def L_nonsecular_old(H_vib, A, eps, Gamma, T, J, time_units='cm', silent=False):
#Construct non-secular liouvillian
ti = time.time()
d = H_vib.shape[0]
evals, evecs = H_vib.eigenstates()
X1, X2, X3, X4 = 0, 0, 0, 0
for i in range(int(d)):
for j in range(int(d)):
eps_ij = abs(evals[i]-evals[j])
A_ij = A.matrix_element(evecs[i].dag(), evecs[j])
A_ji = (A.dag()).matrix_element(evecs[j].dag(), evecs[i])
Occ = Occupation(eps_ij, T, time_units)
IJ = evecs[i]*evecs[j].dag()
JI = evecs[j]*evecs[i].dag()
# 0.5*np.pi*alpha*(N+1)
if abs(A_ij)>0 or abs(A_ji)>0:
r_up = 2*np.pi*J(eps_ij, Gamma, eps)*Occ
r_down = 2*np.pi*J(eps_ij, Gamma, eps)*(Occ+1)
X3+= r_down*A_ij*IJ
X4+= r_up*A_ij*IJ
X1+= r_up*A_ji*JI
X2+= r_down*A_ji*JI
L = spre(A*X1) -sprepost(X1,A)+spost(X2*A)-sprepost(A,X2)
L+= spre(A.dag()*X3)-sprepost(X3, A.dag())+spost(X4*A.dag())-sprepost(A.dag(), X4)
if not silent:
print(("It took ", time.time()-ti, " seconds to build the Non-secular RWA Liouvillian"))
return -0.5*L
def L_left_lindblad(H, PARAMS):
I = qeye(PARAMS['N'])
d_h = tensor(PARAMS['A_L'], I)
J_leads = J_Lorentzian
Lambda_up = lambda x : Lamdba_complex_rate(x, J_leads, PARAMS['mu_L'], PARAMS['T_L'], PARAMS['Gamma_L'],
PARAMS['delta_L'], PARAMS['Omega_L'], type='p', real_only=True)
Lambda_down = lambda x : Lamdba_complex_rate(x, J_leads, PARAMS['mu_L'], PARAMS['T_L'], PARAMS['Gamma_L'],
PARAMS['delta_L'], PARAMS['Omega_L'], type='m', real_only=True)
energies, states = H.eigenstates()
L=0
for i in range(len(energies)):
for j in range(len(energies)):
eta_ij = energies[i] - energies[j]
if (abs(eta_ij)>0):# take limit of rates going to zero
d_h_ij = d_h.matrix_element(states[i].dag(), states[j])
d_h_ij_sq = d_h_ij*d_h_ij.conjugate() # real by construction
if (abs(d_h_ij_sq)>0): # No need to do anything if the matrix element is zero
IJ = states[i]*states[j].dag()
JI = states[j]*states[i].dag()
JJ = states[j]*states[j].dag()
II = states[i]*states[i].dag()
rate_up = Lambda_up(-eta_ij) # evaluated at positive freq diffs
rate_down = Lambda_down(-eta_ij)
print(rate_down)
T1 = rate_up*spre(II)+rate_down*spre(JJ)
T2 = rate_up.conjugate()*spost(II)+rate_down.conjugate()*spost(JJ)
T3 = (rate_up*sprepost(JI, IJ)+rate_down*sprepost(IJ,JI))
L += d_h_ij_sq*(0.5*(T1 + T2) - T3)
return -L
def L_full_secular(H_vib, A, eps, Gamma, T, J, time_units='cm', silent=False):
'''
Initially assuming that the vibronic eigenstructure has no
degeneracy and the secular approximation has been made
'''
ti = time.time()
d = H_vib.shape[0]
L = 0
eVals, eVecs = H_vib.eigenstates()
A_dag = A.dag()
terms = 0
for l in range(int(d)):
for m in range(int(d)):
for p in range(int(d)):
for q in range(int(d)):
secular_freq = (eVals[l] - eVals[m]) - (eVals[p] -
eVals[q])
if abs(secular_freq) < 1E-10:
terms += 1
A_lm = A.matrix_element(eVecs[l].dag(), eVecs[m])
A_lm_star = A_dag.matrix_element(
eVecs[m].dag(), eVecs[l])
A_pq = A.matrix_element(eVecs[p].dag(), eVecs[q])
A_pq_star = A_dag.matrix_element(
eVecs[q].dag(), eVecs[p])
coeff_1 = A_lm * A_pq_star
coeff_2 = A_lm_star * A_pq
eps_pq = abs(eVals[p] - eVals[q])
Occ = Occupation(eps_pq, T, time_units)
r_up = np.pi * J(eps_pq, Gamma, eps) * Occ
r_down = np.pi * J(eps_pq, Gamma, eps) * (Occ + 1)
LM = eVecs[l] * eVecs[m].dag()
ML = LM.dag()
PQ = eVecs[p] * eVecs[q].dag()
QP = PQ.dag()
"""
if abs(secular_freq) !=0:
print (abs(secular_freq), r_up, A_lm, A_lm_star,
A_pq, A_pq_star, r_down, l,m,p,q, m==q, l==p)
"""
if abs(r_up * coeff_1) > 0:
L += r_up * coeff_1 * (spre(LM * QP) -
sprepost(QP, LM))
if abs(r_up * coeff_2) > 0:
L += r_up * coeff_2 * (spost(PQ * ML) -
sprepost(ML, PQ))
if abs(r_down * coeff_1) > 0:
L += r_down * coeff_1 * (spre(ML * PQ) -
sprepost(PQ, ML))
if abs(r_down * coeff_2) > 0:
L += r_down * coeff_2 * (spost(QP * LM) -
sprepost(LM, QP))
if not silent:
print(("It took ", time.time() - ti,
" seconds to build the secular Liouvillian"))
print((
"Secular approximation kept {:0.2f}% of total ME terms. \n".format(
100 * float(terms) / (d * d * d * d))))
return -L
def L_EM_lindblad(splitting, col_em, Gamma, T, J, time_units='cm', silent=False):
# col_em is collapse operator
ti = time.time()
L = 0
EMnb = Occupation(splitting, T, time_units)
L+= 2*np.pi*J(splitting, Gamma, splitting)*(EMnb+1)*(sprepost(col_em, col_em.dag())-0.5*(spre(col_em.dag()*col_em) +spost(col_em.dag()*col_em)))
L+= 2*np.pi*J(splitting, Gamma, splitting)*EMnb*(sprepost(col_em.dag(), col_em)-0.5*(spre(col_em*col_em.dag())+ spost(col_em*col_em.dag())))
if not silent:
print(("It took ", time.time()-ti, " seconds to build the electronic-Lindblad Liouvillian"))
return L
def liouvillian_build(H_0, A, gamma, wRC, T_C):
# Now this function has to construct the liouvillian so that it can be passed to mesolve
H_0, A, Chi, Xi = RCME_operators(H_0, A, gamma, beta_f(T_C))
L = 0
L -= spre(A * Chi)
L += sprepost(A, Chi)
L += sprepost(Chi, A)
L -= spost(Chi * A)
L += spre(A * Xi)
L += sprepost(A, Xi)
L -= sprepost(Xi, A)
L -= spost(Xi * A)
return L, Chi + Xi
def channel(self, loss, dephasing, verbose=False):
data_kraus = loss.kraus
ancilla_phase = loss.propagate(self.ancilla.dim, self.code.N,
self.ancilla.N)
nkraus = len(data_kraus)
ndata_meas = len(self.data_meas.povm_elements)
nancilla_meas = len(self.ancilla_meas.povm_elements)
cmat1 = np.zeros((ndata_meas, nkraus, 2, 2), dtype=complex)
cmat2 = np.zeros((nancilla_meas, nkraus, 2, 2), dtype=complex)
# compute c1(x1|k;a;b), c2(x2|k;a;b) matrices
for k, (n, p) in enumerate(zip(data_kraus, ancilla_phase)):
cmat1[:, k, :, :] = np.reshape(
[qt.expect(m, n*dephasing(keta*ketb.dag())*n.dag())
for m in self.data_meas.povm_elements
for keta in [self.code.plus, self.code.minus]
for ketb in [self.code.plus, self.code.minus]],
cmat1[:, k, :, :].shape)
cmat2[:, k, :, :] = np.reshape(
[qt.expect(m, p*keta*ketb.dag()*p.dag())
for m in self.ancilla_meas.povm_elements
for keta in [self.ancilla.plus, self.ancilla.minus]
for ketb in [self.ancilla.plus, self.ancilla.minus]],
cmat2[:, k, :, :].shape)
# compute p1(x1|k;a), p2(x2|k;a) matrices
pmat1 = np.diagonal(cmat1, axis1=2, axis2=3)
pmat2 = np.diagonal(cmat2, axis1=2, axis2=3)
# compute c(x1, x2| a, a', b') and p(x1, x2| a, b)
cmat = np.moveaxis(np.tensordot(cmat1, cmat2, axes=([1], [1])), 3, 1)
pmat = np.moveaxis(np.tensordot(pmat1, pmat2, axes=([1], [1])), 2, 1)
# determine most likely a, b
abmat = np.empty((ndata_meas, nancilla_meas), dtype=object)
for x1 in range(ndata_meas):
for x2 in range(nancilla_meas):
idx = (x1, x2, Ellipsis)
a, b = np.unravel_index(np.argmax(pmat[idx]),
pmat[idx].shape)
abmat[x1, x2] = (a, b)
# construct channel
channel = qt.Qobj()
Paulis = np.array(
[[self.code_out.identity,
self.code_out.logical_Z_allspace],
[self.code_out.logical_X_allspace,
self.code_out.logical_X_allspace
* self.code_out.logical_Z_allspace]],
dtype=type(self.code_out.identity))
for a1, a2 in np.ndindex(2, 2):
for b1, b2 in np.ndindex(2, 2):
for x1 in range(ndata_meas):
for x2 in range(nancilla_meas):
r = Paulis[abmat[x1, x2]].dag()
channel += (cmat[x1, x2, a1, b1, a2, b2]
* qt.sprepost(r*Paulis[a1, a2],
Paulis[b1, b2].dag()*r.dag()))
return 0.25*channel
def __init__(self, H_S, L1, L2, S_matrix=None, c_ops_markov=None,
options=None):
if options is None:
self.options = qt.Options()
else:
self.options = options
self.H_S = H_S
self.sysdims = H_S.dims
if isinstance(L1, qt.Qobj):
self.L1 = [L1]
else:
self.L1 = L1
if isinstance(L2, qt.Qobj):
self.L2 = [L2]
else:
self.L2 = L2
if not len(self.L1) == len(self.L2):
raise ValueError('L1 and L2 has to be of equal length.')
if isinstance(c_ops_markov, qt.Qobj):
self.c_ops_markov = [c_ops_markov]
else:
self.c_ops_markov = c_ops_markov
if S_matrix is None:
self.S_matrix = np.identity(len(self.L1))
else:
self.S_matrix = S_matrix
# create system identity superoperator
self.Id = qt.qeye(H_S.shape[0])
self.Id.dims = self.sysdims
self.Id = qt.sprepost(self.Id, self.Id)
self.store_states = self.options.store_states
def test_sprepost(self):
U1 = rand_unitary(3)
U2 = rand_unitary(3)
S1 = spre(U1) * spost(U2)
S2 = sprepost(U1, U2)
assert_(S1 == S2)
def pauli_channel_2_qubit(num_of_paulis):
paulis_probs = np.random.random_sample(num_of_paulis)
paulis_probs = paulis_probs / sum(paulis_probs)
channel = qt.Qobj()
for i in range(num_of_paulis):
pauli_rand = qt.tensor(g_1Q[rd.randint(0, 3)], g_1Q[rd.randint(0, 3)])
channel += paulis_probs[i] * qt.sprepost(pauli_rand, pauli_rand)
return channel
def encoder(self, kraus=False):
if self._encoder is None:
self._encoder = (self.zero * qt.basis(2, 0).dag() +
self.one * qt.basis(2, 1).dag())
if kraus:
return self._encoder
else:
return qt.sprepost(self._encoder, self._encoder.dag())
def test_sprepost(self):
U1 = rand_unitary(3)
U2 = rand_unitary(3)
S1 = spre(U1) * spost(U2)
S2 = sprepost(U1, U2)
assert_(S1 == S2)
def test_dqd_current():
"Counting statistics: current and current noise in a DQD model"
G = 0
L = 1
R = 2
sz = qutip.projection(3, L, L) - qutip.projection(3, R, R)
sx = qutip.projection(3, L, R) + qutip.projection(3, R, L)
sR = qutip.projection(3, G, R)
sL = qutip.projection(3, G, L)
w0 = 1
tc = 0.6 * w0
GammaR = 0.0075 * w0
GammaL = 0.0075 * w0
nth = 0.00
eps_vec = np.linspace(-1.5 * w0, 1.5 * w0, 20)
J_ops = [GammaR * qutip.sprepost(sR, sR.dag())]
c_ops = [
np.sqrt(GammaR * (1 + nth)) * sR,
np.sqrt(GammaR * (nth)) * sR.dag(),
np.sqrt(GammaL * (nth)) * sL,
np.sqrt(GammaL * (1 + nth)) * sL.dag()
]
current = np.zeros(len(eps_vec))
noise = np.zeros(len(eps_vec))
for n, eps in enumerate(eps_vec):
H = (eps / 2 * sz + tc * sx)
L = qutip.liouvillian(H, c_ops)
rhoss = qutip.steadystate(L)
current[n], noise[n] = qutip.countstat_current_noise(L, [],
rhoss=rhoss,
J_ops=J_ops)
current2 = qutip.countstat_current(L, rhoss=rhoss, J_ops=J_ops)
assert abs(current[n] - current2) < 1e-8
current2 = qutip.countstat_current(L, c_ops, J_ops=J_ops)
assert abs(current[n] - current2) < 1e-8
current_target = (tc**2 * GammaR /
(tc**2 *
(2 + GammaR / GammaL) + GammaR**2 / 4 + eps_vec**2))
noise_target = current_target * (
1 - (8 * GammaL * tc**2 *
(4 * eps_vec**2 * (GammaR - GammaL) + GammaR *
(3 * GammaL * GammaR + GammaR**2 + 8 * tc**2)) /
(4 * tc**2 * (2 * GammaL + GammaR) + GammaL * GammaR**2 +
4 * eps_vec**2 * GammaL)**2))
np.testing.assert_allclose(current, current_target, atol=1e-4)
np.testing.assert_allclose(noise, noise_target, atol=1e-4)
def liouvillian_build_new(H_0, A, gamma, wRC, T_C):
# Now this function has to construct the liouvillian so that it can be passed to mesolve
H_0, A, Chi, Xi = RCME_operators(H_0, A, gamma, beta_f(T_C))
Z = Chi + Xi
Z_dag = Z.dag()
L = 0
#L+=spre(A*Z_dag)
#L-=sprepost(A, Z)
#L-=sprepost(Z_dag, A)
#L+=spost(Z_dag*A)
L -= spre(A * Z_dag)
L += sprepost(A, Z)
L += sprepost(Z_dag, A)
L -= spost(Z * A)
print("new L built")
return L, Z
def L_vib_lindblad(H_vib, A, eps, Gamma, T, J, time_units='cm', silent=False):
'''
Initially assuming that the vibronic eigenstructure has no
degeneracy and the secular approximation has been made
'''
ti = time.time()
d = H_vib.shape[0]
ti = time.time()
L = 0
eig = H_vib.eigenstates()
eVals = eig[0]
eVecs = eig[1] # come out like kets
l = 0
occs = []
for i in range(int(d)):
l = 0
for j in range(int(d)):
t_0 = time.time(
) # initial time reference for tracking slow calculations
lam_ij = A.matrix_element(eVecs[i].dag(), eVecs[j])
#lam_mn = (A.dag()).matrix_element(eVecs[n].dag(), eVecs[m])
lam_ij_sq = lam_ij * lam_ij.conjugate() # real by construction
eps_ij = abs(eVals[i] - eVals[j])
if abs(lam_ij_sq) > 0:
IJ = eVecs[i] * eVecs[j].dag()
JI = eVecs[j] * eVecs[i].dag()
JJ = eVecs[j] * eVecs[j].dag()
II = eVecs[i] * eVecs[i].dag()
Occ = Occupation(eps_ij, T, time_units)
r_up = 2 * np.pi * J(eps_ij, Gamma, eps) * Occ
r_down = 2 * np.pi * J(eps_ij, Gamma, eps) * (Occ + 1)
T1 = r_up * spre(II) + r_down * spre(JJ)
T2 = r_up.conjugate() * spost(II) + r_down.conjugate() * spost(
JJ)
T3 = (r_up * sprepost(JI, IJ) + r_down * sprepost(IJ, JI))
L += lam_ij_sq * (0.5 * (T1 + T2) - T3)
l += 1
if not silent:
print(("It took ", time.time() - ti,
" seconds to build the vibronic Lindblad Liouvillian"))
return -L
def L_wc_analytic(detuning=0.,
Rabi=0,
alpha=0.,
w0=0.,
Gamma=0.,
T=0.,
tol=1e-7):
energies, states = exciton_states(detuning, Rabi)
dark_proj = states[0] * states[0].dag()
bright_proj = states[1] * states[1].dag()
ct_p = states[1] * states[0].dag()
ct_m = states[0] * states[1].dag()
cross_term = (ct_p + ct_m)
epsilon = -detuning
V = Rabi / 2
eta = sqrt(epsilon**2 + 4 * V**2)
# Bath 1 (only one bath)
G = (lambda x: (DecayRate(x,
beta_f(T),
_J_underdamped,
Gamma,
w0,
imag_part=True,
alpha=alpha,
tol=tol)))
G_0 = G(0.)
G_p = G(eta)
G_m = G(-eta)
site_1 = (0.5 / eta) * ((eta + epsilon) * bright_proj +
(eta - epsilon) * dark_proj + 2 * V * cross_term)
Z_1 = (0.5 / eta) * (G_0 * ((eta + epsilon) * bright_proj +
(eta - epsilon) * dark_proj) + 2 * V *
(ct_p * G_p + ct_m * G_m))
L = -qt.spre(site_1 * Z_1) + qt.sprepost(Z_1, site_1)
L += -qt.spost(Z_1.dag() * site_1) + qt.sprepost(site_1, Z_1.dag())
return L
def test_dqd_current():
"Counting statistics: current and current noise in a DQD model"
G = 0
L = 1
R = 2
sz = projection(3, L, L) - projection(3, R, R)
sx = projection(3, L, R) + projection(3, R, L)
sR = projection(3, G, R)
sL = projection(3, G, L)
w0 = 1
tc = 0.6 * w0
GammaR = 0.0075 * w0
GammaL = 0.0075 * w0
nth = 0.00
eps_vec = np.linspace(-1.5*w0, 1.5*w0, 20)
J_ops = [GammaR * sprepost(sR, sR.dag())]
c_ops = [np.sqrt(GammaR * (1 + nth)) * sR,
np.sqrt(GammaR * (nth)) * sR.dag(),
np.sqrt(GammaL * (nth)) * sL,
np.sqrt(GammaL * (1 + nth)) * sL.dag(),
]
I = np.zeros(len(eps_vec))
S = np.zeros(len(eps_vec))
for n, eps in enumerate(eps_vec):
H = (eps/2 * sz + tc * sx)
L = liouvillian(H, c_ops)
rhoss = steadystate(L)
I[n], S[n] = countstat_current_noise(L, [], rhoss=rhoss, J_ops=J_ops)
I2 = countstat_current(L, rhoss=rhoss, J_ops=J_ops)
assert_(abs(I[n] - I2) < 1e-8)
I2 = countstat_current(L, c_ops, J_ops=J_ops)
assert_(abs(I[n] - I2) < 1e-8)
Iref = tc**2 * GammaR / (tc**2 * (2 + GammaR/GammaL) +
GammaR**2/4 + eps_vec**2)
Sref = 1 * Iref * (
1 - 8 * GammaL * tc**2 *
(4 * eps_vec**2 * (GammaR - GammaL) +
GammaR * (3 * GammaL * GammaR + GammaR**2 + 8*tc**2)) /
(4 * tc**2 * (2 * GammaL + GammaR) + GammaL * GammaR**2 +
4 * eps_vec**2 * GammaL)**2
)
assert_allclose(I, Iref, 1e-4)
assert_allclose(S, Sref, 1e-4)
def L_non_rwa(H_vib,
A,
w_0,
Gamma,
T_EM,
J,
principal=False,
silent=False,
alpha=0.,
tol=1e-5):
ti = time.time()
beta = beta_f(T_EM)
eVals, eVecs = H_vib.eigenstates()
#J=J_minimal # J_minimal(omega, Gamma, omega_0)
d_dim = len(eVals)
G = 0
for i in range(d_dim):
for j in range(d_dim):
eta = eVals[i] - eVals[j]
s = eVecs[i] * (eVecs[j].dag())
#print A.matrix_element(eVecs[i].dag(), eVecs[j])
overlap = A.matrix_element(eVecs[i].dag(), eVecs[j])
s *= A.matrix_element(eVecs[i].dag(), eVecs[j])
s *= DecayRate(eta,
beta,
J,
Gamma,
w_0,
imag_part=principal,
alpha=alpha,
tol=tol)
G += s
G_dag = G.dag()
# Initialise liouvilliian
L = qt.spre(A * G) - qt.sprepost(G, A)
L += qt.spost(G_dag * A) - qt.sprepost(A, G_dag)
if not silent:
print(("Calculating non-RWA Liouvilliian took {} seconds.".format(
time.time() - ti)))
return -L
def test_ttmsolve_jc_model():
"""
Checks the output of ttmsolve using an example from Jaynes-Cumming model,
which can also be found in the qutip-notebooks repository.
"""
# Define Hamiltonian and states
N, kappa, g = 3, 1.0, 10
a = qt.tensor(qt.qeye(2), qt.destroy(N))
sm = qt.tensor(qt.sigmam(), qt.qeye(N))
sz = qt.tensor(qt.sigmaz(), qt.qeye(N))
H = g * (a.dag() * sm + a * sm.dag())
c_ops = [np.sqrt(kappa) * a]
# identity superoperator
Id = qt.tensor(qt.qeye(2), qt.qeye(N))
E0 = qt.sprepost(Id, Id)
# partial trace superoperator
ptracesuper = qt.tensor_contract(E0, (1, N))
# initial states
rho0a = qt.ket2dm(qt.basis(2, 0))
psi0c = qt.basis(N, 0)
rho0c = qt.ket2dm(psi0c)
rho0 = qt.tensor(rho0a, rho0c)
superrho0cav = qt.sprepost(qt.tensor(qt.qeye(2), psi0c),
qt.tensor(qt.qeye(2), psi0c.dag()))
# calculate exact solution using mesolve
times = np.arange(0, 5.0, 0.1)
exactsol = qt.mesolve(H, rho0, times, c_ops, [])
exact_z = qt.expect(sz, exactsol.states)
# solve using transfer method
learning_times = np.arange(0, 2.0, 0.1)
Et_list = qt.mesolve(H, E0, learning_times, c_ops, []).states
learning_maps = [ptracesuper * (Et * superrho0cav) for Et in Et_list]
ttmsol = ttmsolve(learning_maps, rho0a, times)
ttm_z = qt.expect(qt.sigmaz(), ttmsol.states)
# check that ttm result and exact solution are close in the learning times
assert np.allclose(ttm_z, exact_z, atol=1e-5)
def liouvillian(epsilon, delta, J, T):
L = 0 # Initialise liouvilliian
Z = 0 # initialise operator Z
#beta = 1 /(T* 0.695)
beta = 7.683/T
eta = np.sqrt(epsilon**2 + delta**2)
# Here I define the eigenstates of the H_s
H = qt.Qobj([[-epsilon/2., delta/2],[delta/2, epsilon/2.]])
eVecs = H.eigenstates()[1]
psi_p = (1/np.sqrt(2*eta))*(np.sqrt(eta-epsilon)*qt.basis(2,0) + np.sqrt(eta+epsilon)*qt.basis(2,1))
psi_m = (-1/np.sqrt(2*eta))*(np.sqrt(eta+epsilon)*qt.basis(2,0) - np.sqrt(eta-epsilon)*qt.basis(2,1))
#print H.eigenstates()
# Jake's eigenvectors
#psi_p = (1/np.sqrt(2*eta))*(np.sqrt(eta+epsilon)*qt.basis(2,0) - np.sqrt(eta-epsilon)*qt.basis(2,1))
#psi_m = (1/np.sqrt(2*eta))*(np.sqrt(eta-epsilon)*qt.basis(2,0) + np.sqrt(eta+epsilon)*qt.basis(2,1))
sigma_z = (1/eta)*(epsilon*(psi_p*psi_p.dag()-psi_m*psi_m.dag()) + delta*(psi_p*psi_m.dag() + psi_m*psi_p.dag()))
Z = (1/eta)*(epsilon*(psi_p*psi_p.dag()-psi_m*psi_m.dag() )*Gamma(0, beta, J) + delta*(Gamma(eta, beta, J)*psi_p*psi_m.dag() + Gamma(-eta,beta, J)*psi_m*psi_p.dag()))
L += qt.spre(sigma_z*Z) - qt.sprepost(Z, sigma_z)
L += qt.spost(Z.dag()*sigma_z) - qt.sprepost(sigma_z, Z.dag())
return -L
def test_sprepost_td(self):
"Superoperator: sprepost, time-dependent"
# left QobjEvo
assert_(sprepost(self.t1, self.q2)(.5) ==
sprepost(self.t1(.5), self.q2))
# left QobjEvo
assert_(sprepost(self.q2, self.t1)(.5) ==
sprepost(self.q2, self.t1(.5)))
# left 2 QobjEvo, one cte
assert_(sprepost(self.t1, self.t2)(.5) ==
sprepost(self.t1(.5), self.t2(.5)))
def test_sprepost_td(self):
"Superoperator: sprepost, time-dependent"
# left QobjEvo
assert_(sprepost(self.t1, self.q2)(.5) ==
sprepost(self.t1(.5), self.q2))
# left QobjEvo
assert_(sprepost(self.q2, self.t1)(.5) ==
sprepost(self.q2, self.t1(.5)))
# left 2 QobjEvo, one cte
assert_(sprepost(self.t1, self.t2)(.5) ==
sprepost(self.t1(.5), self.t2(.5)))
def testMETDDecayAsArray(self):
"mesolve: time-dependence as array with super as init cond"
me_error = 1e-5
N = 10
a = destroy(N)
H = a.dag() * a
psi0 = basis(N, 9)
rho0vec = operator_to_vector(psi0 * psi0.dag())
E0 = sprepost(qeye(N), qeye(N))
kappa = 0.2
tlist = np.linspace(0, 10, 1000)
c_op_list = [[a, np.sqrt(kappa * np.exp(-tlist))]]
out1 = mesolve(H, psi0, tlist, c_op_list, [])
out2 = mesolve(H, E0, tlist, c_op_list, [])
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def testMETDDecayAsArray(self):
"mesolve: time-dependence as array with super as init cond"
me_error = 1e-5
N = 10
a = destroy(N)
H = a.dag() * a
psi0 = basis(N, 9)
rho0vec = operator_to_vector(psi0 * psi0.dag())
E0 = sprepost(qeye(N), qeye(N))
kappa = 0.2
tlist = np.linspace(0, 10, 1000)
c_op_list = [[a, np.sqrt(kappa * np.exp(-tlist))]]
out1 = mesolve(H, psi0, tlist, c_op_list, [])
out2 = mesolve(H, E0, tlist, c_op_list, [])
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def testMETDDecayAsPartFuncList(self):
"mesolve: time-dep. as partial function list with super as init cond"
me_error = 1e-5
N = 10
a = destroy(N)
H = num(N)
psi0 = basis(N, 9)
rho0vec = operator_to_vector(psi0 * psi0.dag())
E0 = sprepost(qeye(N), qeye(N))
tlist = np.linspace(0, 10, 100)
c_ops = [[[a, partial(lambda t, args, k: np.sqrt(k * np.exp(-t)), k=kappa)]] for kappa in [0.05, 0.1, 0.2]]
for idx, kappa in enumerate([0.05, 0.1, 0.2]):
out1 = mesolve(H, psi0, tlist, c_ops[idx], [])
out2 = mesolve(H, E0, tlist, c_ops[idx], [])
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def testMETDDecayAsStrList(self):
"mesolve: time-dependence as string list with super as init cond"
me_error = 1e-6
N = 10 # number of basis states to consider
a = destroy(N)
H = a.dag() * a
psi0 = basis(N, 9) # initial state
rho0vec = operator_to_vector(psi0 * psi0.dag())
E0 = sprepost(qeye(N), qeye(N))
kappa = 0.2 # coupling to oscillator
c_op_list = [[a, 'sqrt(k*exp(-t))']]
args = {'k': kappa}
tlist = np.linspace(0, 10, 100)
out1 = mesolve(H, psi0, tlist, c_op_list, [], args=args)
out2 = mesolve(H, E0, tlist, c_op_list, [], args=args)
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def testMETDDecayAsStrList(self):
"mesolve: time-dependence as string list with super as init cond"
me_error = 1e-6
N = 10 # number of basis states to consider
a = destroy(N)
H = a.dag() * a
psi0 = basis(N, 9) # initial state
rho0vec = operator_to_vector(psi0 * psi0.dag())
E0 = sprepost(qeye(N), qeye(N))
kappa = 0.2 # coupling to oscillator
c_op_list = [[a, "sqrt(k*exp(-t))"]]
args = {"k": kappa}
tlist = np.linspace(0, 10, 100)
out1 = mesolve(H, psi0, tlist, c_op_list, [], args=args)
out2 = mesolve(H, E0, tlist, c_op_list, [], args=args)
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def __init__(self,
H_S,
L1,
L2,
S_matrix=None,
c_ops_markov=None,
integrator='propagator',
parallel=False,
options=None):
if options is None:
self.options = qt.Options()
else:
self.options = options
self.H_S = H_S
self.sysdims = H_S.dims
if isinstance(L1, qt.Qobj):
self.L1 = [L1]
else:
self.L1 = L1
if isinstance(L2, qt.Qobj):
self.L2 = [L2]
else:
self.L2 = L2
if not len(self.L1) == len(self.L2):
raise ValueError('L1 and L2 has to be of equal length.')
if isinstance(c_ops_markov, qt.Qobj):
self.c_ops_markov = [c_ops_markov]
else:
self.c_ops_markov = c_ops_markov
if S_matrix is None:
self.S_matrix = np.identity(len(self.L1))
else:
self.S_matrix = S_matrix
# create system identity superoperator
self.Id = qt.qeye(H_S.shape[0])
self.Id.dims = self.sysdims
self.Id = qt.sprepost(self.Id, self.Id)
self.store_states = self.options.store_states
self.integrator = integrator
self.parallel = parallel
def testMETDDecayAsPartFuncList(self):
"mesolve: time-dep. as partial function list with super as init cond"
me_error = 1e-5
N = 10
a = destroy(N)
H = num(N)
psi0 = basis(N, 9)
rho0vec = operator_to_vector(psi0 * psi0.dag())
E0 = sprepost(qeye(N), qeye(N))
tlist = np.linspace(0, 10, 100)
c_ops = [[[
a, partial(lambda t, args, k: np.sqrt(k * np.exp(-t)), k=kappa)
]] for kappa in [0.05, 0.1, 0.2]]
for idx, kappa in enumerate([0.05, 0.1, 0.2]):
out1 = mesolve(H, psi0, tlist, c_ops[idx], [])
out2 = mesolve(H, E0, tlist, c_ops[idx], [])
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def _generator(k, H, L1, L2, S=None, c_ops_markov=None):
"""
Create a Liouvillian for a cascaded chain of k system copies
"""
id = qt.qeye(H.dims[0][0])
Id = qt.sprepost(id, id)
if S is None:
S = np.identity(len(L1))
# create Lindbladian
L = qt.Qobj()
E0 = Id
# first system
L += qt.liouvillian(None, [_localop(c, 1, k) for c in L2])
for l in range(1, k):
# Identiy superoperator
E0 = qt.composite(E0, Id)
# Bare Hamiltonian
Hl = _localop(H, l, k)
L += qt.liouvillian(Hl, [])
# Markovian Decay channels
if c_ops_markov is not None:
for c in c_ops_markov:
cl = _localop(c, l, k)
L += qt.liouvillian(None, [cl])
# Cascade coupling
c1 = np.array([_localop(c, l, k) for c in L1])
c2 = np.array([_localop(c, l+1, k) for c in L2])
c2dag = np.array([c.dag() for c in c2])
Hcasc = -0.5j*np.dot(c2dag, np.dot(S, c1))
Hcasc += Hcasc.dag()
Lvec = c2 + np.dot(S, c1)
L += qt.liouvillian(Hcasc, [c for c in Lvec])
# last system
L += qt.liouvillian(_localop(H, k, k), [_localop(c, k, k) for c in L1])
if c_ops_markov is not None:
for c in c_ops_markov:
cl = _localop(c, k, k)
L += qt.liouvillian(None, [cl])
E0.dims = L.dims
# return generator and identity superop E0
return L, E0
def _generator(k, H, L1, L2, S=None, c_ops_markov=None):
"""
Create a Liouvillian for a cascaded chain of k system copies
"""
id = qt.qeye(H.dims[0][0])
Id = qt.sprepost(id, id)
if S is None:
S = np.identity(len(L1))
# create Lindbladian
L = qt.Qobj()
E0 = Id
# first system
L += qt.liouvillian(None, [_localop(c, 1, k) for c in L2])
for l in range(1, k):
# Identiy superoperator
E0 = qt.composite(E0, Id)
# Bare Hamiltonian
Hl = _localop(H, l, k)
L += qt.liouvillian(Hl, [])
# Markovian Decay channels
if c_ops_markov is not None:
for c in c_ops_markov:
cl = _localop(c, l, k)
L += qt.liouvillian(None, [cl])
# Cascade coupling
c1 = np.array([_localop(c, l, k) for c in L1])
c2 = np.array([_localop(c, l+1, k) for c in L2])
c2dag = np.array([c.dag() for c in c2])
Hcasc = -0.5j*np.dot(c2dag, np.dot(S, c1))
Hcasc += Hcasc.dag()
Lvec = c2 + np.dot(S, c1)
L += qt.liouvillian(Hcasc, [c for c in Lvec])
# last system
L += qt.liouvillian(_localop(H, k, k), [_localop(c, k, k) for c in L1])
if c_ops_markov is not None:
for c in c_ops_markov:
cl = _localop(c, k, k)
L += qt.liouvillian(None, [cl])
E0.dims = L.dims
# return generator and identity superop E0
return L, E0
def jc_integrate(self, N, wc, wa, g, kappa, gamma, pump, psi0, use_rwa,
tlist):
# Hamiltonian
a = tensor(destroy(N), identity(2))
sm = tensor(identity(N), destroy(2))
# Identity super-operator
E0 = sprepost(tensor(qeye(N), qeye(2)), tensor(qeye(N), qeye(2)))
if use_rwa:
# use the rotating wave approxiation
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm +
a * sm.dag())
else:
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() + a) * (
sm + sm.dag())
# collapse operators
c_op_list = []
n_th_a = 0.0 # zero temperature
rate = kappa * (1 + n_th_a)
c_op_list.append(np.sqrt(rate) * a)
rate = kappa * n_th_a
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * a.dag())
rate = gamma
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm)
rate = pump
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm.dag())
# evolve and calculate expectation values
output1 = mesolve(H, psi0, tlist, c_op_list, [])
output2 = mesolve(H, E0, tlist, c_op_list, [])
return output1, output2
def testMETDDecayAsFuncList(self):
"mesolve: time-dependence as function list with super as init cond"
me_error = 1e-6
N = 10 # number of basis states to consider
a = destroy(N)
H = a.dag() * a
psi0 = basis(N, 9) # initial state
rho0vec = operator_to_vector(psi0 * psi0.dag())
E0 = sprepost(qeye(N), qeye(N))
kappa = 0.2 # coupling to oscillator
def sqrt_kappa(t, args):
return np.sqrt(kappa * np.exp(-t))
c_op_list = [[a, sqrt_kappa]]
tlist = np.linspace(0, 10, 100)
out1 = mesolve(H, psi0, tlist, c_op_list, [])
out2 = mesolve(H, E0, tlist, c_op_list, [])
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def testMETDDecayAsFuncList(self):
"mesolve: time-dependence as function list with super as init cond"
me_error = 1e-6
N = 10 # number of basis states to consider
a = destroy(N)
H = a.dag() * a
psi0 = basis(N, 9) # initial state
rho0vec = operator_to_vector(psi0 * psi0.dag())
E0 = sprepost(qeye(N), qeye(N))
kappa = 0.2 # coupling to oscillator
def sqrt_kappa(t, args):
return np.sqrt(kappa * np.exp(-t))
c_op_list = [[a, sqrt_kappa]]
tlist = np.linspace(0, 10, 100)
out1 = mesolve(H, psi0, tlist, c_op_list, [])
out2 = mesolve(H, E0, tlist, c_op_list, [])
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def jc_integrate(self, N, wc, wa, g, kappa, gamma, pump, psi0, use_rwa, tlist):
# Hamiltonian
a = tensor(destroy(N), identity(2))
sm = tensor(identity(N), destroy(2))
# Identity super-operator
E0 = sprepost(tensor(qeye(N), qeye(2)), tensor(qeye(N), qeye(2)))
if use_rwa:
# use the rotating wave approxiation
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm + a * sm.dag())
else:
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() + a) * (sm + sm.dag())
# collapse operators
c_op_list = []
n_th_a = 0.0 # zero temperature
rate = kappa * (1 + n_th_a)
c_op_list.append(np.sqrt(rate) * a)
rate = kappa * n_th_a
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * a.dag())
rate = gamma
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm)
rate = pump
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm.dag())
# evolve and calculate expectation values
output1 = mesolve(H, psi0, tlist, c_op_list, [])
output2 = mesolve(H, E0, tlist, c_op_list, [])
return output1, output2
def testMETDDecayAsFunc(self):
"mesolve: time-dependence as function with super as init cond"
N = 10 # number of basis states to consider
a = destroy(N)
H = a.dag() * a
rho0 = ket2dm(basis(N, 9)) # initial state
rho0vec = operator_to_vector(rho0)
E0 = sprepost(qeye(N), qeye(N))
kappa = 0.2 # coupling to oscillator
def Liouvillian_func(t, args):
c = np.sqrt(kappa * np.exp(-t)) * a
data = liouvillian(H, [c]).data
return data
tlist = np.linspace(0, 10, 100)
args = {'kappa': kappa}
out1 = mesolve(Liouvillian_func, rho0, tlist, [], [], args=args)
out2 = mesolve(Liouvillian_func, E0, tlist, [], [], args=args)
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def testMETDDecayAsFunc(self):
"mesolve: time-dependence as function with super as init cond"
N = 10 # number of basis states to consider
a = destroy(N)
H = a.dag() * a
rho0 = ket2dm(basis(N, 9)) # initial state
rho0vec = operator_to_vector(rho0)
E0 = sprepost(qeye(N), qeye(N))
kappa = 0.2 # coupling to oscillator
def Liouvillian_func(t, args):
c = np.sqrt(kappa * np.exp(-t)) * a
data = liouvillian(H, [c]).data
return data
tlist = np.linspace(0, 10, 100)
args = {"kappa": kappa}
out1 = mesolve(Liouvillian_func, rho0, tlist, [], [], args=args)
out2 = mesolve(Liouvillian_func, E0, tlist, [], [], args=args)
fid = self.fidelitycheck(out1, out2, rho0vec)
assert_(max(abs(1.0 - fid)) < me_error, True)
def decoder(self, kraus=False):
S = self.encoder(kraus=True)
if kraus:
return S.dag()
else:
return qt.sprepost(S.dag(), S)
def rcsolve(Hsys,
psi0,
tlist,
e_ops,
Q,
wc,
alpha,
N,
w_th,
sparse=False,
options=None):
"""
Function to solve for an open quantum system using the
reaction coordinate (RC) model.
Parameters
----------
Hsys: Qobj
The system hamiltonian.
psi0: Qobj
Initial state of the system.
tlist: List.
Time over which system evolves.
e_ops: list of :class:`qutip.Qobj` / callback function single
Single operator or list of operators for which to evaluate
expectation values.
Q: Qobj
The coupling between system and bath.
wc: Float
Cutoff frequency.
alpha: Float
Coupling strength.
N: Integer
Number of cavity fock states.
w_th: Float
Temperature.
sparse: Boolean
Optional argument to call the sparse eigenstates solver if needed.
options : :class:`qutip.Options`
With options for the solver.
Returns
-------
output: Result
System evolution.
"""
if options is None:
options = Options()
dot_energy, dot_state = Hsys.eigenstates(sparse=sparse)
deltaE = dot_energy[1] - dot_energy[0]
if (w_th < deltaE / 2):
warnings.warn("Given w_th might not provide accurate results")
gamma = deltaE / (2 * np.pi * wc)
wa = 2 * np.pi * gamma * wc # reaction coordinate frequency
g = np.sqrt(np.pi * wa * alpha / 2.0) # reaction coordinate coupling
nb = (1 / (np.exp(wa / w_th) - 1))
# Reaction coordinate hamiltonian/operators
dimensions = dims(Q)
a = tensor(destroy(N), qeye(dimensions[1]))
unit = tensor(qeye(N), qeye(dimensions[1]))
Nmax = N * dimensions[1][0]
Q_exp = tensor(qeye(N), Q)
Hsys_exp = tensor(qeye(N), Hsys)
e_ops_exp = [tensor(qeye(N), kk) for kk in e_ops]
na = a.dag() * a
xa = a.dag() + a
# decoupled Hamiltonian
H0 = wa * a.dag() * a + Hsys_exp
# interaction
H1 = (g * (a.dag() + a) * Q_exp)
H = H0 + H1
L = 0
PsipreEta = 0
PsipreX = 0
all_energy, all_state = H.eigenstates(sparse=sparse)
Apre = spre((a + a.dag()))
Apost = spost(a + a.dag())
for j in range(Nmax):
for k in range(Nmax):
A = xa.matrix_element(all_state[j].dag(), all_state[k])
delE = (all_energy[j] - all_energy[k])
if abs(A) > 0.0:
if abs(delE) > 0.0:
X = (0.5 * np.pi * gamma *
(all_energy[j] - all_energy[k]) * (np.cosh(
(all_energy[j] - all_energy[k]) /
(2 * w_th)) / (np.sinh(
(all_energy[j] - all_energy[k]) /
(2 * w_th)))) * A)
eta = (0.5 * np.pi * gamma *
(all_energy[j] - all_energy[k]) * A)
PsipreX = PsipreX + X * all_state[j] * all_state[k].dag()
PsipreEta = PsipreEta + (eta * all_state[j] *
all_state[k].dag())
else:
X = 0.5 * np.pi * gamma * A * 2 * w_th
PsipreX = PsipreX + X * all_state[j] * all_state[k].dag()
A = a + a.dag()
L = ((-spre(A * PsipreX)) + (sprepost(A, PsipreX)) +
(sprepost(PsipreX, A)) + (-spost(PsipreX * A)) +
(spre(A * PsipreEta)) + (sprepost(A, PsipreEta)) +
(-sprepost(PsipreEta, A)) + (-spost(PsipreEta * A)))
# Setup the operators and the Hamiltonian and the master equation
# and solve for time steps in tlist
rho0 = (tensor(thermal_dm(N, nb), psi0))
output = mesolve(H, rho0, tlist, [L], e_ops_exp, options=options)
return output
def secular_term(state_j, state_k):
jk = state_j*state_k.dag()
kj = jk.dag()
jj = state_j*state_j.dag()
return 2*sprepost(kj, jk) - (spre(jj) + spost(jj))
def commutator_term2(O1, O2):
# [rho*O1, O2]
return spost(O1*O2)-sprepost(O2, O1)
def rcsolve(Hsys, psi0, tlist, e_ops, Q, wc, alpha, N, w_th, sparse=False,
options=None):
"""
Function to solve for an open quantum system using the
reaction coordinate (RC) model.
Parameters
----------
Hsys: Qobj
The system hamiltonian.
psi0: Qobj
Initial state of the system.
tlist: List.
Time over which system evolves.
e_ops: list of :class:`qutip.Qobj` / callback function single
Single operator or list of operators for which to evaluate
expectation values.
Q: Qobj
The coupling between system and bath.
wc: Float
Cutoff frequency.
alpha: Float
Coupling strength.
N: Integer
Number of cavity fock states.
w_th: Float
Temperature.
sparse: Boolean
Optional argument to call the sparse eigenstates solver if needed.
options : :class:`qutip.Options`
With options for the solver.
Returns
-------
output: Result
System evolution.
"""
if options is None:
options = Options()
dot_energy, dot_state = Hsys.eigenstates(sparse=sparse)
deltaE = dot_energy[1] - dot_energy[0]
if (w_th < deltaE/2):
warnings.warn("Given w_th might not provide accurate results")
gamma = deltaE / (2 * np.pi * wc)
wa = 2 * np.pi * gamma * wc # reaction coordinate frequency
g = np.sqrt(np.pi * wa * alpha / 2.0) # reaction coordinate coupling
nb = (1 / (np.exp(wa/w_th) - 1))
# Reaction coordinate hamiltonian/operators
dimensions = dims(Q)
a = tensor(destroy(N), qeye(dimensions[1]))
unit = tensor(qeye(N), qeye(dimensions[1]))
Nmax = N * dimensions[1][0]
Q_exp = tensor(qeye(N), Q)
Hsys_exp = tensor(qeye(N), Hsys)
e_ops_exp = [tensor(qeye(N), kk) for kk in e_ops]
na = a.dag() * a
xa = a.dag() + a
# decoupled Hamiltonian
H0 = wa * a.dag() * a + Hsys_exp
# interaction
H1 = (g * (a.dag() + a) * Q_exp)
H = H0 + H1
L = 0
PsipreEta = 0
PsipreX = 0
all_energy, all_state = H.eigenstates(sparse=sparse)
Apre = spre((a + a.dag()))
Apost = spost(a + a.dag())
for j in range(Nmax):
for k in range(Nmax):
A = xa.matrix_element(all_state[j].dag(), all_state[k])
delE = (all_energy[j] - all_energy[k])
if abs(A) > 0.0:
if abs(delE) > 0.0:
X = (0.5 * np.pi * gamma*(all_energy[j] - all_energy[k])
* (np.cosh((all_energy[j] - all_energy[k]) /
(2 * w_th))
/ (np.sinh((all_energy[j] - all_energy[k]) /
(2 * w_th)))) * A)
eta = (0.5 * np.pi * gamma *
(all_energy[j] - all_energy[k]) * A)
PsipreX = PsipreX + X * all_state[j] * all_state[k].dag()
PsipreEta = PsipreEta + (eta * all_state[j]
* all_state[k].dag())
else:
X = 0.5 * np.pi * gamma * A * 2 * w_th
PsipreX = PsipreX + X * all_state[j] * all_state[k].dag()
A = a + a.dag()
L = ((-spre(A * PsipreX)) + (sprepost(A, PsipreX))
+ (sprepost(PsipreX, A)) + (-spost(PsipreX * A))
+ (spre(A * PsipreEta)) + (sprepost(A, PsipreEta))
+ (-sprepost(PsipreEta, A)) + (-spost(PsipreEta * A)))
# Setup the operators and the Hamiltonian and the master equation
# and solve for time steps in tlist
rho0 = (tensor(thermal_dm(N, nb), psi0))
output = mesolve(H, rho0, tlist, [L], e_ops_exp, options=options)
return output
