### Interaction-based Quantum Metrology Showing Scaling Beyond the Heisenberg Limit

**Mario Napolitano (left) and Morgan W. Mitchell (right)**

**Authors: Mario Napolitano and Morgan W. Mitchell**

Affiliation: ICFO – Institute of Photonic Sciences, 08860 Castelldefels - Barcelona, Spain

Affiliation: ICFO – Institute of Photonic Sciences, 08860 Castelldefels - Barcelona, Spain

The most precise modern measurement techniques are based on related interferometric techniques: whether the application is defining time-standards, measuring accelerations, magnetic fields, or even detecting the space-time distortion caused by gravitational-waves. All interferometers use the quantum superposition principle and wave-particle duality [1]: the probes, photons or atoms depending on the case, follow simultaneously two different evolutions, and these experience a phase difference due to the quantity being measured. When the paths are recombined, the flux of particles is measured, giving the result of the measurement.

The precision of such interferometers improves by using more probing particles provided that technical sources of noise are suppressed. The square-root law in Poisson statistics of random counts fixes the scale of this improvement saying that the sensitivity, namely the minimum phase difference measurable at a certain level of noise, get better as 1/√N, where N is the total number of probing particles in use. Usually, the scientific community refers to this scaling law for the sensitivity as the Standard Quantum Limit (SQL), or Shot-Noise limit.

The quest for ever-more precise measurements has motivated much of the latest research in atomic physics and quantum optics [2]. For example, most of the achievements regarding entanglement or squeezing find a natural framework in the context of quantum metrology. When applied to interferometry, entanglement means that the inherent fluctuations of each probing particle are correlated with those of another particle: in this way, the intrinsic noise of the probing system as a whole is squeezed, removing uncertainty from critical degrees of freedom and putting it into less-critical ones. In this way, the sensitivity can surpass the 1/√N standard quantum limit. In the ideal case of perfect entanglement, a NooN state, the sensitivity scales as 1/N, a law that in the community is known as the 'Heisenberg limit' [3].

Is this the best that can be done? Are there other ways to use better the probing resources? Does the Heisenberg limit apply always? Only a few years ago, theoretical studies discovered that entanglement is not the only resource that can improve measurement precision in the face of quantum noise [4,5]. According to those works, also the dynamics of the measurement process plays a very important role to establish the best achievable precision. For example, nonlinear dynamics, that is to say interaction between the probing particles, can make a difference and extend, in principle, the limit in the sensitivity.

As presented in a recent letter published in Nature [6], we were able to create the appropriate nonlinear dynamics in a polarization-based interferometer where photons are used to probe the magnetization of an ensemble of cold atoms. We investigated the sensitivity of such measurement as function of the number of photons used to probe, looking for a scaling law beyond the Heisenberg limit. In a particular regime of light intensity and detuning, we detected a polarization rotation of the probing photons due to the atomic magnetization, dependent of the photon number itself. Both this nonlinearity, and the high quality of the light-atom coupling we developed, contributed in amplifying the signal from the atoms, keeping at the same time the noise at the level of the light shot noise.

The sensitivity to the magnetization, in the nonlinear probing, was in fact scaling better than the Heisenberg limit as the photon number was increased, over a range of two orders of magnitude, Fig 1. In fact, the measured scaling was very close to the expected 1/N

^{3/2}for two-particle interaction [7]. With more than 20 million photons, however, incoherent processes such as optical pumping damaged to the atomic preparation and the measurement no longer improved.

**Fig.1**

*(click on the image to see higher resolution version)*: Sensitivity of the nonlinear probe versus number of interacting photons. Blue circles indicate the measured sensitivity, curves show results of numerical modeling, and the black lines indicate SQL, HL, and SH scaling for reference. Scaling surpassing the Heisenberg limit is observed over two orders of magnitude. The measured damage to the magnetization, shown as green diamonds, confirms the non-destructive nature of the measurement. Error bars for standard errors would be smaller than the symbols and are not shown.With this experiment we opened the possibility of investigating experimentally nonlinear dynamics and interaction between quantum probes as new fundamental resources in the quest for greater sensitivity in quantum-interference-based measurements. We think that similar interaction-based measurement will soon start to be extended and tested with other techniques [8,9] although, as also we noticed in our experiment, the range of applicability will depend on the specific implementation.

Description of the experiment:

In our lab, we work with a sample of about one million of cold 87Rb atoms, held in a single-beam optical dipole trap. Pulses of polarized laser light propagate through the sample along the trap axis experiencing a very high coupling with the atomic ensemble, Fig 2a. Previous experiments demonstrated an on-resonance optical depth of above 50, which gives the figure of merit of such good coupling [10]. When instead the light is tuned off-resonant, it can probe in a dispersive, non-destructive way the angular momentum of the atomic ensemble. In particular, the polarization experiences a paramagnetic Faraday rotation proportional to the spin component along the light propagation axis.

**Fig.2a**

*(click on the image to see higher resolution version)*: Experimental schematic: an ensemble of 87Rb atoms,held in an optical dipole trap, is polarized by optical pumping (OP). Linear (P_1), nonlinear (P_{NL}), and a second linear (P_2) Faraday rotation probe pulses measure the atomic magnetization, detected by a shot-noise-limited polarimeter (PM). The atom number is measured by quantitative absorption imaging (AI). Fig.2b : Spectral positions of the pumping, probing, and imaging light on the D2 transition.A key point in our experiment was the ability to calibrate the newly developed, nonlinear-probing technique against a well established and tested linear one, Fig 3. Actually, our light-atom interface is very versatile and different regimes of interaction can be addressed: in particular we could easy switch between a linear and a nonlinear interaction case. After the trap loading and an initial atomic state preparation via optical pumping with on-resonance, circular-polarized light, we probed the polarized sample under both in a linear regimes and in a nonlinear one.

**Fig.3**

*(click on the image to see higher resolution version)*: 3a) Ratio of nonlinear rotation, φ_{NL}, to linear rotation, φ_{L}, vs. nonlinear probe photon number, N_{NL}. The data points and error bars indicate best fit and standard errors from a linear regression for a given N_{NL}. The red curve is a fit showing the expected nonlinear behavior, with some saturation for large N_{NL}. 3b)&c) φ_{L}, φ_{NL}correlation plots for two values of N_{NL}. The atom number N_{A}is varied to produce a range of φ_{L}and φ_{NL}. Green squares: no atoms N_{A}= 0, red circles: 1.5 x 10^{5}< N_{A}< 3.5 x 10^{5}, blue triangles N_{A}‚ 7 x 10^{5}. The blue triangles are shown as a check on detector saturation, and are not included in the analysis.In previous experiments, we demonstrated projection noise sensitivity in the linear regime, namely when low intensity and large (GHz) detuning from the resonances are used. Conversely, in this experiment we also operated in a regime of probing designed specifically to show clearly atom-mediated photon-photon interactions. We expected to excite nonlinearities like fast electronic nonlinearities, namely saturation effects in the photon absorption and stimulated emission. We used intense pulses of about 50ns, a time scale short compared with relaxation process, like optical pumping by spontaneous emission, but still long enough to define the frequency of the light within few tens of MHz. At a particular detuning, due to the symmetry of the electronic structure in alkali atoms, all the linear responses from the different atomic transitions compensate, hence only the nonlinear contributions remain, Fig 2b. Moreover, we carefully checked and excluded any possible other source of nonlinearity apart from the atoms, for example, we showed that the photodetectors continue to be linear even for the largest photon numbers.

**References:**

**[1]**Giovannetti, V., Lloyd, S. & Maccone, L. "Quantum metrology". Phys. Rev. Lett. 96, 010401 (2006). Abstract.

**[2]**Lee, H., Kok, P. & Dowling, J. P. "A quantum Rosetta stone for interferometry". J. Mod. Opt. 49, 2325-2338 (2002). Abstract.

**[3]**Holland, M. J., & Burnett, K. "Interferometric detection of optical phase shifts at the Heisenberg limit". Phys. Rev. Lett. 71, 1355 (1993). Abstract.

**[4]**Boixo, S., Flammia, S. T., Caves, C. M. & Geremia, J. "Generalized limits for single-parameter quantum estimation". Phys. Rev. Lett. 98, 090401(2007). Abstract.

**[5]**Sergio Boixo, Animesh Datta, Matthew J. Davis, Steven T. Flammia, Anil Shaji, and Carlton M. Caves, "Quantum metrology: Dynamics versus entanglement". Phys. Rev. Lett. 101, 040403 (2008). Abstract.

**[6]**M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell & M. W. Mitchell, "Interaction-based quantum metrology showing scaling beyond the Heisenberg limit". Nature 471, 486–489 (2011). Abstract.

**[7]**Napolitano, M. & Mitchell, M. W. "Nonlinear metrology with a quantum interface". New J. Phys. 12, 093016 (2010). Abstract.

**[8]**Woolley, M. J., Milburn, G. J. & Caves, C. M. "Nonlinear quantum metrology using coupled nanomechanical resonators". New J. Phys. 10, 125018(2008). Abstract.

**[9]**Choi, S. & Sundaram, B. "Bose-Einstein condensate as a nonlinear Ramsey interferometer operating beyond the Heisenberg limit". Phys. Rev. A 77, 053613 (2008). Abstract.

**[10]**Koschorreck, M., Napolitano, M., Dubost, B. & Mitchell, M. W. "Sub-projection-noise sensitivity in broadband atomic magnetometry". Phys. Rev. Lett. 104, 093602 (2010). Abstract.

Labels: Precision Measurement 2, Quantum Computation and Communication 4

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