“…it seems that the laws of physics present no barrier to reducing the size of computers until bits are the size of atoms, and quantum behavior holds dominant sway.” —R. P. Feynman, 1985 Ion-Trap Quantum Computation

Michael H. Holzscheiter

uantum computation requires its power. Thought to be an unavoid- yin-yang of quantum computation: a very special physical able outcome of the interaction On the one hand, the qubits must Qenvironment. Numerous between the quantum state and the interact weakly with the environment operations must be performed on the environment, decoherence threatens in order to limit decoherence. On quantum states of the qubits (or quan- the life of a quantum system. the other hand, they must be easily tum bits) before those states decohere, Any attempt at building a real accessible from the outside and must or lose the interlocking phase relation- quantum computer therefore leads to interact strongly with each other, or ships that give quantum computation what some scientists refer to as the else we could not manipulate the

264 Los Alamos Science Number 27 2002 quantum state, implement quantum eral important results achieved within who shared the 1989 Nobel Prize in algorithms, and read out the result of the last few years: the on-demand cre- physics with Dehmelt. a calculation in a timely fashion. ation of entangled states of up to four Paul enjoyed telling the following How can we hope to meet such con- ions by the National Institute of anecdote about how he hit upon the tradictory goals? Science and Technology (NIST) group idea for his device. Germans like soft- Ion-trap quantum computers, as in Boulder, Colorado; the development boiled eggs for breakfast, and on a originally proposed by Ignacio Cirac of a novel cooling scheme by a group particular Sunday morning, Paul had and Peter Zoller (1995), offer a possi- at the University of Innsbruck, Austria, prepared two eggs of different sizes ble solution to this dilemma. As its which would allow researchers to and had placed them on a serving tray. name implies, an ion trap confines quickly cool large numbers of trapped When he started to walk, tray in hand, charged particles to a definite region of ions with drastically reduced opera- toward the bedroom to surprise his space with magnetic and electric fields. tional overhead; and the construction wife with breakfast in bed, the eggs In a specific realization of such a trap, of an effective defense against the began to roll. He counteracted their called a linear radio-frequency quadru- forces of decoherence. motion by shaking the tray and was pole (RFQ) trap, or a linear Paul trap able to confine the larger egg to the (Raizen et al. 1992, Walther 1994), center by shaking with a particular time-varying electric fields are used to The Physics of Ion Traps frequency and amplitude. (It was cer- hold a line of ions in place—like pearls tainly not a well-defined harmonic on a string. These ions serve as the Two basic types of devices can shaking.) The smaller egg, however, physical qubits of the quantum com- confine charged particles to well- kept rolling toward the edge, so Paul puter. Immobilized by the trapping defined regions of free space: Penning changed amplitude and frequency and fields and confined inside an ultrahigh traps and Paul traps. The Penning successfully prevented this egg from vacuum chamber, they are effectively trap, which was primarily developed falling, at the expense of allowing the isolated from the environment. by Hans Dehmelt at the University of larger one to wobble toward the edge. However, by addressing individual ions Washington in Seattle, uses a strong Whether he ever reached the bedroom with sharply defined laser beams, we magnetic field and a static electric with both eggs on the tray and enjoyed can initialize the computer, control the field to create a nearly perfect three- a leisurely breakfast with his wife qubit states during the operation of dimensional, harmonic trapping remains unknown, but that morning logic gates, and read out the results at potential (Dehmelt 1967). Some of the Paul realized not only the basic princi- the end of the computation. The inter- most precise tests of fundamental ple of the RFQ trap but also the mass- action between the individual ions is physical symmetries to date have been selective feature of such an instrument. mediated by the Coulomb force conducted with this device, whose At that time, he was keen on develop- between the charged particles. operating principles are described in ing a mass filter for ions, that is, a This article discusses the design the box “The Penning Trap” on the two-dimensional structure that could principles for isolating single ions in a next two pages. transmit an ion with a specific linear Paul trap (Paul et al. 1958), Although Penning traps nicely charge-to-mass ratio and not any other whose operational principles are solve the fundamental problems of ratio. Eventually, Paul’s idea was used described in detail. The individual ele- ion confinement, so far they have not to generate three-dimensional, mass- ments of an ion-trap computer will be been used for quantum computation. selective confinement systems, but introduced, and how to initialize, The trap’s strong magnetic field Cirac and Zoller returned to the origi- manipulate, and interrogate the qubits causes ions to move rapidly in a nal two-dimensional structure and will be explained. Specific schemes circle (the cyclotron motion dis- proposed using it as the basis for a that were implemented in the quantum cussed in the box), whereas we want quantum computer. computation project at Los Alamos the physical qubits to have as little (Hughes et al. 1998) will illustrate the motion as possible. That is why the The Linear Paul Trap. To under- descriptions. Ion-trap quantum compu- favored trap for quantum computa- stand the linear RFQ trap, consider a tation is rapidly evolving, and numer- tion is the Paul trap, in which there positively charged ion floating in free ous groups around the world are is no magnetic field and oscillating space and surrounded by four infi- developing new ideas and experimen- electric fields (as opposed to static nitely long conducting rods, as shown tal techniques. The reader will get a ones) confine the ions. This device in Figure 1. We can give one pair of flavor of this activity in the last section was invented by Wolfgang Paul from opposing rods a positive charge and of the article, which summarizes sev- the University of Bonn in Germany, the other pair a negative charge Continued on page 268

Number 27 2002 Los Alamos Science 265 Ion-Trap Quantum Computation

The Penning Trap

Decades before individual ions were considered as candidates for qubits in a quantum computer, experimental physicists were challenged to realize a simpler Gedanken, or thought, experiment embodied by the statement often made by theorists: “Consider a single (silver) ion in a uniform magnetic field” (Tanoudji et al. 1977). Thought became reality in 1973, when Hans Dehmelt and his col- leagues at the University of Washington in Seattle were able to capture a single charged particle in a Penning trap. The ion that drifted into the central region of that device was trapped by a strong, uniform magnetic field and by the electro- Capacitor plate + static field produced by a set of specially shaped electrodes. The entire device operated under ultrahigh vacuum to limit the interactions between the ion and the background atoms.

+ The University of Washington group refined the technique and used the trap to confine a single electron (Wineland et al. 1973) and later a single barium ion, using an RFQ Paul trap (Neuhauser et al. 1980), and performed precision spec- troscopy on these systems. The special arrangement of fields caused the single + electron to behave as if it were bound to a nucleus for it displayed a set of energy levels, or excited states, similar to those of the hydrogen atom. Dehmelt Figure A. Electrostatic Forces therefore named his electron in a Penning trap “geonium—a single electron The positively charged particle is bound to Earth.” The artificial geonium “atom” was, in a sense, closer to per- repulsed by the capacitor plates but is fection than a real atom. The spacing between energy levels was nearly con- free to move anywhere in the horizontal stant because it reflected the trap’s nearly perfect harmonic-oscillator potential. plane. Dehmelt and coworkers used geonium to perform some of the most precise tests of fundamental symmetries. In a more mundane fashion, Dehmelt called the ion ASTRID (for “a single trapped ion dancing”). (Perhaps, if you keep an ion or electron for such a long time, you may become attached to it.)

To understand the operating principles of the Penning trap, consider a charged B-field line particle (ion) moving freely in space. To confine it to a specific spot in space, we can apply electrical forces to its charge. If we place the ion between two parallel conducting plates that are charged to an electric potential of the same Capacitor plate + sign as the ion, the Coulomb repulsion will keep the particle from moving closer to either plate (see Figure A).

The ion can still move in directions parallel to the conductors. We can try + to remove all escape routes by placing more conductors around the particle. But Michael Faraday discovered more than 150 years ago that an electric field cannot penetrate a closed metal enclosure—hence, the penchant for science + museums to place a person inside a “Faraday cage” that is then exposed to violent lightning bolts. The courageous volunteer remains unharmed because the lightning’s electric field vanishes inside the cage. Similarly, if we fully Figure B. Applying a Magnetic enclose our particle in a cage of conducting plates, the electric field disappears, Field and we lose the forces holding the particle from the walls. A magnetic field causes the ion to circle around a field line (cyclotron motion), A more successful approach is to use the fact that an electric charge moving in thus confining the ion in the horizontal a magnetic field will experience a force perpendicular to the direction of both plane. the magnetic field and the particle’s velocity (the Lorenz force F = qv × B). Therefore, if we apply a magnetic field perpendicular to our parallel conducting plates (see Figure B), we force the ion onto a circular path around the magnetic field line, closing off the sideway escape routes.

266 Los Alamos Science Number 27 2002 Ion-Trap Quantum Computation

Axis of Whereas the system shown in Figure B can confine charged particles (and has symmetry been used for a number of experiments), the special character of the Penning B-field trap is given by the clever shaping and arrangement of the electrodes. As shown arrow + in Figure C, two end caps shaped as hyperbolae of revolution replace the paral- lel plates, and a ring-shaped center electrode defines the electrostatic potential on the edge of the trap. B Trapping region

The arrangement shown in Figure C not only leads to perfect confinement of individual charged particles but also allows the motion of a trapped particle to – be separated into three independent harmonic motions. In order of decreasing frequency, the three motions are (1) the fast “cyclotron” motion of the charge + around the magnetic field lines, (2) a slower oscillation in the direction of the magnetic field that is due to the electrostatic repulsion from the two end caps, and (3) a much slower drift motion that is due to the crossed electric and mag- Figure C. The Penning Trap netic fields (see Figure D). The two endcaps, which are hyperbolae of revolution, replace the flat capacitor plates. The central ring electrode helps define the harmonic potential at the center of the trap.

Figure D. Motion in the Penning Trap The three-dimensional motion of an ion in the trap consists of three harmonic motions: a fast cyclotron motion, a slower up-down oscillation, and a slow circular drift motion.

× The drift motion is easily understood if one focuses on the cyclotron motion. Distorted E drift motion The positively charged particle is accelerated toward the negatively charged cyclotron motion ring electrode as it moves away from the electrical center of the trap. This acceleration increases the radius of curvature for the outer half of the cyclotron motion. As the particle moves back toward the center during the second half of the cyclotron motion, it decelerates, and the radius of curvature decreases. The E net effect is a distortion of the circular cyclotron motion into a spiral that bends B around the electrical center of the trap, as seen in Figure E.

The harmonic motions account for the almost constant spacing between energy levels in Dehmelt’s geonium atom, but this orderliness is hardly noticeable in the roller-coaster-like motion of a trapped particle. If you actually want to expe- rience the particle’s motion yourself, there is a carnival ride in which these three components of motion are present—but watch your stomach! Figure E. Drift Schematic of the drift motion that results from the crossed electric (E) and magnetic (B) fields.

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Continued from page 265

(a) rf

(c) t = t (b) t = t 2 (d) Time Average 1

– + + – +/– +/–

rf rf rf + – – + +/– +/–

+ + +

0 0 0

rf Voltage – rf Voltage – rf Voltage – t t 1 2 Time Time Time

Figure 1. Principles of the Linear Paul Trap (a) The linear Paul trap consists of four conducting rods. (d) If the polarity changes fast enough, a heavy ion becomes Two opposing rods are connected to one pole of a radio- stuck in a rapid back-and-forth motion. Because the electric frequency (rf) voltage source, whereas the remaining two are fields are at a minimum at the trap axis, an effective force connected to the other pole. The axis of symmetry between pushes the ion toward the center, where it becomes trapped the rods is the trap axis. (b) With the rods charged as shown, (although it is still free to move along the axis). The blue the resulting electric force pushes a positive ion to the nega- dots seen between the rods in part (a) represent a string tive rods and repels it from the positive ones. (c) Half an rf of radially trapped ions. The string can be confined axially period later (see graph below), the polarity of all rods is when a positively charged electrode (end cap) is placed at reversed, and the direction of the force also reverses. each end of the rods.

(relative to some arbitrary “zero” conductors and is pulled outwards. Instead, the ion will respond to potential).1 The positive ion feels a If we now reverse the polarity of the time-averaged electric field. repulsive force from the positively our four electrodes, interchanging If we switch the polarity of the charged conductors and is pushed plus for minus and minus for plus, electrodes at a few megahertz toward the center of the trap. The ion the ion’s motion will begin to (or a few million times a second) simultaneously feels an attractive reverse. Where it was moving out, it by applying a radio-frequency (rf) force due to the negatively charged will now be moving in, and vice voltage and if the amplitude is versa. However, if the reversal takes correct, then the time-averaged field place quickly, the “heavy” ion generates a harmonic pseudopoten- 1 For reasons discussed on the previous cannot easily respond; it has too tial with its minimum located at two pages, all four conductors cannot be positively charged, or the electric fields much inertia to follow the fast the trap axis. The ion is pushed to within the trap would disappear. changes in the electric field exactly. the bottom of the pseudopotential

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well and becomes trapped forever— within certain ranges, an ion will potential along some length of the at least in principle—near the center. remain bound to the axis of the device. trap axis. The vibrations are similar to To generate the mass selectivity Furthermore, the magnitude of the those exhibited by a set of pendula sought by Paul, we add a positive restoring force of the pseudopotential connected to each other by springs; direct-current (dc) component to the holding the ions in the radial direction the swinging of one pendulum sets the rf voltage. Positive ions outside a cer- will remain directly proportional to others in motion (see Figure 2). tain mass range feel less of a restoring the distance from the center—the Unlike vibrations of classical pendula, force from the pseudopotential and hallmark of a harmonic potential.2 however, the vibrations exhibited by a are kicked out of the trap by the repul- In other words, to a good approxima- string of ions are quantized; that is, sive dc field. This technique is one of tion, the ions will undergo harmonic the amplitude of the motion depends several that we can use to preferen- oscillations in the radial direction with on the number of quanta (phonons) in ω ω tially retain qubit ions instead of, say, frequency r (or with frequency x the vibrational mode. ω a residual gas ion that may be present and y in case the potential is different For N ions in a trap, there are in the ultrahigh vacuum trap. in the x- and y-directions). This motion N axial vibrational modes and an Of course, the ion’s motion is still is commonly referred to as the secular additional 2N modes for motions unrestricted in the direction parallel to motion. transverse to the axis. Each mode has the trap’s axis. For confinement in this The ions’ motion can become dis- a distinct vibrational frequency. The third dimension, we simply add an torted if the minima of the rf field and lowest-frequency (lowest-energy) electric dc potential to a pair of “end the pseudopotential are misaligned vibration is the so-called common electrodes” that are on either side of within the trap. Misalignment can mode, in which the ions oscillate back the region of interest. This axial field occur because of some small asymme- and forth in unison along the axis. plugs up the escape route along the try in the trap’s construction or This mode figures heavily in the origi- symmetry axis of the system, and because of small dc patch potentials nal quantum-computing scheme of the ion becomes trapped in three on the electrode surfaces. Regardless Cirac and Zoller. Because all ions dimensions. By substantially reducing of the reason, misalignment will cause participate in the common-mode the ion’s energy (cooling), we coax the ions to lie “off center” (that is, off oscillation, when we add (or remove) the ion into lying along the central the line where the rf field vanishes). a quantum of energy to this motion by portion of the trap axis, where the Those ions will experience the strong interacting with one of the ions, we radial and axial confining potentials gradient of the rf field and undergo influence all other ions in the string. are at a minimum. If several very cold rapid oscillations—at the frequency of Any two qubits, regardless of the dis- ions are in the trap, then they all fall to the applied rf field—around their tance between them, can therefore be the center, and the mutual Coulomb time-averaged equilibrium position. coupled together to perform logic repulsion between the ions causes This so-called micromotion is the operations. them to line up neatly along the axis. main source of ion heating. We can Other proposals make use of some suppress the micromotion by adding a of the higher-frequency modes to Motion in the Trap. Although the compensating dc voltage to some of couple qubits together (James 1998a). combination of rf and dc fields within the rf electrodes (or to auxiliary con- These modes have more-complex the trap drives the ion into a complex trol electrodes) and thereby shift the vibrational patterns and relatively radial motion, that motion is fully ions’ positions closer to the actual rf higher excitation energies than the described by a set of differential equa- center. common mode, but it still takes very tions called “Mathieu’s Equations.” In addition to exhibiting secular little energy to excite them. Even a The bound solutions of those equa- motion and the unwanted micromo- string of very cold ions will vibrate tions yield a stability diagram that tion, an ion or, more important, a in some intricate expression of the allows one to evaluate the effective- string of ions will also vibrate in the various modes, a problem that is ness of the trap, given the values of axial direction. The motion will be addressed in the discussion of ion several critical parameters, such as the harmonic because the dc voltage on cooling. amplitudes of the rf and dc compo- the end electrodes creates a harmonic If only a few ions are confined in nents of the voltage, the rf, the ion the trap, the ions will align themselves mass, and the size of the trap linearly along the axis. But increasing 2 (Dawson 1976). To generate a pure harmonic potential, the number of ions or increasing the the four electrodes should have hyperbolic As long as we keep the values of cross sections, but in practice we approxi- dc voltage applied to the end elec- the critical operational parameters mate that ideal shape with cylindrical rods. trodes introduces instabilities because

Number 27 2002 Los Alamos Science 269 Ion-Trap Quantum Computation

(a) Common Mode tightly focused laser beams, initial- ized to an arbitrary state, manipu- lated, and then probed at the end of a calculation. Most important, the isola- tion from the environment afforded by the trap would allow for long coherence times. (b) Breathing Mode One- and Two-Qubit Operations. The logical qubit states |〉 and |〉 of the ion-trap quantum computer must be defined as they always are for any quantum computer. (To stress that the (c) Higher-Order Arial Mode and used to designate the states are notational and have no numerical meaning, we have used a font differ- ent from the one for the numbers 0 and 1.) We simply identify the ion’s electronic ground state with the qubit state |〉 and a long-lived excited state Figure 2. Vibrational Modes with the qubit state |〉. A set of strongly coupled pendula can be used to envision the vibrational motion of We also want to apply a unitary a string of ions in a harmonic potential. These vibrational modes affect all ions transformation to a single qubit, simultaneously. If we set any one of the pendula in motion, the others will move. that is, to implement a one-qubit gate, Similarly, if we grab hold of any pendulum and stop it, all others will stop. (a) The and rotate the qubit in Hilbert space common mode (center-of-mass mode), in which all pendula swing one way and then to an arbitrary superposition of the the other, has the lowest frequency (lowest energy) of all modes. Using this mode | 〉 | 〉 to couple two qubits together in the trap is the basis of the Cirac-Zoller proposal. and states. (Two-level systems (b) The breathing mode, in which pendula at opposite ends move in opposite direc- and the rotation of a qubit in Hilbert tions, has the next highest frequency. For an odd number of pendula, the middle space are discussed in the article one does not move. This mode is less susceptible to heating by external noise “Quantum Information Processing” sources and has also been used to couple qubits. (c) Shown here is another higher- on page 2.) To do so, we subject the order mode. In an ion trap, the ions can vibrate in three dimensions; for N trapped ion to a laser pulse of a specific ions, there are 3N vibrational modes. amplitude, frequency, and duration. Assuming the ion is in its ground the ions are effectively squeezed ter space available for quantum state, the laser pulse will cause the closer together. The Coulomb computing in a linear RFQ ion trap. electron wave function of the target repulsion between neighboring ions ion to evolve to some superposition becomes stronger than the radial of the ground and excited states. (We restoring force, and the ions start Elements of the Ion-Trap cause the electron to undergo part of buckling out into a zigzag pattern. Quantum Computer a Rabi oscillation.) Illuminating the When even more ions are added, the ion with a so-called π-pulse, for zigzag pattern develops into a com- In 1994, inspired by the great suc- example, will evolve the electron plex three-dimensional helical struc- cess of ion traps in the field of preci- wave function through half a Rabi ture (Raizen et al. 1992, Walther sion measurements, Cirac and Zoller oscillation period and leave the ion in 1991, 1994). Some of the ions will proposed that the RFQ ion trap had the excited state. The qubit would move away from the axis and will the right characteristics to support the have rotated from the |〉 to the |〉 experience the strongest micromotion long sequence of precision operations state. If the duration of the pulse is heating—a situation clearly to be required for quantum computation. halved (a so-called π/2-pulse), the ion avoided. We have studied this transi- A string of ions trapped along the would be left in an equally weighted tion from linear to three-dimensional symmetry axis of the trap would be superposition of the ground and structures in some detail (Enzer et al. the quantum register of the computer. excited states. The qubit would have 2000) and have quantified the parame- Each ion could be addressed by rotated to the 1/√2(|〉 + |〉) state.

270 Los Alamos Science Number 27 2002 Ion-Trap Quantum Computation

(a) Resolved Sideband Structure (b) Resonant Transitions

Carrier transition 〉n〉 →〉n〉 〉

Red sideband Blue sideband Carrier 〉n〉 →〉n – 1〉 〉n〉 →〉n + 1〉

Red Blue Energy

〉

n – 1〉 n〉 n + 1〉

ωω ω ω ω Common-mode vibrational states 0 – 1 0 0 + 1

Figure 3. Vibrational Sideband Spectrum (a) An ion trap naturally couples an ion’s electronic excitations to its vibrational motion. Each electronic transition at resonant ω frequency 0, known as the carrier frequency, is therefore accompanied by other sideband transitions. We show the two side- ω ω bands closest in frequency to the carrier: the lower-energy red sideband at frequency ( 0 – 1), and the higher-energy blue side- ω ω band at frequency ( 0 + 1). A laser with a sufficiently narrow linewidth can interact with the ion via a specific sideband or the carrier. (b) Interacting with a particular qubit (ion) via a sideband transition will change the qubit’s internal state and simultane- ously the external state of all the qubits in the trap, either increasing the number of phonons in the common mode by one (exci- tation on the blue sideband) or decreasing the number by one (excitation on the red sideband).

While we can use laser pulses to the ions are not vibrating because the carrier transition is very narrow3 interact with each qubit separately there are no phonons in the common and is less than the energy difference (and excite a qubit’s electronic, or mode. In the state between the carrier and either side- internal, degrees of freedom), we can band. Thus, the sidebands and the car- | 〉| 〉 also use another laser to excite the q1, q2,… qj 1 , (3) rier can be resolved within the cold trap’s vibrational modes and hence to ion’s frequency spectrum. interact with all qubits simultaneously. the common mode contains one Now consider, for example, a pro- The latter process can be viewed as phonon and all the ions are swaying cedure used to place two ions in an interacting with the qubits’ external in unison along the trap axis. entangled state. Assuming that the degrees of freedom. The state of a As mentioned earlier, the common ions are initially in the state |〉|0〉, string of j qubits in the trap is there- mode is used as a “bus” that couples if we were to address the first ion fore explicitly given as different ions together. To better with a π-pulse from a laser detuned to understand this coupling, consider the blue sideband of the internal tran- | 〉| 〉 q1, q2,… qj n . (1) first that the frequency of the transi- sition, both the internal and external tion between the |〉 state and the |〉 states of that ion would be excited. | 〉 ω The first ket, q1, q2,… qj ,refers to state is 0, and that the common- Because an excitation in the common ω the logical qubit states, with qj = or mode vibrational frequency 1 is mode is felt by both ions, the result | 〉 ω | 〉| 〉 . The second ket, n ,refers to the much lower than 0. Similar to the would be the two-qubit state 1 . common-mode vibrational state, and case of two coupled harmonic oscilla- If, instead, we address the first qubit the value of n,say, 0, 1, 2, …, indi- tors, the energy spectrum of the ion cates the number of phonons in the exhibits resonances not only at the 3 ω The metastable excited state has a very common mode. (Although the string “carrier” frequency 0,but also at the long lifetime, which leads to a narrow ω ± ω of qubits may initially be in another “sideband” frequencies 0 1 (see linewidth according to Heisenberg’s mode, we will restrict our attention to Figure 3). The resonance with the uncertainty principle. To take an example from the Los Alamos experiment, a cal- the common mode.) Thus, in the state higher frequency is commonly known 2 cium ion excited to the 3 D3/2 state will as the blue sideband; the one with the decay to the ground state only after an | 〉| 〉 average delay of about 1 second, which q1, q2,… qj 0 , (2) lower frequency, as the red sideband. results in a transition linewidth of about ∆ω For cold ions, the linewidth 0 of 1 hertz.

Number 27 2002 Los Alamos Science 271 Ion-Trap Quantum Computation

(a) Coupling Two Qubits through the Common Mode Internal State

n = 1〉 Excited:〉 Blue sideband 〉  〉 π-pulse on n = 0 Ground: 1st ion 〉0〉 〉1〉

(b) Entangling Two Qubits

+ + Blue Red sideband sideband π/2-pulse on π-pulse on 1st ion 2nd ion 〉0〉 1 1 — (  〉0〉 +〉1〉 ) — (  〉 + 〉 ) 0〉 √2 √2

Figure 4. Using the Common Mode to Entangle Two Qubits The vibrational state is indicated by the position of the ions entangled if we illuminate the first qubit with a π/2-pulse on on the rungs of a ladder in the harmonic potential well. In the blue sideband. The ions are placed in a superposition of this diagram, the electronic ground state of an ion is indi- states: 1/√2(|〉|0〉 + |〉|1〉). If the second ion is then illumi- cated by a solid circle; the excited state, by an open circle. nated by a π-pulse from a laser tuned to the red sideband, (a) Suppose two qubits are initialized to the state |〉|0〉. it can absorb the photon only if energy is available from Addressing the first qubit with a π-pulse from a laser tuned the vibrational mode. Thus, ion 2 is excited only if ion 1 to the blue sideband will excite the ions to the state |〉|1〉. was excited; it remains in the ground state if ion 1 was in The first ion is in its electronic excited state, while the the ground state. The two-ion system therefore exhibits second remains in its electronic ground state. Because the strong correlation of observables, which according to the common mode affects all ions, both ions are excited Bohr, define the condition of entanglement. The end result to the |n = 1〉 vibrational state. (b) Two qubits can be of the operation is the entangled state 1/√2(|〉 + |〉)|0〉. with a π/2-pulse (see Figure 4), both We can no longer describe the system albeit using only a single ion in the qubits are placed in a superposition of as individual ions being in the ground trap. (The two states of the control the two states, namely, or the excited state. The result of a qubit were the two lowest-energy trap measurement on one ion is strongly vibrational states.) Still, because any 1/√2(|〉|0〉 + |〉|1〉) . (4) correlated to the status of the other computation can be performed with a ion. Notice that this procedure works number of two-qubit cnot gates, We then address the second ion equally well if one or more ions are together with some single-qubit gate (which is still in its ground state) placed in between the first and second operations, the realization of a cnot with a π-pulse tuned to the red side- ions because the excitation of the gate in a quantum computer is an band. The laser energy is too low to common mode is shared by all ions. important benchmark. excite directly the ground-to-excited- Besides defining the individual state electronic transition, but the operations just described, Cirac and Readout. At the end of any quan- transition still occurs if extra energy Zoller also spelled out in detail the tum calculation, the individual qubits can be taken from the common steps needed to perform a “controlled- in the quantum register will be in mode. The end result is that all not” (cnot) gate. In such an operation, defined states, which must be read out phonons have been removed from a “target qubit” flips its state only if a with high fidelity. A powerful readout the quantum register at the end of the second qubit, the “control qubit,” is tool makes use of the phenomenon of operation, and we create a two-qubit originally set to its logical |〉 value. quantum jumps (Sauter et al. 1986, entangled state: Dave Wineland’s group at NIST first Bergquist et al. 1986). The readout implemented the cnot gate in an ion method is easily understood when one 1/√2(|〉 + |〉)|0〉 . (5) trap in 1995 (Monroe et al. 1995), examines the generic ion-level scheme

272 Los Alamos Science Number 27 2002 Ion-Trap Quantum Computation

shown in Figure 5. The ion has two (a) Generic Three-Level System (b) states that we identify as the logical qubit states |〉 and |〉. But the ions s〉 used for quantum computation also have a short-lived excited state |s〉 that is accessible from one of the qubit states, the |〉 state, by laser excita- 〉 tion. When the laser drives the Readout |〉 —> |s〉 transition for a long period, transition the ion fluoresces and emits a huge Qubit number of photons. If that transition is transition not accessible because the ion was in | 〉 the state, there will be no fluores- 〉 cence. Detection of a fluorescence signal, therefore, tells us that the qubit is in the |〉 state, and we observe the “jumps” of the ion between the |〉 and the |〉 state as distinct jumps in the intensity of the fluorescence. Figure 5. Readout Using Quantum Jumps This type of readout will destroy any (a) A generic three-level scheme for ions in a trap is illustrated. The qubit states |〉 quantum information contained in the and |〉 are typically the ground state and a long-lived excited state, respectively. The qubit state and will yield a purely For state |s〉 is short lived and is coupled to the ground state. If the ion is in the ground example, suppose the ion is in an equal state, a laser can drive the |〉→|s〉 transition many times per second, and the ion | 〉 superposition of the states |〉 and |〉; will fluoresce. If the ion “jumps” to the state, there will be no fluorescence, so that the presence or absence of a large fluorescence signal reveals the state of the then probing the ion once with the laser qubit. (Alternatively, one can use two long-lived ground-state hyperfine levels as will not reveal the original state of the qubits and construct a similar readout scheme.) (b) This composite image shows qubit. If we want to get a reading on strings of calcium ions that were laser-cooled to near rest in the Los Alamos quan- the ratio of the two different states in a tum computation ion trap. The spacing between the ions is approximately 30 µm. superposition, we will have to repeat About 108 photons are absorbed and reemitted each second during the time the the measurement multiple times and readout laser is irradiating the ion. That photon flux is easily detectable with a resort to a statistical description. charge-coupled device (CCD) camera. The fluorescence is actually bright enough to If we want to maintain the quan- be seen with the naked eye, except that for calcium, the readout transition is at tum character of the ion’s state at the 397 nm and is outside the range of sensitivity for the human eye. end of a particular calculation, we may resort to a different scheme. transferred into another ion—and so, species can serve as qubits, and Consider an ion placed in a high- the quantum Internet is born! numerous qubit schemes are possible. quality optical cavity, which is tuned While the previous section discussed to the resonance of the internal transi- ion-trap quantum computers in gen- tion in the ion. If the ion is in the state The Los Alamos Ion-Trap eral terms, this section describes an |〉,a photon is emitted into the cavity; Quantum Computing experiment developed at Los Alamos, if it is in the state |〉, no photon is Experiment in which calcium ions were used. emitted. If the ion is in a superposi- We initially chose to use calcium tion state, the photon field in the cav- Currently, every implementation of for a number of reasons, including the ity will end up in a superposition ion-trap quantum computing uses following: All the wavelengths needed between the states consisting of one qubits that are composed of two long- for cooling and manipulation are, at photon in the cavity and no photon in lived internal states of the trapped least in principle, accessible by rela- the cavity. Thus, the quantum state of ions (the ground state and a tively inexpensive diode lasers; the an ion or an atom can be transferred metastable excited state, or two hyper- lifetime of the metastable state allows to a photon (Parkins and Kimble fine sublevels of the ground state) and a reasonable number of coherent 1999, Mundt et al. 2002). This state has the qubits communicating with operations to be performed; and the could be transferred through optical each other through the trap’s vibra- calcium isotope of interest is most fibers to a different trap and then tional modes. Many different ion abundant and can easily be loaded

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into the trap. But the basic quantum (a) Trap Built at Los Alamos computational schemes outlined earlier can be implemented with any Conical endcap Conducting rod element that displays an ionic-level structure similar to that of calcium, such as the other alkali-earth elements beryllium, magnesium, strontium, and barium. At this stage of experimenta- tion, all alkali-earth ions are essen- tially interchangeable, and for mostly technical reasons, calcium has recently been replaced with strontium in our quantum computing experi- ment. (Some of that work is described (b) Partial Energy-Level Diagram for Calcium in the article, “Quantum Information with Trapped Strontium Ions” on 42P page 178.) In addition, other ions, 3/2 such as mercury and ytterbium, 42P 854 nm also exhibit level schemes that are 1/2 applicable to quantum computation, 866 nm  〉 32D τ = 1.14 s albeit with slightly different technical 5/2 approaches. As ion-trap quantum 393 nm 32D τ = 1.16 s 3/2 computers become more sophisti- cated, the choice of ion species will 397 nm 729 nm become a larger issue. Our trap is a linear Paul trap, about 42S  〉 1 centimeter in length and 1.7 mil- 1/2 limeters in diameter, with a cylindrical Figure 6. The Los Alamos Linear Paul Trap geometry, as seen in Figure 6(a). (a) The trap built at Los Alamos for quantum computation is about 1 cm in length We create the strong, radial confine- and 1.7 mm in diameter. An electric field of a few hundred volts oscillating at 8 MHz ment fields by applying a few hundred is applied to two of the conducting rods. The other two rods are RF grounded. volts of rf amplitude at approximately About 10 V of a direct current is placed on the conical end caps. (b) This illustration 8megahertz to two opposing rods. shows a partial energy-level diagram for calcium (not to scale) and shows the wave- The remaining two rods are lengths of some transitions relevant to our quantum computing scheme. The qubit rf-grounded. The axial confinement, transition is shown in red. which prevents the ion from leaking out of the trap along the symmetry mize the amount of heating produced from spontaneous emission from the axis, is produced by a direct current of by micromotion. excited state) can destroy the internal about 10 volts applied to each of the Figure 6(b) shows a schematic state of the quantum register. conical end caps. This combination of diagram of the internal-level structure To load the ions into the trap, we the rf and dc fields leads to an axial of calcium ions and gives the wave- cross a beam of calcium atoms that is ω oscillation frequency 1 for the com- lengths of the relevant transitions. produced by heating a small calcium- mon mode of a few hundred kilohertz (Any other alkali-like ion would have filled reservoir with a beam of elec- 2 and a radial oscillation frequency of a similar structure.) The 4 S1/2 ground trons emitted by an “electron gun.” ω ≈ 2 r 1megahertz. state and the metastable 3 D5/2 excited (The electron gun is essentially identi- Additional dc potentials can be state are used to form the logical qubit cal to the one inside a computer moni- applied to four support rods, which states |〉 and |〉,respectively. The tor or a television screen.) These two are not shown in Figure 6(a) but are metastable excited state has a lifetime beams are aligned so that they cross located outside the actual trap elec- of about 1 second, which is long each other within the effective volume trodes. In this way, one can center enough to allow an interesting number of the trap, that is, within the cylindri- the ion string on the electrical sym- of computational steps to be per- cal volume that fits between the four metry axis of the trap and thus mini- formed before decoherence (resulting rods and the two end caps. The atoms

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that are ionized by electron impact (a) Frequency Shifts Due to the Doppler Effect suddenly feel the confining forces of Laser Ion the electric fields and become trapped. ω Cirac and Zoller (1995), as well as Ion at rest “sees” frequency . other authors, proposed initializing the Ion moving into the laser with velocity v “sees” frequency ω = ω (1 + v/c). computer in the state + Ion moving out of the laser with velocity v “sees” ωω | … 〉|0〉 ; (6) frequency – = (1 – v/c). that is, all qubits are in their electronic (b) Detuning (c) Doppler Cooling of Calcium and vibrational ground states. 4 Doppler-broadened However, the temperature of the Laser line emission profile 42P newly trapped ions is very high, since 1/2 their energy is given by a combination 866 nm 〉 of the temperature of the calcium  oven and the energy imparted to the 397 nm 2 ion by the electric field. (The latter 3 D3/2 energy varies, depending on where the τ = 1.16 s ionization occurs.) In order to reach the initial state and then to perform 42S 〉 hω Energy 1/2 quantum logic operations, the ions’ E = E0 temperature must be reduced to its lowest possible value. Cooling the Figure 7. Doppler Cooling of Ions ions takes place in two steps described (a) When interacting with a laser of frequency ω, an ion at rest sees the native laser in the next two sections. frequency. If the ion is moving, this frequency is shifted by the Doppler effect. An ion moving into the laser beam “sees” the laser frequency Doppler-shifted toward a ω Doppler Cooling of Calcium Ions. higher frequency, +, while the ion moving in the direction of the laser beam “sees” the frequency ω . (b) This first-order Doppler effect is eliminated in ion traps because As its name suggests, this first cool- – the average velocity is zero. However, because of its thermal motion, the ion has a ing step makes use of the Doppler probability to absorb photons at any frequency within its Doppler-broadened absorp- effect, whereby the relative motion tion profile. Similarly, it has a probability to emit a photon over a range of frequen- between a source and an observer cies within its emission profile. Suppose the laser is tuned below the ion’s resonance ω ω ω causes a change in the observed fre- frequency 0 so that < 0. When the ion moves into the laser beam, it will absorb a quency of an acoustic or electromag- photon because it sees the laser frequency Doppler-shifted close to its resonance ω ∼ ω hω hω netic wave. For example, the sound of frequency ( + 0). The ion absorbs a laser photon of energy E = < 0,but on a siren on an ambulance or a police average it reemits a photon with higher energy (from the gray region). Because it car changes its pitch depending on loses energy during each cycle of absorption and emission, the ion cools rapidly to the limit of this method, which is imposed by the recoil energy the ion experiences whether the vehicle moves toward upon reemission of the photon. For typical parameters of our trap, calcium will you or away from you. Similarly, an reach a vibrational level of approximately |n = 10〉 to |n = 30〉 at the end of the Doppler ion or atom will absorb or emit cooling. (c) The transitions used to Doppler-cool calcium ions are shown. photons of different frequencies (energies), depending on its motion absorption profile of an ionic transi- the laser can the ion absorb these relative to the light source. Although tion to become much broader than “off-resonance” photons, because only an ion in the trap is localized by elec- the natural linewidth of the transition then does it “see” the laser frequency tric fields and its average velocity is (second-order Doppler broadening). shift into resonance. However, as a zero, the variation of the instanta- For “hot” ions, the Doppler-broad- result of its random jiggling, the ion neous velocity, as the ion jiggles back ened linewidth is typically much has a probability to emit photons at and forth due to thermal motion, greater than the laser linewidth. any frequency within its Doppler- causes the inherent emission and/or To implement Doppler cooling, we broadened emission line profile. One tune a laser to a frequency below the can easily see from Figure 7 that the resonance frequency of a transition in ion has a greater probability to emit a 4 We often refer to ion temperature rather than energy because the ions show the ion (Figure 7). Only when it is photon with a higher frequency than a distribution of energies over time. moving at a certain velocity toward the absorbed photon. On average,

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more energy is emitted than absorbed, tribution of states depends on the Unfortunately, the long lifetime of 2 which leads to a cooling of the ion. ions’ temperature and the frequency of the 3 D5/2 state is now a hindrance. For rapid cooling, a large number each mode.) Here, we will restrict our In principle, we can scatter only one of photons must be absorbed and attention to the common mode. After photon per second using this transi- emitted, and therefore Doppler cool- Doppler cooling, the ions in the trap tion, which would render the process ing is performed on a transition that can typically occupy the states from of cooling from |n =30〉 to |n =1〉 can be cycled rapidly. We use the |n = 10〉 to about |n = 30〉. Getting the unacceptably slow. Heating processes 397-nanometer transition from the qubits into the common-mode ground —micromotion heating or simply 2 2 | 〉 4 S1/2 to the 4 P1/2 state. The lifetime state ( n =0), therefore, requires an radiative heating from other noise 2 of the 4 P1/2 state is about additional cooling scheme. sources in the system—are much 10 nanoseconds, so the ion can absorb faster, and we would be unable to and reemit about 108 photons per Sideband Cooling of Calcium. reach the desired starting point of 2 second. Unfortunately, the 4 P1/2 state We recall that ions in the trap can all qubits being in the internal and has a chance of roughly 1 in 15 to couple their internal degrees of free- external ground states. 2 decay into the metastable 3 D3/2 state, dom with their external motion, which To speed things up, we artificially ω ω 2 which has a lifetime of about 1 sec- leads to sidebands at 0 ± 1,where shorten the lifetime of the 3 D5/2 state ω ond. The ion then takes so long to 1 is the common-mode frequency, in by introducing an alternate decay route 2 return to the ground state that it would the absorption spectrum. For cold ions via the 4 P3/2 state (Marzoli et al. be lost from the cooling cycle. To with a minimal Doppler linewidth, 1994). We irradiate the ion not only avoid this outcome and force ions to these sidebands are resolvable from with a laser tuned to 729 nanometers return from the D3/2 level to the cool- the carrier—see Figure 8(b). Thus, an (to drive the S–D transition), but also ing cycle, we irradiate the ion with ion can absorb photons not only at the with a second laser tuned to ω two lasers: the cooling laser at carrier frequency 0 of their internal 854 nanometers—see Figure 8(d). 397 nanometers and a “repump” laser |〉→|〉 transition but also on the The second laser pumps the ion from at 866 nanometers. upper and lower sidebands at the the D- to the P-state, from which the ω ω Doppler cooling has its limits. frequencies 0 ± 1. ion rapidly returns to the ground state. Conservation of momentum guaran- Assuming all ions are in the state By carefully choosing the amplitude tees that, after emitting a photon in |〉|n〉, we can tune a laser with a suitably of the 854-nanometer laser, we can one direction, the ion recoils in the narrow linewidth to the red sideband— design the effective lifetime of the h ω ω 2 opposite direction. Although this photon energy [E– = ( 0 – 1]—and 3 D5/2 state according to our needs, recoil energy is small, eventually it excite one of the ions to the state and our calcium ion can jump down counteracts any cooling effects. For |〉|n – 1〉. In essence, energy is removed the ladder of harmonic-oscillator calcium ions, the Doppler limit is from the vibrational mode (the occupa- levels in just 3 to 30 milliseconds. equivalent to a temperature of about tion number is reduced by one phonon) In a real system, the cooling power 3microkelvins. At that temperature, and is used to make up the deficit in from the lasers will always be in com- the kinetic energy of the ions is signif- photon energy. After its radiative life- petition with external heating icantly less than the mutual Coulomb time, the ion can decay to one of three processes. Although no clear theoreti- repulsion between ions. Essentially, states: the state |〉|n – 2〉, by emitting a cal explanation of these processes has h ω ω they do not have enough kinetic photon with energy [E+ = ( 0 + 1]; emerged, many possibilities have been energy to leap-frog each other, so the the state |〉|n – 1〉, by emitting a photon discussed in the literature, and the rele- hω cold ions remain frozen in their rela- with energy [E0 = 0]; or a return to its vant scaling laws with trap parameters tive locations and form a string. The initial state, by emitting a photon with have been developed (James 1998b). h ω ω photos in Figure 4 are examples of ion energy [E– = ( 0 – 1]—see Figure Typical candidates—besides micromo- strings that were realized in our trap. 8(c). On average, the ion loses one tion heating, which can be avoided by hω At a 200-kilohertz common-mode vibrational photon of energy E = 1 carefully tuning the trap voltages—are frequency, the spacing between ions for each excitation–decay cycle. fluctuating contact potentials on the is about 30 micrometers. Because we started somewhere around trap electrodes (originating from insu- Even at a temperature of |n = 30〉, we need about 30 cycles to lating deposits on the electrodes), 3microkelvins, however, the ions bring the vibrational mode to its ground which have a frequency component at have enough energy to occupy any of state(provided there are no competing the trap’s resonant frequency. several vibrational modes, with many effects that heat the ions while they In the absence of a proper theoreti- phonons per mode. (The specific dis- are being cooled). cal description of ion heating, we can

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turn to empirical data accumulated (a) Sideband Cooling Transitions (b) Resolved Sideband Structure from a number of different experi- 42P ments. The adopted procedure is to 3/2 cool the ions to as low a temperature 854 nmn as possible and then turn off the lasers 〉 2 responsible for the cooling. After a 3 D5/2 393 nm variable delay time, we measure the ion’s temperature using sideband ω 0 = 7292 nm ω ω ω 1 0 0 1 spectroscopy. Quentin Turchette and coworkers (2000) conducted the most 42S 〉 1/2 complete study of this type when they Sideband cooling looked at heating effects in traps of different sizes. The separate traps had also undergone different preparation (c) Stepping to Lower Vibrational States “rituals.” The studies suggest a strong D dependence on trap size, that is, on 〉 D the distance between the ions and the D trap electrodes. When the studies are D combined with observations made by Rainer Blatt’s group at the University of Insbruck, one is led to believe that S “bigger is better.” But Ralph deVoe S 〉 S with IBM has recently reported that S hardly any heating was observed over n – 3〉 n – 2〉 n – 1〉 n〉 a short period in a miniaturized trap. Clearly, we have much to learn before we can understand the heating (d) Increasing the Cooling Rate of ions in rf traps. The comforting P thought is that, in all cases, the time P scale for heating from the ion’s P D ground state can be kept long, com- P D pared with the time required for a 〉 D D reasonable number of quantum manipulations. Furthermore, heating times are typically longer than times S for other decoherence processes. S 〉 S S Readout of the Calcium Ion. 〉 〉 〉 〉 | 〉 2 n – 3 n – 2 n – 1 n We use the s = 4 P1/2 excited state in calcium for readout, the same state that is used for Doppler cooling. As Figure 8. Sideband Cooling discussed earlier, this state has a life- (a) This partial energy-level diagram shows the transitions we use for sideband time of only 10 nanoseconds and is cooling of calcium ions. (b) When the linewidth of the carrier transition (frequency accessed from the ground state by a ω 0) is very narrow and the Doppler broadening is minimal, the ion’s vibrational side- laser tuned to 397 nanometers. An bands can be resolved. (c) The figure shows several vibrational levels for the | 〉 | 〉→|〉 | 〉| 〉 ion in the state will absorb and carrier transition. If a single ion is initially in the state n , then illuminat- 8 ing the ion with a laser tuned to the red sideband will excite the ion to the state reemit about 10 photons per second | 〉→|〉 |〉|n – 1〉. The latter state will decay to |〉|n – 2〉 or |〉|n – 1〉, or it will go back to |〉|n〉. when the laser drives the s 2 On average, the number of phonons in the mode decreases by 1 after each excita- transition. (Because the 4 P1/2 state tion/emission. (d) The lifetime of the upper level may be artificially shortened if that can also decay to the long-lived 2 level is coupled to an auxiliary one with a higher decay rate. The faster decay will 3 D3/2 state, we simultaneously increase the cooling rate. irradiate the ion with a laser tuned to

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(a) Calcium Readout Transitions for one of the two ions being excited  〉 2 s 4 P1/2 (intermediate fluorescence), and 866 nm 〉 32D finally, for both ions being in the 5/2 ground state (full fluorescence). 32D 397 nm 3/2 Although it is easy to distinguish 729 nm among these cases, determining which of the two ions is in the ground state 42S 〉 1/2 for the intermediate fluorescence level (b) Quantum Jumps, Single Ion is difficult. We must look at the ions individually, by focusing the laser on 6000 one ion at a time, and then convert to the single-ion measurement. 4000 〉 Ferdinand Schmidt-Kaler and his colleagues from the Innsbruck group 2000 have used this readout technique with three ions, which were spaced at 〉 0  about 6 micrometers from each other Fluorescence (counts/s) 0 51015 in the trap. They cooled the ions to Time (s) the |〉|n〉 state, and all three were (c) Quantum Jumps, Two Ions emitting photons on the readout tran- 14000 sition. The scientists then pointed a 12009 sharply focused laser at 729 nanome- 10000 〉 ters onto one of the ions and placed it 8000 in the |〉 state (the dark state). The 6000 〉 or measured crosstalk among neighbor- 4000 〉 ing ions was less than 1 percent, so 2000 〉 the state of the chosen qubit could be 0  Fluorescence (counts/s) 051015 determined with about 99 percent Time (s) fidelity (Nägerl et al. 1999). Figure 9. Readout of Qubits (a) Shown here are the readout transitions for calcium. (b) For this readout experi- ment, a single ion interacts with two lasers: a low-intensity laser that drives the Important Developments qubit transition |〉→|〉 and a second laser that drives the readout transition |〉→|s〉. The fluorescence signal from that transition, nominally around A Popular Mechanics article from 4,000 counts per second, is recorded with a simple rate meter. When the qubit is in 1949 stated, “Where a calculator on the |〉 state, we can drive the readout transition, but if the ion occupies the state |〉, the ENIAC (electronic numerical the fluorescence disappears. We can distinguish between the |〉 and |〉 states with integrator and calculator) is equipped nearly 100% fidelity. (c) The state of two ions can also be distinguished. No count with 18,000 vacuum tubes and | 〉 corresponds to the state ; 8000 to 10,000 counts per second correspond to the weighs 30 tons, computers in the state |, 〉; 4000 counts per second, to either |〉 or |〉. (In the last case, our experi- future may have only 1000 tubes and mental setup does not allow us to distinguish between the two states.) weigh only one and a half tons.” That 866 nanometers to return the ion to of the detected photon counts for a sin- observation did not turn out to be 2 the 4 P1/2 state.) Even with a modest gle ion in the trap. The fluorescence entirely correct. How could anyone photon-collection efficiency of about signal is nominally about 104 counts have foreseen the development of 10–4,which is due to experimental per second. We randomly excite the transistors and integrated solid-state considerations (we cannot bring a lens ion with a laser tuned to 729 nanome- circuitry or the remarkable parallel too close to the ions without blocking ters, and each time it “jumps” from the developments that have culminated in access to the trap), we can easily |〉 state to the |〉 state, the signal dis- today’s supercomputers? detect the photons scattering from the appears. Figure 9 also shows the fluo- We are still in the “vacuum-tube” ion with a charge-coupled device rescence from a set of two ions. The era of quantum computation, and if (CCD) camera. different levels of intensity are for both asked two years ago about the future In Figure 9, we show a sample trace ions being excited (no fluorescence), of ion-trap-based quantum computers,

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↑↑〉0〉 gled pairs per second, but if we look for entanglement of three or even four Laser 2 photons, the likelihood of finding such a state becomes unacceptably ω h 0 ↓↑〉2〉 ↑↓〉2〉 small for practical purposes (30 per second for 3 photons and a few per 〉 〉 〉 〉 ↓↑ 1 hδ ↑↓ 1 hν year for four photons). ↓↑〉0〉 ↑↓〉0〉 Thus, quantum computing took a leap forward when the NIST team in Boulder demonstrated that it could produce an entangled state of up to Laser 1 h ω 0 four ions “on demand” (Sackett et al. 2000). Based on a proposal by Anders Mølmer and Klaus Sørenson (1999) ↓↓〉0〉 from the University of Aarhus in Denmark, the NIST team around Figure 10. Four-Particle Entanglement Chris Monroe and David Wineland The figure shows the relevant energy levels and transition frequencies used to cre- demonstrated the feasibility of entan- ate deterministic multiparticle entanglement. A two-ion scheme is illustrated. The glement of two and four ions in a |↑↑〉| 〉 hω |↑↓〉| 〉 |↓↑〉| 〉 0 excited state has an energy of 2E0 = 2 0. The 1 and 1 excited states, deterministic way. With a single-pulse in which the internal state of one of the ions is excited and both ions go into a operation of two lasers, the desired h ω ν vibrational excited state, has an energy E1 = ( 0 + ). Lasers tuned to energies state could be produced with a high E + δ and E – δ, where δ is a predetermined laser detuning, can directly excite the 1 1 degree of certainty. ions to the |↑↑〉|0〉 state. Pulsing the two lasers for a time t = π/(2Ω), where Ω is an effective Rabi frequency, will place the ions in the entangled state To understand the technique, con- |Ψ 〉 √ |↑↑〉 |↓↓〉 sider two spin-half particles confined 2 =1/2( – i ). The scheme can be generalized to any number of ions and has been used to create entangled states of up to four ions. in a harmonic well and coupled by (Figure reproduced with permission from Nature.) vibrational degrees of freedom. (The spin description is equivalent to our I would have been hesitant to promise achievements and underlying princi- previous picture of two internal states much. I may have argued that the ples but are unpredicted and significant in an ion.) The NIST group used the systems we were looking at were enhancements of available technology. two ground-state hyperfine levels of mere demonstrations, designed to help 9Be+ ions as an effective spin-half |↓〉 | 〉 us understand the fundamental Four-State Entanglement. To system, with = F = 2, mF = –2 |↑〉 | 〉 physics issues behind qubits and that take full advantage of the power of and = F = 1, mF = –1 . The energy the prospects for scaling these devices quantum computation, we need to levels of the system are shown in hω up to a larger number of qubits were generate entanglement between an Figure 10, where 0 is the internal doubtful. Even today I could argue arbitrary number of qubits. But energy splitting of each particle and ν that, while the computing scheme of generating any entangled state is dif- is the oscillation frequency of the par- Cirac and Zoller is in principle scala- ficult. In the case of photons, entan- ticular collective mode of the particles ble (Hughes et al. 1996), it has yet to glement is achieved by means of a in the trap. be realized with two qubits. statistical process. Many pairs of pho- The group used standard laser- However, because much has hap- tons are created by a method known cooling and optical-pumping tech- pened in the ensuing two years, as parametric down-conversion, niques to prepare the particles in their included here are descriptions of just a whereby a high-energy photon, after spin-down internal state and in the few of the many important develop- entering a special type of crystal, has ground state of their collective ments that have put the ion-trap a certain probability to convert into motion: |Ψ〉 = |↓↓〉|0〉. Laser pulses at quantum computer back on the track two photons, each with half the initial ω0 + (ν – δ) and ω0 – (ν – δ), where for scalable technologies. Similar to energy. In a few cases, two photons δ is the detuning from the resonance the transition from vacuum tubes to emerge in an entangled state. (refer to Figure 10), then drive the solid-state devices (even if not quite as (See the article “Quantum State two-step transition from |↓↓〉|0〉 to fundamental), these developments do Entanglement” on page 52.) We can |↑↑〉|0〉. If the detuning δ is sufficiently not invalidate any of the previous typically produce about 1000 entan- large, the intermediate states |↑↓〉|〉

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and |↓↑〉|〉 are negligibly occupied, see that all superpositions of these an extra degree of freedom in the and no motional excitation occurs in maximally entangled states are invari- axial motion, namely, the breathing the process. Applying the laser fields ant under transformations that apply mode, in which ions on opposite sides for a time t = π/(2Ω), where Ω is the the phase change |↑〉→eiς|↑〉 simulta- of the string move 180° out of phase Rabi oscillation frequency for the neously to both ions. This so-called (refer to Figure 2). Each additional transition, results in the final wave collective dephasing is thought to be ion opens up three more vibrational function a major source of decoherence for modes to the ion string. Every mode trapped ions. of frequency ν can be assigned an |Ψ 〉 √ |↑↑〉 |↓↓〉 . 2 = 1/ 2 ( – i ) , (7) In the NIST experiment, an arbitrary average quantum number nν state of one qubit was encoded in the The initial scheme of Cirac and which is the desired maximally decoherence-free subspace of two ions: Zoller requires a mode to have nν = 0 entangled state. in order to be used for computational α|↑〉 β|↓〉→ α|Ψ 〉 β|Ψ 〉 It turns out that this process is + + + – .(10) operations. For small numbers of ions, entirely scalable for an even number we reach this state by the standard of N qubits and can generate the The encoded information was sub- sideband-cooling methods discussed N-particle entangled state jected to engineered dephasing errors earlier. As seen in Figure 11(a), the or ambient errors, and then the encod- ion has a number of transition possi- |Ψ 〉 √ N = 1/ 2 ing procedure was reversed to recover bilities: Excitation on the lower side- × (|↑↑...↑〉–iN+1|↓↓...↓〉) . (8) the original information. The data band will cool the ion, excitation on showed unequivocally that the noise- the upper sideband will cause heating, If N is an odd number, the state less subsystem protects the informa- and transitions on the carrier will |Ψ 〉 N can still be produced, provided tion from collective dephasing errors cause diffusion. In sideband cooling, one rotates each qubit independently. for a time up to ten times longer than we use an ultranarrow laser and The NIST scientists have used this the typical decoherence time and that excite only the lower sideband so method with two and four ions in the collective dephasing is indeed a major that |n〉→|n – 1〉. trap, but they also caution that the source of errors in ion traps. One For a large number of qubits, how- |Ψ 〉 experimentally realized states 2 could imagine that this type of robust ever, the sheer number of higher |Ψ 〉 and 4 are not fully entangled. Each storage might enable the operation of modes makes it technically difficult, state shows some degree of decoher- a quantum computer constructed from if not impossible, to use standard ence. Although the amount of deco- an array of ion traps as opposed to a sideband-cooling methods. Not only |Ψ 〉 herence in 4 was more than what single trap. (For an introduction to the would we have to identify and excite could be tolerated for quantum com- theory of noiseless subsystems, see the lower-sideband transitions for puting, the achievement of reliably the article “Introduction to Quantum each and every mode, but the spec- creating a four-particle entangled state Error Correction” on page 188. A trum becomes so “dense” that the on demand cannot be underestimated. nuclear magnetic resonance experi- upper sidebands of a neighboring In a later development, the NIST ment demonstrating noiseless subsys- internal transition can overlap the group showed that the maximally tems is presented in the article lower sidebands of another. Cooling entangled states of a pair of trapped “Realizing a Noiseless Subsystem in one mode could actually heat another. 9Be+ ions could be used as a decoher- an NMR Quantum Information Furthermore, the “overhead” needed ence-free subspace for protecting one Processor” on page 260.) to control and cool these modes is qubit of information against dephas- daunting: large numbers of laser ing errors (Kielpinski et al. 2001). The Broadband Cooling. The second pulses, constant retuning of the lasers decoherence-free subspace, also called important recent result is the selective from one mode to the next, and tight a noiseless subsystem, is spanned by enhancement of the probability of control of the qubit register through- the two orthogonal states cooling ions by electromagnetically out the cooling stage. induced transparency (EIT). The For efficient (and simultaneous) |Ψ 〉 √ |↓↑〉 |↑↓〉 + = 1/ 2 ( + i ) , and scheme of Cirac and Zoller has the cooling of more than one mode, qubits coupled together by means of broadband cooling would be required, |Ψ 〉 √ |↓↑〉 |↑↓〉 – = 1/ 2 ( – i ) . (9) the common vibrational mode, in even though that would seemingly which all ions oscillate back and forth exacerbate the problem of unwanted These states serve as the logical qubit in unison along the trap axis. excitation. But recent work by Blatt’s for storing information. It is easy to However, even two trapped ions have group at the University of Innsbruck

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(a) Broadband Excitation (b) EIT Scheme (c)

〉m + 1〉 m〉 m – 1〉 〉m〉  〉 〉m –1〉 〉 〉 〉 + 1 Probe Coupling 〉 〉 + 1 m – 1 m k 

laser laser k  k    〉 〉 〉 〉 〉 k k m k m

〉 〉      m + 1  r 〉 〉m〉  〉

〉m – 1〉 Cooling laser absorption (a.u.) -2 -1 0 1 2 Relative detuning of the two lasers Figure 11. EIT Cooling Sideband cooling of a multi-ion string that is accessing many This result is evident in figure (c), where the solid line gives excited vibrational modes is very difficult in that the sideband the absorption profile for the EIT scheme. For proper tuning of structure becomes dense and complicated. EIT cooling the lasers, the absorption strength for the transition |m〉→|m〉 permits broadband cooling of several vibrational modes, is zero and a strong asymmetry between |m〉→|m + 1〉 and |m〉, |k〉, ... , simultaneously. (a) When a broadband probe laser |m〉→|m –1〉 transitions is introduced. This asymmetry in is applied to the |〉|m〉→|〉|m〉 transition, both cooling (red) absorption between the blue and the red sideband also holds and heating (blue) transitions can occur. (b) When a second for higher-frequency vibrational modes (|k〉→|k ±1〉), allowing coupling laser excites the |r〉→|〉 transition, the ion’s simultaneous cooling of several different modes with one absorption profile becomes modified. Proper choice of laser broadband laser. [Figure was adapted from Schmidt-Kaler (2001) with detuning (to the dashed state) suppresses heating transitions. permission from the authors.] may make broadband cooling possible (that is, it is not excited at all because modes (“spectator” modes) are not (Morigi et al. 2000, Roos et al. 2000). of that quantum interference), and the used for the computation directly; The group adopted the EIT technique maximum absorption is chosen to be they are coupled to and may affect the to selectively enhance the probability around the lower sideband frequency. common mode. The group achieved of exciting cooling transitions rather Because the absorption profile ground-state populations of 73 per- than heating transitions in the ion. generated in this manner is fairly cent for the axial and 58 percent for The necessary asymmetry between wide (much wider than the natural the radial mode. Although this result lower and higher sidebands can be width of the transition used for tradi- is certainly not as satisfactory as that achieved as follows: Consider a three- tional sideband cooling), the asym- achieved by sideband cooling level system with two lower levels metry between heating and cooling (because of the smaller absorption and one shared excited state—see transitions exists for many modes. asymmetries), it is certainly sufficient Figure 11(b). Using a strong coupling Several different modes can be for cooling (and thus suppressing) laser between one of the ground states cooled simultaneously with a single those modes. The EIT method and the upper state creates so-called operation. This technique reduces promises the possibility of cooling light shifts (that is, shifted energy lev- the overhead for laser-cooling of all spectator modes of a multiqubit els, as seen by another probe laser). multi-ion strings and also eases the quantum register with a single For a detuning of the coupling laser requirements for laser stability, operation. That would allow the more above the resonance, a probe laser which are very strict for standard elaborate (individual) sideband sees an absorption profile that shows sideband cooling. cooling scheme to be used on only zero absorption for a detuning equal To show that EIT cooling can the mode needed for calculations. to the coupling laser, the so-called simultaneously cool vibrational Fano profile—see Figure 11(c). modes with significantly different fre- Therefore, such a probe would be quencies of oscillation, the Innsbruck Outlook transparent for that exact detuning— group chose to cool the axial mode the EIT phenomenon. In order to and the radial mode of a single ion Many systems have been proposed obtain optimum cooling using these confined in a three-dimensional Paul in the last several years as potential EIT resonances, the detunings are trap at 3.3 megahertz and 1.6 mega- candidates for becoming quantum chosen such that the carrier transition hertz, respectively (Schmidt-Kaler et computers, including laser-cooled is exactly located at the EIT resonance al. 2001). In a linear trap, the nearby trapped ions (Cirac and Zoller 1995),

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nuclear magnetic resonance chy of time scales present in an ion- Acknowledgments (Gershenfeld and Chuang 1997, Cory trap quantum computer is very prom- et al. 1998), cavity quantum electro- ising. Manipulations on quantum The work described in this article dynamics (Ye et al. 1999), and more registers can be done in microseconds, is the result of a close and fruitful col- recently, superconducting devices, while disturbances by the environment laboration among numerous experi- quantum dots, and silicon-based solid- have been shown to be avoidable mental scientists over several years. state devices. for milliseconds. The inherent deco- I wish to thank them all for their help From the preliminary experiments herence time of the quantum state is in the laboratory, as well as for the performed by several groups world- longer still, for it is limited by the many intellectual discussions that wide, it is apparent that the existing lifetime of the upper qubit state, helped me understand in depth the ion traps are adequate to hold and which is about 1 second in calcium. fundamental principles involved. Over manipulate small numbers of qubits. The decoherence time can be the years, I enjoyed working with Although five to ten qubits hardly a increased even more by an appropriate Daphna Enzer, John Gomez, Mark computer make, these numbers are choice of ions (for example, Gulley, Paul Kwiat, Steve Lamoreaux, large enough to make the technology ytterbium) or by stable ground-state Glen Peterson, Vern Sandberg, Martin well worth pursuing. Ion traps will be hyperfine levels used as logical qubit Schauer, Dale Tupa, and Justin a potent tool for exploring, for exam- states. Torgerson on developing capabilities ple, the possibility of creating entan- It is important to point out that for quantum computation with trapped gled states of large numbers of qubits. despite the revolutionary advances in ions. In addition, I had the privilege to Investigations of the type described computers during the last 50 years, travel to many groups pursuing this here will help us identify the relevant the fundamental principle of computa- dream and learned much of what I physics issues that must be addressed tion has not changed. Today’s fastest know today about the ion-trap to achieve computational gains. supercomputer operates according to quantum processor from many We should also expect that many of the same rules as the ENIAC. interactions with my friends and the technologies now being pursued for Quantum computation, however, rep- colleagues Rainer Blatt, Ferdinand quantum computation will be super- resents a paradigm shift in informa- Schmidt-Kaler, Dietrich Leibfried, seded by even more promising ideas. tion processing. Although a future Christoph Nägerl, and many others at One such idea is to scale up to a larger quantum computer may not look any- the University of Innsbruck in Austria; number of qubits by multiplexing sev- thing like our current ion trap, the with Andrew Steane, David Lucas, eral ion traps with a specific trap that experience and knowledge we gain and Derek Stacey from the Clarendon contains a few qubits acting as the cen- now will be of fundamental impor- Laboratory in Oxford, the United tral processor. After implementing part tance to our understanding this new Kingdom; and last but not least, with of a quantum algorithm, the qubits paradigm of computing. all the members in Dave Wineland’s would be shuffled into one of several For some researchers, building a group at NIST, in Boulder (too storage traps, thus allowing new qubits quantum computer to break secure numerous to name here individually), to be loaded into the processor. Recent codes is an important, and certainly who took me on challenging excur- work also suggests that we could challenging, goal. But for me and most sions into the quantum world of transfer the internal quantum states of of my colleagues, performing experi- trapped ions, as well as into the all too a string of ions in a trap to a set of ments that Erwin Schrödinger and classical world of mountain biking. photons in a high-finesse cavity. The Albert Einstein only dreamed of and quantum information could then be thus gaining a deep understanding of transferred through optical fibers into a this “inconceivable” quantum world Further Reading second cavity and fed back into an ion are far larger rewards. Perhaps we will string in a different trap. Developments encounter some failure of conventional Bergquist, J. C., R. G. Hulet, W. M. Itano, and like this will surely continue to happen quantum mechanics, or perhaps the D. J. Wineland. 1986. Observation of Quantum Jumps in a Single Atom. Phys. and will allow us to explore quantum problems of decoherence will forever Rev. Lett. 57: 1699. computation well beyond the current keep the quantum realm out of our Cirac, J. I., and P. Zoller. 1995. Quantum state of the art. classical grasp. In any event, the future Computations with Cold Trapped Ions. As we get closer to realizing a will be exciting for both quantum Phys. Rev. Lett. 74: 4091. Cory, D. G., M. D. Price, and T. F. Havel. 1998. small quantum processor, the “time physics and computation. Nuclear Magnetic Resonance Spectroscopy: scales” of a particular system become An Experimentally Accessible Paradigm for more relevant. In general, the hierar- Quantum Computing. Physica D 120 (1–2): 82.

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Dawson, P. H., ed. 1976. Quadrupole Mass Morigi, G., J. Eschner, and C. H. Keitel. 2000. Walther, H. 1991. In Light Induced Kinetic Spectrometry and Its Applications. Chaps. II Ground State Laser Cooling Using Effects on Atoms, Ions, and Molecules,p. and III. Amsterdam: Elsevier Scientific Electromagnetically Induced Transparency. 261. Edited by L. Moi et al. (Pisa, Italy: Publishing Co. Phys. Rev. Lett. 85 (21): 4458. ETS Editrice). Dehmelt;, H. G. 1967. Radiofrequency Mundt, A. B., A. Kreuter, C. Becher, ———. 1994. Atoms in Cavities and Traps. Spectroscopy of Stored Ions, Part I. Adv. At. D. Leibfried, J. Eschner, F. Schmidt-Kaler, Adv. At. Mol. Opt. Phys. 32: 379. Mol. Phys. 3: 53. and R. Blatt. 2002. Coupling a Single Wineland, D., P. Ekstrom, and H. Dehmelt. ———. 1969. Radiofrequency Spectroscopy of Atomic Quantum Bit to a High Finesse 1973. Monoelectron Oscillator. Phys. Rev. Stored Ions, Part II. Adv. At. Mol. Phys. 5: Optical Cavity. http://eprints.lanl.gov. quant- Lett. 31: 1279. 109. ph/0202112. Ye,J., D. W. Vernooy, and H. J. Kimble. 1999. ———. 1981. Coherent Spectroscopy on Nägerl, H. C., D. Liebfried, H. Rhode, Trapping of Single Atoms in Cavity QED. Single Atomic System at Rest in Free Space G. Thalhammer, J. Eschnerr, F. Schmidt- Phys. Rev. Lett. 83 (24): 4987. 2. J. Phys. (Paris) 42: 299. Kalerr, and R. Blatt. 1999. Laser Addressing ———. 1988. A Single Atomic Particle of Individual Ions in a Linear Ion Trap. Forever Floating at Rest in Free Space: New Phys. Rev. A 60 (1): 145. Value for Electron Radius. Physica Scripta. Neuhauser, W., M. Hohenstatt, P. E. Toschek, T22: 102. and H. Dehmelt. 1980. Localized Visible Michael Holzscheiter received his M.S. and Enzer, D. G., M. M. Schauer, J. J. Gomez, Ba+ Mono-Ion Oscillator. Phys. Rev. A 22: Ph.D. degrees in physics from the University of M. S. Gulley, M. H. Holzscheiter, 1137. Mainz in Germany, where he studied electrons P. G. Kwiat et al. 2000. Observation of Parkins, A. S., and H. J. Kimble. 1999. trapped in Power-Law Scaling for Phase Transitions in Quantum State Transfer between Motion Penning traps. As Linear Trapped Ion Crystals. Phys. Rev. and Light. J. Opt. B, Quantum a postdoctoral Lett. 85 (12): 2466. Semiclassical Opt. 1 (4): 496. researcher at Gershenfeld, N. A., and I. L Chuang. 1997. Paul, W., H. P. Reinhard, U. von Zahn. 1958. Texas A&M Bulk Spin-Resonance Quantum Electrical Mass Filters as Mass University, Computation. Science 275: 350. Spectrometers and Isotope Filters. Z. Phys. Michael used Huang, X.-P., J. J. Bollinger, T. B. Mitchell, and 152: 143. trapped ions to W. M. Itano. 1998. Phase-Locked Rotation Raizen, M. G., J. M. Gilligan, J. C. Bergquist, study collisional of Crystallized Non-Neutral Plasmas by W. M. Itano, and D. J. Wineland. 1992. processes of Rotating Electric Fields. Phys. Rev. Lett. 80: Ionic Crystals in a Linear Paul Trap. Phys. astrophysical rel- 73. Rev. A 45: 6493. evance. Later, as Hughes, R. J., D. F. V. James, E. H. Knill, Roos, C. F., D. Leibfried, A. Mundt, F. an assistant pro- R. Laflamme, and G. F. Petschek. 1996. Schmidt-Kaler, J. Eschner, and R. Blatt. fessor at Texas A&M, he participated in a col- Decoherence Bounds on a Quantum 2000. Experimental Demonstration of laboration with Los Alamos National Computation with Trapped Ions. Phys. Rev. Ground State Laser Cooling with Laboratory on trapping antiprotons. In 1986, he Lett. 77: 3240. Electromagnetically Induced Transparency. joined Los Alamos and pioneered the dynamic Hughes, R. J., D. F. V. James, J. J. Gomez, Phys. Rev. Lett. 85 (26): 5547. trapping of high-energy particles in Penning M. S. Gulley, M. H. Holzscheiter, Sackett, C. A., D. Kielpinski, B. E. King, traps. He became principal investigator of the P. G. Kwiat, et al. 1998. The Los Alamos C. Langer, V. Meyer, C. J. Myatt, et al. Los Alamos antiproton experiment, which was Trapped Ion Quantum Computer 2000. Experimental Entanglement of Four installed at CERN in 1992. On the basis of his Experiment. Fortschr. Phys. 46 (4–5): 329. Particles. Nature 404: 256. technique’s success, he formed an international James, D. F. V. 1998a. Quantum Dynamics of Sauter, Th., W. Neuhauser, R. Blatt, and collaboration, ATHENA, to create antihydrogen Cold Trapped Ions with Application to P. E. Toschek. 1986. Observation of atoms at rest for ultrahigh precision studies of Quantum Computation. Appl. Phys. B Quantum Jumps. Phys. Rev. Lett. 57: 1696. the symmetries between matter and antimatter 66 (2): 181. Schmidt-Kaler, F., J. Eschner, G. Morigi, and served as spokesman for this collaboration ———. 1998b. Theory of Heating of the C. F. Roos, D. Liebfried, A. Mundt, and from 1995 through 1999. Back in Los Alamos, Quantum Ground State of Trapped Ions. R. Blatt. 2001. Laser Cooling with Michael applied his expertise in ion traps Pys. Rev. Lett. 81: 317. Electromagnetically Induced Transparency: toward a Los Alamos trapped-ion quantum Kielpinski, D., V. Meyer, M. A. Rowe, Application to Trapped Samples of Ions or computer, for which he designed the C. A. Sackett, W. M. Itano, C. Monroe, and Neutral Atoms. Appl. Phys. B 73 (8): 807. radio-frequency quadrupole trap and helped D. J. Wineland. 2001. A Decoherence-Free Steane, A., C. F. Roos, D. Stevens, A. Mundt, build and operate the first-generation experi- Quantum Memory Using Trapped Ions. D. Leibfried, F. Schmidt-Kaler, and ment until the fall of 2001. His research Science 291: 1013. R. Blatt. 2000. Speed of Ion-Trap interest continues in the application of the Marzoli, I., J. I. Cirac, R. Blatt, and P. Zoller. Quantum-Information Processors. ion-trap technology to a wide variety of 1994. Laser Cooling of Trapped Three- Phys. Rev. A 62: 042305. physics problems. Level Ions: Designing Two-Level Systems Tanoudji, C. C., B. Diu, and F. Laloe. 1977. for Sideband Cooling. Phys. Rev. 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