[Paper Review] ON PHOTON STATISTICS IN VARIABLE MEDIA

Sergey Kryuchkov, Erwin Suazo+1 more|arXiv (Cornell University)|Jan 13, 2014

Quantum Mechanics and Applications79 references3 citations

TL;DR

This paper derives explicit solutions for photon statistics in variable media using a quadratic Hamiltonian with multi-parameter squeezed inputs, solving Heisenberg equations of motion via an Ermakov-type system. The key contribution is a formal unitary transformation and extended squeeze/evolution operator that describe time evolution in abstract Hilbert space, enabling full statistical characterization of photons in non-uniform environments.

ABSTRACT

We find explicit solutions of the Heisenberg equations of motion for a general quadratic Hamiltonian in a variable medium in the case of multi-parameter squeezed input photons. The corresponding photon statistics are also derived in the Schrodinger picture in an abstract Hilbert space operator settings. Their time evolution is given in terms of solutions of certain Ermakov- type system. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. 1. An Introduction

Motivation & Objective

• To model photon statistics in variable media with time-varying optical properties.
• To solve the Heisenberg equations of motion for a general quadratic Hamiltonian in such media.
• To derive time evolution of photon statistics in the Schrödinger picture using abstract Hilbert space operators.
• To introduce a formal unitary transformation and extended squeeze/evolution operator for non-uniform systems.
• To establish a framework for multi-parameter squeezed input states in variable media.

Proposed method

• Solving the Heisenberg equations of motion for a quadratic Hamiltonian in a medium with spatial and temporal variations.
• Employing an Ermakov-type system to describe the time evolution of the system's dynamical invariants.
• Formulating the photon statistics in the Schrödinger picture using abstract Hilbert space operator formalism.
• Introducing a unitary transformation to map the time-dependent system to a time-independent reference frame.
• Extending the standard squeeze/evolution operator to account for multi-parameter squeezed input states in variable media.
• Using operator algebra and dynamical invariants to derive explicit expressions for photon statistics.

Experimental results

Research questions

• RQ1 How do photon statistics evolve in a variable medium governed by a general quadratic Hamiltonian?
• RQ2 What is the role of multi-parameter squeezed inputs in shaping the time evolution of photon states in non-uniform media?
• RQ3 How can the Ermakov-type system be used to solve the Heisenberg equations of motion in such systems?
• RQ4 What is the structure of the unitary transformation that preserves the dynamics in variable media?
• RQ5 How does the extended squeeze/evolution operator generalize standard quantum optical evolution in non-uniform environments?

Key findings

• Explicit solutions to the Heisenberg equations of motion are derived for a general quadratic Hamiltonian in variable media with multi-parameter squeezed inputs.
• The time evolution of photon statistics is fully characterized in the Schrödinger picture using abstract Hilbert space operators.
• The dynamics are governed by solutions to an Ermakov-type system, which captures the non-adiabatic and time-dependent behavior of the system.
• A formal unitary transformation is introduced that maps the time-dependent system to a reference system with constant parameters.
• The squeeze/evolution operator is extended to accommodate multi-parameter squeezed states, enabling full statistical description in variable media.
• The framework provides a complete operator-theoretic description of photon statistics in non-uniform optical environments.