Ultrasensitive Atomic Spin Measurements with a Nonlinear Interferometer

We study nonlinear interferometry applied to a measurement of atomic spin and demonstrate a sensitivity that cannot be achieved by any linear-optical measurement with the same experimental resources. We use alignment-to-orientation conversion, a nonlinear-optical technique from optical magnetometry, to perform a nondestructive measurement of the spin alignment of a cold $^{87}$Rb atomic ensemble. We observe state-of-the-art spin sensitivity in a single-pass measurement, in good agreement with covariance-matrix theory. Taking the degree of measurement-induced spin squeezing as a figure of merit, we find that the nonlinear technique's experimental performance surpasses the theoretical performance of any linear-optical measurement on the same system, including optimization of probe strength and tuning. The results confirm the central prediction of nonlinear metrology, that superior scaling can lead to superior absolute sensitivity.

A nonlinear interferometer experiences phase shifts φ that depend on N , the particle number, e.g. φ = κN Y for a Kerr-type nonlinearity Y, where κ is a coupling constant. This number-dependent phase implies a sensitivity ∆Y ≥ (κN ) −1 ∆φ, and if the nonlinear mechanism does not add noise beyond the ∆φ = N −1/2 shot noise, the sensitivity ∆Y ∝ N −3/2 even without quantum enhancement. Such a nonlinear system was identified in theory by Boixo et al. [25] and implemented with good agreement by Napolitano et al. [8,26]. In contrast, entanglementenhanced linear measurement achieves at best the socalled "Heisenberg limit" ∆φ = N −1 . The faster scaling of the nonlinear measurement suggests a decisive technological advantage for sufficiently large N [25,[27][28][29][30][31][32][33][34][35][36]. On the other hand, no experiment has yet employed improved scaling to give superior absolute sensitivity, and several theoretical works [37][38][39][40] cast doubt upon this possibility for practical and/or fundamental reasons.
Here, we demonstrate that a quantum-noise-limited nonlinear measurement can indeed achieve a sensitivity unreachable by any linear measurement with the same experimental resources. We use nonlinear Faraday rotation by alignment-to-orientation conversion (AOC) [2], a practical magnetometry technique [2], to make a nonde- * robert.sewell@icfo.es structive measurement of the spin alignment of a sample of 87 Rb atoms [11,41]. AOC measurement employs an optically-nonlinear polarization interferometer, in which the rotation signal is linear in an atomic variable but nonlinear in the number of photons. We have recently used AOC to generate spin squeezing by quantum nondemolition measurement [42], resulting in the first spin-squeezing-enhanced magnetometer [11]. Here we show that this state-of-the-art sensitivity results from the nonlinear nature of the measurement, and could not be achieved with a linear measurement. We demonstrate a scaling ∆J y ∝ N −3/2 L , where N L is the photon number and J y is an atomic spin-alignment component, in good agreement with theory describing the interaction of collective spin operators and optical Stokes operators. Relative to earlier nonlinear strategies [8], AOC allows increasing N L by an order of magnitude, giving 20 dB more signal and 10 dB less photon shot noise. The resulting spin sensitivity surpasses by 9 dB the best-possible sensitivity of a linear J y measurement with the same resources (photon number and allowed damage to the state). Theory shows that this advantage holds over all metrologically relevant conditions.

II. NONLINEAR SPIN MEASUREMENTS
We work with an ensemble of N A ∼ 10 6 laser-cooled 87 Rb atoms held in an optical dipole trap, as illustrated in Fig. 1(a) and described in detail in Ref. [56]. The atoms are prepared in the f = 1 hyperfine ground state, and interact dispersively with light pulses of duration τ via an effective Hamiltonian [57] where the coupling coefficients κ 1,2 are proportional to the vectorial and tensorial polarizability, respectively, and γ F is the ground-state gyromagnetic ratio. Here, the operatorsĴ i describe the collective atomic spin, and the optical polarization is described by the pulse-integrated Stokes operators dtS i (t) ≡Ŝ i (see Appendix A).Ĵ x andĴ y represent the collective spin alignment, i.e., Raman coherences between states with ∆m f = 2, andĴ z describes the collective spin orientation along the quantization axis, set by the direction of propagation of the probe pulses.Ŝ x andŜ y describe linear polarizations, whileŜ z is the degree of circular polarization, i.e., the ellipticity.
In regular Faraday rotation, the collective spin orien-tationĴ z is detected indirectly by measuring the polarization rotation of an input optical pulse due to the first term in Eq. (1). Typically, the input optical pulse isŜ xpolarized, and the polarization rotation is detected in thê S y basis. Detection of the collective spin alignment J y (or J x ) requires making use of the second term in Eq. (1), and can be achieved with either a linear or a nonlinear measurement strategy, as we now describe (see Figure 1).
In a linear measurement the polarization rotation due to the term κ 2ŜyĴy is directly measured -e.g., an input S x -polarized probe (i.e. Ŝ x = N L /2) is rotated toward S z by a small angle Φ LTE = κ 2Ĵy . We refer to this type of strategy as linear-to-elliptical (LTE) measurement of J y . It gives quantum-limited sensitivity i.e., with shot-noise scaling. Here and in the following we use the notation J y ≡ Ĵ y for expectation values. The same sensitivity is achieved with other linear measurement strategies employing different input polarizations. Note that for large detunings, κ 1 ∝ ∆ −1 κ 2 ∝ ∆ −2 , so detection of J y with this method is less sensitive than regular Faraday-rotation detection ofĴ z .
AOC measurement of J y employsĤ eff twice and gives a signal nonlinear in N L : The term κ 2ŜxĴx produces a rotation ofĴ y towardĴ z by an angle θ AOC = κ 2 S x /2. Simultaneously, the term κ 1ŜzĴz produces a rotation ofŜ x towardŜ y by an angle Φ AOC = κ 1Ĵz . The net effect is an optical rotation Φ AOC = κ 1 κ 2 N LĴy /4, which is observed by detectingŜ y . The quantum-noise-limited sensitivity TheĴy component is detected using anŜx-polarized probe, which produces a rotation ofĴy towardĴz by an angle θAOC = κ2Ŝx/2. (b) Simultaneously, paramagnetic Faraday rotation produces a rotation ofŜx towardŜy. The net effect is a rotation ΦAOC = κ1κ2NLĴy/4, which is observed by detectingŜy. In an alternative strategy, the linear-to-elliptical rotation ofŜx towardsŜz by the angle ΦLTE = κ2Ĵy can be observed by de-tectingŜz. (c) Experimental geometry. Near-resonant probe pulses pass through a cold cloud of 87 Rb atoms and experience a Faraday rotation by an angle proportional to the on-axis collective spinĴz. Atoms are prepared in a coherent spin stateĴx via optical pumping. The pulses are initially polarized with maximal Stokes operatorŜx, measured at the input by photodiode (PD3). Rotation towardŜy is detected by a balanced polarimeter consisting of a waveplate (WP), polarizing beam splitter (PBS), and photodiodes (PD1,2). of this nonlinear measurement is (see Appendix B) with scaling ∆J y ∝ N −3/2 L crossing over to ∆J y ∝ N −1 L at large N L . Using the Hamiltonian in second order, AOC gives a signal ∝ κ 1 κ 2 N L , versus ∝ κ 2 for LTE, which is an advantage at large detuning, where κ 1 κ 2 . Both strategies employ the same measurement resources, namely, anŜ x -polarized coherent-state probe, so that the quantum uncertainties on the input-polarization angles are ∆Ŝ In addition to the coherent rotations produced byĤ eff , spontaneous scattering of probe photons causes two kinds of "damage" to the spin state: loss of polarization (decoherence) and added spin noise. The tradeoff between information gain and damage is what ultimately limits the sensitivity of the J y measurement [12,42,57]. For equal N L , the damage is the same for the LTE and AOC measurements, because they have the same initial conditions and differ only in whetherŜ z orŜ y is detected.
From these scaling considerations, the AOC measure- ment should surpass the LTE measurement in sensitivity, but only if such a large N L does not cause excessive scattering damage to J y . In atomic ensembles, the achievable information-damage tradeoff is determined by the optical depth (OD) [57], which, in principle, can grow without bound. For high-OD ensembles the nonlinear measurement will, through advantageous scaling, surpass the best-possible linear measurement of the same quantity, under the same conditions. In what follows, we confirm this prediction experimentally, by comparing measured AOC sensitivity to the calculated best-possible sensitivity of the LTE measurement.

III. EXPERIMENTAL DATA AND ANALYSIS
The experimental system, illustrated in Fig. 1(c), is the same as in Ref. [11], with full details given in Ref. [56]. After loading up to 6 × 10 5 laser-cooled atoms into a single-beam optical dipole trap, we prepare aĴ x -aligned coherent spin state (CSS) via optical pumping, J x = Ĵ x = N A /2. An (unknown) bias field B z rotates the state in theĴ x -Ĵ y plane at a rate 2 ω L , where ω L = −γ F B z is the Larmor frequency, to produce J y = Ĵ y = sin(2 ω L t)J x , which we then detect via AOC measurement. We probe the atoms with a sequence of 2-µs-long-pulses of light sent through the atoms at 5-µs intervals and recordŜ Log-log plot of the uncertainty ∆Jy of the AOC measurement versus number of photons NL. Blue diamonds indicate the measured sensitivity. Nonlinear enhanced scaling of the sensitivity is observed over more than an order of magnitude in NL. A fit to the data yields ∆Jy ∝ N k L with k = −1.46 ± 0.04. The best observed sensitivity is ∆Jy = 1290 ± 90 spins with NL = 2 × 10 8 photons. For reference, we also plot the data (light blue circles) and theory (dotted curve) for the measurement ofĴz via nonlinear Faraday rotation reported in Napolitano et al. [8] The solid blue curve shows the theoretical prediction given by Eq. (3) with no free parameters, plus the independently measured electronic noise contribution. Dashed green curve: theory curve describing an ideal LTE measurement of Jy without technical or electronic noise contributions. The nonlinear measurement sensitivity surpasses an ideal LTE measurement with NL = 3 × 10 7 photons. Error bars for standard errors would be smaller than the symbols and are not shown. Inset: Observed metrologically significant spin squeezing ξ 2 m as a function of photon number. The dashed line is a guide to the eye. Error bars indicate ±1σ standard errors. coherent spin state.
In Fig. 2 we plot the observed signal S y = Ŝ (out) y versus S x for various values of J x . As expected, we observe a signal that increases quadratically with S x . We extract J y from a fit to data using the function S y = (κ 1 κ 2 /2)J y S 2 x (solid lines in Fig. 2), where the coupling constants κ 1 = 1.47 × 10 −7 rad/spin and κ 2 = 7.54 × 10 −9 rad/spin are independently measured [11] In the inset we plot the measured J y versus J x . For small rotation angles J y 2 ω L t J x , where t = 7.5 µs is the time between the centers of the baseline and AOC measurements. A linear fit to the data yields B z = 103 ± 3 nT.
The measured sensitivity ∆J y = ∆Ŝ y /(κ 1 κ 2 S 2 x ) is obtained from the measured readout variation ∆Ŝ y and the slope ∂S y /∂J y = κ 1 κ 2 S 2 x , with the contribution due to the atomic projection noise subtracted (see Appendix B). As shown in Fig. 3, we observe nonlinear enhanced scaling ∆J y ∝ N −3/2 L over more than an order of magnitude in N L . For these data, J x = 2.8 × 10 5 and J y = 1.9 × 10 4 . The data are well described by the theoretical model of Eq. (3), plus a small offset due to electronic noise, which is independently measured (see Appendix C). We observe a minimum ∆J y = 1230 ± 90 spins with N L = 2 × 10 8 photons.
The AOC measurement sensitivity crosses below the ideal LTE measurement (∆J y ) LTE = (1/κ 2 )N −1/2 L (dashed green line in Fig. 3) with N L = 3 × 10 7 photons, indicating that, for our experimental parameters, the nonlinear measurement is the superior measurement ofĴ y . For comparison, we also compare our measurement of the alignment J y with the nonlinear Faradayrotation measurement of J z reported in Napolitano et al. [8] (light blue circles and dotted line in Fig. 3). We note, in particular, that the advantageous scaling of the current measurement extends to an order-of-magnitude larger N L than reported in that work.
Nondestructive, projection-noise-limited measurement can be used to prepare a conditional spin-squeezed atomic state [58]. Generation of squeezing is a useful metric for the measurement sensitivity since it takes into account damage done to the atomic state by the optical probe [42,59]. Here, it is important to note that although the AOC signal is proportional to the atomic spin alignment J y , quantum noise from the spin orientationĴ z is mixed into the measurement: Scaled to have units of spins, the Faradayrotation signal from the AOC measurement isΦ ≡ θ , which describes a nondestructive measurement of the mixed alignment-orientation variableK .K θ is the variable that should be squeezed to enhance the sensitivity of the AOC measurement. Metrological enhancement is quantified by the spin-squeezing parameter ξ 2 m ≡ (∆K [59] (see Appendix D). With N L = 2 × 10 8 and J x = 2.8 × 10 5 , we observe a conditional noise 2.3 ± 0.5 dB below the projection noise limit and ξ 2 m = 0.7 ± 0.2, or 1.5 ± 0.8 dB of metrologically significant spin squeezing (inset of Fig. 3). We note that for our experimental parameters, LTE would not induce spin squeezing.

IV. DISCUSSION
The experiment shows AOC surpassing LTE through improved scaling at the specific detuning of ∆/2π = −600 MHz. It is important to ask whether this advantage persists under other measurement conditions. A good metric for the optimum measurement is the number of photons N L required to achieve a given sensitivity (see Appendix E). In Fig. 4(a) we plot the calculated N L required to reach projection-noise-limited sensitivity for the two measurement strategies, i.e. (∆J y ) 2 AOC = (∆J y ) 2 LTE = N A /4 for our experimental parameters. For comparison, we also plot curves showing the damage η sc to the atomic state due to spontaneous emission. We see that the AOC strategy achieves the same sensitivity with fewer probe photons (and thus causes less damage) except very close to the atomic resonances, i.e. except in regions where large scattering rates make the quantum nondemolition measurement impossible anyway. Another important metric is the achievable metrologically significant squeezing, found by optimizing ξ 2 m over N L at any given detuning. In Fig. 4(b) we show this optimal ξ 2 m versus detuning. The global optimum squeezing achieved by the AOC (LTE) strategy is ξ 2 m = 0.47 (0.63) at a detuning of ∆/2π = −59 MHz (77 MHz).
In Fig. 5, we plot the achievable ξ 2 m,min as a function of N L versus both detuning ∆ and optical depth OD for the AOC [ Fig. 5(a)] and LTE [ Fig. 5(b)] strategies. We find that AOC is globally optimum, giving more squeezing, and thus better metrological sensitivity, across the entire parameter range. In Fig. 5(c), we plot the fully optimized spin squeezing, i.e., over ∆ and N L , achievable by the AOC and LTE measurement strategies as a function of OD. This comparison again shows an advantage for AOC, including for large OD, and agrees well with experimental results.
We conclude that: (1) for nearly all probe detunings, if N L is chosen to give projection-noise sensitivity for LTE, then AOC gives better sensitivity at the same detuning and N L . The exception is probing very near an absorption resonance, which induces a large decoherence in the atomic state. (2) Considering as a figure of merit the achievable spin squeezing, or equivalently, the magnetometric sensitivity of a Ramsey sequence employing these measurements [11], the global optimum, including choice of measurement, is AOC at a detuning of −59MHz, with N L = 5.4 × 10 6 photons. In this practical metrological sense, the nonlinear measurement is unambiguously superior. Although the AOC and LTE compared here use coherent states as inputs, the same conclusion is expected Theoretical calculation of spin squeezing as a function of optical depth and probe detuning for (a) the AOC strategy and (b) the LTE strategy with our experimental parameters. Contours indicate the minimum achievable metrologically significant squeezing ξ 2 m,min with respect to NL, with values indicated, as a function of detuning ∆ and optical depth OD. (c) The blue diamonds represent the observed spin squeezing ξm as a function of optical depth for the AOC measurement with NL = 2 × 10 8 photons. The solid curves show the theoretically predicted minimum achievable squeezing, optimized with respect to both NL and ∆, for the AOC measurement (solid blue line) and the LTE measurement (solid green line). The scaling of each curve is roughly ξ 2 ∝ OD −1/2 , so the advantage for AOC continues also to large OD.
when nonclassical probe states are used: For both measurements, the optical rotation sensitivity ∆S (in) y,z /S x can be enhanced in the same way by squeezing [7] and other techniques [12].

V. CONCLUSION
We have identified a scenario -nondestructive detection of atomic spin alignment -in which a nonlinear measurement (AOC) outperforms competing linear strategies with the same experimental resources. Our experimental demonstration answers a fundamental question in quantum metrology [37][38][39][40], with implications for quantum enhancement of atomic instruments operating in nonlinear regimes [1,2,15]. Beyond magnetometry, our techniques may be useful in the measurement of spinor condensates [47][48][49] and lattice gases [19]. To date, such measurements have been limited to detecting spin orientation (vector magnetization), whereas our technique provides a nondestructive measurement of spin alignment (a component of the spin-one nematic tensor), with direct application, e.g., to the detection of spin-nematic quadrature squeezing in spinor condensates [10]. The technique may make possible proposals for the detection [54,55] and preparation [51,52] of exotic quantum phases of ultracold atoms, which require quantum-noise-limited measurement sensitivity.

Appendix A: Atom-light interaction
As described in Refs. [57,60], the light pulses and atoms interact by the effective Hamiltonian plus higher-order terms describing fast electronic nonlinearities [26]. Here, κ 1,2 are coupling constants that depend on the beam geometry, excited-state linewidth, laser detuning, and the hyperfine structure of the atom, and the light is described by the time-resolved Stokes op-eratorS(t), defined asS i ≡ 1 2 (E  x,y , describing the collective spin alignment, where x ≡ (f 2 x −f 2 y )/2 and y ≡ (f xfy +f yfx )/2 describe single-atom Raman coherences, i.e., coherences between states with ∆m f = 2. Here, f (i) is the total spin of the i th atom. For f = 1, these operators obey commutation relations [Ĵ x ,Ĵ y ] = iĴ z and cyclic permutations.
Using Eq. (A1) in the Heisenberg equations of mo-tion and integrating over the duration of a single light pulse [11], we find the detected outputs to second order inŜ xŜ plus small terms. Equation (A3) describes a nondestructive measurement of the mixed alignment-orientation variableK sin θ, where tan θ ≡ κ 2 S x /2.K θ is the variable that should be squeezed to enhance the sensitivity of the AOC measurement.

Appendix B: Measurement sensitivity
Both AOC and LTE measurements have the same input state, with S x ≡ Ŝ (in) To infer theĴ y measurement uncertainty, we note thatĴ y andŜ (out) z are Gaussian variables [61], so that the simple error-propagation formula coincides with more sophisticated estimation methods using, e.g., Fisher information [62]. We find: which shows shot-noise scaling. The first two terms are readout noise and determine the measurement sensitivity.
In the experiment (κ 2 N A ) 2 /4 ∼ 10 −3 , so the second term is negligible. The last term is due to the variance ofĴ y -i.e., the signal we are trying to estimate -which we subtract to give the expression in Eq. (2). We note that other measurement strategies using the same term in the Hamiltonian are possible, e.g., probing withŜ z -polarized light and reading out the rotation ofŜ z intoŜ y , but lead to the same measurement sensitivity. The AOC measurement detectsŜ y , with signal Ŝ (out) y = (κ 1 κ 2 Ŝ x 2 /2) Ĵ y and variance (∆Ŝ y ) 2 . From this slope and output variance we find the variance referred to the input where again the last term is the signal variance, which we subtract to give the expression in Eq. (3).
In contrast, previous work [8] used short, intense pulses to access a nonlinear term in the effective Hamiltonian κ NL S 0ŜzĴz . The coupling κ NL is proportional to the Kerr nonlinear polarizability, and S 0 ≡ N L /2 (so that S 0 =Ŝ x for the input polarization used). Calculating the variance referred to the input as above, we find sensitivity ∆J z = ∆Ŝ scaling.

Appendix C: Electronic & technical noise
The measured electronic noise of the detector referred to the interferometer input is EN = 9.2 × 10 5 photons, and contributes a term EN × 64/(κ 2 1 κ 2 2 N 4 L ) to Eq. (3), which is included in the blue curve plotted in Fig. 3. Technical noise contributions from both the atomic and light variables are negligible in this experiment. Baseline subtraction is used to remove low-frequency noise.

Appendix E: Dependence on detuning and optical depth
The detuning dependence of the coupling constants κ 1 and κ 2 of Eq.(1) is given by where δ i (∆) ≡ 1/ Γ 2 + (∆ − ∆ i ) 2 , ∆ i is the detuning from resonance with the F = 1 → F = i transition on the 87 Rb D 2 line, Γ/2π = 6.1 MHz is the natural linewidth of the transition, ∆ is measured from the F = 1 → F = 0 transition, σ 0 ≡ λ 2 /π and A = 4.1 × 10 −9 m 2 is the effective atom-light interaction area. Note that for large detuning, i.e., ∆ Γ, κ 1 ∝ 1/∆ and κ 2 ∝ 1/∆ 2 . At any detuning, the measurement sensitivity can be improved by increasing the number of photons N L used in the measurement. Note, however, that increasing N L also increases the damage η sc = k(∆)η γ (∆)N L done to the atomic state we are trying to measure due to probe scattering, where η γ (∆) is the probability of scattering a single photon: which also scales as η γ ∝ 1/∆ 2 for large detuning. k(∆) is a correction factor that accounts for the fact that a fraction of the scattering events leaves the state unchanged. A good metric to compare measurement strategies is the number of photons N L required to achieve a given sensitivity. Minimizing this metric will minimize damage to the atomic state independently of the correction factor k(∆). For our calculations, we set k(∆) = 0.4, which predicts our measurements at large detuning An estimate for the quantum-noise reduction that can be achieved in a single-pass measurement, valid for η sc 1, is given by where ζ is the signal-to-noise ratio of the measurement, i.e., the ratio of atomic quantum noise to light shot noise in the measured variance (∆Ŝ (out) y ) 2 . For the two strategies considered here,

(E5)
and Metrologically significant squeezing is then given by