Copyright (C) 2011 and later, Paul D. Nation & Robert J. Johansson
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%pylab inline
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from qutip import *
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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 we write the stochastic master equation on the form:
$\displaystyle d\rho(t) = -i[H, \rho] dt + \mathcal{D}[B] \rho dt + D_{1}[A]\rho(t) dt + D_{2}[A]\rho(t) d\xi$
where the first two term gives the deterministic master equation (Lindblad form with collapse operator $B$). Here $A = \sqrt{\eta\gamma} a$ and $B = \sqrt{(1-\eta)\gamma} $a.
We can identify
$\displaystyle D_{1}[A]\rho(t) = - \frac{1}{2}\eta\gamma \mathcal{H}[a^\dagger a] \rho(t) = - \frac{1}{2}\mathcal{H}[A^\dagger A] \rho(t) = -\frac{1}{2}\left( A^\dagger A\rho + \rho A^\dagger A - \mathrm{Tr}[A^\dagger A\rho + \rho A^\dagger A] \rho \right)$
$\displaystyle D_{2}[A]\rho(t) = \mathcal{G}[\sqrt{\eta\gamma}a] \rho = \mathcal{G}[A] \rho = \frac{A\rho A^\dagger}{\mathrm{Tr}[A\rho A^\dagger]} - \rho$
and $d\xi = dN(t)$ with a Poisson distribution.
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N = 15
w0 = 0.5 * 2 * pi
times = 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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# The argument A in the d1 and d2 callback functions is a list with the following
# precomputed superoperators, where c is the stochastic collapse operator given
# to the solver (called once for each operator, if more than one is given)
#
# A[0] = spre(c)
# A[1] = spost(c)
# A[2] = spre(c.dag())
# A[3] = spost(c.dag())
# A[4] = spre(n)
# A[5] = spost(n)
# A[6] = (spre(c) * spost(c.dag())
# A[7] = lindblad_dissipator(c)
$\displaystyle D_{1}[A]\rho(t) = -\frac{1}{2}\left( A^\dagger A\rho + \rho A^\dagger A - \mathrm{Tr}[A^\dagger A\rho + \rho A^\dagger A] \rho \right)$
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def d1_rho_func(A, rho_vec):
n_sum = A[4] + A[5]
return 0.5 * (- n_sum * rho_vec + expect_rho_vec(n_sum, rho_vec, False) * rho_vec)
$\displaystyle D_{2}[A]\rho(t) = \frac{A\rho A^\dagger}{\mathrm{Tr}[A\rho A^\dagger]} - \rho$
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def d2_rho_func(A, rho_vec):
e1 = expect_rho_vec(A[6], rho_vec, False) + 1e-16 # add a small number to avoid division by zero
return [(A[6] * rho_vec) / e1 - rho_vec]
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eta = 0.7
c_ops = [sqrt(1-eta) * sqrt(gamma) * a] # collapse operator B
sc_ops = [sqrt(eta) * 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 = smesolve(H, rho0, times, c_ops=c_ops, sc_ops=sc_ops, e_ops=e_ops,
ntraj=1, nsubsteps=100, d1=d1_rho_func, d2=d2_rho_func,
distribution='poisson', store_measurement=True)
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result2 = smesolve(H, rho0, times, c_ops=c_ops, sc_ops=sc_ops, e_ops=e_ops,
ntraj=10, nsubsteps=100, d1=d1_rho_func, d2=d2_rho_func,
distribution='poisson', store_measurement=True)
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fig, axes = 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(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 = [sqrt(1-eta) * sqrt(gamma) * a] # collapse operator B
sc_ops = [sqrt(eta) * 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 = smesolve(H, rho0, times, c_ops=c_ops, sc_ops=sc_ops, e_ops=e_ops,
ntraj=1, nsubsteps=100, d1=d1_rho_func, d2=d2_rho_func,
distribution='poisson', store_measurement=True)
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result2 = smesolve(H, rho0, times, c_ops=c_ops, sc_ops=sc_ops, e_ops=e_ops,
ntraj=10, nsubsteps=100, d1=d1_rho_func, d2=d2_rho_func,
distribution='poisson', store_measurement=True)
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fig, axes = 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(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, sqrt(5))
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eta = 0.95
c_ops = [sqrt(1-eta) * sqrt(gamma) * a] # collapse operator B
sc_ops = [sqrt(eta) * 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=125, nsubsteps=100,
method='homodyne', store_measurement=True)
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plot_expectation_values([result, result_ref]);
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fig, ax = subplots(figsize=(8,4))
M = sqrt(eta) * sqrt(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, 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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def d1_rho_func(A, rho_vec):
return A[7] * 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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def d2_rho_func(A, rho_vec):
n_sum = A[0] + A[3]
e1 = expect_rho_vec(n_sum, rho_vec, False)
return [sqrt(eta) * (n_sum * rho_vec - e1 * rho_vec)]
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c_ops = []
sc_ops = [sqrt(gamma) * 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, [], sc_ops, e_ops,
ntraj=125, nsubsteps=100, d1=d1_rho_func, d2=d2_rho_func,
m_ops=[[a + a.dag()]], dW_factors=[1/sqrt(gamma * eta)],
distribution='normal', store_measurement=True)
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plot_expectation_values([result, result_ref]);
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fig, ax = 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, 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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