How to use the qutip.expect.expect function in qutip

Position for the annotaion.
            Qobj of a qubit or a vector of 3 elements.

        text : str/unicode
            Annotation text.
            You can use LaTeX, but remember to use raw string
            e.g. r"$\\langle x \\rangle$"
            or escape backslashes
            e.g. "$\\\\langle x \\\\rangle$".

        **kwargs :
            Options as for mplot3d.axes3d.text, including:
            fontsize, color, horizontalalignment, verticalalignment.
        """
        if isinstance(state_or_vector, Qobj):
            vec = [expect(sigmax(), state_or_vector),
                   expect(sigmay(), state_or_vector),
                   expect(sigmaz(), state_or_vector)]
        elif isinstance(state_or_vector, (list, ndarray, tuple)) \
                and len(state_or_vector) == 3:
            vec = state_or_vector
        else:
            raise Exception("Position needs to be specified by a qubit " +
                            "state or a 3D vector.")
        self.annotations.append({'position': vec,
                                 'text': text,
                                 'opts': kwargs})

jump_times = []
    jump_op_idx = []

    for t_idx, t in enumerate(times):

        if e_ops:
            for e_idx, e in enumerate(e_ops):
                data.expect[e_idx, t_idx] += expect_rho_vec(e, rho_t)
        else:
            states_list.append(Qobj(vec2mat(rho_t), dims=dims))

        for j in range(N_substeps):

            if sigma_t.norm() < r_jump:
                # jump occurs
                p = np.array([expect(c.dag() * c, rho_t) for c in c_ops])
                p = np.cumsum(p / np.sum(p))
                n = np.where(p >= r_op)[0][0]

                # apply jump
                rho_t = c_ops[n] * rho_t * c_ops[n].dag()
                rho_t /= expect(c_ops[n].dag() * c_ops[n], rho_t)
                sigma_t = np.copy(rho_t)

                # store info about jump
                jump_times.append(times[t_idx] + dt * j)
                jump_op_idx.append(n)

                # get new random numbers for next jump
                r_jump, r_op = prng.rand(2)

            # deterministic evolution wihtout correction for norm decay

for e_idx, e in enumerate(e_ops):
                data.expect[e_idx, t_idx] += expect_rho_vec(e, rho_t)
        else:
            states_list.append(Qobj(vec2mat(rho_t), dims=dims))

        for j in range(N_substeps):

            if sigma_t.norm() < r_jump:
                # jump occurs
                p = np.array([expect(c.dag() * c, rho_t) for c in c_ops])
                p = np.cumsum(p / np.sum(p))
                n = np.where(p >= r_op)[0][0]

                # apply jump
                rho_t = c_ops[n] * rho_t * c_ops[n].dag()
                rho_t /= expect(c_ops[n].dag() * c_ops[n], rho_t)
                sigma_t = np.copy(rho_t)

                # store info about jump
                jump_times.append(times[t_idx] + dt * j)
                jump_op_idx.append(n)

                # get new random numbers for next jump
                r_jump, r_op = prng.rand(2)

            # deterministic evolution wihtout correction for norm decay
            dsigma_t = spmv(L.data, sigma_t) * dt

            # deterministic evolution with correction for norm decay
            drho_t = spmv(L.data, rho_t) * dt

            rho_t += drho_t

Qobj of a qubit or a vector of 3 elements.

        text : str/unicode
            Annotation text.
            You can use LaTeX, but remember to use raw string
            e.g. r"$\\langle x \\rangle$"
            or escape backslashes
            e.g. "$\\\\langle x \\\\rangle$".

        **kwargs :
            Options as for mplot3d.axes3d.text, including:
            fontsize, color, horizontalalignment, verticalalignment.

        """
        if isinstance(state_or_vector, Qobj):
            vec = [expect(sigmax(), state_or_vector),
                   expect(sigmay(), state_or_vector),
                   expect(sigmaz(), state_or_vector)]
        elif isinstance(state_or_vector, (list, ndarray, tuple)) \
                and len(state_or_vector) == 3:
            vec = state_or_vector
        else:
            raise Exception("Position needs to be specified by a qubit " +
                            "state or a 3D vector.")
        self.annotations.append({'position': vec,
                                 'text': text,
                                 'opts': kwargs})

def _spectrum_es(H, wlist, c_ops, a_op, b_op):
    """
    Internal function for calculating the spectrum of the correlation function
    :math:`\left`.
    """
    if debug:
        print(inspect.stack()[0][3])

    # construct the Liouvillian
    L = liouvillian(H, c_ops)

    # find the steady state density matrix and a_op and b_op expecation values
    rho0 = steadystate(L)

    a_op_ss = expect(a_op, rho0)
    b_op_ss = expect(b_op, rho0)

    # eseries solution for (b * rho0)(t)
    es = ode2es(L, b_op * rho0)

    # correlation
    corr_es = expect(a_op, es)

    # covariance
    cov_es = corr_es - a_op_ss * b_op_ss
    # tidy up covariance (to combine, e.g., zero-frequency components that cancel)
    cov_es.tidyup()

    # spectrum
    spectrum = esspec(cov_es, wlist)

# Calculate the Liouvillian
    if (c_op_list is None or len(c_op_list) == 0) and isket(rho0):
        L = H
    else:
        L = liouvillian(H, c_op_list)

    es = ode2es(L, rho0)

    # evaluate the expectation values
    if n_expt_op == 0:
        results = [Qobj()] * n_tsteps
    else:
        results = np.zeros([n_expt_op, n_tsteps], dtype=complex)

    for n, e in enumerate(e_ops):
        results[n, :] = expect(e, esval(es, tlist))

    data = Result()
    data.solver = "essolve"
    data.times = tlist
    data.expect = [np.real(results[n, :]) if e.isherm else results[n, :]
                   for n, e in enumerate(e_ops)]

    return data

else:
            # calculate all the expectation values, or output rho if
            # no operators
            if n_expt_op == 0:
                if floquet_basis:
                    output.states.append(Qobj(rho))
                else:
                    f_modes_table_t, T = f_modes_table
                    f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
                    output.states.append(Qobj(rho).transform(f_modes_t, True))
            else:
                f_modes_table_t, T = f_modes_table
                f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
                for m in range(0, n_expt_op):
                    output.expect[m][t_idx] = \
                        expect(e_ops[m], rho.transform(f_modes_t, False))

        r.integrate(r.t + dt)
        t_idx += 1

    return output

solver options class. `ntraj` is taken as a two-element list because
        the `mc` correlator calls `mcsolve()` recursively; by default,
        `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
        the `mc` correlator; by default, `mc_corr_eps=1e-10`.

    Returns
    -------
    g2, G2 : tuple
        The normalized and unnormalized second-order coherence function.

    """

    # first calculate the photon number
    if state0 is None:
        state0 = steadystate(H, c_ops)
        n = np.array([expect(state0, a_op.dag() * a_op)])
    else:
        n = mesolve(H, state0, taulist, c_ops, [a_op.dag() * a_op], args=args).expect[0]

    # calculate the correlation function G2 and normalize with n to obtain g2
    G2 = correlation_3op_1t(H, state0, taulist, c_ops,
                            a_op.dag(), a_op.dag()*a_op, a_op,
                            solver=solver, args=args, options=options)
    g2 = G2 / (n[0] * n)

    return g2, G2

Returns
    -------
    corr_mat : ndarray
        A 2-dimensional *array* of correlation values or operators.


    """

    if rho is None:
        # return array of operators
        return np.array([[op1 * op2 for op1 in basis]
                         for op2 in basis], dtype=object)
    else:
        # return array of expectation values
        return np.array([[expect(op1 * op2, rho)
                          for op1 in basis] for op2 in basis], dtype=object)