Blog & Notes | Orjan Ameye

My Setup

Sat, 19 Jul 2025 00:00:00 +0000

My OS & Software Stack

Operating System

Aurora (Universal Blue) - An immutable Linux distribution based on Fedora Silverblue with KDE Plasma desktop. Provides atomic updates and a reliable, modern desktop experience.

Desktop Environment: KDE Plasma with Sweet theme for a clean, dark interface.

Development Environment

Editors & IDEs

VS Code - Primary code editor with excellent Julia and Python support
Jupyter Lab - Interactive computing environment for data analysis and prototyping

Terminal

Konsole - KDE’s terminal emulator
Yakuake - Drop-down terminal for quick access
Bash - Reliable shell for daily tasks

Programming Languages

Julia - Primary language for scientific computing and numerical analysis
Python - Secondary language for broader ecosystem access
Mathematica/Wolfram Language - Symbolic computation and mathematical analysis
Typst - Modern markup language for typesetting and document preparation

Key Libraries & Packages

SciML ecosystem (Julia) - Scientific machine learning and differential equations
Makie (Julia) - High-performance data visualization
HarmonicBalance.jl (Julia) - Harmonic balance method for nonlinear dynamics
Sneg (Mathematica) - Symbolic calculation for quantum many-body systems

Development Tools

Version Control & Containers

Git - Version control and repository management
Distrobox - Containerization for isolated development environments

Productivity & Research

Zotero - Reference management for academic papers
Notion - Project organization and structured notes
Obsidian - Knowledge management and interconnected notes
Nextcloud - Self-hosted cloud storage and file synchronization

Graphics & Design

Inkscape - Vector graphics creation and scientific diagrams
GIMP - Raster image editing and figure preparation

Web Browser

Brave - Privacy-focused browser with built-in ad blocking

Communication

Element - Matrix-based secure messaging
Mattermost - Team collaboration platform
Zulip - Organized team chat with threading
Slack - General workplace communication

Open Source Contributions

Mon, 14 Jul 2025 00:00:00 +0000

A summary of my recent open source contributions, including merged pull requests.

Note: This document is automatically generated using the GitHub CLI and updated regularly to reflect the latest contributions.

Generated on July 19, 2025

🔥 Recent Merged Contributions (External Repositories)

- fix: swept_parameter with BasicSymbolic (merged 2025-07-19)
- fix: streamline sorting options (merged 2025-07-19)
- docs: add forward transmission example and update documentation API (merged 2025-07-19)
- build: bump HarmonicSteadyState compat (merged 2025-07-19)
- implement suggestions to QFI tutorial (merged 2025-07-14)
- build: tag patch release (merged 2025-07-10)
- export get_branches (merged 2025-07-10)
- build: tag patch version (merged 2025-07-08)
- feat: add get_branches (merged 2025-07-08)
- fix: hopf definition (merged 2025-07-07)

🚀 My Open Source Projects

Julia Packages

⭐ 8
- Perform wick contraction on a Keldysh contour
- Created: 2025-04-17

⭐ 7
- ab-initio continuous path integral Monte Carlo
- Created: 2022-07-09

⭐ 3
- Symbolic calculation of the (van Vleck) Floquet recursion formula
- Created: 2025-06-01

⭐ 3
- Generate all combinations of graphs and their symmetry factor
- Created: 2025-04-30

⭐ 2
- Created: 2022-03-15

⭐ 1
- Created: 2025-06-08

⭐ 1
- An opinionated layer on top of FFTW.jl. Fork of EasyFFTs.jl
- Created: 2024-11-20

⭐ 1
- Created: 2024-08-08

Tools & Utilities

⭐ 1
- zotero engines.json
- Created: 2024-10-01

Personal Projects

⭐ 1
- Created: 2022-03-15

📝 All Merged Pull Requests by Organization (External Repositories)

QuantumEngineeredSystems (254 PRs)

As main maintainer of QuantumEngineeredSystems packages:

254 merged PRs
Active development across 3 repositories:
- HarmonicBalance.jl: 192 PRs
- HarmonicSteadyState.jl: 33 PRs
- QuestBase.jl: 29 PRs

Recent contributions (last 5):

- fix: swept_parameter with BasicSymbolic (merged 2025-07-19)
- fix: streamline sorting options (merged 2025-07-19)
- docs: add forward transmission example and update documentation API (merged 2025-07-19)
- build: bump HarmonicSteadyState compat (merged 2025-07-19)
- build: tag patch release (merged 2025-07-10)

JuliaDynamics (61 PRs)

CriticalTransitions.jl (55 PRs) - Co-creator and main contributor

Recent CriticalTransitions.jl contributions (last 3):

- Revert “docs: add logo and update flowchart (#171)” (merged 2025-06-21)
- chore: update copyright years in LICENSE file (merged 2025-06-21)
- docs: sync example folder with docs using Literate (merged 2025-06-21)

Other JuliaDynamics contributions:

- add CoupledSDEs(ds::CoupledSDEs, diffeq) method (merged 2024-12-26)
- fix: set max time span to 1e16 for CoupledSDEs (merged 2024-12-01)
- feat: MTK Jacobian for CoupledDEs (merged 2024-11-23)
- let diffusion_matrix(::CoupledSDEs) export a matrix (merged 2024-10-04)
- add coupledSDEs type (merged 2024-07-24)
- Update real_world.md (merged 2022-11-17)

qojulia (36 PRs)

SecondQuantizedAlgebra.jl (25 PRs) - Major contributor and maintainer

Recent SecondQuantizedAlgebra.jl contributions (last 3):

- build: bump patch version (merged 2025-07-02)
- fix: update unnecessary ModelingToolkit strict compat (merged 2025-07-02)
- feat: remove MTK (merged 2025-06-29)

Other qojulia contributions:

- build: bump compat to SymbolicUtils > v3.26 (merged 2025-06-23)
- refactor: make import and export explicit (merged 2025-06-22)
- test: add potential code quality tests (merged 2025-06-21)
- build: update Documenter workflow (merged 2025-06-21)
- build: add spelling checker (merged 2025-06-21)
- build: update Tests workflow (merged 2025-06-21)
- fix: unsupported SymbolicUtils.SmallVector (merged 2025-06-06)
- seperate quantum algebra from cumulant expansion (merged 2025-05-10)
- remove adjoint method (merged 2024-09-05)
- modify compatibility for SymbolicUtils (merged 2024-07-18)
- add SymbolicsUtils.expand function for QAdd (merged 2024-05-05)

SciML (7 PRs)

- fix: add Complex to check floating point sym (merged 2025-06-29)
- fix: spelling typos (merged 2025-06-24)
- fix: enable support for complex ODEProblem again (merged 2025-06-24)
- build: replace OrdinaryDiffEq with OrdinaryDiffEqCore (merged 2025-01-22)
- add additative noise trait (merged 2024-08-29)
- add g kwarg to remake(::SDEProblem) (merged 2024-07-21)
- add covariance field to NoiseProcess struct (merged 2024-07-16)

qutip (2 PRs)

- implement suggestions to QFI tutorial (merged 2025-07-14)
- add QFI example (merged 2025-06-13)

control-toolbox (2 PRs)

- fix: typo in mam tutorial (merged 2025-04-24)
- docs: add oc_mam (merged 2025-01-15)

Homebrew (1 PRs)

- dvisvgm 3.5 (new formula) (merged 2025-06-06)

JuliaRegistries (1 PRs)

- change url HarmonicBalance due to organization name change (merged 2025-03-19)

JuliaLLVM (1 PRs)

- Update README.md (merged 2025-01-11)

JuliaHomotopyContinuation (1 PRs)

- fix: fix total degree with many parameters threaded (#599) (merged 2024-11-03)

LuxDL (1 PRs)

- Fix typo in docs (merged 2024-08-12)

JuliaSymbolics (1 PRs)

- fix typo of docs rule (merged 2024-08-02)

julia-actions (1 PRs)

- set julia-cache action to v2 (merged 2024-06-12)

JuliaDocs (1 PRs)

- update to setup-julia@v2 in the Github action example (merged 2024-04-18)

bifurcationkit (1 PRs)

- Update gettingstarted.md (merged 2023-10-21)

rokzitko (1 PRs)

- add installation section (merged 2023-03-05)

quarto-dev (1 PRs)

- Correcting error in website-basics.qmd (merged 2022-10-08)

This document was automatically generated using the GitHub CLI.

Preparation of nonclassical states in cavities with a moving mirror

Mon, 14 Jul 2025 00:00:00 +0000

%matplotlib inline

from numpy import *

import matplotlib.pyplot as plt

from qutip import *

Introduction

In Bose et al. PRA 56, 4175 (1997), the authors consider the preparation of nonclassical states in cavities with moving mirror. The nonclassical states are achived by letting the system evolve for a specific duration of time. Here we reproduce some of the results of Bose et al.using numberical simulations of the Schrodinger and master equations using QuTiP.

Hamiltonian and initial state

The Hamiltonian of the cavity with a moving mirror is

$H = \hbar\omega_0 a^\dagger a + \hbar\omega_m b^\dagger b - \hbar g a^\dagger a (b + b^\dagger)$

where $\omega_0$ is the cavity frequency, $\omega_m$ is the frequency of the oscillation mode of the mirror and $g$ is the opto-mechanical coupling strength. The creation operators of the cavity mode and the mirror mode are $a$ and $b$, respectively.

The initial states of the cavity mode and the mirror mode are in the coherent states $|\alpha\rangle$ and $|\beta\rangle$, respectively.

Undamped dynamics

# number of fock states in the cavity and mirror modes
N = 15
M = 30
r = 1.0

def solve_dynamics_undamped(r, k, alpha, beta, t, N, M):

 wm = 1
 w0 = r * wm
 g = k * wm

 a = tensor(destroy(N), identity(M))
 b = tensor(identity(N), destroy(M))

 H = w0 * a.dag() * a + wm * b.dag() * b - g * a.dag() * a * (b + b.dag())

 psi0 = tensor(coherent(N, alpha), coherent(M, beta))

 options = {"nsteps": 15_000}
 result = mesolve(H, psi0, t, options=options)

 return result, expect(commutator(a, a.dag()), result.states), expect(commutator(b, b.dag()), result.states)
 #return result, expect(a.dag() * a, result.states), expect(b.dag() * b, result.states)

Figure 1 in Bose et al.

t = linspace(0, 2 * pi, 50)

alpha = beta = 2

result, C1_A, C1_B = solve_dynamics_undamped(r, 0.1, alpha, beta, t, N, M)
S1 = [entropy_linear(ptrace(rho, 1)) for rho in result.states]

result, C2_A, C2_B = solve_dynamics_undamped(r, 0.5, alpha, beta, t, N, 2*M)
S2 = [entropy_linear(ptrace(rho, 1)) for rho in result.states]

result, C3_A, C3_B = solve_dynamics_undamped(r, 1.0, alpha, beta, t, N, 10*M)
S3 = [entropy_linear(ptrace(rho, 1)) for rho in result.states]

fig, ax = plt.subplots(figsize=(10,5))
ax.plot(t, S1, label=r'$k = 0.1$')
ax.plot(t, S2, label=r'$k = 0.5$')
ax.plot(t, S3, label=r'$k = 1.0$')
ax.legend()
ax.set_title('Linear entropy of the mirror')
ax.set_xlabel(r'$t$', fontsize=18)
ax.set_ylabel(r'$S$', fontsize=18)
ax.set_xlim(0, t.max());

Since the dynamics here is undamped the quantum state is pure, and a nonzero entropy of a the subsystem of the mirror demonstrates that the mirror is entangled with the cavity field.

Check Hilbert space truncation

As a check to make sure that we have use a sufficiently large Hilbert space in the calcuation above we plot the expectation value of the commutators $[a, a^\dagger]$ and $[b , b^\dagger]$ which should be very close to 1 if the Fock-state basis truncation that we used above is OK.

fig, axes = plt.subplots(1, 2, figsize=(16,5))
axes[0].plot(t, C1_A, label=r'$k = 0.1$')
axes[0].plot(t, C2_A, label=r'$k = 0.5$')
axes[0].plot(t, C3_A, label=r'$k = 1.0$')
axes[0].legend()
axes[0].set_title('[a, a.dag()]')
axes[0].set_xlabel(r'$t$', fontsize=18)
axes[0].set_ylabel(r'$S$', fontsize=18);
axes[0].set_ylim(0, 1.1);

axes[1].plot(t, C1_B, label=r'$k = 0.1$')
axes[1].plot(t, C2_B, label=r'$k = 0.5$')
axes[1].plot(t, C3_B, label=r'$k = 1.0$')
axes[1].legend()
axes[1].set_title('[b, b.dag()]]')
axes[1].set_xlabel(r'$t$', fontsize=18)
axes[1].set_ylabel(r'$S$', fontsize=18)
axes[1].set_ylim(0, 1.1);

Figure 2 in Bose et al.

The wigner function of the mirror for various times $t$.

t = [0, pi/2, pi, 3*pi/2, 2*pi]

result, C_A, C_B = solve_dynamics_undamped(r, 0.3, alpha, beta, t, N, M)

fig, axes = plt.subplots(len(t), 2, figsize=(8,16))

for idx, rho in enumerate(result.states):
 rho_mirror = ptrace(rho, 1)
 plot_fock_distribution(rho_mirror, fig=fig, ax=axes[idx, 0])
 plot_wigner(rho_mirror, fig=fig, ax=axes[idx, 1])

fig.tight_layout()

The mirror wigner functions are always positive and the mirror therefore always in a classical state.

Figure 3 in Bose et al.

The wigner function of the cavity field at $t = 2\pi$, for three different values of $k$.

t = [0, 2*pi]

k_vec = [0.5, 0.4082, 0.3536]

states = [solve_dynamics_undamped(r, k, alpha, beta, t, 3*N, 3*M)[0].states[-1] for k in k_vec]

fig, axes = plt.subplots(len(k_vec), 2, figsize=(8,12))

for idx, rho in enumerate(states):
 rho_cavity = ptrace(rho, 0)
 plot_fock_distribution(rho_cavity, fig=fig, ax=axes[idx, 0])
 plot_wigner(rho_cavity, fig=fig, ax=axes[idx, 1])

fig.tight_layout()

For these particular parameters the cavity field is prepared in superpositions of two, three and four coherent states, respectively.

Damped dynamics

# number of fock states in the cavity and mirror modes
N = 10
M = 10

def solve_dynamics_damped(r, k, gamma, alpha, beta, t, N, M):

 wm = 1
 w0 = r * wm
 g = k * wm

 a = tensor(destroy(N), identity(M))
 b = tensor(identity(N), destroy(M))

 H = w0 * a.dag() * a + wm * b.dag() * b - g * a.dag() * a * (b + b.dag())

 psi0 = tensor(coherent(N, alpha), coherent(M, beta))

 c_ops = [sqrt(gamma) * b]

 options = {"nsteps": 15_000, "progress_bar": "enhanced"}
 return mesolve(H, psi0, t, c_ops, options=options)

Figure 6 in Bose et al.

gamma = 1.0
alpha = 2
beta = 0

t = linspace(0, 2 * pi, 25)

S1 = [entropy_linear(ptrace(rho, 1)) for rho in solve_dynamics_damped(r, 0.1, gamma, alpha, beta, t, N, M).states]

 [** 8% ] Elapsed 0.02s / Remaining 00:00:00:00
Total run time: 0.18s*] Elapsed 0.18s / Remaining 00:00:00:00

S2 = [entropy_linear(ptrace(rho, 1)) for rho in solve_dynamics_damped(r, 0.5, gamma, alpha, beta, t, N, M).states]

 Total run time: 0.34s*] Elapsed 0.34s / Remaining 00:00:00:00

S3 = [entropy_linear(ptrace(rho, 1)) for rho in solve_dynamics_damped(r, 1.0, gamma, alpha, beta, t, N, M).states]

 Total run time: 0.69s*] Elapsed 0.69s / Remaining 00:00:00:00

fig, ax = plt.subplots(figsize=(10,5))
ax.plot(t, S1, label=r'$k = 0.1$')
ax.plot(t, S2, label=r'$k = 0.5$')
ax.plot(t, S3, label=r'$k = 1.0$')
ax.legend()
ax.set_title('Linear entropy of the mirror')
ax.set_xlabel(r'$t$', fontsize=18)
ax.set_ylabel(r'$S$', fontsize=18)
ax.set_xlim(0, t.max());

Figure 7 in Bose et al.

t = [0, 2*pi]
k = 0.5
alpha = beta = 2

gamma_vec = [0.001, 0.01, 0.1, 1.0]

states = [solve_dynamics_damped(r, k, gamma, alpha, beta, t, 3*N, 2*M).states[-1] for gamma in gamma_vec]

 Total run time: 38.77s*] Elapsed 38.77s / Remaining 00:00:00:00
Total run time: 45.80s*] Elapsed 45.80s / Remaining 00:00:00:00
Total run time: 51.90s*] Elapsed 51.90s / Remaining 00:00:00:00
Total run time: 53.20s*] Elapsed 53.20s / Remaining 00:00:00:00

fig, axes = plt.subplots(len(gamma_vec), 2, figsize=(8,16))

for idx, rho in enumerate(states):
 rho_cavity = ptrace(rho, 0)
 plot_fock_distribution(rho_cavity, fig=fig, ax=axes[idx, 0])
 plot_wigner(rho_cavity, fig=fig, ax=axes[idx, 1])

fig.tight_layout()

Versions

%reload_ext version_information
%version_information numpy, scipy, matplotlib, qutip

Software	Version
Python	3.13.5 64bit [GCC 15.1.1 20250521 (Red Hat 15.1.1-2)]
IPython	9.2.0
OS	Linux 6.14.11 300.fc42.x86\_64 x86\_64 with glibc2.41
numpy	2.3.1
scipy	1.16.0
matplotlib	3.10.3
qutip	5.2.0
Mon Jul 14 09:01:53 2025 CEST

Quantum Fisher Information with automatic differentiation

Sun, 13 Jul 2025 00:00:00 +0000

This notebook demonstrates the computation of Quantum Fisher Information (QFI) for a driven-dissipative Kerr Parametric Oscillator (KPO) using automatic differentiation. The QFI quantifies the ultimate precision limit for parameter estimation in quantum systems.

We import the necessary packages for quantum simulations and automatic differentiation:

using QuantumToolbox # Quantum optics simulations
using DifferentiationInterface # Unified automatic differentiation interface
using SciMLSensitivity # Allows for ODE sensitivity analysis
using FiniteDiff # Finite difference methods
using LinearAlgebra # Linear algebra operations
using CairoMakie # Plotting
CairoMakie.activate!(type = "svg")

System Parameters and Hamiltonian

The KPO system is governed by the Hamiltonian:

$$H = -p_1 a^\dagger a + K (a^\dagger)^2 a^2 - G (a^\dagger a^\dagger + a a)$$

where:

$p_1$ is the parameter we want to estimate (detuning)
$K$ is the Kerr nonlinearity
$G$ is the parametric drive strength
$\gamma$ is the decay rate

function final_state(p, t)
 G, K, γ = 0.002, 0.001, 0.01

 N = 20 # cutoff of the Hilbert space dimension
 a = destroy(N) # annihilation operator

 coef(p,t) = - p[1]
 H = QobjEvo(a' * a , coef) + K * a' * a' * a * a - G * (a' * a' + a * a)
 c_ops = [sqrt(γ)*a]
 ψ0 = fock(N, 0) # initial state

 tlist = range(0, 2000, 100)
 sol = mesolve(H, ψ0, tlist, c_ops; params = p, progress_bar = Val(false), saveat = [t])
 return sol.states[end].data
end

final_state (generic function with 1 method)

Quantum Fisher Information Calculation

The QFI is computed using the symmetric logarithmic derivative (SLD). For a density matrix $\rho(\theta)$ parametrized by $\theta$:

$$F_Q = \text{Tr}[\partial_\theta \rho \cdot L]$$

where $L$ is the SLD satisfying: $\partial_\theta \rho = \frac{1}{2}(\rho L + L \rho)$

function compute_fisher_information(ρ, dρ)
 reg = 1e-12 * I # Add small regularization to avoid numerical issues with zero eigenvalues
 ρ_reg = ρ + reg

 L = sylvester(ρ_reg, ρ_reg, -2*dρ) # This is a Sylvester equation: ρL + Lρ = 2*dρ
 F = real(tr(dρ * L)) # Fisher information F = Tr(dρ * L)
 return F
end

compute_fisher_information (generic function with 1 method)

Automatic Differentiation Setup

We use finite differences through DifferentiationInterface.jl to compute the derivative of the quantum state with respect to the parameter. This is a key step that enables efficient QFI computation without manual derivative calculations.

final_state([0], 100) # Test the system

state(p) = final_state(p, 2000) # Define state function for automatic differentiation

ρ, dρ = DifferentiationInterface.value_and_jacobian(state, AutoFiniteDiff(), [0.0])

dρ = QuantumToolbox.vec2mat(vec(dρ)) # Reshape the derivative back to matrix form

qfi_final = compute_fisher_information(ρ, dρ) # Compute QFI at final time
println("QFI at final time: ", qfi_final)

QFI at final time: 19795.821946511165

Time Evolution of Quantum Fisher Information

Now we compute how the QFI evolves over time to understand the optimal measurement time for parameter estimation:

ts = range(0, 2000, 100)

QFI_t = map(ts) do t
 state(p) = final_state(p, t)
 ρ, dρ = DifferentiationInterface.value_and_jacobian(state, AutoFiniteDiff(), [0.0])
 dρ = QuantumToolbox.vec2mat(vec(dρ))
 compute_fisher_information(ρ, dρ)
end

println("QFI computed for ", length(ts), " time points")

QFI computed for 100 time points

Visualization

Plot the time evolution of the Quantum Fisher Information:

fig = Figure()
ax = Axis(
 fig[1,1],
 xlabel = "Time",
 ylabel = "Quantum Fisher Information"
)
lines!(ax, ts, QFI_t)
fig

Version Information

using InteractiveUtils
InteractiveUtils.versioninfo()

Julia Version 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 UTC)
Build Info:
 Official https://julialang.org/ release
Platform Info:
 OS: Linux (x86_64-linux-gnu)
 CPU: 12 × AMD Ryzen 5 5600X 6-Core Processor
 WORD_SIZE: 64
 LIBM: libopenlibm
 LLVM: libLLVM-15.0.7 (ORCJIT, znver3)
Threads: 10 default, 0 interactive, 5 GC (on 12 virtual cores)
Environment:
 JULIA_EDITOR = code
 JULIA_VSCODE_REPL = 1
 JULIA_NUM_THREADS = 10

using Pkg
Pkg.status()

Status `/var/home/oameye/Documents/website/content/posts/Quantum_Fisher_Information/Project.toml`
⌃ [13f3f980] CairoMakie v0.13.10
 [a0c0ee7d] DifferentiationInterface v0.7.2
 [6a86dc24] FiniteDiff v2.27.0
 [98b081ad] Literate v2.20.1
 [6c2fb7c5] QuantumToolbox v0.32.1
 [1ed8b502] SciMLSensitivity v7.87.0
 [37e2e46d] LinearAlgebra
Info Packages marked with ⌃ have new versions available and may be upgradable.

This page was generated using .

Notes: Physics Facts

Thu, 15 Feb 2024 00:00:00 +0000

Bohr–Van Leeuwen theorem

The Bohr–Van Leeuwen theorem states that when statistical mechanics and classical mechanics are applied consistently, the thermal average of the magnetization is always zero. This makes magnetism in solids solely a quantum mechanical effect and means that classical physics cannot account for paramagnetism, diamagnetism and ferromagnetism.

Groenewold-van Hove theorem

Groenewold’s theorem, also known as the Groenewold-van Hove theorem, is a fundamental result in the theory of quantization, which demonstrates the impossibility of a consistent mapping from classical observables to quantum operators that preserves the Poisson bracket structure.

It shows that it is impossible to consistently quantize all classical observables while maintaining the correspondence between Poisson brackets and quantum commutators. The theorem has significant implications for the foundations of quantum mechanics, as it highlights the limitations of canonical quantization and the challenges of formulating a consistent quantum theory from classical mechanics.

Groenewold’s theorem also has connections to the Wigner-Weyl transform, which provides a way to map between phase-space functions and quantum operators. However, Groenewold’s theorem asserts that no such map can have all the ideal properties one would desire for a consistent quantization scheme. Despite this, the Wigner-Weyl transform remains a valuable tool for understanding quantum mechanics in phase space.

groenewold van hove theorem

Notes: Optimal Control

Thu, 01 Feb 2024 00:00:00 +0000

Optimal Control and Physics

Cool Things in Optimal Control and Physics

Academic Life Resources

Mon, 15 Jan 2024 00:00:00 +0000

Resource aggregators

PhD Resources of Guillaume Dalle: Collection of cool resources, websites and software to improve your workflow and make your life easier.
Awesome PhD by Helena Hartmann: A curated list of carefully selected tools and resources I wish I knew when starting my PhD.

Survival of the Firstest

Survival of the Firstest by Peter Seiler.

Textext Inkscape Flathub

Mon, 01 Jan 2024 00:00:00 +0000

python setup.py --inkscape-extensions-path /home/oameye/.var/app/org.inkscape.Inkscape/config/inkscape/extensions/ --skip-requirements-check --pdflatex-executable /home/linuxbrew/.linuxbrew/bin/pdflatex --typst-executable /home/linuxbrew/.linuxbrew/bin/typst --lualatex-executable /home/linuxbrew/.linuxbrew/bin/lualatex --xelatex-executable /home/linuxbrew/.linuxbrew/bin/xelatex

Bloch oscillation in a dynamically modulated ring resonator

Mon, 23 Oct 2023 00:00:00 +0000

using LatticeModels, Plots; LM = LatticeModels
using QuantumOptics
using Plots.Measures

default(fmt=:png,
 fontfamily="computer modern", titlefont="computer modern", tickfont="computer modern";
 linewidth=2, legend_position=:best
 )

We want to reproduce the Bloch oscillation in a ring resonator. A ring resonator with a dynamic refractive index modulation can exhibit Bloch oscillation of light along the frequency axis, as a result of an effective force for photons induced by the modulation, as is shown in .

Furthermore, we want to show that one can deduce directional transport on the frequency axis by switching the detuning periodically.

We will implement the model ad deduced in :

$$ \tilde{H}=g \sum_m\left(c_m^{\dagger} c_{m+1}+c_{m+1}^{\dagger} c_m\right)+\sum_m m \Delta(t) c_m^{\dagger} c_m, $$

where $m$ is the resonant mode of the waveguide in the resonator, $g$ is the strength of the modulation, and $\Delta$ is the detuning of the resonant frequency of the waveguide from the modulation frequency.

We start by implementing a lattice of 100 sites, i.e., we consider 100 modes of the waveguide. The lattice will have reflective boundary conditions. Furthermore, the tight binding system has one particle and a chemical potential of zero. No extra particles can be added to the system.

size=100
lattice = SquareLattice(size, 1)
x_basis = PositionBasis(0, size, size)

sample = Sample(lattice, boundaries=BoundaryConditions(1 => true, 2 => true))
system = System(sample, μ=0.0, statistics=LM.OneParticle)

LatticeModels.FilledZones{Sample{Nothing, SquareLattice{2}, Nothing, BoundaryConditions{Tuple{TwistedBoundary, TwistedBoundary}}}}(Sample{Nothing, SquareLattice{2}, Nothing, BoundaryConditions{Tuple{TwistedBoundary, TwistedBoundary}}}(nothing, SquareLattice{2}((100, 1), LatticeModels.Bravais{2, 1}([1.0 0.0; 0.0 1.0], [0.0; 0.0;;]), Bool[1, 1, 1, 1, 1, 1, 1, 1, 1, 1 … 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), BoundaryConditions{Tuple{TwistedBoundary, TwistedBoundary}}((TwistedBoundary(1, 0.0), TwistedBoundary(2, 0.0))), nothing), 0.0, LatticeModels.OneParticle)

We implement the time dependent Hamiltonian where switch the detuning periodically between -$\Delta$ and +$\Delta$ with a period of $T=1$.

g = 2; Δ = 2
function h(t, g, Δ; switch = t -> isodd(floor(t)) ? -1 : 1)
 build_hamiltonian(lattice,
 lattice .|> (site -> Δ* switch(t)*site.coords[1]),
 g => SiteOffset(axis=1)
 )
end
H0 = h(0, g, Δ)

Hamiltonian(dim=100x100)
basis: LatticeBasis{SquareLattice{2}}(100, SquareLattice{2}((100, 1), LatticeModels.Bravais{2, 1}([1.0 0.0; 0.0 1.0], [0.0; 0.0;;]), Bool[1, 1, 1, 1, 1, 1, 1, 1, 1, 1 … 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]))
sparse([1, 2, 1, 2, 3, 2, 3, 4, 3, 4 … 97, 98, 97, 98, 99, 98, 99, 100, 99, 100], [1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 28, 29, 29, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 42, 43, 43, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49, 50, 50, 50, 51, 51, 51, 52, 52, 52, 53, 53, 53, 54, 54, 54, 55, 55, 55, 56, 56, 56, 57, 57, 57, 58, 58, 58, 59, 59, 59, 60, 60, 60, 61, 61, 61, 62, 62, 62, 63, 63, 63, 64, 64, 64, 65, 65, 65, 66, 66, 66, 67, 67, 67, 68, 68, 68, 69, 69, 69, 70, 70, 70, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 74, 75, 75, 75, 76, 76, 76, 77, 77, 77, 78, 78, 78, 79, 79, 79, 80, 80, 80, 81, 81, 81, 82, 82, 82, 83, 83, 83, 84, 84, 84, 85, 85, 85, 86, 86, 86, 87, 87, 87, 88, 88, 88, 89, 89, 89, 90, 90, 90, 91, 91, 91, 92, 92, 92, 93, 93, 93, 94, 94, 94, 95, 95, 95, 96, 96, 96, 97, 97, 97, 98, 98, 98, 99, 99, 99, 100, 100], ComplexF64[2.0+0.0im, 2.0+0.0im, 2.0-0.0im, 4.0+0.0im, 2.0+0.0im, 2.0-0.0im, 6.0+0.0im, 2.0+0.0im, 2.0-0.0im, 8.0+0.0im … 194.0+0.0im, 2.0+0.0im, 2.0-0.0im, 196.0+0.0im, 2.0+0.0im, 2.0-0.0im, 198.0+0.0im, 2.0+0.0im, 2.0-0.0im, 200.0+0.0im], 100, 100)

As initial condition, we choose a Gaussian lattice density profile with a width of $\sigma=5$:

ψx = gaussianstate(x_basis, div(size, 3), 0.0, 5)
ψl = Ket(H0.basis_r, ψx.data)

P0 = Operator(ψl.basis, ψl.data * ψl.data')
d = lattice_density(P0)
plot(d.values)

g = 2; Δ = 2
density_rec = []
@evolution show_progress=false {H := h(t, g, Δ), P0 --> H --> P} for t in 0:0.01:20
 d = lattice_density(P) .|> real
 push!(density_rec,d.values)
end
p1 = heatmap(hcat(density_rec...)', xlabel="lattice site", ylabel="time", title="Δ=±$(Δ)", colorbar=false)

density_rec = []
@evolution show_progress=false {H := h(t, g, Δ, switch= x-> 1), P0 --> H --> P} for t in 0:0.01:20
 d = lattice_density(P) .|> real
 push!(density_rec,d.values)
end
p2 = heatmap(hcat(density_rec...)', xlabel="lattice site", ylabel="time", title="Δ=$(Δ)", colorbar=false)

density_rec = []
@evolution show_progress=false {H := h(t, g, Δ, switch= x-> -1), P0 --> H --> P} for t in 0:0.01:20
 d = lattice_density(P) .|> real
 push!(density_rec,d.values)
end
p3 = heatmap(hcat(density_rec...)', xlabel="lattice site", ylabel="time", title="Δ=-$(Δ)");

l = @layout [a b c{0.42w}]
plot(p1, p2, p3, layout=l, size=(1200,400), plot_title="Normalised lattice density",
 margin=5mm, plot_titlevspan=0.1)
# savefig(plotsdir("lattice_density.png"))

plot(p2, xlim=(20, 50))

function evolution_operator(h, t)
 H = h(t)
 ev_op = Operator(basis(H), H.data)
 LM.evolution_operator!(ev_op, H, t)
 return ev_op
end
time_hamiltonian = t -> h(t, 2, 2)
ev_op = evolution_operator(time_hamiltonian, 2.1)

Operator(dim=100x100)
basis: LatticeBasis{SquareLattice{2}}(100, SquareLattice{2}((100, 1), LatticeModels.Bravais{2, 1}([1.0 0.0; 0.0 1.0], [0.0; 0.0;;]), Bool[1, 1, 1, 1, 1, 1, 1, 1, 1, 1 … 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]))sparse([1, 2, 3, 4, 5, 6, 1, 2, 3, 4 … 97, 98, 99, 100, 95, 96, 97, 98, 99, 100], [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 46, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 56, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 58, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 61, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 63, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 70, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 71, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 74, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 78, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 79, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 83, 83, 83, 83, 83, 83, 83, 83, 83, 83, 83, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 86, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 87, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 93, 93, 93, 93, 93, 93, 93, 93, 93, 93, 93, 94, 94, 94, 94, 94, 94, 94, 94, 94, 94, 94, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 97, 97, 97, 97, 97, 97, 97, 97, 97, 98, 98, 98, 98, 98, 98, 98, 98, 99, 99, 99, 99, 99, 99, 99, 100, 100, 100, 100, 100, 100], ComplexF64[0.9999327113815735 - 0.00820266501446412im, -0.00010093141833268 - 0.00820229703303101im, -3.364034731125e-5 + 5.5196054197e-7im, 1.88653881e-9 + 9.197841102e-8im, 1.8861004e-10 - 4.64309e-12im, 0.0 - 3.0954e-13im, -0.00010093141833268 - 0.00820229703303101im, 0.9997981396159296 - 0.01640441008695316im, -0.00016821022641636 - 0.00820110113353606im, -3.363449908477e-5 + 8.2789113194e-7im … -2.334602203862e-5 + 2.422660081674e-5im, -0.00592992240212792 - 0.00566766043323879im, 0.6879228134802496 - 0.7256911377053686im, -0.00597628217655804 - 0.00561888913526103im, -2.2117e-13 - 2.1053e-13im, 1.3091519e-10 - 1.3586203e-10im, 6.650629582e-8 + 6.356493028e-8im, -2.314663406714e-5 + 2.441743145251e-5im, -0.00597628217655804 - 0.00561888913526103im, 0.6819696779377588 - 0.731334444272082im], 100, 100)

Reproduced Papers

Sun, 01 Jan 2023 00:00:00 +0000

A growing archive of Jupyter notebooks that reproduce results from selected papers across quantum optics, mesoscopic physics and quantum control. Each .ipynb file is self-contained and downloadable directly from this page.

Notebooks

From harmonic oscillator to the parametron

Wed, 02 Nov 2022 00:00:00 +0000

In this notebook, we try to understand the parametron, also known as Kerr Parametric Oscillator (KPO). We will begin with a system we already understand very well and build up are knowledge by slowly encreasing the complexity of the system.

The sources used in the endeavour are:

T. Heugel, Parametrons: From Sensing to Optimization Machines, Doctoral thesis at ETH Zürich, 2022.
J. Košata, J. del Pino, T. Heugel, and O. Zilberberg, HarmonicBalance.Jl: A Julia Suite for Nonlinear Dynamics Using Harmonic Balance, SciPost Phys. Codebases 6 (2022).
O. Zilberberg and A. Eichler, Classical and Quantum Parametric Phenomena (2022).
J. Košata, Spatial and Temporal Mode Engineering in Nonlinear Media, Doctoral thesis at ETH Zürich, 2022.

To run this notebook the following julia environment was used:

using DrWatson
versioninfo()

Julia Version 1.9.0
Commit 8e63055292 (2023-05-07 11:25 UTC)
Platform Info:
OS: Windows (x86_64-w64-mingw32)
CPU: 12 × AMD Ryzen 5 5600X 6-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-14.0.6 (ORCJIT, znver3)
Threads: 10 on 12 virtual cores
Environment:
JULIA_NUM_THREADS = 10

And the following packages are needed:

using HarmonicBalance, OrdinaryDiffEq, ModelingToolkit
HB_settings = (:threading => true, :show_progress => false)

using Plots, LaTeXStrings, Latexify, Measures
import Base.@kwdef
default(fontfamily="computer modern", titlefont="computer modern", tickfont="computer modern",
 legend = :best, linewidth = 2, show = true, fmt = :svg)
plot_size = (750, 350); margin = 5mm;

using Pkg;
Pkg.status(
 ["HarmonicBalance", "OrdinaryDiffEq", "ModelingToolkit", "Plots",
 "Latexify", "DrWatson", "LaTeXStrings", "Measures"]
)

 DrWatson v2.12.5
 HarmonicBalance v0.6.4
 LaTeXStrings v1.3.0
 Latexify v0.15.21
 Measures v0.3.2
 ModelingToolkit v8.46.1
 OrdinaryDiffEq v6.51.2
 Plots v1.38.12

Harmonic Oscillator

We begin with the simple harmonic oscillator and slowly build up to the parametron. We make use of the differential equation solver of Julia to analyse the system.

@variables t x(t) # symbolic variables
@parameters ω ω₀ F γ
Dₜ = Differential(t)
@named sys = ODESystem([Dₜ(Dₜ(x)) ~ - ω₀^2*x - γ*Dₜ(x) + F*cos(ω*t)], t, [x], [ω₀, ω, F, γ])
sys = ode_order_lowering(sys)

$$ \begin{aligned} \frac{\mathrm{d} xˍt\left( t \right)}{\mathrm{d}t} =& F \cos\left( t \omega \right) - \gamma xˍt\left( t \right) - \omega_0^{2} x\left( t \right) \\ \frac{\mathrm{d} x\left( t \right)}{\mathrm{d}t} =& xˍt\left( t \right) \end{aligned} $$

Simple Harmonic Oscillator

Let us first plot the resonator without any damping ($\gamma=0$) or external drive ($F=0$):

x0 = [x => 1.0, Dₜ(x) => 0]; tspan = (0.0, 200.0)
fixed = [ω₀ => 1.0, γ => 0.0, F => 0.0, ω => 0.0]

4-element Vector{Pair{Num, Float64}}:
ω₀ => 1.0
γ => 0.0
F => 0.0
ω => 0.0

prob = ODEProblem(sys, x0, tspan, fixed)
tr = solve(prob, Tsit5(), saveat=0.1);

plot1 = plot(tr.t , tr[1,:], xlabel="t", ylabel="x(t)", legend=false, rightmargin=0mm)
plot2 = plot(tr, idxs=(1,2), xlabel="x(t)", ylabel="v(t)", legend=false, framestyle = :origin,
 aspect_ratio=1, leftmargin=0mm)
plot(plot1, plot2, size = plot_size, margin = margin, legend=false)

In general, it is useful the look at the system in phase space. Note, the path is periodic.

Damping

One can add damping to make it make realistic ($\gamma > 0$). In general, we have three regimes:

$$ \begin{cases} \omega_0 > \gamma/2 \quad \text{(underdamped)} \\ \omega_0 = \gamma/2 \quad \text{(citically damped)} \\ \omega_0 < \gamma/2 \quad \text{(overdamped)}\\ \end{cases} \qquad \underbrace{\quad\Longrightarrow\quad}_{\gamma = \omega_0/Q} \qquad \begin{cases} Q > 1/2 \quad \text{(underdamped)} \\ Q = 1/2 \quad \text{(citically damped)} \\ Q < 1/2 \quad \text{(overdamped)}\\ \end{cases} $$

$Q$ is called the quality factor which characterizes the damping.

u0 = [x => 1.0, Dₜ(x) => 0]; tspan = (0.0, 200.0);
p1 = [ω₀ => 1.0, γ => 0.1, F => 0.0, ω => 1.0]
p2 = [ω₀ => 1.0, γ => 2.0, F => 0.0, ω => 1.0]
p3 = [ω₀ => 1.0, γ => 5.0, F => 0.0, ω => 1.0]
prob = [ODEProblem(sys, u0, tspan, p) for p in [p1, p2, p3]]
tr = [solve(p, Tsit5(), saveat=0.1) for p in prob];

plot1 = plot(;xlabel="t", ylabel="x(t)", legend=false)
plot2 = plot(;xlabel="x(t)", ylabel="v(t)", framestyle = :origin, aspect_ratio=1)
for (sol, lab) in zip(tr, ["underdamped", "criticaly damped", "overdamped"])
 plot!(plot1, sol.t , sol[1,:])
 plot!(plot2, sol, idxs=(1,2), lims = :symmetric, label = lab)
end
plot(plot1, plot2, size = plot_size, margin = margin)

We observe a ringdown process, i.e., exponential weakening of the oscillation amplitude.

Damped driven harmonic oscillator

Let us make the harmonic oscillator complete by adding external driving ($F>0$) with frequency $\omega$.

u0 = [x => 1.0, Dₜ(x) => 0]; tspan = (0.0, 100);
p1 = [ω₀ => 1.0, γ => 0.1, F => 0.01, ω => 1.0]
p2 = [ω₀ => 1.0, γ => 0.1, F => 0.1, ω => 1.0]
p3 = [ω₀ => 1.0, γ => 0.1, F => 0.2, ω => 1.0]
prob = [ODEProblem(sys, u0, tspan, p) for p in [p1, p2, p3]]
tr = [solve(p, Tsit5(), saveat=0.1) for p in prob];

plot1 = plot(;xlabel="t", ylabel="x(t)", legend=false)
plot2 = plot(;xlabel="x(t)", ylabel="v(t)", framestyle = :origin, aspect_ratio=:equal)
for (sol, lab) in zip(tr, ["F = 0.01", "F = 0.1", "F = 0.2"])
 plot!(plot1, sol.t , sol[1,:])
 plot!(plot2, sol, idxs=(1,2), lims = :symmetric, label = lab)
end
plot(plot1, plot2, size = plot_size, margin = margin)

The amplitude of the steady state oscillation depends on $F$, as expected.

u0 = [x => 1.0, Dₜ(x) => 0]; tspan = (0.0, 2000);
p1 = [ω₀ => 1.0, γ => 0.2, F => 0.08, ω => 1.0]
p2 = [ω₀ => 1.0, γ => 0.2, F => 0.08, ω => 0.7]
p3 = [ω₀ => 1.0, γ => 0.2, F => 0.08, ω => 0.5]
prob = [ODEProblem(sys, u0, tspan, p) for p in [p1, p2, p3]]
tr = [solve(p, Tsit5(), saveat=0.1) for p in prob];

plot1 = plot(;xlabel="t", ylabel="x(t)", legend=false)
plot2 = plot(;xlabel="x(t)", ylabel="v(t)", framestyle = :origin, aspect_ratio=1)
for (sol, lab) in zip(tr, ["ω = 1.0", "ω = 0.7", "ω = 0.2"])
 plot!(plot1, sol.t , sol[1,:])
 plot!(plot2, sol, idxs=(1,2), lims = :symmetric, label = lab)
end

plot(plot1, plot2, size = plot_size, margin = margin)

But also on the detuning, i.e., $|\omega_0-\omega|$, of the natural frequency $\omega_0$ and driving frequency $\omega$.

Frequency sweep and harmonic reponse

The harmonic oscillator has only one frequency response, in particular that of the drive $\omega_d$. Indeed:

$$ \ddot{x}(t)+\gamma\dot{x}(t)+\omega_0^2x=F\cos(\omega_d t + \theta) \quad \underbrace{\quad\Longrightarrow\quad}_{\text{Fourier w.r.t. $\omega$}} \quad \tilde{x}(\omega) = \frac{F}{\omega_0^2-\omega^2+i\omega\gamma}\left[\delta(\omega+\omega_d)+\delta(\omega-\omega_d)\right] $$

Let us look at the steady state. For this we use Harmonic Balance method. Clearly, in the long-time limit, our system will behave as a harmonic, i.e., we use the ansatz $x(t)=A\cos(\omega t + \phi)$. Hence, we can write:

$$ x(t) = A \cos(\phi) \cos(\omega t) - A \sin(\phi) \sin(\omega t) = u(A, \phi)\cos(\omega t) + v(A, \phi) \sin(\omega t) $$

where we used some the sum-to-product and defined $u$ and $v$. From, know one we examine our system in terms of the quadratures, i.e., in the rotating frame.

Clearly, for the whole time evolution the amplitude $A$ and phase $\phi$ will not be constants. Hence, in general, we say that the quadratures depend on the slow time $T$, where the $\cos$ and $\sin$ are dependent on fast time $t$.

@parameters ω ω₀ F γ λ θ α ψ η
natural_equation = d(d(x, t), t) + γ*d(x, t) + ω₀^2*(1-λ*cos(2*ω*t + ψ))*x + α*x^3 + η*d(x, t)*x^2
forces = F*cos(ω*t + θ)
diff_eq = DifferentialEquation(natural_equation ~ forces, x)

add_harmonic!(diff_eq, x, ω);
harmonic = get_harmonic_equations(diff_eq);

x0 = [1.0, 0.0]; tspan = (0.0, 2000.0)
fixed = (ω => 1.1, ω₀ => 1.0, γ => 0.01, F => 0.01, α => 0.0, λ => 0.0, η => 0.0, θ => 0.0, ψ => 0.0)

ode_problem = ODEProblem(harmonic, fixed, x0 = x0, timespan = tspan)
tr = solve(ode_problem, Tsit5(), saveat=0.1);

p1 = plot(tr, "u1", harmonic, size = plot_size)
plot!(tr, "v1", harmonic)
p2 = plot(tr, ["u1", "v1"], harmonic, framestyle=:origin, aspect_ratio=1, lims=:symmetric)
plot!([x0[1]],[x0[2]], markers=:true, line=false)
plot!([tr.u[end][1]],[tr.u[end][2]], markers=:true, line=false, size = plot_size, margin = margin)
plot(p1, p2)

Notice that one of the quadratures, $u_1$, is non-zero for large $t$, which means the system oscillates with frequency $\omega_0$.

In experiment, the system is often analysed with frequency sweeps, i.e., we look at the response of the oscillators when changing the external driving frequency adiabatically.

varied = ω => range(0.9, 1.1, 500);

listγ = pairs([0.01, 0.02, 0.03]);
results = Vector(undef, length(listγ))
plotγ = plot()
for (i,gamma) in listγ
 fixed = (ω₀ => 1.0, γ => gamma, F => 0.01, α => 0.0, λ => 0.0, η => 0.0, θ => 0.0, ψ => 0.0)
 results[i] = get_steady_states(harmonic, varied, fixed)
 plot!(results[i], x="ω/ω₀", y="sqrt(u1^2 + v1^2)", color = i == 1 ? :black : i, label="γ = $gamma")
end

listF = pairs([0.01, 0.02, 0.03])
results = Vector(undef, length(listF))
plotF = plot()
for (i,force) in listγ
 fixed = (ω₀ => 1.0, γ => 0.01, F => force, α => 0.0, λ => 0.0, η => 0.0, θ => 0.0, ψ => 0.0)
 results[i] = get_steady_states(harmonic, varied, fixed)
 plot!(results[i], x="ω/ω₀", y="sqrt(u1^2 + v1^2)", color = i == 1 ? :black : i, label="F = $force")
end
plot!(plotF, plotγ;legend = :best, size = plot_size, margin = margin)

The resonant peak has a Lorentzian shape, where the height is affected by $\gamma$ and $F$ and the peak maximum is at $\omega_d=\omega_0 \sqrt{1-\gamma^2 / 2 \omega_0^2}$, i.e., the peak shifts depending on $\gamma$.

Duffing Oscillator

@variables t x(t) # symbolic variables
@parameters ω ω₀ F γ α
Dₜ = Differential(t)
eq = Dₜ(Dₜ(x)) ~ - ω₀^2*x - γ*Dₜ(x) -α*x^3 + F*cos(ω*t)
@named sys = ODESystem([eq], t, [x], [ω₀, ω, F, γ, α])
sys = ode_order_lowering(sys)

$$ \begin{align} \frac{\mathrm{d} xˍt\left( t \right)}{\mathrm{d}t} =& F \cos\left( t \omega \right) - \left( x\left( t \right) \right)^{3} \alpha - \gamma xˍt\left( t \right) - \omega_0^{2} x\left( t \right) \\ \frac{\mathrm{d} x\left( t \right)}{\mathrm{d}t} =& xˍt\left( t \right) \end{align} $$

Harmonics and bifurcations

The equation of motion for the displacement $x(t)$ reads

\begin{align*} \underbrace{\ddot{x}(t)+\gamma \dot{x}(t)+\omega_0^2 x(t)}{\text {damped harmonic oscillator }}+\underbrace{\alpha x(t)^3}{\text {Duffing coefficient }}=\underbrace{F \cos (\omega t)}_{\text {periodic drive }} \end{align*}

For $\alpha=0$, the system becomes linear and responds precisely at the drive frequency, so that $x(t)=X \cos (\omega t+\phi)$, where $X$ and $\phi$ can be found analytically. For $\alpha \neq 0$ this is no longer possible. We can obtain some intuition by treating $\alpha$ perturbatively, i.e., by solving \begin{align*} \ddot{x}(t)+\gamma \dot{x}(t)+\omega_0^{2 x(t)+\epsilon \alpha x(t)^{3=F \cos (\omega t)
\end{align*}
for small $\epsilon$. To zeroth order, the response of the system is $x_0(t)=X_0 \cos \left(\omega t+\phi_0\right)$. Expanding $x(t)=x_0(t)+\epsilon x_1(t)$, we find that the perturbation $x_1(t)$ satisfies to first order
\begin{align*}
\ddot{x}_1(t)+\gamma \dot{x}_1(t)\left[\omega_0}2+\frac{3 \alpha X_0}2}{4}\right] x_1(t)=-\frac{\alpha X_0^3}{4} \cos \left(3 \omega t+3 \phi_0\right), \end{align*} which gives a response of the form $x_1(t)=X_1 \cos \left(3 \omega t+\phi_1\right)$. Clearly, the oscillator now responds not only at frequency $\omega$, but also at $3 \omega$ ! This effect is known as high harmonic generation or more generally frequency conversion. By continuing the procedure to higher orders, we eventually obtain an infinity of harmonics present in the response. In general, there is no analytical solution to such problems.

x0 = [1.0, 0.0]; tspan = (0.0, 1500.0)
fixed = (ω => 1.1, ω₀ => 1.0, γ => 0.01, F => 0.01, α => 0.0, λ => 0.0, η => 0.0, θ => 0.0, ψ => 0.0)

ode_problem = ODEProblem(harmonic, fixed, x0 = x0, timespan = tspan)
tr = solve(ode_problem, Tsit5(), saveat=0.1);

p1 = plot(tr, "u1", harmonic)
plot!(tr, "v1", harmonic)
p2 = plot(tr, ["u1","v1"], harmonic, legend=false, framestyle=:origin, aspect_ratio=1, lims=:symmetric)
plot!([x0[1]],[x0[2]], markers=:true, line=false)
plot!([tr.u[end][1]],[tr.u[end][2]], markers=:true, line=false)
plot(p1, p2, size = plot_size, margin = margin)

Although, at first sight looking at the response nothing changed much but when again performing frequency sweeps it becomes interesting.

varied = ω => range(0.75, 1.25, 500)

listβ = pairs([1.0, 0.5, 0.0, -0.5, -1.0])
results = Vector(undef, length(listβ))
plot1 = plot(;legend = false)
for (i,beta) in listβ
 fixed = Dict(ω₀ => 1.0, γ => 0.01, F => 0.01, α => beta, λ => 1e-8, η => 1e-8, θ => 1e-8, ψ => 1e-8)
 results[i] = get_steady_states(harmonic, varied, fixed; HB_settings...)
 plot!(results[i], x="ω", y="sqrt(u1^2 + v1^2)", color = beta == 0.0 ? :black : i+1, legend=false)
end

displ = -1:0.01:1
plot2 = plot(legend= :outerright, ylabel = "Potential energy", xlabel = "x")
for (i,beta) in listβ
 pot = fixed[ω₀] .* displ.^2 + beta .* displ.^4
 plot!(displ, pot, color = beta == 0.0 ? :black : i+1, label = "α = $beta")
end

l = @layout [a{0.45w} b]
plot(plot1, plot2, layout = l, legendfontsize=10, size = plot_size, margin = margin)

Depending on $\alpha$ the resonance frequency of the oscillator shifts.
The larger the discrepancy of the driving frequency compared to the natural frequency (detuning), the larger the amplitude. Until it drops for really large detuning.
The behaviour w.r.t. is asymmetric around the origin. For negative $\alpha$, we observe instability for high displacements.
We have a bifurcation when changing $\alpha$ or $\omega$, where we go from one state (stable) to three state (two stable, one unstable). Where the initial condition decides in which branch we will end up.

Instability for $x_0\approx \omega_0^2/\alpha_3$

Let us go to the complete time response picture again.

x0 = [x => 1.0, Dₜ(x) => 0.0]; tspan = (0.0, 1000.0)
p = [ω₀ => 1.0, γ => 0.01, F => 0.01, ω => 1.0, α => -0.99]
prob = ODEProblem(sys, x0, tspan, p)
tr = solve(prob, Tsit5(), saveat=0.1);

plot1 = plot(tr.t, tr[1,:], xlabel="t", ylabel="x(t)", legend=false)
plot2 = plot(tr, idxs=(1,2), xlab="x(t)", ylab="v(t)", legend=false, frame=:origin, lims=:symmetric)
plot(plot1, plot2, size = plot_size, margin = margin)

#| fig-align: center
x0 = [x => 1.0, Dₜ(x) => 0.0]; tspan = (0.0, 2.0)
p = [ω₀ => 1.0, γ => 0.01, F => 0.01, ω => 1.0, α => -1.0]
prob = ODEProblem(sys, x0, tspan, p)
tr = solve(prob, Tsit5(), saveat=0.1);

plot1 = plot(tr.t , tr[1,:], xlabel="t", ylabel="x(t)", legend=false)

displ = -1:0.01:1
plot2 = plot(legend= :outerright, ylabel = "Potential energy", xlabel = "x")
for (i,beta) in listβ
 pot = fixed[ω₀] .* displ.^2 + beta .* displ.^4
 plot!(displ, pot, color = beta == 0.0 ? :black : i+1, label = "α = $beta")
end
plot(plot1, plot2, layout = l, legendfontsize=10, size = plot_size, margin = margin)

The system becomes unstable for $x_0\approx \omega_0^2/\alpha_3$. In general, we need that $x_0\ll \omega_0^2/\alpha_3$

Hysteresis

varied = ω => range(0.9, 1.4, 150)
fixed = Dict(ω₀ => 1.0, γ => 0.01, F => 0.01, α => 1.0, λ => 0.0, η => 0.0, θ => 0.0, ψ => 0.0)
result = get_steady_states(harmonic, varied, fixed; random_warmup=true, HB_settings...)
plot(result, "sqrt(u1^2+v1^2)")

# linearly interpolate between two values at two times
right_sweep = ParameterSweep(ω => (0.9, 1.4), (0.0, 2e4))
ode_problem_right = ODEProblem(harmonic, fixed, sweep=right_sweep, x0=[0.01;0.0], timespan=(0.0,2e4))

left_sweep = ParameterSweep(ω => (1.4, 0.9), (0.0, 2e4))
ode_problem_left = ODEProblem(harmonic, fixed, sweep=left_sweep, x0=[0.01;0.0], timespan=(0.0,2e4))

sol_right = solve(ode_problem_right, Tsit5(), saveat = 100);
amplitude_right = sqrt.(sol_right[1,:].^2 + sol_right[2,:].^2)
sol_left = solve(ode_problem_left, Tsit5(), saveat = 100);
amplitude_left = sqrt.(sol_left[1,:].^2 + sol_left[2,:].^2)

plot!(right_sweep[ω].(sol_right.t), amplitude_right, style=:dash, label="left to right sweep", linewidth=1)
plot!(left_sweep[ω].(sol_left.t), amplitude_left, style=:dash, label="right to left sweep", linewidth=1)

We observe hysteresis, i.e., the system takes different paths from high to low driving frequncy or visa versa.

Paramatric modulation

Let us for a second assume nor driving or damping. We do introduce the periodic modulation of the natural frequency $\omega_0$ such that are e.o.m. becomes:

$$\ddot{x}(t)+\omega_0^2(1-\lambda\cos(2\omega t + \psi))x =0.$$

@variables t x(t) # symbolic variables
@parameters λ, ψ
Dₜ = Differential(t)
eq = Dₜ(Dₜ(x)) ~ - ω₀^2*(1-λ*cos(2*ω*t + ψ))*x - γ*Dₜ(x) + F*cos(ω*t)
@named sys = ODESystem([eq], t, [x], [ω₀, ω, F, γ, α, λ, ψ])
sys = ode_order_lowering(sys)

$$ \begin{align} \frac{\mathrm{d} xˍt\left( t \right)}{\mathrm{d}t} =& F \cos\left( t \omega \right) - \gamma xˍt\left( t \right) - \omega_0^{2} \left( 1 - \lambda \cos\left( \psi + 2 t \omega \right) \right) x\left( t \right) \\ \frac{\mathrm{d} x\left( t \right)}{\mathrm{d}t} =& xˍt\left( t \right) \end{align} $$

In the potential picture this effect changes the steepness of the harmonic potential periodically.

Instability of the linear parametrically driven harmonic oscillator

x0 = [x => 1.0, Dₜ(x) => 0.0]; tspan = (0.0, 1000.0)
fixed = [ω₀ => 1.0, γ => 0.01, F => 0.0, ω => 1.0, α => 0.0, λ => 0.1, ψ => 0.0]
prob = ODEProblem(sys, x0, tspan, fixed)
tr = solve(prob, Tsit5(), saveat=0.1);

plot1 = plot(tr.t , tr[1,:], xlabel="t", ylabel="x(t)", legend=false)
plot2 = plot(tr, idxs=(1,2), xlabel="x(t)", ylabel="v(t)", legend=false, framestyle = :origin, lims = :symmetric)
plot(plot1, plot2, size = plot_size, margin = margin)

For different detuning’s $|\omega-\omega_0|$, we observe that the system becomes unstable or stable, where in the stable regions the steady states has always a zero amplitude. This corresponds with our intuition of the potential picture. Indeed, think about it.

Notice how there is a resonance not at one value for $\omega_p$, but for an infinite values of $\omega_p$, i.e. in multiples of $\omega$. This is in stark contract to normal external driving.

We cal the instability lobes Arnold tongues:

They appear at $2\omega_0/\omega_p=n$ with $n\in \mathbb{N}$
They broaden for increasing $\lambda$
Linear dissipation $\gamma$ can only stabilize the system for small $\lambda$

The parametric instability threshold $\lambda_{\text {th }}=2 \sqrt{(\Omega \bar{\gamma})^2+\left(1-\Omega^2\right)^2}$ with $\Omega = \omega/\omega_0$, is where the system becomes unstable in the first instability lobe.

Parametric damping and amplification

Again thinking in the potential picture, the phase $\psi$ of the parametric drive becomes quiet important.

x0 = [x => 1.0, Dₜ(x) => 0.0]; tspan = (0.0, 10.0)
Q = 100
omega0 = 100
λₜₕ = 2/Q
fixed = [ω₀ => omega0, γ => ω₀/Q, F => 0.0, ω => omega0, α => 0.0, λ => 0.9*λₜₕ, ψ => -π/2]
prob = ODEProblem(sys, x0, tspan, fixed)
tr = solve(prob, Tsit5(), saveat=0.001);

plot1 = plot(tr.t , tr[1,:], xlabel="t", ylabel="x(t)", legend=false)
plot2 = plot(tr, idxs=(1,2), xlabel="x(t)", ylabel="v(t)", legend=false, framestyle = :origin, lims = :symmetric)
plot(plot1, plot2, size = plot_size, margin = margin)

x0 = [1.0, 0.0]; tspan = (0.0, 10.0); Q = 100; ω0 = 100; λₜₕ = 2/Q

p1 = plot()
for (lambda, psi) = [(0.0*λₜₕ, 0.0), (0.9*λₜₕ, -π/2), (0.9*λₜₕ, π/2)]
 fixed = (ω => ω0, ω₀ => ω0, γ => ω0/Q, F => 0.00, α => 0.0, λ => lambda, η => 0.0, θ => 0.0, ψ => psi)
 ode_problem = ODEProblem(harmonic, fixed, x0 = x0, timespan = tspan)
 tr = solve(ode_problem, Tsit5(), saveat=0.1);
 plot!(tr, "u1", harmonic)
end


varied = ω => range(95, 105, 500)
p2 = plot(;legend = false)
for (i, (lambda, psi)) = pairs([(0.0*λₜₕ, 0.0), (0.9*λₜₕ, -π/2), (0.9*λₜₕ, π/2)])
 fixed = (ω₀ => omega0, γ => omega0/Q, F => 80.00, α => 0.0, λ => lambda, η => 0.0, θ => 0.0, ψ => psi)
 result = get_steady_states(harmonic, varied, fixed)
 label = "λ = $(round(lambda; digits=2)), ψ = $(round(psi; digits=2))"
 plot!(result, x="ω", y="sqrt(u1^2 + v1^2)", color=i, label=label, legend=:best)
end

l = @layout [a{0.45w} b]
plot(p1, p2, layout = l, legendfontsize = 10, size = plot_size, margin = margin)

Indeed, we can cool or amplify the resonant peak observed in the stable non-zero amplitude steady states created with only linear damping (small $\lambda$).

Stable Parametron without driving

We need the nonlinear terms to stabilize the system. This can be done with the duffing $\alpha$ term, studied before, and the non-linear damping $\eta$:

$$ \ddot{x}(t)+\gamma\dot{x}(t)+\omega_0^2(1-\lambda\cos(2\omega t + \psi))x + \alpha x^3 +\eta x^2 \dot{x}=0 $$

Note that, neither linear ($\gamma$) nor nonlinear damping ($\eta$) is needed to stabilize the system. However, we will include it for completeness. Actually, one might also say, that the instability (i.e. the state of infinitely large amplitude) is just shifted to infinitely large detuning (of course other things of approximations probably break down earlier).

varied1 = ω => range(0.98, 1.02, 300)
fixed1 = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0, α => 1.0, λ => 0.03, η => 0.0, θ => 0.0, ψ => 0.0)
result1 = get_steady_states(harmonic, varied1, fixed1; kwargs...)
plot1 = plot(result1, "sqrt(u1^2+v1^2)", legend=:best)

fixed2 = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0, α => 1.0, η => 0.0, θ => 0.0, ψ => 0.0)
varied2 = (ω => range(0.98, 1.02, 50), λ => range(1e-6, 0.06, 50))
result_2D = get_steady_states(harmonic, varied2, fixed2; random_warmup=false, HB_settings...);
plot2 = plot_phase_diagram(result_2D, class="stable")
hline!([fixed1[λ]], style = :dash, color = :red, legend = false)

plot(plot1, plot2, size = plot_size, margin = margin)

The presence of a nonlinear force $β_3$ leads to an effective detuning of the resonance frequency as a function of the amplitude. This detuning causes a decrease of the pumping efficiency and a saturation of the growth. We observe that for small detuning the zero amplitude stable state is unstable, hence, not observed in nature. Nevertheless, in the region, there are two non-zero stable steady states degenerate in amplitude. These states we call the phase states. When increasing the driving frequency $\omega$ the phase states keep existing, however, the zero amplitude state becomes stable again.

The region where the zero amplitude steady states is unstable is bounded by

$$\omega_{\pm}^2=\omega_0^2\left(1 \pm \sqrt{\frac{\lambda^2}{4}-\frac{1}{Q^2}}\right).$$

Note, that $\alpha$ and $\eta$ have no influence on the boundaries as x is small at A ($\omega_-$) and B ($\omega_+$).

varied = ω => range(0.98, 1.02, 300)
fixed = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0, α => 1.0, λ => 0.03, η => 0.0, θ => 0.0, ψ => 0.0)
result = get_steady_states(harmonic, varied, fixed; random_warmup=true, HB_settings...)
plot_sweep = plot(result, "sqrt(u1^2+v1^2)")

x0_1 = [0.3, 0.2]; x0_2 = [-0.3, -0.2]; tspan = (0.0, 2000.0)
fixed_tr = Dict(ω => 1.0, ω₀ => 1.0, γ => 0.01, F => 0.00, α => 1.0, λ => 0.03, η => 0.0, θ => 0.0, ψ => 0.0)

ode_problem1 = ODEProblem(harmonic, fixed_tr, x0 = x0_1, timespan = tspan)
ode_problem2 = ODEProblem(harmonic, fixed_tr, x0 = x0_2, timespan = tspan)
tr1 = solve(ode_problem1, Tsit5(), saveat=0.1);
tr2 = solve(ode_problem2, Tsit5(), saveat=0.1);

plot_tr = plot(;framestyle = :origin, aspect_ratio=1, legend=false, lims = :symmetric)
plot!(tr1, ["u1", "v1"], harmonic, color = 1)
plot!([x0_1[1]],[x0_1[2]], markers=:true, line=false, color=:green)
plot!([tr1.u[end][1]],[tr1.u[end][2]], markers=:true, line=false, color=:orange)
plot!(tr2, ["u1", "v1"], harmonic, color = 1)
plot!([x0_2[1]],[x0_2[2]], markers=:true, line=false, color=:green)
plot!([tr2.u[end][1]],[tr2.u[end][2]], markers=:true, line=false, color=:orange)

vline!(plot_sweep, [fixed_tr[ω]], style = :dash, color = :red, legend = false)
plot(plot_sweep, plot_tr, size = plot_size, margin = margin)

Ending up in one of the two phase states purely depends on the initial condition (I.C.).

Note that the period of the response is twice as large as the drive. This is called periodicity doubling (time translation symmetry breaking).

varied1 = ω => range(0.98, 1.02, 300)
fixed1 = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0, α => 1.0, λ => 0.03, η => 0.5, θ => 0.0, ψ => 0.0)
result1 = get_steady_states(harmonic, varied1, fixed1; random_warmup=true, HB_settings...)
plot1 = plot(result1, "sqrt(u1^2+v1^2)")

fixed2 = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0, α => 1.0, η => 0.5, θ => 0.0, ψ => 0.0)
varied2 = (ω => range(0.98, 1.02, 50), λ => range(1e-6, 0.06, 50))
result_2D = get_steady_states(harmonic, varied2, fixed2; random_warmup=true, HB_settings...);
plot2 = plot_phase_diagram(result_2D, class="stable")
hline!([fixed1[λ]], style = :dash, color = :red, legend = false)

plot(plot1, plot2, size = plot_size, margin = margin)

When adding nonlinear damping $\eta$ the phase states disappear for a large detuning.

varied = ω => range(0.98, 1.02, 300)
fixed = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0, α => 1.0, λ => 0.03, η => 0.0, θ => 0.0, ψ => 0.0)
result1 = get_steady_states(harmonic, varied, fixed; random_warmup=true, HB_settings...)

varied = ω => range(0.98, 1.02, 300)
fixed = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0, α => 1.0, λ => 0.03, η => 0.5, θ => 0.0, ψ => 0.0)
result2 = get_steady_states(harmonic, varied, fixed; random_warmup=true, HB_settings...)

plot(result1, "sqrt(u1^2+v1^2)")
plot!(result2, "sqrt(u1^2+v1^2)")

Driven Symmetry breaking in a Paramatron

$$\ddot{x}(t)+\gamma\dot{x}(t)+\omega_0^2(1-\lambda\cos(2\omega t + \psi))x + \alpha x^3 +\eta x^2 \dot{x}=F\cos(\omega t + \theta)$$

fixed = (ω₀ => 1.0, γ => 0.01, λ => 0.05, F => 0.001, α => 1.0, η => 0.3, θ => 0, ψ => 0)
varied = ω => range(0.92, 1.08, 500)
result = get_steady_states(harmonic, varied, fixed)
p = plot(result, x="ω", y="sqrt(u1^2 + v1^2)")

timespan = (0.0, 2e4)

 # linearly interpolate between two values at two times
right_sweep = ParameterSweep(ω => (0.92, 1.1), timespan)
ode_problem_right = ODEProblem(harmonic, fixed, sweep=right_sweep, x0=[0.01;0.0], timespan=timespan)

left_sweep = ParameterSweep(ω => (1.1, 0.92), timespan)
ode_problem_left = ODEProblem(harmonic, fixed, sweep=left_sweep, x0=[0.01;0.0], timespan=timespan)

sol_right = solve(ode_problem_right, Tsit5(), saveat = 100);
amplitude_right = sqrt.(sol_right[1,:].^2 + sol_right[2,:].^2)
sol_left = solve(ode_problem_left, Tsit5(), saveat = 100);
amplitude_left = sqrt.(sol_left[1,:].^2 + sol_left[2,:].^2)

plot!(right_sweep[ω].(sol_right.t), amplitude_right, style=:dash, label="left to right sweep", linewidth=1)
plot!(left_sweep[ω].(sol_left.t), amplitude_left, style=:dash, label="right to left sweep", linewidth=1)

$\textcolor{red}{\text{Help:}}$ Why is does the sweep from right to left hop to the highest amplitude branch? In Oded’s book it is stated that it should hop to the lowest amplitude branch.

When we drive the parametron with an extra external drive $F$ we can break the amplitude degeneracy of the phase states. This is called parametric symmetry breaking.

Spaghetti plots

varied_sweep = ω => range(0.98, 1.02, 600)
fixed_sweep1 = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0, α => 1.0, λ => 0.03, η => 0.5, θ => 0.0, ψ => 0.0)
fixed_sweep2 = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0001, α => 1.0, λ => 0.01, η => 0.5, θ => 0.0, ψ => 0.0)
fixed_sweep3 = Dict(ω₀ => 1.0, γ => 0.01, F => 0.0001, α => 1.0, λ => 0.03, η => 0.5, θ => 0.0, ψ => 0.0)

fixed_sweep = [fixed_sweep1, fixed_sweep2, fixed_sweep3]
result_sweep = broadcast(
 p -> get_steady_states(harmonic, varied_sweep, p; HB_settings...),
 fixed_sweep
 )
plots = broadcast(
 res -> plot(plot(res, x="ω", y="sqrt(u1^2 + v1^2)"), plot_spaghetti(res, x="u1", y="v1", z="ω"),
 title = savename(res.fixed_parameters, connector = " ", accesses = (F, λ)), legend=:best),
 result_sweep
 )
plot(plots..., size = (720, 1200), layout = (3,1))