## Quantum enhanced metrology

Exploiting the quantum-mechanical properties of quantum systems offer the possibility of developing devices for enhanced precision measurement and sensing applications. These devices have, however, required low-noise detection capabilities that have hampered their development. Hosten *et al.* describe a method that manipulates a coherent cloud of cold rubidium atoms in a way that relaxes the ultrasensitive detection requirements. The general method may be applied to other coherent quantum systems.

*Science*, this issue p. 1552

## Abstract

Quantum metrology exploits entangled states of particles to improve sensing precision beyond the limit achievable with uncorrelated particles. All previous methods required detection noise levels below this standard quantum limit to realize the benefits of the intrinsic sensitivity provided by these states. We experimentally demonstrate a widely applicable method for entanglement-enhanced measurements without low-noise detection. The method involves an intermediate quantum phase magnification step that eases implementation complexity. We used it to perform squeezed-state metrology 8 decibels below the standard quantum limit with a detection system that has a noise floor 10 decibels above the standard quantum limit.

The prospect of using quantum entanglement to improve the precision of atomic and optical sensors has been a topic of discussion for more than two decades. Examples of recent work using atomic ensembles include the preparation of spin-squeezed states (*1*–*12*), Dicke states (*13*–*15*), and other states with negative Wigner functions (*16*). An assumption common to all this work is that low-noise detection methods are required to properly measure and make use of the prepared quantum states. In fact, detection noise has thus far been the bottleneck in the performance of these systems. To this end, there has been dedicated work on improving state-selective detection of atoms with both optical cavity–aided measurements (*17*, *18*) and fluorescence imaging (*19*, *20*).

Here, we describe the concept and the implementation of a quantum phase magnification technique that relaxes stringent requirements in detection sensitivity for quantum metrology. This method is a generalization of a recent proposal for approaching the Heisenberg limit in measurement sensitivity without single-particle detection (*21*). We demonstrate the method in an ensemble of ^{87}Rb atoms. As in a typical atomic sensor or clock, the goal is to measure a differential phase shift accumulated between a pair of quantum states during a time interval. Making this measurement requires the phase shift to be converted into a population difference (*12*, *22*), after which the population difference is measured. Our scheme magnifies this population difference before the final detection, in effect magnifying the initial phase shift. The atomic ensemble is first spin-squeezed using atomic interactions aided by an optical cavity, and then small rotations—to be sensed—are induced on the atomic state. These rotations are magnified by stretching the rotated states (Fig. 1A), using cavity-aided interactions, and are finally detected via fluorescence imaging. Magnification allows for substantial reduction in the noise requirements for the final detection. Although the method is demonstrated in an atom/cavity system, it is broadly applicable to any quantum system that has a suitable nonlinear interaction [see below and (*23*)].

The collective state of an ensemble of *N* two-level atoms—here, the clock states of ^{87}Rb—can be described using the language of a pseudo–spin-*N/*2 system. The *z*-component of the spin, *J _{z}*, represents the population difference, and the orientation in the

*J*-

_{x}*J*plane represents the phase difference between the two states. As these angular momentum components do not commute, both the population and the phase possess uncertainties. For a state with 〈

_{y}*J*〉 ≈

_{x}*N/*2, the uncertainties satisfy Δ

*J*· Δ

_{z}*J*≥

_{y}*N/*4, where

*J*is now identified with the phase of the ensemble. Coherent spin states (CSS) with noise establish the standard quantum limit (SQL) to minimum resolvable phase or population difference.

_{y}The magnification procedure (Fig. 1A) starts with a mapping of *J _{z}* onto

*J*(Fig. 1B) via a shearing interaction. A rotation of

_{y}*J*into

_{y}*J*follows to complete the sequence. The interaction leading to the mapping (shearing) generates a rotation of the state about the

_{z}*J*axis, with the rotation rate depending on

_{z}*J*, and is represented by the one-axis twisting Hamiltonian (

_{z}*24*) , where ℏ︀ is the Planck constant divided by 2π and χ is the shearing strength.

The Heisenberg equations of motion for the vector operator **J** yield , where the rotation vector is also an operator, and is a unit vector in the *z*-direction. We assume that the angular shifts we seek to measure are small (otherwise, they would readily be measurable without magnification) and that the uncertainties of the states after magnification occupy a small fraction of the Bloch sphere. With these assumptions and working with near-maximal initial *x*-polarization *J _{x}* ≈

*J*=

*N/*2, we can linearize the problem and focus our attention to a planar patch of the spherical phase space (Fig. 1, B to D). The equations of motion then yield

*J*(

_{z}*t*) =

*J*(0) and

_{z}*J*(

_{y}*t*) =

*J*(0) +

_{y}*MJ*(0) with . Thus, the initial

_{z}*J*is mapped onto

_{z}*J*with a magnification factor

_{y}*M*. This is analogous to free expansion of a gas if one identifies

*J*with a particle’s momentum and

_{z}*J*with its position.

_{y}We implement the one-axis twisting Hamiltonian through a dispersive interaction between atoms and light in an optical cavity (*1*) (Fig. 2A). The underlying mechanism is a coupling between the intracavity power and atomic populations. The atom-cavity detuning is set such that the shift in the cavity resonance due to the atoms is proportional to *J _{z}* (Fig. 2C). Thus,

*J*sets the cavity-light detuning, which in turn sets the intracavity power (Fig. 2D), which in turn provides a

_{z}*J*-dependent ac-Stark shift—hence the interaction. Implemented this way, there is also a

_{z}*J*-independent part of the ac-Stark shift, causing global rotations of the state about the

_{z}*J*axis even for

_{z}*J*= 0. The rotation angle ϕ

_{z}_{AC}due to this effect is proportional to the pulse area of the interaction light incident on the cavity. By directly measuring ϕ

_{AC}and offsetting the phase of the microwave oscillator by the same amount, we effectively work in a frame where the center of the states remains in the

*J*-

_{x}*J*plane (

_{z}*23*). The magnification parameter (1)obtained in this implementation directly relates to the measured quantity ϕ

_{AC}. Here, δ

_{c}= 5.5 Hz is the cavity frequency shift per unit

*J*, κ = {8.0, 10.4} kHz is the cavity full linewidth at

_{z}*N*= {0, 5 × 10

^{5}}, and δ

_{0}is the empty cavity-light detuning. The decay of the cavity field results in back-action noise that is not taken into account in the simple Hamiltonian analysis above. However, these effects are negligible in the parameter range we use for the magnification protocol and can be ignored (

*23*).

The details of the experimental apparatus are described in (*12*, *25*). We load up to 5 × 10^{5} atoms at 25 μK into an optical lattice inside the high-finesse (1.75 × 10^{5}) cavity (*23*). A 780-nm standing-wave cavity mode is used for generating the collective interactions and probing the atoms. The lattice holds the atoms at the intensity maxima of this 780-nm mode, ensuring uniform atom-cavity coupling (Fig. 2, A and B). The 1560-nm lattice light, whose frequency is stabilized to the cavity, generates the 780-nm light through frequency doubling, guaranteeing its frequency stability with respect to the cavity. By measuring the phase of a reflected probe pulse with homodyne detection, we can determine the empty cavity frequency down to a *J _{z}* equivalent of three spin-flips.

In our procedure, the low-noise cavity probe is used to obtain reference information about the states before magnification, which is then compared with the noisy fluorescence measurements after magnification.

We first prepare a CSS aligned with the *J _{x}* axis of the Bloch sphere using 2 × 10

^{5}atoms (

*23*), then apply a small microwave-induced rotation about the

*J*axis (±2 mrad) to displace the center of the CSS to a

_{y}*J*value of ±200. As characterized by cavity measurements, the widths of the resulting

_{z}*J*distributions read within 0.5 dB of the calibrated CSS noise level (

_{z}*12*) (Fig. 3A). We illustrate the magnification protocol (Fig. 3) using these characterized states. We implement the following sequence (

*23*): excite the cavity with a 200-μs light pulse, detuned by δ

_{0}= 36 kHz, to shear the state; apply a microwave π/2 rotation about the

*J*axis; and count the atoms state-selectively using fluorescence imaging (

_{x}*23*). The fluorescence detection setup has a technical noise floor of 1200 atoms RMS, which is ~{15, 11} dB above the SQL for {2 × 10

^{5}, 5 × 10

^{5}} atoms.

A comparison of the *J _{z}* distributions obtained separately by cavity and fluorescence measurements (Fig. 3A) allows for extraction of magnification parameters (

*23*). The magnification increases linearly with incident shearing light power (Fig. 3B), quantified by ϕ

_{AC}, and has the expected dependence on cavity-light detuning (Fig. 3C). In the large

*M*limit, the signal-to-noise ratio (SNR) associated with the two states after magnification approaches the value measured by the cavity (Fig. 3D), set by the intrinsic sensitivity of the quantum state. Here, SNR is defined as the separation between the centroids of the two

*J*distributions divided by the RMS width of their distributions.

_{z}For the mapping to be accurate in this protocol, the magnified *J _{z}* noise

*M*Δ

*J*(0) should exceed the initial

_{z}*J*noise Δ

_{y}*J*(0). If we magnify a

_{y}*J*-squeezed state that has Δ

_{z}*J*(0) = Δ

_{z}_{CSS}ξ and Δ

*J*(0) = Δ

_{y}_{CSS}ξ′ (ξ′ · ξ ≥ 1, ξ < 1), the final noise becomes (2)Setting the fractional noise contribution of the second term to 1 – ε requires a magnification of

*M*

_{ε}≈ (2ε)

^{–1/2}ξ′/ξ. This quantity grows unfavorably (at least quadratically) with the squeezing factor ξ.

The unfavorable scaling can be eliminated using a noise-refocusing version of the protocol (Fig. 1D), which enables, in principle, perfect mapping at a chosen magnification. By adding a small rotation θ before magnification, the *J _{y}* noise can be made to focus down through the course of magnification. The action of the small rotation is formally analogous to that of a lens on a beam of light. The final

*J*noise in this version is (3)(

_{y}*23*). For θ = θ

_{0}, the initial

*J*noise becomes the sole noise contribution at

_{z}*M*=

*M*

_{0}≡ 1/θ

_{0}.

We demonstrate the noise-refocusing protocol using squeezed spin states with 5 × 10^{5} atoms (Fig. 4). The states are generated with the same shearing interaction later used for magnification (*1*, *23*). We start with states that are 8-dB squeezed in *J _{z}* and 32-dB antisqueezed in

*J*(ξ ~ 0.4,

_{y}^{}ξ′ ~ 40)—the best we can currently achieve without measurement-based methods. We apply a small microwave rotation θ about the

*J*axis, and investigate the noise measured at the end of the magnification protocol (Fig. 4A). We observe a different optimal magnification value for each θ. The shown family of model curves (using Eq. 3) is a fit to the entire data set with only two free parameters, and the small deviations from these curves are attributable to slow drifts in the initial squeezing level (~1 dB). For the specific example of θ = 29 mrad (Fig. 4B), we explicitly show that the optimal magnification

_{x}*M*~ 30 replicates the SNR of the initially prepared states. Had we not used noise refocusing, the required magnification would have been

*M*

_{0.05 }~ 320 (for an infidelity ε = 0.05), which would have started wrapping the states around the Bloch sphere.

In assessing metrological gain obtained from spin squeezing, the degree of Bloch vector length (coherence) preservation is essential to prevent degradation of signal levels. Throughout all state preparation and magnification, the coherence of the states measured by Ramsey fringe contrasts remains above 96%. The small reduction arises from residual atom-cavity coupling inhomogeneities.

If the magnification technique developed here is used as the readout stage for the more effective measurement-based squeezing methods (*12*), we expect improvements in verifiable squeezing because the technique avoids degradation due to photon losses in the readout (*23*). The magnification protocol can be used on any kind of exotic initial state to ease characterization. Examples include the not yet demonstrated Schrödinger-cat spin states (*26*, *27*), where the spacing of inherent interference fringes can be magnified, or other states that possess negative Wigner functions (*16*). Because the only required key element is a nonlinear phase shift, the method could find broad use in systems that use, for example, collisional interactions in Bose-Einstein condensates (*11*, *28*, *29*), Rydberg blockade interactions in neutral atoms (*21*), Ising interactions in ion traps (*30*), nonlinearities in superconducting Josephson junctions, and nonlinearities in optics. In (*23*) we describe a photonic analog of the phase magnification concept using self-phase modulation.

## Supplementary Materials

## References and Notes

**Acknowledgments:**Supported by grants from the Defense Threat Reduction Agency and the Office of Naval Research. We thank M. Schleier-Smith for crucial discussions over the course of this work.