TimeAxis#

Class representing time in time dependent calculations.

User level function of the Quantarhei package. To be used as:

>>> import quantarhei as qr
>>> time = qr.TimeAxis()

The TimeAxis class stands in a close relation to FrequencyAxis class of the quantarhei package. FrequencyAxis represents the frequencies one obtains in the Fast Fourier transform of a function of the TimeAxis. By default, TimeAxis is of the type upper-half which means that by specifying the start, length and step we represent the upper half of the interval <start-length*step, start+(length-1)*step>. The Fourier transform of a time dependent object defined on the TimeAxis will then have twice as many points as the TimeAxis. This is usefull when the time dependent object has some special symmetries. One example is the so-called quantum bath correlation function which fulfills the relation (in LaTeX)

C(-t) = C^{*}(t)

Examples

Default TimeAxis is of the ‘upper-half’ type

>>> ta = TimeAxis(0.0,100,0.1)
>>> ta.atype
'upper-half'

It is defined between the start and start+(length-1)*step

>>> '%.3f' % ta.min
'0.000'

>>> '%.3f' % ta.max
'9.900'

However, when we ask for the corresponding FrequencyAxis, we get an array of frequencies which is twice as long and contains 2*length points

>>> wa = ta.get_FrequencyAxis()
>>> print(wa.length)
200

The frequency step is therefore twice shorter than one would normally expect.

>>> print(wa.step == wa.step)
True
>>> import numpy
>>> print(numpy.allclose(wa.step,2.0*numpy.pi/(0.1*200)))
True

This definition of the TimeAxis is used in conjunction with the symmetries of the corelation functions, lineshape functions and response functions.

In some case you want the ‘complete’ type of the TimeAxis.

>>> ta = TimeAxis(0.0,100,1.0,atype="complete")
>>> ta.atype
'complete'

In this case the relation of the TimeAxis and the FrequencyAxis is more straightforward. Now, the TimeAxis represents the interval on which a time dependent object is defined completely. The number of points in the corresponding FrequencyAxis is the same as in the TimeAxis and the frequency step is as expected from a normal FFT.

>>> wa = ta.get_FrequencyAxis()
>>> print(wa.length)
100

>>> print(wa.step == wa.step)
True

>>> import numpy
>>> print(numpy.allclose(wa.step,2.0*numpy.pi/(1.0*100)))
True

No other types than complete and upper-half are defined at the moment. The following with throw an Exception

>>> ta = TimeAxis(0.0,100,1.0,atype="lower-half")
Traceback (most recent call last):
...
quantarhei.exceptions.QuantarheiError: Unknown time axis type

Relation between TimeAxis and FrequencyAxis#

Complete TimeAxis and even number of points

>>> ta = TimeAxis(0.0,10,0.1,atype="complete")
>>> times = [0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9]
>>> print(numpy.allclose(ta.data,times))
True

>>> wa = ta.get_FrequencyAxis()
>>> freques = 2.0*numpy.pi*numpy.fft.fftshift(numpy.fft.fftfreq(10,0.1))
>>> print(numpy.allclose(wa.data,freques))
True

>>> print(numpy.allclose(wa.step,freques[1]-freques[0]))
True

>>> tb = wa.get_TimeAxis()
>>> print(numpy.allclose(tb.data,times))
True

>>> wb = tb.get_FrequencyAxis()
>>> print(numpy.allclose(wb.data,freques))
True

Complete TimeAxis and odd number of points

>>> ta = TimeAxis(0.0,11,0.1,atype="complete")
>>> times = [0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]
>>> print(numpy.allclose(ta.data,times))
True

>>> wa = ta.get_FrequencyAxis()
>>> freques = 2.0*numpy.pi*numpy.fft.fftshift(numpy.fft.fftfreq(11,0.1))
>>> print(numpy.allclose(wa.data,freques))
True

>>> print(numpy.allclose(wa.step,freques[1]-freques[0]))
True

>>> tb = wa.get_TimeAxis()
>>> print(numpy.allclose(tb.data,times))
True

>>> wb = tb.get_FrequencyAxis()
>>> print(numpy.allclose(wb.data,freques))
True

Upper-half TimeAxis and even number of points

>>> ta = TimeAxis(0.0,10,0.1)
>>> print(ta.atype=="upper-half")
True

>>> times = [0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9]
>>> print(numpy.allclose(ta.data,times))
True

>>> wa = ta.get_FrequencyAxis()
>>> freques = 2.0*numpy.pi*numpy.fft.fftshift(numpy.fft.fftfreq(20,0.1))
>>> print(numpy.allclose(wa.data,freques))
True

>>> print(numpy.allclose(wa.step,freques[1]-freques[0]))
True

>>> tb = wa.get_TimeAxis()
>>> print(numpy.allclose(tb.data,times))
True

>>> wb = tb.get_FrequencyAxis()
>>> print(numpy.allclose(wb.data,freques))
True

Upper-half TimeAxis and odd number of points

>>> ta = TimeAxis(0.0,11,0.1)
>>> times = [0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]
>>> print(numpy.allclose(ta.data,times))
True

>>> wa = ta.get_FrequencyAxis()
>>> freques = 2.0*numpy.pi*numpy.fft.fftshift(numpy.fft.fftfreq(2*11,0.1))
>>> print(numpy.allclose(wa.data,freques))
True

>>> print(numpy.allclose(wa.step,freques[1]-freques[0]))
True

>>> tb = wa.get_TimeAxis()
>>> print(numpy.allclose(tb.data,times))
True

>>> wb = tb.get_FrequencyAxis()
>>> print(numpy.allclose(wb.data,freques))
True

Class Details#

class quantarhei.core.time.TimeDependent[source]#

Bases: object

cutoff_time: float | None = None#
set_cutoff_time(time: float) → None[source]#
class quantarhei.core.time.TimeAxis(start: float = 0.0, length: int = 1, step: float = 1.0, atype: str = 'upper-half', frequency_start: float = 0.0)[source]#

Bases: ValueAxis

Linear time grid for time-dependent calculations.

Closely related to FrequencyAxis: the FFT of a function defined on a TimeAxis lives on the corresponding FrequencyAxis. The default type 'upper-half' exploits conjugation symmetry of bath correlation functions; 'complete' gives a straightforward full-range grid.

Parameters:

  • start (float) – First value of the time axis (in internal units, i.e. fs).

  • length (int) – Number of time points.

  • step (float) – Spacing between consecutive time points (fs).

  • atype (str, optional) – Axis type. Must be 'upper-half' (default) or 'complete'.

  • frequency_start (float, optional) – Offset applied to the first frequency of the corresponding FrequencyAxis. Default is 0.0.

Raises:

Exception – If atype is not 'upper-half' or 'complete'.

shift_to_zero() → None[source]#

Shifts the values so that the first one is zero

get_FrequencyAxis() → FrequencyAxis[source]#

Returns corresponding FrequencyAxis object