Two implementations of heterodyne detection: direct heterodyne and as two homodyne measurements

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

In [1]:
%matplotlib inline
In [2]:
import numpy as np
In [3]:
import matplotlib.pyplot as plt
In [4]:
from qutip import *

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 [5]:
N = 15
w0 = 1.0 * 2 * np.pi
A = 0.1 * 2 * np.pi
times = np.linspace(0, 15, 150)
gamma = 0.25

ntraj = 250
nsubsteps = 250

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 [6]:
result_ref = mesolve(H, rho0, times, c_ops, e_ops)
In [7]:
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 [8]:
from qutip.expect import expect_rho_vec
In [9]:
def d1_rho_func(t, rho_vec, A, args):
    return A[7] * 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 [10]:
def d2_rho_func(t, rho_vec, A, args):

    B = A[0] + A[3]
    e1 = expect_rho_vec(B, rho_vec, False)
    drho1 = B * rho_vec - e1 * rho_vec

    B = A[0] - A[3]
    e1 = expect_rho_vec(B, rho_vec, False)
    drho2 = B * rho_vec - e1 * rho_vec

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

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

$J_x(t) = \sqrt{\gamma}\left + \sqrt{2} \xi(t)$

$J_y(t) = \sqrt{\gamma}\left + \sqrt{2} \xi(t)$

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 [11]:
result = smesolve(H, rho0, times, [], c_ops, e_ops, 
                  ntraj=ntraj, nsubsteps=nsubsteps,
                  d1=d1_rho_func, d2=d2_rho_func, d2_len=2,
                  dW_factors=[np.sqrt(2/gamma), np.sqrt(2/gamma)], m_ops=[[x, y]],
                  distribution='normal', store_measurement=True,
                  map_func=parallel_map)
10.0%. Run time:  80.60s. Est. time left: 00:00:12:05
20.0%. Run time: 134.52s. Est. time left: 00:00:08:58
30.0%. Run time: 191.49s. Est. time left: 00:00:07:26
40.0%. Run time: 247.64s. Est. time left: 00:00:06:11
50.0%. Run time: 304.30s. Est. time left: 00:00:05:04
60.0%. Run time: 362.20s. Est. time left: 00:00:04:01
70.0%. Run time: 420.10s. Est. time left: 00:00:03:00
80.0%. Run time: 480.71s. Est. time left: 00:00:02:00
90.0%. Run time: 537.85s. Est. time left: 00:00:00:59
100.0%. Run time: 592.92s. Est. time left: 00:00:00:00
Total run time: 592.98s
In [12]:
plot_expectation_values([result, result_ref]);
In [13]:
fig, ax = plt.subplots(figsize=(8,4))

for m in result.measurement:
    ax.plot(times, m[:, 0, 0].real, 'b', alpha=0.05)
    ax.plot(times, m[:, 0, 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(-25, 25)
ax.set_xlim(0, times.max())
ax.set_xlabel('time', fontsize=12)
ax.plot(times, np.array(result.measurement).mean(axis=0)[:,0,0].real, 'k', lw=2);
ax.plot(times, np.array(result.measurement).mean(axis=0)[:,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

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

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

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

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

In [14]:
result = smesolve(H, rho0, times, [], [np.sqrt(gamma/2) * a, -1.0j * np.sqrt(gamma/2) * a],
                  e_ops, ntraj=ntraj, nsubsteps=nsubsteps,
                  m_ops=[[x], [y]], dW_factors=[np.sqrt(2/gamma)],
                  method='homodyne', store_measurement=True,
                  map_func=parallel_map)
10.0%. Run time:  90.68s. Est. time left: 00:00:13:36
20.0%. Run time: 149.96s. Est. time left: 00:00:09:59
30.0%. Run time: 208.58s. Est. time left: 00:00:08:06
40.0%. Run time: 266.32s. Est. time left: 00:00:06:39
50.0%. Run time: 328.32s. Est. time left: 00:00:05:28
60.0%. Run time: 394.28s. Est. time left: 00:00:04:22
70.0%. Run time: 461.11s. Est. time left: 00:00:03:17
80.0%. Run time: 526.44s. Est. time left: 00:00:02:11
90.0%. Run time: 594.06s. Est. time left: 00:00:01:06
100.0%. Run time: 658.45s. Est. time left: 00:00:00:00
Total run time: 658.51s
In [15]:
plot_expectation_values([result, result_ref]);
In [16]:
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(-15, 15)
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 [17]:
result = smesolve(H, rho0, times, [], [np.sqrt(gamma) * a],
                  e_ops, ntraj=ntraj, nsubsteps=nsubsteps,
                  method='heterodyne', store_measurement=True,
                  map_func=parallel_map)
10.0%. Run time:  92.69s. Est. time left: 00:00:13:54
20.0%. Run time: 157.56s. Est. time left: 00:00:10:30
30.0%. Run time: 220.78s. Est. time left: 00:00:08:35
40.0%. Run time: 284.41s. Est. time left: 00:00:07:06
50.0%. Run time: 347.93s. Est. time left: 00:00:05:47
60.0%. Run time: 411.42s. Est. time left: 00:00:04:34
70.0%. Run time: 470.95s. Est. time left: 00:00:03:21
80.0%. Run time: 529.69s. Est. time left: 00:00:02:12
90.0%. Run time: 589.72s. Est. time left: 00:00:01:05
100.0%. Run time: 645.43s. Est. time left: 00:00:00:00
Total run time: 645.53s
In [18]:
plot_expectation_values([result, result_ref]);
In [19]:
fig, ax = plt.subplots(figsize=(8,4))

for m in result.measurement:
    ax.plot(times, m[:, 0, 0].real / np.sqrt(gamma), 'b', alpha=0.05)
    ax.plot(times, m[:, 0, 1].real / np.sqrt(gamma), '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(-15, 15)
ax.set_xlabel('time', fontsize=12)
ax.plot(times, np.array(result.measurement).mean(axis=0)[:, 0, 0].real / np.sqrt(gamma), 'k', lw=2);
ax.plot(times, np.array(result.measurement).mean(axis=0)[:, 0, 1].real / np.sqrt(gamma), 'k', lw=2);

Versions

In [20]:
from qutip.ipynbtools import version_table

version_table()
Out[20]:
SoftwareVersion
IPython2.3.1
matplotlib1.4.2
SciPy0.14.1
OSposix [linux]
Cython0.21.2
Python3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2]
QuTiP3.1.0
Numpy1.9.1
Tue Jan 13 15:56:15 2015 JST