def plot_spectrum(sys):
"""
Plot the High Harmonic Generation spectrum
"""
# Power spectrum emitted is calculated using the Larmor formula
# (https://en.wikipedia.org/wiki/Larmor_formula)
# which says that the power emitted is proportional to the square of the acceleration
# i.e., the RHS of the second Ehrenfest theorem
N = len(sys.P_average_RHS)
k = np.arange(N)
# frequency range
omegas = (k - N / 2) * np.pi / (0.5 * sys.t)
# spectra of the
spectrum = np.abs(
# used windows fourier transform to calculate the spectra
# rhttp://docs.scipy.org/doc/scipy/reference/tutorial/fftpack.html
fftpack.fft((-1)**k * blackman(N) * sys.P_average_RHS))**2
spectrum /= spectrum.max()
plt.semilogy(omegas / sys.omega_laser, spectrum)
plt.ylabel('spectrum (arbitrary units)')
plt.xlabel('frequency / $\\omega_L$')
plt.xlim([0, 45.])
plt.ylim([1e-15, 1.])
def test_Ehrenfest2(sys):
"""
Test the second Ehenfest theorem for the specified quantum system
"""
times = sys.dt * np.arange(len(sys.P_average))
dP_dt = np.gradient(sys.P_average, sys.dt)
print("%.2e" % np.linalg.norm(dP_dt - sys.P_average_RHS))
plt.plot(times, dP_dt, '-r', label='$d\\langle\\hat{p}\\rangle / dt$')
plt.plot(times,
sys.P_average_RHS,
'--b',
label='$\\langle -U\'(\\hat{x})\\rangle$')
plt.legend()
plt.ylabel('force')
plt.xlabel('time $t$ (a.u.)')
def test_Ehrenfest1(sys):
"""
Test the first Ehenfest theorem for the specified quantum system
"""
times = sys.dt * np.arange(len(sys.x_average))
dx_dt = np.gradient(sys.x_average, sys.dt)
print("{:.2e}".format(np.linalg.norm(dx_dt - sys.x_average_rhs)))
plt.plot(times, dx_dt, '-r', label='$d\\langle\\hat{x}\\rangle / dt$')
plt.plot(times,
sys.x_average_rhs,
'--b',
label='$\\langle\\hat{p}\\rangle$')
plt.legend()
plt.ylabel('momentum')
plt.xlabel('time $t$ (a.u.)')
def frft(x, alpha):
"""
Implementation of the Fractional Fourier Transform (FRFT)
:param x: array of data to be transformed
:param alpha: parameter of FRFT
:return: FRFT(x)
"""
k = np.arange(x.size)
y = np.hstack([
x * np.exp(-np.pi * 1j * k**2 * alpha),
np.zeros(x.size, dtype=np.complex)
])
z = np.hstack([
np.exp(np.pi * 1j * k**2 * alpha),
np.exp(np.pi * 1j * (k - x.size)**2 * alpha)
])
G = fftpack.ifft(fftpack.fft(y, overwrite_x=True) *
fftpack.fft(z, overwrite_x=True),
overwrite_x=True)
return np.exp(-np.pi * 1j * k**2 * alpha) * G[:x.size]
def get_hhg_spectrum(F,
omega_0=0.06,
omega_1=0.12,
omega_2=0.18,
data=np.zeros(10),
get_omega=False,
test_mode=False):
"""
Evaluate the HHG spectrum
:param F: the amplitude of the laser field strength
:param get_omega: boolean flag whether to return the omegas
:param coords: tuple of data to be encoded into laser
:return:
"""
####################################################################################################################
#
# Define parameters of an atom (a single-electron model of Ar) in the external laser field
#
####################################################################################################################
# laser field frequency
omega_laser = omega_0
omegas = np.linspace(omega_1, omega_2, len(data))
# the final time of propagation (= 8 periods of laser oscillations)
t_final = 12 * 2. * np.pi / omega_laser
# the amplitude of grid
x_amplitude = 300.
# the time step
dt = 0.05
@njit
def laser(t):
"""
The strength of the laser field.
Always add an envelop to the laser field to avoid all sorts of artifacts.
We use a sin**2 envelope, which resembles the Blackman filter
"""
# laser = 2*np.sin(omega_laser * t)
laser = np.sin(omega_laser * t)
for freq, dat in zip(omegas, data):
laser += dat * np.sin(freq * t)
laser *= F * np.sin(np.pi * t / t_final)**2
return laser
@njit
def v(x, t=0.):
"""
Potential energy.
Define the potential energy as a sum of the soft core Columb potential
and the laser field interaction in the dipole approximation.
"""
return -1. / np.sqrt(x**2 + 2.37) + x * laser(t)
@njit
def diff_v(x, t=0.):
"""
the derivative of the potential energy
"""
return x * (x**2 + 2.37)**(-1.5) + laser(t)
@njit
def k(p, t=0.):
"""
Non-relativistic kinetic energy
"""
return 0.5 * p**2
@njit
def diff_k(p, t=0.):
"""
the derivative of the kinetic energy for Ehrenfest theorem evaluation
"""
return p
@njit
def abs_boundary(x):
"""
Absorbing boundary similar to the Blackman filter
"""
return np.sin(0.5 * np.pi * (x + x_amplitude) /
x_amplitude)**(0.05 * dt)
sys_params = dict(
dt=dt,
x_grid_dim=2 * 1024,
x_amplitude=x_amplitude,
k=k,
diff_k=diff_k,
v=v,
diff_v=diff_v,
abs_boundary=abs_boundary,
)
####################################################################################################################
#
# propagate
#
####################################################################################################################
sys = ImgTimePropagation(**sys_params)
# Set the ground state wavefunction as the initial condition
sys.get_stationary_states(1)
sys.set_wavefunction(sys.stationary_states[0])
if test_mode:
plt.title("No absorbing boundary, $|\\Psi(x, t)|^2$")
test_propagation(sys, t_final)
plt.show()
plt.subplot(121)
print("\nError in the first Ehrenfest relation: ")
test_Ehrenfest1(sys)
plt.subplot(122)
print("\nError in the second Ehrenfest relation: ")
test_Ehrenfest2(sys)
plt.show()
else:
steps = int(np.ceil(t_final / sys.dt))
sys.propagate(steps)
print("F = {} Completed!".format(F))
####################################################################################################################
#
# Evaluate the spectrum
#
####################################################################################################################
# N = len(sys.x_average)
# k = np.arange(N)
#
# # frequency range
# omega = (k - N / 2) * np.pi / (0.5 * sys.t)
#
# # spectra of the
# spectrum = np.abs(
# # used windows fourier transform to calculate the spectra
# # rhttp://docs.scipy.org/doc/scipy/reference/tutorial/fftpack.html
# fftpack.fft((-1) ** k * blackman(N) * sys.x_average)
# ) ** 2
# spectrum /= spectrum.max()
N = len(sys.p_average_rhs)
k = np.arange(N)
# generate the desired momentum grid
omega_amplitude = 2.
domega = 2. * omega_amplitude / N
omega = (k - N / 2) * domega
delta = sys.dt * domega / (2. * np.pi)
spectrum = np.abs(
frft(
blackman(N) * sys.p_average_rhs *
np.exp(np.pi * 1j * k * N * delta), delta))**2
spectrum /= spectrum.max()
return (spectrum, omega / omega_laser) if get_omega else spectrum