QuTiPv5 Paper Notebook: smesolve

tutorials-v5/time-evolution/022_v5_paper-smesolve.md

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+
+# QuTiPv5 Paper Example: Stochastic Solver - Homodyne Detection
+
+Authors: Maximilian Meyer-Mölleringhof ([email protected])
+
+## Introduction
+
+When modelling an open quantum system, stochastic noise can be used to simulate a large range of phenomena.
+In the `smesolve()` solver, noise appears because of continuous measurement.
+It allows us to generate the trajectory evolution of a quantum system conditioned on a noisy measurement record.
+Historically speaking, such models were used by the quantum optics community to model homodyne and heterodyne detection of light emitted from a cavity.
+However, this solver is of course quite general and can thus also be applied to other problems.
+
+In this example we look at an optical cavity whose output is subject to homodyne detection.
+Such a cavity obeys the general stochastic master equation
+
+$d \rho(t) = -i [H, \rho(t)] dt + \mathcal{D}[a] \rho (t) dt + \mathcal{H}[a] \rho dW(t)$
+
+with the Hamiltonian
+
+$H = \Delta a^\dag a$
+
+and the Lindblad dissipator
+
+$\mathcal{D}[a] = a \rho a^\dag - \dfrac{1}{2} a^\dag a \rho - \dfrac{1}{2} \rho a^\dag a$.
+
+The stochastic part
+
+$\mathcal{H}[a]\rho = a \rho + \rho a^\dag - tr[a \rho + \rho a^\dag]$
+
+captures the conditioning of the trajectory through continious monitoring of the operator $a$.
+The term $dW(t)$ is the increment of a Wiener process that obeys $\mathbb{E}[dW] = 0$ and $\mathbb{E}[dW^2] = dt$.
+
+```python
+import numpy as np
+from matplotlib import pyplot as plt
+from qutip import about, coherent, destroy, mesolve, smesolve
+
+%matplotlib inline
+```
+
+## Problem Parameters
+
+```python
+N = 20  # dimensions of Hilbert space
+delta = 10 * np.pi  # cavity detuning
+kappa = 1  # decay rate
+A = 4  # initial coherent state intensity
+```
+
+```python
+a = destroy(N)
+x = a + a.dag()  # operator for expectation value
+H = delta * a.dag() * a  # Hamiltonian
+mon_op = np.sqrt(kappa) * a  # continiously monitored operators
+```
+
+## Solving for the Time Evolution
+
+We calculate the predicted trajectory conditioned on the continious monitoring of operator $a$.
+This is compared to the regular `mesolve()` solver for the same model but without resolving conditioned trajectories.
+
+```python
+rho_0 = coherent(N, np.sqrt(A))  # initial state
+times = np.arange(0, 1, 0.0025)
+num_traj = 500  # number of computed trajectories
+opt = {"dt": 0.00125, "store_measurement": True}
+```
+
+```python
+me_solution = mesolve(H, rho_0, times, c_ops=[mon_op], e_ops=[x])
+```
+
+```python
+stoc_solution = smesolve(
+    H, rho_0, times, sc_ops=[mon_op], e_ops=[x], ntraj=num_traj, options=opt
+)
+```
+
+## Comparison of Results
+
+We plot the averaged homodyne current $J_x = \langle x \rangle + dW / dt$ and the average system behaviour $\langle x \rangle$ for 500 trajectories.
+This is compared with the prediction of the regular `mesolve()` solver that does not include the conditioned trajectories.
+Since the conditioned expectation values do not depend on the trajectories, we expect that this reproduces the result of the standard `me_solve`.
+
+```python
+stoc_meas_mean = np.array(stoc_solution.measurement).mean(axis=0)[0, :].real
+```
+
+```python
+plt.figure()
+plt.plot(times[1:], stoc_meas_mean, lw=2, label=r"$J_x$")
+plt.plot(times, stoc_solution.expect[0], label=r"$\langle x \rangle$")
+plt.plot(
+    times,
+    me_solution.expect[0],
+    "--",
+    color="gray",
+    label=r"$\langle x \rangle$ mesolve",
+)
+
+plt.legend()
+plt.xlabel(r"$t \cdot \kappa$")
+plt.show()
+```
+
+## About
+
+```python
+about()
+```
+
+## Testing
+
+```python
+assert np.allclose(
+    stoc_solution.expect[0] == me_solution.expect[0]
+), "The smesolve and mesolve do not preoduce the same trajectory."
+```