# Stochastic Solver

When a quantum system is subjected to continuous measurement, through homodyne detection for example, it is possible to simulate the conditional quantum state using stochastic Schrodinger and master equations. The solution of these stochastic equations are quantum trajectories, which represent the conditioned evolution of the system given a specific measurement record.

In general, the stochastic evolution of a quantum state is calculated in QuTiP by solving the general equation

(1)\[d \rho (t) = d_1 \rho dt + \sum_n d_{2,n} \rho dW_n,\]

where \(dW_n\) is a Wiener increment, which has the expectation values \(E[dW] = 0\) and \(E[dW^2] = dt\). Stochastic evolution is implemented with the `qutip.stochastic.general_stochastic` function.

## Stochastic Schrodinger Equation

The stochastic Schrodinger equation is given by (see section 4.4, [Wis09])

(2)\[d \psi(t) = - i H \psi(t) dt - \sum_n \left( \frac{S_n^\dagger S_n}{2} -\frac{e_n}{2} S_n + \frac{e_n^2}{8} \right) \psi(t) dt + \sum_n \left( S_n - \frac{e_n}{2} \right) \psi(t) dW_n,\]

where \(H\) is the Hamiltonian, \(S_n\) are the stochastic collapse operators, and \(e_n\) is

(3)\[e_n = \left<\psi(t)|S_n + S_n^\dagger|\psi(t)\right>\]

In QuTiP, this equation can be solved using the function `qutip.stochastic.ssesolve`, which is implemented by defining \(d_1\) and \(d_{2,n}\) from Equation (1) as

(4)\[d_1 = -iH - \frac{1}{2} \sum_n \left(S_n^\dagger S_n - e_n S_n + \frac{e_i^2}{4} \right),\]

and

(5)\[d_{2, n} = S_n - \frac{e_n}{2}.\]

The solver `qutip.stochastic.ssesolve` will construct the operators \(d_1\) and \(d_{2,n}\) once the user passes the Hamiltonian (`H`) and the stochastic operator list (`sc_ops`). As with the `qutip.mcsolve`, the number of trajectories and the seed for the noise realisation can be fixed using the arguments: `ntraj` and `noise`, respectively. If the user also requires the measurement output, the argument `store_measurement=True` should be included.

Additionally, homodyne and heterodyne detections can be easily simulated by passing the arguments `method='homodyne'` or `method='heterodyne'` to `qutip.stochastic.ssesolve`.

Examples of how to solve the stochastic Schrodinger equation using QuTiP can be found in this development notebook.

## Stochastic Master Equation

When the initial state of the system is a density matrix \(\rho\), the stochastic master equation solver `qutip.stochastic.smesolve` must be used. The stochastic master equation is given by (see section 4.4, [Wis09])

(6)\[ d \rho (t) = -i[H, \rho(t)] dt + D[A]\rho(t) dt + \mathcal{H}[A]\rho dW(t)\]

where

(7)\[D[A] \rho = \frac{1}{2} \left[2 A \rho A^\dagger - \rho A^\dagger A - A^\dagger A \rho \right],\]

and

(8)\[\mathcal{H}[A]\rho = A\rho(t) + \rho(t) A^\dagger - \tr[A\rho(t) + \rho(t) A^\dagger].\]

In QuTiP, solutions for the stochastic master equation are obtained using the solver `qutip.stochastic.smesolve`. The implementation takes into account 2 types of collapse operators. \(C_i\) (`c_ops`) represent the dissipation in the environment, while \(S_n\) (`sc_ops`) are monitored operators. The deterministic part of the evolution, described by the \(d_1\) in Equation (1), takes into account all operators \(C_i\) and \(S_n\):

(9)\[d_1 = - i[H(t),\rho(t)] + \sum_i D[C_i]\rho + \sum_n D[S_n]\rho,\]

The stochastic part, \(d_{2,n}\), is given solely by the operators \(S_n\)

(10)\[d_{2,n} = S_n \rho(t) + \rho(t) S_n^\dagger - \tr \left(S_n \rho (t) + \rho(t) S_n^\dagger \right)\rho(t).\]

As in the stochastic Schrodinger equation, the detection method can be specified using the `method` argument.

### Example

Below, we solve the dynamics for an optical cavity at 0K whose output is monitored using homodyne detection. The cavity decay rate is given by \(\kappa\) and the \(\Delta\) is the cavity detuning with respect to the driving field. The measurement operators can be passed using the option `m_ops`. The homodyne current \(J_x\) is calculated using

(11)\[J_x = \langle x \rangle + dW,\]

where \(x\) is the operator passed using `m_ops`. The results are available in `result.measurements`.

    import numpy as np
    import matplotlib.pyplot as plt
    import qutip as qt

    # parameters
    DIM = 20             # Hilbert space dimension
    DELTA = 5*2*np.pi    # cavity detuning
    KAPPA = 2            # cavity decay rate
    INTENSITY = 4        # intensity of initial state
    NUMBER_OF_TRAJECTORIES = 500

    # operators
    a = qt.destroy(DIM)
    x = a + a.dag()
    H = DELTA*a.dag()* a

    rho_0 = qt.coherent(DIM, np.sqrt(INTENSITY))
    times = np.arange(0, 1, 0.0025)

    stoc_solution = qt.smesolve(H, rho_0, times,
                                c_ops=[],
                                sc_ops=[np.sqrt(KAPPA) * a],
                                e_ops=[x],
                                ntraj=NUMBER_OF_TRAJECTORIES,
                                nsubsteps=2,
                                store_measurement=True,
                                dW_factors=[1],
                                method='homodyne')

    fig, ax = plt.subplots()
    ax.set_title('Stochastic Master Equation - Homodyne Detection')
    ax.plot(times, np.array(stoc_solution.measurement).mean(axis=0)[:].real,
            'r', lw=2, label=r'$J_x$')
    ax.plot(times, stoc_solution.expect[0], 'k', lw=2,
            label=r'$\langle x \rangle$')
    ax.set_xlabel('Time')
    ax.legend()

For other examples on `qutip.stochastic.smesolve`, see the following notebook, as well as these notebooks available at QuTiP Tutorials page: heterodyne detection, inneficient detection, and feedback control.