arc.alkali_atom_functions.AlkaliAtom.getC6term#

AlkaliAtom.getC6term(n: int, l: int, j: float, n1: int, l1: int, j1: float, n2: int, l2: int, j2: float, s: float = 0.5) float[source]#

C6 interaction term for the given two pair-states

Calculates \(C_6\) intaraction term for \(|n,l,j,n,l,j \rangle \leftrightarrow |n_1,l_1,j_1,n_2,l_2,j_2\rangle\). For details of calculation see Ref. [1].

Parameters:
  • n (int) – principal quantum number

  • l (int) – orbital angular momentum

  • j (float) – total angular momentum

  • n1 (int) – principal quantum number

  • l1 (int) – orbital angular momentum

  • j1 (float) – total angular momentum

  • n2 (int) – principal quantum number

  • l2 (int) – orbital angular momentum

  • j2 (float) – total angular momentum

  • s (float) – optional, total spin angular momentum of state. By default 0.5 for Alkali atoms.

Returns:

\(C_6 = \frac{1}{4\pi\varepsilon_0} \frac{|\langle n,l,j |er|n_1,l_1,j_1\rangle|^2| \langle n,l,j |er|n_2,l_2,j_2\rangle|^2} {E(n_1,l_1,j_2,n_2,j_2,j_2)-E(n,l,j,n,l,j)}\) (\(h\) Hz m \({}^6\)).

Return type:

float

Example

We can reproduce values from Ref. [1] for C3 coupling to particular channels. Taking for example channels described by the Eq. (50a-c) we can get the values:

from arc import *

channels = [[70,0,0.5, 70, 1,1.5, 69,1, 1.5],\
            [70,0,0.5, 70, 1,1.5, 69,1, 0.5],\
            [70,0,0.5, 69, 1,1.5, 70,1, 0.5],\
            [70,0,0.5, 70, 1,0.5, 69,1, 0.5]]

print(" = = = Caesium = = = ")
atom = Caesium()
for channel in channels:
    print("%.0f  GHz (mu m)^6" % ( atom.getC6term(*channel)
                                  / C_h * 1.e27 ))

print("\n = = = Rubidium  = = =")
atom = Rubidium()
for channel in channels:
    print("%.0f  GHz (mu m)^6" % ( atom.getC6term(*channel)
                                  / C_h * 1.e27 ))

Returns:

 = = = Caesium = = =
722  GHz (mu m)^6
316  GHz (mu m)^6
383  GHz (mu m)^6
228  GHz (mu m)^6

 = = = Rubidium  = = =
799  GHz (mu m)^6
543  GHz (mu m)^6
589  GHz (mu m)^6
437  GHz (mu m)^6

which is in good agreement with the values cited in the Ref. [1]. Small discrepancies for Caesium originate from slightly different quantum defects used in calculations.

References