Laser cooling and trapping of potassium at magic wavelengths

0.52 MB·English

by  M. S. Safronova

Laser cooling and trapping of potassium at magic wavelengths
M. S. Safronova1,2, U. I. Safronova3,4, and Charles W. Clark2
1Department of Physics and Astronomy, 217 Sharp Lab,
University of Delaware, Newark, Delaware 19716
2Joint Quantum Institute, National Institute of Standards and Technology
and the University of Maryland, Gaithersburg, Maryland 20899-8410, USA
3Physics Department, University of Nevada, Reno, Nevada 89557
4Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556
3
1 Wecarryoutasystematicstudyofthestaticanddynamicpolarizabilities ofthepotassiumatom
0 using a first-principles high-precision relativistic all-order method in which all single, double, and
2 partialtripleexcitationsoftheDirac-Fockwavefunctionsareincludedtoallordersofperturbation
theory. Recommended values are provided for a large number of electric-dipole matrix elements.
n
Static polarizabilities of the 4s, 4p , 5s, 5p , and 3d states are compared with other theory and
a j j j
J experiment where available. We use the results of the polarizability calculations to identify magic
wavelengths for the4s−np transitions for n=4,5, i.e. those wavelengths for which thetwo levels
4
have the same ac Stark shifts. These facilitate state-insensitive optical cooling and trapping. The
1
magic wavelengthsforthe4s−5p transitions areofparticular interest forattaining aquantumgas
] of potassium at high phase-space density. We find 20 such wavelengths in the technically interest
h region of 1050−1130 nm. Uncertainties of all recommended values are estimated.
p
- PACSnumbers: 31.15.ac, 37.10.De,31.15.ap,31.15.bw
m
o
t I. INTRODUCTION geous for trapping and controllingatoms in high-Q cavi-
a
ties in the strong coupling regime, so as to minimize de-
.
s
coherence in quantum computation and communication
c Duetotheirapplicationsinultra-preciseatomicclocks,
si degeneratequantumgasesandquantuminformation,the protocols[19],andintheimplementationoftheRydberg
gateforquantumcomputingwithneutralatoms[20,21].
y magic wavelengths of atoms have become a subject of
h great interest in both experiments [1–4] and theory [5– Variations on the magic wavelength idea include the
p
15]. The energy levels of atoms trapped in a light field useofmultiple lightfieldstoattainacStarkshifttargets
[
are shifted by an amount that is proportional to their [22]ortomaximizedifferentialresponsebetweendifferent

1 frequency-dependent polarizability, so the difference in atomicspecies–forexample,the“tune-out”wavelengths
v the energiesofanytwolevelsdependsuponthe trapping thattraponespeciesbutnotanother[23,24]. Designand
1 field. This difference is often called the “ac Stark shift”. evaluationofalltheseapplicationsrequiresaccuratedata
8
The idea of a “magic” wavelength, λ , at which onatomicfrequency-dependentpolarizabilities. Onegoal
1 magic
3 there is no relative shift of a given pair of energy lev- ofourpresentworkis to providea listofallmagicwave-
. els, was first proposed in Refs. [16, 17] in the context of lengths for potassium UV 4s−5pj transitions in regions
1
optical atomic clocks. An atom confined in a trap con- that are convenient for laser cooling of ultracold gases
0
3 structed of light with a magic wavelength for the clock to high phase-space densities. For example, in 2011,
1 transition will, to lowest order, have the same transition low-temperature high-density magneto-optical trapping
: energy as it does in free space. of potassium using the open 4s1/2 →5p3/2 transition at
v 405 nm was performed by McKay et al. [2]. Fermionic
i This simple idea has a number of other applications.
X 40K was captured using a magneto-optical trap (MOT)
Aproblemarisesincoolingandtrappingschemes,where
on the closed 4s→ 4p transition at 767 nm and then
r the ac Stark shift of the cooling or trapping transition 3/2
a may lead to heating. Recent experiments in 6Li [18] and transferred, with high efficiency, to a MOT on the open
40K[2]degeneratequantumgasesinopticaltrapsdemon- 4s→5p3/2 transitionat405nm. Becausethe5p3/2state
hasasmallerlinewidththanthe4p state,theDoppler
strated temperature reductions by a factor of about five 3/2
limit is reduced from 145µK to 24µK, and temperatures
and phase-space density increases by at least a factor of
as low as 63(6)µK were observed.
ten by laser cooling using ultraviolet (UV) transitions
(2s − 3p and 4s − 5p, respectively) compared to con- In this paper we provide a list of magic wavelengths
ventional cooling with the visible or infrared D and D for the 4s − 4p and 4s − 5p transitions, calculate dc
1 2
transitions. However,theacStarkshiftsduetotraplight and ac polarizabilities for several low-lying states, and
must be nearly the same for both levels in the transition provide recommended values for a number of relevant
toallowforefficientanduniformcooling[18]. Thisisac- electric-dipole transitions which are of interest to appli-
complished by building the optical trap using light with cations such as those described above. Where possible,
the magic wavelength for the corresponding UV transi- we compare our results with available experimental [25]
tions. The use of the magic wavelengthsis also advanta- and high-precision theoretical values [26].
2
Someofthecalculationsreportedhererequiredevalua- where three designs for blue-detuned dipole traps were
tionoftheelectric-dipolematrixelementsforveryhighly presented.
excitedstates,suchas14s. Thesestatesareneededsince Magic wavelengthsfor the alkali-metalatoms fromNa
the ac polarizabilities for the magic wavelengths of par- to Cs, for which the ns and np or np atomic levels
1/2 3/2
ticularexperimentalinterest(around1050nm)aredom- have the same ac Stark shifts, were evaluated by Arora
inated by the 5p−nl transitions with n=12−14. Such et al. [14]. The case of circular polarization was con-
states were previously beyond the capabilities of the all- sidered in [30, 31]. McKeever et al. [19] demonstrated
order method used here due to the large spatial extent state-insensitive trapping of Cs atoms at 935 nm while
of the orbitals. In this work, we resolved the numerical maintaining a strongcoupling for the 6s −6p tran-
1/2 3/2
problems associated with such calculations and success- sition. Abichromaticschemeforstate-insensitiveoptical
fully demonstrated the stability of our new approach. trappingofRbatomwasexploredinRef.[22]. Inthecase
We begin with a brief review of recent researchon the ofRb,the magicwavelengthsassociatedwithmonochro-
applications of magic wavelength concepts in Section II. matic trapping were sparse and relatively inconvenient.
The calculation of electric-dipole matrix elements, static Thebichromaticapproachyieldedanumberofpromising
anddynamicpolarizabilitiesaswellastheiruncertainties magic wavelength pairs. The precise magic wavelengths
is discussed in Section III. The magic wavelengths are for Li 2s − 2p and 2s − 3p transitions in convenient
j j
discussed in Section IV. wavelength regions were recently calculated in [32]. The
results were presented for both 6Li and 7Li to illustrate
the possibilities for differential light shifts between the
II. REVIEW OF MAGIC WAVELENGTH two isotopes.
STUDIES
Up to the present, most work on magic wavelengths B. Group II
has been done on group I and group II atoms, which
arethespeciesmosteasilycooledandtrappedbyoptical The magic wavelengths for the Sr 5s2 1S −5s5p 3P◦
0 0
methods. We summarize these in turn. The following and 5s2 1S −5s5p 3P◦ transitions have been measured
0 1
examples are representativeof significant applications of in[33,34]. TheYbclock6s2 1S −6s6p3P◦ magicwave-
0 0
the magic wavelength concept, however, these examples
length was predicted to be 752 nm in [35] and measured
are not intended to constitute an exhaustive review.
to be 759.355nm in Ref. [36]. The magic wavelengthfor
the ultraviolet 6s2 1S ↔ 6s6p 3P◦ clock transition in
0 0
Hg wasrecentlyreportedby Yi et al. [1]. The Stark-free
A. Group I
(magic) wavelength was found to be 362.53(0.21) nm,
in excellent agreement with the theoretical prediction
The cancellation of the differential ac Stark shift of 360 nm from [37], calculated using a method that com-
themicrowavehyperfineclocktransitionintrapped87Rb bines configuration interaction and many-body pertur-
atoms was demonstrated in [3]. The technique had im- bation theory. The magic wavelengths of other group II
plications for experiments involving the precise control and group IIb atoms have been predicted in [38]. The
of optically trapped neutral atoms, but the cancellation magic wavelengthsare very sensitive to the values of the
comesattheexpenseofasmallmagnetic-fieldsensitivity. acpolarizabilitiesandallowforprecisetestsofthetheory
“Doublymagic”conditionsinmagic-wavelengthtrapping [39,40]. Moreover,themagicwavelengthscanbeusedto
ofultracoldalkali-metalatomswereinvestigatedbyDere- determine the values of important electric-dipole matrix
vianko [5]. This work demonstrated that the microwave elements which are difficult to obtain by direct experi-
transitionsinalkali-metalatomsmaybeindeedmadeim- mentaltechniques. Forexample,the5s5p3P◦−5s6s3S
0 1
pervioustobothtrappinglaserintensityandfluctuations matrix element in Sr was recently determined using the
of magnetic fields. experimental value of the Sr 5s2 1S −5s5p 3P◦ magic
0 0
The issueofthe mismatchofthe polarizabilitiesofthe wavelength with 0.5% precision [40]. Dammalapati et
ground and excited states has also arisen in the Ryd- al. [9] investigated light shifts of heavy alkaline earth
berg gate approach to quantum information processing elements barium (Ba) and radium (Ra), which are of in-
[27,28],inwhichthequbitisbasedontwogroundhyper- terest for development of optical lattice clocks and for
finestatesofneutralatomsconfinedinanopticallattice. permanent electric dipole moment searches. The wave-
AnatominaRydbergstatewill,ingeneral,moveinadif- length dependence of light shifts of the ns2 1S ground
0
ferent optical lattice potential than that experienced by state,thensnp3P◦ andns(n−1)d1D excitedstatesin
1 2
the ground state. Therefore, the vibrational state of the Ba (n = 6) and the ns2 1S ground state, the nsnp 3P◦
0 1
atominthelatticemaychangeafterthegateoperationis and ns(n−1)d 3D excited states in Ra (n = 7) were
2
completed, leading to decoherencedue to motionalheat- calculated. Several magic wavelengths in the visible and
ing. This problem may be resolved by the use of magic infraredregionsaccessiblewithcommerciallasersforop-
wavelengths [20, 21]. Use of magic wavelengths in op- tical dipole trapping of Ba and Ra were identified [9].
tical traps for Rydberg atoms was also discussed in [8], Magic wavelengths of an optical clock transition of bar-
3
TABLE I: Absolute values of the reduced electric-dipole matrix elements in K and their uncertainties in a.u.. The present
all-order values are given unless noted otherwise. (a)Expt.[29], (b)determined from Stark shift data in [14]. The uncertainties
are estimated where possible (see text).
Transition Value Transition Value Transition Value Transition Value
4s−4p 4.106(4)(a) 5s−4p 3.885(8) 4s−4p 5.807(7)(a) 5s−4p 5.54(1)
1/2 1/2 3/2 3/2
4s−5p 0.2755 5s−5p 9.49(3) 4s−5p 0.4060 5s−5p 13.40(4)
1/2 1/2 3/2 3/2
4s−6p 0.0855 5s−6p 0.90(1) 4s−6p 0.1302 5s−6p 1.30(2)
1/2 1/2 3/2 3/2
4s−7p 0.0390 5s−7p 0.3347 4s−7p 0.0614 5s−7p 0.4907
1/2 1/2 3/2 3/2
4s−8p 0.0225 5s−8p 0.183(3) 4s−8p 0.0364 5s−8p 0.271(4)
1/2 1/2 3/2 3/2
4s−9p 0.0147 5s−9p 0.120(2) 4s−9p 0.0244 5s−9p 0.178(3)
1/2 1/2 3/2 3/2
4s−10p 0.0105 5s−10p 0.087(1) 4s−10p 0.0177 5s−10p 0.129(2)
1/2 1/2 3/2 3/2
4p −6s 0.903(4) 5p −6s 8.79(2) 4p −6s 1.279(5) 5p −6s 12.50(2)
1/2 1/2 3/2 3/2
4p −7s 0.476(2) 5p −7s 1.801(8) 4p −7s 0.673(3) 5p −7s 2.54(1)
1/2 1/2 3/2 3/2
4p −8s 0.314(2) 5p −8s 0.912(5) 4p −8s 0.444(2) 5p −8s 1.287(7)
1/2 1/2 3/2 3/2
4p −9s 0.230(1) 5p −9s 0.592(3) 4p −9s 0.325(2) 5p −9s 0.834(4)
1/2 1/2 3/2 3/2
4p −10s 0.1791(9) 5p −10s 0.430(2) 4p −10s 0.253(1) 5p −10s 0.607(3)
1/2 1/2 3/2 3/2
4p −11s 0.1452(8) 5p −11s 0.334(2) 4p −11s 0.205(1) 5p −11s 0.471(3)
1/2 1/2 3/2 3/2
4p −3d 7.979(35)(b) 5p −3d 7.2(1) 5p −3d 3.19(5)
1/2 3/2 1/2 3/2 3/2 3/2
4p −4d 0.1121(8) 5p −4d 17.04(6) 4p −4d 0.0400(1) 5p −4d 7.64(3)
1/2 3/2 1/2 3/2 3/2 3/2 3/2 3/2
4p −5d 0.333(2) 5p −5d 0.931(4) 4p −5d 0.155(1) 5p −5d 0.398(2)
1/2 3/2 1/2 3/2 3/2 3/2 3/2 3/2
4p −6d 0.341(3) 5p −6d 0.063(6) 4p −6d 0.157(1) 5p −6d 0.039(3)
1/2 3/2 1/2 3/2 3/2 3/2 3/2 3/2
4p −7d 0.298(2) 5p −7d 0.219(5) 4p −7d 0.136(1) 5p −7d 0.105(3)
1/2 3/2 1/2 3/2 3/2 3/2 3/2 3/2
4p −8d 0.254(2) 5p −8d 0.236(4) 4p −8d 0.116(1) 5p −8d 0.111(2)
1/2 3/2 1/2 3/2 3/2 3/2 3/2 3/2
4p −9d 0.218(2) 5p −9d 0.222(3) 4p −9d 0.0995(8) 5p −9d 0.103(2)
1/2 3/2 1/2 3/2 3/2 3/2 3/2 3/2
4p −3d 10.734(47)(b) 5p −3d 9.6(1)
3/2 5/2 3/2 5/2
4p −4d 0.1170(4) 5p −4d 22.93(8) 3d −5p 7.2(1) 3d −5p 9.6(1)
3/2 5/2 3/2 5/2 3/2 1/2 5/2 3/2
4p −5d 0.467(3) 5p −5d 1.188(7) 3d −6p 1.03(1) 3d −6p 1.39(1)
3/2 5/2 3/2 5/2 3/2 1/2 5/2 3/2
4p −6d 0.471(4) 5p −6d 0.119(8) 3d −7p 0.497(5) 3d −7p 0.673(7)
3/2 5/2 3/2 5/2 3/2 1/2 5/2 3/2
4p −7d 0.409(3) 5p −7d 0.318(7) 3d −8p 0.317(3) 3d −8p 0.428(4)
3/2 5/2 3/2 5/2 3/2 1/2 5/2 3/2
4p −8d 0.349(3) 5p −8d 0.335(6) 3d −9p 0.228(3) 3d −9p 0.308(4)
3/2 5/2 3/2 5/2 3/2 1/2 5/2 3/2
4p −9d 0.299(2) 5p −9d 0.312(4) 3d −10p 0.176(2) 3d −10p 0.238(3)
3/2 5/2 3/2 5/2 3/2 1/2 5/2 3/2
3d −4p 3.578(16)(b) 3d −4f 12.3(2) 3d −4f 14.6(2) 3d −4f 3.27(4)
3/2 3/2 3/2 5/2 5/2 7/2 5/2 5/2
3d −5p 3.19(5) 3d −5f 4.92(2) 3d −5f 5.88(3) 3d −5f 1.315(6)
3/2 3/2 3/2 5/2 5/2 7/2 5/2 5/2
3d −6p 0.464(5) 3d −6f 2.899(8) 3d −6f 3.465(0) 3d −6f 0.775(2)
3/2 3/2 3/2 5/2 5/2 7/2 5/2 5/2
3d −7p 0.224(2) 3d −7f 2.001(5) 3d −7f 2.392(6) 3d −7f 0.535(1)
3/2 3/2 3/2 5/2 5/2 7/2 5/2 5/2
3d −8p 0.143(1)
3/2 3/2
3d −9p 0.103(1)
3/2 3/2
3d −10p 0.079(1)
3/2 3/2
ium were presented in [10]. Dipole polarizabilities of forthedeterminationofapproximatevaluesofthemagic
ns2 1S and nsnp 3P◦ states and relevant magic wave- wavelengthswithoutcalculatingthedynamicpolarizabil-
0 0
lengthsofSr,Yb,Zn,Cd,andHgatomswerestudiedby ities of the excited states near resonances [7].
a semiempirical approach in Refs. [12, 13].
The magic wavelength conditions that can make opti-
cal lattice clocks insensitive to atomic motion were pre- III. MATRIX ELEMENTS AND
sented by Katori et al. [6]. This work demonstrated POLARIZABILITIES
thatthespatialmismatchofthe interactionsintheclock
transitioncanbe treatedasaspatiallyconstantoffsetδν The magic wavelengths for a specific transition are
for specific lattice geometries. Numerical estimates were located by calculating the frequency-dependent polar-
made for Sr [6]. izabilities of the lower and upper states and finding
Theoretical study of the dynamic scalar polarizabili- their crossing points. The all-order approach to the
ties of the ground and selected long-lived excited states calculation of atomic polarizabilities was discussed in
of dysprosium was recently carried out by Dzuba et al. Refs. [14, 24, 26, 32, 43, 44], and we provide only a brief
[7]. Asetofthemagicwavelengthsoftheunpolarizedlat- summary of the methods here. Unless stated otherwise,
tice laser field for each pair of states, which includes the all specific data refers to the K atom, and we use the
ground state and one of these excited states was given. conventionalsystemofatomicunits,a.u.,inwhiche,m ,
e
The authors presented an analytical formula that allows 4πǫ andthereducedPlanckconstant¯hhavethenumer-
0
4
phase approximation(RPA) [45]. We find the contribu-
TABLE II: Values of scalar (α0) and tensor (α2) polarizabil- tion from the K+ ionic core to be α = 5.457 a3. A
ities in K. The present results are compared with theoretical core 0
counter term α compensating for excitation from the
andexperimentalvalues. Ref.[26]didnotincludeuncertainty vc
core to the valence shell which violates the Pauli princi-
estimates. All valuesare in atomic units.
ple is very small. For example, it is α =−0.00015a.u.
vc
Present Theory Expt.
for the 4p states of K.
j
α (4s ) 290.4(6) 290.2(8)[41] 290.8(1.4)[25]
0 1/2 We use the linearized version of the coupled clus-
α (5s ) 4961(22)
0 1/2 ter approach (also referred to as the all-order method),
α0(4p1/2) 611(6) 604.1 [26] 587(87) [42] which sums infinite sets of many-body perturbation the-
α0(4p3/2) 620(5) 614.1[26] 613(103) [42] oryterms,forallsignificanttermsintheequationsabove.
α0(5p1/2) 7053(70) The4s−np,4p−nl,5s−nl,5p−nl,and3d−nltransitions
α (5p ) 7230(61)
0 3/2 with n≤26 are calculated using this approach[44, 46].
α0(3d3/2) 1420(30) As we noted in the Introduction, the present calcu-
α (3d ) 1412(31)
0 5/2 lation required evaluation of the electric-dipole matrix
α (4p ) -109.4(1.1) -107.9[26] elements for highly excited states, since the frequency-
2 3/2
α2(5p3/2) -1065(18) dependent polarizabilities for the 4s − 5p magic wave-
lengthsofparticularexperimentalinterestaredominated
α (3d ) -482(19)
2 3/2
α (3d ) -673(23) by the 5p−nl transitions with n = 12−14. The diffi-
2 5/2
culty with the applications of the all-order method for
these states results fromthe use a complete set ofDirac-
Fock (DF) wave functions on a nonlinear grid gener-
ical value 1. Polarizability in a.u. has the dimension of
ated using B-splines constrained to a spherical cavity.
volume, and its numerical values presented here are ex-
A large cavity with radius of R = 220 a is needed to
pressed in units of a3, where a ≈ 0.052918 nm is the 0
0 0 accommodate all valence orbitals with ns = 4s− 10s,
Bohr radius. The atomic units for α can be converted
to SI units via α/h [Hz/(V/m)2]=2.48832×10−8α [a.u.], np = 4p−10p, and nd = 3d−9d. A cavity radius of
400 a was chosen to accommodate additional valence
where the conversion coefficient is 4πǫ a3/h and the 0
0 0 orbitals with ns = 11s − 14s, np = 11p − 13p, and
Planck constant h is factored out.
nd=10d−12d. Ourbasissetconsistsof70splinesofor-
Thefrequency-dependentscalarpolarizability,α(ω),of
der 11 for eachvalue of the relativistic angular quantum
analkali-metalatominthestatev maybeseparatedinto
number κ for R=220 a and 100 splines of order 13 for
a contribution from the ionic core, α , a core polariz- 0
core R=400 a . We have conducted test comparisons of the
abilitymodificationduetothevalenceelectron,α ,and 0
vc basis set energies with the actual DF values to demon-
a contributionfromthe valence electron, αv(ω). The va-
stratethe numericalstability ofthis calculation. We can
lence contribution to frequency-dependent scalar α and
0 useavailableexperimentalenergiesforthens=4s−11s,
tensor α polarizabilities is evaluated as the sum over
2 np = 4p−10p, and nd = 3d−12d states from [47] and
intermediate k states allowedby the electric-dipoletran-
theoreticalall-orderenergiesforotherstateswithn≤26.
sition rules [45]
The remaining small contributions with n > 26 are cal-
2 culated in the DF approximation. For example, the con-
2 hkkdkvi (E −E )
αv(ω) = k v , tributions from states with n > 26 give only 0.075 a.u.
0 3(2j +1) (E −E )2−ω2
v Xk k v to the polarizability of the 4p1/2 state. We note that
states with n > 19 in our basis have positive energies
j 1 j
αv2(ω) = −4CXk (−1)jv+jk+1(cid:26) 1v jv 2k (cid:27) anTdhperoevviadleuaatidoinscroeftethreepurnesceenrttaaitniotyn oofftthheecmonattirniuxuemle.-
2
hkkdkvi (E −E ) mentsinthisapproachwasdescribedindetailin[32,44].
k v
× , (1)
(E −E )2−ω2 Four all-order calculations were carried out, including
k v
two ab initio all-order calculations with and without
where C is given by the inclusion of the partial triple excitations and two
other calculations that incorporated semiempirical esti-
1/2
5j (2j −1) mates of high-ordercorrelationcorrectionsstarting from
v v
C =(cid:18)6(j +1)(2j +1)(2j +3)(cid:19) both ab initio runs. The spread of these four values
v v v
for each transition defines the estimated uncertainty in
and hkkdkvi are the reduced electric-dipole matrix ele- the final results when considered justified based on the
ments. In these equations, ω is assumed to be at least dominant correlation contributions to the E1 matrix el-
several linewidths off resonance with the corresponding ements [32, 44]. We note that this procedure does not
transitions. Linearpolarizationisassumedinallcalcula- work in the small number of cases where we can not
tion. The ionic core polarizability and α term depend estimate uncertainty in the dominant contributions us-
vc
weakly on ω for the frequencies treated here and are ap- ing the procedure described above. No uncertainties are
proximated by their dc values calculated in the random- listed for such matrix elements, however, their contribu-
5
(’ (cid:4))*+, (cid:5)’ (cid:3))*+, (cid:4)’ (cid:1))*+, (cid:3)’ &)*+, &’
(cid:2)(cid:1)(cid:0)(cid:0)
&’
(cid:21)
(cid:19)(cid:2)(cid:0)(cid:0)(cid:0)
(cid:19)(cid:20) (cid:1)-.+,/01/2(cid:2)34
(cid:10)
(cid:18) (cid:1)-.+,/01/2534
(cid:17)(cid:1)(cid:0)(cid:0)
(cid:16)
(cid:15)
(cid:12)
(cid:9)
(cid:12)
(cid:14)
(cid:10) (cid:0)
(cid:13)
(cid:12)
(cid:11)
(cid:10)
(cid:9)
(cid:8)(cid:6)(cid:1)(cid:0)(cid:0)
(cid:7)
(cid:6)(cid:2)(cid:0)(cid:0)(cid:0)
(cid:6)(cid:2)(cid:1)(cid:0)(cid:0)
(cid:1)(cid:0)(cid:0) (cid:1)(cid:1)(cid:0) (cid:3)(cid:0)(cid:0) (cid:3)(cid:1)(cid:0) (cid:4)(cid:0)(cid:0) (cid:4)(cid:1)(cid:0) (cid:5)(cid:0)(cid:0)
(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:25)(cid:27)(cid:28)(cid:29)(cid:30) (cid:31) (cid:27) (cid:24)(cid:23)!""# $(cid:27)#%
FIG. 1: (Color online) The frequency-dependent polarizabilities of the K 4s and 4p states. The magic wavelengths are
3/2
marked with circles. The approximate positions of the 4p −nl resonances are indicated by vertical lines with small arrows
3/2
on top of thegraph, together with thecorresponding nl.
tions are small, leading to negligible effects on the final
FLg FJf
uncertainties of the polarizabilities. The absolute val-
ues of the reduced electric-dipole matrix elements used JGGG
inoursubsequentcalculationsandtheiruncertaintiesare
listedina.u. inTable I.We listonly the mostimportant Y[ FHGG ]^bF^GcHFJ_IL‘KHa_‘Ja ]^FGHKIcLd‘Ja
subset of the severalhundred matrix elements that were PYZ FGGG b^cFd‘Ha
calculated in this work. X
W
4pOeuxrcriteesdulsttsaftoerssocfaplaortaasnsdiutmenasroercpoomlapraizraebdilwitiitehscoafltchue- RORUV HGG
j T G
P
lations of [48, 49] and with experimental measurements RS
Q
reportedbyMarrusandYellin[42]inTableII.TheBates- OP \HGG
N
DamgaardmethodwasusedbySchmiederetal.[48]and M \FGGG ]^FGHKIHeK‘Ja Lg
the time-dependent gauge-invariant variational method b^cFd‘Ha Hhijklmnl^FoJ
was used by Marrus and Yellin [42]. The uncertainty \FHGG Hhijklmnl^KoJ
in the experimental measurement [42] of the scalar po-
\JGGG
larizability is too large to reflect on the accuracy of the FGHFIH FGHJIG FGHJIH FGHKIG FGHKIH FGHLIG FGHLIH FGHHIG
present calculations. Extensive comparison of the the-
6789:9;<=>?@;87ABBCD;CE
oretical and experimental static polarizabilities for the FIG.2: (Coloronline) Thefrequency-dependentpolarizabili-
alkali-metalatomswas recentlygiveninthe review [45]. tiesof theK4s and5p3/2 states. Themagicwavelengths are
marked with circles and arrows. The approximate positions
of the 5p −14s and 5p −12d resonances are indicated
3/2 3/2
byvertical lines with small arrows on top of thegraph.
IV. MAGIC WAVELENGTHS
We define the magic wavelength λ as the wave- intersect as a function of the wavelength λ. The total
magic
length for which the ac polarizabilities of two states in- polarizabilityforthenp statesisgivenbyα=α −α
3/2 0 2
volved in the atomic transition are the same, leading to form =±1/2andα=α +α forthe m =±3/2case.
j 0 2 j
a vanishing ac Stark shift of that transition. For the Therefore, the total polarizability of the np state de-
3/2
ns−np transitions, a magic wavelength is represented pendsuponitsm quantumnumberandthemagicwave-
j
by the point at which two curves, α (λ) and α (λ), lengths need to be determined separately for the cases
ns np
6
pr(cid:151) pu(cid:152)(cid:153)(cid:154)(cid:155) p(cid:156)(cid:151) pp(cid:152)(cid:153)(cid:154)(cid:155) pu(cid:151) pq(cid:152)(cid:153)(cid:154)(cid:155) pp(cid:151) (cid:157)(cid:152)(cid:153)(cid:154)(cid:155)
pqqq
wqq
(cid:132)(cid:134) r(cid:151)
(cid:132)(cid:133)
(cid:131){ w(cid:158)(cid:159)(cid:154)(cid:155)
(cid:130) q
(cid:129)
(cid:128)
}
z
}
(cid:127)
~{ vwqq
}
|
{
z
y
x
vpqqq
vpwqq
pqrq pqsq pqtq ppqq ppuq pprq
(cid:135)(cid:136)(cid:137)(cid:138)(cid:139)(cid:138)(cid:140)(cid:141)(cid:142)(cid:143)(cid:144)(cid:145)(cid:140) (cid:137)(cid:136)(cid:146)(cid:147)(cid:147)(cid:148)(cid:149)(cid:140)(cid:148)(cid:150)
FIG. 3: (Color online) The frequency-dependent polarizabilities of the K 4s and 5p states. The magic wavelengths are
1/2
marked with circles. The approximate positions of the 5p −nl resonances are indicated by vertical lines with small arrows
1/2
on top of thegraph, together with thecorresponding nl.
(cid:160)¢˙ (cid:160)¥¨(cid:201)˚¸ (cid:160)(cid:204)˙ (cid:160)(cid:160)¨(cid:201)˚¸ (cid:160)¥˙ (cid:160)¡¨(cid:201)˚¸ (cid:160)(cid:160)˙ ˝¨(cid:201)˚¸
(cid:160)¡¡¡
·¶ §¡¡
·(cid:181)
‡« ¢˙
† §˛(cid:201)˚¸ˇ—(cid:209)ˇ(cid:210)(cid:160)(cid:211)¥
– ¡
›(cid:176) §˛(cid:201)˚¸ˇ—(cid:209)ˇ(cid:210)(cid:204)(cid:211)¥
“
›
fl
«
fi
›
‹ ƒ§¡¡
«
“
'
¤
ƒ(cid:160)¡¡¡
ƒ(cid:160)§¡¡
(cid:160)¡¢¡ (cid:160)¡£¡ (cid:160)¡⁄¡ (cid:160)(cid:160)¡¡ (cid:160)(cid:160)¥¡ (cid:160)(cid:160)¢¡
•‚„”»”…‰(cid:190)¿(cid:192)`…„‚´ˆˆ˜¯…˜˘
FIG. 4: (Color online) The frequency-dependent polarizabilities of the K 4s and 5p states. The magic wavelengths are
3/2
marked with circles. The approximate positions of the 5p −nl resonances are indicated by vertical lines with small arrows
3/2
on top of thegraph, together with thecorresponding nl.
7
withm =±1/2andm =±3/2forthens−np tran-
TABLE III: Magic wavelengths for the 4s−np transitions j j 3/2
j sitions, owing to the presence of the tensor contribution
inK.The500−1227 nmand1050−1130 wavelengthranges
to the total polarizability of the np state. The uncer-
were considered for the 4s−4p and 4s −5p transitions, 3/2
j j
tainties in the values of magic wavelengths are found as
respectively. The corresponding polarizabilities are given in
the maximum differences between the central value and
a.u. The resonance near the magic wavelengths are listed in
thefirst column. the crossings of the αns±δαns and αnp ±δαnp curves,
where the δα are the uncertainties in the correspond-
Resonance K λ α
magic ing ns and np polarizability values. All calculations are
4s−4p Transition
1/2 carried out for linear polarization. Several magic wave-
4p −9s 508.12(1) -215(2)
1/2
lengths were calculated for the 4s−4p and 4s−4p
4p1/2−7d3/2 509.47(1) -220(3) 1/2 3/2
4p −8s 531.80(1) -256(2) transitionsinKinRef.[14]usingtheall-orderapproach.
1/2
4p −6d 533.99(1) -260(3) Onlythemagicwavelengthswithλ>600nmwerelisted.
1/2 3/2
4p −7s 577.37(1) -365(3) Inthis work,wepresentseveralothermagicwavelengths
1/2
4p1/2−5d3/2 581.05(1) -375(4) for these D1, D2 transitions above 500 nm.
4p −6s 690.17(1) -1195(10)
1/2 Thefrequency-dependentpolarizabilitiesofthe4sand
4p −4s 768.41(1) 21200(400)
1/2 4p states for λ = 500−800 nm are plotted in Fig. 1.
4p −5s,3d 1227.63(2) 475(40) 3/2
1/2 3/2
The magic wavelengths are marked with circles. The
4s−4p , |m |=1/2 Transition
3/2 j approximate positions of the 4p −nl resonances are
4p −9s 509.38(1) -217(2) 3/2
3/2
indicated by vertical lines with small arrows on top of
4p −7d 511.04(1) -220(2)
3/2 5/2
4p −8s 533.07(1) -259(2) the graph, together with the corresponding nl. For ex-
3/2
4p −6d 535.72(1) -264(2) ample, the arrow labelled 7s indicates the position of
3/2 5/2
4p3/2−7s 578.71(1) -369(3) the 4p3/2 − 7s resonance. The corresponding magic
4p −5d 583.07(1) -383(3) wavelengths are listed in Table III. We note that the
3/2 5/2
4p3/2−6s 692.35(1) -1237(12) 4p3/2 − 5s resonance wavelength is outside of the plot
4p3/2−4s 769.43(1) -27400(200) region at λ = 1253 nm). While there are 8 magic wave-
4p −5s,3d 1227.61(1) 475(45)
3/2 j lengths for the 4s−4p |m | = 1/2 transition in the
3/2 j
4s−4p3/2, |mj|=3/2 Transition wavelength region shown on the plot, there are only 4
4p3/2−7d5/2 510.75(1) -319(2) magic wavelengths for the 4s−4p3/2 |mj| = 3/2 tran-
4p −6d 535.38(1) -263(3)
3/2 5/2 sition since there are no corresponding crossings near
4p −5d 582.80(1) -383(3)
3/2 5/2 the 4p −ns resonances as in the case of |m | = 1/2.
4s−4p 768.98(2) -367 3/2 j
j The 769 nm magic wavelength for the |m | = 1/2 is
j
4s−5p Transition
1/2 not shown on the plot since the corresponding polariz-
5p −14s 1050.238(2) 620(5)
1/2 ability (-27400 a.u.) is outside of the plot y-axis range.
5p −12d 1051.528(2) 620(5)
1/2 3/2 There is only one magic wavelength above 800 nm for
5p −13s 1067.326(2) 600(5)
1/2
the 4s−4p |m | = 1/2 transition due to 4p −5s
5p1/2−11d3/2 1069.017(2) 598(5) 3/2 j 3/2
5p1/2−12s 1090.784(4) 574(5) resonance and none for the |mj| = 3/2 case. The magic
5p1/2−10d3/2 1093.057(3) 572(5) wavelengths for the 4s−4p1/2 transition are very close
5p −11s 1124.419(6) 543(5) to those for 4s−4p |m | = 1/2. They are also given
1/2 3/2 j
5p1/2−9d3/2 1127.560(6) 540(5) in Table III.
4s−5p3/2, |mj|=1/2 Transition The magic wavelengths for the UV 4s − 5pj transi-
5p −14s 1052.438(2) 618(5)
3/2 tions are completely different than those for the D , D
1 2
5p −12d 1053.647(2) 617(5)
3/2 3/2 linesowingtocompletelydifferentsetofresonances. The
5p −13s 1069.656(3) 597(5)
3/2 K case is also significantly different from that of Li [32]
5p −11d 1071.224(3) 595(5)
3/2 3/2
duetodifferencesintheresonanttransitionwavelengths.
5p −12s 1093.312(4) 571(5)
3/2
5p3/2−10d3/2 1095.393(4) 569(5) We list the magic wavelengths for the 4s − 5p1/2 and
5p3/2−11s 1127.275(8) 540(5) 4s−5p3/2transitionsintherangeof1050-1130nm,which
5p −9d 1130.092(8) 538(5) isofparticularexperimentalinterestinTableIII.Wefind
3/2 3/2
4s−5p , |m |=3/2 Transition 20magicwavelengthsinthetechnicallyinterestregionof
3/2 j
5p −12d 1053.593(2) 617(5) 1050−1130 nm accessible by a number of widely used
3/2 3/2
5p −11d 1071.144(2) 595(5) lasers. The magic wavelengths for the 4s−5p transi-
3/2 3/2 3/2
5p3/2−10d3/2 1095.274(3) 569(5) tionnear1053nmwavelengthareillustratedinFig.2. As
5p3/2−9d3/2 1129.908(4) 538(5) inthe caseofthe 4s−4p3/2 transition,there is no magic
wavelengthforthe|m |=1/2casenearthensresonance.
j
All magic wavelengths for the 4s−5p and 4s−5p
1/2 3/2
transitions in the range of 1050-1140 nm are illustrated
inFigs.3andFigs.4. Thesamedesignationsareusedas
inthepreviousgraphs. Comparingthesefigureswiththe
similar plots for Li (see Figs. 3 and 4 of Ref. [32]) shows
8
that K magic wavelengths near 1050-1130 nm originate tify the magic wavelengths for the 4s−4p and 4s−5p
from crossings near much higher resonances (n=9−14 transitions. The magic wavelengths for the ultraviolet
vs. n= 6−7 for Li) making the calculation for K more resonance lines is of particular interest for laser cooling
complicated due to the very large cavity size required to of ultracold gases with high phase-space densities.
accommodate such highly excited orbitals.
Acknowledgement
V. CONCLUSION
This researchwas performed under the sponsorshipof
We have calculated the ground 4s, 4p, and 5p state the US Department of Commerce, National Institute of
ac polarizabilities in K using the relativistic linearized StandardsandTechnology,andwassupportedbytheNa-
coupled-cluster method and evaluated the uncertainties tionalScienceFoundationunderPhysicsFrontiersCenter
of these values. We have used our calculations to iden- Grant PHY-0822671.
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