def compute_rabi_coupling(self):
'''
Rabi couplings, see Leibfried (2003), eq:70
'''
eta = self.eta
if self.sideband_order == 0:
coupling_func = lambda n: np.exp(-1. / 2 * eta**2) * laguerre(
n, 0, eta**2)
elif self.sideband_order == 1:
coupling_func = lambda n: np.exp(-1. / 2 * eta**2) * eta**(1) * (
1. / (n + 1.))**0.5 * laguerre(n, 1, eta**2)
elif self.sideband_order == 2:
coupling_func = lambda n: np.exp(-1. / 2 * eta**2) * eta**(2) * (
1.0 / ((n + 1.) * (n + 2)))**0.5 * laguerre(n, 2, eta**2)
elif self.sideband_order == -1:
coupling_func = lambda n: 0 if n == 0 else np.exp(
-1. / 2 * eta**2) * eta**(1) * (1. / (n))**0.5 * laguerre(
n - 1, 1, eta**2)
elif self.sideband_order == -2:
coupling_func = lambda n: 0 if n <= 1 else np.exp(
-1. / 2 * eta**2) * eta**(2) * (1.0 / (
(n) * (n - 1.)))**0.5 * laguerre(n - 2, 2, eta**2)
else:
raise NotImplementedError(
"Can't calculate rabi couplings sideband order {}".format(
self.sideband_order))
return np.array([coupling_func(n) for n in range(self.nmax)])
def test_l_roots():
weightf = orth.laguerre(5).weight_func
verify_gauss_quad(orth.l_roots, orth.eval_laguerre, weightf, 0., np.inf, 5)
verify_gauss_quad(orth.l_roots,
orth.eval_laguerre,
weightf,
0.,
np.inf,
25,
atol=1e-13)
verify_gauss_quad(orth.l_roots,
orth.eval_laguerre,
weightf,
0.,
np.inf,
100,
atol=1e-12)
x, w = orth.l_roots(5, False)
y, v, m = orth.l_roots(5, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
muI, muI_err = integrate.quad(weightf, 0, np.inf)
assert_allclose(m, muI, rtol=muI_err)
assert_raises(ValueError, orth.l_roots, 0)
assert_raises(ValueError, orth.l_roots, 3.3)
def compute_rabi_coupling(self):
'''
Rabi couplings, see Leibfried (2003), eq:70
'''
eta = self.eta
if self.sideband_order == 0:
coupling_func = lambda n: np.exp(-1./2*eta**2) * laguerre(n, 0, eta**2)
elif self.sideband_order == 1:
coupling_func = lambda n: np.exp(-1./2*eta**2) * eta**(1)*(1./(n+1.))**0.5 * laguerre(n, 1, eta**2)
elif self.sideband_order == 2:
coupling_func = lambda n: np.exp(-1./2*eta**2) * eta**(2)*(1./((n+1.)*(n+2)))**0.5 * laguerre(n, 2, eta**2)
elif self.sideband_order == 3:
coupling_func = lambda n: np.exp(-1./2*eta**2) * eta**(3)*(1./((n+1)*(n+2)*(n+3)))**0.5 * laguerre(n, 3 , eta**2)
elif self.sideband_order == 4:
coupling_func = lambda n: np.exp(-1./2*eta**2) * eta**(4)*(1./((n+1)*(n+2)*(n+3)*(n+4)))**0.5 * laguerre(n, 4 , eta**2)
elif self.sideband_order == 5:
coupling_func = lambda n: np.exp(-1./2*eta**2) * eta**(5)*(1./((n+1)*(n+2)*(n+3)*(n+4)*(n+5)))**0.5 * laguerre(n, 5 , eta**2)
elif self.sideband_order == -1:
coupling_func = lambda n: 0 if n == 0 else np.exp(-1./2*eta**2) * eta**(1)*(1./(n))**0.5 * laguerre(n - 1, 1, eta**2)
elif self.sideband_order == -2:
coupling_func = lambda n: 0 if n <= 1 else np.exp(-1./2*eta**2) * eta**(2)*(1./((n)*(n-1.)))**0.5 * laguerre(n - 2, 2, eta**2)
elif self.sideband_order == -3:
coupling_func = lambda n: 0 if n <= 2 else np.exp(-1./2*eta**2) * eta**(3)*(1./((n)*(n-1.)*(n-2)))**0.5 * laguerre(n -3, 3, eta**2)
elif self.sideband_order == -4:
coupling_func = lambda n: 0 if n <= 3 else np.exp(-1./2*eta**2) * eta**(4)*(1./((n)*(n-1.)*(n-2)*(n-3)))**0.5 * laguerre(n -4, 4, eta**2)
elif self.sideband_order == -5:
coupling_func = lambda n: 0 if n <= 4 else np.exp(-1./2*eta**2) * eta**(5)*(1./((n)*(n-1.)*(n-2)*(n-3)*(n-4)))**0.5 * laguerre(n -5, 5, eta**2)
else:
raise NotImplementedError("Can't calculate rabi couplings sideband order {}".format(self.sideband_order))
return np.array([coupling_func(n) for n in range(self.nmax)])
def _displaced_thermal(cls, alpha, nbar, n):
"""
returns the motional state distribution with the displacement alpha, motional temperature nbar for the state n
"""
# this is needed because for some inputs (larrge n or small nbar, this term is 0 while the laguerre term is infinite. their product is zero but beyond the floating point precision
try:
old_settings = np.seterr(invalid="raise")
populations = (
1.0
/ (nbar + 1.0)
* (nbar / (nbar + 1.0)) ** n
* laguerre(n, 0, -alpha ** 2 / (nbar * (nbar + 1.0)))
* np.exp(-alpha ** 2 / (nbar + 1.0))
)
except FloatingPointError:
np.seterr(**old_settings)
print "precise calculation required", alpha, nbar
populations = [
mp.fprod(
(
1.0 / (nbar + 1.0),
mp.power(nbar / (nbar + 1.0), k),
mp.laguerre(k, 0, -alpha ** 2 / (nbar * (nbar + 1.0))),
mp.exp(-alpha ** 2 / (nbar + 1.0)),
)
)
for k in n
]
print "done computing populations"
populations = np.array(populations)
print "returned array"
return populations
def _displaced_thermal(cls, alpha, nbar, n):
'''
returns the motional state distribution with the displacement alpha, motional temperature nbar for the state n
'''
#this is needed because for some inputs (larrge n or small nbar, this term is 0 while the laguerre term is infinite. their product is zero but beyond the floating point precision
try:
old_settings = np.seterr(invalid='raise')
populations = 1. / (nbar +
1.0) * (nbar / (nbar + 1.0))**n * laguerre(
n, 0, -alpha**2 /
(nbar *
(nbar + 1.0))) * np.exp(-alpha**2 /
(nbar + 1.0))
except FloatingPointError:
np.seterr(**old_settings)
print 'precise calculation required', alpha, nbar
populations = [
mp.fprod((1. / (nbar + 1.0), mp.power(nbar / (nbar + 1.0), k),
mp.laguerre(k, 0, -alpha**2 / (nbar * (nbar + 1.0))),
mp.exp(-alpha**2 / (nbar + 1.0)))) for k in n
]
print 'done computing populations'
populations = np.array(populations)
print 'returned array'
return populations
def _displaced_thermal(cls, alpha, nbar, n):
return (
1.0
/ (nbar + 1.0)
* (nbar / (nbar + 1.0)) ** n
* laguerre(n, 0, -alpha ** 2 / (nbar * (nbar + 1.0)))
* np.exp(-alpha ** 2 / (nbar + 1.0))
)
def _displaced_thermal(cls, alpha, nbar, n):
'''
returns the motional state distribution with the displacement alpha, motional temperature nbar for the state n
'''
#this is needed because for some inputs (larrge n or small nbar, this term is 0 while the laguerre term is infinite. their product is zero but beyond the floating point precision
try:
old_settings = np.seterr(invalid='raise')
populations = 1./ (nbar + 1.0) * (nbar / (nbar + 1.0))**n * laguerre(n, 0 , -alpha**2 / ( nbar * (nbar + 1.0))) * np.exp( -alpha**2 / (nbar + 1.0))
except FloatingPointError:
print 'using coherent state'
# np.seterr(**old_settings)
# populations = np.exp(-alpha**2) * alpha**(2*n) / factorial(n)
populations = (2 * np.pi * alpha**2)**(-.5) * np.exp(-(n - alpha**2)**2/(2*alpha**2))
return populations
def rabi_coupling(n, m, eta):
'''return the Rabi matrix elements <m+n| exp(1j*eta*(a+a.dag)) |n> for n = 0...n-1
@ var n: size of the Hilbert space
@ var m: order m of the sideband
@ var eta: Lamd-Dicke parameter
returns: Wnm Rabi matrix elements for n = 0,...n-1
'''
L = np.array([laguerre(ii, abs(m), eta**2) for ii in range(n)])
fctr = np.array([factorial_simplified(ii, abs(m))**(0.5) for ii in range(n)])
Wmn = exp(-1./2.*eta**2) * eta**abs(m) * fctr * L
if m<0:
Wmn = np.concatenate((np.zeros(-m), Wmn[:m]))
return abs(Wmn)
def __init__(self, date, w0 = 2*np.pi*500e3, phi=np.pi/4, nmax=3000,
basepath='/home/viellieb/work_resonator/datavault/Experiments.dir'):
self.date = date
self.basepath = basepath
k = 2 * np.pi / 729.14e-9
eta0 = k * np.sqrt(const.hbar/(2*40*const.m_p*w0))
self.eta = eta0 * np.cos(phi)
self.delay = 6.7
self.nmax = nmax
print "Generating coupling tables"
coupling_func = lambda n: np.exp(-1./2*self.eta**2) * laguerre(n, 0, self.eta**2)
coupling_array = np.array([coupling_func(n) for n in range(self.nmax)])
self.rabi_coupling = coupling_array
self.n_array = np.arange(self.nmax)
def rabi_coupling(n, m, eta):
'''return the Rabi matrix elements <m+n| exp(1j*eta*(a+a.dag)) |n> for n = 0...n-1
@ var n: size of the Hilbert space
@ var m: order m of the sideband
@ var eta: Lamd-Dicke parameter
returns: Wnm Rabi matrix elements for n = 0,...n-1
'''
L = np.array([laguerre(ii, abs(m), eta**2) for ii in range(n)])
fctr = np.array(
[factorial_simplified(ii, abs(m))**(0.5) for ii in range(n)])
Wmn = exp(-1. / 2. * eta**2) * eta**abs(m) * fctr * L
if m < 0:
Wmn = np.concatenate((np.zeros(-m), Wmn[:m]))
return abs(Wmn)
def test_l_roots():
weightf = orth.laguerre(5).weight_func
verify_gauss_quad(orth.l_roots, orth.eval_laguerre, weightf, 0.0, np.inf, 5)
verify_gauss_quad(orth.l_roots, orth.eval_laguerre, weightf, 0.0, np.inf, 25, atol=1e-13)
verify_gauss_quad(orth.l_roots, orth.eval_laguerre, weightf, 0.0, np.inf, 100, atol=1e-12)
x, w = orth.l_roots(5, False)
y, v, m = orth.l_roots(5, True)
assert_allclose(x, y, 1e-14, 1e-14)
assert_allclose(w, v, 1e-14, 1e-14)
muI, muI_err = integrate.quad(weightf, 0, np.inf)
assert_allclose(m, muI, rtol=muI_err)
assert_raises(ValueError, orth.l_roots, 0)
assert_raises(ValueError, orth.l_roots, 3.3)
def _displaced_thermal(cls, alpha, nbar, n):
'''
returns the motional state distribution with the displacement alpha, motional temperature nbar for the state n
'''
#this is needed because for some inputs (larrge n or small nbar, this term is 0 while the laguerre term is infinite. their product is zero but beyond the floating point precision
try:
old_settings = np.seterr(invalid='raise')
populations = 1. / (nbar +
1.0) * (nbar / (nbar + 1.0))**n * laguerre(
n, 0, -alpha**2 /
(nbar *
(nbar + 1.0))) * np.exp(-alpha**2 /
(nbar + 1.0))
except FloatingPointError:
print 'using coherent state'
# np.seterr(**old_settings)
# populations = np.exp(-alpha**2) * alpha**(2*n) / factorial(n)
populations = (2 * np.pi * alpha**2)**(-.5) * np.exp(
-(n - alpha**2)**2 / (2 * alpha**2))
return populations
def _displaced_thermal(cls, alpha, nbar, n):
'''
returns the motional state distribution with the displacement alpha, motional temperature nbar for the state n
'''
#this is needed because for some inputs (larrge n or small nbar, this term is 0 while the laguerre term is infinite. their product is zero but beyond the floating point precision
try:
old_settings = np.seterr(invalid='raise')
populations = 1. / (nbar +
1.0) * (nbar / (nbar + 1.0))**n * laguerre(
n, 0, -alpha**2 /
(nbar *
(nbar + 1.0))) * np.exp(-alpha**2 /
(nbar + 1.0))
except FloatingPointError:
np.seterr(**old_settings)
def fib():
a, b = 0, 1
while 1:
yield a
a, b = b, a + b
import time
t1 = time.time()
print 'precise calculation required', alpha, nbar
expon_term = mp.exp(-alpha**2 / (nbar + 1.0))
populations = [
float(
mp.fprod((mp.power(nbar / (nbar + 1.0), k),
mp.laguerre(k, 0,
-alpha**2 / (nbar * (nbar + 1.0))),
expon))) for k in n
]
populations = np.array(populations) / (nbar + 1)
print time.time() - t1
print 'done'
return populations
def _displaced_thermal(cls, alpha, nbar, n):
'''
returns the motional state distribution with the displacement alpha, motional temperature nbar for the state n
'''
#this is needed because for some inputs (larrge n or small nbar, this term is 0 while the laguerre term is infinite. their product is zero but beyond the floating point precision
try:
old_settings = np.seterr(invalid='raise')
populations = 1./ (nbar + 1.0) * (nbar / (nbar + 1.0))**n * laguerre(n, 0 , -alpha**2 / ( nbar * (nbar + 1.0))) * np.exp( -alpha**2 / (nbar + 1.0))
except FloatingPointError:
np.seterr(**old_settings)
def fib():
a, b = 0, 1
while 1:
yield a
a, b = b, a + b
import time
t1 = time.time()
print 'precise calculation required', alpha, nbar
expon_term = mp.exp(-alpha**2 / (nbar + 1.0))
populations = [float(mp.fprod((mp.power(nbar / (nbar + 1.0), k), mp.laguerre(k, 0, -alpha**2 / ( nbar * (nbar + 1.0))), expon))) for k in n]
populations = np.array(populations) / (nbar + 1)
print time.time() - t1
print 'done'
return populations
def compute_rabi_coupling(cls, eta, sideband_order, nmax):
'''
Rabi couplings, see Leibfried (2003), eq:70
'''
if sideband_order == 0:
coupling_func = lambda n: np.exp(-1. / 2 * eta**2) * laguerre(
n, 0, eta**2)
elif sideband_order == 1:
coupling_func = lambda n: np.exp(-1. / 2 * eta**2) * eta**(1) * (
1. / (n + 1.))**0.5 * laguerre(n, 1, eta**2)
elif sideband_order == 2:
coupling_func = lambda n: np.exp(-1. / 2 * eta**2) * eta**(2) * (
1. / ((n + 1.) * (n + 2)))**0.5 * laguerre(n, 2, eta**2)
elif sideband_order == 3:
coupling_func = lambda n: np.exp(-1. / 2 * eta**2) * eta**(3) * (
1. / ((n + 1) * (n + 2) *
(n + 3)))**0.5 * laguerre(n, 3, eta**2)
elif sideband_order == 4:
coupling_func = lambda n: np.exp(-1. / 2 * eta**2) * eta**(4) * (
1. / ((n + 1) * (n + 2) * (n + 3) *
(n + 4)))**0.5 * laguerre(n, 4, eta**2)
elif sideband_order == 5:
coupling_func = lambda n: np.exp(-1. / 2 * eta**2) * eta**(5) * (
1. / ((n + 1) * (n + 2) * (n + 3) * (n + 4) *
(n + 5)))**0.5 * laguerre(n, 5, eta**2)
elif sideband_order == -1:
coupling_func = lambda n: 0 if n == 0 else np.exp(
-1. / 2 * eta**2) * eta**(1) * (1. / (n))**0.5 * laguerre(
n - 1, 1, eta**2)
elif sideband_order == -2:
coupling_func = lambda n: 0 if n <= 1 else np.exp(
-1. / 2 * eta**2) * eta**(2) * (1. / (
(n) * (n - 1.)))**0.5 * laguerre(n - 2, 2, eta**2)
elif sideband_order == -3:
coupling_func = lambda n: 0 if n <= 2 else np.exp(
-1. / 2 * eta**2) * eta**(3) * (1. / (
(n) * (n - 1.) *
(n - 2)))**0.5 * laguerre(n - 3, 3, eta**2)
elif sideband_order == -4:
coupling_func = lambda n: 0 if n <= 3 else np.exp(
-1. / 2 * eta**2) * eta**(4) * (1. / (
(n) * (n - 1.) * (n - 2) *
(n - 3)))**0.5 * laguerre(n - 4, 4, eta**2)
elif sideband_order == -5:
coupling_func = lambda n: 0 if n <= 4 else np.exp(
-1. / 2 * eta**2) * eta**(5) * (1. / (
(n) * (n - 1.) * (n - 2) * (n - 3) *
(n - 4)))**0.5 * laguerre(n - 5, 5, eta**2)
else:
raise NotImplementedError(
"Can't calculate rabi couplings sideband order {}".format(
sideband_order))
return np.array([coupling_func(n) for n in range(nmax)])
def _displaced_thermal(cls, alpha, nbar, n):
return 1. / (nbar + 1.0) * (nbar / (nbar + 1.0))**n * laguerre(
n, 0, -alpha**2 / (nbar * (nbar + 1.0))) * np.exp(-alpha**2 /
(nbar + 1.0))
