Theoretical Descriptions

CollectiveSpins.jl provides several different possibilities to simulate multi-spin systems. A full quantum description is available but only possible for small numbers of spins. Additionally, approximations of different orders are implemented using a cumulant expansion approach:

quantum - descriptions-quantum
independent - descriptions-cumulant0
meanfield - descriptions-cumulant1
mpc - descriptions-cumulant2

All variants provide a unified interface wherever possible:

blochstate(phi, theta)
densityoperator(state)
sx(state)
sy(state)
sz(state)
timeevolution(T, system, state0; fout=nothing)
rotate(axis, angles, state)
squeeze(axis, χT, state)
squeezingparameter(state)

The following example should give a first idea how these implementations are used:

using QuantumOptics, CollectiveSpins
const cs = CollectiveSpins

# System parameters
const a = 0.18
const γ = 1.
const e_dipole = [0,0,1.]
const T = [0:0.05:5;]
const N = 5
const Ncenter = 3

const system = SpinCollection(cs.geometry.chain(a, N), e_dipole, γ)


# Define Spin 1/2 operators
spinbasis = SpinBasis(1//2)
sx = sigmax(spinbasis)
sy = sigmay(spinbasis)
sz = sigmaz(spinbasis)
sp = sigmap(spinbasis)
sm = sigmam(spinbasis)
I_spin = identityoperator(spinbasis)

# Initial state (Bloch state)
const phi = 0.
const theta = pi/2.

# Time evolution

# Independent
state0 = cs.independent.blochstate(phi, theta, N)
tout, state_ind_t = cs.independent.timeevolution(T, system, state0)

# Meanfield
state0 = cs.meanfield.blochstate(phi, theta, N)
tout, state_mf_t = cs.meanfield.timeevolution(T, system, state0)

# Meanfield + Correlations
state0 = cs.mpc.blochstate(phi, theta, N)
tout, state_mpc_t = cs.mpc.timeevolution(T, system, state0)

# Quantum: master equation
sx_master = Float64[]
sy_master = Float64[]
sz_master = Float64[]

td_ind = Float64[]
td_mf  = Float64[]
td_mpc = Float64[]

embed(op::Operator) = QuantumOptics.embed(cs.quantum.basis(system), Ncenter, op)

function fout(t, rho)
    i = findfirst(isequal(t), T)
    rho_ind = cs.independent.densityoperator(state_ind_t[i])
    rho_mf  = cs.meanfield.densityoperator(state_mf_t[i])
    rho_mpc = cs.mpc.densityoperator(state_mpc_t[i])
    push!(td_ind, tracedistance(rho, rho_ind))
    push!(td_mf,  tracedistance(rho, rho_mf))
    push!(td_mpc, tracedistance(rho, rho_mpc))
    push!(sx_master, real(expect(embed(sx), rho)))
    push!(sy_master, real(expect(embed(sy), rho)))
    push!(sz_master, real(expect(embed(sz), rho)))
    return nothing
end

Ψ₀ = cs.quantum.blochstate(phi,theta,N)
ρ₀ = Ψ₀⊗dagger(Ψ₀)
cs.quantum.timeevolution(T, system, ρ₀, fout=fout)

# Expectation values
mapexpect(op, states) = map(s->(op(s)[Ncenter]), states)

sx_ind = mapexpect(cs.independent.sx, state_ind_t)
sy_ind = mapexpect(cs.independent.sy, state_ind_t)
sz_ind = mapexpect(cs.independent.sz, state_ind_t)

sx_mf = mapexpect(cs.meanfield.sx, state_mf_t)
sy_mf = mapexpect(cs.meanfield.sy, state_mf_t)
sz_mf = mapexpect(cs.meanfield.sz, state_mf_t)

sx_mpc = mapexpect(cs.mpc.sx, state_mpc_t)
sy_mpc = mapexpect(cs.mpc.sy, state_mpc_t)
sz_mpc = mapexpect(cs.mpc.sz, state_mpc_t)

Quantum

The time evolution of the $N$ spins in a rotating frame corresponding to $\sum_i \omega_0 \sigma^z_i$ is then governed by a master equation

\[\dot{\rho} = -\frac{i}{\hbar} \big[H, \rho\big] + \mathcal{L}[\rho]\]

with the Hamiltonian

\[H = \sum_{ij;i \neq j} \hbar \Omega_{ij} \sigma_i^+ \sigma_j^-\]

and Lindblad-term

\[\mathcal{L}[\rho] = \frac{1}{2} \sum_{i,j} \Gamma_{ij} (2\sigma_i^- \rho \sigma_j^+ - \sigma_i^+ \sigma_j^- \rho - \rho \sigma_i^+ \sigma_j^-).\]

The dipole-dipole interaction $\Omega_{ij} = \frac{3}{4} \gamma G(k_0 r_{ij})$ and the collective decay $\Gamma_{ij} = \frac{3}{2} \gamma F(k_0 r_{ij})$ can be obtained analytically with

\[\begin{aligned} F(\xi) &= \alpha \frac{\sin \xi}{\xi} + \beta \left( \frac{\cos \xi}{\xi^2} - \frac{\sin \xi}{\xi^3} \right) \\ G(\xi) &= -\alpha \frac{\cos \xi}{\xi} + \beta \left( \frac{\sin \xi}{\xi^2} + \frac{\cos \xi}{\xi^3} \right) \end{aligned}\]

with $\alpha = 1 -\cos^2 \theta$ and $\beta = 1-3 \cos^2 \theta$, where $\theta$ represents the angle between the line connecting atoms $i$ and $j$ and the common atomic dipole orientation.

0th order: Independent spins

Each spin evolves independently according to

\[\begin{aligned} \langle\dot{\sigma_k^x}\rangle &= -\frac{1}{2} \gamma \langle\sigma_k^x\rangle \\ \langle\dot{\sigma_k^y}\rangle &= -\frac{1}{2} \gamma \langle\sigma_k^y\rangle \\ \langle\dot{\sigma_k^z}\rangle &= \gamma \big(1 - \langle\sigma_k^z\rangle\big) \end{aligned}\]

1st order: Meanfield

\[\begin{aligned} \langle\dot{\sigma_k^x}\rangle &= \sum_{i;i \neq k} \Omega_{ki} \langle\sigma_i^y\sigma_k^z\rangle -\frac{1}{2} \gamma \langle\sigma_k^x\rangle -\frac{1}{2} \sum_{i;i \neq k} \Gamma_{ki} \langle\sigma_i^x\sigma_k^z\rangle \\ \langle\dot{\sigma_k^y}\rangle &= -\sum_{i;i \neq k} \Omega_{ki} \langle\sigma_i^x\sigma_k^z\rangle -\frac{1}{2} \gamma \langle\sigma_k^y\rangle -\frac{1}{2} \sum_{i;i \neq k} \Gamma_{ki} \langle\sigma_i^y\sigma_k^z\rangle \\ \langle\dot{\sigma_k^z}\rangle &= - \sum_{i;i \neq k} \Omega_{ki} \Big(\langle\sigma_k^x\sigma_i^y\rangle - \langle\sigma_i^x\sigma_k^y\rangle\Big) +\gamma \big(1 - \langle\sigma_k^z\rangle\big) +\frac{1}{2} \sum_{i;i \neq k} \Gamma_{ki} \Big(\langle\sigma_k^x\sigma_i^x\rangle + \langle\sigma_i^y\sigma_k^y\rangle\Big) \end{aligned}\]

2nd order: Meanfield plus Correlations (MPC)

\[\begin{aligned} \langle\dot{\sigma_k^x\sigma_l^x}\rangle &= \sum_{j;j \neq k,l} \Omega_{kj} \langle\sigma_k^z\sigma_l^x\sigma_j^y\rangle + \sum_{j;j \neq k,l} \Omega_{lj} \langle\sigma_k^x\sigma_l^z\sigma_j^y\rangle \\&\qquad - \gamma \langle\sigma_k^x\sigma_l^x\rangle + \Gamma_{kl} \Big( \langle\sigma_k^z\sigma_l^z\rangle - \frac{1}{2} \langle\sigma_k^z\rangle - \frac{1}{2} \langle\sigma_l^z\rangle \Big) \\&\quad - \frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{kj} \langle\sigma_k^z\sigma_l^x\sigma_j^x\rangle - \frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{lj} \langle\sigma_k^x\sigma_l^z\sigma_j^x\rangle \\ \langle\dot{\sigma_k^y\sigma_l^y}\rangle &= - \sum_{j;j \neq k,l} \Omega_{kj} \langle\sigma_k^z\sigma_l^y\sigma_j^x\rangle - \sum_{j;j \neq k,l} \Omega_{lj} \langle\sigma_k^y\sigma_l^z\sigma_j^x\rangle \\&\qquad - \gamma \langle\sigma_k^y\sigma_l^y\rangle + \Gamma_{kl}\Big( \langle\sigma_k^z\sigma_l^z\rangle -\frac{1}{2} \langle\sigma_k^z\rangle -\frac{1}{2} \langle\sigma_l^z\rangle \Big) \\&\quad -\frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{kj} \langle\sigma_k^z\sigma_l^y\sigma_j^y\rangle -\frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{lj} \langle\sigma_k^y\sigma_l^z\sigma_j^y\rangle \\ \langle\dot{\sigma_k^z\sigma_l^z}\rangle &= \sum_{j;j \neq k,l} \Omega_{kj} \Big( \langle\sigma_k^y\sigma_l^z\sigma_j^x\rangle - \langle\sigma_k^x\sigma_l^z\sigma_j^y\rangle \Big) \\&\qquad +\sum_{j;j \neq k,l} \Omega_{lj} \Big( \langle\sigma_k^z\sigma_l^y\sigma_j^x\rangle -\langle\sigma_k^z\sigma_l^x\sigma_j^y\rangle \Big) \\&\quad - 2 \gamma \langle\sigma_k^z\sigma_l^z\rangle + \gamma \big(\langle\sigma_l^z\rangle + \langle\sigma_k^z\rangle\big) \\&\quad +\Gamma_{kl}\Big( \langle\sigma_k^y\sigma_l^y\rangle + \langle\sigma_k^x\sigma_l^x\rangle \Big) \\&\quad +\frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{kj} \Big( \langle\sigma_k^x\sigma_l^z\sigma_j^x\rangle +\langle\sigma_k^y\sigma_l^z\sigma_j^y\rangle \Big) \\&\qquad +\frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{lj} \Big( \langle\sigma_k^z\sigma_l^x\sigma_j^x\rangle +\langle\sigma_k^z\sigma_l^y\sigma_j^y\rangle \Big) \end{aligned}\]

\[\begin{aligned} \langle\dot{\sigma_k^x\sigma_l^y}\rangle &= \Omega_{kl}\Big( \langle\sigma_k^z\rangle - \langle\sigma_l^z\rangle \Big) +\sum_{j;j \neq k,l} \Omega_{kj} \langle\sigma_k^z\sigma_l^y\sigma_j^y\rangle \\&\qquad -\sum_{j;j \neq k,l} \Omega_{lj} \langle\sigma_k^x\sigma_l^z\sigma_j^x\rangle - \gamma \langle\sigma_k^x\sigma_l^y\rangle \\&\quad - \frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{kj} \langle\sigma_k^z\sigma_l^y\sigma_j^x\rangle - \frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{lj} \langle\sigma_k^x\sigma_l^z\sigma_j^y\rangle \\ \langle\dot{\sigma_k^x\sigma_l^z}\rangle &= \Omega_{kl} \langle\sigma_l^y\rangle +\sum_{j;j \neq k,l} \Omega_{kj} \langle\sigma_k^z\sigma_l^z\sigma_j^y\rangle \\&\quad +\sum_{j;j \neq k,l} \Omega_{lj} \Big( \langle\sigma_k^x\sigma_l^y\sigma_j^x\rangle -\langle\sigma_k^x\sigma_l^x\sigma_j^y\rangle \Big) \\&\quad - \frac{3}{2} \gamma \langle\sigma_k^x\sigma_l^z\rangle + \gamma \langle\sigma_k^x\rangle - \Gamma_{kl}\Big( \langle\sigma_k^z\sigma_l^x\rangle -\frac{1}{2} \langle\sigma_l^x\rangle \Big) \\&\quad - \frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{kj} \langle\sigma_k^z\sigma_l^z\sigma_j^x\rangle \\&\quad + \frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{lj} \Big( \langle\sigma_k^x\sigma_l^x\sigma_j^x\rangle +\langle\sigma_k^x\sigma_l^y\sigma_j^y\rangle \Big) \end{aligned}\]

\[\begin{aligned} \langle\dot{\sigma_k^y\sigma_l^z}\rangle &= -\Omega_{kl} \langle\sigma_l^x\rangle -\sum_{j;j \neq k,l} \Omega_{kj} \langle\sigma_k^z\sigma_l^z\sigma_j^x\rangle \\&\quad +\sum_{j;j \neq k,l} \Omega_{lj} \Big( \langle\sigma_k^y\sigma_l^y\sigma_j^x\rangle -\langle\sigma_k^y\sigma_l^x\sigma_j^y\rangle \Big) \\&\quad - \frac{3}{2} \gamma \langle\sigma_k^y\sigma_l^z\rangle + \gamma \langle\sigma_k^y\rangle - \Gamma_{kl}\Big( \langle\sigma_k^z\sigma_l^y\rangle - \frac{1}{2}\langle\sigma_l^y\rangle \Big) \\&\quad - \frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{kj} \langle\sigma_k^z\sigma_l^z\sigma_j^y\rangle \\&\quad + \frac{1}{2} \sum_{j;j \neq k,l} \Gamma_{lj} \Big( \langle\sigma_k^y\sigma_l^x\sigma_j^x\rangle +\langle\sigma_k^y\sigma_l^y\sigma_j^y\rangle \Big) \end{aligned}\]