Author: J. R. Johansson (robert@riken.jp), https://jrjohansson.github.io/

This lecture series was developed by J.R. Johannson. The original lecture notebooks are available here.

This is a slightly modified version of the lectures, to work with the current release of QuTiP. You can find these lectures as a part of the qutip-tutorials repository. This lecture and other tutorial notebooks are indexed at the QuTiP Tutorial webpage.

In [1]:

```
import matplotlib.font_manager
import matplotlib.pyplot as plt
import numpy as np
from IPython.display import Image
from qutip import about, basis, destroy, expect, mcsolve, mesolve, steadystate
%matplotlib inline
```

The Quantum Monte-Carlo trajectory method is an equation of motion for a single realization of the state vector $\left|\psi(t)\right>$ for a quantum system that interacts with its environment. The dynamics of the wave function is given by the Schrodinger equation,

where the Hamiltonian is an effective Hamiltonian that, in addition to the system Hamiltonian $H(t)$, also contains a non-Hermitian contribution due to the interaction with the environment:

Since the effective Hamiltonian is non-Hermitian, the norm of the wavefunction is decreasing with time, which to first order in a small time step $\delta t$ is given by $\langle\psi(t+\delta t)|\psi(t+\delta t)\rangle \approx 1 - \delta p\;\;\;$, where

The decreasing norm is used to determine when so-called quantum jumps are to be imposed on the dynamics, where we compare $\delta p$ to a random number in the range [0, 1]. If the norm has decreased below the randomly chosen number, we apply a "quantum jump", so that the new wavefunction at $t+\delta t$ is given by

for a randomly chosen collapse operator $c_n$, weighted so the probability that the collapse being described by the nth collapse operator is given by

This is a Monte-Carlo simulation showing the decay of a cavity Fock state $\left|1\right>$ in a thermal environment with an average occupation number of $n=0.063$ .

Here, the coupling strength is given by the inverse of the cavity ring-down time $T_c = 0.129$ .

The parameters chosen here correspond to those from S. Gleyzes, et al., Nature 446, 297 (2007), and we will carry out a simulation that corresponds to these experimental results from that paper:

In [2]:

```
Image(filename="images/exdecay.png")
```

Out[2]:

In [3]:

```
N = 4 # number of basis states to consider
kappa = 1.0 / 0.129 # coupling to heat bath
nth = 0.063 # temperature with <n>=0.063
tlist = np.linspace(0, 0.6, 100)
```

Here we create QuTiP `Qobj`

representations of the operators and state that are involved in this problem.

In [4]:

```
a = destroy(N) # cavity destruction operator
H = a.dag() * a # harmonic oscillator Hamiltonian
psi0 = basis(N, 1) # initial Fock state with one photon: |1>
```

In [5]:

```
# collapse operator list
c_op_list = []
# decay operator
c_op_list.append(np.sqrt(kappa * (1 + nth)) * a)
# excitation operator
c_op_list.append(np.sqrt(kappa * nth) * a.dag())
```

Here we start the Monte-Carlo simulation, and we request expectation values of photon number operators with 1, 5, 15, and 904 trajectories (compare with experimental results above).

In [6]:

```
ntraj = [1, 5, 15, 904] # list of number of trajectories to avg. over
mc = mcsolve(H, psi0, tlist, c_op_list, [a.dag() * a], ntraj)
```

The expectation values of $a^\dagger a$ are now available in array `mc.expect[idx][0]`

where `idx`

takes values in `[0,1,2,3]`

corresponding to the averages of `1, 5, 15, 904`

Monte Carlo trajectories, as specified above. Below we plot the array `mc.expect[idx][0]`

vs. `tlist`

for each index `idx`

.

For comparison with the averages of single quantum trajectories provided by the Monte-Carlo solver we here also calculate the dynamics of the Lindblad master equation, which should agree with the Monte-Carlo simultions for infinite number of trajectories.

In [7]:

```
# run master equation to get ensemble average expectation values
me = mesolve(H, psi0, tlist, c_op_list, [a.dag() * a])
# calulate final state using steadystate solver
final_state = steadystate(H, c_op_list) # find steady-state
# find expectation value for particle number
fexpt = expect(a.dag() * a, final_state)
```

In [8]:

```
leg_prop = matplotlib.font_manager.FontProperties(size=10)
fig, axes = plt.subplots(4, 1, sharex=True, figsize=(8, 12))
fig.subplots_adjust(hspace=0.1) # reduce space between plots
for idx, n in enumerate(ntraj):
axes[idx].step(tlist, mc.expect[idx][0], "b", lw=2)
axes[idx].plot(tlist, me.expect[0], "r--", lw=1.5)
axes[idx].axhline(y=fexpt, color="k", lw=1.5)
axes[idx].set_yticks(np.linspace(0, 2, 5))
axes[idx].set_ylim([0, 1.5])
axes[idx].set_ylabel(r"$\left<N\right>$", fontsize=14)
if idx == 0:
axes[idx].set_title("Ensemble Averaging of Monte Carlo Trajectories")
axes[idx].legend(
("Single trajectory", "master equation", "steady state"),
prop=leg_prop
)
else:
axes[idx].legend(
("%d trajectories" % n, "master equation", "steady state"),
prop=leg_prop
)
axes[3].xaxis.set_major_locator(plt.MaxNLocator(4))
axes[3].set_xlabel("Time (sec)", fontsize=14);
```

In [9]:

```
about()
```