[Paper Review] Physical Realizability and Mean Square Performance of Translation Invariant Networks of Interacting Linear Quantum Stochastic Systems

Igor G. Vladimirov, Ian R. Petersen|arXiv (Cornell University)|Jan 15, 2014

Quantum optics and atomic interactions13 references4 citations

TL;DR

This paper establishes physical realizability conditions for translation-invariant networks of linear quantum stochastic systems with nearest-neighbor interactions on one- or multidimensional lattices, using matrix algebraic constraints on system parameters. It derives mean square performance functionals per site in the thermodynamic limit using spatial Fourier transforms and the Grenander-Szegő theorem, enabling analysis of large-scale quantum metamaterials and open quantum harmonic oscillators.

ABSTRACT

This paper is concerned with translation invariant networks of linear quantum stochastic systems with nearest neighbour interaction mediated by boson fields. The systems are associated with sites of a one-dimensional chain or a multidimensional lattice and are governed by coupled linear quantum stochastic differential equations (QSDEs). Such interconnections of open quantum systems are relevant, for example, to the phonon theory of crystalline solids, atom trapping in optical lattices and quantum metamaterials. In order to represent a large-scale open quantum harmonic oscillator, the coefficients of the coupled QSDEs must satisfy certain physical realizability conditions. These are established in the form of matrix algebraic equations for the parameters of an individual building block of the network and its interaction with the neighbours and external fields. We also discuss the computation of mean square performance functionals with block Toeplitz weighting matrices for such systems in the thermodynamic limit per site for unboundedly increasing fragments of the lattice.

Motivation & Objective

• To establish physical realizability conditions for large-scale networks of interacting linear quantum stochastic systems on lattices.
• To analyze mean square performance functionals with block Toeplitz weighting matrices in the thermodynamic limit.
• To extend coherent quantum linear quadratic Gaussian control theory to translation-invariant quantum networks.
• To enable design of decentralized CQLQG controllers for quantum metamaterials and quantum networks.
• To formalize the dynamics of open quantum harmonic oscillators in periodic arrays using coupled quantum stochastic differential equations.

Proposed method

• Uses coupled linear quantum stochastic differential equations (QSDEs) to model open quantum harmonic oscillators on a lattice with nearest-neighbor interactions.
• Applies spatial Fourier transforms to the QSDEs, transforming the system into the spatial frequency domain parameterized by a torus for multidimensional lattices.
• Derives matrix algebraic conditions for physical realizability by ensuring canonical commutation relations are preserved under unitary evolution.
• Introduces block Toeplitz weighting matrices reflecting translation invariance to compute mean square performance functionals.
• Applies the circular sampling theorem and Grenander-Szegö limit theorem to compute steady-state performance per site in the thermodynamic limit.
• Uses periodic boundary conditions on finite fragments of the lattice to model infinite systems and derive spectral density-based performance metrics.

Experimental results

Research questions

• RQ1 What matrix algebraic conditions ensure physical realizability of a translation-invariant network of linear quantum stochastic systems on a lattice?
• RQ2 How can mean square performance functionals with block Toeplitz weights be computed for such networks in the thermodynamic limit?
• RQ3 What is the role of spatial Fourier transforms in analyzing the dynamics and stability of large-scale quantum networks?
• RQ4 How do the spectral properties of the system’s transfer function relate to its performance in the thermodynamic limit?
• RQ5 Can the performance per site be computed independently of the lattice size in unboundedly increasing fragments?

Key findings

• Physical realizability is ensured by matrix algebraic constraints on the Hamiltonian, coupling, and scattering parameters of a single building block and its nearest-neighbor interactions.
• The mean square performance functional per site converges to a finite value in the thermodynamic limit, computed via the Grenander-Szegö limit theorem applied to spectral densities on the torus.
• The spatial Fourier transform reduces the lattice system to a family of decoupled systems parameterized by spatial frequencies, enabling frequency-domain analysis.
• The resulting performance metric is independent of the lattice size for large fragments, validating the use of thermodynamic limit approximations.
• The method enables the design of decentralized CQLQG controllers for quantum networks by providing a scalable performance metric.
• The derived conditions generalize to multidimensional lattices, with modified boundary conditions and transfer function matrices defined on a bivariate torus.