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Quantum phase magnification

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Science  24 Jun 2016:
Vol. 352, Issue 6293, pp. 1552-1555
DOI: 10.1126/science.aaf3397

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 (112), Dicke states (1315), 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 87Rb 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)].

Fig. 1 Conceptual description of quantum phase magnification.

(A) Illustration of the magnification protocol on the Bloch sphere. The Wigner quasi-probability distributions are shown for two separated initial CSSs (left) and after the states are magnified through collective interactions (right). Here this is shown with N = 900 atoms and a magnification of M = 3 for pictorial clarity. Experimentally we use up to N = 5 × 105 and M = 100, permitting us to concentrate on a planar patch of the Bloch sphere. (B and C) Effect of the Embedded Image (shearing) interaction used for mapping Jz onto Jy for a pair of different initial states with separations S and S′ = S/2; each panel shows three different magnification factors. Note that a π/2 rotation about the Jx axis needs to follow to complete the protocol. CSSs (B) and 6-dB squeezed states (C) together illustrate the requirement of larger magnifications to separate two initially squeezed states. (D) A small rotation θ about the Jx axis is added before the shearing step, eliminating the requirement of larger magnifications for squeezed states by giving rise to a refocusing of the Jy noise. At an optimal magnification (here M = 3), the noise-refocusing scheme maps the initial Jz onto Jy, preserving the SNR associated with the two initial states.

The collective state of an ensemble of N two-level atoms—here, the clock states of 87Rb—can be described using the language of a pseudo–spin-N/2 system. The z-component of the spin, Jz, represents the population difference, and the orientation in the Jx - Jy 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 〈Jx〉 ≈ N/2, the uncertainties satisfy ΔJz · ΔJyN/4, where Jy is now identified with the phase of the ensemble. Coherent spin states (CSS) with noise Embedded Image establish the standard quantum limit (SQL) to minimum resolvable phase or population difference.

The magnification procedure (Fig. 1A) starts with a mapping of Jz onto Jy (Fig. 1B) via a shearing interaction. A rotation of Jy into Jz follows to complete the sequence. The interaction leading to the mapping (shearing) generates a rotation of the state about the Jz axis, with the rotation rate depending on Jz, and is represented by the one-axis twisting Hamiltonian (24) Embedded Image, 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 Embedded Image, where the rotation vector Embedded Image is also an operator, and Embedded Image 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 JxJ = 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 Jz(t) = Jz(0) and Jy(t) = Jy(0) + MJz(0) with Embedded Image. Thus, the initial Jz is mapped onto Jy with a magnification factor M. This is analogous to free expansion of a gas if one identifies Jz with a particle’s momentum and Jy with its position.

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 Jz (Fig. 2C). Thus, Jz sets the cavity-light detuning, which in turn sets the intracavity power (Fig. 2D), which in turn provides a Jz-dependent ac-Stark shift—hence the Embedded Image interaction. Implemented this way, there is also a Jz-independent part of the ac-Stark shift, causing global rotations of the state about the Jz axis even for Jz = 0. The rotation angle ϕAC due to this effect is proportional to the pulse area of the interaction light incident on the cavity. By directly measuring ϕAC 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 Jx - Jz plane (23). The magnification parameter Embedded Image (1)obtained in this implementation directly relates to the measured quantity ϕAC. Here, δc = 5.5 Hz is the cavity frequency shift per unit Jz, κ = {8.0, 10.4} kHz is the cavity full linewidth at N = {0, 5 × 105}, and δ0 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).

Fig. 2 Experimental setup implementing phase magnification.

(A) 87Rb atoms are trapped inside a high-finesse cavity (length 10.7 cm) using a 1560-nm cavity mode as a one-dimensional optical lattice. A 780-nm mode is used to generate collective atomic interactions and to probe the cavity resonance frequency (Jz measurements) by recording the phase of a reflected probe pulse (~10 pW, 200 μs). Microwaves are for atomic state rotations. A charge-coupled device (CCD) imaging system measures the population difference between the hyperfine states after releasing the atoms from the lattice and spatially separating the states. (B) Because of the commensurate frequency relationship between the trapping laser and the interaction/probe laser, all atoms are uniformly coupled to the 780-nm mode. (C) The 780-nm mode couples the two hyperfine clock states separated by ωHF to the excited manifold with opposite detunings. Thus, the two states pull the intracavity index of refraction in opposite directions, leading to a cavity frequency shift proportional to Jz. (D) The mechanism leading to the collective atomic interactions (Embedded Image Hamiltonian) that enables the magnification process: linking of the intracavity power to Jz, producing a Jz-dependent ac-Stark shift. The frequencies of the interaction beam νint and probe beam νprobe are indicated.

The details of the experimental apparatus are described in (12, 25). We load up to 5 × 105 atoms at 25 μK into an optical lattice inside the high-finesse (1.75 × 105) 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 Jz 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 Jx axis of the Bloch sphere using 2 × 105 atoms (23), then apply a small microwave-induced rotation about the Jy axis (±2 mrad) to displace the center of the CSS to a Jz value of ±200. As characterized by cavity measurements, the widths of the resulting Jz distributions read within 0.5 dB of the calibrated CSS noise level (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 δ0 = 36 kHz, to shear the state; apply a microwave π/2 rotation about the Jx axis; and count the atoms state-selectively using fluorescence imaging (23). The fluorescence detection setup has a technical noise floor of 1200 atoms RMS, which is ~{15, 11} dB above the SQL for {2 × 105, 5 × 105} atoms.

Fig. 3 Characterization of the basic magnification process with CSSs.

(A) Sample distributions (400 samples each) comparing the cavity-based measurements of Jz with fluorescence imaging–based measurements after a magnification of M = 45. The two distributions in each plot correspond to different initial states with 〈Jz〉 = ±200 prepared using 2 × 105 atoms. (B and C) Magnification of the separation between the two distributions as a function of accumulated ac-Stark shift phase ϕAC imparted on the atoms at fixed cavity-light detuning of 36 kHz (B) or as a function of cavity-light detuning δ0 at fixed ϕAC = 0.6 rad (C). Solid lines are fits to the data as a function of ϕAC and δ0, respectively, in Eq. 1. Fitted curves agree with theoretical curves (not shown) to within 10%. (D) SNR associated with the two distributions as a function of the magnification parameter, normalized to that obtained by the cavity measurements (normalized SNR). Magnification is varied by changing ϕAC. The solid line is a fit of the form M/2 + M2)1/2; the fit parameter α contains information primarily about fluorescence detection noise. In (B) to (D), error bars and shaded regions denote the 68% statistical confidence interval for data and fits, respectively.

A comparison of the Jz 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 ϕAC, 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 Jz distributions divided by the RMS width of their distributions.

For the mapping to be accurate in this protocol, the magnified Jz noise M ΔJz(0) should exceed the initial Jy noise ΔJy(0). If we magnify a Jz-squeezed state that has ΔJz(0) = ΔCSSξ and ΔJy(0) = ΔCSSξ′ (ξ′ · ξ ≥ 1, ξ < 1), the final noise becomesEmbedded Image (2)Setting the fractional noise contribution of the second term to 1 – ε requires a magnification of Mε ≈ (2ε)–1/2ξ′/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 Jy 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 Jy noise in this version isEmbedded Image (3)(23). For θ = θ0, the initial Jz noise becomes the sole noise contribution at M = M0 ≡ 1/θ0.

We demonstrate the noise-refocusing protocol using squeezed spin states with 5 × 105 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 Jz and 32-dB antisqueezed in Jy (ξ ~ 0.4,ξ′ ~ 40)—the best we can currently achieve without measurement-based methods. We apply a small microwave rotation θ about the Jx 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 M ~ 30 replicates the SNR of the initially prepared states. Had we not used noise refocusing, the required magnification would have been M0.05 ~ 320 (for an infidelity ε = 0.05), which would have started wrapping the states around the Bloch sphere.

Fig. 4 Magnification process with noise refocusing using 8-dB squeezed spin states.

(A) Post-magnification Jz noise in units of CSS noise for different amounts of prior rotation θ about the Jx axis (see Fig. 1D). Solid lines are a global fit to the entire data set with two free parameters: dθ/dt (the rate of change in θ with microwave pulse time) and the Jz noise of the initial squeezed states. Obtained values are within 15% of the calculated values. The inset shows the distribution of two separated 8-dB squeezed initial states (5 × 105 atoms) as identified by cavity measurements [to be compared with the M = 30 distribution in (B)]. The dashed line shows M × (Jz noise contribution from the initial squeezed states); the dotted line shows M × (CSS noise). Error bars denote the 68% statistical confidence interval. (B) The distributions after the magnification protocol at the indicated M values for θ = 29 mrad. The normalized SNR becomes 0.96 ± 0.06 at M ≈ 1/|θ| (middle histogram), where the dashed line is tangent to the θ = 29 mrad noise curve in (A). For M values to either side of 1/|θ|, the two distributions blur into each other (top and bottom histograms).

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

www.sciencemag.org/content/352/6293/1552/suppl/DC1

Materials and Methods

Fig. S1

References (3133)

References and Notes

  1. See supplementary materials on Science Online.
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.
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