Abstract
Weakly bound molecules have physical properties without atomic analogues, even as the bond length approaches dissociation. For instance, the internal symmetries of homonuclear diatomic molecules result in the formation of twobody superradiant and subradiant excited states. Whereas superradiance^{1,2,3} has been demonstrated in a variety of systems, subradiance^{4,5,6} is more elusive owing to the inherently weak interaction with the environment. Here we characterize the properties of deeply subradiant molecular states with intrinsic quality factors exceeding 10^{13} via precise optical spectroscopy with the longest molecule–light coherent interaction times to date. We find that two competing effects limit the lifetimes of the subradiant molecules, with different asymptotic behaviours. The first is radiative decay via weak magneticdipole and electricquadrupole interactions. We prove that its rate increases quadratically with the bond length, confirming quantum mechanical predictions. The second is nonradiative decay through weak gyroscopic predissociation, with a rate proportional to the vibrational mode spacing and sensitive to shortrange physics. This work bridges the gap between atomic and molecular metrology based on latticeclock techniques^{7}, enhancing our understanding of longrange interatomic interactions.
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Simple molecules provide a wealth of opportunities for precision measurements. Their richer internal structure compared to atoms enables experiments that push the boundaries in determinations of the electricdipole moment of the electron^{8}, the electrontoproton mass ratio and its variations^{9,10}, and parity violation^{11}. Diatomic molecules are moving to the forefront of manybody science^{12} and quantum chemistry^{13}, providing glimpses into fundamental laws^{14}. However, this attractive complexity of molecular structure has historically posed difficulties for manipulation and modelling^{15}. This work removes many of these barriers by employing techniques of optical lattice atomic clocks^{16,17} to control the quantum states of weakly bound homonuclear diatomic strontium molecules, in particular by using stateinsensitive optical lattices^{18} for molecular transitions with three types of optical transition moments. We observe strongly forbidden optical transitions in this asymptotic diatomic system, an ideal regime for studying the breakdown of the ubiquitous dipole approximation where the size of the quantum particle is a significant fraction of the resonant wavelength λ. We explain these observations with a stateoftheart ab initio molecular model^{19} and asymptotic scaling laws. The results prove that the quantum mechanical effect of subradiance can be exploited for precision spectroscopy, and demonstrate the promise of combining precise state control, coherent manipulation and accurate ab initio calculations with recently available ultracold molecular systems.
We create Sr_{2} molecules by photoassociation ^{20} from an ultracold cloud of spinless strontium atoms, ^{88}Sr, in an optical lattice satisfying the Lamb–Dicke and resolvedsideband conditions^{21} (Methods). The weak optical coupling of the ground ^{1}S_{0} state to the excited ^{3}P_{1} atomic state (22 μs lifetime^{22}) in Sr atoms enables spectroscopic resolution of molecular structure in the immediate proximity to the ^{1}S_{0} + ^{3}P_{1} atomic threshold without losses from photon scattering. This 689 nm intercombination (spinchanging) transition is electricdipole (E1) allowed, where the photon couples states with opposite parity. The magneticdipole (M1) and electricquadrupole (E2) transitions are strictly forbidden. Owing to quantum mechanical symmetrization, these higherorder transitions become allowed in bound homonuclear dimers, as illustrated in Fig. 1a. In the molecular ground state with the asymptotic electronic wavefunction X^{1}Σ_{g}^{+}〉 ≍ ^{1}S_{0}〉^{1}S_{0}〉, only gerade (even) symmetry is possible, allowing optical E1 transitions only to ungerade (odd) excited molecular states. However, M1 and E2 transitions are possible from X^{1}Σ_{g}^{+} to gerade molecular states such as those near the ^{1}S_{0} + ^{3}P_{1} threshold, because these higher moments couple states of the same symmetry. Such transitions are very weak owing to their spin and electricdipoleforbidden nature. As a result, the gerade molecular states are subradiant, whereas the ungerade states are superradiant. That is, if the E1 atomic radiative decay rate of ^{3}P_{1} to ^{1}S_{0} is Γ, then the equivalent rates are approximately 2Γ and 0 for the superradiant and subradiant molecular states. Asymptotically, these states correspond to the superpositions and of atomic states, respectively. In this work, the subradiant states belong to the excited 1_{g} molecular potential (where ‘1’ refers to the total electronic angular momentum projection onto the molecular axis and ‘g’ to the symmetry of the electronic wavefunction), and couple to the ground state only via the higherorder M1 and E2 transitions. We probe optical transition strengths to the subradiant molecular states to establish their asymptotic quadratic dependence on R, the classical expectation value of the bond length^{23}. This behaviour is in stark contrast to the asymptotic E1 transition strengths of the superradiant states, which are constant with R.
We have precisely quantified the optical transition oscillator strengths from X^{1}Σ_{g}^{+} to the subradiant states. The 1_{g} levels have vibrational quantum numbers v′ between −1 and −4 (counting from the continuum) and total angular momenta J′ = 1,2. The oscillator strengths were measured via optical absorption spectra, with areas normalized by the probe light power P and pulse time τ. For each transition, the experimentally obtained quantity is Q ≡ B_{12}/(cπ^{2}w_{0}^{2}) = A/(τP), where B_{12} is an Einstein B coefficient, w_{0} is the waist of the probe beam and A is the Lorentzian area of the natural logarithm of the absorption spectrum (Supplementary Information). In Fig. 1b, the Q values for the M1 and E2 transitions (ΔJ = 1 and 2, respectively, all starting from a J = 0 ground state) are normalized to the Q for an E1 transition near the same atomic threshold, giving ratios of absorption oscillator strengths. We find M1 and E2 Q values that are four to five orders of magnitude suppressed compared to E1, as expected from the Q ~ π^{2}/4(R/λ)^{2} ratio of the M1 and E1 transition moments^{23}. Alternatively, oscillator strengths are proportional to the ratios of the squares of the Rabi frequencies to P, which were measured for M1 in the time domain by observing coherent Rabi oscillations (Supplementary Information). The two methods yield similar results. We performed ab initio calculations of these doubly forbidden transition strengths. The results shown in Fig. 1b are in excellent agreement with measurements, confirming the asymptotic divergence of the M1 and E2 transition moments with R. In the absence of this linear growth, the oscillator strengths would be governed by the rovibrational wavefunction overlaps (Franck–Condon factors), resulting in ratios different from our observations by about an order of magnitude.
We have also measured the lifetimes of the subradiant states. The long molecule–light coherence times enable optical Rabi oscillations as shown in Fig. 2a, with the fringe decay times limited by the natural lifetimes of the 1_{g} states. The Rabi period was used to determine the length of a πpulse needed to excite the groundstate molecules into subradiant states. After a variable wait time, the molecules were returned to the ground state and imaged via excitation to the ^{1}S_{0} + ^{3}P_{1} continuum followed by spontaneous decay ^{20}, as in the cartoon of Fig. 2b. A typical exponential lifetime curve is shown in Fig. 2b. Although this approach was used for lifetime measurements of the 1_{g} states with v′ = −2, −3, −4, the leastbound level allowed a simplified method. The spectrum in Fig. 2c shows two boundfree optical transitions from v′ = −1 to atomic continua. The process at the higher laser frequency corresponds to fragmentation via the doubly excited ^{3}P_{1} + ^{3}P_{1} continuum and is harnessed for direct lifetime measurements, as depicted in Fig. 2d, where a plot of the recovered atom number versus wait time is shown with an exponential fit. Even without an imaging pulse, some of the weakly bound v′ = −1 molecules decay to groundstate atoms, and we subtract this small contribution from the signal. All known systematic effects were controlled (Methods). The lifetime results are presented in Table 1.
Because the molecules are trapped in the Dopplerfree regime, their absorption linewidths can also yield lifetimes. Unlike the direct lifetime measurements in Fig. 2b, d, this technique is sensitive to inhomogeneous broadening from stray magnetic fields and the lattice. Therefore, we engineered stateinsensitive optical lattices for molecular transitions to the deeply subradiant states. The polarization and wavelength were chosen to ensure light shifts ≲1 Hz mW^{−1}, leading to inhomogeneous broadening <50 Hz for 150 mW of lattice light power (Supplementary Information and Supplementary Fig. 1). We nulled the ambient magnetic field to ≲20 mG by using the linear Zeeman effect in Sr_{2}, and applied a bias field of 0.43 G with angle control of ≲2° to define the quantization axis. The four resulting spectra for transitions from X^{1}Σ_{g}^{+} to v′ = −1, −2, −3, −4 are shown in Fig. 2e–h, and are compared with lineshapes expected from direct lifetime measurements. For the narrowest lines, the spectroscopic method overestimates the widths, as a result of broadening caused by the intrinsic linewidth of the probe laser (<200 Hz), magnetic quenching (<90 Hz, discussed below) and the finite probe pulse (<50 Hz).
The radiative lifetimes of the 1_{g} states were calculated from the ab initio model by considering doubly forbidden M1 and E2 transitions to the ground state. The resulting contributions γ_{rad} to the linewidths are in the range ~1–6 Hz (Table 1). Any contributions from decay to other states below the ^{1}S_{0} + ^{3}P_{1} asymptote, as well as from blackbody radiation^{24}, are negligible. Unlike for atoms, the radiative lifetimes alone do not suffice to explain the observed linewidths.
Nonradiative decay is a dominant contributor to the subradiant lifetimes. As shown in Fig. 3a, the 1_{g} bound states can couple to the longlived ^{1}S_{0} + ^{3}P_{0} continuum of the 0_{g}^{−} state. The nature of this coupling is nonadiabatic Coriolis mixing ^{13,19,25} leading to weak gyroscopic predissociation. An estimate of the predissociation rate follows from the Fermi golden rule, , where is the Coriolis interaction and 0_{g}^{−}, E, J′, m′〉 are energynormalized continuum scattering states with energy E. This coupling vanishes at long range owing to the different dissociation thresholds of the 1_{g} and 0_{g}^{−} potentials, but not at short range (Supplementary Information). We calculated the predissociative linewidths from the ab initio model, which was slightly tuned by scaling the ^{3}Π_{g} potential by 1.2% to improve agreement with experiment. Moreover, we can obtain accurate predissociative linewidth ratios without precise knowledge of the shortrange physics. The amplitude of a boundstate rovibrational wavefunction is ψ_{v}(R) ∝ (∂E_{v}/∂v)^{1/2}, where ∂E_{v}/∂v is the known vibrational energy spacing^{26} (Supplementary Information). Thus γ_{pre} = p{(\partial {E}_{v}/\partial v)}_{E={E}_{v}}, where the parameter p can be related to the ^{1}S_{0} + ^{3}P_{1} inelastic collision crosssection^{25,27}. The γ_{pre} values were obtained both from ab initio theory and by fitting p = 2.48 × 10^{−7} to the measured 1_{g} level linewidths.
The results of the lifetime measurements and calculations are shown in Fig. 3b, where the natural widths are plotted versus R. Note that our R/λ ≲ 0.01, which is less than 0.5% of the range formerly explored with trapped ions^{4}. The four 1_{g} subradiant states are marked, as well as two typical nearby superradiant states (from the 0_{u}^{+} and 1_{u} potentials). The predictions for both nonradiative and radiative contributions are also shown. The radiative contribution exhibits ∝R^{2} asymptotic scaling. The nonradiative contribution shows a change from roughly ∝R^{−4} to ∝R^{−2.5} scaling, reflecting the shift of longrange interaction from a C_{6} to a C_{3} character that occurs near R ~ 80 a_{0} for the 1_{g} potential of Sr_{2} (ref. 22). This scaling can be understood from the LeRoy–Bernstein formula^{28}, relating the inverse density of states to the longrange C_{n}/R^{n} behaviour as ∂E_{v}/∂v ∝ E_{v}^{(n+2)/(2n)} ∝ R^{−(n+2)/2}.
Table 1 summarizes the measurements and ab initio calculations for the 1_{g} levels. Moreover, we found that the lifetimes of the subradiant states are tunable by orders of magnitude with modest magnetic fields up to ~10 G. Figure 4a shows the natural linewidths of the four 1_{g} states versus field strength. They broaden with a quadratic coefficient of ~300 Hz G^{−2} (or ~500 Hz G^{−2} for v′ = −2). This broadening could be qualitatively explained via Zeeman mixing with nearby evenJ′ levels that seem to be shortlived owing to their more complex mixing dynamics, which is further substantiated by the narrowing trend of the J′ = 2 widths as shown in Fig. 4b–e.
The Sr_{2} state with the narrowest natural linewidth (v′ = −1) has a measured lifetime longer than that of the atomic ^{3}P_{1} state by an unprecedented factor of nearly 300, opening the door to ultrahighresolution molecular metrology. Our precise determinations of the binding energies and Zeeman coefficients of molecular states in this deeply subradiant regime (Supplementary Information and Supplementary Table 1) should allow fine tuning of parameters in the ab initio molecular model to reach agreement with measurements at the experimental accuracy, which would be a major achievement of quantum chemistry. Furthermore, Fig. 2c hints at the intriguing possibility of using longlived states for ultracold molecule photodissociation^{29}. The shown transition from the leastbound subradiant excited state to the groundstate continuum should have an ultimate width limited by the subradiant state lifetime, corresponding to excess fragment energies of only a nanokelvin.
Methods
^{88}Sr atoms were lasercooled in a twostage magnetooptical trap (MOT) and loaded into a onedimensional optical lattice with a depth of 30 μK and a wavelength near 900 nm. The lattice was generated by a diode laser and semiconductor tapered amplifier, where a diffraction grating removed any amplified spontaneousemission light. Atoms were photoassociated into 3 μK molecules with a density of ≲10^{12} cm^{−3}, which were optically imaged by a photodissociation pulse with a high spectral resolution^{20}. The molecules can be selectively created in either of the two leastbound vibrational levels (v = −1 or −2) of the electronic ground state. They are distributed among two rotational levels with the total angular momentum J = 0 or 2, which are well resolved spectroscopically. These molecules near the ^{1}S_{0} + ^{1}S_{0} groundstate atomic threshold are the starting point for probing electronically excited molecules near the ^{1}S_{0} + ^{3}P_{1} asymptote. Narrow molecular transitions were induced with a laser that was phaselocked to the narrowlinewidth 689 nm cooling laser. The trapping magnetic coils were pulsed off during spectroscopy, and other sources of magneticfield gradients and noise were eliminated. For lifetime measurements, the following parameters were systematically controlled: lattice light power, molecule density by adjusting the photoassociation light pulse detuning, magnetic field by adjusting the current in a set of Helmholtz coils, and probe light power (for spectroscopic linewidth measurements). No systematic shifts of the lifetime values were detected for the accessible densities and lattice intensities, which were each varied by roughly a factor of two. Magnetic fields quench the lifetimes as in Fig. 4, so the ambient fields were carefully nulled.
The ab initio potentials for the ^{3}Π_{g} (^{1}S + ^{3}P), ^{3}Σ_{g}^{+} (^{1}S + ^{3}P) and ^{1}Π_{g} (^{1}S + ^{1}P) electronic states (Supplementary Information and Supplementary Fig. 2) were calculated using linear response theory within the coupledcluster singles and doubles framework. The groundstate X^{1}Σ_{g}^{+} empirical potential was used ^{30}. Excitedstate potentials were fitted to analytical functions^{19}. Spin–orbit couplings between the nonrelativistic states were fixed at their asymptotic values related to the atomic fine structure. Rovibrational level calculations were set up in the Hund’s case (a) framework by including the ^{3}Π_{g}, ^{3}Σ_{g}^{+} and ^{1}Π_{g} electronic states for the 1_{g} symmetry, and ^{3}Π_{g} and ^{3}Σ_{g}^{+} states for the 0_{g}^{−} symmetry. Diagonalization of the multisurface Hamiltonian for a given J′ was performed via the discrete variable representation method.
References
Dicke, R. H. Coherence in spontaneous radiation processes. Phys. Rev. 93, 99–110 (1954).
Eberly, J. H. Superradiance revisited. Am. J. Phys. 40, 1374–1383 (1972).
Gross, M. & Haroche, S. Superradiance: An essay on the theory of collective spontaneous emission. Phys. Rev. 93, 301–396 (1982).
DeVoe, R. G. & Brewer, R. G. Observation of superradiant and subradiant spontaneous emission of two trapped ions. Phys. Rev. Lett. 76, 2049–2052 (1996).
Zhou, W. & Odom, T. W. Tunable subradiant lattice plasmons by outofplane dipolar interactions. Nature Nanotech. 6, 423–427 (2011).
Takasu, Y. et al. Controlled production of subradiant states of a diatomic molecule in an optical lattice. Phys. Rev. Lett. 108, 173002 (2012).
Katori, H. Optical lattice clocks and quantum metrology. Nature Photon. 5, 203–210 (2011).
Baron, J. et al. The ACME collaboration: Order of magnitude smaller limit on the electric dipole moment of the electron. Science 343, 269–272 (2014).
Bressel, U. et al. Manipulation of individual hyperfine states in cold trapped molecular ions and application to HD^{+} frequency metrology. Phys. Rev. Lett. 108, 183003 (2012).
Shelkovnikov, A., Butcher, R. J., Chardonnet, C. & AmyKlein, A. Stability of the protontoelectron mass ratio. Phys. Rev. Lett. 100, 150801 (2008).
Tokunaga, S. K. et al. Probing weak forceinduced parity violation by highresolution midinfrared molecular spectroscopy. Mol. Phys. 111, 2363–2373 (2013).
Yan, B. et al. Observation of dipolar spinexchange interactions with latticeconfined polar molecules. Nature 501, 521–525 (2013).
McGuyer, B. H. et al. Nonadiabatic effects in ultracold molecules via anomalous linear and quadratic Zeeman shifts. Phys. Rev. Lett. 111, 243003 (2013).
Dickenson, G. D. et al. The fundamental vibration of molecular hydrogen. Phys. Rev. Lett. 110, 193601 (2013).
Carr, L. D., DeMille, D., Krems, R. V. & Ye, J. Cold and ultracold molecules: Science, technology and applications. New J. Phys. 11, 055049 (2009).
Hinkley, N. et al. An atomic clock with 10^{−18} instability. Science 341, 1215–1218 (2013).
Bloom, B. J. et al. An optical lattice clock with accuracy and stability at the 10^{−18} level. Nature 506, 71–75 (2014).
Ye, J., Kimble, H. J. & Katori, H. Quantum state engineering and precision metrology using stateinsensitive light traps. Science 320, 1734–1738 (2008).
Skomorowski, W., Pawłowski, F., Koch, C. P. & Moszynski, R. Rovibrational dynamics of the strontium molecule in the A^{1}Σu^{+}, c^{3}Πu, and a^{3}Σu^{+} manifold from stateoftheart ab initio calculations. J. Chem. Phys. 136, 194306 (2012).
Reinaudi, G., Osborn, C. B., McDonald, M., Kotochigova, S. & Zelevinsky, T. Optical production of stable ultracold ^{88}Sr2 molecules. Phys. Rev. Lett. 109, 115303 (2012).
Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75, 281–324 (2003).
Zelevinsky, T. et al. Narrow line photoassociation in an optical lattice. Phys. Rev. Lett. 96, 203201 (2006).
BusseryHonvault, B. & Moszynski, R. Ab initio potential energy curves, transition dipole moments and spin–orbit coupling matrix elements for the first twenty states of the calcium dimer. Mol. Phys. 104, 2387–2402 (2006).
Farley, J. W. & Wing, W. H. Accurate calculation of dynamic Stark shifts and depopulation rates of Rydberg energy levels induced by blackbody radiation. Hydrogen, helium, and alkalimetal atoms. Phys. Rev. A 23, 2397–2424 (1981).
Mies, F. H. & Julienne, P. S. A multichannel quantum defect analysis of twostate couplings in diatomic molecules. J. Chem. Phys. 80, 2526–2536 (1984).
Mies, F. H. A multichannel quantum defect analysis of diatomic predissociation and inelastic atomic scattering. J. Chem. Phys. 80, 2514–2525 (1984).
Ido, T. et al. Precision spectroscopy and densitydependent frequency shifts in ultracold Sr. Phys. Rev. Lett. 94, 153001 (2005).
LeRoy, R. J. & Bernstein, R. B. Dissociation energy and longrange potential of diatomic molecules from vibrational spacings of higher levels. J. Chem. Phys. 52, 3869–3879 (1970).
Wells, N. & Lane, I. C. Prospects for ultracold carbon via charge exchange reactions and laser cooled carbides. Phys. Chem. Chem. Phys. 13, 19036–19051 (2011).
Stein, A., Knöckel, H. & Tiemann, E. The ^{1}S + ^{1}S asymptote of Sr2 studied by Fouriertransform spectroscopy. Eur. Phys. J. D 57, 171–177 (2010).
Acknowledgements
We gratefully acknowledge NIST award 60NANB13D163 and ARO grant W911NF0910504 for partial support of this work. M.M. acknowledges NSF IGERT DGE1069260, and R.M. the Polish Ministry of Science and Higher Education for grant NN204215539. R.M. also thanks the Foundation for Polish Science for support within the MISTRZ program.
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B.H.M., M.M., G.Z.I. and T.Z. designed the experiments. B.H.M., M.M., G.Z.I., M.G.T. and T.Z. carried out the measurements and interpreted the data. B.H.M. and M.M. coled the experimental efforts. W.S. and R.M. carried out the theoretical calculations and interpreted the data. All authors contributed to the manuscript.
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McGuyer, B., McDonald, M., Iwata, G. et al. Precise study of asymptotic physics with subradiant ultracold molecules. Nature Phys 11, 32–36 (2015). https://doi.org/10.1038/nphys3182
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DOI: https://doi.org/10.1038/nphys3182
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