def deriv(self, t, yt):
rho, q, p = self.unpack(yt)
cq = np.einsum('nk,nk->n',self._c,q)
ham_t = self.ham.sys + np.einsum('nab,n->ab',self._hamsb,cq)
F_avg = np.einsum('nab,ba->n',self._hamsb,rho).real
dq = p.copy()
dp = - self._omegasq*q \
- np.einsum('nk,n->nk',self._c,F_avg)
drho = -1j/const.hbar * utils.commutator(ham_t,rho)
return self.pack(drho, dq, dp)
def deriv(self, t, yt):
rho, q, p = self.unpack(yt)
cq = np.einsum('nk,nk->n', self._c, q)
ham_t = np.einsum('nab,n->ab', self._hamsb, cq)
ham_t = self.ham.to_interaction(self.ham.site2eig(ham_t), t)
hamsb_int = list()
for n, hamsb in enumerate(self._hamsb):
hamsb_int_n = self.ham.to_interaction(self.ham.site2eig(hamsb), t)
hamsb_int.append(hamsb_int_n)
hamsb_int = np.array(hamsb_int)
F_avg = np.einsum('nab,ba->n', hamsb_int, rho).real
dq = p.copy()
dp = - self._omegasq*q \
- np.einsum('nk,n->nk',self._c,F_avg)
drho = -1j / const.hbar * utils.commutator(ham_t, rho)
return self.pack(drho, dq, dp)
def heom_deriv(self, t, rho):
hbar = const.hbar
drho = []
if self.ham.sd[0].sd_type == 'ohmic-lorentz':
# Chen Eq. (15)
cjk_over_gammajk = np.einsum('jk,jk->j', self.c, 1./self.gamma)
for n, nmat in enumerate(self.nmats):
L = np.sum(nmat)
rho_n = rho[n]
drho_n = -1j/hbar * utils.commutator(self.ham.sys,rho_n)
njk_gammajk = np.sum(nmat*self.gamma)
for j in range(self.ham.nbath):
Fj = self.ham.sysbath[j] # this is like |j><j|
lamdaj = self.ham.sd[j].lamda
omega_cj = self.ham.sd[j].omega_c
for k in range(self.Kmax+1):
if self.is_scaled:
scale_minus = np.sqrt(nmat[j,k]/np.abs(self.c[j,k]))
else:
scale_minus = nmat[j,k]
rho_njkminus = self.rho_minus(rho,n,j,k)
drho_n -= ( (1j/hbar)*scale_minus
*(self.c[j,k]*np.dot(Fj,rho_njkminus)
-self.c[j,k].conjugate()*np.dot(rho_njkminus,Fj)) )
# Note: The real part is taken because the Ishizaki-Tanimura
# truncation yields a sum over excluded Matsubara frequencies,
# which are purely real; the difference between the *real*
# finite K expansion and 2 lamda kT / omega_c gives the
# neglected contribution (in Markovian "TNL" approximation).
if self.truncation_type == 'Ishizaki-Tanimura':
drho_n -= ( (2*lamdaj*const.kT/(hbar**2*omega_cj)
- cjk_over_gammajk[j]/hbar).real
*utils.commutator(Fj, utils.commutator(Fj, rho_n)) )
if L < self.Lmax:
# If L == self.Lmax, then rho_nkplus = 0
rho_njkplus_sum = np.zeros_like(rho_n)
for k in range(self.Kmax+1):
if self.is_scaled:
scale_plus = np.sqrt((nmat[j,k]+1)*np.abs(self.c[j,k]))
else:
scale_plus = 1.
rho_njkplus_sum += scale_plus*self.rho_plus(rho,n,j,k)
drho_n -= 1j*utils.commutator(Fj, rho_njkplus_sum)
drho_n -= njk_gammajk*rho_n
drho.append(drho_n)
# TODO(TCB): Finish this, i.e. HEOM for single-oscillator(s)
# elif self.ham.sd[0].sd_type == 'oscillators':
# # Liu App. C, Eq. (C1)
# for m in range(len(self.nmats)):
# mmat = self.nmats[m]
# Lm = np.sum(mmat)
# for n in range(len(self.nmats)):
# nmat = self.nmats[n]
# Ln = np.sum(nmat)
# rho_mn = rho[m,n]
# drho_mn = np.zeros_like(rho_mn)
# drho_mn = -1j/hbar * utils.commutator(self.ham.sys,rho_mn)
# njk_gammajk = 0.
#
# # more here
# drho.append(drho_mn)
return np.array(drho)
def heom_deriv(self, t, rho):
hbar = const.hbar
drho = list()
if self.ham.sd[0].sd_type == 'ohmic-lorentz':
# Chen Eq. (15)
cjk_over_gammajk = np.einsum('jk,jk->j', self.c, 1. / self.gamma)
for n, nmat in enumerate(self.nmats):
L = np.sum(nmat)
rho_n = rho[n]
drho_n = -1j / hbar * utils.commutator(self.ham.sys, rho_n)
njk_gammajk = np.sum(nmat * self.gamma)
for j in range(self.ham.nbath):
Fj = self.ham.sysbath[j] # this is like |j><j|
lamdaj = self.ham.sd[j].lamda
omega_cj = self.ham.sd[j].omega_c
for k in range(self.Kmax + 1):
if self.is_scaled:
scale_minus = np.sqrt(nmat[j, k] /
np.abs(self.c[j, k]))
else:
scale_minus = nmat[j, k]
rho_njkminus = self.rho_minus(rho, n, j, k)
drho_n -= ((1j / hbar) * scale_minus *
(self.c[j, k] * np.dot(Fj, rho_njkminus) -
self.c[j, k].conjugate() *
np.dot(rho_njkminus, Fj)))
# Note: The real part is taken because the Ishizaki-Tanimura
# truncation yields a sum over excluded Matsubara frequencies,
# which are purely real; the difference between the *real*
# finite K expansion and 2 lamda kT / omega_c gives the
# neglected contribution (in Markovian "TNL" approximation).
if self.K_truncation == 'Ishizaki-Tanimura':
drho_n -= (
(2 * lamdaj * const.kT / (hbar**2 * omega_cj) -
cjk_over_gammajk[j] / hbar).real *
utils.commutator(Fj, utils.commutator(Fj, rho_n)))
if L < self.Lmax:
# If L == self.Lmax, then rho_nkplus = 0
rho_njkplus_sum = np.zeros_like(rho_n)
for k in range(self.Kmax + 1):
if self.is_scaled:
scale_plus = np.sqrt(
(nmat[j, k] + 1) * np.abs(self.c[j, k]))
else:
scale_plus = 1.
rho_njkplus_sum += scale_plus * self.rho_plus(
rho, n, j, k)
drho_n -= 1j * utils.commutator(Fj, rho_njkplus_sum)
if self.L_truncation == 'TL' and L == self.Lmax:
E, evec = utils.diagonalize(self.ham.sys)
Q_j = np.zeros((self.ham.nsite, self.ham.nsite),
dtype=np.complex)
for a in range(self.ham.nsite):
aouter = np.outer(evec[:, a], evec[:, a])
for b in range(self.ham.nsite):
bouter = np.outer(evec[:, b], evec[:, b])
if self.is_scaled:
num = np.sqrt(nmat[j, k] + 1) * self.c[
j, :] * (1. - np.exp(
-self.gamma[j, :] * t - 1j *
self.ham.omega_diff[a, b] * t))
else:
num = self.c[j, :] * (1. - np.exp(
-self.gamma[j, :] * t -
1j * self.ham.omega_diff[a, b] * t))
denom = self.gamma[
j, :] + 1j * self.ham.omega_diff[a, b]
Q_j += np.sum(num / denom) * np.dot(
aouter, np.dot(self.ham.sysbath[j],
bouter))
rho_njkplus_sum = -(1j / hbar) * (
np.dot(Q_j, rho_n) - np.dot(rho_n,
np.conj(Q_j).T))
drho_n -= 1j * utils.commutator(Fj, rho_njkplus_sum)
# for k in range(self.Kmax+1):
# if self.is_scaled:
# scale_plus = np.sqrt((nmat[j,k]+1)*np.abs(self.c[j,k]))
# else:
# scale_plus = 1.
# rho_njkplus_sum += scale_plus*self.rho_plus(rho,n,j,k)
#
#
drho_n -= njk_gammajk * rho_n
drho.append(drho_n)
# TODO(TCB): Finish this, i.e. HEOM for single-oscillator(s)
# elif self.ham.sd[0].sd_type == 'oscillators':
# # Liu App. C, Eq. (C1)
# for m in range(len(self.nmats)):
# mmat = self.nmats[m]
# Lm = np.sum(mmat)
# for n in range(len(self.nmats)):
# nmat = self.nmats[n]
# Ln = np.sum(nmat)
# rho_mn = rho[m,n]
# drho_mn = np.zeros_like(rho_mn)
# drho_mn = -1j/hbar * utils.commutator(self.ham.sys,rho_mn)
# njk_gammajk = 0.
#
# # more here
# drho.append(drho_mn)
return np.array(drho)
def heom_deriv(self, t, rho):
hbar = const.hbar
drho = list()
if self.ham.sd[0].sd_type == 'ohmic-lorentz':
# Chen Eq. (15)
cjk_over_gammajk_re = np.einsum('jk,jk->j', self.c, 1./self.gamma).real
for n, nmat in enumerate(self.nmats):
L = np.sum(nmat)
rho_n = rho[n]
drho_n = -1j/hbar * utils.commutator(self.ham.sys,rho_n)
drho_n -= np.sum(nmat*self.gamma)*rho_n
for j in range(self.ham.nbath):
Fj = self.ham.sysbath[j] # this is like |j><j|
lamdaj = self.ham.sd[j].lamda
omega_cj = self.ham.sd[j].omega_c
if self.K_truncation == 'Ishizaki-Tanimura':
drho_n -= (1./hbar**2)*(
(2*lamdaj*const.kT/omega_cj - cjk_over_gammajk_re[j])
*utils.commutator(Fj, utils.commutator(Fj, rho_n)) )
if L < self.Lmax:
# If L == self.Lmax, then rho_nkplus = 0
rho_njkplus_sum = np.zeros_like(rho_n)
for k in range(self.Nk):
rho_njkplus_sum += self.rho_plus(rho,n,j,k)
drho_n -= 1j*utils.commutator(Fj, rho_njkplus_sum)
if self.L_truncation == 'TL' and L==self.Lmax:
E, evec = utils.diagonalize(self.ham.sys)
Q_j=np.zeros((self.ham.nsite,self.ham.nsite),dtype=np.complex)
for a in range(self.ham.nsite):
aouter = np.outer(evec[:,a],evec[:,a])
for b in range(self.ham.nsite):
bouter = np.outer(evec[:,b],evec[:,b])
num = self.c[j,:]*(1.-np.exp(-self.gamma[j,:]*t
-1j*self.ham.omega_diff[a,b]*t))
denom = self.gamma[j,:]+1j*self.ham.omega_diff[a,b]
Q_j += np.sum(num/denom)*np.dot(aouter,np.dot(self.ham.sysbath[j],bouter))
rho_njkplus_sum = -(1j/hbar)*(np.dot(Q_j,rho_n)-np.dot(rho_n,np.conj(Q_j).T))
drho_n -= 1j*utils.commutator(Fj, rho_njkplus_sum)
for k in range(self.Nk):
rho_njkminus = self.rho_minus(rho,n,j,k)
drho_n -= (1j/hbar**2)*( nmat[j,k]
*(self.c[j,k]*np.dot(Fj,rho_njkminus)
-self.c[j,k].conjugate()*np.dot(rho_njkminus,Fj)) )
drho.append(drho_n)
elif self.ham.sd[0].sd_type == 'ohmic-exp' or self.ham.sd[0].sd_type == 'cubic-exp':
# Chen Eq. (15)
beta = 1./const.kT
cjk_over_gammajk_re = np.einsum('jk,jk->j', self.c, 1./self.gamma).real
for nab, nabmat in enumerate(self.nmats):
L = np.sum(nabmat)
nmat = nabmat[:,:self.Nk]
amat = nabmat[:,self.Nk:self.Nk+self.nlorentz]
bmat = nabmat[:,self.Nk+self.nlorentz:self.Nk+2*self.nlorentz]
rho_nab = rho[nab]
drho_nab = -1j/hbar * utils.commutator(self.ham.sys,rho_nab)
drho_nab -= np.sum((amat+bmat)*self.Gamma)*rho_nab
drho_nab -= 1j*np.sum((amat-bmat)*self.Omega)*rho_nab
drho_nab -= np.sum(nmat*self.gamma)*rho_nab
for j in range(self.ham.nbath):
Fj = self.ham.sysbath[j] # this is like |j><j|
lamdaj = self.ham.sd[j].lamda
omega_cj = self.ham.sd[j].omega_c
#if self.truncation_type == 'Ishizaki-Tanimura':
# drho_nab -= (1./hbar**2)*(
# (2*lamdaj*const.kT/omega_cj - cjk_over_gammajk_re[j])
# *utils.commutator(Fj, utils.commutator(Fj, rho_nab)) )
if L < self.Lmax:
# If L == self.Lmax, then rho_nabkplus = 0
rho_nabjkplus_sum = np.zeros_like(rho_nab)
for k in range(self.Nk+2*self.nlorentz):
rho_nabjkplus_sum += self.rho_plus(rho,nab,j,k)
drho_nab -= 1j*utils.commutator(Fj, rho_nabjkplus_sum)
for k in range(self.Nk):
rho_nabjkminus = self.rho_minus(rho,nab,j,k)
drho_nab -= (1j/hbar**2)*( nmat[j,k]*self.c[j,k]
*utils.commutator(Fj, rho_nabjkminus) )
for l in range(self.nlorentz):
rho_nabjkminus = self.rho_minus(rho,nab,j,self.Nk+l)
drho_nab -= (1j/hbar)*( amat[j,l]
* self.p[j,l]/(8*self.Omega[j,l]*self.Gamma[j,l])
* (self._coth_b_OminusG[j,l] * utils.commutator(Fj, rho_nabjkminus)
+ utils.anticommutator(Fj, rho_nabjkminus)) )
for l in range(self.nlorentz):
rho_nabjkminus = self.rho_minus(rho,nab,j,self.Nk+self.nlorentz+l)
drho_nab -= (1j/hbar)*( bmat[j,l]
* self.p[j,l]/(8*self.Omega[j,l]*self.Gamma[j,l])
* (self._coth_b_OplusG[j,l] * utils.commutator(Fj, rho_nabjkminus)
- utils.anticommutator(Fj, rho_nabjkminus)) )
drho.append(drho_nab)
return np.array(drho)