Prospects of detecting meÕmp variance using vibrational transition

PHYSICAL REVIEW A 77, 012511 共2008兲

Prospects of detecting me Õ mp variance using vibrational transition frequencies of 2⌺-state molecules Masatoshi Kajita* National Institute of Information and Communications Technology, Nukui-Kitamachi, Koganei, Japan 共Received 15 October 2007; revised manuscript received 10 November 2007; published 24 January 2008兲 This paper discusses the possibility of measuring the variance in the electron-to-proton mass ratio ␤ using the 兩X 2⌺, nv = 0, N = 0, J = 1 / 2, F = 1, M F = 1典 → 兩X 2⌺, nv = 1 or 2, N = 0, J = 1 / 2, F = 1 , M F = 1典 transition frequencies of magnetically trapped cold 24MgH or 40CaH molecules. The uncertainty in this transition frequency can potentially be lower than 10−15, which makes it possible to measure the variance in ␤. The 24MgH transition is more advantageous for measurement than the 40CaH transition, because of the lower collision-loss rate, light shift, and collision shift. DOI: 10.1103/PhysRevA.77.012511

PACS number共s兲: 33.20.⫺t, 06.30.Ft, 34.10.⫹x

I. INTRODUCTION

Possible variations in nature’s fundamental constants are currently a very popular topic in research. Theories unifying gravity and other interactions suggest the possibility of spatial and temporal variations in the physical constants of the universe 关1兴. Webb et al. found variance in the fine structure constant, ␣, in quasars 共the absorption red shift was between 0.5 and 3.5兲 as ⌬␣ / ␣ = 共−0.72⫾ 0.18兲 ⫻ 10−5 关2兴. Reinhold et al. found variance in the electron-to-proton mass ratio, ␤共=me / m p兲, in quasars 共the absorption red shift was between 2.59 and 3.02兲 as ⌬␤ / ␤ = 共2 ⫾ 0.59兲 ⫻ 10−5 关3兴. By comparing highly accurate atomic 共or molecular兲 clocks, variations in the fundamental constants can be detected in a laboratory. The accuracy of atomic clocks has been improved by using ultracold atoms or ions. The frequency uncertainty of the Cs atomic fountain frequency standard is presently less than 10−15 关4兴. The frequency uncertainties of S-D transitions in 199 Hg+ 关5兴, 171Yb+ 关6兴, and 88Sr+ 关7兴 ions are presently 9.1 ⫻ 10−16, 3.8⫻ 10−15, and 3.4⫻ 10−15, respectively. The 87Sr 1 S0 − 3 P0 transition frequency was measured by Boyd et al. within an accuracy of 9 ⫻ 10−16 关8兴. The value of 共d␣ / dt兲 / ␣ is presently 共−0.3⫾ 2兲 ⫻ 10−15 years 关9兴, and its uncertainty is expected to be further reduced as the accuracies of atomic clocks are improved. Hudson et al. measured the microwave transition frequencies of cold OH molecules within an accuracy of 3 ⫻ 10−9, aiming at the measurement of 共d␣ / dt兲 / ␣ through comparisons with measurements from OH megamasers in interstellar space 关10兴. Calmet and Fritzsch have shown 共d␤ / dt兲 / ␤ = Rc共d␣ / dt兲 / ␣, where Rc is a value between −20 and −40 关11兴. Variances in ␣ and ␤ provide very important information on grand unification theory 共GUT兲, because the actual value of Rc depends on the details of this theory. However, variations in ␤ cannot be measured using atomic transitions, because the atomic energy states are only determined by the motion of electrons 共not nuclei兲. Variance in ␤ can be observed by comparing molecular-vibrational or rotationaltransition frequencies. Schiller and Korobov showed the the-

*[email protected] 1050-2947/2008/77共1兲/012511共11兲

oretical dependence of the vibrational-rotational-transition frequencies of H+2 ions on ␤ 关12兴. Assuming that 共d␤ / dt兲 is constant for 1010 years, 共d␤ / dt兲 / ␤ can be estimated to be on the order of 2 ⫻ 10−15 years−1. Therefore, molecularvibrational 共⬀␤1/2兲 or rotational-transition 共⬀␤兲 frequencies with uncertainties lower than 10−15 are useful for measuring the variance in ␤. However, molecular-transition frequencies have mostly been measured at room temperature, as it has been difficult to obtain cold molecules. The uncertainty of molecular transitions could not be lower than 10−11 关13兴 because of the short interaction time between molecules and probing electromagnetic radiation and the Doppler effect. Roth et al. observed the rovibrational spectrum of HD+ ions after sympathetic cooling with laser-cooled Be+ ions 关14兴. The influence of the electric field on the transition frequency can be significant with this method, unless all HD+ ions are positioned at zero-electric field positions. Van Veldhoven et al. improved the resolution of the ND3 inversion spectrum using a decelerated molecular beam 关15兴. The interaction time between molecules and the probing microwave cannot be longer than 10 ms 共the linewidth cannot be narrower than 16 Hz兲 with this method. Our last paper, Ref. 关16兴, proposed the use of the 14 NH 兩X 3⌺ , nv = 0 , N = 0 , J = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典 →兩X 3⌺ , nv = 1 , N = 0 , J = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典 transition frequency to measure the variance in ␤, where nv is the vibrational quantum number, N is the quantum number of molecular rotation, and J is N + S 共S is electron spin兲. Here, F1 is J + I共H兲 关I共H兲: H nuclear spin兴, F is F1 + I共N兲 关I共N兲: N nuclear spin兴, and M D is the component of D共=F , N , S , I兲 parallel to the magnetic field. The NH molecules are trapped by the magnetic field after cooling them with cold buffer gas 关17兴. The interaction time between molecules and the probe laser light can be longer than 1 s. Assuming pure rotational states, M S and M I共N,H兲 are deterministic values in the 兩N = 0 , M F = S + I共N兲 + I共H兲典 state. The Zeeman frequency shift is smaller than 10−15 for the 兩X 3⌺ , nv = 0 , N = 0 , J →兩X 3⌺ , nv = 1 , N = 0 , J = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典 = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典 transition. This treatment 共also as quoted in Ref. 关16兴兲 poses some problems, because spin-spin interaction 关18兴 induces the mix between 兩N = 0 , M S = 1典 and 兩N = 2 , M S = 1 , 0 , −1典 states, that depends on the rotational constant in each vibrational state. Therefore,

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©2008 The American Physical Society

PHYSICAL REVIEW A 77, 012511 共2008兲

MASATOSHI KAJITA

the dependence of the Zeeman coefficient on the vibrational state is actually much more significant than estimated in Ref. 关16兴. The Zeeman frequency shift of the 兩X 3⌺ , nv = 0 , N = 0 , J = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典 →兩X 3⌺ , nv = 1 , N = 0 , J = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典 transition is on the order −18 Hz/ G 共see Appendix A兲. The complicated hyperfine structure is also a problem, because only molecules in one of eighteen hyperfine substates are used for measurement. This paper discusses the potential accuracies of the 兩X 2⌺ , nv = 0 , N = 0 , J = 1 / 2 , F = 1 , M F = 1典 → 兩X 2⌺ , nv = 1 or 2 , N = 0 , J = 1 / 2 , F = 1 , M F = 1典 transition frequencies of 24 MgH or 40CaH molecules. The Doyle group at Harvard succeeded in the magnetic trapping of CaH molecules after precooling using buffer gas 关19兴. There is no spin-spin interaction by molecules in the 2⌺ state 关18兴 and the Zeeman shift in the transition frequency is much lower than the one for NH transition. Also note that 24Mg and 40Ca atoms have no nuclear spin and there are only four hyperfine substates for 24 MgH and 40CaH molecules. 24MgH and 40CaH molecules are more advantageous for obtaining a high S/N ratio for the transition spectrum than 14NH molecules, because a higher fraction of molecules is used for measurement. Section II discusses the elastic and inelastic collisions between trapped molecules, which indicate the possibility of evaporative cooling. Section III presents estimates of frequency shifts induced by electric-magnetic-field or Doppler effects. Section IV presents the experimental procedure to measure the transition frequency. II. COLLISION BETWEEN TRAPPED MOLECULES

Evaporative cooling is a useful method of reducing the temperature of trapped molecules. Fried et al. observed Bose-Einstein condensation with H atoms 关20兴 by only using evaporative cooling. Evaporative cooling is achieved by irradiating a microwave 共frequency f ev兲 to the magnetically trapped molecules 共the magnetic field is zero at the trap center兲. Then, molecules with trapping vibrational energies Etrap 共kinetic energy + Zeeman potential energy兲 higher than Emax = h␮B f ev / 2 are transformed to the M S = −1 / 2 state and repelled from the trap area. The energy distribution is not thermal equilibrium at this moment. Assuming Emax  kBTin 共 Tin is the initial molecular temperature兲, the mean value of Zeeman potential energy at this moment is 共selection of low energy molecules兲: noneq = kBTin − 共Etrap兲av

h␮B f ev 2

1 Emax . ⬇ 2 h␮B f ev exp −1 2kBTin

冉

冊

共1兲 When the magnetic field distribution is given by B ⬀ RnB 共R is the molecular displacement from the trap center兲, the mean values of kinetic energy K and Zeeman potential energy EZ are given by the virial law as 共K兲av ⬇

nB Emax , nB + 2 2

共EZ兲av ⬇

Emax . nB + 2

共2兲

The selection of low energy molecule is possible also by reducing the trapping potential depth, as Weinstein et al. performed with Cr atoms 关21兴. There is also adiabatic cooling effect with this method. The energy distribution is transformed to the thermal equilibrium state with temperature Teq. Molecules, whose Zeeman potential energy increased up to Emax, are extracted with this procedure and 共Etrap兲av of trapped molecules is further reduced 共evaporative cooling兲. Teq is actually on the order of Emax / 10kB 关22兴. This thermalization time ␶th is much longer than 1 / ⌫ec, where ⌫ec is the elastic collision rate. The trap-loss rate ⌫loss should be sufficiently small so that ⌫loss / ⌫ec  ␶th⌫loss ⬍ 1 is satisfied. Monroe et al. found that evaporative cooling is only effective when ⌫loss / ⌫ec is smaller than 1/150 关23兴. Giving Emax initially higher than f kBTin and reducing it slowly down to the final valueEmax 共taking longer time than ␶th兲, much more molecules remain f from the beginin the trapping region than when Emax = Emax ning 关22兴. From molecules in 兩S = 1 / 2 , M S = ⫾ 1 / 2典 states, only molecules in the M S = 1 / 2 state are trapped in a field minimum of a dc magnetic field. The M S = 1 / 2 → −1 / 2 transition 共induced by the collision with other trapped molecules or background gas, pumping by blackbody radiation 关24兴, or Majorana transition 关25兴兲 causes the trap loss. When the density of the trapped molecules is sufficiently high, ⌫loss / ⌫ec converges to the ratio of the inelastic collision cross section to the elastic collision cross section. We estimated the elastic and inelastic collision cross sections between trapped 24MgH and between 40 CaH molecules in the 2⌺ N = 0 state. The values for permanent dipole moment and rotational constants in the vibrational ground state are listed in Table I 关26–30兴. Elastic collision is mainly induced by electric dipoleinduced dipole interaction, and the elastic-collision cross section ␴e is obtained from the distortion of the incident wave function 关31兴. The M S-changing collisional transition is caused by magnetic dipole-dipole or spin-rotation interactions. Spin-rotation interaction does not exist for the pure N = 0 state. The mixture of different rotational states is only caused by electric field E, because spin-spin interaction does not exist for molecules in the 2⌺ state. The mixture of rotational states is estimated by cmix = ␮nvE / hBnv, where ␮nv and Bnv are the permanent dipole moment and the rotational constant in the nvth vibrational state, respectively. Using the values of ␮0 and B0 listed in Table I, cmix is estimated to be smaller than 10−4 when E ⬍ 10 V / cm; thus the M S-changing inelastic collision is mainly caused by the magnetic dipoledipole interaction. The inelastic-collision cross sections ␴i were obtained using Born approximation taking the distortion in the wave function caused by electric dipole-induced dipole interaction into account 共distorted wave Born approximation兲. The minimum intermolecular distance d was assumed to be 0.35 nm 共the wave function is zero at r ⬍ d兲 to avoid divergence. The L = 0 → 2 共L is quantum number of angular momentum of the relative motion兲 scattering term is dominant for this inelastic collision. The author here explains

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PROSPECTS OF DETECTING me / m p VARIANCE…

PHYSICAL REVIEW A 77, 012511 共2008兲

TABLE I. Listed are molecular rotational constants with nv = 0 共B0兲, change ratio of the rotational constant by changing nv 关␩ = 共Bnv − Bnv+1兲 / B0兴, permanent dipole moment with nv = 0 共␮0兲, nv = 0 → 1 vibrational transition dipole moment 共␮⬘兲, nv = 0 → 1 transition frequency 共f 1兲, nv = 0 → 2 transition frequency 共f 2兲, and the natural linewidth of the nv = 0 → 1 , 2 transitions 共⌬f N1,2兲 are shown. The spectrum is observed using two-photon transition given the probe laser frequency, f 1,2 / 2. Superscript “c” denotes calculated values. ␮⬘ was estimated using the values of B0, ␩, and ␮0 with the method described in Appendix B. ⌬f N was estimated using the estimated value of ␮⬘.

␩

B0 共GHz兲

␮0 共D兲

␮⬘ 共D兲

the details on the calculation procedure in Ref. 关31兴. Figure 1 plots the elastic- and inelastic-collision cross sections between 24MgH 共boson兲, 40CaH 共boson兲, and 15NH 共3⌺ N = 0 state, boson兲 molecules as a function of the collision kinetic energy. Magnetic field B is 50 G in Fig. 1共a兲 and 500 G in Fig. 1共b兲. Krems and Dalgarno expected that inelastic collision would generally be more significant for molecules in the 3 ⌺ N = 0 state than molecules in the 2⌺ N = 0 state 关18兴. The ␴i between 24MgH molecules is actually smaller than that between 15NH molecules by a ratio of 1/3. However, the ␴i between 40CaH molecules is much larger than those between






10-11 15

CaH Elastic

NH Elastic





24

MgH Elastic

CaH Inelastic








40








10-15

NH Inelastic

10-17















15



Cross Sections (cm2)



















40

10-13

MgH Inelastic












24



10-1








 



10-2


 

-3

10








10

-4

-5














10











10

(a)

-6










10-19

Kinetic Energy (K)








10-11 15

CaH Elastic

NH Elastic

MgH Elastic










24



40

CaH Inelastic

















10-15

15

NH Inelastic

















10-17 

Cross Sections (cm2)



















40

10-13

MgH Inelastic



24

(b)





10-1










10-2






 



10-3








10-4






 



10-5











10

-6










10-19

f 2 共THz兲

CaH 126.7 关26,27兴 0.023 关26兴 2.94 关29兴 0 . 42c 37.8 关26兴 74.4 关26兴 MgH 174.5 关26,28兴 0.032 关26,28兴 1 . 23c 关30兴 0 . 21c 42.9 关26,28兴 83.9 关26,28兴

40 24

f 1 共THz兲

Kinetic Energy (K)

FIG. 1. 共Color online兲 Elastic and inelastic collision cross sections between magnetically trapped 40CaH and 24MgH molecules in 兩N = 0 , J = 1 / 2 , F = 1 , M F = 1典 state and 15NH molecules in 兩N = 0 , J = 1 , F1 = 3 / 2 , F = 2 , M F = 2典 state as functions of collision kinetic energy. Magnetic field is 共a兲 50 G and 共b兲 500 G. Solid 共dotted兲 lines denote elastic 共inelastic兲 collision cross sections. Black, red, and blue lines 共online兲 correspond to 15NH, 40CaH, and 24MgH molecules.

⌬f N1 共Hz兲 ⌬f N2 共Hz兲 2 . 23c 0 . 82c

4 . 46c 1 . 64c

15

NH or between 24MgH molecules, as Krems also anticipated 关32兴. This is because the intermolecular attractive force 共electric dipole-induced dipole interaction兲 is much stronger for 40CaH molecules than those for 24MgH and 15NH molecules 共larger value of ␮0兲. Then, magnetic interaction also becomes more significant for 40CaH molecules than for 24 MgH and 15NH molecules because of the shorter intermolecular distance. As B increases, the inelastic collision cross sections increase with ultralow kinetic energy region 共⬍10−4 K兲 and decrease with kinetic energy region higher than 10−3 K. This result is explained as follows. The energy discrepancy between M S = 1 / 2 and −1 / 2 states 共⌬EMS兲 is proportional to the magnetic field B 共see Sec. III兲. With ultralow kinetic energy, the distortion of the incident 共scattering兲 wave function is small. Then each incident 共scattering兲 wave function ␾L共r兲 is proportional to kL+1rL at kr ⬍ 1 and each L → L⬘ scattering term is proportional to 共k / k⬘兲2L−1, where k and k⬘ are the wave numbers of incident and scattering waves 关31兴. Considering k ⬀ K1/2 and k⬘ ⬀ 共K + ⌬EMS兲1/2, the L = 0 → 2 scattering term is proportional to 关共K + ⌬EMS兲 / K兴1/2 and ␴i increases as B becomes higher. With higher kinetic energy, the distortion of ␾L共r兲 by the intermolecular attractive force is significant and ␾L共r兲 ⬀ k−Lr−共L+1兲 关31兴. Then each L → L⬘ scattering term is proportional to k−共2L+3兲k⬘−共2L⬘+1兲. The L = 0 → 2 scattering term is proportional to K−3/2共K + ⌬EMS兲−5/2 and ␴i decreases as B becomes higher. Considering the ratio of the inelastic-collision cross section to the elastic-collision cross section Q, evaporative cooling easily occurs for 24MgH molecules with any kinetic energy region 共Q ⱕ 10−3兲. For 40CaH molecules, as Q ⬎ 10−2 with the kinetic energy region between 10−4 K and 10−2 K with B = 50 G, evaporative cooling is more difficult for 40 CaH molecules than for 24MgH molecules. When the kinetic energy region is higher than 10−2 K, Q共⬍10−3兲 is almost same order for 40CaH and 24MgH molecules with any value of B 共confirmed taking K / kB = 50 mK, B = 50,500, 1000, 2000, and 3000 G兲. When the molecular density is lower than 1011 cm−3, ␶th is longer than 10 s. III. FREQUENCY SHIFT

Figure 2 shows the energy structure of XH molecules in the 2⌺ state 共X: 40Ca or 24Mg兲. This section discusses the

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PHYSICAL REVIEW A 77, 012511 共2008兲

MASATOSHI KAJITA

The mean Zeeman shift in the 兩nv , C典 → 兩n⬘v , C⬘典 共兩C典 = 兩N , J , F1 , F , M F典兲 transition frequency is given by

nv = 1 or 2, N = 0, F =1

共⌬f Z兲av =

nv = 1 or 2, N = 0, F = 0 Zeeman shift

µB(ge + gI(H))B/2

䋨

=

䋩

Two photon absorption Doppler free

EZ共n⬘v,C⬘兲 − EZ共nv,C兲 h

冉

冊

av

3kBT EZ共n⬘v,C⬘兲 − EZ共nv,C兲 hnB EZ共nv,C兲

冊

.

共4兲

av

Here, let us consider the dependence of EZ on the magnetic field and 兩nv , C典. Assuming pure N states, all 兩C典 states of XH molecules 共e.g., X: 24Mg or 40Ca兲 are described as 共X atoms have no nuclear spin兲

nv = 0, N = 0, F =1

M F = M N + M S + M I共H兲 ,

nv = 0, N = 0, F = 0 M = -1

冉

M=0

Magnetic field

M=1

FIG. 2. Energy structure of 40CaH and 24MgH molecules in 2⌺ N = 0 state. 共F = 1 , M = 1兲 and 共F = 1 , M = 0兲 are low field seeking states 共trapped兲 and 共F = 1 , M = −1兲 and 共F = 0 , M = 0兲 states are high field seeking states 共not trapped兲. Zeeman energy shift of molecules in the 共F = 1 , M = 1兲 state is also shown.

shift in the XH 兩 ⌺ , nv = 0 , N = 0 , J = 1 / 2 , F = 1 , M F = 1典 →兩 2⌺ , nv = p , N = 0 , J = 1 / 2 , F = 1 , M F = 1典 共p = 1 , 2兲 transition frequencies f p. The values of f p, the vibrational transition dipole moment between nv = 0 and 1 states ␮⬘, and the natural linewidth 共⌬f Np兲 of 24MgH or 40CaH molecules are listed in Table I. As shown in Appendix B, ␮⬘ was estimated 关33兴 by

兩C典 = 兺 ␥共M N,M S,M I共H兲兲兩N,M N,M S,M I共H兲典,

and the Zeeman coefficient ZC,pure 关=共⳵EZ / ⳵B兲 / h : B is magnv netic field兴 is given by ZC,pure = ␮B 兺 兩␥共M N,M S,M I共H兲兲兩2 n v

⫻关gNM N + gSM S + gI共H兲M I共H兲兴,

2

␮⬘ = 冑␩␮0 ,

␩=

Bnv − Bnv+1 B0

.

共3兲

The vibrational transition dipole moment 具nv 兩 ␮ 兩 nv − 1典 is given by 冑nv␮⬘. These transitions are dipole forbidden and can be observed using two-photon absorption by means of two counterpropagating laser lights with a frequency of f 1,2 / 2; therefore, the observed spectrum is free from the first order Doppler effect. This section discusses estimates of frequency shifts caused by Zeeman shift 共induced by a trapping magnetic field兲, the second order Doppler effect, ac-Stark shift 共induced by the irradiated probe laser and black body radiation兲, and the collision between trapped molecules.

共6兲

where ␮B is the Bohr magneton and gN,S,I are the g factors for molecular rotation 共 gN ⬍ 10−3 关34兴兲, electron spin 共gS = 2.003兲, and nuclear spin 共gI共H兲 ⬇ 3.0⫻ 10−3 关35兴兲, respectively. Generally, ␥共M N , M S , M I共H兲兲 depends on the magnetic field and the splitting of hyperfine states; therefore, EZ is a nonlinear function of B. If hyperfine splitting depends on the vibrational state, Zeeman shift is also significant with pure vibrational-transition frequencies 共兩nv , C典 → 兩n⬘v , C典兲. However, the Zeeman shift in the 兩Ca典 = 兩N = 0 , J = S , F1 = S + I共H兲 , F = S + I共H兲 , M F = S + I共H兲典 state does not depend on hyperfine splitting since 兩Ca典 = 兩N = 0,M N = 0,M S = S,M I共H兲 = I共H兲典, Za,pure = ␮B关gSS + gI共H兲I共H兲兴, n v

共7兲

S = I共H兲 = 1/2.

The Zeeman energy shift is strictly linear to B for molecules in the 兩Ca典 state 共see Fig. 2兲. The mean Zeeman shift in the 兩nv , Ca典 → 兩n⬘v , Ca典 transition frequency is given by

冉

EZ共nv,C兲 共⌬f Z兲av = h

A. Zeeman shift

The Zeeman shift in transition frequency is caused by the difference in Zeeman energy shifts in upper and lower energy states. First, let us consider the Zeeman energy shift EZ of trapped molecules, whose energy distribution is thermal equilibrium with temperature T. We assumed that the molecules would only be trapped by the Zeeman trapping force and that the magnetic field would be zero at the trap center. When the magnetic field distribution is given by B ⬀ RnB, the mean value of EZ is obtained using the virial law as 2K / nB = 3kBT / nB 共K is the mean kinetic energy兲.

共5兲

冊

a,pure

Z n⬘ v

av

− Za,pure n v

Za,pure nv

a,pure

a,pure 3kBT Zn⬘v − Znv = . hnB Za,pure n

共8兲

v

Actually gS is independent of the vibrational state and 共⌬f Z兲av is actually very small. Therefore, the 兩nv , Ca典 → 兩n⬘v , Ca典 transition frequency can be measured without significant Zeeman shift, also when molecules are magnetically trapped. 共⌬f Z兲av is nonzero because of the slight dependence of gI on the vibrational state, as explained below. Nuclei

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PROSPECTS OF DETECTING me / m p VARIANCE…

PHYSICAL REVIEW A 77, 012511 共2008兲

imbedded in the molecules are subject to the magnetic shielding effect and the value of gI共X,H兲 is slightly different from the value for bare nuclei 共chemical shift兲. The chemical 0 0 0 shift ␴ 关=共gI共H兲 − gI共H兲 兲 / gI共H兲 ; gI共H兲 is gI共H兲 for H atom兴 in each vibrational state is given by 关36兴

␴共nv兲 = ␴0 − ␴⬘

rd共nv兲 − rd共0兲 . rd共0兲

共9兲

Using 关rd共nv兲 − rd共0兲兴 / rd共0兲 = ␩ / 2 共see Appendix B兲,

␴ ⬘␩ n v gI共H兲共0兲 2

gI共H兲共nv兲 − gI共H兲共0兲 = −

共10兲

and Zan v

−

Za0

Za0

for HF molecule 共presumably with a factor of 1/2 or 1/3兲, H Mg H F H because 兩xCa p − x p 兩 and 兩x p − x p 兩 are smaller than 兩x p − x p 兩 H F Ca Mg 共x p = 2.1, x p = 4.0, x p = 1.0, and x p = 1.2兲. Therefore, the estimate given by Eq. 共13兲 is rather conservative. The 兩Ca典 state is generally mixed with other rotational states by spin-spin interaction or the electric field, which can make the Zeeman frequency shift more significant than the estimate given by Eq. 共13兲 共see Appendix A兲. Spin-spin interaction does not exist for molecules in the 2⌺ state, and the mixture between different N states is only induced by the electric field. When an ac 共frequency f ac兲 or dc 共f ac = 0兲 electric field is applied, the 兩N , M N , M S , M I共H兲典 = 兩0 , 0 , 1 / 2 , 1 / 2典 state is mixed with the 兩N = 1典 state as indicated by 兩N,M N,M S,M I共H兲典 = 兩0,0,1/2,1/2典 = 兩0,0,1/2,1/2典0

⬇−

␴ ⬘␩ n v gI共H兲共0兲 2关gS + gI共H兲共0兲兴

+ p储兩1,0,1/2,1/2典0 + p+兩1,1,1/2,1/2典0

共11兲

+ p−兩1,− 1,1/2,1/2典0 ,

are derived. To our knowledge, the values of ␴⬘ for CaH or 24 MgH molecules have not been measured or calculated. Here, let us estimate the Zeeman frequency shift using the value of ␴⬘ for HF molecule 共␴⬘ = 3.7⫻ 10−5 关36,37兴兲. Assuming a pure 兩Ca典 state, the difference in Zeeman coeffia from Za0 is given by cient Z1,2 40

40

24

CaH

MgH

冏 冏 冏 冏 Za1 − Za0 Za0

Za1 − Za0 Za0

冏 冏 冏 冏

= 5.2 ⫻ 10−10

= 7.2 ⫻ 10−10

Za2 − Za0 Za0

Za2 − Za0 Za0

= 1.04 ⫻ 10−9 ,

= 1.44 ⫻ 10−9 . 共12兲

The mean Zeeman shift in the 兩nv = 0 , Ca典 → 兩nv = 1 or 2, pure 兲av, is given by Ca典 transition frequency, 共⌬f Z1,2 40

CaH

pure 兩⌬f Z1 共Hz兲兩av =

3kBT共Za1 − Za0兲

pure 共Hz兲兩av = 兩⌬f Z,2

冏 冏 pure ⌬f Z1,2 f 1,2

24

MgH

p⫾ =

v

+ f ac兲

␮ nvE ⫾

2冑6h共2Bnv + f ac兲

+

+

␮ nvE 储

2冑3h共2Bn

v

− f ac兲

␮ nvE ⫾

,

2冑6h共2Bnv − f ac兲

,

= 8.7 ⫻ 10−13 av

Zan = Za,pure + Za,mix , n n

33T共K兲 = , nB

v

v

v

1 = ␮B共gS + gI共H兲兲nv , Za,pure nv 2 = ␮BgN,nv共p+2 − p−2兲nv . Za,mix n

av

T共K兲 , nB

45T共K兲 , nB

90T共K兲 , nB

= 1.0 ⫻ 10−12

T共K兲 . nB

共14兲

where the subscript 0 denotes a wave function with a zeroelectric field, and E储 represents the electric field components parallel to the direction of the magnetic field. E+ and E− are the electric field components rotating right and left on the plane perpendicular to the magnetic field 共for a dc electric field, E+ = E−兲. The Zeeman coefficient of 40CaH and 24MgH molecules in the 兩nv , Ca典 states under the electric field is approximately given by

66T共K兲 , nB

pure 兩⌬f Z,1 共Hz兲兩av =

冏 冏

2冑3h共2Bn

共15兲

v

pure 兩⌬f Z,2 共Hz兲兩av =

pure ⌬f Z1,2 f 1,2

hnBZa0

␮ nvE 储

p储 =

共13兲

When a dc electric field is applied, Za,mix = 0 because of nv E+ = E− 共p+ = p−兲. The ac-electric field is given by blackbody radiation and the linearly polarized probe laser, where E+ = E− 共p+ = p−兲 is also satisfied. However, let us also consider the case of E+  E− 共p+  p−兲. Assuming f ac  Bnv, p+ = 2 ⫻ 10−5E共V / cm兲 for 40CaH and 7 ⫻ 10−6E共V / cm兲 for 24MgH molecules. As the molecular rotation angular velocity is proportional to 冑Bnv, gN ⬀ 冑Bnv and 共gN,nv+1 − gN,nv兲 / gN,nv 2 ⬇ −␩ / 2 are derived. Also considering 共p+,n − p2+,n 兲 / p2+,n v+1 v v 2 ⬇ 共Bnv␮nv+1 / Bnv+1␮nv兲 − 1 ⬇ 3␩ 共see Appendix B兲, and gN ⬍ 10−3 关34兴:

The chemical shift of XH molecule becomes more significant is the Pauling’s electroneas 兩xXp − xHp 兩 increases, where xX,H p gativity of X and H atoms. The values of ␴⬘ for 40CaH and 24 MgH molecules are expected to be smaller than the value 012511-5

40

CaH

冏

a,mix Za,mix n +1 − Zn v

v

Za,pure n v

冏

2 = 兩gN,nv+1 p+,n

v+1

− gN,nv p2+,n 兩

⬍ 2.3 ⫻ 10−14E+2共V/cm兲2 ,

v

PHYSICAL REVIEW A 77, 012511 共2008兲

MASATOSHI KAJITA

冏 冏 mix ⌬f Z1,2 f Z1,2

⬍ 1.2 ⫻ 10−17E+2共V/cm兲2 av

T共K兲 , nB

ES共2兲 = −

a,mix

a,mix 3kBT Zn⬘v − Znv mix , 共⌬f Z兲av = hnB Za,pure n

24

MgH

冏

a,mix Za,mix n +1 − Zn v

v

Za,pure n v

冏 冏 mix ⌬f Z1,2 f Z1,2

−

⬍ 4 ⫻ 10−15E+2共V/cm兲2 ,

⬍ 2 ⫻ 10−18E+2共V/cm兲2 av

T共K兲 . nB

共16兲

mix / f Z1,2兩 f ac  2Bnv, 兩⌬f Z1,2

becomes smaller than the esWhen timate given by Eq. 共16兲 with a factor of 共2Bnv / f ac兲2. Actually the electric field is much lower than 100 V / cm and mix pure  ⌬f Z1,2 . ⌬f Z1,2 Here we considered the case where molecules are trapped by an anti-Helmholz coil. To reduce 兩⌬f Z1,2 / f Z1,2兩 lower than 10−15 with this situation, the molecular temperature should be reduced down to 3 mK by evaporative or adiabatic cooling. The Zeeman frequency shift is much more significant for the 14NH 兩X 3⌺ , nv = 0 , N = 0 , J = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典→ 兩X 3⌺ , nv = 1 , N = 0 , J = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典 transition, because of the mixture between 共N = 0 , M S = 1兲 and 共N = 2 , M S = ⫾ 1 , 0兲 states induced by the spin-spin interaction 共see Appendix A兲. B. Second order Doppler effect 40

When CaH and 24MgH molecules are cooled down to 10 mK, their velocities v are on the order of 2 m / s and 2.8 m / s, respectively. The second order Doppler shift is given by SD =− ⌬f 1,2

冉 冊

v2 f 1,2 2c2

2 3h共4B22 − f ac 兲

ES共1兲 = − −

−

2 f ac 兲

2B1␮21E2 2 3h共4B21 − f ac 兲

共18兲

,

1. Blackbody radiation

For blackbody radiation, f ac ⬍ f 1, when the temperature of chamber TC is lower than 300 K. Here we use approximations 共f 1 ⫾ 2B1兲 ⬇ 共f 2 − f 1 ⫾ 2B1,2兲 ⬇ 共f 3 − f 2 + 2B3兲 ⬇ f 1. The Stark shifts in f 1 and f 2 caused by blackbody radiation 共⌬f BB1 , ⌬f BB2兲 are given by ⌬f BB1 ⬇

2共2f 1 − f 2兲␮⬘2E2 2 3h2共f 21 − f ac 兲

⌬f BB2 ⬇

+

2B0␮20E2 2 3h2共4B20 − f ac 兲

共5f 2 − 3f 3 − f 1兲␮⬘2E2 3h −

2

共f 21

−

2 f ac 兲

2B2␮22E2 2 3h2共4B22 − f ac 兲

+

−

2B1␮21E2 2 3h2共4B21 − f ac 兲

,

2B0␮20E2 2 3h 共4B20 − f ac 兲 2

.

共19兲

The average wavelength of blackbody radiation is 18 ␮m 共1.8⫻ 1013 Hz兲 when TC ⬇ 300 K 关39兴, and the first term of Eq. 共19兲 is dominant. The average quadratic electric field strength of blackbody radiation is given by 69.2 共V / cm兲2 关TC共K兲 / 300兴4. The values of blackbody radiation shift are obtained as 40

CaH

共17兲

⌬f BB1 ⬇ 7.35 ⫻ 10−4Hz,

⌬f BB2 ⬇ 2.94 ⫻ 10−3Hz, ⌬f BB1/f 1 ⬇ 1.9 ⫻ 10−17 , ⌬f BB2/f 2 ⬇ 3.8 ⫻ 10−17 ,

When an ac electric field is applied, Stark energy shifts in 兩nv , N = 0典 states are induced by coupling with the 兩nv , N = 1典 and 兩nv ⫾ 1 , N = 1典 states. The off-diagonal matrix elements of the dipole moment are 具nv , N = 0 兩 ␮ 兩 nv , N = 1典 = ␮nv / 冑3, 具nv , N = 0 兩 ␮ 兩 nv + 1 , N = 1典 ⬇冑nv + 1␮⬘ / 冑3, and 具nv , N = 0 兩 ␮ 兩 nv − 1 , N = 1典 ⬇ 冑nv␮⬘ / 冑3. The Stark energy shifts in 兩nv , N = 0典 states, ES共nv兲, are then given by Ref. 关38兴 as 3h共4B20

2 3h关共f 2 − f 1 − 2B1兲2 − f ac 兴

2 h关共f 3 − f 2 + 2B3兲2 − f ac 兴

C. ac Stark effect

2B0␮20E2

2共f 2 − f 1 − 2B1兲␮⬘2E2

共f 3 − f 2 + 2B3兲␮⬘2E2

SD and 兩⌬f 1,2 / f 1,2 兩 ⱕ 4 ⫻ 10−17.

ES共0兲 = −

+

where f ac is the frequency of the ac-electric field, B3 is the rotational constant in the nv = 3 state, and f 3 is the nv = 0 → 3 transition frequency. The ac electric field is given by blackbody radiation and the probing laser light.

v

冏

2B2␮22E2

−

+

24

3h关共f 1 + 2B1兲 − 2

共f 1 − 2B0兲␮⬘2E2 2 3h关共f 1 − 2B0兲2 − f ac 兴

2共f 2 − f 1 + 2B2兲␮⬘2E2 2 3h关共f 2 − f 1 + 2B2兲2 − f ac 兴

,

⌬f BB1 ⬇ 2.1 ⫻ 10−4Hz,

⌬f BB2 ⬇ 8.4 ⫻ 10−4Hz, ⌬f BB1/f 1 ⬇ 4.9 ⫻ 10−18 ,

共f 1 + 2B1兲␮⬘2E2

2 , f ac 兴

MgH

⌬f BB2/f 2 ⬇ 9.8 ⫻ 10−18 .

共20兲

Molecules are actually trapped inside the cryogenic chamber 共TC ⬍ 1 K兲 关19兴 and the average of f ac is lower than 5 ⫻ 1010 Hz. Therefore, the first term of Eq. 共19兲 is negligibly small in comparison with the other two terms. The quadratic electric field strength of blackbody radiation is less than 8.6⫻ 10−9 共V / cm兲2 and the Stark shift caused by blackbody radiation is less than 1.7⫻ 10−9 Hz. 012511-6

PROSPECTS OF DETECTING me / m p VARIANCE…

PHYSICAL REVIEW A 77, 012511 共2008兲 2. Light shift

共a兲nv = 0 → 1 transition. When nv = 0 → 1 transition is observed, f ac = f 1 / 2共B0兲 and the light shift, ⌬f L1, which is then approximately given by ⌬f L1 ⬇

8共2f 1 − f 2兲␮⬘2E2 9h2 f 21

共21兲

.

A typical value of E 共Esat兲, where the two-photon-transition rate has the same value as the spontaneous emission rate from the nv = 1 state 2␲⌬f N1, is estimated as 2␲具nv = 0,N = 0兩␮兩nv = 0,N = 1典具nv = 0,N = 1兩␮兩nv = 1,N = 0典 2 Esat = 2␲⌬f N1 , h2共f 1/2 − 2B0兲 2 = Esat

3h2 f 1⌬f N1 . 2 ␮ 0␮ ⬘

Using Eq. 共22兲, Eq. 共21兲 can be rewritten as ⌬f L1 ⬇

冉 冊

4␮⬘共2f 1 − f 2兲⌬f N1 E 3␮0 f 1 Esat

2

=

共22兲

served, f ac = f 2 / 2 共B0兲 and the light shift, ⌬f L2, which is approximately given by

4␮⬘共2f 1 − f 2兲⌬f N1 I . 3␮0 f 1 IS

⌬f L1 ⬇ −

共23兲 Here, I is the power density of the probe laser light and IS 共listed in Table II兲 is the saturation power density. ⌬f L1 is reduced taking lower value of I / IS. When E = Esat, ⌬f L1 = 6.7 共4.2兲 mHz and ⌬f L1 / f 1 = 1.7⫻ 10−16 共9.8⫻ 10−17兲 for 40 CaH 共 24MgH兲 molecules. The light shift is negligible small, also when the power of the probe laser light is high enough to obtain a high S/N ratio. 共b兲 nv = 0 → 2 transition. When nv = 0 → 2 transition is ob-

冋

1 1 ␮ ⬘2E 2 − 2 h 2f 3 − 3f 2 + 4B3 3f 2 − 6f 1 − 12B1

册

using 2f 3 − 3f 2 = 3f 2 − 6f 1 共second term of vibrational energy兲 and B0 ⬇ B1 ⬇ B3 ⬇

4 ␮ ⬘2E 2 B0 . 9 h2 共f 1 − f 2/2兲2

共24兲

A typical value of E 共Esat兲, where the two-photon-transition rate has the same value as the spontaneous emission rate from the nv = 2 state 2␲⌬f N2, is estimated as

2␲具nv = 0,N = 0兩␮兩nv = 1,N = 1典具nv = 1,N = 1兩␮兩nv = 2,N = 0典 2 Esat = 2␲⌬f N2 , h2共f 1 − f 2/2兲 2 = Esat

3h2共f 1 − f 2/2兲

冑2 ␮ ⬘ 2

TABLE II. Listed are the wavelength 共␭probe兲 and saturation power density 共Is兲 of the probe laser, the wavelength of the pump laser 共␭d兲, and the wavelength of observed fluorescence 共␭ f 兲 are shown. ␭probe 共␮m兲 Is 共W / cm2兲 ␭d 共nm兲 MgH nv = 0 → 1 nv = 0 → 2 40 CaH nv = 0 → 1 nv = 0 → 2 24

14.02 7.15 15.87 8.06

2.2 0.89 1.1 0.34

478 493 691 692

关26兴 关26兴 关26兴 关26兴

关26兴 关26兴 关26兴 关26兴

共25兲

Using Eq. 共25兲, Eq. 共24兲 can be rewritten as ⌬f L1 ⬇

4B0⌬f N2

冉 冊

E 冑 3 2共f 1 − f 2/2兲 Esat

2

=

4B0⌬f N2

I

. 3冑2共f 1 − f 2/2兲 IS 共26兲

␭ f 共nm兲 446 459 634 635

⌬f N2 .

⌬f L2 is reduced taking a lower value of I / IS. When I = IS, ⌬f L1 = 0.88 共0.28兲 Hz and ⌬f L2 / f 2 = 1.2⫻ 10−14 共3.4⫻ 10−15兲 for 40CaH 共 24MgH兲 molecules. To reduce ⌬f L2 / f 2 lower than 10−15, I / IS should be lower than 0.1 共0.3兲 for 40CaH 共 24MgH兲 molecules. The fraction of molecules excited to the nv = 2

012511-7

PHYSICAL REVIEW A 77, 012511 共2008兲

MASATOSHI KAJITA

state is on the order of 共I / IS兲2. Therefore, the laser power density should be chosen with a trade off between low light shift and a high S/N ratio. D. Collision shift

When molecules get close to other molecules, intermolecular interaction causes energy shift Ec. The collision shift of f 1,2 共⌬f C1,2兲 is caused by the difference in Ec in the vibrational ground and excited states. The collision shift is ⌬f C1,2 = nv␴1,2 ,

共27兲

where n denotes the molecular density, v is the mean relative velocity, and ␴1,2 represents the cross sections of the collision shift. Collision partner molecules are mostly in the 兩nv = 0 , N = 0典 state. Intermolecular interaction is actually dominated by electric dipole-induced dipole interaction, as described in Sec. II. The ␴1,2 for 40CaH and 24MgH molecules 共bosons兲 are given by 关16兴 as

␴1,2 =

4␲ ␹1,2 2␥Lk2L+1 共2L + 1兲 , 兺 k2 L=even 兩␹1,2兩 1 + ␥L2 k4L+2

␥L=0 =

␥Lⱖ1 =

冉

m兩␹1,2兩 8ប2

冊

1/4

⌫共3/4兲 , ⌫共5/4兲

冋

6m兩␹1,2兩 L 共L + 1兲共2L − 1兲 ! ! 共2L + 1兲 ! ! ប2L共L + 1兲

␹1,2 =

冉

册

共2L+1兲/4

,

冊

2 ␮20 ␮20 ␮1,2 − 共1 + 3 cos2␰兲, B1 96␲2␧20h B0

using ␮2n = 共1 + nv␩兲␮2n and Bnv = 共1 − nv␩兲B0 共see Appendix v v B兲

␹1 ⬇ −

␮40 ␩共1 + 3 cos2␰兲, 48␲2␧20hB0

␹2 ⬇ −

␮40 ␩共1 + 3 cos2␰兲, 24␲2␧20hB0 k=

2␲mv , h

共28兲

where ␰ is the angle between the vectors of the dipole moment and the molecular relative position and m is the reduced mass. When the collision kinetic energy, K / kB, is lower than 10 mK, the collision cross section is mostly dominated by the L = 0 scattering term and ␴s is inversely proportional to v. Therefore, ⌬f C1,2 does not depend on K and ⌬f C1 / n and ⌬f C2 / n are roughly estimated to be on the order of −2.4⫻ 10−12 and −4.8⫻ 10−12 共−1.7⫻ 10−13 and −3.4⫻ 10−13兲 Hz cm3 for 40CaH 共 24MgH兲molecules at K / kB ⬍ 10 mK. Taking n ⬍ 109 / cm3, ⌬f C1,2 / f 1,2 is smaller than 10−16共10−17兲 for 40CaH 共 24MgH兲 molecular transition.

IV. EXPERIMENTAL PROCEDURE

The transition frequency should be measured using the following procedure. 共1兲 XH molecules are produced by the laser ablation of XH2 共X: 24Mg or 40Ca兲. Weinstein et al. produced more than 1010 40CaH molecules 关19兴. 共2兲 XH molecules in the M S = 1 / 2 state are loaded into the magnetic trap area after cooling with buffer gas. The magnetic field distribution is given by B共R兲 = PRnB. This loading takes about 0.4 s. The buffer gas should be quickly extracted out 共⬍0.1 s兲, opening a gate to another chamber. More than 108 molecules are expected to be trapped 关19兴. 共3兲 The molecular temperature should be reduced using evaporative cooling. Irradiating a microwave with frequency f ev, molecules with energies higher than Emax = h␮B f ev / 2 are transformed to the M S = −1 / 2 state and repelled from the trap. The mean trapping energy is Emax / 2 and the fraction 共Emax / kBTin兲 of initially trapped molecules remain in the trapping region 共selection of low energy molecules兲. The noneq = Emax / 共nB + 2兲 mean Zeeman potential energy is 共EZ兲av 关see Eq. 共2兲兴. The mean Zeeman potential energy is further reduced with a procedure where the energy distribution is transformed to the Boltzmann distribution with a temperature Teq 共⬇Emax / 10kB 关22兴兲. The final value of mean threedimensional Zeeman potential energy is in the order of ev = 3kBTeq / nB ⬇ 3Emax / 10nB. When Emax is initially 共EZ兲av taken higher than kBTin and reduced slowly down to the final f 共taking longer time than ␶th兲, the number of value Emax trapped molecules at the end of this procedure is much larger f / kBTin兲 共evaporative cooling兲. than 共Emax 共4兲 The magnetic gradient is adiabatically reduced 共P = Pi → P f Pi ⬎ P f 兲, so that the molecular trapping energy Etrap 共=kinetic energy +Zeeman potential energy兲 is reduced with a factor of 共P f / Pi兲2/共nB+2兲 共adiabatic cooling兲. The molecular density is also reduced down to the value, where the collision shift is negligible small 共⬍109 / cm3兲. This procedure should take about 0.1 s, that is much longer than the period of molecular vibrational motion inside the trap area 共in the order of 1 ms兲. 共5兲 Molecules in vibrationally excited states are transformed to the ground state, irradiating the pump laser 共resonant with the 兩X 2⌺ , nv = 1 or 2典 → 兩B 2⌺, nv = 0 or 1典 transition兲 for a few ␮s. The wavelength of pump lasers are listed in Table II. The pump-laser power density is higher than 1 mW/ cm2 and the irradiating time is in the order of 10 ␮s. It is preferable to use this procedure to improve the S/N ratio, because 10−1 − 10−2 of trapped molecules are initially in the nv = 1 or 2 states 关19兴. 共6兲 The IR probe laser is irradiated to counterpropagate in two directions for a period longer than ␶ ⬇ 1 / 2␲⌬f N1,2. The wavelengths of probe laser 共␭ p = 2c / f 1,2兲 are listed in Table II. The power density of the probe laser I should be lower than the saturation power density 共Is: listed in Table II兲, so that the light shift of the transition frequency is lower than 10−15. 共7兲 The pump laser is irradiated again to induce the 兩X 2⌺ , nv = 1典 → 兩B 2⌺ , nv = 0典 共or 兩X 2⌺ , nv = 2典 → 兩B 2⌺ , nv

012511-8

PROSPECTS OF DETECTING me / m p VARIANCE…

PHYSICAL REVIEW A 77, 012511 共2008兲

= 1典兲 transition. The pump-laser power density is higher than 1 mW/ cm2 and the irradiating time is in the order of 10 ␮s. Frequency shifted fluorescence 兩B 2⌺ , nv = 0典 → 兩X 2⌺ , nv = 0典 共or 兩B 2⌺ , nv = 1典 → 兩X 2⌺ , nv = 1典兲 is detected. The wavelengths of observed fluorescence are shown in Table II. The frequency uncertainty is dominated by the mean Zeeman frequency shift, which is proportional to the mean Zeeman potential energy 关see Eq. 共8兲兴. When the mean Zeeman potential energy is 3 ⫻ nB mK 共temperature1 ⫻ nB mK兲, the uncertainty of observed frequency is lower than 10−15 关see Eqs. 共8兲 and 共13兲兴. Weinstein et al. reduced the temperature of magnetically trapped Cr atoms 共initial density 1011 cm3兲 from 1 K down to 10 mK, using evaporative and adiabatic cooling 关21兴. 24MgH and 40CaH molecules are in principle much more advantageous for evaporative cooling than Cr atoms, because the ratio of the inelastic collision cross section to the elastic collision cross section Q is much smaller than that for Cr atoms 共for Cr atoms, Q ⬎ 1 with kinetic energy lower than 10 mK and Q ⬇ 1 / 10 with kinetic energy between 50– 200 mK兲 关21兴. However, the initial density of 40 CaH molecule is presently so low as 8 ⫻ 107 cm3 关19兴 and evaporative cooling takes time longer than 104 s 共3 h兲. When the molecular density at procedure 共2兲 is higher than 1011 cm3, evaporative cooling is possible within 10 s. The initial molecular density can be increased by 共i兲 giving higher magnetic field gradient and 共ii兲 producing XH molecules at a position close to the trap center. Here we consider the possibility to reduce the mean Zeeman potential energy without evaporative cooling. The mean Zeeman potential energy is reduced also by the adiabatic cooling effect at procedure 共4兲. We assume nB = 1 共trapped by anti-Helmholz coil兲. When the Zeeman potential depth is adiabatically reduced from 2 K down to 9 mK, Etrap is reduced with a factor of 0.027 关=共9 / 2000兲2/3兴 and molecules, whose initial value of Etrap / kB is lower than 330 mK, remain in the trap after procedure 共4兲. Assuming that the initial temperature is 200 mK 共mean value of Etrap / kB is 900 mK兲, 18% of initially trapped molecules remain in the trap. The mean Zeeman potential energy after procedure 共4兲 is 3 mK and the Zeeman frequency shift is lower than 10−15. Therefore, the measurement with the frequency uncertainty lower than 10−15 is possible also without evaporative cooling. Here we consider the frequency stability without evaporative cooling. The expected frequency stability is estimated by the Allan variance 关40兴:

␴y =

⌬f 1 ␲ f 1,2 冑Nd

冑

␶e , ␶

in the 兩nv = 0 , N = 0 , J = 1 / 2 , F = 1 , M F = 1典 state. The instability of 24MgH nv = 0 → 2 transition frequency is on the order of 3 ⫻ 10−16/冑␶ 共s兲, taking Nd = 105 and ␶e = 1 s. When the initial molecular density is increased up to 1011 cm3 and evaporative cooling is performed within 10 s, the frequency stability is improved because of the larger value of Nd 共although ␶e becomes on the order of 10 s兲. V. CONCLUSION

The 兩nv = 0 , N = 0 , J = 1 / 2 , F = 1 , M F = 1典 → 兩nv = 1 or 2,N = 0 , J = 1 / 2 , F = 1 , M F = 1典 transition frequencies of magnetically trapped molecules in the 2⌺ state are useful for observing variations in the electron-to-proton mass ratio, mainly because of small Zeeman frequency shift. The uncertainty of observed transition is sufficiently low 共10−15兲 to measure variations in an electron-to-proton mass ratio in a laboratory. 24 MgH molecules are more advantageous than 40CaH molecules for trapping and evaporative cooling, because of their smaller inelastic-collision cross sections. Also note that collisional frequency shift and light shift in 24MgH transition are smaller than those in 40CaH transition. Therefore, 24MgH transition is more advantageous for precise measurement than 40CaH molecules. 24MgH transition is more advantageous than 40CaH transition to also obtain a high S/N ratio because of the shorter wavelength 共larger photon energy兲 of observed fluorescence.

APPENDIX A

This appendix estimates the Zeeman frequency shift of the 14NH 兩X 3⌺ , nv = 0 , N = 0 , J = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典 → 兩 X 3⌺ , nv = 1 , N = 0 , J = 1 , F1 = 3 / 2 , F = 5 / 2 , M F = 5 / 2典 transition. Spin-spin interaction is significant for molecules in the 3⌺ state, for example 14NH molecules. Because of this interaction, the 兩N = 0 , M N = 0 , M S = 1典 state is mixed with the 兩N = 2 , M N = 0 , M S = 1典, 兩N = 2 , M N = 1 , M S = 0典 and 兩N = 2 , M N = 2 , M S = −1典 states 关18兴. The mix can roughly be estimated by snv = ␭ / 6Bnv, where ␭ is the spin-spin interaction parameter in the nvth vibrational state. Taking ␭ = 27 GHz and B0 = 490 GHz in the vibrational ground state 关42兴, s0 ⬇ 0.009. As Bnv关=B0共1 − nv␩兲兴 depends on the vibrational state, s also depends on the vibrational state. Taking ␩ = 0.039 关26兴, Zan − Za0

共29兲

where Nd is number of detected molecules, ␶e is the time for one measurement cycle and ␶ is the measurement time. Without evaporative cooling, ␶e is in the order of 1 s. Nd is estimated as follows. The number of trapped molecules after procedure 共4兲 is larger than 107 molecule兲. Assuming also 共I / IS兲 ⱖ 10−1, more than 105 molecules are transformed to the vibrational excited state. The S/N ratio is then on the same order with an 87Sr lattice clock using 104 atoms 关41兴 because the two-photon spectrum is free from the first order Doppler effect and 1/4 of all molecules 共1/2 of trapped molecules兲 are 012511-9

v

Za0

v

共⌬f Zmix兲av ⬇

s21 − s20 =

−2 B−2 1 − B0

B−2 0

= 共s20 − s2n 兲, 2 兲 3kBT共s20 − s1,2 , hnB

s20 ⬇ 2␩s20 = 6.3 ⫻ 10−6 ,

mix 兲av共Hz兲 ⬇ − 3.9 ⫻ 105 共⌬f Z,1

T共K兲 , nB

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MASATOSHI KAJITA

s22 − s20 ⬇ 4␩s20 = 1.3 ⫻ 10−5 , mix 兲av共Hz兲 ⬇ − 7.8 ⫻ 105 共⌬f Z,2

共⌬r兲0 = 具1兩rd兩0典 =

T共K兲 . nB

The 兩⌬f Zmix兩av is much larger than 兩⌬f Zpure兩av and 兩⌬f Z1,2 / f 1,2兩av ⬇ 3.9⫻ 10−9T共K兲 / nB. Assuming T ⬇ 1 mK and nB = 1 共trapped by anti-Helmholz coil兲, 兩⌬f Z / f兩av is on the order of 3.9⫻ 10−12. APPENDIX B

This appendix estimates the internuclear distance, permanent dipole moment, and vibrational transition dipole moment of a diatomic molecule, XY, in all vibrational states 共nv兲 using a parameter ␩关=共Bnv − Bnv+1兲 / B0兴. When X and Y atoms have electric charge ⫾q, dipole moment of XY molecule is given by

冑

h , 2mXY f 1

where mXY is the reduced mass between X and Y atoms and rd共nv兲 is the mean value of rd in each vibrational state. The value of rd共nv兲 is given by r2d共nv兲 = r2d共0兲 + nv共⌬r兲20 . Considering Bnv = ប2 / 2mXYrd共nv兲2, the following formulas using ␩关=共Bnv − Bnv+1兲 / B0兴 共see Table I兲 are derived as

␩=

−2 r−2 d 共nv兲 − rd 共nv + 1兲

r−2 d 共nv兲

=1−

冋

r2d共nv兲 r2d共nv + 1兲

册

.

As can be seen from Table I, ␩  1 and the following approximations are valid:

␮ = qrd ,

共⌬r兲20 =

where rd is relative position between two nuclei. The dipole matrix element is given by

冉

rd共nv兲 − rd共0兲 = rd共0兲

␮nv = 具nv兩␮兩nv典 = qrd共nv兲, 具nv兩␮兩nv − 1典 = 冑nv␮⬘ ,

冊

1 − 1 r2d共0兲 ⬇ ␩r2d共0兲, 1−␩

冑

1 + n v␩

␮2n − ␮20 v

␮20

␮⬘ = q共⌬r兲0 ,

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1 2␲

冉 冊 共⌬r兲0 rd共0兲

2

−1⬇

n v␩ , 2

= n v␩ .

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