PHYSICAL REVIEW A 83, 032502 (2011)

Dynamic polarizabilities and magic wavelengths for dysprosium

V. A. Dzuba and V. V. Flambaum

School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia

Benjamin L. Lev

Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA

(Received 22 November 2010; published 4 March 2011)

We theoretically study dynamic scalar polarizabilities of the ground and select long-lived excited states of

dysprosium, a highly magnetic atom recently laser cooled and trapped. We demonstrate that there is a set of

magic wavelengths of the unpolarized lattice laser ﬁeld for each pair of states, which includes the ground state

and one of these excited states. At these wavelengths, the energy shift due to laser ﬁeld is the same for both

states, which can be useful for resolved sideband cooling on narrow transitions and precision spectroscopy. We

present an analytical formula that, near resonances, allows for the determination of approximate values of the

magic wavelengths without calculating the dynamic polarizabilities of the excited states.

DOI: 10.1103/PhysRevA.83.032502 PACS number(s): 31.15.am, 32.70.Cs, 31.30.jg, 37.10.De

I. INTRODUCTION

The dysprosium atom has many unique features, which

makes it useful for studying fundamental problems of mod-

ern physics. This is a heavy atom that has many stable

Bose and Fermi isotopes (from A = 156 to A = 164) and

a pair of almost-degenerate states of opposite parity at

E = 19 798 cm

−1

. These features were used to study the

parity nonconservation [1–5] and possible time variation of

the ﬁne-structure constant [6–12].

Fermionic Dy has the largest magnetic moment among all

atoms, and only Tb is as magnetic as bosonic Dy. This opens

important opportunities for studying strongly correlated matter

when gases of Dy atoms are cooled to ultracold temperatures

[13]. Recent progress in Doppler and sub-Doppler cooling is an

important step in this direction [13–17]. In addition to narrow-

line magneto-optical trapping (MOT) [18], further cooling on

narrow optical transitions might be possible using resolved-

sideband cooling [19,20].

In this method, vibrational states of the atom may be cou-

pled such that successive photon absorption and spontaneous

emission cycles reduce the vibrational quanta by one, until

the atoms are in the motional ground state of their optical

potential [19]. It is important that this resolved-sideband

cooling is performed at the magic wavelength of the laser

lattice ﬁeld [21,22]. At this wavelength, the energy (ac Stark)

shift due to the laser ﬁeld is the same for both states used in

the cooling. This results in a trap potential that is the same

for both states, and optical transitions between vibrational

states can be well resolved. This allows spectral selection

of cooling transitions, those which remove one vibrational

quanta, without contamination by heating transitions, which

add vibrational quanta. Other beneﬁts to optical trapping at

magic wavelengths include enhanced precision spectroscopy

and longer-lived quantum memory for quantum information

processing (QIP) [21].

In this paper we calculate dynamic polarizabilities of

the ground and three long-lived excited states of Dy and

present a number of magic wavelengths for the transitions

between them. We also present an analytical formula that

allows the determination of approximate values of the magic

wavelengths near resonances without calculating the dynamic

polarizabilities of excited states. The optical ﬁeld is assumed to

be unpolarized, although we estimate that polarization would

induce only small shifts in the magic wavelengths.

II. CALCULATIONS

A. Ab initio calculations

The dynamic scalar polarizability α

of atomic state a is

given by (we use atomic units: ¯h = 1,m

= 1,|e|=1)

(ω) =−

3(2J

+ 1)





− E

+ ω

− E

− ω



a||D||n

(1)

where J

is total momentum of state a, E

is its energy, and

D =−



is the electric dipole operator. Summation goes

over the complete set of excited states n.

We use the relativistic conﬁguration interaction (CI) tech-

nique described in our previous papers [5,11,23] to perform

the calculations. The single-electron and many-electron basis

sets, the ﬁtting parameters, and other details of present

calculations are exactly the same as in Ref. [5]. This simple

method provides a good accuracy for low-lying states of

a many-electron atom. However, it does not allow for the

saturation of the summation in Eq. (1) over a complete set

of many-electron states. On the other hand, the contribution

of the higher-lying states in the dynamic polarizability does

not depend on frequency at small frequencies. Therefore, for

small frequencies we can rewrite Eq. (1)as

(ω) = ˜α

−

3(2J

+ 1)







− E



+ ω

− E



− ω



a||D||n





(2)

032502-1

1 2 3 4 5 6

Résumé du contenu

Page 1

PHYSICAL REVIEW A 83, 032502 (2011)Dynamic polarizabilities and magic wavelengths for dysprosiumV. A. Dzuba and V. V. FlambaumSchool of Physics, Unive

Page 2

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-lyingnear-reso

Page 3

DYNAMIC POLARIZABILITIES AND MAGIC ... PHYSICAL REVIEW A 83, 032502 (2011)TABLE II. Electric dipole transition amplitudes (reduced ma-trix elements in

Page 4

V. A. DZUBA, V. V. FLAMBAUM, AND BENJAMIN L. LEV PHYSICAL REVIEW A 83, 032502 (2011)FIG. 3. Dynamic polarizability α of the ground state of Dy (solidl

Page 5

DYNAMIC POLARIZABILITIES AND MAGIC ... PHYSICAL REVIEW A 83, 032502 (2011)but also for some other transitions. The main condition forit to work is tha

Page 6

V. A. DZUBA, V. V. FLAMBAUM, AND BENJAMIN L. LEV PHYSICAL REVIEW A 83, 032502 (2011)[14] N. Leefer, A. Cing¨oz, D. Budker, S. J. Ferrell, V. V. Yashch

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