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 independent particles to measure a parameter can achieve at best the “standard quantum limit” (SQL) of sensitivity . The same interferometer using entangled particles can achieve in principle the “Heisenberg limit” , 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[4, 5, 6]. Specifically, a -particle interaction will produce sensitivity with appropriate entangled states and even without entanglement. Here we demonstrate this “super-Heisenberg” scaling in a nonlinear, non-destructive[8, 9] measurement of the magnetisation[10, 11] of an atomic ensemble. We use fast optical nonlinearities to generate a pairwise photon-photon interaction () while preserving quantum-noise-limited performance[14, 7], to produce . We observe super-Heisenberg scaling over two orders of magnitude in , limited at large 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.
ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain.
Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot et CNRS,
UMR 7162, Bât. Condorcet, 75205 Paris Cedex 13, France.
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 , and measured in agreement with quantum measurement theory. The complementarity of quantum measurements 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. A collection of photons, with circular plus/minus polarisations is described by single-photon Stokes operators , where the are the Pauli matrices and is the identity. In traditional quantum metrology, a Hamiltonian of the form uniformly and independently couples the photons to , the parameter to be measured. If the input state consists of independent photons, the possible precision scales as , the shot-noise or standard quantum limit (SQL). The factor reflects the statistical averaging of independent results. In contrast, entangled states can be highly, even perfectly, correlated, giving precision limited by , the Heisenberg limit (HL).
The above Hamiltonian is conveniently written , where is a collective variable describing the net polarisation of the photons. The independence of the photons manifests itself in the linearity of this Hamiltonian. Recently Boixo et al. have shown that interactions among particles, or equivalently nonlinear Hamiltonians, can contribute to measurement sensitivity and give scaling beyond the Heisenberg limit. For example, a Hamiltonian , i.e., with a -order nonlinearity in , contains -photon interaction terms . The number of such terms, and thus the signal strength, grows as , while the quantum noise from the input states is unchanged. As a result, a sensitivity limit of applies when entanglement is used, and in the absence of entanglement. For , this already gives a scaling better than the Heisenberg limit, so-called “super-Heisenberg” (SH) scaling. 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 metrologically-relevant interactions include Kerr nonlinearities, cold collisions in condensed atomic gases, Duffing nonlinearity in nano-mechanical resonators and a two-pass effective nonlinearity with an atomic ensemble. Topological excitations in nonlinear systems may also give advantageous scaling.
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 while others give SH scaling but fall short of the ideal under realistic conditions[7, 22]. We use a cold atomic ensemble as a light-matter quantum interface to produce quantum-noise-limited interactions and a Hamiltonian of the form . This Hamiltonian gives a polarisation rotation growing with the photon number, without increasing quantum noise. The experiment, shown schematically in Fig. 1, uses pulses of near-resonant light to measure the collective spin of an ensemble of cold rubidium-87 atoms, probed on the D line. The experimental system is described in detail in the references[23, 8]. The on-axis atomic magnetisation , which plays the role of in this measurement, is prepared in the initial state by optical pumping with resonant circularly polarised light propagating along the trap axis . A weak on-axis magnetic field is applied to preserve during the measurements.
Pulses of polarised, but not entangled, photons pass through the ensemble and experience an optical rotation proportional to . The light-atom interaction Hamiltonian describes this paramagnetic Faraday rotation. Both the linear term and the nonlinear term cause rotation of the plane of polarisation from (vertical) toward (diagonal). Detection of then allows estimation of . As described in the Supplementary Information, and depend on the optical detuning relative to the transition, in particular for the specific detuning , allowing a purely nonlinear estimation to be studied.
The rotation angle is where and 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 . A contribution from initial number fluctuations is negligible for small rotation angles. This gives a measurement uncertainty
indicating a transition from SQL scaling to SH scaling with increasing .
Two regimes of probing are used: the linear probe consists of forty pulses (total illumination time ) spread over with detuning . This gives , i.e., linear estimation and, as described by Koschorreck et al., provides a projection-noise-limited quantum-non-demolition (QND) measurement of , with uncertainty at the parts-per-thousand level. The nonlinear probe consists of a single FWHM, Gaussian-shaped, high-intensity pulse with photons and detuning , so that . 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 via a rotation . Then with the nonlinear probe, contributing with a rotation , 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 caused by the nonlinear probe.
The linear probe is calibrated using quantitative absorption imaging to measure , and we find 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 in the range to to generate a correlation plot for a given . Since both and are linear in , we use linear regression to find the slope for that value of . The experiment is repeated varying the number of photons in the nonlinear pulse.
The observed vs. , shown in Fig. 2a, is well fit by a simple model including saturation of the nonlinear response:
with a saturation parameter and the nonlinear coupling strength per atom per photon.
The noise in the nonlinear probe, again as a function of , is determined from the correlation plots. As illustrated in Fig. 2b-c, the residual standard deviation of the fits indicates the observed uncertainty , which includes the intrinsic uncertainty and a small contribution from electronic noise. In Fig. 3 we plot the fractional sensitivity vs. , calculated using equation (2) and considering the whole polarised ensemble, . In agreement with equation (1), the log-log slope indicates the scaling to within experimental uncertainties in the range to , and SH scaling, i.e., steeper than , over two orders of magnitude to .
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 , 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 . The model is described in detail in the Supplementary Information.
For photon numbers above , the saturation of the nonlinear rotation alters the slope. This can be understood as optical pumping of atoms into states other than by the nonlinear probe. The damage to the atomic magnetisation , shown in Fig. 3 remains small, confirming the non-destructive nature of the measurement. The finite damage even for small is possibly due to stray light and/or magnetic fields disturbing the atoms during the period between the two linear measurements. At large , 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 . Indeed, when the measurement bandwidth is taken into account, the nonlinear probe overtakes the linear one at where both achieve a sensitivity of . 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 at 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 . 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. to achieve metrological sensitivity beyond the Heisenberg limit 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 with super-Heisenberg sensitivity . To rigorously quantify the photon-photon interaction and the sensitivity, we calibrate against a precise, non-destructive, linear measurement of the same atomic quantity, 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.
0.1 Linear & nonlinear probe light.
The probe beam is aligned to the axis of the trap with a waist of , chosen to match the radial dimension of the cloud. In the linear probing regime we use a train of forty pulses, pulse period , each containing photons detuned from the () transition. The maximum intensity is . The signals are summed and can be considered a single, modulated pulse.
The nonlinear probe consists of a single Gaussian-shaped pulse with a FWHM of . The maximum intensity of the nonlinear probe is for a pulse with photons. Theory predicts at a detuning in free space. This is modified by trap-induced light shifts, and we use the empirical value , which gives zero rotation at low probe intensity.
0.2 Instrumental noise.
The instrumental noise is quantified by measuring vs. input photon number ( or ), in the absence of atoms, to find contributions from electronic noise , shot-noise , and technical noise , as described in the Supplementary Information. We find and are and per pulse, respectively, while the technical noise is negligible. The instrumentation is thus shot-noise-limited over the full range of used in the experiment. The intrinsic rotation uncertainty of the nonlinear probe is calculated from the measured as . The correction is at most 5%.
0.3 Instrumental linearity.
The linearity of the experimental system and analysis is verified using a wave-plate in place of the atoms to produce a linear rotation equal to the largest observed nonlinear rotation. Over the full range of photon numbers used in the experiment, the detected rotation angle is constant to within , and SQL scaling is observed.
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We thank I. H. Deutsch and F. Illuminati, for helpful comments. We thank C. M. Caves and A. D. Codorníu for inspiration. This work was supported by the Spanish Ministry of Science and Innovation through the Consolider-Ingenio 2010 project QOIT, Ingenio-Explora project OCHO (Ref. FIS2009-07676-E/FIS), project ILUMA (Ref. FIS2008-01051) and by the Marie-Curie RTN EMALI.
All authors contributed equally to the work presented in this paper.
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to Mario Napolitano (email: firstname.lastname@example.org).
Supplementary Information for “Interaction-based quantum metrology showing scaling beyond the Heisenberg limit.”
M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell & M. W. Mitchell
ICFO-Institut de Ciencies Fotoniques, 08860 Castelldefels (Barcelona), Spain.
Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot et CNRS,
UMR 7162, Bât. Condorcet, 75205 Paris Cedex 13, France.
Appendix A Supplementary Discussion
a.1 Interaction Hamiltonian.
The atom-light interaction is described using collective continuous variables and degenerate perturbation theory in reference  of the main text. We repeat essential results:
The electric dipole interaction , taken in second order perturbation theory, gives rise to an effective (single-atom) Hamiltonian of the form
plus terms in which do not alter the optical polarisation. Here describe the vectorial and tensorial components of the interaction respectively, and the atomic collective variable is where the superscript indicates the ’th atom and , , and . For our case of the F=1 ground state, the operators, defined starting from the angular momentum operators , represent a pseudo-spin 1/2 system involving the states . In this representation, the measurement of , described in the main text, is equally to a measurement of .
For atoms, the fourth-order contribution (again ignoring terms depending only on ) is:
For our input state, consisting of vertically-polarized photons, i.e., , we can drop all but the and terms, because 1) terms in and , leave the initial state unchanged, 2) terms in and commute with the measured variable, giving no measurable signal and 3) the terms in make a contributions smaller than the term by a factor .
The coefficients and depend strongly on the probe frequency due to the excited state hyperfine structure. For the line of Rb, from the ground state, they are shown graphically in Supp. Fig. 1.
We note also that is sensitive to more spin degrees of freedom than is . The population of the state , i.e., , appears in proportional to and produces polarization self-rotation. In contrast, has no dependence on this population, which cannot be detected by any linear measurement.
a.2 Atomic State Preparation.
The atomic ensemble contains up to atoms held in an optical dipole trap formed by a weakly-focused () beam of a Yb:YAG laser at with of optical power. The trap is loaded from a conventional magneto-optical trap (MOT) and cooled to with sub-Doppler cooling. The system has demonstrated high effective optical depth () and sub-projection-noise sensitivity of 500 spins with the linear probe.
The ensemble is polarised by optical pumping with circularly polarised light resonant with the transition, sent along the longitudinal axis of the trap. Repump light resonant with is simultaneously applied via the 6 directions of the MOT beams to prevent accumulation in the hyperfine level. A small bias magnetic field of is applied along the axis to preserve .
Before each polarisation step, the state of the ensemble is reset to a fully-mixed state by repeated pumping from to and back, using resonant lasers from the MOT beams as described in Koschorreck et al.. During the reset process, about 10% of the atoms escape from the trap, allowing measurement with different during a single loading cycle.
a.3 Shot noise limited detection.
Before the ensemble, a beamsplitter and calibrated fast photodiode are used to detect the input pulse energy . After the ensemble, pulses are analyzed in the basis with an ultra-low-noise balanced photo-detector, giving a direct measure of . Both signals are recorded on a digital storage oscilloscope, and rotation angles calculated as where are the measured transmission coefficients for the system optics (vacuum cell, lenses and dichroic mirrors to separate the dipole trap beam). Supp. Fig. 2 shows the noise vs. power curve for generation and detection of nonlinear probe pulses, indicating an electronic noise contribution to of per pulse. This electronic noise is subtracted when calculating in Fig. 3 of the main text.
a.4 System linearity.
To check for systematic errors, both in the apparatus and in the analysis, we repeat the experiment under identical conditions but with no atoms present. We mimic the Faraday rotation signal by rotating the wave-plate used to balance the polarimeter to give a signal equal to the largest signal seen with atoms. The measured rotation is independent of , and gives shot-noise scaling of the sensitivity, plotted in Supp. Fig. (3) over the range of used in the experiment.
We model the nonlinear rotation by integrating the Maxwell-Bloch equations in three spatial dimensions plus time . This semiclassical model describes the average rotation , which remains , while the quantum noise is given by .
In retarded coordinates and , the field envelope and atomic state obey the coupled equations
where is the differential operator of the paraxial wave equation (PWE), is the wave-number, is the Liouvillian describing relaxation and the polarization envelope is
where is the local atomic number density and is the dipole operator describing downward transitions. For the atom distribution, we take a Gaussian with FWHM and in the transverse and longitudinal directions, respectively:
We solve to first order in as follows. We identify a solution to the zero-atom equation as the input field . Specifically, we take where is the unit vector in the direction,
is a Gaussian pulse with FWHM , and
where , , and is the wave-front phase. This describes a Gaussian beam with effective area .
where is the Green function for the PWE.
The detected signal is , where the spatial integral is taken over the surface of the detector and subscripts indicate polarization components. It can be shown, e.g., using Green function techniques, that
Independently determined values for the model parameters and are used, leaving only as a free parameter, found by fitting to the data. We note that 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 , and thus rotation signal, is mostly due to spontaneous decay into the F=2 ground level, as seen in Supp. Fig. 4.
a.6 Sensitivity in time- and number-limited scenarios
When time is limiting, the relevant sensitivity is , where as in Equation 1, and the measurement duration is or for the linear or nonlinear measurement, respectively. The sensitivity can be calculated from the measured values , and , using , and . We find , and . Given an equal number of photons , the nonlinear technique surpasses the linear at , well within the super-Heisenberg portion of the curve in Figure 3. In contrast, when time is not a limited resource, the sensitivity-per-measurement is , and . Extrapolating, the nonlinear technique would surpass the linear at , which is however outside the super-Heisenberg portion of the curve in Figure 3.
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