49. M. V. Medvedyeva, T. Prosen, M. Žnidarič, Phys. Rev. B 93, 094205(2016).

50. M. H. Fischer, M. Maksymenko, E. Altman, Phys. Rev. Lett. 116,160401 (2016).

51. Y. Bar Lev, G. Cohen, D. R. Reichman, Phys. Rev. Lett. 114,100601 (2015).

52. K. Agarwal, S. Gopalakrishnan, M. Knap, M. Müller, E. Demler,Phys. Rev. Lett. 114, 160401 (2015).

53. Y. B. Lev, D. R. Reichman, Europhys. Lett. 113, 46001 (2016).54. A. Chandran, I. H. Kim, G. Vidal, D. A. Abanin, Phys. Rev. B 91,

085425 (2015).

55. R. Islam et al., Nature 528, 77–83 (2015).56. A. M. Kaufmann et al., http://arxiv.org/abs/1603.04409

(2016).57. M. Goihl, M. Friesdorf, A. H. Werner, W. Brown, J. Eisert, http://

arxiv.org/abs/1601.02666 (2016).

ACKNOWLEDGMENTS

We thank M. Fischer, E. Altman, M. Knap, P. Bordia, H. Lüschen,A. Rosch, E. Demler, and S. Sondhi for discussions. D.A.H. isthe A. and H. Broitman Member at the Institute for AdvancedStudy. We acknowledge funding by Max-Planck-Gesellschaft,

Deutsche Forschungsgemeinschaft, the European Union (UQUAM,Marie Curie Fellowship to J.C.).

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/352/6293/1547/suppl/DC1Supplementary TextFigs. S1 to S4Reference (58)

14 April 2016; accepted 31 May 201610.1126/science.aaf8834

REPORTS◥

QUANTUM PHYSICS

Quantum phase magnificationO. Hosten, R. Krishnakumar, N. J. Engelsen, M. A. Kasevich*

Quantum metrology exploits entangled states of particles to improve sensing precisionbeyond the limit achievable with uncorrelated particles. All previous methods requireddetection noise levels below this standard quantum limit to realize the benefits of theintrinsic sensitivity provided by these states. We experimentally demonstrate a widelyapplicable method for entanglement-enhanced measurements without low-noisedetection. The method involves an intermediate quantum phase magnification step thateases implementation complexity. We used it to perform squeezed-state metrology8 decibels below the standard quantum limit with a detection system that has a noise floor10 decibels above the standard quantum limit.

Theprospect of using quantumentanglementto improve the precision of atomic and op-tical sensors has been a topic of discussionfor more than two decades. Examples of re-cent work using atomic ensembles include

the preparation of spin-squeezed states (1–12),Dicke states (13–15), and other states with nega-tive Wigner functions (16). An assumption com-mon to all this work is that low-noise detectionmethods are required to properly measure andmake use of the prepared quantum states. In fact,detection noise has thus far been the bottleneckin the performance of these systems. To this end,there has beendedicatedwork on improving state-selective detection of atoms with both opticalcavity–aided measurements (17, 18) and fluores-cence imaging (19, 20).Here, we describe the concept and the im-

plementation of a quantum phase magnificationtechnique that relaxes stringent requirements indetection sensitivity for quantummetrology. Thismethod is a generalization of a recent proposalfor approaching the Heisenberg limit in measure-ment sensitivity without single-particle detection(21). We demonstrate the method in an ensembleof 87Rb atoms. As in a typical atomic sensor orclock, the goal is to measure a differential phaseshift accumulated between a pair of quantumstates during a time interval.Making thismeasure-ment requires the phase shift to be converted into

a population difference (12, 22), after which thepopulation difference is measured. Our schememagnifies this population difference before thefinal detection, in effect magnifying the initialphase shift. The atomic ensemble is first spin-squeezed using atomic interactions aided by anoptical cavity, and then small rotations—to besensed—are induced on the atomic state. Theserotations are magnified by stretching the rotatedstates (Fig. 1A), using cavity-aided interactions,and are finally detected via fluorescence imaging.Magnification allows for substantial reduction inthe noise requirements for the final detection.Although themethod is demonstrated in an atom/cavity system, it is broadly applicable to anyquantum system that has a suitable nonlinearinteraction [see below and (23)].

The collective state of an ensemble ofN two-level atoms—here, the clock states of 87Rb—canbe described using the language of a pseudo–spin-N/2 system. The z-component of the spin, Jz,represents the population difference, and the ori-entation in the Jx - Jy plane represents the phasedifference between the two states. As these angularmomentum components do not commute, boththe population and the phase possess uncer-tainties. For a state with hJxi ≈ N/2, the un-certainties satisfy DJz · DJy ≥ N/4, where Jy isnow identified with the phase of the ensemble.Coherent spin states (CSS) with noise DJz ¼ DJy ¼ffiffiffiffiN

p=2 ≡ DCSS establish the standardquantum limit

(SQL) to minimum resolvable phase or popula-tion difference.

The magnification procedure (Fig. 1A) startswith a mapping of Jz onto Jy (Fig. 1B) via a shear-ing interaction. A rotation of Jy into Jz follows tocomplete the sequence. The interaction leading tothe mapping (shearing) generates a rotation ofthe state about the Jz axis, with the rotation ratedepending on Jz, and is represented by the one-axis twisting Hamiltonian (24)H ¼ ℏcJ2z , whereℏ is the Planck constant divided by 2p and c isthe shearing strength.The Heisenberg equations of motion for the vec-

tor operator J yield dJ=dt ¼ 1=2ðW� J−J�WÞ,where the rotation vector W ¼ z2cJz is also anoperator, and z is a unit vector in the z-direction.We assume that the angular shifts we seek to mea-sure are small (otherwise, they would readily bemeasurable without magnification) and that theuncertainties of the states after magnification oc-cupy a small fraction of the Bloch sphere. Withthese assumptions and working with near-maxi-mal initial x-polarization Jx ≈ J = N/2, we canlinearize the problem and focus our attention toa planar patch of the spherical phase space (Fig. 1,B to D). The equations ofmotion then yield Jz(t) =Jz(0)andJy(t)=Jy(0)+MJz(0)withM¼ N ∫t0dt ′cðt ′Þ.Thus, the initial Jz is mapped onto Jywith a mag-nification factorM. This is analogous to free expan-sion of a gas if one identifies Jz with a particle’smomentum and Jy with its position.We implement theone-axis twistingHamiltonian

through a dispersive interaction between atomsand light in an optical cavity (1) (Fig. 2A). Theunderlyingmechanism is a coupling between theintracavity power and atomic populations. Theatom-cavity detuning is set such that the shift inthe cavity resonance due to the atoms is propor-tional to Jz (Fig. 2C). Thus, Jz sets the cavity-lightdetuning, which in turn sets the intracavity power(Fig. 2D), which in turn provides a Jz-dependentac-Stark shift—hence the J2z interaction. Imple-mented this way, there is also a Jz-independentpart of the ac-Stark shift, causing global rotationsof the state about the Jz axis even for Jz = 0. Therotation angle fAC due to this effect is propor-tional to the pulse area of the interaction lightincident on the cavity. By directly measuring fACand offsetting the phase of the microwave oscil-lator by the same amount, we effectively work ina frame where the center of the states remains inthe Jx - Jzplane (23). Themagnification parameter

M ¼ Ndcd0

d20 þ ðk=2Þ2fAC ð1Þ

obtained in this implementation directly relates tothemeasured quantity fAC. Here, dc = 5.5 Hz is the

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Department of Physics, Stanford University, Stanford, CA94305, USA.*Corresponding author. Email: kasevich@stanford.edu

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cavity frequency shift per unit Jz,k = {8.0, 10.4} kHzis the cavity full linewidth at N = {0, 5 × 105},and d0 is the empty cavity-light detuning. Thedecay of the cavity field results in back-actionnoise that is not taken into account in the simpleHamiltonian analysis above. However, these ef-fects are negligible in the parameter range weuse for the magnification protocol and can beignored (23).The details of the experimental apparatus are

described in (12, 25). We load up to 5 × 105 atomsat 25 mK 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 col-

lective interactions and probing the atoms. Thelattice holds the atoms at the intensitymaxima ofthis 780-nmmode, ensuring uniform atom-cavitycoupling (Fig. 2, A and B). The 1560-nm latticelight, whose frequency is stabilized to the cavity,generates the 780-nm light through frequencydoubling, guaranteeing its frequency stabilitywith respect to the cavity. Bymeasuring the phaseof a reflected probe pulse with homodyne detec-tion, we can determine the empty cavity frequencydown to a Jz equivalent of three spin-flips.In our procedure, the low-noise cavity probe is

used to obtain reference information about thestates before magnification, which is then com-

pared with the noisy fluorescencemeasurementsafter magnification.We first prepare a CSS aligned with the Jx axis

of the Bloch sphere using 2 × 105 atoms (23), thenapply a small microwave-induced rotation aboutthe Jy axis (±2 mrad) to displace the center ofthe CSS to a Jz value of ±200. As characterizedby cavity measurements, the widths of the result-ing Jz distributions read within 0.5 dB of the ca-librated CSS noise level (12) (Fig. 3A).We illustratethe magnification protocol (Fig. 3) using thesecharacterized states. We implement the follow-ing sequence (23): excite the cavity with a 200-mslight pulse, detuned by d0 = 36 kHz, to shear the

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Fig. 1. Conceptual description of quantum phasemagnification. (A) Illustration of themagnification pro-tocol on theBloch sphere.TheWignerquasi-probabilitydistributions are shown for two separated initial CSSs(left) and after the states are magnified throughcollective interactions (right). Here this is shown withN = 900 atoms and a magnification of M = 3 forpictorial clarity. Experimentally we use up to N = 5 ×105 and M = 100, permitting us to concentrate on aplanar patch of the Bloch sphere. (B and C) Effect of

the J2z (shearing) interaction used for mapping Jz ontoJy for a pair of different initial states with separations Sand S´ = S/2; each panel shows three differentmagnification factors. Note that a π/2 rotation aboutthe Jx axis needs to follow to complete the protocol.CSSs (B) and6-dBsqueezed states (C) together illustratethe requirement of larger magnifications to separatetwo initially squeezed states. (D) A small rotation qabout the Jx axis is added before the shearing step,eliminating the requirement of larger magnificationsfor squeezed states by giving rise to a refocusing of theJy noise. At an optimal magnification (hereM = 3), thenoise-refocusing scheme maps the initial Jz onto Jy,preserving the SNR associated with the two initial states.

M=3

M=6

M=3

M=6

M=3

M=6

Jx Jx

Jz Jz

JyJy

Interaction

M×S

S

~M×S’

S’

θJy

Jz

M=0M=0 M=0S’ Jy

Jz

Jy

Jz

Fig. 2. Experimental setup implementing phase mag-nification. (A) 87Rb atoms are trapped inside a high-finessecavity (length 10.7 cm) using a 1560-nm cavity mode as aone-dimensional optical lattice. A 780-nmmode is used togenerate collective atomic interactions and to probe thecavity resonance frequency (Jzmeasurements) by record-ing the phase of a reflected probe pulse (~10 pW, 200 ms).Microwaves are for atomic state rotations. A charge-coupleddevice (CCD) imaging systemmeasures the population dif-ference between the hyperfine states after releasing theatoms from the lattice and spatially separating the states.(B) Because of the commensurate frequency relationshipbetween the trapping laser and the interaction/probelaser, all atoms are uniformly coupled to the 780-nmmode.(C) The 780-nm mode couples the two hyperfine clockstates separated by wHF to the excited manifold with op-posite detunings. Thus, the two states pull the intracavityindexof refraction in opposite directions, leading to a cavityfrequency shift proportional to Jz. (D) Themechanism lead-

ing to the collective atomic interactions (J2z Hamiltonian)that enables the magnification process: linking of the intra-cavity power to Jz, producing a Jz-dependent ac-Starkshift. The frequencies of the interaction beam nint andprobe beam nprobe are indicated.

87Rb atoms

Jz=0Jz=-200 Jz=200

Frequency

Intr

a-ca

vity

pow

er

Interactionlight

δ0

κ

1560 nm optical lattice

780 nm probe & interaction

To homodynedetection

To CCDMicrowaves

Lattice potentialProbe & interactionintensity

Atoms

|F=1,mF=0⟩

87Rb5P3/2

5S1/2

|F=2,mF=0⟩

π−polarized

νint νprobe

ωHF

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state; apply a microwave p/2 rotation about theJx axis; and count the atoms state-selectively usingfluorescence imaging (23). The fluorescence detec-tion setup has a technical noise floor of 1200 atomsRMS, which is ~{15, 11} dB above the SQL for {2 ×105, 5 × 105} atoms.A comparison of the Jz distributions obtained

separately by cavity and fluorescence measure-ments (Fig. 3A) allows for extraction of magnifica-tion parameters (23). The magnification increaseslinearly with incident shearing light power (Fig.3B), quantified by fAC, and has the expected de-pendence on cavity-light detuning (Fig. 3C). Inthe largeM limit, the signal-to-noise ratio (SNR)associated with the two states after magnificationapproaches the value measured by the cavity (Fig.3D), set by the intrinsic sensitivity of the quantumstate. Here, SNR is defined as the separation be-tween the centroids of the two Jz distributionsdivided by the RMS width of their distributions.For the mapping to be accurate in this proto-

col, themagnified Jz noiseM DJz(0) should exceedthe initial Jy noise DJy(0). If we magnify a Jz-squeezed state that has DJz(0) = DCSSx and DJy(0)= DCSSx′ (x′ · x ≥ 1, x < 1), the final noise becomes

DJy ¼hðDCSSx

′Þ2 þ ðM DCSSxÞ2i1=2

ð2Þ

Setting the fractional noise contribution of thesecond term to 1 – e requires a magnification ofMe ≈ (2e)–1/2x′/x. This quantity grows unfavorably(at least quadratically) with the squeezing factor x.The unfavorable scaling can be eliminated using

a noise-refocusing version of the protocol (Fig. 1D),which enables, in principle, perfect mapping at achosenmagnification. By adding a small rotationq before magnification, the Jy noise can be madeto focus down through the course of magnifica-tion. The action of the small rotation is formallyanalogous to that of a lens on a beam of light.The final Jy noise in this version is

DJy ≈hð1 −MqÞ2ðDCSSx

′Þ2 þ ðMDCSSxÞ2i1=2

ð3Þ

(23). For q = q0, the initial Jz noise becomes thesole noise contribution at M =M0 ≡ 1/q0.We demonstrate the noise-refocusing protocol

using squeezed spin states with 5 × 105 atoms(Fig. 4). The states are generated with the sameshearing interaction later used for magnification(1, 23).We start with states that are 8-dB squeezedin Jz and 32-dB antisqueezed in Jy (x ~ 0.4, x′ ~40)—the best we can currently achieve withoutmeasurement-based methods. We apply a smallmicrowave rotation q about the Jx axis, and in-vestigate the noise measured at the end of themagnification protocol (Fig. 4A). We observe adifferent optimal magnification value for each q.The shown family ofmodel curves (usingEq. 3) is afit to the entire data set with only two free pa-rameters, and the small deviations from thesecurves are attributable to slow drifts in the initialsqueezing level (~1 dB). For the specific exampleof q = 29 mrad (Fig. 4B), we explicitly show thatthe optimal magnification M ~ 30 replicates theSNR of the initially prepared states. Had we notused noise refocusing, the required magnification

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-1 -0.5 0 0.5 1Jz (× 103)

0

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ized

SN

R

-40 -20 0 20 40Jz (×103)

Fluorescence

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 correspondto different initial states with hJzi = ±200 prepared using 2 × 105 atoms. (B and C) Magnification ofthe separation between the two distributions as a function of accumulated ac-Stark shift phase fACimparted on the atoms at fixed cavity-light detuning of 36 kHz (B) or as a function of cavity-lightdetuning d0 at fixed fAC = 0.6 rad (C). Solid lines are fits to the data as a function of fAC and d0,respectively, in Eq. 1. Fitted curves agree with theoretical curves (not shown) to within 10%. (D) SNRassociated with the two distributions as a function of the magnification parameter, normalized tothat obtained by the cavity measurements (normalized SNR). Magnification is varied by changingfAC. The solid line is a fit of the form M/(a2 + M2)1/2; the fit parameter a contains informationprimarily about fluorescence detection noise. In (B) to (D), error bars and shaded regions denote the68% statistical confidence interval for data and fits, respectively.

0 20 40 60 80 100Magnification

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(θ=29 mrad)

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0

0.05

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0.2

Pro

babi

lity

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Fluorescence

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 q about the Jx axis(see Fig. 1D). Solid lines are a global fit to the entire data set with two free parameters: dq/dt (therate of change in q 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 twoseparated 8-dB squeezed initial states (5 × 105 atoms) as identified by cavity measurements [to becompared with the M = 30 distribution in (B)].The dashed line showsM × (Jz noise contribution fromthe 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 indicatedM values for q = 29 mrad.The normalized SNR becomes 0.96 ± 0.06 at M ≈ 1/|q| (middle histogram),where the dashed line is tangent to the q = 29 mrad noise curve in (A). For M values to either side of1/|q|, the two distributions blur into each other (top and bottom histograms).

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would have beenM0.05 ~ 320 (for an infidelity e =0.05), which would have started wrapping thestates around the Bloch sphere.In assessing metrological gain obtained from

spin squeezing, the degree of Bloch vector length(coherence) preservation is essential to preventdegradation of signal levels. Throughout all statepreparation and magnification, the coherence ofthe states measured by Ramsey fringe contrastsremains above 96%. The small reduction arisesfrom residual atom-cavity coupling inhomogeneities.If the magnification technique developed here

is used as the readout stage for themore effectivemeasurement-based squeezing methods (12), weexpect improvements in verifiable squeezing be-cause the technique avoids degradation due tophoton losses in the readout (23). The magnifica-tion protocol can be used on any kind of exoticinitial state to ease characterization. Examples in-clude the not yet demonstrated Schrödinger-catspin states (26, 27), where the spacing of inherentinterference fringes can bemagnified, or other statesthat possess negativeWigner functions (16). Becausethe only required key element is a nonlinear phaseshift, the method could find broad use in systemsthat use, for example, collisional interactions inBose-Einstein condensates (11, 28, 29), Rydbergblockade interactions in neutral atoms (21), Isinginteractions in ion traps (30), nonlinearities insuperconducting Josephson junctions, and non-linearities in optics. In (23) we describe a photon-ic analog of the phase magnification concept usingself-phase modulation.

REFERENCES AND NOTES

1. I. D. Leroux, M. H. Schleier-Smith, V. Vuletić, Phys. Rev. Lett.104, 073602 (2010).

2. M. H. Schleier-Smith, I. D. Leroux, V. Vuletić, Phys. Rev. Lett.104, 073604 (2010).

3. J. G. Bohnet et al., Nat. Photonics 8, 731–736 (2014).4. K. C. Cox, G. P. Greve, J. M. Weiner, J. K. Thompson, Phys. Rev.

Lett. 116, 093602 (2016).5. C. Gross, T. Zibold, E. Nicklas, J. Estève, M. K. Oberthaler,

Nature 464, 1165–1169 (2010).6. C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Bookjans,

M. Chapman, Nat. Phys. 8, 305–308 (2012).7. J. Appel et al., Proc. Natl. Acad. Sci. U.S.A. 106, 10960–10965

(2009).8. R. J. Sewell et al., Phys. Rev. Lett. 109, 253605 (2012).9. W. Muessel, H. Strobel, D. Linnemann, D. B. Hume,

M. K. Oberthaler, Phys. Rev. Lett. 113, 103004 (2014).10. T. Takano, M. Fuyama, R. Namiki, Y. Takahashi, Phys. Rev. Lett.

102, 033601 (2009).11. M. F. Riedel et al., Nature 464, 1170–1173 (2010).12. O. Hosten, N. J. Engelsen, R. Krishnakumar, M. A. Kasevich,

Nature 529, 505–508 (2016).13. B. Lücke et al., Phys. Rev. Lett. 112, 155304 (2014).14. B. Lücke et al., Science 334, 773–776 (2011).15. F. Haas, J. Volz, R. Gehr, J. Reichel, J. Estève, Science 344,

180–183 (2014).16. R. McConnell, H. Zhang, J. Hu, S. Ćuk, V. Vuletić, Nature 519,

439–442 (2015).17. H. Zhang et al., Phys. Rev. Lett. 109, 133603

(2012).18. J. Volz, R. Gehr, G. Dubois, J. Estève, J. Reichel, Nature 475,

210–213 (2011).19. D. B. Hume et al., Phys. Rev. Lett. 111, 253001

(2013).20. I. Stroescu, D. B. Hume, M. K. Oberthaler, Phys. Rev. A 91,

013412 (2015).21. E. Davis, G. Bentsen, M. Schleier-Smith, Phys. Rev. Lett. 116,

053601 (2016).22. I. D. Leroux, M. H. Schleier-Smith, V. Vuletić, Phys. Rev. Lett.

104, 250801 (2010).23. See supplementary materials on Science Online.

24. M. Kitagawa, M. Ueda, Phys. Rev. A 47, 5138–5143(1993).

25. J. Lee, G. Vrijsen, I. Teper, O. Hosten, M. A. Kasevich, Opt. Lett.39, 4005–4008 (2014).

26. R. McConnell et al., Phys. Rev. A 88, 063802(2013).

27. S. Massar, E. S. Polzik, Phys. Rev. Lett. 91, 060401(2003).

28. H. Strobel et al., Science 345, 424–427(2014).

29. W. Muessel et al., Phys. Rev. A 92, 023603(2015).

30. J. G. Bohnet et al., Science 352, 1297–1301(2016).

ACKNOWLEDGMENTS

Supported by grants from the Defense Threat Reduction Agencyand the Office of Naval Research. We thank M. Schleier-Smith forcrucial discussions over the course of this work.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/352/6293/1552/suppl/DC1Materials and MethodsFig. S1References (31–33)

28 January 2016; accepted 17 May 201610.1126/science.aaf3397

ORGANIC CHEMISTRY

Allosteric initiation and regulation ofcatalysis with a molecular knotVanesa Marcos, Alexander J. Stephens, Javier Jaramillo-Garcia, Alina L. Nussbaumer,Steffen L. Woltering, Alberto Valero, Jean-François Lemonnier,Iñigo J. Vitorica-Yrezabal, David A. Leigh*

Molecular knots occur in DNA, proteins, and other macromolecules. However, the benefitsthat can potentially arise from tying molecules in knots are, for the most part, unclear.Here, we report on a synthetic molecular pentafoil knot that allosterically initiates orregulates catalyzed chemical reactions by controlling the in situ generation of acarbocation formed through the knot-promoted cleavage of a carbon-halogen bond. Theknot architecture is crucial to this function because it restricts the conformations that themolecular chain can adopt and prevents the formation of catalytically inactive speciesupon metal ion binding. Unknotted analogs are not catalytically active. Our results suggestthat knotting molecules may be a useful strategy for reducing the degrees of freedomof flexible chains, enabling them to adopt what are otherwise thermodynamicallyinaccessible functional conformations.

Molecular knots are found in circular DNA(1) and approximately 1% of proteins inthe Protein Data Bank (PDB) (2) andform spontaneously in polymer chains ofsufficient length and flexibility (3). Mol-

ecules with the topology of some of the simplestprime knots have been synthesized (4–13), but,althoughknots are fundamental elements of struc-ture, the potential benefits that could arise fromtying molecules in knots are mostly unclear(14–18). Recently, Fe(II) complexes of some syn-thetic molecular knots and links were found tostrongly and selectively bind halide anions (X–)within their central cavities (19). This binding ac-tivity resembles a key feature of dehalogenaseenzymes, which contain halide binding sites thatfacilitate the cleavage of carbon-halogen bonds(20, 21). Here, we show that as little as 1 mole %(mol %) of a synthetic molecular pentafoil knotcan induce Lewis acid–catalyzed reactions by thein situ generation of a carbocation by carbon-halogen bond scission promoted by CH···X– hy-drogen bonding and long range metal-cation···X–

electrostatic interactions.

We previously reported the synthesis of a mo-lecular pentafoil knot [a 51 knot in Alexander-Briggs notation (22)], inspired by Lehn’s cyclicdouble helicates (23) but modified so as to be as-sembled, and the end groups connected, throughimine chemistry (24, 25). The knot binds a chlo-ride or bromide anion extremely strongly withinits central cavity (KCl–= 3.6 × 1010M−1,KBr–= 1.7 ×1010 M−1 in MeCN) through a combination ofCH···X– hydrogen bonding and long-range Fe(II)···X– electrostatic interactions (19). However,attempts to remove the metal ions while main-taining the pentafoil knot topology proved un-successful because of the lability of uncoordinatedimine bonds (25). We therefore investigated analternative route to a metal-free pentafoil knotbased on ring-closing olefin metathesis (RCM) ofa tris(bipyridine) [tris(bipy)] ligand strand (1) thathad proved successful in the synthesis of a relatedStar of David catenane (26) (Fig. 1). [Detailed ex-perimental procedures and full characterizationdata are given in sections S1 and S4 to S8 of thesupplementary materials (SM).]Heating ligand strand 1 with 1.25 equivalents

(equiv.) of FeCl2 in dimethylsulfoxide (DMSO)at 60°C resulted in an intensely colored red-purple solution indicative of the formation oflow-spin Fe(II)–tris(bipy) complexes (Fig. 1, step i).

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School of Chemistry, University of Manchester, Oxford Road,Manchester, M13 9PL, UK.*Corresponding author. Email: david.leigh@manchester.ac.uk

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Quantum phase magnificationO. Hosten, R. Krishnakumar, N. J. Engelsen and M. A. Kasevich

DOI: 10.1126/science.aaf3397 (6293), 1552-1555.352Science 

, this issue p. 1552Scienceto other coherent quantum systems.cold rubidium atoms in a way that relaxes the ultrasensitive detection requirements. The general method may be applied

describe a method that manipulates a coherent cloud ofet al.capabilities that have hampered their development. Hosten enhanced precision measurement and sensing applications. These devices have, however, required low-noise detection

Exploiting the quantum-mechanical properties of quantum systems offer the possibility of developing devices forQuantum enhanced metrology

ARTICLE TOOLS http://science.sciencemag.org/content/352/6293/1552

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