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Interaction-based quantum metrology showing scaling beyond the Heisenberg limit

Quantum metrology studies the use of entanglement and other quantum resources to improve precision measurement. An interferometer using N independent particles to measure a parameter X can achieve at best the"standard quantum limit"(SQL) of sensitivity {\delta}X \propto N^{-1/2}. The same interferometer using N entangled particles can achieve in principle the"Heisenberg limit"{\delta}X \propto N^{-1}, using exotic states. Recent theoretical work argues that interactions among particles may be a valuable resource for quantum metrology, allowing scaling beyond the Heisenberg limit. Specifically, a k-particle interaction will produce sensitivity {\delta}X \propto N^{-k} with appropriate entangled states and {\delta}X \propto N^{-(k-1/2)} even without entanglement. Here we demonstrate this"super-Heisenberg"scaling in a nonlinear, non-destructive measurement of the magnetisation of an atomic ensemble. We use fast optical nonlinearities to generate a pairwise photon-photon interaction (k = 2) while preserving quantum-noise-limited performance, to produce {\delta}X \propto N^{-3/2}. We observe super-Heisenberg scaling over two orders of magnitude in N, limited at large N by higher-order nonlinear effects, in good agreement with theory. For a measurement of limited duration, super-Heisenberg scaling allows the nonlinear measurement to overtake in sensitivity a comparable linear measurement with the same number of photons. In other scenarios, however, higher-order nonlinearities prevent this crossover from occurring, reflecting the subtle relationship of scaling to sensitivity in nonlinear systems. This work shows that inter-particle interactions can improve sensitivity in a quantum-limited measurement, and introduces a fundamentally new resource for quantum metrology.

nonlinear effects, in good agreement with theory 13 . For a measurement of limited duration, super-Heisenberg scaling allows the nonlinear measurement to overtake in sensitivity a comparable linear measurement with the same number of photons. In other scenarios, however, higher-order nonlinearities prevent this crossover from occurring, reflecting the subtle relationship of scaling to sensitivity in nonlinear systems. This work shows that inter-particle interactions can improve sensitivity in a quantum-limited measurement, and introduces a fundamentally new resource for quantum metrology.
The best instruments are interferometric in nature, and operate according to the laws of quantum mechanics. A collection of particles, e.g., photons or atoms, is prepared in a superposition state, allowed to evolve under the action of a Hamiltonian containing an unknown parameter X , and measured in agreement with quantum measurement theory. The complementarity of quantum measurements 15 determines the ultimate sensitivity of these instruments.
Here we describe polarisation interferometry, used for example in optical magnetometry to detect atomic magnetisation 11,16,17 ; similar theory describes other interferometers 2 . A collection of N photons, with circular plus/minus polarisations |+ , |− is described by single-photon Stokes where the σ i are the Pauli matrices and σ 0 is the identity.
In traditional quantum metrology, a Hamiltonian of the formĤ = X N j=1ŝ (j) z uniformly and independently couples the photons to X , the parameter to be measured 1 . If the input state consists of independent photons, the possible precision scales as δX ∝ N −1/2 , the shot-noise or standard quantum limit (SQL). The N −1/2 factor reflects the statistical averaging of independent results.
In contrast, entangled states can be highly, even perfectly, correlated, giving precision limited by δX ∝ N −1 , the Heisenberg limit (HL).
The above Hamiltonian is conveniently writtenĤ = XŜ z , whereŜ i ≡ N i.e., with a k-order nonlinearity inŜ, contains k-photon interaction termsŝ The number of such terms, and thus the signal strength, grows as N k , while the quantum noise from the input states is unchanged. As a result, a sensitivity limit of δX ∝ N −k applies when entanglement is used, and δX ∝ N −(k−1/2) in the absence of entanglement 7 . For k ≥ 2, this already gives a scaling better than the Heisenberg limit, so-called "super-Heisenberg" (SH) scaling 7 . Note that interactions and entanglement are compatible and both improve the scaling. The predicted advantage applies generally to quantum interferometry, and proposed mechanisms to produce metrologicallyrelevant interactions include Kerr nonlinearities 18 , cold collisions in condensed atomic gases 7 , Duffing nonlinearity in nano-mechanical resonators 19 and a two-pass effective nonlinearity with an atomic ensemble 20 . Topological excitations in nonlinear systems may also give advantageous scaling 21 .
In this Letter, we study interaction-based quantum metrology using unentangled probe particles. One challenge in demonstrating SH scaling is to engineer a suitable nonlinear Hamiltonian.
Some nonlinearities have been shown to be intrinsically noisy 14 while others give SH scaling but fall short of the ideal N −(k−1/2) under realistic conditions 7,22 . We use a cold atomic ensemble as a light-matter quantum interface 12 to produce quantum-noise-limited interactions and a Hamiltonian of the formĤ = XŜ zŜ0 = XŜ z N/2. This Hamiltonian gives a polarisation rotation growing with the photon number, without increasing quantum noise 7 . The experiment, shown schematically in Fig. 1, uses pulses of near-resonant light to measure the collective spinF of an ensemble of N A ∼ 10 6 cold rubidium-87 atoms, probed on the 5S 1/2 → 5P 3/2 D 2 line. The experimental system is described in detail in the references 8,23 . The on-axis atomic magnetisation F z , which plays the role of X in this measurement, is prepared in the initial state F z = N A by optical pumping with resonant circularly polarised light propagating along the trap axis z. A weak on-axis magnetic field is applied to preserveF z during the measurements.
The rotation angle is φ = F z [A(∆) + B(∆)N ]/2 where A ∝ α (1) and B ∝ β (1) account for the temporal pulse shape and geometric overlap between the atomic density and the spatial mode of the probe. The shot-noise limited uncertainty in the rotation angle, due to quantum uncertainty in the initial angle, is δφ = N −1/2 /2. A contribution F z B(∆)δN/2 from initial number fluctuations δN = N −1/2 is negligible for small rotation angles. This gives a measurement uncertainty indicating a transition from SQL scaling δF z ∝ N −1/2 to SH scaling δF z ∝ N −3/2 with increasing N .
Two regimes of probing are used: the linear probe consists of forty 1 µs pulses (total illumination time τ L = 40 µs) spread over 400 µs with detuning ∆ L ∆ 0 . This gives A N L B, i.e., linear estimation and, as described by Koschorreck et al. 8 , provides a projection-noise-limited quantum-non-demolition (QND) measurement 24 ofF z , with uncertainty at the parts-per-thousand level 8 . The nonlinear probe consists of a single τ NL = 54 ns FWHM, Gaussian-shaped, highintensity pulse with N NL photons and detuning ∆ 0 , so that A N NL B. Crucially, having two probes allows us to precisely calibrate the nonlinear measurement using a highly sensitive and well characterised independent measurement of the same sample.
We probe the same sample three times for each preparation: First with the linear probe, which gives a precise and non-destructive measurement of F z via a rotation φ L . Then with the nonlinear probe, contributing with a rotation φ NL , which is calibrated against the "true" value (i.e., with negligible error) provided by the previous linear probe. Third, a second linear probe is used to estimate the damage to the atomic magnetisation η ≡ 1 − φ L /φ L caused by the nonlinear probe.
The linear probe is calibrated using quantitative absorption imaging to measure N A , and we find A(∆ L ) = 3.3(1) × 10 −8 rad per atom. The calibration of the nonlinear probe against the first linear probe is shown in Fig. 2: We repeat the above pump/probe sequence while varying N A in the range N A = 1.5 × 10 5 to N A = 3.5 × 10 5 to generate a φ L , φ NL correlation plot for a given N NL . Since both φ L and φ NL are linear in N A , we use linear regression to find the slope with a saturation parameter N The noise in the nonlinear probe, again as a function of N NL , is determined from the φ L , φ NL correlation plots. As illustrated in Fig. 2b-c, the residual standard deviation of the fits indicates the observed uncertainty ∆φ NL , which includes the intrinsic uncertainty δφ NL and a small contribution from electronic noise. In Fig. 3 we plot the fractional sensitivity δF (NL) z / F z vs. N NL , calculated using equation (2) and considering the whole polarised ensemble, F z = 7 × 10 5 . In agreement with equation (1), the log-log slope indicates the scaling δF to within experimental uncertainties in the range N NL = 10 6 to N NL = 10 7 , and SH scaling, i.e., steeper than N −1 , over two orders of magnitude N NL = 5 × 10 5 to N NL = 5 × 10 7 .
Results of numerical modelling using the Maxwell-Bloch equations to describe the nonlinear light-atom interaction are also shown in Fig. 3. Two curves are shown, for detunings ∆ 0 ± 2π(200 kHz), covering the combined uncertainty in ∆ due to the probe laser linewidth and inhomogeneous light shifts in the optical dipole trap. As expected from equation (1), this alters the sensitivity only at low N NL . The model is described in detail in the Supplementary Information.
For photon numbers above N NL 2 × 10 7 , the saturation of the nonlinear rotation alters the slope. This can be understood as optical pumping of atoms into states other than |F = 1, m F = 1 by the nonlinear probe. The damage to the atomic magnetisation η = 1 − φ L /φ L , shown in Fig. 3 remains small, confirming the non-destructive nature of the measurement. The finite damage even for small N NL is possibly due to stray light and/or magnetic fields disturbing the atoms during the 20 ms period between the two linear measurements. At large N , high-order nonlinear effects including optical pumping limit the range of SH scaling.
The experimental results illustrate the subtle relationship of scaling to sensitivity in a nonlinear system. For an ideal nonlinear measurement, the improved scaling would guarantee better absolute sensitivity for sufficiently large N . Indeed, when the measurement bandwidth is taken into account, the nonlinear probe overtakes the linear one at N = 3.2 × 10 6 where both achieve a sensitivity of 1.1 × 10 2 spins Hz −1/2 . As a consequence, the nonlinear technique performs better in fast measurements. In contrast, when measurement time is not a limited resource, the comparison can be made on a "sensitivity-per-measurement" basis, and the ideal crossover point of 3.2 × 10 3 spins at N = 8.7 × 10 7 is never actually reached, due to the higher-order nonlinearities.
Evidently SH scaling enables but does not guarantee enhanced sensitivity: for the nonlinear to overtake the linear, it is also necessary that the scaling extend to large enough N . The comparison shows also that resource constraints dramatically influence the linear vs. nonlinear comparison.
See also the Supplementary Information.
We have realised a scenario proposed by Boixo et al. 4 to achieve metrological sensitivity beyond the Heisenberg limit δX ∝ N −1 using metrologically-relevant interactions among particles. To generate pairwise photon-photon interactions, we use fast nonlinear optical effects in a cold atomic ensemble and measure the ensemble magnetisation F z with super-Heisenberg sensitivity δF z ∝ N −3/2 . To rigorously quantify the photon-photon interaction and the sensitivity, we calibrate against a precise, non-destructive, linear measurement of the same atomic quantity 8 , demonstrate quantum-noise-limited performance of the optical instrumentation, and place an upper limit on systematic, i.e., non-atomic, nonlinearities at the few-percent level. The experiment demonstrates the use of inter-particle interactions as a new resource for quantum metrology.
While possible applications to precision measurement will require detailed study, this first experiment shows that interactions can produce super-Heisenberg scaling and improved precision in a quantum-limited measurement. Author Contributions All authors contributed equally to the work presented in this paper.

Methods
Competing Interests The authors declare that they have no competing financial interests.
Correspondence Correspondence and requests for materials should be addressed to Mario Napolitano (email: mario.napolitano@icfo.es).

Supplementary Discussion
Interaction Hamiltonian. The atom-light interaction is described using collective continuous variables and degenerate perturbation theory in reference [13] of the main text. We repeat essential results: The electric dipole interaction h int = −E·d, taken in second order perturbation theory, gives rise to an effective (single-atom) Hamiltonian of the form plus terms inŜ 0 which do not alter the optical polarisation. Here α (1) , α (2) describe the vectorial and tensorial components of the interaction respectively, and the atomic collective vari- j y ≡ f xfy +f yfx /2,ĵ z ≡f z /2 andĵ 0 ≡f 2 z /2. For our case of the F=1 ground state, theĵ operators, defined starting from the angular momentum operatorsf , represent a pseudo-spin 1/2 system involving the states m f = ±1. In this representation, the measurement ofF z , described in the main text, is equally to a measurement of 2Ĵ z =F z .
The coefficients α (1) and β (1) depend strongly on the probe frequency due to the excited state hyperfine structure. For the D 2 line of 87 Rb, from the F = 1 ground state, they are shown graphically in Supp. Fig. 1.
We note also that H (4) eff is sensitive to more spin degrees of freedom than is H Before each polarisation step, the state of the ensemble is reset to a fully-mixed state by repeated pumping from F = 1 to F = 2 and back, using resonant lasers from the MOT beams as described in Koschorreck et al. 8 . During the reset process, about 10% of the atoms escape from  x , which remains 1, while the quantum noise is given by In retarded coordinates ζ ≡ z and τ ≡ t − z/c, the field envelope E E E(x, τ ) and atomic state ρ(x, τ ) obey the coupled equations where D ≡ ∂ 2 x + ∂ 2 y + 2ik∂ ζ is the differential operator of the paraxial wave equation (PWE), k is the wave-number, L is the Liouvillian describing relaxation and the polarization envelope P P P(x, τ ) is where n is the local atomic number density and d ↓ is the dipole operator describing downward transitions. For the atom distribution, we take a Gaussian with FWHM 2σ T √ ln 2 and 2σ L √ ln 2 in the transverse and longitudinal directions, respectively: where r 2 ≡ x 2 + y 2 .
We find numerically the solution to Eq. (S4) with E E E = E E E (in) as the lowest-order atomic response ρ (1) . A representative case is shown in Supp. Fig. 4. The evolving atomic state generates the field where G is the Green function for the PWE.
H + c.c., where the spatial integral is taken over the surface of the detector and subscripts H, V indicate polarization components. It can be shown, e.g., using Green function techniques 26 , that and thus Independently determined values for the model parameters σ L , σ T , w 0 and τ are used, leaving only N A as a free parameter, found by fitting to the data. We note that N A determines the vertical position of the curve in Figure 3, and has no effect on the sensitivity scaling. In this sense, the model confirms the scaling behaviour with no adjustable parameters.
Simulations indicate that loss of polarization in F = 1, and thus rotation signal, is mostly due to spontaneous decay into the F=2 ground level, as seen in Supp. Fig. 4.