Abstract
Precision spectroscopy of atomic and molecular ions offers a window to new physics, but is typically limited to species with a cycling transition for laser cooling and detection. Quantum logic spectroscopy has overcome this limitation for species with longlived excited states. Here we extend quantum logic spectroscopy to fast, dipoleallowed transitions and apply it to perform an absolute frequency measurement. We detect the absorption of photons by the spectroscopically investigated ion through the photon recoil imparted on a cotrapped ion of a different species, on which we can perform efficient quantum logic detection techniques. This amplifies the recoil signal from a few absorbed photons to thousands of fluorescence photons. We resolve the line centre of a dipoleallowed transition in ^{40}Ca^{+} to 1/300 of its observed linewidth, rendering this measurement one of the most accurate of a broad transition. The simplicity and versatility of this approach enables spectroscopy of many previously inaccessible species.
Introduction
Precision optical spectroscopy of broad transitions provides information on the structure of molecules^{1}, it allows tests of quantum electrodynamics^{2}, and, through comparison with astrophysical data, probes for a possible variation of fundamental constants over cosmological scales^{3,4}. Nuclear properties are revealed through isotope shift measurements^{5,6,7,8,9}, or absolute frequency measurements^{10,11}. Trapped ions are particularly well suited for such high precision experiments. The ions are stored in an almost fieldfree environment and can be lasercooled to eliminate Doppler shifts. These features have enabled record accuracies in optical clocks^{12,13,14,15}. For longlived excited states such as in atoms with clock transitions, the electronshelving technique amplifies the signal from a single absorbed photon by scattering many photons on a closed transition through selective optical coupling of one of the two spectroscopy states to a third electronic level^{16}. The invention of quantum logic spectroscopy (QLS)^{12,17} removed the need to detect the signal on the spectroscopically investigated ion (spectroscopy ion) by transferring the internal state information through a series of laser pulses to the cotrapped, socalled logic ion where the signal is observed via the electronshelving technique. However, this original implementation of QLS requires longlived spectroscopy states to implement the transfer sequence. For transitions with a shortlived excited state, spectroscopy of trapped ions is typically implemented through detection of scattered photons in laserinduced fluorescence^{7,18,19,20,21,22} or detection of absorbed photons in laser absorption spectroscopy^{23}. Neither of the two techniques reaches the fundamental quantum projection noise limit as in the electronshelving technique^{24} due to low light collection efficiency in laserinduced fluorescence and small atomlight coupling in laser absorption spectroscopy. In a variation of absorption spectroscopy, the detuningdependent effect of spontaneous light forces on the motional state of trapped ions is employed to obtain a spectroscopy signal. This has been investigated by observing macroscopic heating on a Dopplercooled twoion crystal^{25}, and through a measurement of motional phase change due to photon scattering^{26}.
Here, we present and demonstrate an extension of QLS, which combines the advantages of these techniques and provides a highly sensitive, quantum projection noiselimited signal for the spectroscopy of broad transitions. It is based on detection of momentum kicks from a few absorbed photons near the resonance of a single spectroscopy ion through a cotrapped logic ion. We demonstrate photon recoil spectroscopy (PRS) on the ^{2}S_{1/2}↔^{2}P_{1/2} transition in a ^{40}Ca^{+} ion, resolving the line by a factor of 1/300 of its observed linewidth of ~34 MHz and achieve a statisticslimited fractional instability of σ_{y}(τ)=5.1 × 10^{−9}, corresponding to 100 absorbed photons for a resolution of 7.2 MHz. This proofofprinciple experiment demonstrates the potential of PRS and its applicability to different species.
Results
Spectroscopy scheme
The principle of PRS is shown in Fig. 1. Starting from the axial motional ground state of the twoion crystal (Fig. 1a), momentum transfer from photon absorption on the spectroscopy ion results in motional excitation (Fig. 1b). Owing to the strong Coulomb interaction, this excitation is shared by the logic ion where it is converted into a longlived internal excitation by applying a socalled redsideband laser pulse, which changes the internal state conditioned on the presence of motional excitation (Fig. 1c)^{27,28,29}. Subsequent electronshelving detection on the logic ion amplifies the signal of a few absorbed photons on the spectroscopy ion to thousands of scattered photons on the logic ion, providing quantum projection noiselimited detection of motional excitation. This signal is backgroundfree, since spectroscopy and detection are separated in time.
During interaction with spectroscopy pulses, absorption of a photon displaces the motional state in momentum by ħk, where k is the wavenumber of the spectroscopy light. Efficient excitation of a chosen normal mode is accomplished through application of short (compared with the oscillation period T_{m} in the trap) spectroscopy laser pulses synchronized to the respective normal mode frequency ω_{m}=2π/T_{m} (refs 26, 30, 31). The momentum displacement of this mode on absorption of a single photon is given by the Lamb–Dicke factor η (in units of the zeropoint momentum fluctuations of the normal mode)^{32}. This motional excitation is shared by the logic ion. It is then converted into an electronic excitation of the logic ion via a transfer pulse on the redsideband. In a simple model, the excitation signal on the logic ion after this pulse is proportional to the probability of leaving the motional ground state, which is approximately given by
where N_{a} is the number of absorbed photons. Choosing p_{e}=0.5 yields a SNR of 1 in a single experiment, the best allowed by quantum projection noise^{24}. For typical Lamb–Dicke parameters of 0.1, this requires absorption of around 10 photons.
An analytical treatment of the observed signal based on a Fokker–Planck equation qualitatively confirms the favourable scaling with the number of absorbed photons and can be found in Supplementary Note 2.
Implementation
We have implemented PRS using ^{25}Mg^{+} as the logic ion to perform an absolute frequency measurement of the ^{2}S_{1/2}↔^{2}P_{1/2} transition in ^{40}Ca^{+} with a natural linewidth of 21.6 MHz. A sketch of the experimental setup together with the level scheme of the spectroscopy ion is shown in Fig. 2. Details of the system have been described previously^{33,34} and are summarized in the Methods section.
Resonance line and sensitivity
Figure 3a shows the average of 12 scans across the resonance of the ^{2}S_{1/2}↔^{2}P_{1/2} transition in ^{40}Ca^{+} using PRS as described in Fig. 1. Typically 70 spectroscopy pulses per cycle are applied and a cycle is repeated 250 times for each frequency point. After each spectroscopy pulse, a repump pulse is applied to clear out the metastable ^{2}D_{3/2} state. The observed linewidth of the transition is ~34 MHz, compatible with Zeeman broadening in the applied bias magnetic field of 5.84(1) × 10^{−4} T and Fourier broadening due to the 50ns short spectroscopy pulses. An additional line broadening due to saturation of the excitation signal for large motional displacements (groundstate depletion) is avoided by choosing a small number of absorbed photons. The line is a numerical simulation of the spectroscopy signal using a full master equation model (Supplementary Note 3) and corrected for experimental imperfections, such as a reduced signal amplitude (typically ≥0.85) from limited sideband transfer efficiency and a signal offset of typically ≤0.15 due to incomplete groundstate cooling. The photon sensitivity of the signal was determined by taking advantage of the excited ^{2}P_{1/2} state branching ratio (Fig. 2b). For this, the transition was excited on resonance and the signal amplitudes with and without repumping pulses on ^{40}Ca^{+} were compared. Figure 3b shows the result as a function of the number of spectroscopy pulses. The Rabi frequency of the spectroscopy laser and the signal per scattered photon is extracted from a numerical simulation of the spectroscopy signal, which takes the excitedstate branching ratio of 14.5:1 (ref. 35) into account, corresponding to 15.5 absorbed photons in the absence of any repumping. An excitation of p_{e}=0.5 corresponding to SNR=1 in a single experiment is achieved by absorbing 9.5(1.2) photons, in qualitative agreement with equation 1 for our experimental parameters and confirmed by the analytical model (Supplementary Note 2). This high sensitivity even for a small Lamb–Dicke factor of η=0.108 is a consequence of the exponential scaling with the square of the number of absorbed photons (equation 1).
Absolute frequency measurement
The main result of this work is shown in Fig. 4, where we demonstrate that PRS enables precision spectroscopy at an unprecedented level. It shows the results of a frequency measurement of the ^{2}S_{1/2}↔^{2}P_{1/2} transition in ^{40}Ca^{+}, which we have implemented using a twopoint sampling technique^{24}. For this, we set the probe laser frequency close to the centre of the atomic resonance and probe the excitation probability for constant frequency steps towards the left and right of the resonance centre. The centre frequency is determined from the average of the excitation probabilities p_{1,r} on each side, multiplied with the frequency discriminant given by the experimentally determined slope of the resonance curve. These steps are repeated until the desired frequency uncertainty is achieved. The frequency measurements shown in Fig. 4a have been performed on 5 days within 11 weeks. Fig. 4b shows the relative frequency uncertainty as a function of averaging time τ in form of an overlapping Allan deviation σ_{y}(τ) (ref. 36) of the concatenated five measurement runs. The observed instability σ_{y}(τ)=5.1 × 10^{−9} (red line) is in agreement with the expected Allan deviation of 5.8(9) × 10^{−9} given the measured SNR for a single ion using the true measurement time, including dead time from groundstate cooling and detection. This confirms that the measurement is statisticslimited. The required averaging time can be reduced by at least a factor of ten through speedoptimized groundstate cooling^{37}. For chosen excitation probabilities p_{1,r}, the mean number of scattered photons for each probe cycle can be determined from the calibration shown in Fig. 3b. This allows us to express the relative frequency uncertainty as a function of the number N_{a} of absorbed photons, which is given by σ_{y}(N_{a})=9.5 × 10^{−8}/, corresponding to an uncertainty of 7.2 MHz for 100 absorbed photons.
Table 1 shows a summary of the evaluated systematic effects for the absolute frequency measurement. Details can be found in the Supplementary Notes 1 and 3. Lineshape asymmetries from an excitation imbalance of Zeemanshifted line components are strongly suppressed by symmetrical driving of the transition by linearly polarized light. Offresonant coupling of the spectroscopy laser to other states can cause frequencydependent AC Stark shifts. We have evaluated both shifts by measuring the line centre frequency as a function of the magnetic field and spectroscopy laser intensity. As expected, the results are compatible with vanishing shifts, limited by the statistical uncertainty of the measurement. Shifts arising from micromotioninduced AC Stark and secondorder Doppler effects, as well as lineshape and frequency uncertainties arising from the spectroscopy laser stabilization have been estimated to be well below 1 kHz. The spectral bandwidth of the 125 ns spectroscopy pulses used in the frequency measurements, together with the limited bandwidth of the switching device (an acoustooptical modulator), can cause a shift in the centre frequency of the spectroscopy pulses on the order of ±50 kHz. We calibrate this shift for each experimental run to within 16 kHz.
Probefrequencydependent Doppler heating and cooling effects during excitation result in an asymmetric lineshape of the motional excitation signal. Starting from the motional ground state in our system, we have a precise knowledge of the Dopplerinduced shifts through the excitation signal. This allows an accurate analytical and numerical calculation of the shift. From a simple model taking into account Dopplerinduced (anti)damping and associated narrowing (spread) of the wave packet in momentum space, this frequency shift is given by
where we have substituted equation 1 for p_{1,r} in the last step. The shift increases for smaller excitation probabilities p_{1,r} used in the twopoint sampling technique, corresponding to a larger detuning of the two probe frequencies from resonance and a smaller number of scattered photons. It is worthwhile noting that within the approximations of the model, this shift is independent of the choice of other parameters, such as spectroscopy laser Rabi frequency or the transition linewidth. The result from this simple model agrees very well for typical excitation probabilities between 0.1 and 0.9 with analytical and numerical models including diffusion and drift in position (Supplementary Notes 2 and 3). For smaller excitation probability, the shift is bounded by the trap frequency. From conservatively estimated fluctuations of the detection signal of ±0.1 around p_{1,r}=0.5, we account for a shift of 151(20) kHz. The uncertainty of this residual Doppler shift can be significantly reduced, while at the same time improving the photon sensitivity, by lowering the trap frequency during spectroscopy. A similar, but much stronger effect from heating/cooling processes during spectroscopy of Dopplercooled ions has been reduced by sympathetic cooling^{38} and interleaved cooling/spectroscopy pulses^{20}.
The absolute frequency averaged over five individual measurements is 755,222,765,896(88) kHz, of which 78 kHz are from systematic and 42 kHz from statistical uncertainty, in agreement with a previous measurement^{20}, but with an accuracy improved by more than an order of magnitude. To our knowledge, this result represents the most accurate absolute frequency measurement of a broad transition in a trapped ion system, resolving the line centre by a factor of more than 300 of the transition’s observed linewidth^{7,20,38}.
Discussion
The high photon sensitivity and absolute frequency accuracy of PRS together with its simplicity, requiring only groundstate cooling, enables spectroscopy of atomic and molecular species inaccessible with other techniques. The method is particularly well suited for systems with nonclosed transitions or intrinsically low fluorescence rates, such as molecular ions^{39,40,41}, astrophysically relevant metal ions with a complex level structure^{3,4,42} and highly charged ions^{43}. PRS is also an efficient replacement for the original QLS implementation for longlived excited states, as it does not require the driving of sideband transitions on the spectroscopy ion, thus lowering the requirements on the spectroscopy laser. Combined with the singlephoton sensitivity for high photon energies and the backgroundfree detection, PRS enables direct frequency comb^{44,45}, XUV^{46,47} and Xray spectroscopy,^{48,49} which offer only a small number of photons per spectral bandwidth. This will pave the way for spectroscopic applications ranging from chemistry to fundamental questions in atomic, molecular and nuclear physics.
Methods
Experimental methods
We trap a twoion crystal consisting of ^{25}Mg^{+} and ^{40}Ca^{+} in a linear ion trap at single^{25}Mg^{+} radial and axial trap frequencies of ~5 MHz and 2.22 MHz, respectively. Using resolved sideband cooling^{33} on ^{25}Mg^{+}, we cool the inphase (ip) and outofphase (op) axial modes with frequencies f_{ip}=1.92 MHz and f_{op}=3.51 MHz to their ground state of motion with a typical residual mean occupation of ≈0.07 and ≈0.02. We have chosen the ipmode as the spectroscopy mode, as it has the largest Lamb–Dicke parameter of =η=0.108 for the calcium ion. The RSB pulse on the logic ion is implemented by a stimulated Raman transition that is red detuned from the F=3, m_{F}=3=↓↔↑= F=2, m_{F}=2 transition in the ^{2}S_{1/2} state (Fig. 5) by the normal mode frequency. As the required πtime for optimum transfer depends on the initial motional state^{27}, we employ a stimulated Raman adiabatic passage (STIRAP)like pulse by linearly ramping down the πRaman beam intensity during 100 μs, while ramping up the intensity of the σRaman beam. Although not fully adiabatic, we achieve a more than 80% transfer efficiency of all motionally excited states from ↓ to ↑, independent of their initial state. Internal state discrimination is performed by applying light resonant with the cycling transition ↓ →e=^{2}P_{3/2}, F=4, m_{F}=4› (Fig. 5) using the πdetection technique^{34}.
The 397nm spectroscopy laser consists of a frequencydoubled extended cavity diode laser. The laser light can be frequency tuned via a doublepass acoustooptical modulator (AOM). An additional AOM is used to provide short (50…125 ns) pulses of spectroscopy light. The intensitystabilized light is delivered to the atom via a singlemode optical fibre to avoid beamsteering errors. The laser is phaselocked to an optical frequency comb, which is referenced to one of the hydrogen masers at the German National Metrology Institute (PTB), providing a direct link to the SI second (PTB) with a relative frequency uncertainty of better than 10^{−12} for averaging times of ~1 h. The spectroscopy and repump light are linearly polarized in a plane spanned by the magnetic field and the trap axis. Under ideal conditions, this results in equal amounts of σ^{+} and σ^{−} polarized light and an additional π component, providing a symmetric spectrum, independent of the magnetic field.
Frequency measurements
For the final frequency measurement, we employ the twopoint sampling method^{24}, where the spectroscopy laser probes the excitation signal at the two frequencies (v_{1}=v_{probe}−Δv_{FWHM}/2, v_{r}=v_{probe}+Δv_{FWHM}/2), which ideally provides an excitation of half the maximum excitation. In every single scan, the experiment is repeated on both frequencies for 5 × 250=1,250 times. To reduce the detection error, we employ the πdetection method^{34} with threshold at 0.5 to extract the excitation probability. This detection scheme is based on postselection, which inherently reduces the effective number of cycles to a typical value of =1,039(12). This corresponds to a quantum projection noise of σ^{(p)}=≈0.0155. The difference of excitation at the two probe frequencies δ p=p(v_{1})−p(v_{r})=p_{1}−p_{r} is used as an error signal to steer the probe frequency v_{probe} towards the line centre in the following scan. The uncertainty for the quantity δ p rescales to σ^{(δp)}=≈0.0220.
The excitation difference δp is turned into an absolute frequency v by the following relation
where the frequency discriminant D is calibrated for each measurement day experimentally through a measurement of the signal slope around the probe frequencies v_{1,r}.
The final statistical uncertainty for the frequency measurement propagates as
where the first term contains the quantum projection noise and the second one the calibration error for the frequency discriminant.
Concatenating the timeresolved frequency uncertainty from the five measurements shown in Fig. 4a, the overlapping Allan deviation is derived as shown in Fig. 4b. This gives an inferred instability of 5.1 × 10^{−9} for an averaging time of 1 s. This result is in agreement with the quantum projection noise limit of 5.8(9) × 10^{−9} estimated according to
where T (=70.5 s) refers to the time for repeating the experimental cycles 1250 times.
The data shown in Fig. 4 are based on a selection of data that have passed statistical tests to check its validity. A total of seven measurement runs, each performed on a different day, have passed normality tests (Kolmogorov–Smirnov test, Lillifors test, Jarque–Bera test, Shapiro–Wilk test). In a second step, twosample tests (twosample Kolmogorov–Smirnov test, twosample ttest) were performed on every combination of two measurements to verify that all the measurements come from the same normal distribution. Two of them did not pass these tests and were excluded from Fig. 4. We suspect that the data was contaminated by drifts in the experimental parameters, such as STIRAP and groundstate cooling efficiency, and calibration of the pulse spectrum.
After excluding the data that fail the distribution tests, the weighted average of all the frequencies measured on a single day is computed according to
with the weight factor w^{(i)}=1/()^{2}. The variance of the mean equals
with N as the number of scans on that measurement day.
A weighted fit of the centre frequencies from different measurement days with the weights 1/ as in Fig. 4a gives the centre frequency v=755, 222, 765, 896 kHz with a statistical uncertainty of 42 kHz.
Additional information
How to cite this article: Wan, Y. et al. Precision spectroscopy by photonrecoil signal amplification. Nat. Commun. 5:3096 doi: 10.1038/ncomms4096 (2014).
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Acknowledgements
We acknowledge the support of DFG through QUEST and grant SCHM2678/31. Y.W. acknowledges support from IGSM, J.B.W. acknowledges support from HALOSTAR and Studienstiftung des deutschen Volkes, I.D.L. acknowledges support from the Alexander von Humboldt Foundation. This work was supported by the European Metrology Research Programme (EMRP) in project SIB04. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. We thank A. Bauch for providing the maser signal, H. Schnatz for help with calibrations, H. Telle for stimulating discussions, and P. Carstens for technical support.
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The experimental setup was built by B.H., F.G., Y.W. and P.O.S. The spectroscopy laser was set up and characterized by J.B.W., N.S., I.D.L. and S.A. Y.W. and F.G. performed the experiments, calibrations, and analysed the data together with P.O.S. The numerical simulations were performed by Y.W., N.L. and P.O.S.; the analytical model was developed by N.L. and K.H. The experiment was conceived and supervised by P.O.S. All authors contributed to the discussion and interpretation of the results, and the writing of the manuscript.
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Supplementary Figure 1, Supplementary Notes 13 and Supplementary References (PDF 101 kb)
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Wan, Y., Gebert, F., Wübbena, J. et al. Precision spectroscopy by photonrecoil signal amplification. Nat Commun 5, 3096 (2014). https://doi.org/10.1038/ncomms4096
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DOI: https://doi.org/10.1038/ncomms4096
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