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 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 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 _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 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 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 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 _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 _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 _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 _rho_expect(oper, state): prod = spmv(oper.data, oper.indices, oper.indptr, state) return sum(vec2mat(prod).diagonal())