arc.divalent_atom_functions.DivalentAtom.getC6term#

DivalentAtom.getC6term(n, l, j, n1, l1, j1, n2, l2, j2, s=0.5)#

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

1(1,2,3)

T. G. Walker, M. Saffman, PRA 77, 032723 (2008) https://doi.org/10.1103/PhysRevA.77.032723