Make spectrum and correlation align with QuTiP

[ytdHuang]

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Description

The current implementation of spectrum does not align with QuTiP (see the documentation here). The spectrum should only return the value of spectrum corresponding to the user specified ω_list. Also, the peudo-inverse method is missing.

Furthermore, in QuTiP, the spectra obtained from fft of the correlation function is done by a function called spectrum_correlation_fft instead of spectrum.

Finally, we can support nothing input state for correlation functions, that will automatically solve the steadystate first.

TODO list:

• separate the source code of spectrum and correlation functions into independent files
• adjust the parameter order of spectrum and correlation functions to align with QuTiP (also add the deprecated methods to deprecated.jl)
• add time list checks for correlation functions
• adjust the return correlation Matrix dimension order to align with QuTiP
• remove the return of ω_list in spectrum function
• support nothing::Nothing input state for correlation functions [only when Hamiltonian is constant (Qobj)]
• allow time-dependent Hamiltonian (QobjEvo) as an input for correlation functions.
• remove spectrum solver FFTCorrelation (move it to deprecated.jl)
• introduce a new function called spectrum_correlation_fft(tlist, corr) to replace the functionality of FFTCorrelation
• introduce PseudoInverse <: SpectrumSolver for spectrum
• introduce correlation_3op_1t
• write the document page: Two-time correlation functions

[ytdHuang]

I have already done the function spectrum_correlation_fft:

function spectrum_correlation_fft(tlist::AbstractVector, corr::AbstractVector; inverse::Bool=false)
    N = length(tlist)
    dt_list = diff(tlist)
    dt = dt_list[1]
    
    all(≈(dt), dt_list) || ArgumentError("tlist must be equally spaced for FFT.")

    F = inverse ? N * ifft(corr) : fft(corr)
    Flist = fftshift(F)
    
    flist = fftshift(fftfreq(N, 1 / dt))

    return 2 * π * flist, 2 * dt * real(Flist)
end

This gives the same result in QuTiP

[albertomercurio]

Great

[albertomercurio]

I would also change the order of the arguments, since I think it is a bit different from QuTiP.

[ytdHuang]

I would also change the order of the arguments, since I think it is a bit different from QuTiP.

I also did it in this PR~

[codecov]

Codecov Report

Attention: Patch coverage is 94.44444% with 5 lines in your changes missing coverage. Please review.

Project coverage is 93.28%. Comparing base (9a9a80d) to head (508d4d1).
Report is 4 commits behind head on main.

Files with missing lines	Patch %	Lines
src/spectrum.jl	93.22%	4 Missing ⚠️
src/correlations.jl	96.15%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #330      +/-   ##
==========================================
+ Coverage   93.19%   93.28%   +0.09%     
==========================================
  Files          36       38       +2     
  Lines        2602     2637      +35     
==========================================
+ Hits         2425     2460      +35     
  Misses        177      177              

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