Quantum metrology beyond Heisenberg limit with entangled matter wave solitons

D. V. Tsarev; S. M. Arakelian; You-Lin Chuang; Ray-Kuang Lee; A. P. Alodjants

doi:10.1364/OE.26.019583

1. Introduction

Nowadays quantum metrology has become one of the fascinating areas in modern quantum physics, which deals with new approaches to the measurement, control, and estimation of physical parameters for achieving ultimate accuracy and exploring all facilities of current quantum technologies [1–9]. Apart from classical measurement theory, a quantum approach predicts the so-called quantum Cramer-Rao (QCR) bound 〈(δϕ_est)〉_ϕ ≥ [νF(ϕ)]⁻¹ for arbitrary physical parameter ϕ within a set of ν trials through Fisher information F(ϕ) [3,4]. In particular, phase estimation requires high precision measurement, which can be realized both in optical [10–12] and atomic systems [13, 14]. The existence of standard quantum limit (SQL) sets a constraint on the linear phase shift (ϕ) measured with the error σ_ϕ ~ N^−1/2. Here, N is the average number of particles.

Surpassing SQL in the phase measurement has been demonstrated experimentally with two-mode systems, such as Mach-Zehnder interferometers (MZI), gyroscopes, and lithography devises, where non-classical squeezed or correlated states are applied as the input states [15–19]. For the linear phase measurement, one can achieve the Heisenberg limit with the accuracy

σ_{ϕ} \geq C_{0} N^{- 1},

which gives the limiting case on QCR bound related to the single mode passing [5, 6], C₀ is some constant here. According to the Heisenberg limit approach with non-classical states such as squeezed states, it is required a very large amount of squeezing. Hence, the generation of squeezed states in optical [10–12, 20] or atomic [21] physics domain represents one possible solution to improve quantum measurements beyond SQL.

On the other hand, in theory it is proved that for arbitrary two-mode quantum interferometers one can saturate the Heisenberg limit shown in Eq. (1) with maximally entangled N-particle state, coined as N00N-state [18, 19, 22, 23]. The N00N-states with few photons have been recently observed in quantum optics domain under the spontaneous parametric down conversion process [24–26]. Alternatively, circuit (or, cavity)-QED devices might be used for these purposes cf. [27–29]. Indeed, atomic N00N-states with N = 2 atoms have been proposed cf. [29]. Notably, these proposals are based on multi-step quantum state engineering protocols which are sensitive enough to losses and decoherence.

As it is shown in [30–32], in the presence of losses which are modeled by fictitious two beam splitters (BS) with transmissivities η₁_,₂, the N00N-states are not optimal for approaching Heisenberg limit. The optimal (two-component) state, and the N00N-state demonstrate approximately equal measurement accuracy, only if η ≥ e^−1/^N, where η ≡ η₁ is transmissivity of BS in one of the arms; η₂ = 1. Though in [30, 31] the authors proposed a specific model of losses for photonic interferometers, the fragility problem of N00N-states because of the losses become important for a very large particle number N. Thus, the preparation of efficient (robust) mesoscopic N00N-states containing a relatively large number of particles and the exploration of them for metrological purposes, are a great challenge and nontrivial task both in theory and experiment [33].

In this work, we propose a method to create N00N-states, which are maximally entangled states in path, by means of matter wave bright solitons in Bose-Einstein condensates (BECs). The BEC phenomenon is already demonstrated for the systems in condensed matter and solid state physics, see e.g. [34, 35]. Superfluid properties that occur due to the interaction between condensate particles represent the important feature of the condensates that allows to keep for some time coherence in macroscopically large system regardless of environment even in the presence of weak dissipation [36].

Quantum features of solitons has been extensively studied in optics some times ago [37–44]. In order, squeezing and quantum correlations effects for solitons in optical fibers were demonstrated. However, solitons in such experiments contain huge number of photons (10⁸ photons in experiment described by S. Friberg et al [43]) which makes practically impossible to explore them for N00N-states formation purposes in the presence of realistic fiber losses.

Remarkably, atomic condensates with negative scattering length might form bright solitons with small number of particles (from several tens to thousands) that allows to treat them quantum mechanically and to consider as suitable candidates for N00N-states formation, cf. [45]. The atomic scattering length, that characterizes atom-atom interaction, might be tuned and enhanced by using Feshbach resonance approach [46, 47].

Second, we suggest to use low branch (LB) exciton polariton BECs to produce photonic solitons in quantum domain with the moderate number of photons for quantum metrology purposes. The exciton polaritons are bosonic quasiparticles representing coherent superposition of excitons and photons which occur under the strong coupling condition for a quantum matter-field interaction in high quality semiconductor microstructures. Exciton polariton condensates emit photons at the output of the sample with quantum state that exactly reflects polariton condensate properties. Strong interaction between excitons provides large Kerr-like nonlinearity of polaritonic system that several orders larger than in VCSELs. The number of photons in such solitons might be several tens as is demonstrated in recent experiments with exciton polariton condensate solitons [48]. Notably polariton-polariton interaction might be tuned by Feschbach resonance approach as well [49]. This feature seems to be practically important for quantum measurement applications even in the present of losses, cf. [30, 31].

Relying on the general form of Gross-Pitaevskii equation (GPE) in Heisenberg representation for a condensate in a double-well potential [45], we describe the corresponding quantum field model for coupled bright solitons occurring in two trapped condensates.

Noticing that two condensates admit well-defined relative phase and the total number of particles N which have been examined in a number of experiments and described in the theory, see e.g. [50–55] and cf. [56].

In the framework of quantum field theory [57] we derive the equations of motion for the condensate’s parameters, i.e., the relative phase and population imbalance between two solitons. Then, we show that the ground state of the system can be a quantum superposition state forming Schrödinger-cat or N00N-state. Utilization of these states focused on linear phase shift measurements is revealed for quantum metrology.

Scaling beyond Heisenberg limit, referred as super-Heisenberg scaling, can be achieved in the framework of interaction-based (nonlinear) quantum metrology [7–9, 58–60]. The saturation of the linear Heisenberg limit is demonstrated for nonlinear relative phase parameter estimations for coupled solitons. Our results provide some possible quantum metrology applications beyond linear Heisenberg limit with entangled matter wave solitons.

2. Model for coupled quantum matter bright solitons

Let us consider two BECs, consisting of N particles, trapped in a double-well potential and weakly coupled to each other due to the Josephson effect. This model has been applied for the studies of several effects, i.e. quantum squeezing, entanglement, and related metrology applications for continuous variables within the tight binding approximation [57, 61–65]. Experimentally, such a system can be implemented in atomic optics domain with the help of highly asymmetric potentials, i.e., a cigar-shaped potential [66]. Without the loss of generality, the spatial distribution of the condensates are denoted along z-direction. In addition to the atomic systems, exciton-polariton condensates in the microcavity are also a possible platform for our model [48,67].

The total Hamiltonian $\hat{H}$ for BECs in a double-well potential can be described by

\hat{H} = {\hat{H}}_{1} + {\hat{H}}_{2} + {\hat{H}}_{i n t},

where

{\hat{H}}_{j} (j = 1, 2)

is the Hamiltonian for condensate particles in j-th well; while

{\hat{H}}_{i n t}

accounts for the inter-well coupling between two sites. In the second quantization form we explicitly have

\begin{array}{l} {\hat{H}}_{j} = \int d z {\hat{a}}_{j} {(z)}^{†} (- \frac{1}{2 M} \frac{\partial^{2}}{\partial z^{2}} + \frac{U}{2} {\hat{a}}_{j} {(z)}^{†} {\hat{a}}_{j} (z)) {\hat{a}}_{j} (z) \\ {\hat{H}}_{i n t} = κ \int d z {\hat{a}}_{2} {(z)}^{†} {\hat{a}}_{1} (z) + H . C . \end{array}

Here, parameter U characterizes two-body interactions, M = sgn[m_eff] = ±1 is used as the normalized effective particle mass m_eff, and κ denotes the inter-well tunneling rate.

In practice bright matter wave solitons with M = 1, U < 0 are obtained for atomic condensates with the negative scattering length that corresponds to attractive particles [46, 47]. Contrary, if M = −1 and U > 0 bright solitons might be formed by the assistance of 1D periodical potential for condensate positive scattering length which is relevant to exciton polariton BECs, or to the atoms with repulsive interaction between the particles [48, 66]. In this case we deal with the negative effective mass of the particles that appear at the edges of Brillouin zone.

The corresponding annihilation (creation) operators of bosonic fields are denoted as ${\hat{a}}_{j} ({\hat{a}}_{j}^{†})$ with j = 1, 2, and obey the commutation relations:

[{\hat{a}}_{i} (z), {\hat{a}}_{j}^{†} (z^{'})] = δ (z - z^{'}) δ_{i j}; i, j = - 1, 2 .

For Hamiltonian (2), we suppose that the ground state of this bosonic system is the product of N single particle states [57]. Physically, this assumption is valid for BECs taken in equilibrium at zero temperature. Thus, the collective ground state for the whole system can be written as:

{| Ψ 〉}_{N} = \frac{1}{\sqrt{N!}} {[\int_{- \infty}^{\infty} d z (Ψ_{1} {\hat{a}}_{1}^{†} + Ψ_{2} {\hat{a}}_{2}^{†})]}^{N} | 0 〉,

with |0〉 ≡ |0〉₁|0〉₂ being a two-mode vacuum state. It is noted that the state vector shown in Eq. (4) relates to the Hartree approach for bosonic systems [37,38,68], which is valid for a large number of particles N. If we apply the variational approach based on the ansatz Ψ₁ ≡ Ψ₁(z, t) and Ψ₂ ≡ Ψ₂(z, t), with the unknown z-dependent wave-functions, one can have the corresponding Lagrangian density in the form [69]:

\begin{array}{l} L = \sum_{j = 1}^{2} (\frac{i}{2} [Ψ_{j}^{*} {\dot{Ψ}}_{j} - {\dot{Ψ}}_{j}^{*} Ψ_{j}] + \frac{1}{2 M} Ψ_{j}^{*} \frac{\partial^{2} Ψ_{j}}{\partial z^{2}} - \frac{U}{2} {| Ψ_{j} |}^{4}) \\ - κ (Ψ_{1}^{*} Ψ_{2} + Ψ_{1} Ψ_{2}^{*}) . \end{array}

In the limit of vanishing coupling constant κ = 0, Eq. (5) leads to the well-known GPE, which supports bright soliton solution when MU < 0, i.e.,

Ψ_{j} = \frac{N_{j}}{2} \sqrt{| U |} s e c h (\frac{N_{j} | U |}{2} z) e^{i M θ_{j}} .

Here N_j is the number of particles in the j-th well and θ_j is the phase, respectively. Below, we take the soliton solutions given in Eq. (6) as our variational ansatz, but imposing time dependent parameters for N_j and θ_j when a weak coupling between the condensates is nonzero. Then, we can obtain the effective Lagrangian by integrating the Lagrangian density (5):

\begin{array}{l} L = \int_{- \infty}^{\infty} L d z = - M (N_{1} {\dot{θ}}_{1} + N_{2} {\dot{θ}}_{2}) \\ + \frac{U^{2}}{24 M} (N_{1}^{3} + N_{2}^{3}) - \frac{4 κ N_{1} N_{2}}{N} I (p) \cos [θ] . \end{array}

Here, we have defined p = (N₂ − N₁)/N and θ = θ₂ − θ₁ as a population imbalance and phase difference between solitons, respectively. The total number of particles is denoted by N = N₁ +N₂. Moreover, we also introduce

I (p) = \int_{0}^{\infty} \frac{d z^{'}}{\cosh^{2} (z^{'}) + \sinh^{2} (z^{'} p)} .

It is important for analytic analysis that integral defined in Eq.(8) can be approximated by a parabola

I (p) \approx 1 - α p^{2}

with α = 0.21. The direct comparison of (8a) and (8b) with p ∈ [0; 1] has revealed that the maximal relative error of the approximation is about 0.7% for p =0.9.

Basing on Eq. (7) we derived the equation of motions for the population imbalance and phase difference, i.e., p and θ in the form,

\dot{p} = - \frac{1}{M} (1 - p^{2}) (1 - α p^{2}) \sin [θ],

\dot{θ} = Λ p + \frac{2 p}{M} \cos [θ] [1 + α - 2 α p^{2}] .

Here, the dots denote the derivative with respect to the dimensionless time t′ = 2|κ|t. In Eqs. (9), the dimensionless parameter $Λ = \frac{U^{2} N^{2}}{16 | κ |}$ is also introduced, which defines various regimes for BEC behavior in double-well trap.

Two sets of nontrivial stationary solutions can be found for Eqs. (9). For the first set we have

p_{0}^{2} = \frac{1}{2 α} [1 + α - \frac{Λ}{2}],

\cos (θ_{0}) = - M;

and for the second set

p_{0}^{2} = 1,

\cos θ_{0} = - \frac{M Λ}{2 (1 - α)} .

The first set of nontrivial solutions given in Eqs. (10) is similar to the one obtained under two-mode approximation, and relevant to the tight binding model [57]. However, a vital parameter of system Λ that we have introduced above is proportional to N² instead of N which occurs in two-mode limit, cf. [65]. This fact seems to be very important in practice when we consider the limit of the large particle number N, cf. [63]. In the following, we show that this set of solutions can be used to construct Schrödinger-cat state with solitons.

As for the second set of solutions given in Eqs. (11), there is no analogy from the results obtained under two-mode approximation [57, 61–64]. Physically, such a set of solutions implies the formation of N00N-state from coupled solitons.

As for the imbalance parameter 0 ≤ |p| ≤ 1, the corresponding Λ parameter lies between 2(1 − α) and 2(1 + α), resulting in the first set of solutions existing only in 1.58 ≤ Λ ≤ 2.42. However, for the phase difference 0 ≤ |cos θ₀| ≤ 1, the second set of solutions can exist for 0 < Λ ≤ 1.58 only. One can see that there is a critical value for Λ_cr = 2(1 − α) ≈ 1.58, at which we have the state with p² = 1 and cos(θ₀) = −M. To be more specific, thereafter, we assume M = 1.

3. Superposition states of quantum solitons

3.1. Schrödinger-cat state (SC-state)

The SC-state exist at 1.58 ≤ Λ ≤ 2.42. The state vector of solitons (4) corresponding to Eqs. (10) has the form

| Ψ^{(\pm)} 〉 = \frac{1}{\sqrt{N!}} {[\int_{- \infty}^{\infty} d z (Ψ_{\mp} {\hat{a}}_{1}^{†} - Ψ_{\mp} {\hat{a}}_{2}^{†})]}^{N} | 0 〉,

with

Ψ_{\pm} = \frac{\sqrt{N U}}{4} (1 \pm | p_{0} |) s e c h (\frac{N U}{4} (1 \pm | p_{0} |) z),

and

| p_{0} | = \sqrt{\frac{1}{2 α} (1 + α - \frac{Λ}{2})}

. By defining macroscopic superposition of states from Eqs. (12), we can construct Schrödinger-cat state (SC-state) from coupled solitons, cf. [57]:

| Ψ 〉 = C (| Ψ^{(+)} 〉 + | Ψ^{(-)} 〉) .

Here, C = [2(1 + X^N)]^−1/2 is a normalization factor, and $X = \frac{1 - p_{0}^{2}}{2} \int_{- \infty}^{\infty} \frac{d x}{\cosh [x] + \cosh [p_{0} x]} \approx (1 - p_{0}^{2}) (1 - α p_{0}^{2})$ with the same α = 0.21. Notice that the modes of SC-state (12) are not orthogonal to each other, but follow the relation:

ϵ = | Ψ^{(\pm)} 〉 | Ψ^{(\mp)} 〉 = X^{N} .

Physically, the size of the cat can be defined by 1/ϵ (see Fig. 1(a)). For macroscopic SC-state, we require ϵ ≪ 1, which implies the maximally achievable cat size obtained with |p₀| → 1 and X → 0.

Fig. 1 (a) The dependence of the “cat size” 1/ϵ (14) on the population imbalance p₀ for different numbers of particles N. One can see that “cat size” tends to infinity when |p₀| tends to 1. Also 1/ϵ ≈ 0 when |p₀| ≈ 0. Infinite “cat size” corresponds to macroscopic SC-state and can be approximately taken as a N00N-state. Zero “cat size” corresponds to microscopic SC-sate which means almost no entanglement. (b) Illustration of the precision measurement of the phase shift, based on a Mach-Zehnder interferometer (MZI). Here, QSPD denotes a quantum state preparation device, ϕ₁ and ϕ₂ are two resulting phases accumulated at the arms of interferometer, BS is a beam splitter, and D is a parity detector that runs in the particle counting regime.

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3.2. N00N-state

The N00N-state exist at 0 < Λ ≤ 1.58. The second set of solutions given in Eqs. (11) presumes the state vector of solitons (4) in the form

| Φ^{(\pm)} 〉 = \frac{1}{\sqrt{N!}} {[\int_{- \infty}^{\infty} d z (Φ {\hat{a}}_{2, 1}^{†})]}^{N} | 0 〉,

with

Φ = \frac{\sqrt{N U}}{2} s e c h (\frac{N U}{2} z) .

Here we have replaced in notation |Ψ〉 → |Φ〉 just for simplicity of the reader’s perception. Also the sign ± here determines modes for p₀ = ±1. The superposition state constructed from Eqs. (15) is:

| Φ 〉 = \frac{1}{\sqrt{2}} (| Φ^{(+)} 〉 + e^{- i θ_{N}} | Φ^{(-)} 〉),

which clear gives us a N00N-state of solitons. Here, we also introduce

θ_{N} = N θ_{0} = N \arccos (- \frac{Λ}{2 (1 - α)})

.

At the critical value of Λ = Λ_cr = 1.58, the SC-state shown in Eq. (13) transformes into the N00N-state described by Eq. (16), with θ₀ = π phase difference between two solitons.

4. Quantum measurements with superposition states

In this section we propose a precision measurement experiment with SC-state and N00N-state. The Mach-Zehnder interferometer (MZI) is illustrated in Fig. 1(b). The device coined as a quantum state preparation device (QSPD) represents the medium with two coupled BECs producing entangled soliton states (may be the superposition state, SC-state or N00N-state) into the input of a MZI. The measured parameter is a linear phase shift ϕ = ϕ₂ − ϕ₁ accumulated in the arms of MZI.

Experimental realization of the scheme in Fig. 1(b) with atomic condensates is based on the so-called method of nonlinear Ramsey interferometry that is performed in time domain, cf. [14]. First the method implies the formation of two entangled solitons composing N00N-state in our case. Then, the total state undergoes free evolution. BS combines modes for the readout by applying π/2 microwave pulse to atomic cloud which couples two internal atomic states.

As it is demonstrated in [70], the current micro- and nanotechnologies allow to design semiconductor based MZI operating with exciton polariton condensates and possessing optically controlable phase shift, polarization, output intensity that opens the door for quantum optical metrology purposes being under discussion.

The sensitivity of the phase parameter ϕ for the scheme in Fig. 1(b) is determined by (cf. [71])

〈 {(Δ ϕ)}^{2} 〉 = \frac{〈 {(Δ \hat{P})}^{2} 〉}{{| \frac{\partial 〈 \hat{P} 〉}{\partial ϕ} |}^{2}},

where,

\hat{P}

is a Hermitian operator suitable for the measurement of the phase ϕ. We propose to use a parity detection procedure with the operator taken for the second mode:

\hat{P} \equiv {\hat{P}}_{{\hat{a}}_{2}} = \exp [i π \int_{- \infty}^{\infty} {\hat{a}}_{2}^{†} {\hat{a}}_{2} d z]

. For this purpose, for parity measurement shown in Fig. 1(b), two matter waves are combined at the BS after phase-shifting operations, and then one of the detectors counts the even or odd number of particles.

Though, at present, parity measurement experimentally represents a non-trivial task requiring high efficiency particle-number counting (resolving) detectors, it is absolutely imperative to achieve Heisenberg scaling by phase measurement in our scheme, cf. [73]. Notably, in the recent experiment it was demonstrated that super-resolution phase measurement was 144 times better than the Rayleigh limit for coherent photonic states in MZI, obtaned by measuring photon number parity as a readout [74]. To describe the parity measurement, one may introduce the following spin operators

{\hat{S}}_{0} = \frac{1}{2} \int_{- \infty}^{\infty} ({\hat{a}}_{1}^{†} {\hat{a}}_{1} + {\hat{a}}_{2}^{†} {\hat{a}}_{2}) d z,

{\hat{S}}_{1} = \frac{1}{2} \int_{- \infty}^{\infty} ({\hat{a}}_{1}^{†} {\hat{a}}_{1} - {\hat{a}}_{2}^{†} {\hat{a}}_{2}) d z,

{\hat{S}}_{2} = \frac{1}{2} \int_{- \infty}^{\infty} ({\hat{a}}_{1}^{†} {\hat{a}}_{2} + {\hat{a}}_{2}^{†} {\hat{a}}_{1}) d z,

{\hat{S}}_{3} = \frac{i}{2} \int_{- \infty}^{\infty} ({\hat{a}}_{2}^{†} {\hat{a}}_{1} - {\hat{a}}_{1}^{†} {\hat{a}}_{2}) d z .

These operators obey SU(2) algebra and to commutation relations: $[{\hat{S}}_{i}, {\hat{S}}_{j}] = i ϵ_{i j k} {\hat{S}}_{k}$ , with i, j, k = 1, 2, 3. Having ${\hat{S}}_{j}$ operators we can define unitary operators for the transformations of quantum state in the beam splitter and phase shift, i.e., ${\hat{U}}_{B S} = \exp [i \frac{π}{2} {\hat{S}}_{2}]$ and ${\hat{U}}_{P S} = \exp [- i ϕ {\hat{S}}_{1}]$ , respectively. Then, the action of MZI on initial quantum state can be described by MZI-operator, i.e., ${\hat{U}}_{M Z I} = {\hat{U}}_{B S} {\hat{U}}_{P S} = \exp [i \frac{π}{2} {\hat{S}}_{2}] \exp [- i ϕ {\hat{S}}_{1}]$ . The parity operator ${\hat{P}}_{a_{2}}$ in this formalism has the form:

{\hat{P}}_{{\hat{a}}_{2}} \equiv \exp [i π ({\hat{S}}_{0} - {\hat{S}}_{1})] .

Thus, for the scheme shown in Fig. 1(b), the resulting expectation value of parity operator ${\hat{P}}_{{\hat{a}}_{2}}$

can be calculated as

〈 {\hat{P}}_{{\hat{a}}_{2}} 〉 = 〈 {\hat{U}}_{M Z I}^{†} {\hat{P}}_{{\hat{a}}_{2}} {\hat{U}}_{M Z I} 〉 = 〈 e^{i π {\hat{S}}_{0}} e^{i ϕ {\hat{S}}_{1}} e^{i π {\hat{S}}_{3}} e^{- i ϕ {\hat{S}}_{1}} 〉 .

It is also more convenient to use the angular momentum state representation instead of the particle number representation. Here, we consider the following substitution |N₁, N₂〉 → |j, m〉, where N₁, N₂ are numbers of particles in the first and second wells. The quantum numbers for angular momenta j and m are introduced as j = N/2 and m = (N₁ − N₂)/2, respectively. The states |j, m〉 are eigenstates of the spin operators ${\hat{S}}_{0, 1}$ with the conditions ${\hat{S}}_{1} | j, m 〉 = m | j, m 〉; {\hat{S}}_{0} | j, m 〉 = j | j, m 〉$ and exp [iπS₃] |N₁, N₂〉 = exp [iπN₁]|N₂, N₁〉.

In terms of the angular momentum we can rewrite SC-state in Eq. (13) and N00N-state Eq. (16) as

| Ψ 〉 = C (| j, - m 〉 + | j, m 〉),

| Φ 〉 = \frac{1}{\sqrt{2}} (| j, - j 〉 + e^{- i θ_{N}} | j, j 〉) .

Then the resulting average value $〈 {\hat{P}}_{a_{2}} 〉$ for initial SC-state and N00N-state, respectively, have the form:

〈 Ψ | {\hat{P}}_{a_{2}} | Ψ 〉 = {(- 1)}^{N} \cos [(ϕ - \frac{π}{2}) N | p_{0} |],

〈 Φ | {\hat{P}}_{{\hat{a}}_{2}} | Φ 〉 = {\begin{matrix} {(- 1)}^{\frac{N}{2}} \cos [ϕ N + θ_{N}]; & N is even \\ {(- 1)}^{\frac{N + 1}{2}} \sin [ϕ N + θ_{N}]; & N is odd \end{matrix}

with the variation

〈 {(Δ {\hat{P}}_{{\hat{a}}_{2}})}^{2} 〉

:

〈 Ψ | {(Δ {\hat{P}}_{{\hat{a}}_{2}})}^{2} | Ψ 〉 \sin^{2} [(ϕ - \frac{π}{2}) N | p_{0} |],

〈 Φ | Δ {\hat{P}}_{{\hat{a}}_{2}} | Φ 〉 = {\begin{matrix} \sin^{2} [ϕ N + θ_{N}]; & N is even \\ \cos^{2} [ϕ N + θ_{N}]; & N is odd \end{matrix}

From the results above, we can see that quantum interference effects arise in the parity measurement scheme depending on the even or odd particle numbers N. As for the sensitivity of interferometer, we immediately obtain from Eq. (17)

〈 Ψ | {(Δ ϕ)}^{2} | Ψ 〉 = \frac{1}{N^{2} {| p_{0} |}^{2}},

〈 Φ | {(Δ ϕ)}^{2} | Φ 〉 = \frac{1}{N^{2}} .

One can see that the Heisenberg limit is achieved for a maximally entangled N00N-state and the precision for SC-state has an extra 1/|p₀|² factor. In Fig. 2(a), we plot a normalized error in the phase measurement $σ_{ϕ} = \sqrt{〈 {(Δ ϕ)}^{2} 〉}$ as a function of particle number N for SC-state. The value σ_ϕ = N^−1/2 characterizes SQL of the phase measurement with classical states, which can be achieved without QSPD. One can see that the accuracy of measurement tends to the Heisenberg limit as the cat size grows and is saturated by |p₀| = 1 at the input (the yellow curve in Fig. 2(a)). On the contrary, a microscopic SC-state obtained when |p₀| → 0 is not suitable to perform the measurements.

Fig. 2 (a) Reduced phase uncertainty $\sqrt{N} σ_{ϕ}$ against total particle number N, for an initial SC-state used in the measurement procedure. The value $\sqrt{N} σ_{ϕ} = 1$ corresponds to SQL limit. (b) The dependence of σ_Θ on Θ demonstrating a second-order like phase transition from the state possessing non-zero σ_Θ beyond the linear Heisenberg limit (gray area) to the state nonapplicable for such measurements. The number of particles N = N_c = 6000 is taken for Lithium atomic condensates with negative scattering length, as example.

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5. Measurements beyond the Heisenberg scaling

The accuracy of measurement can be improved even more by using parameters with nonlinear particle number dependence. In the framework of nonlinear interferometry, the arbitrary Θ-parameter measurement procedure uses transformation |Ψ〉_Θ = exp(iΘG)|Ψ〉 for input state |Ψ〉, where G is the generator of transformation that describe nonlinear phase dependence, cf. [7–9,58,60].

In the general case for G = N^k the ultimate sensitivity of the Θ-parameter measurement in a nonlinear interferometer is bound by the value σ_Θ ≃ 1/N^k, which corresponds to the so-called super-Heisenberg limit for the phase measurement in quantum metrology, cf. [59].

To be more specific, we focus on so-called interaction-based quantum metrology approach where we use light-matter (nonlinear) quantum interface for quantum noise-limited interactions, cf. [59]. The Θ-parameter that characterizes relative strength U of nonlinear inter-particle interaction in each well with respect to the linear inter-well coupling coefficient κ is the subject of high precision measurement beyond Heisenberg limit: $Θ = \frac{Λ}{N^{2}} = \frac{U^{2}}{16 | κ |}$ . Noticing that Θ does not depend on number of particle N which means resistance of such experiment to particle losses. The Θ- parameter measurement can be performed in the same way as for phase-shift ϕ measurement in Fig. 1(b) by accounting only soliton phase difference

θ_{N} = N \arccos (- \frac{Θ N^{2}}{2 (1 - α)}),

for prepared initially N00N-state, see Eqs. (16, 17).

For a sufficiently small Θ, which implies large value of N, we can apply the Taylor expansion

θ_{N} = \frac{π}{2} N + \frac{N^{3}}{2 (1 - α)} Θ + O (Θ^{3}),

which is valid as long as we take into account only a linear dependence on Θ. Setting ϕ = 0 for neglecting unimportant phase shift, we have for the N00N-state at the input of the MZI:

〈 Φ | {\hat{P}}_{{\hat{a}}_{2}} | Φ 〉 = {\begin{matrix} {(- 1)}^{\frac{N}{2}} \cos [θ_{N}]; & N is even \\ {(- 1)}^{\frac{N + 1}{2}} \sin [θ_{N}]; & N is odd \end{matrix}

〈 Φ | {(Δ {\hat{P}}_{{\hat{a}}_{2}})}^{2} | Φ 〉 = {\begin{matrix} \sin^{2} [θ_{N}]; & N is even \\ \cos^{2} [θ_{N}]; & N is odd \end{matrix}

for the average value of

{\hat{P}}_{a_{2}}

and the corresponding variance, respectively. The resulting sensitivity of Θ can be found to be:

〈 {(Δ Θ)}^{2} 〉 = \frac{4 {(1 - α)}^{2}}{N^{6}} .

From Eq. (28), the error in Θ is $σ_{Θ} = \sqrt{〈 {(Δ Θ)}^{2} 〉} ~ N^{- 3}$ that looks quite promising for improving measurement sensitivity which currently achieved with atomic condensates, cf. [59, 65, 66].

In Fig. 2(a)), we show the dependence of σ_Θ as a function of measured Θ-parameter, for different particle numbers N. The blue dashed-curve in Fig. 2(a) corresponds to the ultimate measurements with one particle. The shadowed region in Fig. 2(a))reveals the capacity for the measurements with particle number 1 ≤ N ≤ 6000. The maximal number of particles in the condensate is limited by the number N_c that corresponds to particle number in collapsing condensate possessing negative scattering length, cf. [45]. Obviously, for N > N_c the system become dynamically unstable especially for macroscopic quantum states superposition discussed in the paper.

6. Conclusion

In summary, accepting a quantum field theory approach to the problem of bright matter wave soliton formation in weakly coupled double-well potentials, we reveal the ground states in the Schrödinger-cat superposition state (SC-state) and maximally path entangled N00N-state. With the variational method, we derive the equation of motions for SC-state and N00N-state. Then, within the Mach-Zehnder interferometer we examine quantum phase measurement with these superposition states, in order to have the accuracy beyond the standard quantum limit and the linear Heisenberg limit. We perform the ${\hat{P}}_{{\hat{a}}_{2}}$ operator measurements by applying a parity measurement procedure. Heisenberg-limited phase shift measurements are demonstrated to be saturated for maximally path entangled state containing N particles. A vital combination of condensate parameters $Θ = \frac{U^{2}}{16 | κ |}$ is shown to surpass the linear Heisenberg limit in terms of the nonlinear metrology approach, when scaling is proportional to N⁻³. These results applied to atomic N00N-states represent a promising tool for atomic clocks and atomic gyroscopes [65, 66].

Notably, decoherence effects play an important role for the schemes operating with SC-states and/or N00N-states, cf. [72]. From the practical point of view it is more important to identify characteristic time scales when superposition states and – more generally – two component macroscopic condensates might be implemented for quantum operations. Contrary to standard (single particle) qubits, as it is shown in [75], the required time of gate operation in condensates for producing entanglement is inversely proportional to the particle number N. This enhancement is achieved due to bosonic stimulation effect and implies a fast quantum gate operation. Obviously, decoherence effects occurring in condensate macroscopic states should appear at longer time.

Remarkably, condensate quantum solitons pose some specific peculiarities during their propagation in the presence of decoherence, via one-, two-, and/or three-body losses cf. [76]. In the paper we are examining bright solitons at rest. Obviously, the analysis of motional degree of freedom seems to be important due to the uncertainty relation for soliton momentum and position [37, 38]. One of the possible realizations of the scheme represented in Fig. 2 is connected with exciton-polariton bright solitons propagation in high-Q semiconductor microcavities [48, 70]. The lifetime of solitons is several tens of picoseconds that is large enough as compared to the possible quantum operation. Moreover, recently there were proposed by Y. Sun et al. in [77] few hundred picoseconds lifetime for LB exciton polaritons in semiconductor microstructures. This enables to avoid decoherence and undesirable spreading effects for characteristic long lifetimes [37, 38, 76]. In other words, long-lived exciton polariton condensates can be a new platform for designing maximally entangled states with moving solitons. These problems will be a subject of intensive study both in theory and experiment in forthcoming papers.

Funding

Government of Russian Federation, Grant No. 08-08 and the Ministry of Science and Technology of Taiwan under Grant No. 105-2628-M-007-003-MY4.

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Quantum metrology beyond Heisenberg limit with entangled matter wave solitons

Abstract