Magic-wave-induced $^1S_0-^3P_0$ transition in even isotopes of alkaline-earth-like atoms

0.15 MB·English

by  Vitaly D. Ovsiannikov

Magic-wave-induced 1S 3P transition in even isotopes of alkaline-earth-like atoms
0 0
−
Vitaly D. Ovsiannikov∗
Physics Department, Voronezh State University, Universitetskaya pl. 1, 394006, Voronezh, Russia
Vitaly G. Pal’chikov†
Institute of Metrology for Time and Space at National Research Institute for Physical–Technical
and Radiotechnical Measurements, Mendeleevo, Moscow Region, 141579 Russia
7
0
0 Alexey V. Taichenachev and Valeriy I. Yudin‡
2 Institute of Laser Physics SB RAS, Lavrent’ev Avenue 13/3, Novosibirsk 630090, Russia
Novosibirsk State University, Pirogova st. 2, Novosibirsk 630090, Russia
n
a
J Hidetoshi Katori and Masao Takamoto§

1 Department of Applied Physics, School of Engineering,
1 The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
(Dated: February 2, 2008)

]
h The circular polarized laser beam of the “magic” wavelength may be used for mixing the 3P
1
p stateintothelong-livingmetastable state3P ,thusenablingthestrictly forbidden1S 3P ”clock”
0 0 0
- transition in even isotopes of alkaline-earth-like atoms, without change of the transitio−n frequency.
m
Inoddisotopes thelaser beam may adjust toanoptimum valuethelinewidth ofthe”clock” tran-
o sition, originally enabled bythehyperfinemixing. Wepresent adetailed analysis of various factors
t influencing resolution and uncertainty for an optical frequency standard based on atoms exposed
a
. simultaneously to the lattice standing wave and an additional ”state-mixing” wave, including esti-
s mations of the“magic” wavelengths, Rabifrequencies for the”clock” and state-mixing transitions,
c
ac Stark shifts for theground and metastable states of divalent atoms.
i
s
y
PACSnumbers: 32.70.Jz, 32.80.-t,42.62.Eh,42.62.Fi
h
p
[
Extremely narrow atomic line corresponding to a menta, without hyperfine structure splitting and with-
1 strictlyforbidden1S0 3P0transitionbetweengroundand outantisymmetricandtensorincrementstotheacdipole
−
v metastable states of alkaline-earth-like atoms (such as polarizabilities and to the Stark effect. This makes the
4 Mg, Ca, Sr, Yb, Zn, Cd), currently considered as worth- Stark shift of the upper and lower levels independent of
3
while candidates for an optical frequency standard, may polarization of external fields. Meanwhile the circular
1
beobservedeitheronfreeoddisotopes[1,2,3]oroneven polarization of a laser wave allows for the second-order
1
0 isotopes in external fields [4, 5, 6, 7]. The mixing of the dipole-dipole mixing of the 3P1 state to the metastable
7 3P and3P statesbythehyperfineinteractionintheodd 3P state, which is strictly forbidden for the linearpolar-
1 0 0
0 isotopes and by an external field in the even isotopes is ization.
/
s the basic effect which removes the general selection-rule So, the role of the optical lattice field consists in trap-
c
restrictions on the 0 0 radiation transition. Intensive ping neutral atoms effectively free from collisions and
i −
s investigations of even alkaline-earth-like isotopes during Doppler effect (Lamb-Dicke regime) as well as from the
y
h the last few years were stimulated by a possibility to light field perturbations [1], whereas an additional beam
p design a new frequency standard based on an oscillator of the magic frequency, but with compulsory circular
: withtherecordhighqualityQ-factor. Inallthemethods (elliptic) polarization, will enable the strictly forbidden
v
i basedoninterrogationofthe stronglyforbidden1S0−3P0 radiation transitions via mixing the 3P1 state to the
X transition in the even isotopes embedded into an opti- metastable 3P state. The two waves may be generated
0
r cal lattice, engineered so as to equalize the upper- and byoneandthesamelaserorbecompletelyindependent,
a
lower-level Stark shifts, some additional radiation [4, 5] eachproperlyadjustedtosomeparticularconditions. So
or static [6, 7] fields were applied. they may have different intensities, polarizations, wave
In this article, we propose to use a circularly (ellipti- vectorsandevendifferentwavelengths,subject,however,
cally) polarized wave of the “magic” wavelength λ to the Stark-cancellation regime. In contrast with other
mag
(corresponding to the so-called “Stark-cancellation” methods [4]–[7], in this approach the atoms are exposed
regime, see e.g. [1, 8]), in addition to the optical lattice to onlythe magic-wavelengthradiation,no additionalac
field,inordertomix the3P statetothe 3P state. Since ordcfieldisusedandthereforenoadditionalshiftofthe
1 0
inevenisotopesthenuclearmomentumequalszero,both clock frequency can arise.
the initial and the final states of the frequency standard The origin of the laser radiation-induced mixing con-
transition (the “clock” transition) have zero total mo- sists in the possibility of the second-order dipole transi-
2
tion between the 3P1 state and the metastable 3P0 state where αs(ωm)= α1S0(ωm)= α3P0(ωm) is the ac polariz-
in ac field of a “magic” frequency ω=ω =2πc/λ (c abilityofthe “clock”levels;F representsthe amplitude
m mag L
is the speed of light) with a circular (elliptical) polariza- neartheantinodeofthelatticestandingwave,oscillating
tion. Tothisend,togetherwiththe standingwaveofthe with the “magic” frequency ω .
m
optical lattice, a circularly polarized wave of the magic The Rabi frequency for the running-wave-induced
frequency should be used, which we further consider as transition (2) is directly proportional to the product of
the running wave with the electric field vector the wave intensity I =cF2/8π and to the antisymmet-
r r
ric polarizability αa , and may be presented in MHz, as
F (r,t)=F Re e exp[i(k r ω t)] , (1) 3P
r r { · · − m } follows
where F is a real scalar amplitude, e is a complex unit
r
polarization vector, k = nωm/c is the wave vector with W10 =−0.01915ξαa3P(ωm)Ir, (5)
the unit vector n which should have a non-zero com-
ponent at right angle to the optical lattice beam in or- where Ir is taken in MW/cm2 and αa3P in atomic units.
The value of W determines the magnitude of the coef-
der that interrogation wave could travel along the lat- 10
ficient
tice in compliance with the Doppler-cancellation con-
ditions (for simplicity, we assume a 1D lattice here).
W
10
The contribution of the 3P1-state wave function into a1 = ∆ (6)
the metastable 3P -state wave function is determined by 10
0
the ratio of the field-induced 3P0−3P1 transition ampli- for the running-wave-induced contribution of the 3P1
tude (Rabifrequency)W10 tothe fine-structuresplitting state to the wave function of an atom initially (when
∆10=E3P1−E3P0. In the nonrelativistic dipole approxi- the field (1) is off) in the metastable 3P0 state
mation, the lowest non-vanishing (second) order in F
r
amplitude(theatomicunitsareusedinthispaper,ifnot ψ = 3P +a 3P
0 1 1
| i | i | i
otherwise indicated) = 3P +a a3P(0) +b1P(0) , (7)
F2 | 0i 1(cid:16) | 1 i | 1 i(cid:17)
W = r ξαa (ω ) (2)
10 −4√6 3P m wherethe superscript(0)indicatesapureLS-state. The
singlet-triplet mixing coefficients a and b in (7) may be
isdirectlyproportionaltothecircularpolarizationdegree
ξ =i(n [e e∗])andtotheantisymmetricpartαa (ω ) calculated using the ratio of the lifetimes τ(1P1), τ(3P1)
· × 3P m of singlet and triplet levels and the wavelengths λ(1P
ofthe3PJ tripletstateacpolarizability,whiche.g. forthe 1S ), λ(3P 1S ) of photons emitted in their radiat1io−n
state with maximal total momentum J = L+S = 2 is 0 1− 0
decay, as follows:
(see [9, 10, 11]):
M b2 τ(1P )λ3(3P 1S )
α3PJM(ω) = αs3P(ω)+ 2Jξαa3P(ω) a2 = τ(3P1)λ3(1P1− 1S0), a2+b2 =1. (8)
1 1 0
3M2 J(J +1) −
− 2J−(2J 1) αt3P(ω), (3) For Ir in MW/cm2, ∆10 in cm−1 and αa3P in atomic
− units the rate of the laser field-induced radiation transi-
here M=(n J) is the magnetic quantum number; the tion 3P 1S may be written as
· 0 0
superscripts (s) and (t) indicate the scalar and tensor →
paArtcstoufaltlhy,etahcepaomlaprilzitaubdileit(y2α)3mPJaMy(bωe).compared to the w = a1 2wic =0.4080 10−12 ξαa3P(ωm)Ir 2wic, (9)
| | · (cid:18) ∆ (cid:19)
10
amplitude of the hyperfine interaction, which mixes the
states in the odd isotopes [8, 10], or to the magnetic- where w =1/τ(3P ) is the field-free 3P 1S intercom-
ic 1 1 0
field-induced amplitude when the atoms experience the bination transition rate, the data for wh→ich is presented
action of a magnetic field, which may also be used for in Table II (see e.g. [12, 13]).
the3P0 3P1 statemixing[6,7]. Numericalcomputations As follows from the data of Table I, at the inten-
−
carriedoutinthesingle-electronapproximationwiththe sity I =0.5MW/cm2, the absolute value of the magic-
r
use of the model potential method for analytical presen- wave-induced amplitude (5) may amount to 10 MHz for
tation of the radial wave functions [9, 10], gave the nu- atoms of Ca, Sr and Yb, that is equivalent to the am-
merical values of the antisymmetric polarizabilities pre- plitude induced by a magnetic field of 1 mT [6]. With
sentedinTableIforMg,Ca,Sr,Yb,ZnandCdatomsat account of the data for the spin-orbit splitting of the
the “magic” wavelength corresponding to equal second- lowest (metastable) triplet state 3P (see e.g. [14]) the
J
order ac Stark shifts ∆E(3P0)=∆E(1S0)=EL(2) of the admixture of the 3P1 state in the 3P0-state wave func-
metastable and ground states (the lattice depth) tion at these conditions does not exceed 10−5. Similar
EL(2) =−14αs(ωm)FL2, (4) iensti(m7)ataetstihnedsiecactoentdhiatitonthseis1Py1etsi4ngtloet5stoartdeerasdmsmixatlulerre.
3
TABLEI:Numericalvaluesofthe“magic”wavelengthλ , TABLE II: Numerical values of the “clock” wavelength λ ,
mag c
3izPa1b−il3iPty0 αspa3Plitatinndgt∆he10la=ttiEce3P-fi1e−ld-Ein3Pd0u,caedntsi-escyomndm-oertdriecrpSotlaarrk- cfioeeldffiacniedntqsuaκd(p1r)ataicndinκin(2t)enosfitlyinoefarthienciinrctuenlasritlyypooflatrhiezepdrolabte-
shift(latticedepth)EL(2) fortheground-stateandmetastable tice wave and/or mixing wave Stark shifts (12), the rate wic
alkaline-earth-like atoms in the optical lattice of the magic of spontaneous intercombination transition 3P1 1S0 and the
→
wavelength λ and intensity I =10kW/cm2. The transi- coefficient β for the Rabi frequency (10). The number in
mag L
tionmatrixelementW isgivenforthemixing-waveintensity parentheses determines thepower of ten.
10
I =1MW/cm2.
r Atom λ κ(1)(ω ) κ(2)(ω ) w β
c c m ic
Atom λmag ∆10 αa3P(ωm) EL(2) W10/ξ nm mWm/Hczm2 (MWH/czm2)2 s−1 MW/cmm2|√H|mzW/cm2
nm cm−1 a.u. kHz MHz
Mg 458 4.27 -176 2.78(2) 32.7
Mg 432 20.06 538.5 49.3 10.3 Ca 660 4.50 -255 2.94(3) 137.5
Ca 680 52.16 1054 −102 −20.2 Sr 698 −44.2 -61.5 4.70(4) 176.9
Sr 813.42a 186.83 −1044 −116 20.0 Yb 578 −24.5 -16.8 1.15(6) 180.6
b − −
Yb 759.35 703.57 1084 78.7 20.8 Zn 309 0.816 -6.96 4.0(4) 15.2
− −
Zn 382 190.08 329.4 21.3 6.31 Cd 332 23.0 -10.3 4.17(5) 22.6
− −
Cd 390 542.1 390.6 25.8 7.48
− −
atheexperimentallydeterminedvalue[1,2]
btheexperimentallydeterminedvalue[7] thesplitting∆10 isincm−1. Accordingtothecalculated
numerical values of β (see Table II), the Rabi frequency
in Sr and Yb atoms (10) may achieve 0.3 Hz for the
However, the 3P1-state admixture may be sufficient to field (1) intensity Ir=0.5MW/cm2 and the probe field
enable the radiation transition between the ground and of I =10mW/cm2.
p
metastable states and to amplify the magnitude of the In the Stark-cancellation regime, when the second-
3P0 level width by 7 to 9 orders (in comparison with a orderacStarkshifts(4)oftheclocklevelsaremadeequal
two-photon E1-M1 or three-photon E1 spontaneous ra- to one another, the clock frequency may be distorted by
diation decay width [10]), up to several mHz, making theprobe-field-inducedquadraticacStarkshift(linearin
thinhaenbdcol,sooacnnkidctoranatnotsmhietsiooetnshs1eeSrn0th→iaanl3lPdy0,srmweteaalllilnedirnettgehtcahtnaebi3nPle0,felorenmvetilohwneiicdo.nthe oiInfttehnesFitcy2lo,Icckpor∝lerveFseplps2o)(nqaduninadgdltryha)te,icfioniundruitcnhet-deonrbdsiyetrietashceILSlat∝tatrikFceL2shfiaienfltdds
Together with the radiative decay rate (9), the impor- arnd∝thermixing wave, also including the bilinear in the
tant characteristic of the magic-wave-induced 1S0−3P0 intensities IL and Ir fourth-order correction. This shift
dipole transition, probed by the clock-frequency radia- may be written as
tion,istheamplitude(Rabifrequency)oftheclocktran-
sitionwhichafterintegrationinangularvariablesmaybe ∆ωc = κ(1)(ωc)Ip+κ(2)(eL,ωm)IL2 +κ(2)(e,ωm)Ir2
written as: + κ(2)(e ,e,ω )I I , (12)
L m L r
Ω= ψ vˆp 1S0 =βIr Ip (i[e e∗] ep), (10) where the constant κ(1)(ωc) is determined by the differ-
h | | i × ·
p ence of the upper- and lower-level polarizabilities at the
where vˆp = Ip(ep·r) is the Hamiltonian of the dipole clock-transition frequency ωc = 2πc/λc. For κ(1) in the
interaction pbetween atom and probe field of intensity units of mHz/(mW/cm2) the relation is
I and the unit polarization vector e which, evidently,
p p
shouldbeparalleltotherunningwavevectork i[e e∗], κ(1)(ωc)=−0.0469[α3P0(ωc)−α1S0(ωc)], (13)
∝ ×
thusthemaximalvalueofΩwillbefororthogonalpropa-
gationtotheprobebeam. So,inthecaseofa1Doptical wherepolarizabilitiesα3P0(ωc)andα1S0(ωc)areinatomic
units.
latticetheDoppler-freeinterrogationispossiblewhenthe Thecoefficientsκ(2)aredeterminedbythedifferenceof
probebeampropagatesalongthelatticeandispolarized
the clock-state hyperpolarizabilities at the ”magic” fre-
along the mixing beam wave vector.
quency ω (similar to polarizabilities, the hyperpolariz-
m
The coefficient β includes all radial integrals of the
abilitiesforstateswiththetotalmomentumJ=0include
matrixelement(10)whichmaybepresentedintheunits
only scalar parts, which, however, depend on the wave
of mHz/( mW/cm2 MW/cm2), as follows: polarization vector e [9, 11]),
·
p
β =204.9αa3P(ωm)h1P1(0)|r|1S0ib, (11) κ(2)(e,ωm) = −8.359·10−8
∆10 × [γ3P0(e,ωm)−γ1S0(e,ωm)], (14)
withtheantisymmetricpolarizabilityandtheradialpart whereκ(2) isintheunitsofHz/(MW/cm2)2,thehyper-
of the dipole transition matrix element in atomic units, polarizabilitiesγ1S0(e,ωm)andγ3P0(e,ωm)areinatomic
4
units. Although we assume the same ”magic” frequency catenearantinodesthenthefieldamplitude”seen”byan
ω for the lattice and running waves, the hyperpolar- atom is double what it is in the incident wave, therefore
m
izabilities for the linear polarization may differ essen- the coefficients in the right-hand sides of equations (2),
tially from those for the circular polarization [9, 11], i.e. (5) and (10) can be multiplied by 4. It means that the
κ(2)(e ,ω )=κ(2)(e,ω ) for different polarization vec- Rabi frequencies (5) and (10) for the given laser input
L m m
tors e and e6 . The clock-level hyperpolarizabilities de- intensity may become 4 times greater for the standing
L
termining the coefficient κ(2)(e ,e,ω ) of the interfer- wave in comparison with the running wave, in particu-
L m
ence term, bilinear in the lattice-waveand running-wave lar, in Sr and Yb atoms, W =40MHz and Ω=1.2Hz
10
intensities, depend on the relative orientation (and the for I =0.5MW/cm2 and I =10mW/cm2.
r p
type – linear or circular) of polarization vectors eL and VDOacknowledgesthesupportfromtheCRDF(USA)
e. That is why the fourth-order corrections from the and MinES of Russia (BRHE program, award VZ-010),
both waves should be taken into account together with AVT and VIYu were supported by RFBR (grants 05-02-
the mixed bilinear correction κ(2)(eL,e,ωm)ILIr. 17086, 05-08-01389, 07-02-01230, 07-02-01028) INTAS-
Thenumericalestimatesofκ(1)(ωc)andκ(2)(e,ωm)for SB RAS (grant 06-1000013-9427),and by Presidium SB
thecircularpolarizationofthelaserbeamearepresented RAS.
in table II. The hyperpolarizabilities for the metastable
3P levelsoftheMg,ZnandCdatomsarecomplexvalues
0
with imaginary parts (determining the two-photon ion-
ization width) negligible in comparison with real parts.
In estimating real parts of the hyperpolarizabilities in ∗ Electronic address: [email protected]
† Electronic address: [email protected]
Mg,ZnandCd,wetookintoaccountonlythe“resonant”
‡ Electronic address: [email protected]
terms, which may be determined by the antisymmetric
§ Electronic address: [email protected]
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(2003).
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http://Physics.nist.gov
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