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Benasque Lectures on Gaussian Bosonic Systems and Analogue Gravity

Anthony J. Brady

Abstract

These notes are adapted from six lectures that I delivered at Analogue Gravity in Benasque 2023. They present the unified Gaussian (phase-space) framework to describe linear bosonic quantum systems, the standard tool in quantum optics and continuous-variable quantum information, emphasizing its simplicity and platform independence, with applications to semi-classical black holes and analogue gravity. Parts (I-III) develop the formalism: from harmonic dynamics and Gaussian transformations to state characterization via moments, Wigner functions, and entanglement measures. Part (IV) applies these tools to semi-classical black holes, discussing Hawking radiation and quantum superradiance in rotating black holes, and laboratory analogues in light-matter systems via toy models.

Benasque Lectures on Gaussian Bosonic Systems and Analogue Gravity

Abstract

These notes are adapted from six lectures that I delivered at Analogue Gravity in Benasque 2023. They present the unified Gaussian (phase-space) framework to describe linear bosonic quantum systems, the standard tool in quantum optics and continuous-variable quantum information, emphasizing its simplicity and platform independence, with applications to semi-classical black holes and analogue gravity. Parts (I-III) develop the formalism: from harmonic dynamics and Gaussian transformations to state characterization via moments, Wigner functions, and entanglement measures. Part (IV) applies these tools to semi-classical black holes, discussing Hawking radiation and quantum superradiance in rotating black holes, and laboratory analogues in light-matter systems via toy models.
Paper Structure (17 sections, 1 theorem, 225 equations, 27 figures, 1 table)

This paper contains 17 sections, 1 theorem, 225 equations, 27 figures, 1 table.

Table of Contents

  1. Feel the Vibrations
  2. Quantum Oscillators
  3. Linear Evolution
  4. Normal Modes and Gibbs Form
  5. It's All About the Moments
  6. Mean and Covariance
  7. Characteristic and Wigner Functions
  8. Quantum Measurements
  9. Entangled
  10. Preliminary Remarks
  11. Entropy, Extremality, and Modewise Entanglement
  12. PPT Criterion and the Logarithmic Negativity
  13. Separability via Cauchy-Schwarz Witnesses
  14. Gaussian Physics on the Horizon
  15. Digression: Mode Reduction for Quantum Fields
  16. ...and 2 more sections

Key Result

Theorem 1

There exist local symplectic transformations $\bm{S}_A\in\rm{Sp}(2N,\mathbb{R})$ and $\bm{S}_B\in\rm{Sp}(2M,\mathbb{R})$ such that where each $\bm{S}_{G_i}$ is a two-mode–squeezing transformation between modes $A_i$ and $B_i$ [see Eq. eq:tms_transform], and $\bm{I}_{2(M-N)}$ acts on uncorrelated modes.

Figures (27)

  • Figure 1: A network of coupled quantum harmonic oscillators, Hawking radiation from a black-hole event horizon, and photonic quantum fluctuations propagating near an analogue white-black hole in a dielectric medium---all fall under the umbrella of linear bosonic systems and can be described through the phase-space formalism.
  • Figure 2: Network of interacting quantum harmonic oscillators. Each mode has equally spaced energy levels, with the ground state (vacuum) containing zero quanta. Bilinear coupling, external driving, excitation loss, heating, and linear measurements, such as homodyne detection, are all described within the phase-space framework.
  • Figure 3: Symplectic diagrams: (a) Single-mode squeezer $\textbf{Sq}(r)$, (b) phase rotation $\bm R(\phi)$, (c) two-mode squeezer $\bm S(r)$, (d) beampslitter $\bm B(\theta)$, (d) SUM-gate $\textbf{SUM}_{12}$. Single- and two-mode squeezing lead to (phase-sensitive and phase-insensitive) amplification; our triangle notation is hence borrowed from classical circuit notation for linear amplifiers. Convention for input-output relations of the beamsplitter is shown, where we follow the modes through their transmission path. The SUM-gate is the CV equivalent of the CNOT gate. Input to output flows from left to right.
  • Figure 4: Wigner functions of Gaussian states. Left: vacuum state. Right: squeezed vacuum. Gaussian states are described by multivariate Gaussian distributions in phase space.
  • Figure 5: Unitary dilation of single-mode Gaussian channels: (a) thermal loss channel, (b) thermal amplifier channel, (c) additive Gaussian noise channel (a.k.a., random displacements). A purification of the environment thermal state $\rho_{\rm th}$ is also shown as a two-mode squeezed vacuum for (a, b).
  • ...and 22 more figures

Theorems & Definitions (1)

  • Theorem 1: Modewise Entanglement reznik2003modewise