Representation theory of Gaussian unitary transformations for bosonic and fermionic systems
- URL: http://arxiv.org/abs/2409.11628v1
- Date: Wed, 18 Sep 2024 01:22:38 GMT
- Title: Representation theory of Gaussian unitary transformations for bosonic and fermionic systems
- Authors: Tommaso Guaita, Lucas Hackl, Thomas Quella,
- Abstract summary: We analyze the behavior of the sign ambiguity that one needs to deal with when moving between the groups of the symplectic and special annihilation group.
We 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.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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.
Related papers
- Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum [0.0]
We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian.
This two-dimensional quantum model exhibits higher-order integrals of motion within the enveloping algebra of the Heisenberg algebra.
arXiv Detail & Related papers (2025-02-04T17:06:54Z) - Gaussian quantum Markov semigroups on finitely many modes admitting a normal invariant state [0.0]
Gaussian quantum Markov semigroups (GQMSs) are of fundamental importance in modelling the evolution of several quantum systems.
We completely characterize those GQMSs that admit a normal invariant state and we provide a description of the set of normal invariant states.
We study the behavior of such semigroups for long times: firstly, we clarify the relationship between the decoherence-free subalgebra and the spectrum of $mathbfZ$.
arXiv Detail & Related papers (2024-12-13T10:01:18Z) - Small Circle Expansion for Adjoint QCD$_2$ with Periodic Boundary Conditions [0.0]
Supersymmetry is found at the adjoint mass-squared $g2 hvee/ (2pi)$, where $hvee$ is the dual Coxeter number of $G$.
We generalize our results to other gauge groupsG$, for which supersymmetry is found at the adjoint mass-squared $g2 hvee/ (2pi)$, where $hvee$ is the dual Coxeter number of $G$.
arXiv Detail & Related papers (2024-06-24T19:07:42Z) - Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space [45.9982965995401]
We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators.
We find conditions for their strong continuity and establish properties of their generators.
arXiv Detail & Related papers (2024-06-15T17:39:32Z) - Discrete-coordinate crypto-Hermitian quantum system controlled by
time-dependent Robin boundary conditions [0.0]
unitary quantum mechanics formulated in non-Hermitian (or, more precisely, in hiddenly Hermitian) interaction-picture representation is illustrated via an elementary $N$ by $N$ matrix Hamiltonian $H(t)$ mimicking a 1D-box system with physics controlled by time-dependent boundary conditions.
Our key message is that contrary to the conventional beliefs and in spite of the unitarity of the evolution of the system, neither its "Heisenbergian Hamiltonian" $Sigma(t)$ nor its "Schr"odingerian Hamiltonian" $G(
arXiv Detail & Related papers (2024-01-19T13:28:42Z) - Towards a complete classification of non-chiral topological phases in 2D fermion systems [29.799668287091883]
We argue that all non-chiral fermionic topological phases in 2+1D are characterized by a set of tensors $(Nij_k,Fij_k,Fijm,alphabeta_kln,chidelta,n_i,d_i)$.
Several examples with q-type anyon excitations are discussed, including the Fermionic topological phase from Tambara-gami category for $mathbbZ_2N$.
arXiv Detail & Related papers (2021-12-12T03:00:54Z) - Quantum double aspects of surface code models [77.34726150561087]
We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry.
We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
arXiv Detail & Related papers (2021-06-25T17:03:38Z) - The Geometry of Time in Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
We continue the study of nonrelativistic quantum gravity associated with a family of Ricci flow equations.
This topological gravity is of the cohomological type, and it exhibits an $cal N=2$ extended BRST symmetry.
We demonstrate a standard one-step BRST gauge-fixing of a theory whose fields are $g_ij$, $ni$ and $n$, and whose gauge symmetries consist of (i) the topological deformations of $g_ij$, and (ii) the ultralocal nonrelativistic limit of space
arXiv Detail & Related papers (2020-11-12T06:57:10Z) - Sub-bosonic (deformed) ladder operators [62.997667081978825]
We present a class of deformed creation and annihilation operators that originates from a rigorous notion of fuzziness.
This leads to deformed, sub-bosonic commutation relations inducing a simple algebraic structure with modified eigenenergies and Fock states.
In addition, we investigate possible consequences of the introduced formalism in quantum field theories, as for instance, deviations from linearity in the dispersion relation for free quasibosons.
arXiv Detail & Related papers (2020-09-10T20:53:58Z) - Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range.
For a large part of the presentation, infinite scattering length will be considered.
arXiv Detail & Related papers (2020-06-03T17:54:43Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.