generators.md

Dynamical Generators

The [propagate](@ref QuantumPropagators.propagate) routine simulates the dynamics of a state |Ψ⟩ or ρ̂ under the assumption that the dynamics are described by a generator object. The exact equation of motion is implicit in the [Propagator](@ref QuantumPropagators.AbstractPropagator), but all propagators implemented in the QuantumPropagators package assume that the generator is the time-dependent Hamiltonian Ĥ(t) in the Schrödinger equation (ħ=1)

<a id="eq-se"></a>

$$\tag{SE} i \frac{∂}{∂t} |Ψ⟩ = Ĥ(t) |Ψ⟩\,.$$

Operators

When evaluating the right-hand-side of Eq. (SE), the time-dependent generator Ĥ(t) is first evaluated into a static operator object Ĥ for a specific point in time via the QuantumPropagators.Controls.evaluate function. The 5-argument LinearAlgebra.mul! then implements the application of the operator to the state |Ψ⟩.

Hamiltonians

The built-in hamiltonian function initializes a [Generator](@ref QuantumPropagators.Generators.Generator) object encapsulating a generator of the most general form

<a id="eq-generator"></a>

$$\tag{Generator} Ĥ(t) = Ĥ_0 + \sum_l a_l(\{ϵ_{l'}(t)\}, t) \, Ĥ_l\,.$$

The Hamiltonian consists of an (optional) drift term Ĥ_0 and an arbitrary number of control terms that separate into a scalar control amplitude a_l(\{ϵ_{l'}(t)\}, t) and a static control operator Ĥ_l. Each control amplitude may consist of one or more control function ϵ_{l'}(t). Most commonly, a_l(t) ≡ ϵ_l(t), and thus the Hamiltonian is of the simpler form

$$Ĥ(t) = Ĥ_0 + \sum_l ϵ_l(t) \, Ĥ_l\,.$$

The [evaluate](@ref QuantumPropagators.Controls.evaluate) function evaluates time-dependent [Generator](@ref QuantumPropagators.Generators.Generator) instances into static [Operator](@ref QuantumPropagators.Generators.Operator) objects.

[Control Amplitudes](@id ControlAmplitudes)

The distinction between control amplitudes and control functions becomes important only in the context of optimal control, where the control functions are directly modified by optimal control, whereas the control amplitudes determine how the control functions couple to the control operators, or account for explicit time dependencies in the Hamiltonian.

Just as the [evaluate](@ref QuantumPropagators.Controls.evaluate) function evaluates time-dependent generators into static operators, it also evaluates control amplitudes or control functions to scalar values.

Liouvillians

In an open quantum system, the equation of motion is assumed to take the exact same form as in Eq. (SE),

$$i \frac{∂}{∂t} ρ̂ = L(t) ρ̂\,,$$

where L is the Liouvillian up to a factor of i.

The object representing L should be constructed with the liouvillian function, with convention=:TDSE. Just like hamiltonian, this returns a [Generator](@ref QuantumPropagators.Generators.Generator) instance that [evaluate](@ref QuantumPropagators.Controls.evaluate) turns into a static [Operator](@ref QuantumPropagators.Generators.Operator) to be applied to a vectorized (!) state ρ̂.

Arbitrary Generators

For an "unusual" generator, first decide at which level to address the issue:

• If there is no optimal control being done, it may be sufficient to define a new control object directly. The required interface is defined in QuantumPropagators.Interfaces.check_control.
• For e.g. non-linear controls, it is enough to define a new [control amplitude](@ref ControlAmplitudes) type (representing the a_l in Eq. (Generator)). See QuantumPropagators.Interfaces.check_amplitude for the required interface. Some amplitudes that are useful in the context of optimal control defined in QuantumPropagators.Amplitudes. Moreover, QuantumControl.PulseParametrizations provides the tools to define amplitudes of the form a_l(t) = a(ϵ_l(t)). That is, anything where the value of a_l(t) depends directly on the value of ϵ_l(t)). This includes nonlinear controls such as ϵ^2(t).
• For any Hamiltonian or Liouvillian that is more general than the form in Eq. (Generator), a fully custom generator type would have to be implemented, see below.

In general, the [methods defined in the QuantumPropagators.Controls module](@ref QuantumPropagatorsControlsAPI) (respectively QuantumControl.Controls in the broader context of optimal control) determine the relationship between generators, operators, amplitudes, and controls and must be implemented for any custom types.

In particular,

• QuantumPropagators.Controls.get_controls extracts the time-dependent control functions from a generator.
• QuantumPropagators.Controls.evaluate and [evaluate!](@ref QuantumPropagators.Controls.evaluate!) convert time-dependent generators into static operators, or control amplitudes/functions into scalar values.
• QuantumPropagators.Controls.substitute substitute control amplitudes or control functions for other control amplitudes or control functions.

See QuantumPropagators.Interfaces.check_generator for the full required interface (or the extended QuantumControl.Interfaces.check_generator that also includes the methods required for optimal control).