and coordinate independent notation) towards some non-standard and up-to-date applications such as modular variables and weak values.

MASSIMO RUDAN

Moein Naseri, Sadegh Raeisi

One of the most elusive problems in quantum mechanics is the transition between classical and quantum physics. This problem can be traced back to Schr\"odinger's cat thought experiment. A key element that lies at the center of this problem is the lack of a clear understanding and characterization of macroscopic quantum states. Our understanding of macroscopic quantumness relies on states such as the Greenberger-Horne-Zeilinger (GHZ) or the NOON state. Here we take a first-principle approach to this problem. We start from coherence as the key quantity that captures the notion of quantumness and require the quantumness to be collective and macroscopic. To this end, we introduce macroscopic coherence which is the coherence between macroscopically distinct quantum states. We construct a measure that quantifies how global and collective the coherence of the state is. Our work also provides a first-principle way to derive well-established states like the GHZ and the NOON state as the states that maximize our measure. Our approach paves the way towards a better understanding of the quantum-to-classical transition.

MASSIMO RUDAN