diff --git a/docs/src/getting_started/logo.md b/docs/src/getting_started/logo.md index 587e732fa..154ad0c26 100644 --- a/docs/src/getting_started/logo.md +++ b/docs/src/getting_started/logo.md @@ -14,23 +14,23 @@ A cat state, often referred to as a Schrödinger cat state, is a quantum state t where ``| \alpha \rangle`` is a coherent state with amplitude ``\alpha``. -The triangular cat state is a generalization of the standard cat state. It is a superposition of three coherent states with phases ``\theta_0, \theta_1, \theta_2`` separated by ``120^\circ``(or ``2\pi/3``radians): +The triangular cat state is a generalization of the standard cat state. It is a superposition of three coherent states with phases ``\theta_0, \theta_1, \theta_2`` separated by ``120^\circ`` (or ``2\pi/3``radians): ```math | \psi_{\text{tri-cat}} \rangle = \frac{1}{\sqrt{3}} \left( | \alpha_0 \rangle + | \alpha_1 \rangle + | \alpha_2 \rangle \right) ``` -where ``\alpha_j = \rho e^{i\theta_j}``with ``\theta_j = \frac{\pi}{2} + \frac{2\pi j}{3}``and ``j = 0, 1, 2``. +where ``\alpha_j = \rho e^{i\theta_j}`` with ``\theta_j = \frac{\pi}{2} + \frac{2\pi j}{3}`` and ``j = 0, 1, 2``. ### Wigner Function -The Wigner function ``W(x, p)``is a quasi-probability distribution used in quantum mechanics to represent quantum states in phase space. It is defined as: +The Wigner function ``W(x, p)`` is a quasi-probability distribution used in quantum mechanics to represent quantum states in phase space. It is defined as: ```math W(x, p) = \frac{1}{\pi \hbar} \int_{-\infty}^{\infty} \psi^*(x + y) \psi(x - y) e^{2ipy / \hbar} \, dy ``` -where ``\psi(x)``is the wave function of the quantum state, ``x``is the position, ``p``is the momentum, and ``\hbar``is the reduced Planck constant. Unlike classical probability distributions, the Wigner function can take negative values, which indicates non-classical behavior. +where ``\psi(x)`` is the wave function of the quantum state, ``x`` is the position, ``p`` is the momentum, and ``\hbar`` is the reduced Planck constant. Unlike classical probability distributions, the Wigner function can take negative values, which indicates non-classical behavior. ## Generating the Logo @@ -91,7 +91,7 @@ The figure obtained above coulb be already a potential logo for the package. How \frac{d \hat{\rho}}{dt} = -i [\hat{H}, \hat{\rho}] + \gamma \left( 2 \hat{a} \hat{\rho} \hat{a}^\dagger - \hat{a}^\dagger \hat{a} \hat{\rho} - \hat{\rho} \hat{a}^\dagger \hat{a} \right) ``` -where ``\hat{\rho}`` is the density matrix, ``\hat{H} = \omega \hat{a}^\dagger \hat{a}``is the Hamiltonian of the harmonic oscillator (``\hbar = 1``), ``\hat{a}``and ``\hat{a}^\dagger``are the annihilation and creation operators, and ``\gamma``is the damping rate. Thus, we initialize the system in the triangular cat state and evolve it under the Lindblad master equation, using the [`mesolve`](@ref) function. +where ``\hat{\rho}`` is the density matrix, ``\hat{H} = \omega \hat{a}^\dagger \hat{a}`` is the Hamiltonian of the harmonic oscillator (``\hbar = 1``), ``\hat{a}`` and ``\hat{a}^\dagger`` are the annihilation and creation operators, and ``\gamma`` is the damping rate. Thus, we initialize the system in the triangular cat state and evolve it under the [Lindblad master equation](@ref doc-TE:Lindblad-Master-Equation-Solver), using the [`mesolve`](@ref) function. ```@example logo γ = 0.012 @@ -166,7 +166,7 @@ cmap3 = cgrad(vcat(fill(julia_blue, n_repeats), fill(julia_purple, n_repeats))) ### Normalizing the Wigner function and applying the custom colormap -The colormaps require the input to be in the range ``[0, 1]``. We normalize the Wigner function such that the maximum value is ``1``and the zeros are set to ``0.5``. +The colormaps require the input to be in the range ``[0, 1]``. We normalize the Wigner function such that the maximum value is ``1`` and the zeros are set to ``0.5``. ```@example logo vmax = maximum(wig) diff --git a/src/correlations.jl b/src/correlations.jl index ae9598f98..0df4375f2 100644 --- a/src/correlations.jl +++ b/src/correlations.jl @@ -69,7 +69,7 @@ end C::QuantumObject; kwargs...) -Returns the one-time correlation function of three operators ``\hat{A}``, ``\hat{B}`` and ``\hat{C}``: ``\left\langle \hat{A}(0) \hat{B}(\tau) \hat{C}(0) \right\rangle`` for a given initial state ``|\psi_0\rangle``. +Returns the two-time correlation function (with only one time coordinate ``\tau``) of three operators ``\hat{A}``, ``\hat{B}`` and ``\hat{C}``: ``\left\langle \hat{A}(0) \hat{B}(\tau) \hat{C}(0) \right\rangle`` for a given initial state ``|\psi_0\rangle``. If the initial state `ψ0` is given as `nothing`, then the [`steadystate`](@ref) will be used as the initial state. Note that this is only implemented if `H` is constant ([`QuantumObject`](@ref)). """ @@ -142,7 +142,7 @@ end reverse::Bool=false, kwargs...) -Returns the one-time correlation function of two operators ``\hat{A}`` and ``\hat{B}`` : ``\left\langle \hat{A}(\tau) \hat{B}(0) \right\rangle`` for a given initial state ``|\psi_0\rangle``. +Returns the two-time correlation function (with only one time coordinate ``\tau``) of two operators ``\hat{A}`` and ``\hat{B}`` : ``\left\langle \hat{A}(\tau) \hat{B}(0) \right\rangle`` for a given initial state ``|\psi_0\rangle``. If the initial state `ψ0` is given as `nothing`, then the [`steadystate`](@ref) will be used as the initial state. Note that this is only implemented if `H` is constant ([`QuantumObject`](@ref)).