Stochastic Solver: Mixing stochastic and deterministic equations¶
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"
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.
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¶
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: 0.08s
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: 0.08s. Est. time left: 00:00:00:00 20.0%. Run time: 0.17s. Est. time left: 00:00:00:00 30.0%. Run time: 0.25s. Est. time left: 00:00:00:00 40.0%. Run time: 0.34s. Est. time left: 00:00:00:00 50.0%. Run time: 0.42s. Est. time left: 00:00:00:00 60.0%. Run time: 0.50s. Est. time left: 00:00:00:00 70.0%. Run time: 0.59s. Est. time left: 00:00:00:00 80.0%. Run time: 0.67s. Est. time left: 00:00:00:00 90.0%. Run time: 0.75s. Est. time left: 00:00:00:00 Total run time: 0.84s
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¶
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: 0.08s
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: 0.08s. Est. time left: 00:00:00:00 20.0%. Run time: 0.16s. Est. time left: 00:00:00:00 30.0%. Run time: 0.24s. Est. time left: 00:00:00:00 40.0%. Run time: 0.32s. Est. time left: 00:00:00:00 50.0%. Run time: 0.40s. Est. time left: 00:00:00:00 60.0%. Run time: 0.48s. Est. time left: 00:00:00:00 70.0%. Run time: 0.56s. Est. time left: 00:00:00:00 80.0%. Run time: 0.64s. Est. time left: 00:00:00:00 90.0%. Run time: 0.72s. Est. time left: 00:00:00:00 Total run time: 0.81s
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
rho0 = coherent(N, np.sqrt(5))
Form 1: Standard homodyne with deterministic dissipation on Lindblad form¶
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: 1.08s. Est. time left: 00:00:00:09 20.0%. Run time: 2.06s. Est. time left: 00:00:00:08 30.7%. Run time: 3.06s. Est. time left: 00:00:00:06 40.0%. Run time: 3.90s. Est. time left: 00:00:00:05 50.7%. Run time: 4.91s. Est. time left: 00:00:00:04 60.0%. Run time: 5.83s. Est. time left: 00:00:00:03 70.7%. Run time: 6.83s. Est. time left: 00:00:00:02 80.0%. Run time: 7.68s. Est. time left: 00:00:00:01 90.7%. Run time: 8.73s. Est. time left: 00:00:00:00 100.0%. Run time: 9.61s. Est. time left: 00:00:00:00 Total run time: 9.61s
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);