def _pseudo_inverse_dense(L, rhoss, method='direct', **pseudo_args):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
dense matrix methods. See pseudo_inverse for details.
"""
if method == 'direct':
rho_vec = np.transpose(mat2vec(rhoss.full()))
tr_mat = tensor([identity(n) for n in L.dims[0][0]])
tr_vec = np.transpose(mat2vec(tr_mat.full()))
N = np.prod(L.dims[0][0])
I = np.identity(N * N)
P = np.kron(np.transpose(rho_vec), tr_vec)
Q = I - P
LIQ = np.linalg.solve(L.full(), Q)
R = np.dot(Q, LIQ)
return Qobj(R, dims=L.dims)
elif method == 'numpy':
return Qobj(np.linalg.pinv(L.full()), dims=L.dims)
elif method == 'scipy':
return Qobj(la.pinv(L.full()), dims=L.dims)
elif method == 'scipy2':
return Qobj(la.pinv2(L.full()), dims=L.dims)
else:
raise ValueError("Unsupported method '%s'. Use 'direct' or 'numpy'" %
method)
def _pseudo_inverse_dense(L, rhoss, method='direct', **pseudo_args):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
dense matrix methods. See pseudo_inverse for details.
"""
if method == 'direct':
rho_vec = np.transpose(mat2vec(rhoss.full()))
tr_mat = tensor([identity(n) for n in L.dims[0][0]])
tr_vec = np.transpose(mat2vec(tr_mat.full()))
N = np.prod(L.dims[0][0])
I = np.identity(N * N)
P = np.kron(np.transpose(rho_vec), tr_vec)
Q = I - P
LIQ = np.linalg.solve(L.full(), Q)
R = np.dot(Q, LIQ)
return Qobj(R, dims=L.dims)
elif method == 'numpy':
return Qobj(np.linalg.pinv(L.full()), dims=L.dims)
elif method == 'scipy':
return Qobj(la.pinv(L.full()), dims=L.dims)
elif method == 'scipy2':
return Qobj(la.pinv2(L.full()), dims=L.dims)
else:
raise ValueError("Unsupported method '%s'. Use 'direct' or 'numpy'" %
method)
def test_vec_to_eigbasis():
"BR Tools : vector to eigenbasis"
dimension = 10
for _ in range(50):
H = qutip.rand_herm(dimension, 0.5)
basis = H.eigenstates()[1]
R = qutip.rand_dm(dimension, 0.5)
target = qutip.mat2vec(R.transform(basis).full()).ravel()
flat_vector = qutip.mat2vec(R.full()).ravel()
calculated = _test_vec_to_eigbasis(H.full('F'), flat_vector)
np.testing.assert_allclose(target, calculated, atol=1e-12)
def test_eigvec_to_fockbasis():
"BR Tools : eigvector to fockbasis"
dimension = 10
for _ in range(50):
H = qutip.rand_herm(dimension, 0.5)
basis = H.eigenstates()[1]
R = qutip.rand_dm(dimension, 0.5)
target = qutip.mat2vec(R.full()).ravel()
_eigenvalues = np.empty((dimension, ), dtype=np.float64)
evecs_zheevr = _test_zheevr(H.full('F'), _eigenvalues)
flat_eigenvectors = qutip.mat2vec(R.transform(basis).full()).ravel()
calculated = _test_eigvec_to_fockbasis(flat_eigenvectors, evecs_zheevr,
dimension)
np.testing.assert_allclose(target, calculated, atol=1e-12)
def test_vector_roundtrip():
"BR Tools : vector roundtrip transform"
dimension = 10
for _ in range(50):
H = qutip.rand_herm(dimension, 0.5).full('F')
vector = qutip.mat2vec(qutip.rand_dm(dimension, 0.5).full()).ravel()
assert np.allclose(vector, _test_vector_roundtrip(H, vector))
def _pseudo_inverse_dense(L, rhoss, w=None, **pseudo_args):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
dense matrix methods. See pseudo_inverse for details.
"""
rho_vec = np.transpose(mat2vec(rhoss.full()))
tr_mat = tensor([identity(n) for n in L.dims[0][0]])
tr_vec = np.transpose(mat2vec(tr_mat.full()))
N = np.prod(L.dims[0][0])
I = np.identity(N * N)
P = np.kron(np.transpose(rho_vec), tr_vec)
Q = I - P
if w is None:
L = L
else:
L = 1.0j*w*spre(tr_mat)+L
if pseudo_args['method'] == 'direct':
try:
LIQ = np.linalg.solve(L.full(), Q)
except:
LIQ = np.linalg.lstsq(L.full(), Q)[0]
R = np.dot(Q, LIQ)
return Qobj(R, dims=L.dims)
elif pseudo_args['method'] == 'numpy':
return Qobj(np.dot(Q, np.dot(np.linalg.pinv(L.full()), Q)),
dims=L.dims)
elif pseudo_args['method'] == 'scipy':
# return Qobj(la.pinv(L.full()), dims=L.dims)
return Qobj(np.dot(Q, np.dot(la.pinv(L.full()), Q)),
dims=L.dims)
elif pseudo_args['method'] == 'scipy2':
# return Qobj(la.pinv2(L.full()), dims=L.dims)
return Qobj(np.dot(Q, np.dot(la.pinv2(L.full()), Q)),
dims=L.dims)
else:
raise ValueError("Unsupported method '%s'. Use 'direct' or 'numpy'" %
method)
def testMatrixVecMat(self):
"""
Superoperator: Conversion matrix to vector to matrix
"""
M = scipy.rand(10, 10)
V = mat2vec(M)
M2 = vec2mat(V)
assert_(la.norm(M - M2) == 0.0)
def testVecMatVec(self):
"""
Superoperator: Conversion vector to matrix to vector
"""
V = scipy.rand(100) # a row vector
M = vec2mat(V)
V2 = mat2vec(M).T # mat2vec returns a column vector
assert_(norm(V - V2) == 0.0)
def testMatrixVecMat(self):
"""
Superoperator: Conversion matrix to vector to matrix
"""
M = scipy.rand(10, 10)
V = mat2vec(M)
M2 = vec2mat(V)
assert_(norm(M - M2) == 0.0)
def _pseudo_inverse_dense(L, rhoss, w=None, **pseudo_args):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
dense matrix methods. See pseudo_inverse for details.
"""
rho_vec = np.transpose(mat2vec(rhoss.full()))
tr_mat = tensor([identity(n) for n in L.dims[0][0]])
tr_vec = np.transpose(mat2vec(tr_mat.full()))
N = np.prod(L.dims[0][0])
I = np.identity(N * N)
P = np.kron(np.transpose(rho_vec), tr_vec)
Q = I - P
if w is None:
L = L
else:
L = 1.0j*w*spre(tr_mat)+L
if pseudo_args['method'] == 'direct':
try:
LIQ = np.linalg.solve(L.full(), Q)
except:
LIQ = np.linalg.lstsq(L.full(), Q)[0]
R = np.dot(Q, LIQ)
return Qobj(R, dims=L.dims)
elif pseudo_args['method'] == 'numpy':
return Qobj(np.dot(Q, np.dot(np.linalg.pinv(L.full()), Q)),
dims=L.dims)
elif pseudo_args['method'] == 'scipy':
# return Qobj(la.pinv(L.full()), dims=L.dims)
return Qobj(np.dot(Q, np.dot(la.pinv(L.full()), Q)),
dims=L.dims)
elif pseudo_args['method'] == 'scipy2':
# return Qobj(la.pinv2(L.full()), dims=L.dims)
return Qobj(np.dot(Q, np.dot(la.pinv2(L.full()), Q)),
dims=L.dims)
else:
raise ValueError("Unsupported method '%s'. Use 'direct' or 'numpy'" %
method)
def testVecMatVec(self):
"""
Superoperator: Conversion vector to matrix to vector
"""
V = scipy.rand(100) # a row vector
M = vec2mat(V)
V2 = mat2vec(M).T # mat2vec returns a column vector
assert_(la.norm(V - V2) == 0.0)
def testVecMatIndexCompability(self):
"""
Superoperator: Compatibility between matrix/vector and
corresponding index conversions.
"""
N = 10
M = scipy.rand(N, N)
V = mat2vec(M)
for I in range(N * N):
i, j = vec2mat_index(N, I)
assert_(V[I][0] == M[i, j])
def testVecMatIndexCompability(self):
"""
Superoperator: Compatibility between matrix/vector and
corresponding index conversions.
"""
N = 10
M = scipy.rand(N, N)
V = mat2vec(M)
for I in range(N * N):
i, j = vec2mat_index(N, I)
assert_(V[I][0] == M[i, j])
def two_tone(vg, listening=False, use_pinv=True):
phi, wd = vg
H = make_H(phi, Omega, wd * nvec, 0.0)
if listening:
final_state = steadystate(H, c_op_list)
return expect(a, final_state)
L = liouvillian(H, c_op_list)
#tr_mat = tensor([qeye(n) for n in L.dims[0][0]])
#N = prod(L.dims[0][0])
A = L.full()
#tr_vec = transpose(mat2vec(tr_mat.full()))
rho_ss = steadystate(L)
rho = transpose(mat2vec(rho_ss.full()))
#P = kron(transpose(rho), tr_vec)
#Q = I - P
w = wp - wd
if 1:
D, U = eig(A)
Uinv = inv(U)
#spectrum = zeros(len(wlist))
#for idx, w in enumerate(wlist):
if 1:
if use_pinv:
MMR = pinv(-1.0j * w * I + A)
#MMR = A*MMR*inv(A)
elif use_pinv == 1:
MMR = dot(Q, solve(-1.0j * w * I + A, Q))
else:
MMR = diag(1.0 / (1.0j * w * diag(I) + diag(D)))
Lint = U * MMR * Uinv
#Chi_temp[i,j] += (1/theta_steps)*1j*trace(reshape(tm_l*Lint*(p_l - p_r)*rho_ss_c,dim,dim))
#rho_tr=transpose(rho)
#p1=dot(b_sup, rho_tr)
#p2=dot(rho_tr, b_sup)
#s = dot(tr_vec,
# dot(a_sup, dot(MMR, p1-p2)))
s1 = dot(tr_vec, dot(a_sup, dot(MMR, dot(b_sup, transpose(rho)))))
s2 = dot(tr_vec, dot(b_sup, dot(MMR, dot(a_sup, transpose(rho)))))
s = s1 - s2
#spectrum[idx] =
return -2 * real(s[0, 0])
return spectrum
def _spectrum_pi(H, wlist, c_ops, a_op, b_op, use_pinv=False):
"""
Internal function for calculating the spectrum of the correlation function
:math:`\left<A(\\tau)B(0)\\right>`.
"""
#print issuper(H)
L = H if issuper(H) else liouvillian(H, c_ops)
tr_mat = tensor([qeye(n) for n in L.dims[0][0]])
N = prod(L.dims[0][0])
A = L.full()
b = spre(b_op).full()
a = spre(a_op).full()
tr_vec = transpose(mat2vec(tr_mat.full()))
rho_ss = steadystate(L)
rho = transpose(mat2vec(rho_ss.full()))
I = identity(N * N)
P = kron(transpose(rho), tr_vec)
Q = I - P
spectrum = zeros(len(wlist))
for idx, w in enumerate(wlist):
if use_pinv:
MMR = pinv(-1.0j * w * I + A)
else:
MMR = dot(Q, solve(-1.0j * w * I + A, Q))
s = dot(tr_vec,
dot(a, dot(MMR, dot(b, transpose(rho)))))
spectrum[idx] = -2 * real(s[0, 0])
return spectrum
p_r=spost(p)#.full() print p_l.shape #raise Exception p_l_m_p_r=p_l-p_r #tm_l=spre(a).full() nvec=arange(N) Ecvec=-Ec*(6.0*nvec**2+6.0*nvec+3.0)/12.0 Omega_op=[0.5*Omega*(a*exp(-1.0j*theta)+adag*exp(1.0j*theta)) for theta in [0.0]] tr_mat = tensor(qeye(N)) #N = np.prod(L.dims[0][0]) tr_vec = transpose(mat2vec(tr_mat.full())) p_sup = spre(p)#.full() a_sup = spre(a)#.full() I=identity(N*N) Idiag=diag(I) # Excitation (at T>0). #Lindblad_deph = 0.5*gamma_phi*(kron(conj(tdiag),tdiag) -... # 0.5*kron(Itot,ctranspose(tdiag)*tdiag) -... # 0.5*kron(transpose(tdiag)*conj(tdiag),Itot)); print "yo" #print eye(N).shape
def find_expect(phi=0.1, Omega_vec=3.0, wd=wd, use_pinv=True):
Ej = (phi**2)/(8*Ec) #Ejmax*absolute(cos(pi*phi)) #; % Josephson energy as function of Phi.
wTvec = -Ej + sqrt(8.0*Ej*Ec)*(nvec+0.5)+Ecvec
#wdvec=nvec*sqrt(8.0*Ej*Ec)
#wdvec=nvec*wd
wT = wTvec-wdvec
transmon_levels = Qobj(diag(wT[range(N)]))
H=transmon_levels +Omega_vec #- 0.5j*(Omega_true*adag - conj(Omega_true)*a)
L=liouvillian(H, c_op_list) #ends at same thing as L = H_comm + Lindblad_tm ;
rho_ss = steadystate(L) #same as rho_ss_c but that's in column vector form
tr_mat = tensor([qeye(n) for n in L.dims[0][0]])
#N = prod(L.dims[0][0])
A = L.full()
#D, U= L.eigenstates()
#print D.shape, D
#print diag(D)
b = spre(p).full()-spost(p).full()
a2 = spre(a).full()
tr_vec = transpose(mat2vec(tr_mat.full()))
rho = transpose(mat2vec(rho_ss.full()))
I = identity(N * N)
P = kron(transpose(rho), tr_vec)
Q = I - P
#wp=4.9
#w=-(wp-wd)
spectrum = zeros(len(wlist2), dtype=complex)
for idx, w in enumerate(wlist2):
if use_pinv:
MMR = pinv(-1.0j * w * I + A) #eig(MMR)[0] is equiv to Dint
else:
MMR = dot(Q, solve(-1.0j * w * I + A, Q))
#print diag(1.0/(1j*(wp-wd)*ones(N**2)+D)) #Dint = diag(1./(1i*(wp-wd)*diag(eye(dim^2)) + diag(D)))
#print 1.0/(1j*(wp-wd)*ones(N**2)+D) #Dint = diag(1./(1i*(wp-wd)*diag(eye(dim^2)) + diag(D)))
#U2=squeeze([u.full() for u in U]).transpose()
#Dint=eig(MMR)[0]
#print "MMR", eig(MMR)[1]
#print "Umult", U2*Dint*inv(U2)
s = dot(tr_vec,
dot(a2, dot(MMR, dot(b, transpose(rho)))))
#spectrum[idx] = -2 * real(s[0, 0])
spectrum[idx]=1j*s[0][0] #matches Chi_temp result #(1/theta_steps)
return spectrum
#final_state = steadystate(H, c_op_list)
#print H.shape
#print dir(H)
#U,D = eig(H.full())
#print D
#Uinv = Qobj(inv(D))
#U=Qobj(D)
# Doing the chi integral (gives the susceptibility)
#Dint = 1.0/(1.0j*(wp-wd)) #Qobj(1.0/(1.0j*(wp-wd)*diag(qeye(N**2))))# + diag(D))))
#Hint = H.expm() #U*H*Uinv
#Chi_temp(i,j) += (1.0/theta_steps)*1j*trace(reshape(tm_l*Lint*(p_l - p_r)*rho_ss_c,dim,dim))
#exp1=correlation_2op_1t(H, None, wlist2, c_op_list, a, p, solver="es")
#exp2=correlation_2op_1t(H, None, wlist2, c_op_list, p, a, solver="es", reverse=True)
#exp1=correlation_2op_1t(H, None, wlist2, c_op_list, a, p_l-p_r, solver="es")
#exp1=correlation_ss(H, wlist2, c_op_list, a, p)
#exp2=correlation_ss(H, wlist2, c_op_list, p, a, reverse=True)
exp1=spectrum(H, wlist2, c_op_list, a, p, solver="pi", use_pinv=False)
exp2=spectrum(H, wlist2, c_op_list, p, a, solver="pi", use_pinv=False)
return exp1-exp2
return expect( a, final_state) #tm_l
def two_tone(vg):
phi, wd=vg
Ej = Ejmax*absolute(cos(pi*phi))#(phi**2)/(8*Ec) # #; % Josephson energy as function of Phi.
wTvec = -Ej + sqrt(8.0*Ej*Ec)*(nvec+0.5)+Ecvec
wdvec=nvec*wd
wT = wTvec-wdvec
transmon_levels = Qobj(diag(wT[range(N)]))
for Om in Omega_op:
H = transmon_levels + Om
exp1=spectrum(H, wlist2, c_op_list, a, p, solver="pi", use_pinv=False)
exp2=spectrum(H, wlist2, c_op_list, p, a, solver="pi", use_pinv=False)
return exp1-exp2
#H_comm = -1j*(kron(It,H) - kron(transpose(H),It));
#L = H_comm + Lindblad_tm + Lindblad_tp #+ Lindblad_deph;
#print L
#L2 = [reshape(eye(N),1,N**2); L]; %Add the condition trace(rho) = 1
#rho_ss = L2\[1;zeros(dim^2,1)]; % Steady state density matrix
#rho_ss_c = rho_ss; %Column vector form
L = liouvillian(H, c_op_list)
#D, V=L.eigenstates()
#print "D", D.shape
#print "V", V.shape
#print L.diag()
#print L.diag().shape
#raise Exception
A = L.full()
rho_ss = steadystate(L)
rho = transpose(mat2vec(rho_ss.full()))
P = kron(transpose(rho), tr_vec)
Q = I - P
if 1:
w=wp-wd
#MMR = pinv(-1.0j * w * I + A)
MMR = dot(Q, solve(-1.0j * w * I + A, Q))
#Lexp=L.expm()
#MMR=Lexp*MMR*Lexp.dag()
# MMR = np.dot(Q, np.linalg.solve(-1.0j * w * I + A, Q))
#print p_l.shape
#print p_l
#print transpose(rho).shape
#print dot(p_l.full(), transpose(rho))
s = dot(tr_vec,
dot(tm_l, dot(MMR, dot(p_l_m_p_r, transpose(rho)))))
return -2 * real(s[0, 0])
D,U = eig(L.full())
Uinv = inv(U)
Dint = diag(1.0/(1.0j*(wp-wd)*Idiag + diag(D)))
Lint = U*Dint*Uinv
#H_comm = -1.0j*(kron(It,H) - kron(transpose(H),It))
#print H
#print H.shape, H.full().shape
#print H_comm, H_comm.shape
#L = H_comm + Lindblad_tm + Lindblad_tp + Lindblad_deph
#p_l=spre(p)
#p_r=spost(p)
#tm_l=spre(a)
#print L.shape, A.shape,
#print dir(L)
#print L.eigenenergies().shape
#print L.eigenstates()#.shape
#L2 = [reshape(eye(dim),1,dim^2);L]; %Add the condition trace(rho) = 1
#rho_ss = L2\[1;zeros(dim^2,1)]; % Steady state density matrix
#rho_ss_c = rho_ss; %Column vector form
#print "tm_l", tm_l.full().shape
#print "Lint", Lint.shape
#print "pl", p_l.full().shape
#print "pr", p_r.full().shape
#print rho.shape, to_super(rho_ss).full().shape
#print "rho", to_super(rho_ss) #rho_ss.full().shape #rho.shape
#print (tm_l.full()*Lint*(p_l.full() - p_r.full())*rho).shape
#raise Exception
return 1j*trace(tm_l*Lint*(p_l_m_p_r)*rho_ss)
def hsolve(H, psi0, tlist, Q, gam, lam0, Nc, N, w_th, options=None):
"""
Function to solve for an open quantum system using the
hierarchy model.
Parameters
----------
H: Qobj
The system hamiltonian.
psi0: Qobj
Initial state of the system.
tlist: List.
Time over which system evolves.
Q: Qobj
The coupling between system and bath.
gam: Float
Bath cutoff frequency.
lam0: Float
Coupling strength.
Nc: Integer
Cutoff parameter.
N: Integer
Number of matsubara terms.
w_th: Float
Temperature.
options : :class:`qutip.Options`
With options for the solver.
Returns
-------
output: Result
System evolution.
"""
if options is None:
options = Options()
# Set up terms of the matsubara and tanimura boundaries
# Parameters and hamiltonian
hbar = 1.
kb = 1.
# Set by system
dimensions = dims(H)
Nsup = dimensions[0][0] * dimensions[0][0]
unit = qeye(dimensions[0])
# Ntot is the total number of ancillary elements in the hierarchy
Ntot = int(round(factorial(Nc+N) / (factorial(Nc) * factorial(N))))
c0 = (lam0 * gam * (_cot(gam * hbar / (2. * kb * w_th)) - (1j))) / hbar
LD1 = (-2. * spre(Q) * spost(Q.dag()) + spre(Q.dag()*Q) + spost(Q.dag()*Q))
pref = ((2. * lam0 * kb * w_th / (gam * hbar)) - 1j * lam0) / hbar
gj = 2 * np.pi * kb * w_th / hbar
L12 = -pref * LD1 + (c0 / gam) * LD1
for i1 in range(1, N):
num = (4 * lam0 * gam * kb * w_th * i1 * gj/((i1 * gj)**2 - gam**2))
ci = num / (hbar**2)
L12 = L12 + (ci / gj) * LD1
# Setup liouvillian
L = liouvillian(H, [L12])
Ltot = L.data
unit = sp.eye(Ntot,format='csr')
Lbig = sp.kron(unit, Ltot)
rho0big1 = np.zeros((Nsup * Ntot), dtype=complex)
# Prepare initial state:
rhotemp = mat2vec(np.array(psi0.full(), dtype=complex))
for idx, element in enumerate(rhotemp):
rho0big1[idx] = element[0]
nstates, state2idx, idx2state = enr_state_dictionaries([Nc+1]*(N), Nc)
for nlabelt in state_number_enumerate([Nc+1]*(N), Nc):
nlabel = list(nlabelt)
ntotalcheck = 0
for ncheck in range(N):
ntotalcheck = ntotalcheck + nlabel[ncheck]
current_pos = int(round(state2idx[tuple(nlabel)]))
Ltemp = sp.lil_matrix((Ntot, Ntot))
Ltemp[current_pos, current_pos] = 1
Ltemp.tocsr()
Lbig = Lbig + sp.kron(Ltemp, (-nlabel[0] * gam * spre(unit).data))
for kcount in range(1, N):
counts = -nlabel[kcount] * kcount * gj * spre(unit).data
Lbig = Lbig + sp.kron(Ltemp, counts)
for kcount in range(N):
if nlabel[kcount] >= 1:
# find the position of the neighbour
nlabeltemp = copy(nlabel)
nlabel[kcount] = nlabel[kcount] - 1
current_pos2 = int(round(state2idx[tuple(nlabel)]))
Ltemp = sp.lil_matrix((Ntot, Ntot))
Ltemp[current_pos, current_pos2] = 1
Ltemp.tocsr()
# renormalized version:
ci = (4 * lam0 * gam * kb * w_th * kcount
* gj/((kcount * gj)**2 - gam**2)) / (hbar**2)
if kcount == 0:
Lbig = Lbig + sp.kron(Ltemp, (-1j
* (np.sqrt(nlabeltemp[kcount]
/ abs(c0)))
* ((c0) * spre(Q).data
- (np.conj(c0))
* spost(Q).data)))
if kcount > 0:
ci = (4 * lam0 * gam * kb * w_th * kcount
* gj/((kcount * gj)**2 - gam**2)) / (hbar**2)
Lbig = Lbig + sp.kron(Ltemp, (-1j
* (np.sqrt(nlabeltemp[kcount]
/ abs(ci)))
* ((ci) * spre(Q).data
- (np.conj(ci))
* spost(Q).data)))
nlabel = copy(nlabeltemp)
for kcount in range(N):
if ntotalcheck <= (Nc-1):
nlabeltemp = copy(nlabel)
nlabel[kcount] = nlabel[kcount] + 1
current_pos3 = int(round(state2idx[tuple(nlabel)]))
if current_pos3 <= (Ntot):
Ltemp = sp.lil_matrix((Ntot, Ntot))
Ltemp[current_pos, current_pos3] = 1
Ltemp.tocsr()
# renormalized
if kcount == 0:
Lbig = Lbig + sp.kron(Ltemp, -1j
* (np.sqrt((nlabeltemp[kcount]+1)
* abs(c0)))
* (spre(Q) - spost(Q)).data)
if kcount > 0:
ci = (4 * lam0 * gam * kb * w_th * kcount
* gj/((kcount * gj)**2 - gam**2)) / (hbar**2)
Lbig = Lbig + sp.kron(Ltemp, -1j
* (np.sqrt((nlabeltemp[kcount]+1)
* abs(ci)))
* (spre(Q) - spost(Q)).data)
nlabel = copy(nlabeltemp)
output = []
for element in rhotemp:
output.append([])
r = scipy.integrate.ode(cy_ode_rhs)
Lbig2 = Lbig.tocsr()
r.set_f_params(Lbig2.data, Lbig2.indices, Lbig2.indptr)
r.set_integrator('zvode', method=options.method, order=options.order,
atol=options.atol, rtol=options.rtol,
nsteps=options.nsteps, first_step=options.first_step,
min_step=options.min_step, max_step=options.max_step)
r.set_initial_value(rho0big1, tlist[0])
dt = tlist[1] - tlist[0]
for t_idx, t in enumerate(tlist):
r.integrate(r.t + dt)
for idx, element in enumerate(rhotemp):
output[idx].append(r.y[idx])
return output
def find_expect(phi=0.1, Omega_vec=3.0, wd=wd, use_pinv=True):
Ej = (phi**2) / (
8 * Ec
) #Ejmax*absolute(cos(pi*phi)) #; % Josephson energy as function of Phi.
wTvec = -Ej + sqrt(8.0 * Ej * Ec) * (nvec + 0.5) + Ecvec
#wdvec=nvec*sqrt(8.0*Ej*Ec)
#wdvec=nvec*wd
wT = wTvec - wdvec
transmon_levels = Qobj(diag(wT[range(N)]))
H = transmon_levels + Omega_vec #- 0.5j*(Omega_true*adag - conj(Omega_true)*a)
L = liouvillian(
H, c_op_list) #ends at same thing as L = H_comm + Lindblad_tm ;
rho_ss = steadystate(L) #same as rho_ss_c but that's in column vector form
tr_mat = tensor([qeye(n) for n in L.dims[0][0]])
#N = prod(L.dims[0][0])
A = L.full()
#D, U= L.eigenstates()
#print D.shape, D
#print diag(D)
b = spre(p).full() - spost(p).full()
a2 = spre(a).full()
tr_vec = transpose(mat2vec(tr_mat.full()))
rho = transpose(mat2vec(rho_ss.full()))
I = identity(N * N)
P = kron(transpose(rho), tr_vec)
Q = I - P
#wp=4.9
#w=-(wp-wd)
spectrum = zeros(len(wlist2), dtype=complex)
for idx, w in enumerate(wlist2):
if use_pinv:
MMR = pinv(-1.0j * w * I + A) #eig(MMR)[0] is equiv to Dint
else:
MMR = dot(Q, solve(-1.0j * w * I + A, Q))
#print diag(1.0/(1j*(wp-wd)*ones(N**2)+D)) #Dint = diag(1./(1i*(wp-wd)*diag(eye(dim^2)) + diag(D)))
#print 1.0/(1j*(wp-wd)*ones(N**2)+D) #Dint = diag(1./(1i*(wp-wd)*diag(eye(dim^2)) + diag(D)))
#U2=squeeze([u.full() for u in U]).transpose()
#Dint=eig(MMR)[0]
#print "MMR", eig(MMR)[1]
#print "Umult", U2*Dint*inv(U2)
s = dot(tr_vec, dot(a2, dot(MMR, dot(b, transpose(rho)))))
#spectrum[idx] = -2 * real(s[0, 0])
spectrum[idx] = 1j * s[0][0] #matches Chi_temp result #(1/theta_steps)
return spectrum
#final_state = steadystate(H, c_op_list)
#print H.shape
#print dir(H)
#U,D = eig(H.full())
#print D
#Uinv = Qobj(inv(D))
#U=Qobj(D)
# Doing the chi integral (gives the susceptibility)
#Dint = 1.0/(1.0j*(wp-wd)) #Qobj(1.0/(1.0j*(wp-wd)*diag(qeye(N**2))))# + diag(D))))
#Hint = H.expm() #U*H*Uinv
#Chi_temp(i,j) += (1.0/theta_steps)*1j*trace(reshape(tm_l*Lint*(p_l - p_r)*rho_ss_c,dim,dim))
#exp1=correlation_2op_1t(H, None, wlist2, c_op_list, a, p, solver="es")
#exp2=correlation_2op_1t(H, None, wlist2, c_op_list, p, a, solver="es", reverse=True)
#exp1=correlation_2op_1t(H, None, wlist2, c_op_list, a, p_l-p_r, solver="es")
#exp1=correlation_ss(H, wlist2, c_op_list, a, p)
#exp2=correlation_ss(H, wlist2, c_op_list, p, a, reverse=True)
exp1 = spectrum(H, wlist2, c_op_list, a, p, solver="pi", use_pinv=False)
exp2 = spectrum(H, wlist2, c_op_list, p, a, solver="pi", use_pinv=False)
return exp1 - exp2
return expect(a, final_state) #tm_l
wlist = linspace(-500.0, 500.0, 201) use_pinv=True tr_mat = tensor([qeye(n) for n in L.dims[0][0]]) N = prod(L.dims[0][0]) A = L.full() D, U= L.eigenstates() print D.shape, D print diag(D) b = spre(p).full()-spost(p).full() a = spre(a).full() tr_vec = transpose(mat2vec(tr_mat.full())) rho_ss = steadystate(L) rho = transpose(mat2vec(rho_ss.full())) I = identity(N * N) P = kron(transpose(rho), tr_vec) Q = I - P wp=4.9 w=-(wp-wd) MMR = pinv(-1.0j * w * I + A) #eig(MMR)[0] is equiv to Dint #MMR = pinv(-1.0j * w * I) # + A) MMR = dot(Q, solve(-1.0j * w * I + A, Q)) #MMR = pinv(-1.0j * w * I + D)
fexpt = parallel_map(find_expect, vg, progress_bar=True)
fexpt = reshape(fexpt, (31, 101))
pcolormesh(phi_arr, sample_power_sim_dBm, absolute(fexpt), cmap="RdBu_r")
show()
#w01_vec(i) = wT_vec(2)- wT_vec(1); % Transition energies.
#w12_vec(i) = wT_vec(3)- wT_vec(2);
#w23_vec(i) = wT_vec(4)- wT_vec(3);
#w34_vec(i) = wT_vec(5)- wT_vec(4);
H = make_H(0.2, 190, 4000.0)
print H
L = liouvillian(H, c_op_list)
print L
tr_mat = tensor([qeye(n) for n in L.dims[0][0]])
print tr_mat
tr_vec = transpose(mat2vec(tr_mat.full()))
print tr_vec
N = prod(L.dims[0][0])
print N
A = L.full()
a_sup = spre(a).full()
b_sup = spre(p).full()
D, U = eig(A)
print "D", D
I = identity(N * N)
w = 3.0
MMR1 = pinv(-1.0j * w * I + A)
print A * MMR1 * inv(A)
print U * MMR1 * inv(U)
