# Source code for qutip.scattering

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"""
Photon scattering in quantum optical systems

This module includes a collection of functions for numerically computing photon
scattering in driven arbitrary systems coupled to some configuration of output
waveguides. The implementation of these functions closely follows the
mathematical treatment given in K.A. Fischer, et. al., Scattering of Coherent
Pulses from Quantum Optical Systems (2017, arXiv:1710.02875).
"""
# Author:  Ben Bartlett
# Contact: benbartlett@stanford.edu

import numpy as np
from itertools import product, combinations_with_replacement
from qutip import propagator, Options, basis, tensor, zero_ket, Qobj

__all__ = ['temporal_basis_vector',
'temporal_scattered_state',
'scattering_probability']

class Evolver:
"""
A caching class which takes a Hamiltonian and a list of times to calculate
and memoize propagators for the system between any two times as demanded.

Parameters
----------
H : :class: qutip.Qobj or list
System-waveguide(s) Hamiltonian or effective Hamiltonian in Qobj or
list-callback format. If construct_effective_hamiltonian is not
specified, an effective Hamiltonian is constructed from H and c_ops.
times : list-like
List of times to evaluate propagators over.
options : :class: qutip.Options
Solver options to use when computing propagators.

Attributes
----------
H : :class: qutip.Qobj or list
System-waveguide(s) Hamiltonian, may be time-dependent.
tlist : list-like
List of times to evaluate propagators over.
propagators : (dict of float: (dict of float: :class: qutip.Qobj))
Dictionary of dictionaries of propagator objects with keys of
evaluation times, e.g. propagators[t2][t1] returns U[t2,t1].
"""
def __init__(self, H, tlist, options=None):
self.H = H
self.tlist = tlist
if options is None:
self.options = Options(nsteps=10000, normalize_output=False)
else:
self.options = options
# Make a blank nested dictionary to store propagators
self.propagators = dict.fromkeys(tlist)
for t in tlist:
self.propagators[t] = dict.fromkeys(tlist)

def prop(self, tf, ti):
"""Compute U[t2,t1] where t2 > t1 or return the cached operator.

Parameters
----------
tf : float
Final time to compute the propagator U[tf, ti].
ti : float
Initial time to compute the propagator U[tf,ti].

Returns
-------
propagator : :class: qutip.Qobj
The propagation operator.
"""
left, right = np.searchsorted(self.tlist, [ti, tf], side='left')
t1, t2 = self.tlist[left], self.tlist[right]
if self.propagators[t2][t1] is None:
self.propagators[t2][t1] = propagator(self.H, [t1, t2],
options=self.options,
unitary_mode='single')
# Something is still broken about batch unitary mode (see #807)
return self.propagators[t2][t1]

def set_partition(collection, num_sets):
"""
Enumerate all ways of partitioning collection into num_sets different lists,
e.g. list(set_partition([1,2], 2)) = [[[1, 2], []], [[1], [2]], [[2], [1]],
[[], [1, 2]]].

Parameters
----------
collection : iterable
Collection to generate a set partition of.
num_sets : int
Number of sets to partition collection into.

Returns
-------
partition : iterable
The partitioning of collection into num_sets sets.
"""
for partitioning in product(range(num_sets), repeat=len(collection)):
partition = [[] for _ in range(num_sets)]
for i, set_index in enumerate(partitioning):
partition[set_index].append(collection[i])
yield tuple(tuple(indices) for indices in partition)

def photon_scattering_operator(evolver, c_ops, taus_list):
"""
Compute the scattering operator for a system emitting into multiple
waveguides.

Parameters
----------
evolver : :class: qutip.scattering.Evolver
Evolver-wrapped Hamiltonian describing the system.
c_ops : list
list of collapse operators for each waveguide; these are assumed to
include spontaneous decay rates, e.g.
:math:\\sigma = \\sqrt \\gamma \\cdot a
taus_list : list-like
List of (list of emission times) for each waveguide.

Returns
-------
omega : :class: qutip.Qobj
The temporal scattering operator with dimensionality equal to the
system state.
"""
omega = 1

# Extract the full list of taus
taus = [(0.0, -1)]  # temporal "ground state" for arbitrary waveguide
for i, tau_wg in enumerate(taus_list):
for tau in tau_wg:
taus.append((tau, i))
taus.sort(key = lambda tup: tup[0])  # sort taus by time

# Compute Prod Ueff(tq, tq-1)
for i in range(1, len(taus)):
tq, q = taus[i]
tprev, _ = taus[i - 1]
omega = c_ops[q] * evolver.prop(tq, tprev) * omega

# Add the <0|Uff(TP, tm)|0> term
tmax = evolver.tlist[-1]
taumax, _ = taus[-1]
# if taus[-1] < tmax:
omega = evolver.prop(tmax, taumax) * omega

return omega

[docs]def temporal_basis_vector(waveguide_emission_indices, n_time_bins):
"""
Generate a temporal basis vector for emissions at specified time bins into
specified waveguides.

Parameters
----------
waveguide_emission_indices : list or tuple
List of indices where photon emission occurs for each waveguide,
e.g. [[t1_wg1], [t1_wg2, t2_wg2], [], [t1_wg4, t2_wg4, t3_wg4]].
n_time_bins : int
Number of time bins; the range over which each index can vary.

Returns
-------
temporal_basis_vector : :class: qutip.Qobj
A basis vector representing photon scattering at the specified indices.
If there are W waveguides, T times, and N photon emissions, then the
basis vector has dimensionality (W*T)^N.
"""
# Cast waveguide_emission_indices to list for mutability
waveguide_emission_indices = [list(i) for i in waveguide_emission_indices]

# Calculate total number of waveguides
W = len(waveguide_emission_indices)

# Calculate total number of emissions
num_emissions = sum([len(waveguide_indices) for waveguide_indices in
waveguide_emission_indices])
if num_emissions == 0:
return basis(W * n_time_bins, 0)

# Pad the emission indices with zeros
offset_indices = []
for i, wg_indices in enumerate(waveguide_emission_indices):
offset_indices += [index + (i * n_time_bins) for index in wg_indices]

# Return an appropriate tensor product state
return tensor([basis(n_time_bins * W, i) for i in offset_indices])

[docs]def temporal_scattered_state(H, psi0, n_emissions, c_ops, tlist,
system_zero_state=None,
construct_effective_hamiltonian=True):
"""
Compute the scattered n-photon state projected onto the temporal basis.

Parameters
----------
H : :class: qutip.Qobj or list
System-waveguide(s) Hamiltonian or effective Hamiltonian in Qobj or
list-callback format. If construct_effective_hamiltonian is not
specified, an effective Hamiltonian is constructed from H and
c_ops.
psi0 : :class: qutip.Qobj
Initial state density matrix :math:\\rho(t_0) or state vector
:math:\\psi(t_0).
n_emissions : int
Number of photon emissions to calculate.
c_ops : list
List of collapse operators for each waveguide; these are assumed to
include spontaneous decay rates, e.g.
:math:\\sigma = \\sqrt \\gamma \\cdot a
tlist : array_like
List of times for :math:\\tau_i. tlist should contain 0 and exceed
the pulse duration / temporal region of interest.
system_zero_state : :class: qutip.Qobj
State representing zero excitations in the system. Defaults to
:math:\\psi(t_0)
construct_effective_hamiltonian : bool
Whether an effective Hamiltonian should be constructed from H and c_ops:
:math:H_{eff} = H - \\frac{i}{2} \\sum_n \\sigma_n^\\dagger \\sigma_n
Default: True.

Returns
-------
phi_n : :class: qutip.Qobj
The scattered bath state projected onto the temporal basis given by
tlist. If there are W waveguides, T times, and N photon emissions, then
the state is a tensor product state with dimensionality T^(W*N).
"""
T = len(tlist)
W = len(c_ops)

if n_emissions == 0:
phi_n = zero_ket(W * T)
else:
phi_n = tensor([zero_ket(W * T)] * n_emissions)

if construct_effective_hamiltonian:
# Construct an effective Hamiltonian from system hamiltonian and c_ops
if type(H) is Qobj:
Heff = H - 1j / 2 * sum([op.dag() * op for op in c_ops])
elif type(H) is list:
Heff = H + [-1j / 2 * sum([op.dag() * op for op in c_ops])]
else:
raise TypeError("Hamiltonian must be Qobj or list-callback format")
else:
Heff = H

evolver = Evolver(Heff, tlist)

all_emission_indices = combinations_with_replacement(range(T), n_emissions)

if system_zero_state is None:
system_zero_state = psi0

# Compute <omega_tau> for all combinations of tau
for emission_indices in all_emission_indices:
# Consider unique partitionings of emission times into waveguides
partition = tuple(set(set_partition(emission_indices, W)))
# Consider all possible partitionings of time bins by waveguide
for indices in partition:
taus = [[tlist[i] for i in wg_indices] for wg_indices in indices]
omega = photon_scattering_operator(evolver, c_ops, taus)
phi_n_amp = system_zero_state.dag() * omega * psi0
# Add scatter amplitude times temporal basis to overall state
phi_n += phi_n_amp * temporal_basis_vector(indices, T)

return phi_n

[docs]def scattering_probability(H, psi0, n_emissions, c_ops, tlist,
system_zero_state=None,
construct_effective_hamiltonian=True):
"""
Compute the integrated probability of scattering n photons in an arbitrary
system. This function accepts a nonlinearly spaced array of times.

Parameters
----------
H : :class: qutip.Qobj or list
System-waveguide(s) Hamiltonian or effective Hamiltonian in Qobj or
list-callback format. If construct_effective_hamiltonian is not
specified, an effective Hamiltonian is constructed from H and
c_ops.
psi0 : :class: qutip.Qobj
Initial state density matrix :math:\\rho(t_0) or state vector
:math:\\psi(t_0).
n_emissions : int
Number of photons emitted by the system (into any combination of
waveguides).
c_ops : list
List of collapse operators for each waveguide; these are assumed to
include spontaneous decay rates, e.g.
:math:\\sigma = \\sqrt \\gamma \\cdot a.
tlist : array_like
List of times for :math:\\tau_i. tlist should contain 0 and exceed
the pulse duration / temporal region of interest; tlist need not be
linearly spaced.
system_zero_state : :class: qutip.Qobj
State representing zero excitations in the system. Defaults to
basis(systemDims, 0).
construct_effective_hamiltonian : bool
Whether an effective Hamiltonian should be constructed from H and c_ops:
:math:H_{eff} = H - \\frac{i}{2} \\sum_n \\sigma_n^\\dagger \\sigma_n
Default: True.

Returns
-------
scattering_prob : float
The probability of scattering n photons from the system over the time
range specified.
"""
phi_n = temporal_scattered_state(H, psi0, n_emissions, c_ops, tlist,
system_zero_state,
construct_effective_hamiltonian)
T = len(tlist)
W = len(c_ops)

# Compute <omega_tau> for all combinations of tau
all_emission_indices = combinations_with_replacement(range(T), n_emissions)
probs = np.zeros([T] * n_emissions)

# Project scattered state onto temporal basis
for emit_indices in all_emission_indices:
# Consider unique emission time partitionings
partition = tuple(set(set_partition(emit_indices, W)))
# wg_indices_list = list(set_partition(indices, W))
for wg_indices in partition:
projector = temporal_basis_vector(wg_indices, T)
amplitude = (projector.dag() * phi_n).full().item()
probs[emit_indices] += np.real(amplitude.conjugate() * amplitude)

# Iteratively integrate to obtain single value
while probs.shape != ():
probs = np.trapz(probs, x = tlist)
return np.abs(probs)