letters to nature 16. Lee, P. C. Physical Properties and Processing of Asteroid Regoliths and Interiors. Thesis, Cornell Univ. ment between two separate samples of atoms containing 1012 atoms (1977). each, along the lines of a recent proposal7. Not only do we 17. Thomas, P. C., Veverka, J., Robinson, M. S. & Murchie, S. Shoemaker crater as the source of most ejecta blocks on the asteroid 433 Eros. Nature 413, 394±396 (2001). demonstrate a quantum entanglement at the level of macroscopic 18. Criswell, D. R. in Photon and Particle Interactions with Surfaces in Space (ed. Grard, R. J. L.) 545±556 objects, our experiment proves feasible a new approach to the (Reidel, Dordrecht, 1973). quantum interface between light and atoms suggested in refs 7, 17. It is a step towards the other protocols proposed7,17, such as the Acknowledgements teleportation of atomic states and quantum memory. The entangle- We thank the mission design, mission operations, and spacecraft teams of the NEAR ment is generated through a non-local Bell measurement on the two Project at the Applied Physics Laboratory of Johns Hopkins University and the navigation samples' spins, performed by transmitting a pulse of light through team at the Jet Propulsion Laboratory for their efforts that resulted in making NEAR Shoemaker the ®rst spacecraft to orbit and land on an asteroid. The NEAR-Shoemaker the sample. mission and this work were made possible by NASA. We thank A. Rivkin and M. Cintala The ideal EPR entangled state of two sub-systems described by for suggestions. continuous non-commuting variables ÃX1;2 and ÃP1;2Ðsuch as the Correspondence and requests for materials should be addressed to M.S.R. positions and momenta of two particles, for exampleÐis the state (e-mail: robinson@earth.northwestern.edu). for which ÃX1 ÃX2 ! 0; ÃP1 2 ÃP2 ! 0. Recently5,6, the necessary and suf®cient condition for the entanglement or inseparability for such gaussian quantum variables has been cast in the form of an inequality involving only the variances of variables: h d ÃX1 à ................................................................. X2 2i h d ÃP1 2 ÃP2 2i,2.Inourexperiment,thequantumvari- ables that are analogous to the position and momentum operators Experimental long-lived are two projections of the collective spin (total angular momentum) of an atomic sample. The analogy is evident from the commutation entanglement of two relation ÃJz; ÃJy iÃJ p x, which can be rewritten as ÃX; ÃP i, p where ÃX ÃJz= Jx and ÃP ÃJy= Jx if the atomic sample is macroscopic objects spin-polarized along the x axis with JÃx having a large classical value Jx. For two spin-polarized atomic samples with Jx1 2 Jx2 Jx the above entanglement condition translates into Brian Julsgaard, Alexander Kozhekin & Eugene S. Polzik dJ212 [ dJ2z12 dJ2y12 , 2Jx 1 Institute of Physics and Astronomy, University of Aarhus, 8000 Aarhus, Denmark where we introduce the notations dJ2z12 h d ÃJz1 ÃJz2 2i and .............................................................................................................................................. dJ2y12 h d ÃJy1 ÃJy2 2i. The interpretation of condition (1) comes Entanglement is considered to be one of the most profound from the recognition of the fact that for both atomic samples features of quantum mechanics1,2. An entangled state of a in coherent spin states (CSS) the equality dJ2y;z Jx =2 holds, that system consisting of two subsystems cannot be described as a is, the inequality (1) becomes the equality. Entanglement between product of the quantum states of the two subsystems3±6. In this atoms of the two samples is, therefore, according to condition (1), sense, the entangled system is considered inseparable and non- equivalent to spin variances that are smaller than in samples in a local. It is generally believed that entanglement is usually manifest CSS18,19 characterized by uncorrelated individual atoms. The in systems consisting of a small number of microscopic particles. entangled state of this type is a two-mode `squeezed' state for the Here we demonstrate experimentally the entanglement of two continuous spin variables5,7. A spin squeezed state of a single macroscopic objects, each consisting of a caesium gas sample macroscopic atomic ensemble has been generated previously21,22. containing about 1012 atoms. Entanglement is generated via Entanglement is produced via interaction of atoms with polar- interaction of the samples with a pulse of light, which performs ized light. A polarized pulse of light is described by Stokes operators a non-local Bell measurement on the collective spins of the obeying the same commutation relation as spin operators samples7. The entangled spin-state can be maintained for 0.5 ÃSy; ÃSz iÃSx. ÃSx is the difference between photon numbers in x milliseconds. Besides being of fundamental interest, we expect and y linear polarizations, ÃSy is the difference between polarizations the robust and long-lived entanglement of material objects at 6458, and ÃSz is the difference between the left- and right-hand demonstrated here to be useful in quantum information proces- circular polarizations along the propagation direction, z. In our sing, including teleportation8±10 of quantum states of matter and experiment light is linearly polarized along the x axis. Hence the two quantum memory. pairs of continuous quantum variables engaged in the entanglement In 1935 Einstein, Podolsky and Rosen (EPR) formulated1 what protocol are ÃJz and ÃJy for atoms and ÃSz and ÃSy for light. they perceived as a paradox created by quantum mechanics. Since Here we report on the generation of a state of two separate then, the EPR correlations and other types of entanglement have caesium gas samples (Fig. 1), which obey the entanglement condi- been extensively analysed, notably by Bell2. Entangled or inseparable tion (1). As shown in refs 7 and 17, when an off-resonant pulse is states are fundamental to the ®eld of quantum information, transmitted through two atomic samples with opposite mean spins speci®cally to quantum teleportation of discrete8,9 and continuous10 Jx1 2 Jx2 Jx, the light and atomic variables evolve as variables and to quantum dense coding of discrete11 and ÃSout continuous12,13 variables, to name a few examples. The majority of y ÃSiny aÃJz12; ÃSout z ÃSinz 2 experiments on entanglement until now deal with entangled states ÃJout of light3,4,8±11,14. Entangled states of discrete photonic variables y1 à Jiny1 bÃSinz; ÃJout y2 à Jiny2 2 bÃSinz; ÃJout z1 à Jinz1; ÃJout z2 à Jinz2 (spin-half systems)3,4 as well as entangled states of continuous where a and b are constants. The ®rst line describes the Faraday variables (quadrature-phase operators) of the electro-magnetic effect (polarization rotation of the probe). The second line shows ®eld14 have been generated experimentally. Entangled states of the back action of light on atoms, that is, spin rotation due to the material particles are much more dif®cult to generate experimen- angular momentum of light. According to equations (2), the tally; however, such states are vital for storage and processing of measurement of ÃSout y reveals the value of ÃJz12 ÃJz1 ÃJz2 (provided quantum information. Recently, entangled states of four trapped a is large enough, so that SÃiny is relatively small) without changing ions have been produced15, and two atoms have been entangled via this value. It follows from equations (2) that the total y spin interaction with a microwave photon ®eld16. projection for both samples is also conserved, ÃJout y1 à Jout y2 Here we describe an experiment on the generation of entangle- ÃJin 7 y1 à Jiny2 . The procedure can be repeated with another pulse of 400 © 2001 Macmillan Magazines Ltd NATURE | VOL 413 | 27 SEPTEMBER 2001 | www.nature.com letters to nature light measuring the sum of y components, ÃJy1 ÃJy2, again in a non- and n 1013 as the number of photons in the 0.45 ms probe pulse demolition way, while at the same time leaving the previously with power 5 mW. measured value of ÃJz1 ÃJz2 intact. As a result, the sum of the z The spectral variance data, ¢ dSout y cos ­ 2 dSout y sin ­ 2 components and the sum of the y components of the spins of the dS2 kdJ2z12 kdJ2y12 [ dS2 kdJ212, is plotted in Fig. 2. The linear two samples are known exactly in the ideal case, and therefore the ®t to it, ¢ Jx , is the quantum limit of noise corresponding to the two samples are entangled according to condition (1), since the coherent spin state of samples (see Fig. 2 legend for details). uncertainties on the left-hand side become negligible. Using ¢ Jx as a reference level we can now devise a measurement An important modi®cation of the above protocol is the addition procedure, which will verify the presence of an entangled state. of a magnetic ®eld oriented along the direction x, which allows us to Namely, if for a certain state of the two ensembles the spectral use a single entangling pulse to measure both z and y spin variance of the signal ¢EPR dS2 kdJ2EPR obeys the inequality: projections, as described in the Methods section. The schematic of the experimental set-up is shown in Fig. 1. The ¢EPR dS2 kdJ2EPR , dS2 2kJx ¢ Jx 4 two cells are coated from inside with a paraf®n coating which then apparently dJ2EPR , 2Jx holds for such a state and therefore this enhances the ground-state coherence time T2 up to 5±30 ms state is entangled in accordance with condition (1). It is, of course, depending on the density of atoms. jJxj and T2 are measured by necessary to use otherwise identical conditions for measurements of the magneto-optical resonance method (ref. 20 and references ¢EPR and ¢ Jx , that is, the same value of 2Jx and the same probe therein). intensity and detuning in order to keep the constant k unchanged After the samples are prepared in CSS, as described in the legend throughout the entire experiment. As mentioned above, the value of to Fig. 1, optical pumping is switched off and a probe pulse is sent 2Jx has been controlled by a magneto-optical resonance measure- through. Its Stokes operator Sy is measured by a polarizing beam ment to better than 5%. In the actual experiment described below splitter with two balanced detectors. The differential photocurrent the measurements of ¢EPR and ¢ Jx have been conducted in turn at from the detectors is split in two and its cos ­t and sin ­t power identical conditions at the repetition rate of 500 Hz. spectral components Sout y cos ­ 2 and Sout y sin ­ 2 are measured by The measurement sequence aimed at the generation and veri®ca- lock-in ampli®ers. By repeating this sequence many times we obtain tion of the entanglement consists of the optical pumping pulses, the variances for these components, in short, spectral variances, which induces a CSS, the entangling pulse, which induces an which according to equation (3) in the Methods section are entangled state (pulse I) in the samples, and the verifying pulse, dSout y cos ­ 2 dSin y cos ­ 2 1=2a2dJ2z12 [ 1=2dS2 kdJ2z12, where which occurs after a delay time t and veri®es the entanglement k 1 a2, and similarly for dSout (pulse II). These pulses have the same duration and optical 2 y sin ­ 2 with substitution Jz ! Jy. The coef®cient a jgn=4FA¢ < 2:5 is estimated17 using j < l2=2p frequency as the probe pulse used for the CSS measurements. as the resonant dipole cross-section, g 5 MHz as the full width of Between the two pulses the joint spin state of the two samples is the optical transition, F 4 as total angular momentum of the subject to decoherence. The photocurrents from the two pulses are hyper®ne ground state, A 2 cm2 as the probe beam cross-section, subtracted electronically and the variance of the difference, ¢EPR, is 700 MHz 6P PBS 3/2 Entangling and out verifying beams Jx2 Sy ­ m = 4 m = 4 F = 4 = 325 kHz Pumping 6S1/2 beams F = 3 Jx1 + Optical Entangling Verifying pumping pulse pulse Entangling and B-field verifying pulses 0.5 ms Time Figure 1 The experimental set-up, atomic level structure and the sequence of optical to a perfect spin polarization, p 1. The jJxj for the two cells is adjusted to be pulses. Fluorescence of atoms interacting with light is seen inside the cells (arti®cial equal to or better than 5%. After the optical pumping is completed 0.45 ms long colour). The two atomic samples in glass cells 3 3 3 cm at approximately room entangling and verifying pulses separated by a 0.5-ms delay are sent through. Both temperature are placed in a highly homogenous magnetic ®eld of 0.9 G (­ 325 kHz) pulses are blue-detuned by ¢ 700 MHz from the closest hyper®ne component of surrounded by a magnetic shield. Caesium atoms are optically pumped into F 4, the D2 line at 852 nm. The thermal atomic motion is not an obstacle in this experiment mF 4 ground state in the ®rst cell and into F 4, mF 2 4 in the second cell to but is rather helpful. The probe beam covers most but not the whole volume of the form coherent spin states oriented along the x axis for cell 1 and along -x for cell 2. atomic sample, but the duration of the pulses is longer than the transient time of an Optical pumping is achieved with circularly polarized 0.45 ms pulses at 852 nm and atom across the cell. Therefore each light pulse effectively interacts with and 894 nm with opposite helicity for the two cells. The mean spin value for each sample is entangles all atoms of the samples. In addition, a large detuning of the probe makes given by Jx NS4m 24mrmm 4Np ! 4N, where N is the number of atoms and rmm the Doppler broadening insigni®cant. is the probability of an atom being in the m state, with the limiting value corresponding NATURE | VOL 413 | 27 SEPTEMBER 2001 | www.nature.com © 2001 Macmillan Magazines Ltd 401 letters to nature measured as discussed in the Methods section. The vanishing ¢EPR The imperfect entanglement comes from several factors. First, the corresponds to two repeated measurements on the total spin state of vacuum noise of the entangling pulse prohibits a perfect prepara- the two samples producing the same results, that is, it corresponds tion of the spin state, especially for a small number of atoms. to a perfect knowledge of both ÃJz12 and ÃJy12 and therefore to a Second, the deviation of the initial spin state from CSS, which is due perfectly entangled state. In the experiment the minimal value of to the classical noise of the lasers, becomes more pronounced as the ¢EPR is 2dS2 owing to the quantum noise of the entangling and number of atoms grows. Finally, losses of light on the way from one verifying pulses. cell to another, as well as the spin-state decay between the two The results of measurements with t 0:5 ms are shown in Fig. 3. measurements preclude perfect entanglement at all atomic densi- The results are normalized to the CSS limit ¢ Jx (the linear ®t in ties. The spin-state decay is caused by collisions and the quadratic Fig. 2). This limit thus corresponds to the unity level in Fig. 3. The Zeeman effect. raw experimental data ¢EPR=¢ Jx for the entangled state are shown Measurements with delay times longer than 0.8 ms demonstrate as stars. The values below the unity level verify that the entangled no entanglement. The decoherence process can be illustrated by state of the two atomic samples has been generated and maintained visualizing two ellipses in the phase space, corresponding to the two for 0.5 ms. The derivation of the degree of entanglement from the samples, with the size along the long axes of the order of three, in data in Fig. 3 is given in the Methods section. The degree of units of the coherent state (experimentally measured value). When entanglement calculated operationally from the data without addi- the ellipses start to dephase (rotate) the total noise along the short tional assumptions is y 35 6 7 %. The degree of entanglement axis becomes equal to the coherent state noise (zero degree of useful for teleportation that was calculated using an additional, entanglement) after the time of the order of 1.2 ms if the initial experimentally proved assumption of the initially CSS for both entanglement of 65% (maximal degree allowed by the experimental samples is higher, y9 52 6 7 %. The predicted teleportation light noise) is assumed. After a 0.6-ms delay a degree of entangle- ®delity performed with the same two pulses as used in the present ment of 35% should be expected, given the experimentally mea- paper (see Methods section) is F 55%, which is above the classical sured dephasing time of 5 ms. These numbers agree reasonably well limit of 50%. The factors limiting the ®delity are of a technical with observations. nature, and we expect higher ®delity to be well within reach. It is instructive to analyse the difference between the degree of entanglement and the degree of classical correlations. For uncorre- lated atomic samples the normalized variance of the difference between the two photocurrents would be equal to two, using the notation of Fig. 3. The pure atomic part of this variance Vuncorr would be approximately 1.5 for medium atomic densities. For the 40 actual data in Fig. 3 the atomic part of the variance is approximately (a.u.) Vcorr 0:25. The degree of correlation, 1 2 Vcorr=Vuncorr, is 83% for this example. This number is much higher than the degree of 30 obe pulse, 20 2.0 10 1.5 Spectral variance of the pr 0 0 2 4 6 1.0 Collective spin of the atomic sample, Jx (× 1012) Figure 2 Determination of the coherent spin state limit for entanglement. The total 0.5 measured variance of the two quadratures of the photocurrent Normalized spectral variance ¢ dSout ycos ­ 2 dS out ysin ­ 2 dS 2 kdJ 2z12 kdJ 2y12 [ dS 2 kdJ 212 is plotted as a function of Jx. Jx is measured independently by magneto-optical resonance method and is varied by heating the cells and by adjusting optical pumping. In the absence 0 of atoms in the cells the measured spectral variance is due to the initial probe state 0 2 4 6 variance dS2. dS2 is at the vacuum (shot) noise level, which has been veri®ed Collective spin of the atomic sample, Jx (× 1012) experimentally by checking its characteristic linear dependence on the probe power. With atoms present the measured variance grows linearly with Jx at low densities, which proves Figure 3 Demonstration of the entangled spin state for two atomic samples. The that there is no classical contribution to the spin noise and that therefore the observed results are plotted as a function of Jx and are normalized to the CSS limit ¢ Jx (the atomic ¯uctuations at these densities are entirely due to quantum atomic noise21. The linear ®t in Fig. 2). This limitÐthe boundary between entangled (below the line) and degree of the spin polarization for the data in the ®gure (shown as squares) is nearly separable statesÐthus corresponds to the unity level in the ®gure (solid line). The raw perfect, p $ 95%, which means that the spin state is very close to the coherent spin state experimental data ¢EPR=¢ Jx Ðdata for the entangled spin state, which has lived for (CSS). We therefore conclude that a linear ®t to the observed data corresponds to the CSS t 0:5 msÐare shown as stars. The values below the unity level verify that the of both atomic samples for which dJ 212 2Jx and hence this line, which we denoted by entangled state of the two atomic samples has been generated and maintained for ¢ Jx , establishes the noise level corresponding to the right-hand side of the inequality 0.5 ms. The minimal possible level for ¢EPR=¢ Jx (maximum possible entanglement) (4). Deviations of the observed variance from the linear ®t at high atomic numbers are due is equal to 2dS2=¢ Jx (dotted line), that is, it is set by the total quantum noise of to the nonlinearly growing contribution of classical technical noise of the spin state the two pulses. The normalized shot noise level of the verifying pulse, dS2=¢ Jx because of the technical noise of lasers, the non-ideal cancellation of the back action of (dashed line), which is used for calculations of the degree of entanglement, is also the probe on the two samples, and so on. shown. 402 © 2001 Macmillan Magazines Ltd NATURE | VOL 413 | 27 SEPTEMBER 2001 | www.nature.com letters to nature entanglement, implying that entanglement requires something the best entanglement possible for a given ratio of the light noise and the atomic spin stronger than classical correlations. The reason for this difference noise contribution. This value of entanglement would correspond to the minimal is that in quantum mechanics the measuring device (light in this possible difference between the two pulses, ¢min EPR 2dS2. We note that this value is closer to unity (stronger entanglement) than the degree of entanglement y 1 2 ¢min case) becomes entangled with the measured object and therefore the EPR 2 dS2 = ¢ Jx 2 dS2 1 2 dS2= ¢ Jx 2 dS2 calculated in the previous paragraph. This noise of these two subsystems cannot be treated independently. is because the entanglement in ref. 7 is calculated assuming that the samples prior to Thus, we have demonstrated on-demand generation of entangle- entanglement are known to be in the CSS. This fact has been experimentally veri®ed in the ment of two separate macroscopic objects, which can be maintained present paper for up to intermediate atomic densities, so we may use this way of calculating the degree of entanglement at these densities as well. However, we are for more than 0.5 ms. The state we have demonstrated is not a interested in the degree of entanglement, which has survived the delay time and which has maximally entangled `SchroÈdinger's cat' state, which for 1012 atoms been measured by the verifying pulse. For various reasons, including the decoherence would not survive even for a femtosecond under the conditions of during the delay time, the differential noise between the two pulses does not reduce our experiment. Our state is similar to a two-mode squeezed state, to its minimal value, ¢EPR . ¢min EPR 2dS2. Therefore the actual degree of entanglement, and is an example of a non-maximally entangled state which is as witnessed by the measurement, is lower than the maximum possible: y9 1 2 h suitable for a particular purpose, for example, atomic teleportation. exper , ytheory . Here the degree of squeezing, hexper ¢EPR 2 dS2 =¢ Jx . htheory obtained from the data in Fig. 3 is worse than the theoretical best value, htheory, owing in The long lifetime of this multi-particle entanglement is due to a part to the diffusion of the state during the delay time between the entangling and the high symmetry of the generated state. Entanglement manifests itself verifying pulses. From Fig. 3 we ®nd hexper 0:48 and therefore y9 52% for only in the collective properties of the two ensembles. Therefore a Jx < 3:5 3 1012. loss of coherence for a single atom makes a negligible effect on Expected ®delity of teleportation the entanglement, unlike in a maximally entangled multi-particle It is also possible to estimate the ®delity of teleportation which would be achieved if state. The entanglement is generated by means of light propa- the second, verifying pulse were (instead of veri®cation) used for the Bell measure- gating through the two samples and therefore the samples can be ment on one of the entangled samples and a sample to-be-teleported. Using the value distant, as is required for communications. The off-resonant of y9 and equation (3) from ref. 7 we obtain a value for the ®delity of teleportation character of the interaction used for the creation of entangle- of F 55% for Jx < 3:5 3 1012. This is higher than the classical boundary of F 50%. ment allows for the potential extension of this method to other We note that the entangled state reported here has a random element in it, namely, every media, possibly including solid-state samples with long-lived (ideal) entangling measurement creates the state Jz1 Jz2 x;Jy1 Jy2 p with x;p spin states. M random measured values. However, this state is as ef®cient for teleportation, for example, as is a state Jz1 Jz2 0; Jy1 Jy2 0. Methods Received 11 June; accepted 27 July 2001. Entangling spin components with a single light pulse 1. Einstein, A., Podoslky, B. & Rosen, N. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777±780 (1935). The Larmor precession of the Jz; Jy components with a common frequency ­ does not 2. Bell, J. S. Speakable and Unspeakable in Quantum Mechanics (Cambridge Univ. Press, Cambridge, change their mutual orientation and size and therefore does not affect the entanglement. 1988). On the other hand, the precession allows us to extract information about both z and y 3. Aspect, A., Dalibard, J. & Roger, G. Experimental test of Bell's inequalities using time-varying components from a single probe pulse as described below. Moreover, in the laboratory analyzers. Phys. Rev. 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Lett. 84, behaviour is described by the following equations: ÇÃJz t ­ÃJy t ; ÇÃJy t 2 ­ÃJz t 2726±2729 (2000). bÃSz t , whereas the Stokes operators still evolve according to equation (2). Solving the spin 7. Duan, L. M., Cirac, J. I., Zoller, P. & Polzik, E. S. Phys. Rev. Lett. 85, 5643±5646 (2000). equations and using equation (2) we obtain 8. Bouwmeester, D., Pan, J. W., Mattle, K., Eibl, M., Weinfurter, H. & Zeilinger, A. Experimental quantum teleportation. Nature 390, 575±579 (1997). à Sout y t à Siny t a ÃJz12 cos ­t ÃJy12 sin ­t 3 9. Boschi, D., Branca, S., De Martini, F., Hardy, L. & Popescu, S. Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. The ÃJz;y components are now de®ned in the frame rotating with the frequency ­ around Lett. 80, 1121±1125 (1998). the magnetic ®eld direction x. It is clear from equation (3) that by measuring the 10. Furusawa, A. et al. Unconditional quantum teleportation. Science 282, 706±709 (1998). cos ­t =sin ­t component of ÃSout y t we can acquire the knowledge of the z/y spin 11. Mattle, K., Weinfurter, H., Kwiat, P. G. & Zeilinger, A. Dense coding in experimental quantum projections. Simultaneous measurement of both spin components of the two atomic communication. Phys. Rev. Lett. 76, 4656±4659 (1996). samples is possible because they commute: bÃJ 12. Braunstein, S. L. & Kimble, H. J. Dense coding for continuous variables. Phys. Rev. A 61, 042302 z1 à Jz2; ÃJy1 ÃJy2c i Jx1 Jx2 0. (2000). Degree of entanglement 13. Ban, M. Quantum dense coding via a two-mode squeezed-vacuum state. J. Opt. B 1, L9±L11 (1999). The differential noise for the entangling (pulse I) and verifying (pulse II) pulses can be 14. Ou, Z. Y., Pereira, S. F., Kimble, H. J. & Peng, K. C. Realization of the Einstein-Podolsky-Rosen expressed using equation (3) as: paradox for continuous variables. Phys. Rev. Lett. 68, 3663±3666 (1992). 15. Sackett, C. A. et al. Experimental entanglement of four particles. Nature 404, 256±259 (2000). ¢EPR dÅS2cos dÅS2sin [ dSIout y cos ­ 2 dSIIout y cos ­ 2 dSIout y sin ­ 2 dSIIout y sin ­ 2 16. Rauschenbeutel, A. et al. Step-by-step engineered multiparticle entanglement. Science 288, 2024± n o 2028 (2000). dS2 2 2 II k k21dS2I d à Jz12 I 2 d ÃJz12 II d ÃJy12 I 2 d ÃJy12 II 5 17. Kuzmich, A. & Polzik, E. S. Atomic quantum state teleportation and swapping. Phys. Rev. Lett. 85, 5639±5642 (2000). [ dS2II kdJ2EPR 18. Kitagawa, M. & Ueda, M. Spin squeezed states. Phys. Rev. A 47, 5138±5143 (1993). 19. Wineland, D. J., Bollinger, J. J., Itano, W. M., Moore, F. L. & Jeomzem. D. J. Spin squeezing and reduced The variance dJ2 quantum noise in spectroscopy. Phys. Rev. A 46, R6797±R6800 (1992). EPR determines the uncertainty of the spin components of the state prepared by the entangling pulse and measured after time t. It contains two contributions; 20. Kupriyanov, D. V. & Sokolov, I. M. Optical detection of magnetic resonance by classical and squeezed dS2 light. Quant. Opt. 4, 55±70 (1992). I , due to the quantum noise of the entangling pulse, and all other terms in the curly brackets, which are due to decoherence during the time t. The verifying pulse in our 21. Hald, J., Sùrensen, J. L., Schori, C. & Polzik, E. S. Spin squeezed atoms: a macroscopic entangled experiment is also not noiseless and therefore its variance, dS2 ensemble created by light. Phys. Rev. Lett. 83, 1319±1322 (1999). II, also contributes to ¢EPR. In the experiment dS2 22. Kuzmich, A., Mandel, L. & Bigelow, N. P. Generation of spin squeezing via continuous quantum I dS2II dS2 . The exact entanglement condition, ¢EPR , ¢ Jx , is obtained by substituting equation (5) into equation (4). The degree of entanglement can nondemolition measurement. Phys. Rev. Lett. 85, 1594±1597 (2000). be de®ned as y 1 2 dJ2EPR=2Jx 1 2 ¢EPR 2 dS2 = ¢ Jx 2 dS2 . y varies from 0 (separable state) to 1 (perfect entanglement). The highest degree of entanglement calculated operationally from the data is y 35 6 7 %. Acknowledgements An alternative calculation of the degree of entanglement can be carried out7, which takes We gratefully acknowledge the contributions of J. Hald, J. L. Sùrensen, C. Schori and into account that the initial state of both samples is characterized by the CSS noise level. A. Verchovski. We also thank I. Cirac, A. Kuzmich, A. Sùrensen and P. Zoller for Equation (2) of ref. 7, using the notation of the present paper, yields for the highest discussions. possible degree of entanglement (de®ned again as y 1 2 dJ2EPR=2Jx ) and created by the ®rst pulse in our protocol, the value ytheory 1 2 htheory where the degree of the ``two- Correspondence and requests for materials should be addressed to E.P. mode squeezing'' is, according to ref. 7, htheory dS2= dS2 2kJx dS2=¢ Jx . This is (e-mail: polzik@ifa.au.dk). NATURE | VOL 413 | 27 SEPTEMBER 2001 | www.nature.com © 2001 Macmillan Magazines Ltd 403