166 MANIPULATING ATOMS WITH PHOTONS Nobel Lecture, December 8, 1997 by Ciaupe N. Conen-Taxnounyt Collége de France et Laboratoire Kastler Brossel* de I’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France Electromagnetic interactions play a central role in low energy physics. They are responsible for the cohesion of atoms and molecules and they are at the origin of the emission and absorption of light by such systems. This light is not only a source of information on the structure of atoms. It can also be used to act on atoms, to manipulate them, to control their various degrees of freedom. With the development of laser sources, this research field has con- siderably expanded during the last few years. Methods have been developed to trap atoms and to cool them to very low temperatures. This has opened the way to a wealth of new investigations and applications. Two types of degrees of freedom can be considered for an atom: the in- ternal degrees of freedom, such as the electronic configuration or the spin polarization, in the center of mass system; the external degrees of freedom, which are essentially the position and the momentum of the center of mass. The manipulation of internal degrees of freedom goes back to optical pump- ing [1], which uses resonant exchanges of angular momentum between atoms and circularly polarized light for polarizing the spins of these atoms. These experiments predate the use of lasers in atomic physics. The manipu- lation of external degrees of freedom uses the concept of radiative forces re- sulting from the exchanges of linear momentum between atoms and light. Radiative forces exerted by the light coming from the sun were already in- voked by J. Kepler to explain the tails of the comets. Although they are very small when one uses ordinary light sources, these forces were also investiga- ted experimentally in the beginning of this century by P. Lebedev, E. F. Nichols and G. F. Hull, R. Frisch. For a historical survey of this research field, we refer the reader to review papers [2, 3, 4, 5) which also include a discus- sion of early theoretical work dealing with these problems by the groups of A. P. Kazantsey, V. S. Letokhoy, in Russia, A. Ashkin at Bell Labs, S. Stenholm in Helsinki. It turns out that there is a strong interplay between the dynamics of inter- nal and external degrees of freedom. This is at the origin of efficient laser cooling mechanisms, such as “Sisyphus cooling” or “Velocity Selective Cohe- * Laboratoire Kastler Brossel is a Laboratoire associé au CNRS et a l'Université Pierre et Marie Curie,167 rent Population Trapping”, which were discovered at the end of the 80's (for a historical survey of these developments, see for example [6]). These me- chanisms have allowed laser cooling to overcome important fundamental limits, such as the Doppler limit and the single photon recoil limit, and to reach the microKelvin, and even the nanoKelvin range. We devote a large part of this paper (sections 2 and 3) to the discussion of these mechanisms and to the description of a few applications investigated by our group in Paris (section 4). There is a certain continuity between these recent developments in the field of laser cooling and trapping and early theoretical and experi- mental work performed in the 60’s and the 70's, dealing with internal de- grees of freedom. We illustrate this continuity by presenting in section 1 a brief survey of various physical processes, and by interpreting them in terms of two parameters, the radiative broadening and the light shift of the atomic ground state. 1, BRIEF REVIEW OF PHYSICAL PROCESSES: To classify the basic physical processes which are used for manipulating atoms by light, it is useful to distinguish two large categories of effects: dissipative (or absorptive) effects on the one hand, reactive (or dispersive) effects on the other hand. This partition is relevant for both internal and external degrees of freedom. 1.1 EXISTENCE OF TWO TYPES OF EFFECTS IN ATOM-PHOTON INTERAC- TIONS Consider first a light beam with frequency @, propagating through a medium consisting of atoms with resonance frequency @,. The index of refraction de- scribing this propagation has an imaginary part and a real part which are as- sociated with two types of physical processes. The incident photons can be ab- sorbed, more precisely scattered in all directions. The corresponding attenuation of the light beam is maximum at resonance. It is described by the imaginary part of the index of refraction which varies with @,-@, as a Loren absorption curve. We will call such an effect a dissipative (or absorptive) ef fect. The speed of propagation of light is also modified. The corresponding dispersion is described by the real part n of the index of refraction whose dif- ference from 1, n-I, varies with @,-«, as a Lorentz dispersion curve. We will call such an effect a reactive (or dispersive) effect. Dissipative effects and reactive effects also appear for the atoms, as a result of their interaction with photons. They correspond to a broadening and toa shift of the atomic energy levels, respectively. Such effects already appear when the atom interacts with the quantized radiation field in the vacuum state. It is well known that atomic excited states get a natural width I, which is also the rate at which a photon is spontaneously emitted from such states. Atomic energy levels are also shifted as a result of virtual emissions and reab- sorptions of photons by the atom. Such a radiative correction is simply the Lamb shift [7].168 Physics 1997 Similar effects are associated with the interaction with an incident light beam. Atomic ground states get a radiative broadening I’, which is also the rate at which photons are absorbed by the atom, or more precisely scattered from the incident beam. Atomic energy levels are also shifted as a result of virtual absorptions and reemissions of the incident photons by the atom. Such energy displacements iA’ are called light shifts, or ac Stark shifts [8, 9]. In view of their importance for the following discussions, we give now a brief derivation of the expressions of I” and A’, using the so-called dressed- atom approach to atom-photon interactions (see for example [10], chapter VI). In the absence of coupling, the two dressed states |g, N) (atom in the ground state gin the presence of N photons) and | ¢ N-J) (atom in the exci- ted state ¢ in the presence of N-J photons) are separated by a splitting #4, where 6 = @,-@, is the detuning between the light frequency @, and the atomic frequency @,. The atom-light interaction Hamiltonian V,, couples these two states because the atom in gcan absorb one photon and jump to ¢ The corresponding matrix element of V,, can be written as AQ/2, where the so-called Rabi frequency Q is proportional to the transition dipole moment and to VN. Under the effect of such a coupling, the two states repel each other, and the state | g, N) is shifted by an amount fA’, which is the light shift of g. The contamination of |g, N) by the unstable state | ¢ N-J) (having a width P) also confers to the ground state a width I’, In the limit where QI’. Such a dependence of 7,,, on the Rabi frequen-Claude N. Cohen-Tannoudji 175 cy 2 and on the detuning 6 has been checked experimentally with cesium atoms [40]. Fig.3 presents the variations of the measured temperature Twith the dimensionless parameter Q2/T[6|. Measurements of T'versus intensity for different values of 6 show that T depends linearly, for low enough intensities, ona single parameter which is the light shift of the ground state Zeeman sub- levels. 0 05 U otair Figure 8. Temperature measurements in cesium optical molasses (from reference [40]). The left part of the figure shows the fluorescence light emitted by the molasses observed through a ‘window of the vacuum chamber. The horizontal bright line is the fluorescence light emitted by the atomic beam which feeds the molasses and which is slowed down by a frequency chirped laser beam. Right part of the figure : temperature of the atoms measured by a time of flight tech- nique versus the dimensionless parameter 2/|5|T proportional to the light shift (Q is the op- tical Rabi frequency, Sthe detuning and I” the natural width of the excited state) 2.2 THE LIMITS OF SISYPHUS COOLING At low intensity, the light shift U, « #2/5is much smaller than #I. This ex- plains why Sisyphus cooling leads to temperatures much lower than those achievable with Doppler cooling. One cannot however decrease indefinitely the laser intensity. The previous discussion ignores the recoil due to the spontaneously emitted photons which increases the kinetic energy of the atom by an amount on the order of Ey, where Ep = hPk?/2M (3) is the recoil energy of an atom absorbing or emitting a single photon. When U, becomes on the order or smaller than Ep, the cooling due to Sisyphus cooling becomes weaker than the heating due to the recoil, and Sisyphus cooling no longer works. This shows that the lowest temperatures which can176 Physics 1997 be achieved with such a scheme are on the order of a few Ep/ky This result is confirmed by a full quantum theory of Sisyphus cooling [41, 42] and is in good agreement with experimental results. The minimum temperature in Fig.3 is on the order of LOE p/kg 2.3 OPTICAL LATTICES For the optimal conditions of Sisyphus cooling, atoms become so cold that they get trapped in the quantum vibrational levels of a potential well (see Fig. 4). More precisely, one must consider energy bands in this perodic struc ture [43]. Experimental observation of such a quantization of atomic motion Probe transmission Figure 4. Probe absorption spectrum of a LD optical lattice (from reference (44]). The upper ‘part of the figure shows the two counterpropagating laser beams with frequency @ and orthogo- nal linear polarizations forming the 1D-optical lattice, and the probe beam with frequency @, whose absorption is measured by a detector. The lower part of the figure shows the probe trans- mission versus @,-@. The two lateral resonances corresponding to amplification or absorption of the probe are due to stimulated Raman processes between vibrational levels of the atoms trapped in the light field (see the two insets). The central narrow structure is a Rayleigh line due to the antiferromagnetic spatial order of the atoms.Claude N. Cohen-Tannoudji 177 in an optical potential was first achieved in one dimension [44] [45]. Atoms are trapped in a spatial periodic array of potential wells, called a “1D-optical lattice”, with an antiferromagnetic order, since two adjacent potential wells correspond to opposite spin polarizations. 2D and 3D optical lattices have been realized subsequently (see the review papers [46] [47]). 3. SUBRECOIL LASER COOLING 3.1 THE SINGLE PHOTON RECOIL LIMIT. HOW TO CIRCUMVENT IT In most laser cooling schemes, fluorescence cycles never cease. Since the ran- dom recoil ik communicated to the atom by the spontaneously emitted pho- tons cannot be controlled, it seems impossible to reduce the atomic momen- tum spread &p below a value corresponding to the photon momentum fik. Condition 5p = ik defines the “single photon recoil limit”. It is usual in laser cooling to define an effective temperature Tin terms of the half-width 6p (at 1/V¢) of the momentum distribution by k,7/2= 6p2/2M. In the temperature scale, condition 5p = fk defines a “recoil temperature” Ty by: kpT; PK? “y= on = ER “ The value of Tp ranges from a few hundred nanoKelvin for alkalis to a few microKelvin for helium. 0 v Figure 5. Subrecoil laser cooling. The random walk of the atom in velocity space is supposed to be characterized by a jump rate Rwhich vanishes for v= 0 (a). As a result of this inhomogeneous random walk, atoms which fall in a small interval around v = 0 remain trapped there for a long time, on the order of (R(v)}~!, and accumulate (b). It is in fact possible to circumvent this limit and to reach temperatures T lower than T,, a regime which is called “subrecoil” laser cooling. The basic idea is to create a situation where the photon absorption rate I’, which is also the jump rate Rof the atomic random walk in velocity space, depends on the atomic velocity v = p/M and vanishes for v= 0 (Fig.5a). Consider then an atom with v = 0. For such an atom, the absorption of light is quenched. Consequently, there is no spontaneous reemission and no associated random recoil. One protects in this way ultracold atoms (with v ~0) from the “bad” ef- fects of the light. On the other hand, atoms with v# 0 can absorb and remit light. In such absorption-spontaneous emission cycles, their velocities change in a random way and the corresponding random walk in vspace can transfer178 Physics 1997 atoms from the v# 0 absorbing states into the v~ 0 dark states where they re- main trapped and accumulate (see Fig. 5b). This reminds us of what hap- pens in a Kundt’s tube where sand grains vibrate in an acoustic standing wave and accumulate at the nodes of this wave where they no longer move. Note however that the random walk takes place in velocity space for the situ- ation considered in Fig. 5b, whereas it takes place in position space in a Kundt’s tube. Up to now, two subrecoil cooling schemes have been proposed and de- monstrated. In the first one, called “Velocity Selective Coherent Population Trapping” (VSCPT), the vanishing of R(v) for v = 0 is achieved by using destructive quantum interference between different absorption amplitudes [48]. The second one, called Raman cooling, uses appropriate sequences of stimulated Raman and optical pumping pulses for tailoring the appropriate shape of R(v) [49]. 3.2 BRIEF SURVEY OF VSCPT We first recall the principle of the quenching of absorption by “coherent po- pulation trapping”, an effect which was discovered and studied in 1976 [50, 51]. Consider the 34evel system of Fig.6, with two ground state sublevels g, and g and one excited sublevel ¢,, driven by two laser fields with frequencies ),, and ®,», exciting the transitions g, © 4 and g, ¢ é, respectively. Let iA be the detuning from resonance for the stimulated Raman process consisting of the absorption of one @,, photon and the stimulated emission of one @» photon, the atom going from g, to g,. One observes that the fluorescence 2] Figure 6. Coherent population trapping. A three-level atom g;, f, is driven by owo laser fields with frequences @,, and 0, exciting the transitions g, ¢ and g, + 4 respectively. hd is the de~ tuning from resonance for the stimulated Raman process induced between g, and g, by the two laser fields «@,, and @,, When 4= 0, atoms are optically pumped in a linear superposition of g, and g, which no longer absorbs light because of a destructive interference between the two ab- sorption amplitudes g, —> 4 and g~> é,Claude N. Cohen-Tannoudji 179 rate R vanishes for A = 0. Plotted versus A, the variations of R are similar to those of Fig.5a with v replaced by A. The interpretation of this effect is that atoms are optically pumped into a linear superposition of g, and g, which is not coupled to @ because of a destructive interference between the two ab- sorption amplitudes g, —> ¢ and g, — @. The basic idea of VSCPT is to use the Doppler effect for making the detuning A of the stimulated Raman process of Fig.6 proportional to the atomic velo- city v. The quenching of absorption by coherent population trapping is thus made velocity dependent and one achieves the situation of Fig. 5.a. This is ob- tained by taking the two laser waves @,, and @;. counterpropagating along the zaxis and by choosing their frequencies in such a way that A = 0 for an atom at rest. Then, for an atom moving with a velocity v along the zaxis, the opposite Doppler shifts of the two laser waves result in a Raman detuning A= (k, +k.) v proportional to v. A more quantitative analysis of the cooling process [52] shows that the dark state, for which R = 0, is a linear superposition of two states which differ not only by the internal state (g, or g,) but also by the momentum along the = kbp) = e1 |g, Tiki) + ¢2 192, +k) 6) This is due to the fact that g, and gy must be associated with different mo- menta, ~hk, and +iky in order to be coupled to the same excited state | @ p= 0) by absorption of photons with different momenta +/k, and ~fiky. Furthermore, when A= 0, the state (5) is a stationary state of the total atom + laser photons system. As a result of the cooling by VSCPT, the atomic mo- mentum distribution thus exhibits two sharp peaks, centered at —ik, and 600; 400: 200 °%30°30°40 0 10 2030 v (cm/s) Figure 7, One-dimensional VSCPT experiment. The left part of the figure shows the experimen- tal scheme. The cloud of precooled trapped atoms is released while the two counterpropagating VSCPT beams with orthogonal circular polarizations are applied during a time 6 = Ims. The atoms then fall freely and their positions are detected 6.5 cm below on a microchannel plate. ‘The double band pattern is a signature of the 1D cooling process which accumulates the atoms in a state which is a linear superposition of two different momenta. The right part of the figure gives the velocity distribution of the atoms detected by the microchannel plate. The width dv of the two peaks is clearly smaller than their separation 2v, where vq ~ 9.2 cm/s is the recoil velocity. This is a clear signature of subrecoil cooling,180 Physics 1997 +fiky, with a width 6p which tends to zero when the interaction time @ tends to infinity. The first VSCPT experiment [48] was performed on the 28S, metastable state of helium atoms. The two lower states g, and g, were the M=~-1 and M = +1 Zeeman sublevels of the 2°, metastable state, ¢ was the M = 0 Zeeman sublevel of the excited 25P, state. The two counterpropagating laser waves had the same frequency @, , = @,)= @, and opposite circular polariza- tions. The two peaks of the momentum distribution were centered at 2hk, with a width corresponding to T = Tp /2. The interaction time was then in- creased by starting from a cloud of trapped precooled helium atoms instead of using an atomic beam as in the first experiment [53]. This led to much lower temperatures (see Fig. 8). Very recently, temperatures as low as Tp/800 have been observed [54]. 10) os. os. 04. 02. 00+ ° ‘yy eer) ‘860 1000” 1800 2000 2800” 3000 t(us) ets) Figure 8. Measurement of the spatial correlation function of atoms cooled by VSCPT (from re- ference [54]). After a cooling period of duration @, the two VSCPT beams are switched off dur ing a dark period of duration ,, The two coherent wave packets into which atoms are pumped fly apart with a relative velocity 2v, and get separated by a distance a = 2vgf,. Reapplying the two VSCPT beams during a short probe pulse, one measures a signal S which can be shown to be equal to [1 + G(a)]/2 where G(«) is the spatial overlap between the two identical wave packets se~ arated by a. From G(a), which is the spatial correlation function of each wave packet, one de- termines the atomic momentum distribution which is the Fourier transform of G(a). The left part of the figure gives Sversus a. The right part of the figure gives T,/T versus the cooling time 8, where T, is the recoil temperature and T the temperature of the cooled atoms determined from the width of G(a). The straight line is a linear fit in agreement with the theoretical predic- tions of Lévy statistics. The lowest temperature, on the order of T,/800, is equal to 5 nK. In fact, it is not easy to measure such low temperatures by the usual time of flight techniques, because the resolution is then limited by the spatial extent of the initial atomic cloud. A new method has been developed [54] which consists of measuring directly the spatial correlation function of the atoms Gla) = Jt%_ dzp*(z + @)9(z), where 9(z) is the wave function of the atomic wave packet. This correlation function, which describes the degree of spatial coherence between two points separated by a distance a, is simply the Fourier transform of the momentum distribution | 9(f)|?. This method is analogous to Fourier spectroscopy in optics, where a narrow spectral line (#) is more easily inferred from the correlation function of the emitted electric field G(t)Claude N. Cohen-Tannoudji 181 ‘2 duh¥(t+ 7).B(2), which is the Fourier transform of 1(q). Experimentally, the measurement of G(a) is achieved by letting the two coherent VSCPT wave packets fly apart with a relative velocity 2up = 2hk/m during a dark peri- od t,, during which the VSCPT beams are switched off. During this dark pe- tiod, the two wave packets get separated by a distance a = 2up/p, and one then measures with a probe pulse a signal proportional to their overlap. Fig.8a shows the variations with 4, of such a signal S (which is in fact equal to [1 + G(a)]/2). From such a curve, one deduces a temperature T ~ Tg/625, cor- responding to 5p =hk/25. Fig.8b shows the variations of T,,/Twith the VSCPT interaction time @, As predicted by theory (see next subsection), Ty/T varies linearly with @ and can reach values as large as 800. LPS FS se (a) 300 15 Ov (emis) “15 1s v, (cms) 15 v, (emis) 1s Figure 9. Two-dimensional VSCPT experiment (from reference [58]). The experimemtal scheme is the same as in Fig, 7, but one uses now four VSCPT beams in a horizontal plane and atoms are pumped into a linear superposition of four different momentum states giving rise to four peaks in the two-dimensional velocity distribution (a). When three of the four VSCPT laser beams are adiabatically switched off, the whole atomic population is transferred into a single wave packet (b). VSCPT has been extended to two [55] and three [56] dimensions. Fora J, = 1+ J,= 1 transition, it has been shown [57] that there is a dark state which is described by the same vector field as the laser field. More precisely, if the la- ser field is formed by a linear superposition of N plane waves with wave vec- torsk, (i= 1,2,...N) having the same modulus k, one finds that atoms are cool- ed in a coherent superposition of N wave packets with mean momenta fik, and with a momentum spread 6p which becomes smaller and smaller as the imteraction time @ increases. Furthermore, because of the isomorphism between the de Broglie dark state and the laser field, one can adiabatically change the laser configuration and transfer the whole atomic population in- toa single wave packet or two coherent wave packets chosen at will [58]. Fig. 9 shows an example of such a coherent manipulation of atomic wave packets in two dimensions. In Fig.9a, one sees the transverse velocity distribution182 Physics 1997 associated with the four wave packets obtained with two pairs of counterpro- pagating laser waves along the x and y-axis in a horizontal plane; Fig.9b shows the single wave packet into which the whole atomic population is transferred by switching off adiabatically three of the four VSCPT beams. Similar results have been obtained in three dimensions. 3.3 SUBRECOIL LASER COOLING AND LEVY STATISTICS Quantum Monte Garlo simulations using the delay function [59, 60] have provided new physical insight into subrecoil laser cooling [61]. Fig.10 shows for example the random evolution of the momentum of an atom in a 1D- VSCPT experiment. Each vertical discontinuity corresponds to a spontaneous emission process during which p changes in a random way. Between two suc- cessive jumps, p remains constant. It clearly appears that the random walk of the atom in velocity space is anomalous and dominated by a few rare events whose duration is a significant fraction of the total interaction time. A simple analysis shows that the distribution P(t) of the trapping times 7 in a small trapping zone near v = 0 is a broad distribution which falls as a power-law in the wings. These wings decrease so slowly that the average value (7) of 7 (or the variance) can diverge. In such cases, the central limit theorem (CLT) can obviously no longer be used for studying the distribution of the total trapping time after N entries in the trapping zone separated by Nexits. Toor yy eo i n 3 ° Momentum (hk) ° -10 4 -20 4 . ea q _30 Time(I"*) 4 Sr -0 200000 400000 600000 Figure 10, Monte Carlo wave function simulation of one-dimensional VSCPT (from reference [61]). Momentum p characterizing the cooled atoms versus time. Each vertical discontinuity cor- responds to a spontaneous emission jump during which changes in a random way. Between two successive jumps, p remains constant. The inset shows a zoomed part of the sequence.Claude N. Cohen-Tannoudji 183 It is possible to extend the CLT to broad distributions with power-law wings [62, 63]. We have applied the corresponding statistics, called “Lévy statistics”, to subrecoil cooling and shown that one can obtain in this way a better un- derstanding of the physical processes as well as quantitative analytical predic- tions for the asymptotic properties of the cooled atoms in the limit when the interaction time 6 tends to infinity (61, 64]. For example, one predicts in this way that the temperature decreases as 1/@ when @ — ©, and that the wings of the momentum distribution decrease as 1/#2, which shows that the shape of the momentum distribution is closer to a Lorentzian than a Gaussian. This is in agreement with the experimental observations represented in Fig.8. (The fit in Fig.8a is an exponential, which is the Fourier transform of a Lorentzian). ‘One important feature revealed by this theoretical analysis is the non-ergo- dicity of the cooling process. Regardless of the interaction time @, there are always atomic evolution times (trapping times in the small zone of Fig.5a around v= 0) which can be longer than 6. Another advantage of such a new approach is that it allows the parameters of the cooling lasers to be optimized for given experimental conditions. For example, by using different shapes for the laser pulses used in one-dimensional subrecoil Raman cooling, it has been possible to reach for Cesium atoms temperatures as low as 3 nK [65]. 4, AFEW EXAMPLES OF APPLICATIONS The possibility of trapping atoms and cooling them at very low temperatures, where their velocity can be as low as a few mm/s, has opened the way to a wealth of applications. Ultracold atoms can be observed during much longer times, which is important for high resolution spectroscopy and frequency standards. They also have very long de Broglie wavelengths, which has given rise to new research fields, such as atom optics, atom interferometry and Bose-Einstein condensation of dilute gases. It is impossible to discuss here all these developments. We refer the reader to recent reviews such as [5]. We will just describe in this section a few examples of applications which have been recently investigated by our group in Paris. 4.1 CESIUM ATOMIC CLOCKS Cesium atoms cooled by Sisyphus cooling have an effective temperature on the order of 1 4K, corresponding to a rms. velocity of 1 cm/s. This allows them to spend a longer time Tin an observation zone where a microwave field induces resonant transitions between the two hyperfine levels g, and g, of the ground state. Increasing T decreases the width Av ~ 1/T of the micro- wave resonance line whose frequency is used to define the unit of time. The stability of atomic clocks can thus be considerably improved by using ultra- cold atoms [66. 67]. In usual atomic clocks, atoms from a thermal cesium beam cross two microwave cavities fed by the same oscillator. The average velocity of the atoms is several hundred m/s, the distance between the two cavities is on the184 Physics 1997 order of 1 m. The microwave resonance between g, and g, is monitored and is used to lock the frequency of the oscillator to the center of the atomic line. The narrower the resonance line, the more stable the atomic clock. In fact, the microwave resonance line exhibits Ramsey interference fringes whose width Avis determined by the time of flight Tof the atoms from one cavity to another. For the longest devices, T, which can be considered as the observation time, can reach 10 ms, leading to values of Av~ 1/T on the order of 100 Hz. Much narrower Ramsey fringes, with sub-Hertz linewidths can be obtained in the so-called “Zacharias atomic fountain” [68]. Atoms are captured in a magneto-optical trap and laser cooled before being launched upwards by a la- ser pulse through a microwave cavity. Because of gravity they are decelerated, they return and fall back, passing a second time through the cavity. Atoms therefore experience two coherent microwave pulses, when they pass through the cavity, the first time on their way up, the second time on their way down. The time interval between the two pulses can now be on the order of 1 sec, i.e, about two orders of magnitude longer than with usual clocks. Atomic fountains have been realized for sodium [69] and cesium [70]. A short-term relative frequency stability of 1.3 x 10-!°r ~'/2, where Tis the integration time, has been recently measured for a one meter high Cesium fountain [71, 72]. For T= 104s, Av/v ~ 1.3 x 10°! and for t= 3 x 104s, Av/v~ 8 x 10-'6 has been. measured, In fact such a stability is most likely limited by the Hydrogen maser which is used as a reference source and the real stability, which could be more precisely determined by beating the signals of two fountain clocks, is ex- pected to reach Av/v~ 10-16 for a one day integration time. In addition to the stability, another very important property of a frequency standard is its ac- curacy. Because of the very low velocities in a fountain device, many system- atic shifts are strongly reduced and can be evaluated with great precision. With an accuracy of 2 x 10718, the BNM-LPTF fountain is presently the most accurate primary standard [73]. A factor 10 improvement in this accuracy is expected in the near future. To increase the observation time beyond one second, a possible solution consists of building a clock for operation in a reduced gravity environment. Such a microgravity clock has been recently tested in a jet plane making parabolic flights. A resonance signal with a width of 7 Hz has been recorded in a 10-%g environment. This width is twice narrower than that produced on earth in the same apparatus. This clock prototype (see Fig.11) is a compact and transportable device which can be also used on earth for high precision frequency comparison. Atomic clocks working with ultracold atoms could not only provide an im- provement of positioning systems such as the GPS. They could be also used for fundamental studies. For example, one could build two fountains clocks, ‘one with cesium and one with rubidium, in order to measure with a high ac- curacy the ratio between the hyperfine frequencies of these two atoms. Because of relativistic corrections, the hyperfine splitting is a function of Za. where qs the fine structure constant and Zis the atomic number [74]. Since Zis not the same for cesium and rubidium, the ratio of the two hyperfineClaude N. Cohen-Tannoudji 185 Figure 11. The microgravity clock prototype. The left part is the 60 cm x 60 em x 15 cm optical bench containing the diode laser sources and the various optical components. The right part is the clock itself (about one meter long) containing the optical molasses, the microwave cavity and the detection region. structures depends on @. By making several measurements of this ratio over long periods of time, one could check Dirac’s suggestion concerning a possi- ble variation of o with time. The present upper limit for6./at in laboratory tests [74] could be improved by two orders of magnitude. Another interesting test would be to measure with a higher accuracy the gravitational red shift and the gravitational delay of an electromagnetic wave passing near a large mass (Shapiro effect [75]). 4.2 GRAVITATIONAL CAVITIES FOR NEUTRAL ATOMS We have already mentioned in section 1.3 the possibility of making atomic mirrors for atoms by using blue detuned evanescent waves at the surface of a piece of glass. Concave mirrors (Fig.12a) are particularly interesting because the transverse atomic motion is then stable if atoms are released from a point located below the focus of the mirror. It has been possible in this way to ob- serve several successive bounces of the atoms (Fig-12b) and such a system can be considered as a “trampoline for atoms” [37]. In such an experiment, it is a good approximation to consider atoms as classical particles bouncing off a concave mirror. In a quantum mechanical description of the experiment, one must consider the reflection of the atomic de Broglie waves by the mirror. Standing de Broglie waves can then be introduced for such a “gravitational cavity”, which are quite analogous to the light standing waves for a Fabry- Perot cavity [76]. By modulating at frequency 0/27 the intensity of the eva- nescent wave which forms the atomic mirror, one can produce the equivalent of a vibrating mirror for de Broglie waves. The reflected waves thus have a modulated Doppler shift. The corresponding frequency modulation186, Physics 1997 of these waves has been recently demonstrated [77] by measuring the energy change AE of the bouncing atom, which is found to be equal to 7Q, where n = 0,4 142, ... (Figs.12c and d). The discrete nature of this energy spectrum isa pure quantum effect. For classical particles bouncing off a vibrating mir- ror, AE would vary continuously in a certain range. 107 % % Number of atoms a 0 0102 03 04 05 Bouncing time (s) (a) (b) Probe absorption (arb. un.) time 06 10-15 20 (ce) (d) Figure 12. Gravitational cavity for neutral atoms (from references [37] and [77]). Trampoline for atoms (a) - Atoms released from a magneto-optical trap bounce off a concave mirror formed by ‘a blue detuned evanescent wave at the surface of a curved glass prism. Number of atoms at the initial position of the trap versus time after the trap has been switched off (b). Ten successive bounces are visible in the figure. Principle of the experiment demonstrating the frequency modulation of de Broglie waves (c). The upper trace gives the atomic trajectories (vertical posi- tion zversus time). The lower trace gives the time dependence of the intensity of the evanescent wave, The first pulse is used for making a velocity selection. The second pulse is modulated in in- tensity. This produces a vibrating mirror giving rise to a frequency modulated reflected de Broglie wave which consists in a carrier and sidebands at the modulation frequency. The energy spectrum of the reflected particles is thus discrete so that the trajectories of the reflected partic- les form a discrete set. This effect is detected by looking at the time dependence of the absorp- tion ofa probe beam located above the prism (d).Claude N. Cohen-Tannoudji 187 4.3 BLOCH OSCILLATIONS In the subrecoil regime where 6p becomes smaller than fk, the atomic co- herence length h/&p becomes larger than the optical wavelength A= h/hk = 2m /k of the lasers used to cool the atom. Consider then such an ultracold atom in the periodic light shift potential produced by a non resonant laser standing wave. The atomic de Broglie wave is delocalized over several periods of the periodic potential, which means that one can prepare in this way quasi-Bloch states. By chirping the frequency of the two counterpropagating laser waves forming the standing wave, one can produce an accelerated stand- ing wave. In the rest frame of this wave, atoms thus feel a constant inertial force in addition to the periodic potential. They are accelerated and the de Broglie wavelength Ay, = h/M(v) decreases. When Ayp = 4,,,., the de Broglie wave is Bragg-reflected by the periodic optical potential. Instead of increasing linearly with time, the mean velocity (x) of the atoms oscillates back and forth. Such Bloch oscillations, which are a textbook effect of solid-state physics, are more easily observed with ultracold atoms than with electrons in condensed matter because the Bloch period can be much shorter than the relaxation time for the coherence of de Broglie waves (in condensed matter, the relaxa- tion processes due to collisions are very strong). Fig.13 shows an example of Bloch oscillations [78] observed on cesium atoms cooled by the improved subrecoil Raman cooling technique described in [65]. 9 a = 3 = e OQ S &S =-05 ~2 -1 QO 1 2 Time Figure 13. Bloch oscillations of atonis in a periodic optical potential (from reference [78]. Mean velocity (in units of the recoil velocity) versus time (in units of half the Bloch period) for ultra- cold cesium atoms moving in a periodic optical potential and submitted in addition to a constant force. 5, CONCLUSION We have described in this paper a few physical mechanisms allowing one to manipulate neutral atoms with laser light. Several of these mechanisms can be simply interpreted in terms of resonant exchanges of energy, angular and linear momentum between atoms and photons. A few of them, among the188 Physics 1997 most efficient ones, result from a new way of combining well known physical effects such as optical pumping, light shifts, coherent population trapping. We have given two examples of such cooling mechanisms, Sisyphus cooling and subrecoil cooling, which allow atoms to be cooled in the microKelvin and nanoKelvin ranges. A few possible applications of ultracold atoms have been also reviewed. They take advantage of the long interaction times and long de Broglie wavelengths which are now available with laser cooling and trapping techniques. One can reasonably expect that further progress in this field will be made in the near future and that new applications will be found. Concerning fundamental problems, two directions of research at least look promising. First, a better control of “pure” situations involving a small number of atoms in well-defined states exhibiting quantum features such as very long spatial coherence lengths or entanglement. In that perspective, atomic, molecular and optical physics will continue to play an important role by providing a “testing bench” for improving our understanding of quantum phenomena. A second interesting direction is the investigation of new systems, such as Bose condensates involving a macroscopic number of atoms in the same quantum state. One can reasonably hope that new types of coherent atomic sources (sometimes called “atom lasers”) will be realized, opening the way to interest- ing new possibilities. Itis clear finally that all the developments which have occurred in the field of laser cooling and trapping are strengthening the connections which can be established between atomic physics and other branches of physics, such as condensed matter or statistical physics. The use of Lévy statistics for analyzing subrecoil cooling is an example of such a fruitful dialogue. The interdiscipli- nary character of the present researches on the properties of Bose conden- sates is also a clear sign of the increase of these exchanges. REFERENCES, [1] A Kastler, J. Phys. Rad. 11, 255 (1950). [2] A. Ashkin, Science, 210, 1081 (1980). [3] V.S. Letokhov and V.G. Minogin, Phys.Reports, 73, 1 (1981). [4] S. Stenholm, Rev.Mod.Phys. 58, 699 (1986) [5] CS. Adams and E. Riis, Prog. Quant. Electr. 21, 1 (1997). [6] C. Cohen-Tannoudji and W. Phillips, Physics Today 43, No 10, 33 (1990). [7] W. Heitler, “The quantum theory of radiation”, 3rd ed. (Clarendon Press, Oxford, 1954) [8] JP. Barrat and C,Cohen-Tannoudji, J.Phys.Rad. 22, 329 and 443 (1961). [9] C.Cohen-Tannougji , Ann. Phys. Paris 7, 423 and 469 (1962). [10] C. Cohen-Tannoudji, J. Dupont Roc and G. Grynberg, “Atom-photon interactions Basic processes and applications”, (Wiley, New York, 1992). [11] SH. Autler and C.H. Townes, Phys.Rev. 100, 703 (1955). [12] C. Cohen-Tannoudji, C.RAcad Sci. (Fr) 252, 394 (1961). [13] C. Cohen-Tannoudji and J. Dupont Roc, Phys.Rev. A5, 968 (1972). [14] W.D.Phillips and H.Metcalf, Phys.Rev.Lett. 48, 596 (1982). [15] J.VProdan, WD Phillips and H.Metcalf, Phys.Rev.Lett. 49, 1149 (1982). [16] W. Ertmer, R. Blatt, J.L. Hall and M. Zhu, Phys.Rev.Lett. 54, 996 (1985).a7) (18) 19] [20] [21] [22] [23] [24] [25] [26] [27] (28) [29} [30} [31 [32] [33] [34] (35) [36] (37) (38) 39] (40} (4) (42) (431 (44) (45) [46] [47] (48) [491 (50] (51) Claude N. Cohen-Tannoudji 189 G.A. Askarian, Zh.Eksp.TeorFiz, 42, 1567 (1962) (SovPhysJETP, 15, 1088 (1962)]. C. Cohen-Tannoudji, “Atomic motion in laser light”, in “Fundamental systems in quantum optics”, J. Dalibard, J.-M. Raimond, and J. Zinn-Justin eds, Les Houches ses sion LIII (1990), (North-Holland, Amsterdam 1992) p.1 J. Dalibard and C. Cohen-Tannoudji, J.Opt.Soc.Am. B2, 1707 (1985). ‘.W. Hansch and A.L. Schawlow, Opt. Commun. 18, 68 (1975). D,Wineland and H.Dehmelt, Bull.Am.PhysSoc. 20, 637 (1975) S. Chu, L. Hollberg, J.E. Bjorkholm, A. Cable and A. Ashkin, Phys. Rev. Lett. 55, 48 (1985). VS. Letokhov, V.G. Minogin and B.D. Pavlik, Zh.Eksp.Teor-Fiz. 72, 1828 (1977) [Sov.Phys JETP, 45, 698 (1977)]. DJ. Wineland and W. Itano, Phys.Rev. A20, 1521 (1979). J.P. Gordon and A. Ashkin, Phys.Rev. A21, 1606 (1980). PD. Lett, RN. Watts, C.1. Westbrook, W. Phillips, P.L. Gould and HJ. Metcalf, Phys. Rev. Lett. 61, 169 (1988). ELL. Raab, M. Prentiss, A. Cable, S. Chu and D-E. Pritchard, Phys.Rev.Lett. 59, 2631 (1987). C. Monroe, W. Swann, H. Robinson and C.E. Wieman, PhysRevLett. 65, 1571 (1990). S. Chu, J.E. Bjorkholm, A. Ashkin and A. Cable, Phys.Rev.Lett. 57, 314 (1986). C.S. Adams, HJ. Lee, N. Davidson, M. Kasevich and S. Chu, Phys. Rev. Lett. 74, 3577 (1995). A. Kuhn, H. Perrin, W. Hansel and C. Salomon, OSA TOPS on Ultracold Atoms and BEG, 1996, Vol. 7, p.58, Keith Burnett (ed.), (Optical Society of America, 1997). VS. Letokhov, Pis'ma.Eksp.Teor Fiz. 7, 348 (1968) (JETP Lett, 7, 272 (1968) ]. C. Salomon, J. Dalibard, A. Aspect, H. Metcalf and C. Cohen-Tannoudji, Phys. Rev. Lett. 59, 1659 (1987). RJ. Cook and RK. Hill, Opt. Commun. 43, 258 (1982) VIL Balykin, V.S. Letokhoy, Yu. B. Ovchinnikov and A.I. Sidorov, Phys. Rev. Lett. 60, 2137 (1988) M.A. Kasevich, D.S. Weiss and S. Chu, Opt. Lett. 15, 607 (1990). C.G. Aminoff, AM. Steane, P. Bouyer, P. Desbiolles, J. Dalibard and C. Cohen- ‘Tannoudji, Phys. Rev. Lett. 71, 3083 (1993). J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B6, 2023 (1989). PJ. Ungar, D.S. Weiss, E. Riis and S. Chu, JOSA B6, 2058 (1989). C. Salomon, J. Dalibard, W. Phillips, A. Clairon and S. Guellati, Europhys. Lett. 12, 683 (1990) Y. Castin, These de doctorat, Paris (1991). Y. Castin and K. Molmer, Phys. Rev. Lett. 74, 3772 (1995). Y. Castin and J. Dalibard, Europhys.Lett. 14, 761 (1991). P, Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, Grynberg, Phys. Rev. Lett. 68, 3861 (1992) PS. Jessen, C. Gerz, PD. Lett, W.D. Phillips, S.L. Rolston, RJ.C. Spreeuw and C1. Westbrook, Phys. Rev. Lett. 69, 49 (1992). G. Grynberg and C. Triché, in Proceedings of the International School of Physics “Enrico Fermi”, Course CXXXI, A. Aspect, W. Barletta and R. Bonifacio (Eds), p.243, IOS Press, Amsterdam (1996); A. Hemmerich, M. Weidemiiller and T.W. Hansch, same Proceedings, p.503. PS. Jessen and LH. Deutsch, in Advances in Atomic, Molecular and Optical Physics, 37, 95 (1996), ed. by B. Bederson and H. Walther. ‘A Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988). M. Kasevich and S. Chu, Phys. Rev. Lett. 69, 1741 (1992). Alzetta G., Gozzini A., Moi L., Orriols G., Il Nuovo Cimento 36B, 5 (1976). Arimondo E., Orriols G., Lett. Nuovo Cimento 17, 383 (1976). |-¥. Courtois and G.190 Physics 1997 [52] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B6, 2112 (1989). [53] F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, O. Emile, C. Westbrook, A. Aspect, C. Cohen-Tannoudji, C. R. Acad. Sci. Paris 318, 87-885 (1994) [54] B. Saubamea, T.W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc and C. Cohen- Tannoudji, Phys.Rev.Lett. 79, 3146 (1997) [55] J. Lawall, F. Bardou, B. Saubamea, K. Shimizu, M. Leduc, A. Aspect and C. Cohen- ‘Tannoudji, Phys. Rev. Lett. 73, 1915 (1994). [56] J. Lawall, S. Kulin, B. Saubamea, N. Bigelow, M. Leduc and C. Cohen-Tannoudji, Phys. Rev. Lett. 75, 4194 (1995). [57] M.A. Ol’shanii and V.G. Minogin, Opt. Commun. 89, 393 (1992). [58] S. Kulin, B. Saubamea, E. Peik, J. Lawall, T:W. Hijmans, M. Leduc and G .Cohen- ‘Tannoudji, Phys. Rev. Lett. 78, 4185 (1997). [59] C. Cohen-Tannoudji and J. Dalibard, Europhys. Lett. 1, 441 (1986). [60] P. Zoller, M. Marte and D.F. Walls, Phys. Rev. A35, 198 (1987). (61) F. Bardou, J-P. Bouchaud, O. Emile, A. Aspect and C. Cohen-Tannoudji, Phys. Rev. Lett. 72, 203 (1994). [62] B.V. Gnedenko and A.N. Kolmogorov, “Limit distributions for sum of independent random variables” ( Addison Wesley, Reading, MA, 1954). [63] J.P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990). [64] F. Bardou, Ph. D. Thesis, University of Paris XI, Orsay (1995) [65] J. Reichel, F. Bardou, M. Ben Dahan, E. Peik, S. Rand, C. Salomon and C. Cohen- ‘Tannoudji, Phys. Rev. Lett. 75, 4575 (1995). [66] K Gibble and S. Chu, Metrologia, 29, 201 (1992) [67] S.N. Lea, A. Clairon, C. Salomon, P. Laurent, B. Lounis, J. Reichel, A. Nadir, and G. Santarelli, Physica Scripta T51, 78 (1994) [68] J. Zacharias, Phys.Rev. 94, 751 (1954). See also : N. Ramsey, Molecular Beams, Oxford University Press, Oxford, 1956. [69] M. Kasevich, E. Riis, S. Chu and R. de Voe, Phys. Rev. Lett. 63, 612 (1989) [70] A. Clairon, C. Salomon, S. Guellati and W.D. Phillips, Europhys. Lett. 16, 165 (1991). [71] S. Ghezali, Ph. Laurent, $.N. Lea and A. Clairon, Europhys. Lett. 36, 25 (1996) [72] S. Ghezali, Thése de doctorat, Paris (1997). [73] E. Simon, P. Laurent, C. Mandache and A. Clairon, Proceedings of EFTF 1997, Neuchatel, Switzerland. [74] J. Prestage, R. Tjoelker and L. Maleki, Phys. Rev. Lett. 74, 3511 (1995). [75] LShapiro, Phys. Rev. Lett. 18, 789 (1964). [76] H. Wallis, J. Dalibard and C. Cohen-Tannoudji, Appl. Phys. B54, 407 (1992). (77] A. Steane, P. Szriftgiser, P. Desbiolles and J. Dalibard, Phys. Rev. Lett. 74, 4972 (1995). [78] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).