QuantumOptics.jl is a numerical framework written in the Julia programming language that makes it easy to simulate various kinds of open quantum systems. It is inspired by the Quantum Optics Toolbox for MATLAB and the Python framework QuTiP.

QuantumOptics.jl optimizes processor usage and memory consumption by relying on different ways to store and work with operators.

The framework comes with a plethora of pre-defined systems and interactions making it very easy to focus on the physics, not on the numerics.

Every function in the framework has been severely tested with all tests and their code coverage presented on the framework's GitHub page.

To get started with Julia, check out Julia's setup instructions. For plotting we recommend matplotlib in Python, which plays nicely with Julia. Before you can execute any of the framework's functions, you will need to add the `QuantumOptics`

package to Julia, as shown below. Plotting with matplotlib is then enabled by adding the `PyPlot`

package.

```
Pkg.add("QuantumOptics") # Install QuantumOptics.jl package
Pkg.add("PyPlot") # Support for matplotlib from within Julia
```

```
using QuantumOptics
basis = PositionBasis(-2, 2, 200)
x = position(basis)
p = momentum(basis)
H = p^2/4 + 2*full(x^2)
energies, states = eigenstates((H+dagger(H))/2, 5)
using PyPlot
xpoints = samplepoints(basis)
plot(xpoints, 2*xpoints.^2)
fill_between(xpoints, 0., 2*xpoints.^2, alpha=0.5)
for i=1:length(states)
plot(xpoints, abs2.(states[i].data)*40 + energies[i])
end
xlabel("Position")
ylabel("Energy")
tight_layout()
savefig("particle.svg")
```

```
using QuantumOptics
b = FockBasis(50)
a = destroy(b)
at = create(b)
H = 0.5*(a^2 + at^2)
psi0 = fockstate(b, 3)
tout, psit = timeevolution.schroedinger([0:0.25:1;], psi0, H)
using PyPlot
x = [-5:0.1:5;]
for i in 1:4
subplot(2, 2, i)
Q = qfunc(psit[i], x, x)
pcolor(x, x, Q)
end
tight_layout()
savefig("fock.png")
```

```
using QuantumOptics
b = SpinBasis(3//2)
sm = sigmam(b)
H = 2*sigmaz(b)
J = [sm]
τ = [0:0.025:5;]
ω = [-5:0.05:25;]
ρ0 = dm(spinup(b))
corr = timecorrelations.correlation(τ, ρ0, H, J, sigmap(b), sm)
ω, S = timecorrelations.spectrum(ω, H, J, sm)
using PyPlot
subplot(2, 1, 1)
plot(τ, corr)
xlabel(L"\tau")
ylabel(L"\langle \sigma_+(\tau) \sigma_-(0)\rangle")
subplot(2, 1, 2)
plot(ω, S)
xlabel(L"\omega")
ylabel(L"S(\omega)")
tight_layout()
savefig("spin.svg")
```

```
using QuantumOptics
b = NLevelBasis(3)
t12 = transition(b, 1, 2)
t23 = transition(b, 2, 3)
t31 = transition(b, 1, 3)
H = 10*(t31 + dagger(t31))
J = [1.2*t23, 0.6*t12]
psi0 = basisstate(b, 1)
T = [0:0.01:10;]
tout, psit = timeevolution.mcwf(T, psi0, H, J; seed=2)
using PyPlot
plot(tout, expect(dm(basisstate(b, 3)), psit), label=L"$|3\rangle$")
plot(tout, expect(dm(basisstate(b, 2)), psit), label=L"$|2\rangle$")
plot(tout, expect(dm(basisstate(b, 1)), psit), label=L"$|1\rangle$")
xlabel("Time")
ylabel("Probability")
legend()
tight_layout()
savefig("nlevel.svg")
```

```
using QuantumOptics
b_spin = SpinBasis(1//2)
b_fock = FockBasis(200)
sp = sigmap(b_spin)
sm = sigmam(b_spin)
a = destroy(b_fock)
at = create(b_fock)
H = sp⊗a + sm⊗at
T = [0:0.01:50;]
ψ0 = spindown(b_spin) ⊗ coherentstate(b_fock, 6)
tout, ψt = timeevolution.schroedinger(T, ψ0, H)
using PyPlot
plot(tout, expect(1, sp*sm, ψt))
xlabel("Time")
ylabel("Spin excitation")
tight_layout()
savefig("composite.svg")
```

QuantumOptics.jl is developed in Helmut Ritsch's CQED group at the Institute for Theoretical Physics of the University of Innsbruck. Development is lead by Sebastian Krämer.

QuantumOptics.jl is open source and hosted on GitHub. All community contributions are very welcome and coordinated by the head developer. If you want to join our effort, fork the repository and send us your pull requests!