Quantum Dynamics of Single Trapped Ions

REVIEWS OF MODERN PHYSICS, VOLUME 75, JANUARY 2003 Quantum dynamics of single trapped ions

D. Leibfried University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80305-3328

R. Blatt Institut fu¨r Experimentalphysik, Universita¨t Innsbruck, A-6020 Innsbruck, Austria

C. Monroe FOCUS Center and Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120

D. Wineland National Institute of Standards and Technology, Boulder, Colorado 80305-3328 (Published 10 March 2003)

Single trapped ions represent elementary quantum systems that are well isolated from the environment. They can be brought nearly to rest by laser cooling, and both their internal electronic states and external motion can be coupled to and manipulated by light ﬁelds. This makes them ideally suited for quantum-optical and quantum-dynamical studies under well-controlled conditions. Theoretical and experimental work on these topics is reviewed in the paper, with a focus on ions trapped in radio-frequency (Paul) traps.

CONTENTS C. Electromagnetically induced transparency cooling 300 1. Cooling in the Lamb-Dicke regime 301 I. Introduction 282 2. Scattering rates in EIT cooling 302 II. Radio-Frequency Traps for Single Charged Particles 283 3. Experimental results 303 A. Classical motion of charged particles in rf traps 283 V. Resonance Fluorescence of Single Ions 303 1. Classical equations of motion 283 A. Excitation spectroscopy, line shapes 303 2. Lowest-order approximation 284 B. Nonclassical statistics, antibunching, and 3. Typical realizations 285 squeezing 304 B. Quantum-mechanical motion of charged C. Spectrum of resonance fluorescence, homodyne particles in rf traps 285 detection of fluorescence 305 1. Quantum-mechanical equations of motion 286 VI. Engineering and Reconstruction of Quantum States 2. Lowest-order quantum approximation 287 of Motion 307 C. Special quantum states of motion in ion traps 287 A. Creation of special states of motion and 1. The number operator and its eigenstates 287 internal-state/motional-state entanglement 307 2. Coherent states 288 3. Squeezed vacuum states 288 1. Creation of number states 307 4. Thermal distribution 289 2. Creation of coherent states 308 III. Trapped Two-Level Atoms Coupled to Light Fields 289 3. Creation of squeezed states 310 A. The two-level approximation 290 4. ‘‘Schro¨ dinger-cat’’ states of motion 310 B. Theoretical description of the coupling 290 5. Arbitrary states of motion 312 1. Total Hamiltonian and interaction B. Full determination of the quantum state of Hamiltonian 290 motion 312 2. Rabi frequencies 291 1. Reconstruction of the number-state density 3. Lamb-Dicke regime 292 matrix 313 4. Resolved sidebands 292 2. Reconstruction of s-parametrized 5. Unresolved sidebands 293 quasiprobability distributions 313 6. Spectrum of resonance fluorescence 293 3. Experimental state reconstruction 314 C. Detection of internal states 294 VII. Quantum Decoherence in the Motion of a Single 1. The electron shelving method 294 Atom 315 2. Experimental observations of quantum A. Decoherence background 316 jumps 295 B. Decoherence reservoirs 316 D. Detection of motional-state populations 295 1. High-temperature amplitude reservoir 316 IV. Laser Cooling of Ions 296 2. Zero-temperature amplitude reservoir 318 A. Doppler cooling 296 3. High-temperature phase reservoir 319 B. Resolved-sideband cooling 298 C. Ambient decoherence in ion traps 320 1. Theory 298 VIII. Conclusions 320 2. Experimental results 299 Acknowledgments 320

Appendix: Couplings of Light Fields to The Internal Electronic tectors covering a small solid angle and having low State 320 quantum efficiencies. Moreover, several variations of the 1. Dipole coupling 321 electron shelving technique suggested by Dehmelt 2. Quadrupole coupling 321 3. Raman coupling 321 (1975) can distinguish internal electronic states of the References 322 trapped ion with a detection efficiency close to unity. The first single-particle trapping experiments in the ‘‘...we never experiment with just one electron or atom present context were those on electrons confined in a or (small) molecule.’’—E. Schro¨ dinger, 1952 Penning trap at the University of Washington by Wine- land, Ekstrom, and Dehmelt (1973). The same paper I. INTRODUCTION contained a proposal to store a single atomic ion, an idea further discussed by Dehmelt (1973) in the same year. It In the last 30 years, experiments with single trapped took another seven years until Neuhauser et al. (1980) ions or charged fundamental particles have provided key reported the first experiment with a single barium ion in contributions to many fields in physics. This review does an rf trap. This experiment was followed by one of not attempt to cover all these fields and will therefore Wineland and Itano (1981), who used a single magne- concentrate on the use of trapped ions in quantum op- sium ion in a Penning trap. To date there are about 20 tics and the closely related field of coherent control of groups all over the world that work with single ions in rf the internal state and the motional state of ions in the traps. trap potential. With the availability of spectrally narrow An important ingredient for work with atomic systems light sources based on lasers, these fields have prospered was the advent of laser cooling. It was independently in the last 15 years and are still in rapid development, proposed by Ha¨nsch and Schawlow (1975) for free par- gaining momentum as a result of the new challenges en- ticles and by Wineland and Dehmelt (1975a) for trapped countered in quantum information processing. While there are several review papers and books covering the particles. The first atomic laser cooling experiments more traditional work in ion traps [see, for example, were reported independently by Wineland, Drullinger, and Walls (1978) on Mgϩ ions and by Neuhauser et al. Brown and Gabrielse (1986); Paul (1990); Gosh (1995)], ϩ those on ion-trap-based frequency standards (Diddams (1978) on Ba ions. In the following years laser cooling et al., 2001, and references therein), and those that con- of collections of ions or single ions found widespread use centrate on quantum information processing (Steane, in many different research groups around the world. La- 1997; Wineland et al., 1998), there is no comprehensive ser cooling to the ground state of the trapping potential ϩ work covering quantum optics, coherent control of the was first achieved on a single Hg ion by the National motion, and motional-state reconstruction with trapped Institute of Standards and Technology (NIST) group ions. The purpose of this review is to fill this gap. (Diedrich et al., 1989). One seemingly obvious approach to understanding Many experiments with single ions have the goal of the interaction of atoms and light is to isolate and con- using a narrow optical or microwave transition between fine a single atomic system, put it to rest or at least into electronic states of the ion(s) for frequency standards a well-characterized state of motion, and then direct (Diddams et al., 2001, and references therein). Others light fields onto that isolated system in a precisely con- deal with the quantum-mechanical aspects of the light trolled manner. This idea seems quite straightforward, emitted by a single ion and the quantum dynamics of the but can be difficult to convert into a feasible experiment. ions’ motion in the trapping potential. The latter field Traps for neutral atoms often have a rather shallow trap- gained considerable interest after it was realized that ping potential that can also depend on the electronic these dynamics closely resemble the Jaynes-Cummings state of the atom, thus perturbing the internal states and model that was well known from cavity QED (Blockley, entangling them with the motion. With ion traps, which Walls, and Risken, 1992). These are the experiments couple to the excess charge of the trapped particle, po- that will be preferentially covered in this review. In re- tential wells that are up to several electron volts deep cent years a number of groups have also started working and do not depend on the internal electronic state of the ion can be realized. The most popular forms of ion traps with single ions in an effort to implement quantum in- are the Penning trap (Penning, 1936), in which the formation processing following the proposal by Cirac charged particles are held in a combination of electro- and Zoller (1995), a field that is very closely related to static and magnetic fields, and the traps developed by our subject (Steane, 1997; Wineland et al., 1998). Wolfgang Paul (Paul, 1990), in which a spatially varying Following this Introduction, the second section con- time-dependent field, typically in the radio-frequency sists of a discussion of the classical and quantum motion (rf) domain, confines the charged particles in space. In of single ions in rf traps, including the driven motion due this review only the latter type will be considered. to the trapping rf field (micromotion). Section III intro- The intrinsically low signal levels one faces in the de- duces two-level atoms and describes the coupling of tection of single particles can be overcome by laser- trapped ions to the light field. With these tools at hand, induced fluorescence. On a dipole allowed transition a we study laser cooling in Sec. IV and resonance fluores- single ion can scatter several million photons per second cence in Sec. V. In Sec. VI we review experimentally and a sufficient fraction may be detected even with de- realized methods for engineering and reconstructing

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quantum states of motion. Section VII summarizes ex- A. Classical motion of charged particles in rf traps periments on decoherence and reservoir engineering with single trapped ions. 1. Classical equations of motion

II. RADIO-FREQUENCY TRAPS FOR SINGLE CHARGED The classical equations of motion of a particle with PARTICLES mass m and charge Z͉e͉ in a potential of the form given by Eq. (1) were first studied by Paul, Osberghaus, and In this section the equations of motion of a charged Fischer (1958). They are decoupled in the spatial coor- particle in different common types of rf traps are dis- dinates. Only the motion in the x direction will be discussed. Only trap types that lead to an electric potential cussed below; the other directions can be treated analo- ⌽(x,y,z,t) of approximately quadrupolar spatial shape gously. The equation of motion is ⌽ Z͉e͉ץ in the center of the trapping region are considered here. Z͉e͉ It is further assumed that the potential can be decom- ϭϪ ϭϪ ͓ ␣ϩ ˜ ͑␻ ͒␣Ј͔ (U U cos rft x (5 ץ ¨x posed into a time-dependent part that varies sinusoidally m x m ␻ at the rf drive frequency rf and a time-independent and can be transformed to the standard form of the static part: Mathieu differential equation 1 d2x ⌽͑x,y,z,t͒ϭU ͑␣x2ϩ␤y2ϩ␥z2͒ ϩ͓a Ϫ2q cos͑2␰͔͒xϭ0 (6) 2 d␰2 x x 1 by the substitutions ϩ ˜ ͑␻ ͒ ͑␣ 2ϩ␤ 2ϩ␥ 2͒ U cos rft Јx Јy Јz . 2 ␻ t 4Z͉e͉U␣ 2Z͉e͉U˜ ␣Ј ␰ϭ rf ϭ ϭ (1) , ax ␻2 , qx ␻2 . (7) 2 m rf m rf The condition that this potential has to fulfill the ⌬⌽ϭ The Mathieu equation belongs to the general class of Laplace equation 0 at every instant in time leads differential equations with periodic coefficients. The to restrictions in the geometric factors, namely, general form of the stable solutions follows from the ␣ϩ␤ϩ␥ϭ0, (2) Floquet theorem (McLachlan, 1947; Abramowitz and Stegun, 1972), ␣Јϩ␤Јϩ␥Јϭ 0. ϱ ␤ ␰ ␰ ͑␰͒ϭ i x i2n From these restrictions it is obvious that no local three- x Ae ͚ C2ne dimensional minimum in free space can be generated, so nϭϪϱ the potential can only trap charges in a dynamical way. ϱ Ϫ ␤ ␰ Ϫ ␰ As we shall see below, the drive frequency and voltages ϩ i x i2n Be ͚ C2ne , (8) can be chosen in such a way that the time-dependent nϭϪϱ potential will give rise to stable, approximately har- where the real-valued characteristic exponent ␤ and monic motion of the trapped particles in all directions. x the coefficients C are functions of a and q only and One choice for the geometric factors is 2n x x do not depend on initial conditions. A and B are arbi- ␣ϭ␤ϭ␥ϭ0, (3) trary constants that may be used to satisfy boundary conditions or normalize a particular solution. By insert- ␣Јϩ␤ЈϭϪ␥Ј , ing Eq. (8) into Eq. (6) one obtains a recursion relation, leading to three-dimensional confinement in a pure os- C ϩ ϪD C ϩC Ϫ ϭ0, cillating field. A second choice is 2n 2 2n 2n 2n 2 D ϭ͓a Ϫ͑2nϩ␤ ͒2͔/q , (9) Ϫ͑␣ϩ␤͒ϭ␥Ͼ0, (4) 2n x x x that connects the coefficients and ␤ to a and q . ␣ЈϭϪ␤Ј x x x , Simple rearrangements and recursive use of Eq. (9) leading to dynamical confinement in the x-y plane and yield continued fraction expressions for the C2n , static potential confinement for positively charged par- C2n ticles in the z direction as used in linear traps (Paul, C ϩ ϭ , (10) 2n 2 1 1990). D Ϫ First we give an overview of the classical equations of 2n 1 D ϩ Ϫ motion and their solutions, and approximations to these 2n 2 ... solutions are studied. A quantum-mechanical picture of ions trapped in rf fields following the approach of C2nϪ2 C ϭ , Glauber (1992) is then derived and it is shown that in 2n 1 D Ϫ the range of trapping parameters used in the experi- 2n 1 ments discussed here, the quantized motion of trapped D Ϫ Ϫ 2n 2 ... ions can be modeled by static potential harmonic oscil- ␤2 lators to a very good approximation. and for x,

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FIG. 1. Trap stability: (a) Stability diagram for a cylindrically symmetric trap with rf conﬁnement in all three axes (␣ϭ␤ ϭϪ␥/2,␣Јϭ␤ЈϭϪ␥Ј/2); (b) stability diagram for a linear trap (␣ϩ␤ϭϪ␥,␣ЈϭϪ␤Ј,␥Јϭ0).

1 1 Figure 1(a) shows a plot of the lowest stability region for ␤2ϭa Ϫq ϩ . x x x 1 1 the cylindrically symmetric Paul trap. The axes dis- Ϫ Ϫ played are a and q , and the corresponding values for ͩ D0 D0 ͪ z z 1 1 the x and y dimensions can be found from Eq. (12). The D Ϫ DϪ Ϫ 2 ... 2 ... borderlines of stability in the x and y direction are iden- (11) tical. For the linear trap, the parameters have the rela- ␤ tions Numerical values for x and the coefficients can be extracted by truncating the continued fractions after the ϭϪ ϭ qy qx , qz 0. (14) desired accuracy is reached. The contributions of higher orders in the continued fraction rapidly drop for typical In this case the first stability region is symmetric around the q axis, since the borderlines of stability in the two values of ax and qx used in the experiments described x here. directions are mirror images of each other [see Fig. 1(b)]. General traps with no intrinsic symmetry can have The region of stability in the ai-qi plane (i ෈ even more complicated stability diagrams, because the ͕x,y,z͖) is bounded by pairs of ai and qi that yield ␤ ϭ ␤ ϭ borderlines of stability might not be connected by either i 0or i 1 (Paul, Osberghaus, and Fischer, 1958; Gosh, 1995). The stable region that contains the simple relations such as in the two cases discussed here ϭ ෈ [see Eq. (4)]. The radially defocusing effects of the static points (ai ,qi) (0,0) for all i ͕x,y,z͖ is often called the lowest stability region. The traps relevant for the potential along the z axis in a linear Paul trap can also experiments discussed here work inside this lowest lead to modifications in the stability diagram, especially Ϸ if the confinement along this axis becomes comparable stable region with ai 0. The exact shape of the stability regions depends on in strength to the radial confinement (Drewsen and the actual parameters in Eq. (1). For three-dimensional Brøner, 2000). (3D) rf confinement, for example, with the parameters as in Eq. (3), the trap electrodes are often cylindrically 2. Lowest-order approximation symmetric around one axis (usually labeled the z axis), leading to the parameter relations ␣Јϭ␤ЈϭϪ␥Ј/2 and The lowest-order approximation to the ion trajectory ␣ϭ␤ϭϪ␥ ͉ ͉ 2 Ӷ /2. The parameters ai and qi in the Mathieu x(t) in the case ( ax ,qx) 1 can be found by assuming Ӎ equations along the different axes will then obey CϮ4 0. Then, together with the initial condition B ϭ ϭϪ ϭϪ ϭϪ ϭϪ A, Eq. (9) yields az 2ax 2ay , qz 2qx 2qy . (12) ␤ Ϸͱ ϩ 2 A trapped particle will be stable in all three dimensions x ax qx/2, if ␻ q ͑ ͒Ϸ ͩ ␤ rf ͪͫ Ϫ x ͑␻ ͒ͬ р␤ р ෈ x t 2AC0 cos x t 1 cos rft , (15) 0 i 1, for all i ͕x,y,z͖. (13) 2 2

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FIG. 2. Schematic drawing of the electrodes for a cylindrically Ӎ& symmetric 3D rf trap. Typical dimensions are r0 z0 Ӎ100 ␮m– 1 cm with U˜ Ӎ100– 500 V, ͉U͉Ӎ0 – 50 V, and ␻ ␲Ӎ rf/2 100 kHz– 100 MHz. FIG. 3. Schematic drawing of the electrodes for a linear rf ˜ ␻ trap. A common rf potential U cos( rft) is applied to the dark identical to the solution found by the pseudopotential electrodes; the other electrodes are held at rf ground through approximation (Dehmelt, 1967; Gosh, 1995). The trajec- capacitors (not shown) connected to ground. The lower right tory consists of harmonic oscillations at frequency ␯ portion of the figure shows the x-y electric fields from the ϭ␤ ␻ Ӷ␻ applied rf potential at an instant when the rf potential is posi- x rf/2 rf , the secular motion, superposed with driven excursions at the rf frequency ␻ . The driven tive relative to the ground. A static electric potential well is rf created (for positive ions) along the z axis by applying a posi- excursions are 180° out of phase with the driving field tive potential to the outer segments (gray) relative to the cen- and a factor q /2 smaller than the amplitude of the secu- x ter segments (white). Typical dimensions are r lar motion. These fast, small oscillations are therefore 0 Ӎ100 ␮m– 1 cm with U˜ Ӎ100– 500 V, ͉U͉Ӎ0 – 50 V, and dubbed micromotion. If micromotion is neglected, the ␻ /2␲Ӎ100 kHz– 100 MHz. secular motion can be approximated by that of a har- rf monic oscillator with frequency ␯. Most theoretical papers covering the subject of this review assume this ap- also operated in the left-hand portion of the stability Ͻ proximation. In the course of this paper we shall see that diagram [Fig. 1(b)] with qx 0.5. it is justified in most cases if the ions are at reasonably low kinetic energy, even if we treat the center-of-mass motion of the ion quantum mechanically. B. Quantum-mechanical motion of charged particles in rf traps 3. Typical realizations Even a simple account of the cooling process in ion One of the most popular trap configurations is the traps, as well as the description of nonclassical states, ϭϪ cylindrically symmetric 3D rf trap (az 2ax relies on a quantum-mechanical picture of the motion. ϭϪ ϭϪ ϭϪ 2ay ; qz 2qx 2qy). It can be realized with the Since the trapping potential is not a static, but rather a electrode configuration shown in Fig. 2 where ␣ϭ␣Ј time-dependent potential, it cannot be taken for granted ϭ␤ϭ␤ЈϭϪ2␥ϭϪ2␥Ј and where, to a good approxima- that quantization of the motion in the effective time- ␣ϭ 2ϩ 2 tion, 2/(r0 2z0). This last expression holds exactly averaged potential already gives an adequate picture. In when the electrode surfaces coincide with equipotentials the first quantum-mechanical treatment of the time- of Eq. (1) and holds reasonably well with truncated elec- dependent potential by Cook, Shankland, and Wells trodes as shown in Fig. 2. Typically, the traps are oper- (1985), the authors derived an approximate solution of ated in the left-hand portion of the stability diagram the Schro¨ dinger equation and concluded that the stabil- Ͻ [Fig. 1(a)], where qz 0.5; however, the entire stability ity regions of classical and quantum-mechanical motion diagram has been experimentally explored, including are identical. They also found that the dominant effect parametric instabilities, in an impressive series of experi- of the time-dependent potential is to multiply the wave ments by the group of Werth (Alheit et al., 1996). function of the static pseudopotential by a time- A second very useful trap electrode configuration is dependent phase factor. The essence of these findings ϭ␥ ϭ ϭϪ that for the linear rf trap (qz Ј 0; qy qx) shown was confirmed and further elaborated upon by Combes- schematically in Fig. 3. This trap is essentially a linear cure (1986), who first derived an exact solution, and quadrupole mass filter (Paul, 1990) that has been later by Brown (1991) and Glauber (1992). All these plugged on the ends with a static axial z potential. If the treatments are semiclassical in the sense that the trap- axial potential is made fairly weak compared to the x,y ping rf field is not quantized, but rather represented as a potentials, two or more trapped ions will line up along classical electromagnetic potential of the form of Eq. the trap axis. This can be useful for addressing indi- (1). The treatment given here follows the elegant ap- vidual ions with laser beams. Typically these traps are proach of Glauber (1992).

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1. Quantum-mechanical equations of motion u*͑t͒u˙ ͑t͒Ϫu͑t͒u˙ *͑t͒ϭu*͑0͒u˙ ͑0͒Ϫu͑0͒u˙ *͑0͒ For the quantum-mechanical treatment of the motion ϭ2i␯. (24) we assume that the time-dependent potential is qua- The unknown coordinates xˆ (t) and u(t) satisfy the same dratic in each of the three Cartesian coordinates of the differential equation, so the complex linear combination center of mass of the trapped particle. Then, as in the classical motion, the problem is separable into three m Cˆ ͑t͒ϭͱ i͕u͑t͒xˆ˙ ͑t͒Ϫu˙ ͑t͒xˆ ͑t͖͒ (25) one-dimensional problems. In one dimension and re- 2ប␯ placing the coordinate x by the respective operator xˆ , is proportional to their Wronskian identity and also con- we can write the time-dependent potential V(t)as stant in time: m V͑t͒ϭ W͑t͒xˆ 2, (16) 1 2 Cˆ ͑t͒ϭCˆ ͑0͒ϭ ͓m␯xˆ ͑0͒ϩipˆ ͑0͔͒. (26) ͱ2mប␯ where Moreover the right-hand side is exactly the annihilation ␻2 rf operator of a static potential harmonic oscillator of mass W͑t͒ϭ ͓a ϩ2q cos͑␻ t͔͒ (17) ␯ 4 x x rf m and frequency , can be thought of as a time-varying spring constant that Cˆ ͑t͒ϭCˆ ͑0͒ϭaˆ , (27) 2 will play a role similar to ␻ in the static potential har- which immediately implies the commutation relation monic oscillator. With these definitions, H(m), the † † Hamiltonian of the motion, takes a form very similar to ͓Cˆ ,Cˆ ͔ϭ͓aˆ ,aˆ ͔ϭ1. (28) the familiar Hamiltonian of a static potential harmonic This static potential oscillator will be called the reference oscillator: oscillator in the remainder of this section. pˆ 2 m The Heisenberg operators xˆ (t) and pˆ (t) can be reex- Hˆ (m)ϭ ϩ W͑t͒xˆ 2, (18) pressed in terms of u(t) and the operators of the refer- 2m 2 ence oscillator using Eq. (25): and we can immediately write down the equations of ប motion of these operators in the Heisenberg picture: xˆ ͑t͒ϭͱ ͕aû*͑t͒ϩaˆ †u͑t͖͒, 2m␯ 1 pˆ ˙ ϭ ͓ ˆ (m)͔ϭ xˆ ប xˆ ,H , បm i m pˆ ͑t͒ϭͱ ͕aû˙ *͑t͒ϩaˆ †u˙ ͑t͖͒, (29) 2␯ 1 ˙ ϭ ͓ ˆ (m)͔ϭϪ ͑ ͒ pˆ ប pˆ ,H mW t xˆ , (19) so their entire time dependence is given by the special i solution u(t) and its complex conjugate. which can be combined into For later calculations it is convenient to have expressions for a basis of time-dependent wave functions in the xˆ¨ ϩW͑t͒xˆ ϭ0. (20) Schro¨ dinger picture. Again the reference oscillator used It is easy to verify that this equation is equivalent to the above is very helpful in this task. In analogy to the static potential case we shall consider a set of basis states ͉n,t͘ Mathieu equation (6) if one replaces the operator xˆ with ϭ ϱ a function u(t). This fact can be used to find solutions to in which n 1,2, ..., . These states are the dynamic Eq. (20) by utilizing a special solution of the Mathieu counterpart of the harmonic-oscillator number (Fock) equation subject to the boundary conditions states. The ground state of the reference oscillator ͉nϭ0͘␯ obeys the condition u͑0͒ϭ1, u˙ ͑0͒ϭi␯. (21) aˆ ͉nϭ0͘␯ϭCˆ ͑t͉͒nϭ0͘␯ϭ0, (30) This solution can be constructed from Eq. (8) with A ϭ1, Bϭ0, but since the Heisenberg operator Cˆ is connected to the ϱ ˆ ˆ Schro¨ dinger picture counterpart CS by C(t) ␤ ␻ ␻ ␤ ␻ ͑ ͒ϭ i x rft/2 in rftϵ i x rft/2⌽͑ ͒ ϭ ˆ † ˆ ˆ ˆ ϭ ͓Ϫ ប ˆ (m)͔ u t e ͚ C2ne e t , (22) U (t)CSU(t), with U(t) exp (i/ )H , we imme- nϭϪϱ diately get ⌽ where (t) is a periodic function with period T ˆ ͑ ͒ ˆ ͑ ͉͒ ϭ ͘ ϭ ˆ ͑ ͉͒ ϭ ͘ϭ ϭ ␲ ␻ CS t U t n 0 ␯ CS t n 0,t 0 (31) 2 / rf . In terms of the coefficients of this solution, Eq. (21) takes the form by multiplying Eq. (30) with Uˆ (t) from the left and not- ϱ ϱ ing that Uˆ (t)͉nϭ0͘␯ is the Schro¨ dinger state of the ͑ ͒ϭ ϭ ␯ϭ␻ ͑␤ ϩ ͒ u 0 ͚ C2n 1, rf ͚ C2n x/2 n . time-dependent oscillator that evolves from the ground ϭϪϱ ϭϪϱ n n state of the static potential reference oscillator. Since the (23) time dependence of the Schro¨ dinger operator CS(t)is This solution and its complex conjugate are linearly in- due entirely to the explicit time dependence of u(t), Eq. dependent; they therefore obey the Wronskian identity (31) is equivalent to

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͓u͑t͒pˆ Ϫmu˙xˆ ͔͉nϭ0,t͘ϭ0, (32) 2. Lowest-order quantum approximation or reexpressed in coordinate space The lowest-order approximation for the quantum- mechanical states will be studied by ﬁrst deriving an ap- .(proximate expression for the special solution u(t ץ ប ͭ u͑t͒ Ϫmu˙ ͑t͒xЈͮ ͗xЈ͉nϭ0,t͘ϭ0. (33) ͉ ͉ 2Ӷ ϭ xЈ Again ax ,qx 1 and CϮ4 0 is assumed. Together withץ i the initial conditions of Eq. (21) one ﬁnds The normalized solution is ␤ Ϸͱ ϩ 2 ␯Ϸ␤ ␻ x ax qx/2, x rf/2, ␯ 1/4 ͑ ͒ m 1 im u˙ t ϩ͑ ͒ ͑␻ ͒ ͗xЈ͉nϭ0,t͘ϭͩ ͪ expͫ xЈ2ͬ. 1 qx/2 cos rft ␲ប ͕u͑t͖͒1/2 2ប u͑t͒ u͑t͒Ϸexp͑i␯t͒ , (37) 1ϩq /2 (34) x essentially the lowest-order classical solution found ear- In complete analogy to the static potential harmonic os- lier in Eq. (15). It still must be stressed that the fre- cillator, all other states of a complete orthonormal base quency of the reference oscillator ␯ is equal to the char- can be created by repeated operation on the ground ␤ ␻ acteristic exponent x rf/2 only in this lowest-order ˆ † ␹ state with the creation operator CS(t): approximation. The periodic breathing of n(t) with period T is now obvious, as one can see in the approxi- ͓ ˆ †͑ ͔͒n rf CS t mate expression ␹ (t) for the ground-state wave func- ͉n,t͘ϭ ͉nϭ0,t͘ (35) 0 ͱn! tion: ␯ 1/4 ϩ m 1 qx/2 expressed in coordinate space, and by rewriting u(t) ␹ ͑t͒ϭͩ ͪ ͱ 0 ␲ប 1ϩ͑q /2͒cos͑␻ t͒ such as in Eq. (22), these states are x rf m␻ sin͑␻ t͒ m␯ ϫ ͩ rf rf Ϫ Ј2 ͪ 1 exp ͭ i ប͓ ϩ ͑␻ ͔͒ ប ͮ x , ͗ Ј͉ ͘ϭ ͫϪ ͩ ϩ ͪ ␯ ͬ␹ ͑ ͒ 2 2/qx cos rft 2 x n,t exp i n t n t , (36) 2 (38) with while the phase factor in Eq. (36) is governed by the ground-state pseudoenergy ប␯/2. This expression is 1 m␯ 1/4 exp͕Ϫin arg͓⌽͑t͔͖͒ identical to the static harmonic potential ground-state ␹ ͑ ͒ϭ ͩ ͪ n t 1/2 ␻ ϭ ͱ2nn! ␲ប ͕⌽͑t͖͒ wave function if one sets rf 0. m␯ 1/2 ϫH ͭͫ ͬ xЈͮ C. Special quantum states of motion in ion traps n ប͉⌽͑t͉͒2

m␯ i⌽˙ ͑t͒ In this section various classes of motional states in ion ϫ ͭ ͫ Ϫ ͬ Ј2ͮ traps will be discussed, some nonclassical in nature and exp ប 1 ␯⌽͑ ͒ x , 2 t some more reminiscent of classical motion. For each of the classes theoretical proposals on how to create them where Hn is the Hermitian polynomial of order n. The classical micromotion appears in the wave functions as a in an ion trap have been brought forward and all have pulsation with the period of the rf driving field. For a been created and observed experimentally. static potential harmonic oscillator the evolution of the energy eigenstates only multiplies the wave function by 1. The number operator and its eigenstates a phase factor (which is why they are called stationary To exploit the close analogy between the confinement states). In the time-dependent potential studied here, in an rf ion trap and that in a static harmonic potential, the same is true, but only for times that are integer mul- it is advantageous to express the motional states in the ϭ ␲ ␻ tiples of the rf period Trf 2 / rf . The states given by basis of the eigenstates of the reference oscillator num- Eq. (36) are not energy eigenstates (they periodically ber operator. We shall first do this in the Heisenberg exchange energy with the driving field in analogy to the picture. Since Cˆ (t) is time independent [see Eq. (27)], classical micromotion), but they are the closest approxi- the operator mation to stationary states possible in the time- dependent potential. Therefore they are often called Nˆ ϭCˆ †͑t͒Cˆ ͑t͒ϭaˆ †aˆ (39) quasistationary states. These features will be illustrated in the next few sec- is also time independent and the eigenstates are just the tions, where we shall find the lowest-order corrections to familiar number or Fock states of the static potential the static potential oscillator picture in close analogy to harmonic reference oscillator with the usual ladder alge- the classical pseudopotential solution presented in Sec. bra, ͉ ϭͱ ͉ Ϫ †͉ ϭͱ ϩ ͉ ϩ II.A.2. We also discuss the analogous operator to the aˆ n͘␯ n n 1͘␯ , aˆ n͘␯ n 1 n 1͘␯ , number operator for the static potential harmonic oscil- Nˆ ͉n͘␯ϭn͉n͘␯ . (40) lator and some special classes of motional states in the ion trap. Transforming to the Schro¨ dinger picture we get

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Uˆ †͑t͒Nˆ Uˆ ͑t͒ϭUˆ †͑t͒Cˆ †͑t͒Uˆ ͑t͒Uˆ †͑t͒Cˆ ͑t͒Uˆ ͑t͒ The term ‘‘coherent state’’ was first used by Glauber (1963, 1964) in connection with quantum states of a light ϭ ˆ †͑ ͒ ˆ ͑ ͒ CS t CS t . (41) field. There are different ways to define a coherent state (see Klauder and Skagerstam, 1985, for a review); for The eigenstates and eigenvalues of these operators example, they are the eigenstates of the annihilation op- are easily inferred from Eq. (35) in the last section: erator with complex eigenvalue ␣: Cˆ ͑t͉͒n,t͘ϭͱn͉nϪ1,t͘, S ˆ ͑ ͉͒␣ ϭ␣͉␣ CS t ͘ ͘. (45) Cˆ †͑t͉͒n,t͘ϭͱnϩ1͉nϩ1,t͘, (42) S It is easy to prove that states with coefficients in the implying number-state basis expansion Eq. (44), ˆ ͑ ͉͒ ͘ϭ ͉ ͘ NS t n,t n n,t . (43) ␣n ϭ ͑Ϫ͉␣͉2 ͒ These Schro¨ dinger-picture eigenstates can therefore be cn exp /2 , (46) ͱn! used in complete analogy to the static potential harmonic oscillator, and all algebraic properties of the static are eigenstates of this operator, and the probability dis- ˆ potential ladder operators carry over to CS(t) and tribution among number states is Poissonian, Cˆ † (t). The only difference is that these states are not S ϭ͉ ͉2ϭ͉͗ ͉␣͉͘2ϭ͑ n Ϫ¯n͒ ϭ͉␣͉2 energy eigenstates of the system, since the micromotion Pn cn n ¯n e /n! with ¯n . periodically changes the total kinetic energy of the ion. (47) Nevertheless, due to the periodicity of the micromotion, Another popular choice is to represent coherent states it makes sense to connect the quantum number n to the as the action of a displacement operator, ϭ ␲ ␻ energy of the ion averaged over a period Trf 2 / rf of the drive frequency. This connection will be further ex- Dˆ ͑␣͒ϭexp͓␣Cˆ †͑t͒Ϫ␣*Cˆ ͑t͔͒, (48) plored in Sec. IV on laser cooling. S S Any motional state can be expressed as a superposi- on the vacuum state, namely, tion of the number states ϱ Dˆ ͑␣͉͒0͘ϭ͉␣͘. (49) ⌿ϭ ͉ ͚ cn n,t͘, (44) 0 The action of successively applied displacement opera- and a number of these expansions will be used in what tors is also additive up to phase factors, follows in this paper. For convenience we shall set ͉n,t͘ ϭ͉ ͘ ␣␤ Ϫ␣ ␤ n and only write the time dependence explicitly if it Dˆ ͑␣͒Dˆ ͑␤͒ϭDˆ ͑␣ϩ␤͒e1/2( * * ), (50) helps to clarify matters. so the displacements form a group with Dˆ (0)ϭIˆ as a 2. Coherent states neutral element. Note that the extra phase on the right- In a static potential harmonic oscillator, a coherent hand side makes the displacement operations noncom- state of motion ͉␣͘ of the ion corresponds to a Gaussian mutative in general. minimum-uncertainty wave packet in the position representation whose center oscillates classically in the har- 3. Squeezed vacuum states monic well and retains its shape. The wave packet has the same shape as the ground-state wave function. In any quantum state the product of the variance in Glauber has shown that the states that evolve out of an position and momentum has a lower bound of ប2/4, initial coherent state in the dynamic trapping potential given by the Heisenberg uncertainty relation. The are also displaced forms of the Gaussian ground-state ground state of a static potential harmonic oscillator and Eq. (34) (Glauber, 1992). The displaced Gaussian does all other coherent states are minimum-uncertainty states the same breathing as the ground state, but does not in which the variance in position is (⌬x)2ϭ͗x2͘Ϫ͗x͘2 spread, and its center of gravity follows the classical tra- ϭ1/(m␯)ប/2 and the variance in momentum is (⌬p)2 jectory of an ion in the trap (now secular motion and ϭ(m␯)ប/2. If one now ‘‘squeezes’’ the position variance micromotion). States of this type were first considered the momentum variance must become wider, so the by Schro¨ dinger (1926) when he tried to construct wave Heisenberg uncertainty relation is still fulfilled. In the packets that reflected the classical motion of a harmonic course of the time evolution the squeezed position wave oscillator.1 packet will not retain its shape, but will become wider for half an oscillation period before it contracts back to the original width after a full period. The momentum 1One curiosity in his paper is that he considered the real part wave packet contracts and expands accordingly so that of the constructed wave packet to reflect the physical state, at any time the uncertainty is minimal (Walls 1986; Walls since the probability interpretation of quantum mechanics was and Milburn, 1995). The coefficients in expansion (44) not yet established then. for the so-called squeezed vacuum state are

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ͱ␤ 1/2 ␤ Ϫ n/2 ͱ 2 s s 1 n! ͩ ͪ ͩ ͪ ͑Ϫ1/2͒n/2 ein␾, n even ϭ ␤ ϩ ␤ ϩ ͑ ͒ cn ͭ s 1 s 1 n/2 ! (51) 0, n odd.

␤ The parameter s describes the squeezing of the state, ប␯ Tϭ . (55) namely, the position variance of the squeezed state is ¯nϩ1 reduced at certain times by k lnͩ ͪ B ¯n ⌬x ϭ⌬x /␤ , (52) s 0 s In considering an ensemble it is appropriate to charac- ⌬ where x0 is the variance of the ground state. The terize the state by a density matrix. Moreover, in the ␤ ϭ ground state is recovered for s 1 (therefore the name spirit of choosing the density matrix with the maximum ␤ Ͼ ‘‘squeezed vacuum state’’). For s 1 the position wave ignorance (and therefore maximum entropy), the off- function is narrower than that of the coherent state, diagonal elements have to be zero. This makes it impos- Ͻ␤ Ͻ while for 0 s 1 the momentum wave function has sible to write the thermal distribution in the form of Eq. this property. The angle ␾ describes the alignment of the (44) that would correspond to a density matrix with non- squeezed state with respect to the position and momen- zero off-diagonal elements for TϾ0. So, even if the term tum directions. This can best be visualized in phase ‘‘thermal state’’ is often used in the literature, ‘‘thermal space. The Wigner function of squeezed states has ellip- tical equicontour lines (Walls and Milburn, 1995). If one distribution’’ seems to be a more appropriate reminder of the major axes of these ellipses is aligned with the of the ensemble nature of this entity. position coordinate axis, ␾ is equal to zero. The center After some minor algebra to normalize the trace of of mass of the Wigner function of a squeezed vacuum the states weighted by the Boltzmann factors, the den- state coincides with the origin of phase space. sity matrix may be written as ϱ The probability distribution Pn for a squeezed 1 ¯n n vacuum state is independent of ␾ and again restricted to ϭ ͩ ͪ ͉ ͗͘ ͉ %ˆ th ϩ ͚ ϩ n n , (56) the even states, ¯n 1 nϭ0 ¯n 1 ͱ␤ ␤ Ϫ n with level population probability 2 s s 1 n! P ϭ ͩ ͪ 2Ϫn , n even. (53) n ␤ ϩ1 ␤ ϩ1 ͓͑n/2͒!͔2 ¯nn s s ϭ Pn nϩ1 . (57) For strong squeezing this distribution has a tail that ͑¯nϩ1͒ ␤ ϭ reaches to very high n; for example, with s 40, 16% of the population of the squeezed vacuum is in states III. TRAPPED TWO-LEVEL ATOMS COUPLED TO LIGHT above nϭ20. FIELDS Squeezed vacuum states, like coherent states, have a very compact operator representation. They are gener- With the help of suitable electromagnetic fields the ated from the ground state by the operator internal levels of trapped ions can be coherently coupled 1 1 to each other and the external motional degrees of free- ˆ ͑␧͒ϭ ͭ ␧ ˆ ͑ ͒2Ϫ ␧͓ ˆ ͑ ͒†͔2ͮ dom of the ions. For strongly confined ions and a suit- S exp *CS t CS t , (54) 2 2 able tuning the coupling is formally equivalent to the ␧ϭ i␾ ␤ ␤ ϭ 2r where re and r is related to s by s e . Jaynes-Cummings Hamiltonian (Jaynes and Cummings, 1963). Consequently much of the work devoted to coherent interaction of trapped ions has been inspired by the important role this coupling plays in quantum optics. 4. Thermal distribution Beyond this special case there are many possibilities connected to the interchange of multiple motional If the ion is in thermal equilibrium with an external quanta, in close analogy to multiphoton transitions in reservoir at temperature T the average weight of the quantum optics. Moreover, the light field inducing the excitation of the state ͉n͘ will be proportional to the coupling can act as a source of energy, so that energy ͓Ϫ ប␯ ͔ Boltzmann factor exp n /(kBT) , where kB is the conservation implicit in atom-photon couplings does not Boltzmann constant. Of course it does not make sense have to be fulfilled in the interaction of internal states to assign a temperature to a single realization of a and the motion of trapped ions, allowing interactions in cooled ion. However, if the ion is coupled to that reser- which both the internal state of the atom and its motion voir and the number operator Nˆ is measured many undergo a transition to a higher-energy level. Finally, if times (making sure that the ion reequilibrates after each the full quantum-mechanical picture of the motion, in- measurement), one can extract a temperature from the cluding corrections due to micromotion, is considered, average result ¯n from this ensemble of many different another class of transitions becomes possible that in- realizations according to volves exchange of motional quanta at integer multiples

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 290 Leibfried et al.: Quantum dynamics of single trapped ions of the rf driving field or combinations of integer mul- this review. The generalized description of the coupling tiples of the driving field and the secular motion (micro- of internal states and motion follows the approaches of motion sidebands). Cirac, Garay, et al. (1994) and Bardroff et al. (1996). It is also assumed that it is sufficient to treat the light A. The two-level approximation field in the lowest order in its multipole expansion that yields a nonvanishing matrix element between the near- In most of this review the internal electronic structure resonant electronic states in question. This assumption is of the ion will be approximated by a two-level system justified by the fact that the extension of the electronic ͉ ͉ ប␻ϭប ␻ wave function is much less than the wavelength of the with levels g͘ and e͘ of energy difference ( e Ϫ␻ coupling field. For dipole allowed transitions the field g). This is justified for real ions if the frequencies of the electromagnetic fields that induce the coupling are will be treated in the familiar dipole approximation, only close to resonance for two internal levels and if the while for dipole forbidden transitions only the quadru- Rabi frequencies describing the coupling strength are pole component of the field is considered. For Raman always much smaller than the detuning relative to off- transitions, the near-resonant intermediate level will be resonant transitions. Such a reduction is appropriate for adiabatically eliminated, making these transitions for- most of the experimental situations described in this pa- mally equivalent to the other transition types (see be- per. low). The corresponding two-level Hamiltonian Hˆ (e) is ˆ (e)ϭប͑␻ ͉ ͉ϩ␻ ͉ ͉͒ H g g͗͘g e e͗͘e ␻ ϩ␻ 1. Total Hamiltonian and interaction Hamiltonian e g ϭប ͉͑g͗͘g͉ϩ͉e͗͘e͉͒ 2 The total Hamiltonian Hˆ of the systems considered ␻ here can be written as ϩប ͉͑e͗͘e͉Ϫ͉g͗͘g͉͒. (58) 2 Hˆ ϭHˆ (m)ϩHˆ (e)ϩHˆ (i), (61) Since any operator connected to a two-level system can where Hˆ (m) is the motional Hamiltonian along one trap (e) be mapped onto the spin-1/2 operator basis, Hˆ and axis, Eq. (18), as discussed in Sec. II. Hˆ (e) describes the related operators can be conveniently expressed using internal electronic level structure of the ion (see Sec. the spin-1/2 algebra that is represented by Iˆ, the 2ϫ2 III.A), and Hˆ (i) is the Hamiltonian of the interactions unity matrix, and the three Pauli matrices. In the par- mediated by the applied light fields. ticular case at hand the mapping is As summarized in the Appendix, electric dipole allowed transitions, electric quadrupole allowed transi- , ˆ␴ۋIˆ, ͉g͗͘e͉ϩ͉e͗͘g͉ۋg͗͘g͉ϩ͉e͗͘e͉͉ x tions, and stimulated Raman transitions can be de- ␴ۋ͉ ␴ ͉ ͉Ϫ͉ۋ͉͒ Ϫ͉͉ ͉͑ i g͗͘e e͗͘g ˆ y, e͗͘e g͗͘g ˆ z. (59) scribed in a unified framework that associates a certain on-resonance Rabi frequency ⍀, effective light fre- (e) With this mapping Hˆ is reexpressed as quency ␻, and effective wave vector k with each of these ␻ transition types. The effective light frequencies and Hˆ (e)ϭប ␴ , (60) wave vectors are identical to the frequency and wave 2 z vector of the coupling light field for dipole and quadru- Ϫប ␻ ϩ␻ where the energy is rescaled by ( e g)/2 to sup- pole transitions, but equal to the frequency difference ␻ϭ␻ Ϫ␻ ϭ Ϫ press the state-independent energy contribution in Eq. 1 2 and wave vector difference k k1 k2 of the (58). two light fields driving the stimulated Raman transitions. For running wave light fields all three transition types B. Theoretical description of the coupling can be described by a coupling Hamiltonian of the form Hˆ (i)ϭ͑ប/2͒⍀͉͑g͗͘e͉ϩ͉e͗͘g͉͒ To describe the interaction of the trapped atom with ˆ Ϫ␻ ϩ␾ Ϫ ˆ Ϫ␻ ϩ␾ light fields in a simple but sufficient way, it is assumed, ϫ͓ei(kxS t )ϩe i(kxS t )͔. (62) such as in the preceding section, that the motion of the In the spin-1/2 algebra we can reexpress atom bound in the trap is harmonic in all three dimen- ͒ ␴ ϭ ͑␴ ϩ ␴ۋ͉ ͉ sions. The descriptions presented below will include the e͗͘g ˆ ϩ 1/2 ˆ x i ˆ y , explicit time dependence of the trapping potential, but ͒ ␴ ϭ ͑␴ Ϫ ␴ۋ͉ ͗͘ ͉ in many cases it is sufficient to model the motion of the g e ˆ Ϫ 1/2 ˆ x i ˆ y . (63) ion as a three-dimensional static potential harmonic os- For simplicity the discussion is again restricted to one cillator, because the general theory introduces only very dimension, and the effective wave vector k is chosen to minor changes if the modulus of the dimensionless Paul- lie along the x axis of the trap. The generalization to 2 trap parameters ax and qx related to the static and rf more dimensions is straightforward. potential (see Sec. II.A) is much smaller than 1. This is The simplest picture of the dynamics induced by the true for the traps used in the experiments discussed in light field arises after transformation into the interaction

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ˆ ϭ ˆ (m)ϩ ˆ (e) dominant contribution in the time evolution of the picture with the free Hamiltonian H0 H H and the interaction Vˆ ϭHˆ (i). trapped and illuminated ion will come from this term, ˆ ϭ ͓Ϫ ប ˆ ͔ while the others can be neglected in a second application With U0 exp (i/ )H0t the transformed interaction Ј Hamiltonian is of the rotating-wave approximation. For a given l and l one would speak of a detuning to the lth secular side- ˆ ϭ ˆ † ˆ (i) ˆ Hint U0H U0 band of the lЈth micromotion sideband, a terminology coming from the classical picture of the ion vibrating in ϭ͑ប ͒⍀ ͑i/ប͒Hˆ (e)t͑␴ ϩ␴ ͒ /2 e ϩ Ϫ the trap’s well with secular and micromotion frequen- ˆ (e) ˆ (m) ϫeϪ ͑i/ប͒H te͑i/ប͒H t͓ei(kxˆ Ϫ␻tϩ␾) cies. In the frame of reference of the ion the monochro- matic light field is therefore phase modulated at these ˆ (m) ϩeϪi(kxˆ Ϫ␻tϩ␾)͔eϪ ͑i/ប͒H t two frequencies. For example, if one of the modulation ␻ ␻ sidebands coincides with the transition frequency 0 of ϭ͑ប ͒⍀͑␴ i 0t /2 ϩe the ion at rest, this sideband can induce internal state Ϫi␻ t ͑i/ប͒Hˆ (m)t i(kxˆ Ϫ␻tϩ␾) transitions. ϩ␴Ϫe 0 ͒e ͓e The exact general form of the resonant term can be ˆ (m) ϩeϪi(kxˆ Ϫ␻tϩ␾)͔eϪ ͑i/ប͒H t. (64) calculated, in principle, by performing a polynomial expansion of the above expression, but this is unnecessary Multiplying the time-dependent factors in the above ex- for most practical cases since often ␩Ӷ1, (͉a ͉,q2)Ӷ1, ͓Ϯ ␻Ϯ␻ ͔ x x pressions leads to exp i( 0)t . Two terms are rap- so the coupling strength of higher orders in l and lЈ idly oscillating because ␻ and ␻ add up, while the two 0 vanishes quickly. The coupling strength for some special other terms oscillate with frequency ␦ϭ␻Ϫ␻ Ӷ␻ . 0 0 cases will be calculated in Sec. III.B.2. In all experiments Since the contribution of the rapidly oscillating terms covered in this review only sidebands with lЈϭ0 were hardly affects the time evolution, they can be neglected. ͉ Ј͉у Doing so is called the rotating-wave approximation for driven. Terms with l 1 could then be neglected. Fur- ͉ ͉ 2 Ӷ ␤ ␻ Ϸ␯ historical reasons. ther, it is assumed that ( ax ,qx) 1, so x rf and Ϸ ϩ Ϫ1 The transformation of the motional part of the Hamil- C0 (1 qx/2) , as in Eq. (37). Then the interaction tonion into the interaction picture is equivalent to a Hamiltonian simplifies to ¨ transformation of this part from the Schrodinger to the ˆ ͑ ͒ϭ͑ប ͒⍀ ␴ ␩͑ Ϫi␯tϩ † i␯t͒ i(␾Ϫ␦t) Hint t /2 0 ϩ exp͕i aê aˆ e ͖e Heisenberg picture. The position operator xˆ S will be re- placed by its Heisenberg-picture version xˆ (t) as given in ϩH.c., (69) Eq. (29). Introducing the Lamb-Dicke parameter ␩ with the scaled interaction strength ⍀ ϭ⍀/(1ϩq /2). ϭkx , where x ϭͱប/(2m␯) is the extension along the 0 x 0 0 This scaling reflects the reduction in coupling due to the x axis of the ground-state wave function of the reference wave packet’s breathing at the rf drive frequency. oscillator mentioned in Sec. II.B, yields kxˆ ͑t͒ϭ␩͕aû*͑t͒ϩaˆ †u͑t͖͒, (65) and the interaction Hamiltonian in the rotating-wave approximation takes its final form as 2. Rabi frequencies ˆ ͑ ͒ϭ͑ប ͒⍀␴ ͕␾ϩ␩͓ ͑ ͒ϩ † ͑ ͔͒Ϫ␦ ͖ ␦ Hint t /2 ˆ ϩ exp„i aû* t aˆ u t t … Depending on the detuning , the interaction Hamil- tonian (69) will couple certain internal and motional ϩ H.c. (66) states. If the exponent containing the ladder operators in The time dependence of the exponent is governed by Eq. (69) is expanded in ␩ this will result in terms con- the frequency difference ␦ and the time dependence of taining a combination of ␴Ϯ , with laˆ -operators and m u(t). Recalling the form of the solution Eq. (22) and aˆ †-operators rotating with a frequency of (lϪm)␯ϭs␯. expanding part of the exponent in the Lamb-Dicke pa- If ␦Ϸs␯ these combinations will be resonant, coupling rameter the manifold of states ͉g͉͘n͘ with ͉e͉͘nϩs͘. The coupling strength, often called the ͉s͉th blue (red) sideband ͕␾ϩ␩͓ *͑ ͒ϩ † ͑ ͔͒Ϫ␦ ͖ exp„i aû t aˆ u t t … Rabi frequency for sϾ0(sϽ0), is then (Cahill and ϱ ϱ ͑i␩͒m Glauber, 1969; Wineland and Itano, 1979) i(␾Ϫ␦t) Ϫi␤ ␻ t ϭe ͚ ͭ aê x rf ͚ C* † 2n ⍀ ϭ⍀ ϭ⍀ ͉ ϩ ͉ i␩(aϩa )͉ ͉ mϭ0 m! nϭϪϱ n,nϩs nϩs,n 0 ͗n s e n͘ m nϽ! Ϫ ␻ Ϫ␩2/2 ͉s͉ ͉s͉ 2 ϫ in rftϩ ͮ ϭ⍀ e ␩ ͱ L ͑␩ ͒, (70) e H.c. (67) 0 n nϾ! Ͻ

where nϽ (nϾ) is the lesser (greater) of nϩs and n, and it is easily veriﬁed that anytime the detuning satisﬁes ␣ L (X) is the generalized Laguerre polynomial ͑ ϩ ␤ ͒␻ ϭ␦ n lЈ l x rf , (68) n Xm with l and lЈ as integers and lÞ0iflЈÞ0, two of the ␣͑ ͒ϭ ͑Ϫ ͒m͑ nϩ␣ ͒ Ln X ͚ 1 nϪm . (71) terms in the Hamiltonian will be slowly varying. The mϭ0 m!

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3. Lamb-Dicke regime sibility of driving higher-order sidebands (when ␩ is not too small), trapped-ion experiments can yield inherently The interaction Hamiltonian [Eq. (69)] and the Rabi richer dynamics than cavity QED experiments. With frequencies [Eq. (70)] are further simplified if the ion is suitable light pulses on the carrier, red and blue side- confined to the Lamb-Dicke regime where the extension bands, a classical driving force to create coherent or of the ion’s wave function is much smaller than 1/k.In squeezed states, and the ability to cool to the ground ␩ͱ ϩ † 2 Ӷ this regime the inequality ͗(a a ) ͘ 1 must hold state, one can engineer and analyze a variety of states of for all times. The exponent in Eq. (69) can then be ex- motion as discussed in more detail in Sec. VI. panded to the lowest order in ␩, By choosing larger detunings, ␦ϭl␯ with ͉l͉Ͼ1, in principle, one can obtain a number of nonlinear cou- ˆ ͑ ͒ϭ͑ប ͒⍀ ␴ ϩ ␩͑ Ϫi␯tϩ † i␯t͒ i(␾Ϫ␦t) HLD t /2 0 ϩ͕1 i aê aˆ e ͖e plings, for example, a ‘‘two-phonon’’ coupling, ϩH.c., (72) ˆ ϭ͑ប ͒⍀ ͑␩2 ͓͒ 2␴ i␾ϩ͑ †͒2␴ Ϫi␾͔ Htp /2 0 /2 a ϩe a Ϫe (78) and will contain only three resonances. The first reso- for lϭϪ2. We note, however, that this interaction and nance for ␦ϭ0 is called the carrier resonance and has the those with ͉l͉Ͼ2 are not easy to realize in the laboratory form since efficient ground-state cooling has been achieved only in the Lamb-Dicke regime. There the coupling i␾ Ϫi␾ strength will be significantly reduced compared to car- Hˆ ϭ͑ប/2͒⍀ ͑␴ϩe ϩ␴Ϫe ͒. (73) car 0 rier and first sidebands, because ␩Ӷ1. This Hamiltonian will give rise to transitions of the type ͉ ͉ ↔͉ ͉ ⍀ n͘ g͘ n͘ e͘ with Rabi frequency 0 . These transitions will not affect the motional state. 4. Resolved sidebands ␦ϭϪ␯ The resonant part for is called the first red So far we have assumed that the two internal levels of sideband and has the form the ions have an infinite lifetime, leading to arbitrarily narrow carrier and sideband resonances. In practice this ˆ ϭ͑ប ͒⍀ ␩͑ ␴ i␾ϩ †␴ Ϫi␾͒ 2 Hrsb /2 0 aˆ ϩe aˆ Ϫe . (74) is only approximately true. If the frequency of a suffi- ͉⍀ ͉ ciently stable laser beam has intensity such that n,m This Hamiltonian will give rise to transitions of the type Ӷ␯ for all n,m it is possible to observe well-resolved ͉n͉͘g͘↔͉nϪ1͉͘e͘ with Rabi frequency carrier and sideband resonances. For a detuning ␦ϭl␯ ϩ␦Ј with ␦ЈӶ␯ the ion’s dynamics is governed by the ⍀ ϭ⍀ ͱ ␩ n,nϪ1 0 n (75) few resonant terms in Eq. (69) to which the laser happens to be tuned. This allows for clean manipulation of that will entangle the motional state with the internal the internal and motional states and also for cooling to state of the ion. Indeed this Hamiltonian is formally the ground state, as will be described in Sec. IV.B. If we equivalent to the Jaynes-Cummings Hamiltonian, the neglect dissipative terms, the time evolution of the gen- workhorse of quantum optics, and it is this analogy that eral state inspired many workers originally coming from quantum ϱ optics to do investigations in the field of trapped ions. It ͉⌿͑ ͒ ϭ ͑ ͉͒ ϩ ͑ ͉͒ t ͘ ͚ cn,g t n,g͘ cn,e t n,e͘ (79) is also responsible for the remarkable similarity of inves- nϭ0 tigations done in cavity QED (Englert et al., 1998; ¨ Varcoe et al., 2000; Raimond et al., 2001) to some of the is governed by the Schrodinger equation experiments done in ion traps presented below. This in- (t͉⌿͑t͒͘ϭHˆ ͉⌿͑t͒͘, (80ץiប teraction removes one quantum (phonon) of the secular int motion while the ion goes to the excited state, similarly which is equivalent to the set of coupled equations to the absorption of a light quantum in cavity QED. The counterpart of this interaction is the first blue ϭϪ (1Ϫ͉l͉) i(␦ЈtϪ␾)͑⍀ ͒ c˙ n,g i e nϩl,n/2 cnϩl,e , (81) sideband, resonant for ␦ϭϩ␯: ϭϪ (1ϩ͉l͉) Ϫi(␦ЈtϪ␾)͑⍀ ͒ c˙ nϩl,e i e nϩl,n/2 cn,g . (82) † i␾ Ϫi␾ Hˆ ϭ͑ប/2͒⍀ ␩͑aˆ ␴ϩe ϩaˆ ␴Ϫe ͒. (76) bsb 0 This set of equations may be solved by the method of Laplace transforms, yielding the solution This Hamiltonian will give rise to transitions of the type ͉n͉͘g͘↔͉nϩ1͉͘e͘ with Rabi frequency c ϩ ͑t͒ c ϩ ͑0͒ ͫ n l,e ͬϭ l ͫ n l,e ͬ ͑ ͒ Tn ͑ ͒ , (83) cn,g t cn,g 0 ⍀ ϩ ϭ⍀ ͱnϩ1␩. (77) n,n 1 0 with It has no direct counterpart in the atom-photon realm because such a process would violate energy conservation and is sometimes called anti-James-Cummings cou- 2For example, for quadrupole transitions with a lifetime on pling. Because of this interaction and the additional pos- the order of 1 s, linewidths would be limited to about 1 Hz.

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␦Ј ⍀ Ϫ ␦Ј nϩl,n ␾ϩ͉ ͉␲ Ϫ␦Ј e i( /2)tͫ cos͑fl t/2͒ϩi sin͑fl t/2͒ͬ Ϫi ei( l /2 t/2) sin͑fl t/2͒ n fl n fl n l ϭ n n Tn ⍀ ␦Ј (84) Ά nϩl,n Ϫ ␾ϩ͉ ͉␲ Ϫ␦Ј l ␦Ј l l · Ϫ i( l /2 t/2) ͑ ͒ i( /2)tͫ ͑ ͒Ϫ ͑ ͒ͬ i l e sin fnt/2 e cos fnt/2 i l sin fnt/2 fn fn

l ϭͱ␦Ј2ϩ⍀2 ⌫ and fn nϩl,n. The solution describes a general- Ld␳ϭ ͑2␴Ϫ˜␳␴ϩϪ␴ϩ␴Ϫ␳Ϫ␳␴ϩ␴Ϫ͒, (87) ized form of sinusoidial Rabi flopping between the states 2 ͉n,g͘ and ͉e,nϩl͘ and is essential for the quantum-state ⌫ preparation and analysis experiments described later. where is the spontaneous emission rate. To account for the recoil of spontaneously emitted photons the first term of the Liouvillian contains 5. Unresolved sidebands 1 1 ˜␳ϭ ͵ dz⌼͑z͒eikxˆ z␳eϪikxˆ z, (88) Allowed optical electric dipole transitions for ions 2 Ϫ1 have a linewidth of around several tens of MHz, typi- ⌼ ϭ ϩ 2 cally beyond the highest secular motional frequencies with (z) 3(1 z )/4 the angular distribution pattern observed in ion traps. In this case spontaneous emission of spontaneous emission for a dipole transition. cannot be neglected when considering the dynamics of In most cases of interest the master equation has been the internal and motional states of an ion driven by the solved numerically, often assuming the Lamb-Dicke light field. Spontaneous emission will have two conse- limit as an additional restriction. One example of this quences: the coherent evolution of the internal two-level work will be discussed in the context of laser cooling of system is interrupted, and the recoil of the emitted pho- a trapped ion. ton leads to randomized momentum kicks in the external motion. 6. Spectrum of resonance fluorescence A convenient picture of the dynamics is provided by the master-equation formalism (Walls and Milburn, 1995; The master equation (86) can also be used to derive a Schleich, 2001). The trapped ion is modeled as being formal expression of the light emitted by a trapped ion coupled to a zero-temperature reservoir of optical in the Lamb-Dicke limit. Lindberg (1986) has studied modes in the vacuum state. This is a fair assumption the spectrum of light emitted by a harmonically trapped since the average occupation number of optical modes two-level atom and its interaction with a traveling-wave at room temperature is extremely small. The master laser field. Later, Cirac, Parkins, et al. (1993) derived equation is an equation of motion for the reduced den- general relations applicable to multilevel systems, gen- sity matrix ␳ that describes the time evolution of the eral trapping potentials, and traveling-wave and internal and motional states of the ion only, the reduc- standing-wave configurations. The expressions for the tion of the total problem brought about by tracing over motional sidebands appearing in the spectrum can be the many degrees of freedom of the environment or res- written in terms of correlation functions of internal op- ervoir modes. The exact derivation is very technical, so erators and of steady-state expectation values of exter- the reader is referred to Stenholm (1986) and Cirac et al. nal operators. The internal operators can be evaluated (1992) for details. In brief, the reservoir modes are as- with the optical Bloch equations for an atom at rest, sumed to be coupled to the ion by an interaction of the while the external operators are derived from the ap- Jaynes-Cummings form, plied trapping forces using the formalism introduced by Cirac et al. (1992). ⌬ ϩ Ϫ ⌬ ϭ͑ប ͒ ͑ ͒͑␴ † i ltϩ␴ i lt͒ Specifically, for a single harmonically trapped ion in a Hrc /2 ͚ gl x aˆ l e aˆ le , (85) l traveling-wave field configuration, and for the low- ⍀ Ӷ⌫ ⌬ intensity limit at which 0 , one obtains for the mo- where aˆ l , gl(x), and l are the destruction operator, position-dependent coupling strength, and detuning of tional sidebands the following spectral contributions: ⌰͑␺͒␥ ⍀ 2 the lth reservoir mode, respectively. If we expand the ms 0 ␳ Sbsb͑␻͒ϭ ␩2ͩ ͪ ͗a†a͘ equation of motion for to second order in this reser- ␥2 ϩ͑␻Ϫ␯͒ 2 ss voir interaction and assume the reservoir modes that are ms resonant with the two-level atom to be in zero- cos ␺ 1 2 ϫͯ ϩ ͯ , (89) temperature distributions, we arrive at the master equa- ␦Ϫi␥ ͑␦ϩ␯͒Ϫi␥ tion ⌰͑␺͒␥ ⍀ 2 ms 0 d␳ i Srsb͑␻͒ϭ ␩2ͩ ͪ ͑͗a†a͘ ϩ1͒ ϭϪ ͓Hˆ (m)ϩHˆ (e)ϩHˆ (i),␳͔ϩLd␳. (86) ␥2 ϩ͑␻ϩ␯͒ 2 ss dt ប ms cos ␺ 1 2 Damping by spontaneous emission is contained in the ϫͯ ϩ ͯ . (90) Liouvillain ␦Ϫi␥ ͑␦Ϫ␯͒Ϫi␥

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⌰ ␺ ⍀ ␯ Here, ( ), 0 , and denote the angular distribution for the dipole radiation, the Rabi frequency, and the mo- ␥ tional frequency, respectively. The width ms of the motional sidebands is given by ⍀ 2 0 ␥ ϭ␩2ͩ ͪ ͓P͑␯ϩ␦͒ϪP͑␯Ϫ␦͔͒ (91) ms 2 and the final (steady-state) excitation energy is FIG. 4. V-type three-level system for electron shelving. In ad- P͑␯Ϫ␦͒ϩ␣P͑␦͒ dition to transitions ͉g͘↔͉e͘, transitions to a third level ͉r͘ can ͗a†a͘ ϭ , (92) ss P͑␯ϩ␦͒ϪP͑␯Ϫ␦͒ be independently driven by laser fields. The lifetimes of ͉g͘ and ͉e͘ are usually much longer than that of ͉r͘. with ␥ ͑␦͒ϭ in a crystal show up as bright dots with a size determined P ␥2ϩ␦2 (93) by the resolution limit of the imaging optics, that is, and ␣ϭ2/5 for a dipole transition and the spontaneous around 1 ␮m with a good imaging lens. decay rate ⌫ϭ␥/2. Photomultipliers usually do not offer spatial informa- Thus, aside from the coherent peak, the spectrum of tion directly, but rather pick up the collected fluores- resonance fluorescence consists of two Lorentzian func- cence with better quantum efficiency than CCD cameras tions centered at ␻ӍϮ␯ with different heights which or imagers. With a total detection efficiency of 10Ϫ3 (in- depend on the observation angle ␺. The width of the cluding solid angle and quantum efficiency) of the fluo- motional sidebands is given by half the cooling rate, rescence rate, even a single ion will lead to about 50 000 ⍀ Ӎ⌫ which in the Lamb-Dicke limit is narrower than 0 counts/s on a dipole allowed transition with a 10-ns up- ␩2Ӷ by a factor of the order 1 (Lindberg, 1986). The per state lifetime. asymmetry of the motional sidebands reflects the popu- This signal can be used not only to detect the ion ͉ ͘ lation of the trap levels n and thus mirrors the ion’s itself, but also to distinguish between internal states of residual excitation energy in the trap potential. the ion with extremely high detection efficiency. This For large intensities and on resonance the spectrum technique was suggested by Dehmelt (1975) as a tool for exhibits ac Stark splitting with a center line at the laser spectroscopy and was dubbed electron shelving because frequency, two symmetric sidebands at the Rabi fre- the ion’s outer electron can be ‘‘shelved’’ into a state in quency ⍀ , and width proportional to ⌫ (the Mollow 0 which it does not fluoresce. triplet; Mollow, 1969). The ion motion then leads to additional (narrow) sidebands such as in the low intensity case. For an ion in a standing-wave field configuration, expressions similar to Eq. (89) are found with an explicit 1. The electron shelving method dependence on the ion position with respect to the standing-wave phase. For details we refer the reader to Electron shelving makes use of an internal level struc- the work by Cirac et al. (1992). ture that is well described by a V-type three-level system. In addition to ͉g͘ and ͉e͘ there is a third level ͉r͘, and it is assumed that the transitions ͉g͘↔͉e͘ and C. Detection of internal states ͉g͘↔͉r͘ can be independently driven by laser fields (see Fig. 4). If the lifetimes of ͉g͘ and ͉e͘ are much longer The ions used in the experiments described here can than that of ͉r͘ the transition ͉g͘↔͉r͘ may be strongly be detected by laser-induced fluorescence on an electric driven, resulting in many scattered photons if the ion is dipole allowed transition, usually identical to the transi- projected into ͉g͘ by the first scattering event. On the tion used for Doppler cooling. On such a transition a other hand, if no photons are scattered, the ion was pro- single ion can scatter several millions of photons per jected into state ͉e͘ by the interaction with the driving second and a sufficient fraction may be detected even field. with detectors covering a small solid angle and having Depending on the background light scattered into the 5–20 % quantum efficiencies. The fluorescence is either photomultiplier and its dark count rate, ͉g͘ and ͉e͘ can detected in a spatially resolved manner on charge- be distinguished with high confidence in rather short decoupled-device (CCD) cameras or imager tubes, or on tection periods (typically on the order of a few tens of photomultipliers. Cameras and imager tubes have been ␮s). Apart from this convenient way of measuring the used in a number of experiments to provide pictures of quantum state, Dehmelt’s proposal triggered a number ion clouds and crystals (see, for example, Diedrich et al., of theoretical papers on ‘‘quantum jumps’’ between ͉g͘ 1987; Wineland et al., 1987; Drewsen et al., 1998; Mitch- and ͉e͘ that would manifest themselves by extended pe- ell et al., 1998; Na¨gerl et al., 1998). Since the spatial ex- riods of time with few or many photons detected, re- tension of a cold ion’s wave function is typically smaller spectively, if both transitions are driven simultaneously. than the wavelength of the fluorescence light, single ions The fact that one describes the behavior in the time of a

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FIG. 5. Quantum jumps of a single 138Baϩ ion. If the ion makes the transition to ͉ ϵ e͘ D5/2 , the ﬂuorescence drops. After a mean time equal to the excited-state lifetime (ca. 32 s in this example), a spontaneous tran- ͉ ϵ sition returns the ion to g͘ S1/2 and the ﬂuo- ͉ ↔͉ ϵ rescence on the g͘ r͘ P1/2 transition returns to a higher level.

single atom, undergoing random transitions,3 rendered ion [see Eqs. (83) and (84)]. In general the internal state the usual density-matrix approach inadequate, because may get entangled with the motional state for interac- it yields only ensemble averages, not individual trajecto- tions on the carrier or any sideband. The dependence ries. A number of different correlation functions were can also be used to map the motional state of the ion finally used by different workers to attack these prob- onto the internal state, which can subsequently be mea- lems. They are all related to g(2)(␶), the probability of sured with high efficiency as described above. This map- detecting another photon, originating from the ͉r͘→͉g͘ ping is straightforward in the resolved sideband regime, spontaneous emission, at time tϭ␶, if one was detected where dissipation plays no role. at tϭ0. A review of both theoretical and experimental Practically, since ␩Ӷ1 and the ions are in the Lamb- work on this problem is presented by Blatt and Zoller Dicke regime, the most interesting transitions are the (1988). first red and blue sidebands, since their coupling to low- 2. Experimental observations of quantum jumps est order is linear in ␩ [see Eq. (70)]. The first blue sideband proved to be very useful in characterizing the Experimental observations of quantum jumps of number-state distribution of various states of motion. By single trapped ions were first reported at about the same resonantly driving an ion in the starting state time in three different laboratories (Bergquist et al., ϱ 1986; Nagourney et al., 1986; Sauter et al., 1986). ͉⌿͑ ͒͘ϭ͉ ͘ ͉ ͘ In all three experiments the ͉g͘↔͉r͘ transition was a 0 g ͚ cn n (94) nϭ0 dipole allowed transition also used for Doppler cooling, while the weak transition was a dipole forbidden quad- on the blue sideband for various times t and measuring rupole transition that was excited by an incoherent hol- ϭ ⌿ ͉ ͉ ͉ ˆ ͉⌿ the probability Pg(t) ͗ (t) ( g͗͘g Im) (t)͘ to find low cathode lamp (Nagourney et al., 1986), was far off- the ion in the ground state after the interaction, where resonance scattering and collisional excitation (Sauter Iˆ is the identity operator in the motional-state space, et al., 1986), or was excited by a coherent laser source m one obtains the following signal, which can be easily de- (Bergquist et al., 1986). Indeed the observed fluores- rived from Eq. (83) for the case lϭϩ1: cence showed the random intensity jumps (see Fig. 5) ϱ and all statistical properties to be in accord with theo- 1 ͑ ͒ϭ ͫ ϩ ͑⍀ ͒ͬ retical predictions (see Blatt and Zoller, 1988, and refer- Pg t 1 ͚ Pn cos n,nϩ1t . (95) 2 nϭ0 ences therein). ϭ͉ ͉2 Here Pn cn is the probability of finding the atom in ⍀ D. Detection of motional-state populations the nth motional number state. As long as n,nϩ1 are distinct frequencies, the occupation of all number states ␩ A finite Lamb-Dicke parameter implies that Pn can be found by Fourier transforming the signal ͉ ͘↔͉ ͘ ⍀ g e transitions depend on the motional state of the equation (95). The Rabi frequencies n,nϩ1 are distinct inside the Lamb-Dicke regime, where the blue sideband frequencies scale as ͱnϩ1 to lowest order in ␩ [see Eqs. 3Indeed it was proposed to use the length of light and dark (70) and (76)]. Outside this regime the frequencies will periods for the generation of perfect random numbers (Erber not rise monotonically as a function of n, so distinguish- and Puttermann, 1985). ing them is more complicated.

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The Pn correspond to the diagonal elements of the pler cooling, before introducing the basic building ␳ density matrix nm of the motional state. One can also blocks and results of the general approach, including mi- extend the method described above, using additional cromotion. manipulations of the motional state, to yield the off- In the simple picture micromotion is neglected and ␳ diagonal elements of nm and characterize the complete the trapping potential is approximated by the time- quantum state of motion. Theoretical methods and ex- independent pseudopotential perimental implementation of two such methods will be described in Sec. VI.B. 1 V ͑x͒ϭ m␯2x2. (96) p 2 This description applies to axial motion in a linear trap IV. LASER COOLING OF IONS [see Eq. (4)], where micromotion is absent. If the motion of the trapped ion is taken to be classical, its veloc- The general goal of cooling is to reduce the kinetic ity obeys energy of an ion after it was loaded into the trap, ideally v͑t͒ϭv cos͑␯t͒. (97) to a point at which the ion is in the ground state of the 0 trapping potential with very high probability. Ion traps If the radiative decay time is much shorter than one typically confine ions up to a temperature that corre- oscillation period, ␯Ӷ⌫, one cycle of absorption and sponds to about one-tenth of their well depth, about 1 spontaneous emission occur in a time span in which the eV or 10 000 K. Cooling from these starting tempera- ion does not appreciably change its velocity. In this case tures requires a high scattering rate of the cooling light, the averaged radiation pressure can be modeled as a so it is advantageous to use a dipole transition to a fairly continuous force that depends on the ion’s velocity. If short-lived upper level for this stage. For most traps and the cooling laser field is a single traveling wave along the ions commonly used the first cooling stage will therefore ion’s direction of motion, every absorption will give the occur in the unresolved sideband regime, since the life- ion a momentum kick ⌬pϭបk in the wave-vector direc- time of the upper state is considerably shorter than one tion of the light field, while the emission will generally period of oscillation in the trap. In this case, cooling in a be symmetric about some point. Emission will then lead trap or cooling free atoms is essentially the same (Wine- to zero-momentum transfer on average, but to a random land and Itano, 1979). For example, the limiting kinetic walk in momentum space, similar to Brownian motion. energy under this type of cooling turns out to be the The rate of absorption-emission cycles is given by the same as the limit of Doppler cooling for free atoms. decay rate ⌫ times the probability of being in the excited ␳ ϭ ͉␳͉ To reach the motional ground state with high prob- state ee ͗e ˆ e͘. Therefore the average force is ability a second stage of cooling is necessary, typically dp with a lower scattering rate, but now in the resolved- ͩ ͪ ϷF ϭបk⌫␳ , (98) dt a ee sideband regime. For cooling to the ground state, three a different methods have been used to date, namely, cool- and the excited-state probability is given by (Loudon, ing on a dipole forbidden quadrupole transition, cooling 1973) by stimulated Raman transitions, and cooling utilizing electromagnetically induced transparency (EIT cooling). s/2 ␳ ϭ All these methods will be briefly discussed in this sec- ee ϩ ϩ͑ ␦ ⌫͒2 , (99) 1 s 2 eff / tion. For a more comprehensive treatment the reader is ϭ ͉⍀͉2 ⌫2 referred to the literature cited below and a review of where s 2 / is the saturation parameter propor- cooling techniques in ion traps by Itano et al. (1992). tional to the square of the on-resonance Rabi frequency ⍀. The detuning is composed of the detuning ⌬ϭ␻ Ϫ␻ A. Doppler cooling 0 of the light wave with respect to the resonance frequency of the atom at rest and the Doppler shift: Cooling in a trap was examined in conjunction with ␦ ϭ⌬Ϫ eff k•v. For small velocities, close to the final tem- the first Doppler cooling experiments (Neuhauser et al., perature reached by laser cooling, where the Doppler ⌫ 1978; Wineland et al., 1978; Wineland and Itano, 1979; broadening is small compared to , Fa can be linearized Itano and Wineland, 1981). A semiclassical picture in v: based on the master equation was developed by Sten- Ϸ ͑ ϩ␬ ͒ holm and co-workers, as reviewed by Stenholm (1986). Fa F0 1 v , (100) In these treatments a purely harmonic secular motion with with no micromotion of the ion was assumed. This was s/2 somewhat unsatisfactory from the point of view of ex- ϭប ⌫ F0 k 2 (101) periments in which additional cooling and heating ef- 1ϩsϩ͑2⌬/⌫͒ fects related to micromotion were observed (Blu¨ mel the averaged radiation pressure that displaces the ion et al., 1989; DeVoe et al., 1989). Finally, the most com- slightly from the trap center and plete picture of cooling to date, including micromotion, ⌬ ⌫2 was derived by Cirac, Garay, et al. (1994). We first 8k / ␬ϭ (102) present a very simple, more qualitative picture of Dop- 1ϩsϩ͑2⌬/⌫͒2

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the ‘‘friction coefficient’’ of the cooling force that will reached for a laser detuning ⌬ϭ⌫ͱ1ϩs/2. provide viscous drag if ⌬ is negative. The cooling rate, The simple picture presented so far provides some in- averaged over many oscillation periods, is then sight into the cooling mechanism but neglects several ˙ ϭ ϭ ͑ ϩ␬ 2 ͒ϭ ␬ 2 aspects of cooling in rf traps that can be important in Ec ͗Fav͘ F0 ͗v͘ ͗v ͘ F0 ͗v ͘, (103) practice. In many experiments using Doppler cooling, ͗ ͘ϭ since v 0 for a trapped particle. Thus, without taking sidebands due to micromotion are observed in fluores- the random nature of the light-scattering events into ac- cence spectra. Especially when the rf drive frequency is count, the ion would cool to zero energy. In practice this comparable to or larger than the natural linewidth of the cannot happen since even if the ion has zero velocity it cooling transition, one might wonder what effects this will continue to absorb and emit photons. The emission ϭ ⌫␳ ϭ has on the cooling process. rate for v 0is ee(v 0), with the recoil taking directions dictated by the emission pattern of the transition For example, when including micromotion, the kinetic used (typically an electric dipole transition). Since the energy of the trapped particle has to be reconsidered. ␻ emission pattern is symmetric the average momentum The forced oscillations at the rf drive frequency rf add change is ͗⌬p͘ϭ0, but the momentum undergoes diffu- kinetic energy in excess of the energy in the secular mo- sion, ͗⌬p2͘Þ0. As usual in random-walk processes the tion. However, for the cooling dynamics, which evolve average distance covered by the diffusion is proportional on a much slower time scale than this fast oscillation, it to the square root of the number of recoil kicks or, in is appropriate to look at the kinetic energy averaged ␲ ␻ other words, the second moment of the distribution of over one period 2 / rf of the micromotion (denoted by random processes is proportional to the number N of the overbar): recoils: ͗⌬p2͘ϰ(បk)2N. Not only the emission, but also ͗pˆ ͑t͒2͘ the random times of absorption of photons lead to mo- E ϭ . (107) mentum kicks, but this time only along the axis defined kin 2m by the wave vector of the cooling beam. This still gives The process of cooling can then be defined as an ap- rise to diffusion due to the discreteness of the absorption proach to minimizing this quantity. processes. Again the diffusion will be proportional to For Doppler cooling, the spatial extension of the final the square root of the number of absorptions. Further- more, unless the cooling transition is driven weakly, s motional state is usually small compared to the cooling Ӷ1, absorption and emission will be correlated, leading light wavelength, so it is appropriate to limit a study of to an altered diffusion. While all these effects are in- cooling dynamics to the Lamb-Dicke regime. It is also ͉ ͉ 2 Ӷ cluded in the more general approach discussed later and assumed that the trap operates in the ( ax ,qx) 1 re- also discussed in the literature (Lindberg, 1984; Sten- gime of micromotion. Assuming that the cooling light is holm, 1986, and references therein), this correlation will a traveling wave and following Cirac, Garay, et al. be neglected for the simple picture here. The momen- (1994), one may expand the interaction Hamiltonian tum kicks due to absorption and emission will then have (66) to first order in ␩ (since the time origin for cooling the same rate but different directionality. This can be is unimportant, the phase ␾ is irrelevant and set to zero taken into account by scaling the emission term with a in the following): geometry factor ␰ that reflects the average component of ˆ LD͑ ͒ϭ͑ប ͒⍀͓␴ ͑Ϫ ␦ ͒ϩ ͔ the emission recoil kick along the x axis and takes the Hint t /2 ˆ ϩexp i t H.c. value ␰ϭ2/5 for dipole radiation (Stenholm, 1986). For ϱ the final stage of cooling v will be close to zero, so the ϩ͑ប ͒⍀ͭ ͚ ␩ ␴ Ϫi␦t /2 i C2n ˆ ϩe heating rate is approximately nϭϪϱ 1 d 2 E˙ ϭ ͗p ͘ϭE˙ ϩE˙ Ϫ ␯ϩ ␻ ␯ϩ ␻ h 2m dt abs em ϫ͓aê i( n rft)ϩaˆ †ei( n rft)͔ϩH.c.ͮ . ϭE˙ ͑1ϩ␰͒ abs (108) 1 Ӎ ͑ប ͒2⌫␳ ͑ ϭ ͒͑ ϩ␰͒ k ee v 0 1 . The interpretation is straightforward. The first term rep- 2m resents the strong carrier excitation with Rabi frequency (104) ⍀, while the other terms represent pairs of combined Equilibrium will be reached if heating and cooling pro- secular and micromotion sideband excitations at detun- Ϯ ␯ϩ ␻ ϭ Ϯ Ϯ ceed at an equal rate, so one can infer the final tempera- ings ( n rf)(n 0, 1, 2,...) with weaker Rabi ␩⍀͉ ͉ ture from equating Eqs. (103) and (104): frequencies C2n . For the assumed conditions the magnitude ͉C ͉ rapidly drops with increasing ͉n͉. If all ប⌫ ⌫ 2⌬ 2n 2 ͉␻ ͉ӷ␯ӷ⌫ ͗ ͘ϭ ϭ ͑ ϩ␰͒ͫ͑ ϩ ͒ ϩ ͬ sidebands are resolved, rf , one may choose the m v kBT 1 1 s ⌬ ⌫ , (105) 8 2 detuning in such a way that only one term in Eq. (108) is with k the Boltzmann constant. For this simple model, resonant. This case will be described in the next section. B ⌫у␻ ␯ which neglects the correlation between absorption and In the case in which rf , , the dynamics are more emission, the minimum temperature will be involved and for quantitative insight the master equation connected to Hamiltonian (108) has to be solved. ប⌫ͱ1ϩs ϭ ͑ ϩ␰͒ The sidebands and their relative strength are sketched Tmin 1 , (106) 4kB in Fig. 6. The extra sidebands lead to additional channels

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FIG. 6. Spectrum of the secular and micromotion sidebands involved in the cooling process. The height of the bars represents the coupling strength.

of absorption and emission. Absorption on the red secu- To reach this regime on a dipole allowed transition lar sidebands will lead to a decrease in kinetic energy, one can use either a very stiff trap with high motional while absorption on the blue secular sidebands will lead frequencies or a weakly allowed dipole transition. Both ␻ ϩ␯ to an increase. Note that all sidebands at n rf will approaches have been pursued (Jefferts et al., 1995; Peik lead to heating even if nϽ0. This means that although et al., 1999), but so far cooling to the ground state in the the laser might be red detuned from the resonance of sense that the state ͉g,nϭ0͘ is produced with high prob- the atom at rest (carrier), the ion can be heated, espe- ability has involved either dipole forbidden transitions cially if the excess micromotion due to unwanted static or Raman transitions. potentials in the trap is not compensated (Blu¨ mel et al., 1989; DeVoe et al., 1989; Berkeland et al., 1998; Peik et al., 1999). For good compensation the coupling 1. Theory ͉ ͉ strength quickly drops with n and the effect of these The cooling process in the resolved-sideband limit has extra resonances is not very dramatic. The ‘‘simple’’ previously been described (Neuhauser et al., 1978; Wine- Doppler limit stated in connection with Eq. (106) might land and Itano, 1979; Stenholm, 1986; Cirac et al., 1992; not be reached in all experiments due to these additional Cirac, Garay, et al., 1994). The essential physics can be heating contributions, but it still is a good rule of thumb. 9 ϩ understood with a simple model. To avoid unwanted In the NIST Be experiments the rf drive frequency motional-state diffusion due to transitions of high order ␻ ␲ Ϸ was quite high, rf /(2 ) 230 MHz, compared to a 20- in ␩ it is very advantageous to start resolved-sideband → MHz natural linewidth of the S1/2 P3/2 transition. The cooling inside the Lamb-Dicke regime. In fact all suc- cooling laser was detuned by about half a linewidth, so cessful experiments on ground-state cooling have fea- in this experiment micromotion sidebands did not play a tured an initial Doppler cooling stage that achieved the role. Indeed, at a trap frequency of 11.2 MHz, a Doppler Lamb-Dicke regime. For a typical linewidth of several ϭ cooling limit of ¯nexp 0.47(5) was reached, consistent tens of MHz on the Doppler cooling transition a suffi- ϭ with the theoretical limit of ¯n 0.484 (Monroe et al., ciently high motional frequency greater than about 1 1995). MHz is necessary. In the Lamb-Dicke regime and with no other heating mechanisms present, the cooling laser is detuned to ␦ ϭϪ␯ ␩ B. Resolved-sideband cooling , the first red sideband. To first order in , Eq. (108) is reduced to the resonant red sideband (nϭ0, In a regime where the effective linewidth due to decay since the micromotion sidebands are far off resonance and can be neglected), the carrier detuned by ␯, and the from the excited state with rate ˜⌫ is lower than the mo- blue sideband detuned by 2␯: tional frequency ␯, the individual motional sidebands become resolved.4 One can then tune the cooling laser ˆ LD͑ ͒ϭ͑ប ͒⍀͓␴ i␯tϩ␴ Ϫi␯tϩ ␩͑␴ ϩ␴ †͒ Hint t /2 ˆ ϩe ˆ Ϫe i ˆ ϩaˆ ˆ Ϫaˆ to the first red sideband and cool the ion close to the † i2␯t Ϫi2␯t motional ground state if no heating mechanisms other ϩi␩͑␴ˆ ϩaˆ e ϩ␴ˆ Ϫaê ͔͒. (109) than the recoils of the cooling transition are present. One way to find the final motional state would be to insert this interaction Hamiltonian into the master equation (86) and solve it numerically, as was done by Cirac, 4The effective rate ˜⌫ can be modified to be different from the Garay, et al. (1994), but since every cooling cycle in- ⌫ spontaneous decay rate of the excited state nat ; see, for ex- volves spontaneous emission, coherences never play a ample, Eq. (116) below. strong role and the problem can be approximately

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 Leibfried et al.: Quantum dynamics of single trapped ions 299 solved with rate equations. Every cooling cycle (absorp- to the simplest and most robust method used so far, the tion followed by spontaneous emission) involves a tran- comparison of the probability Pe(t) to end up in the sition from the ground state ͉g͘ to the excited state ͉e͘ excited electronic state ͉e͘ after excitation of the ion on on the red sideband and a subsequent decay on the car- the red and blue sidebands. From Eq. (95) for the first rier (in the Lamb-Dicke regime the ion will predomi- blue sideband and the analogous expression for the red nantly decay on the carrier; therefore other decay chan- sideband [derived from Eq. (83) for lϭϪ1], using ϭ Ϫ nels can be neglected to lowest order in ␩). The cooling Pe(t) 1 Pg(t) and the assumption that the final mo- rate is then given by the product of the excited-state tional states after cooling have a thermal distribution ͉ (see Sec. II.C.4), one gets occupation probability pe(n) of a motional state n͘ and ϱ its decay rate ˜⌫. As in Eq. (99) a Lorentzian line shape ¯n m rsb͑ ͒ϭ ͩ ͪ 2͑⍀ ͒ is assumed for this excitation, but now with detuning Pe t ͚ sin m,mϪ1t mϭ1 ¯nϩ1 Ϫ␯: ϱ ¯n ¯n m ͑␩ͱ ⍀͒2 ϭ ͩ ͪ 2͑⍀ ͒ n ͚ sin mϩ1,mt ϭ˜⌫ ͑ ͒ϭ˜⌫ ϩ ϭ ϩ Rn Pe n . (110) ¯n 1 m 0 ¯n 1 2͑␩ͱn⍀͒2ϩ˜⌫2 ¯n This rate depends on n and vanishes once the ground ϭ Pbsb͑t͒ (113) ¯nϩ1 e state is reached. The ground state is therefore a dark ⍀ ϭ⍀ state of the red sideband excitation and the ion would be using mϩ1,m m,mϩ1 . This means that the ratio of pumped into that state and reside there without any these probabilities is competing mechanisms. In the absence of any other Prsb ¯n heating sources the dominant channel out of the ground ϭ e ϭ R bsb ϩ (114) state is off-resonant excitation of the carrier and the first Pe ¯n 1 blue sideband. Actually both these processes contribute independent of drive time t, carrier Rabi frequency ⍀, to the heating on the same order. The carrier is excited or Lamb-Dicke parameter ␩ (Turchette, Kielpinski, with a probability of ͓⍀/(2␯)͔2 [see Eq. (99) with ˜⌫ et al., 2000). The ratio R can be inferred from a fre- ϭ⌫ ⍀Ӷ␦ ϭ␯ , eff ] but will mostly decay back on the carrier quency scan over both sidebands while keeping the light transition. Decay on the blue sideband after carrier ex- intensity and excitation time constant and will directly citation that leads to heating only occurs with a rate of yield the mean occupation ͓⍀ ␯ ͔2␩2˜⌫ ␩ /(2 ) ˜ . Note that the Lamb-Dicke factor ˜ for R this decay is not equal to the one of the excitation, be- ϭ ¯n Ϫ (115) cause the emitted photon can go in any direction, not 1 R only along the wave vector of the cooling beam, and of the thermal motional state. some experimental arrangements use a three-level system in which the emitted photon does not have the same 2. Experimental results wavelength as the cooling light (see below). The second dominant heating process is excitation on the first blue Several groups have reported experiments that cool sideband with probability ͕␩⍀/͓2(2␯)͔͖2 (see above, up to four ions close to the ground state. The first report but now ␦ ϭ2␯) followed by decay on the carrier with a of ground-state cooling was by the Boulder group, cool- eff 198 ϩ rate of ͕␩⍀/͓2(2␯)͔͖2˜⌫. For the final stage of the cooling Hg on a quadrupole allowed transition. The trap ing the problem may be restricted to the ground state was adjusted to be nearly spherical, with a secular fre- and the first excited state with rate equations quency of 2.96 MHz using an appropriate positive dc bias on the ring electrode. After 20 ms of Doppler cool- 2 2 2 2 →2 ͑␩⍀͒ ⍀ ␩⍀ ing on the strong S1/2 P1/2 transition this laser was ϭ Ϫ ͫͩ ͪ ␩2˜⌫ϩͩ ͪ ˜⌫ͬ 2 2 p˙ 0 p1 p0 ˜ , (111) shut off and another laser on the S → D first red ˜⌫ 2␯ 4␯ 1/2 5/2 sideband transition was turned on for 200–500 ms. Since 2 ϭ͉ ͘ p˙ ϭϪp˙ the natural lifetime of the D5/2 e state limits the 1 0 ⌫ ϭ scatter rate to about (1/2) nat 6 photons/s, the upper for the probabilities p0 ,p1 to be in the respective states. level was intentionally broadened by coupling it to the In the steady state, p˙ ϭ0,p ϭ1Ϫp , this yields 2 ϭ͉ i 1 0 short-lived P3/2 aux͘ level with an additional laser 2 source (repumper). This effectively broadens the line- ˜⌫ ˜␩ 2 1 ¯nϷp Ϸͩ ͪ ͫͩ ͪ ϩ ͬ. (112) width to (Marzoli et al., 1994) 1 2␯ ␩ 4 ⍀2 aux The factor in square brackets is of order one and ␯ ˜⌫ϭ (116) ͑⌫ ϩ⌫ ͒2ϩ4⌬2 ӷ˜⌫, so the particle is cooled to the ground state with aux nat aux probability p Ϸ1Ϫ͓˜⌫/(2␯)͔2 very close to one. if the Rabi frequency on the ͉e͘→͉aux͘ transition is 0 ⍀ Several methods have been used in experiments to aux and the auxiliary laser is detuned from resonance ⌬ determine the final mean excitation number ¯n after by aux . The choice of this Rabi frequency and detuning resolved-sideband cooling. This review will be restricted will determine the effective linewidth ˜⌫ and in turn the

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 300 Leibfried et al.: Quantum dynamics of single trapped ions cooling rate and final thermal distribution. After shut- ting off the sideband cooling laser, the repumper was left on for another 5 ms to ensure that the ion returned to the ground state ͉g͘. Following the cooling cycle another laser pulse with roughly saturation intensity was 2 swept over the red and blue sidebands of the S1/2 →2 D5/2 transition to record the excitation probability on the sidebands. The probability was found with the elec- 2 ↔2 tron shelving method by observing the S1/2 P1/2 fluorescence (see Sec. III.C) and averaging over about 40 sweeps consisting of one measurement for each frequency setting. From the strength of the sidebands the final mean occupation number of the motional state was ϭ → determined as outlined in Sec. IV.B.1 to be ¯n 0.05 FIG. 7. Sideband absorption spectrum on the S1/2 D5/2 tran- Ϯ0.012. In this experiment the final temperature could sition of a single calcium ion (Roos et al., 1999) in a trap with be determined in only two dimensions. 4.51-MHz motional frequency along the cooled axis: ᭺, after Ground-state cooling in all three dimensions was first Doppler cooling; ᭹, after resolved-sideband cooling; (a) red achieved by the NIST group (Monroe et al., 1995). The sideband; (b) blue sideband. Each data point represents 400 trap used in this experiment had motional frequencies of experiments. ␯ ␯ ␯ ϭ ␲ 9 ϩ ( x , y , z) 2 (11.2,18.2,29.8) MHz. The Be ion was first Doppler cooled well into the Lamb-Dicke re- level. The most efficient cooling occurred when the red 2 →2 gime in all three dimensions on the S1/2 P3/2 transi- ⍀ sideband Rabi frequency 1,0 on the quadrupole transition. After Doppler cooling alone the motional states tion was roughly equal to the effective linewidth ˜⌫ [see had (¯n ,¯n ,¯n )ϭ(0.47,0.30,0.18), measured with the x y z Eq. (116)]. sideband ratio method outlined above. Then a total of The ion was Doppler cooled into the Lamb-Dicke re- 15 cycles of interspersed resolved-sideband cooling gime with a 2.6-ms pulse of light on the S →P tran- pulses (order xyz xyz ...) were applied, five cycles in 1/2 1/2 sition. A second laser resonant with the D →P tran- each of the three directions. Each cycle consisted of a 3/2 1/2 sition prevented optical pumping to the metastable D3/2 pulse on the red sideband of the stimulated Raman tran- ϩ level. A short pulse of ␴ polarized light on the S sition from the (2S ,Fϭ2,m ϭϪ2)ϭ͉g͘ to the 1/2 1/2 F →P transition optically pumped the ion to ͉g͘. Fol- (2S ,Fϭ1,m ϭϪ1)ϭ͉e͘ state using the 2P state as 1/2 1/2 F 1/2 lowing this the two lasers for sideband cooling were a virtual intermediate state [see also the Appendix, Eq. switched on for varying times between 3 and 7 ms. An- (A3)]. The pulse time was adjusted to make a ␲ pulse on ϩ other ␴ pulse ensured that the ion was prepared in ͉g͘ the ͉g͉͘nϭ1͘→͉e͉͘nϭ0͘ transition, typically taking 1–3 Ϫ after the cooling, counteracting the possibility of the ␮s. Following this a resonant ͉e͘→2P ␴ repump pulse 3/2 ion’s being pumped into the S (mϭϩ1/2) state by the of about 7-␮s length optically pumped the ion to ͉g͘ via 1/2 cooling. For the polarizations and branching ratios the 2P level. After the complete set of cooling pulses a 3/2 present in the experiment this would happen on average probe pulse interrogated the transition probability to after about 90 (͉g͘→͉e͘→P ) cooling cycles (Roos ͉e͘. Cooling and probe pulses made up one experiment, 3/2 et al., 1999). typically repeated 1000 times with the same settings. The Finally the cooling result was detected by comparing probe pulse was swept over all six red and blue side- the red and blue sideband transition probabilities on the bands, thus mapping out all relevant transition prob- ͉g͘→͉e͘ transition as described above. Figure 7 shows abilities. From the ratio of corresponding sidebands the the results for a single ion cooled at a 4.51-MHz mo- final average motional quantum numbers were inferred ϭ tional frequency. From the residual noise around the red to be (¯nx ,¯ny ,¯nz) (0.033,0.022,0.029). ϩ sideband transition frequency an upper limit of 99.9% Peik et al. (1999) report cooling a single 115In ion to ground-state occupation was inferred. We note that ¯nϭ0.7(3) corresponding to 58% ground-state occupa- ground-state cooling using resolved-sideband transitions tion. The cooling transition was the weakly allowed was also experimentally demonstrated for up to four 5s21S →5s5p 3P intercombination line with an ex- 0 1 ions (King et al., 1998; Sackett et al., 2000; Rohde et al., perimentally observed linewidth of ⌫ϭ(2␲)650 kHz. 2001) and for neutral atoms in optical lattices (Hamann The trap frequency ␯ was about (2␲)1 MHz. et al., 1998; Perrin et al., 1998; Vuletic et al., 1998). At the University of Innsbruck, ground-state cooling was achieved on a single 40Caϩ ion for various motional frequencies from ␯ϭ(2␲)2 MHz up to 4 MHz (Roos C. Electromagnetically induced transparency cooling et al., 1999). The method used was very similar to the ϩ experiment in 198Hg . The cooling transition was the From the previous sections it might have become in- ͉ ϭ ϭϪ well-resolved red sideband on the g͘ S1/2(m 1/2) tuitively clear that any type of laser cooling in a trap →͉ ϭ ϭϪ e͘ D5/2(m 5/2) quadrupole transition. The 1.045 depends on the balance of absorption and emission of s lifetime of the upper level was shortened by an addi- photons on the red and blue sidebands. Every ͉ tional laser that coupled e͘ to the quickly decaying P3/2 absorption-emission cycle may be viewed as a scattering

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␳ event, with some events leading to cooling, some to population ex in the excited state. The scattering rate is heating, and some to no change in the motional state. then this population times the total decay rate ⌫ of the The ion will be cooled on average if the events that ⌬ ϭ⌫␳ excited state, W( ) ex . In the Lamb-Dicke regime dissipate motional energy are more probable than the absorption and emission will be dominated by carrier heating events. Typically the likelihood of cooling events and first-order red (blue) sidebands with transition prob- decreases as the kinetic energy decreases, while the abilities proportional to ⍀2 and ␩2⍀2n ͓␩2⍀2(nϩ1)͔, heating events settle at a fixed (possibly very low) rate. respectively, and we can neglect all processes of higher Equilibrium is reached when cooling and heating events order in ␩. Scattering on the carrier (͉g,n͘→͉g,n͘) will on average balance each other. The derivation of the be the most frequent process but will not change the Doppler cooling limit, Eq. (105), in Sec. IV.A is an ex- probabilities Pn to be in a certain n state. Motional- ample of that mechanism in which the velocity- ͉ ͘→͉ independent heating rate Eq. (104) is balanced with the state-changing events are limited to g,n g,n Ϯ ͘ ͉ ͘ ͉ Ϯ ͘ velocity- (and therefore kinetic-energy-) dependent (0,1) via the intermediate states e,n or e,n 1 .By cooling rate Eq. (103). comparing the Rabi frequencies on carrier and sideband Inspection of Eq. (106) and its derivation reveals that we can see that the scattering rate on a path involving the minimum Doppler temperature is dictated by the one red (blue) sideband transition (for example, ͉g,n͘ 2 line shape of the cooling transition [see Eq. (99)]. This →͉e,n͘→͉g,nϩ1͘) will be suppressed by the factor ␩ n implies that one might influence the cooling process by ͓␩2(nϩ1)͔ compared to the carrier transition. In more n tailoring the line shape of the cooling transition. In the detail, the rates RnϮ1 will go as preceding sections our discussion was limited to effec- Rn ϭW͑⌬͒␩2͑nϩ1͒ϩW͑⌬Ϫ␯͒␩2͑nϩ1͒, tive two-level systems and laser intensities around or be- nϩ1 low saturation with not too strong coupling of the atom n 2 2 R Ϫ ϭW͑⌬͒␩ nϩW͑⌬ϩ␯͒␩ n, (117) to the light field. The basic idea of electromagnetically n 1 induced transparency (EIT) cooling is to go beyond this where the first contribution comes from the scattering scenario and utilize the strong coupling of one laser to path through ͉e,n͘ and the second from the path ͉e,n ⌳ Ϯ ͘ n the atom in a three-level -type scheme to create an 1 . For example, in the second path on Rnϩ1 the atom absorption profile for the second (weaker) laser that is absorbs at detuning ⌬Ϫ␯ since the remaining energy ប␯ advantageous for cooling (Morigi et al., 2000). goes into the motion. Knowing these rates we can imme- We shall first derive a generalized treatment of the diately write down rate equations for the motional-level cooling that works for an arbitrary scattering rate on the populations cooling transition. The cornerstones of this treatment are outlined in Stenholm (1986). We shall derive some d nϩ1 nϪ1 n n P͑n͒ϭR P ϩ ϩR P Ϫ Ϫ͑R ϩR ͒P general statements about what kind of dependence of dt n n 1 n n 1 nϪ1 nϩ1 n the scattering rate on the relative detuning of atom and ϭ ͓ ͑ ϩ ͒Ϫ ͑ ͔͒ laser is most useful. We shall then derive the scattering AϪ Pnϩ1 n 1 Pn n rate in the ⌳ system for an atom at rest and set the ϩ ͓ Ϫ ͑ ϩ ͔͒ Aϩ PnϪ1n Pn n 1 , (118) parameters according to the design principles found ear- lier. with the n-independent coefficients 2 AϮϭ␩ ͓W͑⌬͒ϩW͑⌬ϯ␯͔͒. (119) 1. Cooling in the Lamb-Dicke regime The rate equation (118) can be converted into an equa- We shall now calculate the cooling and heating rates tion of motion for the average motional quantum num- for a laser-driven, trapped ion with arbitrary scattering ber ⌬ rate W( ). A general scattering (absorption-emission) ϱ cycle will proceed from ͉g,n͘ over ͉e,nЈ͘ to ͉g,nЉ͘, and, d d ϭ ϭϪ͑ Ϫ ͒ ϩ ¯n ͚ n Pn AϪ Aϩ ¯n Aϩ (120) in principle, one could find the scattering rates for all dt nϭ1 dt possible combinations (n,nЈ,nЉ) and derive rate equations for the probability P(n) to be in a certain state n by appropriate pairing and resumming of the respective based on all these scattering rates. Here we shall limit populations in connection with the AϮ coefficients. As possible scattering paths by assuming that some sort of long as the cooling rate Ϫ(AϪϪAϩ)Ͻ0, ¯n will evolve precooling (for example, Doppler cooling) has left the towards the final state of cooling, the steady state of Eq. ion in a state with thermal distribution in or close to the (120) given by ␩2 р Lamb-Dicke regime with ¯n 1. In this approximation Aϩ W͑⌬͒ϩW͑⌬Ϫ␯͒ ϭ ϭ the treatment cannot describe the complete cooling dy- ¯nf . (121) AϪϪAϩ W͑⌬ϩ␯͒ϪW͑⌬Ϫ␯͒ namics, but as long as the cooling method in question ⌬ reaches the Lamb-Dicke regime, it will yield useful ex- It is instructive to plug the scattering rates WL( )ofa pressions for the cooling limit and the cooling rate to- Lorentzian line into this formalism. Then the Doppler ␯Ӎ⌫ ⌫Ͼ␯ Ӷ ⌫ wards the final state. We assume that we know the scat- limit ¯nf /2 is recovered for while ¯n 1 for tering rate W(⌬) at laser detuning ⌬ of the atom at rest. Ӷ␯. For best cooling results with a general scattering This quantity can usually be found by solving Bloch rate we want to minimize ¯nf , which happens naturally if equations for the atom and calculating the steady-state the scattering rate on the red sideband W(⌬ϩ␯)is

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FIG. 8. Notation of levels, detunings, Rabi frequencies, and decay rates used in the calculation of an electromagnetically induced transparency (EIT) line shape. FIG. 9. Qualitative scattering rate on the ͉g͘↔͉e͘ transition as ⌬ ⌬ Ͼ g is varied for r 0. In addition to the broad resonance ⌬ Ϸ much bigger than the carrier and blue sideband rates, around g 0, a narrow bright resonance appears to the right ⌬ ϭ⌬ W(⌬) and W(⌬Ϫ␯), respectively. of the dark resonance at g r . The distance between the dark and the bright resonance is equivalent to the ac Stark shift of ͉e͘ caused by the strong beam. 2. Scattering rates in EIT cooling The general idea of EIT cooling in a ⌳ system is to ϭ ⌬2⍀2⍀2⌫ϩ ⌬2⌫2͑⍀2⌫ ϩ⍀2⌫ ͒ D 8 r g 4 r g g r use a dark resonance to completely suppress the carrier ϩ ⌬2͓⌬2⍀2⌫ ϩ⌬2⍀2⌫ ͔ϩ ⌬ ⌬⍀4⌫ scattering (Morigi et al., 2000). With a prudent choice of 16 r g r g r g 8 r g r parameters of the two light fields driving the system we Ϫ ⌬ ⌬⍀4⌫ ϩ͑⍀2ϩ⍀2͒2͑⍀2⌫ ϩ⍀2⌫ ͒ shall then also be able to fulfill W(⌬Ϫ␯)ӶW(⌬ϩ␯). 8 g r g r g r g g r . (124) We denote the levels, detunings, and Rabi frequencies of This rather complicated expression quickly simplifies if ⌫ ϭ␣⌫ ⌬ Ϸ⌬ ⍀ Ӷ ⍀ ⌬ the two transitions as indicated in Fig. 8. The equations we set g and assume r g and g ( r , r), of motion for the density matrix (Bloch equations) can, following the idea that the strong laser field on the in principle, be derived from Eq. (86) or by adding phe- ͉r͘↔͉e͘ transition optically pumps the internal state to nomenological damping terms to the unitary evolution ͉g͘ and also modifies the scattering rate of a comparably that correctly reflect the decay of the ͉e͘ state. The weaker beam on the ͉g͘↔͉e͘ transition into W(⌬) ϭ⌫␳ ⌬ Bloch equations in our system are ee( ), which is advantageous for cooling. This ␳ ⍀ yields d rr r ϭi ͑␳ Ϫ␳ ͒ϩ⌫ ␳ , dt 2 re er r ee ⌬2⍀2⌫ ͑⌬͒Ϸ g W ␣͓⌬2⌫2ϩ ͑⍀2 Ϫ⌬⌬ ͒2͔ . (125) ␳ ⍀ 4 /4 g d gg g r ϭi ͑␳ Ϫ␳ ͒ϩ⌫ ␳ , dt 2 ge eg g ee Figure 9 shows the qualitative behavior of the scattering rate vs detuning ⌬ for (⌬ ,⌬ )Ͼ0. Indeed W(⌬) van- ␳ ⍀ ⍀ r g d rg g r ⌬ϭ ϭiͫ͑⌬ Ϫ⌬ ͒␳ ϩ ␳ Ϫ ␳ ͬ, ishes at 0, so the carrier is completely suppressed. dt g r rg 2 re 2 eg The position of the two maxima is given by d␳ ⍀ ⍀ ⌫ 1 re ϭ ͫ r ͑␳ Ϫ␳ ͒ϩ g ␳ Ϫ⌬ ␳ ͬϪ ␳ ⌬ ϭ ͑Ϯͱ⌬2ϩ⍀2Ϫ⌬ ͒ i , Ϯ . (126) dt 2 rr ee 2 rg r re 2 re 2 r r r ␳ ⍀ ⍀ ⌫ d ge g r We want the narrow bright resonance at positive detun- ϭiͫ ͑␳ Ϫ␳ ͒ϩ ␳ Ϫ⌬ ␳ ͬϪ ␳ , ⌬ ϭ␯ dt 2 gg ee 2 gr g ge 2 ge ing to coincide with the red sideband, ϩ : (122) ␯ϭ ͑ͱ⌬2ϩ⍀2Ϫ⌬ ͒ 1/2 r r r . (127) ⌫ϭ⌫ ϩ⌫ with g r . We shall assume that reequilibration of The reader may recognize that this condition is equiva- the internal-state dynamics is much faster than the ex- lent to saying that the Stark shift of the ͉r͘→͉g͘ reso- ternal motional dynamics, so we can solve for the steady nance has to be equal to the motional frequency. For this state of the Bloch equations to describe the scattering choice of parameters, W(ϩ␯) takes the largest value events at all times. Doing so and using the conservation Ϫ␯ ␳ ϩ␳ ϩ␳ ϭ possible, while W( ) then assumes a comparatively of probability rr gg ee 1 one can (after some al- small value from the wing of the broad bright resonance ␳ gebra) derive the steady-state solution for ee (Janik to the left of the origin. To find the cooling limit quan- et al., 1985): titatively, we can start from Eq. (121) with W(0)ϭ0. ⌬2⍀2⍀2⌫ Using Eq. (127) we get after some algebra 4 g r ␳ ͑⌬͒ϭ , (123) ee D W͑Ϫ␯͒ ⌫ 2 ¯n ϭ ϭͩ ͪ . (128) ⌬ϭ⌬ Ϫ⌬ s W͑␯͒ϪW͑Ϫ␯͒ 4⌬ where g r and r

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Ӷ ⌬ ϭ⌬ ӷ⌫ To ensure that ¯ns 1 we have to choose g r , sity of the strong beam was fine-tuned by observing the ⍀ while r has to be set to accommodate Eq. (127). resolved-sideband excitation on the red sideband of ei- Finally we should note that our treatment of EIT ther mode and minimizing it. The lowest observed mean ϭ cooling is only an approximation, since we neglected all vibrational quantum number was ¯ny 0.18, correspond- recoils happening when the system is relaxing back into ing to 84% ground-state probability. On the z mode at ␯ ϭ ␴ϩ the steady state after scattering a photon on the side- z 3.3 MHz, after the intensity of the beam was band transitions. Fortunately, since we start in the increased to adjust the ac Stark shift, a minimum mean ϭ Lamb-Dicke regime, the relaxation proceeds predomi- vibrational number of ¯nz 0.1 was observed, corre- nantly through carrier scattering, thus not altering the sponding to a 90% ground-state probability. The cooling average motional quantum number. Morigi et al. (2000) results were largely independent of the intensity of the ␲ have done a Monte Carlo simulation that also takes beam as long as it was kept much smaller than the ␴ϩ these effects into account. Indeed the cooling dynamics intensity. The intensity ratio I␴ /I␲Ӎ100 was varied by a are slightly slower in this simulation, but very close to factor of 4, with no observable effect on the final ¯n. the approximate treatment. From the dependence of the mean vibrational quantum number on the EIT pulse length, an initial cooling rate of 1 quantum per 250 ␮s was found for the y direction. 3. Experimental results In addition, both modes were simultaneously cooled The Innsbruck group has demonstrated EIT cooling by setting the intensity of the strong beam for an ac with a single calcium ion (Roos et al., 2000). The ⌳ sys- Stark shift roughly halfway between the two mode fre- tem was implemented within the S →P transition, quencies. Both modes were cooled simultaneously with 1/2 1/2 ϭ ϭ whose Zeeman sublevels mϭϩ1/2 and mϭϪ1/2 consti- (p0)y 58% and (p0)z 74% ground-state probability. tute a four-level system. By applying the strong laser on ␴ϩ ͉ ϵ͉ ϭϪ ͉ ϵ͉ the transition ( r͘ S1/2 ,m 1/2͘, e͘ P1/2 ,m V. RESONANCE FLUORESCENCE OF SINGLE IONS ϭ1/2͘) and the weaker cooling laser on a ␲ transition ͉ ϵ͉ ϭϩ (from g͘ S1/2 ,m 1/2͘), they ensured that very The observation and analysis of resonance fluores- ͉ little population would ever be in the extra P1/2 ,m cence emitted from atomic systems provide an impor- ϭϪ1/2͘ state, so this was effectively a three-level sys- tant tool for investigation of the interaction between tem. An additional laser is used to repump the ion from matter and radiation. Essentially there are three the D3/2 level, but the branching ratio to that level was complementary strategies available for these measure- so small that the above conclusion was not seriously ments: compromised. (i) Excitation spectroscopy comprises the observa- The two beams were generated by splitting frequency- tion of resonance fluorescence as a function of the doubled light from a Ti:sapphire laser near 397 nm into detuning of the exciting electromagnetic radia- two suitable beams with the help of two acousto-optic tion; modulators (the same beam was also used for Doppler (ii) the spectrum of resonance fluorescence is ob- precooling of the ion). The second-order Bragg reflexes tained by measuring the spectral distribution of of two acousto-optical modulators driven at around 90 the emitted resonance fluorescence at a fixed de- MHz had a blue detuning of about ⌬␴ϭ⌬␲Ӎ75 MHz tuning of the exciting radiation; relative to the S ↔P line center (natural linewidth of 1/2 1/2 (iii) quantum-dynamical information can be obtained the transition ⌫Ϸ20 MHz). The beams (typically about by measuring the time interval statistics of the 50 ␮W in the strong and 0.5 ␮W in the weak beam) were photon-counting events from resonance fluores- then focused into a Ϸ60-␮m waist onto the single ion ␯ ␯ ␯ cence, usually in the form of correlation functions. in a trap with oscillation frequencies ( x , y , z) ϭ (1.69,1.62,3.32) MHz. The k vectors of the two beams All three techniques have been used to investigate enclosed an angle of 125° and their k-vector difference single-ion resonance fluorescence. ⌬k had a component along all three trap axes. The beam intensity was controlled with the power of the acousto- A. Excitation spectroscopy, line shapes optical modulator rf drive. The ion was first Doppler precooled. The Doppler A single two-level atom at rest in free space (Deh- Ϸ cooling limit on this was (¯nx ,¯ny ,¯nz) (6,6,3) in the trap melt, 1973) is the ideal object for the observation of used. The EIT cooling beams were applied for periods resonance fluorescence. The outcome of corresponding between 0 and 7.9 ms, with ¯n reaching its asymptotic experiments can be completely described in terms of the limit after about 1.8 ms. The final mean occupation num- optical Bloch equations, as given by Eq. (86), where the ␳ ber was probed on the resolved sidebands of the excited-state population ee as a function of the detun- ↔ ⌬ S1/2 D5/2 transition, either using the technique de- ing of the exciting radiation describes the observed scribed in Sec. IV.B.1 or fitting experimental Rabi oscil- line shapes. However, since the detuning also sensitively lations to those expected for a small state with thermal influences the cooling and heating dynamics, the ob- distribution (Meekhof et al., 1996; Roos et al., 1999). served line shape in excitation spectroscopy usually ex- In this manner EIT cooling of the y and the z oscilla- hibits this influence as well. More precisely, for weak tions at 1.62 and 3.32 MHz was investigated. The inten- confinement (i.e., trap frequency ␯Ͻ⌫, the natural line-

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 304 Leibfried et al.: Quantum dynamics of single trapped ions width of the excited state) and a laser detuning far be- served at ␶ϭ0. Whereas classical fields show correlation low resonance (i.e., ⌬Ͻ0 or ‘‘red detuning’’), the functions g(2)(0)у1 and g(2)(␶)рg(2)(0), the nonclas- Doppler-broadened line shape resembles that of a ther- sical resonance fluorescence of a single atom exhibits the mal ensemble of atoms; however, the Doppler broaden- so-called antibunching behavior, i.e., g(2)(0)ϭ0 and ing results from a convolution of overlapping sidebands. g(2)(␶)Ͼg(2)(0). This condition entails sub-Poissonian For a red detuning close to resonance, cooling sets in emission probability for ␶ϭ0. Both antibunching and and the line shape can be described very well by the sub-Poissonian statistics have been observed in Lorentzian of an atom at rest. For a positive (or ‘‘blue’’) quantum-optical experiments with atoms (Kimble, Da- detuning ⌬Ͼ0, heating dominates and the large number genais, and Mandel, 1977; Short and Mandel, 1983); of resulting sidebands broadens the transition to such an however, corrections for the fluctuating atom numbers in extent that the absorption sharply drops near resonance the atomic beam had to be applied. ⌬ϭ0. Therefore excitation spectra of a single two-level For a single two-level atom the intensity correlation ion exhibit an asymmetric (‘‘half-Lorentzian’’) shape function is due to the cooling and heating dynamics (Nagourney ␳ ͑␶͒ et al., 1983). For three- and more-level systems with sev- (2)͑␶͒ϭ ee g ␳ ͑ϱ͒ , (130) eral lasers contributing to the cooling and heating dy- ee namics the observed line shapes can become quite in- ␳ where ee represents the population of the excited state volved and difficult to interpret (Reiß et al., 1996). of the atom. Since g(2)(␶) is a conditional probability as Especially well-investigated cases are the three-level ϩ ϩ a function of time, it allows one to observe directly the (eight-sublevel) systems of single trapped Ba and Ca quantum-dynamical behavior of the matter-radiation in- ions (Janik et al., 1985; Siemers et al., 1992). These teraction. Likewise, any motional effects affecting the ions must be modeled by a ⌳-shaped three-level ↔ ↔ resonance fluorescence can be characterized from an S1/2 P1/2 D3/2 (or, accounting for all Zeeman sublev- analysis of g(2)(␶). els, eight-level) system. The excitation usually involves With a single trapped Mgϩ ion Diedrich and Walther two dipole transitions, sharing the fluorescing level P1/2 , (1987) observed antibunching and sub-Poissonian statis- and excitation spectroscopy is obtained by tuning only tics of an individual two-level atom. In that experiment one exciting laser across resonance while the second one antibunching in the resonance fluorescence of one, two, is kept fixed below resonance. Thus net cooling usually and three trapped ions was observed and the antibunch- can be observed even for a certain range of blue detuning property decreased as predicted for increasing ion ings. Three-level (eight-sublevel) systems can exhibit numbers, since for an increasing number of independent line shapes that agree very well with the theoretical de- atoms the photon counts become more and more uncor- scriptions given by the optical Bloch equations for an related. The dynamical interaction between matter and atom at rest. In fact, the richness of these systems allows radiation leads to Rabi oscillations and for a single ion one to investigate in quantitative detail the effects of the observed intensity correlation agrees very well with optical pumping, the appearance of dark resonances, the theoretically predicted function (Carmichael and and Raman processes (Schubert et al., 1992, 1995; Siem- Walls, 1976), ers et al., 1992). The excellent quantitative agreement between theory and experiment allows one to precisely 3⌫ g(2)͑␶͒ϭ1ϪeϪ3⌫␶/2ͫcos ⍀␶ϩ sin ⍀␶ͬ, (131) derive all spectroscopic parameters, such as the detun- 2⍀ ings, the Rabi frequencies, and the orientation of the ⍀2ϭ⍀2ϩ⌬2Ϫ ␥ 2 quantization axis required for more detailed quantum- where 0 ( /4) , with the Rabi frequency ⍀ ⌫ optical calculations, e.g., correlation functions, as dis- 0 at resonance, the natural linewidth , and the detun- ⌬ cussed below. ing . The single-atom resonance fluorescence clearly exhibited sub-Poissonian statistics. The deviation of the distribution from Poisson statistics is usually described B. Nonclassical statistics, antibunching, and squeezing in terms of Mandel’s Q parameter, A powerful way to acquire detailed information about ͗͑⌬n͒2͘ Ϫ͗n͘ ϭ T T the quantum dynamics of an atom is the statistical analy- Q , (132) ͗n͘T sis of the measured stream of photon counts. In particu- ͗ ͘ ϭ͚ϱ lar, nonclassical features of the resonance fluorescence where n T nϭ0npn(T) denotes the mean number of are observed in the second-order correlation functions detected photons, with pn(T) the probability for detect- (intensity correlations) ing n photons within the time T. This parameter is related to the photon correlation function (Mandel, 1979; ͗EϪ͑t͒ EϪ͑tϩ␶͒ Eϩ͑tϩ␶͒ Eϩ͑t͒͘ (2) • • • Merz and Schenzle, 1990) by g ͑␶͒ϭ Ϫ ϩ . (129) ͓͗E ͑t͒ E ͑t͔͒͘2 • 2Z T (2) Here the scattered light field of the atomic resonance Q͑T͒ϭ ͵ d␶͑TϪ␶͓͒g ͑␶͒Ϫ1͔, (133) T 0 fluorescence is described by the electric-field operators ϩ Ϫ ϭ E and E (Glauber, 1963, 1964) and for ␶у0. This where Z ͗n͘T /T is the mean count rate. While for any function is proportional to the probability of detecting a classical radiation Q(T)у0 at all T, the measurement Ϫ second photon at time ␶ after a first one has been ob- of Diedrich and Walther revealed QϭϪ7ϫ10 5. In that

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 Leibfried et al.: Quantum dynamics of single trapped ions 305 experiment the classical residual (driven) micromotion C. Spectrum of resonance fluorescence, homodyne of the trapped ion was observed as a modulation in the detection of fluorescence intensity correlation; however, secular motion could not be detected. Information about the dynamics of the interaction be- In an experiment with one and two trapped Hgϩ ions tween matter and radiation can also be gathered by Itano et al. (1988) were able to observe antibunching spectral analysis of the emitted resonance fluorescence. ↔ The spectrum of resonance fluorescence emitted by a and sub-Poissonian statistics on the P1/2 D3/2 transition. Spontaneous emission on this transition is very free two-level atom interacting with a traveling-wave la- weak (⌫ϭ52 sϪ1); however, each decay can be observed ser field was studied by Mollow (1969). For low excita- since it interrupts the otherwise steady photon flow on tion intensities the fluorescence spectrum exhibits an ⌫ϭ ϫ 8 Ϫ1 ↔ elastic peak centered at the incident-laser frequency ␻ , the strong ( 4 10 s ) P1/2 S1/2 transition that is L used for excitation and detection. Thus each spontane- while for higher intensities an inelastic component be- ous decay on the weak transition is detected by the ab- comes dominant, with contributions centered at the fre- ␻ ␻ Ϯ⍀ sence of many photons due to electron shelving (see Sec. quencies L and L 0 (for resonant excitation), ⍀ III.C). Using this technique, antibunching on the weak where 0 denotes the Rabi frequency. This so-called transition was observed and, due to the greater detec- ‘‘Mollow triplet’’ arises from the dynamic (or ac) Stark tion efficiency of the electron shelving technique (as splitting of the two-level transition. As outlined in Sec. compared to usual photon counting in the experiment III.B.6 above, the observation of the fluorescence spectrum provides an alternative method for detecting the by Diedrich and Walther), a Q value of QϭϪ0.25 for internal quantum dynamics as well as the (quantum) one and two ions could be measured. motion in the trap. In a subsequent experiment at the University of Ham- ϩ With a single trapped and laser-cooled Ba ion, Stal- burg, Schubert et al. (1992, 1995) observed antibunching gies et al. (1996) analyzed the spontaneously emitted ϭϪ ϫ Ϫ4 and sub-Poissonian statistics (Q 7 10 due to lim- light on the P ↔S transition at 493 nm with a Fabry- ited efficiency of photon counting) in the resonance 1/2 1/2 ϩ Perot interferometer and recorded emission spectra of fluorescence of a single Ba ion, which cannot be mod- the single-ion fluorescence. The piezo-tuned confocal fil- eled as a two-level system. Aside from antibunching and ter resonator provided a spectral resolution of about sub-Poissonian statistics, the observed intensity correla- 16.7 MHz. This was sufficient to resolve the spectral tions in this case revealed maximum photon correlations components of the transition, which has a natural width much larger than what is possible with two-level atoms, of about 20 MHz. The observed spectra show the ex- as well as photon antibunching with much larger time pected dynamical Stark effect and are in agreement with constants of the initial photon anticorrelation. Thus a theoretical predictions based on calculations using opti- detailed study of the internal dynamics due to optical cal Bloch equations and parameters that were derived pumping and the preparation of Zeeman coherences be- from excitation spectra as outlined above in Sec. V.A. came possible. Furthermore, the exact form of the ob- Refined spectroscopic information on the fluorescence served g(2) functions allowed the direct observation and spectrum can be gained by beating the fluorescent light quantitative description of the preparation of superposi- with a local oscillator and subsequent homodyne or het- tion states vs mixed states of a single trapped particle erodyne analysis. The elastic component scattered by an (Schubert et al., 1995). ensemble of cold-trapped atoms was first detected by Aside from the antibunching property, which directly Westbrook et al. (1990) with atoms confined in an optical lattice. With a single trapped and laser-cooled Mgϩ reflects the photon nature of light, other nonclassical ef- ↔ fects are observable in resonance fluorescence. An ex- ion, Rayleigh scattering on the S1/2 P3/2 transition was observed by Ho¨ ffges, Baldauf, Eichler, et al. (1997) and ample is the squeezing property, which is usually ob- Ho¨ ffges, Baldauf, Lange, and Walther (1997). Hetero- served in a homodyne detection scheme (Slusher et al., dyne detection allowed for a spectral resolution of the 1985; Wu et al., 1986) and which describes an asymmet- elastic component with a linewidth of 0.7 Hz. ric noise behavior of the quadrature components of the As shown above in Sec. V.A, the motion of trapped electromagnetic radiation. Squeezing has also been pre- particles in an external potential shows up as sidebands dicted to appear in the resonance fluorescence of single to the elastically scattered component. The height of the two-level atoms (Walls and Zoller, 1981) and three-level sidebands then presents a measure of the motional am- atoms (Vogel and Blatt, 1992). Due to limited detection plitude, and their width reflects the cooling rate. Such efficiency, Mandel (1982) has shown that the observable motional sidebands were first detected in the fluores- effect from squeezing in the photon statistics of the reso- cence spectrum of ensembles of cold atoms in optical nance fluorescence of a single atom is extremely weak. lattices (Jessen et al., 1992) and they were only recently Alternatively, the detection of homodyne intensity cor- observed in the homodyne spectra of single trapped relations is expected to offer an easier way to observe ions. With a single trapped and laser-cooled Ybϩ ion, squeezing in the resonance fluorescence of a single Bu¨ hner and Tamm observed motional sidebands in the ↔ trapped ion, as was proposed by Vogel (1991) and Blatt resonance fluorescence of the P1/2 S1/2 transition at et al. (1993). However, no experimental results are cur- 369 nm of 172Ybϩ at a radial secular motion of 785 kHz rently available. (Bu¨ hner and Tamm, 2000; Bu¨ hner, 2001). The measured

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maximum width of the sidebands was about 1.5 kHz at saturation intensity and for an optimal detuning of the cooling laser of ⌬Ӎ⌫. Taking into account residual broadening by the detection technique (i.e., detection bandwidth of the spectrum analyzer and fluctuating trap voltages), the authors concluded that the observed cooling rate agreed well with predictions based on the two- level cooling calculations. With a single trapped 171Ybϩ ion that has a hyperfine splitting, a weak spontaneous decay to a hyperfine component of the excited D3/2 state leads to intermittent ↔ resonance fluorescence on the P1/2 S1/2 transition due to electron shelving. This results in an additional inelastic component of the observed fluorescence spectrum (Hegerfeldt and Plenio, 1995), and its height and width are uniquely determined by the shelving conditions. This leads to a Lorentzian component centered on the elastic scattering and is interpreted as a broadening due to a FIG. 10. Heterodyne signal on a secular sideband of the un- driven motion of a single Baϩ ion (Raab et al., 2001). The fit- ‘‘stochastic intensity modulation.’’ In the experiment this Ϫ ted width corresponds to a cooling rate of 2␲ 900 s 1. feature was observed and found to be in agreement with the theoretical prediction (Bu¨ hner and Tamm, 2000). With a single trapped Baϩ ion, heterodyne measure- a part of the emitted resonance fluorescence (Eschner ments were also performed by the Innsbruck group. The et al., 2001). For this, part of the emitted light (about 4% measured width of the elastic component was limited by of the solid angle) is collimated with a lens to a plane the resolution bandwidth of 64 MHz of the spectrum mirror about 25 cm away from the ion and then retrore- analyzer. Since the phase of the driven micromotion at flected (via the same lens) onto the ion. The fluores- ␻ ϭ ␲ rf 2 18.53 MHz is well defined, the corresponding cence and its mirror image are then observed with a micromotion sidebands were observed with the same photomultiplier opposite the mirror collecting both the linewidth. From the height of the sidebands a modula- directly emitted and the back-reflected part. Moving the tion index corresponding to a residual micromotion am- mirror using a piezoelectric transducer and overlapping plitude of 26 nm was derived. This measurement allows the direct and reflected parts allows one to observe in- one to detect and compensate for residual micromotion, terference fringes that result from the interfering beam and it is sensitive to micromotion amplitudes of about 1 paths (direct and reflected paths) and the visibility of the nm (Raab et al., 2000). Furthermore, sidebands due interference fringes can be calculated from the first- ␻ ␻ ␻ ␲ to secular motion at frequencies ( x , y , z)/2 order correlation function ϭ Ϫ ϩ (0.62,0.65,1.3) MHz were observed using an excitation ͗E ͑tϪ␶͒E ͑t͒͘ (1)͑␶͒ϭ technique: By exciting the ion motion with a weak rf g ͗ Ϫ͑ ͒ ϩ͑ ͒͘ . (134) field (applied to nearby electrodes) one can detect co- E t E t herent sidebands with a high signal-to-noise ratio, since The emitted field EϮ(t) can be expressed in terms of the this results in a driven motion and thus is similar to the atomic polarization, which in turn can be calculated detection of micromotion sidebands. Scanning the exci- from the optical Bloch equations. As expected, the vis- tation across the sideband frequency results in a reso- ibility decreases with increasing laser intensity due to nant enhancement of the ion motion, making the side- the increasing ratio of inelastic to elastic scattering; that bands easily observed. Extrapolating the observed is, the internal dynamics of the interaction can be inves- linewidth to zero amplitude of the exciting rf field even- tigated through a study of the interference fringes. In tually reveals the cooling rate limiting the width of the addition, external atomic motion influences the interfer- motional sidebands. The experimentally observed line- ence since the ion oscillation results in a phase modula- widths of about 750 Hz agree within the error limits with tion of the field EϮ(t). This effect allows one to measure the calculated cooling rates (Raab et al., 2000). These the amplitude of any residual ion motion via an obser- findings were corroborated by direct observation of the vation of the interference fringes. Moreover, the ion po- sidebands as shown in Fig. 10 (Raab et al., 2001). No sition with respect to the mirror can be determined on ϩ asymmetric sidebands were observed in either the Ba the scale of the extension of the wave function, which is ϩ experiment (Innsbruck) or the Yb experiment (Braun- several nm. schweig), since in both cases only Doppler cooling was Including the back-reflected part of the emitted fluo- available and the residual quantum numbers of the har- rescence in the optical Bloch equations reveals that the monic motion were between 10 and 30 (depending on reflected part actually acts back on the ion and changes the ion and the respective secular frequencies), leading the excited-state populations and thus the overall reso- to almost equal sideband amplitudes. nance fluorescence. This back action has been detected Using a new technique, the Innsbruck group recently by an independent observation of the excited-state ϩ studied the interaction of a single trapped Ba ion with population as a function of the mirror position (i.e.,

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 Leibfried et al.: Quantum dynamics of single trapped ions 307

l ϭ͉⍀ ͉ ϭ␲ whether there is constructive or destructive interfer- Eq. (84) is fnt␲,n nϩl,n t␲,n . Here, resonant ence). This can be considered inhibition and enhance- means ␦Јϭ0. The phase factors Ϫi and ei␾ are physi- ment of the resonance fluorescence by the reflected (and cally irrelevant in the present context since they factor phase-sensitive) presence of the self-emitted photon. out of the wave function; they are just kept for math- This back action mediated by the external mirror has ematical consistency. For these conditions the solution also been observed with the radiation of two ions; that l matrices Tn take the simple forms is, two ions interfering with the mirror image of each Ϫ i␾ other show the same effect (Eschner et al., 2001). This 0 ie T0 ϭͩ ͪ (135) technique opens the way for quantum feedback sce- n ϪieϪi␾ 0 narios in which single emitted fluorescence quanta can for the carrier and be used to influence and eventually control the internal i␾ and external quantum dynamics of trapped ions. Ϯ 0 e T 1ϭͩ ͪ (136) n ϪeϪi␾ 0 VI. ENGINEERING AND RECONSTRUCTION OF for the red (lϭϪ1) and blue (lϭϩ1) sidebands. For QUANTUM STATES OF MOTION creating pure number states, the phase factors of succes- A. Creation of special states of motion and internal-state/ sive pulses will just add up to an overall phase factor motional-state entanglement that is irrelevant, therefore one can set these phase factors equal to one for all these pulses without losing gen- Fueled by the strong analogy between cavity QED erality. When generating superpositions of number and the trapped-ion system, various theoretical propos- states these phases will determine the relative phase of als have been made on how to create nonclassical and the number-state components and cannot be disre- arbitrary states of motion of trapped ions or even how to garded. Starting from the state ͉g,0͘, with a blue side- 1͉ ͘ϭ͉ ͘ create states in which the internal degree of freedom is band pulse, T0 g,0 e,1 , the first excited motional entangled with the external motion in a highly nonclas- state connected with the ͉e͘ internal state is created. Ϫ ͉ ͘ϭ 0͉ ͘ sical way (Schro¨dinger-cat states). The theoretical efforts This may either be converted to i g,1 T1 e,1 or were complemented by the first experiments on nonclas- walked higher up the number-state ladder with a red Ϫ1͉ ͘ϭ͉ ͘ sical states of motion by the groups at NIST and the sideband pulse T1 e,1 g,2 . In this manner one can University of Innsbruck. In the NIST experiments, co- step through to high number states, as was done in an herent, quadrature squeezed number states and super- experiment of the NIST group using Raman transitions ϩ positions of number states of trapped Be were pre- between two hyperfine levels of Beϩ (Meekhof et al., pared (Meekhof et al., 1996). The preparation of 1996). Once the number state is created, the signature of ϩ number states was also achieved for trapped Ca at the the state can be found by driving transitions on the first University of Innsbruck (Roos et al., 1999). The NIST blue sideband as described in Sec. III.D. The rate of the ⍀ group succeeded in entangling coherent motional states Rabi flopping, n,nϩ1 in Eq. (70), depends on the value with opposite phases to the two internal states of n of the number state occupied. According to Eq. (Schro¨ dinger-cat state; Monroe et al., 1996). (95) the expected signal is proportional to 1 ϩ ⍀ The common starting point for all motional-state en- cos( n,nϩ1t). The experimentally observed signal also gineering experiments so far has been the ground state turned out to be damped, probably due to a combina- of motion, reached by resolved-sideband cooling meth- tion of uncontrolled noise sources (see below). This was ods as described in Sec. IV.B. Since resolved-sideband accounted for by introducing n-dependent exponential ␥ cooling has only been successfully applied inside the damping with constants n in the fits of the blue side- Lamb-Dicke regime, most of the state preparation also band excitation curves, took place in this regime. The description of interactions 1 will be restricted to this limit in the following and only fit P ͑t͒ϭ ͓1ϩcos͑⍀ ϩ t͒exp͑Ϫ␥ t͔͒. (137) generalized where appropriate. g 2 n,n 1 n The observed flopping curves are approximately de- 1. Creation of number states scribed by Eq. (137), as can be seen in Fig. 11(a) for an Several techniques for the creation of number states initial ͉g,nϭ0͘ number state created in this experiment. of motion have been proposed, using quantum jumps The fit to Eq. (137) is drawn as a solid line and yielded ⍀ ␲ ϭ ␥ ϭ Ϫ1 (Cirac, Blatt, et al., 1993; Eschner et al., 1995), adiabatic 0,1 /(2 ) 188(2) kHz and 0 11.9(4) s . passage (Cirac, Blatt, and Zoller, 1994), or trapping In this experiment Pg(t) was recorded for a series of states (Blatt et al., 1995). Despite the possibilities de- number states ͉g,n͘ up to nϭ16 and the Rabi-frequency ⍀ ⍀ scribed in these papers, a simple technique involving ratios n,nϩ1 / 0,1 were extracted. They are plotted in multiple ␲ pulses is the only one to have been used in Fig. 11(b), showing very good agreement with the theo- experiments so far. The ion is initially cooled to the ͉g,0͘ retical frequencies, which include the trap’s finite Lamb- number state. Higher-n number states are created by Dicke parameter ␩ϭ0.202 [solid line in Fig. 11(b)]. ␲ ␥ applying a sequence of resonant pulses of laser radia- The damping constant n was also extracted and ob- tion on the blue sideband, red sideband, or carrier. A ␲ served to increase with n [the data are consistent with ␥ Ϸ␥ ϩ 0.7 pulse corresponds to a time t␲,n so that the argument in n 0(n 1) ]. The source of the damping was be-

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FIG. 12. Number states in the Innsbruck experiment (Roos et al., 1999): (a) Rabi oscillations on the blue sideband for the ground state (nϭ0); (b) Rabi oscillations as in (a) but now for the number state ͉nϭ1͘.

D5/2 state) via the P3/2 level with light at 854 nm. Since the repumping was done inside the Lamb-Dicke regime,5 it did not change the motional state for most of the experiments, thus leaving the ion in the ͉g͉͘nϭ1͘ number state with high probability. Figure 12 shows the Rabi flopping dynamics on the blue sideband for the two prepared number states [(a) nϭ0 and (b) nϭ1]. The ⍀ Rabi frequency 0,1 is 21(1) kHz and the frequency ratio ⍀ ⍀ Ϸ & is 1,2 / 0,1 1.43, close to , with the asymptotic value of Eq. (77) for the Lamb-Dicke parameter approaching FIG. 11. Number states in the NIST experiment (Meekhof zero. In this experiment a contrast over 0.5 was main- et al., 1996): (a) Rabi oscillations on the blue sideband for the ground state in the trap (nϭ0); (b) ᭹, measured ratio of Rabi tained for about 20 periods. Since a heating rate of frequencies for different number states; solid lines, theoretical about one quantum in 190 ms (1/70 ms) was measured predictions for several Lamb-Dicke parameters. along the axial (radial) direction for the Innsbruck trap, heating was not believed to play a leading role during Ϸ lieved to be in part due to uncontrolled magnetic-field the 1-ms duration of individual experiments. The de- fluctuations at the position of the ion and frequency and cay in contrast was mainly attributed to magnetic-field intensity fluctuations of the two Raman light fields used fluctuations in the laboratory and intensity fluctuations in the experiment. One example of such noise sources, in the laser beams. the magnetic fields due to currents switching at the 2. Creation of coherent states 60-Hz line frequency and its harmonics, was successfully suppressed by producing a compensation field of the Coherent states of motion can be produced from the right amplitude and phase with extra coils close to the ͉nϭ0͘ state by a spatially uniform classical driving field trap. (Carruthers and Nieto, 1965), by a ‘‘moving standing Another source of dissipation was an unexpectedly wave’’ (Wineland et al., 1992), by pairs of standing waves high heating rate out of the motional ground state in (Cirac, Blatt, and Zoller, 1994), or by a nonadiabatic these experiments. The observed heating rate was about shift of the trap center (Janszky and Yushin, 1986; Yi one quantum per millisecond, about three orders of and Zaidi, 1988; Heinzen and Wineland, 1990). In ex- magnitude higher than expected from thermal electronic periments of the NIST group, the first two approaches noise (Wineland et al., 1998; Turchette, Kielpinski, et al., were taken. For the classical drive, a sinusoidally varying 2000). potential at the trap oscillation frequency was applied to The n0.7 scaling of the damping constants with number one of the trap compensation electrodes for a fixed time state was extracted from fits to the data in this experi- of typically 10 ␮s with varying amplitudes. This gave rise ment. Theoretical work has incorporated spontaneous to an approximately spatially homogeneous force on the emission and heating to explain the observed scaling ion of the form (Plenio and Knight, 1996, 1997; Bonifacio et al., 2000; Di ͑ ͒ϭ ͑␻ Ϫ␸͒ Fidio and Vogel, 2000; Budini, de Matos Filho, and F t eE0 sin drivet , (138) Zagury, 2002). which can be associated with an interaction potential The second experiment on number states was carried ប out at the University of Innsbruck (Roos et al., 1999). H ϭϪxˆF͑t͒ϭϪͱ ͕aû*͑t͒ϩaˆ †u͑t͖͒F͑t͒. There a single 40Caϩ ion was cooled to the ground state I 2m␯ → on the S1/2 D5/2 quadrupole transition (see also Sec. (139) IV.B), leaving the ion in the nϭ0 number state 99.9% of the time. The nϭ1 number state was created by apply- ␲ 5␩ Ϸ → ␩ Ϸ ing a pulse on the blue sideband and then incoherently 854 0.03 for the D5/2 P3/2 transition and 393 0.09 for the repumping the excited electronic state ͉e͘ (in this ex- spontaneous decay, so the probability for two carrier transi- ϭϪ Ϫ␩2 Ϫ␩2 ϭ periment corresponding to the mf 5/2 sublevel of the tions is (1 854)(1 393) 0.99.

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Putting in the lowest-order approximation, Eq. (15), Ϸ ␯ ͓ ϩ ␻ ͔ ϩ u(t) exp(i t)͕ 1 (qx/2)cos( rft) /(1 qx/2)͖, already displays the general behavior. As was pointed out by Carruthers and Nieto (1965), this interaction will change the quantum state of the oscillator only if the drive is resonant with its motion. In the lowest-order approxi- ␻ Ϸ␯ mation this would mean a resonance at drive and ␻ Ϸ␯Ϯ␻ two weaker resonances at drive rf . To lowest order, the presence of micromotion will then just rescale ϩ the coupling strength by a factor 1/(1 qx/2) for a drive ␯ ␻ ϭ␯ frequency close to . For drive there are two stationary terms in Eq. (139) and two rotating at 2␯.Ifwe neglect the nonstationary terms (rotating-wave approximation) HI becomes time independent and can easily be integrated, yielding the evolution operator FIG. 13. Rabi oscillations for a coherent state (Meekhof et al., 1996). The data (dots) are displayed together with a fit to a ͑ ͒ϭ ͓͑⍀ ͒ †Ϫ͑⍀ ͒ ͔ϭ ͑⍀ ͒ U t exp dt aˆ d*t aˆ D dt , (140) sum of number states having a coherent-state population distribution (line). The fitted value for the mean quantum num- with ber is ¯nϭ3.1Ϯ0.1. The inset shows the amplitudes of the 1 ϪeE ប number-state components (bars) with a fit to a Poisson distri- ⍀ ϭ 0ͱ i␸ ϭ Ϯ d ϩ ប ␯e . (141) bution, corresponding to ¯n 2.9 0.1 (line). 1 qx/2 2 2m

The drive coherently displaces an initial motional state ⍀2 ͉⌿͘ to ϭប 0,s ⌬ ͑⌬ ϩ⌬␻ ϩ⌬␸͒ Fdip ⌬ k sin kx t (144) drive ͉⌿Ј͘ϭDˆ ͑⍀ t͉͒⌿͘. (142) d ⌬␻ ⍀ Ӷ⌬ for ( , 0,s) drive . If the oscillation amplitude of the The coherent displacement is proportional to the time ion remains in a small enough range, ⌬k͗x͘Ӷ1, we can the drive is applied and the amplitude of the driving approximate sin(⌬k͗x͘ϩ⌬␻tϩ⌬␸)Ϸsin(⌬␻tϩ˜␸) for a field. Experimentally the drive was calibrated by apply- suitable ˜␸. Then the dipole force is of the form dis- ing it to the motional ground state with a certain voltage cussed in the context of Eq. (139) and will result in a amplitude and fitting the internal-state evolution on the coherent displacement of the motional wave function blue sideband Pg(t) to the expected signal of a coherent that grows linearly with the coupling time. Note that the ͉␣͘ ͉␣͉2ϭ state (see below) with the magnitude ¯n as the range where this assumption is valid coincides with the only free parameter. definition of the Lamb-Dicke regime, so this drive will In the ‘‘moving standing-wave’’ approach, two laser closely approximate a coherent drive while ␩2¯nӶ1. ⌬␻ beams with a frequency difference of and detuned By an appropriate choice of the internal levels and ⌬ ϷϪ → by about drive 12 GHz from the S1/2 P1/2 transition ϩ beam polarization, the dipole force can even create a in Be were used. The detuning was much bigger than state-dependent mechanical drive that allows one to en- ⌫ the linewidth of the P1/2 state, so this state was popu- tangle the internal states of the ion with the motion. As lated with only an extremely small probability, an example we shall discuss the level scheme used in ϩ ⍀ 2 Be in the NIST experiment. Here the ͉g͘ state was р ͩ 0,s ͪ ϭ ϭϪ Pp 4 ⌬ , (143) chosen to be the (F 2,mF 2) hyperfine substate and drive ͉ ϭ ϭϪ 2 e͘ was (F 1,mF 1) of the S1/2 manifold. The light ⍀ ϭ ͦ͗ ͉ ͉ ͦ͘Õប ␴Ϫ Ϫ where 0,s 2 s eϪ(r•E0) P1/2 is the on-resonance fields were polarized and detuned 12 GHz from ͉ ͘ 2 Rabi frequency of an ion in the hyperfine state s and is the P1/2 level. This resulted in a driving dipole force for driven by one of the equally strong beams E1,2 ͉e͘ with the (1,Ϫ1) state coupling to the (2,Ϫ2) of the ϭ Ϫ␻ ϩ␸ 2 Ϫ E0 cos(k1,2x 1,2t 1,2), in complete analogy to the P1/2 manifold, but there exists no (3, 3) substate in Appendix. At the position of the ion the interference of this manifold that could couple to ͉g͘. This level there- the two off-resonant light fields resulted in a time- fore remains unaffected by the drive. The dependence of ϭ ϩ 2 varying light-field intensity I ͗(E1 E2) ͘ with a cross the drive on the internal state was crucial for entangling ⌬␻ϭ␻ Ϫ␻ term modulated at 1 2 (here the averaging internal and motional states as described in Sec. VI.A.4. ͗•••͘ ͉␻ brackets denote averaging over a time T with 1 As in the predictions of the Jaynes-Cummings model, Ϫ␻ ͉Ӷ Ӷ␻ 2 1/T 1,2). The other components are time inde- the internal-state evolution Pg(t) driven on the blue pendent and will not lead to any consequences for the sideband, Eq. (95), will undergo collapses and revivals ion’s motion. The difference frequency part leads to a (von Foerster, 1975; Eberly et al., 1980), a purely quan- spatially and time-varying intensity with a corresponding tum effect (Shore and Knight, 1993). Figure 13 shows an ͉ ͘ ac Stark shift of level s that can resonantly drive the example of Pg(t) after the creation of a coherent state ⌬␻ϭ␻ Ϫ␻ ϭ␯ motion if 1 2 . The dipole force on the ion of motion. Similar behavior has been observed in the is proportional to the spatial derivative of this ac Stark context of cavity QED (Brune, Schmidt-Kaler, et al., shift, 1996). The inset of Fig. 13 shows the probabilities of the

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 310 Leibfried et al.: Quantum dynamics of single trapped ions

levels could therefore no longer be well distinguished in frequency, so it was no longer possible to extract the level populations by Fourier transform of Pg(t).

4. ‘‘Schro¨dinger-cat’’ states of motion In Schro¨ dinger’s original thought experiment (Schro¨ - dinger, 1935) he describes how one could, in principle, transform a superposition inside an atom to a large-scale FIG. 14. Rabi oscillations for a squeezed state (Meekhof et al., superposition of a live and dead cat by coupling cat and 1996). The data (dots) are displayed together with a fit to a atom with the help of a ‘‘diabolical mechanism.’’ Subse- sum of number states having a squeezed-state population dis- ¨ tribution with ␤ ϭ40Ϯ10, which corresponds to ¯nϷ7.1. quently the term ‘‘Schrodinger’s cat’’ developed a more s general meaning, first referring to superposition states of macroscopic systems (the poor cat would be in such a number components, extracted by Fourier transforma- state in the original paper) and later, especially in the tion of the signal. The probabilities of different motional context of quantum optics, referring to a superposition levels display the expected Poissonian distribution over of two coherent states ͉␣͘ and ͉␤͘ (see Sec. II.C.2) with a n (see Sec. II.C.2). In the experiment the observed re- separation in phase space ͉␣Ϫ␤͉ much larger than the vival for higher-¯n coherent states was attenuated due to variance of a coherent state. Such a state would allow progressively faster decay rates of the higher-n number one to distinguish the positions of the two coherent con- states (see Sec. VI.A.1); for states with ¯nտ7 no revival tributions by a position measurement with very high re- was observed. liability, and they are macroscopic in the sense that their maximum spatial separation is much larger than the 3. Creation of squeezed states single-component wave-packet extension. In an experiment of the NIST group, an analogous A ‘‘vacuum squeezed state’’ of motion can be created state for a single trapped ion was engineered (Monroe by a parametric drive at 2␯ (Janszky and Yushin, 1986; et al., 1996). Out of an equal superposition of internal Yi and Zaidi, 1988; Heinzen and Wineland, 1990), by a states (͉e͘ and ͉g͘) a combined state of the motional combination of standing- and traveling-wave laser fields and internal degrees of freedom of the form ͉g͉͘␣͘ (Cirac, Blatt, and Zoller, 1994), or by a nonadiabatic ϩ͉e͉͘␣ei␾͘ was created in which the two coherent mo- change in the trap spring constant (Janszky and Yushin, tional components had the same amplitude but different 1986; Yi and Zaidi, 1988; Heinzen and Wineland, 1990). phases. The motional components of the superposition In the experiment of the NIST group the ion was cooled were indeed separated in space by distances much to the ground state and then irradiated with two Raman greater than their respective wave-packet spread for beams that differed in frequency by 2␯, driving Raman suitable phases ␾. transitions between the even-n levels within the same This situation is interesting from the point of view of hyperfine state. In analogy to the coherent drive de- the quantum measurement problem associated with scribed in Sec. VI.A.2 the interaction can be thought of wave-function collapse, which was historically debated as a parametric drive induced by an optical dipole force by Einstein and Bohr, among others. One practical ap- ␯ ␤ modulated at 2 . The squeeze parameter s (defined as proach toward resolving this controversy is the introduc- the factor by which the variance of the squeezed quadra- tion of quantum decoherence, or the environmentally ture is decreased) grows exponentially with the driving induced reduction of quantum superpositions into statis- time. tical mixtures and classical behavior (Zurek, 1991, 2001). Figure 14 shows Pg(t) for a squeezed state prepared Decoherence is commonly interpreted as a way of quan- by the NIST group in this way. The data were fitted to a tifying the elusive boundary between classical and quan- ␤ vacuum squeezed-state distribution, allowing only s to tum worlds, and almost always precludes the existence vary. The fit of the data in Fig. 14 is consistent with a of macroscopic ‘‘Schro¨ dinger-cat’’ states, except at ex- ␤ ϭ Ϯ squeezed state with a squeeze parameter s 40 10 tremely short-time scales. The creation of mesoscopic (corresponding to a noise level 16 dB below the zero- Scho¨ dinger-cat states in controlled environments allows point variance in the squeezed quadrature component). one to do controlled studies of quantum decoherence For this squeezing parameter, the average motional and the quantum/classical boundary (Brune, Hagley, quantum number is ¯nϷ7.1. et al., 1996; Myatt et al., 2000; Turchette, Myatt, et al., The probability distribution for a vacuum squeezed 2000). Some of this work is described in Sec. VII. state is restricted to the even states, as can be seen from The creation of Schro¨ dinger-cat states of a single ion ␤ ϭ Eq. (53). This distribution is very flat; with s 40, 16% in the experiments cited above relied on a sequence of of the population is in states above nϭ 20 (see Sec. laser pulses. The coherent states were excited with the II.C.3). The NIST experiment was conducted with a use of a pair of Raman laser beams. The key to the Lamb-Dicke parameter of 0.202. The Rabi-frequency experiment was that the displacement beams were both differences of these high-n levels were small [see Fig. polarized ␴ϩ, so that they did not affect the ͉e͘ internal 11(b)] and began to decrease with n after nϭ20. The state (see Sec. VI.A.2). It is this selectivity that allowed

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 Leibfried et al.: Quantum dynamics of single trapped ions 311

FIG. 15. Steps for creation of a Schro¨ dinger-cat state (Monroe et al., 1996). For detailed explanation, see text.

the components of a superposition of internal states to be associated to different motional states. The evolving state of the system during the sequence is summarized in Fig. 15. Following (a) laser cooling to the ͉g͉͘nϭ0͘ state, the Schro¨ dinger-cat state was created by applying several sequential pulses of the Raman beams: (b) A ␲/2 pulse on the carrier transition split the wave function into an equal superposition of states ͉g͉͘0͘ and ͉e͉͘0͘. (c) The selective dipole force of the coherent displacement beams excited the motion correlated with the ͉e͘ component to a state ͉␣͘. (d) A ␲ pulse on the carrier transition then swapped the internal states of the superposition. (e) Next, the displacement beams excited the motion correlated with the new ͉e͘ component to a second coherent state ͉␣ei␾͘. (f) A final ␲/2 pulse on the carrier combined the two coherent states. The relative phases [␾ and the phases of steps (b), (d), and (f)] of the steps above were controlled by phase locking the rf sources that created the frequency splitting of the Raman or displacement beams, respectively. The state created after step (e) is a superposition of two independent coherent states, each correlated with FIG. 16. Phase signal of a Schro¨ dinger-cat state for different an internal state of the ion (for ␾ϭ␲), magnitudes of the coherent displacement (Monroe et al., ͉␣͉͘e͘ϩ͉Ϫ␣͉͘g͘ 1996). ͉⌿͘ϭ . (145) & or decoherence. The experiment was continuously In this state, the widely separated coherent states re- repeated—cooling, state preparation, detection—while place the classical notions of ‘‘dead’’ and ‘‘alive’’ in slowly sweeping the relative motional phase ␾ of the Schro¨ dinger’s original thought experiment. The coher- coherent states. ␾ ence of this mesoscopic superposition was verified by Figure 16 shows the measured Pg( ) for a few differ- recombining the coherent wave-packet components in ent values of the coherent-state amplitude ␣, which is set the final step (f). This resulted in different degrees of by changing the duration of application of the displace- interference of the two wave packets as the relative ment beams [steps (c) and (e) from above]. The unit phase ␾ of the displacement forces [steps (c) and (e)] visibility (CӍ1) of the interference feature near ␾ϭ0 was varied. The nature of the interference depended on verifies that superposition states were produced instead the phases of steps (b), (d), and (f) and was set to cause of statistical mixtures, and the feature clearly narrows as destructive interference of the wave packets in the ͉g͘ ␣ increases. The amplitude of the Schro¨ dinger-cat state state. The interference was directly measured by detect- was extracted by fitting the interference data to the ex- ␾ ͉ ing the probability Pg( ) that the ion was in the g͘ pected form of the interference fringe. The extracted internal state for a given value of ␾. The signal for par- values of ␣ agreed with an independent calibration of ticular choices of the phases in (b), (d), and (f) is the displacement forces. Coherent-state amplitudes as high as ␣Ӎ2.97(6) were measured, corresponding to an 1 2 P ͑␾͒ϭ ͓1ϪCeϪ␣ (1Ϫcos ␾) cos͑␣2 sin ␾͔͒, (146) average of ¯nӍ9 vibrational quanta in the state of mo- g 2 tion. This indicates a maximum spatial separation of ␣ ϭ where ␣ is the magnitude of the coherent states and C 4 x0 83(3) nm, which was significantly larger than the ϭ ϭ1 is the expected visibility of the fringes in the absence single wave-packet size characterized by x0 7.1(1) nm of any mechanisms decreasing the contrast, such as, for as well as a typical atomic dimension (Ӎ0.1 nm). The example, imperfect state preparation, motional heating, individual wave packets were thus clearly spatially sepa-

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 312 Leibfried et al.: Quantum dynamics of single trapped ions rated and also separated in phase space. state and subsequently measuring the occupation of the Of particular interest is the fact that as the separation motional ground state with a method involving adiabatic of the cat state is made larger, the decay from superpo- passage (Cirac, Blatt, and Zoller, 1994). D’Helon and sition to statistical mixture accelerates. In the experi- Milburn (1996) proposed to tomographically reconstruct ment, decoherence due to coupling to a thermal reser- the Wigner function of the vibrational state by measur- voir is expected to result in the loss of visibility in the ing the electronic state population inversion after apply- Ϫ␣2␭ interference pattern of Cϭe t, where ␭ is the cou- ing a standing-wave pulse to the ion that must reside at pling constant and t the coupling time. This exponential a node of the standing wave. Bardroff et al. (1996) pro- reduction of coherence with the square of the separation posed using a technique they called quantum-state en- (␣2 term) is thought to be the basic reason that bigger doscopy, exploiting the explicit time dependence of the ‘‘cats’’ decay faster (Zurek, 1991). In Fig. 16(d), the ob- rf trapping potential. The technique relies on driving the served loss of contrast at the largest observed separation ion on combined secular and micromotion sidebands already indicates the onset of decoherence. and monitoring the internal-state evolution. Leibfried The precise control of quantum wave packets in the et al. (1996) proposed and experimentally realized two ion-trap version of Schro¨ dinger’s cat provides a very sen- techniques that reconstruct either the density matrix in sitive indicator of quantum decoherence, whose charac- the number-state basis or the Wigner function with the terization is of great interest to quantum measurement help of coherent displacements and monitoring of the theory and applications such as quantum computing internal-state evolution on the blue sideband, similar to (Ekert and Josza, 1996) and quantum cryptography the number-state population measurements described (Bennett, 1996). It was employed in a series of experi- above. These techniques and the experimental results ments that are described in Sec. VII. will be discussed in more detail below. Lutterbach and Davidovich (1997) proposed a very elegant method to 5. Arbitrary states of motion directly measure the Wigner function from the inversion of the electronic states after a coherent displacement of The methods presented above for creating a the original motional state and a driving pulse of appro- ¨ Schrodinger-cat state can be generalized to arbitrary priate length on the blue sideband. Freyberger (1997) states of motion. The general strategy for creating such describes a scheme for measuring the motional density states was first described by Law and Eberly (1996) in matrix in the number-state basis, which, like the pro- the context of cavity QED and later generalized to su- posal of Poyatos, Walser, et al. (1996), relies on coherent perpositions of internal states and arbitrary motional displacement and a filtering measurement to determine states of a trapped ion by Gardiner et al. (1997) and the population in the motional ground state. Finally, Kneer and Law (1998). No experiments to realize these Bardroff et al. (1999) proposed a scheme to measure the proposals have been reported so far. characteristic function of the motional state by applying internal-state changing and coherent driving light pulses, B. Full determination of the quantum state of motion very similar to the procedure for preparing a Schro¨ - dinger cat described in Sec. VI.A.4. The controlled interaction of light and rf electromag- The two reconstruction methods described by Leib- netic fields with trapped ions allows one not only to pre- fried et al. (1996) were specially developed to fit the pare very general states of motion, but also to com- technical possibilities and constraints of the NIST appa- pletely determine these quantum-mechanical states ratus and could therefore be experimentally realized. using novel techniques. The ability to prepare a variety We shall restrict the more detailed discussion to these of nonclassical input states in ion traps that can, for ex- methods. The first method reconstructs the density ma- ample, exhibit negative values of the Wigner function trix in the number-state basis, while the second yields makes state reconstruction in ion traps an attractive the Wigner function of the motional state of a single goal. In addition, comparing the results of the state de- trapped atom. termination with the intended state allows one to quan- Both measurement techniques rely on the ability to itfy the accuracy of preparation and detrimental effects displace the input state to several different locations in such as, for example, decoherence and motional heating. phase space. Specifically, a coherent displacement (see The first proposal for motional-state reconstruction in Sec. VI.A.2) of the form U(Ϫ␣)ϭU†(␣)ϭexp(␣*a ion traps was put forward by Wallentowitz and Vogel Ϫ␣a†)(Ϫ␣ simplifies the notation below) controlled in (1995). Their method encodes the information on the phase and amplitude is used in those schemes. In a man- motional state in the ground-state occupation of an ner similar to the characterization of motional-state atom driven on the first red and blue sidebands simulta- populations described in the sections above, radiation neously. A Fourier integral transform of the measured on the blue sideband is then applied to the atom to ex- ␣ data then yields the motional density matrix in a gener- tract the signal Pg(t, ) (notice that this signal will vary alized position representation. Poyatos, Walser, et al. for different displacements). The internal state at tϭ0is (1996) proposed to measure either the Husimi Q func- always prepared to be ͉g͘ for the various motional input tion or the quadrature distribution function P(x,␪)by states, so, according to Eq. (95), modified for the rates of applying a suitable sequence of phase shifts, coherent loss of contrast for different number states (see above), displacements, and squeezing operations to the motional the signal averaged over many measurements is

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 Leibfried et al.: Quantum dynamics of single trapped ions 313

ϱ NϪ1 1 Ϫ␥ 1 Ϫ ␲ ͑ ␣͒ϭ ͭ ϩ ͚ ͑␣͒ ͑⍀ ͒ ktͮ (l)ϵ ͚ ͑␣ ͒ il( /N)p Pg t, 1 Qk cos k,kϩ1t e , Qk Qk p e 2 kϭ0 2N pϭϪN (147) ϱ ␣ ϭ ␥(l)␳ where Qk( ) denotes the population of the coherently ͚ kn n,nϩl , (151) displaced number state U(␣)͉k͘. For no coherent dis- nϭmax(0,Ϫl) placement the signal, Eq. (95), is recovered and contains with matrix elements information about the populations of the number states 2 ϩ eϪ͉␣͉ ͉␣͉2k min(k,n) min(k,l n) only. But since these measurements are repeated for sev- ␥(l)ϭ ͚ ͚ ͉␣͉2(nϪjϪjЈ)ϩl kn ϭ eral magnitudes and phases of the coherent displace- k! jЈϭ0 j 0 ment, information can be extracted about the coherence ͱ͑ ϩ ͒ Ϫ Ϫ Ј k k l n !n! properties of the state, especially the off-diagonal ele- ϫ͑Ϫ1͒ j j ͩ ͪͩ ͪ (152) ments of the density matrix—or the Wigner function can j jЈ ͑lϩnϪj͒!͑nϪjЈ͒!

be reconstructed from the measured displaced popula- for every diagonal ␳ ϩ of the density matrix. To keep ␣ n,n l tions Qk( ). the matrix dimension ﬁnite, a cutoff for the maximum n in Eq. (151) is introduced, based on the magnitude of the input state. For an unknown input state, an upper bound on n could be extracted from the populations ␣ 1. Reconstruction of the number-state density matrix Qk( ). If these are negligible for k higher than a certain ␣ kmax and all displacements , they are negligible in the As outlined above, the readily measurable quantities input state as well, and it is convenient to truncate Eq. ϭ in an experiment are the number-state populations (151) at nmax kmax . The resulting matrix equation is ␣ ␳ Qk( ) of the coherently displaced unknown initial state overcomplete for some l, but the diagonals n,nϩl can characterized by the density matrix ␳. The quantities still be reconstructed by a general linear least-squares ␣ ␳ Qk( ) are related to by method (Press et al., 1992). ͑␣͒ϭ ͉ †͑␣͒␳ ͑␣͉͒ ϭ ␣ ͉␳͉␣ Qk ͗k U U k͘ ͗ ,k ,k͘, (148) ͉␣ ͘ where ,k is a coherently displaced number state 2. Reconstruction of s-parametrized quasiprobability ␣ (Moya-Cessa and Knight 1993). Hence every Qk( )is distributions the population of the displaced number state ͉␣,k͘ for an ensemble characterized by the input density matrix ␳. As pointed out by several authors, all s-parametrized Rewriting Eq. (148) in terms of number-state density- quasiprobability distributions F(␣,s) have a particularly ␳ matrix elements n,m yields simple representation when expressed in populations of displaced number states Q (␣) (Royer, 1984; Moya- 1 k Q ͑␣͒ϭ ͗0͉akU†͑␣͒␳U͑␣͒͑a†͒k͉0͘ Cessa and Knight, 1993; Banaszek and Wodkiewicz, k k! 1996; Wallentowitz and Vogel, 1996): ϱ n 1 1 n ϭ ͗␣͉͑aϪ␣͒k␳͑a†Ϫ␣*͒k͉␣͘ ͑␣ ͒ϭ ͓͑ ϩ ͒ ͔n ͑Ϫ ͒kͩ ͪ ͑␣͒ F ,s ͚ s 1 /2 ͚ 1 Qk . k! ␲ nϭ0 kϭ0 k Ϫ͉␣͉2 ϱ k Ϫ Ϫ Ј (153) e ͉␣͉2k ͑␣*͒n j␣m j ϭ ͚ ͚ For sϭϪ1 the sum breaks down to one term and ϭ k! n,m 0 j,jЈϭ0 n!m! ␣ Ϫ ϭ ␣ ␲ F( , 1) Q0( )/ gives the value of the Husimi Q quasiprobability distribution at the complex coordinate Ϫ Ϫ Ј k ϫ͑Ϫ1͒ j j ͩ ͪ ␣ j . The reconstruction scheme of Poyatos, Walser, et al. (1996) is based on the Husimi Q distribution. For sϭ0 k ␣ ϭ ␣ ␣ ϫͩ ͪͱ͑ ϩ ͒ ͑ ϩ Ј͒ ␳ the Wigner function F( ,0) W( ) for every point in m j ! n j ! ϩ ϩ . (149) jЈ n jЈ,m j the complex plane can be determined by the single sum ϱ To separate the contributions of different matrix ele- 2 ␳ ͑␣͒ϭ ͚ ͑Ϫ ͒n ͑␣͒ ments n,m , one can choose the coherent displacements W 1 Qn . (154) ␲ nϭ0 to lie on a circle in phase space, ␣ ϭ͉␣͉ ͓ ͑␲ ͒ ͔ In the reconstruction performed by the NIST group, the p exp i /N p , (150) sum was carried out only to a ﬁnite nmax , as described where p෈͕ϪN,...,NϪ1͖. The number of angles 2N above. Since truncation of the sum leads to artifacts in on that circle determines the maximum number of states the quasiprobability distributions (Collett, 1996), the ex- ϭ Ϫ nmax N 1 that can be included in the reconstruction. perimental data were averaged over different nmax . This ␣ With a full set of measured populations Qk( p) of the smoothes out the artifacts to a high degree. state displaced along 2N points on a circle, a discrete In contrast to the density-matrix method described in Fourier transform of Eq. (149), evaluated at the values Sec. VI.B.1, summing the displaced probabilities with ␣ p , results in the following matrix equations: their weighting factors provides a direct and local

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 314 Leibfried et al.: Quantum dynamics of single trapped ions method of obtaining the quasiprobability distribution at the point ␣ in phase space, without the need to measure at other values of ␣. This also distinguishes the method from experiments that determines the Wigner function by inversion of integral equations (tomography) (see, for example, Smithey et al., 1993 and Dunn et al., 1995). Finally, for a known density matrix the Wigner function can be derived by expanding Eq. (154) in the number-state basis, ϱ ϱ 2 ͑␣͒ϭ ͑Ϫ ͒n ␣ ͉ ␳ ͉␣ W ͚ 1 ͚ ͗ ,n k͘ kl͗l ,n͘, (155) ␲ nϭ0 k,lϭ0 with the matrix elements given by (lуn) (Cahill and Glauber, 1969) ͗ ͉␣ ͘ϭͱ ␣lϪn Ϫ1/2͉␣͉2L (lϪn)͉͑␣͉2͒ l ,n n!/l! e n , (156) L (lϪn) where n is a generalized Laguerre polynomial. Us- FIG. 17. Surface and contour plots of the reconstructed ing this approach a plot of the Wigner function using Wigner function W(␣) of an approximate nϭ1 number state reconstructed density-matrix data may be created (Leib- (Leibfried et al., 1996). The negative values of W(␣) around fried, Meekhof, et al., 1998). the origin highlight the nonclassical nature of this state.

3. Experimental state reconstruction measured phases per radius. The white contour represents W(␣)ϭ0. The negative values around the origin In the experiments of the NIST group (Leibfried highlight the nonclassical character of this state. The et al., 1996; Leibfried, Monroe, and Pfau, 1998), the co- ␣ herent displacement needed for the reconstruction map- measured Wigner function W( ) was rotationally sym- ping was provided by a spatially uniform classical driv- metric within experimental errors. The minimum of the ing field at the trap oscillation frequency ␯ (see Sec. large negative part of the Wigner function around the Ϫ VI.A.2). The field was applied on one of the trap com- origin has a value of 0.25, close to the largest negative pensation electrodes for a time of about 10 ␮s. The rf value possible for a Wigner function in the chosen oscillators that created and displaced the states were phase-space coordinates of Ϫ1/␲. This highlights the phase locked to control their relative phase. Different fact that the prepared input state is very nonclassical. displacements were realized by varying the amplitude In contrast to the number state, the state closest to a and the phase of the displacement oscillator. For every classical state of motion in a harmonic oscillator is a ␣ ␣ ␣ displacement , Pg(t, ) was recorded. Qn( ) was then coherent state. The reconstruction of the number-state extracted from the measured traces by a numerical density matrix of a small coherent state with amplitude singular-valued decomposition method (Press et al., ͉␤͉Ϸ0.67 is depicted in Fig. 18. The reconstructed off- 1992). To determine the amplitude ͉␣͉ of each displace- diagonal elements are generally slightly smaller than ment, the same driving field was applied to the ͉nϭ0͘ those expected from the theory of a pure coherent state. ground state and the resulting collapse and revival trace In part this is attributed to decoherence during the mea- (see Sec. VI.A.2) were fitted to that of a coherent state surement, so the reconstruction shows a mixed-state with the amplitude ␣ as the only variable. character rather than a pure coherent-state signature. The accuracy of the reconstruction was limited by the The reconstructed Wigner function of a second coher- uncertainty in the applied displacements, the errors in ent state with amplitude ͉␤͉Ϸ1.5 is shown in Fig. 19. The the determination of the displaced populations, motional heating, and decoherence during the measurement. The value of the Wigner function was found by the sum, Eq. (154), with simple error propagation rules. The density matrix was reconstructed by a linear least- squares method, and the error was estimated with the help of a covariance matrix (Press et al., 1992). As the size of the input state increases, state preparation and the relative accuracy of the displacements become more critical, thereby increasing their uncertainties. Surface and contour plots of the Wigner function of an approximate ͉nϭ1͘ number state are shown in Fig. ␳ ⌰ 17. The plotted surface is the result of fitting a linear FIG. 18. Experimental amplitudes nm and phases nm of the interpolation between the actual data points to a 0.1 number-state density-matrix elements of an ͉␣͉Ϸ0.67 coherent ϫ0.1 grid. The octagonal shape is an artifact of the eight state (Leibfried et al., 1996).

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 Leibfried et al.: Quantum dynamics of single trapped ions 315

FIG. 19. Surface and contour plots of the reconstructed ␣ Wigner function W( ) of an approximate coherent state FIG. 21. Reconstructed density-matrix amplitudes of a ther- (Leibfried et al., 1996). The approximately Gaussian mal distribution of states after imperfect Doppler cooling (¯n minimum-uncertainty wave packet is centered around a coher- Ϸ1.3). As one would expect for a thermal distribution, no ent amplitude of about 1.5 from the origin. The half width at coherences are present within experimental uncertainties and half maximum is about 0.6, in accordance with the minimum- ͱ the populations drop exponentially with n (Leibfried et al., uncertainty half width of (1/2)ln(2)Ϸ0.59. 1996).

plotted surface is the result of ﬁtting a linear interpola- The NIST group also created and reconstructed a co- tion between the actual data points to a 0.13ϫ0.13 grid. herent superposition of ͉nϭ0͘ and ͉nϭ2͘ number The approximately Gaussian minimum-uncertainty states. This state is ideally suited to demonstrate the sen- wave packet is centered around a coherent amplitude of sitivity of the reconstruction to coherences. The only ␳ ␳ about 1.5 from the origin. The half width at half maxi- nonzero off-diagonal elements should be 02 and 20 , ͉␳ ͉ϭ͉␳ ͉ϭͱ␳ ␳ Ϸ mum is about 0.6, in accordance with the minimum- with a magnitude of 02 20 00 22 0.5 for a su- uncertainty half width of ͱln(2)/2Ϸ0.59 in the chosen perposition with about equal probability of being mea- ͉ ϭ ͘ ͉ ϭ ͘ phase-space coordinates. To suppress truncation arti- sured in the n 0 or n 2 state. In the reconstruction shown in Fig. 20 the populations ␳ and ␳ are some- facts in the Wigner function summation (154), the data 00 22 ϭ ϭ what smaller, due to imperfections in the preparation, were averaged over nmax 5 and nmax 6. ͉␳ ͉ϭ͉␳ ͉ but the coherence has the expected value of 20 02 Ϸͱ␳ ␳ 00 22. Finally, a thermal distribution was generated by only Doppler cooling the ion. The reconstruction of the resulting state is depicted in Fig. 21. As expected, there are no coherences, and the diagonal, which gives the number-state occupation, shows an exponential behavior within the experimental errors, indicating a mean occupation number ¯nϷ1.3.

VII. QUANTUM DECOHERENCE IN THE MOTION OF A SINGLE ATOM

Decoherence in quantum systems (for an overview see Zurek, 2001, and references therein) has been a subject of enduring interest because it relates to the fundamental distinction between quantum and classical behavior. More recently, quantum decoherence has come to the forefront because it is the most signiﬁcant ob- FIG. 20. Reconstructed density-matrix amplitudes of an ap- stacle for applications in quantum information science proximate (1/&)(͉0͘Ϫi͉2͘) number-state superposition. The (DiVincenzo, 2001). We shall review the decoherence amplitudes of the coherences indicate that the reconstructed studies of motional quantum states of a single harmoni- density matrix is close to that of a pure state (Leibfried et al., cally bound atom. Since this problem is closely related 1996). to decoherence of superposition states of a single mode

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of the electromagnetic field, a large body of theoretical It is sometimes useful to incorporate a quantum mea- research in quantum optics can be applied.6 suring device or quantum ‘‘meter’’ into the scheme As was briefly discussed in Sec. VI.A.5, arbitrary above using a von Neumann chain (Zurek, 1991). Here states of motion of a single atom can be prepared (in it is assumed that the quantum system, initially given by ͉␺ ϩ ͉␺ principle) using a variety of techniques. Following the c1 1͘ c2 2͘, is first coupled to a quantum meter and discussion in Sec. VI.A.4, the motional and internal the combination is then coupled to the environment. In states can be manipulated with coherent radiation, gen- the first stage of coupling, erating operations of the form ͑ ͉␺ ϩ ͉␺ ͉␺ ͉␾ c1 1͘ c2 2͘) M͘ e͘ ͉ ͉͘ ͘→ ␪͉ ͉͘ ͘ϩ i␾ ␪͉ ͉͘ Ј͘ g n cos g n e sin e n . (157) →͑ ͉␺Ј͉͘␺ ͘ϩ ͉␺Ј͉͘␺ ͘ ͉␾ ͘ c1 1 M1 c2 2 M2 ) e . (159) This operation faithfully maps the motional state under Upon coupling to the environment, the evolution is study to the internal state, which can subsequently be ͑ ͉␺Ј͉͘␺ ͘ϩ ͉␺Ј͉͘␺ ͘ ͉␾ ͘ measured with high accuracy. Before this mapping/ c1 1 M1 c2 2 M2 ) e measurement step, superpositions of motional states can →c ͉␺Љ͉͘␺Ј ͉͘␾ ͘ϩc ͉␺Љ͉͘␺Ј ͉͘␾ ͘, (160) decay naturally due to ambient fluctuating fields. Apply- 1 1 M1 e1 2 2 M2 e2 ␾ ͉␾ Ӎ ing larger fields that dominate over the ambient level and, as before, if ͗ e2 e1͘ 0 and the final environmen- improves experimental control and increases the data tal states are not measured, then the final state of the rate. More importantly, applying different types of deco- system and meter is expressed by the density matrix hering fields before mapping/measurement allows the ͉c ͉2͉␺Љ͗͘␺Љ͉ ͉␺Ј ͗͘␺Ј ͉ϩ͉c ͉2͉␺Љ͗͘␺Љ͉ ͉␺Ј ͗͘␺Ј ͉. study of a more general class of reservoir couplings. 1 1 1 M1 M1 2 2 2 M2 M2 (161) Thus the correlation between the system and meter states is established, yielding the expected classical re- A. Decoherence background sult, but the quantum coherence is lost. Including the quantum meter more closely describes the ion experi- Consider the decoherence of a superposition state ments discussed below because the system (the ion’s mo- ͉␺ ϩ ͉␺ ͉␺ ͉␺ tion) is not directly measured but is instead coupled to c1 1͘ c2 2͘, where 1͘ and 2͘ are states of a single mode of motion. This mode of motion will be taken as the ion’s internal state, which acts as the meter. the quantum ‘‘system’’ under investigation. In a basic model of decoherence (Zurek, 1991, 2001), it is assumed B. Decoherence reservoirs that the mode couples to an external environment having initial state ͉␾ ͘. The total initial state of the motion e Below, several simple decoherence reservoirs are dis- and environment is thus ␺ ϭ(c ͉␺ ͘ϩc ͉␺ ͘) ͉␾ ͘. 0 1 1 2 2 e cussed, and experimental demonstrations of decoher- Due to the coupling between the system and the envi- ence in these reservoirs are summarized. Decoherence ronment, the joint state evolves to of harmonic-oscillator superposition states into a variety ␺ ϭ͑ ͉␺ ϩ ͉␺ ͉␾ →␺ 0 c1 1͘ c2 2͘) e͘ final of reservoir environments has been investigated extensively in theory; see, for example, Caldeira and Leggett ϭ ͉␺Ј͉͘␾ ͘ϩ ͉␺Ј͉͘␾ ͘ c1 1 e1 c2 2 e2 . (158) (1985), Walls and Milburn (1985, 1995), Collett (1988), To illustrate how this causes decoherence, we assume Zurek (1991), Buzˇek and Knight (1995), and Poyatos, ␨ ͉␺Ј͘ϭ i 1͉␺ ͘ ͉␺Ј͘ Cirac, and Zoller (1996). Typically a system harmonic the coupling is such that 1 e 1 and 2 ␨ ϭ i 2͉␺ ␺ oscillator is coupled to a bath of environmental quantum e 2͘; then, after the interaction, final ␨ Ϫ␨ ϭ ͉␺ ͉␾ ϩ i( 2 1) ͉␺ ͉␾ oscillators, with particular couplings determined by the c1 1͘ e1͘ e c2 2͘ e2͘, with the initial states of the system correlated with different states of type of reservoir. the environment. For strong coupling, the environmen- ␾ ͉␾ Ӎ tal states will typically be distinct, with ͗ e2 e1͘ 0. 1. High-temperature amplitude reservoir Now if the final environmental states are uncontrolled The interaction Hamiltonian for amplitude damping is and unmeasured, these degrees of freedom must be ignored or mathematically traced over. In this case, ϱ ␺ ϭ͑ប ͒ ⌫ †ϩ the pure state final becomes a statistical mixture HI /2 ͚ iabi H.c., (162) ␳ ϭ͉ ͉2͉␺ ␺ ͉ iϭ0 expressed by the density matrix final c1 1͗͘ 1 ϩ͉ ͉2͉␺ ␺ ͉ c2 2͗͘ 2 . Hence the off-diagonal or coherence where a is the lowering operator for the system oscilla- ͉␺ ͘ † terms of the pure-state density matrix (c1 1 tor, bi is the raising operator for the ith environmental ϩ ͉␺ ͘ ͗␺ ͉ϩ ͗␺ ͉ ⌫ c2 2 )(c1* 1 c2* 2 ) are lost to the environment. oscillator, and i gives the strength of coupling between the system oscillator and the ith environmental oscillator. The case in which the system oscillator is a single 6See, for example, Caldeira and Leggett, 1985; Walls and Mil- mode of the electromagnetic field has been investigated burn, 1985; Collett, 1988; Zurek, 1991; Vogel and Welsch, 1994; in the experiments of Brune, Hagley, et al. (1996). This Buzˇek and Knight, 1995; Poyatos, Cirac, and Zoller, 1996; model and similar ones specific to trapped-ion experi- Schleich, 2001. ments are discussed theoretically (Murao and Knight,

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FIG. 22. Phase-space representation of interference experiment (Ramsey, 1963) for coherent-state superpositions coupled to an amplitude reservoir (from Myatt et al., 2000 and Turchette, Myatt, et al., 2000). Starting from the initial system/ meter state ͉␣ϭ0͉͘g͘, the Schro¨ dinger-cat state shown in panel (2) is created by applying a ␲/2 pulse on the internal FIG. 23. Loss of coherence of a Schro¨ dinger-cat state caused state followed by a state-dependent optical dipole force. Noise by coupling to an amplitude reservoir (from Myatt et al., 2000). is then applied to the motion simulating coupling to a hot re- Amplitude noise was applied for a fixed duration of 3 ␮s but sistor. This causes a random displacement ␤ as shown in panel with varying amplitude. The fringe contrast for all experiments (3). The steps used to create the cat state are then reversed, ␾ ␲ is normalized to that observed in the absence of applied noise. but with a phase shift R on the final /2 pulse. Finally, the ͉ ͘ The scaling shows that the decoherence rate is proportional to probability Pg of finding the ion in the g internal state is ⌬␣ ϭ 1 ͕ Ϫ ͓␾ the square of the phase-space separation of the superposi- measured. Ideally, this results in Pg 2 1 cos R ϩ ␤ ⌬␣ ͔ ␾ tion components. 2Im( * ) ͖, having the sinusoidal oscillations with R characteristic of Ramsey interferometry. Noise reduces the Ramsey fringe contrast because ␤ must be averaged over a After preparation, the superposition was exposed to distribution of values. For Gaussian noise with variance ␴␤ , the fluctuating field for a fixed time [Fig. 22, part (3)]. 2 2 this yields a contrast of exp(Ϫ4␴␤͉⌬␣͉ ). The components of the motional-state superposition were then recombined by reversing the steps that cre- 1998; Schneider and Milburn, 1998, 1999; Wineland ated it [Fig. 22, part (4)] and a final ␲/2 pulse was ap- et al., 1998; Bonifacio et al., 2000). One model assumes plied to the internal states. Finally, the internal state was that the motion of a trapped ion couples to the measured. This sequence was repeated many times for ␾ (noisy) uniform electric field E caused by the environ- various values of the relative phase R between the cre- mental oscillators through the potential U ation and reversal steps. The contrast of the resulting ϭϪZ͉e͉x"E, where Z͉e͉ is the ion’s charge and x its interference fringe characterizes the amount of coher- displacement from the equilibrium position. Such a ence remaining in the final state. The results are dis- model corresponds to the case of a noisy electric field played in Fig. 23. The contrast of the fringe decays as Ϫ␬͉␣Ϫ␣Ј͉2͗ 2͘ ͱ͗ 2͘ due to a resistor coupled between the trap electrodes exp( En ), where En is the root-mean- (Wineland and Dehmelt, 1975b; Wineland et al., 1998) square value of the applied noise and ␬ is a constant. and is described by the Hamiltonian in Eq. (162). Similar scaling was also observed for the ambient fluc- In the experiments, a hot resistor is simulated by ap- tuating fields after exposure for varying times (Myatt plying a random uniform electric field between the elec- et al., 2000; Turchette, Myatt, et al., 2000). The exponen- trodes that has some spectral amplitude at the ion’s mo- tial dependence of the decoherence rate on the size of tional frequency ␯ (Myatt et al., 2000; Turchette, Myatt, the cat state ⌬␣ϵ͉␣Ϫ␣Ј͉ agrees with theoretical predic- et al., 2000). This is achieved using a commercial functions and indicates why it is difficult to preserve super- tion generator producing pseudorandom voltages that is positions on a macroscopic scale. connected through a bandpass filter to one of the trap The experimental situation here is slightly different electrodes. The initial coherence and subsequent deco- from the theoretical models outlined above. First, the herence of the motional-state superpositions were mea- environment acted only on the system (motional state) ͉␺Ј ͘ sured using single-atom interferometry, as described in and not the meter (internal state), so that M1 ϭ͉␺ ͘ ͉␺Ј ͘ϭ͉␺ ͘ detail by Myatt et al. (2000) and Turchette, Myatt, et al. M1 and M2 M2 in Eq. (160). In addition, in (2000). this case the action of the system on the environment is ͉␾ Ϸ͉␾ Ϸ͉␾ Briefly, an initial state of the type shown in the left- nearly negligible, so that e1͘ e2͘ e͘. Because ͉␺Ј͘ hand side of Eq. (160) was first created, with 1 the noise voltage En in this experiment could, in prin- ϭ͉␣͘ ͉␺ ͘ϭ͉ ͘ ͉␺Ј͘ϭ͉␣Ј͘ ͉␺ ͘ϭ͉ ͘ ϭ ͉␾ , M1 e , 2 , M2 g , and c1 c2 ciple, be measured classically without disturbing e͘ ap- ϭ1/&. This is illustrated schematically in part (2) of Fig. preciably one could subsequently apply an operation to 22. Here ͉␣͘ and ͉␣Ј͘ are coherent states of amplitudes ␣ the ion that reverses the environmental effect. In prac- and ␣Ј. tice this could be readily done for the engineered noise

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 318 Leibfried et al.: Quantum dynamics of single trapped ions

gestion by Poyatos, Cirac, and Zoller (1996). The technique relies on laser cooling, as illustrated in Fig. 24. Coherent Raman beams drive the ͉g͉͘n͘↔͉e͉͘nϪ1͘ transition with Rabi rate ⍀. At the same time, an optical pumping beam causes spontaneous Raman transitions from ͉e͉͘n͘ to ͉g͉͘n͘ at rate ␥. From the diagram in Fig. 24, it is clear that all populations tend towards the state ͉g͉͘0͘, which is characteristic of a Tϭ0 reservoir. By varying the intensities of the lasers, the reservoir parameters can be experimentally controlled. In these experiments, the evolution of the initial state ␺ϭ 1/& (͉0͘ϩ͉2͘)͉g͘ was observed for varying lengths FIG. 24. Technique for implementing a zero-temperature en- of reservoir interaction time. The coherence was mea- vironment (Myatt et al., 2000). A pair of laser beams drives the sured using an interference experiment similar to that of ͉g͉͘n͘↔͉e͉͘nϪ1͘ transition at rate ⍀, while spontaneous Ra- the previous section. However, here the internal state man scattering is used to make transitions ͉e͉͘n͘ to ͉g͉͘n͘ at serves several functions: it allows the preparation of the rate ␥. When ␥ӷ⍀, the internal states together with the spon- motional superposition and is the quantity ultimately ϭ taneous events act as a T 0 reservoir for motional states. measured, but also, in the middle of any particular experiment, it acts as part of the environment due to the but not for the ambient noise. Without performing such coupling of Fig. 24. The resulting data are shown in Fig. a measurement, however, it is necessary to average over 25, where each point is the contrast of the interference ͉␾ the possible environmental states ͕ e͖͘, and this leads fringes after interacting with the reservoir for the given to the decoherence exhibited in Fig. 23. time. Two cases are shown, ␥Ͼ⍀ and ␥Ͻ⍀. In the ﬁrst case, the coherence simply decays due to coupling to the 2. Zero-temperature amplitude reservoir reservoir. Here it is natural to take the reservoir to be In the cavity QED decoherence experiments of the internal state plus the rest of the environment, which Brune, Hagley, et al. (1996), the system oscillator fre- includes the spontaneously scattered photons. The initial quency was around 50 GHz and the ambient tempera- nonexponential decay is a manifestation of the quantum ture was TϷ0.6 K, so the quantum number of the equi- Zeno effect (Myatt et al., 2000; Turchette, Myatt, et al., librium oscillator was around 0.05 and the reservoir 2000) and arises because the condition ␥ӷ⍀ is not rig- temperature was effectively near zero. In the ion experi- orously satisﬁed. In contrast, when ␥Ͻ⍀ the coherence ments, the oscillator frequency was around 5 MHz and between the ͉0͘ and ͉2͘ states disappears and reappears the ambient temperature was 300 K or greater, implying over time, with an overall decay of the fringe contrast. a very large equilibrium oscillator number (Turchette, The underlying effect is population transfer back and Kielpinski, et al., 2000). Nevertheless, it is possible to forth (Rabi oscillation) between the states ͉g͉͘2͘ and simulate a Tϭ0 reservoir for the ions, following a sug- ͉e͉͘1͘. Indeed, for ␥→0, near-perfect revival of the

FIG. 25. Decoherence of the motional superposition state 1/& (͉0͘ϩ͉2͘) coupled to the engineered zero-temperature reservoir (from Myatt et al., 2000 and Turchette, Myatt, et al., 2000). The superposition is prepared with a sequence of laser pulses and the reservoir applied for a variable time. The preparation pulses are then reversed with a variable phase shift, and the ﬁnal state is recorded as a function of this phase. The data show the contrast of the resulting interference pattern for two cases of the relative values of ⍀ and ␥.The initial contrast is not unity due to imperfections in the state preparation and reversal pulses.

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FIG. 26. Loss of coherence of various Fock- state superpositions caused by coupling to a phase reservoir (from Myatt et al., 2000). Phase noise was applied for a ﬁxed duration of 20 ␮s but with varying amplitude. The fringe contrast for all experiments is normalized to that observed in the absence of applied noise. The scaling shows that the decoherence rate is proportional to the square of the Fock-state number ⌬n of the superposition components.

fringe contrast is obtained (Turchette, Myatt, et al., seems that decoherence is needed only to describe situ- 2000), since in this case the environment is restricted to ations in which, for practical or technical reasons, infor- just the internal states and the apparent decoherence is mation pertaining to the overall system is lost. These easily reversed. A scheme for observing a similar rever- limitations seem only to be practical and not fundamen- sal in the context of cavity QED is discussed in Raimond tal unless some mechanism that is so far missing in quan- et al. (1997). tum mechanics is found to cause intrinsic decoherence. In the zero-temperature experiment with ␥Ͼ0, once For a summary of such alternatives, see Leggett (1999). the atom scatters a photon through the spontaneous Ra- man process, no measurement can be made even in prin- 3. High-temperature phase reservoir ciple to restore the initial superposition. That is, the The interaction Hamiltonian for a phase damping is emission (and subsequent absorption by a measuring ap- paratus or the environment) of a spontaneous photon ϱ ϭ͑ប ͒ † ⌫ †ϩ projects the atom into a definite state (͉g͉͘1͘ in the con- HI /2 aa ͚ ib H.c. (163) ϭ i text of Fig. 24) and phase information is irreversibly lost. i 0 This situation is identical to that of the cavity experi- This interaction does not change the energy of the sys- ments of Brune, Hagley, et al. (1996). The decoherence tem oscillator and can be considered a model for in this second ion experiment can be contrasted with the quantum-nondemolition measurements (Walls and Mil- first, in which the environment could, in principle, be burn, 1985). measured and its effects reversed. The data for ␥Ͻ⍀ do In the experiments, a phase reservoir is realized by illustrate how coherence lost to the environment can be modulating the trap frequency, thus advancing (or re- recovered, and an alternative explanation states that tarding) the phase of harmonic motion in the trap. transferring the ͉g͉͘2͘ component of the superposition Gaussian noise is symmetrically applied to the trap elec- to the ͉e͉͘1͘ state provides ‘‘which-path’’ information in trodes to produce a noisy electric-field gradient, and the the interferometer—the paths being the ͉g͉͘0͘ and noise is uniform up to a cutoff frequency well below the ͉g͉͘2͘ parts of the superposition. The oscillation in trap frequency so that no energy is transferred to the ion which-path information is analogous to that seen in the motion. experiments of Chapman et al. (1995), Du¨ rr et al. (1998), Motional decoherence caused by this phase noise is Bertet et al. (2001), and Kokorowski et al. (2001). most clearly demonstrated in a superposition of Fock ͉ ϩ ͉ The zero-temperature experiment also illustrates a states of the form c1 n͘ c2 nЈ͘. Similar techniques to fundamental dilemma in explaining decoherence. If the that described above were used to characterize phase environment is restricted to be the internal state by tak- decoherence in a variety of Fock-state superpositions ing ␥→0, then the coherence information can be recov- (Myatt et al., 2000; Turchette, Myatt, et al., 2000), and ered. Even in the case of spontaneous emission, coher- the results are plotted in Fig. 26. In analogy to the case ence need not be lost if the photon is emitted into a of amplitude damping, we find that the decoherence rate high-quality cavity from which it can later be recovered, scales with the square of the distance between the super- as has been observed in the cavity QED experiments position’s constituents, here meaning the squared differ- (Maıˆtre et al., 1997; Varcoe et al., 2000). Therefore it ence in Fock-state indices.

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C. Ambient decoherence in ion traps ing the electrodes from direct deposition, one can achieve a decrease in heating rate by a factor on the The ambient motional decoherence observed in all ex- order of 100 (Rowe et al., 2002). In any case, we conjec- periments can be characterized by high-temperature am- ture that for clean metal electrode surfaces, free from plitude damping: its characteristics are the same as those oxides or adsorbed gases that could support mobile elec- caused by thermal electronic noise in the resistance of trons, the heating should approach that predicted by Ϫ the electrodes or resistance coupled to the electrodes. thermal electronic noise ͗dn/dt͘Ӎ1s 1. At the relatively low ion oscillation frequencies, where the characteristic wavelength is much larger than the VIII. CONCLUSIONS electrode spacing, the ambient decoherence is ad- equately described by thermal (Johnson) noise associ- Single ions confined in rf traps offer two nearly ideal ated with lumped circuit elements attached to the elec- basic quantum systems, a two-level system, represented trodes, or equivalently, thermally fluctuating dipole by two of the internal electronic states, and an approxi- oscillators in the electrode bulk (Turchette, Kielpinski, mate harmonic oscillator, represented by the motion. et al., 2000). With appropriate light fields these two subsystems can Typical heating rates (expressed as quanta per second be coupled in a number of ways, leading to a wealth of from the motional ground state) are observed to be possible studies. In this review the underlying theory Ӎ 3Ϫ 4 Ϫ1 ␯ Ӎ and a number of experiments utilizing this system have around ͗dn/dt͘ 10 10 s for z 10 MHz, and the distance from the ion to the nearest electrode surface is been discussed, with special emphasis on laser cooling, around 150 ␮m (Turchette, Kielpinski, et al., 2000). resonance fluorescence, quantum-state engineering, However, given the estimated electrode resistance quantum-state reconstruction, and motional decoher- (Wineland et al., 1998) and attached circuit elements, an ence of single ions. electrode temperature of 106 K or greater is needed to This system will continue to be studied and, very explain most of the heating results (Turchette, Kielpin- likely in the future, much more complicated superposi- ski, et al., 2000). The principle cause of the anomalously tion states will be realized. Although an important goal large heating is not understood at this time, but some of of ion-trap experiments is to realize arbitrary entangled its characteristics have been determined. The fluctuating states for many ions, as required in quantum computing, field has no sharp spectral features in the range from 2 for example, the single-ion experiments discussed here to 20 MHz, and it seems to be emanating from the elec- will continue to be the test bed for studies of operation trodes themselves. More local-field sources such as col- fidelity and decoherence. lisions with background gas or free electrons are ruled out because these collisional sources would heat the in- ACKNOWLEDGMENTS ternal modes of two ions at nearly the same rate as the center-of-mass modes. However, the internal mode heat- We want to thank James Bergquist, John Bollinger, ing is observed to be negligible compared to the heating Ignacio Cirac, Ju¨ rgen Eschner, Wayne Itano, Steven Jef- of the center-of-mass modes (King et al., 1998), indicat- ferts, David Kielpinski, Brian King, Christopher Langer, ing that the fields at the site of the ions are approxi- Dawn Meekhof, Volker Meyer, Giovanna Morigi, Chris- mately uniform spatially. [We note that the experiments topher Myatt, Hanns-Christoph Na¨gerl, Christian Roos, of Rohde et al. (2001) disagree with this finding.] In con- Mary Rowe, Cass Sackett, Ferdinand Schmidt-Kaler, trast, a more distant field source, such as electrical noise Quentin Turchette, Christopher Wood, and Peter Zoller from laboratory equipment, is contradicted by observa- for their cooperation and many stimulating discussions. tions that the heating rate depends sensitively on the Work at the University of Innsbruck was supported by trap size R, scaling typically as RϪ␬ for ␬ on the order of the Austrian Fonds zur Fo¨ rderung der wissenschaftli- 5. Thermal noise arising from circuit resistance should chen Forschung (Grant No. SFB15 and START Grant scale as ␬ϭ2 (Wineland et al., 1998), but noise from No. Y147-PHY), by the European Commission [TMR fluctuating patch potentials on the electrode surfaces is networks ‘‘Quantum Information’’ (Grant No. ERB- predicted to scale as ␬Ϸ4 (Turchette, Kielpinski, et al., FRMX-CT96-0087) and ‘‘Quantum Structures’’ (Grant 2000). The scatter of the data for 9Beϩ ions is rather No. ERB-FMRX-CT96-0077)], and by the Institut fu¨ r large but is consistent with the patch potential model. Quanteninformation GmbH. Work at NIST was sup- Data on other traps (Diedrich et al., 1989; Rohde et al., ported by NSA/ARDA, ONR, ARO, and NRO. Work 2001) also appear to be consistent with this value of ␬. at Michigan was supported by NSA/ARDA, ARO, and Previous experiments (Turchette, Kielpinski, et al., NSF. 2000) have given some indication that beryllium deposition onto the electrodes causes a higher heating rate. APPENDIX: COUPLINGS OF LIGHT FIELDS TO THE Such deposition occurs because these ions are created INTERNAL ELECTRONIC STATE by ionization neutral beryllium atoms that pass near the center of the trap after being emitted from a wide-angle The coupling of electromagnetic fields to charges is a source. In this process, some of the atoms from the complex subject. It has been extensively studied, for ex- source miss the trap and are deposited on the electrodes. ample, by Cohen-Tannoudji et al. (1989). Deriving the Preliminary evidence indicates that, by physically mask- relevant interaction Hamiltonians HI from first prin-

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ciples is beyond the scope of this review, so we shall 1 1 q ϭ ͩ x x Ϫ ␦ x2 ͪ . (A8) restrict ourselves to stating these Hamiltonians and list- ij 2 i j 3 ij ing their properties as far as they are necessary to describe the atom-field couplings studied here. In all cases For a plane-wave electric field like Eq. (A1) the Hamil- the electromagnetic field(s) will not be quantized, but tonian simplifies to treated as a classical plane-wave field of the form 1 ϭ ͑ ͓͒ i(kxϪ␻t)ϩ ͔ ͑ ͒ϭ ͓ i(kxϪ␻t)ϩ ͔ HQ eϪkx E0 x ie c.c. (A9) E x,t E0 e c.c. , (A1) 2 •

with the real field amplitude E0 . We shall assume (a) and the matrix element is ប␻Ϸ Ϫ ͉ that Ee Eg so all electronic states except g͘ and ͉ ͘ 1 e can be neglected and (b) that all ac Stark shifts, rep- ͗g͉H ͉e͘ϭ eϪk͗g͉x͑E x͉͒e͘ ͉ ͉ ϭ Q 0• resented by the diagonal elements ͗j HI j͘, j ͕g,e͖, are 2 ϭ lumped into the definitions of Ej , namely, Ej Ej0 ϫ͓ i(kxϪ␻tϩ␲/2)ϩ ͔ ϩ ͉ ͉ e c.c. . (A10) ͗j HI j͘, where Ej0 is the energy of level j in absence of the coupling. We can then expand HI in the remain- Comparison of this equation with Eq. (62) yields ing off-diagonal terms, ͑ប ͒⍀ϭ ͑ ͒͗ ͉ ͑ ͉͒ ͘ /2 eϪ k/2 g x E0•x e (A11) ϭ͉͑ ͗͘ ͉ϩ͉ ͗͘ ͉͒͗ ͉ ͉ ͘ HI g e e g g HI e , (A2) for quadrupole transitions, where the phase factor ␲/2 is where we have chosen a convention in which the matrix lumped into ␾ in Eq. (62). Since the quadrupole inter- ͉ ͉ element ͗g HI e͘ is real. action is an even function of position, only matrix ele- In the remainder of this appendix we shall study these ments between states of the same parity differ from matrix elements for the types of transitions used in the zero. Again the actual numerical value of the matrix el- described experiments. Doppler cooling and fluores- ement depends on the angular momentum values of ͉g͘ cence experiments relied on dipole transitions. Experi- and ͉e͘ and the field polarization [see, for example, ments with resolved sidebands relied on the excitation James (1998)]. To relate the order of magnitude of quad- of quadrupole transitions or stimulated Raman transi- rupole transition matrix elements to those of the more tions between two long-lived states. familiar dipole transitions, one can deduce from Eqs. (A10) and (A5) that their approximate ratio is a0k Ӎ Ϫ3 Ϫ4 10 – 10 , where a0 is the Bohr radius, so quadru- 1. Dipole coupling pole transitions have a much weaker decay and higher For dipole coupling to a single outer-shell electron the saturation intensity when driven by a laser source. interaction Hamiltonian is ϭ ͑ ͒ HD eϪx•E x,t , (A3) with the electron charge eϪ . For a plane-wave electric 3. Raman coupling field of the form of Eq. (A1) this becomes An alternative way to create an effective two-level ϭ ͓ i(kxϪ␻t)ϩ ͔ HD eϪx•E0 e c.c. (A4) system is to couple two ground-state levels by two- and the matrix element is photon stimulated Raman transitions (Heinzen and Wineland, 1990; Monroe et al., 1995). The Raman tran- ͗ ͉ ͉ ͘ϭ ͗ ͉͑ ͉͒ ͓͘ i(kxϪ␻t)ϩ ͔ g HD e eϪ g E0•x e e c.c. . (A5) sitions are induced by two light fields whose frequency Comparison of this equation with Eq. (62) yields difference matches the separation of the two ground- state levels (plus the relatively small detunings to, for ͑ប ͒⍀ϭ ͗ ͉͑ ͉͒ ͘ /2 eϪ g E0•x e (A6) example, a sideband). Each beam is close to resonance for dipole transitions. Since the dipole interaction is an with an allowed dipole transition to a short-lived excited odd function of position, only matrix elements between electronic state ͉3͘ but sufficiently detuned to make states of opposite parity differ from zero. The actual nu- population of that state negligible. While the coupling is merical value of the matrix element depends on the an- enhanced, the near-resonant excited state can be adia- gular momentum values of ͉g͘ and ͉e͘ and the field po- batically eliminated in the theoretical treatment (Wine- larization. Details on this can be found, for example, in land et al., 1998), leaving an effective two-level coupling James (1998). between the two ground states. The coupling is formally equivalent to a narrow single-photon transition if one makes the following identifications: 2. Quadrupole coupling ␻↔␻ Ϫ␻ 1 2 , For quadrupole coupling to a single outer-shell elec- ↔⌬ ϭ Ϫ tron the interaction Hamiltonian is k k k1 k2 . (A12) ␻ ␻ Here 1 ,k1 ( 2 ,k2) are the frequency and wave vector ץ ϭ ͉ ͉ ͉ Ej , (A7) of the light fields coupling e͘ ( g͘)to 3͘. If both fields ץ HQ ͚ eϪqij i,j xi ⌬ are detuned from resonance by R , the coupling with the quadrupole tensor strength is given by

Rev. Mod. Phys., Vol. 75, No. 1, January 2003 322 Leibfried et al.: Quantum dynamics of single trapped ions

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