Progress and perspectives on composite laser-pulses spectroscopy
for high-accuracy optical clocks.
Probing an atomic resonance without disturbing it is an ubiquitous issue in physics. This problem is critical in high-accuracy spectroscopy or for the next generation of atomic optical clocks. Ultra-high resolution frequency metrology requires sophisticated interrogation schemes and robust protocols handling pulse length errors and residual frequency detuning offsets . This review reports recent progress and perspective in such schemes, using sequences of composite laser-pulses tailored in pulse duration, frequency and phase, inspired by NMR techniques and quantum information processing. After a short presentation of Rabi technique and NMR-like composite pulses allowing efficient compensation of electromagnetic field perturbations to achieve robust population transfers, composite laser-pulses are investigated within Ramsey’s method of separated oscillating fields in order to generate non-linear compensation of probe-induced frequency shifts. Laser-pulses protocols such as Hyper-Ramsey (HR), Modified Hyper-Ramsey (MHR), Generalized Hyper-Ramsey (GHR) and hybrid schemes are reviewed. These techniques provide excellent protection against both probe induced light-shift perturbations and laser intensity variations. More sophisticated schemes generating synthetic frequency-shifts are presented. They allow to reduce or completely eliminate imperfect correction of probe-induced frequency-shifts even in presence of decoherence due to the laser line-width. Finally, two universal protocols are presented which provide complete elimination of probe-induced frequency shifts in the general case where both decoherence and relaxation dissipation effects are present by using exact analytic expressions for phase-shifts and the clock frequency detuning. These techniques might be applied to atomic, molecular and nuclear frequency metrology, mass spectrometry as well as precision spectroscopy.
- I Introduction
- II Ramsey and Hyper-Ramsey interrogation schemes
- III Composite Ramsey spectroscopy with phase-step protocols
- IV Protocols based on free evolution time combinations
- V Composite laser-pulses protocols robust against dissipation
- VI Implementation in quantum metrology
- VII Conclusion and perspectives
- VII.1 Envelopes and phase for the HR protocol
- VII.2 Envelopes and reduced elements for the GHR protocol
- VII.3 Time-dependent components of the rotation matrix
The history of very high precision spectroscopy started in the 1930s with the atomic and molecular beam resonance method proposed by I.I. Rabi to improve the resolution of nuclear moment measurements Rabi et al. (1938). The original experiment was based on a combination of inhomogeneous fields and a rotating field to flip nuclear spins and measure the magnetic moment of the species. With enough interaction time and under a quasi-resonant irradiation, coherent oscillations can be achieved between targeted quantum states which are manipulated by an external coherent field with tunable amplitude and frequency. In such a case, the final two-level occupation probabilities can be controlled with an adequate selection of a radio-frequency field and pulse duration. By scanning the frequency of the electromagnetic excitation around the exact resonance, a narrow spectroscopic transition is observed which can be used to obtain a very narrow discriminator, stabilizing the frequency of a local oscillator on the atomic frequency for instance, thus achieving an atomic clock. The Rabi technique provided plenty of information not only on atomic and molecular structure, but also on nuclear properties Rabi et al. (1939).
To improve the frequency resolution, Ramsey designed a scheme, where he replaced the single oscillatory field by a double microwave excitation pulse separated by a free evolution time Ramsey (1950). The probing electromagnetic field perturbation on the atomic transition itself was reduced by averaging the probe induced frequency-shift over the entire sequence of pulses separated in space or time Ramsey (1956). Such a technique has drastically impacted time and frequency metrology with microwave atomic clocks since the 1950s Essen and Parry (1955); Ramsey (1990); Vanier and Audoin (1989). This protocol also provides the highest resolution for evaluation and reduction of systematic frequency shifts perturbing an atomic transition.
Atomic optical clocks are today recognized to be ideal platforms for highly accurate frequency measurements, leading to very stringent tests for physical theories and variations of the fundamental constants with time, and also for quantum simulation investigations, as reviewed in Ludlow et al. (2015). Depending on the selected atomic species used to achieve stable and accurate optical frequency standards, single trapped ion clocks Rosenband et al. (2008); Margolis (2009); Chou et al. (2010) and neutral atoms lattice clocks Ye et al. (2008); Derevianko and Hidetoshi (2011); Katori (2011) have been characterized over many years, reducing systematic uncertainties to a value surpassing current microwave atomic frequency standards. Very long storage time of Doppler and recoil-free quantum particles have been obtained using laser cooling techniques and a relative accuracy level below will most likely be achieved in the near future. This uncertainty reduction will be obtained thanks to a combination of technological advances, but also to the development of ad-hoc protocols.
The present work reports the recent advances on the development of those protocols, where the interrogation process is composed by a sequence of laser pulse. The basic idea is to drive the quantum system by a sequence of pulses whose composite action produces the planned target state. A tuning of the pulse parameters leads to the compensation of the quantum imperfections. Few composite pulse schemes were inspired by nuclear magnetic resonance (NMR) Levitt (1982) and quantum information processing Vandersypen and Chuang (2005); Braun and Glaser (2014). As tested in the experiments of refs. Taichenachev et al. (2009); Yudin et al. (2010); Huntemann et al. (2012a); Huntemann et al. (2016); Hobson et al. (2016), this approach improves the performance of optical clocks based on ultra-narrow atomic transitions, by relaxing the sensitivity to clock interrogation disturbances.
Even if a clock optical transition has a very narrow linewidth, the clock interrogation process may limit the final accuracy. The limitations have different sources as the laser probe frequency/intensity instabilities, light-shifts associated to the very weak excitation of additional optical levels of the atom/ion, decoherence and relaxation of the probed atomic system. In weakly allowed transitions with fermionic species, the frequency light-shift is usually small and does not represent an important contribution at the relative level of accuracy presently reached Hinkley et al. (2013); Nicholson et al. (2015). However, for clocks operating on strongly forbidden transitions and very long natural lifetimes, or for clocks using less stable local oscillators, shorter and more intense pulses are a necessity and may generate an important light-shift of the clock transition. These light-shifts represent a non-negligible issue for clocks either based on a single trapped ion, or on bosonic neutral atoms with forbidden dipole transitions activated by mixing a static magnetic field with a single laser Taichenachev et al. (2006); Barber et al. (2006), magic-wave induced transition in even isotopes Ovsiannikov et al. (2007), or an E1-M1 two-photon laser excitation Santra et al. (2005); Zanon-Willette et al. (2006, 2014). In order to eliminate those systematic frequency-shift induced by the probe laser below the accuracy level, it is necessary to develop new and very robust spectroscopic techniques schemes.
This elimination of systematic frequency shifts based on new spectroscopic techniques is characterized by an important historical evolution. In NMR, the key issue is to tackle systematic effects responsible for imperfect rotations of the nuclear spins, because of the use of a non uniform electromagnetic field for instance Levitt (1982). Composite rotations manipulating the quantum system have been extensively developed to get rid of dual imperfections from pulse length error and resonant offset detuning. The composite pulse approach has been also applied in quantum computation to correct imperfect operations on qubits. It was theoretically investigated for a scalable quantum computer, based on trapped electrons in vacuum, where qubits are encoded in the external (cyclotron motion) and internal (spin) degrees of freedom Stortini and Marzoli (2005). This approach allowed manipulation of the cyclotron motion without modifying the spin evolution. In Dunning et al. (2014) several sequences of NMR-type composite pulses were applied to manipulate a thermal cold atom cloud for interferometric applications. In Lin et al. (2016) composite pulse sequences have been applied to laser and microwave excitation of trapped ions to produce entanglement. In ref. Vitanov et al. (2015) the Ramsey interrogation of ytterbium trapped ions was modified by adding a central off-resonant approximate pulse, in a way reminiscent of the spin-echo technique. Playing with the relative detuning between applied pulses, this composite-like method eliminated small-to-moderate fluctuations in detuning, thereby greatly enhancing the fringe contrast in the presence of laser detuning drifts.
The optical-clock protocols are the results of a theoretical effort of deriving ad-hoc time-dependent Hamiltonians compensating the clock limitations listed above. Historically the first clock synthetic Hamiltonian is the well-known Ramsey protocol with separate oscillating fields creating an interference pattern in the clock spectral response. Within the last few year, some new protocols were derived in order to bypass the limitations previously listed and also to match the potential high accuracy of the optical clock new generation. In analogy with NMR techniques presented above, a composite pulse Ramsey (R) spectroscopy, denoted as hyper-Ramsey spectroscopy (HR), modified hyper-Ramsey spectroscopy (MHR) or generalized hyper-Ramsey (GHR) including laser phase steps have been introduced in frequency metrology in order to provide highly efficient correction of the clock light-shift induced by the probing laser Yudin et al. (2010); Zanon-Willette et al. (2015, 2016a, 2016b); Hobson et al. (2016). Additional perturbations associated with decoherence due to finite laser spectral width and atomic relaxation by spontaneous emission are also corrected by composite pulses, and in particular by those based on a phase-step during the pulse sequence Yudin et al. (2016).
This effort may be considered as a rewriting of composite pulses for optical clocks. However the goal is quite different: a resolution increase for NMR, a better accuracy for clocks. This difference implies an accurate control of state populations in the first case. The second case instead requires a precise determination of the free evolution of the quantum system. Therefore the basic tools are similar, but the final protocols different. Let’s also point out the strong similarity between the design of time-dependent Hamiltonians for optical clocks and the realization of artificial magnetism for ultra-cold neutral atoms, as reviewed in Dalibard et al. (2011). In this case the atomic center-of-mass is controlled by applying an Hamiltonian with proper space-dependence, and the successive application of different Hamiltonian may improve the target of reaching a specific final state. For the optical-clock case, different time-dependent Hamiltonians are applied within a sequence designed for a very accurate measure of the quantum state under exploration. The sequence target is to improve the accuracy and robustness of the measurement itself.
A complementary approach, denoted as synthetic protocol, is based on independent and parallel measurements of several clock-frequency shifts for different free evolution times and an appropriate combination of those measurements to generate the so-called synthetic frequency-shift, that produces the optical clock frequency with a high immunity to the laser probe perturbations. This approach, presenting large advantages in presence of an atomic decoherence, may be considered as a different composite pulse protocol, where a different parameter of the Ramsey’s interrogation scheme is properly varied: the free evolution time. The a posteriori treatment of the independent measurements applied within the synthetic protocol constitutes also an element of the phase-step protocols.
Let’s mention here few original features of the composite pulse strategy for the optical clocks. As key point in the determination of the clock proper frequency by eliminating all the shifts produced by different source, the resonant frequency of an isolated atomic or molecular transition is a symmetry point (isolated meaning without a perturbation produced by the presence of a neighbouring transition). That symmetry is associated to the unperturbed atomic response and this symmetry feature characterizes the Ramsey’s free evolution time within the pulse sequence. While the laser excitation and interrogation parts of the composite pulse sequence excite the atoms with different parameters (frequency, phase, and so on), the probe signal produces an atomic response ”averaged” over all the pulse sequence. The central symmetry point should appear in that response: the composite pulse protocol reaches this target applying properly chosen parameters.
In most atomic frequency standards, the laser probe is stabilized to the atomic transition by a frequency modulation technique. Probe-induced shifts introduce a distortion of the absorption line-shape and modify the clock operating frequency. In order to eliminate the asymmetry effect on the true clock position, a phase-step modulation was proposed and tested in Ramsey and Silsbee (1951); Morinaga et al. (1989); Klipstein et al. (2001); Letchumanan et al. (2004, 2006). The phase-step composite pulse protocol introduces a similar but more sophisticated approach: the error signal is given by the difference between two different transition probabilities in presence of phase steps within the composite pulse sequence. Here a technical feature introduced in order to improve the signal quality is translated into the construction of ad-hoc Hamiltonians.
Even if the probe laser is pre-stabilized on a high-finesse Fabry-Perot cavity, the resulting finite line-width of the laser is often limited by thermal noise Numata et al. (2004); Ludlow et al. (2007). For the clock atoms this limiting line-width represents a dephasing process which deteriorates the clock interrogation, reduces the contrast and compromises the robustness of any error signal. Fast improvements in the design of very high finesse Fabry-Perot cavities used to stabilize clock lasers should offer in the future very narrow line-widths below a few 100 mHz Jiang et al. (2011); Kessler et al. (2012); Amairi et al. (2013) for a new generation of frequency standards. Actually, this decoherence issue has been treated within the contest of the composite pulse protocols. Let’s point out that the decoherence produced by the laser line-width dephasing does not influence the free evolution time, that represents the most important element in the precise recovery of the clock transition frequency. Instead it influences the atomic evolution within the excitation periods. The operation of most composite protocols is heavily compromised by the decoherence presence. However the construction of ad-hoc Hamiltonian protocols produces a very large reduction of its role, greatly compensating the laser probe shifts and the dependence on the laser excitation parameters, i.e., intensity and interaction time. This is an important result because decoherence are usually considered as a strong limit on the reachable accuracy, and because the decoherence role is circumvent by Hamiltonian interactions. This result is a part of the present large interest into the control over dissipative processes and to the realization that the coupling to the environment can be manipulated to drive the system into desired quantum states Breuer (2007); Müller et al. (2012).
This work reviews the theoretical and experimental efforts performed so far on the optical clock composite pulse schemes for a robust compensation of features such as light-shifts, laser probe frequency and intensity induced instabilities, decoherence and relaxation. Section II, after introducing the wave-function formalism for a two-level atomic system in coherent interaction with a laser, investigates the NMR-like composite pulses and their simplest R and HR composite pulse counterparts. The benefits and limits of the R and HR schemes are presented. Section III emphasizes the laser phase manipulation of each individual pulse through laser steps as a key parameter to improve clock operations. Probing schemes like HR, MHR, GHR containing phase-steps are described. Section IV presents optical Bloch-equations describing coherent interaction between laser and atoms, including several dissipative processes disrupting clock operation. It is shown how the combination of GHR error signals can provide immunity to both decoherence and relaxation. Section V discusses the synthetic frequency approach. A recent experimental modification of the Ramsey configuration, denoted as auto-balanced Ramsey spectroscopy, is also briefly discussed. Section VI reports two recent experimental implementations based on HR and MHR composite laser-pulses protocols on a single Yb ion Huntemann et al. (2012a) and on Sr bosonic atoms optically trapped at a magic wavelength Hobson et al. (2016), respectively. Conclusions and perspectives terminate our review.
Ii Ramsey and Hyper-Ramsey interrogation schemes
A frequency standard, shown in Fig. 1, requires locking and stabilizing the phase or the frequency of an external oscillator to some atomic or ionic transition by means of laser spectroscopy. Higher frequencies and ultra-narrow lines improve measurement precision. This is the reason why the field of frequency metrology has moved over several decades from microwave transitions to optical narrow lines Ludlow et al. (2015). Weakly allowed or forbidden clock transitions are now widely investigated for the next generation of high-accuracy frequency standards based on a single trapped ion Margolis (2009) and neutral atoms in optical lattices Katori et al. (2003); Ye et al. (2008); Derevianko and Hidetoshi (2011). Depending on the nature of the quantum absorbers, i.e fermionic or bosonic particles Akatsuka et al. (2008), using either a single stabilized clock laser Ovsiannikov et al. (2007), a combination of static magnetic field and laser Taichenachev et al. (2006); Barber et al. (2006); Baillard et al. (2007); Kulosa et al. (2015) or several oscillating laser fields may be exploited to probe the atomic transition Santra et al. (2005); Zanon-Willette et al. (2006, 2014). In the following we consider a two-level system interacting with a single EM field without loss of generality.
A Doppler recoil-free atomic clock as depicted in Fig. 2 can be implemented as a two-level quantum system with energy splitting . We start with a typical description of coherent atom-light interaction where external perturbations like decoherence, collisions between particles and all atomic level relaxations are neglected. The two-level system is probed by a clock laser at frequency . Within rotating wave approximation (RWA), the interaction is governed by the complex Rabi frequency including the laser phase , where the th subscript will denote the number of the applied laser pulse in an interrogation pulse sequence. The clock laser detuning from the unperturbed transition frequency is:
A two-level system is a good approximation for most spectroscopic investigations, but is not suitable for high-resolution spectroscopy in the case of optical clocks. With a two-level transition, the virtual excitation of external non-resonant states leads to light shifts of energy levels Cohen-Tannoudji and Guéry-Odelin (2011). We define as the probe-induced frequency shift altering atomic energies while the probe laser is switched on. It is proportional to the laser probe intensity. It can sometimes be useful to compensate for that light shift by stepping the laser frequency during pulses by a fixed amount . Therefore we introduced as the effective total detuning:
The above equation introduces the residual uncompensated frequency shift as the difference between the external laser probe-induced frequency shift and the laser frequency step used to cancel it (see Fig. 2). Because the and are very small compared to the detuning of the non-resonant states, is constant over the whole clock interrogation process if the probe laser intensity is constant.
It is important here to stress the operational difference between a laser spectroscopy experiment and an optical (or microwave) clock. In a spectroscopic investigation, the frequency is controlled independently from the atom (or sample) to be explored. Within a clock, a servo-loop system is applied as depicted in the upper left of Fig. 1 producing the probe laser oscillation at clock frequency . The purpose of the servo-loop system, including the composite pulse interrogation, is to compensate light shifts and produce , that is a zero offset . An imperfect (or real!) clock operates with an arbitrary small offset, or equivalently the atoms are probed by a laser which misses resonance, that is . In the following we investigate several protocols aiming to cancel even in presence of a probe-induced shift correction different from zero.
ii.1 Single or adjacent pulses: Rabi interrogation and NMR-like composite pulses for robust population transfer
The superposition of clock states induced by a laser (or RF) pulse is described by the following linear combination:
where and are probability amplitudes related to states and respectively. The parameter is here defined as the effective pulse area :
with being the laser pulse duration and where we introduce a generalized Rabi frequency for convenience. If the total atom-laser interaction time is shorter than the damping times for the dissipation mechanisms discussed in Sec. IV, clock state dynamics are described by the following set of Schrödinger’s equations:
Using the solution of Schrödinger’s equation, the matrix solution for transition amplitudes can be written as:
Applying the above matrix with initial conditions , a final complex amplitude is obtained, leading to the well-known Rabi transition probability :
In Rabi’s original experiment Rabi et al. (1938, 1939), a thermal molecular beam passes through a coil excited by a radio frequency (RF) field with interaction time . The time of interaction and the field amplitude are chosen such that the product , a so-called pulse, corresponds to a pulse area. If the radio frequency is tuned to the transition, at time all particles with are detected in their excited state. In this configuration, the single field has to be perfectly controlled and homogeneous in the interrogation zone to achieve good sensitivity. Fig. (a)a shows the Rabi spectrum associated with the single pulse excitation scheme obtained by scanning the frequency detuning. As shown by the Fourier transforms of Fig. (a)a, the Rabi line-shape exhibits a decreasing width with increasing excitation time. The resolution of this spectroscopy is limited by the flight time of particles through the coil. The FWHM width of a pulse is (in Hz). An increase of the duration is possible to a certain extent, but for experimental reasons, it becomes difficult to maintain the appropriate microwave field for long periods or large pulse areas. It is important to note that in this case, the clock frequency-shift of the transition probability is always linearly dependent on the residual uncompensated part of the light-shift. In modern optical ion and lattice clocks, the Rabi spectroscopy has been mostly implemented to obtain very high resolution measurements of various metrological clock transitions with low systematics Ludlow et al. (2015).
If several adjacent pulses are used, the single matrix of Eq. (6) is replaced by a product of several matrices in order to explore various sequences based on NMR-like composite pulse excitations. Tab. 1 reports examples of Rabi composite pulses inducing a robust population transfer between two targeted quantum states required either for offset detuning or RF field compensation Levitt (1986). The Table first column introduces a compact indication of the composite pulse composition. The lineshapes for different NMR-like composite pulses are shown in Fig. (b)b and Fig. (c)c. The corresponding contour plots in the right column of Fig. 3 evidence the optimized robustness of the population inversion by using several adjacent pulses optimized against uncompensated residual frequency-shifts in Fig. (b)b, and against important pulse area variations in Fig. (c)c.
ii.2 Pulses with an interleaved free evolution time: Ramsey and Hyper-Ramsey schemes
Within the original R spectroscopy configuration, atoms are probed by two successive pulses separated by a free evolution time T Ramsey (1950). During free evolution time T where the probe laser is switched-off, the matrix given by Eq. (7) does not contain the light-shift from external off-resonant states. The free-evolution transfer matrix reduces to:
because no additional laser frequency step is applied during free evolution time.
In the HR interrogation three pulses are applied, with a free evolution time after the first or second pulse Yudin et al. (2010), see an example in Fig. 4 with the free evolution time applied after the first pulse. Note that another composite sequence with pulse order reversal can be also used as proposed in Zanon-Willette et al. (2015). The coherent population transfer induced by such pulse sequences is given by a simple product of matrices, each of them being individually tailored in frequency, duration and phase. If pulses are labeled by , the corresponding transition probability is given by:
where we introduce . The composite pulse sequence may
includes a laser phase-step during each pulse which can be manipulated to control the resonance shape. This applies to the HR- protocol of Fig. 4 where the pulse sequence includes a laser phase sign inversion during the second interaction. The transition probability describing the coherent population transfer between atomic states depends on pulse areas and phase jumps over the entire laser probing sequence. By scanning the detuning between the laser and the two-level resonant frequency, a HR resonance
is constructed containing information about perturbations induced by the laser probe on the line-shape.
Ignoring for simplicity the additional phase step, the expression can be written in a compact form as:
where is the clock frequency detuning during free evolution time. The envelopes are given by
and the phase
The envelopes , and the phase driving the resonance amplitude are given in Appendix VII.1. This formula is valid only for phase jumps within the pulse sequence. However note that the general form valid for arbitrary phase steps, described in the following by Eq. (28), has the same structure of the above one. Using Eq. (11), and its generalization in presence of phase-steps derived in ref. Zanon-Willette et al. (2015), the population transfer efficiency and the frequency-shift affecting the resonance can both be evaluated accurately under various experimental laser pulse conditions including R and HR schemes Ramsey (1950); Yudin et al. (2010).
When the second pulse area vanishes, i.e , i.e. , and , , , the generalized transition probability takes the following form:
with in Eq.(11), and for the phase
By applying a trigonometrical transformation, we recover the standard expression for the transition probability derived by Ramsey in 1950 for a spin interacting with a radio-frequency field Ramsey (1950, 1956) as:
This expression established by Ramsey Ramsey (1950) was the initial version of the method of separated oscillating fields in molecular beams.
A remarkable information is the frequency-shift generated by Eq. (14) that determines the central fringe position sensitivity to a detuning fluctuation. From a geometrical point of view, this Ramsey phase-shift is exactly two times the Euler angle accumulated by a Bloch’s vector projection of rotating components in the complex plane using a two dimensional Cauley-Klein representation of the spin 1/2 rotational group Bloch and Rabi (1945).
The separated oscillating fields method invented by Ramsey and presented in Fig. (a)a
effectively reduces the clock sensitivity to light-shift effects. Interference fringes in the population transfer, as shown in
Fig. (a)a, are observed versus the clock laser detuning and the central feature is used to lock
the local oscillator to the atomic or molecular transition. It has been widely applied in high precision measurements for atomic clocks
based on atomic beams crossing a microwave cavity twice Vanier and Audoin (1989); Ramsey (1990); Essen and Parry (1955) and was extended to Zacharias-type
fountain geometries where laser cooled atoms are thrown up vertically Kasevich et al. (1989); Clairon et al. (1991). In the last device, a cold
atomic cloud experiences a first pulse () during its rise when passing through a microwave
resonator, then freely evolves without light interaction during its free launch and free fall. Finally it undergoes a second
pulse in the same cavity before detection.
The resolution of such a clock configuration is only limited by the atomic cloud time of flight T between microwave interactions
Campbell and Phillips (2011). The resonance width (in Hz) is when . The reduction of the Ramsey clock frequency-shift
reported in Fig. (a)a by the factor was observed in molecular beam experiments
with RF fields Shirley (1963); Fabjan and Pipkin (1972); Code and Ramsey (1971); Greene (1978).
Despite its great resolution, the original Ramsey method remains too sensitive to perturbations from the optical probe laser field itself. Some spatial laser beam configurations where proposed in the 1980’s by Bordé canceling first-order Doppler-shifts to observe optical Ramsey fringes Bordé (1983). However, if external AC Stark-shifts are not reduced or potentially eliminated, the Ramsey central fringe is pulled away from resonance and the fringes themselves become asymmetric around the maximum Marrocco et al. (1998). This asymmetry was observed for single ion clock using an ultra-narrow electric electric octupole (E3) optical transition Margolis (2009) and in some alkaline-earth neutral bosonic clocks with completely forbidden transitions Ludlow et al. (2015).
Few techniques were proposed to solve such a problem. A spectroscopic laser probe configuration based on a pulsed EIT (Electromagnetically Induced Transparency)/Raman two-photon excitation suggested in 2006 introduces internal ac Stark-shifts of a three-level system in order to counteract the light-shift contribution from external off-resonant states. This approach restores the Ramsey fringes at the unperturbed clock frequency Zanon-Willette et al. (2006). A modified Ramsey method was also proposed to cancel the overall light-shift with deliberate application of a laser frequency step during light pulses Taichenachev et al. (2009). However, all these methods require exact knowledge of the light-shift correction or an excellent control of laser power variations to efficiently compensate frequency shifts.
The HR spectroscopy was proposed in 2010 to relax the constraint on laser power control and to eliminate the probe induced frequency-shifts Yudin et al. (2010). The scheme is based on pulses that can have different lengths, frequencies, and possibly phase inversion. The initial version is based on a sequence of two different pulses, a first pulse of length as in the Ramsey’s technique, and a second pulse of length , tailored in two parts and . A laser frequency step for a basic pre-compensation of the light-shift is also introduced during pulses to correct the expected external light-shift from off-resonant states of the probed two level system. The transition probability describing the HR resonance, reported in Fig. (b)b, has a fringe inversion at the resonance because of the pulse area. A discriminator slope to lock the laser frequency is obtained by a phase modulation on one of the pulses. Even if the phase is discussed in detail within the following SubSection, let’s point out here the presence of a small sensitivity to pulse area variation, limiting the method’s efficiency. This result is shown by the shaded area on the right panel of Fig. (b)b. That sensitivity can be compensated by applying a laser phase step during the -length pulse, as in the HR- composite pulse of Fig. 4. Therefore the HR- sequence can be seen as an echo pulse Yudin et al. (2010). The important result is a strong non-linear cubic dependence of the central fringe frequency shift with the uncompensated light-shift , see on the right in Fig. (c)c. This drastically reduces residual uncompensated light-shift contribution to a very low order.
ii.3 Canonical form of the clock frequency-shift
This subsection examines the clock-frequency shift which impacts the Ramsey interference pattern as described by Eq. (11). A good approximation of the shift for the central fringe extremum is given by the following simple relation:
It is thus possible to eliminate the frequency shift of the central fringe by engineering with special choices of laser step frequency, pulse duration, and phase inversion.
A more sophisticated expression for the composite clock-frequency shift than Eq. (17) is needed if the line-shape is perturbed by weak distortions due to decoherence, as derived in Section V.3. The central fringe frequency-shift is thus calculated by applying a first-order expansion to Eq. (11) around the unperturbed frequency clock detuning of the resonance. The result takes the form:
where is the partial derivative with respect to the unperturbed clock detuning . The main term is modified by two high-order phase-shifts as follows:
The high-order expressions given by Eq. (20a) and Eq. (20b) account for a possible distortion of the line-shape when the free evolution time T is not very large compared to each pulse duration (). This is shown in Fig. 6 for the HR- protocol and various Ramsey free evolution times and a fixed pulse duration.
and in an alternative expression,
to be compared to the following one for the three-pulse scheme.
In the case of three different pulse areas, the HR and HR- phase shifts are derived in Yudin et al. (2010); Zanon-Willette et al. (2015). Following Abramowitz and Stegun (1968), they can be rewritten into a closed form solution as:
where the reduced notation of Eq. (57) was inserted within the last term.
For the R, HR and HR- protocols the clock frequency phase-shift based on Eq. (18) is plotted on the right column of Fig. 5 versus the uncompensated residual frequency shift . Here and in following figures they are plotted in the case of during each laser pulse.
In the R case, the clock frequency shift is linearly dependent as shown in
Fig. (a)a. As in Fig. (b)b for the HR technique based
on the combination and , to be inserted in Eq. (21), the clock becomes non linear. A relative variation of for all pulse
areas affects the protocol and requires a careful control to avoid significant shifts. Fig. (c)c presents the clock frequency-shifts of the HR- technique including a laser phase inversion coupled with a pulse. By inserting in Eq. (23) the pulse areas
and while fixing the intermediate laser field phase
to (), a very good compensation of the relative pulse area variation is obtained.
These plots clearly show that composite pulses are really efficient to extend the region where both pulse area variations and residual
light-shifts are simultaneously rejected to a very low level of perturbations. HR spectroscopy is now implemented in single ion clocks
based on ultra-narrow transitions Huntemann
et al. (2012a); Huntemann et al. (2016), as presented in Section VI.
Fig. 6 shows the comparison between clock frequency shifts computed from Eq. (17), Eq. (18) and the numerical tracking of the extremum of the central fringes. The high-order corrections given by Eq. (18) are in very good agreement with numerical trackings for all free evolution times. On the contrary, differences exist between the results based on Eq. (17) analytical expression and those of the numerical tracking, becoming more pronounced when the free evolution time is comparable to the pulse duration.
Iii Composite Ramsey spectroscopy with phase-step protocols
In most atomic frequency standards, the laser probe is stabilized to the atomic transition by a standard frequency modulation technique applied at the half-height of the central Ramsey fringe. But if some AC Stark-shifts are present due to non-resonant atomic states, the line-shape is distorted and shifted from the correct clock frequency, leading also to errors and instabilities in the frequency lock point. The proper strategy to eliminate the asymmetry effect on the true position of the central fringe and to generate a robust and stable lock-point for the local laser probe oscillator is the phase-step modulation, as proposed and tested in Ramsey and Silsbee (1951); Morinaga et al. (1989); Letchumanan et al. (2006, 2004). In addition this phase modulation technique produces an error signal with enhanced immunity to potential offset variations Morinaga et al. (1989); Klipstein et al. (2001). A corresponding composite pulse approach is characterized by the presence of appropriate phase-step modulations within specific areas of the pulse sequence. The measured signal is based on a difference of properly chosen generalized transition probabilities following the application of the phase-step modulation. The signal becomes anti-symmetric with respect to the clock laser detuning . This Section presents the composite pulse phase-step protocols of Table 2. Frequency lock points generated from these configurations are well protected against large laser pulse area variations and potential errors in the frequency shift compensations, because they decouple the unperturbed frequency measurement from laser intensity variations Hobson et al. (2016); Zanon-Willette et al. (2016b).
iii.1 Transition probabilities including laser phase-steps and error signal definition
Within a GHR sequence, the second or first Ramsey pulse is divided into subsections with individual manipulation of frequency, duration and laser phase, as shown in Fig. 7 for two types of laser pulsed sequences, GHR and GHR. The calculations of different error signals require to explicitly include the laser phase dependence, produced by the phase steps, within the matrix elements in Eq. (7). The GHR transition probability is expressed by:
where the envelopes and of the two sequences are given in Appendix VII.2, taking into account both laser phases and initial atomic preparation. The composite phase-shift represents the atomic phase accumulated by the wave-function during the -th laser interrogation sequence. It is expressed for two different pulse sequences as follows.
For the GHR sequence:
And for the GHR sequence:
in both cases, reduced matrix components are given in Appendix VII.2.
In the case of phase-step protocols, the dispersive-shape of the error signal is computed by taking the difference between two spectroscopic signals with opposite phase and as:
It may be written as
with A and B related to and by Eq. (12).
This laser frequency stabilization scheme synthesizes an anti-symmetric error signal, i.e., a dispersion line-shape, to lock the laser frequency to the center of the unperturbed clock transition. The frequency lock point shift from the error signal due to an imperfect light-shift compensation is directly given by the relation:
iii.2 Error signals of R and HR schemes
This Subsection, as well the next one, reviews the robustness of different dispersive errors signals to some residual light-shifts and pulse area variations. Some relevant phase-step protocols are reported in Table 2 for application in R spectroscopy Ramsey and Silsbee (1951); Letchumanan et al. (2004, 2006) and HR spectroscopy Yudin et al. (2010); Huntemann et al. (2012a); Huntemann et al. (2016).
The first laser phase-step configuration based on the R protocol in Tab. 2 was initially proposed in Ramsey and Silsbee (1951). Following Eq. (27), the error signal is produced by combining two Ramsey transition probabilities in presence of phase shifts
This approach leads to dispersive-like resonance lineshapes with increased sensitivity to detect the clock resonance frequency. This phase-step modulation was applied to single ion clock devices in order to produce better control of the frequency lock point stabilizing the local laser oscillator Letchumanan et al. (2004, 2006). The error signal of the R phase-step protocol is plotted on the top left part Fig. (a)a. Its associated dispersive signal is plotted for several values of the uncompensated residual light-shift on the top right part.
The error signal of the second phase-step configuration based on the HR- protocol, presented in Fig. (b)b, is based on the same phase-steps in presence of an additional phase reversal of the laser field during the intermediate pulse. Following Eq. (27), the error signal is
The main advantage of such a protocol is to generate a frequency lock point which is driven by a cubic non-linear
sensitivity to the uncompensated residual light-shifts leading to a much better control of the frequency discriminant as shown on the right
of Fig. (b)b.
The HR phase-step protocol was proposed in Yudin et al. (2010) and experimentally implemented on a single ion clock in Huntemann et al. (2012a) to strongly reduce the residual probe-induced frequency shift by four orders of magnitude. Recently, using this spectroscopic technique Huntemann et al. (2016), the single ion clock achieved a systematic relative uncertainty of probe induced shifts. This protocol still suffers from small residual light-shifts when exploring a wider range of uncompensated frequency offsets. The HR protocol has a residual uncompensated frequency-shift of around 1-2 mHz over a 220 mHz residual offset which may compromise the access to a clock fractional accuracy below .
iii.3 Error signals of MHR and GHR schemes
Spectroscopic schemes using different phase-step protocols have been recently introduced in order to completely eliminate residual
light-shift corrections on the central fringe over large uncompensated residual light-shifts.
These are the MHR Hobson et al. (2016) and GHR Zanon-Willette
et al. (2016b) schemes.
These new dispersive error signals are centered at the unperturbed atomic resonance with steep discriminants which are impervious
to variations in laser probe induced clock frequency shifts.
The error signals are built from a combination of different transition probabilities as in Hobson et al. (2016); Zanon-Willette et al. (2016b):
The MHR protocol described by Eq. (35a) is based on a superposition of two HR- transition
probabilities. For error signal generation, the phase-step modulation is obtained by interleaving a HR- protocol from
Tab. 2, where during the first pulse, with a HR- protocol
where an opposite phase is used during the last pulse.
The calculated dispersive error signal shape is presented in Fig. (c)c along its frequency
lock point response to residual probe-induced shifts.
This non standard protocol was the first to synthesize an error signal yielding full immunity to residual probe light-shifts and great
robustness to pulse area errors originated from laser power variations. It has been successfully tested in a neutral atom optical lattice clock
based on magnetically induced spectroscopy Hobson et al. (2016) demonstrating suppression of a sizable probe Stark
shift to below even with very large errors in shift compensation.
The GHR() and GHR() protocols from Tab. 2, proposed in Zanon-Willette et al. (2016b) and shown in Fig. (a)a and Fig. (b)b, respectively, use a single either or a phase-step modulation during the intermediate pulse. The associated error signals computed using Eq. (35b) are presented in the same figure. The plots of their lock point sensitivity shown on the right in Fig. (a)a and Fig. (b)b, evidence that they are fully protected from errors in residual probe-induced light-shifts. Another stabilization scheme can be generated by combining the error signals of the GHR() and GHR() protocols, presenting opposite slopes of the error signal. This hybrid scheme denoted GHR() is defined by the following normalized difference between two error signals:
iii.4 Robustness of error signal slopes
This subsection presents parameters useful to setup optimal working conditions in interrogation
protocols in order to achieve a very efficient lock of the local oscillator to the unperturbed optical clock transition Riis and Sinclair (2004).
For simplicity, and because out of scope of this work, we neglect statistical fluctuation in the measured signals produced by external time-dependent perturbation or phase noise contribution Kabytayev et al. (2014); Chen et al. (2012).
We study the stability and robustness of error signal slopes at against residual uncompensated light-shift associated to small variations of pulse areas. The frequency stability scales as the inverse of the slope and a reduced slope results in a clock instability. Error signal slopes against residual shifts for all protocols are shown in Fig. 10 for a pulse area variation. Compared to standard error signals based on R protocol in Fig. (a)a and HR protocol in Fig. (b)b, the new protocols expand the possible range for the uncompensated shift, i.e., the range between two zero crossings of the error slope. Both residual uncompensated offsets and pulse area variations are directly transferred to a slope reduction of error signals with no change in their frequency lock point. However the MHR protocol shown in Fig. (c)c presents a small shape asymmetry and a tiny deviation of the maximum slope value from .
Note that the GHR protocol, shown in Fig. (f)f, eliminates unstable operation by an undesired sign inversion of the slope when the uncompensated light shift is too large.
If the light shift is not constant over the pulse sequences, and therefore a large residual offset is present, a strong distortion or rotation of the error signal slope is originated (see for example Fig. (b)b, Fig. (c)c and Fig. 9) and it may compromise the lock point stability and degrade the clock operating condition. Then, the laser compensation step should be checked and steered to the point where .
Various implementations have been experimentally tested. For the single ion clock frequency standard Huntemann et al. (2012a), a stabilization using the HR phase-step protocol scheme was combined with a second interleaved servo system where Rabi spectroscopy with the same probe light intensity is used. Then, the frequency difference between the two interrogation techniques was used to control laser frequency step . This method ensures that slow drifts of the light-shift will not degrade its suppression. It is also possible to use some clock frequency-shift symmetries offered by the MHR protocol with the uncompensated part of the probe shift to design an efficient steering process to Hobson et al. (2016).
Iv Protocols based on free evolution time combinations
This section reviews some combinations of R and HR interrogation protocols with different free evolution times. The target is to generate a clock frequency-shift strongly protected against residual light-shifts over larger offset clock detunings. Some non-linear clock frequency-shifts can be synthesized specifically to be extremely robust against decoherence and relaxation.
iv.1 Synthetic frequency protocol for HR spectroscopy
The synthetic shift technique reduces the sensitivity to pulse area variations and extends the non-linear efficiency of HR protocols to larger uncompensated residual light-shifts. It is based on both independent and parallel measurements of several clock-frequency shifts for different free evolution times and careful combination of those measurements to generate the so-called synthetic frequency-shift. This approach is more robust than previous phase-step locking protocols, reducing both residual uncompensated light-shifts and laser power variations even in presence of decoherence.
The synthetic frequency method, discussed in detail in Yudin et al. (2016), is based on a polynomial serie expansion of the clock’s residual frequency shift on its dependence to free evolution time T under frequency stabilization:
The coefficients depend on pulse parameters (durations, amplitudes, phases) and uncompensated frequency shift . This method was originally designed to allow suppression of the black-body radiation shift experienced by atomic clocks Yudin et al. (2011), but it can easily be extended to handle arbitrary systematic shifts (Stark shift, Zeeman shift, and so on).
The basic idea is to build a synthetic frequency using multiple HR sequences with specific choices of free evolution times (but same pulse parameters, assuming we can enforce exact ratios between Ramsey free evolution times) to cancel contributions up to a given order in Eq. (37). For example, using two sequences with free evolution times and , and stabilized frequencies and shifted by and , respectively, the synthetic frequency at the lowest order is defined as
where is the offset detuning ratio. Considering the case of and , it can be shows that the frequency shift can be written as:
Similarly, using three different HR sequences with free evolution times , , and , the shift of the the synthetic frequency at the following order is given by:
Following this pattern to higher orders, it results that the higher orders expansions follow binomial coefficient laws. Using Eq. (56c) for the HR phase shift, the clock frequency shift is calculated on the basis of Eq. (18). Thus, under , the calculations show the following general character of dominating dependencies on :
without synthetic frequency approach, and
for the synthetic frequency approaches at different orders.
Fig. 11 reports the calculations for the above quantities. For the synthetic frequencies, higher-order (more than cubic) non-linearities appear. This character is not changed under variations of , , and T, i.e., we do not need the rigorous condition =. Because in real experiments the value of can be controlled only at the level of 1-10, this method can be very successful in atomic clock implementations.
It is also very important to notice that the combination of the synthetic frequency protocol applied to the HR- scheme is quite stable to decoherence. Indeed, Fig. 12, showing graphs for the above clock shifts against the ratio in the presence of decoherence described by the parameter, demonstrates strong and robust suppression of the shift.
The main advantage of the synthetic method is to reduce the decoherence perturbation without destroying an efficient light-shift compensation using the HR- protocol even when all pulse areas are modified by . The synthetic frequency protocol is also better by one to three orders of magnitude than MHR and GHR() protocols when laser-induced decoherence is considered.
Apart from the combination with Ramsey and hyper-Ramsey spectroscopy for two-level systems, the synthetic frequency protocol can be applied to the Ramsey spectroscopy of coherent population trapping (CPT) resonances (e.g., see Zanon et al. (2005); Chen et al. (2010); Blanshan et al. (1991)). Note that CPT clocks are one of the prospective variants of compact RF clocks with relatively valuable metrological characteristics. Because the probe-induced shift for CPT-Ramsey resonance satisfies the general dependence of Eq. (37) on the free evolution interval T (see Hemmer et al. (1989), where the dependence was found), one can expect good efficiency of the synthetic frequency protocol in this case too. The same approach can also be applied to so-called pulsed optical pumping (POP) clocks Micalizio et al. (2012). All these examples demonstrate the universality of the synthetic frequency protocol, which can be used in any type of clocks based on Ramsey spectroscopy.
iv.2 Auto-balanced Ramsey spectroscopy
A variant of the synthetic protocol approach, denoted as auto-balanced Ramsey spectroscopy, was very recently presented in ref. Sanner et al. (2017). It is based on the combination of two Ramsey sequences, with short and long free evolution times, whence T is the control parameter. The originality of the method is the use of two interconnected control loops. The first feedback loop uses the error signal provided by the short Ramsey sequence to lock an additional phase step correction between the Ramsey pulses, while the second loop locks the mean frequency from the error signal of the long Ramsey sequence. Notice that this sequence contains the phase step correction as an additional control parameter. To demonstrate the efficiency of the auto-balancing approach, the Yb clock transition was experimentally probed with 3 different technical pulse defects. The auto-balanced Ramsey probing technique was thus able to recover the undisturbed clock transition against pulse areas delivered with 97 of the nominal intensity for the last 3 ms of their 15 ms on-time, against weak phase step excursion of the local laser oscillator and finally against phase lag. A final reduction by about of the light shift was experimentally observed in an Yb ion clock operating on the E3 transition.
V Composite laser-pulses protocols robust against dissipation
This Section introduces new protocols dealing with the decoherence associated to the finite line-width of the probe laser, which disturbs the clock interrogation by reducing the contrast while also compromising the robustness of any error signal. For the aimed relative accuracy, the decoherence induced by clock laser line-width degrades the robustness of clock lock points for GHR protocols of the last Section. This issue will be mitigated by fast improvements in the design of very high finesse Fabry-Perot cavities used to stabilize clock lasers, thus offering very narrow line-widths below a few 100 mHz Jiang et al. (2011); Kessler et al. (2012); Amairi et al. (2013) for a new generation of frequency standards. However new composite pulse protocols represent an alternative approach to this issue. Relaxation processes for the clock populations are also included into the presented analysis. Notice also that the MHR and GHR() protocols are not fully equivalent in the presence of decoherence produced by the finite laser linewidth Hobson et al. (2016); Zanon-Willette et al. (2016b).
v.1 Matrix solution to optical Bloch equations
The relevant analysis has to be performed within a formalism based on the two-level density matrix where the atomic decoherence can be treated properly Tabatchikova et al. (2013, 2015). The direct numerical integration of density matrix equations includes dephasing of the off-diagonal elements Allen and Eberly (1975); Berman and Malinovsky (1975) and relaxation terms of populations, in order to describe dissipative processes such as spontaneous emission, dephasing and decoherence within a closed two-level configuration. The atomic evolution includes a decoherence term , a spontaneous emission rate denoted and a population difference relaxation induced by collisions. The Bloch variables , and are used to describe the atomic excitation after the -th optical pulse of the composite pulse sequence. The general set of time-dependent optical Bloch equations is given by Zanon-Willette et al. (2017):