index.md

CurrentModule = QuantumStatePlots

QuantumStatePlots

Documentation for QuantumStatePlots.

About

Render Wigner function

Based on the definition of Wigner function in Fock basis:

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

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

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

Example

The quantum state and its wigner function:

julia> using QuantumStateBase

julia> ρ = SqueezedState(0.8, π/8, Matrix, dim=100)

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

Wigner function

Surface

julia> using QuantumStatePlots, Plots

julia> surface(w)

Heatmap

julia> using QuantumStatePlots, Plots

julia> heatmap(w)

Contour

julia> using QuantumStatePlots, Plots

julia> contour(w)

Density matrix

Real part

julia> using QuantumStatePlots, Plots

julia> plot_real(ρ, 35)

Imag part

julia> using QuantumStatePlots, Plots

julia> plot_imag(ρ, 35)

Reference

• Quantum mechanics as a statistical theory

Index

APIs

Modules = [QuantumStatePlots]