Pulses and Waveforms

What you will learn:

• how Pulses are used to define the driving Hamiltonian;

• what are Waveforms and which options you can choose from;

• how to create a Pulse;

• some helpful tips to keep in mind when creating a Pulse.

Programming the driving Hamiltonian

To program the driving Hamiltonian of a system with \(N\) atoms in Pulser, one needs to combine two objects:

• The Channel, which defines

  • the addressed basis, i.e. the states \(|a\rangle\) and \(|b\rangle\)) and

  • which atoms are targeted. i.e. for which atom(s) in \(i \in \{1,...,N\}\) is \(\Omega_i\), \(\delta_i\) and \(\phi_i\) defined.

• The Pulse, which defines, over a given duration \(\Delta t\),

  • the Rabi frequency, \(\Omega(t \rightarrow t+\Delta t)\) (given as a Waveform);

  • the detuning, \(\delta(t \rightarrow t+\Delta t)\) (also given as a Waveform);

  • the phase, \(\phi\), which is constant from \(t\) to \(t+\Delta t\).

Note

By adding pulses to channels, the full driving Hamiltonian is defined over the entire duration of the Sequence.

Waveforms

The Waveform base class

In Pulser, a Waveform defines some time-dependent parameter over a certain duration. Every Waveform has two key properties:

• its duration, which is an integer value in \(ns\);

• its samples, which define the Waveform’s value at each \(ns\).

Note

In Pulser, samples are always defined with a time step of 1 ns. This means that, to access a value at t=x #ns, one can simply get samples[x].

Available Waveforms

To create a Waveform, one must use one of its subclasses:

ConstantWaveform(duration, value)	A waveform of constant value.
RampWaveform(duration, start, stop)	A linear ramp waveform.
BlackmanWaveform(duration, area)	A Blackman window of a specified duration and area.
InterpolatedWaveform(duration, values[, …])	A waveform created from interpolation of a set of data points.
CustomWaveform(samples)	A custom waveform.
KaiserWaveform(duration, area[, beta])	A Kaiser window of a specified duration and beta parameter.
CompositeWaveform(*waveforms)	A waveform combining multiple smaller waveforms.

Pulses

Standard definition

To create a Pulse, one must define \(\Omega(t)\), \(\delta(t)\) and \(\phi\) over a duration \(\Delta t\). While the phase \(\phi\) must be constant in each Pulse, \(\Omega\) and \(\delta\) are time-dependent; as such, they are defined as waveforms.

Note

In a Pulse, \(\Omega(t)\) and \(\delta(t)\) are always in units of \(rad/\mu s\), while \(\phi\) is in \(rad\).

As an example, below is a 500 \(ns\) Pulse, with amplitude given by a BlackmanWaveform of area \(\pi\), detuning given by a RampWaveform from -10 to 10 \(rad/\mu s\) and a phase of \(\pi/2\).

Note

In Pulser, Rabi frequency and amplitude are equivalent terms for \(\Omega\) and are used interchangeably.

Caution

In the same Pulse, the amplitude and detuning waveforms must have the same duration.

import numpy as np
import pulser

pulse = pulser.Pulse(
    amplitude=pulser.BlackmanWaveform(500, np.pi),
    detuning=pulser.RampWaveform(500, -10, 10),
    phase=np.pi / 2,
)
pulse.draw()  # Draws the Pulse

Shortcuts for constant parameters

When the amplitude or detuning are constant, these class methods avoid having to use ConstantWaveform:

Pulse.ConstantAmplitude	Pulse with a constant amplitude and a detuning waveform.
Pulse.ConstantDetuning	Creates a Pulse with an amplitude waveform and a constant detuning.
Pulse.ConstantPulse	Pulse with a constant amplitude and a constant detuning.

Note

• In Pulse.ConstantAmplitude() and Pulse.ConstantDetuning(), the pulse duration is taken from the Waveform parameter.

• In Pulse.ConstantPulse(), duration must be explicitly given.

Below is an example of these methods in action, all of them creating the same 1000 \(ns\) pulse with \(\Omega=1~rad/\mu s\), \(\delta=-1~rad/\mu s\) and \(\phi=0\).

import pulser

const1 = pulser.ConstantWaveform(1000, 1)  # A constant waveform of 1000 ns
pulse1 = pulser.Pulse.ConstantAmplitude(
    amplitude=1,  # float
    detuning=-const1,  # Waveform
    phase=0,  # float
)
pulse2 = pulser.Pulse.ConstantDetuning(
    amplitude=const1,  # Waveform
    detuning=-1,  # float
    phase=0,  # float
)
pulse3 = pulser.Pulse.ConstantPulse(
    duration=1000,  # int
    amplitude=1,  # float
    detuning=-1,  # float
    phase=0,  # float
)
assert pulse1 == pulse2 == pulse3

Tips for Pulse creation

Keep the Channel constraints in mind

Some Channel parameters dictate what is allowed in a Pulse once it’s added to the Sequence, so it is often useful to have these limitations in mind when first designing the Pulse. In particular, you should keep in mind:

• Channel.max_abs_detuning, the maximum absolute value of detuning allowed;

• Channel.max_amp, the maximum amplitude allowed;

• Channel.min_avg_amp, the minimum average amplitude allowed (when not zero);

• Channel.min_duration, the minimum pulse duration allowed;

• Channel.clock_period, which dictates that every pulse’s duration must be a multiple of this value.

Remember that waveforms can be concatenated

When programming \(\Omega(t)\) and \(\delta(t)\) with Pulser, it’s usually preferable to divide these quantities into a stream of simple pulses. However, this is not always convenient, as the natural breaking point in the amplitude and detuning waveforms may not always match. In these cases, the CompositeWaveform allows for the creation of a more complex waveform by concatenation of multiple, smaller waveforms. Take a look a this page to see how CompositeWaveform might help you.