Combinatorics and Boson normal ordering: A gentle introduction

Abstract

We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick's theorem. The methodology can be straightforwardly generalized from the simple example given herein to a wide class of operators.

8 pages, 1 figure

References in corpus (9)

• Normal Ordering for Deformed Boson Operators and Operator-valued Deformed Stirling Numbers
• The general boson normal ordering problem
• Representations of Monomiality Principle with Sheffer-type Polynomials and Boson Normal Ordering
• Boson Normal Ordering via Substitutions and Sheffer-type Polynomials
• Combinatorics of boson normal ordering and some applications
• Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem
• Deformed Bosons: Combinatorics of Normal Ordering
• Normal Order: Combinatorial Graphs
• Dobinski-type relations: Some properties and physical applications

Cited by in corpus (36)

• Bosonic continuum theory of one-dimensional lattice anyons
• Quantum metrology of spatial deformation using arrays of classical and quantum light emitters
• Designing Kerr Interactions for Quantum Information Processing via Counterrotating Terms of Asymmetric Josephson-Junction Loops
• Exact $k$-body representation of the Jaynes-Cummings interaction in the dressed basis: Insight into many-body phenomena with light
• Laguerre-type derivatives: Dobinski relations and combinatorial identities
• Quantum mechanical bound for efficiency of quantum Otto heat engine
• On the use of shot noise for photon counting
• A simple combinatorial interpretation of certain generalized Bell and Stirling numbers
• Generalized Fock spaces and the Stirling numbers
• Photon-number moments and cumulants of Gaussian states
• Performance optimization of continuous variable quantum teleportation with generalized photon-varying non-Gaussian operations
• Combinatorics of chemical reaction systems
• On Urn Models, Non-commutativity and Operator Normal Forms
• Graph model of the Heisenberg-Weyl algebra
• Ladder Operators and Endomorphisms in Combinatorial Physics
• Fractional Fourier Transform and Geometric Quantization
• Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension
• Combinatorially interpreting generalized Stirling numbers
• Deciding finiteness of bosonic dynamics with tunable interactions
• Newton series expansion of bosonic operator functions
• Non-linear ladder operators and coherent states for the 2:1 oscillator
• Dispersive regime of multiphoton qubit-oscillator interactions
• Fock-space projection operators for semi-inclusive final states
• A combinatorial Hopf algebra for the boson normal ordering problem
• Combinatorial interpretation and proof of Glaisher-Crofton identity
• On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks
• Exact kinetic propagators for coherent state complex Langevin simulations
• Combinatorics of the three-parameter PASEP partition function
• Heisenberg-Weyl algebra revisited: Combinatorics of words and paths
• Unified boson sampling
• Approximate substitutions and the normal ordering problem
• Quantization due to the time evolution, with applications to Quantum Yang-Mills theory, Quantum Gravity and Classical Statistical Field Theory
• Asymptotic normality of some graph sequences
• Combinatorial Approach to Boson Anti-Normal Ordering Problem
• Relating the wave-function collapse with Euler's formula, with applications to Classical Statistical Field Theory
• Glass and jamming transition of simple liquids: static and dynamic theory