Quantifying the mesoscopic quantum coherence of approximate NOON states and spin-squeezed two-mode Bose-Einstein condensates
QQuantifying the mesoscopic quantum coherence of approximate NOON states andspin-squeezed two-mode Bose-Einstein condensates
B. Opanchuk, L. Rosales-Zárate, R. Y. Teh and M. D. Reid
Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne, Australia
We examine how to signify and quantify the mesoscopic quantum coherence of approximate two-mode NOON states and spin-squeezed two-mode Bose-Einstein condensates (BEC). We identify twocriteria that verify a nonzero quantum coherence between states with quantum number different by n . These criteria negate certain mixtures of quantum states, thereby signifying a generalised n -scopicSchrodinger cat-type paradox. The first criterion is the correlation (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 (here ˆ a and ˆ b arethe boson operators for each mode). The correlation manifests as interference fringes in n -particledetection probabilities and is also measurable via quadrature phase amplitude and spin squeezingmeasurements. Measurement of (cid:104) ˆ a † n ˆ b n (cid:105) enables a quantification of the overall n -th order quantumcoherence, thus providing an avenue for high efficiency verification of a high-fidelity photonic NOONstates. The second criterion is based on a quantification of the measurable spin-squeezing parameter ξ N . We apply the criteria to theoretical models of NOON states in lossy interferometers and double-well trapped BECs. By analysing existing BEC experiments, we demonstrate generalised atomic“kitten” states and atomic quantum coherence with n (cid:39) atoms. I. INTRODUCTION
In 1935 Schrodinger considered the preparation of amacroscopic system in a quantum superposition of twomacroscopically distinguishable states [1]. Such systemsare called “Schrodinger cat-states” after Schrodinger’s ex-ample of a cat in a superposition of dead and alive states.The preparation of such states in the laboratory is dif-ficult due to the existence of external couplings, whichcause the superposition state to decohere to a classicalmixture [2, 3]. While for the mixture the “cat” is prob-abilistically “dead” or “alive”, the paradox is that for thesuperposition the “cat” is apparently neither “dead” or“alive”. Developments in quantum optics and the coolingof atoms and mechanical oscillators have made the gen-eration of mesoscopic cat-states feasible [4, 5]. This isinteresting for atomic systems where a superposition of amassive system being in two states at different locationsmight be created. Ghirardi, Rimini, Weber [6] and Diosi[7] and Penrose [8] have proposed that for such systemsdecoherence mechanisms would prevent the formation ofthe Schrodinger cat superposition states. To carry outtests, firm proposals are required for the creation anddetection of cat-states.A major consideration for cat-state experiments is thatthe generation of the cat-state is not likely to be ideal.This is especially true for larger N . One of the mostwell-studied cat-states is the NOON state [9–14] | ψ NOON (cid:105) = 1 √ {| N (cid:105) a | (cid:105) b + e iφ | (cid:105) a | N (cid:105) b } (1)where N particles or photons are superposed as beingin the spatial mode a or the spatial mode b . Ideally, acat-state requires N → ∞ but “ N -scopic kitten-state”realisations focus on finite N > . Here | n (cid:105) a ( | m (cid:105) b ) arethe eigenstates of particle number ˆ n a = ˆ a † ˆ a ( ˆ n b = ˆ b † ˆ b ),respectively, and ˆ a, ˆ a † ( ˆ b , ˆ b † ) are the boson operatorsfor mode a ( b ). The NOON states have been generated in optics for N up to [13] and with atoms for N =2 [15]. At low N however photon detection efficienciesare usually very low and results are often obtained bypostselection processes.Generating for higher N is challenging. Proposals ex-ist to exploit the nonlinear interactions formed from BoseEinstein condensates (BEC) trapped in the spatially sep-arated wells of an optical lattice [16–24]. Under someconditions, theory shows that the atoms can tunnel be-tween the wells, resulting in the formation of a NOONsuperposition. However, it is known that for realistic pa-rameters, the states generated are in fact of the type | ψ (cid:105) = N (cid:88) m =0 d m | N − m (cid:105) a | m (cid:105) b (2)where there exist nonzero probabilities for numbers otherthan or N [19, 21–23, 25]. (The d m are probabilityamplitudes). Oscillation between two BEC states withsignificantly different mode numbers has been experimen-tally observed [26], presumably resulting in the formationof a superposition of type (2) at intermediate times.A key question (raised by Leggett and Garg [27]) ishow to rigorously signify the Schrodinger cat-like prop-erty of the state in such a non-ideal scenario. In thispaper we propose quantifiable “catness” signatures thatcan be applied to nonideal NOON-type states generatedin photonic and cold atom experiments. The signaturesthat we examine exclude all classical mixtures of suffi-ciently separated quantum states, so that it is possibleto exclude all classical interpretations where the “cat”is “dead” or “alive” (see the Conclusion for a qualifica-tion). For the cat-system that can be found in one oftwo macroscopically distinguishable states ρ D and ρ A , arigorous signature must negate all mixtures of the form ρ mix = P D ρ D + P A ρ A (3)where P D and P A are probabilities and P D + P A = 1 .In our treatment, the ρ D ( ρ A ) are density operators for a r X i v : . [ qu a n t - ph ] M a r quantum states otherwise unspecified except that theygive macroscopically distinct outcomes (“alive” or “dead”)for a measurement of quantum number ˆ n a − ˆ n b .The first step is to generalise this approach for the non-ideal case (2), where there are nonzero probabilities forobtaining outcomes in an intermediate (“sleepy”) domainover which the cat cannot be identified as either dead oralive. In Sections II and III, we follow Refs. [27, 29, 30]and consider states ρ DS and ρ SA that give outcomes inthe combined “dead/ sleepy” and “sleepy/ alive” regionsrespectively. The states have overlapping outcomes indis-tinguishable over a range n . We explain how the negationof all mixtures of the type ρ mix = P − ρ DS + P + ρ SA (4)(where P − and P + are probabilities and P − + P + = 1 )will imply a generalised n -scopic cat-type paradox, in thesense that the system cannot be explained by any mixtureof quantum superpositions of states different by up to n quanta. It is proved that the observation of the nonzero n -scopic quantum coherence term (cid:104) |(cid:104) n | ρ | (cid:105)| n (cid:105) (cid:54) = 0 (5)( ρ is the density operator) will negate all mixtures oftype (4), thus signifying a generalised n -scopic “kitten”quantum superposition. We identify two criteria thatverify the n -scopic quantum coherence (5). In a separatepaper, we examine a third criterion based on uncertaintyrelations and Einstein-Podolsky-Rosen steering [31].Previous studies have proposed signatures (criteria ormeasures) for mesoscopic “cat” states (see for instance[3, 4, 22, 25, 29–39]). These include proposals basedon interference fringes, entanglement measures, uncer-tainty relations, negative Wigner functions and state fi-delity. Not all of these signatures however provide a di-rect negation of all the mixtures (3), (4). Further, mostof these studies do not address the nonideal case wherethere may be a range of outcomes not binnable as either“dead” or “alive”. Exceptions include the work of Refs.[22, 27, 29, 30, 32–35, 40] which (like the work of this pa-per) are based on the observations of a nonzero quantumcoherence.The first criterion that we consider is a nonzero n thorder correlation (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 (6)This criterion is necessary and sufficien t for the n th or-der quantum coherence (5). While normally evidencedfor NOON states by fringe patterns formed from n -foldphoton count coincidences, we show in Section VII howthis moment can also be measured for small N usinghighly efficient homodyne detection and Schwinger-spinmoments. In Section IV, we show that the value of thecoherence (cid:104) ˆ a † n ˆ b n (cid:105) when suitably normalised translatesto an effective fidelity measure of the n -scopic “catness”property of the state (which is not given directly by the state fidelity). We provide (in Section V) a theoreticalmodel for the NOON state with losses, thus examiningthe degradation of the fidelity measure in that case. Wealso show how the fidelity measure can be applied toquantify mesoscopic quantum coherence for the case ofnumber states | N (cid:105) incident on a beam splitter (the lin-ear beam splitter model). The quantum coherence (5) isoptimally robust with respect to losses when n (cid:28) N , butthis can be achieved for high n .The criterion (6) was proposed by Haigh et al to signifyNOON-type superposition states created from nonlinearinteractions in two-well BECs [22]. In Section VI, weevaluate (cid:104) ˆ a † n ˆ b n (cid:105) in dynamical regimes suitable for theformation of approximate NOON states, using a two-mode Josephson model (the nonlinear beam splitter ). InSection VII, we analyse the measurement strategy usingmulti-particle interferometry. For the case of n = 2 , ,the moment (6) is readily measured in terms of Schwingerspin observables. In fact by analysing spin squeezing datareported from the atomic BEC experiment of Esteve et al[41], we infer (in Section VIII) the existence of two-atom( n = 2 ) generalised (sometimes called “embedded” [21])kitten-states.The second criterion that we consider for an n -scopicquantum coherence (5) is based on spin squeezing [42,43]. The amount of squeezing observed for a given num-ber of atoms N is quantified by a squeeze parameter ξ N < [41, 44–46]. In Section IIIb, we apply the meth-ods of Ref. [30] and prove that a given measured amountof squeezing places a lower bound of √ Nξ N on the value of n for which the quantum coherence (cid:104) |(cid:104) n | ρ | (cid:105)| n (cid:105) is nonzero: n > √ Nξ N (7)This criterion requires (cid:104) ˆ a † ˆ b (cid:105) (cid:54) = 0 and is not therefore use-ful to identify ideal NOON states. However the squeezingsignature (7) is very effective in confirming a high degreeof mesoscopic quantum coherence for states (2) where ad-jacent d m and d m +1 are nonzero. This occurs in systemswith high losses or linear couplings. In Section IIIb weapply this signature to published experimental data, andconfirm a mesoscopic coherence (with n ∼ atoms) intwo-mode BEC systems. We note this is consistent withthe recent work of Ref. [32, 35] which proposes quanti-fiers (measures) of mesoscopic quantum coherence basedon Fisher information and reports significant values ofatomic coherence for BEC systems. II. n -SCOPIC QUANTUM COHERENCE We begin by considering the outcomes of observable J Z = (ˆ a † ˆ a − ˆ b † ˆ b ) . For the ideal NOON state (1) theseare − N and N . In the limit of large N , we identify thetwo outcomes as “dead ( D )” and “alive ( A )” in order tomake a simplistic analogy with the Schrodinger cat ex-ample. How does one signify the superposition nature ofthe NOON state? The density matrix ρ for the superpo-sition | ψ NOON (cid:105) has nonzero off-diagonal coherence terms (cid:104) |(cid:104) N | ρ | (cid:105)| N (cid:105) (cid:54) = 0 that distinguish it from the classicalmixture ( P D and P A are probabilities, P D + P A = 1 ) ρ mix = P D | (cid:105)| N (cid:105)(cid:104) N |(cid:104) | + P A | N (cid:105)| (cid:105)(cid:104) |(cid:104) N | (8)Thus, the detection of the nonzero coherence (cid:104) |(cid:104) N | ρ | (cid:105)| N (cid:105) serves to signify an N -scopic cat-state inthis case.The ultimate objective of a “Schrodinger cat” experi-ment is to negate classical realism at a macroscopic level.The accepted definition of macroscopic realism is thata system must be in a classical mixture of two macroscop-ically distinguishable states [27]. Similarly, we take as thedefinition of N -scopic realism is that a system must bein a classical mixture of two states that give predictionsdifferent by N quanta. In this paper, the meaning of classical mixture is in the quantum sense only, that thedensity operator ρ for the system is equivalent to a clas-sical mixture of the two (quantum) states, as in (8). −50 0 50 u u ( u u ) (a) −10 −u0 u 10 u u ( u u ) u uu u uu (b) Figure 1:
Nonideal scenarios for NOON generation:
Proba-bility P (2 j z ) of an outcome of J z for the NOON state afterattenuation as modelled by a beam splitter coupling. (a) N = 50 , η = 0 . (b) η = 0 . . Similar plots are obtained forpure NOON-type states (2) generated via a Josephson two-mode interaction. The right graph shows how to confirm an n -scopic quantum coherence, as explained in the text. More generally, the states generated in the experimentsgive outcomes for ˆ n a and ˆ n b different to and N , as aresult of even a small amount of loss or noise in the sys-tem (real predictions are illustrated in Figure 1). Thequestion becomes how to confirm by experiment that thesystem is indeed in a superposition of two mesoscopi-cally distinguishable quantum states, as opposed to anyalternative classical description where there would be nomesoscopic “cat” paradox. This question was examinedin Ref. [29, 30] for continuous outcomes realised fromquadrature phase amplitude measurements and we ap-ply the approach given there. The following is a resultfound in that paper as applied to this case. Result 1 : − An n -scopic quantum coherence andgeneralised n -scopic cat-paradox : Consider the fol-lowing mixture sketched in Figure 1b (for j c = 0 ). ρ mix = P − ρ DS + P + ρ SA (9)where P − and P + are probabilities and P − + P + = 1 .Here, ρ DS is a quantum state whose two-mode number state expansion may only include eigenstates with out-come J z < j c + n ; and ρ SA is a quantum state whoseexpansion only includes eigenstates with J z > j c − n .The outcome J z ≥ j c + n is interpreted as “alive” andthe outcome J z ≤ j c − n is interpreted as “dead”. Theintermediate overlapping regime is “sleepy”. The nega-tion of all mixtures of the type (9) will imply a gen-eralised n -scopic “cat-type” paradox, in the sense thatthe system cannot be viewed as either “dead/ sleepy” or“alive/ sleepy” − and cannot therefore be explained byany mixture of superpositions of states different by upto n quanta. If (9) can be negated, then there is an n -scopic generalised quantum coherence (cat-type paradox) .The negation of (9) implies that for some n (cid:48) , m (cid:48) b (cid:104) n + m (cid:48) | a (cid:104) n (cid:48) | ρ | n + n (cid:48) (cid:105) a | m (cid:48) (cid:105) b (cid:54) = 0 (10)(in fact j c = n (cid:48) − m (cid:48) ). The converse is also true. Condi-tions that negate (9) equivalently demonstrate (10), andwe refer to these conditions as signatures of n - scopicquantum coherence, or of an n -scopic generalised catparadox. Proof:
The justification is that if (9) fails, then thesystem cannot be thought of as being in one quantumstate ρ or the other ρ . We have not constrained the ρ or ρ , except to say they cannot include both “dead” and“alive” states (each one is orthogonal to either the deador alive state). Hence there is a negation of the premisethat the system must always be either “ dead or alive”.In that sense we have an analogy with the Schrodinger“cat” paradox but where the “dead” and “alive” states areseparated by n quanta. That the coherence is nonzerofollows on expanding the density matrix in the numberstate basis. (cid:3) We note that the nonzero n -scopic quantum coherencehas a physical significance, in that it is then possible (inprinciple) to filter out the intermediate “sleepy states” us-ing measurements of | ˆ J Z | > n/ to create a conditionalcat-state where the separation in J Z of the “dead” and“alive” states is of order n . This method of preparationhas been carried out experimentally [10]. However, wherethe n -scopic coherences are small, the heralding probabil-ity for the cat-state also becomes small, making the statesincreasingly difficult to generate. This motivates SectionV which examines how to quantify the quantum coher-ence through experimental signatures. First, we identifytwo criteria for the condition Eq. (10). III. TWO CRITERIA FOR n -SCOPICQUANTUM COHERENCEA. Correlation test It is well known that higher order correlations can de-tect NOON states. We clarify with the following result.
Result 2 : − The n -th order correlation test: Re-stricting to two-mode quantum descriptions for ρ , theobservation of (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 (11)is a signature of the n -scopic quantum coherence (10).The Result can be proved straightforwardly by expand-ing the operator ˆ a † n ˆ b n in terms of the Fock basis elements | n a (cid:105)| n b (cid:105)(cid:104) m b |(cid:104) m a | or equivalently by considering an ar-bitrary density matrix ρ written in the two-mode Fockbasis and noting that the condition (11) is equivalentto (10). We will find it useful to note the following: Ifthe moment (cid:104) ˆ a † n ˆ b n (cid:105) is nonzero then there is a nonzeroprobability that the system is in the following generalised n -scopic superposition state: | ψ n (cid:105) = a ( n ) n (cid:48) m (cid:48) | n (cid:48) (cid:105)| m (cid:48) + n (cid:105) + b ( n ) n (cid:48) m (cid:48) | n (cid:48) + n (cid:105)| m (cid:48) (cid:105) + d | ψ (cid:105) (12)The a ( n ) n (cid:48) m (cid:48) , b ( n ) n (cid:48) m (cid:48) , d are probability amplitudes satisfy-ing a ( n ) n (cid:48) m (cid:48) ,b ( n ) n (cid:48) m (cid:48) (cid:54) = 0 , the d being unspecified. | ψ (cid:105) isan unspecified quantum state orthogonal to the states | n (cid:48) (cid:105)| m (cid:48) + n (cid:105) and | n (cid:48) + n (cid:105)| m (cid:48) (cid:105) . The meaning of “nonzeroprobability that the system is in” in this context is thatthe density operator for the quantum system is necessar-ily of the form ρ = (cid:80) R P R | ψ R (cid:105)(cid:104) ψ R | where at least oneof the states | ψ (cid:105) R with nonzero P R is an n -scopic super-position state | ψ n (cid:105) . The Appendix A gives a detailedexplanation of this last result. (cid:3) B. Spin squeezing test and application toexperiment
Significant n th order quantum coherence can also insome cases be detected by observation of spin squeezing.We define the standard Schwinger operators ˆ J X = (cid:16) ˆ a † ˆ b + ˆ a ˆ b † (cid:17) / j Y = (cid:16) ˆ a † ˆ b − ˆ a ˆ b † (cid:17) / (2 i )ˆ J Z = (cid:16) ˆ a † ˆ a − ˆ b † ˆ b (cid:17) / N = ˆ a † ˆ a + ˆ b † ˆ b (13)We consider a system described by a superposition oftwo-mode number states as in (2). Thus we specify ageneralised superposition as | ψ (cid:105) = (cid:88) i,j c ij | n i (cid:105)| m j (cid:105)≡ (cid:88) k d k | ψ k (cid:105) (14)where the last line relabels (for convenience) all states ofthe ij array by an index k . In (2) we have a superposition | ψ (cid:105) = (cid:80) n =0 d n | n (cid:105)| N − n (cid:105) where N (the total number of particles) is fixed. This case for large N and where d n (cid:54) = 0 for some n (cid:54) = 0 , N has been described as a superpositionof “dead”, “alive” and “sleepy” cats. Considering the gen-eral case (14), we can define for each term | ψ k (cid:105) such that d k (cid:54) = 0 the spin number difference j k = ( n i − m j ) / . Theaim is to put a lower bound on the spread of possible j k values (depicted in Figure 1). We define the spread as δ = max {| j k − j k (cid:48) |} (15)such that for j k and j k (cid:48) , the coefficients d k , d k (cid:48) (cid:54) = 0 . Forthe ideal NOON state, δ = N . Here max denotes themaximum of the set.We can show that a certain amount of squeezing in J Y determines a lower bound in the spread of eigen-states of J Z . The method is similar to that given in Ref.[30] which studied quadrature phase amplitude squeez-ing. The spin Heisenberg uncertainty relation is (∆ ˆ J Y )(∆ ˆ J Z ) ≥ |(cid:104) ˆ J X (cid:105)| / (16)Spin squeezing is obtained when [42, 43] (∆ ˆ J Y ) < |(cid:104) ˆ J X (cid:105)| / (17)It is clear that in that case a low variance (∆ ˆ J Y ) will al-ways imply a high variance in ˆ J Z . For many spin squeez-ing experiments, (cid:104) ˆ J X (cid:105) ∼ (cid:104) ˆ N (cid:105) / which means the Blochvector lies on the surface or near the surface of the Blochsphere, so that the system is close to a pure state. Squeez-ing is then obtained when (∆ ˆ J Y ) < (cid:104) ˆ N (cid:105) / .For pure states, the high variance in ˆ J Z is associatedwith a minimum spread of the superposition of eigen-states of ˆ J Z . Thus, there is a lower bound on the bestamount of squeezing determined by the maximum spread(extent) δ of the superposition. In the Appendix, follow-ing the methods of Refs. [30], this connection is gener-alised for mixed states. We prove the following result. Result : − Spin squeezing test for n -th or-der quantum coherence: An experimentally measuredamount of spin squeezing in J Y is defined in terms of a“squeezing parameter” ξ N = (cid:16) ∆ ˆ J Y (cid:17)(cid:113) |(cid:104) ˆ J X (cid:105)| / → (cid:16) ∆ ˆ J Y (cid:17) (cid:104) N (cid:105) / / (18)where ξ N < implies spin squeezing and ξ N = 0 isthe optimal possible squeezing (achievable as N → ∞ ).Here we have taken the case where (cid:104) ˆ J X (cid:105) ∼ (cid:104) ˆ N (cid:105) / .We can conclude that there exists a nonzero coherence (cid:104) |(cid:104) n | ρ | (cid:105)| n (cid:105) (cid:54) = 0 for a value n where n > √ Nξ N (19) Proof:
The proof is given in the Appendix. (cid:3)
The particular test given by Result 3 requires (cid:104) ˆ J X (cid:105) (cid:54) =0 . This would imply nonzero single atom coherence termsgiven as (cid:104) ˆ a † ˆ b (cid:105) (cid:54) = 0 . We note that the final result (19)indicates that the coherence size is of order √ N . Spinsqueezing with a considerable number N of atoms hasbeen observed in several atomic experiments and excel-lent agreement has been obtained for N ∼ with atwo-mode model [41, 44, 45]. Typically, the number ofatoms is N ∼ or more, indicating values of quantumcoherence of order n > atoms. IV. MEASURABLE QUANTIFICATION OFTHE MESOSCOPIC QUANTUM COHERENCEA. Catness fidelity and quantum coherence
The observation of (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 certifies the existence ofthe (nonzero) n -scopic quantum coherence, but does notspecify the magnitude of the quantum coherence (QC),originating from terms like C ( n (cid:48) ,m (cid:48) ) n = 2 | b (cid:104) n + m (cid:48) | a (cid:104) n (cid:48) | ρ | n + n (cid:48) (cid:105) a | m (cid:48) (cid:105) b | (20)taken from Eq. (10). In fact, we can easily identify states(such as | α (cid:105)| β (cid:105) ) for which the n -scopic quantum coher-ence vanishes as n → ∞ (for any m (cid:48) , n (cid:48) ), but for whichthe moment (cid:104) ˆ a † n ˆ b n (cid:105) increases. Put another way, the ob-servation (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 does not tell us the probability P R that the system will be found in an associated n -scopicsuperposition Eq. (12), nor the values of the probabilityamplitudes a ( n ) n (cid:48) m (cid:48) ,b ( n ) n (cid:48) m (cid:48) .We explain in this Section that the measured corre-lation (cid:104) ˆ a † n ˆ b n (cid:105) when suitably normalised places a lowerbound on the sum of the magnitudes of the n th orderquantum coherences, defined as C n = N (cid:88) n (cid:48) ,m (cid:48) C ( n (cid:48) ,m (cid:48) ) n (21)Here N is a normalisation factor that ensures the max-imum value of C n = 1 for the optimal case. The nor-malised correlation thus gives measurable informationabout C n which is an effective “ catness-fidelity ”.The “catness-fidelity” contrasts with the standardstate-fidelity measure F (defined as the overlap betweenan experimental state ρ exp and the desired superpositionstate [47]). The standard measure is not directly suffi-cient to quantify a cat-state since it may be possible formixtures that are not cat-type superpositions to give ahigh absolute F as N → ∞ [3]. B. General Result for two-mode mixed states
Defining a suitable catness-fidelity is straightforwardfor pure states. Any two-mode state | ψ (cid:105) can be expandedin the number state basis and can thus be written interms of a superposition of the states (12) but with a ( n ) n (cid:48) m (cid:48) , b ( n ) n (cid:48) m (cid:48) arbitrary. The state fidelity F of | ψ (cid:105) with respectto the symmetric n -scopic superposition | ψ sup (cid:105) = ( | n (cid:48) (cid:105)| m (cid:48) + n (cid:105) + e iφ | n (cid:48) + n (cid:105)| m (cid:48) (cid:105) ) / √ is F = |(cid:104) ψ sup | ψ (cid:105)| = 12 (cid:16) | a ( n ) n (cid:48) m (cid:48) | + | b ( n ) n (cid:48) m (cid:48) | + 2 | a ( n ) n (cid:48) m (cid:48) b ( n ) ∗ n (cid:48) m (cid:48) | (cid:17) (22)where the phase φ is chosen to maximise F . We see thatthe magnitude of the quantum coherence of the pure statedensity operator with respect to the states | n (cid:48) (cid:105)| m (cid:48) + n (cid:105) and | n (cid:48) + n (cid:105)| m (cid:48) (cid:105) is directly related to the fidelity F : C ( n (cid:48) ,m (cid:48) ) n = 2 |(cid:104) m (cid:48) + n |(cid:104) n (cid:48) | ρ | n + n (cid:48) (cid:105)| m (cid:48) (cid:105)| = 2 | a ( n ) n (cid:48) m (cid:48) b ( n ) ∗ n (cid:48) m (cid:48) | (23)We note that F = 1 if and only if a ( n ) n (cid:48) m (cid:48) b ( n ) ∗ n (cid:48) m (cid:48) = 1 / , whichimplies C ( n (cid:48) m (cid:48) ) n = 1 . Similarly, C ( n (cid:48) ,m (cid:48) ) n = 1 implies F =1 . An arbitrary two-mode pure state is a superposition ofstates over different n (cid:48) , m (cid:48) and we may define as the total“ n -scopic catness fidelity” the sum of the magnitudes ofthe n th order coherences i.e. C n = N (cid:80) n (cid:48) ,m (cid:48) C ( n (cid:48) ,m (cid:48) ) n = 2 N (cid:88) n (cid:48) ,m (cid:48) | a ( n ) n (cid:48) m (cid:48) b ( n ) ∗ n (cid:48) m (cid:48) | (24)where N is a normalisation factor to ensure the maximumvalue of C n = 1 . A pure two-mode state with fixed N asgiven in the Introduction can be written | ψ (cid:105) = N (cid:88) m =0 d m | N − m (cid:105) a | m (cid:105) b = (cid:88) m (cid:48) The proof follows from (27) using the definition(21). (cid:3) Realistically, it is difficult in an experiment totruly verify that the probability for obtaining a certainmode number is zero. In light of this, we deduce in theAppendix C a correction term to the Result 4, assumingthe experimentalist is at least able to verify that the “non-relevant” probabilities P m (cid:48) ,n (cid:48) + n , P m (cid:48) + n,n (cid:48) are sufficientlysmall, and that there is a practical upper bound to themode numbers (defined by an energy or atom numberbound). C. Ideal NOON case In the ideal NOON case, an experimentalist would ob-serve N particles in mode a or N particles in mode b .Consider an experiment where indeed only such proba-bilities are nonzero. This is not unrealistic for photonicexperiments with small N that use postselection. Theexperimentalist could deduce that the most general formof the density operator in this case is ρ = P N ρ N + P alt ρ alt (29)where ρ N is the density operator of a NOON superpo-sition (12) (with n (cid:48) = m (cid:48) = d = 0 ), and ρ alt is an al-ternative density operator describing classical mixturesof number states (namely | N (cid:105)| (cid:105) and | (cid:105)| N (cid:105) ). Here, P N + P alt = 1 and P N , P alt are probabilities.We see from (12) that the quantity C N defined as C N = 2 | a ( N )00 b ( N )00 | P N (30)gives an effective fidelity measure of the state ρ relativeto the NOON cat state. We call the quantity C N the catness-fidelity , and note that ≤ C N ≤ . Clearly,the value of C N = 1 is optimal and can only occur ifthe system ρ is the pure symmetric NOON state (1) forwhich | a ( N )00 | = | b ( N )00 | = √ . For the ideal NOON state,the prediction is (cid:104) a † n b n (cid:105) = δ Nn N ! / and S = N ! so thatthe catness fidelity is indeed : c N = C N = 2 N ! |(cid:104) (cid:0) a † (cid:1) n b n (cid:105)| = 1 (31)The value of C N reduces for asymmetric NOON statesor for mixed states where P N < . V. EXAMPLES OF QUANTIFICATIONA. Attenuated NOON states Photonic NOON states have been reported experimen-tally for up to N = 5 . For a rigorous detection of a cat-like state, it is necessary to account for losses that mayarise as a result of processes including detection ineffi-ciencies. To model loss, we use a simple beam splitterapproach [3]. We calculate the moments of final detectedfields ˆ a det , ˆ b det given by a det = √ ηa + √ − ηa v , b det = √ ηb + √ − ηb v where ˆ a , ˆ b are the boson operators forthe incoming field modes, prepared in a NOON state, and a v , b v are boson operators for vacuum modes associatedwith the environment. Here η is the probability that anincoming photon/ particle is detected. We find (cid:104) ˆ a † ndet ˆ b ndet (cid:105) = η n (cid:104) ˆ a † n ˆ b n (cid:105) = η n δ nN N ! / (32)The system is a mixture of type ρ = P N ρ N + P alt ρ alt defined in (29). The catness-fidelity signature C N of Eq.(29) is measurable as c N ( S = N !) defined by (28) andis plotted in Figure 2. Comparing with the distribu-tions of Figure 1 which are generated for the attenuatedNOON state, we see that only the extremes n = N havea nonzero coherence. As loss increases, the N th quan-tum coherence remains (in principle) rigorously certifi-able since it is predicted that (cid:104) ˆ a † N ˆ b N (cid:105) (cid:54) = 0 for all values η . However, the fidelity C N is greatly reduced with de-creasing η , particularly for larger N (Figure 2). u u u u = 1u = 2u = 5u = 10u = 20u = 100 Figure 2: The N th order catness-fidelity C N (Eq. (30)) for theattenuated NOON state versus detection efficiency η . Here C N = c N = 2 (cid:104) ˆ a † n ˆ b n (cid:105) /N ! . c n and (cid:104) ˆ a † n ˆ b n (cid:105) = 0 for n < N . B. States formed from number states incident on alinear beam splitter Next we consider a two-mode number state | N (cid:105)| (cid:105) in-cident at the two single-mode input ports of a beam split-ter, so that N quanta are incident on one arm only. Theoutput state is the N -scopic superposition (2) but withbinomial coefficients: | out (cid:105) = N (cid:88) m =0 d m | m (cid:105) a | N − m (cid:105) b , (33)where d m = √ N ! / (cid:112) N m !( N − m )! . Different to theNOON states, nonzero quantum coherences (cid:104) ˆ a † m ˆ b m (cid:105) (cid:54) = 0 exist for all m ≤ N .Evaluation gives that the pure state n -scopic catness-fidelity (24) (defined as the sum of the magnitude of allthe n th order coherences) is C n = N n,N N − n (cid:88) m =0 | d m d ∗ m + n | (34)where N n,N is a normalisation constant to ensure themaximum value of C n is . For this system, the normal-isation N n,N is determined by the bounds on the coher-ences of the density matrix for a pure state. For exam-ple, where n = N , d d ∗ N ≤ / and hence N N,N = 2 .The general results for the normalisation N n,N are givenin the Appendix C. Using Result 4, a measurable lowerbound to the catness-fidelity given by (28) is c n = N n,N |(cid:104) a † n b n (cid:105)| S (35)where S = max { B ( N,n ) m } (for N fixed) with B ( N,n ) m = (cid:115) ( m + n )! ( N − m )! m ! ( N − m − n )! The value of m that gives the maximum value of B ( N,n ) m is given by: m = ( N − n ) / if N and n have the sameparity, and m = ( N − n ± / if n and N does not havethe same parity. u u u , u u u u u u u = 5u = 10 (a) u u u , u u u u u u 90 100u = 50u = 100 u (b) Figure 3: Measures of n -th order quantum coherence (catness-fidelity) for the output state of the linear beam splitter with N particles incident in one arm. C N and c n vs n , for N = 5 ,10 and 100. C n ≥ c n as expected. The expression S can be determined from the valuesof N, m, n which are known for the experiment. Onecan then experimentally measure the moment (cid:104) ˆ a † n ˆ b n (cid:105) toobtain a value for c n . The prediction is (cid:104) ˆ a † n ˆ b n (cid:105) = N − n (cid:88) m =0 d ∗ m + n d m B ( N,n ) m = N !2 n ( N − n )! (36)A comparison is given between the actual catness-fidelity C n and the estimated one c n in Figure 3 for this beamsplitter case. We see that in this instance the lower boundis a good estimate of the actual fidelity. As might beexpected for this system, the first order quantum coher-ence is significant whereas the highest order coherencegiven by n = N is small. In fact all values of fidelity for n > N/ are insignificant. We also note that for a fixed n , a higher fidelity can be obtained by increasing N tobe much greater than n .With attenuation present for each mode (as describedin the previous section), we evaluate the final detectedmoments. The solutions are (cid:104) ˆ a † ndet ˆ b ndet (cid:105) = η n (cid:104) ˆ a † n ˆ b n (cid:105) where (cid:104) ˆ a † n ˆ b n (cid:105) is given by (36). The density matrix has thesame dimensionality as without losses, and the boundson the coherences and the normalisation N n,N are asabove. Figure 4 plots the values of the catness-fidelity c n versus efficiency η . We note that the first order co-herence n = 1 is much more robust with respect to loss,as compared to the higher order coherences. Interestingis that for a fixed n , the robustness with respect to lossimproves quite dramatically if one increases the value of N . At high N , the highest order coherences are almostimmeasurable e.g. for N = 100 , the quantum coherencebecomes measurable at n < . We note also that thecut-off for a measurable n increases with increasing N ,making generation of n -scopic cat-states in this gener-alised sense quite feasible. u u u (a) u u u (b) Figure 4: Measures of n th order quantum coherence (catness-fidelity) for the output of the linear beam splitter versus de-tection efficiency η . Definitions as for Figure 3. Left N = 5 ,Right N = 100 . VI. MESOSCOPIC QUANTUM COHERENCEIN DYNAMICAL TWO-WELL BOSE-EINSTEINCONDENSATESA. Hamiltonian and Model A mesoscopic NOON state can in principle be createdfrom the nonlinear interaction modelled by the two-modeJosephson (LMG) Hamiltonian [48, 49] H = κ ˆ a † ˆ b + κ ˆ b † ˆ a + g a † ˆ a ] + g b † ˆ b ] (37)( (cid:126) = 1 ). This Hamiltonian is well described in the lit-erature and models a Bose-Einstein condensate (BEC)constrained to two potential wells of an optical lattice[16, 17, 19, 21–23, 41, 45, 50]. The occupation of eachwell is modelled as a single mode (boson operators ˆ a † , ˆ a and ˆ b † , ˆ b respectively). The nonlinearity is quantified by g and the tunnelling between wells by κ . We consider asystem prepared with a definite number N of atoms inone mode (well) (that denoted by ˆ a ). Since the numberof particles is conserved, the state at any later time isof the form (2). The Hamiltonian can be represented inmatrix form and the time dependence of the d m solvedas explained in Refs. [16, 21, 22, 50]. B. Two-state oscillation and creation ofNOON-states Solutions give the probability P ( m ) = | d m | of mea-suring m particles in the well A at a given time. For someparameters, the population oscillates between wells andthere is an almost complete transfer to the well B at sometunnelling time T N . For larger nonlinearity g , the systemcan approximate a dynamical two-state system, showingoscillations between the two distinguishable states | N (cid:105)| (cid:105) and | (cid:105)| N (cid:105) over long timescales (Figure 5). At inter-mediate times ( ∼ T N / ) before the complete tunnellingfrom one state to the other, approximate NOON statescan be formed. Figure 6 depicts the probabilities P ( m ) at the intermediate times T N / and T N / that violatea Leggett-Garg inequality [27, 28]. It is known howeverthat even for moderate N , the predicted tunnelling times T N are typically much longer than practical decoherencetimes [19, 21, 51, 52]. For instance, Carr et al reportimpossibly long times for the typical parameters of Rbatoms [21]. u u ( u ) u = 0 u = u u /6 u = u u /3 u / u u u u ( u ) (a) u u ( u ) u = 0 u = u u /6 u = u u /3 u / u u u u ( u ) (b) Figure 6: Mesoscopic two-state oscillation and generation ofNOON-type states: Top: N = 100 , g = 2 , n L = 10 . Below: N = 20 , g = 4 , n L = 4 . Time t is in units κ . u u ( u ) u = 0 u = u u /6 u = u u /3 u / u u u u ( u ) Figure 5: Two-state mesoscopic dynamics: The creation ofNOON states. Top: Probability P ( m ) for the number ofatoms in well a at times t = 0 , t = T N / , t = T N / .Here j z = 2 m − N . Beneath shows the two-state oscillation.We use N = 100 , g = 1 . Time t is in units κ . u/u u u u (a) u/u u u u (b) Figure 7: Signifying the creation of NOON-type states underthe Hamiltonian (37): The n -th order quantum coherencemeasure c n versus time t in units κ . Left: N = 5 , g = 10 , n L = 0 . The NOON state N = 5 is signified by c = 1 , c i ∼ ( i (cid:54) = 5 ) at t = T N / . Right: N = 20 , g = 4 , n L = 4 asfor Figure 6b. The large quantum coherence c n for n = 12 signifies the superposition (38) at t = T N / . u u ( u ) u u u (a) (b) Figure 8: Plot of P ( m ) and the n th order quantum coherence c n for the state of Figure 6b at t = T N / (as in Figure 7b). C. Creation of n -scopic quantum superpositions It is possible however to generate states with a sig-nificant mesoscopic coherence by preparing the systemin an initial state | n L (cid:105)| N − n L (cid:105) where n L (cid:54) = 0 , N . Aspointed out by Gordon and Savage [19] and Carr et al[21], the Hamiltonian (37) predicts (in some parameterregimes) an approximate two-state oscillation betweenthe two states | n L (cid:105)| N − n L (cid:105) and | N − n L (cid:105)| n L (cid:105) . At ap-proximately half the time for oscillation from one stateto the other, an n -scopic superposition state of the typegiven by (12) where m (cid:48) , n (cid:48) (cid:54) = 0 is formed i.e. | ψ (cid:105) = 1 √ {| n L (cid:105)| N − n L (cid:105) + | N − n L (cid:105)| n L (cid:105)} (38)Here, n = N − n L . Such n -scopic superposition stateshave been called “embedded” cat-states [21]. These em- bedded cat-states are identical to those superpositions(12) discussed in the previous section. Calculations re-veal that for some parameters, the period of oscillationreduces to practical values [19, 21]. Two-state oscillationof the BEC has been experimentally observed [26]. Wepresent in Figure 6 predictions for this type of oscillationwith N = 20 and n L = 2 where the solutions indicatestates (38) with a separation of n = 16 atoms.The question becomes how to certify the quantum co-herence of the embedded cat-states (38) that may begenerated in the experiment where such oscillation is ob-served. The value of the catness-fidelity signature c n iscalculated and given in Figure 7, for the parameters ofFigure 6. The c n for moderate n would feasibly be mea-surable using higher order interference in multi-atom de-tection, as described in Section VII. VII. MEASUREMENT OF MESOSCOPICQUANTUM COHERENCE VIA (cid:104) ˆ a † n ˆ b n (cid:105) Finally, we address how one may measure the corre-lation (cid:104) ˆ a † n ˆ b n (cid:105) . The measurement of J Z is a photon oratom number difference, achievable with counting de-tectors or imaging. Schwinger spin operators J X =( a † b + b † a ) / , J Y = (cid:0) a † b − b † a (cid:1) / i are measured simi-larly as a number difference, after rotating to a differentmode pair using polarisers [53]; or Rabi rotations with π/ pulses [41, 44, 45]; or beam splitters and phase shifts. A. Interferometric detection For instance, we consider the measurable output num-ber difference I D after transforming the incoming modes a , b to new modes c , d via a 50/ 50 beam splitter andphase shift ϕ : I D = ˆ c † ˆ c − ˆ d † ˆ d = ˆ a † ˆ be iφ + ˆ a ˆ b † e − iφ = 2 J X cos φ − J Y sin φ (39)Here the transformed boson operators for the new modesare ˆ c = (ˆ a +ˆ b exp iφ ) / √ , ˆ d = (ˆ a − ˆ b exp iφ ) / √ . Selecting φ = 0 or φ = − π/ measures J X or J Y . For N = 1 , (cid:104) ˆ a † ˆ b (cid:105) = (cid:104) J X + iJ Y (cid:105) . The first order moment (cid:104) ˆ a † ˆ b (cid:105) is thusmeasurable via the fringe visibility in I D as one varies φ i.e. that (cid:104) a † b (cid:105) is nonzero is detectable via first orderinterference. Similar transformations using atom inter-ferometry give the same results as explained in Ref. [54].If we have a NOON state incident on the interferometer,the nonzero value of (cid:104) ˆ a † N ˆ b N (cid:105) can be deduced by observa-tion of higher order interference fringes that are signifiedby an e iNφ oscillation. This is the usual method for de-tecting NOON states [11–13, 22].The method can also be used to detect and quantifythe n th order quantum coherence c n . We consider thatwe have a fixed total number N of particles so the input0state is of the form (2). The probability of detecting N quanta at the output denoted by mode c is (cid:104) ˆ c † N ˆ c N (cid:105) /N ! .The probability of obtaining M particles at the port c isa calculable function of the correlation functions (cid:104) ˆ c † n ˆ c n (cid:105) where n ≥ M . Suppose we measure (cid:104) ˆ c † n ˆ c n (cid:105) for a givenfixed n . Expanding we find (cid:104) ˆ c † n ˆ c n (cid:105) = 12 n (cid:104) (ˆ a † + ˆ b † e − iφ ) n (ˆ a + ˆ be iφ ) n (cid:105) = 12 n n (cid:88) m =0 (cid:18) nm (cid:19) n (cid:88) (cid:96) =0 (cid:18) n(cid:96) (cid:19) ×(cid:104) (ˆ a † ) n − m (ˆ b † ) m (ˆ a ) n − (cid:96) ˆ b (cid:96) (cid:105) e iφ ( (cid:96) − m ) (40)The terms that oscillate as e inφ are proportional to the n th order moment (cid:104) ˆ a † n ˆ b n (cid:105) . Hence, if we measure (cid:104) ˆ c † n ˆ c n (cid:105) ,the fringe visibility associated with this oscillation al-lows determination of the magnitude of (cid:104) ˆ a † n ˆ b n (cid:105) . Where (cid:104) ˆ a † n ˆ b n (cid:105) is the only nonzero moment (as for the idealNOON states with n = N ), only the oscillation e inφ will contribute and the higher order interference enable aclear signature and quantification of the n th order quan-tum coherence (cid:104) ˆ a † n ˆ b n (cid:105) .For nonideal NOON states the interference methodbecomes less precise. However, the rapidly oscillat-ing terms can only arise from moments that indicatea higher order of quantum coherence. This is evidentby the last line of (40). The moments are of form (cid:104) (ˆ a † ) n − m ˆ a n − (cid:96) ˆ b † m ˆ b (cid:96) (cid:105) e iϕ ( (cid:96) − m ) so the oscillation frequencywhere l − m = n requires l = n and m = 0 and thereforehas a nonzero amplitude only if (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 , which is asignature for a quantum coherence of order n . Similarly,the oscillation frequency l − m = n − requires a nonzeroquantum coherence of order n − . While the (cid:104) ˆ c † n ˆ c n (cid:105) canbe evaluated from the probabilities for particle counts,in practice for large numbers N , resolution of atom orphoton number is difficult. Here one can measure theprobability that n is in a binned region the n e.g. theregion n > M . This probability is given by P ( n ≥ M ) = (cid:88) n ≥ M ς (cid:104) ˆ c † n ˆ c n (cid:105) /M ! where ς are calculable constants. Here, measurement ofa nonzero amplitude for oscillations e iMφ with frequency M or greater is evidence of quantum coherence of order (cid:38) M . The high frequency oscillation can only arise fromthe high order quantum coherence terms. In Figure 9, weplot P ( n > M ) and the Fourier analysis for the two-modeexample given in Figure 6b. B. Spin squeezing observables and quadraturephase amplitudes An alternative method is given in Ref. [54] for N = 2 .We note that (cid:104) ˆ a † ˆ b (cid:105) = (cid:104) ˆ J X (cid:105) − (cid:104) ˆ J Y (cid:105) + i (cid:104){ ˆ J X , ˆ J Y }(cid:105) (41) u u ( u ≥ u ) u = 10u = 11u = 12u = 13u = 14 (a) u (peaks) | F [ u ( u ≥ u ) ]| u = 10u = 11u = 12u = 13u = 14 (b) Figure 9: (a) The probability of measuring more or equal to M photons, for N = 20 , g = 4 , n L = 4 at t = T N / (same asin Fig. 7b) after a rotation. (b) The Fourier transform of thecurves from (a), plotted against angular frequency (which isequivalent to the number of oscillations in the range of π ),showing a significant peak at ω = 12 (the expected separationof the state, ( | (cid:105)| (cid:105) + | (cid:105)| (cid:105) ) / √ ) for all M . where { A, B } ≡ AB + BA . The real part of (cid:104) ˆ a † ˆ b (cid:105) can be evaluated by measurement of (cid:104) ˆ J X (cid:105) and (cid:104) ˆ J Y (cid:105) . Weshow that the moment is nonzero if we can show that (cid:104) ˆ J X (cid:105) (cid:54) = (cid:104) ˆ J Y (cid:105) . If necessary, the imaginary part can bedetermined by measurement of suitably rotated spin ob-servables defined by ˆ J θ = ˆ J X cos θ − ˆ J Y sin θ .For N = 3 manipulation gives (see Appendix for de-tails) (cid:68) ˆ a † ˆ b (cid:69) = 2 (cid:68) ˆ J X (cid:69) − √ (cid:104) ˆ J π (cid:105) + (cid:104) ˆ J π (cid:105) ) − i (cid:68) ˆ J Y (cid:69) + i √ (cid:104) ˆ J π (cid:105) + (cid:104) ˆ J π (cid:105) ) (42)where (cid:104) ˆ J θ (cid:105) are measurable by standard interferometry/atom interferometry techniques.We note that similar expansions can be made express-ing the a and b operators in terms of quadrature phaseamplitudes X and P . For optical NOON states, thismay be a useful way to accurately measure the moments (cid:104) ˆ a † M ˆ b M (cid:105) since quadrature phase amplitudes can be mea-sured with high efficiency. Specifically, we define the am-plitudes ˆ X and ˆ P by ˆ a = ˆ X A + i ˆ P A and ˆ b = ˆ X B + i ˆ P B .Hence (we drop the “hats” for convenience) (cid:104) ˆ a † ˆ b (cid:105) = (cid:104) ˆ X A ˆ X B (cid:105) + (cid:104) ˆ P A ˆ P B (cid:105) − i (cid:104) ˆ P A ˆ X B + ˆ X A ˆ P B (cid:105) (43)1which is readily measurable. Continuing (cid:104) ˆ a † ˆ b (cid:105) = (cid:104) ( ˆ X A − ˆ P A )( ˆ X B − ˆ P B ) (cid:105) + (cid:104){ ˆ X A , ˆ P A }{ ˆ X B , ˆ P B }(cid:105)− i (cid:104){ ˆ X A , ˆ P A } ( ˆ X B − ˆ P B ) (cid:105) + i (cid:104) ( ˆ X A − ˆ P A ) { ˆ X B , ˆ P B }(cid:105) (44)The anticommutator is measurable by rotation of thequadratures. We define the measurable rotated quadra-ture phase amplitudes as ˆ X θ = ˆ X cos( θ ) + ˆ P sin( θ ) and ˆ P θ = − ˆ X sin( θ ) + ˆ P cos( θ ) . Hence, ˆ X π/ = √ { ˆ X + ˆ P } and ˆ P π/ = √ {− ˆ X + ˆ P } and we note that (cid:104) ˆ X π/ (cid:105) = (cid:104) ˆ X + ˆ P + ˆ X ˆ P + ˆ P ˆ X (cid:105) / . Thus, we can deduce ei-ther { ˆ X, ˆ P } by measuring the moments (cid:104) ˆ X (cid:105) , (cid:104) ˆ P (cid:105) and (cid:104) ˆ X π/ (cid:105) . C. Experimental certification of atomic quantumcoherence n ∼ by inferring the correlation (cid:104) ˆ a † n ˆ b n (cid:105) from spin squeezing Esteve et al. experimentally realise the system mod-elled by the two-mode Hamiltonian [41]. The groundstate solutions have been solved and studied in Ref [23].Esteve et al report data obtained on cooling their two-well system, including measurements for the spin mo-ments (cid:104) ˆ J θ (cid:105) associated with ultra-cold atomic mode pop-ulations of two wells of the optical lattice [41]. Theirobservations analyse the variances of the Heisenberg un-certainty principle ∆ ˆ J z ∆ ˆ J y ≥ |(cid:104) ˆ J x (cid:105)| / ∼ N/ (45)They report spin squeezing in ˆ J z with enhanced noise in ˆ J y . They also report (cid:104) ˆ J z (cid:105) ∼ and (cid:104) ˆ J y (cid:105) ∼ . Hence wecan conclude (cid:104) ˆ J z (cid:105) < N/ < (cid:104) ˆ J y (cid:105) (46)Thus we deduce (cid:104) ˆ J y (cid:105) − (cid:104) ˆ J z (cid:105) (cid:54) = 0 (47)which implies (cid:104){ ˆ J cx , ˆ J cy }(cid:105) (cid:54) = 0 where ˆ J cx , ˆ J cy areSchwinger operators defined for the rotated modes ˆ c =(ˆ a + ˆ b ) / √ and ˆ d = e − iπ/ √ (ˆ a − ˆ b ) . Hence we conclude |(cid:104) ˆ c † ˆ d (cid:105)| (cid:54) = 0 (48)which (using the Results of Section III) gives evidence intheir BEC system of a two-atom coherence i.e. a gener-alised n -scopic superpositions with n = 2 of type | ψ (cid:105) = c | (cid:105) c | (cid:105) d + c | (cid:105) c | (cid:105) d + c | (cid:105) c | (cid:105) d + ψ (49)where the coefficients satisfy c (cid:54) = 0 and c (cid:54) = 0 andwhere ψ is orthogonal to each of | (cid:105) c | (cid:105) d , | (cid:105) c | (cid:105) d and | (cid:105) c | (cid:105) d . We note that this is consistent with the predic-tions of [23] for the nonzero moments (cid:104) ˆ c † ˆ d (cid:105) (cid:54) = 0 for the populations of modes c, d in atomic systems with κ < .The observation of (cid:104) ˆ a † ˆ b (cid:105) (cid:54) = 0 would be evidence of asuperposition of atoms constrained to the modes of thewells | ψ (cid:105) = c | (cid:105) a | (cid:105) b + c | (cid:105) a | (cid:105) b + c | (cid:105) a | (cid:105) b + ψ (50)where c (cid:54) = 0 and c (cid:54) = 0 . This is predicted for atomicBEC with κ > [23]. Three-atom superpositions (forwhich (cid:104) ˆ a † ˆ b (cid:105) (cid:54) = 0 ) and higher are also predicted (up to N ) and should be evident via higher order fringe pat-terns, or else directly via the J θ measurements as above. D. Entanglement The observation of the n th order quantum coherence (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 is not in itself sufficient to imply entangle-ment . For instance ψ in the expression (49) might in-clude contributions from terms such as | (cid:105)| (cid:105) and | (cid:105)| (cid:105) .This means that a separable form for | ψ (cid:105) e.g. | ψ (cid:105) = 12 ( | (cid:105) c + | (cid:105) c )( | (cid:105) d + | (cid:105) d ) may be possible. The separable state contrasts withthe “dead here-alive there” entangled superposition statewhose ideal form is precisely the NOON state e.g. for N = 2 | ψ (cid:105) = 1 √ {| (cid:105) c | (cid:105) d + | (cid:105) c | (cid:105) d } In this paper, we are only concerned with how to cer-tify an n -scopic quantum superposition, without regardto entanglement. However, the entangled case is of spe-cial interest, especially where the two modes are spatiallyseparated. For the ideal NOON case, we therefore pointout that one can make simple measurements to confirmthe entanglement. If one measures the individual modenumbers n a and n b , the results or N are obtained foreach mode. The observations would be correlated, so thatthere is only a nonzero probability to obtain | N (cid:105)| (cid:105) or | (cid:105)| N (cid:105) . This eliminates the possibility of nonzero contri-butions from terms in ψ and it remains only to confirmthe nonzero quantum coherence in order to confirm theentanglement. The observation of (cid:104) ˆ a † N ˆ b N (cid:105) (cid:54) = 0 then be-comes sufficient to certify the entanglement of the NOONstate. While simple in principle, this procedure is not souseful in practice. For example, the attenuated NOONstate of Section V would predict a nonzero probability forobtaining | (cid:105)| (cid:105) and a more careful analysis is necessaryto deduce entanglement. VIII. CONCLUSION We have examined how to rigorously confirm andquantify the mesoscopic quantum coherence of non-ideal2NOON states. In this paper, we link the observation ofquantum coherence to the negation of certain types ofmixtures, given as (3) and (4). However it is stressed weare restricting to mixtures where the “dead” and “alive”states are quantum states that can therefore be repre-sented by density operators ( ρ A and ρ D in the equation(3)). This contrasts with other possible signatures of acat-state where the dead and alive states might also behidden variable states, as in Ref. [27].In this paper, we have focused on two criteria for the n -th order quantum coherence , defined as a quantum coher-ence between number states different by n quanta. Thefirst criterion is a nonzero n th order moment (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 and the second is a quantifiable amount of Schwingerspin squeezing. We have shown how the first criterioncan be a quantifier of the overall n th order coherence.The second criterion can be a robust and effective sig-nature for large n , and can verify high orders of coher-ence in existing atomic experiments, but does not sig-nify all cases of n -scopic quantum coherence. In Sec-tions V-VI, we have illustrated the use of the criteriawith the examples of attenuated NOON states, numberstates | N (cid:105) that pass through beam spitters, and approxi-mate NOON states formed from N particles via nonlinearinteractions. These examples model recent photonic andatomic BEC experiments.In Section VII, we have examined how the moments (cid:104) ˆ a † n ˆ b n (cid:105) might be measured. Optical NOON states arenormally verified by n th order interference fringes, whichimply (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 . Direct photon detection normally in-troduces high losses which creates low fidelities that maymake significant statistics difficult, except with post se-lection. We suggest that to obtain higher cat-fidelities themoments can be measured via high efficiency quadraturephase amplitude detection.Finally, in Section VIII, we analyse data from exper-iments, noting that the signatures do not directly proveentanglement i.e. do not distinguish between a local su-perpositions of type | N (cid:105) + | (cid:105) for one mode, and the en-tangled superposition of the NOON state. Hence we can-not conclude a superposition of states with different masslocations, although we believe this could be possible us-ing for instance the entanglement criteria presented inRefs. [22, 54, 55]. Acknowledgements We thank P. Drummond, B. Dalton, Q. He and thoseat the 2016 Heraeus Seminar on Macroscopic Entangle-ment for discussions on topics related to this paper. Weare grateful to the Australian Research Council for sup-port through its Discovery Projects program. Appendix A: Result 2 To explain the connection between the condition (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 and the superposition state (12) in detail,consider the most general two-mode quantum state forthis two-mode system that cannot be a superposition oftwo states distinct by n quanta. We note that any purestate | ψ ent (cid:105) can be expanded in the two-mode number(Fock) state basis: | ψ ent (cid:105) = (cid:88) n,m c nm | n a (cid:105)| m b (cid:105) = c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) + ... We see that if (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 , then the state is necessarily ofthe form (12)which involves a superposition of two statesdistinct by n quanta. We note that when (cid:104) ˆ a † n ˆ b n (cid:105) (cid:54) = 0 , thedensity operator ρ for the system cannot be written in analternative form except to provide a nonzero coherence(10) between states | n (cid:48) (cid:105)| m (cid:48) + n (cid:105) and | n (cid:48) + n (cid:105)| m (cid:48) (cid:105) . We con-clude that the diagonal elements b (cid:104) m (cid:48) + n | a (cid:104) n (cid:48) | ρ | n (cid:48) (cid:105) a | m (cid:48) + n (cid:105) b and b (cid:104) m (cid:48) | a (cid:104) n (cid:48) + n | ρ | n (cid:48) + n (cid:105) a | m (cid:48) (cid:105) b are also nonzero.Thus, there is a nonzero probability P D that the systemis found in state | n (cid:48) (cid:105)| m (cid:48) + n (cid:105) (that we call “dead”) andalso a nonzero probability P A that the system is found instate | n (cid:48) + n (cid:105)| m (cid:48) (cid:105) (that we call “alive”). Yet, the superpo-sition state (12) cannot be given as a classical mixture (9)which has a zero coherence between the states | n (cid:48) (cid:105)| m (cid:48) + n (cid:105) and | n (cid:48) + n (cid:105)| m (cid:48) (cid:105) whose J z values are different by n . Appendix B: Proof of Result 3 for Spin squeezingtest We follow from the main text and generalise to con-sider a two-mode description of the state as given by fora mixed state by a density operator ρ . We expand interms of pure states | ψ R (cid:105) so that ρ = (cid:80) R P R | ψ R (cid:105)(cid:104) ψ R | for some probabilities P R . Each pure state | ψ R (cid:105) can beexpressed as a superposition of number eigenstates givenby (14). We know that the variance (∆ ˆ J Y ) of any mix-ture satisfies (∆ ˆ J Y ) ≥ (cid:80) R P R (∆ ˆ J Y ) R . Thus (∆ ˆ J Y ) ≥ (cid:88) R P R (∆ ˆ J Y ) R ≥ (cid:88) R P R |(cid:104) ˆ J X (cid:105) R | J Z ) R For all the possible mixtures denoted by a choice of set {| ψ R (cid:105)} (where P R (cid:54) = 0 ) we can determine the spread δ R for each state | ψ R (cid:105) and then select the maximum of theset δ R and call it δ . We select the mixture set consistentwith the density operator that has the minimum possiblevalue of δ : That is, we determine that the density oper-ator cannot be expanded in a set | ψ R (cid:105) with a smaller δ .Then for the pure states of this set | ψ R (cid:105) , the maximumvariance in ˆ J Z is (∆ ˆ J Z ) = δ / i.e. (∆ ˆ J Z ) R ≤ δ / .3Then we see that the uncertainty relation (16) implies aminimum value for the variance in ˆ J Y : (∆ ˆ J Y ) R ≥ |(cid:104) ˆ J X (cid:105) R | J Z ) R ≥ δ |(cid:104) ˆ J X (cid:105) R | Simplification gives (∆ ˆ J Y ) ≥ δ (cid:88) R P R |(cid:104) ˆ J X (cid:105) R | ≥ δ | (cid:88) R P R (cid:104) ˆ J X (cid:105) R | = 1 δ |(cid:104) ˆ J X (cid:105)| Taking the case of the spin squeezing experiments wheremeasurements give (cid:104) ˆ J X (cid:105) ∼ (cid:104) N (cid:105) / , we see that (∆ ˆ J Y ) ≥ δ |(cid:104) ˆ J X (cid:105)| = (cid:104) N (cid:105) δ (B1)Thus there is a lower bound on the best amount of squeez-ing determined by the maximum spread (extent) δ ofthe superposition. We can now prove the Result 3: Themeasured amount of squeezing places a lower bound onthe extent δ of the broadest superposition: Thus if themeasured squeezing is ξ N , then from (B1) the underly-ing state has a minimum breadth δ of superposition (inthe eigenstates of ˆ J Z ) given by δ > (cid:104) N (cid:105) J Y ) = √ Nξ N . Thewidth δ of the superposition gives the extent or size ofthe coherence i.e. the value of n in the expression (12). Appendix C: Catness-fidelity quantifier for mixedstates Discussion in terms of superposition states: We givethe proof of Result (4) in terms of the superpositionstates. The experiment may confirm a range of values of j z for J z for which (cid:104) ˆ a † j z ˆ b j z (cid:105) (cid:54) = 0 . Take one such value: j z = N . Then we know there is a nonzero probability P N that the system be in a superposition of form | ψ N (cid:105) nm = a ( n,m ) N | n (cid:105)| m + N (cid:105) + b ( n,m ) N | n + N (cid:105)| m (cid:105) + c | ψ (cid:105) (C1)where a ( n,m ) N ,b ( n,m ) N (cid:54) = 0 . Based on the measured mo-ments, we can write the density operator in the generalform ρ = (cid:88) n,m P ( n,m ) N ρ ( n,m ) N + P mix ρ mix + P n ρ n (C2)where ρ N = | ψ N (cid:105)(cid:104) ψ N | , ρ mix is a mixture of states | n (cid:105)| m + N (cid:105) and | n + N (cid:105)| m (cid:105) , and ρ n is a state thatgives predictions different to j z = ± N / . Only the firstterm will contribute a nonzero value of (cid:104) a † N b N (cid:105) . Thefirst term can also include superpositions of the differ-ent | ψ N (cid:105) with different n, m but evaluation of the mo-ment (cid:104) a † N b N (cid:105) will be the same as if the system were in a mixture of those states (due to the orthogonality).The relevant ( Re ) values of n , m such that probabilitiesare nonzero can be determined from the measurementsof mode number and we assume the sums only includesthose nonzero contributions. We note that the first termis written as a mixture of the NOON-type states. In somecases, such a mixture can be equivalent to (and thereforerewritten as) a classical mixture ρ mix , but the nonzeromoment (cid:104) ˆ a † N ˆ b N (cid:105) cannot arise in this case. The valueof (cid:104) ˆ a † N ˆ b N (cid:105) is zero for any ρ mix , and the prediction for (cid:104) ˆ a † N ˆ b N (cid:105) given by ρ is |(cid:104) ˆ a † N ˆ b N (cid:105)| = | (cid:88) n,m a ( n.m ) N b ( n,m ) ∗ N P ( n,m ) N × (cid:114) ( m + N )! m ! (cid:114) ( n + N )! n ! |≤ S (cid:88) n,m | a ( n,m ) N b ( n,m ) ∗ N P ( n,m ) N | (C3)We have used the prediction for (cid:104) a † N b N (cid:105) for the state | ψ N (cid:105) and the definitions of S as in the main text. Themeasurement of the moment (cid:104) ˆ a † N ˆ b N (cid:105) thus allows thedetermination of a lower bound on an effective fidelityfor the Schrodinger cat NOON state. Correction term: Now we consider that the experimen-talist can only confirm that the total probability of the“nonrelevant” ( N Re ) outcomes is less than or equal to (cid:15) . The contribution of the “nonrelevant” terms to the C N (the sum of the N -th order coherences) is boundedby the probabilities. For any density matrix, the off-diagonal elements are bounded by the diagonal elementsthat give the probabilities: Always a ( n.m ) N b ( n,m ) ∗ N ≤ andassuming (cid:80) NRe P ( n,m ) N ≤ (cid:15) , we find (cid:88) NRe a ( n.m ) N b ( n,m ) ∗ N P ( n,m ) N ≤ (cid:15)/ Using that ( n + N )! n ! ≤ ( n + N ) N , this implies (cid:104) ˆ a † N ˆ b N (cid:105) ≤ (cid:88) Re a ( n.m ) N b ( n,m ) ∗ N P ( n,m ) N × (cid:114) ( m + N )! m ! (cid:114) ( n + N )! n !+ (cid:88) NRe a ( n.m ) N b ( n,m ) ∗ N P ( n,m ) N ( N up + N ) N ≤ S (cid:88) n,m | a ( n,m ) N b ( n,m ) ∗ N P ( n,m ) N | + (cid:15) N up + N ) N (C4)Thus we know that C N ≥ (cid:88) Re a ( n.m ) N b ( n,m ) ∗ N P ( n,m ) N ≥ [ (cid:104) ˆ a † N ˆ b N (cid:105) − (cid:15) N up + N ) N ] /S (C5)4where N up is the upper bound for the mode numbers,given that the system cannot have infinite mode or par-ticle (atom) number. For the cases of interest to us onthis paper, the total mode number is the atom number N , which is fixed. Appendix D: Evaluation of Normalisation We consider the state | out (cid:105) = N (cid:88) m =0 d m | m (cid:105) a | N − m (cid:105) b , (D1)We quantify the n -th order quantum coherence by the pa-rameter C n (that we have also called the catness-fidelity) C n = N n,N N − n (cid:88) m =0 | d m d ∗ m + n | (D2)where N n,N is a normalisation constant to ensure themaximum value of C n is . The normalisation N n,N isdetermined by the bounds on the coherences of the den-sity matrix for a pure state. For example, where n = N ,the maximum | d d ∗ N | is obtained for d = d N = √ with all other amplitudes zero. Hence | d d ∗ N | ≤ / and N N,N = 2 . Similarly, for n = N − and N ≥ (so that the d terms in d d ∗ N − + d d ∗ N areall different), we find d d ∗ N − + d d ∗ N ≤ / where inthis case the maximum (cid:80) N − nm =0 | d m d ∗ m + n | is found taking d = d N − = d = d N = . The maximum value formore general n and N can be found numerically.(1) We start by analyzing n = N . Then C n = N n,N | d d ∗ N | . There is only one term in the sum andtherefore only two amplitudes contributing to the sum.The number of terms is independent of N . We canshow that the maximum value of the sum of the co-herences (namely (cid:80) N − nm =0 | d m d ∗ m + n | ) is given when d = d N = √ , and all other amplitudes zeros. Hence C n ≤N n,N | d d ∗ N | = N n,N and the optimal normalisation is N n,N = 2 .(2) Next we consider n > N/ . Here (cid:80) N − nm =0 | d m d ∗ m + n | = d d n + d d n +1 + ..d N − n d N andsince n > N − n the terms in the summation involvedifferent d i ’s which can be therefore be chosen indepen-dently apart from normalisation requirements. Takingthe N − n + 1) contributing amplitudes as equal, andall other as zero, (cid:80) N − nm =0 | d m d ∗ m + n | = ( N − n +1)2( N − n +1) = which we verify is the maximum value.(3) For the remaining values, we determine the boundsnumerically. We analyse all these cases and fit an expres-sion for the maximum value of (cid:80) N m =0 | d m d ∗ m + N | . On nu-merically analysing the cases n < N/ , we find that to agood approximation (cid:80) N − nm =0 | d m d ∗ m + n | ≤ cos (cid:16) π [ N/n ]+2 (cid:17) , where [ N/n ] denotes the integer part of N/n and hence N n,N = 1 / cos (cid:16) π [ N/n ]+2 (cid:17) . We numerically verified thisbound for all N up to . Appendix E: Evaluation of (cid:104) (cid:16) ˆ a † ˆ b (cid:17) (cid:105) For N = 3 , we would like to measure the expectationvalue of the following observable (cid:16) ˆ a † ˆ b (cid:17) = ( J x + iJ y ) = J x − iJ y + i (cid:0) J x J y J x + J y J x + J x J y (cid:1) − (cid:0) J y J x + J x J y + J y J x J y (cid:1) . In the expansion, we have dropped the “hats” and usedlower case x and y in the subscripts of the ˆ J X and ˆ J Y defined in (13) to simplify notation. The first and secondterms can be measured in experiments. However, we needto express J x J y J x + J y J x + J x J y and J y J x + J x J y + J y J x J y in terms of some other measurements that can be carriedout in experiments. To this end, we define a rotatedSchwinger operators as follows: J θ = J x cos θ + J y sin θJ θ + π ≡ G θ = J x cos (cid:16) θ + π (cid:17) + J y sin (cid:16) θ + π (cid:17) = − J x sin θ + J y cos θ. For θ = π , these rotated operators correspond to: J π = 1 √ (cid:2)(cid:0) J y J x + J x J y + J y J x J y (cid:1) + (cid:0) J x J y J x + J y J x + J x J y (cid:1) + J x + J y (cid:3) G π = 1 √ (cid:2) − (cid:0) J y J x + J x J y + J y J x J y (cid:1) + (cid:0) J x J y J x + J y J x + J x J y (cid:1) − J x + J y (cid:3) . 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