Bose-Einstein condensate soliton qubit states for metrological applications

By utilizing Bose-Einstein condensate solitons, optically manipulated and trapped in a double-well potential, coupled through nonlinear Josephson effect, we propose novel quantum metrology applications with two soliton qubit states. In addition to steady-state solutions in different scenarios, phase space analysis, in terms of population imbalance - phase difference variables, is also performed to demonstrate macroscopic quantum self-trapping regimes. Schr\"odinger-cat states, maximally path-entangled ($N00N$) states, and macroscopic soliton qubits are predicted and exploited for the distinguishability of obtained macroscopic states in the framework of binary (non-orthogonal) state discrimination problem. For arbitrary phase estimation in the framework of linear quantum metrology approach, these macroscopic soliton states are revealed to have a scaling up to the Heisenberg limit (HL). The examples are illustrated for HL estimation of angular frequency between the ground and first excited macroscopic states of the condensate, which opens new perspectives for current frequency standards technologies.


Introduction
Nowadays, nonlinear collective mode formation and interaction in Kerr-like medium represent an indispensable platform for various practically important applications in time and frequency metrology [1,2], spectroscopy [3,4], absolute frequency synthesis [5], distance ranging [6]. In photonic settings frequency combs are proposed for these purposes [7]. The combs occur due to the nonlinear mode mixing in special (ring) microcavities, which possess some certain eigenmodes. Notably, bright soliton formation emerges with vital phenomena accompanying micro-comb generation [8]. Physically, such a soliton arises due to the purely nonlinear effect of temporal self-organization pattern occurring in an open (driven-dissipative) photonic system. However, because of the high level of various noises in the system they can be hardly explored for purely quantum metrological purposes.
Instead, atomic optics, which operates with Bose-Einstein condensates (BECs) at low temperatures, provides a suitable platform for various quantum devices that may be useful for metrology and sensing tasks [9]. In particular, so-called Bosonic Josephson junction (BJJ) systems, established through two weakly linked and trapped atomic condensates, are at the heart of the current quantum technologies in atomtronics that considers atom condensates and aims to design (on-chip) quantum devices. The condensates in this case represent low dimensional systems and may be manipulated by magnetic and laser field combinations. In this sense, they represent an alternative to optical analogues.
The nonlinear coupled-mode theory admits solution of (1) that simply represents a quantummechanical superposition where wave functions Ψ 1 ( ) and Ψ 2 ( ) characterize the condensate in two wells. For weakly interacting atoms one can assume that where Φ 1 ( ) and Φ 2 ( ) are ground-and first-order excited mode state wave functions possessing energies 1 and 2 , respectively; 1 ( ) and 2 ( ) are time-dependent functions. If the particle number is not too large, GPE (1) may be integrated in spatial ( -dimension) leaving only two condensate variables 1,2 ( ), cf. [30]. In particular, Φ 1 ( ) and Φ 2 ( ) may be timeindependent Gaussian-shape wave functions obeying different symmetry. Practically, this two-mode approximation is valid for the condensates of several hundreds of particles [55]. The condensate in this limit is effectively described by two (macroscopically populated) modes as a result.

Quantization of coupled solitons
The sketch in Fig. 1 explains the two-soliton system described in our work. If trapping potential ( ) is weak enough and the condensate particles interact not so weakly, the ansatz solution (3) is no more suitable. Especially we would like to mention condensates with a negative scattering length which admit a bright soliton solution for Ψ 1,2 ( , ) in (2). In fact, in this case one can speak about two-soliton solution problem for GPE (1) without trapping potential ( ) known in classical theory of solitons [56].
In quantum theory we deal with a bosonic field operatorˆ( , ) ∝ˆ1 +ˆ2 instead of (2), wherê 1,2 ≡ˆ1 ,2 ( , ) are field operators which correspond to mean-field amplitudes Ψ 1,2 ( , ) defined in (2). We assume that experimental conditions allow the formation of atomic bright solitons in each well of harmonic double-well potential. In particular, these conditions may be realized be means of manipulation with weak trapping potential ( ). In an experiment this manipulation may be performed with the help of a dipole trap and laser field.
Then, considering linear superposition state like (4) we can write the total Hamiltonianˆfor two BEC solitons in the formˆ=ˆ1 +ˆ2 +ˆ, whereˆ( = 1, 2) is the Hamiltonian for condensate particles in -th well; whileˆaccounts the coupling between two wells due to the soliton overlapping. In the second quantization form we explicitly haveˆ= The annihilation (creation) operators of bosonic fields denoted asˆ(ˆ †) with = 1, 2 obey the commutation relations: Fig. 1. Sketch of probability density distribution |Ψ| 2 versus spatial coordinates and , 2D projection of the 3D coupled condensates trapped in a double-well (dashed green curve) and harmonic (dashed magenda curve) potentials, respectively. Shadow regions display 1D condensate wave packets projections; they represent a secant-shape in -direction, and Gaussian-shape in the transverse directions.
In the Hartree approximation for a large particle number, >> 1, one can assume that the quantum -particle two-soliton state is the product of two-soliton states and can be written as [57][58][59] where Ψ ( , ) are unknown wave functions, |0 ≡ |0 1 |0 2 is a two-mode vacuum state. The state (6) is normalized as Ψ Ψ = 1, and the bosonic field-operatorsˆact on it aŝ Applying variational field theory approach based on the ansatz Ψ ( , ) we obtain the Lagrangian density in the form: where we suppose − 1 ≈ and omit the common term .
Noteworthy, from (8) one can obtain the coupled GPEs for Ψ -functions as where = 2 − 1 is the energy (frequency) spacing.
Set of Eqs. (9) lead to the known problem for transitions between two lowest self-trapped states of condensates in the nonlinear coupled mode approach if we account (3) for Ψ ( , ) condensate wave functions representation [29,30].
On the other hand, Eqs. (9) can be recognized in the framework of soliton interaction problem that may be solved by means of perturbation theory for solitons [56]. In particular, in accordance with Karpman's approach we can recognize in (9) terms proportional to = Ψ * Ψ 2 + 2|Ψ | 2 Ψ , , = 1, 2, ≠ as perturbations for two fundamental bright soliton solutions. Physically, implies the nonlinear Josephson coupling between the solitons.
In this work we establish a variational approach for solution of Eqs. (9), cf. [38]. For the weakly coupled condensate states, i.e. for 0, set of Eqs. (9) reduces to two independent GPEs: which possess bright (non-moving) soliton solutions In the case of ≠ 0 and for non-zero inter-soliton distance , we examine ansatzes for Ψ ( , ) in the form In particular, our approach presumes the existence of two well distinguished solitons (separated by the small distance , with the shape preserved) interacting through dynamical variation of the particle numbers ( ≡ ( )) and phases ( ≡ ( )), which occurs in the presence of weak coupling between the solitons. In other words, and should be considered as time-dependent (variational) parameters. Setting (12) in (8) where = ( 2 − 1 )/ ( 1,2 = 2 (1 ∓ )) is the particle number population imbalance; Physically, Ω is an angular frequency spacing between the ground and first excited macroscopic states of the condensate; it represents a vital (measured) parameter for metrological purposes in this work. In (13) we introduce the notation Λ = 2 2 /16 and define the functionals where Δ ≡ 4 is a normalized distance between solitons. Finally, by using Eq. (13) for the population imbalance and phase-shift difference, and Θ, we obtain the set of equations where dots denote derivatives with respect to renormalized time = Λ . In contrast to the problem with coupled Gaussian-shape condensates (cf. [29,30]), the solutions of Eqs. (15) crucially depend on features of governing functionals ( , Δ) and ( , Δ). In Appendix we represent some analytical approximations for ( , Δ) and ( , Δ), which we exploit further.

SS solution for
The SS solutions of (15) play a crucial role for metrological purposes with coupled solitons cf. [37]. We start from the SS solution 2 = 1 of Eq. (15a) by setting the time-derivatives to zero. As seen from (14), in the limit of maximal population imbalance, 2 = 1, and do not depend on Δ and approach (16b) Notably, in quantum domain the SS solutions (17) admit the existence of quantum states with maximal population imbalance = ±1 and phase difference. The latter depends on the frequency spacing Ω, which is the subject of precise measurement with maximally path-entangled 00 -states in this paper. Below we perform the analysis of the SS solutions of Eqs. (15) in two limiting cases Ω ≠ 0, Δ 0 and Ω 0, Δ ≠ 0.

SS solutions for Θ = 0, and Δ 0
To find the SS solutions we rewrite (15b) as for Θ = 0 and for Θ = , respectively. In Appendix we represent a polynomial approximation for , functionals (14). Since the equations obtained from (18) and (19) are quite cumbersome, here we just briefly analyze the results.
In the limit of closely spaced solitons and Θ = 0, the population imbalance at equilibrium depends only on Ω and obeys Similarly, for fixed soliton phase difference Θ = we have We plot the graphical solutions of Eqs. (20), (21)   (22a) = 0, Θ = ; (22b) 2 ≈ 0.17 Θ = . (22c) As seen from Eq. (22b), at relative phase Θ = (19) possesses three solutions: a parametrically unstable solution occurs at = 0 and two degenerate SS solutions appear for = ± 0 . 0 varies from 0.41 at Δ ≈ 0 to 0.64 at Δ ≈ 2.8 for non-zero soliton inter-distance, respectively. For Δ > 2.8 these SS solutions do not exist. In Fig. 3 we represent the more general analysis of SS solutions for Θ = 0 as functions of inter-soliton distance Δ for different Ω. For that we exploit the sixth-order polynomial approximation, see Appendix. In particular, at Ω 0 there exists one solution at = 0, stable at Δ ≤ Δ ≈ 0.5867. For Δ > Δ this solution becomes parametrically unstable.
On the other hand, for Δ > Δ Eqs. (18) possesses the degenerate SS solutions similar to the ones at Θ = . The bifurcation for population imbalance occurs at Δ = Δ ; in Fig. 3 the + (upper,positive) and − (lower, negative) branches characterize this bifurcation. In the vicinity of Δ we can consider ± = ± 0 , where At Ω ≠ 0 the behavior of SS solutions depending on distance Δ complicates -see green curves in Fig. 3. The solid curves correspond to SS solutions for different Δ, while the dotted ones describe the unstable solutions. From Fig. 3 it is clearly seen that for |Ω| > 0 there is no bifurcation for population imbalance and two stationary solution branches ± occur with At relatively large values of parameter Ω/Λ only one SS solution exists -see red curve in Fig. 3.

Small amplitude oscillations
We start our analysis here from small amplitude oscillations close to the SS solutions (22). For that we linearize Eqs. (15) in the vicinity of (22), assuming 0 ≤ Δ < 0.6 and Ω << 1. The first assumption allows to use the approximation of , -functionals by the fourth-degree polynomial, see Appendix.
In the vicinity of SS points determined by Eq. (22c), we obtain -phase oscillations characterized by , and 0 determined in (22c). For Ω 0 and Δ = 0 angular frequency is ≈ 1.42 that is much smaller than in zero-phase regime. The analysis of (15) in the vicinity of (22b) reveals that this solution is parametric unstable, and highly nonlinear behavior is expected. Indeed, direct numerical simulation demonstrates anharmonic dynamics plotted in Fig. 4. For 0 < | | < 0.5 the nonlinear regime of self-trapping is observed, which turns into nonlinear oscillations at | | > 0.5.
The analysis of SS solution (17) reveals the strong sensitivity to -perturbation, when the condition 2 = 1 is violated, the high-amplitude nonlinear oscillations occur. On the other hand, the solution (17) is robust to phase perturbations, which is an important property for metrology. , → 0 and Eqs. (15) i.e. the population imbalance is a constant in time and the running-phase regime establishes. For large but finite Δ, SS solution = ± 0 with 0 → 1 exists for the zero-phase regime, Θ = 0; for example, for Δ = 10 the SS population imbalance is 0 ≈ 0.96.

Phase-space analysis
The dynamical behavior of the coupled soliton system can be generalized in terms of a phase portrait of two dynamical variables and Θ, as shown in Figs. 5 and 6.
In Fig. 5 we represent − Θ phase-plane for Ω = 0 and for different (increasing) values of distance Δ. We distinguish three different dynamic regimes. Solid curves correspond to oscillation regime when ( ) and Θ( ) are some periodic functions of normalized time, see e.g. (25) and red curve in Fig. 4. The dashed curves in Fig. 5 indicate the self-trapping regime, when ( ) is periodic and the sign of does not change, c.f. (28) and blue curve in Fig. 4. Physically, this is the macroscopic quantum self-trapping (MQST) regime characterized by a nonzero average population imbalance, when the most of particles are "trapped" within one of the solitons. At the same time, the behavior of phase Θ( ) may be quite complicated but periodic in time. On the other hand, for the running-phase regime depicted by dashed-dotted curves Θ( ) grows infinitely, see green curve in Fig. 5(b). Due to the symmetry that takes place at Ω = 0, the running-phase can be achieved only with the MQST regime, see Fig. 5.
As seen from Fig. 5, central area of nonlinear Rabi-like oscillations between the ground and first excited macroscopic states happen for a relatively small inter-soliton distance Δ and are inherent to zero-phase oscillations, see Fig. 5(a). As we discussed before, at Δ = Δ ≈ 0.5867 this area splits into two regions characterized by the MQST regimes, Fig. 5(b). This splitting occurs due to the bifurcation of population imbalance, cf. black curve in Fig. 3. These regions are moving away from each other with growing Δ, see Fig. 5(c-f). It is worth noticing the bifurcation effect and occurrence of MQST states at zero-phase regime for coupled solitons in Fig. 1 disappear for the condensates described by Gaussian states, cf. [29,30].
The phase trajectories inherent to -phase region 2 < Θ < 3 2 stay weakly perturbed until the second critical value Δ ≈ 2, when the MQST regime in Fig. 5(d) changes to Rabi-like oscillations (Fig. 5(e)) and then approaches the running-phase at Δ ≈ 6, see Fig. 5(f). At large enough Δ the particle tunneling vanishes and the zero-phase MQST domains arise in the vicinity of population imbalance = ±1, Fig. 5(f). The phase dynamics corresponds to the running-phase regime with = const, see Fig. 5(f) and (29). For non-zero Ω, the phase portrait becomes asymmetric, Fig. 6. To elucidate the role of Ω we study the soliton interaction for a given inter-soliton distance Δ = 0.75 > Δ that corresponds to the one after the bifurcation. As seen from Fig. 6(a), the phase portrait does not change significantly for small Ω, cf. Fig. 5(b).
One of the SS solutions for zero and -phase regimes disappears with increasing Ω; the running-phase regime establishes, Fig. 6(b). Further increasing of Ω leads to vanishing of the SS solution for zero-phase, Fig. 6(c).
Thus, phase portraits in Figs. 5, 6 provide the existence of degenerate SSs for coupled solitons by varying inter-soliton distance Δ and Ω. Such solutions, as we show below, may be exploited for the macroscopic superposition soliton states formation in the quantum approach.

Phase estimation with macroscopic qubit states.
Suppose that some quantum system is prepared in state | , which carries information about some parameter Γ that we would like to estimate; in this work we are interested in fundamental bound for POVM measurements and consider pure states of the quantum system. In quantum metrology the sensitivity of some parameter Γ estimation is described by the error propagation formula given as (cf. [28]) where |(ΔΠ) 2 | = |Π 2 | − |Π| 2 is the variance of fluctuations of some operator Π that corresponds to the measurement procedure. Typically, such procedures are based on appropriate interferometric schemes and use quantum superpositions, which contain required information about estimated parameter Γ. In the case of SC-states, which presume macroscopic (non-orthogonal in general) states, the measurement procedure requires some specification. In particular, we assume that quantum system may be prepared in the state | that we represent as In (31) is a relative (estimated) phase between states | 0 and | 1 , which are defined as |Φ 1 and |Φ 2 are two macroscopic states representing two "halves" of the SC-states. In particular, operatorsΠ = | | realize a projection onto the superposition of states |Φ 1,2 , which generally are not orthogonal to each other obeying the condition Simultaneously, we require the states in Eqs. (32) to fulfill the normalization condition = , , = 0, 1.
On the other hand, the situation with = 0 characterizes in (33), (35) completely orthogonal states |Φ 1,2 that become possible if |Φ 1,2 approaches two-mode Fock states. In other words, this is a limit of the 00 -state for which coupled solitons are examined.
Then, we define a complete set of operatorsΣ , = 1, 2, 3 (cf. [50]) which obey the SU(2) algebra commutation relation. The meaning of sigma-operators is evident from their definitions (36). Due to properties (34) the states | are suitable candidates for macroscopic qubit states, which we can define by mapping | 0 → |0 and | 1 → |1 , respectively, cf. [43,60]. In this form we can use them for POVM measurements defined with operators [61] Importantly, current quantum (photonic) technologies permit POVM tomography, cf. [44]. Average values of sigma-operators in (36) may be obtained by means of (31), (34) and look like From (38) it follows that only Σ 2,3 contain the information about desired phase . To estimate the sensitivity of phase measurement it is possible to assume that = Γ and use Eq. (30) with measured operatorΠ ≡Σ 2 . Taking into account Σ 2 2 = 1 for the variance of fluctuations (ΔΣ 2 ) 2 we obtain Finally, from Eqs. (30), (39) for the phase error propagation we obtain that clearly corresponds to HL of arbitrary ( -linearly dependent) phase estimation and explores the sigma-operator measurement procedure. Notice this procedure can be mapped onto the parity measurement, cf. [37,50].

Soliton SC-qubit states.
The phase estimation procedure described above enables to use two-soliton quantum states (6) for frequency quantum metrology purposes. It is instructive to represent soliton wave functions (12) in the form Degenerate SS solutions of population imbalance obtained before (see e.g. (17) and Fig. 3) enable to prepare various superposition soliton states for quantum metrology purposes. In particular, for Θ = 0 from (6) we obtain for two "halves" of the SC-state, where In Eqs. (43) + and − are two SS solutions corresponding to upper and lower branches in Fig. 3, respectively. In (43) we omit the common unimportant term − ( /2+ 1 ) . In particular, for Ω ≈ 0, we have ± → ± 0 .
The scalar product for state (42) is where characterizes solitons wave functions overlapping. Assuming non-zero and positive Ω for one can obtain In Fig. 7 we establish the principal features of coefficients (35) and parameter (see the inset in Fig. 7) as functions of Δ. The value of Ω plays a significant role in the distinguishability problem for states |Φ 1 and |Φ 2 . In particular, at Ω = 0, in the bifurcation point Δ = Δ = 0.5867 we have = 1 that implies indistinguishable states |Φ 1 and |Φ 2 , see red curve in the inset of Fig. 7. The coefficients 1,2 → ∞ in this limit.
From Fig. 7 it is evident that the coefficients 1,2 rapidly approach (due to the factor ) the levels 1 = 1, 2 = 0 (completely distinguishable macroscopic SC soliton states), when Δ increases. In this limit, as seen from Fig. 3, ± approaches ± 0 , and from (45) we obtain Practically, in this limit red and green curves coincide in Fig. 7. Thus, we can exploit states (42) for metrological measurement purposes for arbitrary phase estimation that we describe in Sec. 5.1. The phase may be created after soliton SC-state formation by means of some additional soliton interaction or collisions.

Frequency measurement, Γ ≡ Ω.
Now we represent a particularly important case of measurement of angular frequency Ω that characterizes energy spacing between the ground and first excited macroscopic states. The SS solutions (17), which correspond to the maximal population imbalance 2 = 1, allow to prepare the maximally path-entangled superposition state, a.k.a. 00 -state. As seen from (17b), the solution with = 1 exist, when −2( − 1) ≤ Ω/Λ ≤ 2( + 1). Similarly, the domain of solution = −1 is −2( + 1) ≤ Ω/Λ ≤ 2( − 1). To achieve the superposition 00 -state formation we require both solutions to exist simultaneously. This restricts the domain of allowed Ω as −2( − 1) ≤ Ω/Λ ≤ 2( − 1). Substituting = ±1 into (41) we obtain which are relevant to the 00 -state's two "halves" defined as Considering the superposition of states (48) and omitting unimportant common phase (0.5 (+) − 2 ) we arrive to (cf. (31)) that represents the 00 -state of coupled BEC solitons for our problem. In (49) is the phase shift that contains Ω-parameter required for estimation. Comparing Eq. (50) with (31) we can conclude that the 00 -state's "halves" | 0 and |0 in (50) may be associated with states | 0 and | 1 , respectively. To estimate the sensitivity of Ω measurement we use (30) with measured operatorΠ ≡Σ defined as (cf.(36)(c)) Since the states (48) are orthogonal, the mean value of (51) is Fig. 8 demonstrates Σ as a function of Ω/Λ. Notice, the interference pattern in Fig. 8 exhibits essentially nonlinear behavior for measured Σ . The variance of fluctuations (ΔΣ) 2 for the measured sigma-operator reads as Now, by using (30), (53) we can easily find the propagation error for frequency Ω estimation as Equation (54) is non-applicable for Ω = 0 since the denominator in (54) turns to zero. We choose the optimal estimation area for the frequency Ω, where the best sensitivity is reached, in the vicinity of the domain border at Ω/Λ → 2( − 1). In this limit (54) Equation (55) exhibits one of the important results of this work: for a given Λ that characterizes atomic condensate peculiarities Eq. (55) demonstrates Heisenberg scaling for frequency measurement sensitivity, cf. (40).

Conclusion
In summary, we have considered the problem of two-soliton formation for 1D BECs trapped effectively in a double-well potential. These soliton Josephson junctions analytical solutions and corresponding phase portraits exhibit the occurrence of novel macroscopic quantum selft-trapping (MQST) phases in contrast to the condensates with only Gaussian wave functions. With these soliton states, we have also explored the formation of the Schrödinger-cat (SC) state in the framework of the Hartree approximation. In particular, we have analyzed the distinguishabiltiy problem for binary (non-orthogonal) macroscopic states. Compared to the results known in the literature, see e.g. [37], finite frequency spacing Ω leads to the distinguishable macroscopic states for condensate solitons. This circumstance may be important for the experimental design of the SC-states.
The important part of this work is devoted to the applicability of predicted states for quantum metrology. By utilizing the macroscopic qubits problem with interacting BEC solitons, one can apply the sigma-operators to elucidate the measurement and subsequent estimation of arbitrary phase, that linearly depends on the particle number, up to the HL. It is worth mentioning the sigma-operators relate to the POVM detection tomography procedure. On the other hand, the phase estimation procedure for the phase-dependent sigma-operator can be realized by means of the parity measurement technique that produces the same accuracy for phase estimation. We have shown that in the limit of soliton state solution with the population imbalance | | = ±1 the coupled soliton system admits the maximally path-entangled 00 -state formation. The feasibility of frequency Ω estimation at the Heisenberg level is also demonstrated.
In this paper we have not examined the loses and decoherence effects for the quantum soliton system depicted in Fig. 1. Recently in [39] we examined this problem for quantum solitons possessing simple Josephson coupling. From the experimental point of view, the recent BEC soliton experiments with lithium condensates demonstrated, that the collisions may be recognized as one of the most detrimental effects [35]. In particular, as we established in [39], the three-body and one-body losses may be unimportant at the time scales of few tens of milliseconds that relevant to experimental conditions in [35]. Moreover, the purely quantum analysis of the problem demonstrated that the superposition of Fock states, occurring at some specific parameters of the system, behaves robust to few particle losses. The Detailed analysis of this problem for interaction of solitons depicted in Fig. 1 we will publish elsewhere.

Disclosures
The authors declare no conflicts of interest.

Appendix: Approximation of functionals and .
Solutions of Eqs. (15) strictly depend on functionals and and their derivatives ≡ / and ≡ / defined in (14). In Fig. 9 we represented them as two-dimension surfaces given in − Δ plane. From Fig. 9 (a,b) it is seen that , approaches zero for large Δ excluding edge domains where | | 1. The behavior of , , as it is follows from Fig. 9 (a,b), is not so evident for small Δ inherent to 0 ≤ Δ < 1.5. Thus for numerical estimations we use the polynomial approximations of and .