qml.pulse.rydberg_drive¶
-
rydberg_drive(amplitude, phase, detuning, wires)[source]¶ Returns a
ParametrizedHamiltonianrepresenting the action of a driving laser field\[\frac{1}{2} \Omega(t) \sum_{q \in \text{wires}} (\cos(\phi(t))\sigma_q^x - \sin(\phi(t))\sigma_q^y) - \delta(t) \sum_{q \in \text{wires}} n_q\]where \(\Omega/(2\pi)\), \(\phi\) and \(\delta/(2\pi)\) correspond to the amplitude, phase, and detuning of the laser, \(q\) corresponds to the wire index, and \(\sigma_q^\alpha\) for \(\alpha = x,y\) are the Pauli matrices on the corresponding qubit. Finally, \(n_q=\frac{1}{2}(\mathbb{I}_q-\sigma_q^z)\) is the number operator on qubit \(q\).
Note
For hardware execution, input time is expected to be in units of \(\mu\text{s}\), and the frequency in units of MHz. It is recommended to also follow this convention for simulation, as it avoids numerical problems due to using very large and very small numbers. Frequency inputs will be converted internally to angular frequency, such that
amplitude\(= \Omega(t)/ (2 \pi)\) anddetuning\(= \delta(t) / (2 \pi)\).This driving term can be combined with an interaction term to create a Hamiltonian describing a driven Rydberg atom system. Multiple driving terms can be combined by summing them (see example).
- Parameters
amplitude (Union[float, Callable]) – Float or callable representing the amplitude of a laser field. This should be in units of frequency (MHz), and will be converted to amplitude in angular frequency, \(\Omega(t)\), internally where needed, i.e. multiplied by \(2 \pi\).
phase (Union[float, Callable]) – float or callable representing the phase (in radians) of the laser field
detuning (Union[float, Callable]) – Float or callable representing the detuning of a laser field. This should be in units of frequency (MHz), and will be converted to detuning in angular frequency, \(\delta(t)\), internally where needed, i.e. multiplied by \(2 \pi\).
wires (Union[int, List[int]]) – integer or list containing wire values for the Rydberg atoms that the laser field acts on
- Returns
a
ParametrizedHamiltonianrepresenting the action of the laser field on the Rydberg atoms.- Return type
Example
We create a Hamiltonian describing a laser acting on 4 wires (Rydberg atoms) with a fixed detuning and phase, and a parametrized, time-dependent amplitude. The Hamiltonian includes an interaction term for inter-atom interactions due to van der Waals forces, as well as the driving term for the laser driving the atoms.
We provide all frequencies in the driving term in MHz (conversion to angular frequency, i.e. multiplication by \(2 \pi\), is taken care of internally where needed). Phase (in radians) will not undergo unit conversion.
For the driving field, we specify a detuning of \(\delta = 1 \times 2 \pi \text{MHz}\), and an amplitude \(\Omega(t)\) defined by a sinusoidal oscillation, squared to ensure a positve amplitude (a requirement for some hardware implementations). The maximum amplitude will dependent on the parameter
ppassed to the amplitude function later, and should also be passed in units of MHz. We introduce a small phase shift as well, on the order of 1 rad.atom_coordinates = [[0, 0], [0, 4], [4, 0], [4, 4]] wires = [0, 1, 2, 3] H_i = qml.pulse.rydberg_interaction(atom_coordinates, wires) amplitude = lambda p, t: p * jnp.sin(jnp.pi * t) ** 2 phase = 0.25 detuning = 1. H_d = qml.pulse.rydberg_drive(amplitude, phase, detuning, wires)
>>> H_i HardwareHamiltonian: terms=6 >>> H_d HardwareHamiltonian: terms=3
The first two terms of the drive Hamiltonian
H_dcorrespond to the first sum (the sine and cosine terms), describing drive between the ground and excited states. The third term corresponds to the shift term due to detuning from resonance. This drive term corresponds to a global drive that acts on all 4 wires of the device.The full Hamiltonian evolution and expectation value measurement can be executed in a
QNode:dev = qml.device("default.qubit.jax", wires=wires) @qml.qnode(dev, interface="jax") def circuit(params): qml.evolve(H_i + H_d)(params, t=[0, 0.5]) return qml.expval(qml.Z(0))
Here we set a maximum amplitude of \(2.4 \times 2 \pi \text{MHz}\), and calculate the result of running the pulse program:
>>> params = [2.4] >>> circuit(params) Array(0.78301974, dtype=float64) >>> jax.grad(circuit)(params) [Array(-0.6250622, dtype=float64)]
We can also create a Hamiltonian with local drives. The following circuit corresponds to the evolution where additional local drives acting on wires
0and1respectively are added to the Hamiltonian:amplitude_local_0 = lambda p, t: p[0] * jnp.sin(2 * jnp.pi * t) ** 2 + p[1] phase_local_0 = jnp.pi / 4 detuning_local_0 = lambda p, t: p * jnp.exp(-0.25 * t) H_local_0 = qml.pulse.rydberg_drive(amplitude_local_0, phase_local_0, detuning_local_0, [0]) amplitude_local_1 = lambda p, t: jnp.cos(jnp.pi * t) ** 2 + p phase_local_1 = jnp.pi detuning_local_1 = lambda p, t: jnp.sin(jnp.pi * t) + p H_local_1 = qml.pulse.rydberg_drive(amplitude_local_1, phase_local_1, detuning_local_1, [1]) H = H_i + H_d + H_local_0 + H_local_1 @jax.jit @qml.qnode(dev, interface="jax") def circuit_local(params): qml.evolve(H)(params, t=[0, 0.5]) return qml.expval(qml.Z(0)) p_global = 2.4 p_local_amp_0 = [1.3, -2.0] p_local_det_0 = -1.5 p_local_amp_1 = -0.9 p_local_det_1 = 3.1 params = [p_global, p_local_amp_0, p_local_det_0, p_local_amp_1, p_local_det_1]
>>> circuit_local(params) Array(0.62640288, dtype=float64) >>> jax.grad(circuit_local)(params) [Array(1.07614151, dtype=float64), [Array(0.36370049, dtype=float64, weak_type=True), Array(0.91057498, dtype=float64, weak_type=True)], Array(1.3166343, dtype=float64), Array(-0.11102892, dtype=float64), Array(-0.02205843, dtype=float64)]