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jaynes_cummings_model.py
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jaynes_cummings_model.py
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import tensorflow as tf
import numpy as np
from tqdm import tqdm
class JC_model:
""" Class provides the tool for Jaynes-Cummings model simulation.
Args:
- sys_dim, system dimention, type: int
- mode_type, defines environment state,
type: string, possible settings:
- Fock state
- Coherent state
- Thermal state
- mode_par, parameter of the environment state,
type: - 'int', for Fock state
- 'complex', for Coherent state
- 'float', for Thermal state
"""
def __init__(self, sys_dim, mode_type, mode_par):
self.sys_dim = sys_dim
self.mode_type = mode_type
self.mode_par = mode_par
self.env_dim = self.minimal_env_dim()
self.env_state = self.env_init()
sigma_x = tf.constant([[0, 1], [1, 0]], dtype=tf.complex128)
sigma_y = tf.constant([[0, -1j], [1j, 0]], dtype=tf.complex128)
sigma_z = tf.constant([[1, 0], [0, -1]], dtype=tf.complex128)
self.pauli = tf.concat([sigma_x[tf.newaxis],
sigma_y[tf.newaxis],
sigma_z[tf.newaxis]], axis=0)
self.generator = None
def minimal_env_dim(self):
""" Return the minimal dimetion of the environmet
with respect to the field type. """
if self.mode_type == 'Fock_state':
return self.mode_par + 2
elif self.mode_type == 'Coherent_state':
return int(3 * abs(self.mode_par) ** 2) + 2
elif self.mode_type == 'Thermal_state':
mean = 1/(np.exp(self.mode_par) - 1)
return int(mean + 3 * mean / (1 + mean)) + 2
else:
print('Unknown mode type')
def field_mode_operators(self):
""" Returns field mode creation (a†) and annihilation (a) operators. """
arr = tf.math.sqrt(tf.range(
0., self.env_dim, dtype=tf.float64))
matr = tf.reshape(tf.concat(
[arr for i in range(self.env_dim)], 0), [self.env_dim, self.env_dim])
annih_oper = tf.cast(tf.linalg.band_part(
matr, 0, 1) - tf.linalg.band_part(
matr, 0, 0), dtype=tf.complex128)
return annih_oper, tf.linalg.adjoint(annih_oper)
def env_init(self):
""" Return the initial state of environment. """
if self.mode_type == 'Fock_state':
fock_density = tf.einsum(
'i,j->ij', tf.one_hot(
int(self.mode_par), self.env_dim), tf.math.conj(
tf.one_hot(int(self.mode_par), self.env_dim)))
return tf.cast(fock_density, dtype=tf.complex128)
elif self.mode_type == 'Coherent_state':
annihilation, creation = self.field_mode_operators()
displacement = tf.linalg.expm(
self.mode_par * creation - np.conj(self.mode_par) * annihilation)
alpha_state = tf.einsum(
'ij, j->i', displacement, tf.cast(
tf.one_hot(0, self.env_dim), dtype=tf.complex128))
coherent_density = tf.einsum(
'i,j->ij', alpha_state, tf.math.conj(alpha_state))
return tf.cast(coherent_density, dtype=tf.complex128)
elif self.mode_type == 'Thermal_state':
thermal_density = tf.reduce_sum(
[tf.math.exp(- self.mode_par * float(n)) * tf.einsum(
'i,j->ij', tf.one_hot(n, self.env_dim), tf.math.conj(
tf.one_hot(n, self.env_dim))) for n in range(self.env_dim)], axis=0)
return tf.cast(
thermal_density / tf.linalg.trace(
thermal_density), dtype=tf.complex128)
else:
print('Unknown mode type')
def lindblad_generator(self, alpha, omega, gamma):
""" Return Lindblad generator of Jaynes-Cummings model
with dissipation.
Args:
- alpha, system Hamiltonian decomposition
- omega, field oscillation frequency
- gamma, damping amplitude """
# TODO simulation for qudit system
dim = self.sys_dim * self.env_dim
id_env = tf.eye(self.env_dim, self.env_dim, dtype=tf.complex128)
id_sys = tf.eye(self.sys_dim, self.sys_dim, dtype=tf.complex128)
identity = tf.eye(dim, dim, dtype=tf.complex128)
# Creation & Annihilation operators
annihilation, creation = self.field_mode_operators()
# System Hamiltonian
h_sys = tf.reshape(
tf.einsum('ij,kl->ikjl', tf.reduce_sum(
[alpha[i] * self.pauli[i] for i in range(3)], axis=0), id_env), (dim, dim))
# Field Hamiltonian
h_field = tf.reshape(tf.einsum(
'ij,kl->ikjl', id_sys, creation @ annihilation), (dim, dim))
# Interaction Hamiltonian
h_int = omega * tf.reshape(
tf.einsum('ij,kl->ikjl', 1/2 * (
self.pauli[0] - 1j * self.pauli[1]), annihilation) +\
tf.einsum('ij,kl->ikjl', 1/2 * (
self.pauli[0] + 1j * self.pauli[1]), creation), (dim, dim))
jc_ham = h_sys + h_field + h_int
# Commutator vectorization
ham_id = tf.einsum('ij,kl->ikjl', jc_ham, identity)
id_ham = tf.einsum('ij,lk->ikjl', identity, jc_ham)
commutator = tf.reshape(
1j * (id_ham - ham_id), (dim ** 2, dim ** 2))
# Jump operators in vectorized form
annihilation_jump = tf.reshape(tf.einsum(
'ij,kl->ikjl', id_sys, annihilation), (dim, dim))
creation_jump = tf.reshape(tf.einsum(
'ij,kl->ikjl', id_sys, creation), (dim, dim))
# Dissipator in vectorized form
dissipator = gamma * (
tf.reshape(tf.einsum('ij,kl->ikjl', tf.transpose(
creation_jump), annihilation_jump), (
dim ** 2, dim ** 2)) -\
1/2 * tf.reshape(tf.einsum(
'ij,kl->ikjl', identity, creation_jump @ annihilation_jump), (
dim ** 2, dim ** 2)) -\
1/2 * tf.reshape(tf.einsum(
'ij,kl->ikjl', tf.transpose(
creation_jump @ annihilation_jump), identity), (
dim ** 2, dim ** 2))
)
self.generator = commutator + dissipator
def sample_spherical(self, npoints, ndim=3):
""" Generates "npoints" uniformly distributed random points in
"ndim"-dimensioanl sphere """
vec = np.random.randn(ndim, npoints)
vec /= np.linalg.norm(vec, axis=0)
return vec
def generate_dynamics(self, number_of_lines,
total_time, time_step, rho_0=None):
""" Generate dynamics
Args:
- number_of_lines, number of different trajectories
- total_time, total simulation time
- time_step, simulation step
- rho_0, initial state of the system,
if not given the system will
be initialized randomly """
dim = self.sys_dim * self.env_dim
if rho_0 == None:
lines = []
for _ in range(number_of_lines):
line = []
vec = self.sample_spherical(1)
sys_state = 1/2 * (tf.cast(
tf.eye(2), dtype=tf.complex128) + tf.reduce_sum(
[vec[i] * self.pauli[i] for i in range(3)], axis=0))
initial_state = tf.reshape(tf.einsum(
'ij,kl->ikjl', sys_state, self.env_state), (dim, dim))
for time in tqdm(np.arange(0., total_time, time_step)):
state = tf.linalg.expm(
time * self.generator) @ tf.reshape(
initial_state, (dim ** 2, 1))
line.append(tf.einsum('ikjk->ij', tf.reshape(
state, (self.sys_dim, self.env_dim, self.sys_dim, self.env_dim))))
lines.append(line)
self.dynamics = tf.convert_to_tensor(lines)
else:
line = []
initial_state = tf.reshape(tf.einsum(
'ij,kl->ikjl', rho_0, self.env_state), (dim, dim))
for time in tqdm(np.arange(0., total_time, time_step)):
state = tf.linalg.expm(
time * self.generator) @ tf.reshape(
initial_state, (dim ** 2, 1))
line.append(tf.einsum('ikjk->ij', tf.reshape(
state, (self.sys_dim, self.env_dim, self.sys_dim, self.env_dim))))
self.dynamics = tf.convert_to_tensor(line)