Abstract
Many modern theories predict that the fundamental constants depend on time, position or the local density of matter. Here we develop a spectroscopic method for pulsed beams of cold molecules, and use it to measure the frequencies of microwave transitions in CH with accuracy down to 3 Hz. By comparing these frequencies with those measured from sources of CH in the Milky Way, we test the hypothesis that fundamental constants may differ between the high and lowdensity environments of the Earth and the interstellar medium. For the fine structure constant we find Δα/α=(0.3±1.1) × 10^{−7}, the strongest limit to date on such a variation of α. For the electrontoproton mass ratio we find Δμ/μ=(−0.7±2.2) × 10^{−7}. We suggest how dedicated astrophysical measurements can improve these constraints further and can also constrain temporal variation of the constants.
Introduction
In the Standard Model of particle physics, the fundamental constants are fixed parameters, but in many extensions of the standard model they may change^{1,2}. In higherdimensional theories for example, which aim at a unified description of gravity and the other forces, the constants depend on the size of the compactified extra dimensions. Their size may be changing in a conceptually similar way to the observed expansion of the Universe, causing the constants to change with position or time. Some theories of dark energy also predict a variation of the constants. These theories hypothesize that a scalar field is responsible for the observed cosmic acceleration. In chameleon models, this field acquires an effective mass that depends strongly on the local matter density^{3,4}. In our highdensity environment, the field has such a short range that it goes undetected in tests of the equivalence principle and searches for fifth forces. In these models, particle masses and coupling constants may also depend on matter density^{5}. Given the potential role of these theories in answering outstanding questions in physics, it is important to test the prediction of varying constants.
Atomic and molecular transition frequencies provide a natural way to look for variation of the fine structure constant, α, and the electron–proton mass ratio, μ. Their presentday time derivatives are strongly constrained by comparisons of atomic clocks^{6,7}, while observations of high redshift objects test changes on cosmological scales. Some analyses of optical absorption spectra at high redshifts suggest a variation of α across the Universe^{8}, but others find no variation at the level of a few partspermillion^{9,10}. High redshift measurements using microwave and mmwave molecular transitions, which can be far more sensitive to changes in the constants than optical transitions, give null results for Δα/α at levels between 8 and 0.8 partspermillion^{11,12,13,14}. For Δμ/μ, a recent study of methanol absorption lines at high redshift found no variation at the 0.1 partspermillion level^{15}.
Measurements within our galaxy test whether constants depend on the local density. For changes in α with matter density the 1σ limit is Δα/α<2 × 10^{−7} (ref. 16). For changes in μ a study of methanol maser lines sets a 1σ upper limit, Δμ/μ<2.9 × 10^{−8} (ref. 17). However, Levshakov et al.^{18,19}, comparing terrestrial and astrophysical microwave transitions in ammonia and other molecules, find an eightstandarddeviation systematic difference. This suggests a fractional change in μ of 2.6 × 10^{−8} when going from the earth to the interstellar medium, tentatively supporting the chameleon hypothesis. This intriguing situation calls out for new, highprecision measurements of the most sensitive molecular transition frequencies and for higherquality astronomical observations.
Here, we measure the frequencies of the 3.3 and 0.7 GHz Λdoublet transitions of CH, which are very sensitive to changes in α and μ (refs 20, 21). With these measurements and existing astronomical observations of CH in the interstellar medium, we constrain variation of the constants in our Galaxy, and open the way for future radioastronomy measurements of higher accuracy to test more stringently whether the constants vary. Our spectroscopic method has Hzlevel accuracy, and takes advantage of the short pulse duration and small velocity spread of cold molecular beams to control the interaction between molecules and an applied microwave field. This method can also be applied to other molecules of interest in chemical physics, metrology, particle physics and cosmology^{22,23,24,25,26}.
Results
The CH molecule
Figure 1 shows the relevant energy levels in the ground electronic and vibrational state of CH, X^{2}Π(v=0). These eigenstates of the Hamiltonian are also eigenstates of the parity operator, P, and of the squared total angular momentum operator, F^{2}. In addition, they are approximate eigenstates of N^{2} and J^{2}, where J=N+S and N=L+R. Here, S and L are the electronic spin and orbital angular momentum operators, and R is the rotational angular momentum operator. The spin–orbit interaction splits the N=1 level into a pair of levels labelled by J=1/2 and J=3/2. Each is split by a Coriolis interaction into a Λdoublet whose components are the parity eigenstates, p=±1=Λ=+1±(−1)^{J−S}Λ=−1, where Λ is the quantum number for the projection of L onto the internuclear axis. It is these Λdoublet frequencies that we measure. The interaction with the hydrogen nuclear spin, I=1/2, produces a hyperfine structure in each of the Λdoublet components. The levels are labelled as (J^{p}, F).
These two Λdoublet transitions have been observed towards numerous clouds in the interstellar medium^{27,28,29,30,31,32,33,34,35,36}, and the 3.3GHz transition has also been observed in other galaxies^{37,38}. Figure 1 gives the sensitivity coefficients of the transitions to changes in the constants, K_{α} and K_{μ} (refs 20, 21), defined such that the rest frequency emitted by molecules in the astronomical source is ω_{lab} (1+K_{α}Δα/α+K_{μ}Δμ/μ), where ω_{lab} is the laboratory frequency. As these coefficients are large and different, a very sensitive test is possible when the transitions are observed together in the same gas. Until now, this test was hindered by a lack of precise laboratory frequencies.
Frequency measurements
Using the apparatus shown in Fig. 2, we measure the laboratory frequencies by the Ramsey method of separated oscillating fields^{39,40}. The source produces short, cold pulses of CH molecules that pass through a state selector, a transmission line resonator and then a detector. The state selector preferentially populates one of the two parity eigenstates of the Λdoublet being measured. Inside the transmission line resonator the molecules interact with a standingwave microwave field. A first microwave pulse (a π/2 pulse), of duration τ=15 μs and angular frequency ω, is applied when the molecules are at antinode m_{1} of the field. This prepares an equal superposition of the two Λdoublet components, which then evolves freely for a time T at the angular frequency ω_{0}, developing a phase difference of δT relative to the microwave oscillator, where δ=(ω−ω_{0}). We choose the free evolution time to be T=mλ/(2v_{0})−τ, where v_{0} is the mean speed, λ is the wavelength in the transmission line and m is an integer, so that the molecules are now at antinode m_{1}+m. A second microwave pulse of equal duration and amplitude then completes the population transfer with a probability
Here, and β represents a possible phase shift of the microwave field (modulo 2π) because of the change in position of the molecules between the two pulses. The population in one of the parity states is measured, as a function of ω, by timeresolved laserinduced fluorescence. From the timeofflight profile we select molecules with arrival times in the range t_{0}±δt, where t_{0} is the most probable arrival time, and δt≃0.02t_{0}. This ensures that all molecules used in the analysis are sufficiently well localized near the antinodes when the pulses are applied.
Figure 3a shows data measured over a wide frequency range around the (1/2^{+},1)–(1/2^{−},1) resonance. The line is a fit to the model b+aP(δ), where b and a are constants and P(δ) is given by equation (1). We set β=0, and τ and T to the values used in the experiment, leaving only the offset b, the amplitude a (which is negative when m is odd) and the resonance angular frequency ω_{0} as fit parameters. The model fits the data well and we use it to determine ω_{0}. Figure 3b shows narrower frequency scans for the same transition and for three different values of T chosen to make m=4, 5 and 6. Here, as in all our data, we see no dependence of ω_{0} on m, so we average together measurements taken using different values of m. We also see no variation of ω_{0} when we subdivide the data according to the arrival time of the molecules.
Our model assumes that β is zero, as for a perfect standing wave. As the standing wave is not perfect, β is not exactly zero and there will be a systematic frequency shift of δf=β/(2πT). Let the amplitudes of the two counterpropagating waves be proportional to (1±Δ), so that Δ is the amplitude imbalance between them. The microwave electric field is
where A is an amplitude, k=2π/λ and (modulo π)
When Δ is small, ϕ changes very slowly around the antinodes. Suppose the first pulse starts when the molecules are at z_{1}=z_{0}+δz_{0}, where z_{0} is the position of an antinode, and the second pulse starts when the molecules are at z_{2}=z_{1}+(1+∈)mλ/2. Expanding the trigonometric functions for small displacements around the antinodes (that is, δz_{0}<<λ, ∈<<1), gives δf≃Δ∈(v_{0}/λ), independent of δz_{0} to first order. For the J=1/2 measurement, taking Δ=0.1 and ∈=0.05 (see Methods), we would expect a velocitydependent frequency shift of 0.06 Hz (m s^{−1})^{−1}.
A second systematic frequency shift is expected because of the motion of the molecules during the two short π/2 pulses. Including the Doppler shift in the expression for the lineshape in the case of perfect π/2 pulses and expanding about the resonance frequency, we find that a travelling wave results in the systematic frequency shift δf_{D}=(1−4/π)(v_{0}/λ)(τ/T). For the counterpropagating waves this is further reduced by the imbalance factor Δ. For the J=1/2 measurement, with Δ=0.1, τ=15 μs and T=450 μs, we expect a shift of−0.01 Hz (m s^{−1})^{−1}.
We control these and any other velocitydependent shifts, by measuring all the frequencies for at least three different values of v_{0}. We find that the measured frequencies shift linearly with v_{0}, and that the gradient, df/dv_{0}, differs for different values of m_{1}. The largest gradients we measure are 0.07±0.01 Hz (m s^{−1})^{−1} for the J=1/2 measurement, and 0.03±0.01 Hz (m s^{−1})^{−1} for the J=3/2 measurement. After extrapolating to zero velocity the results for different values of m_{1} all agree, so we take the weighted mean of the zerovelocity values as the final frequency. This procedure does not rely on a knowledge of Δ.
To minimize systematic shifts due to magnetic fields, the interaction region is magnetically shielded. For a linearly polarized microwave field and a small magnetic field, there is a symmetry in the shifts and amplitudes of the Zeeman subcomponents, and therefore no systematic shift. Circular polarization components of the microwave field are strongly suppressed inside the transmission line. For the J=1/2 level the spin and orbital angular momentum contributions to the magnetic moment very nearly cancel, so Zeeman shifts are small. For example, the shift of the M_{F}=+1 component of the (1/2^{+},1) level is approximately 0.01 Hz nT^{−1}. For the J=3/2 level the Zeeman shifts are far greater, 15.1M_{F} Hz nT^{−1} for F=1 and 9.07M_{F} Hz nT^{−1} for F=2. To measure the residual field in the interaction region, we applied single 780μslong pulses to molecules moving at 570 m s^{−1}, and measured the Zeeman splittings of the J=3/2 levels as a function of applied magnetic field in all three directions. Reversal of the applied field measures the residual field in that direction, averaged over the interaction region. We also look for any broadening of the singlepulse lineshape at zero applied field. The residual fields are consistent with zero in all three directions, with upper limits of 3, 56 and 25 nT along x, y and z, respectively. To measure the size of magneticfield related shifts in the Ramsey data, we apply small homogeneous fields in each direction. For the J=1/2 transitions, we observe no significant shift at the 1 Hz level, even for fields as large as 50 μT. For the J=3/2, ΔF=0 transitions, the fieldinduced frequency shifts are all smaller than 0.04 Hz nT^{−1}. Multiplying this by the largest upper limit to the field gives a systematic uncertainty of 2 Hz. For the J=3/2, ΔF=1 transitions, we could not rule out gradients as large as 0.1 Hz nT^{−1} for x, 0.2 Hz nT^{−1} for y and 0.05 Hz nT^{−1} for z. Multiplying these by the upper limits to the residual field gives a systematic uncertainty of 11 Hz for these transitions.
We measured the Stark shift of the J=1/2 transition by applying electric fields up to 20 V cm^{−1} between the plates of the transmission line. Measuring the change in this shift upon reversing the applied field sets an upper limit to uncontrolled Stark shifts of 0.1 Hz. We checked for systematic shifts depending on microwave power or probe laser detuning, but these were negligible. Frequency shifts due to collisions in the beam, spurious frequency sidebands, blackbody radiation^{41}, the motional Stark shift and the secondorder Doppler shift are all negligible at the current accuracy level of 10^{−9}.
Table 1 shows our measured frequencies along with their uncertainties, which are obtained by adding the statistical and systematic uncertainties in quadrature. The systematic uncertainty is negligible for all but the last two, where it is 11 Hz. The three J=1/2 measurements are consistent with the previous best laboratory^{42} and astrophysical^{30,31} measurements, and are 300 times more precise than these. We know of no previous laboratory measurements of the J=3/2 frequencies. There are astrophysical measurements of the first two J=3/2 transitions in the table^{36}, and our results are consistent with these and about 300 times more precise.
Constraining changes of α and μ in the Milky Way
With these new measurements and existing astronomical data, we can set useful bounds on the possible variations of α and μ with matter density. We study five interstellar sources where the density is typically 10^{19} times smaller than on Earth. It is necessary to compare two spectral lines from each source, so that the Doppler shift can be removed, and for high sensitivity to changes in the constants these two should have very different sensitivity coefficients. In one of the sources the J=1/2 and J=3/2 Λdoublet transitions of CH are observed together, and in the other four the J=1/2 CH transition is observed together with the ground state Λdoublet transition of OH. Astronomical spectra are plotted on a velocity scale that is determined from the Doppler shift of the received radiation. For each selected source the spectral lines are narrow and have similar shapes, and we assume that the molecules producing them have the same velocities. To obtain our bounds, we assume that there is either a change in α or μ, but not in both (this is a limitation of our analysis). Then, the fractional change in α is
with a similar equation for the change in μ. Here, Δv_{12}=v^{(1)}−v^{(2)} is the difference between the measured velocities of lines 1 and 2 and is determined from the published spectra, is the sensitivity coefficient for transition i, is the frequency measured in the laboratory and is the frequency used to define the velocity scale in the astronomical measurement. This is usually the accepted value for the transition frequency at the time of the measurement, is specified along with the published spectra, and carries no uncertainty. The laboratory measurements presented here allow the second term in square brackets to be determined to high accuracy, so the uncertainty in Δα/α is now entirely due to the uncertainty in Δv_{12}.
Table 2 shows the results obtained from the five selected interstellar sources. We find the central velocities by fitting Gaussian profiles to the spectral lines. Where there are multiple velocity components, we fit multiple Gaussians. In each source there are multiple velocity components and/or multiple hyperfine components, so we obtain several different values for the velocity difference. Their standard deviation is typically several times the individual uncertainties of the fits, probably because the underlying velocity distributions are not Gaussian. For each source, we use the mean and standard deviation of the individual results to give a central value and uncertainty for Δv_{12}. The angular size of the observed part of the source (the beam size) is often different for the two species. This can result in systematic velocity differences. This is not significant for CasA or W51, where the sources are smaller or comparable in size to the smallest beam size used. Neither is it significant for Heiles cloud 2 or L134N, because the beam size used for the CH and OH observations were nearly the same, and maps of these clouds show that velocity variation is less than 0.2 km s^{−1} over a much wider angular scale than this beam size^{30}. For the observations of RCW 36, which is an extended source, the beam size for OH was twice that for CH. However, maps of this cloud^{44} show that the observed region is centred on an intense localized clump whose size is approximately the OH beam size, and that any systematic velocity differences due to different beam sizes will be smaller than our quoted uncertainty.
Using Δv_{12}, the measured values of for CH and OH^{45} and the quoted values of , we calculate the quantity in square brackets in equation (4), which we label as Δv′_{12} in Table 2. From these, and the calculated sensitivity coefficients for OH (ref. 20) and CH (ref. 21), we find values for Δα/α and Δμ/μ. Taking the weighted mean of their values, we arrive at the results Δα/α=(0.3±1.1) × 10^{−7} and Δμ/μ=(−0.7±2.2) × 10^{−7}, giving us 1σ upper bounds of Δα/α<1.4 × 10^{−7} and Δμ/μ<2.9 × 10^{−7}.
Discussion
Our analysis demonstrates the high sensitivity of this method. With dedicated astronomical measurements, far higher precision could be reached and the study could be extended to sources at high redshift. The scope is particularly great for studying sources where the 3.3 and 0.7 GHz transitions are observed simultaneously. There is currently a shortage of such data, and the comparison provides excellent sensitivity to variation of the constants and is relatively immune to systematic shifts that can arise when analysing transitions in different species. The relatively low intensity of the CH lines requires the use of large telescopes, but there are several that are suitable. The 100m telescope of the MaxPlanckGesellschaft covers the entire frequency range between the two Λdoublets, whereas the 100m Green Bank Telescope covers the frequency range from 0.29 to 2.6 GHz and so is suitable for measuring both transitions at moderate redshifts. For a higher collecting surface, the 300m telescope at Arecibo can be used. It covers frequencies from 3.3 GHz down to 1.225 GHz and selected frequency bands between 690 and 312 MHz, so could be used to study both transitions over a wide range of redshifts. In a few years, even higher sensitivity will be provided by a new 500 m telescope being built in the Guizhou province of China. To circumvent problems of rf interference, interferometric measurements are efficient. The Very Large Array provides such measurements with high sensitivity and currently covers all frequencies between 1 and 3.3 GHz. Looking further ahead, the Square Kilometre Array, to be constructed in South Africa and Australia, will provide unprecedented sensitivity. Being an interferometric facility located in areas of low interference, this telescope should be able to measure the two Λdoublets of CH in a large number of interstellar clouds, both at low and high redshift. With a reasonable precision of 0.1 km s^{−1} in the measurement of Δv_{12}, and a modest dataset, it will be possible to reach below the 10^{−8} level for both Δα/α and Δμ/μ. The laboratory frequencies are now so accurate that their uncertainties are unlikely to be a limiting factor in any future measurements.
Methods
Frequency measurements
The setup for measuring the frequencies is shown in Fig. 2. A pulsed supersonic beam of CH radicals, with a repetition rate of 10 Hz, is produced by photodissociation of CHBr_{3} (96% purity stabilized in ethanol) in a 4bar carrier gas of either He, Ne, Ar or Kr (refs 46, 47). The measured mean speeds of CH for each of these carrier gases are v_{0}=1710, 810, 570 and 420 m s^{−1}, and the translational temperature is typically 0.4–0.5 K. About 95% of the molecules are formed in the J=1/2 state. The mixture, formed by bubbling the carrier gas through liquid bromoform, expands through the 1mmdiameter nozzle of a pulsed solenoid valve into the vacuum chamber. The light from the photodissociation laser (220 mJ, 20 ns, 248 nm) is focused in front of the nozzle to a spot 1 mm high (along y) and 4 mm wide (along z).
At z=86 mm, the molecules pass through a 2mmdiameter skimmer and enter the second chamber, where the pressure is below 10^{−7} mbar. They pass through the state selector and transmission line resonator described below, and are then detected at z=780 mm by driving the A^{2}Δ(v=0)←X^{2}Π(v=0) transition with a laser and imaging the resulting fluorescence onto a photomultiplier tube. In this way, the timeofflight profile of each molecular pulse is recorded with a temporal resolution of about 5 μs. The probe laser beam comes from a frequencydoubled continuouswave titaniumsapphire laser. The beam propagates along x, is linearly polarized along z, has a wavelength near 430.15 nm and has a power of 5 mW in a rectangularshaped beam, 4 mm high and 1.4 mm wide. We measure the population in the (1/2^{−}, F) levels for the J=1/2 measurement and in the (3/2^{+}, F) levels for the J=3/2 measurement, by driving the transitions to the J=3/2 and J=5/2 levels of A^{2}Δ(v=0, N=2). The frequencies of these transitions, which are labelled R_{22ff}(1/2) and R_{11ff}(3/2), are given in ref. 48. The CH density at the detector is approximately 10^{6} cm^{−3}.
The state selector is situated at z=241 mm. For the J=1/2 measurement it is a laser beam of the same frequency, polarization and propagation direction as the detection laser. This beam depletes the population of the negative parity component. For the J=3/2 measurement, we drive population from (1/2^{−}, F) to the (3/2^{+}, F′) state, using approximately 10 μW of radiation near 533 GHz. This mmwave radiation is the amplified 54th harmonic of a frequency synthesizer. The radiation is collimated by a teflon lens of 30 mm focal length, giving a 10mmdiameter beam, which propagates along x and is polarized along y. The efficiency of the population transfer is about 50%.
The microwave transmission line resonator is built from a pair of 30mmwide parallel copper plates separated by 5 mm along y. At the Λdoublet frequencies of interest, the transmission line can only support a TEM_{00} mode polarized along y. Microwave radiation is launched into one end from a semirigid coaxial cable. The other end is not terminated and so a resonator with a quality factor of approximately 10 is formed. For each frequency component measured, the length was cut so that the transmission line had a resonance at that frequency. The length was approximately 480 mm for the J=1/2 measurement and 440 mm for the J=3/2 measurement. The microwave radiation is generated by a synthesizer phase locked to a GPS frequency reference whose fractional accuracy is better than 10^{−13}. A fast, high isolation switch controls the microwave pulse duration. The transmission line is housed inside a mumetal magnetic shield. For the J=1/2 measurement a single shield is used, whereas for the J=3/2 measurement we use a pair of nested shields. The outer shield extended into the region of the state selector, and a pair of holes cut into this shield allowed the mmwave beam to pass through. Coils inside the shields allow us to apply wellcontrolled magnetic fields.
For each transition, we first make an approximate measurement of the transition frequency by applying a single microwave pulse to the molecules and measuring the population as a function of the frequency. Owing to the two counterpropagating waves, we observe two peaks separated by twice the Doppler shift. Their relative amplitudes indicate that Δ≈0.1. The mean frequency of these peaks is the transition frequency. With the frequency fixed at this value, we map out the standing microwave field by driving the transition with a 15μs pulse and scanning the time when it is applied. The times at which the resonance is largest are the times when the molecules pass each of the antinodes. Next, we apply a short pulse when the molecules are at an antinode, and we scan the microwave power. We observe Rabi oscillations in the population, and thus determine the power needed to drive a πpulse. We do this for each of the antinodes. With the parameters now fixed, we measure the transition frequency using the Ramsey method of separated oscillating fields^{39}, as described in the main text. Compared with the continuous wave excitation, this method has a higher resolution and is relatively immune to the broadening and distortion of the lineshape caused by nonuniform fields.
Additional information
How to cite this article: Truppe, S. et al. A search for varying fundamental constants using hertzlevel frequency measurements of cold CH molecules. Nat. Commun. 4:2600 doi: 10.1038/ncomms3600 (2013).
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Acknowledgements
We thank Ben Sauer and Jony Hudson for their helpful advice. This work was supported in the UK by the EPSRC and the Royal Society.
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Affiliations
Centre for Cold Matter, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, UK
 S. Truppe
 , R.J. Hendricks
 , S.K. Tokunaga
 , E.A. Hinds
 & M.R. Tarbutt
JILA and Department of Physics, University of Colorado, 440 UCB, Boulder, Colorado 803090440, USA
 H.J. Lewandowski
Petersburg Nuclear Physics Institute, Gatchina 188300, Russia
 M.G. Kozlov
MaxPlanckInstitut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
 Christian Henkel
Astronomy Department, King Abdulaziz University, PO Box 80203, Jeddah, Saudi Arabia
 Christian Henkel
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Contributions
The experiment was designed and built by S.T., S.K.T, R.J.H. and M.R.T. The data were collected by S.T., S.K.T., R.J.H. and H.J.L. The laboratory data were analysed by S.T., R.J.H. and M.R.T. The astronomical data were analysed by M.R.T. The manuscript was drafted by S.T., M.R.T. and E.A.H. and the outlook prepared by C.H. The work was inspired by M.G.K. and the team was led by M.R.T and E.A.H. All authors discussed the methodology and the results, and improved the manuscript.
Competing interests
The authors declare no competing financial interests.
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Correspondence to M.R. Tarbutt.
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