Copyright (C) 2011 and later, Paul D. Nation & Robert J. Johansson
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%matplotlib inline
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import matplotlib.pyplot as plt
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import numpy as np
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from qutip import *
from qutip.expect import expect_rho_vec
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from matplotlib import rcParams
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
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.
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N = 15
w0 = 0.5 * 2 * np.pi
times = np.linspace(0, 15, 150)
dt = times[1] - times[0]
gamma = 0.1
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a = destroy(N)
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H = w0 * a.dag() * a
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rho0 = fock(N, 5)
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e_ops = [a.dag() * a, a + a.dag()]
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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
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result_ref = mesolve(H, rho0, times, c_ops+sc_ops, e_ops)
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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)
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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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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()
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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
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result_ref = mesolve(H, rho0, times, c_ops+sc_ops, e_ops)
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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)
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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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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
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rho0 = coherent(N, np.sqrt(5))
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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
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result_ref = mesolve(H, rho0, times, c_ops+sc_ops, e_ops)
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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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plot_expectation_values([result, result_ref]);
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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);
$\displaystyle D_{1}[A]\rho(t) = \mathcal{D}[\kappa a]\rho(t) = \mathcal{D}[A]\rho(t)$
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L = liouvillian(H, np.sqrt(gamma) * a)
def d1_rho_func(t, rho_vec):
return L * rho_vec
$\displaystyle D_{2}[A]\rho(t) = \sqrt{\eta} \mathcal{H}[\sqrt{\kappa} a]\rho(t) = \sqrt{\eta} \mathcal{H}[A]\rho(t) = \sqrt{\eta}(A\rho + \rho A^\dagger - \mathrm{Tr}[A\rho + \rho A^\dagger] \rho) \rightarrow \sqrt{\eta} \left((A_L + A_R^\dagger)\rho_v - \mathrm{Tr}[(A_L + A_R^\dagger)\rho_v] \rho_v\right)$
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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)])
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result_ref = mesolve(H, rho0, times, c_ops+sc_ops, e_ops)
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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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plot_expectation_values([result, result_ref])
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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);
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from qutip.ipynbtools import version_table
version_table()
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