def _ssesolve_single_trajectory(H, dt, tlist, N_store, N_substeps, psi_t,
A_ops, e_ops, data, rhs, d1, d2, d2_len,
homogeneous, distribution, args,
store_measurement=False, noise=None):
"""
Internal function. See ssesolve.
"""
if noise is None:
if homogeneous:
if distribution == 'normal':
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps, d2_len)
else:
raise TypeError('Unsupported increment distribution for homogeneous process.')
else:
if distribution != 'poisson':
raise TypeError('Unsupported increment distribution for inhomogeneous process.')
dW = np.zeros((len(A_ops), N_store, N_substeps, d2_len))
else:
dW = noise
states_list = []
measurements = np.zeros((len(tlist), len(A_ops)), dtype=complex)
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
s = cy_expect(e.data.data, e.data.indices, e.data.indptr, psi_t)
data.expect[e_idx, t_idx] += s
data.ss[e_idx, t_idx] += s ** 2
else:
states_list.append(Qobj(psi_t))
psi_prev = np.copy(psi_t)
for j in range(N_substeps):
if noise is None and not homogeneous:
for a_idx, A in enumerate(A_ops):
dw_expect = norm(spmv(A[0].data, A[0].indices, A[0].indptr, psi_t)) ** 2 * dt
dW[a_idx, t_idx, j, :] = np.random.poisson(dw_expect, d2_len)
psi_t = rhs(H.data, psi_t, t + dt * j,
A_ops, dt, dW[:, t_idx, j, :], d1, d2, args)
# optionally renormalize the wave function: TODO: make the
# renormalization optional througha configuration parameter in
# Odeoptions
psi_t /= norm(psi_t)
if store_measurement:
for a_idx, A in enumerate(A_ops):
phi = spmv(A[0].data, A[0].indices, A[0].indptr, psi_prev)
measurements[t_idx, a_idx] = (norm(phi) ** 2 * dt * N_substeps
+ dW[a_idx, t_idx, :, 0].sum())
return states_list, dW, measurements
def _ssesolve_single_trajectory(H, dt, tlist, N_store, N_substeps, psi_t,
A_ops, e_ops, data, rhs, d1, d2, d2_len,
homogeneous, distribution, args,
store_measurement=False, noise=None):
"""
Internal function. See ssesolve.
"""
if noise is None:
if homogeneous:
if distribution == 'normal':
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps, d2_len)
else:
raise TypeError('Unsupported increment distribution for homogeneous process.')
else:
if distribution != 'poisson':
raise TypeError('Unsupported increment distribution for inhomogeneous process.')
dW = np.zeros((len(A_ops), N_store, N_substeps, d2_len))
else:
dW = noise
states_list = []
measurements = np.zeros((len(tlist), len(A_ops)), dtype=complex)
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
s = cy_expect(e.data.data, e.data.indices, e.data.indptr, psi_t)
data.expect[e_idx, t_idx] += s
data.ss[e_idx, t_idx] += s ** 2
else:
states_list.append(Qobj(psi_t))
psi_prev = np.copy(psi_t)
for j in range(N_substeps):
if noise is None and not homogeneous:
for a_idx, A in enumerate(A_ops):
dw_expect = norm(spmv(A[0].data, A[0].indices, A[0].indptr, psi_t)) ** 2 * dt
dW[a_idx, t_idx, j, :] = np.random.poisson(dw_expect, d2_len)
psi_t = rhs(H.data, psi_t, t + dt * j,
A_ops, dt, dW[:, t_idx, j, :], d1, d2, args)
# optionally renormalize the wave function: TODO: make the
# renormalization optional througha configuration parameter in
# Odeoptions
psi_t /= norm(psi_t)
if store_measurement:
for a_idx, A in enumerate(A_ops):
phi = spmv(A[0].data, A[0].indices, A[0].indptr, psi_prev)
measurements[t_idx, a_idx] = (norm(phi) ** 2 * dt * N_substeps
+ dW[a_idx, t_idx, :, 0].sum())
return states_list, dW, measurements
def _smepdpsolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
c_ops, e_ops, data):
"""
Internal function.
"""
states_list = []
rho_t = np.copy(rho_t)
prng = RandomState() # todo: seed it
r_jump, r_op = prng.rand(2)
jump_times = []
jump_op_idx = []
for t_idx, t in enumerate(tlist):
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)))
for j in range(N_substeps):
if expect_rho_vec(d_op, sigma_t) < r_jump:
# jump occurs
p = np.array([rho_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] * psi_t * c_ops[n].dag()
rho_t /= rho_expect(c.dag() * c, rho_t)
rho_t = np.copy(rho_t)
# store info about jump
jump_times.append(tlist[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.data,
L.data.indices,
L.data.indptr, sigma_t) * dt
# deterministic evolution with correction for norm decay
drho_t = spmv(L.data.data,
L.data.indices,
L.data.indptr, rho_t) * dt
rho_t += drho_t
# increment density matrices
sigma_t += dsigma_t
rho_t += drho_t
return states_list, jump_times, jump_op_idx
def _smepdpsolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
c_ops, e_ops, data):
"""
Internal function.
"""
states_list = []
rho_t = np.copy(rho_t)
prng = RandomState() # todo: seed it
r_jump, r_op = prng.rand(2)
jump_times = []
jump_op_idx = []
for t_idx, t in enumerate(tlist):
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)))
for j in range(N_substeps):
if expect_rho_vec(d_op, sigma_t) < r_jump:
# jump occurs
p = np.array([rho_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] * psi_t * c_ops[n].dag()
rho_t /= rho_expect(c.dag() * c, rho_t)
rho_t = np.copy(rho_t)
# store info about jump
jump_times.append(tlist[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.data,
L.data.indices,
L.data.indptr, sigma_t) * dt
# deterministic evolution with correction for norm decay
drho_t = spmv(L.data.data,
L.data.indices,
L.data.indptr, rho_t) * dt
rho_t += drho_t
# increment density matrices
sigma_t += dsigma_t
rho_t += drho_t
return states_list, jump_times, jump_op_idx
def d1_current(A, psi):
"""
Todo: cythonize, requires poisson increments
"""
n1 = norm(spmv(A[0].data, A[0].indices, A[0].indptr, psi), 2)
return (-0.5 * (spmv(A[2].data, A[2].indices, A[2].indptr, psi)
- n1 ** 2 * psi))
def d1_psi_homodyne(A, psi):
"""
OK
Todo: cythonize
"""
e1 = cy_expect(A[1].data, A[1].indices, A[1].indptr, 0, psi)
return 0.5 * (e1 * spmv(A[0].data, A[0].indices, A[0].indptr, psi) -
spmv(A[2].data, A[2].indices, A[2].indptr, psi) -
0.25 * e1 ** 2 * psi)
def d2_rho_heterodyne(A, rho_vec):
"""
todo: cythonize, docstrings
"""
M = A[0] + A[3]
e1 = cy_expect_rho_vec(M, rho_vec)
d1 = spmv(M.data, M.indices, M.indptr, rho_vec) - e1 * rho_vec
M = A[0] - A[3]
e1 = cy_expect_rho_vec(M, rho_vec)
d2 = spmv(M.data, M.indices, M.indptr, rho_vec) - e1 * rho_vec
return [1.0/np.sqrt(2) * d1, -1.0j/np.sqrt(2) * d2]
def d2_rho_heterodyne(A, rho_vec):
"""
todo: cythonize, docstrings
"""
M = A[0] + A[3]
e1 = cy_expect_rho_vec(M, rho_vec)
d1 = spmv(M.data, M.indices, M.indptr, rho_vec) - e1 * rho_vec
M = A[0] - A[3]
e1 = cy_expect_rho_vec(M, rho_vec)
d2 = spmv(M.data, M.indices, M.indptr, rho_vec) - e1 * rho_vec
return [1.0/np.sqrt(2) * d1, -1.0j/np.sqrt(2) * d2]
def d1_psi_photocurrent(A, psi):
"""
Todo: cythonize.
Note: requires poisson increments
.. math::
D_1(\psi, t) = - \\frac{1}{2}(C^\dagger C \psi - ||C\psi||^2 \psi)
"""
return (-0.5 * (spmv(A[3].data, A[3].indices, A[3].indptr, psi)
-norm(spmv(A[0].data, A[0].indices, A[0].indptr, psi)) ** 2 * psi))
def d1_psi_photocurrent(A, psi):
"""
Todo: cythonize.
Note: requires poisson increments
.. math::
D_1(\psi, t) = - \\frac{1}{2}(C^\dagger C \psi - ||C\psi||^2 \psi)
"""
return (-0.5 * (spmv(A[3].data, A[3].indices, A[3].indptr, psi)
-norm(spmv(A[0].data, A[0].indices, A[0].indptr, psi)) ** 2 * psi))
def d1_psi_heterodyne(A, psi):
"""
not working/tested
Todo: cythonize
"""
e1 = cy_expect(A[0].data, A[0].indices, A[0].indptr, 0, psi)
B = A[0].T.conj()
e2 = cy_expect(B.data, B.indices, B.indptr, 0, psi)
return (e2 * spmv(A[0].data, A[0].indices, A[0].indptr, psi)
- 0.5 * spmv(A[2].data, A[2].indices, A[2].indptr, psi)
- 0.5 * e1 * e2 * psi)
def _rhs_psi_platen(H, psi_t, t, A_ops, dt, dW, d1, d2, args):
"""
TODO: support multiple stochastic increments
.. note::
Experimental.
"""
sqrt_dt = np.sqrt(dt)
dW_len = len(dW[0, :])
dpsi_t = _rhs_psi_deterministic(H, psi_t, t, dt, args)
for a_idx, A in enumerate(A_ops):
# XXX: This needs to be revised now that
# dpsi_t is the change for all stochastic collapse operators
# TODO: needs to be updated to support mutiple Weiner increments
dpsi_t_H = (-1.0j * dt) * spmv(H.data, H.indices, H.indptr, psi_t)
psi_t_1 = psi_t + dpsi_t_H + d1(A, psi_t) * dt + d2(A, psi_t)[0] * dW[a_idx,0]
psi_t_p = psi_t + dpsi_t_H + d1(A, psi_t) * dt + d2(A, psi_t)[0] * sqrt_dt
psi_t_m = psi_t + dpsi_t_H + d1(A, psi_t) * dt - d2(A, psi_t)[0] * sqrt_dt
dpsi_t += 0.50 * (d1(A, psi_t_1) + d1(A, psi_t)) * dt + \
0.25 * (d2(A, psi_t_p)[0] + d2(A, psi_t_m)[0] + 2 * d2(A, psi_t)[0]) * dW[a_idx,0] + \
0.25 * (d2(A, psi_t_p)[0] - d2(A, psi_t_m)[0]) * (dW[a_idx,0] ** 2 - dt) * sqrt_dt
return dpsi_t
def d1_rho_photocurrent(A, rho_vec):
"""
Todo: cythonize, add (AdA)_L + AdA_R to precomputed operators
"""
n_sum = A[4] + A[5]
e1 = cy_expect_rho_vec(n_sum, rho_vec)
return -spmv(n_sum.data, n_sum.indices, n_sum.indptr, rho_vec) + e1 * rho_vec
def d1_rho_photocurrent(A, rho_vec):
"""
Todo: cythonize, add (AdA)_L + AdA_R to precomputed operators
"""
n_sum = A[4] + A[5]
e1 = cy_expect_rho_vec(n_sum, rho_vec)
return -spmv(n_sum.data, n_sum.indices, n_sum.indptr, rho_vec) + e1 * rho_vec
def _rhs_psi_platen(H, psi_t, t, A_ops, dt, dW, d1, d2, args):
"""
TODO: support multiple stochastic increments
.. note::
Experimental.
"""
sqrt_dt = np.sqrt(dt)
dW_len = len(dW[0, :])
dpsi_t = _rhs_psi_deterministic(H, psi_t, t, dt, args)
for a_idx, A in enumerate(A_ops):
# XXX: This needs to be revised now that
# dpsi_t is the change for all stochastic collapse operators
# TODO: needs to be updated to support mutiple Weiner increments
dpsi_t_H = (-1.0j * dt) * spmv(H.data, H.indices, H.indptr, psi_t)
psi_t_1 = psi_t + dpsi_t_H + d1(A, psi_t) * dt + d2(A, psi_t)[0] * dW[a_idx,0]
psi_t_p = psi_t + dpsi_t_H + d1(A, psi_t) * dt + d2(A, psi_t)[0] * sqrt_dt
psi_t_m = psi_t + dpsi_t_H + d1(A, psi_t) * dt - d2(A, psi_t)[0] * sqrt_dt
dpsi_t += 0.50 * (d1(A, psi_t_1) + d1(A, psi_t)) * dt + \
0.25 * (d2(A, psi_t_p)[0] + d2(A, psi_t_m)[0] + 2 * d2(A, psi_t)[0]) * dW[a_idx,0] + \
0.25 * (d2(A, psi_t_p)[0] - d2(A, psi_t_m)[0]) * (dW[a_idx,0] ** 2 - dt) * sqrt_dt
return dpsi_t
def _rhs_rho_deterministic(L, rho_t, t, dt, args):
"""
Deterministic contribution to the density matrix change
"""
drho_t = spmv(L, rho_t) * dt
return drho_t
def d1_psi_homodyne(A, psi):
"""
OK
Todo: cythonize
.. math::
D_1(\psi, t) = \\frac{1}{2}(\\langle C + C^\\dagger\\rangle\\psi -
C^\\dagger C\\psi - \\frac{1}{4}\\langle C + C^\\dagger\\rangle^2\\psi)
"""
e1 = cy_expect(A[1].data, A[1].indices, A[1].indptr, psi)
return 0.5 * (e1 * spmv(A[0].data, A[0].indices, A[0].indptr, psi) -
spmv(A[3].data, A[3].indices, A[3].indptr, psi) -
0.25 * e1 ** 2 * psi)
def _smesolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
A_ops, e_ops, data, rhs, d1, d2, d2_len):
"""
Internal function. See smesolve.
"""
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps, d2_len)
states_list = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
# XXX: need to keep hilbert space structure
data.expect[e_idx, t_idx] += expect(e, Qobj(vec2mat(rho_t)))
else:
states_list.append(Qobj(rho_t)) # dito
for j in range(N_substeps):
drho_t = spmv(L.data.data,
L.data.indices,
L.data.indptr, rho_t) * dt
for a_idx, A in enumerate(A_ops):
drho_t += rhs(L.data, rho_t, A,
dt, dW[a_idx, t_idx, j, :], d1, d2)
rho_t += drho_t
return states_list
def d1_rho_homodyne(A, rho):
"""
not tested
Todo: cythonize
"""
return spmv(A[0].data, A[0].indices, A[0].indptr, rho)
def d1_rho_photocurrent(A, rho_vec):
"""
Todo: cythonize, add (AdA)_L + AdA_R to precomputed operators
"""
n_sum = A[4] + A[5]
e1 = cy_expect_rho_vec(n_sum, rho_vec, 0)
return 0.5 * (e1 * rho_vec - spmv(n_sum, rho_vec))
def _smesolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
A_ops, e_ops, data, rhs, d1, d2):
"""
Internal function. See smesolve.
"""
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps)
states_list = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
# XXX: need to keep hilbert space structure
data.expect[e_idx, t_idx] += expect(e, Qobj(vec2mat(rho_t)))
else:
states_list.append(Qobj(rho_t)) # dito
for j in range(N_substeps):
drho_t = spmv(
L.data.data, L.data.indices, L.data.indptr, rho_t) * dt
for a_idx, A in enumerate(A_ops):
drho_t += rhs(
L.data, rho_t, A, dt, dW[a_idx, t_idx, j], d1, d2)
rho_t += drho_t
return states_list
def d1_psi_homodyne(A, psi):
"""
OK
Todo: cythonize
.. math::
D_1(\psi, t) = \\frac{1}{2}(\\langle C + C^\\dagger\\rangle\\psi -
C^\\dagger C\\psi - \\frac{1}{4}\\langle C + C^\\dagger\\rangle^2\\psi)
"""
e1 = cy_expect(A[1].data, A[1].indices, A[1].indptr, psi)
return 0.5 * (e1 * spmv(A[0].data, A[0].indices, A[0].indptr, psi) -
spmv(A[3].data, A[3].indices, A[3].indptr, psi) -
0.25 * e1 ** 2 * psi)
def d2_current(A, psi):
"""
Todo: cythonize, requires poisson increments
"""
psi_1 = spmv(A[0].data, A[0].indices, A[0].indptr, psi)
n1 = norm(psi_1, 2)
return psi_1 / n1 - psi
def d2_rho_homodyne(A, rho):
"""
not tested
Todo: cythonize
"""
e1 = _rho_expect(A[2], rho)
return spmv(A[1].data, A[1].indices, A[1].indptr, rho) - e1 * rho
def d1_rho_homodyne(A, rho_vec):
"""
D1[a] rho = lindblad_dissipator(a) * rho
Todo: cythonize
"""
return spmv(A[7].data, A[7].indices, A[7].indptr, rho_vec)
def d1_psi_heterodyne(A, psi):
"""
Todo: cythonize
.. math::
D_1(\psi, t) = -\\frac{1}{2}(C^\\dagger C - \\langle C^\\dagger \\rangle C +
\\frac{1}{2}\\langle C \\rangle\\langle C^\\dagger \\rangle))\psi
"""
e_C = cy_expect(A[0].data, A[0].indices, A[0].indptr, psi) # e_C
B = A[0].T.conj()
e_Cd = cy_expect(B.data, B.indices, B.indptr, psi) # e_Cd
return (-0.5 * spmv(A[3].data, A[3].indices, A[3].indptr, psi) +
0.5 * e_Cd * spmv(A[0].data, A[0].indices, A[0].indptr, psi) -
0.25 * e_C * e_Cd * psi)
def d1_psi_heterodyne(A, psi):
"""
Todo: cythonize
.. math::
D_1(\psi, t) = -\\frac{1}{2}(C^\\dagger C - \\langle C^\\dagger \\rangle C +
\\frac{1}{2}\\langle C \\rangle\\langle C^\\dagger \\rangle))\psi
"""
e_C = cy_expect(A[0].data, A[0].indices, A[0].indptr, psi) # e_C
B = A[0].T.conj()
e_Cd = cy_expect(B.data, B.indices, B.indptr, psi) # e_Cd
return (-0.5 * spmv(A[3].data, A[3].indices, A[3].indptr, psi) +
0.5 * e_Cd * spmv(A[0].data, A[0].indices, A[0].indptr, psi) -
0.25 * e_C * e_Cd * psi)
def d2_psi_homodyne(A, psi):
"""
OK
Todo: cythonize
"""
e1 = cy_expect(A[1].data, A[1].indices, A[1].indptr, 0, psi)
return (spmv(A[0].data, A[0].indices, A[0].indptr, psi) - 0.5 * e1 * psi)
def d1_rho_homodyne(A, rho_vec):
"""
D1[a] rho = lindblad_dissipator(a) * rho
Todo: cythonize
"""
return spmv(A[7].data, A[7].indices, A[7].indptr, rho_vec)
def d2_psi_heterodyne(A, psi):
"""
not working/tested
Todo: cythonize
"""
e1 = cy_expect(A[0].data, A[0].indices, A[0].indptr, 0, psi)
return spmv(A[0].data, A[0].indices, A[0].indptr, psi) - e1 * psi
def d2_rho_homodyne(A, rho):
"""
not tested
Todo: cythonize
"""
e1 = _rho_expect(A[2], rho)
return [spmv(A[1].data, A[1].indices, A[1].indptr, rho) - e1 * rho]
def _rhs_rho_deterministic(L, rho_t, t, dt, args):
"""
Deterministic contribution to the density matrix change
"""
drho_t = spmv(L.data,
L.indices,
L.indptr, rho_t) * dt
return drho_t
def _rhs_rho_milstein_homodyne(L, rho_t, t, A_ops, dt, dW, d1, d2, args):
"""
.. note::
Experimental.
Milstein scheme for homodyne detection.
This implementation works for commuting stochastic jump operators.
TODO: optimizations: do calculation for n>m only
"""
A_len = len(A_ops)
M = np.array([A_ops[n][0] + A_ops[n][3] for n in range(A_len)])
e1 = np.array([cy_expect_rho_vec(M[n], rho_t, 0) for n in range(A_len)])
d1_vec = np.sum([spmv(A_ops[n][7], rho_t)
for n in range(A_len)], axis=0)
d2_vec = np.array([spmv(M[n], rho_t)
for n in range(A_len)])
# This calculation is suboptimal. We need only values for m>n in case of
# commuting jump operators.
d2_vec2 = np.array([[spmv(M[n], d2_vec[m])
for m in range(A_len)] for n in range(A_len)])
e2 = np.array([[cy_expect_rho_vec(M[n], d2_vec[m], 0)
for m in range(A_len)] for n in range(A_len)])
drho_t = _rhs_rho_deterministic(L, rho_t, t, dt, args)
drho_t += d1_vec * dt
drho_t += np.sum([(d2_vec[n] - e1[n] * rho_t) * dW[n, 0]
for n in range(A_len)], axis=0)
drho_t += 0.5 * np.sum([(d2_vec2[n, n] - 2.0 * e1[n] * d2_vec[n] +
(-e2[n, n] + 2.0 * e1[n] * e1[n]) * rho_t) * (dW[n, 0] * dW[n, 0] - dt)
for n in range(A_len)], axis=0)
# This calculation is suboptimal. We need only values for m>n in case of
# commuting jump operators.
drho_t += 0.5 * np.sum([(d2_vec2[n, m] - e1[m] * d2_vec[n] - e1[n] * d2_vec[m] +
(-e2[n, m] + 2.0 * e1[n] * e1[m]) * rho_t) * (dW[n, 0] * dW[m, 0])
for (n, m) in np.ndindex(A_len, A_len) if n != m], axis=0)
return rho_t + drho_t
def d2_psi_heterodyne(A, psi):
"""
Todo: cythonize
X = \\frac{1}{2}(C + C^\\dagger)
Y = \\frac{1}{2}(C - C^\\dagger)
D_{2,1}(\psi, t) = \\sqrt(1/2) * (C - \\langle X \\rangle) \\psi
D_{2,2}(\psi, t) = -i\\sqrt(1/2) * (C - \\langle Y \\rangle) \\psi
"""
X = 0.5 * cy_expect(A[1].data, A[1].indices, A[1].indptr, psi)
Y = 0.5 * cy_expect(A[2].data, A[2].indices, A[2].indptr, psi)
d2_1 = np.sqrt(0.5) * (spmv(A[0].data, A[0].indices, A[0].indptr, psi) - X * psi)
d2_2 = -1.0j * np.sqrt(0.5) * (spmv(A[0].data, A[0].indices, A[0].indptr, psi) - Y * psi)
return [d2_1, d2_2]
def d2_rho_homodyne(A, rho_vec):
"""
D2[a] rho = a rho + rho a^\dagger - Tr[a rho + rho a^\dagger]
= (A_L + Ad_R) rho_vec - E[(A_L + Ad_R) rho_vec]
Todo: cythonize, add A_L + Ad_R to precomputed operators
"""
M = A[0] + A[3]
e1 = cy_expect_rho_vec(M, rho_vec)
return [spmv(M.data, M.indices, M.indptr, rho_vec) - e1 * rho_vec]
def d2_psi_heterodyne(A, psi):
"""
Todo: cythonize
X = \\frac{1}{2}(C + C^\\dagger)
Y = \\frac{1}{2}(C - C^\\dagger)
D_{2,1}(\psi, t) = \\sqrt(1/2) * (C - \\langle X \\rangle) \\psi
D_{2,2}(\psi, t) = -i\\sqrt(1/2) * (C - \\langle Y \\rangle) \\psi
"""
X = 0.5 * cy_expect(A[1].data, A[1].indices, A[1].indptr, psi)
Y = 0.5 * cy_expect(A[2].data, A[2].indices, A[2].indptr, psi)
d2_1 = np.sqrt(0.5) * (spmv(A[0].data, A[0].indices, A[0].indptr, psi) - X * psi)
d2_2 = -1.0j * np.sqrt(0.5) * (spmv(A[0].data, A[0].indices, A[0].indptr, psi) - Y * psi)
return [d2_1, d2_2]
def d2_rho_homodyne(A, rho_vec):
"""
D2[a] rho = a rho + rho a^\dagger - Tr[a rho + rho a^\dagger]
= (A_L + Ad_R) rho_vec - E[(A_L + Ad_R) rho_vec]
Todo: cythonize, add A_L + Ad_R to precomputed operators
"""
M = A[0] + A[3]
e1 = cy_expect_rho_vec(M, rho_vec)
return [spmv(M.data, M.indices, M.indptr, rho_vec) - e1 * rho_vec]
def _rhs_rho_milstein_homodyne_single(L, rho_t, t, A_ops, dt, dW, d1, d2, args):
"""
.. note::
Experimental.
Milstein scheme for homodyne detection with single jump operator.
"""
A = A_ops[0]
M = A[0] + A[3]
e1 = cy_expect_rho_vec(M, rho_t, 0)
d2_vec = spmv(M, rho_t)
d2_vec2 = spmv(M, d2_vec)
e2 = cy_expect_rho_vec(M, d2_vec, 0)
drho_t = _rhs_rho_deterministic(L, rho_t, t, dt, args)
drho_t += spmv(A[7], rho_t) * dt
drho_t += (d2_vec - e1 * rho_t) * dW[0, 0]
drho_t += 0.5 * (d2_vec2 - 2 * e1 * d2_vec + (-e2 + 2 * e1 * e1) * rho_t) * (dW[0, 0] * dW[0, 0] - dt)
return rho_t + drho_t
def sop_H(A, rho_vec):
"""
Evaluate the superoperator
H[a] rho = a rho + rho a^\dagger - Tr[a rho + rho a^\dagger] rho
-> (A_L + Ad_R) rho_vec - E[(A_L + Ad_R) rho_vec] rho_vec
Todo: cythonize, add A_L + Ad_R to precomputed operators
"""
M = A[0] + A[3]
e1 = cy_expect_rho_vec(M, rho_vec, 0)
return spmv(M, rho_vec) - e1 * rho_vec
def d2_psi_homodyne(A, psi):
"""
OK
Todo: cythonize
.. math::
D_2(\psi, t) = (C - \\frac{1}{2}\\langle C + C^\\dagger\\rangle)\\psi
"""
e1 = cy_expect(A[1].data, A[1].indices, A[1].indptr, psi)
return [spmv(A[0].data, A[0].indices, A[0].indptr, psi) - 0.5 * e1 * psi]
def sop_H(A, rho_vec):
"""
Evaluate the superoperator
H[a] rho = a rho + rho a^\dagger - Tr[a rho + rho a^\dagger]
-> (A_L + Ad_R) rho_vec - E[(A_L + Ad_R) rho_vec]
Todo: cythonize, add A_L + Ad_R to precomputed operators
"""
M = A[0] + A[3]
e1 = cy_expect_rho_vec(M, rho_vec)
return spmv(M.data, M.indices, M.indptr, rho_vec) - e1 * rho_vec
def d2_psi_homodyne(A, psi):
"""
OK
Todo: cythonize
.. math::
D_2(\psi, t) = (C - \\frac{1}{2}\\langle C + C^\\dagger\\rangle)\\psi
"""
e1 = cy_expect(A[1].data, A[1].indices, A[1].indptr, psi)
return [spmv(A[0].data, A[0].indices, A[0].indptr, psi) - 0.5 * e1 * psi]
def _rhs_rho_euler_homodyne_fast(L, rho_t, t, A, dt, ddW, d1, d2, args): """ fast Euler-Maruyama for homodyne detection """ dW = ddW[:,0] d_vec = spmv(A[0][0], rho_t).reshape(-1, len(rho_t)) e = np.real(d_vec[:-1].reshape(-1,A[0][1],A[0][1]).trace(axis1=1,axis2=2)) drho_t = d_vec[-1] drho_t += np.dot(dW, d_vec[:-1]) drho_t += (1.0 - np.inner(e, dW)) * rho_t return drho_t
def _ssesolve_single_trajectory(H, dt, tlist, N_store, N_substeps, psi_t,
A_ops, e_ops, data, rhs, d1, d2, d2_len,
homogeneous, ssdata):
"""
Internal function. See ssesolve.
"""
if homogeneous:
if ssdata.incr_distr == 'normal':
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps, d2_len)
else:
raise TypeError('Unsupported increment distribution for homogeneous process.')
else:
if ssdata.incr_distr != 'poisson':
raise TypeError('Unsupported increment distribution for inhomogeneous process.')
states_list = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
s = cy_expect(e.data.data, e.data.indices, e.data.indptr, 0, psi_t)
data.expect[e_idx, t_idx] += s
data.ss[e_idx, t_idx] += s ** 2
else:
states_list.append(Qobj(psi_t))
for j in range(N_substeps):
dpsi_t = (-1.0j * dt) * (H.data * psi_t)
for a_idx, A in enumerate(A_ops):
if homogeneous:
dw = dW[a_idx, t_idx, j, :]
else:
dw_expect = norm(spmv(A[0].data, A[0].indices, A[0].indptr, psi_t)) ** 2 * dt
dw = np.random.poisson(dw_expect, d2_len)
dpsi_t += rhs(H.data, psi_t, A, dt, dw, d1, d2)
# increment and renormalize the wave function
psi_t += dpsi_t
psi_t /= norm(psi_t)
return states_list
def sop_G(A, rho_vec):
"""
Evaluate the superoperator
G[a] rho = a rho a^\dagger / Tr[a rho a^\dagger] - rho
-> A_L Ad_R rho_vec / Tr[A_L Ad_R rho_vec] - rho_vec
Todo: cythonize, add A_L + Ad_R to precomputed operators
"""
e1 = cy_expect_rho_vec(A[6], rho_vec)
if e1 > 1e-15:
return spmv(A[6].data, A[6].indices, A[6].indptr, rho_vec) / e1 - rho_vec
else:
return -rho_vec
def sop_G(A, rho_vec):
"""
Evaluate the superoperator
G[a] rho = a rho a^\dagger / Tr[a rho a^\dagger] - rho
-> A_L Ad_R rho_vec / Tr[A_L Ad_R rho_vec] - rho_vec
Todo: cythonize, add A_L + Ad_R to precomputed operators
"""
e1 = cy_expect_rho_vec(A[6], rho_vec, 0)
if e1 > 1e-15:
return spmv(A[6], rho_vec) / e1 - rho_vec
else:
return -rho_vec
def d2_psi_photocurrent(A, psi):
"""
Todo: cythonize
Note: requires poisson increments
.. math::
D_2(\psi, t) = C\psi / ||C\psi|| - \psi
"""
psi_1 = spmv(A[0].data, A[0].indices, A[0].indptr, psi)
n1 = norm(psi_1)
if n1 != 0:
return psi_1 / n1 - psi
else:
return - psi
def d2_psi_photocurrent(A, psi):
"""
Todo: cythonize
Note: requires poisson increments
.. math::
D_2(\psi, t) = C\psi / ||C\psi|| - \psi
"""
psi_1 = spmv(A[0].data, A[0].indices, A[0].indptr, psi)
n1 = norm(psi_1)
if n1 != 0:
return psi_1 / n1 - psi
else:
return - psi
def _rhs_rho_milstein_homodyne_single_fast(L, rho_t, t, A, dt, ddW, d1, d2, args): """ fast Milstein for homodyne detection with 1 stochastic operator """ dW = ddW[:,0] d_vec = spmv(A[0][0], rho_t).reshape(-1, len(rho_t)) e = np.real(d_vec[:-1].reshape(-1, A[0][1], A[0][1]).trace(axis1=1,axis2=2)) e[1] -= 2.0*e[0]*e[0] drho_t = (1.0 - np.inner(e, dW)) * rho_t dW[0] -= 2.0*e[0]*dW[1] drho_t += d_vec[-1] drho_t += np.dot(dW, d_vec[:-1]) return drho_t
def _rhs_rho_milstein_homodyne_fast(L, rho_t, t, A, dt, ddW, d1, d2, args): """ fast Milstein for homodyne detection with >2 stochastic operators """ dW = ddW[:,0] sc_len = len(A) sc2_len = 2*sc_len d_vec = spmv(A[0][0], rho_t).reshape(-1, len(rho_t)) e = np.real(d_vec[:-1].reshape(-1, A[0][1], A[0][1]).trace(axis1=1,axis2=2)) d_vec[sc2_len:-1] -= np.array([e[m]*d_vec[n] + e[n]*d_vec[m] for (n,m) in np.ndindex(sc_len, sc_len) if n > m]) e[sc_len:sc2_len] -= 2.0*e[:sc_len]*e[:sc_len] e[sc2_len:] -= 2.0*np.array([e[n]*e[m] for (n,m) in np.ndindex(sc_len, sc_len) if n > m]) drho_t = (1.0 - np.inner(e, dW)) * rho_t dW[:sc_len] -= 2.0*e[:sc_len]*dW[sc_len:sc2_len] drho_t += d_vec[-1] drho_t += np.dot(dW, d_vec[:-1]) return drho_t
def _rhs_psi_platen(H, psi_t, A, dt, dW, d1, d2):
"""
.. note::
Experimental.
"""
sqrt_dt = np.sqrt(dt)
dpsi_t_H = (-1.0j * dt) * spmv(H.data, H.indices, H.indptr, psi_t)
psi_t_1 = psi_t + dpsi_t_H + d1(A, psi_t) * dt + d2(A, psi_t) * dW
psi_t_p = psi_t + dpsi_t_H + d1(A, psi_t) * dt + d2(A, psi_t) * sqrt_dt
psi_t_m = psi_t + dpsi_t_H + d1(A, psi_t) * dt - d2(A, psi_t) * sqrt_dt
dpsi_t = 0.50 * (d1(A, psi_t_1) + d1(A, psi_t)) * dt + \
0.25 * (d2(A, psi_t_p) + d2(A, psi_t_m) + 2 * d2(A, psi_t)) * dW + \
0.25 * (d2(A, psi_t_p) - d2(A, psi_t_m)) * (
dW ** 2 - dt) * sqrt_dt
return dpsi_t
def _rhs_psi_platen(H, psi_t, A, dt, dW, d1, d2):
"""
TODO: support multiple stochastic increments
.. note::
Experimental.
"""
sqrt_dt = np.sqrt(dt)
dpsi_t_H = (-1.0j * dt) * spmv(H.data, H.indices, H.indptr, psi_t)
psi_t_1 = psi_t + dpsi_t_H + d1(A, psi_t) * dt + d2(A, psi_t)[0] * dW[0]
psi_t_p = psi_t + dpsi_t_H + d1(A, psi_t) * dt + d2(A, psi_t)[0] * sqrt_dt
psi_t_m = psi_t + dpsi_t_H + d1(A, psi_t) * dt - d2(A, psi_t)[0] * sqrt_dt
dpsi_t = 0.50 * (d1(A, psi_t_1) + d1(A, psi_t)) * dt + \
0.25 * (d2(A, psi_t_p)[0] + d2(A, psi_t_m)[0] + 2 * d2(A, psi_t)[0]) * dW[0] + \
0.25 * (d2(A, psi_t_p)[0] - d2(A, psi_t_m)[0]) * (dW[0] ** 2 - dt) * sqrt_dt
return dpsi_t
def _ssesolve_single_trajectory(data, H, dt, tlist, N_store, N_substeps, psi_t,
A_ops, e_ops, m_ops, rhs, d1, d2, d2_len,
dW_factors, homogeneous, distribution, args,
store_measurement=False, noise=None,
normalize=True):
"""
Internal function. See ssesolve.
"""
if noise is None:
if homogeneous:
if distribution == 'normal':
dW = np.sqrt(
dt) * scipy.randn(len(A_ops), N_store, N_substeps, d2_len)
else:
raise TypeError('Unsupported increment distribution for homogeneous process.')
else:
if distribution != 'poisson':
raise TypeError('Unsupported increment distribution for inhomogeneous process.')
dW = np.zeros((len(A_ops), N_store, N_substeps, d2_len))
else:
dW = noise
states_list = []
measurements = np.zeros((len(tlist), len(m_ops), d2_len), dtype=complex)
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
s = cy_expect_psi_csr(e.data.data, e.data.indices, e.data.indptr, psi_t, 0)
data.expect[e_idx, t_idx] += s
data.ss[e_idx, t_idx] += s ** 2
else:
states_list.append(Qobj(psi_t))
psi_prev = np.copy(psi_t)
for j in range(N_substeps):
if noise is None and not homogeneous:
for a_idx, A in enumerate(A_ops):
dw_expect = norm(spmv(A[0], psi_t)) ** 2 * dt
dW[a_idx, t_idx, j, :] = np.random.poisson(dw_expect, d2_len)
psi_t = rhs(H.data, psi_t, t + dt * j,
A_ops, dt, dW[:, t_idx, j, :], d1, d2, args)
# optionally renormalize the wave function
if normalize:
psi_t /= norm(psi_t)
if store_measurement:
for m_idx, m in enumerate(m_ops):
for dW_idx, dW_factor in enumerate(dW_factors):
if m[dW_idx]:
m_expt = norm(spmv(m[dW_idx].data, psi_prev)) ** 2
else:
m_expt = 0
measurements[t_idx, m_idx, dW_idx] = (m_expt +
dW_factor * dW[m_idx, t_idx, :, dW_idx].sum() / (dt * N_substeps))
if d2_len == 1:
measurements = measurements.squeeze(axis=(2))
return states_list, dW, measurements
def d2_rho_photocurrent(A, rho_vec):
"""
Todo: cythonize, add (AdA)_L + AdA_R to precomputed operators
"""
e1 = cy_expect_rho_vec(A[6], rho_vec, 0)
return [spmv(A[6], rho_vec) / e1 - rho_vec] if e1.real > 1e-15 else [-rho_vec]
def d1_rho_heterodyne(A, rho_vec):
"""
todo: cythonize, docstrings
"""
return spmv(A[7].data, A[7].indices, A[7].indptr, rho_vec)
def d2_rho_photocurrent(A, rho_vec):
"""
Todo: cythonize, add (AdA)_L + AdA_R to precomputed operators
"""
e1 = cy_expect_rho_vec(A[6], rho_vec) + 1e-15
return [spmv(A[6].data, A[6].indices, A[6].indptr, rho_vec) / e1 - rho_vec]
def d2_rho_photocurrent(A, rho_vec):
"""
Todo: cythonize, add (AdA)_L + AdA_R to precomputed operators
"""
e1 = cy_expect_rho_vec(A[6], rho_vec) + 1e-15
return [spmv(A[6].data, A[6].indices, A[6].indptr, rho_vec) / e1 - rho_vec]
def _rho_expect(oper, state):
prod = spmv(oper.data, oper.indices, oper.indptr, state)
return sum(vec2mat(prod).diagonal())
def d1_rho_heterodyne(A, rho_vec):
"""
todo: cythonize, docstrings
"""
return spmv(A[7].data, A[7].indices, A[7].indptr, rho_vec)