LazyTensor and FFTOperators ?

[savowe]

Hello.
Is it currently possible to achieve time-propagation of a compoung system with internal and external degrees of freedom ( e.g. Hilbert space of NLevelBasis(2)⊗PositionBasis(x_min, x_max, N_x), with a Hamiltonian for the external parts in the form 1⊗V(x)+1⊗p^2/2) using the FFT Operators and not numerical diagonalization?
The LazyTensor is only able to work on sparse and dense operators and therefore need to be used before application of LazyProduct and LazySum.
However, one can not make a tensorproduct of a sparse and a FFTOperator:

using QuantumOptics
internal_basis = NLevelBasis(2)
position_basis = PositionBasis(-5, 5, 1024)
momentum_basis = MomentumBasis(position_basis)
Tpx = transform(momentum_basis, position_basis)
Txp = dagger(Tpx)
x = position(position_basis)
p = momentum(momentum_basis)
# LazyTensor of internal degree of freedom and kinetic operator in momentum basis (meant as 1⊗V(x)+1⊗p^2/2)
H_kin_p = LazyTensor(internal_basis⊗momentum_basis ,[2], [p])^2/2
# The naive thing does not work:
H_kin_x = LazyProduct(identityoperator(internal_basis) ⊗ Txp, H_kin_p, identityoperator(internal_basis) ⊗ Tpx)

Is there a workaround for that?

[david-pl]

The trick is to create the FFTOperator directly on the composite basis and then use a LazyProduct.
Changing your code like this should work:

using QuantumOptics
internal_basis = NLevelBasis(2)
position_basis = PositionBasis(-5, 5, 1024)
momentum_basis = MomentumBasis(position_basis)
Tpx = transform(internal_basis⊗momentum_basis, internal_basis⊗position_basis)
Txp = dagger(Tpx)
x = position(position_basis)
p = momentum(momentum_basis)
H_kin_p = one(internal_basis)⊗p
H_kin_x = LazyProduct(Txp, H_kin_p, Tpx)

[savowe]

Indeed, it works. Thank you