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arXiv:1510.01469 · Physical Review A

Classical-quantum correspondence in bosonic two-mode conversion systems: Polynomial algebras and Kummer shapes

Eva-Maria Graefe, Hans Jürgen Korsch, Alexander Rush

Published Apr 4, 2016• 6 citations
arXivHTMLPDF

Abstract

Bosonic quantum conversion systems can be modeled by many-particle single-mode Hamiltonians describing a conversion of $m$ molecules of type A into $n$ molecules of type B and vice versa. These Hamiltonians are analyzed in terms of generators of a polynomially deformed su(2) algebra. In the mean-field limit of large particle numbers, these systems become classical and their Hamiltonian dynamics can again be described by polynomial deformations of a Lie algebra, where quantum commutators are replaced by Poisson brackets. The Casimir operator restricts the motion to Kummer shapes, deformed Bloch spheres with cusp singularities depending on $m$ and $n$. It is demonstrated that the many-particle eigenvalues can be recovered from the mean-field dynamics using a WKB-type quantization condition. The many-particle state densities can be semiclassically approximated by the time periods of periodic orbits, which show characteristic steps and singularities related to the fixed points, whose bifurcation properties are analyzed.

DOI
10.1103/physreva.93.042102

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Bosonic quantum conversion systems can be modeled by many-particle single-mode Hamiltonians describing a conversion of mmm molecules of type A into nnn molecules of type B and vice versa. These Hamiltonians are analyzed in terms of generators of a polynomially deformed su(2) algebra. In the mean-field limit of large particle numbers, these systems become classical and their Hamiltonian dynamics can again be described by polynomial deformations of a Lie algebra, where quantum commutators are replaced by Poisson brackets. The Casimir operator restricts the motion to Kummer shapes, deformed Bloch spheres with cusp singularities depending on mmm and nnn. It is demonstrated that the many-particle eigenvalues can be recovered from the mean-field dynamics using a WKB-type quantization condition. The many-particle state densities can be semiclassically approximated by the time periods of periodic orbits, which show characteristic steps and singularities related to the fixed points, whose bifurcation properties are analyzed.
We consider a simple quantum model of atom-molecule conversion where bosonic atoms can combine into diatomic molecules and vice versa. The many-particle system can be expressed in terms of the generators of a deformed su(2)\text{su}(2)su(2) algebra, and the mean-field dynamics takes place on a deformed version of the Bloch sphere, a teardrop shaped surface with a cusp singularity. We analyze the mean-field and many-particle correspondence, which shows typical features of quantum-classical correspondence. We demonstrate that semiclassical methods can be employed to recover full many-particle features from the mean-field description in cold atom systems with atom-molecule conversion, and we derive an analytic expression for the many-particle density of states in the limit of large particle numbers.
Classical simulability is ultimately determined by both the dynamics of a quantum system and the observables being evaluated. Lie-algebraic simulation exploits the latter to make exact polynomial-time classical simulations by propagating observables through low-dimensional invariant operator spaces. However, its conventional formulation in terms of polynomial-dimensional dynamical Lie algebras does not directly accommodate bosonic systems as their algebras are neither compact nor semisimple. In this contribution, we overcome this limitation, making bosonic systems accessible to the Lie-algebraic programme of exact polynomial-time classical simulation. We prove that expectation values, fixed-order correlation functions, including multi-time correlators and out-of-time-ordered correlators, and gradients are efficiently computable whenever their operator modules have polynomial dimension. This recovers Gaussian quantum optics and extends it to non-Gaussian input states, while identifying exact polynomial regimes of interacting non-Gaussian dynamics including bounded-photon Kerr and pair-hopping Hamiltonians and nilpotent polynomial phase dynamics. We show that unlike in the finite-dimensional spin and fermionic setting treated previously, a finite-dimensional bosonic generator algebra alone does not guarantee finite observable dynamics. We further derive a controlled perturbative hierarchy for squeezing beyond exact sector confinement and confirm the predicted error orders numerically. We also evaluate operator spreading on interacting chains of up to 400400400 modes and connect a topological doublon band with flux-reversed edge motion. These results provide a unified formalism for classifying, discovering, and systematically approximating tractable bosonic quantum dynamics with classical polynomial-time simulation.
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Abstract Quantum models of interacting bosons have a wide range of applications, including the propagation of optical modes in nonlinear media, such as the k-photon down-conversion. Many of these models are related to nonlinear deformations of finite group algebras and, in this sense, are exactly solvable. While advanced group-theoretic methods were developed to study the eigenvalue spectrum, in quantum optics, the primary focus is not on the spectrum of the Hamiltonian but rather on the evolution of an initial state—such as the generation of optical signal modes by a strong pump mode propagating through a nonlinear medium. I propose a simple and general method to solve the state evolution problem, applicable to a broad class of quantum models of interacting bosons. For the k-photon down-conversion model and its generalizations, the solution to the state evolution problem is expressed as an infinite series expansion in powers of the propagation time, with coefficients determined by a recursion relation involving only a single polynomial function. This polynomial function is unique to each nonlinear model. As an application, I compare the exact solution of the parametric down-conversion process with the semiclassical parametric approximation.
A new method is proposed for studying the semiclassical limit for model Hamiltonians which are polynomials in the generators of the Lie algebra SU(n)\mathrm{SU}(n)SU(n). The symmetric representation, characterized by a single independent Casimir invariant is mapped onto a subspace of a set of canonical coordinates, and a systematic expansion in the reciprocal of the Casimir invariant is carried out. WKB calculations for a specific example yield improved accuracy compared to previous work.
In this thesis, we study semiclassical phase-space methods for quantum evolution in Hermitian and non-Hermitian systems. We first present the dynamics of Gaussian wave packets under non-Hermitian Hamiltonians and interpret them as a classical dynamics for complex Hamiltonians. We use these to derive exact dynamics for wave packets in the quadratic Swanson oscillator. We show that in the case of unbroken PT-symmetry there can be periodic divergences in this system and relate this to the fact that any operator mapping the system to a Hermitian counterpart is unbounded. We apply the semiclassical wave-packet dynamics to two further anharmonic example systems: a PT-symmetric wave guide, a version of which we propose as a filtering device for optical beams, and a non-Hermitian single-band tight-binding model, for which we use classical equations of motion to model both narrowly and widely distributed initial states. We further develop an exact quantum propagator for non-Hermitian dynamics using lattices of wave packets, whose evolution is governed by the semiclassical equations of motion. We demonstrate that this accurately reproduces quantum dynamics compared to the split operator method. Finally we study a Hermitian two-mode many-particle model for bosonic atom-molecule conversion, for which the classical phase-space structure is an orbifold. We show that standard semiclassical tools may be applied to recover features such as the dynamics, spectrum and density of states of the many-particle system.