Collective Modes

A module that calculates the band structure and decay rates of different collective modes of quasimomentum k with functions:

• Omega_k_chain
• Gamma_k_chain
• Omega_k_2D
• Gamma_k_2D

It is probably best to illustrate the functionality using some examples.

One-dimensional case

The following code reproduces the band strucuture and decay plots from A. Asenjo-Garcia et al, 10.1103/PhysRevX.7.031024 (Figs. 1b and 1c).

using CollectiveSpins

# Define chain parameters
a = 0.2   # spin-spin distance
polarization_par = [1, 0, 0]
polarization_perp = [0, 1, 1]

k_max = 10000
k_list = [iii*pi/(a*k_max) for iii=-k_max:k_max]

band_structure_par = []
band_structure_perp = []
decay_par = []
decay_perp = []
for k in k_list
    append!(band_structure_par, Omega_k_chain(k, a, polarization_par))
    append!(band_structure_perp, Omega_k_chain(k, a, polarization_perp))
    append!(decay_par, Gamma_k_chain(k, a, polarization_par))
    append!(decay_perp, Gamma_k_chain(k, a, polarization_perp))
end

using PyPlot

figure()
subplot(211)
plot(k_list*a/pi, band_structure_par)
plot(k_list*a/pi, band_structure_perp, color="red")
xlabel("k d/pi")
ylabel("Omega(k)/gamma0")

subplot(212)
plot(k_list*a/pi, decay_par)
plot(k_list*a/pi, decay_perp, color="red")
xlabel("k d/pi")
ylabel("Gamma(k)/gamma0")

tight_layout()

┌ Warning: No working GUI backend found for matplotlib
└ @ PyPlot ~/.julia/packages/PyPlot/2MlrT/src/init.jl:153

Two-dimensional case

For the two-dimensional case, consider the following two examples. The first example reproduces the band structure for a 2D square lattice with in-plane circularly polarized and perpendicular (x-axis) polarized atoms, which can be found in E. Shahmoon et al 10.1103/PhysRevLett.118.113601 (Fig. 4a).

For a particularly elegant derivation of collective frequency shifts and decay rates in 2D atomic arrays, see the thesis of Dominik Wild, "Algorithms and Platforms for Quantum Science and Technology".

using CollectiveSpins
using PyPlot

polarization_par = [0, 1, im]
polarization_perp = [1, 0, 0]
a_vec1 = [0.2, 0]
a_vec2 = [0, 0.2]
k_max = 200

k_listy = [iii*pi/(a_vec1[1]*k_max) for iii=k_max:-1:0]
k_listz = copy(k_listy)
k_listy = append!(k_listy, [iii*pi/(a_vec1[1]*k_max) for iii=0:k_max])
k_listz = append!(k_listz, zeros(k_max+1))
k_listy = append!(k_listy, ones(k_max+1)*pi/a_vec1[1])
k_listz = append!(k_listz, [iii*pi/(a_vec1[1]*k_max) for iii=0:k_max])
k_dim = length(k_listy)

bandstructure_par = zeros(k_dim)
bandstructure_perp = zeros(k_dim)
for iii=1:k_dim
    k = [k_listy[iii], k_listz[iii]]
    bandstructure_par[iii] = Omega_k_2D(k, a_vec1, a_vec2, polarization_par)
    bandstructure_perp[iii] = Omega_k_2D(k, a_vec1, a_vec2, polarization_perp)
end

positions = [0, k_max, 2*k_max, 3*k_max]
labels = ["M", "Gamma", "X", "M"]

figure(figsize=(6,4))
plot(collect(1:k_dim), bandstructure_par, "k--")
plot(collect(1:k_dim), bandstructure_perp, color="gold")
xticks(positions, labels)
ylim([-5.2, 5.2])

tight_layout()

The next example shows how one can compute the decay rates for a 2 square lattices with atoms polarized in-plane (y-axis) and perpendicularly (x-axis). This can be found in A. Asenjo-Garcia et al, 10.1103/PhysRevX.7.031024 (Figs. 2c and 2d).

polarization_par = [0, 1, 0]
polarization_perp = [1, 0, 0]
a_vec1 = [0.2, 0]
a_vec2 = [0, 0.2]
k_max = 200

k_list = [iii*pi/(a_vec1[1]*k_max) for iii=-k_max:k_max]
k_dim = length(k_list)
decay_par = zeros(k_dim, k_dim)
decay_perp = zeros(k_dim, k_dim)
y = 1
for ky in k_list
    z = 1
    for kz in k_list
        k = [ky, kz]
        decay_par[z, y] = Gamma_k_2D(k, a_vec1, a_vec2, polarization_par)
        decay_perp[z, y] = Gamma_k_2D(k, a_vec1, a_vec2, polarization_perp)
        z += 1
    end
    global y += 1
end


positions = [0, k_max, 2*k_max]
labels = [-1, 0, 1]

figure(figsize=(10,4))
subplot(121)
imshow(decay_par, cmap="jet")
colorbar()
clim(0, 22)
xlabel("k_y d/pi")
ylabel("k_z d/pi")
plt.xticks(positions, labels)
plt.yticks(positions, labels)

subplot(122)
imshow(decay_perp, cmap="jet")
colorbar()
clim(0, 22)
xlabel("k_y d/pi")
ylabel("k_z d/pi")
plt.xticks(positions, labels)
plt.yticks(positions, labels)

tight_layout()