Author: C. Staufenbiel, 2022

The *Floquet formalism* deals with periodic time-dependent systems. The Floquet approach can be more efficient for such problems than using the standard master equation solver `qutip.mesolve()`

and it has a broader range of validity for periodic driving.

In this notebook, we will discuss the solver functionality of the Floquet formalism implemented in QuTiP using an example quantum system. A more detailed introduction into the Floquet formalism can be found in the documentation.

A more in depth introduction into the internal functions of the Floquet formalism, used also by the solvers `fsesolve`

and `fmmesolve`

, is given in the *floquet formalism notebook*.

In [1]:

```
import numpy as np
from qutip import (about, basis, fmmesolve, fsesolve,
plot_expectation_values, sigmax, sigmaz)
```

In this example we will consider a strongly driven two level system, described by the time-dependent Hamiltonian:

$$ H(t) = -\frac{\Delta}{2} \sigma_x - \frac{\epsilon_0}{2} \sigma_z + \frac{A}{2} sin(\omega t) \sigma_z$$In [2]:

```
# define constants
delta = 0.2 * 2 * np.pi
eps0 = 2 * np.pi
A = 2.5 * 2 * np.pi
omega = 2 * np.pi
# Non driving hamiltoninan
H0 = -delta / 2.0 * sigmax() - eps0 / 2.0 * sigmaz()
# Driving Hamiltonian
H1 = [A / 2.0 * sigmaz(), "sin(w*t)"]
args = {"w": omega}
# combined hamiltonian
H = [H0, H1]
# initial state
psi0 = basis(2, 0)
```

We can now use the `qutip.fsesolve()`

function to solve the dynamics of the system using the Floquet formalism for the SchrĂ¶dinger equation. The arguments are similar to the ones passed to `qutip.sesolve()`

. There is an optional parameter `T`

which defines the period of the time-dependence. If `T`

is not given it is assumed that the passed `tlist`

spans one period. Therefore we always pass `T`

in this tutorial.

The `Tsteps`

argument to `fsesolve()`

can be used to set the number of time steps in one period `T`

for which the Floquet modes are precalculated. Increasing this number should result in a better numerical accuracy. `Tsteps`

should be even!

In [3]:

```
# period time
T = 2 * np.pi / omega
# simulation time
tlist = np.linspace(0, 2.5 * T, 101)
# simulation
result = fsesolve(H, psi0, tlist, T=T, e_ops=[sigmaz()],
args=args, Tsteps=1000)
plot_expectation_values([result], ylabels=["<Z>"]);
```

Similar to `mesolve()`

we can also use the Floquet formalism to solve a master equation for a dissipative quantum system. The corresponding function is `fmmesolve()`

. However, the dissipation process is here described as a noise spectral-density function.

For example we can define a linear noise spectral-density as:

$$ S(\omega) = \frac{\gamma \cdot \omega}{4 \pi} $$where $\gamma$ is the dissipation rate. The system-bath interaction is described by coupling operators, e.g. here we use $\sigma_x$ as a coupling operator. Note that `fmmesolve`

currently only works with one coupling operator and one noise spectrum.

In [4]:

```
# Noise Spectral Density
gamma = 0.1
def noise_spectrum(omega):
return gamma * omega / (4 * np.pi)
# Coupling Operator
c_ops = [sigmax()]
# Solve using Fmmesolve
fme_result = fmmesolve(
H,
psi0,
tlist,
c_ops=c_ops,
spectra_cb=[noise_spectrum],
e_ops=[sigmaz()],
T=T,
args=args,
floquet_basis=False,
)
```

We can observe the dissipation dynamics when comparing the results to the expectation values obtained from `fsesolve()`

.

In [5]:

```
plot_expectation_values([result, fme_result],
ylabels=["<Z>"], show_legend=True);
```

In [6]:

```
about()
```

In [7]:

```
fme_result_nodis = fmmesolve(
H,
psi0,
tlist,
c_ops=c_ops,
spectra_cb=[lambda w: 0.0],
e_ops=[sigmaz()],
T=T,
args=args,
floquet_basis=False,
)
```

In [8]:

```
assert np.allclose(result.expect[0], fme_result_nodis.expect[0], atol=1e-2)
assert not np.allclose(fme_result.expect[0],
fme_result_nodis.expect[0], atol=1e-2)
```