%pylab inline
Populating the interactive namespace from numpy and matplotlib
from qutip import *
from numpy import log2, cos, sin
from scipy.integrate import odeint
from qutip.cy.spmatfuncs import cy_expect_rho_vec, spmv
th = 0.1 # Interaction parameter
alpha = cos(th)
beta = sin(th)
gamma = 1
# Exact steady state solution for Vc
Vc = (alpha*beta - gamma + sqrt((gamma-alpha*beta)**2 + 4*gamma*alpha**2))/(4*alpha**2)
#********* Model ************
NN = 200
tlist = linspace(0,50,NN)
Nsub = 200
N = 20
Id = qeye(N)
a = destroy(N)
s = 0.5*((alpha+beta)*a + (alpha-beta)*a.dag())
x = (a + a.dag())/sqrt(2)
H = Id
c_op = [sqrt(gamma)*a]
sc_op = [s]
e_op = [x, x*x]
rho0 = fock_dm(N,0) #initial vacuum state
# Solution of the differential equation for the variance Vc
y0 = 0.5
def func(y, t):
return -(gamma - alpha*beta)*y - 2*alpha*alpha*y*y + 0.5*gamma
y = odeint(func, y0, tlist)
# list solver
help(stochastic_solvers)
Help on function stochastic_solvers in module qutip.stochastic: stochastic_solvers() Available solvers for ssesolve and smesolve euler-maruyama: A simple generalization of the Euler method for ordinary differential equations to stochastic differential equations. Only solver which could take non-commuting sc_ops. *not tested* -Order 0.5 -Code: 'euler-maruyama', 'euler', 0.5 milstein, Order 1.0 strong Taylor scheme: Better approximate numerical solution to stochastic differential equations. -Order strong 1.0 -Code: 'milstein', 1.0 Numerical Solution of Stochastic Differential Equations Chapter 10.3 Eq. (3.1), By Peter E. Kloeden, Eckhard Platen milstein-imp, Order 1.0 implicit strong Taylor scheme: Implicit milstein scheme for the numerical simulation of stiff stochastic differential equations. -Order strong 1.0 -Code: 'milstein-imp' Numerical Solution of Stochastic Differential Equations Chapter 12.2 Eq. (2.9), By Peter E. Kloeden, Eckhard Platen predictor-corrector: Generalization of the trapezoidal method to stochastic differential equations. More stable than explicit methods. -Order strong 0.5, weak 1.0 Only the stochastic part is corrected. (alpha = 0, eta = 1/2) -Code: 'pred-corr', 'predictor-corrector', 'pc-euler' Both the deterministic and stochastic part corrected. (alpha = 1/2, eta = 1/2) -Code: 'pc-euler-imp', 'pc-euler-2', 'pred-corr-2' Numerical Solution of Stochastic Differential Equations Chapter 15.5 Eq. (5.4), By Peter E. Kloeden, Eckhard Platen platen: Explicit scheme, create the milstein using finite difference instead of derivatives. Also contain some higher order terms, thus converge better than milstein while staying strong order 1.0. Do not require derivatives, therefore usable for :func:`qutip.stochastic.general_stochastic` -Order strong 1.0, weak 2.0 -Code: 'platen', 'platen1', 'explicit1' The Theory of Open Quantum Systems Chapter 7 Eq. (7.47), H.-P Breuer, F. Petruccione rouchon: Scheme keeping the positivity of the density matrix. (smesolve only) -Order strong 1.0? -Code: 'rouchon', 'Rouchon' Efficient Quantum Filtering for Quantum Feedback Control Pierre Rouchon, Jason F. Ralph arXiv:1410.5345 [quant-ph] taylor1.5, Order 1.5 strong Taylor scheme: Solver with more terms of the Ito-Taylor expansion. Default solver for smesolve and ssesolve. -Order strong 1.5 -Code: 'taylor1.5', 'taylor15', 1.5, None Numerical Solution of Stochastic Differential Equations Chapter 10.4 Eq. (4.6), By Peter E. Kloeden, Eckhard Platen taylor1.5-imp, Order 1.5 implicit strong Taylor scheme: implicit Taylor 1.5 (alpha = 1/2, beta = doesn't matter) -Order strong 1.5 -Code: 'taylor1.5-imp', 'taylor15-imp' Numerical Solution of Stochastic Differential Equations Chapter 12.2 Eq. (2.18), By Peter E. Kloeden, Eckhard Platen explicit1.5, Explicit Order 1.5 Strong Schemes: Reproduce the order 1.5 strong Taylor scheme using finite difference instead of derivatives. Slower than taylor15 but usable by :func:`qutip.stochastic.general_stochastic` -Order strong 1.5 -Code: 'explicit1.5', 'explicit15', 'platen15' Numerical Solution of Stochastic Differential Equations Chapter 11.2 Eq. (2.13), By Peter E. Kloeden, Eckhard Platen taylor2.0, Order 2 strong Taylor scheme: Solver with more terms of the Stratonovich expansion. -Order strong 2.0 -Code: 'taylor2.0', 'taylor20', 2.0 Numerical Solution of Stochastic Differential Equations Chapter 10.5 Eq. (5.2), By Peter E. Kloeden, Eckhard Platen ---All solvers, except taylor2.0, are usable in both smesolve and ssesolve and for both heterodyne and homodyne. taylor2.0 only work for 1 stochastic operator not dependent of time with the homodyne method. The :func:`qutip.stochastic.general_stochastic` only accept derivatives free solvers: ['euler', 'platen', 'explicit1.5']. Available solver for photocurrent_sesolve and photocurrent_mesolve: Photocurrent use ordinary differential equations between stochastic "jump/collapse". euler: Euler method for ordinary differential equations. Default solver -Order 1.0 -Code: 'euler' predictor–corrector: predictor–corrector method (PECE) for ordinary differential equations. -Order 2.0 -Code: 'pred-corr'
# Euler-Maruyama
sol_euler = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='euler-maruyama',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 1.09s
# Milstein
sol_mil = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='milstein',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 1.55s
# predictor-corrector
sol_pc_euler = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='pc-euler',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 2.42s
# predictor-corrector-2
sol_pc_euler2 = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='pred-corr-2',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 3.77s
# Default: taylor1.5
sol_taylor15 = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='taylor1.5',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 7.01s
# Taylor 2.0
sol_taylor20 = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='taylor2.0',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 8.04s
plot(tlist,sol_euler.expect[1] - abs(sol_euler.expect[0])**2, label='Euler')
plot(tlist,sol_mil.expect[1] - abs(sol_mil.expect[0])**2, label='Milstein')
plot(tlist,sol_pc_euler.expect[1] - abs(sol_pc_euler.expect[0])**2, label='pc-euler')
plot(tlist,sol_pc_euler2.expect[1] - abs(sol_pc_euler2.expect[0])**2, label='pc-euler-2')
plot(tlist,sol_taylor15.expect[1] - abs(sol_taylor15.expect[0])**2, label='taylor1.5')
plot(tlist,sol_taylor20.expect[1] - abs(sol_taylor20.expect[0])**2, label='taylor2.0')
plot(tlist,Vc*ones(NN),"k:", label='exact steady state solution')
plot(tlist,y,"k", label="exact solution")
legend();
show()
#plot(tlist,sol_euler.expect[1] - abs(sol_euler.expect[0])**2, label='Euler')
plot(tlist,sol_mil.expect[1] - abs(sol_mil.expect[0])**2, label='Milstein')
plot(tlist,sol_pc_euler.expect[1] - abs(sol_pc_euler.expect[0])**2, label='pc-euler')
plot(tlist,sol_pc_euler2.expect[1] - abs(sol_pc_euler2.expect[0])**2, label='pc-euler-2')
plot(tlist,sol_taylor15.expect[1] - abs(sol_taylor15.expect[0])**2, label='taylor1.5')
plot(tlist,sol_taylor20.expect[1] - abs(sol_taylor20.expect[0])**2, label='taylor2.0')
plot(tlist,Vc*ones(NN),"k:", label='exact steady state solution')
plot(tlist,y,"k", label="exact solution")
legend();
xlim([0,25])
ylim([0.3238,0.325])
show()
#plot(tlist,sol_euler.expect[1] - abs(sol_euler.expect[0])**2, label='Euler')
#plot(tlist,sol_mil.expect[1] - abs(sol_mil.expect[0])**2, label='Milstein')
#plot(tlist,sol_pc_euler.expect[1] - abs(sol_pc_euler.expect[0])**2, label='pc-euler')
#plot(tlist,sol_pc_euler2.expect[1] - abs(sol_pc_euler2.expect[0])**2, label='pc-euler-2')
plot(tlist,sol_taylor15.expect[1] - abs(sol_taylor15.expect[0])**2, label='taylor1.5')
plot(tlist,sol_taylor20.expect[1] - abs(sol_taylor20.expect[0])**2, label='taylor2.0')
plot(tlist,Vc*ones(NN),"k:", label='exact steady state solution')
plot(tlist,y,"k", label="exact solution")
legend();
xlim([0,25])
ylim([0.3241,0.32418])
show()
th = 0.1
alpha = cos(th)
beta = sin(th)
gamma = 1
eps = 0.3
VcEps = ((1-2*eps)*alpha*beta - gamma + \
sqrt((gamma-alpha*beta)**2 + 4*gamma*alpha*((1-eps)*alpha + eps*beta)))/(4*(1-eps)*alpha**2)
UcEps = (-(1-2*eps)*alpha*beta - gamma + \
sqrt((gamma-alpha*beta)**2 + 4*eps*beta*gamma*(beta-alpha)))/(4*eps*beta**2)
NN = 200
tlist = linspace(0,3,NN)
Nsub = 200
N = 20
Id = qeye(N)
a = destroy(N)
s = 0.5*((alpha+beta)*a + (alpha-beta)*a.dag())
x = (a + a.dag())/sqrt(2)
H = Id
c_op = [sqrt(gamma)*a]
sc_op = [sqrt(1-eps)*s, sqrt(eps)*1j*s]
e_op = [x, x*x]
rho0 = fock_dm(N,0)
y0 = 0.5
def func(y, t):
return -(gamma - (1-2*eps)*alpha*beta)*y - 2*(1-eps)*alpha*alpha*y*y + 0.5*(gamma + eps*beta*beta)
y = odeint(func, y0, tlist)
def funcZ(z, t):
return -(gamma + (1-2*eps)*alpha*beta)*z - 2*eps*beta*beta*z*z + 0.5*(gamma + (1-eps)*alpha*alpha)
z = odeint(funcZ, y0, tlist)
# Euler-Maruyama
sol_euler = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='euler-maruyama',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 1.49s
# Milstein
sol_mil = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='milstein',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 3.05s
# predictor-corrector
sol_pc_euler = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='pc-euler',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 4.21s
# predictor-corrector-2
sol_pc_euler2 = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='pred-corr-2',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 7.26s
# Default: taylor1.5
sol_taylor15 = smesolve(H, rho0, tlist, c_op, sc_op, e_op,
nsubsteps=Nsub, method='homodyne', solver='taylor1.5',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 19.44s
plot(tlist,sol_euler.expect[1] - abs(sol_euler.expect[0])**2, label='Euler')
plot(tlist,sol_mil.expect[1] - abs(sol_mil.expect[0])**2, label='Milstein')
plot(tlist,sol_pc_euler.expect[1] - abs(sol_pc_euler.expect[0])**2, label='pc-euler')
plot(tlist,sol_pc_euler2.expect[1] - abs(sol_pc_euler2.expect[0])**2, label='pc-euler-2')
plot(tlist,sol_taylor15.expect[1] - abs(sol_taylor15.expect[0])**2, label='taylor1.5')
plot(tlist,Vc*ones(NN), label='exact steady state solution')
plot(tlist,y, label="exact solution")
legend();
show()
plot(tlist,sol_euler.expect[1] - abs(sol_euler.expect[0])**2, label='Euler')
plot(tlist,sol_mil.expect[1] - abs(sol_mil.expect[0])**2, label='Milstein')
plot(tlist,sol_pc_euler.expect[1] - abs(sol_pc_euler.expect[0])**2, label='pc-euler')
plot(tlist,sol_pc_euler2.expect[1] - abs(sol_pc_euler2.expect[0])**2, label='pc-euler-2')
plot(tlist,sol_taylor15.expect[1] - abs(sol_taylor15.expect[0])**2, label='taylor1.5')
plot(tlist,Vc*ones(NN), label='exact steady state solution')
plot(tlist,y, label="exact solution")
legend();
xlim([1.5,3.0])
ylim([0.347,0.356])
(0.347, 0.356)
plot(tlist,sol_euler.expect[1] - abs(sol_euler.expect[0])**2-y.transpose()[0], label='Euler')
plot(tlist,sol_mil.expect[1] - abs(sol_mil.expect[0])**2-y.transpose()[0], label='Milstein')
plot(tlist,sol_pc_euler.expect[1] - abs(sol_pc_euler.expect[0])**2-y.transpose()[0], label='pc-euler')
plot(tlist,sol_pc_euler2.expect[1] - abs(sol_pc_euler2.expect[0])**2-y.transpose()[0], label='pc-euler-2')
plot(tlist,sol_taylor15.expect[1] - abs(sol_taylor15.expect[0])**2-y.transpose()[0], label='taylor1.5')
legend()
show()
plot(tlist,sol_mil.expect[1] - abs(sol_mil.expect[0])**2-y.transpose()[0], label='Milstein')
plot(tlist,sol_pc_euler.expect[1] - abs(sol_pc_euler.expect[0])**2-y.transpose()[0], label='pc-euler')
plot(tlist,sol_pc_euler2.expect[1] - abs(sol_pc_euler2.expect[0])**2-y.transpose()[0], label='pc-euler-2')
plot(tlist,sol_taylor15.expect[1] - abs(sol_taylor15.expect[0])**2-y.transpose()[0], label='taylor1.5')
legend()
show()
sc_ops_multiple = [sc_op[0]*0.5**0.5, sc_op[0]*0.5**0.5]
#Euler-Maruyama
sol_eul = smesolve(H, rho0, tlist, c_op, sc_ops_multiple, e_op,
nsubsteps=Nsub, method='homodyne', solver='euler',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 1.74s
#Milstein
sol_milstein = smesolve(H, rho0, tlist, c_op, sc_ops_multiple, e_op,
nsubsteps=Nsub, method='homodyne', solver='milstein',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 3.56s
#taylor1.5
sol_taylor = smesolve(H, rho0, tlist, c_op, sc_ops_multiple, e_op,
nsubsteps=Nsub, method='homodyne', solver='taylor1.5',
options=Odeoptions(store_states=True, average_states=False))
Total run time: 14.48s
plot(tlist,sol_eul.expect[1] - abs(sol_eul.expect[0])**2, label='Euler')
plot(tlist,sol_milstein.expect[1] - abs(sol_milstein.expect[0])**2, label='Milstein')
plot(tlist,sol_taylor.expect[1] - abs(sol_taylor.expect[0])**2, label='taylor1.5')
plot(tlist,Vc*ones(NN), label='exact steady state solution')
plot(tlist,y, label="exact solution")
legend()
show()
from qutip.ipynbtools import version_table
version_table()
Software | Version |
---|---|
QuTiP | 4.4.0.dev0+21a52e95 |
Numpy | 1.16.0 |
SciPy | 1.2.0 |
matplotlib | 3.0.2 |
Cython | 0.29.3 |
Number of CPUs | 2 |
BLAS Info | OPENBLAS |
IPython | 7.2.0 |
Python | 3.6.7 (default, Oct 22 2018, 11:32:17) [GCC 8.2.0] |
OS | posix [linux] |
Fri Feb 01 13:36:16 2019 JST |