# Mean-Field Dynamics

In this tutorial we discuss to use TEMPO and the process tensor approach to compute the dynamics of a many-body system of the type introduced in [FowlerWright2022] (PhysRevLett.129.173001 (2022) / arXiv:2112.09003).

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**Contents:**

Background and introduction

many-body system and environment Hamiltonians

system Hamiltonian and field equation of motion after the mean-field reduction

Creating time-dependent system with field and bath objects

TEMPO computation for single dynamics

PT-TEMPO computation for multiple sets of dynamics

We firstly import OQuPy and other useful packages:

```
import sys
sys.path.insert(0,'..')
import oqupy
import numpy as np
import matplotlib.pyplot as plt
```

Check the current OQuPy version; mean-field functionality was introduced
in version **0.3.0** and revised in its current format in **0.4.0.**

```
oqupy.__version__
```

```
'0.4.0'
```

The following matrices will be useful below:

```
sigma_z = oqupy.operators.sigma("z")
sigma_plus = oqupy.operators.sigma("+")
sigma_minus = oqupy.operators.sigma("-")
```

## 1. Background and introduction

Our goal will be to reproduce a line from **Fig. 2a.** of [FowlerWright2022]
which shows the photon number dynamics for the driven-dissipative system
of molecules in a single-mode cavity.

### Many-body system and environment Hamiltonian

The Hamiltonian describing the many-body system with one-to-all light-matter coupling is

Together with the vibrational environment of each molecule,

This is taken to be a continuum of low frequency modes with coupling characterized by a spectral density with a ‘Gaussian’ cut-off

where \(\alpha=0.25\) is the unit-less coupling strenght and
\(\hbar \nu_c = 0.15\) eV is a cutoff frequency for the environment
of the BODIPY-Br molecule considered in the Letter. For the numericla
simulation we here choose to express all frequencies as angular
frequencies in the units of \(\frac{1}{\text{ps}}\) (setting
\(\hbar = k_B = 1\)) and all times in units of ps. The parameters
relevant to **Fig. 2a.** given in those units are:

\(\nu_c = 0.15 \text{eV} = 227.9 \frac{1}{\text{ps}}\) … environment cutoff frequency

\(T = 300 \text{K} = 39.3 \frac{1}{\text{ps}}\) … environment temperature

\(\omega_0 = 0.0 \frac{1}{\text{ps}}\) … two-level system frequency

*****\(\omega_c = -0.02 \text{eV} = -30.4 \frac{1}{\text{ps}}\) … bare cavity frequency

\(\Omega = 0.2 \text{eV} = 303.9 \frac{1}{\text{ps}}\) … collective light-matter coupling

together with the rates

\(\kappa = 15.2 \frac{1}{\text{ps}}\) … field decay

\(\Gamma_\downarrow = 15.2 \frac{1}{\text{ps}}\) … electronic dissipation

\(\Gamma_\uparrow \in (0.2\Gamma_\downarrow, 0.8\Gamma_\downarrow)\) … electronic pumping

The latter appear as prefactors for Markovian terms in the quantum master equation for the total density operator

As indicated, it is the pump strength \(\Gamma_\uparrow\) that is
varied to generate the different lines of **Fig. 2a.** In this tutorial
we generate the \(\Gamma_\uparrow=0.8\,\Gamma_\downarrow\) line
using the TEMPO method, and then the Process Tensor approach to
calculate all of the lines efficiently.

The following code box defines each of the above parameters.

*** N.B.** for calculating the dynamics only the detuning
\(\omega_c-\omega_0\) is relevant, so we set \(\omega_0=0\) for
convenience.

```
alpha = 0.25
nu_c = 227.9
T = 39.3
omega_0 = 0.0
omega_c = -30.4
Omega = 303.9
kappa = 15.2
Gamma_down = 15.2
Gamma_up = 0.8 * Gamma_down
```

### System Hamiltonian and field equation of motion after the mean-field reduction

The mean-field approach is based on a product-state ansatz for the total density operator \(\rho\),

where \(\text{Tr}_{\otimes{i}}\) denotes a partial trace taken over
the Hilbert space of all two-level systems and
\(\text{Tr}_{a, \otimes{j\neq i}}\) the trace over the photonic
degree of freedom and all but the \(i^{\text{th}}\) two-level
system. As detailed in the Supplement of the Letter, after rescaling the
field \(\langle a \rangle \to \langle a \rangle/\sqrt{N}\)
(\(\langle a \rangle\) scales with \(\sqrt{N}\) in the lasing
phase), the dynamics are controlled by the mean-field Hamiltonian
\(H_{\text{MF}}\) for a *single molecule,*

together with the equation of motion for the field \(\langle a \rangle\),

Therefore in order to calculate the dynamics we need to encode the
field’s equation of motion in addition to the Hamiltonian for a single
two level-system \(\rho_i\). This is done in OQuPy using the
`MeanFieldSystem`

class.

## 2. Creating time-dependent system with field and bath objects

A `MeanFieldSystem`

object is initialised with a field equation of
motion and one or more `TimeDependentSystemWithField`

which objects in
turn are characterised by Hamiltonians with both time and field
depedence. In the present example, we need only one
`TimeDependentSystemWithField`

, for the single molecule Hamiltonian
\(H_{\text{MF}}\), but other problems may require multiple such
objects e.g. to encode different types of molecules.

`field_eom(t, state_list, a)`

which takes as arguments
time, a *list*of states as square matrices (numpy ndarrays) and a field - a function

`H_MF(t, a)`

which takes a time and a fieldSince positional arguments are used in the definition of these
functions, the order of arguments matter, whereas their names do not. In
particular, both functions must have a time variable for their first
argument, even if there happens to be no explicit time-dependence in the
problem (there is no ‘`SystemWithField`

’ class in OQuPy).

```
def H_MF(t, a):
return 0.5 * omega_0 * sigma_z +\
0.5 * Omega * (a * sigma_plus + np.conj(a) * sigma_minus)
def field_eom(t, state_list, a):
state = state_list[0]
expect_val = np.matmul(sigma_minus, state).trace()
return -(1j * omega_c + kappa) * a - 0.5j * Omega * expect_val
```

Note that the second argument of `field_eom`

must be a list, even in
the case of a single `TimeDependentSystemWithField`

object (this
requirement is a feature of most functionality involving the
`MeanFieldSystem`

class, as we will see below). Thus, in order to
compute the expectation \(\langle \sigma^- \rangle\) we took the
first element of this list - a \(2\times2\) matrix - before
multiplying by \(\sigma^-\) and taking the trace.

It is a good idea to test these functions:

```
test_field = 1.0+1.0j
test_time = 0.01
test_state_list = [ np.array([[0.0,2j],[-2j,1.0]]) ]
print('H_eval =', H_MF(test_time, test_field))
print('EOM_eval =', field_eom(test_time, test_state_list, test_field))
```

```
H_eval = [[ 0. +0.j 151.95+151.95j]
[151.95-151.95j 0. +0.j ]]
EOM_eval = (258.29999999999995+15.2j)
```

In, we need to specify Lindblad operators for the pumping and dissipation processes:

```
gammas = [ lambda t: Gamma_down, lambda t: Gamma_up]
lindblad_operators = [ lambda t: sigma_minus, lambda t: sigma_plus]
```

Here the rates and Lindblad operators must be callables taking a single
argument - time - again, even though in our example there is no explicit
time-dependence. The `TimeDependentSystemWithField`

object is then
constructed as

```
system = oqupy.TimeDependentSystemWithField(
hamiltonian=H_MF,
gammas=gammas,
lindblad_operators=lindblad_operators)
```

and the encompasing `MeanFieldSystem`

as

```
system_list = [system] # a list of TimeDependentiSystemWithField objects
mean_field_system = oqupy.MeanFieldSystem(
system_list=system_list,
field_eom=field_eom)
```

where we note the single system must be placed in a list,
`system_list`

, before being passed to the `MeanFieldSystem`

constructor.

Correlations and a Bath object are created in the same way as in any other TEMPO computation (refer to preceding tutorials), although here we will need the Bath in a list:

```
correlations = oqupy.PowerLawSD(alpha=alpha,
zeta=1,
cutoff=nu_c,
cutoff_type='gaussian',
temperature=T)
bath = oqupy.Bath(0.5 * sigma_z, correlations)
bath_list = [bath]
```

## 3. TEMPO computation for single dynamics

For our simulations we use the same initial conditions for the system and state used in the Letter:

```
initial_field = np.sqrt(0.05) # Note n_0 = <a^dagger a>(0) = 0.05
initial_state = np.array([[0,0],[0,1]]) # spin down
initial_state_list = [initial_state] # initial state must be provided in a list
```

To reduce the computation time we simulate only the first 0.3 ps of the dynamics with much rougher convergence parameters compared to the letter.

```
tempo_parameters = oqupy.TempoParameters(dt=3.2e-3, tcut=64e-3, epsrel=10**(-5))
start_time = 0.0
end_time = 0.3
```

The `oqupy.MeanFieldTempo.compute`

method may then be used to compute
the dynamics in an analogous way a call to `oqupy.Tempo.compute`

is
used to compute the dynamics for an ordinary `System`

:

```
tempo_sys = oqupy.MeanFieldTempo(mean_field_system=mean_field_system,
bath_list=[bath],
initial_state_list=initial_state_list,
initial_field=initial_field,
start_time=start_time,
parameters=tempo_parameters)
mean_field_dynamics = tempo_sys.compute(end_time=end_time)
```

```
--> TEMPO-with-field computation:
100.0% 93 of 93 [########################################] 00:00:06
Elapsed time: 6.3s
```

`MeanFieldTempo.compute`

returns a `MeanFieldDynamics`

object
containing an array of timesteps, the field values at these timesteps,
and a list of ordinary `Dynamics`

objects, one for each of
`TimeDependentSystemWithField`

objects (here only one):

```
times = mean_field_dynamics.times
fields = mean_field_dynamics.fields
system_dynamics = mean_field_dynamics.system_dynamics[0]
states = system_dynamics.states
```

We plot a the square value of the fields i.e. the photon number,
producing the first part of a single line of **Fig. 2a.**:

```
n = np.abs(fields)**2
plt.plot(times, n, label=r'$\Gamma_\uparrow = 0.8\Gamma_\downarrow$')
plt.xlabel(r'$t$ (ps)')
plt.ylabel(r'$n/N$')
plt.ylim((0.0,0.15))
plt.legend(loc='upper left')
```

```
<matplotlib.legend.Legend at 0x7fdb289b2b90>
```

If you have the time you can calculate the dynamics to
\(t=1.3\,\text{ps}\) as in the Letter and check that, even for these
very rough parameters, the results are reasonably close to being
converged with respect to `dt`

, `tcut`

and `epsrel`

.

While you could repeat the TEMPO computation for each pump strength
\(\Gamma_\uparrow\) appearing in **Fig. 2a.**, a more efficient
solution for calculating dynamics for multiple sets of system parameters
(in this case Lindblad rates) is provided by PT-TEMPO.

## 4. PT-TEMPO computation for multiple sets of dynamics

The above calculation can be performed quickly for many-different pump strengths \(\Gamma_\uparrow\) using a single process tensor.

As discussed in the Supplement Material for the Letter, there is no guarantee that computational parameters that gave a set of converged results for the TEMPO method will give converged results for a PT-TEMPO calculation. For the sake of this tutorial however let’s assume the above parameters continue to be reasonable. The process tensor to time \(t=0.3\,\text{ps}\) is calculated using these parameters and the bath via

```
process_tensor = oqupy.pt_tempo_compute(bath=bath,
start_time=0.0,
end_time=0.3,
parameters=tempo_parameters)
```

```
--> PT-TEMPO computation:
100.0% 93 of 93 [########################################] 00:00:01
Elapsed time: 1.1s
```

Refer the Time Dependence and PT-TEMPO tutorial for further discussion of the process tensor.

To calculate the dynamics for the 4 different pump strengths in **Fig.
2a.**, we define a separate `MeanFieldSystem`

object for each pump
strength. Only the `gammas`

array needs to be modified between sets of
constructor calls:

```
pump_ratios = [0.2, 0.4, 0.6, 0.8]
mean_field_systems = []
for ratio in pump_ratios:
Gamma_up = ratio * Gamma_down
# N.B. a default argument is used to avoid the late-binding closure issue
# discussed here: https://docs.python-guide.org/writing/gotchas/#late-binding-closures
gammas = [ lambda t: Gamma_down, lambda t, Gamma_up=Gamma_up: Gamma_up]
# Use the same Hamiltonian, equation of motion and Lindblad operators
system = oqupy.TimeDependentSystemWithField(H_MF,
gammas=gammas,
lindblad_operators=lindblad_operators)
mean_field_system = oqupy.MeanFieldSystem(system_list=[system],
field_eom=field_eom)
mean_field_systems.append(mean_field_system)
```

We can then use `compute_dynamics_with_field`

to compute the dynamics
at each \(\Gamma_\uparrow\) for the particular initial condition
using the process tensor (now in a list) calculated above:

```
t_list = []
n_list = []
for i, mean_field_system in enumerate(mean_field_systems):
mean_field_dynamics = oqupy.compute_dynamics_with_field(
process_tensor_list=[process_tensor],
mean_field_system=mean_field_system,
initial_state_list=[initial_state],
initial_field=initial_field,
start_time=0.0)
t = mean_field_dynamics.times
fields = mean_field_dynamics.fields
n = np.abs(fields)**2
t_list.append(t)
n_list.append(n)
```

```
--> Compute dynamics with field:
100.0% 93 of 93 [########################################] 00:00:04
Elapsed time: 4.2s
--> Compute dynamics with field:
100.0% 93 of 93 [########################################] 00:00:03
Elapsed time: 3.9s
--> Compute dynamics with field:
100.0% 93 of 93 [########################################] 00:00:04
Elapsed time: 4.1s
--> Compute dynamics with field:
100.0% 93 of 93 [########################################] 00:00:04
Elapsed time: 4.0s
```

Finally, plotting the results:

```
for i,n in enumerate(n_list):
ratio = pump_ratios[i]
label = r'$\Gamma_\uparrow = {}\Gamma_\downarrow$'.format(pump_ratios[i])
plt.plot(t_list[i], n_list[i], label=label)
plt.xlabel(r'$t$ (ps)')
plt.ylabel(r'$n/N$')
plt.ylim((0.0,0.15))
plt.legend(loc='upper left')
```

```
<matplotlib.legend.Legend at 0x7fdb28590d00>
```

## 5. Summary

To summarise the classes and methods for calculating mean-field dynamics:

A Hamiltonian with time \(t\) and field \(a\) dependence is used to construct a

`TimeDependentSystemWithField`

objectOne or more

`TimeDependentSystemWithField`

objects and a field equation of motion forms a`MeanFieldSystem`

`oqupy.MeanFieldTempo.compute`

or`.compute_dynamics_with_field`

(process tensor) may be used to calculate`MeanFieldDynamics`

`MeanFieldDynamics`

comprises one of more system`Dynamics`

and a set of field values`fields`

.