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Exact State Evolution and Energy Spectrum in Solvable Bosonic Models

Valery Shchesnovich

TL;DR

The paper addresses the exact solution of state evolution and energy spectra for a broad class of solvable bosonic models relevant to nonlinear quantum optics, such as k-photon down-conversion. It develops an algebraic framework that yields the evolution of arbitrary initial states and explicit eigenstates, expressing spectral data via continued fractions, Möbius-group matrices, and Jacobi determinants. Key contributions include a complete recursion-based construction of evolution coefficients, a Hessenberg-matrix representation of the $g$-factors, and a closed-form stationary zero-energy state in odd-dimensional invariant subspaces. The results provide a rigorous, divergence-free analytical toolkit with broad applicability to exact-solvable bosonic systems and offer a foundation for future asymptotic analyses as subspace dimension grows.

Abstract

Solvable bosonic models provide a fundamental framework for describing light propagation in nonlinear media, including optical down-conversion processes that generate squeezed states of light and their higher-order generalizations. In quantum optics a central objective is to determine the time evolution of a given initial state. Exact analytic solution to the state-evolution problem is presented, applicable to a broad class of solvable bosonic models and arbitrary initial states. Moreover, the characteristic equation governing the energy spectrum is derived and the eigenstates are found in the form of continued fractions and as principal minors of the associated Jacobi matrix. The results provide a solid analytical framework for discussion of exactly solvable bosonic models.

Exact State Evolution and Energy Spectrum in Solvable Bosonic Models

TL;DR

The paper addresses the exact solution of state evolution and energy spectra for a broad class of solvable bosonic models relevant to nonlinear quantum optics, such as k-photon down-conversion. It develops an algebraic framework that yields the evolution of arbitrary initial states and explicit eigenstates, expressing spectral data via continued fractions, Möbius-group matrices, and Jacobi determinants. Key contributions include a complete recursion-based construction of evolution coefficients, a Hessenberg-matrix representation of the -factors, and a closed-form stationary zero-energy state in odd-dimensional invariant subspaces. The results provide a rigorous, divergence-free analytical toolkit with broad applicability to exact-solvable bosonic systems and offer a foundation for future asymptotic analyses as subspace dimension grows.

Abstract

Solvable bosonic models provide a fundamental framework for describing light propagation in nonlinear media, including optical down-conversion processes that generate squeezed states of light and their higher-order generalizations. In quantum optics a central objective is to determine the time evolution of a given initial state. Exact analytic solution to the state-evolution problem is presented, applicable to a broad class of solvable bosonic models and arbitrary initial states. Moreover, the characteristic equation governing the energy spectrum is derived and the eigenstates are found in the form of continued fractions and as principal minors of the associated Jacobi matrix. The results provide a solid analytical framework for discussion of exactly solvable bosonic models.
Paper Structure (16 sections, 5 theorems, 87 equations, 1 figure)

This paper contains 16 sections, 5 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Solvable bosonic models
  3. Evolution of arbitrary state
  4. The states in the expansion of the evolution operator
  5. Solving the recursion; the $g$-factors
  6. The $g$-factors as powers of a Hessenberg matrix
  7. The state evolution: main results
  8. The energy spectrum
  9. The spectrum via continuous fractions
  10. The characteristic equation for the energy eigenvalues
  11. Amplitudes of the eigenstates as Jacobi determinants
  12. The stationary state
  13. Discussion
  14. Conclusion
  15. acknowledgements
  16. ...and 1 more sections

Key Result

Theorem 1

The state $|A_{m,k}\rangle \equiv (\hat{A} +\hat{A}^\dag)^m(\hat{A}^\dag)^k|\Psi_0\rangle$ is given by where $L_{m,k}= \mathrm{min}\left(m,\left[\frac{k+m}{2}\right]\right)$ (with $[\ldots]$ denoting the integer part) and the scalars $g^{(l)}_{n,k}$ are given by Eq. (gln_E) or, alternatively, by Eqs. (gln_B)-(gln_BT).

Figures (1)

  • Figure 1: The non-zero squared amplitudes $\psi_{2p}^2$ of the stationary state in Eq. (\ref{['psi_n0k']}) vs $p$ for $k=2$ and $N=100$ (the value $\psi_0^2=1$ was removed for better visibility of the distribution for $p>0$).

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Corollary 3