`import qutip_jax as qjax
import qutip as qt
import jax
import jax.numpy as jnp
from functools import partial
qjax.use_jax_backend()

# Define time-dependent functions
@partial(jax.jit, static_argnames=("omega",))
def H_1_coeff(t, omega):
    return 2.0 * jnp.pi * 0.25 * jnp.cos(2.0 * omega * t)

# Hamiltonian components
N = 100
with qt.CoreOptions(default_dtype="jaxdia"):
    a = qt.tensor(qt.qeye(2), qt.destroy(N))
    sm = qt.tensor(qt.destroy(2), qt.qeye(N))
    
H_0 = 2.0 * jnp.pi * a.dag() * a + 2.0 * jnp.pi * sm.dag() * sm
H_1_op = sm * a.dag() + sm.dag() * a
H = [H_0, [H_1_op, qt.coefficient(H_1_coeff, args={"omega": 1.0})]]

# Initial state
state = qt.tensor(qt.fock(2, 0, dtype="jax"), qt.fock(N, 8, dtype="jax"))

# Collapse operators and observables
c_ops = [jnp.sqrt(0.1) * a]
e_ops = [a.dag() * a, sm.dag() * sm]

# Time list
tlist = jnp.linspace(0.0, 100, 2)

# Define the function for which we want to compute the gradient
def f(omega):
    H[1][1] = qt.coefficient(H_1_coeff, args={"omega": omega})
    result = qt.mcsolve(H, state, tlist, c_ops, e_ops, ntraj=10, options={"method": "diffrax"})
    # Return the expectation value of the number operator at tlist[1]
    return result.expect[0][1].real

# Compute the gradient
gradient = jax.grad(f)(1.0)

print("Gradient:", gradient)
`