cosine_pulse

boulderopal.signals.cosine_pulse(duration, amplitude, drag=0.0, start_time=0.0, end_time=None, flat_duration=0.0)

Create a Signal object representing a cosine pulse.

Parameters:

duration (float) – The duration of the pulse.
amplitude (float or complex) – The amplitude of the pulse, \(A\).
drag (float, optional) – The DRAG parameter, \(\beta\). Defaults to 0, in which case there is no DRAG correction.
start_time (float, optional) – The time at which the cosine pulse starts, \(t_\mathrm{start}\). Defaults to 0.
end_time (float or None, optional) – The time at which the cosine pulse ends, \(t_\mathrm{end}\). Defaults to the duration.
flat_duration (float, optional) – The amount of time that the pulse remains constant after the peak of the cosine, \(t_\mathrm{flat}\). If passed, it must be nonnegative and less than the difference between end_time and start_time. Defaults to 0, in which case no constant part is added to the cosine pulse.

Returns:

The cosine pulse.

Return type:

Signal

See also

boulderopal.signals.gaussian_pulse: Create a Signal object representing a Gaussian pulse.
boulderopal.signals.hann_series: Create a Signal object representing a sum of Hann window functions.
boulderopal.signals.sech_pulse: Create a Signal object representing a hyperbolic secant pulse.
boulderopal.signals.sinusoid: Create a Signal object representing a sinusoidal oscillation.
boulderopal.signals.square_pulse: Create a Signal object representing a square pulse.
Graph.signals.cosine_pulse_pwc: Graph operation to create a Pwc representing a cosine pulse.

Notes

The cosine pulse is defined as

\[\begin{split}\mathop{\mathrm{Cos}}(t) = \begin{cases} 0 &\mathrm{if} \quad t < t_\mathrm{start} \\ \frac{A}{2} \left[1+\cos \left(\omega \{t-\tau_-\} \right) + i\omega\beta \sin \left(\omega \{t-\tau_-\}\right)\right] &\mathrm{if} \quad t_\mathrm{start} \le t < \tau_- \\ A &\mathrm{if} \quad \tau_- \le t \le \tau_+ \\ \frac{A}{2} \left[1+\cos \left(\omega\{t-\tau_+\}\right) + i\omega \beta\sin \left(\omega \{t-\tau_+\}\right)\right] &\mathrm{if} \quad \tau_+ < t \le t_\mathrm{end} \\ 0 &\mathrm{if} \quad t > t_\mathrm{end} \\ \end{cases},\end{split}\]

where \(\omega=2\pi /(t_\mathrm{end}-t_\mathrm{start} - t_\mathrm{flat})\), \(\tau_\mp\) are the start/end times of the flat segment, with \(\tau_\mp=(t_\mathrm{start}+t_\mathrm{end} \mp t_\mathrm{flat})/2\).

If the flat duration is zero (the default setting), this reduces to

\[\mathop{\mathrm{Cos}}(t) = \frac{A}{2} \left[1+\cos \left(\omega \{t-\tau\} \right) + i\omega\beta \sin \left(\omega \{t-\tau\}\right)\right] \theta(t-t_\mathrm{start}) \theta(t_\mathrm{end}-t),\]

where now \(\omega=2\pi /(t_\mathrm{end}-t_\mathrm{start})\), \(\tau=(t_\mathrm{start}+t_\mathrm{end})/2\) and \(\theta(t)\) is the Heaviside step function.

Examples

Define a cosine pulse.

>>> pulse = bo.signals.cosine_pulse(duration=3.0, amplitude=1.0)
>>> pulse.export_with_time_step(time_step=0.5)
array([0.0669873+0.j, 0.5      +0.j, 0.9330127+0.j, 0.9330127+0.j,
       0.5      +0.j, 0.0669873+0.j])

Define a flat-top cosine pulse with a DRAG correction.

>>> pulse = bo.signals.cosine_pulse(
...     duration=3.0, amplitude=1.0, drag=0.1, flat_duration=0.6
... )
>>> pulse.export_with_sampling_rate(sampling_rate=2.0)
array([0.10332333-0.07968668j, 0.69134172-0.12093555j,
       1.        +0.j        , 1.        +0.j        ,
       0.69134172+0.12093555j, 0.10332333+0.07968668j])