Copyright (C) 2011 and later, Paul D. Nation & Robert J. Johansson
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"
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 $E[dN(t)] = \eta \langle a^\dagger a\rangle (t)$.
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 $B = \sqrt{(1-\eta)\gamma} a$
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()]
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
result_ref = mesolve(H, rho0, times, c_ops + sc_ops, e_ops)
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: 0.04s
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,
)
10.0%. Run time: 0.04s. Est. time left: 00:00:00:00 20.0%. Run time: 0.09s. Est. time left: 00:00:00:00 30.0%. Run time: 0.13s. Est. time left: 00:00:00:00 40.0%. Run time: 0.17s. Est. time left: 00:00:00:00 50.0%. Run time: 0.21s. Est. time left: 00:00:00:00 60.0%. Run time: 0.26s. Est. time left: 00:00:00:00 70.0%. Run time: 0.30s. Est. time left: 00:00:00:00 80.0%. Run time: 0.34s. Est. time left: 00:00:00:00 90.0%. Run time: 0.38s. Est. time left: 00:00:00:00 Total run time: 0.43s
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()
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
result_ref = mesolve(H, rho0, times, c_ops + sc_ops, e_ops)
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: 0.04s
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,
)
10.0%. Run time: 0.04s. Est. time left: 00:00:00:00 20.0%. Run time: 0.09s. Est. time left: 00:00:00:00 30.0%. Run time: 0.13s. Est. time left: 00:00:00:00 40.0%. Run time: 0.17s. Est. time left: 00:00:00:00 50.0%. Run time: 0.22s. Est. time left: 00:00:00:00 60.0%. Run time: 0.26s. Est. time left: 00:00:00:00 70.0%. Run time: 0.30s. Est. time left: 00:00:00:00 80.0%. Run time: 0.35s. Est. time left: 00:00:00:00 90.0%. Run time: 0.39s. Est. time left: 00:00:00:00 Total run time: 0.43s
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()
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
rho0 = coherent(N, np.sqrt(5))
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
result_ref = mesolve(H, rho0, times, c_ops + sc_ops, e_ops)
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,
)
10.7%. Run time: 0.49s. Est. time left: 00:00:00:04 20.0%. Run time: 0.95s. Est. time left: 00:00:00:03 30.7%. Run time: 1.40s. Est. time left: 00:00:00:03 40.0%. Run time: 1.86s. Est. time left: 00:00:00:02 50.7%. Run time: 2.31s. Est. time left: 00:00:00:02 60.0%. Run time: 2.77s. Est. time left: 00:00:00:01 70.7%. Run time: 3.22s. Est. time left: 00:00:00:01 80.0%. Run time: 3.49s. Est. time left: 00:00:00:00 90.7%. Run time: 3.95s. Est. time left: 00:00:00:00 100.0%. Run time: 4.33s. Est. time left: 00:00:00:00 Total run time: 4.34s
plot_expectation_values([result, result_ref]);
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);
$\displaystyle D_{1}[A]\rho(t) = \mathcal{D}[\kappa a]\rho(t) = \mathcal{D}[A]\rho(t)$
L = liouvillian(H, np.sqrt(gamma) * a)
def d1_rho_func(t, rho_vec):
return L * rho_vec
$\displaystyle D_{2}[A]\rho(t) = \sqrt{\eta} \mathcal{H}[\sqrt{\kappa} a]\rho(t) = \sqrt{\eta} \mathcal{H}[A]\rho(t) = \sqrt{\eta}(A\rho + \rho A^\dagger - \mathrm{Tr}[A\rho + \rho A^\dagger] \rho) \rightarrow \sqrt{\eta} \left((A_L + A_R^\dagger)\rho_v - \mathrm{Tr}[(A_L + A_R^\dagger)\rho_v] \rho_v\right)$
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)])
result_ref = mesolve(H, rho0, times, c_ops + sc_ops, e_ops)
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,
)
10.7%. Run time: 4.20s. Est. time left: 00:00:00:35 20.0%. Run time: 8.04s. Est. time left: 00:00:00:32 30.7%. Run time: 12.04s. Est. time left: 00:00:00:27 40.0%. Run time: 16.02s. Est. time left: 00:00:00:24 50.7%. Run time: 20.04s. Est. time left: 00:00:00:19 60.0%. Run time: 23.93s. Est. time left: 00:00:00:15 70.7%. Run time: 27.93s. Est. time left: 00:00:00:11 80.0%. Run time: 30.41s. Est. time left: 00:00:00:07 90.7%. Run time: 34.44s. Est. time left: 00:00:00:03 100.0%. Run time: 37.80s. Est. time left: 00:00:00:00 Total run time: 37.81s
plot_expectation_values([result, result_ref])
(<Figure size 800x400 with 2 Axes>, array([[<AxesSubplot:>], [<AxesSubplot:xlabel='time'>]], dtype=object))
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);
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 Giguère, 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. Previous lead developers: Chris Granade & A. Grimsmo. Currently developed through wide collaboration. See https://github.com/qutip for details. QuTiP Version: 4.7.5 Numpy Version: 1.22.4 Scipy Version: 1.8.1 Cython Version: 0.29.37 Matplotlib Version: 3.5.2 Python Version: 3.10.4 Number of CPUs: 4 BLAS Info: Generic OPENMP Installed: False INTEL MKL Ext: False Platform Info: Linux (x86_64) Installation path: /usr/share/miniconda3/envs/test-environment/lib/python3.10/site-packages/qutip ================================================================================ Please cite QuTiP in your publication. ================================================================================ For your convenience a bibtex reference can be easily generated using `qutip.cite()`