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qubit_models.py
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qubit_models.py
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import numpy as np
from scipy.misc import factorial
from scipy.integrate import quad, simps
from scipy.special import hermite, eval_hermite
from scipy.sparse.linalg import eigsh
import itertools
sx = np.array([[0, 1], [1, 0]], complex)
sy = np.array([[0 , -1j], [1j, 0]])
sz = np.array([[1, 0], [0, -1]], complex)
s0 = np.array([[1, 0], [0, 1]], complex)
def inner(v1, v2):
return np.dot(np.conj(v1), v2)
def dispersion(spectrum, k=0):
if spectrum.ndim == 1:
return np.ptp(spectrum)
else:
return np.ptp(spectrum[k], axis = -1)
def average(spectrum, k=0):
if spectrum.ndim == 1:
return np.average(spectrum)
else:
return np.average(spectrum[k], axis = -1)
def difference(spectrum, i, j):
return spectrum[j] - spectrum[i]
def row_indices(N, M):
return np.outer(np.arange(N), np.ones(M)).astype(int)
def column_indices(N, M):
return row_indices(N, M).T
def hermite_norm(n):
return (np.pi**0.25 * np.sqrt(2.**n * factorial(n)))
def optimize(qubit, K, tol=1e-10):
# Finds optimal size of basis for convergence of first K eigenvalues
qubit.N = K
evals = np.linalg.eigvalsh(qubit.hamiltonian())[:K]
error = 1
count = 0
while error > tol:
count += 1
qubit.N += 1
new_evals = np.linalg.eigvalsh(qubit.hamiltonian())[:K]
error = np.linalg.norm(new_evals - evals)
evals = new_evals
qubit.K = K
qubit.optimized = True
qubit.tol = tol
class Fluxonium:
def __init__(self, parameters):
for (key, value) in parameters.items():
setattr(self, key, value)
self.optimized = False
self.D = 1000
self.phimax = 10
if not hasattr(self, 'majoranas'):
self.majoranas = False
def omega_LC(self):
return np.sqrt(8 * self.Ec * self.El)
def rescaling_factor(self):
return (8 * self.Ec / self.El)**0.25
def potential_energy(self, phi):
return self.potential(phi + self.flux) + 0.5 * self.El * phi**2
def number_operator(self):
sqrts = np.sqrt(np.arange(1, self.N))
s = self.rescaling_factor()
nop = (np.sqrt(2) * 1j / s) * (np.diag(sqrts, k=1) - np.diag(sqrts, k=-1))
return np.kron(nop, s0) if self.majoranas else nop
def phase_operator(self):
sqrts = np.sqrt(np.arange(1, self.N))
s = self.rescaling_factor()
phase_op = (s / np.sqrt(2)) * (np.diag(sqrts, k=1) + np.diag(sqrts, k=-1))
return np.kron(phase_op, s0) if self.majoranas else phase_op
def potential_operator(self, potential):
grid = np.linspace(-self.phimax, self.phimax, self.D)
r = np.tile(row_indices(self.N, self.N), (self.D, 1, 1))
c = np.tile(column_indices(self.N, self.N), (self.D, 1, 1))
phis = np.rollaxis(np.tile(grid, (self.N, self.N, 1)), 2)
fs = np.exp(-phis**2) * eval_hermite(r, phis) * eval_hermite(c, phis)
fs *= potential(self.rescaling_factor() * phis + self.flux)
fs /= hermite_norm(r) * hermite_norm(c)
vop = simps(fs, np.linspace(-self.phimax, self.phimax, self.D), axis=0)
return vop
def hamiltonian(self):
ham = self.omega_LC() * np.diag(np.arange(self.N) + 0.5)
ham += self.potential_operator(self.potential)
if self.majoranas:
ham = np.kron(ham, s0)
vx, vy, vz = self.majorana_couplings()
ham += vx * np.kron(np.eye(self.N), sx)
ham += np.kron(self.potential_operator(vy), sy)
ham += np.kron(self.potential_operator(vz), sz)
return ham
def majorana_couplings(self):
P = -1. if self.parity =='odd' else 1.
E = self.couplings
S = self.shifts
vx = - (E[0,1] + P * E[2, 3])
vy = lambda phi: (E[0,2] * np.cos((phi - S[0,2])/2)
+ P * E[1,3] * np.cos((phi - S[1,3])/2))
vz = lambda phi: - (E[1,2] * np.cos((phi - S[1,2])/2)
+ P * E[0,3] * np.cos((phi - S[0,3])/2))
return vx, vy, vz
def energies(self, return_evecs=False):
if return_evecs:
evals, evecs = np.linalg.eigh(self.hamiltonian())
return evals, evecs
else:
evals = np.linalg.eigvalsh(self.hamiltonian())
return evals
def energies_vs_flux(self, fluxes, subtract_minimum=True):
energies = []
for flux in fluxes:
self.flux = flux
energies.append(self.energies())
energies = np.vstack(energies).T
if subtract_minimum == True:
energies -= np.min(energies)
return energies
def number_matrix_elements(self, evecs):
nop = self.number_operator()
return inner(evecs.T, nop.dot(evecs))
def phase_matrix_elements(self, evecs):
phase = self.phase_operator()
return inner(evecs.T, phase.dot(evecs))
class ChargeQubit:
def __init__(self, parameters):
for (key, value) in parameters.items():
setattr(self, key, value)
self.D = 1001 # number of samples for numerical integration with simps
def size(self):
return 2 * self.N + 1
def numbers(self):
return np.arange(-self.N, self.N + 1, 1, dtype=int)
def number_operator(self):
nop = np.diag(self.numbers()) + 0j
return nop
def josephson_potential_coefficients(self, tol=1e-10):
grid = np.linspace(0, 2*np.pi, self.D)
S = np.ceil((self.N+1)/2).astype(int) # Number of coefficients we take into account
ns = np.tile(np.arange(S), (self.D, 1)).T
phis = np.tile(grid, (S, 1))
re = simps(self.potential(phis) * np.cos(ns * phis) / (2*np.pi),
grid, axis=1)
im = simps(self.potential(phis) * np.sin(ns * phis) / (2*np.pi),
grid, axis=1)
re[np.abs(re) < tol] = 0
im[np.abs(im) < tol] = 0
return re - 1j * im
def josephson_potential_operator(self):
cs = self.josephson_potential_coefficients()
vop = np.diag(np.full(self.size(), cs[0]))
js = np.nonzero(cs[1:])[0] + 1
for j in js:
vop += cs[j] * np.eye(self.size(), k=-2*j)
vop += np.conj(cs[j]) * np.eye(self.size(), k=2*j)
return vop
def majorana_hamiltonian(self):
P, E, S = self.parity, self.couplings, self.shifts
C = E * np.exp(1j * S)
signs = (-1)**np.abs(self.numbers())
HM = 0.5 * (E[0,1] + P * E[2,3]) * np.diag(signs) + 0j
HM -= 0.5 * (C[1,2] + P * C[0,3]) * np.eye(self.size(), k=-1)
HM -= 0.5 * (C[1,2] + P * C[0,3]) * np.eye(self.size(), k=1)
signs = (-1)**np.abs(self.numbers()[1:])
HM -= 0.5 * 1j * (C[0,2] + P * C[1,3]) * np.diag(signs, k=-1)
HM += 0.5 * 1j * (C[0,2] + P * C[1,3]) * np.diag(signs, k=1)
return HM
def hamiltonian(self):
N = self.number_operator() - self.ng * np.eye(self.size())
V = self.josephson_potential_operator()
HM = self.majorana_hamiltonian() if self.majoranas else 0
return self.Ec * N.dot(N) + V + HM
def dipole_matrix_elements(self, evecs):
nop = self.number_operator()
return inner(evecs.T, nop.dot(evecs))
def use_cosine_potential(self):
self.potential = lambda phi: self.Ej * (1 - np.cos(phi))
def use_abs_potential(self):
def abs_potential(phi):
V = 0
for T in self.transmission:
V += self.gap * (1 - np.sqrt(1 - T * np.sin(phi/2)**2))
return V
self.potential = abs_potential
def energies(self, branch='none', return_evecs=False):
if return_evecs:
evals, evecs = np.linalg.eigh(self.hamiltonian())
else:
evals = np.linalg.eigvalsh(self.hamiltonian())
if branch is not 'none':
ns = np.arange(self.size())
es = np.where(ns[::2] % 4 != 0, ns[::2] + 1, ns[::2])[:-1]
os = ns[np.isin(ns, es, invert=True)][:-1]
ngmin = 0 if branch == 'even' else 1
evals = evals[es] if np.rint(self.ng) % 2 == ngmin else evals[os]
if return_evecs:
evecs = evecs[es] if np.rint(self.ng) % 2 == ngmin else evecs[os]
return (evals, evecs) if return_evecs else evals
def energies_vs_ng(self, ngs, branch='none', subtract_minimum=True):
energies = []
for (i, ng) in enumerate(ngs):
self.ng = ng
energies.append(self.energies(branch))
energies = np.vstack(energies).T
if subtract_minimum == True:
energies -= np.min(energies)
return energies
def energies_vs_flux(self, fluxes, branch='none', subtract_minimum=True):
energies = []
for (i, flux) in enumerate(fluxes):
self.flux = flux
if self.majoranas == True:
self.shifts[0,3] = flux/2
energies.append(self.energies(branch))
energies = np.vstack(energies).T
if subtract_minimum == True:
energies -= np.min(energies)
return energies
def frequency_spectrum(self, ngs, freqs, initial_states=[0],
kappa=0.1, branch='none'):
"""
Returns an image of the frequency spectrum of the system
as a function of ng.
Parameters:
-----------
ngs:
list of induced charges to be included in the plot.
freqs:
list of frequencies to be included in the plot.
initial_states:
list of initial states to be included in the frequency spectrum.
For instance, if [0,1] the plot includes all transitions starting
from ground and excited states.
kappa:
Linewidth (same for every transition, for now
it's just a convenience parameter with no physical input).
"""
spectrum = np.zeros((len(ngs), len(freqs)))
for (n, ng) in enumerate(ngs):
self.ng = ng
evals, evecs = self.energies(branch, return_evecs=True)
g = self.dipole_matrix_elements(evecs)
for i in initial_states:
omegas = evals[i+1:] - evals[i]
for (j, omega) in enumerate(omegas):
spectrum[n] += (0.25 * np.abs(g[i,i+j+1])**2 * kappa**2
/ ((freqs-omega)**2 + 0.25 * kappa**2))
return spectrum
def frequency_spectrum_vs_flux(self, fluxes, freqs, initial_states=[0],
kappa=0.1, branch='none'):
spectrum = np.zeros((len(fluxes), len(freqs)))
for (n, flux) in enumerate(fluxes):
self.flux = flux
self.shifts[0,3] = flux/2
evals, evecs = self.energies(branch, return_evecs=True)
g = self.dipole_matrix_elements(evecs)
for i in initial_states:
omegas = evals[i+1:] - evals[i]
for (j, omega) in enumerate(omegas):
spectrum[n] += (0.25 * np.abs(g[i,i+j+1])**2 * kappa**2
/ ((freqs-omega)**2 + 0.25 * kappa**2))
return spectrum