Combined error signal in Ramsey spectroscopy of clock transitions

We have developed a universal method to form the reference signal for the stabilization of arbitrary atomic clocks based on Ramsey spectroscopy. Our approach uses an interrogation scheme of the atomic system with two different Ramsey periods and a specially constructed combined error signal (CES) computed by subtracting two error signals with the appropriate calibration factor. CES spectroscopy allows for perfect elimination of probe-induced light shifts and does not suffer from the effects of relaxation, time-dependent pulse fluctuations and phase-jump modulation errors and other imperfections of the interrogation procedure. The method is simpler than recently developed auto-balanced Ramsey spectroscopy techniques (Sanner et al 2018 Phys. Rev. Lett. 120 053602; Yudin et al 2018 Phys. Rev. Appl. 9 054034), because it uses a single error signal that feeds back on the clock frequency. The use of CES is a general technique that can be applied to many applications of precision spectroscopy.


Introduction
Atomic clocks based on high-precision spectroscopy of isolated quantum systems are currently the most precise scientific instruments, with fractional frequency instabilities and accuracies at the 10 −18 level [1][2][3][4][5]. Frequency measurements at this level enable improved tests of fundamental physics, as well as new applications like chronometric geodesy [6,7].
For many promising clock systems, probe-field-induced frequency shifts can limit the clock frequency instabilities and accuracies. In the case of magnetically-induced spectroscopy [8,9], ac-Stark shifts can limit the achievable clock stability, and for ultranarrow electric octupole [10] and two-photon transitions [11,12], the large off-resonant ac-Stark shift can completely prevent high-accuracy clock performance. Similarly, the large number of off-resonant laser modes present in clocks based on direct frequency comb spectroscopy [13,14] induce large ac-Stark shifts. Probe-field-induced shifts also cause instability for microwave atomic clocks based on coherent population trapping (CPT) [15][16][17][18][19][20]. Compact microwave cold-atom clocks [21,22] and hot-cell devices like the pulsed optical pumping (POP) clock [23,24] that are based on direct microwave interrogation can also be affected by probe-induced frequency shifts.
Probe-induced shifts can be suppressed through the use of Ramsey spectroscopy [25] in combination with cleverly devised modifications. In contrast to continuous-wave spectroscopy, Ramsey spectroscopy has a large number of extra degrees of freedom associated with many parameters that can be precisely controlled: the durations of Ramsey pulses 1 t and 2 t , the dark time T, the phase composition of composite Ramsey pulses [26], variations in Ramsey sequences including the use of three or more Ramsey pulses, different error signal variants, and so on. Some modified Ramsey schemes for the suppression of the probe-field-induced shifts in atomic

Theoretical model
In this section, we follow the analysis developed in [38], which we repeat here for completeness. We consider a two-level atom with unperturbed frequency ω 0 of the clock transition g e ñ « ñ | | (see figure 1), which interacts with a Ramsey sequence of two completely arbitrary pulses (with durations 1 t and 2 t ) of the resonant probe field with frequency ω: The pulses are separated by a free evolution interval (dark time) T, during which the atom-field interaction is absent (see figure 1). We emphasise that the Ramsey pulses with arbitrary durations 1 t and 2 t do not depend on the dark time T and can have an arbitrary shape and amplitude (i.e. during 1 t and 2 t an amplitude t ( ) can be an arbitrary real function), and an arbitrary phase function j(t) (e.g. the Ramsey pulses can be composite pulses, chirped pulses, and so on). In a given sequence of Ramsey measurements, the pulse shape and amplitude must be consistent from one measurement to another. We assume only one restriction: aside from phase jumps applied to generate the error signal (discussed below), the phase function should be constant during the dark time T, , as is typical for Ramsey spectroscopy. Our main goal is to develop a universal one-loop method, which allows us to stabilize the probe field frequency ω at the unperturbed frequency of the clock transition, ω=ω 0 , in the presence of decoherence, arbitrary relaxation and light shifts. For this purpose, we will use the formalism of the density matrix r , which has the following form in the basis of states gñ | and eñ | . In the resonance approximation, the density matrix components t jk r ( ) satisfy the following differential equations: Here the time dependencies Ω(t) and t d( ) are determined by the following: ( ) during the action of the Ramsey pulses 1 t and 2 t , but Ω(t)=0 and t d d =( ) during the dark time T. d á ñ is a matrix element of the atomic dipole moment, δ=ω−ω 0 is the detuning of the probe field from the unperturbed atomic frequency ω 0 , and t sh D ( ) is an actual probe-field-induced shift (see figure 1) of the clock transition during the Ramsey pulses [e.g. it can be the ac-Stark shift, which is proportional to the t 2  | ( )| and does not depend on the phase j (t)]. Also equation (3) contains five relaxation constants, { e g , e g g  , g g , g e g  , Γ}: e g is a decay rate (e.g. spontaneous) of the exited state e ; ñ | e g g  is a transition rate (e.g. spontaneous) to the ground state g ; ñ | g g is a decay rate of the ground state gñ | (e.g. due to black-body radiation and/or collisions); g e g  is a transition rate from the ground state gñ | to the exited state eñ | . Note that e g e g g =  and g e g g g =  in the case of closed two-level system, while e g e g g <  and/or g e g g g <  in the case of open system. The constant 2 e g g g G = + + G  ( ) describes the total rate of decoherence: spontaneous as well as all other processes, which are included in the parameter G  (e.g. an influence of the nonzero spectral width of the probe field). Equations (3) can be rewritten in the vector form Left part: schematic illustration of a sequence of two arbitrary Ramsey pulses (with durations 1 t and 2 t ) separated by the dark time T, during which the phase jumps a  are applied to create an error signal (10). Aside from phase jumps a  , the phase j (t) (see in equation (1)) should not be changed during the dark time T (i.e. j (t)=const), while during the Ramsey pulses 1 t and 2 t the function j (t) can have an arbitrary behavior. Right part: scheme of the clock transition g e ñ « ñ | | (with unperturbed frequency ω 0 ) interacting with the probe field at the frequency ω, where t sh D ( ) is an actual probe-field-induced shift during Ramsey pulses 1 t and 2 t .
In this case, a spectroscopic Ramsey signal can be presented in the following general form, which describes Ramsey fringes (as a function of δ) where the scalar product is determined in the ordinary way: x y x y , . Operators W 1 t and W 2 t describe the evolution of an atom during the first ( 1 t ) and second ( 2 t ) Ramsey pulses, respectively, and the operator G T describes free evolution during the dark time T. Vectors in r  and obs r  are initial and observed states, respectively.
For example, if an atom before the Ramsey sequence was in the ground state gñ | , and after the Ramsey sequence we detect the atom in the exited state eñ | , then vectors in r  and obs r  are determined, in accordance with definition For stabilization of the frequency ω we need to form an error signal, which is obtained by using data from two Ramsey sequences, one with phase jump a + between the two pulses and one with phase jump a -(see figure 1), as was proposed in [40]. Note that the phase jumps of the laser field can be easily realized in experiments by the use of an acousto-optical modulator (AOM). We mathematically describe these phase jumps a + and a -(their precise timing does not matter) by the operators F a + and F a -, respectively. In this case, let us introduce the expression of the Ramsey signal in the presence of the phase jump α, described by the operator F â As a result, the error signal can be presented as a difference is typically used. However, in real experiments, we can have a a ¹ + -| | | |due to various technical reasons (e.g. electronics) which will lead to a shift of the stabilised frequency ω in the case of standard Ramsey spectroscopy. Therefore, here we will consider the general case of arbitrary a + and ato demonstrate the robustness of CES technique, where the condition a a ¹ + -| | | |does not lead to an additional frequency shift. Next we consider the structure of the following operators: G T , F a + , F a -, and D F . The operator for the free evolution, G T , has the following general matrix form in the Liouvillian (6). The matrix elements G 11 (T), G 14 (T), G 41 (T), and G 44 (T) depend on four relaxation rates: { e g , e g g  , g g , g e g  }. In particular, for purely spontaneous relaxation of the exited state eñ | , when 0 - Operators for the phase jumps F a + and F ahave the forms As a result, taking into account equation (11), we obtain a formula for the matrix product D G T (15), the error signal (10) can be rewritten in the following form: Note that this result is the same if we apply phase jumps α ± at any arbitrary point during the dark interval T. It is interesting to note that the expression of the error signal in the presence of relaxation is formally different from the error signal in the absence of relaxation only due to the scalar multiplier e T -G , which primarily affects the amplitude, but not the overall shape of the error signal. This is one of the main specific properties of the phase jump technique for Ramsey spectroscopy that makes it robust against relaxation. Indeed, for other well-known methods of frequency stabilization, which use a frequency jump technique between alternating total periods of Ramsey interrogation T (10) does not exist. Thus, the phase jump technique has a fundamental advantage over the frequency jump technique in that it is less sensitive to relaxation. In addition, in the ideal case of a a a = -= + -, the error signal (10) can be expressed as depends only on δT.

CES protocol
In this section we demonstrate the universality and robustness of the CES technique. We use the Ramsey interrogation of the clock transition for two different, fixed intervals of free evolution T 1 and T 2 , where we have two error signals S T described by equation (18). However, for frequency stabilization we introduce the CES as the following superposition, T T T CES err obs in obs in If we apply δ=0 for operators T 1 ¡ d and T 2 ¡ d , then due to equation (17) we have D Fˆˆ. In this case, we obtain from equation (24): since the two terms inside of the square brackets cancel in equation (24). Thus, we have analytically shown that the CES method always leads to zero field-induced shift of the stabilized frequency ω in an atomic clock, 0 clock d = .
From a practical viewpoint, it is most important that the calibration coefficient cal b (equation (23)) does not depend on the values of the phase jumps a  used for error signals, or other parameters (such as: amplitude, shape, duration, phase structure j (t), shift t sh D ( ), etc) of the two Ramsey pulses 1 t and 2 t . Thus, cal b can be considered as a phenomenological parameter, which is fixed for given setup (via the relaxation constant Γ) and for given T 1,2 (via the difference T 1 −T 2 ). In the ideal case with no relaxation (Γ=0), we obtain 1 cal b = for arbitrary T 1,2 . However, in the general case, the value of cal b should be empirically determined before long-term frequency stabilization.
As we see from equation (24), to maximize the slope of S CES err d ( ) ( ) it is necessary to use the condition T 2 =T 1 . Formally we can even use T 2 =0 (with the phase jumps α ± in the virtual point between pulses 1 t and 2 t ).
However, due to technical transient regimes (i.e. in acousto-optic modulators) under switching-off/on of Ramsey pulses in real experiments, we believe that it is necessary to keep some nonzero dark time, T 0 2 ¹ , which significantly exceeds any various transient times. For example, in the case of magnetically-induced spectroscopy [8,9], the transient processes, associated with switching-off/on of magnetic field, can be relatively slow.
In figure 1, we use some abstract shapes of the Ramsey pulses to stress that our exact analytical result (25) is valid for arbitrary Ramsey pulses, and, therefore, the CES method is very robust to different technical nonidealities, which can exist in real experiments. Indeed, in real clock experiments, the Ramsey pulses have rectangular shapes (e.g. see figure 2), which, however, can be deformed due to various technical reasons (e.g. the transient regime in an AOM during switching off/on of the Ramsey pulses, some phase chirping, and so on). In the case of usual Ramsey spectroscopy and hyper-Ramsey methods [5, 27-29, 31, 34, 35], these technical causes can lead to an additional (technical) shift of the stabilized clock frequency ω, while the CES method is insensitive to these non-idealities.
Note that the CES approach has some formal similarity to the two-loop methods in [33,37,38], because of the use of two different dark times T 1 and T 2 . However, the CES technique requires only one feedback loop for frequency stabilization.

CES for different Ramsey sequences
We assume that the main reason for the shift of the stabilized frequency ω arises from the probe-induced shift Δ sh during Ramsey pulses. All calculations are done for the ideal case of the phase jumps: , as is typical for high-precision modern atomic clocks based on strongly forbidden optical transition 1 S 0    3 P 0 in neutral atoms (such as Mg, Ca, Sr, Yb, Hg) and ions (e.g. Al + , In + ), or for the octupole transition in the ion Yb + .
In this section, we compare CES spectroscopy for two different pulse sequences: the usual Ramsey sequence with two equal rectangular π/2-pulses (see figure 2(a)), and the hyper-Ramsey sequence proposed in [27]   Therefore, in our calculations we will use the following expression for the calibration coefficient where the parameter χ determines the deviation of cal b from the ideal value (23). In this case, instead of equation (24) we obtain another formula for the CES T T T CES err obs in obs in  figure 3 we present a comparison of the CES method for two different pulse sequences: a standard Ramsey sequence with two equal pulses (see figure 2(a)) and the original hyper-Ramsey sequence [27] using a composite pulse (see figure 2(b)). These graphs show the shift of the clock frequency clock sh d D ( )versus the probe-induced shift Δ sh during the Ramsey pulses. In the calculations, we have assumed a five-percent deviation of cal b from the ideal value (23), i.e. 0.95 1.05   c in equation (27). As we see, the hyper-Ramsey sequence is more robust and persistent, because the use of this scheme leads to a significant reduction of the residual shift clock d in comparison with the usual Ramsey scheme.
In addition, figure 4 shows that the combination of the CES technique with a hyper-Ramsey sequence significantly exceeds the possibilities of standard hyper-Ramsey spectroscopy [27], even for imperfect determination of the calibration coefficient cal b . For example, typical experimental conditions used in current lattice optical clocks are T 1 ∼1s, τ∼20ms and π/2 excitation pulse (Ω 0 τ=π/2). Figure 4(a) shows that if imperfect CES is used and cal b is offset from its optimal value by 5%, a light shift of   interrogation light shift level) that corresponds to the level of clock 0 d w|¯| 10 −17 . Of course, for the perfect CES spectroscopy (23), this clock shift will be reduced to zero, 0 clock d = .

Generalized CES and the procedure for frequency stabilization
The calibration coefficient cal b can be estimated as a ratio of the amplitudes of the central Ramsey fringes related to the interrogation procedures with T 1 and T 2 dark times. However, in this section we describe a more precise method to determine cal b . For this purpose, we will consider a generalized combined error signal (GCES) where the generalized calibration coefficient b d ( ) is a function of δ, which satisfies the following condition 0 e . 2 9 T T cal In this case, the stabilized frequency (with the use of GCES (28)) will also always be unshifted, 0 There are many different variants of the function b d ( ). For example, b d ( ) can be constructed as following functions where we use an additional measurement in the absence of a phase jump (α=0) before the second Ramsey pulse, A W G W , 0 , is preferable because of 'symmetry' in relation to the phase jumps a  .
In figure 5, we compare signals of CES (21) and GCES (28) for two different pulse sequences (see figure 2) in the presence of the field-induced shift Δ sh (during Ramsey pulses). As we see from figure 5(a), as Δ sh increases the lineshape S CES err d ( ) ( ) becomes significantly non-antisymmetrical, while the lineshape S GCES err d ( ) ( ) (see figure 5(b)) maintains its antisymmetry (especially for the hyper-Ramsey scheme, see the right panel in figure 5(b)). Figure 5(c) shows the dependencies of b d ( ) calculated by the use of equation (31). The procedure of frequency stabilization can be organized in conformity with several scenarios. First, we can continually apply GCES (28) together with equation (31) using six measurements for each frequency point (three different phase jumps, α=±π/2, 0, and two different dark times, T 1,2 ). However, the use of six measurements can reduce the efficiency of the frequency stabilization, because it increases the length of the interrogation procedure. From our viewpoint, more optimal scenario is the following. In the initial period of frequency stabilization, we use GCES with equation (31). It allows us to determine the calibration coefficient cal b (see equation (29)) with satisfactory accuracy, because during measurements we will have the information about the value b d ( ) under δ≈0. Then the procedure of long-term frequency stabilization can be done with the CES technique (21), using only four measurements for each frequency point (two phase jumps, α=±π/2, and two dark times, T 1,2 ). Moreover, we can regularly (but rarely) use GCES again. Indeed, on the one hand, it allows us to do a regular adjustment of the coefficient cal b (to eliminate, for example, an influence of possible slow variations of the parameter Γ in equation (23)). On the other hand, such intermittent application of GCES will not lead to the significant slowing-down of the process of long-term frequency stabilization.
In addition, as we see from figures 3-5, the CES or GCES technique works better if the ratio sh 0 D W | | becomes smaller. Distortions in the error signals arising from this problem can be largely reduced by the use of an additional and well-controllable frequency step Δ step only during the Ramsey pulses 1 t and 2 t [27,41]. In this case, all dependencies presented in figures 4 and 5 will be the same if we will replace sh eff D  D =(Δ sh −Δ step ). Thus, we can always apply a frequency step step sh D » D (e.g. with an acousto-optic modulator) during excitation to achieve the condition 1 eff 0 D W  | | for an effective shift Δ eff , as it was used in experiments [5,28,29,37], where the required value of Δ step was determined empirically. Note The CPT resonance is formed when the difference between optical frequencies ( 1 2 w w -) is varied near the lowfrequency rf transition between lower energy levels 1ñ | and 2ñ | : 2 1 hfs w w -» D (see figure 6(a)). For example, these energy levels can correspond to the hyper-fine structure of alkali atoms (Rb, Cs, etc) In this case, the stabilized rf frequency difference 2 ) is the operating frequency for CPT based clocks, where the hyperfine splitting hfs D plays role of the unperturbed frequency 0 w from previous sections.
The dynamics of the Λ system in the rotating wave approximation are described by the differential equation system for the density matrix components   (35)) during the dark time T. r r r r r ¶ + -= W - where the time integration interval is over the second (detecting) pulse τ 2 , which starts at the time t d (see figure 6(b)). The value α corresponds to the phase jump during the dark time T (see figure 6(b)). In the case of CPT spectroscopy, this phase jump describes a phase difference of the product E E ) during the first Ramsey pulse 1 t and the product E E ) during the second pulse 2 t (e.g. see in [43]): in the formulas (10) and (21)-(31) from the previous sections, we describe a realization of the CES/GCES techniques for CPT Ramsey spectroscopy. However, in this case, it is necessary to use 12 G (instead of Γ in equation (23)) to determine the calibration coefficient | . Calculations presented in figure 7 clearly demonstrate that the CES/GCES technique can be very effective for an rf clock based on CPT Ramsey spectroscopy. Indeed, for an ideal calibration coefficient (36) we see a total absence of the clock shift, 0 clock d = (see horizontal line in figure 7). But even for non-ideal calibration coefficient cal CPT b ( ) , the residual shift for the CES method (see green and red colored dashed lines in figure 7) is much less than for usual Ramsey spectroscopy (see blue colored dashed line in figure 7).

Conclusion
We have developed a universal one-loop method to form the reference signal for stabilization of arbitrary atomic clocks based on Ramsey spectroscopy. This method uses the interrogation of an atomic system for two different Ramsey periods and a specially constructed CES (see equation (21)). The CES technique requires four measurements for each frequency point as well as a preliminary measurement (or estimation) of the calibration coefficient cal b . It was shown that the highest robustness is achieved with the combination of the CES protocol and a hyper-Ramsey pulse sequence (see in [27]). Also a method of GCES was developed (see equation (28)), which requires six measurements for each frequency point and has an exceptional robustness. The CES/GCES spectroscopy allows for perfect elimination of probe-induced light shifts and does not suffer from the effects of relaxation, time-dependent pulse fluctuations and phase-jump modulation errors and other non-idealities of the interrogation procedure. A variant of the frequency stabilization using CES with intermittent GCES protocols has been proposed. In addition, the applicability of CES/GCES techniques for CPT atomic clocks has been described. The implementation of this approach can lead to significant improvement of the accuracy and long-term stability for a variety of types of atomic clocks. Also, it will be interesting to experimentally compare the one-loop CES/GCES method with the two-loop ABRS [37][38][39]. We believe that both methods have comparable efficiency for frequency stabilization, but CES/ GCES is technically simpler because only one feedback loop is required. Moreover, in the case of optical transitions, the CES/GCES protocol with the use of hyper-Ramsey pulse sequence (see in [27]) can be even more efficient in comparison with ABRS.