Acc. Chem. Res. 1996, 29, 585-590
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Manipulating the Motion of a Single Trapped Atom C. MONROE,* D. M. MEEKHOF, B. E. KING, D. LEIBFRIED, W. M. ITANO,
AND
D. J. WINELAND
National Institute of Standards and Technology, Boulder, Colorado 80303 Received May 31, 1996
Introduction A single atom confined in an electromagnetic trap (see Figure 1) has many similarities to a molecule. Spectroscopy reveals both internal electronic structure and external vibrational structure. In the case of a trapped atom, the vibrational states can be controlled to a high degree, primarily because the effective vibrational potential is externally applied and can be made almost purely harmonic. By coupling the electronic and vibrational states of a trapped atom with laser radiation, motional wavepackets of the atom can be manipulated and interesting states of motion can be created which exhibit highly nonclassical behavior. In this Account, we describe experiments which control the bound states of a “molecule” formed by a single trapped atomic ion bound to a harmonic trap. We create1 (i) Fock states (eigenstates of vibration) including the n ) 0 zero-point vibrational ground state, (ii) coherent states, where the wavepacket orbit corresponds most closely to classical oscillation in the trap, (iii) squeezed states, in which the wavepackets undergo a “breathing” mode in the trap, and (iv) “Schro¨dinger cat” states of motion,2 where the trapped ion is placed in a coherent superposition of spatially separated coherent states. Such states of motion are useful for a number of applications, including issues involving quantum measurement theory, sensitive electronic detection,3 spectroscopy below the quantum C. Monroe (born in 1965 in Detroit, MI) received his S.B. in physics from the Massachusetts Institute of Technology in 1987, and his M.S. and Ph.D. in physics from the University of Colorado, Boulder, in 1989 and 1992, respectively, where he was a Feldkamp Fellow during his research on optical and magnetic trapping of neutral atoms with the goal of Bose−Einstein condensation in a vapor. He was a National Research Council postdoctoral fellow in the National Institute of Standards and Technology (NIST) Ion Storage Group from 1992 to 1994, and has been a staff physicist in the same group since 1995. D. M. Meekhof (born in 1965 in Seattle, WA) received her B.S. in physics from the California Institute of Technology in 1987, and her Ph.D. from the University of Washington in 1993 on the measurement of parity nonconservation in atomic thallium and lead. She is currently a National Research Council postdoctoral fellow at NIST. B. E. King (born in 1969 in New Westminster, Canada) received his B.Sc. in physics from Simon Fraser University in 1992, and M.S. in physics from the University of Colorado, Boulder. He is currently a doctoral candidate at the University of Colorado, Boulder, working in the Ion Storage Group at NIST. D. Leibfried (born in 1965 in Stuttgart, Germany) received his diploma at the University of Munich and the Max Planck Institute for Quantum Optics in Munich, Germany, and received his Ph.D. there in 1995 on the high-precision spectroscopy of hydrogen. He has been a guest researcher with the NIST Ion Storage Group since 1995 with a Deutsche Forschungsgemeinschaft fellowship. W. M. Itano (born in 1951 in Pasadena, CA) received his B.S. from Yale University in 1973 and his M.A. and Ph.D. from Harvard University in 1975 and 1979, respectively. Since 1979 he has been a staff physicist in the NIST Ion Storage Group. D. J. Wineland (born in 1944 in Milwaukee, WI) received his B.S. in physics from the University of California, Berkeley, in 1965. He was a National Science Foundation Graduate Fellow from 1965 to 1970 at Harvard University where he performed research on an atomic deuterium maser and received the M.A. and Ph.D. degrees in 1966 and 1970, respectively. From 1970 to 1975 he was a postdoctoral research associate at the University of Washington, Seattle, where he worked on the rf spectroscopy of trapped electrons. In 1975, he joined the Time and Frequency Division of NBS (now NIST) where, since 1978, he has been the Ion Storage Group leader. He is a fellow of NIST, the American Physical Society, and the Optical Society of America and a member of Sigma Xi, AAAS, and the National Academy of Sciences. Also, he has a very nice personality and drives a 1975 Dodge van.
Figure 1. Image of one, two, and three beryllium (9Be+) ions held in a rf (Paul) electromagnetic trap. The ions are illuminated by resonant laser light, and the scattered fluorescence is collected. The resolution (a few micrometers) is limited by the optical imaging system and is much larger than the actual size of the atomic wavepackets (a few nanometers). All of the experiments reported here deal with a single trapped ion.
limit,3 quantum computers,4,5 and the study of quantum decoherence.6 Internal and External States of a Trapped Ion Since 9Be+ is hydrogenic, the internal energy level structure of its outer 2S electron is fairly simple. We isolate two particular hyperfine energy levels of the electron in its 2S1/2 orbital ground state. The |F ) 2, mF ) 2〉 (denoted by |V〉) and |F ) 1, mF ) 1〉 (denoted by |v〉) levels are separated in frequency by ωHF/2π ≈ 1.25 GHz. The other 2S1/2 Zeeman levels are well resolved from these two states by the application of a (1) Meekhof, D. M.; Monroe, C.; King, B. E.; Itano, W. M.; Wineland, D. J. Phys. Rev. Lett. 1996, 76, 1796. (2) Monroe, C.; Meekhof, D. M.; King, B. E.; Wineland, D. J. Science 1996, 272, 1100. (3) Heinzen, D. J.; Wineland, D. J. Phys. Rev. A 1990, 42, 2977. Wineland, D. J.; Bollinger, J. J.; Itano, W. M.; Heinzen, D. J. Phys. Rev. A 1994, 50, 67. (4) Lloyd, S. Sci. Am. 1995, 273 (Oct), 140. DiVincenzo, D. P. Science 1995, 270, 255. Bennett, C. H. Phys. Today 1995, 48 (Oct), 24. (5) Cirac, J. I.; Zoller, P. Phys. Rev. Lett. 1995, 74, 4091. Monroe, C.; Meekhof, D. M.; King, B. E.; Itano, W. M.; Wineland, D. J. Phys. Rev. Lett. 1995, 75, 4714. (6) Zurek, W. H. Phys. Today 1991, 44 (Oct), 36.
S0001-4842(95)00252-4 This article not subject to U.S. Copyright. Published 1996 by the American Chemical Society
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Monroe et al. and phase. Both beams are near λ ≈ 313 nm and detuned ∆ ≈ 10 GHz from an excited 2P1/2 electronic orbital state (radiative line width γ/2π ≈ 19.4 MHz). Because ∆ . γ, the excited 2P1/2 state can be adiabatically eliminated, resulting in a stimulated Raman coupling between the two 2S ground states (|V〉 and |v〉) exhibiting a linewidth inversely proportional to the interaction time. When R1 and R2 are applied to the ion with wavevector difference δk ) k1 - k2 along the x-direction, the effective coupling Hamiltonian in the rotating-wave approximation is given by1 †
†
Heff ) g(σ+eiη(a +a)-iδt + σ-e-iη(a +a)+iδt)
Figure 2. (a) Electronic (internal) and motional (external) energy levels (not to scale) of the trapped 9Be+ ion, coupled by indicated laser beams R1 and R2. The difference frequency of the Raman beams R1 and R2 is set near ωHF/2π ≈ 1.250 GHz, providing a two-photon Raman coupling between the 2S1/2(F ) 2, mF ) 2) and 2S1/2(F ) 1, mF ) 1) hyperfine ground states (denoted by |V〉 and |v〉, respectively). The motional energy levels are depicted by a ladder of vibrational states separated in frequency by the trap frequency ω. The Raman beams are detuned ∆ ≈ -12 GHz from the 2P1/2(F ) 2, mF ) 2) excited state. As shown, the Raman beams are tuned to the red sideband. (b) Detection of the internal state is accomplished by illuminating the ion with σ+-polarized “detection” beam D2, which drives the cycling 2S1/2(F ) 2, mF ) 2) f 2P3/2(F ) 3, mF ) 3) transition, and observing the scattered fluorescence. The vibrational structure is omitted from the figure since it is not quite resolved (2P radiative line width γ/2π ≈ 19.4 MHz). Beam D1, also σ+-polarized, allows the spontaneous recycling from the |v〉 to |V〉 state.
∼0.2 mT magnetic field. The electronic levels of interest are summarized in Figure 2.9 In order to controllably access the vibrational levels of an atom oscillating in a trap, it is necessary that the vibrational level spacing (trap oscillation frequency) be much larger than any atomic or motional relaxation rates. To this end, we have designed a radiofrequency (rf) (Paul) ion trap7-9 which strongly confines a single 9Be+ ion. We realize harmonic oscillation frequencies of (ωx, ωy, ωz)/2π ≈ (11, 19, 29) MHz of atomic motion along the three principal axes of the trap. As shown in Figure 2, the internal 9Be+ electronic states |V〉 and |v〉 are accompanied by the ladder of external harmonic oscillator levels of energy En ) pω(n + 1/2), where we have only considered the x-dimension of the oscillator (ω ≡ ωx) and its associated quantum number n ≡ nx ∈ (0, 1, 2, ...). Coherent manipulation of the state of motion is accomplished by applying a pair of laser beams (R1 and R2) to the trapped ion (Figure 2a). By tuning the difference frequency of R1 and R2 to be near ωHF, twophoton stimulated Raman transitions between the 2S1/2 electronic ground states can be driven. In general, this Raman process alters both the internal electronic state of the ion and the vibrational state of the ion in the trap. The two Raman beams are generated from a single laser source and modulator, allowing excellent stability of their relative frequency (7) Dehmelt, H. G. Adv. Atom. Mol. Phys. 1967, 3, 53; 1969, 5, 109. (8) Jefferts, S. R.; Monroe, C.; Bell, E. W.; Wineland, D. J. Phys. Rev. A 1995, 51, 3112. (9) Monroe, C.; Meekhof, D. M.; King, B. E.; Jefferts, S. R.; Itano, W. M.; Wineland, D. J.; Gould, P. Phys. Rev. Lett. 1995, 75, 4011.
(1)
In this expression, σ( is the raising (lowering) operator of the internal states |V〉 and |v〉, and a† and a are the creation and annihilation operators for the state of harmonic vibration: a|n〉 ) xn|n - 1〉 and a†|n〉 ) xn + 1|n + 1〉. The parameter g is the coupling strength which depends on ∆ and the intensity of the laser beams,9 η ) |δk|x0 is the Lamb-Dicke parameter (x0 ) (p/2mω)1/2 ≈ 7 nm is the spatial spread of the n ) 0 ground vibrational state), and δ is the relative frequency of the two Raman beams R1 and R2 with respect to the free atom resonance. The Lamb-Dicke parameter controls the amount of coupling between the internal electronic states and the external vibrational states, and has the value η ≈ 0.2 in the experiment.9 By setting δk to be parallel to the x-axis of the trap, there is almost no coupling between the internal states and motion in the y- and z-directions. By tuning the relative frequency to δ ) ω(n′ - n), Raman transitions are driven between the levels |V,n〉 and |v,n′〉. In this case, Heff is dominated by a single stationary term, resulting in the coupling matrix element †
†
〈v,n′|Heff|V,n〉 ) g〈v,n′|σ+eiη(a +a) + σ-e-iη(a +a)|V,n〉 (2) This coupling describes a coherent Rabi flopping process between the states |V,n〉 and |v,n′〉 induced by the laser beams. In the Lamb-Dicke limit, where η〈a† + a〉 , 1, the above matrix element can be expanded to its lowest order, resulting in a Rabi flopping frequency of
(
Ωn,n′ ≈ g
)
η|n′-n| n (n - 1)(n> - 2)...(n< + 1) |n′ - n|!x > > (3)
where |n′ - n| is the sideband order, and n> (n
