QuantumStateBase

Installation

The package can be installed with the Julia package manager. From the Julia REPL, type ] to enter the Pkg REPL mode and run:

pkg> add QuantumStateBase

Quick start

Construct a squeezed thermal state and plot the Wigner function

julia> using QuantumStateBase, Plots

julia> state = SqueezedThermalState(0.5, 3π/2, 0.3, dim=35);

julia> w = wigner(state, LinRange(-3, 3, 101), LinRange(-3, 3, 101));

julia> heatmap(w.x_range, w.p_range,  w.𝐰_surface')

Wigner function

Wigner function is introduced by Eugene Winger, who describe quantum state using revised classical probability theory.

The quasi probability distribution is defined as:

\[W_{mn}(x, p) = \frac{1}{2\pi} \int_{-\infty}^{\infty} dy \, e^{-ipy/h} \psi_m^*(x+\frac{y}{2}) \psi_n(x-\frac{y}{2})\]

Owing to the fact that the Moyal function is a generalized Wigner function. We can therefore implies that

\[W(x, p) = \sum_{m, n} \rho_{m, n} W_{m, n}(x, p)\]

Here, $\rho$ is the density matrix of the quantum state, defined as:

\[\rho = \sum_{m, n, i} \, p_i \, | n \rangle \langle n | \hat{\rho}_i | m \rangle \langle m |\]

\[\hat{\rho}_i = | \psi_i \rangle \langle \psi_i |\]

\[\hat{\rho}_i \, \text{is a density operator of pure state.}\]

And, $W_{m, n}(x, p)$ is the generalized Wigner function

\[W_{m, n} = \{ \begin{array}{rcl} \frac{1}{\pi} exp[-(x^2 + y^2)] (-1)^m \sqrt{2^{n-m} \frac{m!}{n!}} (x-ip)^{n-m} L_m^{n-m} (2x^2 + 2p^2), \, n \geq m \\ \frac{1}{\pi} exp[-(x^2 + y^2)] (-1)^n \sqrt{2^{m-n} \frac{n!}{m!}} (x+ip)^{m-n} L_n^{m-n} (2x^2 + 2p^2), \, n < m \\ \end{array}\]