Quantum Fisher Information As a Predictor of Decoherence in the Preparation of Spin-Cat States for Quantum Metrology

PHYSICAL REVIEW A 95, 043642 (2017)

Quantum Fisher information as a predictor of decoherence in the preparation of spin-cat states for quantum metrology

Samuel P. Nolan1,* and Simon A. Haine2 1School of Mathematics and Physics, University of Queensland, Brisbane, Queensland, Australia 2Department of Physics and Astronomy, University of Sussex, Brighton, United Kingdom (Received 14 November 2016; published 28 April 2017) In its simplest form, decoherence occurs when a quantum state is entangled with a second state, but the results of measurements made on the second state are not accessible. As the second state has effectively “measured” the ﬁrst, in this paper we argue that the quantum Fisher information is the relevant metric for predicting and quantifying this kind of decoherence. The quantum Fisher information is usually used to determine an upper bound on how precisely measurements on a state can be used to estimate a classical parameter, and as such it is an important resource. Quantum-enhanced metrology aims to create nonclassical states with large quantum Fisher information and utilize them in precision measurements. In the process of doing this it is possible for states to undergo decoherence; for instance atom-light interactions used to create coherent superpositions of atomic states may result in atom-light entanglement. Highly nonclassical states, such as spin-cat states (Schrödinger cat states constructed from superpositions of collective spins) are shown to be highly susceptible to this kind of decoherence. We also investigate the required ﬁeld occupation of the second state, such that this decoherence is negligible.

DOI: 10.1103/PhysRevA.95.043642

I. INTRODUCTION condensates [28–31]. There has also been success outside the realm of quantum atom-optics, with NOON states having been Quantum metrology is the science exploiting quantum realized in modestly sized systems such as superconducting correlations to estimate a classical parameter χ, such as a flux qubits [32], optics [33], and in trapped ions [34–36], the phase, beyond the sensitivity available in uncorrelated systems. latter with up to 14 particles. Given a metrological scheme with access to N total particles In any case, to actually do anything useful with such there is an upper bound on the precision, χ 1/N, called a state, it may be necessary to perform a unitary rotation. the Heisenberg limit [1,2]. For a two-mode interferometer This could be, for example, to prepare the state for input with conserved total particle number, called an SU(2) inter- into an interferometer. However, unitary evolution is only an ferometer, the class of states which yield Heisenberg-limited approximation, valid when the system used to perform this sensitivity are spin-cat states, an example of which is the operation is sufficiently large such that it can be considered well-known NOON state, which achieves Heisenberg-limited classical. sensitivities via a parity measurement [3–6]. In this paper we relax this approximation, and investigate As NOON states are highly nonclassical, possibly massive rotations caused by interaction with a quantized auxiliary superpositions, they could also find a number of applications system. As an example, consider a two-component atomic outside quantum metrology. In particular these states could be Bose-Einstein condensate. A rotation of the state on the Bloch well suited for testing macroscopic realism [7,8], gravitational sphere can be implemented by interaction with an optical decoherence [9], and spontaneous wave function collapse field via the ac Stark shift [37]. It is often the case that theories [10–12], as well as realizing the Greenberger, Horne, the number of photons in this state is sufficiently large that and Zeilinger (GHZ) state, which could test local hidden the quantum degrees of freedom of the light are ignored. variable theories [13]. Optical GHZ states could also find However, in metrology we are are often interested in quantum applications in quantum communication and computation states that are particularly sensitive to decoherence, such as [14–17]. spin-cat states; therefore in this paper we investigate the effect A spin-cat state in a Bose-Einstein condensate would be of treating this optical field as a quantized auxiliary system. well suited to a number of these applications, particularly Decoherence in systems such as these has been considered metrology. However, this state has yet to be realized, due to previously [38], using a stochastic wave function approach in the immense challenge of maintaining the quantum coherence small systems. Although quantum Fisher information (QFI) of of the state [18,19]. This is despite a number of proposed quantum states with decoherence has also been considered in methods, previously relying on Josephson coupling between the literature [39–41], the goal of this paper is to employ the two modes [20–22], collisions of bright solitons [23], and approach of defining QFI for the optical field. through the atomic Kerr effect [24–27]. Although the atomic It has been shown that in the presence of entanglement interaction times have been too small to generate spin-cat between the state and some auxiliary system, the metrological states, the Kerr effect has successfully been used to gen- usefulness of the state may be enhanced by allowing mea- erate large numbers of entangled particles in Bose-Einstein surements on the auxiliary system [42–48]. However in this paper we take a different approach, and study the metrological usefulness of a state if measurements of the auxiliary system *[email protected] are forbidden. The goal is not to devise schemes to enhance

2469-9926/2017/95(4)/043642(12) 043642-1 ©2017 American Physical Society SAMUEL P. NOLAN AND SIMON A. HAINE PHYSICAL REVIEW A 95, 043642 (2017) metrological sensitivity, but to study the sensitivity of quantum states (particulary spin-cat states) to this kind of decoherence. (a) After introducing the formalism in which we work in Sec. II, we demonstrate the central idea of this paper in Sec. III by studying the intuitive case of a simple operator product Hamiltonian. In this situation a number of results may be obtained analytically, which we use to understand decoherence in terms of the noise properties of the initial auxiliary state. In (b) Sec. IV we turn our attention to a beam-splitter Hamiltonian, which is less intuitive. In Sec. V we introduce a semiclassical formalism which gives us a simple picture of this decoherence, and also an efficient means of simulating the full composite system. Finally, in Sec. VI we apply this method to study the limits of this decoherence, deriving the required auxiliary field occupation to negate significant entanglement between the systems. FIG. 1. A visual summary of our scheme. (a) The ideal situation in quantum metrology. A state |ψ (0) is prepared unitarily for input II. FORMALISM A into the interferometer Uˆχ = exp(−iGˆ Aχ); i.e., the quantum Fisher The generic problem considered in quantum metrology is information of Uˆprep|ψA(0) with respect to Gˆ A is valuable. (b) If the this: given an initial quantum stateρ ˆ(0) that undergoes unitary preparation involves interaction with some other quantum system, evolution Uˆχ = exp(−iGχˆ ), how precisely can the classical then they may become entangled. If we cannot measure anything parameter χ be estimated? The answer is given by the quantum about system B then system A is now mixed, and we have lost ˆ | Cramér-Rao bound (QCRB)√ which places a lower bound on quantum Fisher information with respect to Uprep ψA(0) .Inthis the sensitivity, i.e., χ 1/ F, where paper we investigate this decoherence, and determine under what circumstances scheme (b) is well approximated by scheme (a). − 2 (λi λj ) 2 F = 2 |ei |Gˆ |ej | (1) λi + λj i,j can be ignored such that Hˆ AB can be replaced with Hˆ A is often referred to as the undepleted pump approximation. While is the QFI [49–52] and λ , |e are the eigenvalues and i i this is usually a good approximation, there are experimentally eigenvectors ofρ ˆ(0). Whenρ ˆ(0) is pure, Eq. (1) reduces to accessible regimes where it becomes invalid [53–55]. If we do the variance of Gˆ , specifically F = 4V (Gˆ ). The QFI does not not permit measurements on B, then the entanglement between depend on the choice of a particular measurement signal, only the two systems will result in decoherence, which we quantify on the input state and the Hermitian operator Gˆ , called the as a reduction in the QFI of the probe system F , because generator of χ. A F 4V (Gˆ ). The system we consider in this paper is illustrated in Fig. 1. A A In what follows we work in the standard formalism for We begin with some quantum state |ψ , called the probe state, A SU(2) interferometers [56], whereby our probe system consists which could be used to probe a classical parameter χ. Before † † of a conserved total number of N =aˆ aˆ + aˆ aˆ bosons this happens the state must be prepared in some way. Ideally, A 1 1 2 2 each in one of two modes, with bosonic annihilation operators this would occur by performing some unitary operation Uˆprep aˆ1 and aˆ2, respectively. Collective observables are represented on the initial state |ψ (0): |ψ =Uˆ |ψ (0) [Fig. 1(a)]. † † A A prep A as pseudospin operators Jˆ = 1 (aˆ aˆ )σ (aˆ aˆ )T , where σ However in practice, treating this preparation step as unitary k 2 1 2 k 1 2 k is usually an approximation, as the physical mechanism to is the kth Pauli matrix; hence the system is described by the well-known SU(2) algebra. The auxiliary system has a mean achieve this preparation can involve entanglement with some = auxiliary subsystem B. In this case we replace Uˆ (that field occupation of nˆB NB bosons, which for simplicity are prep confined to a single mode bˆ, with number operator nˆ = bˆ†bˆ. is assumed to operate only on subspace A) with Uˆ = B AB In the absence of quantum correlations between particles, an exp(−iHˆ t/h¯), which can potentially cause entanglement AB ensemble of two-level particles is well described by a coherent between subsystems A and B, and therefore cause decoherence spin state (CSS) [57,58], which can be thought of as a rotation in subsystem A when system B is ignored. In this case system of the maximal J eigenstate: |CSS=|θ,φ=Rˆ(θ,φ)|N ,0 A is described by the stateρ ˆ = Tr {|ψ ψ |}, where z A A B AB AB where (up to a global phase) Rˆ(θ,φ) is the rotation operator, |ψAB =UÂB |ψA(0)⊗|ψB (0) [Fig. 1(b)]. − ˆ − ˆ To illustrate this concept, consider the example of an Rˆ(θ,φ) = e iJzφe iJx θ , (2) optical parametric oscillator (OPO), used to create the well- known squeezed vacuum states by creating pairs of photons and the state |N1,N2 indicates N1(2) bosons in mode 1 † † via a Hamiltonian Hˆ A = η(aâˆ + aˆ aˆ )[37]. The physical (2). Coherent spin states have the useful property that mechanism that achieves this process involves the annihilation they are an extreme eigenstate of the rotated pseudospin † of a photon of twice the frequency from a pump beam, operator Jˆθ,φ = Rˆ (θ,φ)JˆzRˆ(θ,φ) = Jˆz cos(θ) + [Jˆx sin(φ) + which we label system B. This process is described by the Jˆy cos(φ)] sin(θ). † † † Hamiltonian Hˆ AB = g(bˆ aâˆ + bâˆ aˆ ). In this context, the We are interested in a class of states called spin-cat (SC) approximation that the entanglement between the systems states, which are an equal superposition of opposite coherent

043642-2 QUANTUM FISHER INFORMATION AS A PREDICTOR OF . . . PHYSICAL REVIEW A 95, 043642 (2017)

|C |2 spin states, i.e., the maximum and minimum Jˆθ,φ eigenstates, m,n , as states with lower purity usually have reduced QFI. Expanding the magnitude of Cm,n to second order in even 1 iϑ |SC=√ (|max+e |min). (3) powers of m,n = (λm − λn)τ (odd powers do not contribute) 2 reveals a link between the QFI of the auxiliary system with ˆ These states are highly nonclassical and have the maximum respect to GB and the resultant decoherence in the probe F = 2 ˆ = system: QFI for an SU(2) interferometer, A NA, so long as GA Jˆ . In contrast, coherent spin states are shot-noise limited, F θ,φ 2 B 2 4 F ˆ = ˆ |Cm,n| = 1 − + O , (9) with A NA with respect to GA Jθ,φ. 4 m,n m,n When Jˆ = Jˆ , the spin-cat state is the well-known θ,φ z √ where F = 4V (Gˆ )istheQFIof|ψ (0) associated with NOON state, |NOON=(|N ,0+|0,N )/ 2[59,60]. B B B A A measuring some classical parameter η under evolution Uˆ = NOON states are particularly relevant as many experiments η exp(−iGˆ η). For short times at least, we identify this QFI would be limited to performing measurements on the probe B as being the relevant parameter to predict the decay of the system in the number basis. However another relevant basis off-diagonal matrix elements ofρ ˆ . is Jˆ , as it is straightforward to show that the well-known A y Such an identification is particularly intuitive for consid- one-axis twisting interaction will eventually lead to a spin-cat ering the role of |ψ (0) in the decoherence of system A: in the Jˆ basis, i.e., |SC=exp(−iJˆ2π/2)|π/2,0. B y z if one considers the possibility that the outgoing state of system B could be measured by an observer, then if this III. SEPARABLE INTERACTIONS state carries information which can distinguish between the In this section we consider the case where the interaction eigenvalues λm and λn, we no longer expect there to be a Hamiltonian between systems A and B is a separable tensor coherent superposition of these components. The interaction product of operators acting on each Hilbert space. Specifically, with system B effectively measures system A. That is, states with high QFI with respect to their ability to estimate the Hˆ = hg¯ Hˆ ⊗ Gˆ . (4) AB A B physical observable corresponding to HÂ cause the most rapid Such an interaction may arise when the Hermitian opera- decoherence. Even if Cm,n is known, calculating the QFI of the probe tor HÂ is required in the state preparation of system A, system requires diagonalization of the reduced density matrix but is moderated by the Hermitian operator Gˆ B acting on subspace B. [see Eq. (1)]. Fortunately for evolution under a separable Hamiltonian, some simple analytic results exist for some initial states. For any state that is initially a spin-cat state of extreme A. Some general results Jˆθ,φ = HÂ eigenstates, there is a simple relationship between For an initially separable and pure state |ψAB (0)= the probe QFI and the purity of the reduced density matrix: |ψ (0)⊗|ψ (0), in terms of the dimensionless time τ = gt A B F = 2 − the evolved state is A NA(2γ 1), (10) − ˆ iλmGB τ and from Eq. (8) the purity is given by |ψAB (τ)= cm|m⊗(e |ψB (0)). (5) m = 1 +|C |2 γ 2 (1 max ), (11) In terms of the eigenstates |m and eigenvalues λ of Hˆ ,the ∗ m A where C = C + = (C + ) is the extreme off- = {| |} max NA 1,1 1,NA 1 reduced density matrixρ Â TrB ψAB (τ) ψAB (τ) takes diagonal term of the coherence matrix; i.e., for a spin-cat in the simple form ˆ Jθ,φ, the QFI of the auxiliary system depends only on the purity ρˆ (τ) = c c∗C (τ)|mn|, (6) ofρ Â, which at least for short times depends only on the QFI A m n m,n of the initial auxiliary state |ψ (0); i.e., F is a function only m,n B A of FB and time. | = 2 where j ψA(0) cj is the initial state. We have defined From these relations, to second order in |Cmax| [Eq. (9) with λ − λ = N ]wehave −i(λm−λn)Gˆ B τ m n A Cm,n(τ) =ψB (0)|e |ψB (0), (7) γ ≈ exp − 1 F N 2 τ 2 (12) which we call the coherence matrix of the probe system, as it 8 B A is responsible for the decay of the off-diagonal terms of ρA. and This term is a direct consequence of a partial trace over the F ≈ N 2 exp − 1 F N 2 τ 2 . (13) auxiliary system B. A A 4 B A C = C = If m,n 1 thenρ Â remains pure, and if m,n δm,n then Although these relations only hold for small time, they do not | | ρÂ is a completely incoherent mixture of eigenstates m m . assume anything about the input state of the auxiliary system. More generally the relationship between the purity of the probe For any |ψB (0), the QFI with respect to Gˆ A = HÂ and purity = { 2} C system γ Tr (ˆρA) and m,n is of a Jˆθ,φ = HÂ spin-cat state simply decay exponentially with F 2 = | |2| |2|C |2 B and time squared, at a rate proportional to NA, as one γ cm cn m,n . (8) might expect for a state capable of reaching the Heisenberg m,n limit. This kind of scaling has been seen in previous studies of As we are interested in maintaining states with high values Heisenberg-limited states under decoherence [18,19], but not of FA, we are particularly interested in the magnitude of in this context.

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1 1 (a) Fock (b)

A coherent amplitude

/N 0.5 0.5 γ

A phase F

0 0 0 0.1 0.2 τ 0.3 0.4 0.5 0 0.1 0.2 τ 0.3 0.4 0.5

1 1 1 (c) 1 (d) 2 A 0.5 0.5 0.5 /N γ

A 0.5

F 0 0 0.01 0.02 0 0.01 0.02 0 0 0 0.1 0.2 τ 0.3 0.4 0.5 0 0.1 0.2 τ 0.3 0.4 0.5

FIG. 2. Time evolution under Hˆ AB = hg¯ Jˆz ⊗ nˆB of the quantum Fisher information FA and purity γ ofρ ˆA(τ), for a variety of initial states with average particle number NA = 20 and NB = 100 and dimensionless time parameter τ = gt. Panels (a) and (b) have initial CSS |ψA(0)=|π/2,0, and (c) and (d) have |ψA(0)=|NOON. Note that the normalization of FA is different for (a) compared to (c). Insets in (c) and (d) show short evolution times. Each plot also varies the initial auxiliary state for the cases discussed in the text, i.e., a Glauber-coherent state (solid black), an amplitude-squeezed state with r = 1 (dashed blue), a phase-squeezed state with r =−1 (dashed red), and a Fock state (magenta asterisks); the Fock states have been included for completeness, although they have FB = 0 so no coherence is lost. For all auxiliary input states we chose β to be real. Time scales can vary greatly between experimental systems, for instance g ≈ 102 rad/s[72], 104 rad/s[73], up to values as high as 106 rad/s[74].

B. An example squeezed states, which have the form We will now study the decoherence imparted on a probe |ψB (0)=Dˆ (β)Sˆ(r)|0, (15) after evolution under a HÂ = Jˆz rotation, specifically where Dˆ (β) = exp(βbˆ† − β∗bˆ) is the coherent displacement 2 Hˆ AB = hg¯ Jˆz ⊗ nˆB . (14) operator with coherent amplitude β, and Sˆ = exp{r[bˆ − (bˆ†)2]/2} is the single mode squeezing operator with real squeezing parameter r and optical vacuum |0. In particular This kind of interaction describes a number of systems, for we focus on three cases, the Glauber-coherent state (r = 0), instance superconducting qubits coupled to a microwave cav- the amplitude-squeezed state (r>0), and the phase-squeezed ity [61–63], or the weak probing of an ensemble of two-level state (r<0), which, for a fixed mean photon number, have atoms with light detuned far from resonance [31,42,47,64–71]. F phase > F coherent > F amplitude. This Hamiltonian generates a Jˆz rotation, which corresponds B B B to a relative phase being imparted between the two levels It is possible to evaluate the coherence matrix [Eq. (7)] ˆ = ˆ available to the probe system. Although we are agnostic about analytically for these states. Because we have HA Jz the specific system being studied, for convenience we will the spin-cat states that obey the relations Eq. (10) and ˆ = ˆ adopt the language of atom-light interactions, and will often Eq. (11) are NOON states. The generator GA Jz has integer − = refer to the quanta of the auxiliary field as photons. eigenvalues, and so we make the substitution λm λn − An interaction of the form Eq. (14) leads to entanglement m n. For simplicity we will restrict ourselves to real β, although it is not necessary to do so. By observing between the Jz spin projection of system A and the the phase that exp[−i(m − n)nˆB τ]Dˆ (β)Sˆ(r)|0=Dˆ (β )Sˆ(r )|0, where of system B,asnˆB is the generator of phase. Identifying β = β exp[−i(m − n)τ] and r = r exp[−2i(m − n)τ], the Gˆ B = nˆB , it is immediately obvious that the optimal choice problem is reduced to evaluating the overlap of two squeezed for |ψB (0) is a Fock state (i.e., an nˆB eigenstate) as this state coherent states, see for instance [75,76]. We obtain has FB = 4V (nˆB ) = 0, and the operation can be performed without generating any entanglement between the systems, β2[1+coth(r)][e−i(m−n)τ −1] 2 exp − − i.e., |Cm,n| = 1, which is illustrated in Fig. 2. This is consistent e i(m n)τ +coth(r) Cm−n(β,r,τ) = (16) with our view of system B carrying away information about cosh2(r) − e−2i(m−n)τ sinh2(r) Jz, as Fock states have an entirely undefined phase, so cannot be used to make a measurement via the interaction Eq. (14). as the coherence matrix for a squeezed coherent state. For However, as Fock states are difficult to engineer, it is important completeness we also provide the result for a Glauber-coherent to consider the behavior of other states. state, obtained by simply taking the limit r → 0, Throughout this paper we will focus on commonly acces- −i(m−n)τ sible states such as Glauber coherent states and quadrature Cm−n(β,τ) = exp[NB (e − 1)]. (17)

043642-4 QUANTUM FISHER INFORMATION AS A PREDICTOR OF . . . PHYSICAL REVIEW A 95, 043642 (2017)

2 1 at a rate directly proportional to NA, which is clearly not the coherent (a) case for coherent spin states [see Eq. (12) and Eq. (13)]. As

A amplitude an example, using the experimental parameters of the system /N 0.5 phase A demonstrated in [73], for the situation considered in Fig. 2 F with coherent light, the QFI of the NOON state would halve 0 −6 0 0.2 0.4 0.6 0.8 1 in approximately 10 seconds, while the QFI of the coherent τ spin state would take roughly an order of magnitude longer to decay by the same amount. Other time scales are discussed in 1 (b) the ﬁgure legend. 2 A In Fig. 3 we demonstrate that FB remains an excellent

/N 0.5 A predictor of decoherence where simple analytic expressions F are unavailable. In Fig. 3(a) we plot the QFI as a function 0 −4 −3 −2 −1 0 of time for a CSS for three different input states, resulting in 10 10 10 10 10 identical dynamics even for large times. This seems to indicate FB that our results are not restricted to spin-cat states. Up until 1 now we have neglected the contribution of nˆB to the magnitude 1.r =0 (c)

A ˆ 2.r =0.2 of the rotation; i.e., to rotate the state about Jz by some angle φ /N 0.5 3.r = −0.2 = A we require an interaction time τ φ/NB . This must be taken F 4.|β|2 = 100 into account in order to meaningfully compare the ability of 0 different states |ψB (0) to perform some fixed rotation φ,soin −4 −3 −2 −1 0 10 10 10 10 10 Figs. 3(b) and 3(c) we take our generator to be Gˆ B = nˆB /NB . FB In this case our full expression for FA [Eq. (18)] does not hold, and although it is straightforward to obtain a more general ˆ FIG. 3. Time evolution under HAB = hg¯ Jˆz ⊗ nˆB of the quantum expression from the coherence matrix, it is not particularly F = Fisher information A and purity γ ofρ Â(τ), with NA 20. In (a) enlightening. Because FB (r) is not one-to-one, when the state | =| we plot the QFI for the CSS ψA(0) π/2,0 for three different is oversqueezed there is a turning point in Figs. 3(b) and 3(c). F = initial states, all with B 100. We evolve√ from different auxiliary We also see revivals which are predicted by Eq. (18). These states, a Glauber-coherent state with√ β = 25, (solid black), an revivals occur as a result of the quantization of the fields; amplitude-squeezed state with β = 50, r ≈ 0.352 (blue asterisk), √ for instance, if |ψA(0)=|NOON they will occur when τ = and a phase-squeezed state with β = 20, r ≈−0.111 (red circles). 2πk/NA, where k = 1,2,3,.... The excellent agreement indicates that FB rather than |ψB (0) Within the limitations discussed above, FB entirely de- is the relevant quantity to consider when predicting this kind of termines the subsequent dynamics. The starting point for this decoherence, not only for NOON states. In (b) and (c) we plot entire analysis was identifying an operator Gˆ , which we were F against F for a NOON state (b), normalized to N 2 ,andCSS B A B A able to do because the reduced density matrix could be written (c), normalized to NA. The QFI of the auxiliary system is varied in C different ways for a number of initial states, (1)–(3) varying |β|2 and in terms of m−n [Eq. (6)], which was a direct consequence of (4) varying r, both with arg(β) = 0. Each point was evolved for a the operator product form of the Hamiltonian. We now turn our attention to a beam-splitter Hamiltonian, where this is not fixed rotation angle φ = τNB (β,r) = π and as such in (b), (c) we the case. take FB = 4V (nˆB /NB ). Arrow indicates oversqueezed regime.

Evaluating the coherence matrix analytically for a squeezed IV. NONSEPARABLE INTERACTIONS (BEAM SPLITTER) coherent input state allows us to extend the short-time A kind of interaction highly relevant to quantum metrology results presented in the previous section to longer times. is an atomic beam splitter: a non-photon-conserving process | =| ˆ = For ψA(0) NOON with generator GB nˆB [not, for that transfers population between our atomic modes. A ˆ = instance, G nˆB /NB , which we consider in Figs. 3(b) and common method for atomic interferometry is a Mach-Zehnder 2 2 3(c)], when β sinh (r) we obtain interferometer, which may be realized by performing two of F ≈ N 2 exp 1 F [cos(N τ) − 1] . (18) these pulses, separated by a phase shift. This kind of evolution A A 2 B A is also highly relevant to the preparation of spin-cat states; for Expanding this to second order in NAτ recovers Eq. (13), instance it may be useful to rotate a Jˆy spin-cat state, perhaps but this expression also predicts revivals in the QFI. Because generated via the atomic Kerr effect, to a NOON state, which this result was derived from Cm−n(β,r,τ), we emphasize that would require a rotation about Jˆx . If we perform this rotation unlike Eq. (13) it is not general in |ψB (0); it only holds for without assuming classical light, how might this decohere our squeezed coherent states and Fock states, the latter simply atomic system? because FB = 0. In particular, as the transfer of an atom is correlated with In Fig. 2 we show the probe QFI and purity for a NOON the creation or annihilation of a photon, the number of photons state compared to a coherent spin state, for a number of input in the optical beam carries information about the number states, and clearly see the QFI of the auxiliary state correctly of transferred atoms, thus destroying the coherence of the predicts the rate of decoherence. It is evident that coherent superposition. This will be particularly relevant when creating spin states are more robust to this kind of decoherence. As we NOON states, as the creation of this state results in the creation have shown, the QFI of NOON states decays exponentially of ∼±NA/2 photons which, depending on the initial state, may

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√ √ √ β = 10 β = 100 β = 1000 1 1 1 2 F/NA 2 0.8 Fmax/N 0.8 0.8 γ A 0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 θ/π θ/π θ/π

FIG. 4. Loss of QFI and purity from evolution fully quantized beam-splitter Hamiltonian Hˆ AB = hg¯ (Xˆ ⊗ Jˆx + Yˆ ⊗ Jˆy ) as a function of time. Time is normalized to the beam-splitter angle θ = 2|β|τ,for|ψA(0)=|NOON with NA = 20 and |ψB (0)=|β, with arg(β) = 0. The QFI is calculated with respect to Gˆ A = Jˆz, such that after a π/2 rotation about the Jˆx axis the state approaches the QFI for a NOON state. Fmax is the QFI of the probe system in the limit of classical light, i.e., as the coherent amplitude becomes sufficiently large that we may substitute bˆ → β in the beam-splitter Hamiltonian [Eq. (20)]. be easily distinguishable. If the Hamiltonian for this process pictures coincide with is not separable, i.e., of the form Eq. (4), can we identify a Hˆ = ˆ ⊗ ˆ + ˆ ⊗ ˆ generator and corresponding Fisher information which is a AB hg¯ (X Jx Y Jy ), (20) useful predictor for this decoherence? which we call the beam-splitter Hamiltonian, where Xˆ = bˆ + bˆ† and Yˆ =−i(bˆ − bˆ†) are the standard optical amplitude A. The Tavis-Cummings model and phase quadratures. To arrive at this Hamiltonian we have = The fully quantized Hamiltonian for an atomic beam splitter assumed the field is on resonance ω ω0 and have made the generated from atom-light interaction is the Tavis-Cummings rotating wave approximation. As in Sec. II, retaining a quantized description of the Hamiltonian, which describes an ensemble of NA,two-level hω = E − E auxiliary system introduces decoherence to the evolution. atoms (with energy difference ¯ 0 2 1) interacting ˆ ω Figure 4 shows the QFI and the purity of a Jy spin-cat state with a single-mode optical field of frequency through dipole | coupling [77], ( SC ) being rotated by a quantized beam splitter [Eq. (20)] against the evolution time, parametrized by the beam-splitter 1 1 † angle θ = 2|β|τ, compared to evolution under a classical H = + ˆ + ˆ+ + ˆ− ⊗ ˆ + ˆ AB hω¯ nˆB 2 hω¯ 0Jz 2 hg¯ (J J ) (b b). beam splitter Uˆclassical = exp(−iJˆx θ), obtained by taking the (19) classical limit for the optical field bˆ → β. We see that as |β|2 becomes large, the full evolution approximates a classical If we were to ignore the quantum degrees of freedom of the beam splitter. light, the interaction term would simply result in a rotation 1 about Jˆ = (Jˆ+ + Jˆ−). x 2 B. Identifying a generator In typical experimental systems the field is close to resonance, ω ≈ ω0, and the coupling g is small compared to ω0, ω. In Sec. III we found that the QFI of the generator of time Therefore the rotating wave approximation is often made, and evolution for system B was an excellent tool for predicting de- it is a good approximation to neglect the energy-nonconserving coherence. However, the difficulty with using this approach for † terms Jˆ+bˆ and Jˆ−bˆ. decoherence introduced under the beam-splitter Hamiltonian Before throwing away these terms, the interaction part of [Eq. (20)] is that because the evolution is not separable, the † the Hamiltonian can be written as Hint = hg¯ Jˆx ⊗ (bˆ + bˆ ) reduced density matrix cannot be written in the form of Eq. (6). which certainly looks separable; however the evolution caused This means it is unclear how to identify a generator for the Hˆ = + 1 ˆ auxiliary system. Clearly under the beam-splitter Hamiltonian by 0 hω¯ nˆB 2 hω¯ 0Jz cannot be neglected. Moving into the interaction picture allows us to evolve the initial state the optical field quadratures Xˆ and Yˆ are responsible for forward in time under Hˆ int only, but transforming this generating the atom-light entanglement; however the basis in Hamiltonian into the interaction picture and integrating the which the off-diagonal density matrix elements will decay resultant interaction picture Hamiltonian in time gives rise to depends on the argument of the coherent amplitude β.To nonseparable evolution. isolate the role of the quantum fluctuations in each quadrature, However, moving into the interaction picture reveals that we make the approximation that quantum fluctuations in one of quantities such as the purity of the reduced density matrix and the quadratures is negligible. Specifically, restricting ourselves expectation values of any observable that commutes with Jˆz to light with real coherent amplitude β, we compare the full (such as the QFI with Gˆ A = Jˆz) are unchanged by evolution quantum evolution [Eq. (20)] to two cases: ˆ = − ˆ ⊗ ˆ +ˆ ˆ = under Hˆ 0. So long as we are only interested in calculating these (1) classical Y : UX exp [ i(X Jx Y Jy )τ] quantities, we neglect Hˆ 0 and the interaction and Schrödinger exp [−iXˆ ⊗ Jˆx τ],

043642-6 QUANTUM FISHER INFORMATION AS A PREDICTOR OF . . . PHYSICAL REVIEW A 95, 043642 (2017)

1 Thus for large NB we expand in 1/Xˆ to ﬁrst order, giving ˆ ≈ ˆ ˆ ˆ ≈ˆ ≡ (a) φ Y/ X and θ X τ θ. This decouples the part of the 2

A Hamiltonian which generates entanglement from the part that generates the rotation. /N 0.5

F Restricting ourselves to an initial state |ψA(0)=|SC, and θ = π/2, we note that exp(−iφˆJˆz) will cause dephasing of − ˆ 0 off-diagonal terms in the Jz basis. The exp( iθJx ) term will 0 0.5 1 1.5 2 then rotate this state such that it is approximately aligned with θ/π the maximal and minimal Jz eigenstates, before it undergoes 1 − ˆ ˆ Fmax further decoherence due to exp( iφJz). For NA 1, if the (b) analytic fluctuations in φˆ are much less than 1, then this second 2 A ˆ dephasing process will be much more significant than the first, FA(U) /N 0.5 as the off-diagonal terms are significantly more separated after

F F (Uˆ ) A X the π/2 rotation. As such, it is a reasonable approximation to ˆ FA(UY ) neglect the effect of the first dephasing step, and in terms 0 ˆ | = α | −2 −1 0 1 of the pseudospin eigenspectrum Jα m; α λm m; α with r α = x,y,z, the reduced density matrix of system A becomes ∗ ∗ FIG. 5. Comparison of quantized beam-splitter evolution, ne- ≈ Cz ρÂ(θ) cm (cn ) Am,m (An,n ) m−n glecting quantization of different optical quadratures. (a) QFI of m,m,n,n = | | probe system as a function of beam-splitter angle θ 2 β τ for each x x −i λ −λ θ ×| | ( m n ) case with |ψB (0)=|β, and bottom: QFI as a function of squeezing m; z n; z e , (22) = magnitude.√ The system was simulated exactly for NA 20 atoms and = | ˆ β = 100 for an initial Jˆ spin-cat state, rotated about the Jˆ axis. where cj j; x ψA(0) is this initial sate in the Jx eigenbasis y x = | Both plots are a comparison of the fully quantized evolution [Eq. (20)] and Aj,k j; z k; x is a change of basis. In analogy to Eq. (7) we identify (using λz = j) to the two cases “classical X” and “classical Y”. As in Fig. 4, Fmax is j the QFI due to the rotation only, without decoherence. In (b) we fix Cz = | −i(m−n)Y/ˆ Xˆ | θ = π/2 (the optimum value) and vary r. The magenta circles are an m−n ψB (0) e ψB (0) (23) analytic calculation [Eq. (26)], based off the approximate generator ˆ Gˆ B ≈ Y/ˆ Xˆ . as the term responsible for decay of coherence in the Jz eigenbasis. As an example, for Glauber-coherent states (with β real) (2) classical X: UˆY = exp [−i(Xˆ Jˆx + Yˆ ⊗ Jˆy )τ], i.e., (m − n)2 Eq. (20) with the substitution Yˆ →Yˆ =2Im(β) = 0forUˆX Cz = − m−n(β) exp . (24) and Xˆ →Xˆ =2β for UˆY . 8NB Figure 5 showstheQFIofaJˆ spin-cat state evolved y Now, proceeding as in Sec. III we can identify the QFI for the under Eq. (20) compared to the two cases UˆX and UˆY . auxiliary system as FB = 4V (Gˆ B ) with For comparison, we have also shown Fmax.ThisistheQFI F = ˆ ˆ ˆ = A 4V (Jz)foraJy spin-cat state evolved under Uclassical Gˆ B ≈ Y/ˆ Xˆ . (25) exp(−iJˆx θ) which imparts no decoherence; therefore FA < For coherent states, V (Yˆ ) = 1, so F = 1/N , indicating that Fmax. Figure 5(a) indicates that UˆX agrees well with the B B classical evolution and has only a small impact on the increasing the number of photons used to implement the beam splitter reduces the decoherence. A qualitative explanation for coherence, and that UˆY agrees well with the evolution due to Eq. (20). Figure 5(b) varies the squeezing parameter r this is, if the initial state contains a large number of photons, it at the optimum beam-splitter angle θ = 2|β|τ = π/2, and is more difficult to distinguish the creation or annihilation of ∼ ˆ = + shows good agreement with the outcome of Fig. 5(a) for NA/2 photons. Conversely, a Fock state has V (Y ) 2NB ˆ = moderate |r|, i.e., that fluctuations in Yˆ are predominantly 1, and X 0, indicating that it has a very high QFI and will responsible for the decoherence. This picture breaks down for cause extremely rapid decoherence. Again, this fits with our highly phase-squeezed initial auxiliary states, and it becomes intuitive picture, as the creation or annihilation of one photon important to consider quantum fluctuations in Xˆ rather than Yˆ from a Fock state is immediately distinguishable, indicating to correctly describe the system. that it cannot be used to create a coherent superposition of ˆ ≈ ˆ ˆ Motivated by Fig. 5 we continue by studying evolution atomic population. The generator GB Y/ X also indicates that phase-squeezed states should cause less decoherence than under UˆY only. Ignoring the quantum fluctuations in Xˆ allows us to define the commuting operators φˆ = arctan(Y/ˆ Xˆ ) and amplitude-squeezed states. If we are restricted to rotations θ = π/2, such that θˆ =Xˆ τ 1 + (Y/ˆ Xˆ )2, such that exp(−iJˆx π/2)|SC=|NOON, then the reduced density matrix takes the form of Eq. (6), and the results obtained in Sec. III − ˆ ˆ + ˆ ˆ − ˆ ˆ − ˆ ˆ ˆ ˆ i( X Jx Y Jy )τ = iφJz iθJx iφJz e e e e . (21) can be applied here but with Gˆ B = Y/ˆ Xˆ .Wehave

π 1 Evaluating expectation values with√ respect to a squeezed F = ≈ 2 − F 2 2 2 A θ NA exp B NA , (26) coherent state gives Xˆ =2|β|≈2 NB if |β| sinh (r). 2 4

043642-7 SAMUEL P. NOLAN AND SIMON A. HAINE PHYSICAL REVIEW A 95, 043642 (2017) which although similar to Eq. (13), does not depend on time as uses; for instance it is simple in this picture to study the effects we have ﬁxed θ. The purity can be obtained from Eq. (10). This of entanglement generated by Xˆ and Yˆ simultaneously. Addi- result agrees well with the exact (numeric) evolution, shown tionally, it is only ever necessary to manipulate matrices which in Fig. 5, indicating that the generator is well approximated by belong to the probe vector space, rather than constructing and Gˆ B ≈ Y/ˆ Xˆ . evolving the full dim(A) × dim(B) state before performing a partial trace to obtainρ ˆA, which rapidly becomes intractable V. A SEMICLASSICAL PICTURE OF DECOHERENCE even for modest particle numbers.

The results presented in the previous section were obtained A. Jˆz rotation by evolving the full quantum state |ψ , which becomes AB As an example we first show that this method can recover increasing challenging as our basis size increases. We also the results presented in Sec. III. As we have alluded to, found that it was an excellent approximation in most regimes decoherence under the separable Hamiltonian [Eq. (14)] can be to neglect the quantum fluctuations in one quadrature, which understood by averaging over a single parameter X ={φ} with allowed us to treat the interaction as separable such that we Uˆ (φ) = exp(−iJˆ φ), with the noise properties of φ related to could identify a generator for system B. Here we present z the quantum fluctuations of the operator φˆ = nˆ τ.IntheJˆ an approximate, general approach to studying decoherence B z eigenbasis this gives the reduced density matrix arising from the entanglement of a probe with an auxiliary quantum field. This approach does not require us to neglect ∗ − − ρˆ ≈ c c dφP(φ)e i(m n)φ|mn|. (28) quantum fluctuations in Xˆ or Yˆ to identify a generator, and A m n m,n also affords us an efficient way of simulating the system −i(m−n)φ numerically. We make use of this in Sec. VI, where we use If we identify Cm−n = dφP(φ)e , this has the same this method to explore the required auxiliary field occupation form as Eq. (6). If we assume P (φ) is Gaussian with mean φ to negate decoherence arising from the entanglement between and standard deviation σφ, then we can evaluate this integral the systems. to obtain This is done by modeling the reduced density matrix − − − 1 − 2 2 Cz = i(m n)φ 2 (m n) σφ as an average over a set of noisy classical variables X, m,n e e , (29) which have some distribution function P (X), characterized by adding the z superscript to denote decoherence in the Jˆ | | |2 z X ψB (0) . Following a series of measurements, the reduced eigenbasis. For coherent light, this expression agrees with density matrix is well approximated by Eq. (17) by identifying φ =φˆ=|β|2τ and σ 2 = V (φˆ) = φ |β|2τ 2, which is seen easily by expanding exp[−i(m − m)τ] ≈ ˆ | | ˆ † ρÂ dXP (X)U(X) ψA(0) ψA(0) U (X). (27) to second order in τ. We have identified FB = 4V (nˆB ) which tells us that A similar model of decoherence has been considered else- Gˆ B = nˆB and so we associate φ with the mean and noise where, where it was used to prove a general link between the properties of the operator nˆB . Although this description probe QFI and purity [78]. This relation is approximate in the correctly predicts Gˆ B it is only approximate, and because we sense that the quantization of the optical field is neglected; have neglected the quantization of the photon field it will not for instance revivals predicted by Eq. (18) are absent in this capture the revivals seen in Figs. 2(c) or 2(d). picture. Nevertheless, for sufficiently short times, Eq. (27)isan excellent approximation to the exact dynamics followed by a B. Beam splitter partial trace. In the inset of Fig. 7 we compare this method Now we turn our attention to studying the decoherence to an exact calculation for small NA for the beam-splitter generated by evolution under the beam-splitter Hamiltonian Hamiltonian, and find excellent agreement. [Eq. (20)]. As in Sec. IV B we study the approximate Although conceptually similar, we emphasize that this ap- rotation Uˆ (θ,φ) = exp(−iJˆzφ)exp(−iJˆx θ), identifying X = proach is distinct from stochastic phase space methods [79,80] {θ,φ}. Again, we assume the distribution functions for θ and commonly used to model Bose-Einstein condensates beyond φ, P (θ,φ) = Q(θ)Q(φ), are Gaussian. We interpret (θ,φ)as a mean-field treatment, such as the well-known truncated the azimuthal and elevation Bloch sphere angles, respectively, Wigner approximation [81–83]. Significantly, in these phase and identify space methods expectation values of observable quantities θ cos(φ) θ sin(φ) are reconstructed by averaging over phase space trajectories, X = ,Y= . (30) whereas in this method we have full access to the (approxi- τ τ mate) reduced density matrix. This allows us to easily calculate These are to be interpreted as noisy classical variables with =ˆ 2 = ˆ the QFI, which for a mixed state is difficult to obtain via a X X and σX V (X) (with analogous relations for Y ), phase space method. Additionally, although we often evaluate which imply that the angles are related to the coherent theintegralinEq.(27) numerically, we do this by performing amplitude of the light, with θ = 2|β|τ and φ = arg(β). a Riemann sum over P (X) rather than stochastically sampling Following a procedure similar to that in Sec. VAwe arrive from the distribution. at the reduced density matrix, For the two rotations we study, it is useful to choose ∗ ∗ ≈ x x ρÂ cm (cn ) Am,m (An,n ) these noisy, classical variables to be the Bloch sphere angles X ={φ} for the separable Hamiltonian or X ={θ,φ} for the m,m ,n,n × Cz Cx | | beam-splitter Hamiltonian. This approach has a number of m,n(φ,σφ) m,n (θ,σθ ) m,z n,z , (31)

043642-8 QUANTUM FISHER INFORMATION AS A PREDICTOR OF . . . PHYSICAL REVIEW A 95, 043642 (2017) which is of the form of Eq. (22), with the difference that the 1 x phase factor has been replaced by C which directly causes coherent m ,n (a) amplitude decay of the off-diagonal matrix elements in the Jˆx eigenbasis

2 phase also. Both Cx , Cz have the same form as Eq. (29), but in A j,k j,k 0.5 terms of the relevant classical variable. /N A

The reduced density matrix Eq. (31) with the relations F Eq. (30) affords us an understanding of decoherence in terms ˆ ˆ of the noise properties of the optical quadratures X and Y .If 0 β is real, we set φ = 0 (which corresponds to performing our 0 2 4 6 8 rotations about Jˆx only), and we obtain the following noise φ/π relations: 30 2 θ V (Xˆ ) V (Yˆ ) 2 = 2 = coherent σθ ,σφ . (32) (b) 4|β|2 4|β|2 20 amplitude phase TFS Observing that Xˆ =2|β|, these relations agree with result B Eq. (25) that the generator responsible for decay of the off- N 10 ˆ ˆ z ≈ diagonal matrix elements ofρ Â in the Jz eigenbasis is GB ˆ ˆ ˆ x ≈ ˆ ˆ Y/ X , but they also allow us to identify GB θX/ X ,as 0 0 20 40 60 80 100 the generator of decay in the Jˆx eigenbasis. However, Fig. 5 NA indicates that for the rotation of a Jˆy spin-cat state about Jˆx , noise in Yˆ dominates. FIG. 6. (a) FA for a Jˆy spin-cat state under a Jˆz rotation as a function of rotation angle φ = τNB . Simulation was performed = = | |= VI. MITIGATING DECOHERENCE exactly for NA 20 and NB 100, with r 1 for the amplitude TFS = and phase squeezed states. (b) NB after a φ π rotation, which In Fig. 4, it is apparent that as β increases, FA approaches corresponds to the first revival in (a) for r =|0.5|. the classical limit. This agrees with the result that FB = 1/|β|2,asGˆ z ≈ Y/ˆ Xˆ , with Xˆ =2|β| and V (Yˆ ) = 1 (for B making the substitution τ = φ/N we obtain coherent light). We also see this behavior in Figs. 3(b) and 3(c); B F = 2 −2r when comparing A for a fixed rotation angle φ τNB φ e F = 2 ∝ N TFS[|NOON] ≈ N 2 , (33) the decoherence vanishes as B 4V (nˆB )/NB 1/NB goes B ln(2) A to zero, which corresponds to the limit of large photon number. Motivated by these observations, here we study which is valid within the same approximations as Eq. (13), the following question: given evolution under either of the although this does not explicitly depend on FB , as expected entangling Hamiltonians we have considered, what is the states with larger FB per photon (for instance phase-squeezed required auxiliary field occupation to mitigate decoherence states) would require more photons to perform this rotation F 2 in the probe system? while maintaining A NA/2. More specifically, we calculate the required NB , such that This scaling is intuitive if we consider that the information after rotating the probe state by a fixed angle the probe QFI relating to the Jˆz projection of system A is encoded onto F = 2 TFS | has at least A NA/2, which we will call NB . This is the ψB as a phase shift. In order to maintain coherence between QFI of the twin-Fock state, defined as |TFS=|NA/2,NA/2 the maximal and minimal Jz eigenstates, we require that this with respect to Gˆ A = Jˆy . Our motivation for this metric is that information be hidden in the quantum fluctuations of the twin-Fock states are far less exotic than spin-cat states, and can phase of |ψB (0). More specifically, in order to maintain be realized simply by a projective measurement in the Jz basis. indistinguishability, we require that the magnitude of the F ≈ 2 = Superpositions of twin-Fock states also have A NA/2, phase shift after time τ,sayφcat φNA/NB , that each and can be manufactured via any pairwise particle creation component of the superposition causes on |ψB (0) is less than | ∼ process, such as four-wave mixing [84,85] and spin-exchange the characteristic phase√ fluctuations of ψB (0) , V (nˆB /NB ) r ∼ TFS ∼ −2r 2 2 collisions [86–89]. Although a TFS would be less attractive e /β. Setting φcat V (nˆB /NB )givesNB e φ NA. than a NOON state for a number of fundamental tests, it is Interestingly, Jˆy spin-cat states are surprisingly robust to an excellent candidate for quantum metrology. If one had a decoherence arising from this Hamiltonian. Figure 6(a) plots spin-cat state, and were unable to maintain the QFI above the FA for this state as a function of time (parametrized by the what could be achieved with a twin-Fock state (which is much rotation angle φ = NB τ), calculated with respect to Gˆ A = Jˆy . simpler to create), it would be much less challenging to simply The oscillations in FA are a consequence of the rotation; FA use the latter. is maximum when the state is aligned along the Jˆy axis. In TFS = Fig. 6(b) we plot NB for a φ π rotation, which corresponds TFS ∝ to the first revival in Fig. 6(a). The quadratic scaling NB A. Jˆz rotation 2 NA exhibited by NOON states under this rotation is not evident TFS As we have analytic results for rotating a NOON state under here; instead we find that NB is approximately independent HAB = hg¯ Jˆz ⊗ nˆB , this is our starting point. From Eq. (13), of NA.

043642-9 SAMUEL P. NOLAN AND SIMON A. HAINE PHYSICAL REVIEW A 95, 043642 (2017)

1000 Crucially, in this relation the squeezing factor exp(2r) has 20 the opposite sign to that of Eq. (33), as we expect from FB ≈ 15 4V (Y/ˆ Xˆ ) that states with small ﬂuctuations in Yˆ will require TFS 800 B 10 the least number of photons. As this result is approximate N we compare it to a numeric solution in Fig. 7, both using 5 exact diagonalization for small NA, and using the semiclassical 600 0 0 10 20 picture presented in Sec. V for a much larger range of NA. N TFS A B We ﬁnd excellent agreement between the exact numerics, the N semiclassical picture, and this analytic result, which uses the 400 approximate generator Gˆ B ≈ Y/ˆ Xˆ .

200 VII. CONCLUSIONS coherent amplitude In quantum metrology it is sometimes necessary to prepare phase a state for input into a metrological device via an operation 0 ˆ 0 200 400 600 800 1000 such as a beam splitter or Jz rotation. This evolution may NA be performed via an interaction with an auxiliary system, and although it is commonplace to assume this auxiliary FIG. 7. The required number of photons such that the rotated system is sufficiently large that any entanglement between spin-cat state has QFI equal to that of a twin-Fock state, for a the two systems may be neglected, here we retain a quantized Glauber-coherent state, and amplitude and phase squeezed states. description of both systems. We find that the QFI associated Lines are a least-squares linear fit; the gradients are for comparison with the auxiliary system’s ability to estimate the Jˆz projection to Eq. (35). Simulation was performed with arg(β) = 0andr = 0.5 of our primary system through the interaction Hamiltonian is for the squeezed states, using the semiclassical picture [Eq. (27)]. NB an excellent predictor of decoherence and loss of metrological | |2 was varied by changing β rather than r. Inset: Comparison of exact usefulness. solution (points) to semiclassical picture (shapes) for small particle It is simple to define this QFI for a separable Hamiltonian numbers. [Eq. (4)], and we also derive an approximate QFI for a beam- splitter Hamiltonian [Eq. (20)]. By introducing an alternative 2 picture of this decoherence, viewing the reduced density matrix TheoriginoftheNA scaling for NOON states is the linear ˆ as an average over an ensemble of noisy classical variables, NA dependence of φcat, which is absent for a Jy spin-cat we are also able to generalize our result for the beam-splitter ˆ rotating about the Jz axis. Here, the coherence is carried by case by defining two generators responsible for the decay of the distinguishability of extreme Jˆ eigenstates. Fluctuations in y off-diagonal coherence in both the Jˆx and Jˆz eigenbases. In ˆ GB = nˆB /NB will cause diffusion of the phase of each branch summary it is desirable to chose initial auxiliary states with of the√ superposition. This phase diffusion will be of order small QFI, especially for NOON states which are particularly −r φ = V (GB ) ∼ e /β.Asφ increases, the separation be- susceptible to this kind of decoherence; see Fig. 2. tween the two branches decreases, becoming indistinguishable We have also estimated the required auxiliary field occupa- when φ ∼ π/2. In this case the nonclassical nature of the tion to negate this kind of decoherence in both situations. As an state is lost, and we expect FA ∼ NA. We expect that the phase example, it would require roughly 1 × 105 coherent photons = 2 diffusion that leads to FA NA/2 will occur well before this to impart a π phase shift on a 100 atom NOON state, or about = 3 at some value φTFS. Setting φ φTFS gives 3 × 10 coherent photons to rotate a 100 atom Jˆy spin-cat state by π/2 about the Jˆ axis, while maintaining the QFI above that x φ 2 of a twin-Fock state. TFS | ≈ −2r NB [ SC ] e , (34) φTFS ACKNOWLEDGMENTS and from Fig. 6(b) we estimate that φTFS ≈ π/3 is a good The authors would like to thank Iulia Popa-Mateiu, Stuart rule of thumb. Szigeti, Joel Corney, Michael Hush, and Murray Olsen for invaluable discussion and feedback. The authors also made use of the University of Queensland School of Mathematics B. Beam splitter and Physics high performance computing cluster “Obelix” and thank Leslie Elliot and Ian Mortimer for computing support. TFS ˆ Likewise, we can use Eq. (26) to estimate NB for a Jy S.A.H. acknowledges the support of Australian Research spin-cat rotating about Jˆx by θ = π/2; we obtain Council (ARC) under Project No. DE130100575. This project has received funding from the European Union’s Horizon 2020 e2r research and innovation program under Marie Sklodowska- N TFS ≈ N 2 . (35) Curie Grant Agreement No. 704672. B 4ln(2) A

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