def nodes(n):
left = mp.mpf(0)
right = mp.mpf(n + (n - 1) * sqrt(n) * 1.01)
i = 2
factor = 2
while True:
l = [(x, mp.laguerre(n, 0, x))
for x in mp.linspace(left, right, n * factor**i)]
intervals = []
for j in range(len(l) - 1):
prod = l[j][1] * l[j + 1][1]
if prod < 0:
intervals.append([l[j][0], l[j + 1][0]])
if len(intervals) == n:
break
i += 1
roots = []
f = lambda x: mp.laguerre(n, 0, x)
for ab in intervals:
a, b = ab
try:
z = mp.findroot(f, (a, b), tol=1e-50, solver='bisect')
except:
z = bisect(f, a, b, tol=1e-50)
roots.append(z)
return roots
def nodes(n):
left = mp.mpf(0)
right = mp.mpf(n+(n-1)*sqrt(n)*1.01)
i = 2
factor = 2
while True:
l = [ (x,mp.laguerre(n,0,x)) for x in mp.linspace(left,right,n*factor**i)]
intervals = []
for j in range(len(l)-1):
prod = l[j][1]*l[j+1][1]
if prod < 0:
intervals.append([l[j][0], l[j+1][0]])
if len(intervals) == n:
break
i += 1
roots = []
f = lambda x: mp.laguerre(n,0,x)
for ab in intervals:
a,b = ab
try:
z = mp.findroot(f, (a, b), tol=1e-50, solver='bisect')
except:
z = bisect(f, a, b, tol=1e-50)
roots.append( z )
return roots
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):
"""
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 ila_integrand_lp1(self, eta, rloc):
k, p, l = self.k, self.p, self.l
kz = mpmath.sqrt(k**2 - eta**2)
res = mpmath.power(eta, np.abs(l) + 1) / mpmath.sqrt(kz) \
* mpmath.laguerre(p, l, self.a(eta) ** 2) * mpmath.exp(-self.a(eta) ** 2 / 2) \
* ((1 - kz / k) * mpmath.besselj(l + 2, eta * rloc / k) + (1 + kz / k) * mpmath.besselj(l, eta * rloc / k))
return res
def integrand(self, eta, l, p, n, kind="A"):
if kind not in ("A", "B", "C", "D", "E"):
raise NotImplementedError("Integrand types supported \
go only from A to E")
if kind == "C":
return (mpmath.sqrt(self.k ** 2 - eta ** 2) / eta \
* self.integrand(eta, l, p, n, kind="A"))
if kind == "D":
return (mpmath.sqrt(self.k ** 2 - eta ** 2) / eta \
* self.integrand(eta, l, p, n, kind="B"))
pm = 1
exponent = n - 1
if kind == "B":
pm = -1
elif kind == "E":
pm = 0
exponent = n
if p == 0:
lgr_factor = 1
else:
lgr_factor = (self.a(eta) ** 2 \
* mpmath.laguerre(p - 1, l, self.a(eta) ** 2))
kz = mpmath.sqrt(self.k**2 - eta**2)
return ( mpmath.power(eta, np.abs(l) + 1) / mpmath.sqrt(kz) \
* mpmath.exp(-self.a(eta) ** 2 / 2) * (1 + pm * kz / self.k) \
* mpmath.power(eta / self.k, exponent) * lgr_factor)
def radial_integrand_lm1(self, eta, r, theta, phi):
k = self.k
kz = mpmath.sqrt(k**2 - eta**2)
l, p = self.l, self.p
a, b = self.alpha, self.beta
result = (mpmath.power(eta, np.abs(l) + 1) / mpmath.sqrt(kz) \
* mpmath.laguerre(p, l, self.a(eta) ** 2) \
* mpmath.exp(-self.a(eta) ** 2 / 2 + 1j * (l - 1) * phi \
+ 1j * kz * r * np.cos(theta)) \
* ((a + 1j * b) / 2 * (1 - kz / k) \
* mpmath.besselj(l - 2, eta * r * np.sin(theta)) \
* np.sin(theta) \
+ eta / k * (1j * a - b) \
* mpmath.besselj(l - 1, eta * r * np.sin(theta)) \
* np.cos(theta) \
+ (a + 1j * b) / 2 * (1 + kz / k) \
* mpmath.besselj(l, eta * r * np.sin(theta)) * np.sin(theta)))
return result
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 func(channel):
channell = []
for i in range(0,len(channel)):
channell.append(float(mpmath.laguerre(degree,0,channel[i])))
return numpy.asarray(channell)
def wk(n, x):
return x/((n+1)*mp.laguerre(n+1,0,x))**2
def wk(n, x):
return x / ((n + 1) * mp.laguerre(n + 1, 0, x))**2
