Spin squeezing of high-spin, spatially extended quantum fields

Investigations of spin squeezing in ensembles of quantum particles have been limited primarily to a subspace of spin fluctuations and a single spatial mode in high-spin and spatially extended ensembles. Here, we show that a wider range of spin-squeezing is attainable in ensembles of high-spin atoms, characterized by sub-quantum-limited fluctuations in several independent planes of spin-fluctuation observables. Further, considering the quantum dynamics of an $f=1$ ferromagnetic spinor Bose-Einstein condensate, we demonstrate theoretically that a high degree of spin squeezing is attained in multiple spatial modes of a spatially extended quantum field, and that such squeezing can be extracted from spatially resolved measurements of magnetization and nematicity, i.e.\ the vector and quadrupole magnetic moments, of the quantum gas. Taking into account several experimental limitations, we predict that the variance of the atomic magnetization and nematicity may be reduced as far as 20 dB below the standard quantum limits.

Spin squeezing describes a form of entanglement in a many-spin system for which the variance in measurements of particular collective spin variables is smaller than the minimum variance achievable in uncorrelated systems [1,2]. Aside from representing a theoretically tractable form of entanglement in many-body quantum systems, spin squeezing also promises to improve measurement precision in a variety of applications such as chronometry, magnetometry, and atom interferometry. The recent achievement of metrologically beneficial pseudo-spin squeezing in cold atomic gases, on the Zeemaninsensitive hyperfine transitions utilized in atomic clocks, is among the first realizations of this promise [3,4].
The conventional form of spin squeezing involves the collective vector spin operator, F, which is the equal-weight sum of the vector spin operators for all of the ensemble's constituent particles. Given an average ensemble spin oriented along theẑ axis, the measurement variance of orthogonal spin projections obey an uncertainty relation, (∆F x ) 2 (∆F y ) 2 ≥ F z 2 /4. Ensemble states of uncorrelated particles are limited by the standard quantum limited variance, (∆F ⊥ ) 2 SQL = | F z |/2, achieved by partitioning the minimum measurement uncertainties equally between the two vector spin projections.
This form of vector operator spin squeezing describes present experiments not only on pseudo-spin 1/2 atomic gases, i.e. where each atom is restricted to just two internal [3,4] or external [5] states, but also on gases of higher-spin atoms (e.g. the f = 4 hyperfine-state manifold of Cs [6]). However, the description of higher-spin atoms by their vector spin alone is incomplete. On the one hand, light-atom interactions that affect higher-order spin moments may impair efforts to create vector-spin squeezing in high-spin atomic gases [7,8]. On the other hand, the deliberate addressing and measurement of higher-order spin moments [9] offers benefits to certain metrological applications, e.g. eliminating directional dependencies of atomic magnetometers [10].
Here, we consider theoretically the characterization and preparation of spinsqueezed states in f = 1 spinor Bose gases. We go beyond existing treatments of spin squeezing by evaluating such squeezing in an ensemble that is explicitly higherdimensional not only in its spin degrees of freedom, but also in its spatial degrees of freedom. Our investigation is motivated by two experimental results: the controlled amplification of spin noise in 87 Rb quantum fluids [11,12,13], and the use of a spinor Bose-Einstein condensate as a spatially resolving magnetometer [14]. As we discuss herein, these two results, respectively, provide a complete means for inducing squeezing of both the spin vector and quadrupole moments of the f = 1 quantum gas, and also a means of applying the subsequent spatially extended non-classical quantum fluid toward spatially resolved magnetometry with sub shot-noise sensitivity.

Single-mode squeezing for arbitrary spin
We begin by introducing a generalized treatment of spin squeezing that applies equally to ensembles of spin-1/2 and also of higher-spin particles. Such squeezing will be defined with respect to a coherent spin state [2], in which all particles are prepared in an identical single-particle spin-f state |ψ . Consider an orthogonal basis of the single-particle spin space that includes the state |ψ and also the 2f states |ξ a . We now define manybody observables M (1) a = α (|ξ a ψ| + |ψ ξ a |) α and M (2) a = α i (|ξ a ψ| − |ψ ξ a |) α where the summation index α is taken over all particles in the N-particle ensemble. These spin fluctuation operators describe the subspace of spin excitations orthogonal to the coherent spin state. With respect to this uncorrelated state, these observables satisfy the commutation relations [M (1) a , M (2) b ] = 2iNδ a,b . Thus, measurements of each pair M (1,2) a independently obey an uncertainty relation (∆M Following the treatment of squeezing for vector spin operators, we identify one mode of spin fluctuations, labeled by a, to be spin squeezed when the measurement variance for one quadrature of the M (1) a plane is below the standard quantum limit, N. Applying this treatment to an ensemble of spin-1/2 particles, we recall that any coherent spin state can be written as the m = +1/2 eigenstate of a projection of the dimensionless vector spin. Taking that projection to lie alongẑ, we identify the one mode of spin fluctuation operators as the Pauli operators σ x and σ y . More generally, for an ensemble of spin-f particles prepared in the |ψ = |m z = f eigenstate of the F z spin operator, the spin fluctuation operators defined by the state |ξ 1 = |m z = f − 1 act on |ψ as 2/f times the spin vector operators F x and F y . For this one mode of spin-fluctuation operators, the commutation relation matches that of the spin-vector operators. Thus, we recover the conventional description of vector-spin squeezing as pertaining to just one mode of spin fluctuations atop a maximum-spin coherent spin state.
To illustrate further the possibility of squeezing in several spin fluctuation modes, we consider spin squeezing of spin-1 particles prepared initially in the |m z = 0 state. Our treatment is facilitated by working in a polar state basis, where |φ e is the zero-eigenvalue state of F · e [15]. The pair of fluctuation operators defined according to the state |φ x (|φ y ) are identified as M (1) = −N xz (−N yz ) and M (2) = +F y (−F x ), where the former is a component of the quadrupole moment tensor [16]. The coherent spin state |m z = 0 is thus regarded as the uncorrelated vacuum state for independent spin fluctuations in the F x -N yz and the F y -N xz planes, and correlated states may be spin squeezed with respect to vector-spin or quadrupole-spin components, or linear combinations thereof. Finally, we note that this unconventional form of spin squeezing, about the polar |m z = 0 state, may be mapped directly onto the more conventional vector-spin squeezing discussed above by a unitary transformation that rotates the polar basis states onto the proper F z eigenstates; experimentally, such generic unitary transformations upon alkali-atom hyperfine spin states are produced by rf-pulses and quadratic Zeeman energy shifts [17].

Spin squeezing in a spatially extended quantum field
We now consider such multi-spin-mode squeezing in a spatially extended, ferromagnetic f = 1 gaseous Bose-Einstein condensate where, as we show below, such squeezing is produced naturally by spin-dependent interactions [18]. As in recent experiments [11,12], these condensed atoms (at rest) are prepared initially in the common spin state |m z = 0 . Within the volume occupied by the condensate, we define position-space spin-fluctuation measurement densities where φ e (r) is now the position-space Bose field operator for the polar state |φ e , and the index a ∈ {x, y} runs over the transverse spin polarizations. These observables represent components of the local nematicity (rank-2 moments of the atomic spin) and magnetization (rank-1 moments of the atomic spin) of the quantum gas, and obey the commutation relation M (1) , and expectation values are taken with respect to the polar condensate.
These measurement density operators are used to construct operators corresponding to spatially resolved measurements of particular spin fluctuations of the quantum field. Defining the real measurement mode functions {A(r), B(r), C(r), D(r)}, we may define mode measurement operators as follows: where the common polarization index is omitted for clarity. In this definition for M is non zero to the extent that the two measurement operators probe non-parallel spinfluctuation quadratures on a common population of atoms. We see that our present treatment subsumes the commonly discussed single-mode description of spin squeezing in atomic ensembles. That is, the equal-weight sum of atomic spin operators is measured by spatial-mode operators defined with the uniform functions A = D = 1 and B = C = 0. However, our treatment easily allows for a multi-mode description of spin squeezing and collective spin measurements of a spatially extended quantum field. For example, spin squeezing in a spatially resolved measurement of the atomic spin, e.g. in the spinor-gas magnetometer of Ref. [14], may be treated by defining measurement mode functions corresponding to every resolved pixel in the magnetization-sensitive image. Our treatment applies also to optical measurements sensitive to spatially varying linear combinations of magnetization and nematicity, which are achieved by measuring different components of the linear optical susceptibility tensor [19].

Generation of spin squeezing in a ferromagnetic f = 1 Bose-Einstein condensate
Now, let us consider the evolution of the paramagnetic condensate under the quantumcoherent spin dynamics produced by the spin-dependent contact interactions between atoms [20,21]. As in previous experiments, we consider the condensate to be prepared in the uncorrelated polar state and then allowed to evolve freely at a fixed value of the quadratic Zeeman shift q, which gives the difference between the average energy of the |m z = ±1 Zeeman sublevels and that of the |m z = 0 state. Within the Bogoliubov approximation, the spin-dependent Hamiltonian may be written as follows: where H 0 = − 2 ∇ 2 /2m+q with m being the atomic mass. The spin-mixing interactions are described through H int = c 2 n(r) where n(r) > 0 is the density of the condensate and c 2 = 4π 2 (a 2 − a 0 )/3m relates to the s-wave scattering lengths a f for binary collisions among atoms with total spin f ∈ {0, 2} [22,23]. For a ferromagnetic Bose gas, c 2 < 0.
Here, we restrict the spin excitations to the volume of the condensate. We find it convenient to consider separately the real and imaginary parts of φ a , i.e. we consider the fields Z a (r) = φ † a + φ a and P a (r) = i(φ † a − φ a ), which commute as [Z a (r), P a (r ′ )] = −2i δ 3 (r − r ′ ). Considering the linearized equations of motion for these operators, obtained from H spin given above, one identifies normal dynamical modes by defining a set of functions ζ n (r), spanning the condensate volume, as solutions to the following differential equation: It follows that E 2 n and ζ n (r) are real and that we may normalize the mode functions so that d 3 r ζ * n H −1 0 ζ m = δ n,m /2. With the aid of these functions, we define the magnonexcitation mode operators (omitting the polarization index) These operators are found to be canonically conjugate, satisfying the equations of motion Ż n = P n and Ṗ n = −E 2 n Z n and commutation relations [Z n , P m ] = −iδ n,m . The canonical evolution of magnon excitation mode n is akin to that of a harmonic oscillator with frequency ω n = E 2 n / , with E 2 n > 0 defining stable magnon excitations and E 2 n < 0 defining unstable modes. The action of this canonical evolution on the initial fluctuations in each normal mode may be discerned by examining the scaled covariance matrix where the mode operators are scaled by their initial rms fluctuations. It is important to note that this scaling circularizes the initial fluctuations of the normal mode operators, but that these initial fluctuations do not have the minimum uncertainty. One finds, however, that the fluctuations do reach their minimum product for excitations of a homogeneous condensate and approximate the minimum product for inhomogeneous condensates with densities varying only on long length scales. The Hermitian covariance matrix may be expressed as C n (t) = R n (t)S n (t)R † n (t) with S n (t) being diagonal with eigenvalues with κ = (χ+χ −1 −2)/4 and χ = E 2 n Z 2 n (0) / P 2 n (0) . We note that κ sin 2 ω n t is positive for both stable and unstable magnon modes.
For stable magnon modes, the initial spin fluctuations rotate in theZ n -P n plane and also experience periodic, bounded dilatations and contractions. In contrast, for unstable magnon modes, the spin fluctuations become progressively squeezed along one quadrature axis and amplified along the other, with the squeezed/amplified axes defined by the rotation matrix R n (t). The fluctuations and dynamical evolution of these mode operators relate directly to those of normalized spatial-mode measurement observables given asM corresponding toZ n andP n , respectively, at t = 0. Spatially resolved spin measurements tailored to detecting the squeezed and amplified quadratures of the nth normal mode are defined as time-dependent linear combinations ofM (1) n andM (2) n through the rotation matrix R n (t).

Evolution of a homogeneous spinor condensate
For a homogeneous condensate, the evolution following a quench of the initially paramagnetic quantum gas is expressed simply. Translational symmetry allows one to identify normal modes a priori as possessing Fourier components only with magnitude k. The normal-mode equation (6) is satisfied by the functions ζ k,θ = 2/V cos(k·r+θ), with V being the condensate volume, and with the magnon-excitation spectrum given as [20]. Unstable modes exist for q below a critical value q 0 = −c 2 n, specifically within the range |ǫ + q − q 0 /2| < q 0 /2.
The initial fluctuations of the normal mode operators defined with respect to the mode functions {ζ k,θ } are given as Z 2 k,θ = (ǫ k + q) −2 and P 2 k,θ = 1, giving κ = q 2 0 /(4E 2 k ). The normalized covariance matrix for modes with wave vector k is given as We see that the dynamically unstable momentum-space modes are progressively squeezed over time (figure 1). The maximally unstable mode, with ω 2 k = −(q 0 /2 ) 2 , is accessed for q < q 0 /2. For this mode, the orientation of the squeezing axis is fixed at an angle ϑ = −π/4 in the M (1) -M (2) plane, and the greatest squeezing is achieved. Away from this maximally squeezed mode, the orientation of the squeezing/amplification axes varies in time. At long times, defined by κ sin 2 (ω k t) = q 2 0 4 2 |ω k | 2 sinh 2 (|ω k |t) ≫ 1, spin fluctuations at dynamically unstable wave vectors are squeezed by a factor that decreases exponentially in time and is given as The orientation of the squeezed quadrature axis is given by the relation tan ϑ = − |ω k |/(ǫ k + q), with ϑ subsequently varying between 0 and −π/2. This momentum-space treatment describes how the dynamical instabilities of a ferromagnetic f = 1 spinor Bose-Einstein condensate may be used as a mode-bymode parametric amplifier, both for the amplification of small spin fluctuations atop the polar condensate, as explored in Refs. [12,13,24], and also for the generation of a spin-squeezed quantum field. This field may be manipulated further to suit a particular metrological application. For example, spin squeezing may first be prepared at a particular wave vector k by allowing the system to evolve for a given time at q = q 1 for which ω 2 k < 0. Thereafter, the quadratic shift may be adjusted to q = q 2 at which the spin modes at wave vector k are dynamically stable. Under such stable evolution, the spin fluctuations undergo rotation in the M (1) -M (2) plane, so that squeezing can be generated along any desired quadrature axis. More general control of the k-dependent rotation angle could be realized by applying a k-dependent optically induced quadratic Zeeman shift, e.g. by taking advantage of the band structure of an appropriately tuned and polarized optical lattice.

Locality in spin squeezing and measurement
It is illuminating to consider the evolution of a spatially extended f = 1 quantum field in position space. The spin-dependent contact interactions to which we have ascribed the generation of a correlated quantum field act locally; thus, we would expect spatial correlations to propagate at a finite rate across the spinor condensate. Yet, the above description for the homogeneous system (section 3.1) dealt with measurement observables that incorporate correlations between particles at infinite range. In contrast, given the local origin of spin correlations, we expect that spin squeezing should be effectively observed by local measurements of the collective atomic spin on a finite number of atoms.
One approach to tracing the spatial evolution of spin correlations is to consider correlations among the position-space spin-fluctuation measurement densities, via the covariance matrix defined as For the uncorrelated condensate prepared at time t = 0, we have C(r, r ′ ; 0) = I n δ 3 (r − r ′ ), i.e. correlations are purely local. At later times, for the initially homogeneous condensate, Now, let us inquire as to the spatial range of spin-squeezing produced by spindependent contact interactions by asking whether finite-volume spatial-mode operators may be defined that will capture the spin squeezing produced in a homogeneous spinor condensate. We consider two possibilities. First, we define "quadrature-tuned" measurement mode functions through their Fourier transform as where f (k) is real. Now, following the evolution time t, evaluating the covariance matrix with respect to these spin-fluctuation observables, we obtain simply We consider also a second, simpler, "uniform quadrature" mode function with Fourier transform where k max is the wave vector for which maximal squeezing is achieved. Now we are in a position to consider forms of the measurement operators appropriate for capturing squeezing achieved by the spinor Bose-Einstein condensate amplifier. We examine the long-time limit (equation 14), and consider amplification under two conditions.

Shallow quench
Under the condition q 0 /2 ≤ q < q 0 , the maximally unstable mode occurs at zero wave vector. Expanding about k = 0, we approximate We note that D, proportional to /m, has the units of a diffusion constant. Let us now define the measurement operator through so that the quantity (AD − BC) > 0 is a 3D Gaussian with peak value of unity at the origin, and with a volume of v = (2πσ 2 ) 3/2 . Evaluating the correlation matrix, we now find fluctuations of the M A.B.C.D operator are squeezed with respect to their variance at t = 0. An estimate for the squeezing achieved using the quadrature-tuned mode function may be obtained analytically. If we assume that σ 2 > Dt, then the reduction in the variance is given approximately as That is, the spin correlations produced in the quenched spinor Bose-Einstein condensate are indeed local, contained within a volume that grows diffusively in time. In contrast, uniform-quadrature measurements must be made on larger volumes to observe the maximum attainable squeezing. effective in capturing the spin squeezing produced in this system within a measurement volume that grows only slowly with evolution time. In contrast, a uniform-quadrature measurement requires larger volumes to capture this squeezing. This requirement is explained by the finite differences between the quadrature axes along which squeezing is achieved at different wave vectors. The higher the degree of squeezing one wishes to detect, the more precisely must the measurement quadrature axes be tuned to the spin-squeezed axis by restriction to a narrow range of wave vectors.

Deep quench
For a deep quench, with q < q 0 /2 the maximum instability occurs on a sphere in kspace, with radius k max defined by ǫ k + q = q 0 /2. About this shell, the gain is given approximately as  To capture this squeezing, we define a measurement mode through the following relation: with X chosen so that AD − BC = 1 at r = 0. For large σ, the volume defined by this function tends toward v = √ 2π 3 σ/k 2 max , so that the effective radius of the measurement volume is r ∼ (σ/k 2 max ) 1/3 . Now we determine the degree of spin squeezing captured by this measurement operator. Repeating calculations as above, with the requirement σ 2 > Dt, we have Again, we see that local measurements will capture the spin squeezing produced by the spinor Bose-Einstein condensate amplifier. Here, the correlation volume grows slower than diffusion, with radius scaling only as r ∝ t 1/6 . Numerical evaluations of S (−) are presented in figure 3, confirming the rapid convergence onto the single-mode maximal degree of squeezing for finite-volume, quadrature-tuned measurement modes.

Experimental considerations
We conclude with a discussion of experimental prospects for realizing and detecting multi-mode spin squeezing in a quenched ferromagnetic f = 1 condensate. We focus on the case of 87 Rb, which has been shown experimentally [25,26,27] and theoretically [28,29] to be ferromagnetic, with ∆a = −1.4(3) a B with a B being the Bohr radius. The behaviour of condensates prepared in the |m z = 0 state and quenched to the regime of dynamic instabilities by a rapid change of q have been studied using in situ magnetization-sensitive imaging. Pertinent to the present discussion, the amplification of spin fluctuations following quenches to different values of q was measured precisely in Ref. [12], and such amplification was found to be roughly consistent with quantumlimited amplification of quantum fluctuations in the initial state, with a gain, measured by the increase in the variance of spin fluctuations, as high as +30 dB.
In contrast to previous studies on the parametric amplification of spin fluctuations [12,24], here we have considered the squeezing that may accompany such amplification. We assume experimental conditions similar to those of Refs. [11,12], i.e. an optically trapped f = 1 87 Rb condensate of N = 2.0 × 10 6 atoms with peak density n = 2.6×10 14 cm −3 . At this density, the characteristic time scale for spin mixing in the spinor condensate is /q 0 = 8 ms and the characteristic length scale is ( 2 /(2mq 0 )) 1/2 = 1.8 µm. Let us estimate the maximum observable degree of spin squeezing, taking into account non-linearities, atom loss, and the spatial inhomogeneity of the condensate.

Single-mode nonlinearities and atom loss
Thus far, we have treated spin dynamics in a quenched f = 1 spinor condensate exclusively through the Bogoliubov approximation (equation 5), i.e. neglecting highorder terms in the spin fluctuation operators and also the depletion of the |m z = 0 condensate due to spin mixing. Within this linear, undepleted-pump approximation, we obtain the unphysical result, expressed in equation 10, that the minimum spinquadrature variance S (−) in each unstable spin mode tends exponentially to zero. As in quantum-optical systems, sensible limits to squeezing are recovered by going beyond the linearized equations of motion and the undepleted-pump approximation.
These limits are partly elucidated in the full dynamics of a single spin mode. We consider therefore the dynamics of a ferromagnetic spinor condensate in the single-mode approximation. Such a treatment would be appropriate for a spinor condensate spanning a volume v ( 2 /2mq 0 ) 3/2 , realized experimentally with an atom number of nv 1000. The full Hamiltonian is now given as where the Bose operators φ e annihilate particles in polar state |φ e and in the single spatial mode of the condensate.
In this expression, the single-polarization term ∝ φ † a 2 φ 2 z + φ † z 2 φ 2 a matches the two-axis counter-twisting Hamiltonian described in Ref. [2]. On its own, this numberconserving Hamiltonian produces squeezing that saturates with time due to depletion of the pump (the population φ † z φ z ), reaching the Heisenberg limit with S (−) ∝ 1/N [30]. However, the Hamiltonian contains additional terms: the externally imposed and pump-dependent quadratic Zeeman shifts and inter-polarization coupling. These nonlinear terms restrict spin squeezing to sub-Heisenberg scaling, with S (−) ∝ 1/ √ N . Nevertheless, the expected single-mode squeezing is significant. We have performed exact single-mode calculations for a 1000-atom sample, and find a reduction of spin fluctuations by as much as 20 dB.
The single-mode limit allows us to discuss an additional limitation arising from the decay of atoms from the trap. Atom loss can be modeled by adding a noise source and a density-dependent loss rate in the coherent evolution of the Bose fields [30]. We have performed numerical calculations including such terms, representing quantum noise in the evolution of a classical field as in the truncated Wigner approximation (TWA) [31]. In accord with experimental observations, we assume an initial per-atom loss rate of 1 s −1 . We find squeezing in a N = 1000 atom sample can still be the level of -20 dB, a result that is not surprising given the very short spin-mixing time as compared to the lifetime of trapped atoms.

Effects of inhomogeneity and multi-mode nonlinearities
Experimental realizations of spinor Bose-Einstein condensates also differ from our idealized treatment in that the condensates are inhomogeneous, and, thus, the adequacy of a linearized normal-mode description to long-time dynamical evolution is less clear than in the homogeneous case. To assess the influence of mixing between normal modes and also atom losses for this situation, we have performed numerical simulations of the evolution of a quasi-one-dimensional quenched spinor Bose-Einstein condensate. Such a condensate is assumed sufficiently tightly confined in two radial dimensions that the spinor wavefunction is radially uniform, while being more weakly confined in the third dimension, along which the wavefunction is allowed to vary. Specifically, we consider a condensate with Thomas-Fermi radii of 3 and 200 µm along the radial and axial dimensions, respectively. The central, radially averaged condensate density n 0 determines the spin mixing interaction strength to be 2|c 2 |n 0 = h × 15 Hz.
The quantum evolution of this spinor condensate is calculated using the TWA to simulate the effects of quantum noise and squeezing [24]. This approximation involves adding random noise to the initial mean-field condensate wavefunction, with the noise variance determined, in this case, by the one-dimensional condensate density. Temporal evolution is thereafter considered according to classical wave equations, which include the full non-linear dynamics given by spin-mixing Hamiltonian (the non-linear Schrödinger equation) and also the effects of atom loss (through additional noise and loss terms, described above). Expectation values, variances, and correlations for various measurement operators are quantified by repeating the calculations for many instances of the initial noise. These calculations were found convergent at a temporal resolution of 3.5 µs and spatial resolution of 0.5 µm.
We present results of a numerical simulation of an initially polar, trapped, 87 Rb f = 1 spinor condensate quenched at t = 0 to a quadratic Zeeman shift of q = h×2.5 Hz. The results for several numerical runs, i.e. for three random instances of initial noise as evaluated according to the TWA, are presented in figure 4a. Dynamical instabilities following the quench to low q result in amplified fluctuations in the transverse components of the magnetization and the related components of the nematicity; only one polarization of the observables, F x and N yz , is shown in the figure. The strong local correlation between these observables is clearly evident, reflecting the ϑ ≃ π/4 angle of the amplified quadrature axis for the maximally unstable normal modes.
Less evident in these position-space plots is the sharp reduction in the initial fluctuations of the normal-mode measurement observables. To reveal this reduction, we identify the normal dynamical modes based on the initial condensate distribution. The temporal gain determined for the first 41 unstable modes is shown in the inset of figure 5, sorted in decreasing order of temporal gain (|ω n |) and labeled by an integer mode index. Normalized spatial-mode measurement operators are then formed as defined in equations 11 and 12, and the values of these operators at different evolution times t are evaluated for an ensemble of 600 numerical runs. From these values, we evaluate the scaled covariance matrices C n numerically and determine therefrom the quadrature axes that are amplified and squeezed, and also the measurement variance along those axes.
The squares of the measurement outcomes along the amplified and squeezed quadrature axes for the three simulation runs discussed above are shown in figure 4b. Aside from pronounced variations according to the initial random values assigned to the quantum noise in these simulations, these data indicate a suppression of spin fluctuations between 20 and 30 dB.
The ensemble averaged reduction S (−) in the initial spin fluctuations, determined for each mode n and at different evolution times, is shown in figure 5. We find the temporal evolution of the dynamically unstable modes causes the measurement variance of the squeezed quadrature operators to diminish exponentially in time for evolution times up to around 60 ms, after which the variance saturates and then begins growing. The minimum variance of around 3.6 × 10 −3 or -24 dB is achieved for the unstable modes with highest gain (lowest mode index) at t = 65 ms. This maximum degree of squeezing matches well with the estimated effects of nonlinearities and atom loss presented above.

Conclusion
We have presented a formalism for the evaluation of spin squeezing in ensembles of high-spin atoms. This formalism identifies several independent planes of spin fluctuations atop an arbitrary coherent spin state, revealing a greater resource for applications of spin squeezing in quantum information and metrology than for ensembles of spin-1/2 particles. Moreover, we developed an understanding of spin squeezing in spatially extended quantum fields, where squeezing in many spatial modes can be separately generated and measured. This formalism is applicable to many contemporary experiments on spin squeezing of atomic gases or of other distributed quantum objects.
Applying this formalism, we consider a source of spin squeezing in an extended spinor Bose-Einstein condensate. Referring to recent experiments studying the evolution of 87 Rb condensates quenched to a regime of dynamical instabilities [11,12,13], we show that the strong amplification of spin fluctuations demonstrated in those experiments is accompanied also by spin squeezing, i.e. that the dynamically unstable normal modes act as quantum parametric amplifiers of spin fluctuations. Treating experimental realities such as atom losses and effects of confinement, we apply analytic and numerical tools to determine that measurable spin squeezing better than 20 dB may be achieved in such systems.
This treatment is relevant also to the question of symmetry breaking in a quantum many-body system. An initially polar-state spinor Bose-Einstein condensate preserves the SO(2) rotational symmetry about the alignment axis. The quantum quench considered in this work, a rapid reduction in the quadratic Zeeman shift to induce dynamical instabilities and parametric amplification of spin fluctuations, traverses the atomic system across a symmetry-breaking quantum phase transition, where the spin-dependent contact interactions favor broken symmetry states of maximum magnetization [11]. In a closed noiseless quantum system, bona fide symmetry breaking does not occur; rather, the system evolves coherently to a symmetric superposition of broken-symmetry states. The spin-squeezed states described in this work indeed represent such a coherent superposition.
The metrological significance of spin squeezed quantum fields, and also the question of symmetry breaking in quantum systems, provide strong motivation for the detection and characterization of squeezing in quenched spinor Bose gases. For this purpose, spatially resolved measurements of the magnetization of optically trapped condensates have been demonstrated [14], and optical means of measuring components of both the magnetization and the nematicity have been suggested [19]. However, sensitivity below the atomic shot noise level, required for the direct detection of squeezing, is experimentally challenging and was not achieved in those measurements.
Alternately, we propose that one could use the condensate itself as a premeasurement amplifier. Following the generation of inhomogeneous spin squeezing with the quadratic shift at q 1 , one may quickly increase the quadratic Zeeman shift. The previously unstable dynamical modes now evolve stably as the squeezed and amplified spin-fluctuation quadratures rotate in their quadrature spaces. After their rotation, the spin fluctuations may be again parametrically amplified by returning the quadratic shift to q 1 . Spin squeezing produced by the first stage of amplification would result in a reduction of spin fluctuations observed after the second amplification stage, as compared to a single-stage amplification of vacuum spin noise. We have performed TWA-based numerical simulations which confirm that this technique is applicable to trapped spinor Bose-Einstein condensates at experimentally accessible settings.