The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d

Péter R. Śurján

Arvind, Biswadeb Dutta, C. L. Mehta +1 more

We present an analysis, based on the metaplectic group Mp(2), of the recently introduced single-mode inverse creation and annihilation operators and of the associated eigenstates of different two-photon annihilation operators. We motivate and obtain a quantum operator form of the classical M\"obius or fractional linear transformation. The subtle relation to the two unitary irreducible representations of Mp(2) is brought out. For problems involving inverse operators the usefulness of the Bargmann analytic function representation of quantum mechanics is demonstrated. Squeezing, bunching, and photon-number distributions of the four families of states that arise in this context are studied both analytically and numerically.

Tommaso Guaita, Lucas Hackl, Thomas Quella

Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form $\langle 0|\exp{(-i\hat{H})}|0\rangle$, where $|0\rangle$ is a Gaussian state and $\hat{H}$ an arbitrary quadratic Hamiltonian. We provide closed formulas for $\langle 0|\exp{(-i\hat{H})}|0\rangle$ and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.

ter Hg Hennie Morsche, P.J. Oonincx

Malkhaz Mumladze

In this article we proposed derivation of the wave function of a quantum system from the functional representation of the state of this system.There is considered representation of creation and annihilation operators by boundary and co-boundary operators of chain and co-chain complexes on the physical space.