Driven-dissipative toroidal Landau levels in cavity arrays

Introduction to the Julia language

Motivation

  • ultracold gases: PRL 111, 185301 (2013), PRL 111, 185302 (2013)
  • classical circuits: arXiv:1309.0878 (2013)
  • classical pendula (two copies of HH model): arXiv:1503.06808
  • solid-state superlattices: Nature 497, 598 (2013), Nat Phys 10, 525 (2014)
  • silicon photonics: Nature Photonics 7, 1001 (2013)

Momentum space magnetism

  • model Hamiltonian

\(H=\mathcal{H}_0+\frac{1}{2}\kappa \sum_{m,n}\left[(m-m_0)^{2}+(n-n_0)^{2}\right]\hat{a}_{m,n}^{\dagger}\hat{a}_{m,n}\)

  • dual in momentum space

\(\widetilde{H} = E_n(\mathbf{p}) + \frac{\kappa}{2} [i\nabla_{\mathbf{p}} + \mathcal{A}_n(\mathbf{p})]^2\)

Harper-Hofstadter model

\(\mathcal{H}_0=-J\sum_{m,n}(e^{i \phi_{m,n}^x}\hat{a}_{m+1,n}^{\dagger}\hat{a}_{m,n}+e^{i \phi_{m,n}^y}\hat{a}_{m,n+1}^{\dagger}\hat{a}_{m,n}) + \text{h.c.}\)

lattice.png

Harper-Hofstadter model

butterfly.jpg

Harper-Hofstadter model

bands.png

\(BW = \max_{\mathbf{p}}(E_1(\mathbf{p})) - \min_{\mathbf{p}}(E_1(\mathbf{p}))\)

\(\Delta E = \langle E_2(\mathbf{p})\rangle_{\mathbf{p}} - \langle E_1(\mathbf{p})\rangle_{\mathbf{p}}\)

Non Abelian correction

\(\epsilon_{n,\beta} = E_n + \left(\beta + \frac{1}{2}\right) \kappa |\Omega_n| + \delta E\)

\(\delta E(\mathbf{p}) = \frac{\kappa}{2} \sum_{n^{\prime} \neq n} |\mathcal{A}_{n,n^{\prime}}(\mathbf{p})|^2\)

nonabcorr.png

\(\eta_{\text{zpe}} = \frac{4\pi\alpha}{\kappa} (E_{\text{ex}} - \epsilon_{1,0})\)

Driving and dissipation

\(i\partial_{t}a_{i,j}(t)+i\gamma a_{i,j}(t)-f_{i,j}e^{-i\omega_{0}t}=\left[a_{i,j}(t),H\right]\)

Selection rules: \(\delta\)–pump

delta.png

Selection rules: Gaussian pump

gaussian.png

Intermezzo: Hitting the edge

edge.png

Homogeneous & random pumping

homogeneous.png

Experimental proposal

cyclotron.png

Experimental proposal

\(\chi_\beta (\mathbf{p}) = \mathcal{N} \sum_j e^{- i p_y j a} e^{ - ( p_x + j a l_{\Omega_n}^2 )^2 / 2 l_{\Omega_n}^2} H_\beta ( p_x / l_{\Omega_n} + j a l_{\Omega_n})\)

analytics.png

Julia

First contact

  • 2 language problem
  • vectorize everything, or else!
function mandel(z)
    c = z
    maxiter = 80
    for n = 1:maxiter
        if abs(z) > 2
            return n-1
        end
        z = z^2 + c
    end
    return maxiter
end
  • C-level performance

Multiple dispatch

  • traditional OO paradigm – dispatch based on one argument only
obj.foo(arg1, arg2, ..)
  • Julia – multiple dispatch
foo(obj, arg1, arg2, ..)

julia> methods(+)
# 139 methods for generic function "+":
+(x::Bool) at bool.jl:33
+(x::Bool,y::Bool) at bool.jl:36
+(y::FloatingPoint,x::Bool) at bool.jl:46
+(x::Int64,y::Int64) at int.jl:14
+(x::Int8,y::Int8) at int.jl:14
+(x::UInt8,y::UInt8) at int.jl:14
+(x::Int16,y::Int16) at int.jl:14
+(x::UInt16,y::UInt16) at int.jl:14
+(x::Int32,y::Int32) at int.jl:14
+(x::UInt32,y::UInt32) at int.jl:14
+(x::UInt64,y::UInt64) at int.jl:14
+(x::Int128,y::Int128) at int.jl:14

User-defined types

  • type declaration not mandatory
  • parametric types
type Wavefunction{T}
    c::Array{T, 2}
    x::Array{T, 2}
end
+{T}(psi1::Wavefunction{T}, psi2::Wavefunction{T}) =
                        Wavefunction{T}(psi1.c + psi2.c, psi1.x + psi2.x)
+{T}(psi::Wavefunction{T}, c::Float64) = Wavefunction{T}(psi.c .+ c, psi.x .+ c) 
*{T}(c::Number, psi::Wavefunction{T}) = Wavefunction{T}(c*psi.c, c*psi.x)
/{T}(psi1::Wavefunction{T}, psi2::Wavefunction{T}) =
                         Wavefunction{T}(psi1.c ./ psi2.c, psi1.x ./ psi2.x)
import Base.abs
abs(psi::Wavefunction{Complex{Float64}}) = Wavefunction{Float64}(abs(psi.c), abs(psi.x))
import Base.maximum
function maximum(psi::Wavefunction{Float64})
    maxc = maximum(psi.c)
    maxx = maximum(psi.x)
    return max(maxc, maxx)
end

Julia is..

  • free, open-source software (MIT license)
  • high level (Python, MATLAB)
  • homoiconic

Interactive computation with Jupyter