def getTTF(fitObject):
if fitObject.fitFunction == FIT_FUNCTIONS.index('Fermi-Dirac'):
TTF = (6.0 * polylog.fermi_poly3(fitObject.fitData[6]))**(-1.0 / 3.0)
TTFErr = 0.5 * ((6.0 * polylog.fermi_poly3(
fitObject.fitData[6] - fitObject.fitDataConf[6]))**(-1.0 / 3.0) -
(6.0 * polylog.fermi_poly3(fitObject.fitData[6] +
fitObject.fitDataConf[6]))**
(-1.0 / 3.0))
return TTF, TTFErr
else:
print('T/TF only available for Fermi-Dirac fit.')
return -1, -1
def cloudsize(ToverTF, N, tof, wr, wz):
"""Calculate the cloud size of an ideal Fermi gas
**Inputs**
* ToverTF: float, temperature in units of the Fermi temperature T_F
* N: float, number of atoms
* tof: float, time-of-flight in seconds
* wr: float, radial trap frequency in Hz
* wz: float, axial trap frequency in Hz
**Outputs**
* bprime: the size of the cloud in meters [1], (p.69, there called Ri)
**References**
[1] "Making, probing and understanding ultracold Fermi gases", W. Ketterle
and M. Zwierlein, arXiv:cond-mat/0801.2500 (2008)
"""
a = fugacity(ToverTF)
fa = np.log(1+np.e**a)*(1+np.e**a)/np.e**a
TF = hbar/kb*(6*N*wr**2*wz)**(1./3)
T = TF*(6*fermi_poly3(a))**(-1./3)
mass = 6*mp
bprime = expansionfactor(tof, wr)*fa*np.sqrt(2*kb*T/(mass*wr**2))
return bprime
def fugacity(ToverTF):
"""Calculate the log of the fugacity e^{\beta\mu}"""
minpoly3 = lambda x, num: fermi_poly3(x) - num
a = sp.optimize.brentq(minpoly3, -1e3, 1e3, args=(ToverTF**(-3)/6))
return a
def cloudsize(ToverTF, N, tof, wr, wz):
"""Calculate the cloud size of an ideal Fermi gas
**Inputs**
* ToverTF: float, temperature in units of the Fermi temperature T_F
* N: float, number of atoms
* tof: float, time-of-flight in seconds
* wr: float, radial trap frequency in Hz
* wz: float, axial trap frequency in Hz
**Outputs**
* bprime: the size of the cloud in meters [1], (p.69, there called Ri)
**References**
[1] "Making, probing and understanding ultracold Fermi gases", W. Ketterle
and M. Zwierlein, arXiv:cond-mat/0801.2500 (2008)
"""
a = fugacity(ToverTF)
fa = np.log(1 + np.e**a) * (1 + np.e**a) / np.e**a
TF = hbar / kb * (6 * N * wr**2 * wz)**(1. / 3)
T = TF * (6 * fermi_poly3(a))**(-1. / 3)
mass = 6 * mp
bprime = expansionfactor(tof, wr) * fa * np.sqrt(2 * kb * T /
(mass * wr**2))
return bprime
def ideal_fermi_numbers(fitparams, pixcal, sigma=None):
"""Determine T/T_F and N for an ideal Fermi gas.
**Inputs**
* fitparams: the result of fitting the image with ideal_fermi_radial
fitparams is a list containing the central optical density,
logarithm of the fugacity and thermal radius of the cloud in
pixels
* pixcal: calibration for the camera in meters per pixel
**Outputs**
* ToverTF: temperature of the Fermi gas T/T_F
* N: number of atoms
**Optional inputs**
* sigma: photon absorption cross-section
the default value is the resonant cross-section for 6Li
"""
if sigma==None:
sigma = 3*671e-9**2/(2*np.pi)
mubeta = fitparams[1]
ToverTF = (6*fermi_poly3(mubeta))**(-1./3)
N = pixcal**2*sp.integrate.quad(n2D_radial, 0, np.infty,\
args=(ideal_fermi_radial, fitparams))[0]/sigma
return ToverTF, N
def fugacity(ToverTF):
"""Calculate the log of the fugacity e^{\beta\mu}"""
minpoly3 = lambda x, num: fermi_poly3(x) - num
a = sp.optimize.brentq(minpoly3, -1e3, 1e3, args=(ToverTF**(-3) / 6))
return a
def ideal_fermi_numbers(fitparams, pixcal, sigma=None):
"""Determine T/T_F and N for an ideal Fermi gas.
**Inputs**
* fitparams: the result of fitting the image with ideal_fermi_radial
fitparams is a list containing the central optical density,
logarithm of the fugacity and thermal radius of the cloud in
pixels
* pixcal: calibration for the camera in meters per pixel
**Outputs**
* ToverTF: temperature of the Fermi gas T/T_F
* N: number of atoms
**Optional inputs**
* sigma: photon absorption cross-section
the default value is the resonant cross-section for 6Li
"""
if sigma==None:
sigma = 3*671e-9**2/(2*np.pi)
mubeta = fitparams[1]
ToverTF = (6*fermi_poly3(mubeta))**(-1./3)
N = pixcal**2*sp.integrate.quad(n2D_radial, 0, np.infty,\
args=(ideal_fermi_radial, fitparams))[0]/sigma
return ToverTF, N
def ideal_fermi_numbers_2D_angled(fitparams, pixcal, sigma=None):
"""Determine T/T_F and N for an ideal Fermi gas.
**Inputs**
* fitparams: array_like
An array containing the central optical density,
logarithm of the fugacity and thermal radius of the cloud in
pixels.
* pixcal: float,
Calibration for the camera in meters per pixel.
**Outputs**
* ToverTF: temperature of the Fermi gas T/T_F
* N: number of atoms
**Optional inputs**
* sigma: photon absorption cross-section
The default value is the resonant cross-section for 6Li.
"""
if sigma == None:
sigma = 3 * 671e-9**2 / (2 * np.pi)
mubeta = fitparams[5]
ToverTF = (6 * fermi_poly3(mubeta))**(-1. / 3)
x, y, sx, sy = fitparams[0:4]
sx = max(sx, 5)
sy = max(sy, 5)
[X, Y] = np.mgrid[x - 5 * sx:x + 5 * sx, y - 5 * sy:y + 5 * sy]
newparams = [
fitparams[0], fitparams[1], fitparams[2], fitparams[3], fitparams[4],
fitparams[5], 0., 0., 0.
]
odimg_fitted = idealfermi_2D(newparams, X, Y)
N = odimg_fitted.sum() * pixcal**2 / sigma
return ToverTF, N
def ideal_fermi_numbers_2D_angled(fitparams, pixcal, sigma=None):
"""Determine T/T_F and N for an ideal Fermi gas.
**Inputs**
* fitparams: array_like
An array containing the central optical density,
logarithm of the fugacity and thermal radius of the cloud in
pixels.
* pixcal: float,
Calibration for the camera in meters per pixel.
**Outputs**
* ToverTF: temperature of the Fermi gas T/T_F
* N: number of atoms
**Optional inputs**
* sigma: photon absorption cross-section
The default value is the resonant cross-section for 6Li.
"""
if sigma==None:
sigma = 3*671e-9**2/(2*np.pi)
mubeta = fitparams[5]
ToverTF = (6*fermi_poly3(mubeta))**(-1./3)
x, y, sx, sy = fitparams[0:4]
sx=max(sx,5)
sy=max(sy,5)
[X, Y] = np.mgrid[x-5*sx:x+5*sx, y-5*sy:y+5*sy]
newparams=[fitparams[0],fitparams[1],fitparams[2],fitparams[3],fitparams[4],fitparams[5],0.,0.,0.]
odimg_fitted = idealfermi_2D(newparams, X, Y)
N = odimg_fitted.sum() * pixcal**2 / sigma
return ToverTF, N
def test_fermi_poly3_float(self):
testfloat = 1.
assert_approx_equal(- lerch.Li(3, -np.exp(testfloat)), \
polylog.fermi_poly3(testfloat), significant=7)
def test_fermi_poly3_array(self):
fp_approx1 = polylog.fermi_poly3(self.xx1)
# the polynomial approximation should be good to 1e-7, fails for
# higher precision test
assert_array_almost_equal(self.exact1, fp_approx1, decimal=7)
def best_fugacity(logfugacity):
return abs(ToverTF - (6*fermi_poly3(logfugacity))**(-1./3))
def best_fugacity(logfugacity):
return abs(ToverTF - (6*fermi_poly3(logfugacity))**(-1./3))
