# cosine_pulse

The Boulder Opal Toolkits are currently in beta phase of development. Breaking changes may be introduced.

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, 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

signals.cosine_pulse_pwc()

Graph operation to create a Pwc representing a cosine pulse.

signals.gaussian_pulse()

Create a Signal object representing a Gaussian pulse.

signals.hann_series()

Create a Signal object representing a sum of Hann window functions.

signals.sech_pulse()

Create a Signal object representing a hyperbolic secant pulse.

signals.sinusoid()

Create a Signal object representing a sinusoidal oscillation.

signals.square_pulse()

Create a Signal object representing a square 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 = qctrl.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 = qctrl.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])