Dynamic polarizabilities and magic wavelengths for

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 modern 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 narrowline magneto-optical trapping (MOT) [18], further cooling on
narrow optical transitions might be possible using resolvedsideband cooling [19,20].
In this method, vibrational states of the atom may be coupled 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
1050-2947/2011/83(3)/032502(6)
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 αa of atomic state a is
given by (we use atomic units: h
¯ = 1,me = 1,|e| = 1)
1
αa (ω) = −
3(2Ja + 1) n
1
1
+
a||D||n2 ,
×
Ea − En + ω Ea − En − ω
(1)
where J
a is total momentum of state a, Ea is its energy, and
D = − i ri is the electric dipole operator. Summation goes
over the complete set of excited states n.
We use the relativistic configuration interaction (CI) technique 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
αa (ω) = α˜ a −
032502-1
1
3(2Ja + 1)
n
1
1
×
+
a||D||n 2 ,
Ea − En + ω Ea − En − ω
(2)
©2011 American Physical Society
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) at ω = 0 provides the correct value of
the polarizability.
Dysprosium ground-state static polarizability is known to
be 166 aB3 [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 closedshell 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 system is given by
αa (0) = −
2 c||D||n2
,
3 cn 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] for 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
d
,
δV (r) = − 4
2 r0 + r 4
(4)
where r0 is a cutoff parameter (we use r0 = 1 aB ) 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 (166a03 [24]), which results in
d = 3.7 aB3 .
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 aB3 . 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 o
H8 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 Io9 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 Ko9 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 polarizability of the Dy ground state 5 I8 .
State n
Configuration
Term
En (cm −1 )
|Ana |(a.u.)a
4f 9 5d6s 2
4f 9 5d6s 2
4f 9 5d6s 2
4f 9 5d6s 2
4f 9 5d6s 2
4f 9 5d6s 2
4f 9 5d6s 2
4f 9 5d6s 2
4f 9 5d6s 2
4f 10 6s6p
4f 10 6s6p
4f 9 5d6s 2
4f 10 6s6p
4f 10 6s6p
4f 9 5d6s 2
4f 9 5d6s 2
4f 10 6s6p
4f 10 6s6p
4f 10 6s6p
4f 9 5d 2 6s
4f 9 5d 2 6s
4f 9 5d6s 2
4f 9 5d 2 6s
4f 10 6s6p
4f 9 5d 2 6s
4f 10 6s6p
Ho8
Ho7
7 o
I9
7 o
I8
7 o
G7
5 o
K9
7 o
I7
5 o
I8
5 o
H7
(8,0)o8
(8,1)o9
7 o
K8
(8,1)o7
(8,1)o8
7 o
K9
7 o
K7
(8,2)o9
(8,2)o8
(8,2)o7
9 o
G7
7 o
H9
5 o
K8
7 o
G9
(7,2)o9
?o9
(8,1)o9
7565
8519
9990
12 007
12 655
13 495
14 367
14 625
15 194
15 567
15 972
16 288
16 693
16 733
16 717
17 687
17 727
18 021
18 433
18 528
19 557
19 688
21 540
21 838
23 271
23 737
0.061
0.124
0.059
0.573
0.108
0.424
0.475
1.828
1.452
0.464
1.365
0.182
1.842
0.633
0.415
0.763
0.897
0.684
0.636
0.067
0.036
0.627
0.523
0.513
0.003
12.277
a
7
7
Ana ≡ 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
032502-2
DYNAMIC POLARIZABILITIES AND MAGIC . . .
PHYSICAL REVIEW A 83, 032502 (2011)
TABLE II. Electric dipole transition amplitudes (reduced matrix elements in atomic units) used for calculating the dynamic
polarizabilities of the selected three long-lived Dy excited states.
|Ana |a (a.u.)
State n
Configuration
4f 10 6s 2
4f 10 6s 2
4f 10 5d6s
4f 10 5d6s
4f 10 5d6s
4f 10 5d6s
4f 10 5d6s
4f 10 5d6s
4f 10 6s 2
4f 10 5d6s
4f 10 5d6s
4f 10 5d6s
4f 10 5d6s
4f 9 6s 2 6p
4f 9 6s 2 6p
4f 10 5d6s
4f 10 5d6s
4f 10 5d6s
4f 10 5d6s
4f 10 5d6s
4f 9 5d6s6p
4f 10 5d6s
Term
5
I8
I7
3
[8]9
3
[7]8
3
[6]7
3
[9]10
3
[8]8
3
[7]7
3
K28
3
[9]9
3
[10]10
3
[9]8
3
[10]9
( 152 , 12 )7
( 152 , 12 )8
3
[8]7
3
[7]8
3
[6]7
1
[9]9
1
[9]10
?9
?7
5
En (cm−1 )
O1b
O2c
O3d
0
4134
17 515
17 613
18 095
18 463
18 903
18 938
19 019
19 241
19 798
20 194
20 209
20 614
20 790
21 074
21 603
21 778
22 046
22 487
23 219
23 361
0.061
0.007
0.033
0.195
0.170
0.059
0.424
0.154
0.321
0.139
0.046
0.060
0.401
0.210
0.268
0.093
0.119
0.049
0.336
0.062
0.064
0.015
0.192
0.288
0.010
4.254
2.911
0.450
0.130
0.081
0.004
3.879
0.116
0.112
0.400
0.300
0.259
0.083
0.079
0.358
0.041
2.164
3.737
0.550
0.362
0.463
0.079
1.313
0.172
Ana ≡ n||D||a.
State a = 4f 9 5d6s 2 , 7 Ho8 , E = 7565.60 cm−1 ; λ = 1322 nm.
c
State a = 4f 9 5d6s 2 , 7 Io9 , E = 9990.95 cm−1 ; λ = 1001 nm.
d
State a = 4f 9 5d6s 2 , 5 Ko9 , E = 13495.92 cm−1 ; λ = 741 nm.
a
b
for dysprosium or to find magic wavelengths for other complex
atoms. We demonstrate that many-body calculations and
simple estimations give close results.
In the case of a narrow resonance, energy denominator in
Eq. (1) is close to zero. This makes it possible to write an
approximate formula for the magic frequency corresponding
to this wavelength. Starting from the condition
αGS (ω∗ ) = αa (ω∗ )
FIG. 1. Dynamic polarizability α of the ground state of Dy (solid
line) and O1 (dotted line) between laser frequencies (wavelengths)
12 500 cm−1 (800 nm) and 13 500 cm−1 (741 nm). Lines cross
at magic frequencies, and large dots correspond to the most useful
frequencies.
amplitude from the excited state a to the resonance state n,
and αGS (ωn ) is the scalar polarizability of the ground state at
ωn . Note that ωn is the resonance frequency for the upper state,
and the polarizability of the ground state usually changes very
little in the vicinity of ωn .
In case of two closely spaced resonances in the upper-state
polarizability, magic frequencies in the vicinity of resonance
energies E1 and E2 can be found using the approximate
(5)
and presenting dynamic polarizability of the excited state in
the vicinity of the resonance n in the form
1
αa (ω) = αa (0) −
3(2Ja + 1)
1
1
+
a||D||n2 ,
×
Ea − En + ω Ea − En − ω
(6)
we arrive at the following expression:
∗
= |Ea − En | +
ωan
Ea − En
δn ,
|Ea − En |
a||D||n2
δn =
,
3(2Ja + 1) [αGS (ωn ) − αa (0)]
1
(7)
where ωn = |Ea − En | is the resonance frequency, n is the
resonance number, a||D||n is the electric dipole transition
FIG. 2. Dynamic polarizability α of the ground state of Dy (solid
line) and O2 (dotted line) between laser frequencies (wavelengths)
8500 cm−1 (1.18 µm) and 13 000 cm−1 (769 nm). Lines cross at magic
frequencies, and large dots correspond to the most useful frequencies.
032502-3
V. A. DZUBA, V. V. FLAMBAUM, AND BENJAMIN L. LEV
PHYSICAL REVIEW A 83, 032502 (2011)
of the ground state and the relevant transition amplitudes.
An approximate solution can be found using the following
rules.
(i) The dynamic polarizability of the ground state is fitted
very well within the energy interval from 0 to 0.05 a.u. by
2.955
2.955
−
αGS (ω) =
ω + 0.108 15 ω − 0.108 15
+110 + 3000ω2 .
FIG. 3. Dynamic polarizability α of the ground state of Dy (solid
line) and O3 (dotted line). Lines cross at magic frequencies, and the
large dot corresponds to the most useful frequency.
formula
∗
=
ωa12
where
δ12 = 12
E1 + E2
+ δ12 ,
2
(8)
(0)
CJ 12 αGS
(ω12 ) − αa(0) (0) + A22 − A21
A22 + A21
(9)
.
Here 12 = (E1 − E2 )/2, CJ = 3(2Ja + 1), ω12 = (E1 +
E2 )/2, A1 = a||D||1, and A2 = a||D||2.
One does not need to know dynamic polarizability of an
excited state to find magic frequencies using Eq. (7) or Eq. (8).
However, one still needs to know the dynamic polarizability
(10)
We keep here numerical parameters for the dominant contribution to αGS , which is due to the transition to the 421-nm state
with energy E = 23 737 cm−1 = 0.108 15 a.u. This transition
dominates due to the largest value of the transition amplitude:
a||D||n = 12.277 a.u. (see Table I). The last two terms in
Eq. (10) fit the contribution of all other transitions in the
energy interval 0 < ω < 0.05 a.u. Resonances within this
interval are ignored since they are too narrow for any practical
importance.
(ii) Transition amplitudes can be estimated using approximate selection rules. If the difference of the total angular
momentum L between two states is larger than 1 or if total
spin S of the states is not equal, then the amplitude is not zero
due to relativistic corrections but is likely to be small. One
can use for a rough estimate A = 0.1 a.u. A similar estimation
can be used if the transition is suppressed by configuration
mixing, i.e., the transition between leading configurations
cannot be reduced to a single-electron allowed electric dipole
transition. If no selection rules are broken, the amplitude is
likely to be large, and one can use A = 3 a.u. as a rough
estimate.
(iii) Polarizability of the excited state at zero frequency
can be estimated using Eq. (1) with experimental energies
and with the amplitude estimated using the procedure in the
previous paragraph.
This procedure can help in estimating magic wavelengths
not only for the transitions considered in the present paper
TABLE III. Magic wavelengths λ∗ , frequencies ω∗ (cm−1 ), and polarizabilities α for the three transitions in Dy.
Transition
GS-O1
Ea = 7566 cm−1
λ = 1322 nm
GS-O2
Ea = 9991 cm−1
λ = 1001 nm
GS-O3
Ea = 13 496 cm−1
λ = 741 nm
Resonance
En − Ea
δn a
δn b
Magic frequencies
Formula
11 337
11 372
13 224
13 136c
7523
7622
8912
4
3
537
1
4
7
11 333
11 368
12 686
13 101
7522
7617
8905
13 227
5407
6302
6697
7293
9722
483
4
2
4
479
8
12 744
5403
6300
6693
6814
9714
−35
∗
Using formula (7), ωan
= En − Ea − δn .
∗
Using formulas (8) and (9), ωa12
= (E1 + E2 )/2 − Ea + δ12 .
c
En = (20 614 + 20 790)/2 − 7566.
a
b
032502-4
Calculations
λ∗
(nm)
α
(a.u.)
11 326
11 366
12 638
13 100
7521
7613
8891
9749
12 817
5401
6297
6671
6812
9716
883
880
791
763
1330
1314
1125
1026
780
1850
1588
1494
1468
1029
192
192
202
208
172
174
178
181
203
163
171
172
172
181
DYNAMIC POLARIZABILITIES AND MAGIC . . .
PHYSICAL REVIEW A 83, 032502 (2011)
but also for some other transitions. The main condition for
it to work is that the magic wavelength should be close to a
resonance so that the resonance term dominates in Eq. (1).
III. RESULTS
Table III shows the magic wavelengths and corresponding
polarizabilities for the three transitions in Dy. We substitute
calculated transition amplitudes from Table II when using
the analytical formulas (7) and (8). The column marked
as calculations presents magic frequencies that come from
numerical calculations. Magic wavelengths in the next column
correspond to calculated frequencies. Most of the magic
frequencies are due to very narrow resonances and might
be inconvenient for practical use due to optical dipole trap
frequency instabilities and enhanced spontaneous emission.
However, there are magic wavelengths for each of the
three transitions where the resonance is not very narrow or
even absent. They are λ = 791 nm (ω∗ = 12 638 cm−1 ) and
λ = 763 nm (ω∗ = 13 100 cm−1 ) for the 1322-nm GS-O1
transition; λ = 1125 nm (ω∗ = 8891 cm−1 ), λ = 1026 nm
(ω∗ = 9749 cm−1 ), and λ = 780 nm (ω∗ = 12 817 cm−1 )
for the 1001-nm GS-O2 transition; and λ = 1029 nm
(ω∗ = 9716 cm−1 ) and λ = 1468 nm (ω∗ = 6812 cm−1 ) for
the 741-nm GS-O3 transition.
The magic frequency ω∗ = 13 100 cm−1 for the GS-O1
transition is between two close resonances at ω = 13 149 cm−1
and ω∗ = 13 224 cm−1 that correspond to transitions from
the 7 Ho8 state at E = 7566 cm−1 to the close states of the
4f 9 6s 2 6p1/2 configuration at E = 20 614 cm−1 and E =
20 789 cm−1 . Using formulas (8) and (9) gives a very
accurate estimate of the magic frequency (see Table III).
It is interesting to note that there is a frequency interval
for the GS-O1 transition where polarizabilities of two states
come very close to each other but do not cross: δα/α 2%
for 12 121 < ω < 12 183 cm−1 .
The magic wavelengths λ = 780 and 1026 nm for the
GS-O2 transition and λ = 1468 nm for the GS-O3 transition
do not correspond to any very close resonance, and the
values of the polarizabilities coincide by chance rather than
due to a resonance. Therefore, these magic wavelengths are
the least sensitive to laser frequency fluctuations and are
the most promising for resolved-sideband cooling, precision
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magic wavelengths can be reached with high optical power
using a Ti:sapphire or tapered amplified diode laser for 780 nm
and a diode laser or fiber laser for 1026 nm. The GS-O3
transition magic wavelength at 1468 nm could be reached with
a low-power diode laser, which could perhaps be doped-fiber
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a fiber laser.
Future work will explore the role of laser polarization on
the magic wavelength position. Such a calculation is beyond
the scope of this present work, but we estimate that the shifts
will be small since the magic wavelengths for unpolarized light
occur in proximity to resonances.
Optical dipole traps at these wavelengths, far from the broad
Dy transitions, would be suitable for lattice confinement in
the Lamb-Dicke regime without undue heating. For example,
one-dimensional lattice confinement at the 780-nm magic
wavelength with 0.5 W provides ample trap depth with
sub-1-Hz scattering rates. With larger laser intensities, suitable
trap depths and low scattering rates can be achieved at the other
magic wavelengths. Vibrational spacing can be many tens of
kilohertz, which is large enough for resolved-sideband cooling
on the 2-kHz-wide 741-nm transition [28]. For rapid cooling,
the 50-Hz-wide 1001-nm transition, much narrower than any
trap frequencies in a typical three-dimensional (3D) optical
lattice, would need to be broadened via a quenching transition
[22], and the optical dipole trap magic wavelength would need
to be adjusted to compensate the Stark shift from the quenching
laser. Resolved-sideband cooling on these narrow transitions
in a 3D optical lattice may provide an alternative route to
quantum degeneracy [29] versus evaporative cooling, which
may fail due to (as yet unmeasured) unfavorable scattering
properties in this highly dipolar gas.
ACKNOWLEDGMENTS
We thank J. Ye, I. Deutsch, and N. Burdick for discussions.
The work was funded in part by the Australian Research
Council (V.A.D., V.V.F.), the NSF (Grant No. PHY08-47469)
(B.L.L.), AFOSR (Grant No. FA9550-09-1-0079) (B.L.L.),
and the Army Research Office MURI award W911NF0910406
(B.L.L.).
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