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V. A. DZUBA, V. V. FLAMBAUM, AND BENJAMIN L. LEV PHYSICAL REVIEW A 83, 032502 (2011)
where the summation is over a limited number of low-lying
near-resonant states and a constant ˜α
a
is chosen in such
a way that Eq. (2)atω = 0 provides the correct value of
the polarizability.
Dysprosium ground-state static polarizability is known to
be 166 a
3
B
[24]. Static polarizabilities of excited states are
not known and need to be calculated. We use an approximate
approach in which the dysprosium atom is treated as a closed-
shell system and the effect of electron vacancies in the open
shells is taken into account via fractional occupation numbers.
The static polarizability of a closed-shell s ystem is given by
α
a
(0) =−
2
3
cn
c||D||n
2
c
−
n
, (3)
where the summation is over a complete set of single-electron
states, including states in the core (c) and states above the
core ( n). Electric dipole matrix elements are calculated using
relativistic Hartree-Fock and Hartree-Fock in external field
approximations [25]. Note that core polarization needs to be
included only in one of two electric dipole matrix elements in
(3) (see, e.g., Ref. [26] f or details).
We use the standard B-spline technique [27] to generate
a complete set of single-electron states. An additional term
is included in the Hartree-Fock Hamiltonian to simulate the
effect of correlations. This term has the form
δV (r) =−
d
2
r
4
0
+ r
4
, (4)
where r
0
is a cutoff parameter (we use r
0
= 1 a
B
) and d
is dipole polarizability of the core. We treat d as a fitting
parameter and choose it to fit the known polarizability of
dysprosium’s ground state (166a
3
0
[24]), which results in
d = 3.7 a
3
B
.
Then we perform similar calculations for the excited states
of the 4f
9
6s
2
5d configuration, resulting in a calculated value
of the static polarizability of 114 a
3
B
. Note that this approach
does not distinguish between different states of the same
configuration. Therefore, static polarizabilities of all these
states are assumed to be equal. This is only true for the
static polarizabilities. Dynamic polarizabilities are different
for different states due to contributions of the near-resonant
states in Eq. ( 2).
In the present paper we consider dynamic polarizabilities
of four states of dysprosium: the even ground state (GS) and
three odd long-lived excited states. The first excited state is
7
H
o
8
at λ = 1322 nm (E = 7565.60 cm
−1
), and we denote it
as O1 for reference. This state is in the telecommunications
band and could be used for hybrid atom-photon telecom
quantum information networks. The second excited state is
the
7
I
o
9
state at 1001 nm (9990.95 cm
−1
), which we denote
as O2. Quantum dots (QDs) emit in this wavelength range,
allowing for the possibility of hybrid quantum circuits of
QD single-photon emitters coupled to neutral atom-based
long-lived quantum memory. O3 is the
5
K
o
9
state at 741 nm
(13 495.92 cm
−1
), which is a closed cycling transition with a
linewidth [28] optimal for creating a narrow-line MOT. States
O2 and O3 could also be useful for resolved-sideband cooling,
as discussed below.
TABLE I. Electric dipole transition amplitudes (reduced matrix
elements in atomic units) used for calculating the dynamic polariz-
ability of the Dy ground state
5
I
8
.
State n
Configuration Term E
n
(cm
−1
) |A
na
|(a.u.)
a
4f
9
5d6s
27
H
o
8
7565 0.061
4f
9
5d6s
27
H
o
7
8519 0.124
4f
9
5d6s
27
I
o
9
9990 0.059
4f
9
5d6s
27
I
o
8
12 007 0.573
4f
9
5d6s
27
G
o
7
12 655 0.108
4f
9
5d6s
25
K
o
9
13 495 0.424
4f
9
5d6s
27
I
o
7
14 367 0.475
4f
9
5d6s
25
I
o
8
14 625 1.828
4f
9
5d6s
25
H
o
7
15 194 1.452
4f
10
6s6p (8,0)
o
8
15 567 0.464
4f
10
6s6p (8,1)
o
9
15 972 1.365
4f
9
5d6s
27
K
o
8
16 288 0.182
4f
10
6s6p (8,1)
o
7
16 693 1.842
4f
10
6s6p (8,1)
o
8
16 733 0.633
4f
9
5d6s
27
K
o
9
16 717 0.415
4f
9
5d6s
27
K
o
7
17 687 0.763
4f
10
6s6p (8,2)
o
9
17 727 0.897
4f
10
6s6p (8,2)
o
8
18 021 0.684
4f
10
6s6p (8,2)
o
7
18 433 0.636
4f
9
5d
2
6s
9
G
o
7
18 528 0.067
4f
9
5d
2
6s
7
H
o
9
19 557 0.036
4f
9
5d6s
25
K
o
8
19 688 0.627
4f
9
5d
2
6s
7
G
o
9
21 540 0.523
4f
10
6s6p (7,2)
o
9
21 838 0.513
4f
9
5d
2
6s ?
o
9
23 271 0.003
4f
10
6s6p (8,1)
o
9
23 737 12.277
a
A
na
≡n||D||a.
We calculate dynamic polarizabilities using Eq. (1)in
which we substitute transition amplitudes found from the CI
calculations [5] and experimental energies. We use theoretical
values in the few cases where experimental energies are not
available. Tables I and II show calculated electric dipole
transition amplitudes (reduced matrix elements) used in the
calculations. The data from Table II can be used to calculate
the lifetimes of the three excited states. The results are 5.2 ms
for O1, 2.7 ms for O2, and 21 µs for O3, although O3 has
recently been measured to be 89.3 µs[28].
Figures 1, 2, and 3 show dynamic polarizabilities of three
pairs of states: GS and O1 (Fig. 1), GS and O2 (Fig. 2), and
GS and O3 (Fig. 3). Lines crossing indicates that energy shifts
of two states in a laser field are identical, and atoms have
the same oscillation frequency in optical dipole traps at this
wavelength regardless of whether they are in their ground or
excited state. These so-called magic wavelengths occur most
often very close to narrow resonances.
B. Simple estimations
In this subsection we present a way of estimating magic
wavelengths for complex atoms in the vicinity of narrow
resonances. Although all magic wavelengths presented in this
work are found by the many-body calculations, the formulas
in this subsection can be used to find more magic wavelengths
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