/
util.py
637 lines (562 loc) · 24.2 KB
/
util.py
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from qutip import qeye, create, destroy, basis, tensor, fock, mesolve, expect, Options
from qutip.parallel import parallel_map
from numpy import sqrt, pi, linspace, arange, save, array, copy, exp, concatenate, flip, transpose
from numpy.linalg import eig, solve
import matplotlib.pyplot as plt
from time import strftime, localtime
from os import makedirs
from os.path import exists, abspath
from functools import partial
from copy import deepcopy
from typing import List
def mkdir(path: str):
if not exists(path): # determine the existance of folder
makedirs(path) # makedirs create folder if the path does not exist
print("--- create new folder... ---")
print("--- path:", path, "---")
else:
print("--- There is a folder! ---")
class molSystem(object):
def __init__(
self,
duration: int,
spacing: int,
cooperativity,
kappa_i,
kappa_e,
gamma=12,
delta_al=0,
delta_cl=0,
f_fc=0.37,
P_in=1,
N_p=3,
N_s=3,
N_m=1,
save=True,
save_path="../data/" + strftime("%Y%m%d_%H%M%S", localtime()) + "/",
):
"""
Unit of kappa_i, kappa_e, gamma, P_in: MHz
"""
self.duration = duration
self.spacing = spacing
self.tlist = arange(0, self.duration, self.spacing)
self.C = cooperativity # Cooperativity (Note: not take into account Franck-Condon factor)
self.kappa_i = 2 * pi * kappa_i # Unit: Mega rad/s, intrinsic resonator loss rate
self.kappa_e = (
2 * pi * kappa_e
) # Unit: Mega rad/s, coupling rate between waveguide and resonator
self.kappa = self.kappa_i + self.kappa_e # Unit:Mega rad/s cavity photon decay rate
self.gamma = 2 * pi * gamma # Unit: Mega rad/s excited molecule spontaneous decay rate
self.g = sqrt(self.C * self.gamma * self.kappa) # Unit: MHz
self.delta_al = (
2 * pi * delta_al
) # difference between atomic transition and light frequency
self.delta_cl = 2 * pi * delta_cl # difference between cavity resonance and light frequency
self.f_fc = f_fc # Franck Condon factor
self.gamma_g = self.gamma * self.f_fc # decay rate to vibrational ground state
self.gamma_s = self.gamma * (1 - self.f_fc) # decay rate to higher vibrational states
self.P_in = P_in # Unit: MHz Photon number generation rate at input side of waveguide
self.E = 1j * sqrt(2 * self.kappa_e * self.P_in) # Intracavity field
self.Omega = -2 * sqrt(2 * self.kappa_e * self.P_in) # Rabi frequency
self.N_p = N_p # maximum photon number in Fock space
self.N_s = N_s # Number of molecule states: |0>: |g>; |1>:|e>; |2>:|s>
self.N_m = N_m # number of molecules
self._path = save_path # saving path of data
self._result = False # flag of simulation
self._save = save
if self._save == True:
mkdir(self._path)
### numerical simulation ###
def _init_simulation(self):
"""
create the Hamiltonian, initial state and collapse operator of the molecule-cavity system
"""
# define initial state
self.psi0 = tensor(
[fock(self.N_p, 0), fock(self.N_p, 0)] + [basis(self.N_s, 0) for _ in range(self.N_m)]
) # CW mode + CCW mode + molecular internal state
# ----------------define operators-------------------
# define identity operator for all molecular internal state
iden = [qeye(self.N_s) for m in range(self.N_m)] # identity operator for all molecules
# ----------------Hamiltonian-----------------------
# annihilation operator of whispering gallery mode
self.aCW = tensor([destroy(self.N_p), qeye(self.N_p)] + iden)
self.aCCW = tensor([qeye(self.N_p), destroy(self.N_p)] + iden)
H0 = (
self.delta_cl * self.aCW.dag() * self.aCW
+ self.delta_cl * self.aCCW.dag() * self.aCCW
+ 1j * (self.E * self.aCW.dag() - self.E.conjugate() * self.aCW)
)
# excited state energy
temp = []
for m in range(self.N_m):
op_temp = iden[:]
op_temp[m] = self.delta_al * (basis(self.N_s, 1) * basis(self.N_s, 1).dag())
temp.append(op_temp)
H1 = sum([tensor([qeye(self.N_p), qeye(self.N_p)] + temp[m]) for m in range(self.N_m)])
# Coupled to cavity
temp = []
for m in range(self.N_m):
op_temp = iden[:]
op_temp[m] = basis(self.N_s, 0) * basis(self.N_s, 1).dag()
temp.append(op_temp)
op = (
1j
* self.g
* sum([tensor([create(self.N_p), qeye(self.N_p)] + temp[m]) for m in range(self.N_m)])
)
op += (
1j
* self.g
* sum([tensor([qeye(self.N_p), create(self.N_p)] + temp[m]) for m in range(self.N_m)])
)
H1 += op
temp = []
for m in range(self.N_m):
op_temp = iden[:]
op_temp[m] = basis(self.N_s, 1) * basis(self.N_s, 0).dag()
temp.append(op_temp)
op = (
-1j
* self.g
* sum([tensor([destroy(self.N_p), qeye(self.N_p)] + temp[m]) for m in range(self.N_m)])
)
op += (
-1j
* self.g
* sum([tensor([qeye(self.N_p), destroy(self.N_p)] + temp[m]) for m in range(self.N_m)])
)
H1 += op
# total Hamiltionian
self.H = H0 + H1
# -----------------collapse operators--------------
self.c_ops = []
if self.kappa > 0:
self.c_ops.append(sqrt(2 * self.kappa) * self.aCW)
self.c_ops.append(sqrt(2 * self.kappa) * self.aCCW)
# spontaneous decay to higher vibrational state molecule
if self.gamma_s > 0:
for m in range(self.N_m):
op_temp = iden[:]
op_temp[m] = basis(self.N_s, 2) * basis(self.N_s, 1).dag()
self.c_ops.append(
sqrt(self.gamma_s) * tensor([qeye(self.N_p), qeye(self.N_p)] + op_temp)
)
# spontaneous decay to vibrational ground state molecule
if self.gamma_g > 0:
for m in range(self.N_m):
op_temp = iden[:]
op_temp[m] = basis(self.N_s, 0) * basis(self.N_s, 1).dag()
self.c_ops.append(
sqrt(self.gamma_g) * tensor([qeye(self.N_p), qeye(self.N_p)] + op_temp)
)
def simulation(self, show_progress=None, save_info=True, save_data=False):
"""
Numerically calculate the evolution of system via Lindblad master equation
The unit of duration, spacing: us
show_progress (None/True): show progress bar for the master equation calculation
if self._save = True
save_info: record system information in a txt file
save_data: save the simulation results in npy files
"""
if self._result == True:
print(
"The system has been simulated. Create a new folder to save data if self._save = True."
)
self._path = ("../data/" + strftime("%Y%m%d_%H%M%S", localtime()) + "/",)
if self._save == True:
mkdir(self._path)
if (save_info and self._save) == True:
# write setting to a txt file
try:
para = open(self._path + "parameters.txt", "a")
para.write(
" P_in = %f MHz\n C = %f\n g = %fMHz\n kappa_i=%fMHz\n kappa_e=%fMHz\n gamma = %fMHz\n duration = %fus\n sampling point number = %d\n f_fc = %f\n N_m = %d\n N_p = %d\n file path: %s\n save path: %s"
% (
self.P_in,
self.C,
self.g / (2 * pi),
self.kappa_i / (2 * pi),
self.kappa_e / (2 * pi),
self.gamma / (2 * pi),
self.duration,
self.spacing,
self.f_fc,
self.N_m,
self.N_p,
abspath(__file__),
self._path,
)
)
except IOError:
print("File error")
finally:
para.close()
self._init_simulation()
# -----------------projection operators------------------
# define identity operator for all molecular internal state
iden = [qeye(self.N_s) for m in range(self.N_m)] # identity operator for all molecules
# (i-1)th element in operator list is the projection operator for i-th molecule
# ground state
temp = []
for m in range(self.N_m):
op_temp = iden[:]
op_temp[m] = basis(self.N_s, 0).proj()
temp.append(op_temp)
s1_proj = [tensor([qeye(self.N_p), qeye(self.N_p)] + temp[m]) for m in range(self.N_m)]
# excited state
temp = []
for m in range(self.N_m):
op_temp = iden[:]
op_temp[m] = basis(self.N_s, 1).proj()
temp.append(op_temp)
s2_proj = [tensor([qeye(self.N_p), qeye(self.N_p)] + temp[m]) for m in range(self.N_m)]
# other states
temp = []
for m in range(self.N_m):
op_temp = iden[:]
op_temp[m] = basis(self.N_s, 2).proj()
temp.append(op_temp)
s3_proj = [tensor([qeye(self.N_p), qeye(self.N_p)] + temp[m]) for m in range(self.N_m)]
exp_list = []
exp_list.append(s1_proj[0]) # ground state (first molecule)
exp_list.append(s2_proj[0]) # excited state (first molecule)
exp_list.append(s3_proj[0]) # other states (first molecule)
exp_list.append(
self.aCW.dag() * self.aCW + self.aCCW.dag() * self.aCCW
) # photon population in cavity
exp_list.append(self.aCW) # expectation value of aCW
exp_list.append(self.aCCW) # expectation value of aCCW
exp_list.append(self.aCW.dag() * self.aCW) # CW mode photon population in cavity
exp_list.append(self.aCCW.dag() * self.aCCW) # CCW mode photon population in cavity
# ------------------calcualte master equation-----------------
# set absolute tolerance for solver. Check qutip.solver.Options
# options = Options(atol=1e-14, nsteps=10000)
output = mesolve(
self.H,
self.psi0,
self.tlist,
self.c_ops,
exp_list,
# options=options,
progress_bar=show_progress,
)
self.gState = output.expect[0]
self.eState = output.expect[1]
self.sState = output.expect[2]
self.cavity_photon = output.expect[3]
self.exp_aCW = output.expect[4]
self.exp_aCCW = output.expect[5]
if (save_data and self._save) == True:
save(self._path + "tlist.npy", self.tlist)
save(self._path + "gState.npy", output.expect[0])
save(self._path + "eState.npy", output.expect[1])
save(self._path + "sState.npy", output.expect[2])
save(self._path + "cavity_photon.npy", output.expect[3])
save(self._path + "aCW.npy", output.expect[4])
save(self._path + "aCCW.npy", output.expect[5])
self._result = True
### theoretical derivation ###
"""
Calculate auxilary functions for theoretical derivation
"""
def _init_theory(self):
"""
output:
N_m + 1 dimensional array: population ratio of expectation of aCW in each manifold
"""
g_list = array([self.g * sqrt(i) for i in range(1, self.N_m + 1)])
# write a program for excited state population
pop31n = list(
-(self.Omega / 2 * 1j)
* (
(g_list ** 2)
+ (1j * self.delta_cl + self.kappa) * (1j * self.delta_al + self.gamma / 2)
)
/ (
2 * (g_list ** 2)
+ (1j * self.delta_cl + self.kappa) * (1j * self.delta_al + self.gamma / 2)
)
/ (1j * self.delta_cl + self.kappa)
) # proportion coefficient between excited state and ground state for n>0
pop0 = [(-1j * self.Omega / 2 / (1j * self.delta_cl + self.kappa))]
# proportion coefficient between excited state and ground state for n=0
self.pop31 = (
pop0 + pop31n
) # N_m + 1 dimensional array: population ratio of expectation of aCW in each manifold
# print("g list:", g_list)
# print("excited state coefficient:", pop)
self.pop41 = list(
-(self.Omega / 2 * 1j)
* (-(g_list ** 2))
/ (
2 * (g_list ** 2)
+ (1j * self.delta_cl + self.kappa) * (1j * self.delta_al + self.gamma / 2)
)
/ (1j * self.delta_cl + self.kappa)
) # N_m dimensional array: population ratio of expectation of aCCW in each manifold (N>0)
"""
calculate the population distribution in n-molecule manifolds.
return: pop_dynamics = [rho0(t), rho1(t), rho2(t), ... , rhoN(t) ] N+1 dimension array
"""
def decayRate(N: int):
"""
N: the number of molecules coupled to cavity
Return decay rate of N-molecule manifold
"""
g_eff = sqrt(N) * self.g
return ((g_eff ** 2) * (self.Omega ** 2) * self.gamma_s) / (
16 * (g_eff ** 4)
+ 8 * (g_eff ** 2) * (-2 * self.delta_al * self.delta_cl + self.gamma * self.kappa)
+ ((self.gamma ** 2) + 4 * (self.delta_al ** 2))
* ((self.delta_cl ** 2) + (self.kappa ** 2))
)
dimension = self.N_m + 1
decayRateList = [decayRate(i) for i in range(1, dimension)]
matrix = []
temp = [0 for _ in range(dimension)]
temp[1] = decayRateList[0]
matrix.append(temp)
for i in range(1, self.N_m):
temp = [0 for _ in range(dimension)]
temp[i] = -decayRateList[i - 1]
temp[i + 1] = decayRateList[i]
matrix.append(temp)
temp = [0 for _ in range(dimension)]
temp[self.N_m] = -decayRateList[self.N_m - 1]
matrix.append(temp)
matrix = array(matrix)
w, v = eig(matrix)
# print("eigenvalues:", w)
# print("eigenvectors:\n", v)
initial_state = array([0 for _ in range(dimension)])
initial_state[-1] = 1
coeffient = solve(v, initial_state)
# print("coefficient:\n", coeffient)
func_coff = copy(v)
for i in range(dimension):
func_coff[:, i] *= coeffient[i]
# print("func_coff:\n", func_coff
self.pop_dynamics = [
sum([exp(w[i] * self.tlist) * coff[i] for i in range(len(w))]) for coff in func_coff
]
### calculate transmitted and reflected photons
def photon_simulated(self, save_data=False):
if not self._result:
print("Error: have not simulated system yet.")
return
# calculate transmission and reflection
self.tran = (
abs(1 + 1j * sqrt(2 * self.kappa_e / self.P_in) * self.exp_aCW) ** 2
) # real-time transmission
self.refl = (
abs(1j * sqrt(2 * self.kappa_e / self.P_in) * self.exp_aCCW) ** 2
) # real-time reflection
# calculate photon number
time_step = self.tlist[1] - self.tlist[0]
self.tran_photon_nums = []
self.refl_photon_nums = []
p_num = 0
for transmission in self.tran:
p_num += transmission * time_step * self.P_in
# transmission photon number at specific time
self.tran_photon_nums.append(p_num)
p_num = 0
for reflection in self.refl:
p_num += reflection * time_step * self.P_in
# reflection photon number at specific time
self.refl_photon_nums.append(p_num)
if self._save and save_data:
save(self._path + "tran.npy", self.tran)
save(self._path + "refl.npy", self.refl)
save(self._path + "tran_photon_nums.npy", self.tran_photon_nums)
save(self._path + "refl_photon_nums.npy", self.refl_photon_nums)
def photon_theory(self, save_data=False):
self._init_theory()
e_pop = self.pop31
g_pop_dynamics = self.pop_dynamics
# calculate transmission and reflection
self.exp_aCW_theory = sum([e_pop[i] * g_pop_dynamics[i] for i in range(self.N_m + 1)])
self.tran_theory = (
abs(1 + 1j * sqrt(2 * self.kappa_e / self.P_in) * self.exp_aCW_theory) ** 2
)
e_pop = self.pop41
self.exp_aCCW_theory = sum([e_pop[i] * g_pop_dynamics[i + 1] for i in range(self.N_m)])
self.refl_theory = abs(1j * sqrt(2 * self.kappa_e / self.P_in) * self.exp_aCCW_theory) ** 2
# calculate photon number
time_step = self.tlist[1] - self.tlist[0]
self.tran_photon_nums_theory = []
self.refl_photon_nums_theory = []
p_num = 0
for transmission in self.tran_theory:
p_num += transmission * time_step * self.P_in
# transmission photon number at specific time
self.tran_photon_nums_theory.append(p_num)
p_num = 0
for reflection in self.refl_theory:
p_num += reflection * time_step * self.P_in
# reflection photon number at specific time
self.refl_photon_nums_theory.append(p_num)
if self._save and save_data:
save(self._path + "tran_theory.npy", self.tran_theory)
save(self._path + "refl_theory.npy", self.refl_theory)
save(self._path + "tran_photon_nums_theory.npy", self.tran_photon_nums_theory)
save(self._path + "refl_photon_nums_theory.npy", self.refl_photon_nums_theory)
# -------------------------drawing-------------------------------------
def draw_population(self):
if not self._result:
print("Error: have not simulated system yet.")
return
fig, axes = plt.subplots(1, 1, figsize=(10, 8))
axes.plot(self.tlist, self.gState, color="k", label=r"$|g\rangle$", linewidth=5)
axes.plot(self.tlist, self.eState, color="b", label=r"$|e\rangle$", linewidth=5)
axes.plot(self.tlist, self.sState, color="r", label=r"$|s\rangle$", linewidth=5)
axes.legend(loc=6, fontsize=20)
# axes.set_xscale("log")
axes.set_xlabel("Time/us", fontsize=28)
axes.set_ylabel("Population", fontsize=28)
axes.tick_params(axis="both", which="major", labelsize=25)
axes.tick_params(axis="both", which="minor", labelsize=12)
if not exists(self._path + "figures"): # detect figures folder
makedirs(self._path + "figures")
plt.savefig(self._path + "figures/Population_Dynamics.png")
def draw_comparison_tran(self):
if not hasattr(self, "tran"):
print("Error: call photon_simulated method first")
return
if not hasattr(self, "tran_theory"):
print("Error: call photon_theory method first")
return
fig, axes = plt.subplots(1, 1, figsize=(10, 8))
axes.plot(self.tlist, self.tran, color="k", label="simulation", linewidth=5)
axes.plot(
self.tlist,
self.tran_theory,
color="b",
label="theoretical",
linestyle="dashed",
linewidth=5,
)
axes.set_xlim(left=0.5)
axes.set_ylim(top=0.3)
axes.legend(loc=1, fontsize=18)
axes.set_xscale("log")
axes.set_xlabel("Time/us", fontsize=28)
axes.tick_params(axis="both", which="major", labelsize=25)
axes.tick_params(axis="both", which="minor", labelsize=12)
if not exists(self._path + "figures"): # detect figures folder
makedirs(self._path + "figures")
plt.savefig(self._path + "figures/Transmission_Comparison.png")
### calculate transmission spectrum
def _helper(delta: float, Mol_original: molSystem, sampletlist: List[float], method: str):
"""
input: method: 'simulation' or 'theory'
output: transmission at sampling time spot
"""
Mol_temp = deepcopy(Mol_original)
Mol_temp.delta_al = Mol_temp.delta_cl = delta
Mol_temp._save = False
if method == "simulation":
Mol_temp._init_simulation()
Mol_temp.simulation(show_progress=None, save_info=False, save_data=False)
Mol_temp.photon_simulated()
tran = Mol_temp.tran
elif method == "theory":
Mol_temp._init_theory()
Mol_temp.photon_theory()
tran = Mol_temp.tran_theory
else:
raise ValueError("method value should be 'simulation' or 'theory'")
average_tran = []
time_step = Mol_temp.tlist[1] - Mol_temp.tlist[0]
t_idx_list = list(array(sampletlist) // time_step)
t_flag = t_idx_list.pop(0)
p_num = 0
for t_idx, transmission in enumerate(tran):
# average tranmission=total transmission photon number / total input photon number
p_num += transmission
if t_idx == t_flag:
# t_idx + 1 represents time
average_tran.append(p_num / (t_idx + 1))
if len(t_idx_list) > 0:
t_flag = t_idx_list.pop(0)
else:
break
del Mol_temp
return average_tran
def tran_spectrum(
Mol: molSystem,
delta_max: float,
delta_step: float,
sampletlist: List[float],
method="theory",
calculation="parallel",
save_data=False,
):
"""
input:
an instance of molSystem (neglect the detuning setting)
delta_max: Unit: MHz. positive maximum detuning frequency.
delta_step: Unit: MHz. Increasing step of detuning frequency from zero
sampletlist: The list of sampling time to draw spectrum
output:
transmission spectrum data
"""
if Mol.tlist[-1] < sampletlist[-1]:
raise ValueError("tlist range does not cover sampletlist!")
if method == "simulation" or method == "theory":
print("The way to derive transmission spectrum is " + method)
else:
raise ValueError("method value should be 'simulation' or 'theory'")
if calculation == "serial" or calculation == "parallel":
print("The way to derive transmission spectrum is " + calculation)
else:
raise ValueError("calculation value should be 'serial' or 'parallel'")
deltalist = arange(0, delta_max, delta_step)
if calculation == "serial":
# serial version
spectrum_data = []
for delta in deltalist:
spectrum_data.append(_helper(2 * pi * delta, Mol, sampletlist, method))
elif calculation == "parallel":
# parallel version
spectrum_data = parallel_map(
_helper,
2 * pi * array(deltalist), # convert the unit to Mega rad/s
task_kwargs=dict(Mol_original=Mol, sampletlist=sampletlist, method=method),
progress_bar=True,
)
if save_data:
save(Mol._path + "tran_spectrum.npy", spectrum_data)
save(Mol._path + "deltalist.npy", array(deltalist))
save(Mol._path + "sampletlist.npy", array(sampletlist))
# draw graph
delta_minus = -copy(deltalist)
delta_data = concatenate([flip(delta_minus), deltalist])
spec_minus = flip(copy(spectrum_data), axis=0)
spec_data = transpose(concatenate([spec_minus, spectrum_data]))
fig, axes = plt.subplots(1, 1, figsize=(10, 8))
for t_idx, spec in enumerate(spec_data):
axes.plot(delta_data, spec, label=str(sampletlist[t_idx]) + r"$\mu s$")
axes.legend(loc=0)
if not exists(Mol._path + "figures"): # detect figures folder
makedirs(Mol._path + "figures")
plt.savefig(Mol._path + "figures/Transmission_Spectrum.png")
if __name__ == "__main__":
Mol = molSystem(duration=201, spacing=0.01, cooperativity=50, kappa_i=50, kappa_e=50)
# Mol.simulation(show_progress=True, save_data=True)
# Mol.photon_simulated(save_data=True)
# Mol.photon_theory(save_data=True)
# Mol.draw_population()
# Mol.draw_comparison_tran()
tran_spectrum(
Mol,
delta_max=1000,
delta_step=10,
sampletlist=[20, 100, 200],
method="theory",
calculation="parallel",
save_data=True,
)
del Mol