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"""
This module provides exact solvers for a system-bath setup using the
hierarchy equations of motion (HEOM).
"""
# Authors: Neill Lambert, Anubhav Vardhan, Alexander Pitchford
# Contact: nwlambert@gmail.com
import timeit
import numpy as np
#from scipy.misc import factorial
import scipy.sparse as sp
import scipy.integrate
from copy import copy
from qutip import Qobj, qeye
from qutip.states import enr_state_dictionaries
from qutip.superoperator import liouvillian, spre, spost
from qutip.cy.spmatfuncs import cy_ode_rhs
from qutip.solver import Options, Result, Stats
from qutip.ui.progressbar import BaseProgressBar, TextProgressBar
from qutip.cy.heom import cy_pad_csr
from qutip.cy.spmath import zcsr_kron
from qutip.fastsparse import fast_csr_matrix, fast_identity
[docs]class HEOMSolver(object):
"""
This is superclass for all solvers that use the HEOM method for
calculating the dynamics evolution. There are many references for this.
A good introduction, and perhaps closest to the notation used here is:
DOI:10.1103/PhysRevLett.104.250401
A more canonical reference, with full derivation is:
DOI: 10.1103/PhysRevA.41.6676
The method can compute open system dynamics without using any Markovian
or rotating wave approximation (RWA) for systems where the bath
correlations can be approximated to a sum of complex eponentials.
The method builds a matrix of linked differential equations, which are
then solved used the same ODE solvers as other qutip solvers (e.g. mesolve)
This class should be treated as abstract. Currently the only subclass
implemented is that for the Drude-Lorentz spectral density. This covers
the majority of the work that has been done using this model, and there
are some performance advantages to assuming this model where it is
appropriate.
There are opportunities to develop a more general spectral density code.
Attributes
----------
H_sys : Qobj
System Hamiltonian
coup_op : Qobj
Operator describing the coupling between system and bath.
coup_strength : float
Coupling strength.
temperature : float
Bath temperature, in units corresponding to planck
N_cut : int
Cutoff parameter for the bath
N_exp : int
Number of exponential terms used to approximate the bath correlation
functions
planck : float
reduced Planck constant
boltzmann : float
Boltzmann's constant
options : :class:`qutip.solver.Options`
Generic solver options.
If set to None the default options will be used
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation.
stats : :class:`qutip.solver.Stats`
optional container for holding performance statitics
If None is set, then statistics are not collected
There may be an overhead in collecting statistics
exp_coeff : list of complex
Coefficients for the exponential series terms
exp_freq : list of complex
Frequencies for the exponential series terms
"""
def __init__(self):
raise NotImplementedError("This is a abstract class only. "
"Use a subclass, for example HSolverDL")
[docs] def reset(self):
"""
Reset any attributes to default values
"""
self.planck = 1.0
self.boltzmann = 1.0
self.H_sys = None
self.coup_op = None
self.coup_strength = 0.0
self.temperature = 1.0
self.N_cut = 10
self.N_exp = 2
self.N_he = 0
self.exp_coeff = None
self.exp_freq = None
self.options = None
self.progress_bar = None
self.stats = None
self.ode = None
self.configured = False
[docs] def create_new_stats(self):
"""
Creates a new stats object suitable for use with this solver
Note: this solver expects the stats object to have sections
config
integrate
"""
stats = Stats(['config', 'run'])
stats.header = "Hierarchy Solver Stats"
return stats
[docs]class HSolverDL(HEOMSolver):
"""
HEOM solver based on the Drude-Lorentz model for spectral density.
Drude-Lorentz bath the correlation functions can be exactly analytically
expressed as an infinite sum of exponentials which depend on the
temperature, these are called the Matsubara terms or Matsubara frequencies
For practical computation purposes an approximation must be used based
on a small number of Matsubara terms (typically < 4).
Attributes
----------
cut_freq : float
Bath spectral density cutoff frequency.
renorm : bool
Apply renormalisation to coupling terms
Can be useful if using SI units for planck and boltzmann
bnd_cut_approx : bool
Use boundary cut off approximation
Can be
"""
def __init__(self, H_sys, coup_op, coup_strength, temperature,
N_cut, N_exp, cut_freq, planck=1.0, boltzmann=1.0,
renorm=True, bnd_cut_approx=True,
options=None, progress_bar=None, stats=None):
self.reset()
if options is None:
self.options = Options()
else:
self.options = options
self.progress_bar = False
if progress_bar is None:
self.progress_bar = BaseProgressBar()
elif progress_bar == True:
self.progress_bar = TextProgressBar()
# the other attributes will be set in the configure method
self.configure(H_sys, coup_op, coup_strength, temperature,
N_cut, N_exp, cut_freq, planck=planck, boltzmann=boltzmann,
renorm=renorm, bnd_cut_approx=bnd_cut_approx, stats=stats)
[docs] def reset(self):
"""
Reset any attributes to default values
"""
HEOMSolver.reset(self)
self.cut_freq = 1.0
self.renorm = False
self.bnd_cut_approx = False
[docs] def run(self, rho0, tlist):
"""
Function to solve for an open quantum system using the
HEOM model.
Parameters
----------
rho0 : Qobj
Initial state (density matrix) of the system.
tlist : list
Time over which system evolves.
Returns
-------
results : :class:`qutip.solver.Result`
Object storing all results from the simulation.
"""
start_run = timeit.default_timer()
sup_dim = self._sup_dim
stats = self.stats
r = self._ode
if not self._configured:
raise RuntimeError("Solver must be configured before it is run")
if stats:
ss_conf = stats.sections.get('config')
if ss_conf is None:
raise RuntimeError("No config section for solver stats")
ss_run = stats.sections.get('run')
if ss_run is None:
ss_run = stats.add_section('run')
# Set up terms of the matsubara and tanimura boundaries
output = Result()
output.solver = "hsolve"
output.times = tlist
output.states = []
if stats: start_init = timeit.default_timer()
output.states.append(Qobj(rho0))
rho0_flat = rho0.full().ravel('F') # Using 'F' effectively transposes
rho0_he = np.zeros([sup_dim*self._N_he], dtype=complex)
rho0_he[:sup_dim] = rho0_flat
r.set_initial_value(rho0_he, tlist[0])
if stats:
stats.add_timing('initialize',
timeit.default_timer() - start_init, ss_run)
start_integ = timeit.default_timer()
dt = np.diff(tlist)
n_tsteps = len(tlist)
for t_idx, t in enumerate(tlist):
if t_idx < n_tsteps - 1:
r.integrate(r.t + dt[t_idx])
rho = Qobj(r.y[:sup_dim].reshape(rho0.shape), dims=rho0.dims)
output.states.append(rho)
if stats:
time_now = timeit.default_timer()
stats.add_timing('integrate',
time_now - start_integ, ss_run)
if ss_run.total_time is None:
ss_run.total_time = time_now - start_run
else:
ss_run.total_time += time_now - start_run
stats.total_time = ss_conf.total_time + ss_run.total_time
return output
def _calc_matsubara_params(self):
"""
Calculate the Matsubara coefficents and frequencies
Returns
-------
c, nu: both list(float)
"""
c = []
nu = []
lam0 = self.coup_strength
gam = self.cut_freq
hbar = self.planck
beta = 1.0/(self.boltzmann*self.temperature)
N_m = self.N_exp
g = 2*np.pi / (beta*hbar)
for k in range(N_m):
if k == 0:
nu.append(gam)
c.append(lam0*gam*
(1.0/np.tan(gam*hbar*beta/2.0) - 1j) / hbar)
else:
nu.append(k*g)
c.append(4*lam0*gam*nu[k] /
((nu[k]**2 - gam**2)*beta*hbar**2))
self.exp_coeff = c
self.exp_freq = nu
return c, nu
def _calc_renorm_factors(self):
"""
Calculate the renormalisation factors
Returns
-------
norm_plus, norm_minus : array[N_c, N_m] of float
"""
c = self.exp_coeff
N_m = self.N_exp
N_c = self.N_cut
norm_plus = np.empty((N_c+1, N_m))
norm_minus = np.empty((N_c+1, N_m))
for k in range(N_m):
for n in range(N_c+1):
norm_plus[n, k] = np.sqrt(abs(c[k])*(n + 1))
norm_minus[n, k] = np.sqrt(float(n)/abs(c[k]))
return norm_plus, norm_minus
def _pad_csr(A, row_scale, col_scale, insertrow=0, insertcol=0):
"""
Expand the input csr_matrix to a greater space as given by the scale.
Effectively inserting A into a larger matrix
zeros([A.shape[0]*row_scale, A.shape[1]*col_scale]
at the position [A.shape[0]*insertrow, A.shape[1]*insertcol]
The same could be achieved through using a kron with a matrix with
one element set to 1. However, this is more efficient
"""
# ajgpitch 2016-03-08:
# Clearly this is a very simple operation in dense matrices
# It seems strange that there is nothing equivalent in sparse however,
# after much searching most threads suggest directly addressing
# the underlying arrays, as done here.
# This certainly proved more efficient than other methods such as stacking
#TODO: Perhaps cythonize and move to spmatfuncs
if not isinstance(A, sp.csr_matrix):
raise TypeError("First parameter must be a csr matrix")
nrowin = A.shape[0]
ncolin = A.shape[1]
nrowout = nrowin*row_scale
ncolout = ncolin*col_scale
A._shape = (nrowout, ncolout)
if insertcol == 0:
pass
elif insertcol > 0 and insertcol < col_scale:
A.indices = A.indices + insertcol*ncolin
else:
raise ValueError("insertcol must be >= 0 and < col_scale")
if insertrow == 0:
A.indptr = np.concatenate((A.indptr,
np.array([A.indptr[-1]]*(row_scale-1)*nrowin)))
elif insertrow == row_scale-1:
A.indptr = np.concatenate((np.array([0]*(row_scale - 1)*nrowin),
A.indptr))
elif insertrow > 0 and insertrow < row_scale - 1:
A.indptr = np.concatenate((np.array([0]*insertrow*nrowin), A.indptr,
np.array([A.indptr[-1]]*(row_scale - insertrow - 1)*nrowin)))
else:
raise ValueError("insertrow must be >= 0 and < row_scale")
return A