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"""Permutational Invariant Quantum Solver (PIQS)
This module calculates the Liouvillian for the dynamics of ensembles of
identical two-level systems (TLS) in the presence of local and collective
processes by exploiting permutational symmetry and using the Dicke basis.
It also allows to characterize nonlinear functions of the density matrix.
"""
# Authors: Nathan Shammah, Shahnawaz Ahmed
# Contact: nathan.shammah@gmail.com, shahnawaz.ahmed95@gmail.com
from math import factorial
from decimal import Decimal
import numpy as np
from scipy.integrate import odeint
from scipy.linalg import eigvalsh
from scipy.special import entr
from scipy.sparse import dok_matrix, block_diag, lil_matrix
from qutip.solver import Options, Result
from qutip import (
Qobj,
spre,
spost,
tensor,
identity,
ket2dm,
)
from qutip import sigmax, sigmay, sigmaz, sigmap, sigmam
from qutip.cy.piqs import Dicke as _Dicke
from qutip.cy.piqs import (
jmm1_dictionary,
_num_dicke_states,
_num_dicke_ladders,
get_blocks,
j_min,
j_vals,
)
__all__ = [
"num_dicke_states",
"num_dicke_ladders",
"num_tls",
"isdiagonal",
"dicke_blocks",
"dicke_blocks_full",
"dicke_function_trace",
"purity_dicke",
"entropy_vn_dicke",
"Dicke",
"state_degeneracy",
"m_degeneracy",
"energy_degeneracy",
"ap",
"am",
"spin_algebra",
"jspin",
"collapse_uncoupled",
"dicke_basis",
"dicke",
"excited",
"superradiant",
"css",
"ghz",
"ground",
"identity_uncoupled",
"block_matrix",
"tau_column",
"Pim",
]
def _ensure_int(x):
"""
Ensure that a floating-point value `x` is exactly an integer, and return it
as an int.
"""
out = int(x)
if out != x:
raise ValueError(f"{x} is not an integral value")
return out
# Functions necessary to generate the Lindbladian/Liouvillian
[docs]def num_dicke_states(N):
"""Calculate the number of Dicke states.
Parameters
----------
N: int
The number of two-level systems.
Returns
-------
nds: int
The number of Dicke states.
"""
return _num_dicke_states(N)
[docs]def num_dicke_ladders(N):
"""Calculate the total number of ladders in the Dicke space.
For a collection of N two-level systems it counts how many different
"j" exist or the number of blocks in the block-diagonal matrix.
Parameters
----------
N: int
The number of two-level systems.
Returns
-------
Nj: int
The number of Dicke ladders.
"""
return _num_dicke_ladders(N)
[docs]def num_tls(nds):
"""Calculate the number of two-level systems.
Parameters
----------
nds: int
The number of Dicke states.
Returns
-------
N: int
The number of two-level systems.
"""
if np.sqrt(nds).is_integer():
# N is even
N = 2 * (np.sqrt(nds) - 1)
else:
# N is odd
N = 2 * (np.sqrt(nds + 1 / 4) - 1)
return int(N)
[docs]def isdiagonal(mat):
"""
Check if the input matrix is diagonal.
Parameters
==========
mat: ndarray/Qobj
A 2D numpy array
Returns
=======
diag: bool
True/False depending on whether the input matrix is diagonal.
"""
if isinstance(mat, Qobj):
mat = mat.full()
return np.all(mat == np.diag(np.diagonal(mat)))
# nonlinear functions of the density matrix
[docs]def dicke_blocks(rho):
"""Create the list of blocks for block-diagonal density matrix in the Dicke basis.
Parameters
----------
rho : :class:`qutip.Qobj`
A 2D block-diagonal matrix of ones with dimension (nds,nds),
where nds is the number of Dicke states for N two-level
systems.
Returns
-------
square_blocks: list of np.array
Give back the blocks list.
"""
shape_dimension = rho.shape[0]
N = num_tls(shape_dimension)
ladders = num_dicke_ladders(N)
# create a list with the sizes of the blocks, in order
blocks_dimensions = int(N / 2 + 1 - 0.5 * (N % 2))
blocks_list = [
(2 * (i + 1 * (N % 2)) + 1 * ((N + 1) % 2))
for i in range(blocks_dimensions)
]
blocks_list = np.flip(blocks_list, 0)
# create a list with each block matrix as element
square_blocks = []
block_position = 0
for block_size in blocks_list:
start_m = block_position
end_m = block_position + block_size
square_block = rho[start_m:end_m, start_m:end_m]
block_position = block_position + block_size
square_blocks.append(square_block)
return square_blocks
[docs]def dicke_blocks_full(rho):
"""Give the full (2^N-dimensional) list of blocks for a Dicke-basis matrix.
Parameters
----------
rho : :class:`qutip.Qobj`
A 2D block-diagonal matrix of ones with dimension (nds,nds),
where nds is the number of Dicke states for N two-level
systems.
Returns
-------
full_blocks : list
The list of blocks expanded in the 2^N space for N qubits.
"""
shape_dimension = rho.shape[0]
N = num_tls(shape_dimension)
ladders = num_dicke_ladders(N)
# create a list with the sizes of the blocks, in order
blocks_dimensions = int(N / 2 + 1 - 0.5 * (N % 2))
blocks_list = [
(2 * (i + 1 * (N % 2)) + 1 * ((N + 1) % 2))
for i in range(blocks_dimensions)
]
blocks_list = np.flip(blocks_list, 0)
# create a list with each block matrix as element
full_blocks = []
k = 0
block_position = 0
for block_size in blocks_list:
start_m = block_position
end_m = block_position + block_size
square_block = rho[start_m:end_m, start_m:end_m]
block_position = block_position + block_size
j = N / 2 - k
djn = state_degeneracy(N, j)
for block_counter in range(0, djn):
full_blocks.append(square_block / djn) # preserve trace
k = k + 1
return full_blocks
[docs]def dicke_function_trace(f, rho):
"""Calculate the trace of a function on a Dicke density matrix.
Parameters
----------
f : function
A Taylor-expandable function of `rho`.
rho : :class:`qutip.Qobj`
A density matrix in the Dicke basis.
Returns
-------
res : float
Trace of a nonlinear function on `rho`.
"""
N = num_tls(rho.shape[0])
blocks = dicke_blocks(rho)
block_f = []
degen_blocks = np.flip(j_vals(N))
state_degeneracies = []
for j in degen_blocks:
dj = state_degeneracy(N, j)
state_degeneracies.append(dj)
eigenvals_degeneracy = []
deg = []
for i, block in enumerate(blocks):
dj = state_degeneracies[i]
normalized_block = block / dj
eigenvals_block = eigvalsh(normalized_block)
for val in eigenvals_block:
eigenvals_degeneracy.append(val)
deg.append(dj)
eigenvalue = np.array(eigenvals_degeneracy)
function_array = f(eigenvalue) * deg
return sum(function_array)
[docs]def entropy_vn_dicke(rho):
"""Von Neumann Entropy of a Dicke-basis density matrix.
Parameters
----------
rho : :class:`qutip.Qobj`
A 2D block-diagonal matrix of ones with dimension (nds,nds),
where nds is the number of Dicke states for N two-level
systems.
Returns
-------
entropy_dm: float
Entropy. Use degeneracy to multiply each block.
"""
return dicke_function_trace(entr, rho)
[docs]def purity_dicke(rho):
"""Calculate purity of a density matrix in the Dicke basis.
It accounts for the degenerate blocks in the density matrix.
Parameters
----------
rho : :class:`qutip.Qobj`
Density matrix in the Dicke basis of qutip.piqs.jspin(N), for N spins.
Returns
-------
purity : float
The purity of the quantum state.
It's 1 for pure states, 0<=purity<1 for mixed states.
"""
f = lambda x: x * x
return dicke_function_trace(f, rho)
[docs]class Dicke(object):
"""The Dicke class which builds the Lindbladian and Liouvillian matrix.
Examples
--------
>>> from piqs import Dicke, jspin
>>> N = 2
>>> jx, jy, jz = jspin(N)
>>> jp = jspin(N, "+")
>>> jm = jspin(N, "-")
>>> ensemble = Dicke(N, emission=1.)
>>> L = ensemble.liouvillian()
Parameters
----------
N: int
The number of two-level systems.
hamiltonian : :class:`qutip.Qobj`
A Hamiltonian in the Dicke basis.
The matrix dimensions are (nds, nds),
with nds being the number of Dicke states.
The Hamiltonian can be built with the operators
given by the `jspin` functions.
emission: float
Incoherent emission coefficient (also nonradiative emission).
default: 0.0
dephasing: float
Local dephasing coefficient.
default: 0.0
pumping: float
Incoherent pumping coefficient.
default: 0.0
collective_emission: float
Collective (superradiant) emmission coefficient.
default: 0.0
collective_pumping: float
Collective pumping coefficient.
default: 0.0
collective_dephasing: float
Collective dephasing coefficient.
default: 0.0
Attributes
----------
N: int
The number of two-level systems.
hamiltonian : :class:`qutip.Qobj`
A Hamiltonian in the Dicke basis.
The matrix dimensions are (nds, nds),
with nds being the number of Dicke states.
The Hamiltonian can be built with the operators given
by the `jspin` function in the "dicke" basis.
emission: float
Incoherent emission coefficient (also nonradiative emission).
default: 0.0
dephasing: float
Local dephasing coefficient.
default: 0.0
pumping: float
Incoherent pumping coefficient.
default: 0.0
collective_emission: float
Collective (superradiant) emmission coefficient.
default: 0.0
collective_dephasing: float
Collective dephasing coefficient.
default: 0.0
collective_pumping: float
Collective pumping coefficient.
default: 0.0
nds: int
The number of Dicke states.
dshape: tuple
The shape of the Hilbert space in the Dicke or uncoupled basis.
default: (nds, nds).
"""
def __init__(
self,
N,
hamiltonian=None,
emission=0.0,
dephasing=0.0,
pumping=0.0,
collective_emission=0.0,
collective_dephasing=0.0,
collective_pumping=0.0,
):
self.N = N
self.hamiltonian = hamiltonian
self.emission = emission
self.dephasing = dephasing
self.pumping = pumping
self.collective_emission = collective_emission
self.collective_dephasing = collective_dephasing
self.collective_pumping = collective_pumping
self.nds = num_dicke_states(self.N)
self.dshape = (num_dicke_states(self.N), num_dicke_states(self.N))
def __repr__(self):
"""Print the current parameters of the system."""
string = []
string.append("N = {}".format(self.N))
string.append("Hilbert space dim = {}".format(self.dshape))
string.append("Number of Dicke states = {}".format(self.nds))
string.append(
"Liouvillian space dim = {}".format((self.nds ** 2, self.nds ** 2))
)
if self.emission != 0:
string.append("emission = {}".format(self.emission))
if self.dephasing != 0:
string.append("dephasing = {}".format(self.dephasing))
if self.pumping != 0:
string.append("pumping = {}".format(self.pumping))
if self.collective_emission != 0:
string.append(
"collective_emission = {}".format(self.collective_emission)
)
if self.collective_dephasing != 0:
string.append(
"collective_dephasing = {}".format(self.collective_dephasing)
)
if self.collective_pumping != 0:
string.append(
"collective_pumping = {}".format(self.collective_pumping)
)
return "\n".join(string)
[docs] def lindbladian(self):
"""Build the Lindbladian superoperator of the dissipative dynamics.
Returns
-------
lindbladian : :class:`qutip.Qobj`
The Lindbladian matrix as a `qutip.Qobj`.
"""
cythonized_dicke = _Dicke(
int(self.N),
float(self.emission),
float(self.dephasing),
float(self.pumping),
float(self.collective_emission),
float(self.collective_dephasing),
float(self.collective_pumping),
)
return cythonized_dicke.lindbladian()
[docs] def liouvillian(self):
"""Build the total Liouvillian using the Dicke basis.
Returns
-------
liouv : :class:`qutip.Qobj`
The Liouvillian matrix for the system.
"""
lindblad = self.lindbladian()
if self.hamiltonian is None:
liouv = lindblad
else:
hamiltonian = self.hamiltonian
hamiltonian_superoperator = -1j * spre(hamiltonian) + 1j * spost(
hamiltonian
)
liouv = lindblad + hamiltonian_superoperator
return liouv
[docs] def pisolve(self, initial_state, tlist, options=None):
"""
Solve for diagonal Hamiltonians and initial states faster.
Parameters
==========
initial_state : :class:`qutip.Qobj`
An initial state specified as a density matrix of
`qutip.Qbj` type.
tlist: ndarray
A 1D numpy array of list of timesteps to integrate
options : :class:`qutip.solver.Options`
The options for the solver.
Returns
=======
result: list
A dictionary of the type `qutip.solver.Result` which holds the
results of the evolution.
"""
if isdiagonal(initial_state) == False:
msg = "`pisolve` requires a diagonal initial density matrix. "
msg += "In general construct the Liouvillian using "
msg += "`piqs.liouvillian` and use qutip.mesolve."
raise ValueError(msg)
if self.hamiltonian and isdiagonal(self.hamiltonian) == False:
msg = "`pisolve` should only be used for diagonal Hamiltonians. "
msg += "Construct the Liouvillian using `piqs.liouvillian` and"
msg += " use `qutip.mesolve`."
raise ValueError(msg)
if initial_state.full().shape != self.dshape:
msg = "Initial density matrix should be diagonal."
raise ValueError(msg)
pim = Pim(
self.N,
self.emission,
self.dephasing,
self.pumping,
self.collective_emission,
self.collective_pumping,
self.collective_dephasing,
)
result = pim.solve(initial_state, tlist, options=None)
return result
[docs] def c_ops(self):
"""Build collapse operators in the full Hilbert space 2^N.
Returns
-------
c_ops_list: list
The list with the collapse operators in the 2^N Hilbert space.
"""
ce = self.collective_emission
cd = self.collective_dephasing
cp = self.collective_pumping
c_ops_list = collapse_uncoupled(
N=self.N,
emission=self.emission,
dephasing=self.dephasing,
pumping=self.pumping,
collective_emission=ce,
collective_dephasing=cd,
collective_pumping=cp,
)
return c_ops_list
[docs] def coefficient_matrix(self):
"""Build coefficient matrix for ODE for a diagonal problem.
Returns
-------
M: ndarray
The matrix M of the coefficients for the ODE dp/dt = Mp.
p is the vector of the diagonal matrix elements
of the density matrix rho in the Dicke basis.
"""
diagonal_system = Pim(
N=self.N,
emission=self.emission,
dephasing=self.dephasing,
pumping=self.pumping,
collective_emission=self.collective_emission,
collective_dephasing=self.collective_dephasing,
collective_pumping=self.collective_pumping,
)
coef_matrix = diagonal_system.coefficient_matrix()
return coef_matrix
# Utility functions for properties of the Dicke space
[docs]def energy_degeneracy(N, m):
"""Calculate the number of Dicke states with same energy.
The use of the `Decimals` class allows to explore N > 1000,
unlike the built-in function `scipy.special.binom`
Parameters
----------
N: int
The number of two-level systems.
m: float
Total spin z-axis projection eigenvalue.
This is proportional to the total energy.
Returns
-------
degeneracy: int
The energy degeneracy
"""
numerator = Decimal(factorial(N))
d1 = Decimal(factorial(_ensure_int(N / 2 + m)))
d2 = Decimal(factorial(_ensure_int(N / 2 - m)))
degeneracy = numerator / (d1 * d2)
return int(degeneracy)
[docs]def state_degeneracy(N, j):
r"""Calculate the degeneracy of the Dicke state.
Each state :math:`\lvert j, m\rangle` includes D(N,j) irreducible
representations :math:`\lvert j, m, \alpha\rangle`.
Uses Decimals to calculate higher numerator and denominators numbers.
Parameters
----------
N: int
The number of two-level systems.
j: float
Total spin eigenvalue (cooperativity).
Returns
-------
degeneracy: int
The state degeneracy.
"""
if j < 0:
raise ValueError("j value should be >= 0")
numerator = Decimal(factorial(N)) * Decimal(2 * j + 1)
denominator_1 = Decimal(factorial(_ensure_int(N / 2 + j + 1)))
denominator_2 = Decimal(factorial(_ensure_int(N / 2 - j)))
degeneracy = numerator / (denominator_1 * denominator_2)
degeneracy = int(np.round(float(degeneracy)))
return degeneracy
[docs]def m_degeneracy(N, m):
r"""Calculate the number of Dicke states :math:`\lvert j, m\rangle` with
same energy.
Parameters
----------
N: int
The number of two-level systems.
m: float
Total spin z-axis projection eigenvalue (proportional to the total
energy).
Returns
-------
degeneracy: int
The m-degeneracy.
"""
jvals = j_vals(N)
maxj = np.max(jvals)
if m < -maxj:
e = "m value is incorrect for this N."
e += " Minimum m value can be {}".format(-maxj)
raise ValueError(e)
degeneracy = N / 2 + 1 - abs(m)
return int(degeneracy)
[docs]def ap(j, m):
r"""
Calculate the coefficient ``ap`` by applying :math:`J_+\lvert j,m\rangle`.
The action of ap is given by:
:math:`J_{+}\lvert j, m\rangle = A_{+}(j, m) \lvert j, m+1\rangle`
Parameters
----------
j, m: float
The value for j and m in the dicke basis :math:`\lvert j, m\rangle`.
Returns
-------
a_plus: float
The value of :math:`a_{+}`.
"""
a_plus = np.sqrt((j - m) * (j + m + 1))
return a_plus
[docs]def am(j, m):
r"""Calculate the operator ``am`` used later.
The action of ``ap`` is given by:
:math:`J_{-}\lvert j,m\rangle = A_{-}(jm)\lvert j,m-1\rangle`
Parameters
----------
j: float
The value for j.
m: float
The value for m.
Returns
-------
a_minus: float
The value of :math:`a_{-}`.
"""
a_minus = np.sqrt((j + m) * (j - m + 1))
return a_minus
[docs]def spin_algebra(N, op=None):
"""Create the list [sx, sy, sz] with the spin operators.
The operators are constructed for a collection of N two-level systems
(TLSs). Each element of the list, i.e., sx, is a vector of `qutip.Qobj`
objects (spin matrices), as it cointains the list of the SU(2) Pauli
matrices for the N TLSs. Each TLS operator sx[i], with i = 0, ..., (N-1),
is placed in a :math:`2^N`-dimensional Hilbert space.
Notes
-----
sx[i] is :math:`\\frac{\\sigma_x}{2}` in the composite Hilbert space.
Parameters
----------
N: int
The number of two-level systems.
Returns
-------
spin_operators: list or :class: qutip.Qobj
A list of `qutip.Qobj` operators - [sx, sy, sz] or the
requested operator.
"""
# 1. Define N TLS spin-1/2 matrices in the uncoupled basis
N = int(N)
sx = [0 for i in range(N)]
sy = [0 for i in range(N)]
sz = [0 for i in range(N)]
sp = [0 for i in range(N)]
sm = [0 for i in range(N)]
sx[0] = 0.5 * sigmax()
sy[0] = 0.5 * sigmay()
sz[0] = 0.5 * sigmaz()
sp[0] = sigmap()
sm[0] = sigmam()
# 2. Place operators in total Hilbert space
for k in range(N - 1):
sx[0] = tensor(sx[0], identity(2))
sy[0] = tensor(sy[0], identity(2))
sz[0] = tensor(sz[0], identity(2))
sp[0] = tensor(sp[0], identity(2))
sm[0] = tensor(sm[0], identity(2))
# 3. Cyclic sequence to create all N operators
a = [i for i in range(N)]
b = [[a[i - i2] for i in range(N)] for i2 in range(N)]
# 4. Create N operators
for i in range(1, N):
sx[i] = sx[0].permute(b[i])
sy[i] = sy[0].permute(b[i])
sz[i] = sz[0].permute(b[i])
sp[i] = sp[0].permute(b[i])
sm[i] = sm[0].permute(b[i])
spin_operators = [sx, sy, sz]
if not op:
return spin_operators
elif op == "x":
return sx
elif op == "y":
return sy
elif op == "z":
return sz
elif op == "+":
return sp
elif op == "-":
return sm
else:
raise TypeError("Invalid type")
def _jspin_uncoupled(N, op=None):
"""Construct the the collective spin algebra in the uncoupled basis.
jx, jy, jz, jp, jm are constructed in the uncoupled basis of the
two-level system (TLS). Each collective operator is placed in a
Hilbert space of dimension 2^N.
Parameters
----------
N: int
The number of two-level systems.
op: str
The operator to return 'x','y','z','+','-'.
If no operator given, then output is the list of operators
for ['x','y','z',].
Returns
-------
collective_operators: list or :class: qutip.Qobj
A list of `qutip.Qobj` representing all the operators in
uncoupled" basis or a single operator requested.
"""
# 1. Define N TLS spin-1/2 matrices in the uncoupled basis
N = int(N)
sx, sy, sz = spin_algebra(N)
sp, sm = spin_algebra(N, "+"), spin_algebra(N, "-")
jx = sum(sx)
jy = sum(sy)
jz = sum(sz)
jp = sum(sp)
jm = sum(sm)
collective_operators = [jx, jy, jz]
if not op:
return collective_operators
elif op == "x":
return jx
elif op == "y":
return jy
elif op == "z":
return jz
elif op == "+":
return jp
elif op == "-":
return jm
else:
raise TypeError("Invalid type")
[docs]def jspin(N, op=None, basis="dicke"):
r"""
Calculate the list of collective operators of the total algebra.
The Dicke basis :math:`\lvert j,m\rangle\langle j,m'\rvert` is used by
default. Otherwise with "uncoupled" the operators are in a
:math:`2^N` space.
Parameters
----------
N: int
Number of two-level systems.
op: str
The operator to return 'x','y','z','+','-'.
If no operator given, then output is the list of operators
for ['x','y','z'].
basis: str
The basis of the operators - "dicke" or "uncoupled"
default: "dicke".
Returns
-------
j_alg: list or :class: qutip.Qobj
A list of `qutip.Qobj` representing all the operators in
the "dicke" or "uncoupled" basis or a single operator requested.
"""
if basis == "uncoupled":
return _jspin_uncoupled(N, op)
nds = num_dicke_states(N)
num_ladders = num_dicke_ladders(N)
jz_operator = dok_matrix((nds, nds), dtype=np.complex128)
jp_operator = dok_matrix((nds, nds), dtype=np.complex128)
jm_operator = dok_matrix((nds, nds), dtype=np.complex128)
s = 0
for k in range(0, num_ladders):
j = 0.5 * N - k
mmax = int(2 * j + 1)
for i in range(0, mmax):
m = j - i
jz_operator[s, s] = m
if (s + 1) in range(0, nds):
jp_operator[s, s + 1] = ap(j, m - 1)
if (s - 1) in range(0, nds):
jm_operator[s, s - 1] = am(j, m + 1)
s = s + 1
jx_operator = 1 / 2 * (jp_operator + jm_operator)
jy_operator = 1j / 2 * (jm_operator - jp_operator)
jx = Qobj(jx_operator)
jy = Qobj(jy_operator)
jz = Qobj(jz_operator)
jp = Qobj(jp_operator)
jm = Qobj(jm_operator)
if not op:
return [jx, jy, jz]
if op == "+":
return jp
elif op == "-":
return jm
elif op == "x":
return jx
elif op == "y":
return jy
elif op == "z":
return jz
else:
raise TypeError("Invalid type")
[docs]def collapse_uncoupled(
N,
emission=0.0,
dephasing=0.0,
pumping=0.0,
collective_emission=0.0,
collective_dephasing=0.0,
collective_pumping=0.0,
):
"""
Create the collapse operators (c_ops) of the Lindbladian in the
uncoupled basis
These operators are in the uncoupled basis of the two-level
system (TLS) SU(2) Pauli matrices.
Notes
-----
The collapse operator list can be given to `qutip.mesolve`.
Notice that the operators are placed in a Hilbert space of
dimension :math:`2^N`. Thus the method is suitable only for
small N (of the order of 10).
Parameters
----------
N: int
The number of two-level systems.
emission: float
Incoherent emission coefficient (also nonradiative emission).
default: 0.0
dephasing: float
Local dephasing coefficient.
default: 0.0
pumping: float
Incoherent pumping coefficient.
default: 0.0
collective_emission: float
Collective (superradiant) emmission coefficient.
default: 0.0
collective_pumping: float
Collective pumping coefficient.
default: 0.0
collective_dephasing: float
Collective dephasing coefficient.
default: 0.0
Returns
-------
c_ops: list
The list of collapse operators as `qutip.Qobj` for the system.
"""
N = int(N)
if N > 10:
msg = "N > 10. dim(H) = 2^N. "
msg += "Better use `piqs.lindbladian` to reduce Hilbert space "
msg += "dimension and exploit permutational symmetry."
raise Warning(msg)
[sx, sy, sz] = spin_algebra(N)
sp, sm = spin_algebra(N, "+"), spin_algebra(N, "-")
[jx, jy, jz] = jspin(N, basis="uncoupled")
jp, jm = (
jspin(N, "+", basis="uncoupled"),
jspin(N, "-", basis="uncoupled"),
)
c_ops = []
if emission != 0:
for i in range(0, N):
c_ops.append(np.sqrt(emission) * sm[i])
if dephasing != 0:
for i in range(0, N):
c_ops.append(np.sqrt(dephasing) * sz[i])
if pumping != 0:
for i in range(0, N):
c_ops.append(np.sqrt(pumping) * sp[i])
if collective_emission != 0:
c_ops.append(np.sqrt(collective_emission) * jm)
if collective_dephasing != 0:
c_ops.append(np.sqrt(collective_dephasing) * jz)
if collective_pumping != 0:
c_ops.append(np.sqrt(collective_pumping) * jp)
return c_ops
# State definitions in the Dicke basis with an option for basis transformation
[docs]def dicke_basis(N, jmm1=None):
r"""
Initialize the density matrix of a Dicke state for several (j, m, m1).
This function can be used to build arbitrary states in the Dicke basis
:math:`\lvert j, m\rangle\langle j, m'\rvert`. We create coefficients for
each (j, m, m1) value in the dictionary jmm1. The mapping for the (i, k)
index of the density matrix to the :math:`\lvert j, m\rangle` values is
given by the cythonized function `jmm1_dictionary`. A density matrix is
created from the given dictionary of coefficients for each (j, m, m1).
Parameters
----------
N: int
The number of two-level systems.
jmm1: dict
A dictionary of {(j, m, m1): p} that gives a density p for the
(j, m, m1) matrix element.
Returns
-------
rho: :class: qutip.Qobj
The density matrix in the Dicke basis.
"""
if jmm1 is None:
msg = "Please specify the jmm1 values as a dictionary"
msg += "or use the `excited(N)` function to create an"
msg += "excited state where jmm1 = {(N/2, N/2, N/2): 1}"
raise AttributeError(msg)
nds = _num_dicke_states(N)
rho = np.zeros((nds, nds))
jmm1_dict = jmm1_dictionary(N)[1]
for key in jmm1:
i, k = jmm1_dict[key]
rho[i, k] = jmm1[key]
return Qobj(rho)
[docs]def dicke(N, j, m):
r"""
Generate a Dicke state as a pure density matrix in the Dicke basis.
For instance, the superradiant state given by
:math:`\lvert j, m\rangle = \lvert 1, 0\rangle` for N = 2,
and the state is represented as a density matrix of size (nds, nds) or
(4, 4), with the (1, 1) element set to 1.
Parameters
----------
N: int
The number of two-level systems.
j: float
The eigenvalue j of the Dicke state (j, m).
m: float
The eigenvalue m of the Dicke state (j, m).
Returns
-------
rho: :class: qutip.Qobj
The density matrix.
"""
nds = num_dicke_states(N)
rho = np.zeros((nds, nds))
jmm1_dict = jmm1_dictionary(N)[1]
i, k = jmm1_dict[(j, m, m)]
rho[i, k] = 1.0
return Qobj(rho)
# Uncoupled states in the full Hilbert space. These are returned with the
# choice of the keyword argument `basis="uncoupled"` in the state functions.
def _uncoupled_excited(N):
"""
Generate the density matrix of the excited Dicke state in the full
:math:`2^N` dimensional Hilbert space.
Parameters
----------
N: int
The number of two-level systems.
Returns
-------
psi0: :class: qutip.Qobj
The density matrix for the excited state in the uncoupled basis.
"""
N = int(N)
jz = jspin(N, "z", basis="uncoupled")
en, vn = jz.eigenstates()
psi0 = vn[2 ** N - 1]
return ket2dm(psi0)
def _uncoupled_superradiant(N):
"""
Generate the density matrix of a superradiant state in the full
:math:`2^N`-dimensional Hilbert space.
Parameters
----------
N: int
The number of two-level systems.
Returns
-------
psi0: :class: qutip.Qobj
The density matrix for the superradiant state in the full Hilbert
space.
"""
N = int(N)
jz = jspin(N, "z", basis="uncoupled")
en, vn = jz.eigenstates()
psi0 = vn[2 ** N - (N + 1)]
return ket2dm(psi0)
def _uncoupled_ground(N):
"""
Generate the density matrix of the ground state in the full\
:math:`2^N`-dimensional Hilbert space.
Parameters
----------
N: int
The number of two-level systems.
Returns
-------
psi0: :class: qutip.Qobj
The density matrix for the ground state in the full Hilbert space.
"""
N = int(N)
jz = jspin(N, "z", basis="uncoupled")
en, vn = jz.eigenstates()
psi0 = vn[0]
return ket2dm(psi0)
def _uncoupled_ghz(N):
"""
Generate the density matrix of the GHZ state in the full 2^N
dimensional Hilbert space.
Parameters
----------
N: int
The number of two-level systems.
Returns
-------
ghz: :class: qutip.Qobj
The density matrix for the GHZ state in the full Hilbert space.
"""
N = int(N)
rho = np.zeros((2 ** N, 2 ** N))
rho[0, 0] = 1 / 2
rho[2 ** N - 1, 0] = 1 / 2
rho[0, 2 ** N - 1] = 1 / 2
rho[2 ** N - 1, 2 ** N - 1] = 1 / 2
spin_dim = [2 for i in range(0, N)]
spins_dims = list((spin_dim, spin_dim))
rho = Qobj(rho, dims=spins_dims)
return rho
def _uncoupled_css(N, a, b):
r"""
Generate the density matrix of the CSS state in the full 2^N
dimensional Hilbert space.
The CSS states are non-entangled states given by
:math:`\lvert a,b\rangle = \prod_i(a\lvert1\rangle_i + b\lvert0\rangle_i)`.
Parameters
----------
N: int
The number of two-level systems.
a: complex
The coefficient of the :math:`\lvert1_i\rangle` state.
b: complex
The coefficient of the :math:`\lvert0_i\rangle` state.
Returns
-------
css: :class: qutip.Qobj
The density matrix for the CSS state in the full Hilbert space.
"""
N = int(N)
# 1. Define i_th factorized density matrix in the uncoupled basis
rho_i = np.zeros((2, 2), dtype=complex)
rho_i[0, 0] = a * np.conj(a)
rho_i[1, 1] = b * np.conj(b)
rho_i[0, 1] = a * np.conj(b)
rho_i[1, 0] = b * np.conj(a)
rho_i = Qobj(rho_i)
rho = [0 for i in range(N)]
rho[0] = rho_i
# 2. Place single-two-level-system density matrices in total Hilbert space
for k in range(N - 1):
rho[0] = tensor(rho[0], identity(2))
# 3. Cyclic sequence to create all N factorized density matrices
# |CSS>_i<CSS|_i
a = [i for i in range(N)]
b = [[a[i - i2] for i in range(N)] for i2 in range(N)]
# 4. Create all other N-1 factorized density matrices
# |+><+| = Prod_(i=1)^N |CSS>_i<CSS|_i
for i in range(1, N):
rho[i] = rho[0].permute(b[i])
identity_i = Qobj(np.eye(2 ** N), dims=rho[0].dims, shape=rho[0].shape)
rho_tot = identity_i
for i in range(0, N):
rho_tot = rho_tot * rho[i]
return rho_tot
[docs]def excited(N, basis="dicke"):
"""
Generate the density matrix for the excited state.
This state is given by (N/2, N/2) in the default Dicke basis. If the
argument `basis` is "uncoupled" then it generates the state in a
2**N dim Hilbert space.
Parameters
----------
N: int
The number of two-level systems.
basis: str
The basis to use. Either "dicke" or "uncoupled".
Returns
-------
state: :class: qutip.Qobj
The excited state density matrix in the requested basis.
"""
if basis == "uncoupled":
state = _uncoupled_excited(N)
return state
jmm1 = {(N / 2, N / 2, N / 2): 1}
return dicke_basis(N, jmm1)
[docs]def superradiant(N, basis="dicke"):
"""
Generate the density matrix of the superradiant state.
This state is given by (N/2, 0) or (N/2, 0.5) in the Dicke basis.
If the argument `basis` is "uncoupled" then it generates the state
in a 2**N dim Hilbert space.
Parameters
----------
N: int
The number of two-level systems.
basis: str
The basis to use. Either "dicke" or "uncoupled".
Returns
-------
state: :class: qutip.Qobj
The superradiant state density matrix in the requested basis.
"""
if basis == "uncoupled":
state = _uncoupled_superradiant(N)
return state
if N % 2 == 0:
jmm1 = {(N / 2, 0, 0): 1.0}
return dicke_basis(N, jmm1)
else:
jmm1 = {(N / 2, 0.5, 0.5): 1.0}
return dicke_basis(N, jmm1)
[docs]def css(
N,
x=1 / np.sqrt(2),
y=1 / np.sqrt(2),
basis="dicke",
coordinates="cartesian",
):
r"""
Generate the density matrix of the Coherent Spin State (CSS).
It can be defined as,
:math:`\lvert CSS\rangle = \prod_i^N(a\lvert1\rangle_i+b\lvert0\rangle_i)`
with :math:`a = sin(\frac{\theta}{2})`,
:math:`b = e^{i \phi}\cos(\frac{\theta}{2})`.
The default basis is that of Dicke space
:math:`\lvert j, m\rangle \langle j, m'\rvert`.
The default state is the symmetric CSS,
:math:`\lvert CSS\rangle = \lvert+\rangle`.
Parameters
----------
N: int
The number of two-level systems.
x, y: float
The coefficients of the CSS state.
basis: str
The basis to use. Either "dicke" or "uncoupled".
coordinates: str
Either "cartesian" or "polar". If polar then the coefficients
are constructed as sin(x/2), cos(x/2)e^(iy).
Returns
-------
rho: :class: qutip.Qobj
The CSS state density matrix.
"""
if coordinates == "polar":
a = np.cos(0.5 * x) * np.exp(1j * y)
b = np.sin(0.5 * x)
else:
a = x
b = y
if basis == "uncoupled":
return _uncoupled_css(N, a, b)
nds = num_dicke_states(N)
num_ladders = num_dicke_ladders(N)
rho = dok_matrix((nds, nds), dtype=np.complex128)
# loop in the allowed matrix elements
jmm1_dict = jmm1_dictionary(N)[1]
j = 0.5 * N
mmax = int(2 * j + 1)
for i in range(0, mmax):
m = j - i
psi_m = (
np.sqrt(float(energy_degeneracy(N, m)))
* a ** (N * 0.5 + m)
* b ** (N * 0.5 - m)
)
for i1 in range(0, mmax):
m1 = j - i1
row_column = jmm1_dict[(j, m, m1)]
psi_m1 = (
np.sqrt(float(energy_degeneracy(N, m1)))
* np.conj(a) ** (N * 0.5 + m1)
* np.conj(b) ** (N * 0.5 - m1)
)
rho[row_column] = psi_m * psi_m1
return Qobj(rho)
[docs]def ghz(N, basis="dicke"):
"""
Generate the density matrix of the GHZ state.
If the argument `basis` is "uncoupled" then it generates the state
in a :math:`2^N`-dimensional Hilbert space.
Parameters
----------
N: int
The number of two-level systems.
basis: str
The basis to use. Either "dicke" or "uncoupled".
Returns
-------
state: :class: qutip.Qobj
The GHZ state density matrix in the requested basis.
"""
if basis == "uncoupled":
return _uncoupled_ghz(N)
nds = _num_dicke_states(N)
rho = dok_matrix((nds, nds), dtype=np.complex128)
rho[0, 0] = 1 / 2
rho[N, N] = 1 / 2
rho[N, 0] = 1 / 2
rho[0, N] = 1 / 2
return Qobj(rho)
[docs]def ground(N, basis="dicke"):
"""
Generate the density matrix of the ground state.
This state is given by (N/2, -N/2) in the Dicke basis. If the argument
`basis` is "uncoupled" then it generates the state in a
:math:`2^N`-dimensional Hilbert space.
Parameters
----------
N: int
The number of two-level systems.
basis: str
The basis to use. Either "dicke" or "uncoupled"
Returns
-------
state: :class: qutip.Qobj
The ground state density matrix in the requested basis.
"""
if basis == "uncoupled":
state = _uncoupled_ground(N)
return state
nds = _num_dicke_states(N)
rho = dok_matrix((nds, nds), dtype=np.complex128)
rho[N, N] = 1
return Qobj(rho)
[docs]def identity_uncoupled(N):
"""
Generate the identity in a :math:`2^N`-dimensional Hilbert space.
The identity matrix is formed from the tensor product of N TLSs.
Parameters
----------
N: int
The number of two-level systems.
Returns
-------
identity: :class: qutip.Qobj
The identity matrix.
"""
N = int(N)
rho = np.zeros((2 ** N, 2 ** N))
for i in range(0, 2 ** N):
rho[i, i] = 1
spin_dim = [2 for i in range(0, N)]
spins_dims = list((spin_dim, spin_dim))
identity = Qobj(rho, dims=spins_dims)
return identity
[docs]def block_matrix(N, elements="ones"):
"""Construct the block-diagonal matrix for the Dicke basis.
Parameters
----------
N : int
Number of two-level systems.
elements : str {'ones' (default),'degeneracy'}
Returns
-------
block_matr : ndarray
A 2D block-diagonal matrix with dimension (nds,nds),
where nds is the number of Dicke states for N two-level
systems. Filled with ones or the value of degeneracy
at each matrix element.
"""
# create a list with the sizes of the blocks, in order
blocks_dimensions = int(N / 2 + 1 - 0.5 * (N % 2))
blocks_list = [
(2 * (i + 1 * (N % 2)) + 1 * ((N + 1) % 2))
for i in range(blocks_dimensions)
]
blocks_list = np.flip(blocks_list, 0)
# create a list with each block matrix as element
square_blocks = []
k = 0
for i in blocks_list:
if elements == "ones":
square_blocks.append(np.ones((i, i)))
elif elements == "degeneracy":
j = N / 2 - k
dj = state_degeneracy(N, j)
square_blocks.append(dj * np.ones((i, i)))
k = k + 1
return block_diag(square_blocks)
# ============================================================================
# Adding a faster version to make a Permutational Invariant matrix
# ============================================================================
[docs]def tau_column(tau, k, j):
"""
Determine the column index for the non-zero elements of the matrix for a
particular row `k` and the value of `j` from the Dicke space.
Parameters
----------
tau: str
The tau function to check for this `k` and `j`.
k: int
The row of the matrix M for which the non zero elements have
to be calculated.
j: float
The value of `j` for this row.
"""
# In the notes, we indexed from k = 1, here we do it from k = 0
k = k + 1
mapping = {
"tau3": k - (2 * j + 3),
"tau2": k - 1,
"tau4": k + (2 * j - 1),
"tau5": k - (2 * j + 2),
"tau1": k,
"tau6": k + (2 * j),
"tau7": k - (2 * j + 1),
"tau8": k + 1,
"tau9": k + (2 * j + 1),
}
# we need to decrement k again as indexing is from 0
return int(mapping[tau] - 1)
[docs]class Pim(object):
"""
The Permutation Invariant Matrix class.
Initialize the class with the parameters for generating a Permutation
Invariant matrix which evolves a given diagonal initial state `p` as:
dp/dt = Mp
Parameters
----------
N: int
The number of two-level systems.
emission: float
Incoherent emission coefficient (also nonradiative emission).
default: 0.0
dephasing: float
Local dephasing coefficient.
default: 0.0
pumping: float
Incoherent pumping coefficient.
default: 0.0
collective_emission: float
Collective (superradiant) emmission coefficient.
default: 0.0
collective_pumping: float
Collective pumping coefficient.
default: 0.0
collective_dephasing: float
Collective dephasing coefficient.
default: 0.0
Attributes
----------
N: int
The number of two-level systems.
emission: float
Incoherent emission coefficient (also nonradiative emission).
default: 0.0
dephasing: float
Local dephasing coefficient.
default: 0.0
pumping: float
Incoherent pumping coefficient.
default: 0.0
collective_emission: float
Collective (superradiant) emmission coefficient.
default: 0.0
collective_dephasing: float
Collective dephasing coefficient.
default: 0.0
collective_pumping: float
Collective pumping coefficient.
default: 0.0
M: dict
A nested dictionary of the structure {row: {col: val}} which holds
non zero elements of the matrix M
"""
def __init__(
self,
N,
emission=0.0,
dephasing=0,
pumping=0,
collective_emission=0,
collective_pumping=0,
collective_dephasing=0,
):
self.N = N
self.emission = emission
self.dephasing = dephasing
self.pumping = pumping
self.collective_pumping = collective_pumping
self.collective_dephasing = collective_dephasing
self.collective_emission = collective_emission
self.M = {}
[docs] def isdicke(self, dicke_row, dicke_col):
"""
Check if an element in a matrix is a valid element in the Dicke space.
Dicke row: j value index. Dicke column: m value index.
The function returns True if the element exists in the Dicke space and
False otherwise.
Parameters
----------
dicke_row, dicke_col : int
Index of the element in Dicke space which needs to be checked
"""
rows = self.N + 1
cols = 0
if (self.N % 2) == 0:
cols = int(self.N / 2 + 1)
else:
cols = int(self.N / 2 + 1 / 2)
if (dicke_row > rows) or (dicke_row < 0):
return False
if (dicke_col > cols) or (dicke_col < 0):
return False
if (dicke_row < int(rows / 2)) and (dicke_col > dicke_row):
return False
if (dicke_row >= int(rows / 2)) and (rows - dicke_row <= dicke_col):
return False
else:
return True
[docs] def tau_valid(self, dicke_row, dicke_col):
"""
Find the Tau functions which are valid for this value of (dicke_row,
dicke_col) given the number of TLS. This calculates the valid tau
values and reurns a dictionary specifying the tau function name and
the value.
Parameters
----------
dicke_row, dicke_col : int
Index of the element in Dicke space which needs to be checked.
Returns
-------
taus: dict
A dictionary of key, val as {tau: value} consisting of the valid
taus for this row and column of the Dicke space element.
"""
tau_functions = [
self.tau3,
self.tau2,
self.tau4,
self.tau5,
self.tau1,
self.tau6,
self.tau7,
self.tau8,
self.tau9,
]
N = self.N
if self.isdicke(dicke_row, dicke_col) is False:
return False
# The 3x3 sub matrix surrounding the Dicke space element to
# run the tau functions
indices = [
(dicke_row + x, dicke_col + y)
for x in range(-1, 2)
for y in range(-1, 2)
]
taus = {}
for idx, tau in zip(indices, tau_functions):
if self.isdicke(idx[0], idx[1]):
j, m = self.calculate_j_m(idx[0], idx[1])
taus[tau.__name__] = tau(j, m)
return taus
[docs] def calculate_j_m(self, dicke_row, dicke_col):
"""
Get the value of j and m for the particular Dicke space element.
Parameters
----------
dicke_row, dicke_col: int
The row and column from the Dicke space matrix
Returns
-------
j, m: float
The j and m values.
"""
N = self.N
j = N / 2 - dicke_col
m = N / 2 - dicke_row
return (j, m)
[docs] def calculate_k(self, dicke_row, dicke_col):
"""
Get k value from the current row and column element in the Dicke space.
Parameters
----------
dicke_row, dicke_col: int
The row and column from the Dicke space matrix.
Returns
-------
k: int
The row index for the matrix M for given Dicke space
element.
"""
N = self.N
if dicke_row == 0:
k = dicke_col
else:
k = int(
((dicke_col) / 2) * (2 * (N + 1) - 2 * (dicke_col - 1))
+ (dicke_row - (dicke_col))
)
return k
[docs] def coefficient_matrix(self):
"""
Generate the matrix M governing the dynamics for diagonal cases.
If the initial density matrix and the Hamiltonian is diagonal, the
evolution of the system is given by the simple ODE: dp/dt = Mp.
"""
N = self.N
nds = num_dicke_states(N)
rows = self.N + 1
cols = 0
sparse_M = lil_matrix((nds, nds), dtype=float)
if (self.N % 2) == 0:
cols = int(self.N / 2 + 1)
else:
cols = int(self.N / 2 + 1 / 2)
for (dicke_row, dicke_col) in np.ndindex(rows, cols):
if self.isdicke(dicke_row, dicke_col):
k = int(self.calculate_k(dicke_row, dicke_col))
row = {}
taus = self.tau_valid(dicke_row, dicke_col)
for tau in taus:
j, m = self.calculate_j_m(dicke_row, dicke_col)
current_col = tau_column(tau, k, j)
sparse_M[k, int(current_col)] = taus[tau]
return sparse_M.tocsr()
[docs] def solve(self, rho0, tlist, options=None):
"""
Solve the ODE for the evolution of diagonal states and Hamiltonians.
"""
if options is None:
options = Options()
output = Result()
output.solver = "pisolve"
output.times = tlist
output.states = []
output.states.append(Qobj(rho0))
rhs_generate = lambda y, tt, M: M.dot(y)
rho0_flat = np.diag(np.real(rho0.full()))
L = self.coefficient_matrix()
rho_t = odeint(rhs_generate, rho0_flat, tlist, args=(L,))
for r in rho_t[1:]:
diag = np.diag(r)
output.states.append(Qobj(diag))
return output
[docs] def tau1(self, j, m):
"""
Calculate coefficient matrix element relative to (j, m, m).
"""
yS = self.collective_emission
yL = self.emission
yD = self.dephasing
yP = self.pumping
yCP = self.collective_pumping
N = float(self.N)
spontaneous = yS * (1 + j - m) * (j + m)
losses = yL * (N / 2 + m)
pump = yP * (N / 2 - m)
collective_pump = yCP * (1 + j + m) * (j - m)
if j == 0:
dephase = yD * N / 4
else:
dephase = yD * (N / 4 - m ** 2 * ((1 + N / 2) / (2 * j * (j + 1))))
t1 = spontaneous + losses + pump + dephase + collective_pump
return -t1
[docs] def tau2(self, j, m):
"""
Calculate coefficient matrix element relative to (j, m+1, m+1).
"""
yS = self.collective_emission
yL = self.emission
N = float(self.N)
spontaneous = yS * (1 + j - m) * (j + m)
losses = yL * (
((N / 2 + 1) * (j - m + 1) * (j + m)) / (2 * j * (j + 1))
)
t2 = spontaneous + losses
return t2
[docs] def tau3(self, j, m):
"""
Calculate coefficient matrix element relative to (j+1, m+1, m+1).
"""
yL = self.emission
N = float(self.N)
num = (j + m - 1) * (j + m) * (j + 1 + N / 2)
den = 2 * j * (2 * j + 1)
t3 = yL * (num / den)
return t3
[docs] def tau4(self, j, m):
"""
Calculate coefficient matrix element relative to (j-1, m+1, m+1).
"""
yL = self.emission
N = float(self.N)
num = (j - m + 1) * (j - m + 2) * (N / 2 - j)
den = 2 * (j + 1) * (2 * j + 1)
t4 = yL * (num / den)
return t4
[docs] def tau5(self, j, m):
"""
Calculate coefficient matrix element relative to (j+1, m, m).
"""
yD = self.dephasing
N = float(self.N)
num = (j - m) * (j + m) * (j + 1 + N / 2)
den = 2 * j * (2 * j + 1)
t5 = yD * (num / den)
return t5
[docs] def tau6(self, j, m):
"""
Calculate coefficient matrix element relative to (j-1, m, m).
"""
yD = self.dephasing
N = float(self.N)
num = (j - m + 1) * (j + m + 1) * (N / 2 - j)
den = 2 * (j + 1) * (2 * j + 1)
t6 = yD * (num / den)
return t6
[docs] def tau7(self, j, m):
"""
Calculate coefficient matrix element relative to (j+1, m-1, m-1).
"""
yP = self.pumping
N = float(self.N)
num = (j - m - 1) * (j - m) * (j + 1 + N / 2)
den = 2 * j * (2 * j + 1)
t7 = yP * (float(num) / den)
return t7
[docs] def tau8(self, j, m):
"""
Calculate coefficient matrix element relative to (j, m-1, m-1).
"""
yP = self.pumping
yCP = self.collective_pumping
N = float(self.N)
num = (1 + N / 2) * (j - m) * (j + m + 1)
den = 2 * j * (j + 1)
pump = yP * (float(num) / den)
collective_pump = yCP * (j - m) * (j + m + 1)
t8 = pump + collective_pump
return t8
[docs] def tau9(self, j, m):
"""
Calculate coefficient matrix element relative to (j-1, m-1, m-1).
"""
yP = self.pumping
N = float(self.N)
num = (j + m + 1) * (j + m + 2) * (N / 2 - j)
den = 2 * (j + 1) * (2 * j + 1)
t9 = yP * (float(num) / den)
return t9