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Experimental Entanglement of Four Particles letters to nature ................................................................. Rabi frequency ­ and the Lamb–Dicke parameter is h. For an excitation involving momentum transfer ~k and a total particle Experimental entanglement 2 1/2 mass of M, h is given by (~k /2Mn) . Entanglement is achieved by applying H for a time t ¼ p=2­˜ , making the spin wavefunction of four particles i ↑↑ i2 ↓↓ i =Î jw2 ¼ðj ij Þ 2. This spin state is in fact created for any C. A. Sackett*, D. Kielpinski*, B. E. King*†, C. Langer*, V. Meyer*, initial motional state |ni, so long as the Lamb–Dicke criterion 2 C. J. Myatt*‡, M. Rowe*, Q. A. Turchette*‡, W. M. Itano*, D. J. Wineland* h n p 1 is satisfied. & C. Monroe* In order for the intermediate states |↑↓ 1i and |↓↑ 1i to be negligibly occupied, the detuning d must be large compared to the transition * Time and Frequency Division, National Institute of Standards and Technology, linewidth h­. However, it is clear from the expression for ­˜ that the Boulder, Colorado 80303, USA entanglement speed is maximized for small d, and in fact the † Atomic Physics Division, NIST, Gaithersburg, Maryland 20899, USA technique can still be applied for d < h­ (refs 14, 16). Although ‡ Research Electro-Optics, 1855 S. 57th Ct., Boulder, Colorado 80301, USA motional excitation does then occur to some degree, for select .................................. ......................... ......................... ......................... ......................... ........ values of d the excitation vanishes at precisely the time that the Quantum mechanics allows for many-particle wavefunctions that entangled spin state is created. The condition for this to occur is p cannot be factorized into a product of single-particle wave- d=h­ ¼ 2 m ð2Þ functions, even when the constituent particles are entirely distinct. Such ‘entangled’ states explicitly demonstrate the non-local char- for any integer m, and the maximum excitation during the pulse acter of quantum theory1, having potential applications in high- then has mean quantum number n¯ ¼ 1=2m. Our experiment is precision spectroscopy2, quantum communication, cryptography operated with m ¼ 1. and computation3. In general, the more particles that can be As discussed in ref. 6, the entanglement method is scalable in the entangled, the more clearly nonclassical effects are exhibited4,5— sense that precisely the same operation can be used to generate the and the more useful the states are for quantum applications. Here N-particle entangled state we implement a recently proposed entanglement technique6 to p i ↑↑… ↑i Nþ1 ↓↓… ↓i= generate entangled states of two and four trapped ions. Coupling jwN ¼ðj þ i j Þ 2 ð3Þ between the ions is provided through their collective motional degrees of freedom, but actual motional excitation is minimized. if N is any even number, while for N odd, |wNi can be generated using Entanglement is achieved using a single laser pulse, and the one entanglement pulse accompanied by a separate independent method can in principle be applied to any number of ions. rotation of each particle’s spin. Most experimental demonstrations of entanglement to date have If the ions are uniformly illuminated, the Mølmer and Sørensen relied on the selection of data from random processes, such as the scheme6,14 requires that they all participate equally in the inter- preparation and detection of photon pairs in parametric down- mediate motional excitation, which implies that the only suitable conversion7–9 or of atoms in a thermal beam10. All methods of this mode for arbitrary N is the centre-of-mass mode. However, this type suffer from inescapable signal degradation when entanglement mode has a practical disadvantage because in our experiments of larger numbers of particles is attempted, as the probability of fluctuating ambient electric fields cause it to heat at a significant randomly generating the appropriate conditions decreases expo- rate. Although for large d the entanglement operation is largely nentially. For instance, in the experiment of ref. 7, two-photon independent of the motion, so that heating is unimportant, in the entangled states could be generated and detected at a rate of roughly small-d case it is necessary that motional decoherence be avoided. 1,000 per second, three-photon states at a rate of 30 per hour, and Modes involving only relative ion motion couple to higher four-photon states at an extrapolated rate of several per year. moments of the field, so heating of these modes is negligible15. Trapped ions have been suggested as a system in which such effects For N ¼ 2 and N ¼ 4, such modes do exist in which each particle might be avoided11, and we have demonstrated two-particle entan- participates with equal amplitude17. In both cases, they are sym- glement in a deterministic way12. By ‘deterministic’ we mean that metric ‘stretch’ modes, in which alternating ions oscillate out of the desired state could be produced with a high degree of certainty at a user-specified time13, which is necessary for avoiding the degrada- tion described above. However, that experiment relied on the particular behaviour of two ions in a quadrupole radio-frequency 0 trap, and could not easily be applied to larger numbers of particles. The entanglement technique proposed by Mølmer and 6,14 ω Sørensen can be understood by considering a pair of spin-half 2 0 charged particles confined together in a harmonic potential. The 1 δ 1 ~ ν energy levels of this system are illustrated in Fig. 1, where q0 is the 0 0 internal energy splitting of each particle, and n is the oscillation frequency of a particular collective mode of the particles in the trap. 15 ω Using laser-cooling and optical-pumping techniques , the particles 0 are initially prepared in their spin-down internal state and in the ground state of their collective motion: jwi ¼j↓↓ 0i. By applying 2 2 0 optical fields oscillating at q0 þ n d and q0 n þ d, the two-step transition from |↓↓ 0i to |↑↑ 0i is driven. For sufficiently large d, the Figure 1 Entanglement scheme for two particles. Each ion is initially prepared in the |↓i intermediate states |↑↓ 1i and |↓↑ 1i are negligibly occupied, so that no internal state, and the collective motion of the pair is cooled to its ground state |0i. Laser 2 ↓↓i ↑↑i motional excitation occurs. The resulting interaction hamiltonian, fields oscillating near q0 þ n and q0 n couple the | and | states as shown. By in the rotating-wave approximation and the Lamb–Dicke limit, is detuning the single transition frequencies by a small amount d, the populations of the |↓↑1i then and |↑↓1i states are kept small. Then, by driving the double transition for the appropriate time, the entangled state ðj ↑↑ i2 ij ↓↓ iÞ=Î2 is created. For four ions, the same procedure ~­˜ H ¼ ðj↑↑ih↓↓jþj↓↓ih↑↑jþj↑↓ih↓↑jþj↓↑ih↑↓jÞ ð1Þ generates the state ðj ↑↑↑↑ i þ ij ↓↓↓↓ iÞ=Î2. We note that in the actual experiment, each 2 of the single transitions shown is itself a two-photon Raman transition, driven by a pair of with ­˜ ¼ h2­2=d when the single particle j↓i↔j↑itransition has laser beams; the entire process therefore consists of a four-photon transition. 256 © 2000 Macmillan Magazines Ltd NATURE | VOL 404 | 16 MARCH 2000 | www.nature.com letters to nature phase. We use these modes here. Excitation of the centre-of-mass Table 1 Characterization of two-ion and four-ion states mode still affects the experiment, as the motion in spectator modes NP0 P1 P2 P3 P4 r(↑↓) modifies the coupling strength to the mode of interest14,15. For this ............................................................................................................................................................................. 2 0.43 0.11 0.46 – – 0.385 reason, we initially cool both the centre-of-mass and stretch modes 4 0.35 0.10 0.10 0.10 0.35 0.215 ............................................................................................................................................................................. to near their ground state. ↓i 9 + N is the number of ions, Pj denotes the probability that j ions were measured to be in | , and r(↑↓) The experiment was performed using Be ions confined in a denotes the amplitude of the density matrix element r↑…↑,↓…↓. Uncertainties in r(↑↓) and the N ¼ 2 miniature linear radio-frequency trap18, with the N ions lying in a populations are 60.01, and uncertainties in the N ¼ 4 populations are 60.02. line along the trap’s weak axis. Two spectrally resolved ground-state hyperfine levels comprise the effective spin-half system, with ↓i[ ; 2 i ↑i[ ; 2 i j jF ¼ 2 mF ¼ 2 and j jF ¼ 1 mF ¼ 1 , where F is that the middle probabilities are non-zero indicates that we do not the total angular momentum quantum number, and ~mF is the generate the entangled states with perfect accuracy. projection of the angular momentum along the quantization axis In order to prove that we are generating a reasonable approxima- defined by an externally applied magnetic field. The hyperfine tion to |wNi, it is necessary to prove that the populations of |↑…↑i splitting q0/2p is approximately 1.25 GHz. Coherent coupling and |↓…↓i are coherent. In terms of the density matrix for the ↓i ↑i between | and | is provided by stimulated Raman transitions. system, r, we must measure the far off-diagonal element r↑…↑,↓…↓, The two Raman laser beams have a wavelength of 313 nm with a whose amplitude will be abbreviated r(↑↓). This can be achieved by difference frequency near q0, and are perpendicular, with their applying a simple analysis pulse to the ions before observing them. difference wave-vector lying along the line of ions. They are detuned If the Raman difference frequency is set to q0 (and the frequency ,80 GHz blue of the 2P1/2 excited state, with intensities giving modulator turned off), each ion i undergoes ordinary Rabi oscilla- ­=2p < 500 kHz.
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