We develop a multi-valued logic for quantum computing for use in multi-level quantum systems, and discuss the practical advantages of this approach for scaling up a quantum computer. Generalizing the methods of binary quantum logic, we establish that arbitrary unitary operations on any number of d-level systems (d > 2) can be decomposed into logic gates that operate on only two systems at a time. We show that such multi-valued logic gates are experimentally feasible in the context of the linear ion trap scheme for quantum computing. By using d levels in each ion in this scheme, we reduce the number of ions needed for a computation by a factor of log 2 d. 03.67. Lx, 03.65.Bz, 89.80.+h

It is often desirable in laser spectroscopy and isotope separation to extract as much as possible of an atomic or molecular population that is distributed among a number of ground-state sublevels and low-lying metastable levels.We describe a form of coherent trapping that occurs when multiple resonant laser beams are used to couple the various ground states to a common upper level. This effect prevents the extraction of the entire population. We then study the effect with two dye lasers and an atomic beam and suggest possible ways to maximize the pumping efficiency.When two cw lasers are tuned so that they couple two different ground-state sublevels to a common upper level, there is no steady-state population in the upper This effect, which holds even for intense laser fields, might be termed coherent trapping of atomic populations. It is due to optical pumping of the ground-state sublevels into a coherent superposition state that is decoupled from the laser fields. The theory of the effect has been worked out by Arimondo and Orriols' and by Whitley.2 The source of the effect is easily seen by writing the wavefunction in the interaction picture, (1) and substituting it into Schrddinger's equation. Here we have taken the atomic-energy eigenfunctions as Oi(r) and the energy of the various states as Ei = hw 1 , i = 1,2,3. The amplitudes then satisfy the equations of motionwhere we have taken the applied field to beand we have neglected the counterrotating terms. The two dipole matrix elements, which are assumed real for convenience, are denoted by gj, i = 1,2, while the detuning of the two lasers from resonance, b6 and 5,, are defined in Fig 1. Examination of Eqs. (2a) and (2c) shows that there is a constant of the motion when Ba = 6b =_ 6, i.e., when the lasers are tuned to the two-photon resonance. The effect of the constant is most easily seen if we introduce two new amplitudes, r(t) = ai(t)cos 0 -a3(t)sin 0, (4a) s(t) = a,(t)sin 0 + a 3 (t)cos 0,where the angle 0 is defined byThe equations of motion for the amplitudes of the two ground states are then expressible in terms of r(t) andwhere R is the generalized Rabi frequency(5b) (6) One linear combination of the ground states is coupled to the excited state by the applied fields, and the other linear combination is decoupled entirely. The population initially in the linear combination,remains there. If the initial states are not prepared coherently, this means that half of the population remains in the ground state and nothing is gained by the application of two lasers rather than one. We have not included spontaneous or collisional damping in the calculations, but these are included in the calculations of Refs. 1 and 2. The result of including damping is that the population initially in the linear combination of states s is optically pumped in a few lifetimes into the decoupled linear combination r. In the steady state there is no population in level 2 or in the linear combination s. All of it is in the linear combination r. The atom is decoupled from the...

Calculations are presented that show the response of an atom to a picosecond laser pulse which resonantly excites a manifold of Rydberg states. The coherent atomic state that is produced is of the form of a spatially localized wave packet. The motion, decay, and reformation of the wave packet are described and related to the complicated quantum beat pattern that appears in the subsequent spontaneous decay.

We report an observation of the long-term evolution of a radially localized electronic wave packet formed by the coherent superposition of Rydberg states of atomic potassium. Initially, the wave packet can be described classically. Subsequent dephasing of the discrete states in the superposition leads to a loss of spatial localization so that the evolution can no longer be described classically. However, the wave packet revives at a later time. Theory and experiment show good agreement including an accurate measurement of the phase shift of the wave packet on revival.PACS numbers: 32.90.+a, 31.50,+w, 32.60.+i Quantum-mechanical wave-packet states can approach the classical ideal of a spatially localized particle traveling along a well-defined trajectory. The uncertainty principle places a limitation on this localization. But, for bound systems excited appreciably above their ground states, this is not a very stringent limitation. A more serious limitation on the quasiclassical nature of the states is the fact that for most systems the wave packet does not remain localized, but rapidly spreads.Normally, as time goes on the wave packet continues to spread and becomes less and less classical in its nature. There are a few cases, however, in which the spreading reverses itself and the wave packet relocalizes, again approximating a particle moving on a classical trajectory. A closely related decay and revival of classical coherence has been predicted, *~3 and observed 4 recently in the micromaser realization of the Jaynes-Cummings problem. There it is the coherent Rabi oscillations of the inversion of a Rydberg atomic transition that decay and revive. In both cases the revival is possible only due to the discreteness of the quantized energy states. In the Jaynes-Cummings problem it is the quantized nature of the cavity field, while in the present case it is the quantized nature of the atomic energy. A second necessary feature in both cases is the fact that the coherent superposition state is made up of frequency components that are almost equally spaced.In this paper, we report the observation of the decay and revival of a spatially localized Rydberg electronic wave packet. Methods for exciting such wave packets have been reported in several papers 5 " 9 and several different types of these wave packets have been observed. 10~13 In this experiment, a radially localized wave packet is formed by the coherent excitation of Rydberg states by a short, optical pulse. The Rydberg states have a range of values for the principal quantum number n with an average value of n. The resulting wave packet has the appearance of a shell oscillating between the nucleus and the outer turning point. The oscillations are at the classical orbital period. The classical orbital period is inversely proportional to the first derivative of the average energy of the wave packet with respect to n (xQ^lKh* a.u.). 5 These oscillations have been observed experimentally for a few periods. u,nThe long-term evolution of the wave packet is more complex. ...

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Abstract. It is shown that when a monochromatic laser couples a single atomic ground level to two closely spaced excited levels the system can be driven into a state in which quantum interference prevents any fluorescence from the excited levels, regardless of the intensity of the exciting field. This steady-state interference occurs only at a particular excitation frequency which depends on the separation of the excited states and the relative size of the two transition dipole matrix elements. The results are derived from the density matrix equations of motion. It is shown that a correct description of the effect requires the inclusion of generalised Einstein A coefficients which are usually neglected in phenomenological damping theories. A dressed-state analysis is introduced to simplify the generalisation to atoms having more complex manifolds of excited states. Analogous interferences in multiphoton absorption and ionisation are also discussed briefly.

We describe the time evolution of a wave function in the infinite square well using a fractional revival formalism and show that at all times the wave function can be described as a superposition of translated copies of the initial wave function. Using the model of a wave form propagating on a dispersionless string from classical mechanics to describe these translations, we connect the reflection symmetry of the square-well potential to a reflection symmetry in the locations of these translated copies, and show that they occur in a ''parity-conserving'' form. The relative phases of the translated copies are shown to depend quadratically on the translation distance along the classical path. We conclude that the time-evolved wave functions in the infinite square well can be described in terms of translations of the initial wave-function shape, without approximation and without any reference to its energy eigenstate expansion. That is, the set of translated initial wave functions forms a Hilbert space basis for the time-evolved wave functions. ͓S1050-2947͑97͒06606-7͔

Rydberg atomic wave packets have demonstrated striking classical properties. In particular, the wave packets display regions in which they evolve like classical atoms. However, between these regions the wave packet is dispersed and its evolution departs from the classical model. In these nonclassical regions, the wave packet can either be dispersed or coalesced into a number of equally spaced sub-wave-packets.The regions in which the sub-wave-packets appear have come to be known as fractional revivals. This paper reports the observation of these fractional revivals and discusses their implications.Observations'