Applied example
In this example, we will create the spin-boson Hamiltonian we have used for open-system research in our paper, for 1 spin and 3 bosonic modes.
The Hamiltonian is as follows: \[ \hat{H} = \hat{H}_S + \hat{H}_B + \hat{H}_C \]
with the spin system Hamiltonian \(\hat{H}_S\) :
\[ \hat{H} = \frac {\hbar \Delta} {2} \sigma^z_0, \]
the bosonic bath Hamiltonian \(\hat{H}_B\) :
\[ \hat{H} = \sum_{k=0}^2 \hbar \omega_k c_k^{\dagger} c_k, \]
and the coupling between system and bath \(\hat{H}_C\) :
\[ \hat{H} = \sigma_0^x \sum_{k=0}^2 \frac {v_k} {2} \left( c_k + c_k^{\dagger} \right) \]
For simplicity, we will set \(\hbar\) to 1.0 for this example.
Rust implementation:
use qoqo_calculator::CalculatorComplex;
use struqture::bosons::BosonProduct;
use struqture::mixed_systems::{
HermitianMixedProduct, MixedHamiltonianSystem,
};
use struqture::prelude::*;
use struqture::spins::PauliProduct;
let mut system = MixedHamiltonianSystem::new([Some(1)], [Some(3)], []);
// Setting up constants:
let delta = 1.0;
let omega_k = [2.0, 3.0, 4.0];
let v_k = [5.0, 6.0, 7.0];
// First, H_S:
let pp = PauliProduct::new().z(1);
let hmp = HermitianMixedProduct::new(
[pp], [BosonProduct::new([], []).unwrap()], []
).unwrap();
system
.add_operator_product(hmp, CalculatorComplex::new(delta / 2.0, 0.0))
.unwrap();
// Second, H_B:
for k in 0..3 {
let bp = BosonProduct::new([k], [k]).unwrap();
let hmp = HermitianMixedProduct::new(
[PauliProduct::new()], [bp], []
).unwrap();
system
.add_operator_product(
hmp, CalculatorComplex::new(v_k[k] / 2.0, 0.0)
).unwrap();
}
// Third, H_C: the hermitian conjugate is implicitly stored,
// we don't need to add it manually
let pp = PauliProduct::new().x(0);
for k in 0..3 {
let bp = BosonProduct::new([], [k]).unwrap();
let hmp = HermitianMixedProduct::new([pp.clone()], [bp], []).unwrap();
system
.add_operator_product(
hmp, CalculatorComplex::new(omega_k[k], 0.0)
).unwrap();
}
// Our resulting H:
println!("{}", system);
Python implementation:
from qoqo_calculator_pyo3 import CalculatorComplex
from struqture_py.bosons import BosonProduct
from struqture_py.mixed_systems import (
HermitianMixedProduct, HermitianMixedProduct, MixedHamiltonianSystem,
)
from struqture_py.spins import (PauliProduct, PauliProduct)
system = MixedHamiltonianSystem([1], [3], [])
# Setting up constants:
delta = 1.0
omega_k = [2.0, 3.0, 4.0]
v_k = [5.0, 6.0, 7.0]
# First, H_S:
pp = PauliProduct().z(1)
hmp = HermitianMixedProduct([pp], [BosonProduct([], [])], [])
system.add_operator_product(
hmp, CalculatorComplex.from_pair(delta / 2.0, 0.0)
)
# Second, H_B:
for k in range(3):
bp = BosonProduct([k], [k])
hmp = HermitianMixedProduct([PauliProduct()], [bp], [])
system.add_operator_product(
hmp, CalculatorComplex.from_pair(v_k[k] / 2.0, 0.0)
)
# Third, H_C: the hermitian conjugate is implicitly stored,
# we don't need to add it manually
pp = PauliProduct().x(0)
for k in range(3):
bp = BosonProduct([], [k])
hmp = HermitianMixedProduct([pp], [bp], [])
system.add_operator_product(
hmp, CalculatorComplex.from_pair(omega_k[k], 0.0)
)
# Our resulting H:
print(system)