Entanglement-enhanced radio-frequency field detection and waveform sensing
F. Martin Ciurana G. Colangelo L. Slodicka R. J. Sewell M. W. Mitchell

































We demonstrate a new technique for detecting components of arbitrarily-shaped radio-frequency waveforms based on stroboscopic back-action evading measurements. 
We combine quantum non-demolition measurements and stroboscopic probing to detect waveform components with magnetic sensitivity beyond the standard quantum limit.
Using an ensemble of 1.5× 10^6 cold rubidium atoms, we demonstrate entanglement-enhanced sensing of sinusoidal and linearly chirped waveforms, with 1.0 (2)dB and 0.8 (3)dB metrologically relevant noise reduction, respectively. We achieve volume-adjusted sensitivity of δB√(V)≈ 11.20 fT√(cm^3/Hz), comparable to the best  magnetometers.



Entanglement-enhanced radio-frequency field detection and waveform sensing 


    M. W. Mitchell
    December 30, 2023
==============================================================================



Quantum noise and quantum coherence both play essential roles in determining the fundamental sensitivity of interferometric instruments such as atomic magnetometers and atomic clocks. 
This is clearly seen in a widely-used rule of thumb for the projection noise limit δ B_ PN of sensing magnetic fields with  spin-f atoms <cit.>: 

    δ B_ PN√()≃ħ/g μ_B1/√(2 f ).

Here  is the total acquisition time including averaging repeated measurements, μ_ B is the Bohr magneton, g is the ground-state Landé factor, and ħ is Planck's constant. 
The factor 1/√(2 f ) reflects the standard quantum limit (SQL) spin projection noise of the atomic precession angle, which scales as 1/√().
The signal accumulation time  is determined by the smaller of the single-measurement duration and spin atomic coherence time. Analogous expressions govern clocks and other atomic instruments. 

Reduction of spin projection noise below the SQL <cit.>, which implies entanglement among atoms <cit.> and/or atomic components <cit.>, has been demonstrated by quantum non-demolition measurement <cit.> including large degrees of squeezing using cavity enhancement <cit.>. 
Use of conditional spin-squeezed states has been demonstrated in magnetometry <cit.> and clock operation <cit.>. 
These works employ a measure-evolve-measure (MEM) sequence, in which a first quantum non-demolition (QND) measurement produces a state with reduced projection noise, a period of free evolution accumulates signal, and a second QND measurement detects the change relative to the first measurement. 
This method exploits the coherence of the atomic system, allowing signal to accumulate prior to readout of the atomic state, which typically destroys coherence. 

Here we demonstrate a generalization of this method, to allow quantification of arbitrarily-shaped signal components, using the same resources of atomic spin squeezing and QND measurement. 
This broadens the scope of entanglement-enhanced sensing to include arbitrary time-varying signals, of which magnetic waveforms due to firings of single neurons <cit.> and event-related fields in magnetoencephalography <cit.> are notable examples. 
While rapid sampling of the same signals can also detect waveforms, it reduces τ and thus the sensitivity of the measurement. 
Our method includes radio-frequency () magnetometry as a special case, and we demonstrate entanglement-enhanced detection of radio fields in a compact, high-sensitivity  magnetometer, with a sensitivity-volume figure of merit comparable to the best demonstrated instruments.

Principle of the method 
We consider an ensemble of atoms, described by a polarization F, precessing in response to a magnetic field B(t) = y B_y(t) + x B_x(t), with |B_y| ≫ |B_x|.
F precesses about y at an experimenter-controlled Larmor angular frequency (t) = γ B_y(t) + O(γ B_x^2/B_y) ≈γ B_y(t), driven transversally by the small unknown perturbation B_x(t). 
The component F_z is assumed accessible to QND measurement. The dynamics of the system are given by

    d/dt(t) = γ(t) ×(t)

where γ is the gyromagnetic ratio of the atomic state. 
As shown in the , for an initial atomic polarization oriented along +y the evolution of the measurable spin component F_z(t) can be expressed as

    (t)     =    (0) cosΘ (t) +(0) sinΘ (t) 
    
           
    + γ⟨ F_y(0) ⟩∫_0^t dt'  (t') cos[ Θ(t) - Θ(t') ] 
    
           
    + O()^2 + O(γ t δ F_y )

where Θ(t) ≡∫_0^t dt'  (t') is the accumulated angle.
We note that the first line describes an operator relation, namely a rotation of the spin components , about y, and contains the quantum noise associated with the spin variables. 
As with other squeezing-enhanced atomic measurements, the noise in F_z(t) can be reduced by squeezing a linear combination of F_z(0) and F_x(0). In particular, QND measurement of F_z(0) can squeeze this component, while subsequent measurements at times {t_i}, chosen so that Θ(t_i)=nπ, n ∈ℤ, excludes noise of F_x from entering the measurement record, achieving back-action evasion <cit.>. 

In contrast, the second line of Eq. (<ref>) has no operator content, and describes a noiseless displacement by an amount proportional to the integral of the waveform B_x(t') multiplied by the pattern function cos[ Θ(t) - Θ(t') ]. 
This describes a coherent build-up of the signal component matching the pattern function, and a cancellation of signal components orthogonal to it <cit.>. 
By proper choice of Θ(t), the pattern function can be made to take on any functional form bounded by ± 1. 
The third line of Eq. (<ref>) is negligible for the interesting case of weak signals and large atom number.


Experimental technique
Our experimental apparatus, illustrated in Fig. <ref> (a), is described in detail in <cit.>. 
Briefly, we trap up to 1.5×10^6 ^87Rb atoms in a weakly focused single beam optical dipole trap. 
The atoms are laser cooled to a temperature of 16 and optically pumped into the f=1 hyperfine ground state. 
A bias magnetic field along y is generated with coils in a near-Helmholtz geometry fed by a programmable current source, and monitored using the atoms as an in-situ DC vector magnetometer <cit.>. 
An  field along x is produced with a low-inductance coil and an arbitrary waveform generator. 
Optical pumping (OP) along the y direction is used to produce an initial atomic polarization, with an efficiency of ∼ 96%, as measured by Faraday rotation <cit.>.

A non-destructive measurement of the atomic state is made using a train of 600 duration pulses of linearly polarized light 700MHz red detuned from the f=1→ f'=0 transition on the D_2 line. 
The interaction between the atoms and the probe pulses is given by the effective Hamiltonian τ_pulseĤ_eff=g_1 where the g_1 is a coupling constant depending on the probe beam geometry and detuning and τ_pulse is the pulse duration, operators F_i describe the atomic spins, and S_i the optical polarization <cit.>.
Light pulses propagating along the trap axis experience a polarization rotation = cosϕ + sinϕ, where S_i^( in/out) are Stokes operators before/after passing the atoms <cit.> and ϕ = g_1 F_z is the Poincaré-sphere rotation angle.
 is detected with a shot-noise-limited balanced polarimeter and = /2 is measured by splitting a constant fraction of the input light to a reference detector before the atoms. 
g_1 is calibrated by independent measurement <cit.>.

 



Calibration of response 
The coherent response of the atoms is illustrated in Fig. <ref>. 
With a constant B_y we apply   excitation of the form (t)=a_cos(ω_0 t) and observe Rabi oscillation, i.e., sinusoidal oscillation of , amplitude modulated at the Rabi frequency as predicted by Eq. (<ref>).
We observe good experiment-theory agreement, and use the data as a calibration of  and . 

Waveform detection
Selective response to chirped waveforms is shown in Fig. <ref>. 
Using a ramped field (t), we produce a Larmor frequency that sweeps linearly from = 2 π×42.2 to = 2 π×47.5 over 800. 
This produces a “chirped” pattern function Θ(t), making the MEM sequence sensitive to (t) signals with similar chirp, but insensitive to other waveforms, e.g. at constant frequency or with the opposite chirp. 
To confirm this selectivity, we use an arbitrary waveform generator to apply transverse fields of the form (t)=a_cos(ω_0 t+ κ t^2), i.e., a linearly chirped waveform, in the time between  and . 

Fig. <ref> shows the resulting signal, i.e., the amplitude of the observed  oscillation, as a function of the chirp κ. 
As expected, we observe a peak in the population transferred by the  drive when κ matches the field ramp. 
Agreement with theory from Eq. (<ref>) is good, and variation in experimental signal is consistent with the independently-measured fluctuations of the magnetic field at the position of the atoms.

 

Stroboscopic QND measurement
To perform one QND measurement,  probe pulses are sent through the atoms at intervals of one-half of the Larmor period, and experience a polarization rotation ϕ_n = g_1 (t_n), where n indexes the pulses.
Because of the inversion of F_z between pulses, the ϕ_n can be aggregated as a single distributed measurement of F_z, quantified by
Φ≡^-1∑_n=1^(-1)^n-1ϕ_n.
This multi-pulse probing has been shown to be a true QND measurement of the collective spin <cit.> with an uncertainty below the standard quantum limit for F_z <cit.>. 

Back-action evading  sensing
With these elements, we demonstrate back-action evasion in a MEM sequence to detect  magnetic fields. 
We load the ODT with = 1.5e6 atoms, measure the bias field as in <cit.> and then repeat the following MEM sequence 16 times: 
dispersive measurement of  as in <cit.>; optical pumping to produce full polarization along +y; QND measurement  with result Φ_1; free evolution for time =300; and a second QND measurement  with result Φ_2, as illustrated in Fig. <ref> (b). 
During the hold time  is held constant, i.e. κ = 0, with =2 π×50.16. The QND measurements  and  are made over 200 and contained 4e8photons, see <cit.> for details.
The 16 repetitions of the MEM sequence allow us to vary  since atoms are lost from the trap during optical pumping.
We repeat the full sequence 463 times to collect statistics.



Projection noise level
Fig. <ref> shows the measured variance of Φ_2 and the conditional variance (Φ_2|Φ_1) as a function of the number of atoms in the trap. 
A linear measurement will show a variance that is quadratic in  <cit.>:

    var (Φ) =var_0(Φ) + g̃^2_1 1/2α + a_2 ^2

where var_0(Φ) is the readout noise, quantified by repeating the measurement without atoms in the trap, a_2 ^2 is atomic technical noise associated with, e.g., fluctuations in state preparation, and the term ∝ corresponds to atomic projection noise. 
The factor 1/2 describes the F_z variance of an f=1 atom polarized along y, and α accounts for the net noise reduction due to off-resonant scattering of probe photons. 
This scattering both reduces the number of noise-contributing atoms by pumping some into the far-off-resonance f=2 ground state, and adds noise as some atoms return to the f=1 state with randomized polarizations. 
We compute the evolution of the state using the covariance matrix methods reported in <cit.> to find α = 0.96.
The coupling g̃^2_1 is different from g_1 due to inhomogeneous atom-field coupling <cit.>.
A fit of Eq. (<ref>) to the data finds g̃^2_1=1.2(2)× 10^-14 rad^2, determining the projection-noise level. 

Squeezing
To study the generation of squeezing we look at the correlation between  and . 
The first train of QND-pulses redistributes the noise to the non-measured component and  is used to evaluate its variance conditioned on the first measurement. 
The measurement noise reduction is quantified by var(Φ_2|Φ_1)=var(Φ_2-χΦ_1), where χ=cov(Φ_1, Φ_2)/var(Φ_1)>0 describes the correlation between Φ_1 and Φ_2. 
As seen in Fig. <ref>, the conditional variance is [parse-numbers=false]2.4 (2)dB below the projection noise level.




Metrological improvement is quantified by the Wineland criterion <cit.> which takes into account the coherence loss and the noise reduction of the measured state: 

    ξ_m^2=1/η^2var(Φ_2|Φ_1)/varΦ_2

where ξ_m^2<1 indicates metrological advantage and η accounts for the loss of coherence of the spin-squeezed state relative to the input coherent spin state.
The coherence after the first measurement is =η, where η≡ (1-η_sc)(1-η_field) and η_sc=0.11 and η_field=0.04 are independently measured coherence loss due to probe scattering and field inhomogeneities, respectively, during .
We find the metrological improvement due to squeezing ξ_m^2=0.79(5), or [parse-numbers=false]1.0 (2)dB.

Entanglement-assisted waveform detection 
To detect chirped waveform components beyond the projection-noise level, we repeat the above measurement and analysis using a chirped waveform with κ = 2π×5.78e6, i.e., we ramp (t) to produce a chirped pattern function during =600. The probe frequencies are matched to the Larmor precession frequency during  and  with 2.9e8photons, details in <cit.>. 
For this experiment, = 1.3e6.
We repeat the MEM sequence 21 times per trap loading to vary , and the experiment 82 times to gather statistics.
Analyzed as above, we find 1.5(3)dB of noise reduction with 0.8(3)dB of metrological advantage. 
The reduced squeezing is due to the smaller , smaller total photon number, and technical noise accumulated during the longer .

Magnetic sensitivity 
The sensitivity to the  drive amplitude a_ is δ B_ = Δϕ/(| d⟨ϕ⟩ /da_|) with ϕ being the observed Faraday rotation signals.
The signal accumulated during the  can be obtained by integration of Eq. (<ref>), resulting in an enhanced single-shot sensitivity  <cit.>:

    δ B_√()   ≃Δϕ_cond/g_1η_hold⟨(0) ⟩2/γ √()

where ⟨(0) ⟩=, η_hold is the total coherence loss during  and , and Δϕ_cond^2=var(Φ_2|Φ_1) is the conditional variance between the Φ_2 and Φ_1 optical signals, including the atomic noise and read-out noise.
Applied for the two  waveforms used, i.e., constant frequency and linearly chirped, we find sensitivities of 2.96pT/√(Hz) and 3.36pT/√(Hz), respectively. 
Focusing on the first, scaling the sensitivity by the volume of the atomic cloud, V =1.43× 10^-5cm^3, we find δB√(V)≈ 11.20 fT√(cm^3/Hz). 
For comparison, the best alkali-vapor  magnetometer <cit.> in this frequency range showed a sensitivity of 0.24fT/√(Hz) with V=96 cm^3 or δB√(V)= 2.35 fT√(cm^3/Hz). 
Thus the  magnetometer demonstrated here has a volume-adjusted sensitivity comparable with the best existing instruments.

We have experimentally demonstrated detection of radio-frequency fields and arbitrarily-shaped radio-frequency waveform components beyond the projection noise limit, using stroboscopic back-action-evading measurements on magnetic atomic ensembles. The combination of QND measurements and stroboscopic probing in a measure-evolve-measure sequence gives this quantum sensing advantage, while also allowing full use of the system coherence, resulting in a sensitivity-volume figure of merit comparable to the best  magnetometers at these frequencies.



§ ACKNOWLEDGEMENTS

We thank C. Abellán and W. Amaya for lending us the  generator and S. Coop for useful feedback on the manuscript.
Work supported by MINECO/FEDER, MINECO projects MAQRO (Ref. FIS2015-68039-P), XPLICA (FIS2014-62181-EXP) and Severo Ochoa grant SEV-2015-0522, Catalan 2014-SGR-1295, by the European Union Project QUIC (grant agreement 641122), European Research Council project AQUMET (grant agreement 280169) and by Fundació Privada CELLEX.













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Supplementary Information










 §.§ Operator definition

We define the collective spin operation ≡∑_i, where  is the spin of the i'th atom. The collective spin obeys commutation relations [,]=i, here and throughout we set ħ=1. The probe pulses are described by the Stokes operator  defined as ≡1/2(,)(,)^T,where the  are the Pauli matrices and  are the annihilation operators for the σ_± polarizations, which obey [,]=i and cyclic permutations. The input pulses are fully  polarized, i.e., with = /2, = = 0 and Δ^2 S_i=/4, i∈{x,y,z} where  is the number of photons in the pulse. 



 §.§ State preparation

The atoms are polarized so that ⟨ F⟩ is oriented along +y via optical pumping under a parallel bias field .
We use a single 50 long pulse of circularly polarized light resonant with the f=1→ f'=1 transition of the D_2 line and propagating along the y-axis, and illuminate the atoms with repumper light resonant with the f=2→ f=2' transition to prevent accumulation of atoms in the f=2 hyperfine level. 
To measure the atomic polarization, the atoms are subsequently rotated into  by slowly rotating the bias field from  to , and then measured with the Faraday probe. 
The amplitude of the transferred atoms is compared to the signal from an ensemble directly polarized by on axis optical pumping in a  field <cit.>, resulting in a input polarized atomic ensemble with ≃, Fig. <ref> (a). 





 §.§ Magnetic field measurement

We make use of the atoms as an in-situ DC vector magnetometer to measure the applied magnetic field as described in <cit.>. We prepare an ⟨ F⟩ along z under an orthogonal bias field  via optical pumping with circular polarized light propagating along the trap axis, and observe the free induction decay signal (FID) of the resulting Larmor precession using the Faraday probe. 
In order to observe the pure magnetic dephasing of the spins we probe semi-continuously with a reduced <10^6 and large detuning, Δ=1.5, to minimize probe scattering. 
We fit the FID traces with the function ϕ(t)=β+αcos( t+ϕ_0)e^-t/T_2 and extract the Larmor angular frequency  and the spin coherence time T_2. 
We minimize the amplitude of β by compensating homogeneous field along x and z with coils in a near-Helmholtz geometry and cancel field gradients along the length of the trap by running current in opposite direction through pairs of wires transversal to the trap. The field is optimized in an iterative routine resulting in typical =[parse-numbers=false]4.8(3) and fields 99.1% along the y-axis. A typical optimized FID signal is shown in Fig. <ref> (b). 




 §.§ Rabi Flopping Calibration

Adding an oscillatory (sinusoidal) driving field perpendicular to a static bias field induces Rabi flops between the magnetic sub-levels of the atoms in the hyperfine levels. 
When the frequency of the RF-signal matches the atomic energy splitting the flopping frequency is minimal and the population transferred by the RF maximal. 
As for the magnetic field measurement, in order to to minimize probe scattering the atoms are probed semi-continuously with a reduced  and Δ=1.5. 
The traces are fitted with the function 

    (t)=a cos( t+ϕ_L)cos(ω_R t+ϕ_R)
 
where  is the Larmor angular frequency and ω_R is the Rabi flopping frequency. 
Fitting the maximal axial polarization ℱ_z/|| as a function of the frequency of the RF excitation with a Lorenztian <cit.> we determine the resonance frequency to be [parse-numbers=false]42.26 (1) and estimate the strength of the RF-field coupled to the atoms to be a=[parse-numbers=false]0.60(5)G. 



 §.§ Ramped bias field

To characterize the ramp of the bias field, we fit the FID signal with a chirped function of the form ϕ(t)=β+αcos( t+κ t^2 +ϕ_0)e^-t/T_2. 
We find κ=2π×[parse-numbers=false]5.6 (1)×10^6 and =[parse-numbers=false]4.5 (4). To confirm the linearity of the chirp we divide the FID-signal in 100 long segments and fit them individually with a function ϕ̃(t)=β'+α' cos(ω_L^(i) t+ϕ_0'), where ω_L^(i) is the Larmor frequency of the i-th segment. 
A quadratic fit to the fit outputs for ω_L^(i) confirms the linearity of the ramp as the quadratic term in negligible, a_2 t/a_1=5.8× 10^-8, where t=800 is the time during the ramp is on, see Fig. <ref>.





 §.§ Stroboscopic Probing

We probe the atoms via off-resonant paramagnetic Faraday-rotation using τ_pulse=600 duration pulses of linearly polarized light with a detuning of 700MHz to the red of the D_2 line, and sent sent every half Larmor period, =/π.
For the experiment with constant bias field during the hold time, =2 π×50.16 and the QND measurements  and  are made at frequencies ==101.01 over a time 200 giving a total of = =20 probe pulses, each with =2×10^7 photons. 
For the detection chirped waveform components the experiment was done ramping (t) to produce a chirped pattern function with κ = 2π×5.78e6 during =600.  The probe frequencies = 84.75  =93.46 matched the Larmor precession frequency during  and  and contained = 16, =18, respectively, with =1.81×10^7 photons per pulse.
During a probe pulse the atomic spins rotate by an angle θ=γ B τ_pulse. For our parameters, τ_pulse and B=||≲70 G we find Θ∼ 30, so we can neglect the rotation of the spins during the probe pulse. 



 §.§ Volume estimation


We use absorption imaging to estimate the volume of the atomic cloud. Atoms originally in the f=1 hyperfine ground state are transferred into the f=2 hyperfine ground state by a 100 pulse of laser light tuned to the f=1→ f'=2 transition. A 100 pulse of circularly polarized light resonant to the f=2→ f'=3 transition cast a shadow on a CCD camera. To avoid spatially dependent light shifts the dipole trap is switched off during the process.
To estimate the radial extension of the atom we use the time-of-flight (TOF) technique <cit.>. Free, thermal and isotropic expansion of the atoms is described by


    ω_a(t)^2=ω_a^2(0)+k_B T/m(t-t_0)^2

where k_B is Boltzmann constant and m the atomic mass.
ω_a(0) is the initial size of the ensemble, T its temperature and t_0 is a time delay for switching off the dipole trap, all 3 free parameters of the fit. 

We determine the width of the radial profile by integrating each frame of the image along the longitudinal direction (the z-axis) and fit the resulting density profile with a Gaussian to get the center and the width of the atomic ensemble. 
The width of the atomic cloud ω_a is plotted on Fig. <ref> (a) as a function of time. 
We fit with Eq. (<ref>) to find ω_a(0)=14.2 (4), from which we determine the radial full width at half maximum (FWHM) of the atomic cloud ρ_radial=2√(2 ln2) ω_a(0)=33 (1).
The temperature of the atomic sample is 15.5 (1)

The axial dimension of the trap is much longer than the spread of the ensemble during the TOF, resulting in ρ_axial(t)≃ρ_axial(0).
The axial shape of the atomic ensemble is calculated by integrating the images along the transverse direction, Fig. <ref> (b). 
The black solid line is a fit with a Lorentzian of the form L(z)=aρ_axial/2((z-z_0)^2+(1/2ρ_axial)^2)+b, where ρ_axial is the FWHM. 
We determine the axial FWHM atomic length ρ_axial=3.28 (6).

We approximate the atomic volume to an ellipsoid with semi-axis given by ρ_axial and ρ_radial and calculate its volume using V=4π/3ρ_radial^2ρ_axial=1.43e-5. 







 §.§ Waveform component detection


The dynamics of the spins are governed by the Heisenberg equations of motion (with ħ = 1):

    d/dtF_i = -i [F_i,H_0(t)+ H'(t)]

where H_0 = - γ F· B = - γ F_y B_y(t), γ is the gyromagnetic ratio for the F=1 ground hyperfine state, and the perturbation H' = - γ F_x B_x(t) describes the RF drive. We use a Dyson series to solve the resulting system of differential equations. We define F^(0)(t) to be the solution to Eq. (<ref>) when H' = 0, i.e. F_y(t) = F_y(0) and 

    ( 
    [ (t); (t) ])    =   ( 
    [  cosΘ(t)  sinΘ(t); -sinΘ(t)  cosΘ(t) ]) ( 
    [ (0); (0) ])

where Θ(t)≡∫_0^t dt'  ω_L(t'), (t) ≡γ B_y(t) is the accumulated angle.
We then use the well-known result <cit.>

    F_i(t) = F_i^(0)(t) - i ∫_0^t dt'   [F_i^(0)(t), H'(t')] + O(H')^2

which, in light of the definition of H'(t'), gives the signal

    (t)     =    cosΘ (t) (0) + sinΘ (t) (0) 
    
           
    + γ∫_0^t dt'   F_y(t') B_x(t') cos[ Θ(t) - Θ(t') ] 
    
           + O(B_x)^2

We note that F_y(t') = F_y(0) + O(B_x)^2 and writing δ F_y^2 = ⟨ F_y^2 ⟩ - ⟨ F_y ⟩^2, we find 

    (t)     =    cosΘ (t) (0) + sinΘ (t) (0) 
    
           
    + γ⟨ F_y(0) ⟩∫_0^t dt'   F_y(t') B_x(t') cos[ Θ(t) - Θ(t') ] 
    
           + O(B_x)^2 + O(γ t B_x δ F_y)^2.

The term O(B_x)^2 describes contributions of higher order in H', and can be neglected in the interesting scenario of weak signals. For our initial state of all atoms pumped into the y direction, δ F_y ≪ F_y, making the last term negligible also.

Dropping the higher other terms and choosing the probing times such that Θ(t_i)=nπ simplifies Eq. (<ref>) further, to 

    (t_n)=(-1)^n ((0) + γ⟨ F_y(0) ⟩∫_0^t_n dt'  (t') cosΘ(t'))




 §.§ Field sensitivity

The accumulated signal during the hold time  is obtained by integration of Eq. (<ref>), where for the case of ramped field, B_x(t') = a_cos(ω_0 t'+κ t'^2), and Θ(t) = ω_L t + κ_L t^2, where κ_L =(ω_f-ω_i)/Δ t describes the change of the Larmor frequency caused by the ramp, applied for a time Δ t. The radio-frequency case is included has a special case when κ=κ_L=0.

The solution to Eq. (<ref>) becomes

    F_z()    = (0) - γ⟨(0) ⟩×
       ∫_0^ dt'   a_cos(ω_0 t'+κ t'^2) cos(ω_L t' + κ_L t'^2)
    
       = (0)+ γ a_⟨(0) ⟩ I(ω_L,ω_0,κ_L,κ, )

Evaluating the integral on resonance, i.e. with ω_0 = ω_L, and κ_L=κ. In cases of interest κ≪ω_0, giving the leading-order in I(ω_L,ω_0,κ_L,κ, )≃/2, with an error smaller than 3% for our parameters.

The sensitivity of a coherent spin state to the  drive amplitude a_ during a time  is

    δ B_^CSS   = Δϕ/| d⟨ϕ⟩ /da_|
       = Δϕ/g_1⟨(0) ⟩1/γ I(ω_L,ω_0,κ_L,κ,)
       ≃Δϕ/g_1⟨(0) ⟩2/γ

where ⟨(0) ⟩=.  For the spin squeezed state 

    δ B_^SSS   ≃Δϕ_cond/g_1η⟨(0) ⟩2/γ

where η is the coherence loss including  and the first measurement and Δϕ_cond^2=var(Φ_2|Φ_1) is the conditional variance between the Φ_2 and Φ_1 optical signals, including the atomic noise and read-out noise.



 §.§ Statistics of probing inhomogeneously-coupled atoms

We consider the statistics of the Faraday rotation measurements of an ensemble of  atoms, described by individual spin operators f_i. To define the SQL, we consider the atomic ensemble to be in a coherent spin state, with independent individual spins and fully polarized with ≃. When the spatial structure of the probe beam is taken into account, the Faraday rotation is described by the input-output relation for the Stokes component 

    ^(out)   =   ^(in) + ^(in)∑_i=1^ g( x_i) ^(i)

where g( x_i) is the coupling strength for the i-th atom, proportional to the intensity at the location x_i of the atom. ^(in) has zero mean and variance /2, where  si the mean photon number in the probe pulse . 
The atomic spin is polarized along  and orthogonal to the measured  direction. The rotation angle ϕ = ^(out)/^(in) has statistics

    ⟨ϕ⟩   =   ⟨⟩∑_i=1^< g( x_i) >_ x_i
       ≡   ⟨⟩⟨⟩ g_1 
    var(ϕ)    =   var(ϕ_0) + var() < ∑_i=1^ g^2( x_i) >_, x_i
       ≡   var(ϕ_0) + var() ⟨⟩g̃_1^2

where ϕ_0 is the polarization angle of the input light, subject to shot-noise fluctuations and assumed independent of , and the angle brackets indicate an average over the number and positions of the atoms. For known ⟨⟩ and var(), measurements of ⟨ϕ⟩ and var(ϕ) versus  give the calibration factors g_1 and g̃_1^2. 



  §.§.§ Calibration of the g_1 factor

We calibrate ⟨ϕ⟩, the measured rotation angle of the dispersive atom number measurement (DAMN) <cit.> against an independent estimate of  made via absorption imaging.
The results are shown in Fig. <ref> (a). 
For this experiment, the atoms are prepared in an -state with a single 50 duration circularly polarized optical pumping pulse on resonance with the f=1→ f'=1 transition of the D_2 line and propagating along the trap axis with an bias field =180mG applied to fix the atomic polarization, and then probed with the Faraday probe. 
The number of atoms is estimated using absorption imaging in the same way as explained in Volume estimation.  
We calculate the resonant interaction cross-section and take into account the finite observable optical depth. The statistical error in the absorption imaging is <3%, including imaging noise and shot-to-shot trap loading variation.



  §.§.§ Calibration of the g̃_1^2 factor




To measure g̃^2_1 we use the the stroboscopic QND data. The atoms are prepared into an -polarized state and then probed stroboscopically at twice the Larmor period, in such a way that the measured variable is ±, evading back-action effects. If ϕ_n is the measured Faraday rotation angle for pulse n, and ϕ_0^(n) is the corresponding input angle, we can define the pulse-train-averaged rotation signal as 

    ϕ   ≡   1/∑_n=1^ (-1)^n+1ϕ_n

with variance

    var(ϕ)    ≡   var( ϕ_0 ) + g̃_1^2 ∑_n=1^var(F_z,n)

where ϕ_0 = 1∑_n=1^ϕ_0^(n), with zero mean and variance var(ϕ_0) = ()^-1, and F_z,n is the value of  at the time of the n-th probe pulse. 

We compute the resulting evolution of the state using the covariance matrix methods described in <cit.> and presented in the following section resulting in 

    var( g_1/∑_n=1^F_z,n)     =    g̃_1^2 1/2α

corresponding to the linear term in  used in Eq. (4) of the main text to determine the atomic quantum noise, i.e.,

    var (Φ) =var(Φ_0) + g̃_1^2 1/2α + a_2 ^2,

where the correction factor α=0.96 accounts for decoherence and noise introduced into the atomic state due to off-resonant probe scattering during the QND measurement <cit.>.











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