Code Documentation

class src.markovanalyzer.polyspectra_calculator.System(transition_dict, measurement_op, gamma_ph=None, gamma_det=None)

Class that will represent the system of interest. It contains the parameters of the system and the methods for calculating and storing the polyspectra.

Parameters

Attributes

freqdict

Stores the frequencies from the analytic spectra, order 2 to 4

Sdict

Stores the analytic spectra, order 2 to 4

numeric_f_datadict

Stores the frequencies from the numeric spectra, order 2 to 4

numeric_spec_datadict

Stores the numeric spectra, order 2 to 4

eigvalsarray

Stores eigenvalues of Liouvillian

eigvecsarray

Stores eigenvectors of Liouvillian

eigvecs_invarray

Stores the matrix inversion of the eigenvector matrix

zero_indint

Contains the index of the steady state in the eigvalues

A_primarray

Stores the measurement superoperator mathcal{A} as defined in 10.1103/PhysRevB.98.205143

rho_steadyarray

Steady state of the Liouvillian

s_karray

Stores small s (Eq. 7) from 10.1103/PhysRevB.102.119901

Nint

Number of points in time series in window for the calculation of numerical spectra

fsfloat

Sampling rate of the simulated signal for numerical spectra

a_warray

Fourier coefficients of simulated signal for numerical spectra

a_w_cutarray

Contains only the frequencies of interest from a_w (to speed up calculations)

enable_gpubool

Set if GPU should be used for analytic spectra calculation

gpu_0int

Stores pointer to zero an the GPU

calc_a_w3(a_w)

Preparation of a_(w1+w2) for the calculation of the bispectrum

Parameters

a_warray

Fourier coefficients of signal

Returns

a_w3array

Matrix corresponding to a_(w1+w2)

calculate_one_spectrum(f_data, order, bar=True, verbose=False, beta_offset=True, enable_gpu=False, cache_trispec=True)

Calculates analytic polyspectra (order 2 to 4) as described in 10.1103/PhysRevB.98.205143 and 10.1103/PhysRevB.102.119901

Parameters

f_dataarray

Frequencies at which the spectra are calculated

orderint {2,3,4}

Order of the polyspectra to be calculated

g_primbool

Set if mathcal_a should be applied twice/squared (was of use when defining the current operator) But unnecessary for standard polyspectra

barbool

Set if progress bars should be shown during calculation

verbosebool

Set if more details about the current state of the calculation are needed

correction_onlybool

Set if only the correction terms of the S4 from erratum 10.1103/PhysRevB.102.119901 should be calculated

beta_offsetbool

Set if constant offset due to deetector noise should be added to the power spectrum

enable_gpubool

Set if GPU should be used for calculation

cache_trispecbool

Set if Matrix multiplication in the calculation of the trispectrum should be cached

Returns

S[order]array

Returns spectral value at specified frequencies

extension_for_single_photon(rates, m_op, gamma_ph, gamma_det)

Adds further connections to the rates dictionary based on the input array m_op and float gamma_det.

Parameters

ratesdict

Existing rates dictionary where keys are in ‘from_state->to_state’ format and values are the rates.

m_oplist

Array of rates used to connect the original levels with their corresponding replicated levels.

gamma_detfloat

Rate used to connect each replicated level back to its original level.

Returns

dict

Extended rates dictionary containing both the original, replicated, and new connection rates.

Example

>>> rates = {'0->1': 'gamma_in', '0->2': 'gamma_in', '1->0': 'gamma_A', '1->2': 'gamma_spin + gamma_R', '2->1': 'gamma_spin'}
>>> m_op = ['m_op[0]', 'm_op[1]', 'm_op[2]']
>>> gamma_det = 'gamma_det'
>>> extension_for_single_photon(rates, m_op, gamma_det)
{
    '0->1': 'gamma_in',
    '0->2': 'gamma_in',
    '1->0': 'gamma_A',
    '1->2': 'gamma_spin + gamma_R',
    '2->1': 'gamma_spin',
    '3->4': 'gamma_in',
    '3->5': 'gamma_in',
    '4->3': 'gamma_A',
    '4->5': 'gamma_spin + gamma_R',
    '5->4': 'gamma_spin',
    '0->3': 'm_op[0]',
    '1->4': 'm_op[1]',
    '2->5': 'm_op[2]',
    '3->0': 'gamma_det',
    '4->1': 'gamma_det',
    '5->2': 'gamma_det'
}

first_matrix_step(rho, omega)

Helper method to move function out of the class. njit is not working within classes

matrix_step(rho, omega)

Helper method to move function out of the class. njit is not working within classes

plot_all(f_max=None)

Method for quick plotting of polyspectra

Parameters

f_maxfloat

Maximum frequencies upto which the spectra should be plotted

Returns

Returns matplotlib figure

replicate_and_extend_rates(rates)

Replicates the rates dictionary by adding the highest state number + 1 to all states. The replicated rates are then added to the original rates dictionary.

Parameters

ratesdict

Original rates dictionary where keys are in ‘from_state->to_state’ format and values are the rates.

Returns

dict

Extended rates dictionary containing both the original and replicated rates.

Example

>>> rates = {'0->1': 'gamma_in', '0->2': 'gamma_in', '1->0': 'gamma_A', '1->2': 'gamma_spin + gamma_R', '2->1': 'gamma_spin'}
>>> replicate_and_extend_rates(rates)
{'0->1': 'gamma_in',
 '0->2': 'gamma_in',
 '1->0': 'gamma_A',
 '1->2': 'gamma_spin + gamma_R',
 '2->1': 'gamma_spin',
 '3->4': 'gamma_in',
 '3->5': 'gamma_in',
 '4->3': 'gamma_A',
 '4->5': 'gamma_spin + gamma_R',
 '5->4': 'gamma_spin'}

save_spec(path)

Save System class with spectral data

Parameters

pathstr

Location of file

second_matrix_step(rho, omega, omega2)

Helper method to move function out of the class. njit is not working within classes

transform_m_op(old_m_op)

Transforms the input array old_m_op to a new array new_m_op. The new array is twice as long, with the original entries set to 0, and the new entries set to 1.

Parameters

old_m_opnumpy.ndarray

Original m_op array.

Returns

new_m_opnumpy.ndarray

Transformed m_op array.

Example

>>> old_m_op = np.array([1, 0, 0])
>>> transform_m_op(old_m_op)
array([0, 0, 0, 1, 1, 1])

src.markovanalyzer.polyspectra_calculator.pickle_save(path, obj)

Helper function to pickle system objects

Parameters

pathstr

Location of saved data

obj : System obj

src.markovanalyzer.polyspectra_calculator.rates_to_matrix(rates)

Convert a dictionary of rates to a continuous-time Markov process transition rate matrix.

Parameters

ratesdict

Dictionary containing the rates for each transition. Keys should be in the format ‘from_state->to_state’ and values should be non-negative floats representing the rates.

Returns

np.ndarray

A NumPy array representing the transition rate matrix.

Raises

ValueError

If the keys in the dictionary are not in the ‘from_state->to_state’ format or if rates are negative.

Example

>>> rates = {"1->2": 0.5, "2->1": 0.7, "2->3": 0.2}
>>> rates_to_matrix(rates)
array([[-0.5,  0.5,  0. ],
       [ 0.7, -0.9,  0.2],
       [ 0. ,  0. ,  0. ]])

src.markovanalyzer.polyspectra_calculator.second_term(omega1, omega2, omega3, s_k, eigvals, enable_gpu)

For the calculation of the erratum correction terms of the S4. Calculates the second sum as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

Parameters

enable_gpubool

Specify if GPU should be used

omega1float

Frequency of interest

omega2float

Frequency of interest

omega3float

Frequency of interest

s_karray

Array calculated with :func:small_s

eigvalsarray

Eigenvalues of the Liouvillian

Returns

out_sumarray

Second correction term as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

src.markovanalyzer.polyspectra_calculator.small_s(rho_steady, a_prim, eigvecs, eigvec_inv, enable_gpu, zero_ind, gpu_zero_mat)

For the calculation of the erratum correction terms of the S4. Calculates the small s (Eq. 7) from 10.1103/PhysRevB.102.119901

Parameters

zero_indint

Index of the steadystate eigenvector

enable_gpubool

Specify if GPU should be used

gpu_zero_mataf array

Zero array stored on the GPU

rho_steadyarray

A @ Steadystate density matrix of the system

a_primarray

Super operator A’ as defined in 10.1103/PhysRevB.98.205143

eigvecsarray

Eigenvectors of the Liouvillian

eigvec_invarray

The inverse eigenvectors of the Liouvillian

reshape_indarray

Indices that give the trace of a flattened matrix.

Returns

s_karray

Small s (Eq. 7) from 10.1103/PhysRevB.102.119901

src.markovanalyzer.polyspectra_calculator.third_term(omega1, omega2, omega3, s_k, eigvals, enable_gpu)

For the calculation of the erratum correction terms of the S4. Calculates the third sum as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

Parameters

enable_gpubool

Specify if GPU should be used

omega1float

Frequency of interest

omega2float

Frequency of interest

omega3float

Frequency of interest

s_karray

Array calculated with :func:small_s

eigvalsarray

Eigenvalues of the Liouvillian

Returns

out_sumarray

Third correction term as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

src.markovanalyzer.fitting_tools.arcsinh_scaling(s_data, arcsinh_const, order, s_err=None, s_err_p=None, s_err_m=None)

Helper function to improve visibility in plotting (similar to a log scale but also works for negative values)

Parameters

s_dataarray

spectral values of any order

arcsinh_constfloat

these parameters sets the rescaling amount (the smaller, the stronger the rescaling)

orderint

important since the error arrays are called differently in the second-order case

s_errarray

spectral errors of order 3 or 4

s_err_parray

spectral values + error of order 2

s_err_marray

spectral values - error of order 2

Returns

src.markovanalyzer.gpu_backend.second_term_gpu(omega1, omega2, omega3, s_k, eigvals)

For the calculation of the erratum correction terms of the S4. Calculates the second sum as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

Parameters

omega1float

Frequency of interest

omega2float

Frequency of interest

omega3float

Frequency of interest

s_karray

Array calculated with :func:small_s

eigvalsarray

Eigenvalues of the Liouvillian

Returns

out_sumarray

Second correction term as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

src.markovanalyzer.gpu_backend.third_term_gpu(omega1, omega2, omega3, s_k, eigvals)

For the calculation of the erratum correction terms of the S4. Calculates the third sum as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

Parameters

omega1float

Frequency of interest

omega2float

Frequency of interest

omega3float

Frequency of interest

s_karray

Array calculated with :func:small_s

eigvalsarray

Eigenvalues of the Liouvillian

Returns

out_sumarray

Third correction term as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

src.markovanalyzer.model_comparison_tools.compare_models(result1, result2)

Compare two lmfit.MinimizerResult objects.

Parameters: - result1: lmfit.MinimizerResult for the simpler model - result2: lmfit.MinimizerResult for the complex model

Returns: - A dictionary with AIC, BIC for each model and the p-value for the Likelihood Ratio Test

src.markovanalyzer.njit_backend.second_term_njit(omega1, omega2, omega3, s_k, eigvals)

For the calculation of the erratum correction terms of the S4. Calculates the second sum as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

Parameters

omega1float

Frequency of interest

omega2float

Frequency of interest

omega3float

Frequency of interest

s_karray

Array calculated with :func:small_s

eigvalsarray

Eigenvalues of the Liouvillian

Returns

out_sumarray

Second correction term as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

src.markovanalyzer.njit_backend.third_term_njit(omega1, omega2, omega3, s_k, eigvals)

For the calculation of the erratum correction terms of the S4. Calculates the third sum as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.

Parameters

omega1float

Frequency of interest

omega2float

Frequency of interest

omega3float

Frequency of interest

s_karray

Array calculated with :func:small_s

eigvalsarray

Eigenvalues of the Liouvillian

Returns

out_sumarray

Third correction term as defined in Eq. 109 in 10.1103/PhysRevB.102.119901.