Ultracold molecules: new probes on the variation of fundamental constants

Ultracold molecules offer brand new opportunities to probe the variation of fundamental constants with unprecedented sensitivity. This paper summarizes theoretical background and current constraints on the variation of fine structure constant and electron-to-proton mass ratio, as well as proposals and experimental efforts to measure the variations based on ultracold molecules. In particular, we describe two novel spectroscopic schemes on ultracold molecules which have greatly enhanced sensitivity to fundamental constants: resonant scattering near Feshbach resonances and spectroscopy on close-lying energy levels of ultracold molecules.


Introduction
Creation of ultracold molecules at temperatures below 1 µK by magneto-association [1,2] or photoassociation [3,4,5] opens exciting prospects to test the variation of fundamental constants on a brand new level. Ultracold molecules offer two major advantages: first, the very low temperature allows the molecular samples to be fully polarized in one single quantum state and recent experiments have developed schemes to access any rovibrational and magnetic levels with very high fidelity. Secondly, the success in spatially confining a large number (> 10 4 ) of ultracold molecules for a long storage time (>1 s) promises that an extremely high frequency resolution of <1mHz/ √ Hz can be reached on molecular transitions. These features well prepare ultracold molecules as an excellent candidate to perform new generation, very high resolution molecular spectroscopy.
Ultracold molecules can also be prepared in new, exotic regimes, which offer new strategies to measure the variation of fundamental constants. For example, Feshbach spectroscopy, which identifies coupling to weakly-bound quantum states in atomic and molecular collisions, can precisely determine the value of s−wave scattering length [6]. Here, scattering length near Feshbach resonances can be an extremely sensitive probe on the variation of the electron-proton mass ratio [7]. As a second example, the variation of fundamental constants can also be strongly enhanced when the molecules are radiatively excited to close-lying, narrow energy levels. Several promising cases have been identified in cold molecule systems, including Cs 2 [8] and Sr 2 [9].
In this review paper, we will provide an overview on the experimental proposals of measure the variation of fundamental constants based on ultracold molecules. As we will show below, molecular energy structure is mostly sensitive to two fundamental dimensionless constants: the fine-structure constant α = e 2 hc , which determines the strength of the electroweak interaction. Here −e is electron's charge, 2πh is Planck's constant, and c is the speed of light; the second fundamental constant is the electronto-proton mass ratio β = m e /m p , which characterizes the strength of strong interaction in terms of the electroweak. At present, NIST lists the following values of these constants [10]: α −1 = 137.035999679(94) and β −1 = 1836.15267247(80).
Remarkably, in quantum chromodynamics (QCD) there is a similar coupling constant α s for strong interaction as is α for electroweak interaction. However, because of the highly nonlinear character of the strong interaction, this constant is not well defined. Instead of α s , the strength of the strong interaction is usually characterized by the parameter Λ QCD , which has the dimension of mass and is defined as the position of the Landau pole in the logarithm for the strong coupling constant, α s (r) = C/ ln (rΛ QCD /hc), where C is a constant and r is the distance between two interacting particles.
In the Standard Model (SM), there is another fundamental parameter with the dimension of mass -the Higgs vacuum expectation value (VEV), which determines the electroweak unification scale. The electron mass m e and quark masses m q are proportional to the Higgs VEV. Consequently, the dimensionless parameters X e = m e /Λ QCD and X q = m q /Λ QCD link the electroweak unification scale with the strong scale. For the light quarks u and d, X q ≪ 1. Because of that, the proton mass m p is proportional to Λ QCD and X e ∝ β. Below we will use β instead of X e because it is more directly linked to the observables in atomic and molecular experiments.
We begin by reviewing the theory and the state of the art of the search for the variation of α and β. Next, we discuss, in some detail, the dependence of molecular energy structure on α and β. Finally, we describe possible laboratory experiments with ultracold molecules on time-variation. Such experiments are a novelty, and the laboratory results obtained so far are not yet competitive. However, there are proposals for significant improvements, and several groups have already started implementing them.

Theoretical motivation
How the variation of the physical constants and the violation of the local position invariance may come about? Light scalar fields very naturally appear in modern cosmological models, affecting parameters of the SM including α and β (for the whole list of SM parameters see Ref. [11]). Cosmological variations of these scalar fields are, in turn, expected to take place because of the drastic changes in the composition of the universe during its evolution.
Theories unifying gravity and other interactions suggest the possibility of spatial and temporal variation of physical "constants" in the universe [12]. Moreover, there exists a mechanism for making all coupling constants and masses of elementary particles both space-and time-dependent, and influenced by the local environment (see review [13]). Variation of the coupling constants could be non-monotonic, such as, e.g., damped oscillations.
These variations are usually associated with the effect of massless (or very light) scalar fields. One such a field is the dilaton: a scalar which appears in string theories together with the graviton, in a massless multiplet of closed-string excitations. Other scalars naturally appear in those cosmological models in which our universe is a "brane" floating in a space of a larger dimension. The scalars are simply brane coordinates in extra dimensions. However, the only relevant scalar field recently discovered, the cosmological dark energy, so far does not show visible variations. Observational limits on the variation of the physical constants given in section 3 are quite stringent, allowing only for scalar couplings, which are smaller than or, in a new class of "chameleon-type models", comparable to gravity [14]. Preliminary observations hint that there may be space variation of β at the level of δβ/β ∼ 10 −8 [15].
A possible explanation of the small observed variation was suggested by Damour et al [16,17], who pointed out that cosmological evolution of scalars naturally leads to their self-decoupling. Damour and Polyakov [17] have further suggested that the variations should take place when the scalars become excited by some physical change in the universe, such as phase transitions, or by other drastic changes in the equation of state of the universe. They considered several phase transitions, but since the publication of their paper a new transition has been discovered, one from a matter dominated (decelerating) era to a dark-energy dominated (accelerating) era. This transition is a relatively recent event, corresponding to a cosmological redshift z ≈ 0.5, or a look-back time of approximately 5 billion years.
The time dependence of the perturbation related to the transition from the decelearting to the accelerating era could be calculated [18,19]. The calculation shows that the self-decoupling process is effective enough to explain why after this transition the variation of the constants is as small as observed in the present-time laboratory experiments. However, the calculated time dependence is also consistent with the observations of the variation of the proton-to-electron mass ratio and the electromagnetic fine-structure constant at z ≥ 1 [20,21,22].

Current constraints on the variation of α and β
The analysis of the data from the Big Bang nucleosynthesis [23], quasar absorption spectra, and the Oklo natural nuclear reactor yields the space-time variation of the constants on the time scale of the lifetime of the universe, i.e. from a few billion to more than ten billion years. In comparison, the frequencies of different atomic and molecular transitions in the laboratory experiments yield a time variation on the timescale from a few months to a few years. There is no model-independent connection between the variations on such different timescales. However, in order to compare the astrophysical and laboratory results, we often assume a linear time dependence of the constants. In this way, we can interpret all the results in terms of the time derivatives of the fundamental constants. Within this assumption, we can use the quasar absorption spectra to obtain the best current limit on the variation of the mass ratio β and of X e , [24,25] Combining this result with the atomic clock results [43], one can obtain upper bounds on the variation of α [26,27,28]. The best upper bound on the variation of α is provided by the single-ion optical clock experiments as [29] α/α = (−1.6 ± 2.3) × 10 −17 yr −1 . ( The measurements at the Oklo natural reactor provide the best bound on the variation of X s = m s /Λ QCD , where m s is the strange quark mass [30,31,32], Note that the Oklo data cannot yield any bound on the variation of α since the effect of α there is much smaller than the effect of X s and should be neglected within the accuracy of the present theory [32].

Dependence of atomic and molecular spectra on α and β
Atomic and molecular spectra are most naturally described in atomic units (h = m e = e = 1), with the energy measured in Hartrees (1 Hartree = e 4 mē h 2 = 2 Rydberg = 219474.6313705(15) cm −1 ). In these units, the nonrelativistic Schrödinger equation for an atom with an infinitely heavy pointlike nucleus does not include any dimensional parameters. The dependence of the spectrum on α appears only through relativistic corrections, which describe the fine-structure, the Lamb shift, etc. The dependence of atomic energies on β is known as the isotope effect and is caused by a finite nuclear mass and volume. There are even smaller corrections to atomic energies, which depend on both α and β and are known as the hyperfine structure.
One could argue that the atomic energy unit itself depends on α as it can be expressed as α 2 m e c 2 , with m e c 2 the rest energy of a free electron. However, an experimental search for a possible variation of the fundamental constants relies on the observation of the time-variations of the ratios of different transition frequencies to one another. In such ratios, the dependence of the units on the fundamental constants cancels out. Below we will use atomic units unless stated otherwise.
The relativistic corrections to the binding energies of the atomic valence electrons are on the order of α 2 Z 2 , where Z is the atomic number, and become quite large for heavy elements. For our purposes, it is convenient to consider the dependence of the atomic transition frequencies on α 2 in the form where x = ( α α 0 ) 2 − 1 ≈ 2δα α and ω 0 is a transition frequency for α = α 0 . Rough estimates of the enhancement factor K = δ(log ω)/δ(log α) can be obtained from simple oneparticle models, but in order to obtain accurate values one has to account for electronic correlations via large-scale numerical calculations. Recently, such calculations have been carried out for many atoms and ions [33,34,35,36,37,38,39,40].
Isotope effects in atoms are on the order of β ∼ 10 −3 and the magnetic hyperfine structure roughly scales as α 2 βZg nuc ∼ 10 −7 Zg nuc , where g nuc is nuclear g-factor. One has to keep in mind that g nuc depends on the quark parameters X q . This dependence has to be considered when comparing, e.g., the frequency of the hyperfine transition in 133 Cs (Cs frequency standard) [26] or the hydrogen 21 cm hyperfine line [41,42] with various optical transitions [26]. At present there are many very accurate experiments comparing different optical and microwave atomic clocks [43,44,45,46,47,48,49,50,51,29]. A detailed discussion of the atomic experiments can be found in recent reviews [52,53].
Molecular spectroscopy opens additional possibilities to study the variation of fundamental constants. It is known that β defines the scales of electronic, vibrational, and rotational intervals in molecular spectra, E el : E vib : E rot ∼ 1 : β 1/2 : β. In addition, molecules have fine and hyperfine structure, Λ-doubling, hindered rotation, etc. All these effects have different dependencies on the fundamental constants.
The sensitivity to temporal variation of the fundamental constants may be strongly enhanced in coupling between molecular levels and continuum, as well as transitions between narrow close-lying levels of different types. In the following, we describe selected cases to illustrate the potential of ultracold molecules in probing the constants: scattering lengths of ultracold atomic and molecular collisions, narrow close-lying levels of diatomic molecules, and the proposed experiment with ultracold Cs 2 and Sr 2 molecules. Enhancement of the relative variation δω/ω can also occur in transitions between nearly degenerate levels of atoms [33,34,36,54,55], thermal molecules [8,56,57,24,58], and nuclei [59,60].

Enhanced sensitivity of scattering length on β in ultracold atomic and molecular collisions near Feshbach resonances
An interesting case of the enhancement of the effect of the variation of fundamental constants arises in collisions of ultracold atoms and molecules near Feshbach resonances [7]. In cold collisions, scattering phase shift φ is very sensitive to the change of the electron-proton mass ratio β. This dependence can be understood as φ depends on the ratio of the molecular potential V and the atomic mass m at , namely, φ ∝ (V /m at ) 1/2 . Since the molecular potential V is predominately electronic and the atomic mass is essentially baryonic, we have φ ∼ (V /m at ) 1/2 ∼ (m e /m p ) 1/2 ∼ β 1/2 . A model potential calculation shows that, among other cold collisions properties, scattering length A can be extremely sensitive to β. A fractional variation of β can be amplified to a change of A according to: where the enhancement factor K = δ(log A)/δ(log β) can be as large as 10 9 ∼ 10 12 near narrow or optical Feshbach resonances [7]. Currently the best measurement on the scattering length has an uncertainty of 10 −4 , which can potentially probe the variation of β on the level of 10 −13 ∼ 10 −16 .
Recently, Gibble group at Penn State University demonstrated that atomic-clock type experiment can determine the scattering phase shift with very high precision [61]. They concluded that 1-p.p.m. (10 −6 ) measurement on the scattering length can be reached [61]. If such a measurement is performed near narrow Feshbach resonances in two consecutive years, variation of β can be probed on the level of 10 −15 ∼ 10 −18 /yr.
Note that the calculation of the factor K in Ref. [7] is based on the analytic formula for the scattering length derived in Ref. [62]. This formula is valid for an arbitrary interatomic potential with an inverse-power long-range tail (−C 6 /r 6 for neutral atoms, where C 6 is the van der Waals potential constant and r is the atomic separation), i.e., it includes all anharmonic corrections.
To the best of our knowledge, it is the only suggested experiment on time-variation where the observable is not a frequency. However, another parameter, L, with the dimension of length is needed to compare A with and thus render it dimensionless. In Ref. [7] the scattering length was defined in atomic units (a B ). It is important, however, that because of the large enhancement in Equation (5), the possible dependence of L on β becomes irrelevant. For example, if we measure A in conventional units, meters, which are linked to the Cs standard, then δL/L = −δβ/β, and As long as K ≫ 1, the dependence of the units used on the fundamental constants can be neglected. Below, we discuss several other experiments with huge enhancement factors, where this argument can also be applied.

Narrow close-lying levels of diatomic molecules
In this section we focus on narrow close-lying levels of varying nature in diatomic molecules. Such levels may occur due to the cancelation between either hyperfine and rotational structures [56], or between the fine and vibrational structures within the electronic ground state [58]. The intervals between the levels correspond to microwave frequencies, which are experimentally accessible, and have narrow linewidths, typically ∼ 10 −2 Hz. The enhancement of the relative variation K can exceed 10 5 in such cases.

Molecules with cancelation between hyperfine structure and rotational intervals
Consider diatomic molecule with the 2 Σ ground state (one unpaired electron). Examples of such molecules include LaS, LaO, LuS, LuO, and YbF [63]. The hyperfine interval ∆ hfs is proportional to α 2 ZF rel (αZ)βg nuc , where F rel is an additional relativistic (Casimir) factor [68]. The rotational interval ∆ rot ∝ β is approximately independent of α. If we find a molecule with ∆ hfs ≈ ∆ rot , the splitting ω between hyperfine and rotational levels will depend on the following combination ω ∝ β α 2 F rel (αZ) g nuc − const .
The relative variation is then given by where the factor K comes from the variation of F rel (αZ), and for Z ∼ 50, K ≈ 1. As long as ∆ hfs /ω ≫ 1, we can neglect the last term in Equation (8).
The data on the hyperfine structure of diatomics are hard to come by and usually not very accurate. Using the data from [63], one can find that ω = (0.002 ± 0.01) cm −1 for 139 La 32 S [56]. Note that for ω = 0.002 cm −1 , the relative frequency shift is As new data on molecular hyperfine constants become available, it is likely that other molecular candidates with the cancelation effect will be found.

Molecules with cancelation between fine-structure and vibrational intervals
The fine-structure interval, ω f , increases rapidly with the nuclear charge Z: In contrast, the vibrational energy quantum decreases with the atomic mass where the reduced mass for the molecular vibration is M r m p . Therefore, we obtain an equation Z = Z(M r , v) for the spectral lines at fixed Z, M r where we can expect an approximate cancelation between the fine-structure and vibrational intervals: Using Eqs. (10)(11)(12), it is easy to find the dependence of the transition frequency on the fundamental constants where the enhancement factor, K = ω f ω , is due to the relative frequency shift for a given change of the fundamental constants. Large values of the factor K are experimentally favorable, as the correspondingly large relative shifts can be more easily detected. However, large value of K do not always guarantee a more sensitive measurement. In some cases of quasi-degenerate levels, this factor may become irrelevant [54]. Thus, it is also important to consider the absolute values of the shifts and compare them with the linewidths of the transitions in question.
Assuming δα/α ∼ 10 −17 and ω f ∼ 500 cm −1 , we obtain δω ∼ 10 −14 cm −1 ∼ 3×10 −4 Hz. In order to obtain a similar sensitivity from a comparison of the hyperfine transition frequencies of Cs or Rb, one would have to measure the shifts with an accuracy of ∼ 10 −7 Hz. Note that discussed here narrow close levels exist, for example, in the molecular ion Cl + 2 and in the molecule SiBr [57].

Proposed experiments with Cs 2 and Sr 2
In this section we discuss two recently proposed experiments with cold diatomic molecules, one with Cs 2 at Yale [8,69] and one with Sr 2 at JILA [9]. The Yale experiment is based on the idea described in Ref. [8] to match the electronic energy with a large number of vibrational quanta. The difference compared with Eqs. (10 -12) is that here the electronic transition is between a 1 Σ + g ground state and a 3 Σ + u excited state and thus, to the first approximation, its frequency is independent of α. The energy of this transition is about 3300 cm −1 and the number of the vibrational quanta needed to match it is on the order of 100. For the vibrational quantum number v ∼ 100, the density of the levels is high due to the anharmonicity of the potential and hence it is possible to find two nearby levels belonging to two different potential energy curves. This leads to an enhanced sensitivity to variation of β, as in Equation (12). Cold Cs 2 molecules in a particular quantum state can be produced by the photoassociation of cold Cs atoms in a trap.
Let us estimate the sensitivity of this proposed experiment to the variation of α and β. If we neglect the anharmonicity, we can write the transition frequency between the closely-spaced vibrational levels of the two electronic terms as where v 2 ≪ v 1 . The dependence of this frequency on α and β is given by where we made use of the inequality ω ≪ ω el,0 . For the ground state of atomic Cs, the q-factor is about 1100 cm −1 , which is close to 1 4 α 2 Z 2 E 6s , where E 6s is the ground-state binding energy. If we assume that the same relation holds for the electronic transition in the molecule, we obtain |q| ∼ 1 4 α 2 Z 2 ω el,0 ∼ 120 cm −1 . Using this rough estimate and Equation (15) where we assumed that the relativistic corrections reduce the dissociation energy of the molecule, as a result of which q is negative. This estimate shows that the experiment with Cs 2 is mostly sensitive to the variation of β.
As noted above, for high vibrational states, the potential is highly anharmonic. This significantly decreases the sensitivity as estimated by Eq. (16). This can be seen either from the WKB approximation [8,69], or from an analytic solution for the Morse potential [9]. The quantization condition for the vibrational spectrum in the WKB approximation yields, by differentiation with respect to β, the following result For the harmonic part of the potential, ρ = const and the shift δE v increases linearly with v, but for vibrational states near the dissociation limit, the level density ρ(E) −→ ∞ and δE v −→ 0. Consequently, the maximum sensitivity ∼ 1000 cm −1 is reached at v ≈ 60, and rapidly drops for higher v. The Yale group has found a conveniently close vibrational level of the upper 3 Σ u state with v = 138. The sensitivity for this level is however only ∼ 200 cm −1 [69]. There are still good prospects for finding other close-lying levels with smaller v, for which the sensitivity may be several times higher. The sensitivity as given by Eq. (16) towards the variation of α is also reduced by the anharmonicity of the potential. For the highest vibrational levels of the electronic ground state as well as for all levels of the upper (weakly bound) electronic state, the separation between the nuclei is large, R ≈ 12 a.u. Thus, both electronic wave functions are close to either symmetric (for 1 Σ + g ) or antisymmetric combination (for 3 Σ + u ) of the atomic 6s functions, As a result, all the relativistic corrections are (almost) the same for both electronic states.
The deleterious effect of the anharmonicity on the sensitivity to the variation of β and α can be also obtained from the analysis of the Morse potential. Its eigenvalues are given by with ω 0 = 2πa 2d/M and d the dissociation energy. The last eigenvalue E N is found from the conditions E N +1 ≤ E N and E N −1 ≤ E N . Clearly, E N is very close to zero and is independent of β and α and thus of their variation. We note that the highest absolute sensitivity can be expected for vibrational levels in the middle of the potential energy curve. However, in this part of the spectrum, there are no close-lying levels of a different state that would allow to maximize the relative sensitivity δω/ω. Zelevinsky et al [9] proposed to measure molecular transition with the highest sensitivity to β using a frequency comb. In the proposed experiment, an optical lattice is used to trap Sr 2 molecules formed in one of the uppermost vibrational levels of the ground electronic states by photoassociation. In the next step, a Raman transition is proposed to create molecules in one of the most sensitive levels in the middle of the potential well. To minimize light shifts on the relevant transition, the optical lattice is formed by a laser operated at the "magic" wavelength [9]. Sensitivity to the external magnetic field can also be greatly reduced by choosing molecular levels with identical magnetic moment.
The frequency comb scheme offers the highest possible absolute sensitivity for a given molecule. Unfortunately, the dissociation energy of Sr 2 is only about 1000 cm −1 , which is 3 times smaller than that of Cs 2 . Consequently, the highest sensitivity for the Sr 2 molecule occurs at about 270 cm −1 , i.e. only slightly higher than for the v = 138 level in Cs 2 . Therefore, it may be useful to try to apply this scheme to another molecule with a larger dissociation energy. Finally, we note that the sensitivity to α-variation in the Sr 2 experiment is additionally reduced by a factor (55/38) 2 ≈ 2 because of the smaller Z.

Conclusions
Astrophysical observations of the spectra of diatomic and polyatomic molecules can reveal a possible variation of the electron-to-proton mass ratio β on a time scale from 6 to 12 billion years. However, the astrophysical results obtained so far are inconclusive. Much of the same can be said about the astrophysical search for an α-variation.
The use of cold molecules holds the promise to dramatically enhance the sensitivity of the molecular experiments. An example is the scattering length in cold collisions of atoms and molecules near narrow Feshbach resonances, which can have a surprisingly high sensitivity on the variation of β [7]. Measurement of scattering phase shift in atomic clock-type experiments [61] can potentially test the fractional variation of β on the level of 10 −15 ∼ 10 −18 /yr. Preliminary spectroscopic experiment with ultracold Cs 2 molecules have been recently performed at Yale [69]. The electronic transition between the 3 Σ + u and 1 Σ − g states of Cs 2 is independent, to the first approximation, of α. On the other hand, the sensitivity to the β-variation may be enhanced because of the large number of the vibrational quanta needed to match the electronic transition. However, the anharmonicity of the potential suppresses this enhancement for very high vibrational levels near the dissociation limit. As a result, the sensitivity to the variation of β for the v = 138 level is about the same as that given in Eq. (13). It is possible that there are other close-lying levels with smaller vibrational quantum number and, consequently, allow for a higher sensitivity. Even if such levels are not found, the experiment with the v = 138 level may improve the current limit on the time-variation of β by several orders of magnitude. An experiment with the Sr 2 molecule has been recently proposed at JILA [9]. This experiment has a similar sensitivity to the time-variation of β as the experiment with Cs 2 ; these experiments are complementary to the experiments with the molecular radicals, which are mostly sensitive to the time-variation of α [58].

Acknowledgments
We want to thank J. Ye, D. DeMille, and S. Schiller for their extremely useful comments. C.C. acknowledges support from the NSF-MRSEC program under No. DMR-0820054 and ARO Grant No. W911NF0710576 from the DARPA OLE Program. V.F. acknowledges support from Masden grant and Australian Research Council. M.K. acknowledges support from RFBR grant 08-02-00460.