Optimal atomic interferometry robust to detection noise using spin-1 atomic condensates

Implementation of the quantum interferometry concept to spin-1 atomic Bose-Einstein condensates is analyzed by employing a polar state evolved in time. In order to identify the best interferometric configurations, the quantum Fisher information is maximized. Three optimal configurations are identified, among which one was not reported in the literature yet, although it gives the highest value of the quantum Fisher information in experimentally achievable short time dynamics. Details of the most optimal configurations are investigated based on the error-propagation formula which includes the interaction-based readout protocol to reduce the destructive effect of detection noise. In order to obtain Heisenberg scaling accessible by present day experimental techniques, an efficient measurement and a method for the inversion of dynamics were developed, as necessary for the protocol's implementation.


I. INTRODUCTION
Quantum interferometry that initially emerged in the quantum optics domain a little while back was successfully applied to systems composed of massive particles. Numerous proof-of-principle experiments have demonstrated potential of ultra-cold atoms in precision measurements based on interferometric techniques [1]. Today, ultra-cold atoms play an important role in measurements of physical quantities that could not be measured with optical devices, or they are measured with weaker precision. Chip-scale inertial sensors for real-time positioning and navigation, ultra-precise atomic clocks or magnetometers operating in Earths magnetic field are good examples.
Spinor Bose-Einstein condensates consist of atoms with the total spin F , in which all internal Zeeman states numerated by the quantum magnetic number m F = 0, ±1, · · · , ±F are trapped by the same optical trap [2]. Among various sensors achievable with ultracold atoms, spinor Bose-Einstein condensates with Zeeman energy levels sensitive to magnetic field can be used to encode information about unknown physical quantities using quantum interferometry techniques [3][4][5][6][7][8][9]. Furthermore, non-linear interactions between atoms allow generating non-classical states, such as squeezed or entangled states [10][11][12][13]. A non-classical state used as an input state of a quantum interferometer allow precision measurement below the shot-noise limit, potentially approaching the ultimate Heisenberg limit with highly entangled states. Utility of the non-classical states in quantum interferometry typically requires detection of particles with very low noise, which is hardly achievable with atomic-based technology. Recently, the concept of interaction-based readout [14][15][16][17][18] were proposed, and verified experimentally [4] for some special case, in order to overcome the detection noise problem. The interaction-based readout is nothing else but a unitary evolution applied to the quantum interferometer after the phase encoding step, but before the measurement takes place. Typically, the unitary evolution is based on the inter-particle non-linear interactions, the same as used for the non-classical state preparation.
The purpose of this paper is to perform comprehensive study of quantum interferometry using spin-1 atomic Bose-Einstein condensates. The system was studied in this context, and the possibility of metrological gain was demonstrated experimentally for two different interferometric configurations [12,18]. On the theoretical level, however, it is interesting to prove and explain which configuration is the most optimal one, i.e. gives the highest possible and practicable precision. We believe that such analysis would help in understanding very foundations of quantum interferometry using spin-1 atomic condensates and further planning of experiments. In what follows, we consider the most general form of linear quantum interferometry [6]. We identify optimal configurations of interferometric rotations by studying the quantum Fisher information (QFI) for the initial polar state. Our results show that the choices of mentioned experiments [12,18] lies among the optimal once, however we found another configuration which determines the highest value of the QFI in short time dynamics accessible by nowadays experiments. We discuss how to achieve experimentally the configuration taking into account the interaction-based readout to protect against detection noise. Finally, we show how that protocol, which is reversed evolution applied after the phase encoding step, can be realized experimentally with a single rotation of the state in the early evolution of the system. The paper is organized as follows. In Section II we present the model and show the analytical solution for the polar state |0, N, 0 evolution, as well as other quantities important in derivation of the QFI value. In Section III we calculate the QFI values and identify the best interferometric configuration. Next, in Section IV we discuss the effect of detection noise for various signals and show how the interaction-based readout can be implemented in the most optimal configuration in order to achieve the highest precision.

II. THE MODEL AND TIME EVOLUTION
We consider a spinor Bose-Einstein condensate, with three internal levels, in the single mode approximation (SMA) where all atoms from different Zeeman states occupy the same spatial mode φ(r), which satisfies the Gross-Pitaevskii equation with chemical potential µ. We assume that the many-body Hamiltonian has the following form [19]: is the operator of atoms number in the Zeeman state m F andĴ is the collective pseudo-spin operator defined within the SU(3) Lie agebra generators in the next section. The c 0 and c 2 coefficients can be expressed in terms of s-wave scattering lengths [20,21]. The Hamiltonian can be engineered in F = 1 [3,10,11,22,23] or F = 2 [4,12] hyperfine manifold using Rb 87 atoms. The characteristic feature of the Hamiltonian (1) is the conservation of the z-component of the collective pseudo-spin operator [Ĥ,Ĵ z ] = 0. The Hamiltonian has a block-diagonal structure with each block labeled by the magnetization M = −N, −N + 1, . . . , N , which is the eigenvalue of thê J z operator.
We start the evolution from the polar state |0, N, 0 , which is a ground state of the Hamiltonian (1) in the high magnetic field limit. Since the initial state has M = 0 and the quadratic Zeeman effect can be compensated with microwave dressing [24][25][26], the time evolution is governed by the Hamiltonian were we skipped constant terms, and is given by witht = tc ′ 2 / which we assume to be positive. Since theĴ 2 operator is diagonal in the total spin momentum basis, i.e.Ĵ 2 |N, J, M = J(J + 1)|N, J, M , it is convenient to decompose the polar state |0, N, 0 into the total spin eigenbasis and then solve the evolution (3) analytically, which gives where for zero magnetization, and the prim after the sum notation ′ indicates summation over even values of J = 0, 2, 4, . . . , N when the total number of atoms N is even, or summation over odd values of J = 1, 3, 5, . . . , N when N is odd, see the Appendix A for explanation. The representation (4) clearly demonstrates that the evolution is periodic with ∆t = π.
As the magnetization is conserved by the Hamiltonian, the time evolution of all quantities of interest considered in the next section can be expressed in terms of the following terms N 0 , N 2 0 and â †2 0â 1â−1 whose evolution can be calcuated analytically for the state (4), they are where and Re â †2 0â 1â−1 = In addition, it is convenient to use also Ĵ 2 = 2N , Ĵ 2 z = 0. The evolution of the state (4), as well as above quantities, are usually considered using the recursion relation [27]. Here, based on the decomposition of the polar state into the total spin eigenbasis, we obtained quite simple to calculate analytical expressions.

III. IDENTIFICATION OF THE BEST INTERFEROMETERIC CONFIGURATIONS
The interferometric protocol we consider consists of four steps in general, see Fig.1. The scheme starts with the dynamical state preparation by the unitary evolu-tionÛ 1 = e −it1Ĵ 2 followed by the phase θ accumulation exp −iθΛ n during an interrogation timeT under generalized generator of interferometric rotationΛ n . The phase θ depends on the physical parameter to measure, e.g. magnetic field, and we assume that it is imprinted onto the state in the most general way. Next, an optional unitary evolution through the operatorÛ 2 can be applied before performing a quantum measurement (QM). In this sectionÛ 2 = 1, however it is non-zero and plays a significant role in the interaction-based readout protocol considered in the next section.
The purpose of this section is to identify the operator Λ n which determines the best precision in the θ estimation for the state (3) at the timet 1 .
In a general linear interferometer, the output state |ψ(θ) can be written as the action of the SU(3) rotation on the input state |ψ(t 1 ) , i.e.
whereΛ n =Λ · n is a scalar product of a unit vector n and the vectorΛ = {Ĵ x ,Q yz ,Ĵ y ,Q zx ,D xy ,Q xy ,Ŷ ,Ĵ z } composed of bosonic SU(3) Lie algebra generators: whereâ mF is the annihilation operator of the particle in the m F Zeeman component. In this scheme, the minimal possible uncertainty of the parameter θ is determined by the inverse of the quantum Fisher information ∆θ 1/ F Q [|ψ(t 1 ) ,Λ n ], which depends on the input state |ψ(t 1 ) and the generator of the interferometric rotationΛ n . We will refer the generator of the interferometric rotationΛ n as an interferometer, for simplicity. The QFI is defined as [28] where Γ[ψ(t 1 ) ] is the covariance matrix The maximal value of the QFI is given by the largest eigenvalue λ max of the covariance matrix, and for the three level system considered here it is F Q = 4λ max . The generator of the optimal interferometric rotation is determined by the eigenvector corresponding to the maximal eigenvalue of the covariance matrix (23). In general, the symmetric matrix (23) has 36 distinct elements, but for the input state |ψ(t 1 ) defined in the equation (3) most of its entries are 0 due to rotational symmetry e −iαĴz |ψ(t 1 ) = |ψ(t 1 ) , which holds for any α ∈ R due to conservation of magnetization. This property results in the block diagonal structure of the covariance matrix in the subspace of zero magnetization, which is the following: where and with averages of particular operators taken att =t 1 , i.e. · = ψ(t 1 )| · |ψ(t 1 ) . Matrices Γ ± share the same eigenvalues, but they have different eigenvectors [29].
There are three different non-zero eigenvalues of the covariance matrix (24), and hence, three different eigenvectors which define three generators of interferometric rotation of practical importance. The first eigenvector iŝ whereĴ x (γ) = (Ĵ x + γQ yz )/ 1 + γ 2 andĴ y (γ) = (Ĵ y − γQ zx )/ 1 + γ 2 . The value of ǫ does not change the value of the resulting QFI, and we will always consider ǫ = 1. The application ofΛ numerically that when γ ≫ 1, thenΛ (I) n ≈Q yz and F (I) Q ≈ 4Γ 22 = 4∆ 2Q yz as can be seen in Fig.2. Therefore, in the further part of the paper we will always takê Λ (I) n =Q yz . The second eigenvector of (24) iŝ and its usages as an interferometer will results in F (II) Q = 4Γ 55 = 4∆ 2D xy , for ǫ = 1. The last non-trivial eigenvector of the covariance matrix iŝ and this interferometer turns out to give the QFI equal to 4Γ 77 , i.e. F (III) Q = 4∆ 2Ŷ . The optimal interferometers recognited by us have a two-mode nature [30], as: In the case ofΛ  A general protocol for the optimal linear entanglement-enhanced quantum interferometry with spinor condensates. The optimal interferometric rotations discussed in the text are (a)Λ In additionΛ (I) n =Ĵ x (γ), with γ chosen such that it maximizes the QFI value, is also shown for comparison. All of them demonstrate Heisenberg-like scaling of the QFI. It is important to stress that the QFI value starts from 4N when the interferometer isΛ n is the most optimal for the experimentally relevant situations wheret ∼ 1/ √ N , at least in the ideal case considered. Time evolution of all QFIs is known analytically, as corresponding covariance matrix elements can be expressed in terms of N 0 , N 2 0 and â † 0â † 0â 1â−1 whose time evolution was presented in the previous section. Notice, other choices of interferometers composed of a linear superposition of SU(3) algebra generators are also possible, but their usage will lead to lower values of the QFI than ones obtained from the three optimal interferometers established by us.
The best interferometric configurations identified in this section are summarized in Fig. 3. They are quite abstract at the moment, however one can associate them with a measurement of physical quantities such as e.g. magnetic field. Let us consider atomic magnetometers based on detection of the Larmor frequency ω induced by a weak magnetic field oriented along the z axis. During the Larmor precession cycle the stateρ is subject to the phase imprinting process and is effectively rotated around the operatorĴ z :ρ(θ) = e −iθĴzρ e iθĴz with θ = ωT . One can employΛ (I) n andΛ (II) n interferometers in that phys-ical situation by a three stage procedure [31]. In order to realize a general rotation one needs to find a unitary transformationR such that The procedure is as follows: (i) after the preparation timē t =t 1 the state is rotated, resulting inρ R =R † ρR, (ii) the rotated stateρ R is subject to the phase imprinting processρ R (θ) = e −iθĴzρ R e iθĴz and (iii) the state is disrotated using the conjugate rotationR † givingρ(θ) = R †ρ R (θ)R. It is straightforward to show that for the first interferometric rotation withΛ (I) n =Q yz one has the unitary transformationR (I) = e −iπĴy/2 e −iπDxy/2 , while forΛ (II) n =D xy one can find thatR (II) = e −iπQxy/4 . Unitary transformationsR (I) andR (II) may be realized experimentally as they involve either spin operators or two extremal modes m F = ±1 [12,32]. In the case of the third optimal interferometer found by us, namelŷ Λ (III) n =Ŷ , the physical interpretation is already understood very well within SU (1, 1) interferometry and was realized experimentally in [18].

IV. IDENTIFICATION OF OPTIMAL OBSERVABLES
In the quantum interferometry scheme, a physical quantity like magnetic field is mapped onto the phase difference θ between internal states of atoms. Then it can be extracted by performing a quantum measurement. The expectation value of the observable P carries information about the unknown value of θ, and thus can be exploited in the estimation procedure. As illustrated in Fig. 4, in the limit of a large number of measurements the precision in the θ estimation is given by the errorpropagation formula [1,33,34]: where ∆ 2P = P 2 − P 2 . We have already mentioned in the previous section that the uncertainty of θ is bounded from below by the QFI, namely ∆ −2 θ(P) ≤ F Q , which is nothing else but the Cramèr-Rao inequality. In principle, it is possible to choose such an observable P which saturates the Cramer-Rao inequality. The natural question arises which P provides the highest precision under the three optimal interferometric rotations we identified in the previous section, i.e.Q yz ,D xy ,Ŷ ? When identifying such observables, it is important to take into account detection imperfections. Entangled states which are generated in time with the Hamiltonian (1) provide limited sensitivity due to the requirement of perfect detection, sometimes on the level of a single particle. In the nowadays experiments, the measurement is burdened with the particle detection noise, therefore imperfect detection of a state causes a significant drop of FIG. 4. The origin of the error-propagation formula (38). The precision in the θ estimation is based on the measurement of a signal P which is illustrated by the orange solid curve. The uncertainty ofP is marked by the gray shadow region. The tangent of the curve P at some value of θ can be determined by its slope ∂ θ P and by the ratio ∆P/∆θ. Their equivalence leads to the error-propagation formula (38).
the QFI value. In a standard ultra-cold atom setup information about various physical quantities is estimated from the measurement of the atom number. In an ideal system, the probability p(N |N ) of detecting N number of atoms given that the true number isN equals δ N,N . In a realistic scenario, that property no longer holds and one can detect N number of atoms even thoughN = N truly hit the detector.
Mathematically, detection noise is modeled by replacing all ideal probabilities p(N |θ) withp(N |θ) = N p(N |N )p(N |θ), where N p(N |N ) = 1 [35,36]. It can be shown by simple algebra that the kth moment of the particle number operator is modified by the Gaussian detection noise p(N |N ) = exp −(N −N ) 2 /2σ 2 /(σ √ 2π) in the following way: where M l (σ) = x l exp(−x 2 /2σ 2 )/(σ √ 2π)dx is the lth central moment of the Normal distribution and N k−l id = NN k−l p(N , θ) is the ideal expectation value without detection noise, e.g. N 2 gdn = N 2 id + σ 2 . In Eq. (39) we assumed that the atom number is large enough so that the difference between the sum N N k p(N |N ) and the integral dN N k p(N |N ) is negligible. Notice, the effect of the Gaussian detection noise on the moments of operatorN is the same as if it was replaced byN =N 0 + δN , whereN 0 denotes an ideal particle number operator and δN is an independent random operator satisfying δN = 0 and (δN ) 2 = σ 2 [37].
The Gaussian detection noise does not modify the derivative entering in the error-propagation formula (38) but adds to the variance typically as follows: where σ 2 P is connected with the second moment ofP. In the case of our model we assume that the width of the Gaussian distribution σ is the same for the probabilities of detecting atoms in all three Zeeman components.
The interaction-based readout, very nicely introduced and explained e.g. in [14][15][16][17], helps to avoid direct detection of entangled states, and therefore, protects against the noise effect. Typically, it is done by the time-reversed evolution and we will start our analysis with this protocol. In the further part of this work we will use these methods while recognizing the observable P which saturates the Cramer-Rao inequality under the optimal interferometric rotations determined by us in the previous section.
A. Interaction-based readout to protect against the detection noise The interaction-based readout protocol is simply a unitary evolution based on the time reversed non-linear interactions applied after the phase imprinting operation [14,38], and is defined in the following way: The protocol considered by us fits to the one sketched in Fig. 1, and contains non-trivial unitary operationÛ 2 = e it2Ĵ 2 , witht 2 =t 1 for simplicity. It turns out that the one among realatively easy to measure experimentally observables, namelyP = J 2 z ,Ŷ ,Q yz , is sufficient to saturate the corresponding QFI value with particular choice of the generator of the interferometric rotation.

The first optimal interferometerΛ
When the first optimal interferometerΛ (I) n is used in the protocol (41), then the inverse of the uncertainty calculated from the error-propagation formula (38) for P =Ĵ 2 z , i.e.
saturates the Cramèr-Rao inequality as long as σ → 0, what is demonstrated in Fig. 5. That fantastic agreement should be possible to prove analytically, however we were not succeeded. Notice, the right hand side of Eq.(42) is 0/0 expression when θ → 0 and σ → 0 be- and ∆ 2 (Ĵ 2 z ) id → 0 in the initial state. A direct consequence thatĴ 2 z commutes with bothÛ 1,2 is a fast drop of the F (I) Q value when σ increases. It means that the detection noise reduces the signal's value, although it is so simple to measure. One can overcome the problem and measureP =Q yz in place ofĴ 2 z . The measurement of Q yz is possible using nowadays technology with the appropriate choice of state rotations which map the value of Q yz onto the value of Ĵ z [10]. Then, as can be seen in Fig. 5, the short time dynamics achievable in nowadays experiments is quite well captured. The disadvantage of that choice ofP is the zero value of ∆ −2 θ (I) gdn (Q yz ) when t → 0 and σ → 0, so one does not start from SQL as it was the case forP =Ĵ 2 z . The resulting inverse of the uncertainty is insensitive to the detection noise from the errorpropagation formula (38) which is demonstrated in the inset of Fig. 5. Finally, the first maximum of ∆ −2 θ (I) gdn (Q yz ) at θ → 0 has the Heisenberg scaling, see Fig. 5. This configuration seems to be the most promising for quantum metrology beyond the SQL and it was not realized experimentally yet, although it is in the range of the present technology.

The second interferometerΛ
The second among optimal interferometers, namelŷ Λ II n =D xy , was already realized experimentally [12] and the measurement ofĴ 2 z was performed. Here, we just demonstrate the origin and rightness of such a choice pointing out its sensitivity to the detection noise. Indeed, one can show numerically that saturates the Cramèr-Rao inequality, as ∆ −2 θ (II) whenever σ → 0, as demonstrated in Fig. 6 by the green dashed line. However, the operatorP =Ĵ 2 z commutes with bothÛ 1 andÛ 2 , which makes the precision very sensitive to the detection noise. The non-zero value of σ decreases the value of ∆ −2 θ (II) gdn (Ĵ 2 z ) as shown in the inset of Fig.6. Nevertheless, we found out that there is another signal that do not share this property and it is as simple to measure experimentally as the previous one, namelyP =Ŷ . The inverse of the uncertainty calculated from the error-propagation formula (38) for this signal is given by Indeed, in the very initial period of time we achieve the same result as in the case ofP =Ĵ 2 z . Although, proceeding with the evolution little further causes a drop of the Fisher information. Nevertheless, initially the value of ∆ −2 θ

3.Λ
(III) n =Ŷ Finally, the third interferometer identified by usΛ n = Y was realized experimentally [4] as well, however in the context of the SU (1, 1) interferometry. The measurement ofŶ was performed. Indeed, our calculations confirm that it is an optimal choice, as saturates the Cramèr-Rao inequality, i.e ∆ −2 θ (III) , whenever σ → 0 as illustrated in Fig. 7. Moreover, our calculations show that the uncertainty from the error propagation (46) formula is insensitive to detection noise.
In summary, for all of the three interferometers we recognized in the previous section it is possible to choose quite simple to measure experimentally signalsP in such a way that they are insensitive to the detection noise, preventing a drop of the signal's value. In addition to the known in the literature configurations, we found out the additional one which gives desired precision much faster in time than the remaining two.

B. Proposal for an experimental realization of interaction-based readout by a single rotation
Now, we concentrate on the first interferometerΛ (I) n = Q yz and the measurement ofP =Q yz . Indeed, the configuration is very promising and, in addition, insensitive 2 ( √ 3Ŷ +Dxy)}. Superimposed white lines show mean-field phase portraits explained in the text. The initial coherent state |0, N, 0 is located along the z axis of the generalized Bloch sphere and is squeezed at some moment of timet =t1. Next, the state is imposed to the phase imprinting process and an extra rotation aroundĴz,S, by the proper angle α, is applied in order to artificially inverse a further evolution.
to the detection noise. However, the main objection for experimental realization would be difficulty of implementation of the time inversion, which is necessary to havê U 2 = e iĴ 2t 2 . In the following we present the method based on a single rotation of the state which works for short enough times.
In order to clearly present our idea, we start consideration by introducing the symmetric and antisymmetric bosonic operatorsĝ S = (â which are symmetric when l = S and anti-symmetric for l = A. The above spin operators have cyclic commutation relations, e.g.
[Ĵ x,l ,Ĵ y,l ] = 2iĴ z,l . The Hamiltonian that we used in the previous subsections expressed in terms of the symmetric and anti-symmetric operators reads:Ĥ The SU(2) subspace spanned by the symmetric spin operators is {Ĵ x,S ,Ĵ y,S ,Ĵ z,S } = {Ĵ x ,Q yz , 1 2 ( √ 3Ŷ +D xy )}, and the SU(2) subspace spanned by the anti-symmetric spin operators is {Ĵ x,A ,Ĵ y,A ,Ĵ z,A } = {Q zx ,Ĵ y , 1 2 ( √ 3Ŷ − D xy )}. Let us concentrate our attention on the symmetric subspace, as both interferometer and measurement are located in it.
The state, as well as its evolution, can be illustrated on the Bloch sphere spanned by the SU(2) symmetric spin operators with the help of the Husimi function where arbitrary spin-coherent state in the symmetric sub-space is defined as The initial coherent state |0, N, 0 is located along the z axis of the Bloch sphere as Ĵ x,S = Ĵ y,S = 0 and Ĵ z,S = −N . The very initial evolution of the state is identical to the one governed by the one-axis twisting model [39]. Hence, one can predict approximate quantum evolution following the mean-field phase portrait [40] which is explained in Appendix E and shown in Fig. 8 by white arrows. The initial coherent state |0, N, 0 evolves along circulating trajectories and is squeezed initially. The squeezed state can be rotated around theĴ z,S axis of the Bloch sphere, so the further evolving state will ideally turn back to the initial coherent state after the timē t = 2t 1 . The rotation allows us to perform backward evolution needed for implementation of the interactionbased readout protocol. Our idea is explained in details in Fig. 8. Therefore, we consider the time evolution of the initial state in the following way: (53) As long as the state is squeezed, or a bit oversqueezed, the angle α can be treated as [41] tan(2α) = Γ 12 where Γ ij are covariance matrix elements calculated in Section III. We show numerically, that such a rotation allows us to inverse the evolution at the very early stage, and thus protects the signal against the detection noise without significant lose of information comparing to the ideal situation which is shown in Fig. 5. The variation of the corresponding ∆ −2 θ versus the total atom number is ∼ N 2 , while the change of the time scale is typically t ∼ N −1/2 for the ideal protocols and t ∼ N −2/3 for the evolution (53) inverted by the rotation, see Fig. 9 for more details.
The interesting question arises if the same trick can be applied in a bimodal system that is more often used for the squeezing generation according to the one-axis model [39]. The answer is positive. One can apply the rotation to invert the evolution of a quantum state in the two mode systems as we have checked numerically for the angle α given by the formula tan(2α) = (γ 12 )/(γ 22 −γ 11 ), where γ 11 = ∆ 2Ŝ x , γ 22 = ∆ 2Ŝ y , γ 12 = − {Ŝ x ,Ŝ y } , and S x,y,z are spin operators defined for the bimodal system. The expressions for γ 11 , γ 22 and γ 12 can be calculated analytically [41]. In fact, the numerical results for ∆ −2 θ from the error-propagation formula withΛ n =Ŝ y andP =Ŝ y show that it behaves like the inverse of the squeezing parameter [42]. The gain is the resistance of such the squeezing parameter inverse against the detection noise.

V. SUMMARY
We implemented the quantum interferometry concept in spinor Bose-Einstein condensates employing the time evolved polar state. We focused on the quantum Fisher information in order to identify the best configurations. We solved analytically the dynamics of the polar state in the total spin eigenbasis, paying special attention to quantities that are important to calculate the QFI value. We found out three optimal generators of the interferometeric rotation that lead to Heisenberg scaling of the QFI, among which two, namelyΛ (II) n =D xy andΛ (III) n =Ŷ , were already successfully implemented experimentally in [12] and [18], respectively. However, we found out that there is even a better choice for the generator of the interferometric rotation, which isΛ (I) n =Q yz , because it gives much higher value of the QFI in early times of the evolution. Experimental realization of this interferometer with nowadays techniques is possible, although will require a three-step procedure as explained in Fig. 3.
Next, we considered optimal observables that would allow one to exploit the potential of particular interferometers based on the error-propagation formula. Indeed, we established that relatively easy to measure observableŝ J 2 z ,Ŷ ,Q yz are sufficient to reach the QFI value. However, it turns out that only some of them are resistant against the detection noise even when the interaction-based readout protocol was employed.
Finally, we showed how to implement the interactionbased readout protocol, which requires inversion of the evolution, by a single rotation of the state. That idea was not implemented yet, up to our knowledge, and can be applied in the case of a bimodal system as well. As an example, we considered the most prominent configuration withΛ (I) n =Q yz . We showed, that variation of the corresponding precision in the θ estimation from errorpropagation formula, assuming thatP =Q yz is measured, has the Heisenberg scaling with the total atom number, it is N 2 , while the time scale goes like N −2/3 . The advantage of our purpose is a resistance of the corresponding precision in the θ estimation against the detection noise.
and demonstrates that the dynamics is periodic with period ∆t = π.
Once we treated the time evolution of the polar state analytically, the evolution of particular quantities of interest can also be calculated analytically using relations shown in Appendixes B, C and D. The final results are quite complex, however they can be expressed by a single sum over the total spin length J as shown in Section I. Appendix B: Some useful relations needed for derivation of actions of annihilation and particle number operators onto the spin states and alsô