SpectralDensity#

The SpectralDensity class represents the spectral density of a bath coupled to a quantum system. It is closely related to CorrelationFunction — the two are related by a Fourier transform.

Quantarhei package (http://www.github.com/quantarhei)

spectraldensities module

class quantarhei.qm.corfunctions.spectraldensities.SpectralDensity(axis: Any = None, params: Any = None, values: Any = None)[source]#

Bases: DFunction, UnitsManaged

Spectral density of a system-bath coupling.

Stores the bath spectral density \(J(\omega)\) as a discrete function on a frequency grid. Can be constructed from the same parameter dictionaries as CorrelationFunction.

Parameters:

axis (TimeAxis or FrequencyAxis) – Frequency grid on which the spectral density is defined, either directly as a FrequencyAxis or derived from a TimeAxis via FFT.
params (dict or list of dict) – Parameter dictionary (or list of dictionaries for composite spectral densities). Each dictionary must contain at least 'ftype' and 'reorg'.
values (numpy.ndarray, optional) – If provided, use these pre-computed values instead of evaluating the analytical form.

Examples

SpectralDensity object can be ctreated with the same parameters as CorrelationFunction. The temperature can be set, but it is not a compulsory parameter.

>>> from quantarhei import TimeAxis
>>> params = dict(ftype="OverdampedBrownian", cortime=100, reorg=20, T=300)
>>> time = TimeAxis(0.0,1000,1.0)
>>> with energy_units("1/cm"):           sd = SpectralDensity(time, params)

>>> cf = sd.get_CorrelationFunction()

>>> print(sd.get_temperature())
300

If we create the same without temperature

>>> parwoT = dict(ftype="OverdampedBrownian", cortime=100, reorg=20)
>>> with energy_units("1/cm"):           sdwoT = SpectralDensity(time, parwoT)

everything is alright. CorrelationFunction, however, cannot be created as above, because temperature must be known. Attempt to create CorrelationFunction as above would lead to an exception. We have to specify temperature as a parameter to the get_CorrelationFunction method.

>>> cf = sdwoT.get_CorrelationFunction(temperature=300)

Reorganization of the spectral density is an input parameter which can be obtained by calling the corresponding method

>>> with energy_units("1/cm"):
...     print(sdwoT.get_reorganization_energy())
20.0

At the same time, reorganization energy can be calculated from the shape of the spectral density by integrating over it. The accuracy of such estimation depends on numerics, hence the relative tolerance of only 1.0e-2 below

>>> lamb_definition = sd.get_reorganization_energy()
>>> lamb_measured = sd.measure_reorganization_energy()
>>> print(numpy.allclose(lamb_definition, lamb_measured, rtol=1.0e-2))
True

energy_params = ('reorg', 'omega', 'freq', 'fcp', 'g_FWHM', 'l_FWHM', 'freq1', 'freq2', 'gamma')#

analytical_types = 'OverdampedBrownian'#

axis: FrequencyAxis#

add_to_data(other: SpectralDensity) → None[source]#: Addition of data from a specified CorrelationFunction to this object

add_to_data2(other: SpectralDensity) → None[source]#: Addition of data from a specified SpectralDensity to this object

is_analytical() → bool[source]#

Returns True if analytical

Returns True if the CorrelationFunction object is constructed by analytical formula. Returns False if the object was constructed by numerical transformation from spectral density.

get_temperature() → float[source]#: Returns the temperature of the correlation function

get_reorganization_energy() → float | ndarray[source]#: Returns the reorganization energy of the cspectral density

measure_reorganization_energy() → float[source]#

Calculates the reorganization energy of the spectral density

Calculates the reorganization energy of the spectral density by integrating over frequency.

copy() → SpectralDensity[source]#: Creates a copy of the current correlation function

get_CorrelationFunction(temperature: float | None = None, ta: Any = None) → CorrelationFunction[source]#: Returns correlation function corresponding to the spectral density. If a TimeAxis object is included, the CorrelationFunction object will be returned with that TimeAxis instance as its time axis.

get_FTCorrelationFunction(temperature: float | None = None) → FTCorrelationFunction[source]#

Returns Fourier transformed correlation function

Fourier transformed correlation function is calculated from the analytical formula connecting spectral density and FT correlation function.

Parameters:: temperature (optional) – Temperature which can be missing among the spectral density parameters