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arXiv:1111.3049v1 [physics.atom-ph] 13 Nov 2011

Effect of α variation on the vibrational spectrum of Sr2

K. Beloy,1A. W. Hauser,1A. Borschevsky,1V. V. Flambaum,2,1and P. Schwerdtfeger1

1Centre for Theoretical Chemistry and Physics, New Zealand Institute

for Advanced Study, Massey University, Auckland 0745, New Zealand

2School of Physics, University of New South Wales, Sydney 2052, Australia

(Dated: November 15, 2011)

We consider the effect of α variation on the vibrational spectrum of Sr2in the context of a planned

experiment to test the stability of µ ≡ me/mp using optically trapped Sr2 molecules [Zelevinsky

et al., Phys. Rev. Lett. 100, 043201; Kotochigova et al., Phys. Rev. A 79, 012504]. We find the

prospective experiment to be 3 to 4 times less sensitive to fractional variation in α as it is to

fractional variation in µ. Depending on the precision ultimately achieved by the experiment, this

result may give justification for the neglect of α variation or, alternatively, may call for its explicit

consideration in the interpretation of experimental results.

PACS numbers: 06.20.Jr, 33.20.Tp

I.INTRODUCTION

In the endeavor to understand nature on its most fun-

damental level, physicists are striving for a description

of all fundamental forces within a single unified theory.

Some promising theories suggest that observable quanti-

ties such as the electron-to-proton mass ratio µ ≡ me/mp

or the fine structure constant α ≡ e2/¯ hc may not have

fixed values [1]. A detected drift in µ or α could thus

provide valuable insight into the fundamental workings

of nature beyond our current understanding. To date,

laboratory measurements have verified the stability of µ

and α on the fractional level of 10−14[2] and 10−17[3] per

annum, respectively. More stringent laboratory tests are

further motivated by recent evidence of a spatial gradient

in the value of α based on an analysis of quasar absorp-

tion spectra [4]. It has been suggested that the Earth’s

motion relative this gradient may lead to measureable

effects in the laboratory [5].

The Ye group at JILA (Boulder) aims to test the sta-

bility of µ to high precision using Sr2molecules confined

in an optical lattice [6]. The experimental protocol, as

outlined by Zelevinsky et al. [7, 8], calls for optical Ra-

man spectroscopy between select vibrational levels of the

X1Σ+

gground electronic potential. A variation in µ al-

ters the vibrational spectrum, and experimental sensitiv-

ity to this change may be optimized with a prudent choice

of levels to incorporate into the spectroscopic scheme. It

was shown in Refs. [7, 9] that, with respect to variation

in µ, the lowest and highest vibrational levels experience

minimal displacement relative to the potential, whereas

levels in the intermediate part of the spectrum experi-

ence a much larger shift. Zelevinsky et al. have focused

on transitions between the n = 27 intermediate level (n

being the vibrational quantum number) and “anchor”

levels at the bottom and top of the spectrum. Fig. 1

illustrates the basic objective of the experiment. A de-

tected drift in the frequency ratio R (see Figure) is to

be interpreted as a drift in the electron-to-proton mass

ratio.

Here we investigate the effect of α variation on this

Energy

Internuclear Separation

X 1?+g

n = 0

n = 27

n = nmax – 2

R = ? – ?'

?????'

h?'

h?

FIG. 1. (color online) Basic illustration of the experiment

described by Zelevinsky et al. [7, 8] to test the stability of

the electron-to-proton mass ratio using the vibrational spec-

trum of88Sr2. The curve represents the X1Σ+

tential, with horizontal dashed lines corresponding to select

vibrational levels supported by this potential. Frequencies ν

and ν′are to be measured by an optical Ramsey scheme in-

volving an intermediate excited electronic state (not shown).

The dimensionless ratio R is sensitive to variations in µ and

is independent of any external reference (clock) frequency.

g electronic po-

promising experiment. The electronic potential depends

on α through relativistic effects of electron motion. A

variation of α alters the potential and, consequently, the

vibrational spectrum supported by it. Thus, a measured

drift in R may be due (or partially due) to α varia-

tion, threatening misinterpretation of the experimental

results.

The experiment described by Zelevinsky et al. is sim-

ilar in spirit to an experiment posed simultaneously by

DeMille et al. [9] to test the stability of µ using diatomic

molecules. A key difference is that, whereas Zelevinsky

et al. focus on a single electronic potential, DeMille et

al. suggest probing the splitting between vibrational lev-

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els supported by different electronic potentials. Recently,

we analyzed the influence of α variation on the experi-

ment of DeMille et al. using a semi-classical (WKB) ap-

proach, specifically focusing on the system Cs2[10]. We

found the experiment to be order-of-magnitude as sen-

sitive to fractional variation in α as it is to fractional

variation in µ. Considering the anticipated precision of

this experiment [9], together with the current laboratory

limit on α variation [3], we concluded that α variation

may not be negligible for the proposed experiment. This

finding largely motivated our present work.

II.PRELIMINARY SET-UP

Strictly speaking, only variations in dimensionless

quantities have physical meaning. For a given vibrational

level n, we will concern ourselves with the normalized en-

ergy En= En/D, where Enis the vibrational energy rela-

tive to the bottom of the potential and D is the potential

depth. Clearly, En is limited to the range 0 < En < 1.

Variations in µ and α induce a shift in En,

δEn= (∂µEn)δµ

µ+ (∂αEn)δα

α,

where we employ the shorthand notation

∂µ≡

∂

∂ lnµ,

∂α≡

∂

∂ lnα.

The quantities ∂µEn and ∂αEn quantify the sensitivity

of the vibrational level n to fractional variations in the

electron-to-proton mass ratio and the fine structure con-

stant, respectively.

We could, if desired, regard δEn = δEn× D as an

“absolute” energy shift, an association which amounts

to arbitrarily assuming the potential depth to be fixed

with respect to any variation. Fixing any other energy

reference—such as the atomic unit of energy, given by

e4me/¯ h2= α2mec2, or the SI unit of energy, which it-

self references the hyperfine frequency of Cs as well as

a platinum-iridium prototype mass held in Paris [11]—

would be equally justified and would generally yield a

different “absolute” energy shift. Here we actively avoid

the possibly slippery notion of absolute energy shift and

quote results for ∂µEnand ∂αEn, as these are unambigu-

ously defined.

III.THE MORSE POTENTIAL

The Morse potential represents an idealized electronic

potential for a diatomic molecule. It is given by

V (r) = D

?

1 − e−a(r−r0)?2

,

where r is the internuclear separation, with r0being the

equilibrium distance, and a−1is directly related to the

0.20.4 0.6 0.81.0

0.1

0.2

0.3

0.4

0.5

f(x)

x

FIG. 2. The function f(x) = 2?x − 1 +√1 − x?. For the

Morse potential, this function modulates the level sensitivi-

ties to both µ and α variation across the vibrational spectrum,

with the argument x taken as En ≡ En/D. The most deeply

(En → 0) and loosely (En → 1) bound levels are insensitive to

variations, while intermediate levels have much larger sensi-

tivities.

width of the potential. The normalized vibrational en-

ergies for the Morse potential are given precisely by the

formula

En= ǫ?n +1

2

?−1

4ǫ2?n +1

2

?2, (1)

where ǫ ≡ ¯ ha?2/DM and M is the reduced nuclear

mass.

From Eq. (1) we see that a variation in Enmay be at-

tributed solely to a variation in the parameter ǫ. Specif-

ically, we may write

δEn= f(En)δǫ

ǫ

= f(En)

?

(∂µlnǫ)δµ

µ+ (∂αlnǫ)δα

α

?

,

(2)

where f(x) = 2?x − 1 +√1 − x?. The function f(x) is

displayed in Fig. 2. This function modulates the sensi-

tivity of the various levels of the vibrational spectrum

to variations in µ and α. Notably it approaches zero

in the limits x → 0 and x → 1 and has a maximum

at x = 3/4. This translates to minimal sensitivities for

the lowest and highest vibrational levels, with the largest

sensitivities occurring for levels in the intermediate part

of the spectrum.

We may go a step further and, based on physical rea-

soning, deduce a numerical value for the factor (∂µlnǫ)

appearing in Eq. (2). This is accomplished most trans-

parently by assuming atomic units, though we reiterate

that ǫ itself is dimensionless. When expressed in atomic

units, the molecular potential (and its depth, width,

etc.) is independent of the electron-to-proton mass ratio,

whereas the reduced mass has a value which is inversely

proportional to µ. From the definition of ǫ, it follows

that (∂µlnǫ) = 1/2.

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In contrast to (∂µlnǫ), there is not a simple analyti-

cal result for the factor (∂αlnǫ). Nevertheless, we may

provide a physically reasonable estimate of (∂αlnǫ) by

realizing that, in atomic units, the electronic potential

is independent of α in the nonrelativistic limit, hav-

ing relativistic corrections which scale as (αZ)2, with

Z being the nuclear charge number. This suggests that

(∂αlnǫ) ∼ (αZ)2. From this reasoning, we may suspect

the vibrational spectrum of Sr2to be nearly as sensitive

to α variation as it is to µ variation.

The exercise of this section provides us with useful

insight which is applicable to real physical systems. A

true potential, of course, is not restricted to the form of

a Morse potential. Nevertheless, the function f(x) dis-

played in Fig. 2 is expected to give a qualitatively accu-

rate depiction of the sensitivities ∂µEnand ∂αEnversus

the normalized energy En. In the vicinity of the equi-

librium distance, the potential resembles that of a har-

monic oscillator. The lower portion of the energy spec-

trum is then well-described by a single term, proportional

to?n +1

gion, ∂µEn vs En and ∂αEn vs En are essentially linear.

Approaching the dissociation limit, anharmonic effects

become important and the remaining terms in the Dun-

ham expansion, proportional to?n +1

then drive the sensitivities back to zero. The fact that

the sensitivities approach zero in the limits En→ 0 and

En → 1 is a consequence of our choice for the zero of

energy (bottom of the potential) and our choice to nor-

malize energy to the dissociation energy.

2

?, in a Dunham-type expansion. Across this re-

2

?2,?n +1

2

?3, etc.,

IV.

AB INITIO CALCULATIONS FOR LEVEL

SENSITIVITIES

We have calculated the X1Σ+

using the relativistic computation chemistry program

DIRAC10 [12]. In order to reduce computational ef-

fort we employed the infinite order two-component rel-

ativistic Hamiltonian obtained after the Barysz-Sadlej-

Snijders (BSS) transformation of the Dirac Hamiltonian

in a finite basis set [13]. This approximation includes

both scalar and spin-orbit relativistic effects to infinite

order and is one of the most computer time efficient and

accurate approximations to the four-component Dirac-

Coulomb Hamiltonian. Electron correlation was taken

into account using closed-shell single-reference coupled-

cluster theory including single, double, and perturbative

triple excitations [CCSD(T)]. The Faegri dual family ba-

sis set [14] was used, augmented by diffuse and high

angular momentum exponents to obtain 21s18p12d6f2g

Gaussian orbitals. Virtual orbitals with energies above

45 a.u. were omitted, and the 56 outer core electrons were

correlated.

We subsequently fed our CCSD(T) potential curve into

a Matlab routine to solve the Schr¨ odinger equation for

the nuclear part of the molecular wave function within

the Born-Oppenheimer approach. A symmetric three-

g

potential of Sr2

point finite difference method was applied to obtain the

nuclear eigenfunctions together with their corresponding

vibrational energies. For the discretization of the inter-

nuclear distance a step size of 7 × 10−3˚ A was chosen.

We may gauge the accuracy of our ab initio method

through direct comparison with experimental results of

Gerber et al. [15]. These authors have tabulated en-

ergies for the n = 0 through n = 35 portion of the

X1Σ+

g vibrational spectrum.

determined the dissociation energy of this state to be

D = 1060(30) cm−1. Our computed dissociation energy,

D = 993 cm−1, is about 2σ lower than the experimen-

tal value. Comparing individual levels, we find that our

computed vibrational energies differ from experimental

values by no more than 2 cm−1for levels spanning the

lower half of the potential depth (n = 0 through n = 15).

Above this, our computed energies steadily diverge from

experiment values, with our values being increasingly

smaller in comparison.For example, for n = 27 our

computed energy En= 789 cm−1is 3% lower than the

experimental value En= 811 cm−1, whereas for n = 35

our computed energy En= 889 cm−1is 5% lower than

the experimental value En= 940 cm−1. This divergence

in the upper part of the spectrum is undoubtedly cor-

related to the fact that our dissociation energy is lower

than the experimental dissociation energy.

To see how the normalized energies change with respect

to variations in µ and α, we recompute the potential en-

ergy curve, as well as the vibrational spectrum supported

by it, for various values of µ and α in the neighborhood

of µ = 1/1836 and α = 1/137. The computational chem-

istry programs assume atomic units; numerical variations

in µ and α are effected by modifying parameter values

for the reduced mass within our Matlab routine and the

speed of light within DIRAC10 (M = 44µ−1a.u. for

88Sr2and c = 1/α a.u., where a.u. denotes the respective

atomic units of mass and velocity). We then obtain the

sensitivities ∂µEn and ∂αEn from numerical differentia-

tion with respect to µ and α. We emphasize that our

method for obtaining these sensitivities treats variations

in µ and α in a similar manner and on equal footing. Fig-

ure 3 displays our results for ∂µEnand ∂αEnfor the levels

n = 0 through n = 35; these level sensitivities are plot-

ted versus the normalized energy En. We note a behavior

which resembles that “predicted” by the Morse potential.

Namely, both sensitivity curves approach the appropri-

ate limits for En→ 0 and En→ 1, while simultaneously

peaking at En∼= 3/4. Moreover, the curves are found

to be essentially proportional, with the ratio ∂αEn/∂µEn

being 0.28±0.02 across the entire range of data (and 0.28

at the common maximum of the two curves).

Furthermore, they have

V. CONCLUSION

Here we have considered the influence of α variation

on the experiment proposed by Zelevinsky et al. [7, 8] to

probe variation in the electron-to-proton mass ratio using

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0.00.2 0.40.60.81.0

0.00

0.05

0.10

0.15

0.20

0.25

Level Sensitivities

∂μƐn

∂αƐn

Ɛn

FIG. 3. (Color online) Level sensitivities ∂µEn and ∂αEn vs

En for the X1Σ+

initio results for individual levels n = 0 through n = 35, with

the solid curves being fits to this data (the fits are extended

through En = 1).

g state of Sr2. The circles correspond to ab

the vibrational spectrum of Sr2. The relevant observable

in this experiment is the frequency ratio R, illustrated in

Fig. 1. With one “anchor” level taken at the bottom of

the spectrum and another at the top, the frequency ratio

is given approximately by

R∼= 2En− 1,

where n labels the intermediate level (e.g., n = 27 in

Fig. 1).Our ab initio computations predict that the

frequency ratio R is only 3 to 4 times less sensitive to

variation in α as it is to variation in µ. Specifically, we

find that variations in µ and α induce a variation in R

according to the relation

δR = K

?δµ

µ+ 0.28δα

α

?

. (3)

We estimate the uncertainty in the factor 0.28 to be on

the order of 10%, based primarily on the discrepancy

of our computed dissociation energy to the experimental

dissociation energy. Equation (3) summarizes the prin-

ciple result of this work. The factor K here is given ap-

proximately by K∼= 2×∂µEn; non-zero shifts in the two

anchor levels amount to small corrections which reduce

K from this value.

As suggested by Eq. (3), a measured drift in the fre-

quency ratio R cannot, by itself, be used to distinguish

between µ variation or α variation. To extract informa-

tion about variations in the respective constants them-

selves requires further experimental input. Optical ion

clocks have been used to test the stability of α, with the

ratio of clock frequencies being insensitive to µ variation.

The current best limit on α variation allows for a drift

on the fractional level of 4×10−17/year [3]. For the pro-

posed experiment of Zelevinsky et al., this result may be

used with Eq. (3) to justify neglect of α variation, which

has been implicitly assumed in previous works [7, 8]. On

the other hand, for high experimental precision—namely,

experimental precision capable of detecting a drift in µ

at the fractional level of 1 × 10−17/year—equation (3)

indicates that α variation should not be neglected. Such

high precision is conceivable; in the related proposal of

DeMille et al. [9], referred to in the Introduction, the

authors argued that their method could plausibly detect

fractional variations in µ at<

For such high experimental precision, additional exper-

imental input could perhaps be obtained by substituting

Sr2with another species, such as Yb2, in the experiment.

Yb has a similar valence structure as Sr and also has

isotopes which lack nuclear spin (168,170,172,174,176Yb).

Moreover, as with Sr, high precision spectroscopy on

optically trapped Yb has become a refined art [16, 17].

From the (αZ)2scaling of the relativistic corrections to

the electronic potential, we may presume that an Yb2

experiment would be about equally sensitive to α varia-

tion as to µ variation, with an estimated sensitivity ratio

0.28×(70/38)2= 0.95. Using Sr2and Yb2results in con-

junction, one could conceivably determine both µ and α

variation to high precision with the proposed experiment

of Zelevinsky et al.

∼10−17.

VI. ACKNOWLEDGEMENTS

This work was supported by the Marsden Fund, ad-

ministered by the Royal Society of New Zealand. VF

further acknowledges support by the ARC.

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