# zeropi.py
#
# This file is part of scqubits.
#
# Copyright (c) 2019, Jens Koch and Peter Groszkowski
# All rights reserved.
#
# This source code is licensed under the BSD-style license found in the
# LICENSE file in the root directory of this source tree.
############################################################################
import os
import warnings
import numpy as np
from scipy import sparse
import scqubits.core.central_dispatch as dispatch
import scqubits.core.constants as constants
import scqubits.core.descriptors as descriptors
import scqubits.core.discretization as discretization
from scqubits.core.noise import NoisySystem, NOISE_PARAMS
import scqubits.core.qubit_base as base
import scqubits.core.storage as storage
import scqubits.io_utils.fileio_serializers as serializers
import scqubits.ui.qubit_widget as ui
import scqubits.utils.plotting as plot
import scqubits.utils.spectrum_utils as spec_utils
# - ZeroPi noise class
class NoisyZeroPi(NoisySystem):
pass
# -Symmetric 0-pi qubit, phi discretized, theta in charge basis---------------------------------------------------------
[docs]class ZeroPi(base.QubitBaseClass, serializers.Serializable, NoisyZeroPi):
r"""Zero-Pi Qubit
| [1] Brooks et al., Physical Review A, 87(5), 052306 (2013). http://doi.org/10.1103/PhysRevA.87.052306
| [2] Dempster et al., Phys. Rev. B, 90, 094518 (2014). http://doi.org/10.1103/PhysRevB.90.094518
| [3] Groszkowski et al., New J. Phys. 20, 043053 (2018). https://doi.org/10.1088/1367-2630/aab7cd
Zero-Pi qubit without coupling to the `zeta` mode, i.e., no disorder in `EC` and `EL`,
see Eq. (4) in Groszkowski et al., New J. Phys. 20, 043053 (2018),
.. math::
H &= -2E_\text{CJ}\partial_\phi^2+2E_{\text{C}\Sigma}(i\partial_\theta-n_g)^2
+2E_{C\Sigma}dC_J\,\partial_\phi\partial_\theta
-2E_\text{J}\cos\theta\cos(\phi-\varphi_\text{ext}/2)+E_L\phi^2\\
&\qquad +2E_\text{J} + E_J dE_J \sin\theta\sin(\phi-\phi_\text{ext}/2).
Formulation of the Hamiltonian matrix proceeds by discretization of the `phi` variable, and using charge basis for
the `theta` variable.
Parameters
----------
EJ: float
mean Josephson energy of the two junctions
EL: float
inductive energy of the two (super-)inductors
ECJ: float
charging energy associated with the two junctions
EC: float or None
charging energy of the large shunting capacitances; set to `None` if `ECS` is provided instead
dEJ: float
relative disorder in EJ, i.e., (EJ1-EJ2)/EJavg
dCJ: float
relative disorder of the junction capacitances, i.e., (CJ1-CJ2)/CJavg
ng: float
offset charge associated with theta
flux: float
magnetic flux through the circuit loop, measured in units of flux quanta (h/2e)
grid: Grid1d object
specifies the range and spacing of the discretization lattice
ncut: int
charge number cutoff for `n_theta`, `n_theta = -ncut, ..., ncut`
ECS: float, optional
total charging energy including large shunting capacitances and junction capacitances; may be provided instead
of EC
truncated_dim: int, optional
desired dimension of the truncated quantum system; expected: truncated_dim > 1
"""
EJ = descriptors.WatchedProperty('QUANTUMSYSTEM_UPDATE')
EL = descriptors.WatchedProperty('QUANTUMSYSTEM_UPDATE')
ECJ = descriptors.WatchedProperty('QUANTUMSYSTEM_UPDATE')
EC = descriptors.WatchedProperty('QUANTUMSYSTEM_UPDATE')
dEJ = descriptors.WatchedProperty('QUANTUMSYSTEM_UPDATE')
dCJ = descriptors.WatchedProperty('QUANTUMSYSTEM_UPDATE')
ng = descriptors.WatchedProperty('QUANTUMSYSTEM_UPDATE')
ncut = descriptors.WatchedProperty('QUANTUMSYSTEM_UPDATE')
def __init__(self, EJ, EL, ECJ, EC, ng, flux, grid, ncut, dEJ=0, dCJ=0, ECS=None, truncated_dim=None):
self.EJ = EJ
self.EL = EL
self.ECJ = ECJ
if EC is None and ECS is None:
raise ValueError("Argument missing: must either provide EC or ECS")
if EC and ECS:
raise ValueError("Argument error: can only provide either EC or ECS")
if EC:
self.EC = EC
else:
self.EC = 1 / (1 / ECS - 1 / self.ECJ)
self.dEJ = dEJ
self.dCJ = dCJ
self.ng = ng
self.flux = flux
self.grid = grid
self.ncut = ncut
self.truncated_dim = truncated_dim
self._sys_type = type(self).__name__
self._evec_dtype = np.complex_
# for theta, needed for plotting wavefunction
self._default_grid = discretization.Grid1d(-np.pi / 2, 3 * np.pi / 2, 100)
self._init_params.remove('ECS') # used in for file Serializable purposes; remove ECS as init parameter
self._image_filename = os.path.join(os.path.dirname(os.path.abspath(__file__)), 'qubit_img/zeropi.jpg')
dispatch.CENTRAL_DISPATCH.register('GRID_UPDATE', self)
[docs] @staticmethod
def default_params():
return {
'EJ': 10.0,
'EL': 0.04,
'ECJ': 20.0,
'EC': 0.04,
'dEJ': 0.0,
'dCJ': 0.0,
'ng': 0.1,
'flux': 0.23,
'ncut': 30,
'truncated_dim': 10
}
[docs] @classmethod
def create(cls):
phi_grid = discretization.Grid1d(-19.0, 19.0, 200)
init_params = cls.default_params()
zeropi = cls(**init_params, grid=phi_grid)
zeropi.widget()
return zeropi
[docs] def supported_noise_channels(self):
"""Return a list of supported noise channels"""
return [
'tphi_1_over_f_cc',
'tphi_1_over_f_flux',
't1_flux_bias_line',
# 't1_capacitive',
't1_inductive',
]
[docs] def set_params(self, **kwargs):
phi_grid = discretization.Grid1d(kwargs.pop('grid_min_val'),
kwargs.pop('grid_max_val'),
kwargs.pop('grid_pt_count'))
self.grid = phi_grid
for param_name, param_val in kwargs.items():
setattr(self, param_name, param_val)
[docs] def receive(self, event, sender, **kwargs):
if sender is self.grid:
self.broadcast('QUANTUMSYSTEM_UPDATE')
def _evals_calc(self, evals_count):
hamiltonian_mat = self.hamiltonian()
evals = sparse.linalg.eigsh(hamiltonian_mat, k=evals_count, sigma=0.0, which='LM', return_eigenvectors=False)
return np.sort(evals)
def _esys_calc(self, evals_count):
hamiltonian_mat = self.hamiltonian()
evals, evecs = sparse.linalg.eigsh(hamiltonian_mat, k=evals_count, sigma=0.0, which='LM',
return_eigenvectors=True)
# TODO consider normalization of zeropi wavefunctions
# evecs /= np.sqrt(self.grid.grid_spacing())
evals, evecs = spec_utils.order_eigensystem(evals, evecs)
return evals, evecs
def get_ECS(self):
return 1 / (1 / self.EC + 1 / self.ECJ)
def set_ECS(self, value):
warnings.warn("It is not possible to directly set ECS (except in initialization). Instead, set EC or ECJ, "
"or use set_EC_via_ECS() to update EC indirectly.", Warning)
ECS = property(get_ECS, set_ECS)
[docs] def set_EC_via_ECS(self, ECS):
"""Helper function to set `EC` by providing `ECS`, keeping `ECJ` constant."""
self.EC = 1 / (1 / ECS - 1 / self.ECJ)
[docs] def hilbertdim(self):
"""Returns Hilbert space dimension"""
return self.grid.pt_count * (2 * self.ncut + 1)
[docs] def potential(self, phi, theta):
"""
Parameters
----------
phi: float
theta: float
Returns
-------
float
value of the potential energy evaluated at phi, theta
"""
return (-2.0 * self.EJ * np.cos(theta) * np.cos(phi - 2.0 * np.pi * self.flux / 2.0)
+ self.EL * phi ** 2 + 2.0 * self.EJ
+ self.EJ * self.dEJ * np.sin(theta) * np.sin(phi - 2.0 * np.pi * self.flux / 2.0))
[docs] def sparse_kinetic_mat(self):
"""
Kinetic energy portion of the Hamiltonian.
Returns
-------
scipy.sparse.csc_matrix
matrix representing the kinetic energy operator
"""
pt_count = self.grid.pt_count
dim_theta = 2 * self.ncut + 1
identity_phi = sparse.identity(pt_count, format='csc')
identity_theta = sparse.identity(dim_theta, format='csc')
kinetic_matrix_phi = self.grid.second_derivative_matrix(prefactor=-2.0 * self.ECJ)
diag_elements = 2.0 * self.ECS * np.square(np.arange(-self.ncut + self.ng, self.ncut + 1 + self.ng))
kinetic_matrix_theta = sparse.dia_matrix((diag_elements, [0]), shape=(dim_theta, dim_theta)).tocsc()
kinetic_matrix = (sparse.kron(kinetic_matrix_phi, identity_theta, format='csc')
+ sparse.kron(identity_phi, kinetic_matrix_theta, format='csc'))
if self.dCJ != 0:
kinetic_matrix -= 2.0 * self.ECS * self.dCJ * self.i_d_dphi_operator() * self.n_theta_operator()
return kinetic_matrix
[docs] def sparse_potential_mat(self):
"""
Potential energy portion of the Hamiltonian.
Returns
-------
scipy.sparse.csc_matrix
matrix representing the potential energy operator
"""
pt_count = self.grid.pt_count
grid_linspace = self.grid.make_linspace()
dim_theta = 2 * self.ncut + 1
phi_inductive_vals = self.EL * np.square(grid_linspace)
phi_inductive_potential = sparse.dia_matrix((phi_inductive_vals, [0]), shape=(pt_count, pt_count)).tocsc()
phi_cos_vals = np.cos(grid_linspace - 2.0 * np.pi * self.flux / 2.0)
phi_cos_potential = sparse.dia_matrix((phi_cos_vals, [0]), shape=(pt_count, pt_count)).tocsc()
phi_sin_vals = np.sin(grid_linspace - 2.0 * np.pi * self.flux / 2.0)
phi_sin_potential = sparse.dia_matrix((phi_sin_vals, [0]), shape=(pt_count, pt_count)).tocsc()
theta_cos_potential = (-self.EJ
* (sparse.dia_matrix(([1.0] * dim_theta, [-1]), shape=(dim_theta, dim_theta)) +
sparse.dia_matrix(([1.0] * dim_theta, [1]), shape=(dim_theta, dim_theta)))).tocsc()
potential_mat = (sparse.kron(phi_cos_potential, theta_cos_potential, format='csc')
+ sparse.kron(phi_inductive_potential, self._identity_theta(), format='csc')
+ 2 * self.EJ * sparse.kron(self._identity_phi(), self._identity_theta(), format='csc'))
if self.dEJ != 0:
potential_mat += (self.EJ * self.dEJ * sparse.kron(phi_sin_potential, self._identity_theta(), format='csc')
* self.sin_theta_operator())
return potential_mat
[docs] def hamiltonian(self):
"""Calculates Hamiltonian in basis obtained by discretizing phi and employing charge basis for theta.
Returns
-------
scipy.sparse.csc_matrix
matrix representing the potential energy operator
"""
return self.sparse_kinetic_mat() + self.sparse_potential_mat()
[docs] def sparse_d_potential_d_flux_mat(self):
r"""Calculates a of the potential energy w.r.t flux, at the current value of flux,
as stored in the object.
The flux is assumed to be given in the units of the ratio \Phi_{ext}/\Phi_0.
So if \frac{\partial U}{ \partial \Phi_{\rm ext}}, is needed, the expression returned
by this function, needs to be multiplied by 1/\Phi_0.
Returns
-------
scipy.sparse.csc_matrix
matrix representing the derivative of the potential energy
"""
op_1 = sparse.kron(self._sin_phi_operator(x=- 2.0 * np.pi * self.flux / 2.0),
self._cos_theta_operator(), format='csc')
op_2 = sparse.kron(self._cos_phi_operator(x=- 2.0 * np.pi * self.flux / 2.0),
self._sin_theta_operator(), format='csc')
return - 2.0 * np.pi * self.EJ * op_1 - np.pi * self.EJ * self.dEJ * op_2
[docs] def d_hamiltonian_d_flux(self):
r"""Calculates a derivative of the Hamiltonian w.r.t flux, at the current value of flux,
as stored in the object.
The flux is assumed to be given in the units of the ratio \Phi_{ext}/\Phi_0.
So if \frac{\partial H}{ \partial \Phi_{\rm ext}}, is needed, the expression returned
by this function, needs to be multiplied by 1/\Phi_0.
Returns
-------
scipy.sparse.csc_matrix
matrix representing the derivative of the Hamiltonian
"""
return self.sparse_d_potential_d_flux_mat()
[docs] def sparse_d_potential_d_EJ_mat(self):
r"""Calculates a of the potential energy w.r.t EJ.
Returns
-------
scipy.sparse.csc_matrix
matrix representing the derivative of the potential energy
"""
return - 2.0 * sparse.kron(self._cos_phi_operator(x=- 2.0 * np.pi * self.flux / 2.0),
self._cos_theta_operator(), format='csc')
[docs] def d_hamiltonian_d_EJ(self):
r"""Calculates a derivative of the Hamiltonian w.r.t EJ.
for calcu
Returns
-------
scipy.sparse.csc_matrix
matrix representing the derivative of the Hamiltonian
"""
return self.sparse_d_potential_d_EJ_mat()
[docs] def d_hamiltonian_d_ng(self):
r"""Calculates a derivative of the Hamiltonian w.r.t ng.
as stored in the object.
Returns
-------
scipy.sparse.csc_matrix
matrix representing the derivative of the Hamiltonian
"""
return -8 * self.EC * self.n_theta_operator()
def _identity_phi(self):
r"""
Identity operator acting only on the `\phi` Hilbert subspace.
Returns
-------
scipy.sparse.csc_matrix
"""
pt_count = self.grid.pt_count
return sparse.identity(pt_count, format='csc')
def _identity_theta(self):
r"""
Identity operator acting only on the `\theta` Hilbert subspace.
Returns
-------
scipy.sparse.csc_matrix
"""
dim_theta = 2 * self.ncut + 1
return sparse.identity(dim_theta, format='csc')
[docs] def i_d_dphi_operator(self):
r"""
Operator :math:`i d/d\phi`.
Returns
-------
scipy.sparse.csc_matrix
"""
return sparse.kron(self.grid.first_derivative_matrix(prefactor=1j), self._identity_theta(), format='csc')
def _phi_operator(self):
r"""
Operator :math:`\phi`, acting only on the `\phi` Hilbert subspace.
Returns
-------
scipy.sparse.csc_matrix
"""
pt_count = self.grid.pt_count
phi_matrix = sparse.dia_matrix((pt_count, pt_count))
diag_elements = self.grid.make_linspace()
phi_matrix.setdiag(diag_elements)
return phi_matrix
[docs] def phi_operator(self):
r"""
Operator :math:`\phi`.
Returns
-------
scipy.sparse.csc_matrix
"""
return sparse.kron(self._phi_operator(), self._identity_theta(), format='csc')
[docs] def n_theta_operator(self):
r"""
Operator :math:`n_\theta`.
Returns
-------
scipy.sparse.csc_matrix
"""
dim_theta = 2 * self.ncut + 1
diag_elements = np.arange(-self.ncut, self.ncut + 1)
n_theta_matrix = sparse.dia_matrix((diag_elements, [0]), shape=(dim_theta, dim_theta)).tocsc()
return sparse.kron(self._identity_phi(), n_theta_matrix, format='csc')
def _sin_phi_operator(self, x=0):
r"""
Operator :math:`\sin(\phi + x)`, acting only on the `\phi` Hilbert subspace.
Returns
-------
scipy.sparse.csc_matrix
"""
pt_count = self.grid.pt_count
vals = np.sin(self.grid.make_linspace() + x)
sin_phi_matrix = sparse.dia_matrix((vals, [0]), shape=(pt_count, pt_count)).tocsc()
return sin_phi_matrix
def _cos_phi_operator(self, x=0):
r"""
Operator :math:`\cos(\phi + x)`, acting only on the `\phi` Hilbert subspace.
Returns
-------
scipy.sparse.csc_matrix
"""
pt_count = self.grid.pt_count
vals = np.cos(self.grid.make_linspace() + x)
cos_phi_matrix = sparse.dia_matrix((vals, [0]), shape=(pt_count, pt_count)).tocsc()
return cos_phi_matrix
def _cos_theta_operator(self):
r"""
Operator :math:`\cos(\theta)`, acting only on the `\theta` Hilbert subspace.
Returns
-------
scipy.sparse.csc_matrix
"""
dim_theta = 2 * self.ncut + 1
cos_theta_matrix = 0.5 * (sparse.dia_matrix(([1.0] * dim_theta, [-1]), shape=(dim_theta, dim_theta)) +
sparse.dia_matrix(([1.0] * dim_theta, [1]), shape=(dim_theta, dim_theta))).tocsc()
return cos_theta_matrix
[docs] def cos_theta_operator(self):
r"""
Operator :math:`\cos(\theta)`.
Returns
-------
scipy.sparse.csc_matrix
"""
return sparse.kron(self._identity_phi(), self._cos_phi_operator(), format='csc')
def _sin_theta_operator(self):
r"""
Operator :math:`\sin(\theta)`, acting only on the `\theta` Hilbert space.
Returns
-------
scipy.sparse.csc_matrix
"""
dim_theta = 2 * self.ncut + 1
sin_theta_matrix = (-0.5 * 1j
* (sparse.dia_matrix(([1.0] * dim_theta, [1]), shape=(dim_theta, dim_theta)) -
sparse.dia_matrix(([1.0] * dim_theta, [-1]), shape=(dim_theta, dim_theta))).tocsc())
return sin_theta_matrix
[docs] def sin_theta_operator(self):
r"""
Operator :math:`\sin(\theta)`.
Returns
-------
scipy.sparse.csc_matrix
"""
return sparse.kron(self._identity_phi(), self._sin_theta_operator(), format='csc')
[docs] def plot_potential(self, theta_grid=None, contour_vals=None, **kwargs):
"""Draw contour plot of the potential energy.
Parameters
----------
theta_grid: Grid1d, optional
used for setting a custom grid for theta; if None use self._default_grid
contour_vals: list, optional
**kwargs:
plotting parameters
"""
theta_grid = theta_grid or self._default_grid
x_vals = self.grid.make_linspace()
y_vals = theta_grid.make_linspace()
return plot.contours(x_vals, y_vals, self.potential, contour_vals=contour_vals,
xlabel=r'$\phi$', ylabel=r'$\theta$', **kwargs)
[docs] def wavefunction(self, esys=None, which=0, theta_grid=None):
"""Returns a zero-pi wave function in `phi`, `theta` basis
Parameters
----------
esys: ndarray, ndarray
eigenvalues, eigenvectors
which: int, optional
index of desired wave function (default value = 0)
theta_grid: Grid1d, optional
used for setting a custom grid for theta; if None use self._default_grid
Returns
-------
WaveFunctionOnGrid object
"""
evals_count = max(which + 1, 3)
if esys is None:
_, evecs = self.eigensys(evals_count)
else:
_, evecs = esys
theta_grid = theta_grid or self._default_grid
dim_theta = 2 * self.ncut + 1
state_amplitudes = evecs[:, which].reshape(self.grid.pt_count, dim_theta)
# Calculate psi_{phi, theta} = sum_n state_amplitudes_{phi, n} A_{n, theta}
# where a_{n, theta} = 1/sqrt(2 pi) e^{i n theta}
n_vec = np.arange(-self.ncut, self.ncut + 1)
theta_vec = theta_grid.make_linspace()
a_n_theta = np.exp(1j * np.outer(n_vec, theta_vec)) / (2 * np.pi) ** 0.5
wavefunc_amplitudes = np.matmul(state_amplitudes, a_n_theta).T
wavefunc_amplitudes = spec_utils.standardize_phases(wavefunc_amplitudes)
grid2d = discretization.GridSpec(np.asarray([[self.grid.min_val, self.grid.max_val, self.grid.pt_count],
[theta_grid.min_val, theta_grid.max_val, theta_grid.pt_count]]))
return storage.WaveFunctionOnGrid(grid2d, wavefunc_amplitudes)
[docs] def plot_wavefunction(self, esys=None, which=0, theta_grid=None, mode='abs', zero_calibrate=True, **kwargs):
"""Plots 2d phase-basis wave function.
Parameters
----------
esys: ndarray, ndarray
eigenvalues, eigenvectors as obtained from `.eigensystem()`
which: int, optional
index of wave function to be plotted (default value = (0)
theta_grid: Grid1d, optional
used for setting a custom grid for theta; if None use self._default_grid
mode: str, optional
choices as specified in `constants.MODE_FUNC_DICT` (default value = 'abs_sqr')
zero_calibrate: bool, optional
if True, colors are adjusted to use zero wavefunction amplitude as the neutral color in the palette
**kwargs:
plot options
Returns
-------
Figure, Axes
"""
theta_grid = theta_grid or self._default_grid
amplitude_modifier = constants.MODE_FUNC_DICT[mode]
wavefunc = self.wavefunction(esys, theta_grid=theta_grid, which=which)
wavefunc.amplitudes = amplitude_modifier(wavefunc.amplitudes)
return plot.wavefunction2d(wavefunc, zero_calibrate=zero_calibrate,
xlabel=r'$\phi$', ylabel=r'$\theta$', **kwargs)