Twist-and-store entanglement in bimodal and spin-1 Bose-Einstein condensates
TTwist-and-store entanglement in bimodal and spin-1 Bose-Einstein condensates
Artur Niezgoda
Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland
Emilia Witkowska and Safoura Sadat Mirkhalaf
Institute of Physics, Polish Academy of Sciences,Aleja Lotnik´ow 32/46, PL-02-668 Warsaw, Poland (Dated: September 17, 2020)A scheme for dynamical stabilization of entanglement quantified by the quantum Fisher infor-mation is analyzed numerically and analytically for bimodal and spin-1 Bose-Einstein condensatesin the context of atomic interferometry. The scheme consists of twisting dynamics followed by asingle rotation of a state which limits further evolution around stable center fixed points in themean-field phase space. The resulting level of entanglement is of the order or larger than at themoment of rotation. It is demonstrated that the readout measurement of parity quantifies the levelof entanglement during entire evolution.
I. INTRODUCTION
Entanglement is a fascinating concept of quantumphysics and, as already well established, a unique re-source for emerging quantum technologies. In metrol-ogy, for example, entangled states such as squeezed statescan improve the sensitivity of interferometric measure-ments [1–3] because they allow overcoming the stan-dard quantum limit, where sensitivity scales as ∼ / √ N for N uncorrelated particles, approaching the ultimateHeisenberg limit with scaling as ∼ /N . Initially, thisconcept emerged in terms of squeezing [4] and very re-cently was applied [5–7] in the optical domain. Lately, itwas also successfully generated and characterized in thesystem composed of massive particles, namely ultra-coldatoms [8].In general, a production of squeezed and entangledstates requires inter-atomic interaction which dynami-cally generates non-trivial quantum correlations betweenatoms. The same interaction might be undesirable af-ter reaching the required level of entanglement becauseit can still dynamically degrade entanglement or inter-atomic correlations. The twisting types of interaction[9, 10] allows a uniform description of dynamical entan-glement generation for many setups composed of coldatoms [8], e.g. for cavity induced spin squeezing [11–13]and from spin-changing collisions in bimodal [14–16] andspin-1 Bose-Einstein condensates [17–21]. In particular,in the latter setup the undesired effect of interaction isdifficult to reduce.In this paper, we propose a simple method for entan-glement stabilization and storage by a single rotation ofa state in bimodal and spin-1 Bose-Einstein condensates.The idea is very simple and, as is illustrated in Fig. 1,it employs a structure of the mean-field phase space ofthe system Hamiltonian. The structure is the same forboth bimodal and spin-1 condensates as we demonstratein Section II. The method considers a generalized Ram-sey protocol with an additional rotation of a state ap-plied after twisting dynamics. Once the initial spin co- herent state, placed around a saddle point, is twistedalong constant energy lines, the single rotation puts thestate around two stable center points where further dy-namics is confined and stabilized. We provide detailsof the scheme in Section III. We observe that the valueof the quantum Fisher information (QFI), which quanti-fies not only the level of the sensitivity of interferometricmeasurements but also the level of entanglement [22], re-mains at least as at the moment of rotation, moreover itcan initially grow. We provide an analytical explainationof this feature of the QFI using a single argument of anenergy conservation in Section V. Therefore, we concludethat the QFI can exhibit Heisenberg scaling with the pre-factor of the order of one during the entire evolution inthe idealized scheme considered in this paper. FIG. 1. Illustration of the method for entanglement stabiliza-tion and storage. The condensate is initialized at the unstablefixed point (a). Initial evolution produces spin squeezing andentanglement along the diverging manifold of the separatrix(b). The quantum state is quickly rotated to locate it aroundthe two stable fixed points (c). Subsequent evolution of therotated state (d) is confined around stable fixed points leadingto the stable value of the quantum Fisher information withHeisenberg scaling.
The best sensitivity, and therefore the QFI value, canbe estimated using the signal-to-noise ratio [23] whenappropriate readout measurement is provided. In gen-eral, identification of a good observable to measure thatgives the highest precision is a difficult task, in partic-ular for non-Gaussian states. It might require measure-ments of higher order correlation functions [24]. Here, inSection VI, we define the parity operators for both the a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p bimodal and spin-1 systems. We show analytically, andconfirm numerically, that the measurement of parity [25]allows the sensitivity to saturate the QFI value. We provethis, by using only the fact of parity conservation. Themeasurement can be robust against phase noise if the op-erator representing the noise commutes with the parityoperator [26]. II. THE MODEL AND STRUCTURE OFCLASSICAL MEAN-FIELD PHASE SPACE
The desired structure of the mean-field phase space iscomposed of two stable center fixed points located sym-metrically on both sides of an unstable saddle fixed point.We concentrate here on the two systems widely exploredtheoretically and experimentally in the ultra-cold atomicgases, namely bimodal and spinor Bose-Einstein conden-sates.
A. Bimodal condensate
We consider here the twisting model enriched by a lin-ear coupling term between the two modes a and b turningthe state along an orthogonal direction of the formˆ H BI = (cid:126) χ ˆ S z − (cid:126) Ω ˆ S x , (1)where ˆ S x = (cid:16) ˆ a † ˆ b + ˆ b † ˆ a (cid:17) , ˆ S y = i (cid:16) ˆ a † ˆ b − ˆ b † ˆ a (cid:17) , ˆ S z = (cid:16) ˆ a † ˆ a − ˆ b † ˆ b (cid:17) are pseudo-spin operators satisfying thecyclic commutation relation [ ˆ S l , ˆ S n ] = i (cid:80) m (cid:15) lnm ˆ S m ,where (cid:15) lnm is the Levi-Civita symbol and ˆ a (ˆ a † ) and ˆ b (ˆ b † )are bosonic mode annihilation (creation) operators of anatom in the mode a ( b ). The above Hamiltonian describestwo weakly-coupled Bose-Einstein condensates interact-ing with the strength χ in the presence of an externalfield of the strength Ω. The model can be realized exper-imentally employing either a double-well trapping poten-tial [15, 27] or internal (e.g. two hyperfine atomic states)degrees of freedom [14].To obtain the mean-field phase space one can calculatean average value of (1) over the spin coherent state | ϕ, θ (cid:105) BI = e − iϕ ˆ S x e − iθ ˆ S y | N, (cid:105) , (2)where ˆ a † N √ N ! | , (cid:105) = | N, (cid:105) and ϕ ∈ [0 , π ] , θ ∈ [0 , π ]. Thespin coherent state is a double rotation of a maximallypolarized state when all atoms are in the state a . This Alternatively, one can substitute the quantum mechanical oper-ators by complex numbers ˆ a → √ N a e iϕ a (ˆ b → √ N b e iϕ b ), where Nz = N a − N b and ϕ = ϕ a − ϕ b corresponds to the relativephase between the two internal states. This procedure is not ob-vious for the spinor system as we will concentrate on symmetricsubspace of the Hamiltonian. leads to H BI = Λ2 z − (cid:112) − z cos ϕ, (3)where z = cos θ and Λ = χN/ Ω [28] while keepingthe leading terms. The parameters ( z, ϕ ) are conju-gate coordinates which draw trajectories in the mean-field phase space resulting from the Hamilton equations˙ z = −√ − z sin ϕ and ˙ ϕ = Λ z + z √ − z cos ϕ . The desiredby our protocol feature of the above mean-field phasespace trajectories is a presence of suitable configurationof stable and unstable fixed points. The position of fixedpoints is a solution of ( ˙ z = 0, ˙ ϕ = 0). The resulting struc-ture of phase space is shown in Fig. 2. The three principalregimes can be distinguished depending on the value ofΛ and characterized by different positions and number offixed points [29, 30]. The first one is the “Rabi” regimefor Λ < →
0, the evolutionis similar to resonant Rabi oscillations with N indepen-dent particles. The two stable center fixed points arelocalized at ( z, ϕ ) = (0 ,
0) and ( z, ϕ ) = (0 , π ). The sec-ond is the “Josephson” regime appearing for Λ >
1. Inthis regime the fixed point localized at ( z, ϕ ) = (0 , π ) be-comes unstable and the two new stable fixed points format ( z, ϕ ) = ( ± (cid:113) − , π ). The change happens just af-ter the bifurcation point at Λ = 1. In this regime, thecharacteristic “ ∞ ” shape is drawn up by trajectories cen-tered around an unstable fixed point at ( z, ϕ ) = (0 , π ),see Fig. 2. The “ ∞ ” shape is the one that allows storingentanglement. Finally, the third “Fock” regime occurs forΛ (cid:29)
1, when the phase portrait has the same structureas the one-axis twisting (OAT) model [9]. It is composedof two stable fixed points at ( z, ϕ ) = ( ± , ϕ ), and theunstable one at ( z, ϕ ) = (0 , ϕ ). FIG. 2. The structure of classical mean-field phase space forthe bimodal system versus Λ. The upper panels show the viewfrom the positive side of x -axis, while the bottom panels showthe view from the negative side. The principal three regimesare distinguished as indicated by a name above the Λ axis,and discussed in the main text. In this paper, we considerΛ = 2 and the initial state located around an unstable fixedpoint located along the x -axis, at the negative side of it. B. Spinor condensate
The same structure of the mean-field phase space canbe realized in spinor Bose-Einstein condensates withthree internal levels instead of two, as discussed above.It can be seen in the single mode approximation (SMA)where all atoms from different Zeeman states occupythe same spatial mode φ ( r ) which satisfies the Gross-Pitaevskii equation. The many-body Hamiltonian isexpressed in terms of annihilation (creation) operatorsˆ a m F (ˆ a † m F ) of an atom in the m F Zeeman state andspin-1 operators, which we collected in the vector (cid:126)
Λ = { ˆ J x , ˆ Q yz , ˆ J y , ˆ Q zx , ˆ D xy , ˆ Q xy , ˆ Y , ˆ J z } (see Appendix A fordefinitions) is ˆ H S c (cid:48) = − N ˆ J + q ˆ N s , (4)after dropping constant terms [31, 32]. Here, the en-ergy unit c (cid:48) = N | c | (cid:82) d r | φ ( r ) | is associated to thespin interaction energy, ˆ J = ˆ J x + ˆ J y + ˆ J z and ˆ N s =ˆ a † ˆ a + ˆ a †− ˆ a − [10, 33, 34]. The last term in (4) isdue to quadratic Zeeman effect which can have contribu-tion from the external magnetic field or microwave lightfield [35]. The value of q can be either positive or nega-tive. The Hamiltonian (4) conserves the z -component ofthe collective angular momentum operator [ ˆ H S , ˆ J z ] = 0;hence, the linear Zeeman energy term is irrelevant andis omitted here. The magnetization M ∈ [ − N, N ], beingthe eigenvalue of the ˆ J z operator, is a conserved quantity.The above Hamiltonian can be engineered e.g. in F = 1hyperfine manifold using Rb atoms [17, 19, 36–38].For our purposes it is convenient to introduce the sym-metric and anti-symmetric bosonic annihilation opera-tors, ˆ g s = (ˆ a + ˆ a − ) / √ g a = (ˆ a − ˆ a − ) / √
2, andthe corresponding pseudo-spin operatorsˆ J x,σ = ˆ a † ˆ g σ + ˆ a ˆ g † σ , (5)ˆ J y,σ = i (ˆ a † ˆ g σ − ˆ a ˆ g † σ ) , (6)ˆ J z,σ = ˆ g † σ ˆ g σ − ˆ a † ˆ a , (7)where indices σ = s and σ = a refer to symmet-ric and anti-symmetric subspace. The above operatorshave cyclic commutation relations, e.g. [ ˆ J x,σ , ˆ J y,σ ] =2 i ˆ J z,σ . Note, the symmetric subspace is spannedby { ˆ J x,s , ˆ J y,s , ˆ J z,s } = { ˆ J x , ˆ Q yz , ( √ Y + ˆ D xy ) } whilethe anti-symmetric subspace by { ˆ J x,a , ˆ J y,a , ˆ J z,a } = { ˆ Q zx , ˆ J y , ( √ Y − ˆ D xy ) } . The spin-1 Hamiltonian (4)can be expressed in terms of symmetric and anti-symmetric operators [39, 40] asˆ H S | c (cid:48) | = − N ˆ J x,s + q J z,s − N ˆ J y,a + q J z,a − N (cid:0) ˆ g † s ˆ g a + ˆ g † a ˆ g s (cid:1) (8) up to constant terms. The Hamiltonian (8) is a sumof two (non-commuting) bimodal Hamiltonians for sym-metric and anti-symmetric operators, as in (1), providedthat they are rotated in respect to each other, plus a mix-ing term which comes from the ˆ J z operator. Therefore,the mean-field phase space of the spinor system in eachsubspace is expected to have the same structure as thebimodal condensate (1).To show this, we concentrate here on the symmetricsubspace spanned by the symmetric pseudo-spin oper-ators ˆ J x,s , ˆ J y,s , ˆ J z,s (the anti-symmetric mean-field sub-space is provided in Appendix B). The structure of mean-field phase space can be obtained by calculating an aver-age value of (4) over the spin coherent state defined forthe symmetric subspace as | ϕ, θ (cid:105) S = e − iϕ ˆ J z,s / e − iθ ˆ J y,s / | N (cid:105) s (9)where | N (cid:105) s = ˆ g † sN √ N ! | (cid:105) and once again ϕ ∈ (0 , π ) , θ ∈ (0 , π ). The spin coherent state (9) can be interpreted as adouble rotation of maximally polarized state | N (cid:105) s in thesymmetric subspace, when all atoms are in the symmetricmode. The state | N (cid:105) s is an eigenstate of ˆ J z,s such thatˆ J z,s | N (cid:105) s = N | N (cid:105) s , and is located on the north pole ofthe Bloch sphere in the symmetric subspace. In terms ofspin-1 operators it reads | N (cid:105) s = e − iπ/ Q xy | N (cid:105) . Onthe contrary, the state with N atoms in the m F = 0mode, ˆ a † N √ N ! | (cid:105) = | N (cid:105) , lies on the south pole of thesame Bloch sphere. In addition, one can show that | ϕ, θ (cid:105) S = 1 √ N ! (cid:20) ˆ g † s cos θ a † sin θ e iϕ (cid:21) N | (cid:105) , (10)up to the constant phase factor. We use the above ex-pression while illustrating an arbitrary state | Ψ (cid:105) on theBloch sphere in the symmetric subspace with the help ofthe Husimi function Q S ( ϕ, θ ) = |(cid:104) Ψ | ϕ, θ (cid:105) S | .An average value of the spin-1 Hamiltonian (4) overthe spin coherent state (9) leads to H S = Λ2 (1 − z ) cos ϕ + z + 1 , (11)by keeping the leading terms and omitting the constantones, and once again z = cos θ while Λ = − /q . Note, thevalues of Λ can be both negative and positive dependingon the value of q . The negative value of Λ does not changethe structure of the mean-field phase space as discussedin Fig. 3.The three different regimes are also present in the caseof the symmetric (anti-symmetric) subspace of the spinorsystem. To find positions of fixed points one shouldstart with Hamilton equations for conjugate variables( z, ϕ ) using (11), they are ˙ ϕ = − Λ z cos ϕ + 1 = 0,˙ z = 2Λ(1 − z ) cos ϕ sin ϕ = 0. Next, one calculatessolutions of ( ˙ z, ˙ ϕ ) = (0 ,
0) which are locations of fixedpoints. The three regimes can be distinguished and theyare listed below for negative values of Λ. The “Rabi”
FIG. 3. The structure of phase portraits of the spinor system versus Λ in the symmetric subspace. The upper panels show aview of the north poles of the Bloch sphere, while the bottom panels show a view of south poles. The structure is the same asthe one for the bimodal system, provided that the latter is rotated by π/ y -axis. The three different regimes appearas well and are indicated above the Λ axis. In this paper, we focus on Λ = − regime is in the limit Λ → z = ±
1. It is true up to the bi-furcation which occurs at Λ = 1. On the other hand, inthe “Josephson” regime, just after bifurcation, the fixedpoint at z = − z, ϕ ) = (1 / Λ ,
0) and( z, ϕ ) = (1 / Λ , π ). In addition, the “Fock” regime takesplace when the interaction term dominates over the linearone. This regime is characterized by the two stable cen-ter fixed points at ( z, ϕ ) = (0 , π/
2) and ( z, ϕ ) = (0 , π/ ϕ = 0 , π .In our work we focus on the Josephson regime for | Λ | = 2. The desired “ ∞ ” shape is draw up by tra-jectories centered around an unstable fixed point. More-over, the angle among constant energy lines incomingand outgoing from the saddle fixed point equals to π/ III. TWIST-AND-STORE PROTOCOL
The interferometric protocol we consider consists offour steps in general, see Fig. 4. The scheme starts withthe dynamical state preparation by the unitary evolu-tion determined by the system Hamiltonian followed bythe state rotation at a given moment of time. The uni- θ accumulation during an interrogationtime T under generalized generator of interferometric rota-tion e − iθ ˆΛ n . Finally, a readout measurement (RM) is per-formed. tary evolution continues and eventually leads to the sta-bilization of dynamics around the two stable fixed pointslocated symmetrically around the unstable saddle fixedpoint. This state can further be used in quantum inter-ferometry protocol, which consists of the phase θ accu-mulation during an interrogation time T under the gen-eralized generator ˆΛ n of interferometric rotation e − iθ ˆΛ n .In particular, this is the phase encoding step in which theunitary transformation e − iθ ˆΛ n describes our interferome-ter in the language of the quantum mechanics. The phase θ depends on the physical parameter to be measured, e.g.a magnetic field, and we assume that it is imprinted ontothe state in the most general way. At the end, a readoutmeasurement (RM) is performed.In this paper, we consider the system at zero tem-perature and therefor its unitary evolution is given bythe ˆ U BI = e − it ˆ H BI operator for the bimodal and byˆ U S = e − it ˆ H S for the spin-1 systems. The initial stateis the spin coherent state located around the unstablesaddle fixed point, | ψ (0) (cid:105) BI = | , π/ (cid:105) BI for the bimodalsystem and | ψ (0) (cid:105) S = | , π (cid:105) S for the spin-1 system. Note,in the latter case the state is located on the south poleof the symmetric Bloch sphere and it is the polar state | , N, (cid:105) . The corresponding Schr¨odinger equations aresolved numerically in the Fock state basis where oper-ators are represented by matrices and states are repre-sented by vectors. IV. QUANTIFYING ENTANGLEMENT
We measure the level of entanglement using the quan-tum Fisher information (QFI) because we consider theprotocol in the context of quantum interferometry, as il-lustrated in Fig. 4. It is already well established thatthe QFI is a good certification of entanglement useful forquantum interferometry [22].In a general linear quantum interferometer, the out-put state | ψ ( θ ) (cid:105) can be considered as the action of therotation performed on the input state | ψ ( t ) (cid:105) , namely | ψ ( θ ) (cid:105) = e − iθ ˆΛ n | ψ ( t ) (cid:105) . The QFI quantifies the minimalpossible precision of estimating the imprinted phase θ in quantum interferometry [23]. The minimal precisionis given by the inverse of the quantum Fisher informa-tion F Q , ∆ θ (cid:62) / (cid:112) F Q . In general, the QFI value de-pends on the input state and generator of an interfero-metric rotation. The generator can be considered as thescalar product ˆΛ n = (cid:126) Λ · n . The vector (cid:126) Λ is composedof bosonic Lie algebra generators describing a given sys-tem. Specifically, it is (cid:126) Λ BI = { ˆ S x , ˆ S y , ˆ S y } for bimodaland (cid:126) Λ S = { ˆ J x , ˆ Q yz , ˆ J y , ˆ Q zx , ˆ D xy , ˆ Q xy , ˆ Y , ˆ J z } for spinorcondensates. The unit vector n determines the directionof rotation in the generalized Bloch sphere.The QFI value is given by the variance F Q = 4∆ ˆΛ n , (12)for pure states [42]. It is possible to find the generatorˆΛ n , for which the QFI reaches its maximum value [2].For pure states, this problem can be solved by noticingthat the variance in (12) can be written in terms of thecovariance matrixΓ ij [ | ψ ( t ) (cid:105) ] = 12 (cid:104) ˆΛ i ˆΛ j + ˆΛ j ˆΛ i (cid:105) − (cid:104) ˆΛ i (cid:105)(cid:104) ˆΛ j (cid:105) , (13)and then F Q = 4 n T · Γ[ | ψ ( t ) (cid:105) ] · n . (14) Therefore, one concludes that the maximal value of theQFI is given by the largest eigenvalue λ max of (13) whilethe direction of rotation n max by the eigenvector corre-sponding to λ max .There are two characteristic limits for the QFI value.The first one is the standard quantum limit (SQL) typ-ical for coherent states where the QFI is equal to N forbimodal system and to 4 N for spinor system [42]. When-ever the QFI value is larger than the SQL, the state isentangled [1]. The second is the Heisenberg limit whichbounds the value of the QFI from above, and it is equalto N for bimodal system and 4 N for spinor system [42].Here, we focus on the maximal value of (12) optimizedover n at a given moment of time t and the given in-put state | ψ ( t ) (cid:105) BI / S . In the case of bimodal system, themaximal QFI is F Q, BI = 4 λ max , BI , (15)where λ max , BI is the maximal eigenvalue of the 3 × i in (13) is replaced by ˆΛ BI ,i .In Appendix C we discuss the direction of interferometricrotation leading to the maximal value of the QFI. In thecase of spinor condensate, the QFI reads F Q, S = 4 λ max , S , (16)where this time λ max , S is the maximal eigenvalue of 8 × i is replaced by ˆΛ S ,i . Although thereare eight possible eigenvalues, only a few of them con-tribute to the maximal QFI value. It is because of theadditional constant of motion, namely magnetization,which introduces symmetry of covariance matrix, simpli-fies its form and diminishes the number of various valuesof λ S and directions of interferometric rotations n , seeAppendix C for details of calculations.In Fig. 5 we show an example of the QFI evolutionin the Josephson regime for | Λ | = 2 when | ψ ( t ) (cid:105) BI / S =ˆ U BI / S | ψ (0) (cid:105) BI / S (without optional rotation discussed inFig. 4 and in Section V). It was shown for bimodal con-densates that for | Λ | = 2 the unitary evolution gener-ates the fastest speed and amount of entanglement [41].This is because of the characteristic “ ∞ ” shape in themean-field phase portrait with the angle between in- andout-going constant energy lines equals to π/ N t/ ln(2 N ) for bimodal system and as t/ ln(8 N/
3) for spinor system (the difference in N comesfrom the energy unit chosen for both systems). This canbe interpreted as the appearance of the first maximum of F Q with Heisenberg scaling at t (cid:39) ln(2 N ) N ( t (cid:39) ln(8 N/ (a) (b) (c)FIG. 5. The scaling of the quantum Fisher information with N vs. time for the bimodal (a) and spinor (b) systems with | Λ | = 2. The values of N are given in the legend. (c) The maximal value of the QFI for spinor system, F max Q , S , versus Λ for N = 100 demonstrating that the maximal value of entanglement is generated for Λ (cid:39) − and spinor systems. The scaling can be explained us-ing a theory developed in [44] under two approximations.The first is the truncation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of mo-tion for expectation values of spin operators’ products.We truncate the hierarchy by keeping the first- and thesecond-order moments, which is equivalent to the Gaus-sian approximation. The second approximation is theshort-time expansion. The details of calculations are pre-sented in Appendix D for the bimodal and in AppendixE for the spinor systems. V. ENTANGLEMENT STABILIZATION ANDSTORAGE AROUND STABLE FIXED POINTS
The regular part of the initial evolution and structureof the mean-field phase space give a possibility of a stabi-lization scheme with nearly stationary value of the QFIat a relatively high level. The scheme consists of threesteps, as discussed in Fig. 4. The first step is unitaryevolution until the QFI reaches the value close to thefirst maximum. Then, an instantaneous pulse rotatesthe state through α BI around the ˆ S x axis, | ψ ( t +1 ) (cid:105) BI = e − iα BI ˆ S x | ψ ( t − ) (cid:105) BI (17)for the bimodal system, and through α S around the ˆ J zs axis, | ψ ( t +1 ) (cid:105) S = e − iα S ˆ J zs | ψ ( t − ) (cid:105) S (18)for the spin-1 system, where t − denotes the time just be-fore and t +1 after the rotation. Shortly before the rotationthe Husimi function of the state is highly stretched. Ro-tation throws the most stretched part of the state aroundstable regions of the phase space. Later on, for t > t +1 ,the state dynamics is governed by the unitary evolu-tion without any manipulations. However, it is trappedaround the two stable fixed points.An example of the QFI is presented in Fig. 6. Ananimation for time evolution of the Husimi function is shown in the Supplement Materials for the spinor sys-tem. A roughly stationary value of the QFI is obtainedin the long time limit. More interestingly, the twofoldincrease of the QFI value can be observed just after therotation. One might expect that the best rotation angleis π/ t up to20% does not spoil the scheme, but rather lowers theQFI value. Finally, we note that the rotation can alsobe performed after the QFI reaches the maximum. Theslight increase of the QFI value is observed as well. Thisis illustrated in Fig. 6. All in all, we conclude, that it isadvantageous to rotate the state in shorter times becauseof the fast gain in the QFI value.It is intuitive that the QFI value stabilizes in the longtime limit. When the state is located around the stablefixed point, the further dynamics are limited in this areaof phase space and are approximately ”frozen”. However,from the mathematical point of view it is non-trivial toshow that indeed the value of the QFI, and therefore theentanglement, does not decrease in time. In below, weprove this for the bimodal condensate.We assume that the direction of interferometric rota-tion just before the rotation is ˆΛ BI , n max ( t − ) ≈ √ ( ˆ S z − ˆ S y ) and therefore F Q, BI ( t − ) = 4∆ (cid:16) ˆ S z − ˆ S y √ (cid:17) , while afterthe rotation for t ≥ t +1 one has ˆΛ BI , n max ( t ) ≈ ˆ S z and F Q, BI ( t ) = 4∆ ˆ S z . This is a fairly good approximation,as one can see in Appendix C 1. The QFI after rotationcan be also written as F Q, BI ( t ) = 4 (cid:126) χ (cid:104) (cid:104) ˆ H BI ( t ) (cid:105) + (cid:126) Ω (cid:104) ˆ S x ( t ) (cid:105) (cid:105) , (19)where we used the relation (cid:126) χ (cid:104) ˆ S z (cid:105) = ˆ H BI + (cid:126) Ω ˆ S x em-ploying (1) and (cid:104) ˆ S z (cid:105) = 0. Next, we note that an averageenergy is conserved after rotation, (cid:104) ˆ H BI ( t ) (cid:105) = (cid:104) ˆ H BI ( t +1 ) (cid:105) , (a) (b) (c)(d) (e) (f)FIG. 6. The figures show QFI without (black lines) and with optional rotation at t = 2 .
37 (a), 3 .
08 (b) for bimodal and t = 1(d), 1 . N = 600 and N = 100, respectively. The two different values of rotation angle are considered α BI / S = π/ α BI / S = π/ π/ π/ π/ N = 600 and N = 100 for bimodal and spinor systems, respectively. while an average value of the ˆ S x operator is boundedfrom below and above, namely N ≥ (cid:104) ˆ S x (cid:105) ≥ − N . Thistwo properties lead to the inequality F Q, BI ( t ) ≥ (cid:126) χ (cid:20) (cid:104) ˆ H BI ( t +1 ) (cid:105) − (cid:126) Ω N (cid:21) . (20)The energy of the bimodal system after the rotation (17)reads (cid:104) ˆ H BI ( t +1 ) (cid:105) = (cid:126) χ (cid:104) ( ˆ S z ( t − )cos α BI + ˆ S y ( t − )sin α BI ) (cid:105)− (cid:126) Ω (cid:104) ˆ S x ( t − ) (cid:105) , and for α BI = π/
4, it equals to (cid:104) ˆ H BI ( t +1 ) (cid:105) = F Q, BI ( t − ) − (cid:126) Ω (cid:104) ˆ S x ( t − ) (cid:105) . Finally, one considers the latterterm in (20) to show that F Q, BI ( t ) ≥ F Q, BI ( t − ) , (21)for t ≥ t +1 as (cid:104) ˆ S x ( t − ) (cid:105) ≥ − N as well.The same reasoning can be used to demonstrate F Q, S ( t ) ≥ F Q, S ( t − ) for spinor system, and we providethe calculation in Appendix F. VI. THE PARITY OPERATOR AS ANEFFICIENT READOUT MEASUREMENT
The precision of estimation of the unknown phase θ can be estimated using the signal-to-noise ratio as δθ = ∆ ˆ S| ∂ θ (cid:104) ˆ S(cid:105)| (22) with ∆ ˆ S = (cid:104) ˆ S (cid:105) − (cid:104) ˆ S(cid:105) representing the variance of thesignal ˆ S of which an average value is to be measured.Generally speaking, the precision in the θ estimation ful-fils 1 δθ ≤ F Q . (23)As mentioned before, the QFI gives the highest possibleprecision on estimation of θ , but its measurement requiresextracting the whole state tomography [8]. On the otherhand, the inverse of the signal-to-noise ratio gives thelowest precision while it needs measurement of the firstand second moments of the observable ˆ S which is a bonusfrom the experimental point of view.On the one hand, in general ˆ S is unknown. On theother hand, in some cases it is known as for examplethe parity operator for the Greenberger-Horne-Zeilinger(GHZ) state [45, 46] or ˆ J z for the spinor system [47].Instead, the nonlinear squeezing parameter was recentlyproposed [24] to saturate the QFI value at short timesfor bimodal condensates. However, the measurement ofnonlinear squeezing parameter is related to the measure-ments of higher order moments and correlations.Here, we show that the inverse of signal-to-noise ratiowith the parity operator in the place of ˆ S in (22) when θ → S BI = ( − ˆ S x − N/ commuteswith the bimodal Hamiltonian, (cid:104) ˆ S BI , ˆ H BI (cid:105) = 0, andalso with the rotation operator (17), (cid:104) ˆ S BI , e − iα BI ˆ S x (cid:105) =0. When the initial state | ψ (0) (cid:105) BI is the eigenstate ofˆ S BI , we have ˆ S BI | ψ (0) (cid:105) BI = | ψ (0) (cid:105) BI , and consequentlyˆ S BI | ψ ( t ) (cid:105) BI = | ψ ( t ) (cid:105) BI . Finally, it is easy to show the re-lation ˆΛ BI , n max ˆ S BI = − ˆ S BI ˆΛ BI , n max even if one considersa general form of the generator of interferometric rota-tion ˆΛ BI , n max = a ˆ S z + b ˆ S y with any a + b = 1, seeAppendix C 1.We use all the above-mentioned properties of the stateand parity operator to calculate (22). To do this weexpand an average value of the parity operator up tothe leading terms in θ , obtaining BI (cid:104) ψ ( θ ) | ˆ S BI | ψ ( θ ) (cid:105) BI =1 − θ (cid:104) ψ ( t ) | ˆΛ , n max | ψ ( t ) (cid:105) BI + 0( θ ). Having that, thevariance in (22) can be expressed as∆ ˆ S BI = 4 θ (cid:104) ˆΛ , n max (cid:105) + 0( θ ) , (24)because (cid:104) ˆ S (cid:105) = 1. The leading terms of the derivativein respect to θ of an average value of the parity is simply ∂ θ (cid:104) ˆ S BI (cid:105) = − θ (cid:104) ˆΛ , n max (cid:105) + 0( θ ) . (25)Therefore, by inserting (24) and (25) into (22) it is pos-sible to show that the leading terms in θ of the inverse ofthe signal-to-noise ratio δθ − | θ =0 = 4∆ ˆΛ BI , n max , (26)saturate the QFI value according to (12) due to the factthat (cid:104) ˆΛ BI , n max (cid:105) = 0. Note, the above derivation holdsalso with optional rotation of the state because the parityand rotation operators commute.In Fig. 7 we demonstrate our finding for the most op-timal interferometer ˆΛ BI , n max given numerically in Ap-pendix C 1 (yellow dotted line) and simpler ˆΛ BI = ˆ S z operator (blue dashed line) without (a) and with (b) op-tional rotation that locates the state around stable fixedpoints. The perfect agreement can be noticed.Exactly the same reasoning can be applied for thespinor system but this time we define the parity as ˆ S S =( − ˆ J z,s − N . One can show, by simple algebra, that par-ity commutes with the spinor Hamiltonian, (cid:104) ˆ S S , ˆ H S (cid:105) = 0,and the optional rotation operator (18), (cid:104) ˆ S S , e − iα S ˆ J z,s (cid:105) =0. The initial state is the eigenstate of the parity oper-ator, ˆ S S | ψ (0) (cid:105) S = | ψ (0) (cid:105) S , and also any other state pro-duced by the unitary evolution, ˆ S S | ψ ( t ) (cid:105) S = | ψ ( t ) (cid:105) S . A (a) (b)(c) (d)FIG. 7. (a-b) QFI (black solid) from the maximal eigen-value of the covariance matrix (13) in bimodal system. Theerror from the inverse of the signal to noise ratio whenˆ S = ( − ˆ S x − N/ the parity measurement with the gener-ator of interferometric rotation given by the eigenvector ofthe maximal eigenvalue of the covariance matrix (yellow dot-ted line) and when ˆΛ BI , n max = ˆ S z (blue dashed line). Nu-merical results without rotation (a), and with rotation (17)when α BI = π/ Nχt / Λ = 2 .
36 (b) for N = 600. (c-d)QFI (black solid) from the maximal eigenvalue of the covari-ance matrix (13) in the spinor system for N = 100. Here,ˆΛ S , n max = ˆ J y,a and the inverse of the signal-to-noise ratio isshown by the orange dashed line in the case without optionalrotation (c), and with the optional rotation (18) by α S at t c (cid:48) / (cid:126) = 1 . S = ˆ J z with the same ˆΛ S , n max = ˆ J y,a . general form of the generator of interferometric rotationfor the spinor system should be ˆΛ S , n max = a ˆ J x,s + b ˆ J y,s with a + b = 1, see Appendix C. One can follow thesame calculations as for the bimodal system and con-sider the leading terms in θ of the inverse of the signal-to-noise ratio obtaining δθ − | θ =0 = 4∆ ˆΛ S , n max . Thelatter saturates the QFI value according to (12) because (cid:104) ˆΛ BI , n max (cid:105) = 0. The derivation also holds true with op-tional rotation of the state (18) as the parity and rotationoperators do commute. We illustrate our finding in Fig. 7without (c) and with (d) the optional rotation that lo-cates the state around stable fixed points using variousinterferometers. In addition, we also illustrates that thesimple signal ˆ S S = ˆ J z saturates the QFI value when theoptional rotation is not taken into account (see dashedgreen lines in Fig. 7 (c) and (d)). The latter readoutmeasurement is effective because the variance of magne-tization is a constant of motion. Therefore, one can usethe same treatment as in the case of parity to see thatindeed the inverse of signal-to-noise ratio with ˆ J z in theplace of ˆ S in (22) saturates the QFI value.Finally, note that the sensitivity from the inverse ofsignal-to-noise ratio might be resistance against phasenoise. This is the case when the operator describingthe phase noise does commute with the parity operator.Then, the sensitivity from (22) does not change even for aconvex mixture of quantum states, see calculation in [26].This fact is not in contradiction with the convexity of theQFI [57], which states that a convex mixture of quantumstates contains fewer quantum correlations than the en-semble average. VII. DISCUSSION AND CONCLUSION
In this work we have investigated theoretically the pos-sibility of the entanglement stabilization in bimodal andspin-1 condensates. Our method utilizes the structureof the mean-field phase space. In particular, twistingdynamics of the spin coherent state initiated around anunstable saddle fixed point is enriched by a single rota-tion which locates the state around stable center fixedpoints. This allows for the generation of non-Gaussianstates with the stable value of the QFI which exhibitsHeisenberg scaling with a pre-factor of the order of one.We analyzed the method numerically and analyticallyproving ( i ) the scaling of the QFI and time with totalatoms number, ( ii ) the lower bound of the QFI afteroptional rotation and ( iii ) the optimal parity enhancedreadout measurement.In this paper, we have ignored the effects arising fromany source of decoherence, such as a dissipative interac-tion with a heat reservoir or atomic losses. The deco-herence effects will degrade the sensitivity in the θ es-timation. If minimized, the entangled state stabilizedby the scheme proposed here yields a higher resolution.However, we must stress that decoherence effects will de-grade all schemes proposed to enhance interferometricmeasurements. Therefore, it might be necessary to makedetailed comparisons of schemes with the incorporationof decoherence.There is one other source of decoherence other than en-vironmental, namely detection noise, which we would liketo address in the context of the parity measurement. Inthe signal-to-noise ratio (22), the effect of detection noiseon moments of the operator ˆ S in the large atoms numberlimit is the same as if it was replaced by ˆ˜ S = ˆ S + ˆ δ S ,where ˆ δ S is an independent Gaussian operator satisfying (cid:104) ˆ δ S (cid:105) = 0 and (cid:104) ˆ δ S (cid:105) = σ [58]. Therefore, it is clear thatthe detection resolution σ (cid:46) ACKNOWLEDGMENTS
We thank K. Paw(cid:32)lowski, A. Smerzi and P. Treut-lein for discussion. EW and SM are supportedby the Polish National Science Center Grants DEC-2015/18/E/ST2/00760. AN is supported by Project no.2017/25/Z/ST2/03039, funded by the National ScienceCentre, Poland, under the QuantERA programme.
Appendix A: Spin-1 operators ˆ J x = 1 √ (cid:0) ˆ a † − ˆ a + ˆ a † ˆ a − + ˆ a † ˆ a +1 + ˆ a † +1 ˆ a (cid:1) , (A1)ˆ Q zx = 1 √ (cid:0) − ˆ a † − ˆ a − ˆ a † ˆ a − + ˆ a † ˆ a +1 + ˆ a † +1 ˆ a (cid:1) , (A2)ˆ J y = i √ (cid:0) ˆ a † − ˆ a − ˆ a † ˆ a − + ˆ a † ˆ a +1 − ˆ a † +1 ˆ a (cid:1) , (A3)ˆ Q yz = i √ (cid:0) − ˆ a † − ˆ a + ˆ a † ˆ a − + ˆ a † ˆ a +1 − ˆ a † +1 ˆ a (cid:1) , (A4)ˆ D xy = ˆ a † − ˆ a +1 + ˆ a † +1 ˆ a − , (A5)ˆ Q xy = i (cid:0) ˆ a † − ˆ a +1 − ˆ a † +1 ˆ a − (cid:1) , (A6)ˆ Y = 1 √ (cid:0) ˆ a † − ˆ a − − a † ˆ a + ˆ a † +1 ˆ a +1 (cid:1) , (A7)ˆ J z = ˆ a † +1 ˆ a +1 − ˆ a † − ˆ a − , (A8)where ˆ a m F is the annihilation operator of the particle inthe m F Zeeman component.
Appendix B: Anti-symmetric mean-field phase space
We will now address the equivalence of the anti-symmetric subspace. Similarly to the symmetric case,we are calculating an average value of (4) over the spincoherent state defined for the anti-symmetric subspaceas | ϕ, θ (cid:105) a = e − iϕ ˆ J z,a / e − iθ ˆ J y,a / | N (cid:105) a (B1)0where | N (cid:105) a = ˆ g † aN √ N ! | (cid:105) with ϕ ∈ (0 , π ) , θ ∈ (0 , π ).The spin coherent state (9) can be interpreted as a dou-ble rotation of the maximally polarized state | N (cid:105) a in theanti-symmetric subspace and is an eigenstate of ˆ J z,a withthe eigenvalue N . Similarly to symmetric sphere, it is lo-cated on the north pole of the Bloch sphere. In termsof spin-1 operators it reads | N (cid:105) a = e − iπ/ Q xy | N (cid:105) .Just as in the symmetric case the state with N atomsin the m F = 0 mode is located on the south poleof the same Bloch sphere. To illustrate an arbitrarystate | Ψ (cid:105) on the Bloch sphere we use Husimi function Q a ( ϕ, θ ) = |(cid:104) Ψ | ϕ, θ (cid:105) a | .An average value of the spin-1 Hamiltonian (4) overthe spin coherent state (B1) leads to H a = Λ2 (1 − z ) sin ϕ + z − , (B2)by keeping the leading and omitting the constant terms,and once again z = cos θ while Λ = − /q . Based on thedifference between (B2) and (11), as well as the form of(8), one can see that phase portrait for anti-symmetricalsubspace will be rotated by π/ z − axis. Appendix C: Structure of covariance matrix
It is interesting to find eigenvalues and eigenvectors forcovariance matrix in the case of both systems. It is 3 × ×
1. Bimodal system
In the case of the bimodal system the optimal gener-ator interferometric rotation can be found analyticallywhen Ω = 0, see [64, 65]. In the general case, theonly analysis can be done numerically and, therefore, wepresent it below.In Fig. 8(a) the black solid line shows the QFI (12)given by the maximal eigenvalue of the covariance ma-trix (13), and variances of various generators of interfer-ometric rotation ˆΛ n in direction n as indicated in thefigure caption. Indeed one can see that in the case with-out optional rotation, Fig. 8(c), initially the generatorof interferometric rotation is a superposition of ˆ S y andˆ S z (purple dot-dashed line in (a)) which saturates theQFI value. On the other hand, we also observe that thevariance of ˆ S z estimates well overall variation of the QFIin time. When the optional rotation (17) is applied, seeFig. 8(b), the optimal rotation axis is also given by ˆ S z (dashed line). Therefore we conclude that the QFI is well estimated by 4∆ ˆ S z while the optimal interferometric ro-tation is the z -axis of the Bloch sphere. (a) (b)(c) (d)FIG. 8. (a-b) The QFI (black solid line) from the maxi-mal eigenvalue of the covariance matrix (13), the QFI from(12) when ˆΛ n = ( ˆ S z − ˆ S y ) / √ n = ( ˆ S z + ˆ S y ) / √ n = ˆ S z (blue dashed line). (c-d) An illustration of the bestdirection of interferometric rotation n max , i.e. i -th compo-nent of the covariance matrix eigenvector (13) correspondingto the highest eigenvalue. The x component is plotted withblack dashed line, y by the dark gray solid line and z with thelight gray dot-dashed line. Left panels: without optional ro-tation of the state (17). Right panels: when optional rotationis applied at t = 2 .
36 with α BI = − π/
6. The calculationsare performed for N = 600.
2. Spinor system: fixed magnetization
The Hamiltonian (4) conserves the magnetizationwhich means that we have [ ˆ H, ˆ J z ] = 0. Thus the fol-lowing occurs for an arbitrary state | Ψ ϕ (cid:105) = e − iϕ ˆ J z e − i ˆ Ht | Ψ (cid:105) = e − i ˆ Ht e − iϕM | Ψ (cid:105) . (C1)An action of the rotation operator e − iϕ ˆ J z results in thephase factor given by the product of rotation angle andmagnetization. On the other hand, the QFI has the samevalue for both | Ψ (cid:105) and | Ψ ϕ (cid:105) , and therefore one has thecondition 4 n T · Γ[ | Ψ (cid:105) ] · n = 4 n T · Γ[ | Ψ ϕ (cid:105) ] · n , and so Γ[ | Ψ (cid:105) ] = Γ[ | Ψ ϕ (cid:105) ]. From the definition of covari-ance matrix (13) one can see thatΓ[ | Ψ ϕ (cid:105) ] ij = (cid:104) Ψ |
12 (ˆ˜Λ i ˆ˜Λ j + ˆ˜Λ j ˆ˜Λ i ) | Ψ (cid:105) − (cid:104) Ψ | ˆ˜Λ i | Ψ (cid:105)(cid:104) Ψ | ˆ˜Λ j | Ψ (cid:105) , (C2)1where ˆ˜Λ i = e iϕ ˆ J z ˆΛ i e − iϕ ˆ J z . The rotation of the vector (cid:126) Λ S components gives: e iϕ ˆ J z ˆ J x e − iϕ ˆ J z = ˆ J x cos ϕ − ˆ J y sin ϕ (C3a) e iϕ ˆ J z ˆ J y e − iϕ ˆ J z = ˆ J y cos ϕ + ˆ J x sin ϕ (C3b) e iϕ ˆ J z ˆ J z e − iϕ ˆ J z = ˆ J z (C3c) e iϕ ˆ J z ˆ Q xy e − iϕ ˆ J z = ˆ Q xy cos 2 ϕ + ˆ D xy sin 2 ϕ (C3d) e iϕ ˆ J z ˆ D xy e − iϕ ˆ J z = ˆ D xy cos 2 ϕ − ˆ D xy sin 2 ϕ (C3e) e iϕ ˆ J z ˆ Q yz e − iϕ ˆ J z = ˆ Q yz cos ϕ + ˆ Q zx sin ϕ (C3f) e iϕ ˆ J z ˆ Q zx e − iϕ ˆ J z = ˆ Q zx cos ϕ − ˆ Q yz sin ϕ (C3g) e iϕ ˆ J z ˆ Y e − iϕ ˆ J z = ˆ Y . (C3h)Therefore, one can distinguish the following groups ofoperators: { ˆ J x , ˆ J y } , { ˆ D xy , ˆ Q xy } , { ˆ Q zx , ˆ Q yz } , { ˆ J z } , { ˆ Y } ,which rotations can be described with the operator:ˆ R ϕ = (cid:18) cos φ − sin φ sin φ cos φ (cid:19) . In fact, we can see thatΓ[ | Ψ ϕ (cid:105) ] = M ϕ · Γ[ | Ψ (cid:105) ] · M Tϕ , (C4)where the rotation matrix M ϕ is equal to cos φ − sin φ φ φ φ φ − cos φ φ φ − sin 2 φ φ cos 2 φ . From the relation (C4) and Γ = Γ T , one obtains a setof equations that determine the possible zero values ofcovariance matrix elements, for example: (cid:40) Γ cos φ + Γ sin ϕ = Γ cos φ − Γ sin ϕ, Γ cos φ − Γ sin ϕ = Γ cos φ − Γ sin ϕ, which shows that Γ = Γ and Γ = 0. Solving allpossible remaining equations will give conditions for allthe elements of the covariance matrix, namely Γ = Γ ,Γ = Γ , Γ = − Γ . Except for Γ , Γ and elementslisted in (C5), all the remaining elements are zero. On theother hand Γ is defined by variance of ˆ J z , which standsfor fluctuations of magnetization, thus this element is 0as well. In the subspace of zero magnetization we ar-rive with the block diagonal structure of the covariancematrix:Γ = Γ s ⊕ Γ a ⊕ [Γ ] ⊕ [Γ ] ⊕ [Γ ] ⊕ [0] , (C5)whereΓ s = (cid:18) Γ Γ Γ Γ (cid:19) , Γ a = (cid:18) Γ − Γ − Γ Γ (cid:19) . (C6) A diagonalization of the above matrix gives four pos-sible generators of interferometric rotation [66]ˆΛ S , = ˆΛ − γ ˆΛ (cid:112) γ , (C7)ˆΛ S , = ˆΛ , (C8)ˆΛ S , = ˆΛ , (C9)where γ ij = (Γ jj − Γ ii − (cid:112) (Γ ii − Γ jj ) + 4Γ ij ) / (2Γ ij ).The corresponding values of the QFI are given by thevariance F Q, S = 4∆ ˆΛ S ,ij . (C10)There are three possible values which depend on time. Itis worth noting that in the short times dynamics, it isˆΛ S , (or ˆΛ S , as they are equivalent) that determinesthe QFI value. Moreover, we observe that it can be ap-proximated by ˆΛ without significant change in the QFIvalue, namely ˆΛ S , (cid:39) ˆΛ = ˆ J x . This is illustrated inFig. 9. FIG. 9. An illustration of optimal generators of interfero-metric rotation ˆΛ S ,ij for spinor system with fixed magnetiza-tion given by (C7)-(C9) calculated for N = 100 atoms. TheQFI optimalized over all directions n is shown by the blacksolid line. The corresponding values of the QFI for a givengenerator derived in the main text are: F Q, S = 4∆ ˆΛ S , (which equals to 4ˆΛ S , ) is marked by the yellow dashedline, F Q, S = 4∆ ˆΛ S , by the dashed brown thin line and F Q, S = 4∆ ˆΛ S , by the thin blue line. The case withˆΛ S ,ij = ˆ J x is also shown for comparison by the dashed redline. In addition, the QFI with ˆΛ S ,ij = √ ( ˆ J xs − ˆ J ys ) andˆΛ S ,ij = √ ( ˆ J xa + ˆ J ya ) are shown by the purple dot-dashedand green dashed-double-dotted lines. The latter illustratesthat the QFI value before the first maximum is given by F Q, S = 4∆ (cid:16) ˆ J xs − ˆ J ys √ (cid:17) .
3. Spinor system: non-zero fluctuations ofmagnetization
We consider here the more general case of the rotatedstate | Ψ ϕ (cid:105) = e − iϕ ˆ J z,s e − i ˆ Ht | Ψ (cid:105) , used by us in the main text to locate dynamics aroundstable fixed points. Here, ˆ J z,s = ( ˆ D xy + √ Y ). Theanalysis presented in the previous subsection is not validbecause [ ˆ J z,s , ˆ H ] (cid:54) = 0. Moreover, the state after rota-tion is no longer in the subspace of zero magnetizationbut it is spread over all subspaces of even magnetization.Therefore, it has non-zero fluctuations of magnetization.To calculate elements of the covariance matrix we usedEq. (13), where an average is taken over a general state | k (cid:105) = (cid:80) M,n C M,n | n, M + N − n, n − M (cid:105) which co-efficients of decomposition in the Fock state basis are C M,n ≡ C n,M + N − n,n − M resulting from the symmetryof rotation around ˆ J z,s . The summation over n de-pends on the sign of the M : from max(0 , M/ , M ) tomin( M, N + M , M + N ) while − N < M < N . Due tothe rotation, the system has non-zero variance of mag-netization ∆ ˆ J z which is constant in time. In addition,the possible eigenvalues of ˆ J z can only be even, i.e. M ∈ {− N, − N + 2 , ..., N − , N } assuming N is even aswell due to symmetry of rotation operator ˆ J z,s . There-fore, C M,n = C − M,n − M .We can distinguish operators that change magneti-zation by ±
1, they are { ˆ J x , ˆ Q yz , ˆ J y , ˆ Q zx } , by ± { ˆ D xy , ˆ Q xy } and by 0: { ˆ Y , ˆ J z } . The mean value of op-erators from the group { ˆ J x , ˆ Q yz , ˆ J y , ˆ Q zx } is zero sincethe state is spread over subspaces of even magnetiza-tion. Moreover, a mean value of product of operatorsthat change magnetization by odd value are zero. We usethis fact while calculating the covariance matrix elementsΓ ij : with subscript i for the operator from the group { ˆ J x , ˆ Q yz , ˆ J y , ˆ Q zx } and j from { ˆ D xy , ˆ Q xy , ˆ Y , ˆ J z } . The sec-ond property that should be taken into account is thesymmetry of the state, namely C M,n = C − M,n − M , whichsets the elements like Γ or Γ to zero.After careful consideration of all covariance matrix el-ements, one can show that it simplifies toΓ S = Γ s ⊕ Γ a ⊕ Γ r , (C11)for the spinor system, whereΓ s = (cid:18) Γ Γ Γ Γ (cid:19) , Γ a = (cid:18) Γ Γ Γ Γ (cid:19) , Γ r = Γ
00 Γ Γ
00 Γ . Diagonalization of (C11) gives the following eigenval-ues: λ ( ± )S ,ij = Γ S ,ii + Γ S ,jj ± (cid:113) (Γ S ,ii − Γ S ,jj ) + 4Γ ,ij , (C12)where the pairs of indexes ( i, j ) are one of(1 , , (3 , , (5 , , (6 , λ (+) S,ij for whichthe four possible generators of interferometric rotationare ˆΛ S ,ij = ˆΛ j − γ ij ˆΛ j (cid:113) γ ij (C13)where γ ij = (Γ jj − Γ ii − (cid:112) (Γ ii − Γ jj ) + 4Γ ij ) / (2Γ ij ).The corresponding values of the QFI determined by(C13), namely F Q, S = 4∆ ˆΛ S ,ij , (C14)are demonstrated in Fig. 10. FIG. 10. An illustration of optimal generators of inter-ferometric rotation ˆΛ S ,ij for spinor system with fluctuatingmagnetization given in (C13) calculated for N = 100 atoms.The relevant example discussed in the main text for statesafter the rotation around ˆ J zs by π/ t = 1 .
6. The QFIoptimized over all n is shown by the black solid line. Thecorresponding values of the QFI for particular generators areshown with ˆΛ S, (light pink dash-dotted line), ˆΛ S, (yel-low dotted line), ˆΛ S, (green dash-double-dotted line), ˆΛ S, (purple dashed line). Finally, the QFI with ˆΛ S ,ij = ˆ J xs andˆΛ S ,ij = ˆ J ya are shown by the red and orange solid lines, re-spectively. The latter demonstrates that the QFI value afterthe rotation can be approximated well by F Q, S = 4∆ ˆ J ya . Appendix D: Scaling of the QFI for bimodal system
In order to analyze scaling of the QFI with the sys-tem size, we use a general theory developed in [44].3One starts with equations of motion for operators ofspin components which involve terms that depend onthe first-order and second-order moments. Then, thetime evolution of the second-order moments depends onsecond- and third-order moments, and so on. It leadsto the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchyof equations of motion for expectation values of operatorproducts. We truncate the hierarchy by keeping the first-and the second-order moments. (cid:104) ˆ S i ˆ S j ˆ S k (cid:105) (cid:39) (cid:104) ˆ S i ˆ S j (cid:105)(cid:104) ˆ S k (cid:105) + (cid:104) ˆ S j ˆ S k (cid:105)(cid:104) ˆ S i (cid:105) + (cid:104) ˆ S k ˆ S i (cid:105)(cid:104) ˆ S j (cid:105)− (cid:104) ˆ S i (cid:105)(cid:104) ˆ S j (cid:105)(cid:104) ˆ S k (cid:105) . (D1)Let us first rotate the Hamiltonian (1) around the x -axis of the Bloch sphere through π/
4. The reason isas follows: there is nonzero angle between the constantenergy line outgoing from the saddle fixed point and the z -axis. This angle is close to π/ y -axis, see Fig. 2. Note, the largest fluctuationsthat determine the QFI value are now located along the y -axis. Next, we introduce a small parameter ε = 1 /N and transform spin components into ˆ h j = √ ε ˆ S j whilethe commutation relations to [ˆ h i , ˆ h j ] = i √ ε ˆ h k (cid:15) ijk . Therotated Hamiltonian (1) isˆ H = 1 √ ε (cid:16) ˆ h z + ˆ h y + ˆ h z ˆ h y + ˆ h y ˆ h z − a ˆ h x (cid:17) , (D2)where a = 2 ε Ω /χ , the energy unit is set to (cid:126) χ/ (2 √ ε ) andwe introduced dimensionless time τ = χt/ (2 √ ε ).Equations of motion for expectation values s j = (cid:104) ˆ h j (cid:105) and second order moments δ jk = (cid:104) ˆ h j ˆ h k + ˆ h k ˆ h j (cid:105) − (cid:104) ˆ h j (cid:105)(cid:104) ˆ h k (cid:105) relevant for our purposes are˙ s x = ( δ zz − δ yy ) , (D3)˙ δ zz = − δ zz s x − aδ yz , (D4)˙ δ yy = 4 δ yy s x + 2 aδ yz , (D5)The initial spin coherent state | , π/ (cid:105) BI gives the fol-lowing initial conditions: s x (0) = 1 / (2 √ ε ) and δ zz (0) = δ yy (0) = 1 / a → δ zz ( τ ) = δ zz (0) e − f ( τ ) with f ( τ ) = 4 (cid:82) τ s x ( t ) dt . The analysis of non-homogeneousequation can be done by setting δ zz ( τ ) = C ( τ ) e − f ( τ ) with C ( τ ) = C (0) − a (cid:82) τ δ yz ( t ) e f ( t ) dt = δ zz (0) + Φ( τ ).The part Φ( τ ) is very small and it can be omitted be-cause of two reasons. Firstly, Φ( τ ) is of the order ofsmall parameter ε . Secondly, in the short time expan-sion (up to the second order) one can indeed see thatΦ( τ ) (cid:39) Φ(0) + ˙Φ(0) τ = 0 due to δ yz (0) = 0. There-fore, we conclude that the solution of (D4) can be wellapproximated by the solution of its homogeneous part. The same analysis can be performed on Eq. (D5) lead-ing to δ yy ( τ ) = δ yy (0) e f ( τ ) . Eq. (D3) takes the form˙ s x ( τ ) = − sinh [ f ( τ )], that has an analytical solutionwhen one expands the function f ( τ ) up to the first orderin Taylor series f ( τ ) (cid:39) f (0)+ ˙ f (0) τ . The self-consistencycondition gives f (0) = 0 and ˙ f (0) = 4 s x (0). The approx-imated solution for s x takes the form [43] s x ( τ ) = s x (0) − cosh(4 s x (0) τ ) − s x (0) , (D6)while the variance in the y direction reads δ yy = δ yy (0) e s x (0) τ − sinh(4 sx (0) τ ) − sx (0) τ [4 sx (0)]2 . (D7)It can be shown by maximization of the QFI overthe time resolves in the scaling of the first maximum as χt max (cid:39) ln(2 N ) /N . The leading term of the QFI max-imum at the best time gives F Q, BI (cid:39) ˆ S y (cid:39) ε δ yy (cid:39) e ε ≈ . N . Appendix E: Scaling of the QFI for spinor system
In the case of spinor system we follow the same track ofcalculations as presented in the previous Appendix. Firstwe rotate the spin-1 Hamiltonian (4) around the ˆ J z,s by π/ J y,s axis of theBloch sphere in the symmetric subspace. However, thistime the angle is two times smaller because commuta-tion relations [ ˆ J i,s , ˆ J j,s ] = i J k,s (cid:15) ijk contain the factor 2.After the rotation of Hamiltonian, one introduces thesmall parameter ε = 1 /N , transforming spin componentsinto ˆ h j = √ ε ˆ J j , ˆ q j = √ ε ˆ Q j . The rotated and re-scaledHamiltonian readsˆ H = − √ ε (cid:20) (cid:16) ˆ h x,s + ˆ h y,s (cid:17) + (cid:16) ˆ h y,a cos π h z,a sin π (cid:17) + (cid:16) ˆ h z cos π q xy sin π (cid:17) + a ˆ n − a ˆ n (cid:21) , (E1)where ˆ n = √ ε ˆ N , ˆ n = √ ε ˆ N , a = 2 q/ε while the energyunit is √ ε | c (cid:48) | / τ = √ εt | c (cid:48) | / (cid:126) .Equations of motion for expectation values s j = (cid:104) ˆ h j (cid:105) and second order moments δ j,k = (cid:104) ˆ h j ˆ h k + ˆ h k ˆ h j (cid:105) − (cid:104) ˆ h j (cid:105)(cid:104) ˆ h k (cid:105) are much more complex as for bimodal con-densates, but one can find the general structure quitesimilar. The relevant for our purposes are˙ s zs = − ( δ ys,ys − δ xs,xs ) − √
24 ( δ ya,ya − δ xa,xa ) , (E2)˙ δ xs,xs = − δ xs,xs s zs − aδ xs,ys , (E3)˙ δ ys,ys = 2 δ ys,ys s zs + aδ xs,ys , (E4)4 TABLE I. List of commutation relations among SU(3) algebra generators and spin components in the symmetric and anti-symmetric subspace used in this paper. T i T j ˆ J x ( ˆ J x,s ) ˆ Q yz ( ˆ J y,s ) ˆ Q zx ( ˆ J x,a ) ˆ J y ( ˆ J y,a ) ˆ J z ˆ D xy ˆ Q xy ˆ Y ˆ J z,s ˆ J z,a ˆ J x ( ˆ J x,s ) 0 2 i ˆ J z,s − i ˆ Q xy i ˆ J z − i ˆ J y − i ˆ Q yz i ˆ Q zx − i √ Q yz − i ˆ Q yz − i ˆ Q yz ˆ Q yz ( ˆ J y,s ) − i ˆ J z,s − i ˆ J z − i ˆ Q xy i ˆ Q zx i ˆ J x i ˆ J y i √ J x i J x i ˆ J x ˆ Q zx ( ˆ J x,a ) i ˆ Q xy i ˆ J z i ˆ J z,a − i ˆ Q yz i ˆ J y − i ˆ J x − i √ J y − i ˆ J y − i ˆ J y ˆ J y ( ˆ J y,a ) − i ˆ J z i ˆ Q xy − i ˆ J z,a i ˆ J x − i ˆ Q zx − i ˆ Q yz i √ Q zx i ˆ Q zx i ˆ Q zx ˆ J z i ˆ J y − i ˆ Q zx i ˆ Q yz − i ˆ J x i ˆ Q xy − i ˆ D xy i ˆ Q xy − i ˆ Q xy ˆ D xy i ˆ J y,s − i ˆ J x − i ˆ J y i ˆ Q zx − i ˆ Q xy i ˆ J z Q xy − i ˆ Q zx − i ˆ J y i ˆ J x i ˆ Q yz i ˆ D xy − i ˆ J z − i ˆ J z i ˆ J z ˆ Y i √ J y,s − i √ J x i √ J y − i √ Q zx J z,s i ˆ J y,s − i J x i ˆ J y − i ˆ Q zx − i ˆ Q xy i ˆ J z J z,a i ˆ J y,s − i ˆ J x i ˆ J y − i ˆ Q zx i ˆ Q xy − i ˆ J z for symmetric operators, and˙ s za = −
12 ( δ ys,ys − δ xs,xs ) − √
22 ( δ ya,ya − δ xa,xa ) , (E5)˙ δ xa,xa = −√ δ xa,xa s za − aδ xa,ya , (E6)˙ δ ya,ya = √ δ ya,ya s za + aδ xa,ya , (E7)for anti-symmetric operators. In Table I we listed com-mutation relations useful to obtain (E2) - (E7).The initial spin coherent state | , π (cid:105) S gives the fol-lowing non-zero initial values for s zσ (0) = − / √ ε and δ xσ,xσ (0) = δ yσ,yσ (0) = 1 for σ = s, a . Equations forexpectation values for first and second moments in theshort-time expansion show that some terms appearingin the above equations are zero if their average valuesare initially zero, e.g. δ z,z = 0 , δ qxy,xy (cid:39)
0. We did notput such terms in the final forms of Eqs. (D3) - (D5).The equations for symmetric and anti-symmetric opera-tors are very similar to the one obtained for the bimodalsystem. There are two differences: (i) s zσ (with σ = s, a )in (E2) and (E5) play the role of s x in (D3) and (ii) sym-metric and anti-symmetric subspaces are coupled to eachother in (E2) and (E5). The coupling makes the scalinganalysis a little more intricate. Taking both into account,one can use solutions from the previous Appendix andfind s zs ( τ ) = s zs (0) −− cosh(2 s zs (0) τ ) − s zs (0) − √
24 cosh( √ s za (0) τ ) − √ s za (0) , (E8) s za ( τ ) = s za (0) −−
12 cosh(2 s zs (0) τ ) − s zs (0) − √
22 cosh( √ s za (0) τ ) − √ s za (0) . (E9)Note, the symmetric and anti-symmetric subspaces arecoupled to each other and this has to be taken into ac-count while explaining the scaling of δ xσ,xσ .In order to explain the scaling of the first maximum,one needs to find a derivative of the variances in respect to time. Now, there are two equations for σ = s and σ = a that help to express relations among cosh hav-ing different arguments. The maximization of the QFIover the time provides the scaling of the maximum tobe | c (cid:48) | t max / (cid:126) = ln(8 N/
3) by keeping leading terms in ε . Finally, the value of the maximum of the QFI gives F Q, S (cid:39) ˆ J xs (cid:39) e − / N ≈ . N when consideringthe leading terms in ε . Appendix F: Explanation of the QFI stabilizationafter rotation in the long times limit for spinorsystem
Here we use the same reasoning as presented in themain text concerning the bimodal system at the end ofSection V. We assume that the direction of interferomet-ric rotation just before the rotation for spinor system isˆΛ S , n max ( t − ) ≈ ˆ J xσ ± ˆ J yσ √ , with sign ”+” for σ = s and sign”-” for σ = a , and therefore F Q, S ( t − ) = 4∆ (cid:16) ˆ J xσ ± ˆ J yσ √ (cid:17) ,while after the rotation for t ≥ t +1 one has ˆΛ S , n max ( t ) ≈ ˆ J ya and F Q, BI ( t ) = 4∆ ˆ J ya . It is a fairly good approx-imation, as demonstrated in Appendix C and in Figs. 9and 10.The QFI after rotation for t ≤ t +1 can be also writtenas F Q, S ( t ) = 4 (cid:34) − N (cid:104) ˆ H S ( t ) (cid:105) c (cid:48) − (cid:104) ˆ J xs ( t ) (cid:105) − (cid:104) ˆ J z ( t ) (cid:105) + q (cid:104) ˆ N ( t ) (cid:105) (cid:35) , (F1)where we used (4). 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