Searching for Coherent States: From Origins to Quantum Gravity

Abstract

We discuss the notion of coherent states from three different perspectives: the seminal approach of Schrödinger, the experimental take of quantum optics, and the theoretical developments in quantum gravity. This comparative study tries to emphasise the connections between the approaches, and to offer a coherent short story of the field, so to speak. It may be useful for pedagogical purposes, as well as for specialists of quantum optics and quantum gravity willing to embed their perspective within a wider landscape.

Taming Fluctuations for Gaussian States in Loop Quantum Cosmology

“…The canonical observables of LQC are V and 1 λ Ŝλ where Ŝλ := i 2 ( ĥλ − ĥ * λ ). We may also define Ĉλ := 1 2 ( ĥλ + ĥ * λ ).These operators are both self-adjoint and if there did exist a unique β such that ĥλ were its exponentiation, then Ŝλ and Ĉλ would look like its sine and cosine and in small λ, 1 λ Ŝλ would approximate β, whence it resembles the conjugate momentum of V .…”

“…It has been argued [1] that the constancy of fluctuations of Gaussian states is also a crucial ingredient their semi-classicality. Time independence ensures that their fluctuations do not spread out under dynamical evolution, and thus remain minimal.…”

“…We now compute the exact solutions for the fluctuations of V and 1 λ Ŝλ for Gaussian states in the volume representation. For a given l, the super-selection sector of the non-separable polymer Hilbert space is the subspace generated by the lattice of volume eigenstates of the form |nl for n ∈ Z. Fixing l = 1 as we have done makes this space unique.…”

“…These two properties -sharpness and minimal fluctuation -may be taken to be constitutive features of semi-classical states (see, for instance, [1][2][3]). In ordinary quantum mechanics, such states are easy to find: Gaussian states -and coherent states more generallysatisfy both of these conditions.…”

Quasienergy operators and generalized squeezed states for systems of trapped ions

“…Recent advances in quantum optics enable trapping of single particles or atoms [19,[37][38][39], while progress in quantum engineering techniques allows preparing these particles in well-defined quantum states [40,41], under conditions of accurate control of the interaction between a quantum system (trapped ions) and the environment [42][43][44][45][46][47][48]. Besides quantum optics CS are of large interest, starting from pure mathematical topics up to physical applications such as quantum gravity [49,50]. In addition, high precision measurements that include engineering of quantum correlated states such as quantum tests of physics beyond the Standard Model (SM) or relativistic geodesy [51,52], are driven by continuous progress recorded in the development of ultrastable optical atomic clocks [53,54].…”