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arXiv:1712.03896v1 [quant-ph] 11 Dec 2017

Institut f¨ ur Theoretische Physik, Leibniz Universit¨ at Hannover, Appelstr. 2, DE-30167 Hannover, Germany 2 QSTAR, INO-CNR, and LENS, Largo Enrico Fermi 2, IT-50125 Firenze, Italy 3 Dipartimento di Fisica e Astronomia, Universit` a degli Studi di Firenze, via Sansone 1, I-50019 Sesto Fiorentino, Italy 4 Institut f¨ ur Quantenoptik, Leibniz Universit¨ at Hannover, Welfengarten 1, DE-30167 Hannover, Germany Recent experiments have demonstrated the generation of entanglement by quasi-adiabatically driving through quantum phase transitions of a ferromagnetic spin-1 Bose-Einstein condensate in the presence of a tunable quadratic Zeeman shift. We analyze, in terms of the Fisher information, the interferometric value of the entanglement accessible by this approach. In addition to the TwinFock phase studied experimentally, we unveil a second regime, in the broken axisymmetry phase, which provides Heisenberg scaling of the quantum Fisher information and can be reached on shorter time scales. We identify optimal unitary transformations and an experimentally feasible optimal measurement prescription that maximize the interferometric sensitivity. We further ascertain that the Fisher information is robust with respect to non-adiabaticity and measurement noise. Finally, we show that the quasi-adiabatic entanglement preparation schemes admit higher sensitivities than dynamical methods based on fast quenches.

I.

INTRODUCTION

Atom interferometry has become an indispensable tool for both the testing of fundamental physics and precision measurements [1]. Without entanglement between the atoms, the attainable sensitivity is fundamentally limited by the standard quantum limit (SQL) [2, 3]. Employing multipartite entanglement allows to shift this bound towards the Heisenberg limit (HL) [2–4]. In view of the high effort required for handling coherent ensembles with a large atom number N , it is crucial that the √ HL replaces the SQL scaling of the sensitivity ∝ N with ∝ N . Entanglement that—in the absence of technical noise—facilitates to surpass the SQL is unambiguously witnessed by the Fisher information (FI). In spinor Bose-Einstein condensates (BEC), entanglement useful to enhance the sensitivity of atom interferometers beyond the SQL can be generated exploiting spin-changing collisions [5]. A common realization relies on the parametric amplification of quantum fluctuations leading to squeezed Gaussian states [6–12]. SubSQL sensitivities [6, 11, 12], entanglement [9, 10], and squeezing of up to 10 dB beyond the SQL [7, 8, 10] have been demonstrated [5]. Furthermore, multipartite entanglement in spinor BECs can be also generated near the ground state of ferromagnetic [13] and antiferromegnetic [14] spin-1 BECs [15]. In the following we focus on the ferromagnetic case, relevant for experiments with 87 Rb. In the presence of an (effective) quadratic Zeeman shift q, the system exhibits three quantum phases [13, 16]. By preparing the BEC in the ground state at q > qc (here qc > 0 and the critical points are located at q = ±qc ) and slowly driving through both quantum phase transitions an entanglement depth of about 900 particles has been witnessed in the Twin-Fock (TF) phase at q < −qc [17], see also [18, 19]. Recently [20], we have shown that the ground state

of a ferromagnetic at q = 0 can be used for heralded generation of highly entangled macroscopic superposition states. In the present paper we extend this study and analyze the interferometric sensitivity of the entangled quantum states that are generated along the (quasi-)adiabatic passage when scanning over different values of q. Their full potential is revealed by considering atom interferometry involving all three modes, which generalizes the two-mode interferometry experimentally implemented, e. g., in [11]. We find that besides the TF state studied in Ref. [13, 17] also the ground state at the center, q = 0, of the broken axisymmetry phase leads to Heisenberg scaling. This state can be reached by (quasi-)adiabatically scanning over only a single critical point, stopping the evolution half-way to the TF state. We identify the interferometric transformations that provide the most sensitive phase imprinting and demonstrate that the measurement of particle numbers, an established experimental technique, is optimal for the phase estimation. Our simulations show that performing the passage within reasonable, finite time does not strongly impair the attainable FI. We further analyze the effect of measurement noise and find that surpassing the SQL with state-of-the-art technology is well feasible. We finally show that, under realistic conditions, quasi-adiabatic schemes produce states with larger interferometric sensitivity than those accessible by parametric amplification.

II.

FISHER INFORMATION AND INTERFEROMETRY

Let us briefly review some concepts used in the paper. In any atom interferometer, a phase θ is imprinted into an initial density matrix ρˆ0 , leading to a ρˆθ which is subsequently measured to determine the phase θ. The resulting estimation of θ has an uncertainty, which is bounded

2 by the (classical) Cram´er-Rao bound, ∆θ ≥ ∆θCR . Here √ ∆θCR = 1/ νF

(1)

and F (θ) =

X µ

1 P (µ|θ)

∂P (µ|θ) ∂θ

2 (2)

is the (classical) FI which depends on ρˆθ and the chosen measurement observable. The sum comprises all possible measurement outcomes µ and P (µ|θ) is the probability to measure µ given that the quantum state is ρˆθ . Finally, ν is the number of measurements [21]. Maximizing the FI over all possible generalized quantum measurements defines the quantum Fisher information (QFI) FQ [21– 23]: F ≤ FQ and the equality F = FQ can always be reached by an optimal measurement [22]. Correspondingly, a quantum Cram´er-Rao bound is introduced as p ∆θQCR = 1/ νFQ (3) with ∆θCR ≥ ∆θQCR . For N qubits, we have FQ ≤ N 2 (HL) [2, 3], and FQ ≤ N (SQL) if ρˆ0 is not entangled [3]. Thus both the classical and quantum FI witness interferometrically useful entanglement: FQ ≥ F > N is equivalent to a ∆θQCR undercutting the SQL. We assume that the phase θ is imprinted by a collective unitary transformation across n modes. Let gˆj be the generators of the defining representation of su(n). We denote the vector comprising these dn = n2 − 1 generators with respect to the i-th of N particles as g ˆ(i) ≡ (i) (i) (ˆ g1 , . . . , gˆdn ). Then the final density matrix acquires ˆ (θ)ˆ ˆ †(θ), with U ˆ (θ) = exp(−iθ u · G), ˆ the form ρˆθ = U ρ0 U PN (i) ˆ where we call G = ˆ the collective g ˆ, and i=1 g u ∈ S dn −1 is the interferometric direction. For a pure initial state, ρˆ0 = |ψihψ|, the QFI due to an interferoˆu ≡ u · G ˆ reads metric transformation generated by R ˆ u ] = 4 (∆R) ˆ 2 = 4 uTΓ ˆ u, FQ [|ψi, R G

(4)

where ΓG ˆ denotes the covariance matrix of the operators ˆiG ˆ j i/2 + ˆ [24], with elements (Γ ˆ )ij = hG composing G G ˆ ˆ ˆ ˆ hGj Gi i/2 − hGi ihGj i. The leading eigenvector of ΓG ˆ identifies the optimal interferometric direction uopt . By convention, in the case of qubits (n = 2) the gˆi are normalized such that (γi max − γi min )2 = 1, γi max and γi min being the maximum and minimum eigenvalues of gˆi , respectively. More generally, the SQL is given by (γi max −γi min )2 N and the HL by (γi max −γi min )2 N 2 [2].

III.

MODEL

In the following we study an optically trapped spin1 BEC of N particles with magnetic sublevels mf ∈ {0, ±1}. In the single-mode approximation, the spinor

dynamics is modeled by the Hamiltonian [25, 26] i h ˆ+ + N ˆ− ) ˆ = λ N ˆ 0 − 1 + q (N H 2 + λ(ˆ a†1 a ˆ†−1 a ˆ20 + a ˆ†2 ˆ1 a ˆ−1 ), 0 a

(5)

where a ˆ†i and a ˆi are the creation and annihilation operˆ0,± = a ators for mf = i, and N ˆ†0,±1 a ˆ0,±1 are the number operators for the respective sublevels. The total number of atoms is equal to N and is assumed fixed here. The interaction coefficient λ (negative for ferromagnetic condensates such as the F = 1 hyperfine groundstate manifold of 87 Rb) depends on the trapping potential and microscopic parameters, namely the scattering lengths and the mass of the atoms [26, 27]. The effective quadratic Zeeman shift q may be controlled by an external magnetic field and near-resonant microwave dressing [16, 26]. Spinchanging collisions, described by the last line of Eq. (5), preserve the total magnetization, i. e., the eigenvalue D of ˆ ≡N ˆ+ − N ˆ− . Hence starting from an initial condensate D in mf = 0 and then quenching—or slowly driving—the magnetic field so to prepare entangled states ensures that the system remains in the subspace of D = 0. The dynamics thus takes place in the Hilbert space spanned by the Fock states |ki ≡ |N− = k, N0 = N − 2k, N+ = ki ˆi . By restricting the dynamwith Ni the eigenvalues of N ics to the magnetization-free subspace, the linear coupling to the magnetic field and its fluctuations (linear Zeeman shift) becomes irrelevant, which leads to phase noise stability. In the magnetization-free subspace, model (5) presents three quantum phases [13, 16] as a function of q with quantum phase transitions at q = ±qc , qc = 2N |λ|: the polar (P) phase (q > qc ), the broken-axisymmetry (BA) phase (|q| < qc ), and the TF phase (q < −qc ). For large N , the respective ground states approach |k = 0i in the P phase and the TF state |TFi ≡ |k = N/2i in the TF phase. In the BA phase, all the three modes stay populated, with an average number of particles in ˆ0 i/N ' (1 + q/qc )/2 [5]. mf = 0 given by hN

IV. USEFUL ENTANGLEMENT IN THE GROUND STATE OF A SPINOR BEC

We first evaluate the QFI of the ground state |ψ0 (q)i of the Hamiltonian (5) in the different phases. Arbitrary collective unitary rotations of a system of indistinguishable spin-1 particles, as considered in this paper, can be ˆ as the 8-dimensional vector of colexpressed by taking G lective Gell-Mann operators, generating the su(3). The covariance matrix ΓG ˆ is discussed in Appendix A 1. We find it convenient to introduce the symmetric (g) and antisymmetric (h) creation and annihilation operators 1 † ˆ † = √1 (ˆ gˆ† = √ (ˆ a1 + a ˆ†−1 ), h a†1 − a ˆ†−1 ) 2 2

(6)

3 explicit expression given by s

P

1.5

TF

BA

1

0.5

0

q / qc

|CBAi ≡

-0.5

-1

-1.5

FIG. 1. (Color online) QFI for the ground state of a nonmagnetized ferromagnetic spin-1 BEC as a function of the ˆ generating the su(3), we quadratic Zeeman shift q. For G depict FQ /N corresponding to the eigendirections of ΓG ˆ for which FQ exceeds N . The quantum phase transitions are indicated by dashed vertical lines. The solid horizontal line is FQ /N = (N + 2)/2.

and present our results in terms of three sets of collective pseudospin- 21 operators, whose Schwinger representation reads a ˆ† gˆ + gˆ† a ˆ0 Sˆx = 0 , 2 a ˆ† gˆ − gˆ† a ˆ0 , Sˆy = 0 2i a ˆ† a ˆ0 − gˆ† gˆ Sˆz = 0 , 2

a ˆ† a ˆ−1 + a ˆ†−1 a ˆ1 Jˆx = 1 , 2 a ˆ† a ˆ−1 − a ˆ†−1 a ˆ1 Jˆy = 1 , 2i a ˆ† a ˆ1 − a ˆ†−1 a ˆ−1 Jˆz = 1 , 2

(7)

ˆ Thus S ˆ ≡ and Aˆi just as Sˆi with gˆ replaced by h. (Sˆ1 , Sˆ2 , Sˆ3 ) generates rotations within the two-level sysˆ ≡ (Aˆ1 , Aˆ2 , Aˆ3 ) tem composed of the modes (ˆ a0 , gˆ), A ˆ ˆ ˆ ˆ corresponds to (ˆ a0 , h), and J ≡ (Jx , Jy , Jˆz ) to (ˆ a1 , a ˆ−1 ). Figure 1 displays, across the three quantum phases P, BA, and TF, FQ /N corresponding to the eigenvectors of ΓG ˆ which provide FQ > N . Large values of the QFI are observed in two cases. First, in the TF phase, h i (TF) ˆ opt FQ |TFi, R = N (N + 2)/2 (8) (TF) ˆ opt where R is given by an arbitrary linear combination of Jˆx and Jˆy . Second, at the center of the BA phase (i. e., for q = 0, we indicate as |CBAi the corresponding ground state) we have h i (CBA) ˆ opt FQ |CBAi, R = N (N + 1)/2, (9) (CBA) ˆ opt where R is an arbitrary linear combination of Sˆx ˆ and Ay . As we show in Appendix A 2, the state has an

bN/2c 2N (N !)3 X 1 p |ki. k k! (N − 2k)! (2N )! 2 k=0

(10)

Hence both |CBAi and |TFi present approximately equal QFI and a Heisenberg scaling FQ ∝ N 2 . It is well known that the ground state of the TF phase approaches a TF state [13] and that the latter exhibits a QFI with Heisenberg scaling with N [5, 6, 28]. For an analysis of the QFI of the ground state of an antiferromagnetic spin-1 BEC see [14]. Conversely, Eq. (9) is a novel and less evident result. To gain some intuition regarding the large amount of useful entanglement found in the BA phase at q = 0, let us rewrite the Hamiltonian (5) in terms of the ˆ and A ˆ operator manifolds of Eq. (7). We obtain, up S to c-numbers, ˆ = 2 λSˆ2 − q Sˆz + 2 λAˆ2 − q Aˆz , H (11) x y 3 3 which is a sum of two (non-commuting) Lipkin-Meshkovˆ and A, ˆ respectively. Since Glick Hamiltonians for S λ < 0, the ground state of the first term in Eq. (11), 2(λSˆx2 − 3q Sˆz ), at q = 0 is a NOON state aligned along the Sx -axis (i. e. a superposition of the maximum and the minimum eigenstates of Sˆx ). Its QFI saturates the Heisenberg limit for rotations generated by Sˆx . Similarly, the ground state of the second term of Eq. (11), at q = 0, is a NOON state aligned along the Ay -axis. This hints at large amounts of entanglement in the CBA state. However, since the symmetric and antisymmetric spin algebras share the same central mode a ˆ0 and therefore do not commute with each other, a more detailed inspection of the ground state is required. To this end, let us trace ˆ mode. This leaves us with the state out the h ρˆ =

N X

P (Nh )|φNh ihφNh |,

(12)

Nh =0

in the modes (ˆ a0 , gˆ), where P (Nh ) is the probability to ˆ mode, and |φN i is a state measure Nh particles in the h h of N − Nh particles in (ˆ a0 , gˆ). In Figure 2(a) we plot P (Nh ) as a function of Nh . The most probable value of Nh is Nh = 0, and P (Nh ) = 0 for odd values of Nh . Since ˆ †h ˆ and hCBA|Sˆx |CBAi = 0, ˆh ≡ h Sˆx commutes with N the QFI decomposes according to N X FQ |CBAi, Sˆx = P (Nh )FQ |φNh i, Sˆx .

(13)

Nh =0

In Figure 2(b) we show FQ [|φNh i, Sˆx ] as a function of Nh . Large values of the QFI are observed up to Nh ' N/2, in accordance with the presence of macroscopic superposition states [20]. As can be seen from the Husimi distributions in Figure 2(c), for Nh > N/2 the |φNh i resemble

4

(a) (c)

0.05 h

P(N P(Nh))

0.04 0.03

0 500 0 h

Sy

100

400

Nh

200

BA

TF

300

(b)

300

Nh=220

200

(a)

100 0

0

100

N N h

200

300

FIG. 2. (Color online) (a) Probability P (Nh ) to find Nh particles in the h mode of |CBAi. (b) QFI FQ [|φNh i, Sˆx ] of the states |φNh i obtained by projecting |CBAi onto a fixed number of particles Nh in the h mode. (c) Husimi distributions of |φNh i for different values of Nh showing the presence of NOON-like states in a very broad range of Nh values. In this Figure N = 500.

NOON states along Sˆx . This explains the Heisenberg scaling of the QFI (9). ˆ manifold can be manipulated exWe note that the S perimentally by radiofrequency pulses coupling the mf = 0 to the mf = ±1 modes. An atomic clock using transˆ manifold of a spin-1 BEC has been formations in the S demonstrated in Ref. [11], see also [7, 8, 10] for squeezing ˆ spin. Our results thus reveal the possibility to of the S attain a sensitivity close to the HL preparing the spin-1 BEC in its ground state at q = 0. Since, when starting with the mf = 0 BEC, q = 0 is reached after an adiabatic variation of q that is half as large as the one required to arrive in the TF regime, implementing |CBAi is less demanding in terms of BEC stability than the experiment reported in [17]. Finally, in Appendix A 3 we prove that a measurement ˆ+ , N ˆ− ) is, at any θ, optimal for both |CBAi and of (N (TF) |TFi. Optimal interferometric transformations Ropt leave the TF state in the N0 = 0 subspace, thus rendering ˆ+ , N ˆ− ) equivalent to D. ˆ A similar argument, see Ap(N pendix A 3, applies to |CBAi. Hence for both states and any phase θ the experimentally relevant measurement of ˆ turns out to be optimal. D

V.

P

Nh=100

0.01

Q

Nh=0

Sx

0.02

FFQ // (N Nh)) (N-N

Sz

QUASI-ADIABATIC STATE PREPARATION

Next we consider an experimental sequence for the variation of q(t) as the one recently discussed in Ref. [17]. We assume a BEC prepared at q/qc = 1.5 in the P phase where |ψ(t = 0)i = |k = 0i. The value of

(b) 1.5

1

0.5

0

-0.5

-1

-1.5

q / qc, Q=0.1 FIG. 3. (Color online) Quasi-adiabatic evolution of a ferromagnetic spin-1 BEC initialized in mf = 0. The quadratic Zeeman shift q(t) is linearly ramped with a slope ∝ Q. The quantum phase transitions are indicated by dashed vertical lines. (a) FQ /N , where the interferometric transformation is generated by Sˆx (green) and Jˆx (red), which are optimal for |CBAi and |TFi, respectively. The solid horizontal line is FQ /N = (N + 2)/2, for comparison. (b) Overlap |hψ(t)|ψ0 (q(t))i|2 of the time evolved state |ψ(t)i with the ground state at q(t).

the quadratic Zeeman term is varied following the ramp q(t)/qc = 1.5 − qc Qt/4, where Q > 0 characterizes the non-adiabaticity of the process. Figure 3 illustrates our observations for N = 500 particles and Q = 0.1. We find that the QFI is hardly affected by the finite ramping speed. This is particularly striking since, as demonstrated in Figure 3(b), the fidelity |hψ(t)|ψ0 (q(t))i|2 with the respective ground state |ψ0 i is dramatically diminished. The oscillations present in both Figure 3(a) and (b) resemble the ones found in Ref. [17] for the conversion ˆ+ + N ˆ− i/N . efficiency hN Note that at the critical points the energy gap between the ground and first excited state scales ∝ N −1/3 [13], and that a larger Q means that the phase boundaries are crossed more rapidly. Hence, enlarging N or Q displaces |ψ(t)i further away from the respective ground state, creating a larger number of excitations. Figure 4 shows which fraction of the QFI for |CBAi and |TFi is accessible within finite time. As expected, it decreases with both N and Q. Note that we vary N at constant qc . In Figure 4(b) we show slices through FQ (N, Q) at

5

(a)

(a) (b)

q = −1.5qc

(b)

q = −1.5qc

(c)

FIG. 4. (Color online) QFI of the states obtained via quasiadiabatic evolution up to q = 0 [panel (a)] and q = −1.5qc [panel (b)] normalized to the QFI of the respective ground states |CBAi and |TFi, indicated as FQ0 . Panel (c) shows the normalized QFI as a function of Q. The green line corresponds to q = 0, the red line to q = −1.5qc . The inset shows the normalized QFI as a function of N . In all panels, the interferometric transformations are generated by Sˆx and Jˆx as is optimal for |CBAi and |TFi, respectively.

fixed N or Q, respectively. The wavy distortions are due to the mentioned oscillations in the QFI, whose frequency depends both on Q and—less pronounced—on N . We find large parts of the QFI conserved during nonadiabatic evolutions with Q . 0.5, and an overall rather small dependence on N . These features significantly ease experiments. Note particularly that at constant Q the overall ramping time scales linearly with N , while the factual (dimensionful) speed of the linear ramp goes even as dq/dt ∝ N 2 . Together with Figure 4(a) this implies that enlarging the particle number reduces the requirements on adiabaticity and BEC stability. A pronounced dependence on whether the ramp of q(t) is terminated at q = 0 (CBA) or q = −1.5qc (TF) is not discernible. This is consistent with the numerical analysis in [17] show-

FIG. 5. (Color online) FI in the presence of Gaussian measurement uncertainty with variance σ 2 for both N± . (a) Dependence of the FI on the interferometric phase θ. (b) Peak value of the FI as a function of σ. The dashed vertical line indicates measurement noise with a spread corresponding to a single particle (σ = 1/2). In this Figure N = 500.

ing that the second phase transition—in contrast to the first one—has but little impact on the amount of created excitations. VI.

FINITE MEASUREMENT PRECISION

To investigate the impact of a finite measurement resˆ± is afolution, we assume that the detection of both N 2 fected by Gaussian noise with variance σ , leading to an imprecise measurement of the number of particles. In this case, the actual measurement probabilities ensue from a convolution of the ideal quantum theoretical result with a Gaussian probability distribution Γ√2σ (x) of variance 2σ 2 and zero mean. We thus determine the classical FI from the effective probability distribution Peff (D|θ) =

N X

Γ√2σ (|D − D0 |)P (D0 |θ),

(14)

D 0 =−N

where P (D0 |θ) is the noiseless probability to find D0 upon ˆ at a phase θ. Figure 5 illustrates how the measuring D FI is affected by the detection uncertainty. Panel (a) shows for both |CBAi and |TFi that, while F (θ) as a whole is strongly damped, pronounced maxima at small

6 t = 0 reaches 1 (g/h) ˆ opt ˆ ± i + ∆N ˆ± FQ [|ψ(t)i, R ] = 1 + 2 hN N

N|λ|t FIG. 6. (Color online) QFI attainable by a quench of the quadratic Zeeman shift to its resonance value qr and subsequent free evolution of the mf = 0 BEC. In the experimentally relevant regime [12], the analytic (red) approximation (15) for small t coincides with the full numeric simulation (green) of Hamiltonian (5). Even at times significantly exceeding current technical feasibility the QFI fails to reach the level of the TF state indicated by the threshold (black).

θ remain far above the SQL, in close analogy to experiments presented in [6]. From panel (b) we infer that for worse than single particle detection (σ = 1/2) these peak values of the FI decrease approximately ∝ σ −2√ . Evaluating the relative standard deviation σmax (N )/ N up to which the FI yet exceeds the SQL we have found that the TF state is slightly less sensitive to particle counting √ noise than the√CBA state: σmax [|TFi]/ N ≈ 0.4 while √ σmax [|CBAi]/ N ≈ 0.2. Both σmax / N are easily undercut by state-of-the-art experiments [6].

VII.

PARAMETRIC AMPLIFICATION

Finally, we compare the (quasi-)adiabatic state preparation with the dynamical generation of entanglement following a quench of q. Such a quench may render the initial mf = 0 condensate dynamically unstable. Spinchanging collisions populate mf = ±1, thereby generating entanglement [26, 29–33]. ˆ± i N , in line with experiments [8, 10– Assuming hN √ 12], we may approximate N0 ' N and a ˆ†0 , a ˆ 0 ' N0 , ˆ = H ˆg + H ˆh which simplifies the Hamiltonian (5) to H 2 † 2 †2 †ˆ ˆ ˆ ˆ ˆ with Hg = αˆ g gˆ + β(ˆ g + gˆ )/2 and Hh = αh h − β(h + ˆ †2 )/2, where α = q + λ(N − 1/2) and β = N λ. h As before, the generated entanglement can be used for interferometric transformations in the symmetric and anti-symmetric subspaces. The corresponding QFI of the state |ψ(t)i obtained by free evolution after a quench at

(15)

ˆ± )2 ˆ± i(hN ˆ± i + 1), hN ˆ± i with (∆N = hN = −1/2 2 2 2 2 2 (N λ) τ sinh (t/τ ), and τ = β − α , where (g/h) ˆ opt R is the respective generator which maximizes the QFI. This expression is valid only for short times, ˆ± i N holds. The as long as the assumption hN (g/h) ˆ opt and a derivation of Eq. (15) are explicit form of R presented in Appendix B. Tuning q allows to arbitrarily choose α. At α = 0 the Hamiltonian is reduced to spin-changing collisions only. As expected, this affords ˆ± i and thus of FQ , entailing the strongest growth of hN the definition of the resonance value qr = −λ(N − 1/2). Applying Eq. (15) to recent spin squeezing experiments [8, 10, 11] provides a relative QFI, FQ /N , which ranges from 5 to 20, thereby corresponding to a FQ /N 2 of less than 2 × 10−2 only. Recall that, in the ideal case, quasi-adiabatic entanglement generation as discussed in this paper allows for FQ /N 2 ≈ 0.5. In the absence of technical noise it would be advantageous to extend parametric amplification protocols to longer evolution times. We therefore numerically simulate the anticipated further evolution of the QFI under the full Hamiltonian (5) at q = qr . Figure 6 indicates that even under ideal experimental conditions parametric amplification is unable to reach the QFI attainable by the quasi-adiabatic approach. In the best scenario, dynamical spin-changing collision creates entangled states with a QFI FQ /N 2 ≈ 0.257 [14]. VIII.

CONCLUSIONS

We have studied the generation of entanglement useful for quantum enhanced interferometry with ferromagnetic spin-1 BECs, focusing on the experimentally relevant case of quasi-adiabatic driving through quantum phase transitions. We have shown that, starting out from a BEC in mf = 0 and a quadratic Zeeman shift of q > qc , at q = 0 and thus halfway to the TF state another highly entangled state, |CBAi, of equal interferometric value (approximately equal QFI) emerges. This allows us to propose an alternative interferometric scheme admitting Heisenberg scaling. For both |TFi and |CBAi, optimal values of the QFI are obtained with interferometric transformations corresponding to common radio-frequency coupling techniques. The optimal measurement procedure is based on the well established counting of particles in the mf = ±1 modes. According to our findings surpassing the standard quantum limit is expected to remain feasible under realistic conditions, when the quasi-adiabatic transition is performed at finite speed, and measurement uncertainty is present. While the TF state is less sensitive to imperfections of atom counting, the CBA state has the

7 advantage of being quasi-adiabatically reachable within half the time. Both regimes favorably compare to squeezing through parametric amplification, thus constituting a promising source of interferometrically useful entanglement.

and corresponding eigenvectors

(2)

u+

(1)

u− ACKNOWLEDGMENTS

(2)

u− We acknowledge support by the SFB 1227 “DQ-mat”, projects A02 and B01, of the German Research Foundation (DFG). M. G. thanks the Alexander von Humboldt foundation for support.

Appendix A: Adiabatic phase transition 1.

k=0

ˆ†−1 a ˆ0 − a ˆ†0 a ˆ−1 ˆ1 = ˆ2 = a G G , 2i ˆ†1 a ˆ−1 + a ˆ†−1 a ˆ1 ˆ3 = ˆ4 = a G G , 2 ˆ†0 a ˆ1 + a ˆ†1 a ˆ0 ˆ5 = ˆ6 = a G G , 2 ˆ†−1 a ˆ−1 + a ˆ†0 a ˆ0 − 2ˆ a†1 a ˆ1 ˆ7 = ˆ8 = a √ G G . 2 3 P (A1) Consider an arbitrary (normalized) state |ψi = k ck |ki ˆ is block ˆ with D|ψi = 0. The covariance matrix of G diagonal, ˆ 2 ˆ 2 ΓG ˆ = Γ(G ˆ 1 ,G ˆ 2 ,G ˆ 6 ,G ˆ 7 ) ⊕ Γ(G ˆ 3 ,G ˆ 8 ) ⊕ (∆G4 ) ⊕ (∆G5 ) (A2) ˆ i )2 the variance of G ˆ i . The twofold degenerate with (∆G eigenvalues of Γ(Gˆ 1 ,Gˆ 2 ,Gˆ 6 ,Gˆ 7 ) are (A3)

with

k=0

bN/2−1c

X

c∗k ck+1 (k + 1)

p

The corresponding eigenvectors read √ √ u0 = 1, 3 , u1 = 3, −1 .

(A7)

Finally, ˆ 4 )2 = (∆G ˆ 5 )2 = (∆G

bN/2c 1 X |ck |2 k(k + 1). 2

(A8)

k=0

Numerically evaluating ΓG ˆ in the ground state |ψ0 (q)i of the Hamiltonian (5) we find that only the (two-fold ˆ 4 )2 = (∆G ˆ 5 )2 , dedegenerate) eigenvalues λ+ and (∆G picted in Figure 1 and Figure 3(a), significantly differ from zero for large N . They √ correspond to the eigen√ ˆ1 + G ˆ 6 )/ 2, Aˆy = (G ˆ2 + G ˆ 7 )/ 2, operators Sˆx = (G ˆ4, which become optimal in the BA phase, and Jˆx = G ˆ ˆ Jy = −G5 , prevalent in the TF phase, respectively. 2.

Properties of the CBA state

In this section, we provide some analytical results on the CBA state, i. e., the ground state at q = 0. a.

Coefficients in the Fock basis

Let us introduce the collective pseudospin-1 operator ˆ composed of L

bN/2c X 1 N+ |ck |2 k(2N − 4k − 1) , A= 4

1 2

k=0

(A6)

a ˆ†−1 a ˆ0 + a ˆ†0 a ˆ−1 , 2 a ˆ†−1 a ˆ−1 − a ˆ†0 a ˆ0 , 2 a ˆ†−1 a ˆ1 − a ˆ†1 a ˆ−1 , 2i a ˆ†0 a ˆ1 − a ˆ†1 a ˆ0 , 2i

λ± = A ± |B|

(A5)

where < and = denote the real and imaginary part, respectively. Note that if ck ∈ R for all k we obtain =(B) = 0 and hence Γ(Gˆ 1 ,Gˆ 2 ,Gˆ 6 ,Gˆ 7 ) = Γ(Gˆ 1 ,Gˆ 6 ) ⊕ Γ(Gˆ 2 ,Gˆ 7 ) . The eigenvalues of Γ(Gˆ 3 ,Gˆ 8 ) are 2 bN/2c bN/2c X X λ0 = 0, λ1 = 3 |ck |2 k 2 − |ck |2 k .

Gell-Mann Covariance Matrix

The collective Gell-Mann operators read

B=

1 ( =(B), <(B), 0, |B|), 2|B| 1 =√ ( <(B), −=(B), |B|, 0), 2|B| 1 =√ (−=(B), −<(B), 0, |B|), 2|B| 1 =√ (−<(B), =(B), |B|, 0), 2|B|

(1) u+ = √

(N − 2k)(N − 2k − 1)

k=0

(A4)

ˆ x = √1 (ˆ ˆ0 + a ˆ†−1 a ˆ0 ) = 2Sˆx , L a†0 a ˆ1 + a ˆ†0 a ˆ−1 + a ˆ†1 a 2 1 ˆy = √ L (ˆ a†0 a ˆ1 − a ˆ†0 a ˆ−1 − a ˆ†1 a ˆ0 + a ˆ†−1 a ˆ0 ) = 2Aˆy , i 2 ˆz = a L ˆ†−1 a ˆ−1 − a ˆ†1 a ˆ1 = −2Jˆz . (A9)

8 This allows us to express the ground state of the Hamiltonian (5) in the subspace of D = 0 at q = 0 as [25] N 1 ˆ+ L |N− = 0, N0 = 0, N+ = N i, |CBAi = p (2N )! √ † ˆ+ = L ˆ x + iL ˆ y = 2(ˆ a a ˆ0 + a ˆ† a ˆ1 ). L −1

ordering of and within pairs we arrive at (2k)!/(2k k!). This completes the proof, since N (2k)! N! = k . (A15) X(k) = 2 k!(N − 2k)! N − 2k 2k k!

0

(A10) ˆN We expand the operator L + , leading to s bN/2c 2N X −k |CBAi = Ck [ˆ a0 , a ˆ†0 ]ˆ a†k ˆN |0, 0, N i −1 a 1 (2N )! k=0 s bN/2c 2N N ! X Ck [ˆ a0 , a ˆ†0 ]|k, 0, ki, (A11) = (2N )!

Applying Lemma 1 to Eq. (A11) gives s |CBAi =

bN/2c 2N N !3 X 1 p |k, N − 2k, ki k (2N )! 2 k! (N − 2k)! k=0 (A16)

as reported in Eq. (10).

k=0

†

where Ck [ˆ a, a ˆ ] denotes the sum over all possible products composed of k operators a ˆ and N −k operators a ˆ† in arbi† trary order, and we have used that a ˆ−1 and a ˆ1 commute † with each other as well as with a ˆ0 and a ˆ0 . Each term in Ck [ˆ a0 , a ˆ†0 ] leads to a total creation of N − 2k particles in ˆN mode a ˆ0 . Since we apply L + to a state containing zero particles in the central mode, only values of k ≤ N/2 contribute to Eq. (A11). An explicit evaluation can be performed by means of the following Lemma 1. For an arbitrary bosonic mode with creation operator a ˆ† and vacuum |0i N! Ck [ˆ a, a ˆ† ]|0i = k (ˆ a† )N −2k |0i ∀k ≤ N/2. 2 k!(N − 2k)! (A12)

b.

Quantum Fisher information

As discussed in the main text, see Figure 1, and in Appendix A 1, the QFI of |CBAi is maximized by any ˆ opt ∈ span{Sˆx , Aˆy }. We consider without loss of genR ˆ opt = Sˆx . Then erality R ˆ opt ] = 4(∆Sˆx )2 FQ [|CBAi, R b(N −1)/2c

X

=

√ 2 √ (k + 1) N − 2k ck + N − 2k − 1 ck+1 ,

k=0

(A17) where s ck =

Proof. Since each term in the sum described by Ck [ˆ a, a ˆ† ] describes the creation of N − 2k particles, we can write Ck [ˆ a, a ˆ† ]|0i = X(k)(ˆ a† )N −2k |0i,

(A13)

which reduces the problem to the identification of the combinatorial factor X(k). We use Wick’s theorem [34], a ˆ|0i = 0, and the fundamental Wick contractions a ˆa ˆ=a ˆ† a ˆ† = ^ a ˆ† a ˆ = 0, ^ ^ † a ˆa ˆ =1 ^

(A14)

ˆ ≡ AˆB− ˆ :AˆB: ˆ and the double dots denote norwhere Aˆ B ^ mal ordering. The contribution of each permutation to X(k) is the number of variants it admits for enclosing all k annihilation operators into a ˆa ˆ† -contractions. Taking into consideration all permutations of a ˆ, a ˆ† reveals that X(k) is the number of possibilities to tag k unsorted dis- N joint tuples in a set of N elements. There are N −2k † different choices for the positions of the a ˆ which are not going to be contracted. Thus, we merely have to count the number of possibilities to pair 2k objects. First arbitrarily arranging them and then compensating for the

1 2N N !3 p k (2N )! 2 k! (N − 2k)!

(A18)

are the Fock-state coefficients of |CBAi from Eq. (A16), and ck>N/2 ≡ 0. Thus √

√ N − 2k ck + N − 2k − 1 ck+1 s 2N N !3 N +1 p = , k+1 (2N )! 2 (k + 1)! (N − 2k − 1)!

(A19)

which, after some rearrangements, leads to ˆ opt ] FQ [|CBAi, R (N + 1)!2 = 2(2N )!

b(N −1)/2c

X

k=0

N k

N − k N −2k−1 (A20) 2 . k+1

The sum ensues from the following Lemma 2. bn/2c

X

k=0

m k

m − k n−2k 2m 2 = ∀n ≤ 2m (A21) n − 2k n

9 Proof. Consider 2m sites grouped into m pairs. For each k the left-hand side of Eq. (A21) is the number of possibilities to distribute 2k + (n − 2k) = n indiscernible objects on these 2m sites in such a way that exactly k pairs are completed. The number of obtained pairs can assume values between max{0, n − m} and bn/2c. Note that k < n − m do not contribute to Eq. (A21). Thus, summing over all 0 ≤ k ≤ n/2 amounts to counting the variants of distributing n identical elements on 2m sites, which gives 2m . n

D=2n

X

(A27)

PˆD Sˆx2m

|CBAi = 0

D=2n+1

and

Choosing n = N − 1 and m = N we obtain ˆ opt ] = N (N + 1)/2. FQ [|CBAi, R

provides both the necessary and sufficient condition for n ˆ opt optimality [22]. We observe that R |ψi, n ∈ N0 has only real coefficients in the Fock basis {|N− , N0 , N+ i}. Furthermore, X PˆD Sˆx2m+1 |CBAi = 0,

X

(A22)

PˆN± Jˆx2m+1 |TFi = 0,

N± =2n

X c.

Mean particle number

PˆN± Jˆx2m

(A28) |TFi = 0

N± =2n+1

We further determine the exact mean particle numbers in the a ˆ±1 modes, using X ˆ± i = hN k|ck |2 , (A23)

for all n, m ∈ N0 . This entails 2m+1 2n ˆ ˆ opt ˆ opt |ψi = 0, hψ|R PN+ , N− R 2m+1 2n ˆ opt |ψi = 0. ˆ opt PˆN , N R hψ|R

k

+

with the coefficients of the CBA state defined in Eq. (A18). Applying Lemma 2 with n = N and m = N − 1, we find N 2N − 2 · 4 2N − 1 N 1 1 = 1− +O . 4 2N N2

ˆ± i = hN

(A24)

ˆ± i = For N 1, we recover the mean field expression hN N/4. 3.

Optimal measurements

We consider the two interferometrically relevant states |ψi ∈ {|CBAi, |TFi} along with the respective optimal generators of the interferometric rotation 1 (CBA) ˆ opt R (φ) = √ e−iφ a ˆ†0 a ˆ1 + eiφ a ˆ†0 a ˆ−1 2 2 (A25) + eiφ a ˆ†1 a ˆ0 + e−iφ a ˆ†−1 a ˆ0 , 1 −iφ † (TF) ˆ opt R (φ) = e a ˆ1 a ˆ−1 + eiφ a ˆ†−1 a ˆ1 2 ˆ

providing |ψ(θ)i = e−iθRopt |ψi. Let us first focus on φ = (CBA) (TF) ˆ opt ˆ opt 0, where R (0) = Sˆx and R (0) = Jˆx , and show ˆ ˆ that a measurement of (N+ , N− ) is optimal at any θ. The projections PˆN+ , N− ≡ |N− , N0 , N+ ihN− , N0 , N+ | with N0 = N − N+ − N− ˆ+ , N ˆ− ) are one-dimensional. onto the eigenstates of (N In addition, hψ| dψi = 0 and hence hψ(θ)| dψ(θ)i = 0 for all θ. Therefore ˆ opt |ψ(θ)i = 0 ∀N+ , N− , θ

(A26)

(A29)

−

Thus ˆ opt |ψ(θ)i hψ(θ)|PˆN+ , N− R ∞ X (−1)j+l θ2(j+l)+1 =i (2j)!(2l)! j,l=0 1 2j+1 ˆ 2l+1 ˆ opt ˆ opt hψ|R PN + , N − R |ψi 2j + 1 1 2j ˆ 2l+2 ˆ opt ˆ opt |ψi PN + , N − R − hψ|R 2l + 1 ∈ iR,

(A30)

which implies Eq. (A26). ˆ still for Next, we consider aPmeasurement of D, φ = 0. Due to PˆD = N+ −N− =D PˆN+ , N− , Eq. (A26) holds also when PˆN+ , N− is substituted by PˆD . Since the PˆD are no longer one-dimensional, this is not sufficient for optimality [22]. However, we are able to show that for both |CBAi and |TFi the Hilbert space H = span{|N+ , N0 , N− i} can be restricted to H0 such n ˆ opt that {R |ψi, n ∈ N0 } ∈ H0 while the dimensionality 0 ˆ of PD H for any D is one. Let us start with the TF ˆ0 ] = 0 and N ˆ0 |TFi = 0, we can state. Since [Jˆx , N 0 choose H = PˆN0 =0 H. Regarding |CBAi, recall that ˆ x /2. Hence [Sˆx , L ˆ 2 ] = 0. Then L ˆ 2 |CBAi = Sˆx = L 0 ˆ N (N + 1)|CBAi [25] suggests to set H = PLˆ 2 =N (N +1) H. ˆ z has a non-degenerate spectrum in H0 . Thus L ˆ z = −D ˆ L establishes the one-dimensionality of all PˆD H0 . To proceed to arbitrary φ we note that ˆ ˆ (CBA) ˆ opt R (φ) = e2iφJz Sˆx e−2iφJz , ˆ ˆ (TF) ˆ opt R (φ) = eiφJz Jˆx e−iφJz

(A31)

10 and recall that the classical FI (2) depends only on P (µ|θ) = hψ(θ)|Pˆµ |ψ(θ)i with µ the possible measureˆ ˆ ment outcomes. Because e−iφJz PˆN+ , N− eiφJz = PˆN+ , N− ˆ ˆ and, thanks to D|ψi = 0, e−iφJz |ψi = |ψi, our results hold for any φ.

Appendix B: Parametric amplification

We start by introducing the generators of su(1, 1)

ˆg i = β 2 τ 2 sinh2 (t/τ ) for |c|2 ez < 1. We first derive hN 2 ˆg ) = 2hN ˆg i(hN ˆg i + 1). Applying N0 ' N and and (∆N √ † ˆ a ˆ0 , a ˆ0 ' N0 also to S gives √

√ N N 1 † Sˆx ' (ˆ g + gˆ ), Sˆy ' (ˆ g − gˆ† ), Sˆz ' (N − gˆ† gˆ). 2 2i 2 (B6) The corresponding covariance matrix ΓSˆ is block diagonal. The eigenvalues of its (x, y)-block evaluate to

ˆ (a) = 1 a ˆ (a) = 1 a ˆ (a) = 1 (2ˆ a† a ˆ + 1), L ˆ†2 , L ˆ2 , (B1) L + − 0 4 2 2 where a ˆ† denotes some bosonic creation operator. √ Unˆ0 ' N0 the der the approximations N0 ' N and a ˆ†0 , a ˆ = Hamiltonian (5) becomes, discarding c-numbers, H ˆ ˆ Hg + Hh with ˆ (g/h) ˆ g/h = 2~αL ˆ (g/h) + ~g/h β L ˆ (g/h) + L (B2) H + − 0 ˆg ≡ gˆ† gˆ and g/h = ±1. Let us denote the eigenstates of N † ˆ h ˆ by |nig/h . The initial state (mf = 0 ˆh ≡ h and N condensate) |ψ(0)i = |0ig ⊗ |0ih evolves as |ψ(t)i = ˆ g/h t/~)|0ig/h . |ψ(t)ig ⊗ |ψ(t)ih with |ψ(t)ig/h ≡ exp(−iH An explicit form of |ψ(t)ig/h is obtained from the disentanglement theorem for su(1, 1) [35]: for t, r ∈ R, z ∈ C ˆ + − z∗L ˆ − + 2irL ˆ0 t exp z L (B3) ˆ + exp p0 L ˆ 0 exp p− L ˆ− = exp p+ L with p+ = zτ sinh(t/τ )/C, p0 = −2 ln C, p− = −z ∗ τ sinh(t/τ )/C, C = cosh (t/τ ) − irτ sinh (t/τ ), and −1/2 τ = |z|2 − r2 . The result for |z|2 = r2 is obtained by taking the corresponding limit [35]. This entails, see also [5], s ∞ 1 X n 2n c n |ψ(t)ig,h = p g,h |2nig,h (B4) 2 n |C| n=0 with c = −iβτ sinh(t/τ )/C, C = cosh (t/τ ) + −1/2 iατ sinh (t/τ ), and τ = β 2 − α2 . ˆg i, ∆N ˆg , and the QFI relies on The evaluation of hN the generating function

(xy)

λ±

=

√ N ˆg i ± 2∆N ˆg , 1 + 2hN 4

(B7)

ˆg )2 /4. Thus, for while the variance of Sˆz is λ(z) = (∆N ˆ N hNg i the maximal QFI attainable with rotations (xy) generated by linear combinations of the Sˆi is 4λ+ . The corresponding (normalized) eigenvector

s

s

! ˆg i √ α 2 hN 1− 2 ,0 ˆg β 2 ∆N (B8) gives the optimal direction of interferometric rotations in the symmetric subspace.

(g) uopt

1 = ±√ 2

ˆg i √ α2 hN 1+ 2 ,− ˆg β 2 ∆N

ˆ † and φ± : a Recall that φgh : gˆ† ↔ ih ˆ†1 ↔ a ˆ†−1 leave the algebraic relations of the creation and annihilation operators, the initial state |ψ(0)i, the Hamiltonian, and thus ˆg ) = N ˆh and φ± (N ˆ+ ) = |ψ(t)i invariant. Hence φgh (N ˆ− imply hN ˆ h i = hN ˆg i, ∆N ˆh = ∆N ˆg , hN ˆ+ i = hN ˆ− i, and N ˆ+ = ∆N ˆ− . Then N ˆg + N ˆh = N ˆ+ + N ˆ− yields hN ˆ+ i = ∆N ˆ ˆ hNg i. Using D|ψ(t)i = 0 and |ψ(t)i = |ψ(t)ig ⊗ |ψ(t)ih √ ˆ + = ∆N ˆg / 2. Finally, we furthermore find that ∆N φgh (Aˆx ) = −Sˆy , φgh (Aˆy ) = Sˆx , and φgh (Aˆz ) = Sˆz entail that the eigenvalues of ΓA ˆ coincide with the ones of ΓS ˆ, while

(h) (g) (g) uopt = −uopt, y , uopt, x , 0

(B9)

(B5)

ˆ± i, the optimal rotation for defines, again for N hN phase imprinting within the antisymmetric subspace.

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