# Stochastic Solver: Heterodyne Detection¶

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

In [1]:
import matplotlib.pyplot as plt
import numpy as np
import scipy as sp
from qutip import (Options, about, coherent, destroy, general_stochastic,
parallel_map, plot_expectation_values, smesolve, spost,
spre, stochastic_solvers)
from qutip.expect import expect_rho_vec

%matplotlib inline


## Introduction¶

Homodyne and hetrodyne detection are techniques for measuring the quadratures of a field using photocounters. Homodyne detection (on-resonant) measures one quadrature and with heterodyne detection (off-resonant) both quadratures can be detected simulateously.

The evolution of a quantum system that is coupled to a field that is monitored with homodyne and heterodyne detector can be described with stochastic master equations. This notebook compares two different ways to implement the heterodyne detection stochastic master equation in QuTiP.

## Deterministic reference¶

In [2]:
N = 15
w0 = 1.0 * 2 * np.pi
A = 0.1 * 2 * np.pi
times = np.linspace(0, 15, 201)
gamma = 0.25

ntraj = 50
nsubsteps = 50

a = destroy(N)
x = a + a.dag()
y = -1.0j * (a - a.dag())

H = w0 * a.dag() * a + A * (a + a.dag())

rho0 = coherent(N, np.sqrt(5.0), method="analytic")
c_ops = [np.sqrt(gamma) * a]
e_ops = [a.dag() * a, x, y]

In [3]:
result_ref = mesolve(H, rho0, times, c_ops, e_ops)

In [4]:
plot_expectation_values(result_ref);


## Heterodyne implementation #1¶

Stochastic master equation for heterodyne in Milburn's formulation

$\displaystyle d\rho(t) = -i[H, \rho(t)]dt + \gamma\mathcal{D}[a]\rho(t) dt + \frac{1}{\sqrt{2}} dW_1(t) \sqrt{\gamma} \mathcal{H}[a] \rho(t) + \frac{1}{\sqrt{2}} dW_2(t) \sqrt{\gamma} \mathcal{H}[-ia] \rho(t)$

where $\mathcal{D}$ is the standard Lindblad dissipator superoperator, and $\mathcal{H}$ is defined as above, and $dW_i(t)$ is a normal distributed increment with $E[dW_i(t)] = \sqrt{dt}$.

In QuTiP format we have:

$\displaystyle d\rho(t) = -i[H, \rho(t)]dt + D_{1}[A]\rho(t) dt + D_{2}^{(1)}[A]\rho(t) dW_1 + D_{2}^{(2)}[A]\rho(t) dW_2$

where $A = \sqrt{\gamma} a$, so we can identify

$\displaystyle D_{1}[A]\rho = \gamma \mathcal{D}[a]\rho = \mathcal{D}[A]\rho$

In [5]:
L = liouvillian(H)
d1_operator = L + D

def d1_rho_func(t, rho_vec):
return d1_operator * rho_vec


$D_{2}^{(1)}[A]\rho = \frac{1}{\sqrt{2}} \sqrt{\gamma} \mathcal{H}[a] \rho = \frac{1}{\sqrt{2}} \mathcal{H}[A] \rho = \frac{1}{\sqrt{2}}(A\rho + \rho A^\dagger - \mathrm{Tr}[A\rho + \rho A^\dagger] \rho) \rightarrow \frac{1}{\sqrt{2}} \left\{(A_L + A_R^\dagger)\rho_v - \mathrm{Tr}[(A_L + A_R^\dagger)\rho_v] \rho_v\right\}$

$D_{2}^{(2)}[A]\rho = \frac{1}{\sqrt{2}} \sqrt{\gamma} \mathcal{H}[-ia] \rho = \frac{1}{\sqrt{2}} \mathcal{H}[-iA] \rho = \frac{-i}{\sqrt{2}}(A\rho - \rho A^\dagger - \mathrm{Tr}[A\rho - \rho A^\dagger] \rho) \rightarrow \frac{-i}{\sqrt{2}} \left\{(A_L - A_R^\dagger)\rho_v - \mathrm{Tr}[(A_L - A_R^\dagger)\rho_v] \rho_v\right\}$

In [6]:
B1 = spre(c_ops[0]) + spost(c_ops[0].dag())
B2 = spre(c_ops[0]) + spost(c_ops[0].dag())

def d2_rho_func(t, rho_vec):
e1 = expect_rho_vec(B1.data, rho_vec, False)
drho1 = B1 * rho_vec - e1 * rho_vec

e1 = expect_rho_vec(B2.data, rho_vec, False)
drho2 = B2 * rho_vec - e1 * rho_vec

return np.vstack([1.0 / np.sqrt(2) * drho1, -1.0j / np.sqrt(2) * drho2])


The heterodyne currents for the $x$ and $y$ quadratures are

$J_x(t) = \sqrt{\gamma}\left<x\right> + \sqrt{2} \xi(t)$

$J_y(t) = \sqrt{\gamma}\left<y\right> + \sqrt{2} \xi(t)$

where $\xi(t) = \frac{dW}{dt}$.

In qutip we define these measurement operators using the m_ops = [[x, y]] and the coefficients to the noise terms dW_factor = [sqrt(2/gamma), sqrt(2/gamma)].

In [7]:
result = general_stochastic(
ket2dm(rho0),
times,
d1_rho_func,
d2_rho_func,
e_ops=[spre(op) for op in e_ops],
len_d2=2,
ntraj=ntraj,
nsubsteps=nsubsteps,
solver="platen",
dW_factors=[np.sqrt(2 / gamma), np.sqrt(2 / gamma)],
m_ops=[spre(x), spre(y)],
store_measurement=True,
map_func=parallel_map,
)

10.0%. Run time:   8.02s. Est. time left: 00:00:01:12
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100.0%. Run time:  63.62s. Est. time left: 00:00:00:00
Total run time:  63.63s

In [8]:
plot_expectation_values([result, result_ref]);

In [9]:
fig, ax = plt.subplots(figsize=(8, 4))

for m in result.measurement:
ax.plot(times, m[:, 0].real, "b", alpha=0.05)
ax.plot(times, m[:, 1].real, "r", alpha=0.05)

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

ax.set_ylim(-10, 10)
ax.set_xlim(0, times.max())
ax.set_xlabel("time", fontsize=12)
ax.plot(times, np.array(result.measurement).mean(axis=0)[:, 0].real, "k", lw=2)
ax.plot(times, np.array(result.measurement).mean(axis=0)[:, 1].real, "k", lw=2);


## Heterodyne implementation #2: using two homodyne measurements¶

We can also write the heterodyne equation as

$\displaystyle d\rho(t) = -i[H, \rho(t)]dt + \frac{1}{2}\gamma\mathcal{D}[a]\rho(t) dt + \frac{1}{\sqrt{2}} dW_1(t) \sqrt{\gamma} \mathcal{H}[a] \rho(t) + \frac{1}{2}\gamma\mathcal{D}[a]\rho(t) dt + \frac{1}{\sqrt{2}} dW_2(t) \sqrt{\gamma} \mathcal{H}[-ia] \rho(t)$

And using the QuTiP format for two stochastic collapse operators, we have:

$\displaystyle d\rho(t) = -i[H, \rho(t)]dt + D_{1}[A_1]\rho(t) dt + D_{2}[A_1]\rho(t) dW_1 + D_{1}[A_2]\rho(t) dt + D_{2}[A_2]\rho(t) dW_2$

so we can also identify

$\displaystyle D_{1}[A_1]\rho = \frac{1}{2}\gamma \mathcal{D}[a]\rho = \mathcal{D}[\sqrt{\gamma}a/\sqrt{2}]\rho = \mathcal{D}[A_1]\rho$

$\displaystyle D_{1}[A_2]\rho = \frac{1}{2}\gamma \mathcal{D}[a]\rho = \mathcal{D}[-i\sqrt{\gamma}a/\sqrt{2}]\rho = \mathcal{D}[A_2]\rho$

$D_{2}[A_1]\rho = \frac{1}{\sqrt{2}} \sqrt{\gamma} \mathcal{H}[a] \rho = \mathcal{H}[A_1] \rho$

$D_{2}[A_2]\rho = \frac{1}{\sqrt{2}} \sqrt{\gamma} \mathcal{H}[-ia] \rho = \mathcal{H}[A_2] \rho$

where $A_1 = \sqrt{\gamma} a / \sqrt{2}$ and $A_2 = -i \sqrt{\gamma} a / \sqrt{2}$.

In summary we have

$\displaystyle d\rho(t) = -i[H, \rho(t)]dt + \sum_i\left\{\mathcal{D}[A_i]\rho(t) dt + \mathcal{H}[A_i]\rho(t) dW_i\right\}$

which is a simultaneous homodyne detection with $A_1 = \sqrt{\gamma}a/\sqrt{2}$ and $A_2 = -i\sqrt{\gamma}a/\sqrt{2}$

Here the two heterodyne currents for the $x$ and $y$ quadratures are

$J_x(t) = \sqrt{\gamma/2}\left<x\right> + \xi(t)$

$J_y(t) = \sqrt{\gamma/2}\left<y\right> + \xi(t)$

where $\xi(t) = \frac{dW}{dt}$.

In qutip we can use the predefined homodyne solver for solving this problem.

In [10]:
opt = Options()
opt.store_states = True
result = smesolve(
H,
rho0,
times,
[],
[np.sqrt(gamma / 2) * a, -1.0j * np.sqrt(gamma / 2) * a],
e_ops,
ntraj=100,
nsubsteps=nsubsteps,
solver="taylor15",
m_ops=[x, y],
dW_factors=[np.sqrt(2 / gamma), np.sqrt(2 / gamma)],
method="homodyne",
store_measurement=True,
map_func=parallel_map,
)

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90.0%. Run time:  28.02s. Est. time left: 00:00:00:03
100.0%. Run time:  31.18s. Est. time left: 00:00:00:00
Total run time:  31.19s

In [11]:
plot_expectation_values([result, result_ref]);

In [12]:
fig, ax = plt.subplots(figsize=(8, 4))

for m in result.measurement:
ax.plot(times, m[:, 0].real, "b", alpha=0.05)
ax.plot(times, m[:, 1].real, "r", alpha=0.05)

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

ax.set_xlim(0, times.max())
ax.set_ylim(-25, 25)
ax.set_xlabel("time", fontsize=12)
ax.plot(times, np.array(result.measurement).mean(axis=0)[:, 0].real, "k", lw=2)
ax.plot(times, np.array(result.measurement).mean(axis=0)[:, 1].real, "k", lw=2);


## Implementation #3: builtin function for heterodyne¶

In [13]:
result = smesolve(
H,
rho0,
times,
[],
[np.sqrt(gamma) * a],
e_ops,
ntraj=ntraj,
nsubsteps=nsubsteps,
solver="taylor15",
method="heterodyne",
store_measurement=True,
map_func=parallel_map,
)

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100.0%. Run time:  16.10s. Est. time left: 00:00:00:00
Total run time:  16.12s

In [14]:
plot_expectation_values([result, result_ref]);