Nonlinear quantum metrology with moving matter-wave solitons

The estimation of some parameters with super-Heisenberg (SH) sensitivity, i.e. beyond Heisenberg limit, is one of the principal problems for current quantum metrology. We propose to use Bose–Einstein condensate quantum bright solitons for this purpose. We have shown that solitons, as quantum nonlinear structured field objects, allow SH phase estimation even with coherent probes in the framework of a nonlinear metrology approach. To achieve ultimate scaling in nonlinear phase estimation, which is 1/N3, we examine soliton phase shift occurring due to interaction of weakly coupled 1D solitons. We have shown that steady states of coupled solitons can be used for the formation of maximally path-entangled N00N-state providing minimal propagation error of solitons’ relative distance, momentum, and atom–atom scattering length parameter.


Introduction
Over the past decade, rapidly growing experimental facilities in quantum technologies have led to an increasing interest in quantum metrology and sensing [1,2]. Many important applications, such as gravitational waves detection interferometers [3], frequency standards and atomic clocks [4,5], magnetometers [6], quantum gyroscopes [7], thermometry of living cells [8], or high resolution imaging in biology [9] require to achieve ultimate sensitivity for some vital parameters, defined by quantum limitations of measurement accuracy. The measuring schemes, which should be highly phase-sensitive, propose an estimation of the phase shift containing all the information about the required parameter, see [10,11].
In quantum optics, measurements for metrological purposes typically base on Mach-Zehnder interferometers operating with few photons (discrete variables) or with bright light beams containing a large average number of photons (so-called continuous variables [9]). In atomic optics domain, matter-wave interferometers commonly explore two internal state operations corresponding to a two-mode optical interferometer design [1].
Previously it was shown (see e.g. [12]) that two weakly coupled Bose-Einstein condensates (BEC) represent a versatile platform for metrological purposes in the framework of linear phase-sensitive measurements. In particular, Josephson and Fock regimes of interacting condensates allow to enhance accuracy of phase measurement beyond standard quantum limit (SQL), N 1 , where N is an average number of particles involved in a measurement scheme. Moreover, it was shown that so-called Heisenberg limit, N −1 , might be saturated with maximally entangled N-particle N00N-state in the framework of two-mode model approach [13,14]. In this sense, N00N-state produced with a moderate number of particles represents a vital problem for current metrology and sensing. Some proposals are realized with a few number of particles [15][16][17][18]. The formation of robust N00N-states with atomic condensates is discussed in [19][20][21] where authors proposed to use two-site Bose-Hubbard model that is valid under some physical conditions for interacting condensates [22].
In [23] we examined weakly coupled one dimensional matter-wave bright solitons operating in a nonmoving regime. We have shown that soliton superposition, N00N-state, is formed in the limit of Schrödingercat state only under certain requirements on parameters of the system. The sensitivity beyond Heisenberg limit was predicted.
In this sense it is important to answer the question: can we obtain any benefits for metrological purposes with interacting condensates considered beyond two-mode approximation? In other words, what is the fundamental limit for the phase estimation and measurement if we use quantum field structures like solitons?
Before we answer these questions in the paper, some important features of solitons in quantum domain are worth mentioning. In particular, quantum field theory predicts that the average value of quantum soliton field amplitude represents superposition of classical solitons having different phase velocities [24]. Physically, it means that nonlinear phase contribution to quantum soliton behavior is non-trivial: the phase self-modulation compensates dispersion only on average, see [24]. The squeezing effect of quantum fluctuations and entanglement manifest this behavior, see [25,26].
Strictly speaking, considering quantum solitons we deal with nonlinear quantum metrology approach based on many-body interactions [27,28]. The nonlinear phase shift is the subject of measurement and estimation in this case [29,30]. It is shown that super-Heisenberg (SH) scaling, which is beyond Heisenberg limit, might be attained even with coherent (semiclassical) probes [31].
The material of the paper is arranged as follows. In section 2 we consider ultimate scaling for nonlinear phase estimation accessible with quantum matter-wave soliton which we map onto single-mode (Goldstone) Hamiltonian. In section 3 we represent quantum field theory of moving solitons suitable for metrological purposes. We analyse various dynamical regimes for that. In section 4 the conditions and prerequisites of N00Nstate formation with matter-wave solitons are established. Soliton coordinate and momentum represent new additional variables in this case which introduce some new important peculiarities in the measurement procedure of phase characteristics in the framework of quantum metrology. The sensitivity of coupled solitons' parameters estimation is discussed in section 5. In section 6 we summarize the results obtained.

Quantum solitons as a metrological tool
We start with the description of BEC given in Heisenberg representation. The dimensionless Hamiltonian describing dilute 1D atomic condensate looks like ) is a quantum field annihilation (creation) operator obeying bosonic commutation relations. We suppose that condensate is strongly trapped in the transverse yz plane that implies 2D harmonic potential, and U tr (x) is a trapping potential in the third x dimension that is of our interest. In equation (1) u a a 2 sc 0 p = | | characterizes Kerr-like atomic nonlinearity, a sc is an s-wave scattering length (in the paper we focus on attractive particles with negative scattering length which is already accounted in (1)), a m 0 0  w = is a characteristic trap size and ω 0 is a harmonic trap frequency in a radial direction. All the variables in (1) are given in dimensionless units. In particular, all space variables are normalized on a 0 : x y z x a y a z a , , , , 0 0 0  , and time is normalized on ω 0 : t t 0 w  . For a moderate number of particles x t , ŷ ( )admits factorization [19] x t a t x , where a t ( ) is a time dependent operator characterizing quantum properties of a single condensate mode, Ψ(x) is a C-number responsible for condensate spatial distribution, which is taken independent on N.
Let us now examine bright (non-moving) soliton solution case. At first, it is instructive to examine the problem in classical limit. Classical (C-number) description implies convenient Gross-Pitaevskii equation , y y  ( ) ( )if condensate fluctuations are completely neglected [32]. The bright (non-moving) soliton solution of equation (7) is (see (2)) and obey normalization condition, , where N N = á ñ is an average particle number. In the framework of field theory it is possible to consider classical Hamilton function corresponding to equation (7) with U tr =0, which looks like (see [33]) Then taking into account quantization condition (3) one can represent corresponding Hamiltonian operator as Thus, equation (11) represents effective (single mode) Hamiltonian in quantum domain for bright soliton solution, see (4)- (6). One can see that the nonlinear structure of soliton field provides much higher power in respect to condensate quantum field (mode) operators â † and â, see (4) and (11).
In the framework of quantum metrology we are interested in precise estimation of some parameter χ bounded by a quantum Cramer-Rao bound [34,35] where ν is a number of experimental runs, F Q is a quantum Fischer information. Without loss of generality, the unknown value χ;Ω j might be inferred from the measurement procedure characterized by a unitary operator , where Ĝ is a generator of state transformation. In the case of pure states, the Fisher information is characterized by dispersion of Ĝ [30]. As a result, one can show that for enough small χ=1 and G N k =ˆ, the χ-parameter estimation scales as Thereafter we remove n -factor for simplicity. It follows from (14), that ultimate precision of measurement a condensate effective energy, χ=Ω 0 , determined by (4) is which clearly corresponds to familiar Heisenberg limit [35]. An accurate measurement of χ=Ω 2 ;u-parameter according to its definition in (6) for nonlinear ('Kerr-like') phase shift implies maximal scaling It is important that one can surpass the Heisenberg limit (15) and obtain N 1 3 2 c D µ measuring nonlinear phase shift with coherent probes, see (16) and [31].
The equation (14) demonstrates one important result of this paper for quantum solitons considering them in a single mode approximation: maximal scaling is

Quantum model of coupled matter-wave solitons
3.1. Quantum field approach From the practical point of view it is more convenient to consider an interferometer consisting of two-mode condensate trapped in a double-well potential [23]. The Hamiltonian of the system under consideration is where H ĵ ( j=1, 2) is the Hamiltonian for condensate particles in jth well, see (1); while H int accounts the interwell coupling between two sites. In the second quantization form we explicitly have where κ denotes the inter-well tunneling rate and depends on the distance between the traps, see figure 1. Schrödinger picture is more suitable for the analysis of the system described by (18). Further we restrict ourselves by the Hartree approach justified in a large particle number limit [23]. The N-particle ground state vector, N Yñ | , can be written in this case as a product of N single-particle states | | is a two-mode vacuum state; Ψ 1 and Ψ 2 are unknown time-dependent soliton 'wavefunctions' for which normalization condition is satisfied. In this case, the bosonic field annihilation operators act on the state vector (19) as follows: We can find the functions Ψ j by variational approach, conventionally used in the field theory [19]. In particular, using (19) and (21) with (18) we obtain the Lagrangian density of coupled condensates in the form: Notice that in (22) we have omitted the common factor, N, describing the mean total number of particles. It is a constant so, as a common factor of Lagrangian terms, it makes no contribution to soliton dynamics. Varying (22) for the independent field variables Ψ 1 and Ψ 2 , we obtain coupled equations Since N?1 in equation (23) we suppose that N−1≈N. ¾  = ¾  = | | | | and relative distance δ which are measured via solitons' nonlinear relative phase estimation. The platform is represented by two highly asymmetric (cigarshape) condensates which are trapped in a double-well potential. d is the distance between trap centers, it can be explored by tailoring tunneling rate, κ.
Equation (23) in the limit of κ=0 admit exact moving soliton solution, see (8): where x 0j and P j play the roles of jth soliton initial average center position and momentum, respectively. In equation (24) we use normalization condition that leads to (20).

Basic equations
We suppose that weak coupling between solitons does not change the form of soliton envelope for which we consider wave-function ansatz, see (24), in the form: where N j (t), θ j (t), X j (t) and P j (t) are jth soliton's particle number, phase, coordinate and momentum, respectively, which now are time-depended (variational) parameters. Thereafter, using Galilean invariance, we examine soliton properties in the reference frame supposing P 1 =−P 2 in (25). By plugging (25) With the Lagrangian (26), one can obtain the following variational equations for canonically conjugate soliton parameters: where dimensionless time τ=2κt is used.

Soliton dynamics 4.1. Small-amplitude oscillations
The set of equations (28) characterizes a model of interacting solitons in the presence of their motion, see [23,36]. This motion introduces some new important features which might be practically useful in the case of quantum measurement realization. We start our analysis from small-amplitude inter-well oscillations occurring for the condensate population imbalance, z, see [37]. Fortunately, in this limit it is possible to approximate functional (27) For numerical estimations we examine 3 7 Li condensate containing N≈10 3 attractive particles. The scattering length, a sc =−1.45 nm, is a tunable variable through Feshbach resonance [38,39]. Trap characteristic length, a 0 ≈7 μm, achieved with ω≈2π×29 Hz. The condensate density in the trap center, n Na 0 3 = -, in this case is about 3×10 12 cm −3 . The particle interaction energy, UN a N ma k ), that is clearly seen in figure 2(b). To examine temporal behavior of the distance between solitons, δ, and momentum, P, we combine (28c) and (28d) obtaining one second-order differential equation for δ: z P 1 3 cos 3 cos 6 sin . 33 In (33) one can omit rapidly oscillating terms characterizing population imbalance, z, and phase difference, θ, see figures 2(a)-(d). As a result, we obtain¨1  Our estimations and figure 2 clearly demonstrate the important relations between characteristic frequencies which are ω 0 /ω π >1, Ω 0 /Ω π <1, see [37].
The analysis of (28) is quite complicated in a general case. Figure 4 reflects essentially nonlinear behavior of solitons' parameters. The dynamics starts from the running regime of the phase difference, figure 4(a). The parameters of the system correspond to self-trapping for population imbalance in this case. Then, the system enters to large amplitude (nonlinear) Rabi-oscillations shown in figure 4(b). The figure 4(a) shows that the phase, θ, admits oscillations in this case.
It is important to emphasize that these features of z and θ are inherent to moving solitons only and caused by the other soliton parameters influence, see [23,36]. Actually, the parameters δ and P behave unstable and possess highly anharmonic but small amplitude persistent oscillations.

The steady-states
The set of equations (28) for static (non-moving) solitons admits steady-state solutions which are important for quantum measurement purposes [23]. Particularly, we are interested in an extreme case of the population imbalance, z=±1. From (28) we have  35 In this limit one can evaluate functional (27) analytically as . Thus, the measurement procedure should be performed within the time t meas <τ cr . The numerical estimations performed for Lithium condensate solitons with δ 0 ;0.1 and P;0.25 lead to time τ cr ;0.08 which corresponds to experimentally accessible values of δ 0 ;0.7 μm, P 0.036 m 1  m - and t cr ;2.16 ms taken in physical units.
As a result, equation (36) can be taken as time-independent for small τ and steady-state solution of (35a) looks like z a 1; 37 Notably, in the limit of non-moving solitons we put P=0 in (37b) and the phase difference takes the form arccos 2 ], see in [23].

Projection measurement procedure
The results obtained in (37) with population imbalances, z=±1, enable to consider a macroscopic quantum superposition N00N-state. Two 'halves' of the N00N-state can be obtained substituting z=±1 into (25) and then into (19) where x x sech . The N00N-state now represents a quantum superposition of (38): where we omitted the common phase factor e N i 2 q and introduced the phase difference between two soliton states; θ 0 is the phase difference defined in (37b).

Note that equation (37b) is valid if inequality
A L < | | is fulfilled. Physically, this condition implies certain restrictions for Λ-parameter that determines N00N-state formation domain. In the limit of non-moving solitons, one gets 0<Λ<π/2. On the other hand, N00N-state formation becomes impossible since A 0  as far as P  ¥.
Consider a projectile measurement procedure with state (39) introducing an operator (see [40]): As a result, for the average value of Σ-operator (40), we obtain N cos 41 As seen from (41), the solitons' phase shift, θ 0 , occurs in combination with a particle number, N, and as a result, is super-resolved. Actually, an arbitrary parameter, χ, relaying to θ 0 can now be estimated with sensitivity determined by the error propagation formula (see [41]): Factor N on the right-hand side in (43) indicates that the Heisenberg limit can be achieved if θ 0 depends on χ linearly. To overcome this limit, we use a nonlinear metrology approach, see [27,28,30].

Solitons' distance and momentum measurements
In general, the solitons' phase difference, θ 0 , is a complicated quantity containing the information about atomic condensate material parameters, solitons' position, and momentum differences. Hence, these parameters might be subjects of estimation in a quantum domain. We start from accuracy of δ 0 or P-parameter measurements. Practically, such measurements seem to be actual in the framework of high resolution imaging and quantum lithography exploiting matter-wave peculiarities, see [13,14]. The SQL of the measurements can be derived from the following simple arguments. Suppose that the phase, θ, between two interfering counter-propagating atomic matter waves is θ=Pδ/ÿ (here we use actual physical units). The uncertainty in θ-measurement implies minimum detectable displacement Δδ=ÿΔθ/P at fixed P. the SQL of phase measurement is , for displacement measurement we obtain Similar arguments bring us to the SQL formulation for momentum difference measurement It is important to stress that since the measurement time is small enough, we do not consider SQL occurring due to time dependent measurements for the soliton position and momentum, see [42].
Taking into account (37b) and definition (43), the solitons' position difference, χ≡δ 0 , can be estimated with precision We can perform an estimation of soliton momentum difference in separate or subsequent measurements due to uncertainty relation that takes place for the quantum soliton coordinate and momentum [24].
is relevant to numerical estimations considered in the paper.  Formally equation (49) predicts better scaling for momentum estimation than (48) if Pδ 0 /2 is close to πk/2 (k=0, 1, 2, K). However in this case A-parameter tends to Λ and domain of accessible N collapses. In practice, equation (48) demonstrates better accuracy that obtained when Pδ 0 /2 approaches πk.

Measurement of condensate material parameters
The soliton N00N-state (39) provides measurement of χ≡u 2 -parameter with error, see [23], The result obtained in (50) is in agreement with maximal scaling predicted in equation (17). Figure 6 demonstrates SH sensitivity (50) of u 2 measurement performed with fixed (known) values of P, δ 0 and κ. It is important to stress that for experimentally accessible condensate parameters, all the curves are situated essentially below the Heisenberg limit. Equation (50) diverges when the denominator goes to zero. In this limit, we are not able to use equation (50) for our purposes. In this limit, we are not able to use equation (50) for our purposes, see figure 6.

Conclusion
In summary, we have theoretically studied the ultimate sensitivity of measurement of matter-wave quantum solitons' parameters using the nonlinear metrology approach. We have shown that nonlinear structured quantum fields, which are bright quantum solitons, provide a unique opportunity to achieve SH scaling ( N 1 3 2 µ ) even with coherent probes. More accurate description of the problem being under discussion is based on the studies of two weakly-coupled elongated BECs trapped in a double-well potential in zy plane. Various dynamical regimes for solitons' parameters: population imbalance, relative phase, momentum, and coordinate, are examined in detail to elucidate experimentally accessible parameters for the estimation procedure. We have shown that steady-state solutions for population imbalance and phase difference exist within some parameters' window (at relatively short times) and provide maximally entangled N00N-state formation with moving quantum solitons. Then, we use soliton N00N-state to perform the estimation procedure of solitons' parameters by using the solitons' nonlinear relative phase. Particularly, the propagation measurement error of solitons' parameters, namely the relative coordinate and momentum, and u 2 -parameter, proportional to atom-atom scattering length squared, scales as 1/N 3 . Applying these parameters' measurements for current quantum metrological purposes looks promising.
In the paper we have focused mostly on (projective) measurement procedure with soliton N00N-states to elucidate phase estimation beyond Heisenberg limit. Notably, the same SH limit might be attained with parity measurement scheme, see [23,43]. Let us briefly discuss how to create N00N-states with solitons described above. We are confident that it is a non-trivial task and it opens new challenges both in theory and experiment. We find the protocol suggested in [20] the most suitable for these purposes. One can prepare N00N-state dynamically starting from the state C N n n 0 e , , where N n n , -ñ | is a two-mode Fock state, C n characterizes initial (binomial) distribution, and θ (0) is an initial phase difference. Next, it is necessary to construct appropriate two-soliton Hamiltonian which is relevant to a two-mode approach in the framework of procedure described in equations (7)-(12) Then, applying an evolution operator with this Hamiltonian after some time interval (defined by characteristic parameters of solitons) one can obtain superposition (Schrödinger-cat or N00N) states like in equations (38), (39) Notably, as it was shown in [20], two halves of N00N-state are shifted by π in θ-phase domain which is in agreement with results that we exploit in equations (38), (39). Actually, the phase difference θ 0 =θ 0 (0) in (37b) represents a steady-state solution and goes to π in the non-moving solitons limit for critical value Λ=1.58, see [23].
Obviously, decoherence and losses represent very important problem at the stage of N00N-state formation [20]. We discussed various aspects of this problem for non-moving quantum bright solitons in [23]. Many-body effects including single and two-body losses should be considered [44]. In forthcoming publications, we plan to examine the influence of these effects on our model in details.