# Stochastic Solver: Mixing stochastic and deterministic equations¶

Copyright (C) 2011 and later, Paul D. Nation & Robert J. Johansson

In [1]:
from matplotlib import rcParams
import matplotlib.pyplot as plt
import numpy as np
from qutip import (about, coherent, destroy, fock, general_stochastic, ket2dm,
liouvillian, mesolve, parallel_map, photocurrent_mesolve,
plot_expectation_values, smesolve, spost, spre)
from qutip.expect import expect_rho_vec
%matplotlib inline

rcParams["font.family"] = "STIXGeneral"
rcParams["mathtext.fontset"] = "stix"
rcParams["font.size"] = "14"


## Direct photo-detection¶

Here we follow an example from Wiseman and Milburn, Quantum measurement and control, section. 4.8.1.

Consider cavity that leaks photons with a rate $\kappa$. The dissipated photons are detected with an inefficient photon detector, with photon-detection efficiency $\eta$. The master equation describing this scenario, where a separate dissipation channel has been added for detections and missed detections, is

$\dot\rho = -i[H, \rho] + \mathcal{D}[\sqrt{1-\eta} \sqrt{\kappa} a] + \mathcal{D}[\sqrt{\eta} \sqrt{\kappa}a]$

To describe the photon measurement stochastically, we can unravelling only the dissipation term that corresponds to detections, and leaving the missed detections as a deterministic dissipation term, we obtain [Eq. (4.235) in W&M]

$d\rho = \mathcal{H}[-iH -\eta\frac{1}{2}a^\dagger a] \rho dt + \mathcal{D}[\sqrt{1-\eta} a] \rho dt + \mathcal{G}[\sqrt{\eta}a] \rho dN(t)$

or

$d\rho = -i[H, \rho] dt + \mathcal{D}[\sqrt{1-\eta} a] \rho dt -\mathcal{H}[\eta\frac{1}{2}a^\dagger a] \rho dt + \mathcal{G}[\sqrt{\eta}a] \rho dN(t)$

where

$\displaystyle \mathcal{G}[A] \rho = \frac{A\rho A^\dagger}{\mathrm{Tr}[A\rho A^\dagger]} - \rho$

$\displaystyle \mathcal{H}[A] \rho = A\rho + \rho A^\dagger - \mathrm{Tr}[A\rho + \rho A^\dagger] \rho$

and $dN(t)$ is a Poisson distributed increment with $E[dN(t)] = \eta \langle a^\dagger a\rangle (t)$.

### Formulation in QuTiP¶

In QuTiP, the photocurrent stochastic master equation is written in the form:

$\displaystyle d\rho(t) = -i[H, \rho] dt + \mathcal{D}[B] \rho dt • \frac{1}{2}\mathcal{H}[A^\dagger A] \rho(t) dt • \mathcal{G}[A]\rho(t) d\xi$

where the first two term gives the deterministic master equation (Lindblad form with collapse operator $B$ (c_ops)) and $A$ the stochastic collapse operator (sc_ops).

Here $A = \sqrt{\eta\gamma} a$ and $B = \sqrt{(1-\eta)\gamma}$a.

In [2]:
N = 15
w0 = 0.5 * 2 * np.pi
times = np.linspace(0, 15, 150)
dt = times[1] - times[0]
gamma = 0.1

a = destroy(N)

H = w0 * a.dag() * a

rho0 = fock(N, 5)

e_ops = [a.dag() * a, a + a.dag()]


### Highly efficient detection¶

In [3]:
eta = 0.7
c_ops = [np.sqrt(1 - eta) * np.sqrt(gamma) * a]  # collapse operator B
sc_ops = [np.sqrt(eta) * np.sqrt(gamma) * a]  # stochastic collapse operator A

In [4]:
result_ref = mesolve(H, rho0, times, c_ops + sc_ops, e_ops)

In [5]:
result1 = photocurrent_mesolve(
H,
rho0,
times,
c_ops=c_ops,
sc_ops=sc_ops,
e_ops=e_ops,
ntraj=1,
nsubsteps=100,
store_measurement=True,
)

Total run time:   0.07s

In [6]:
result2 = photocurrent_mesolve(
H,
rho0,
times,
c_ops=c_ops,
sc_ops=sc_ops,
e_ops=e_ops,
ntraj=10,
nsubsteps=100,
store_measurement=True,
)

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Total run time:   0.67s

In [7]:
fig, axes = plt.subplots(2, 2, figsize=(12, 8), sharex=True)

axes[0, 0].plot(times, result1.expect[0],
label=r"Stochastic ME (ntraj = 1)", lw=2)
axes[0, 0].plot(times, result_ref.expect[0], label=r"Lindblad ME", lw=2)
axes[0, 0].set_title("Cavity photon number (ntraj = 1)")
axes[0, 0].legend()

axes[0, 1].plot(times, result2.expect[0],
label=r"Stochatic ME (ntraj = 10)", lw=2)
axes[0, 1].plot(times, result_ref.expect[0], label=r"Lindblad ME", lw=2)
axes[0, 1].set_title("Cavity photon number (ntraj = 10)")
axes[0, 1].legend()

axes[1, 0].step(times, dt * np.cumsum(result1.measurement[0].real), lw=2)
axes[1, 0].set_title("Cummulative photon detections (ntraj = 1)")
axes[1, 1].step(
times,
dt * np.cumsum(np.array(result2.measurement).sum(axis=0).real) / 10,
lw=2
)
axes[1, 1].set_title("Cummulative avg. photon detections (ntraj = 10)")

fig.tight_layout()


### Highly inefficient photon detection¶

In [8]:
eta = 0.1
c_ops = [np.sqrt(1 - eta) * np.sqrt(gamma) * a]  # collapse operator B
sc_ops = [np.sqrt(eta) * np.sqrt(gamma) * a]  # stochastic collapse operator A

In [9]:
result_ref = mesolve(H, rho0, times, c_ops + sc_ops, e_ops)

In [10]:
result1 = photocurrent_mesolve(
H,
rho0,
times,
c_ops=c_ops,
sc_ops=sc_ops,
e_ops=e_ops,
ntraj=1,
nsubsteps=100,
store_measurement=True,
)

Total run time:   0.07s

In [11]:
result2 = photocurrent_mesolve(
H,
rho0,
times,
c_ops=c_ops,
sc_ops=sc_ops,
e_ops=e_ops,
ntraj=10,
nsubsteps=100,
store_measurement=True,
)

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Total run time:   0.66s

In [12]:
fig, axes = plt.subplots(2, 2, figsize=(12, 8), sharex=True)

axes[0, 0].plot(times, result1.expect[0],
label=r"Stochastic ME (ntraj = 1)", lw=2)
axes[0, 0].plot(times, result_ref.expect[0], label=r"Lindblad ME", lw=2)
axes[0, 0].set_title("Cavity photon number (ntraj = 1)")
axes[0, 0].legend()

axes[0, 1].plot(times, result2.expect[0],
label=r"Stochatic ME (ntraj = 10)", lw=2)
axes[0, 1].plot(times, result_ref.expect[0], label=r"Lindblad ME", lw=2)
axes[0, 1].set_title("Cavity photon number (ntraj = 10)")
axes[0, 1].legend()

axes[1, 0].step(times, dt * np.cumsum(result1.measurement[0].real), lw=2)
axes[1, 0].set_title("Cummulative photon detections (ntraj = 1)")
axes[1, 1].step(
times,
dt * np.cumsum(np.array(result2.measurement).sum(axis=0).real) / 10,
lw=2
)
axes[1, 1].set_title("Cummulative avg. photon detections (ntraj = 10)")

fig.tight_layout()


## Efficient homodyne detection¶

The stochastic master equation for inefficient homodyne detection, when unravaling the detection part of the master equation

$\dot\rho = -i[H, \rho] + \mathcal{D}[\sqrt{1-\eta} \sqrt{\kappa} a] + \mathcal{D}[\sqrt{\eta} \sqrt{\kappa}a]$,

is given in W&M as

$d\rho = -i[H, \rho]dt + \mathcal{D}[\sqrt{1-\eta} \sqrt{\kappa} a] \rho dt + \mathcal{D}[\sqrt{\eta} \sqrt{\kappa}a] \rho dt + \mathcal{H}[\sqrt{\eta} \sqrt{\kappa}a] \rho d\xi$

where $d\xi$ is the Wiener increment. This can be described as a standard homodyne detection with efficiency $\eta$ together with a stochastic dissipation process with collapse operator $\sqrt{(1-\eta)\kappa} a$. Alternatively we can combine the two deterministic terms on standard Lindblad for and obtain the stochastic equation (which is the form given in W&M)

$d\rho = -i[H, \rho]dt + \mathcal{D}[\sqrt{\kappa} a]\rho dt + \sqrt{\eta}\mathcal{H}[\sqrt{\kappa}a] \rho d\xi$

Below we solve these two equivalent equations with QuTiP

In [13]:
rho0 = coherent(N, np.sqrt(5))


### Form 1: Standard homodyne with deterministic dissipation on Lindblad form¶

In [14]:
eta = 0.95
c_ops = [np.sqrt(1 - eta) * np.sqrt(gamma) * a]  # collapse operator B
sc_ops = [np.sqrt(eta) * np.sqrt(gamma) * a]  # stochastic collapse operator A

In [15]:
result_ref = mesolve(H, rho0, times, c_ops + sc_ops, e_ops)

In [16]:
result = smesolve(
H,
rho0,
times,
c_ops,
sc_ops,
e_ops,
ntraj=75,
nsubsteps=100,
solver="platen",
method="homodyne",
store_measurement=True,
map_func=parallel_map,
noise=111,
)

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In [17]:
plot_expectation_values([result, result_ref]);

In [18]:
fig, ax = plt.subplots(figsize=(8, 4))

M = np.sqrt(eta * gamma)

for m in result.measurement:
ax.plot(times, m[:, 0].real / M, "b", alpha=0.025)

ax.plot(times, result_ref.expect[1], "k", lw=2)

ax.set_ylim(-25, 25)
ax.set_xlim(0, times.max())
ax.set_xlabel("time", fontsize=12)
ax.plot(times,
np.array(result.measurement).mean(axis=0)[:, 0].real / M, "b", lw=2);


### Form 2: Combined homodyne with deterministic dissipation for missed detection events¶

$\displaystyle D_{1}[A]\rho(t) = \mathcal{D}[\kappa a]\rho(t) = \mathcal{D}[A]\rho(t)$

In [19]:
L = liouvillian(H, np.sqrt(gamma) * a)

def d1_rho_func(t, rho_vec):
return L * rho_vec


$\displaystyle D_{2}[A]\rho(t) = \sqrt{\eta} \mathcal{H}[\sqrt{\kappa} a]\rho(t) = \sqrt{\eta} \mathcal{H}[A]\rho(t) = \sqrt{\eta}(A\rho + \rho A^\dagger - \mathrm{Tr}[A\rho + \rho A^\dagger] \rho) \rightarrow \sqrt{\eta} \left((A_L + A_R^\dagger)\rho_v - \mathrm{Tr}[(A_L + A_R^\dagger)\rho_v] \rho_v\right)$

In [20]:
n_sum = spre(np.sqrt(gamma) * a) + spost(np.sqrt(gamma) * a.dag())

def d2_rho_func(t, rho_vec):
e1 = expect_rho_vec(n_sum.data, rho_vec, False)
return np.vstack([np.sqrt(eta) * (n_sum * rho_vec - e1 * rho_vec)])

In [21]:
result_ref = mesolve(H, rho0, times, c_ops + sc_ops, e_ops)

In [22]:
result = general_stochastic(
ket2dm(rho0),
times,
e_ops=[spre(op) for op in e_ops],
ntraj=75,
nsubsteps=100,
solver="platen",
d1=d1_rho_func,
d2=d2_rho_func,
len_d2=1,
m_ops=[spre(a + a.dag())],
dW_factors=[1 / np.sqrt(gamma * eta)],
store_measurement=True,
map_func=parallel_map,
noise=111,
)

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In [23]:
plot_expectation_values([result, result_ref])

Out[23]:
(<Figure size 800x400 with 2 Axes>,
array([[<AxesSubplot:>],
[<AxesSubplot:xlabel='time'>]], dtype=object))
In [24]:
fig, ax = plt.subplots(figsize=(8, 4))

for m in result.measurement:
ax.plot(times, m[:, 0].real, "b", alpha=0.025)

ax.plot(times, result_ref.expect[1], "k", lw=2)

ax.set_ylim(-25, 25)
ax.set_xlim(0, times.max())
ax.set_xlabel("time", fontsize=12)
ax.plot(times, np.array(result.measurement).mean(axis=0)[:, 0].real, "b", lw=2);


## Versions¶

In [25]:
about()

QuTiP: Quantum Toolbox in Python
================================
Copyright (c) QuTiP team 2011 and later.
Current admin team: Alexander Pitchford, Nathan Shammah, Shahnawaz Ahmed, Neill Lambert, Eric GiguĂ¨re, Boxi Li, Jake Lishman, Simon Cross and Asier Galicia.
Board members: Daniel Burgarth, Robert Johansson, Anton F. Kockum, Franco Nori and Will Zeng.
Original developers: R. J. Johansson & P. D. Nation.
Currently developed through wide collaboration. See https://github.com/qutip for details.

QuTiP Version:      4.7.1.dev0+9098716
Numpy Version:      1.22.4
Scipy Version:      1.8.1
Cython Version:     0.29.32
Matplotlib Version: 3.5.2
Python Version:     3.10.4
Number of CPUs:     2
BLAS Info:          Generic
OPENMP Installed:   False
INTEL MKL Ext:      False
Platform Info:      Linux (x86_64)
Installation path:  /home/runner/work/qutip-tutorials/qutip-tutorials/qutip/qutip
================================================================================
For your convenience a bibtex reference can be easily generated using qutip.cite()