Effects of mass defect in atomic clocks

Transcription

1 Journal of Physics: Confrnc Sris PAPER OPEN ACCESS Effcts of mass dfct in atomic clocks To cit this articl: A V Taichnachv and V I Yudin 18 J. Phys.: Conf. Sr Viw th articl onlin for updats and nhancmnts. This contnt was downloadd from IP addrss on 6/8/18 at 6:1

2 IOP Conf. Sris: Journal of Physics: Conf. Sris (18) 16 doi :1.188/ /951/1/16 Effcts of mass dfct in atomic clocks A V Taichnachv 1,, V I Yudin 1,,3 1 Institut of Lasr Physics SB RAS, Novosibirsk 639, Russia Physics Dpartmnt, Novosibirsk Stat Univrsity, Novosibirsk 639, Russia 3 Dpartmnt of Physics and Tchnoloy, Novosibirsk Stat Tchnical Univrsity, Novosibirsk 6373, Russia taichnachv@lasr.nsc.ru, viyudin@mail.ru Abstract. W considr som implications of th mass dfct on th frquncy of atomic transitions. W hav found that som wll-known frquncy shifts (such as ravitational and quadratic Dopplr shifts) can b intrprtd as consquncs of th mass dfct, i.., without th nd for th concpt of tim dilation usd in spcial and nral rlativity thoris. Morovr, w show that th inclusion of th mass dfct lads to prviously unknown shifts for clocks basd on trappd ions.. 1. Introduction At th prsnt tim, atomic clocks ar most prcis scintific dvics. Th principl of opration of ths quantum instrumnts is basd on modrn mthods of lasr physics and hih-prcision spctroscopy. In this way, th unprcdntd valu of fractional instability and uncrtainty at th lvl of 1-18 has alrady bn achivd with th oal of 1-19 on th horizon [1]. Frquncy masurmnts at such a lvl could hav a hu influnc on furthr dvlopmnts in fundamntal and applid physics. In particular, w can fors tsts of quantum lctrodynamics and cosmoloical modls, sarchs for drifts of th fundamntal constants, nw typs of chronomtric odsy, and so on (s, for xampl, rviw []). Howvr, this lvl of xprimntal accuracy rquirs a comparabl lvl of thortical support, which would account for systmatic frquncy shifts of atomic transitions du to diffrnt physical ffcts. Thus, modrn atomic clocks ar also at th point of intrwavin diffrnt aras of thortical physics. In this papr w dvlop th mass dfct concpt with rspct to atomic clocks. Historically, considrations of th mass dfct hav bn connctd with nuclar physics, whr th mass dfct xplains th hu nry mittd du to diffrnt nuclar ractions. Howvr, a quit unxpctd rsult is that this ffct has a dirct rlation to frquncy standards, whr it lads to shifts in th frquncis of atomic transitions. Th main ida of our approach is as follows. Lt us considr an arbitrary atomic transition btwn () () () th round () and xcitd () stats with unprturbd frquncy ω ( E ) /, whr E and E () E ar th unprturbd nris of th corrspondin stats (s in Fi.1). Usin Einstin's famous formula, E=Mc, which links th mass M and nry E of a particl (c is th spd of liht in vacuum), w can find th rst masss of our particl, M and M, for th stats () and (), rspctivly: Contnt from this work may b usd undr th trms of th Crativ Commons Attribution 3. licnc. Any furthr distribution of this work must maintain attribution to th author(s) and th titl of th work, journal citation and DOI. Publishd undr licnc by Ltd 1

3 IOP Conf. Sris: Journal of Physics: Conf. Sris (18) 16 doi :1.188/ /951/1/16 () E Mc and () E M c. Th fact that M M is th ssnc of th so-calld mass dfct. In our cas, th connction btwn M and M rads: M c M c ω M M ω c / W show that th rlationship (1) allows us to rintrprt som wll-known systmatic frquncy shifts (.. thos connctd with th tim dilation ffcts) [3,4]. Morovr, our approach prdicts som nw shifts prviously unconsidrd, to our bst knowld, in th scintific litratur.. Gravitational shift As th first xampl, lt us show how th mass dfct allows us to formulat a vry simpl xplanation of th ravitational rdshift vn undr a classical dscription of th ravitational fild (as classical potntial U G ). Indd, bcaus th potntial nry of a particl in a classical ravitational fild is qual to th product MU G (whr U G <), w can writ th nry of j-th stat E j (U G ) as: E j ( UG ) M jc M jug M jc (1 UG / c ), ( j, ) Usin qs.(1) and (), w find th frquncy of th transition () () in th ravitational fild: ω ω (1 U G / c ). (3) This xprssion coincids to th ladin ordrwith th wll-known formula from th nral rlativity thory: ω ω 1 U / c (1 U / ) G G c in th limit of wak ravitational fild U G /c <<1. Fiur 1. Schm of atomic transition () (). Also it is shown th quantization of nry lvls on translational drs of frdom in th confinin potntial U(r), whr du to th mass dfct. 3. Motion-inducd shifts for atoms (ions) trappd in a confinin potntial Th scond xampl concrns frquncy shifts in th prsnc of a confinin potntial U(r), which w tak to b th sam for both stats () and (). Such a situation occurs both for clocks basd on nutral

4 IOP Conf. Sris: Journal of Physics: Conf. Sris (18) 16 doi :1.188/ /951/1/16 atoms in optical lattic at maic wavlnth and thos basd on trappd ions. In this cas, w hav th standard task of quantizin th nry lvls with translational drs of frdom: ˆ (vib) H ˆ j j, ( r) E j, ( r), H j pˆ / M j U( r) whr th Hamiltonian j, Ĥ dscribs th translational motion of th particl in th j-th intrnal stat j (j=,), th wavfunction ( ) corrsponds to th -th vibrational lvl (=,1,,...) and r is j, r coordinat of atomic cntr-of-mass. Thus, takin into account th translational motion, th atomic wav function is dscribd by th pair products j j, ( r). Bcaus of th mass dfct (M M ), (vib ) (vib ) th nry lvls for th lowr and uppr stats diffr: E,α E,α. Consquntly, th frquncy btwn corrspondin lvls of trappd particl is diffrnt from th unprturbd frquncy,, (vib ) (vib ) E,α with a valu (s Fi.1): ω = ω αα ω = E,α /ħ. Lt us now stimat this valu. For this purpos, w writ th Hamiltonian for uppr stat in th followin form: ˆ ˆ ˆ ˆ pˆ pˆ ω H ˆ H H; H p. (6) M M M M whr th oprator H can b considrd as a small prturbation. Thn usin standard prturbation thory, w obtain th followin stimat for th rlativ valu of th shift 1 ˆ j, p j, (7) c M W not that this xprssion coincids with a wll-known rlativistic corrction, which is th quadratic Dopplr shift du to th tim dilation ffct for movin particl [4]. Indd, at th prsnt tim th followin xplanation is convntionally usd. In accordanc with spcial rlativity, th tick rat t in th movin (with vlocity v) coordinat systm chans with rspct to th tick rat t in motionlss (laboratory) coordinat systm by th law: tt (1v /c ) 1/. As a rsult, an atomic oscillation with infrquncy is prcivd by an xtrnal obsrvr to b shiftd to (1v /c ) 1/. In th nonrlativistic limit, (v /c <<1), w hav: (1v /c ) [1(p/M) /c ] (whr p is momntum of particl). Thn, if w tak into account quantum considrations throuh th rplacmnt p pˆ i, w obtain th xprssion for frquncy shift (7). 4. Prviously unconsidrd fild-inducd shifts for trappd ions Bsids th rintrprtation of som wll-known shifts, th mass dfct concpt prdicts additional contributions for fild-inducd shifts that hav not bn prviously discussd in th scintific litratur. W mphasiz that ths additional shifts ar associatd with translational drs of frdom and thy vanish if w will not tak into account th mass dfct. W will considr a trappd ion (with char Z i ) in th prsnc of additional wak lctric fild with potntial ϕ add (r), which dscribs all controlld and uncontrolld filds xcpt th trappin potntial U(r). Lt us show how th additional potntial ϕ add (r) will prturb th vibrational structur formd by U(r). For this purpos w will us vibrational infunctions ( ), which dscrib a j, r spatial localization of th ion in th intrnal stats j (j=,) du to th trap potntial U(r), as a basis for prturbation thory on th small additional intraction U add (r) = Z i ϕ add (r). Th cntr of th ion localization r in th trap is dtrmind by avrain: r j r. In, j, this cas, w will us th followin Taylor sris of th oprator U add (r) at th point of ion localization r : Uadd ( r) Z iadd ( r ) d Eadd ( r ) Q Wadd ( r )... 3

5 IOP Conf. Sris: Journal of Physics: Conf. Sris (18) 16 doi :1.188/ /951/1/16 1 whr E add ( r ) add ( r) r and W ( ) add ( r ) add r r ar th lctric vctor and 6 lctric tnsor, rspctivly, at th point r. Th oprators d and Q corrspond to th msoscopic dipol and quadruplol momnts of ion cloud in a trap: d Z i ( r r ) (9) Q Z 3( r r ) ( r r ) r r kl i k As w s, that apart from th wll-known lctronic dipol and quadrupol momnts th char spatial distribution connctd with motion of ion in a trap lads to msoscopic multipol momnts coupld to th wak additional lctric fild and its radints. Lt us considr now th first ordr shift: whr Q Q j, j, j, j,α = j, U add j, ħ l kl = Q j,α W add r ħ. Th shift of th clock transition frquncy rads, (1) Q ( Wadd r ),, Hr th rsidual quadrupol momnt causd by th mass dfct Q Q Q,,,, has nonzro valu bcaus th wavfunctions dscribin th translational motion in th xcitd and round stats ar diffrnt in nral cas. Estimat of ordr of manitud ivs Q Z R i, whr R is th siz of ion cloud and M M Mc,. Thouh th xprssion (1) contains a vry small multiplir, 1, th siz of ion localization R sinificantly Mc xcds th Bohr radius a. Indd, R a vn for th dply-coold ion to th lowst vibrational lvl in th confind potntial U(r) (i.., for th quantum limit of coolin), and R 1 4 a for th uppr vibrational stats, which ar populatd if th ion is lasr-coold to th usual Dopplr tmpratur (mk ran). As a rsult, th quadrupol shift of th clock transition, modifid by th mass dfct, can b mtroloically sinificant for modrn and futur optical frquncy standards. For xampl, lt us considr an atomic clock basd on th transition 1 S 3 P in th ion 7 Al + [5]. Bcaus of th zro lctronic anular momntum for th clock transition, J = J =, th quadrupol momnt, associatd with intrnal drs of frdom, is vry small ( Q 1 6 a, s Rf. [6]). Howvr, for th rsidual quadrupol momnt causd by mass dfct w hav th stimat 6 Q ~ (1 1 ) a for R a. Such valus of th quadrupol momnt may b important for atomic clocks with fractional uncrtainty at th lvl of Conclusion W hav considrd som manifstations of th mass dfct in atomic clocks. As a rsult, som wllknown systmatic shifts, prviously intrprtd as th tim dilation ffcts in th fram of spcial and nral rlativity thoris, can b considrd as a consqunc of th mass dfct in quantum mchanics. Furthrmor, our approach has prdictd a sris of prviously unknown shifts for ion clocks. Ths rsults ar important for hih prcision optical atomic clocks and can b intrstin for thortical quantum physics. Acknowldmnts This work was supportd by th Russian Scintific Foundation (No ). V.I.Yudin was also supportd by th Ministry of Education and Scinc of th Russian Fdration (No /4.6), and Russian Foundation for Basic Rsarch (No ).. 4

6 IOP Conf. Sris: Journal of Physics: Conf. Sris (18) 16 doi :1.188/ /951/1/16 Rfrncs [1] Schioppo M t al.17 Natur Photonics [] Ludlow A D t al. Rv. Mod. Phys [3] Einstin A Annal. Physik [4] Chou C W t al. Scinc [5] Chou C W t al. Phys. Rv. Ltt [6] Bloy K t al. 17 Phys. Rv. A