Prospects for atom interferometry

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1 Contemporary Physics, 2001, volume 42, number 2, pages 77± 95 Prospects for atom interferometry R. M. GODUN, M. B. D ARCY, G. S. SUMMY and K. BURNETT Atom interferometers were rst realized ten years ago, and since then have evolved from beautiful demonstrations of quantum physics into instruments at the leading edge of precision measurement. In this article we trace the development of atom interferometry, looking at how the physical principles have been put into practice to achieve ground-breaking experiments. We also discuss new atom optical techniques that are becoming available and anticipate the ways in which the consequent improvements will provide new opportunitie s in metrology and the study of fundamental physics. 1. Introduction Wave-like behaviour of both light and matter has been one of the most important areas of study in physics since the beginning of the twentieth century. A crucial aspect of this behaviour is the ability of waves to exhibit interference eœects. These eœects were rst understood in the early nineteenth century through the work of Young [1] and Fresnel [2] with light. Over the past hundred years light interferometers have evolved into powerful measuring devices with a wide range of application s including measurement of rotations, accelerations, distances and spectra. In the 1920s de Broglie [3] showed that matter also had an associated wave-like character which immediately provided the possibility of performing analogou s experiments with particles. This has led to the development of atom interferometry which is now a very active area of research. In this review, we explain the physics behind atom interferometers and discuss the measurements that they can make. Throughout, we draw analogies with light and so it is useful to begin our discussion in section 2 with a description of a light interferometer. In section 3 we will move on to consider the requirements of a generic atom interferometer and in section 4 we will look at the details of some speci c atom interferometers. These have already achieved measurements with a sensitivity equal to the world s best optical interferometers. We will review the current limitations to these instruments in section 5 and Author s address: Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK. consider new techniques and experiments for the future in sections 6 and Interferometry basics Interferometers are devices that use the interference of waves to make measurements. To do this, an incident wave must be split into several components that travel along diœerent paths and are recombined at the point where interference is to be observed. The most commonly encountered example of an interferometer uses light. In this case, the manipulation of the waves is achieved with simple mirrors and beam splitters as shown in gure 1. The beam travelling towards the detector has an electric eld that is a superposition of those from the two arms, E 1 and E 2. These have the same amplitude, E 0, and since the path lengths are x 1 and x 2, the resultant electric eld is given by E total ˆ E 1 E 2 ˆ E 0 exp ikx 1 exp ikx 2 Š ˆ E 0 exp ikx 1 1 exp ik x 2 x 1 Š ˆ E 0 exp ikx 1 1 exp iu Š, where k is the wavelength of the light ( ˆ 2p /k where k is the wavelength of the light) and Uˆ k(x 2 x 1 ) is the phase diœerence between the two superposing waves. The intensity recorded by the detector, I to ta l, is given by I total / je total j 2 / E 2 0 j1 exp iu j2 / 2E cos U Š Contemporary Physics ISSN print/issn online Ó DOI: / Taylor & Francis Ltd

2 78 R. M. Godun et al. Figure 1. waves. A Mach ± Zehnder light interferometer where the beam splitters and mirrors separate and recombine the paths of the light In the situation pictured above, the two paths are the same length and so the interfering waves will add together in phase with each other, that is Uˆ 0. The detector will record a spot of bright light resulting from this constructive interference. If, however, the path length diœerence is changed, the phase diœerence between the superposing waves, U, will vary. The detector will then record the intensity of the light as varying between bright and dark, giving rise to a set of interference fringes as shown by the solid line in gure 2. To look at just one of many examples of how this device can be useful, we consider how the Mach ± Zehnder interferometer in gure 1 can be used as an instrument for measuring rotations. The principle is the same as that of a Sagnac interferometer which is described in [4]. Suppose that the device is subjected to a rotation about an axis perpendicular to the page. When stationary, the light waves from the two interferometer arms add together in phase, as described above. When the instrument is rotating, however, the two paths to the detector acquire an additional phase diœerence, D U, proportional to the rate of rotation. The resulting interference pattern seen at the detector is then I total / 2E cos U D U Š, and will be oœset from the stationary interferometer fringes by D U, as shown by the dotted line in gure 2. The phase shift due to rotation at angular velocity X is given by D U light ˆ 4p X A, 1 kc Figure 2. The solid line shows the interference fringes as the phase diœerence between the two interfering waves is scanned in a stationary interferometer. The dotted line shows the same scan for a rotating interferometer. The phase oœset from the stationary interferometer s fringes is D U. where k and c are the wavelength and speed of the light and A is the area enclosed by the two paths of the interferometer. A measurement of the phase shift D U and a knowledge of k and A allow the rate of rotation X to be determined. The device can therefore be used as a gyroscope with a sensitivity which depends on how precisely the resulting interference fringes allow D U to be measured. Until the 1920s, no one had ever considered making such interferometers with matter sources because the wave-like nature of matter was not known. All this changed in 1924

3 Prospects for atom interferometry 79 when de Broglie postulated that any particle with momentum, p, should have an associated wavelength, k d B, related through his renowned equation k db ˆ h p, where h is Planck s constant. Typical de Broglie wavelengths for atoms at room temperature are about 1000 times smaller than wavelengths of visible light, making it di cult to detect eœects related to this wave-like nature. Nevertheless, experimental evidence of matter waves came just three years later in 1927 when Davisson and Germer [5] demonstrated electron diœraction from a crystal lattice. The wave nature of atoms was rst observed in 1930 by Estermann and Stern who diœracted helium atoms from single crystal surfaces of NaCl [6]. Neutron matter waves were later observed in the 1940s through re ection [7], diœraction [8] and interference [9] of neutron beams. With the ability to manipulate electron, neutron or atom waves it should then be possible to build matter wave analogues of light interferometers, but how do atoms compare with electrons and neutrons for use in interferometry? Atoms have larger masses and hence smaller de Broglie wavelengths (for a given velocity) than electrons and neutrons and this makes atom interferometry technically much more di cult. However, it is advantageou s to use atoms rather than electrons or neutrons for several reasons. Firstly, atomic sources are readily available and atoms can be precisely controlled by electromagnetic elds acting on their internal states. This control is essential to be able to guide the matter waves around precise paths. Although electrons are also readily availabl e and can be manipulated easily, their electric charge results in interactions both with other electrons and with stray external elds. This leads to unwanted phase shifts in an electron interferometer. Neutrons have the advantage over electrons that they do not interact electromagnetically with each other; however, they are not as useful as atoms because they do not interact with applied electric elds. Additionally, neutron sources could not be incorporated into a table-top experiment. Atom interferometers also have several advantages over light interferometers. The range of physical phenomena which can be probed by matter interferometers goes beyond the possibilitie s available to light interferometers. For example, properties of the particles themselves such as electric polarizabilitie s or collision cross-sections can be probed. Gravitational interactions can also be explored since the interfering waves have mass. Additionally, atom interferometers have the potential to measure phase shifts much more accurately than light interferometers. Consider once again the example of a rotating Mach ± Zehnder interferometer. If the interfering waves are light, then the phase shift is given by equation (1). If, however, an interferometer of equivalent area is built using matter waves then the measured phase shift will be D U atom ˆ 4p k db v X A 2 ˆ kc k db v D U light ˆ mc2 ±hx D U light, where k d B is the de Broglie wavelength and v is the atomic speed. The ratio of the measured phase shifts in the matter and light interferometers is therefore mc 2 / ± hx, where m is the particle mass in a matter interferometer and x is the light frequency in an optical interferometer. This ratio suggests that atom interferometers could make measurements with times more accuracy than light interferometers. These huge gains have yet to be realized as atom interferometers have not achieved areas or particle uxes comparable with light interferometers. However, the eld of atom interferometry is still developing and the techniques are rapidly improving. 3. Designing an atom interferometer Just as with light interferometers, the key components of atom interferometers are the source and the elements to manipulate the waves such as the mirrors and beam splitters. Let us begin by considering the source. In light optics, lasers are the ideal source for many interferometry experiments because they have large coherence lengths, are well collimated and have a high photon ux. The coherence length is the maximum distance along the wave over which the phase at all points has a well de ned relationship. Two points separated by more than this distance will not have a xed phase relationship between them and interference cannot be observed with a detector which has a response time longer than the time scale of the phase uctuations. Larger coherence lengths therefore allow larger path diœerences between the interferometer arms before the interference fringes start to disappear. The fringes in gure 2, for example, would wash out after fewer periods if the coherence length of the source were smaller. For accurate measurements of the phase shift, D U, it is desirable to take readings over as many fringes as possible and therefore a large coherence length is required. High photon ux also improves the sensitivity of an interferometer as it increases the signal-to-nois e ratio in the interference fringes. The lower the noise in the results, the more accurately the phase of the interference fringes is de ned and the more precisely D U can be measured. In atom optics, there are two main choices of atomic source: atom beams and cold atom clouds. Atom beams are

4 80 R. M. Godun et al. created, in general, by allowing atoms to emerge from a hole in an oven. The ux of detected atoms is typically 10 9 atom s 1. This is much higher than the uxes attainable with cold atom clouds, typically about 10 7 atom s 1. Using atom beams in interferometry therefore gives better signal-to-nois e ratios and can increase the precision of a phase shift measurement. Beam experiments also allow continuous measurement which can be useful for some interferometer applications. Cold atom clouds such as those from magneto-optic traps (MOTs) [10] or Bose ± Einstein condensates (BECs) [11], on the other hand, have much narrower momentum distributions than beams. The eœect of a momentum distribution in an atomic source is analogous to that of a frequency distribution in a light source. The narrower the distribution, the larger the coherence length. The relationship between the momentum spread, D p, in an atomic source and the coherence length, D x, is given by D x D p ± h. In a MOT, the momentum distribution width after laser cooling is typically 10 ± hk, where k ˆ 2p /k and k is the wavelength of the light cooling the atoms. This leads to a coherence length k/10. A BEC is much colder and the extremely narrow distribution gives a coherence length at least an order of magnitude larger. Atomic beams, on the other hand, have much smaller coherence lengths, typically 100 to 1000 times less than cold atom clouds and so their path diœerences cannot be as large. It is quite common, however, to operate atom interferometers in a con guration with zero path length diœerence. To create the interference fringes, the relative phase, U, between the interfering waves must then be scanned in some way other than altering the path length. Examples of how this relative phase can be changed are given later. The coherence length is then no longer an important characteristic and so it is not immediately obvious why cold atoms might be useful. However, narrower atomic momentum distribution s have two further advantages. The rst is that the atom cloud expands more slowly. This is useful because if the atoms move apart too quickly, they might move out of the beam splitting interaction regions and be lost from the experiment. Atoms in cold clouds can therefore remain together for longer, allowing the interferometer paths to be traced out over a longer time. This can lead to an increased sensitivity in certain types of experiment, examples of which will be seen in section 4. The second advantage of colder atoms is that some beam splitting mechanisms are momentum dependent and hence have a more uniform eœect on narrower momentum distributions. This produces an improvement in the contrast of the interference fringes and so any oœset phases, D U, can be more precisely determined. Some further points to consider when choosing an atomic source are that the vacuum systems for cold atom experiments can be much more compact than in beam experiments. This is because atoms in a beam travel much faster and cover greater distances in the time of the interferometer, typically up to several metres, whereas cold atoms will only travel a few millimetres. Apart from the technical convenience of smaller experiments, they are also more stable over time allowing more accurate measurements to be made. In general, the relative importance of the diœerent source characteristics will depend largely on what is to be measured. Having chosen an appropriate source of matter waves, they need to be manipulated with the analogues of mirrors and beam splitters. The mirrors and beam splitters which are used to de ect light cannot be used for atoms and new elements must be designed. Material diœraction gratings can, however, be used in atom optics in exactly the same way as they are used in light optics [12]. An incident matter wave will be diœracted into a variety of orders, some of which can be selected out to create an interferometer, as shown for example in gure 3. If the detector records the number of atoms emerging from the third grating in the direction shown, then interference fringes will be seen as the phase diœerence between the two paths is changed. This is done by translating the third grating along its length. The material gratings used in atom optics, however, usually require smaller periodicities than those used in light optics because of the small de Broglie wavelengths of atoms. Apart from the di culty in the fabrication of such small structures, these beam splitters have the disadvantag e that they block a large portion of the atoms, reducing the signal. This can also lead to clogging of the small apertures after prolonged use. Atoms have internal structure, however, and this allows the creation of many additional types of mirrors and beam splitters that make use of an atom s interaction with electromagnetic elds. Consider the example shown in gure 4. An atom initially in its ground state exposed to light can absorb a photon and undergo a transition into an excited state. The absorption of the photon will cause the atom to recoil with velocity ± hk/m, where k ˆ 2p /k, k is the wavelength of the absorbed light and m is the mass of the atom. In this way the light acts as a type of `mirror, de ecting atoms into a diœerent momentum state with unit probability. Note that it is not a mirror in the conventiona l sense as the angle of incidence does not necessarily equal the angle of re ection. If, however, the probabilit y of absorbing the photon is between zero and one, the atom will be left in a superposition of the ground and excited states, each with diœerent momenta. The components of the atom in the two states will then spatially separate from one another and can become the two interferometer arms. The light in this con guration has behaved as a beam splitter.

5 Prospects for atom interferometry 81 Figure 3. A Mach ± Zehnder style interferometer where the beam splitters are diœraction gratings which separate and recombine the paths of the atomic matter waves. The crucial attribute of any mirror or beam splitter in an atom interferometer is that it must not disturb the phase of the matter wave; it must be what is known as `coherent. Any disruption to the phase relationship between the two interferometer arms will result in destruction of the interference fringes. The example in gure 4 would therefore be useless if the atom were to decay out of the excited state by spontaneous emission. This would produce a random phase shift, due to the momentum change with random direction and timing, which would destroy the phase relationship between the two interferometer arms. The beam splitter depicted in the gure above could only be used, therefore, in the case of an atom with a metastable excited state so that spontaneous decay is unlikely to occur during the time of the experiment [13]. Similarly, the light itself must not impart random phase shifts onto the atoms. The light beam must therefore have a long coherence length, such as that provided by a laser. For atoms without a metastable excited level, a second, counter-propagatin g laser beam can be used in addition to the rst to cause the atom to return to a ground level (not necessarily the same level as initially) by stimulated emission. The atom gains a further velocity recoil in the process and is left in a state from which spontaneous decay cannot occur. Since the absorption and emission process relies on stimulated transitions, coherence is preserved between the interferometer arms. A Raman transition is a speci c example of this and is shown in gure 5. The frequency diœerence between the two laser beams is set to x 1 3 so that the atom can undergo a simultaneous two-photon transition between the ground states j1i and j3i, gaining momentum kicks from the absorption and emission of the photons equal to ± h (k 1 k 2 ). If the two laser beams are counter-propagating, then k 1. k 2 and the atoms receive a momentum of 2 ± hk. In this way, atoms in states j1i and j3i can be given diœerent momenta and will Figure 4. An atom in one of its ground states (thick lines) can absorb a photon and be transferred into one of its excited states (thin lines). The atom gains momentum from the photon and is de ected in a way similar to re ection from a mirror. Figure 5. Atoms in state j1i will undergo a simultaneous twophoton Raman transition to state j3i when the frequency diœerence between the laser beams is equal to x 1 3. The transition proceeds via a virtual level shown by the dotted line which is a large detuning, D, from the excited state j2i. This ensures a minimization of loss through spontaneous decay.

6 82 R. M. Godun et al. separate spatially from one another if given su cient time. The large detuning, D, of the laser beams from the transition into state j2i reduces the probabilit y of this excited state becoming populated. This in turn inhibit s spontaneous decay, as required to preserve the coherence between the two interferometer arms. Note that if the two counter-propagatin g laser beams are of the same frequency, then they create a standing wave. In this case the momentum kicks described above can also be considered as the result of the atoms being diœracted by an optical diœraction grating formed from the standing light wave. A useful feature of these Raman transitions is that they can be used either as `mirrors or `beam splitters, depending on how long the light is applied. If the atoms are in state j1i when the light is rst applied, they will be transferred into state j3i as described above. Once in state j3i, however, if the light is still applied they will be transferred back into state j1i by the reverse process. The atoms will continue to oscillate between states j1i and j3i for as long as they are exposed to the light. To use the Raman transition as a mirror, the light must be applied for the correct length of time for all the atoms to go from one ground state to the other. They will then all receive the same momentum kick and be de ected equally. An exposure to the laser beams with frequency, intensity and pulse length to cause such a complete transfer between states is known as a p pulse. To act as a beam splitter, the process should create a superposition of states with diœerent momenta. This is accomplished by exposing the atoms to the light for only a fraction of the time required for a complete transfer. If the frequency, intensity and pulse length of the laser light are such that the atom is left in an equal superposition of these states, then the laser beams are said to create a p /2 pulse. Light has proved to be the most widely used mechanism for creating mirrors and beam splitters for atoms because of the relative ease with which the necessary optical potentials can be realized. There are also many other ways in which electromagnetic elds can be used to manipulate the momentum and position of atoms. Examples include atomic mirrors which use either a static magnetic eld [14] or an evanescent light wave [15] to provide a repulsive potential which re ects atoms. Another example is the use of magnetic elds from current carrying wires [16]. We will examine some of these techniques later in section 6. Having seen that many mirrors and beam splitters rely on manipulatin g the atoms internal states, it is important to examine how these processes can be used to build an interferometer. Consider the interferometer in gure 6, where the beam splitting mechanism is Raman pulses and the atom is always in its ground states j1i and j3i (the excited state j2i is not depicted). Notice that the middle two beams form a p pulse and hence act as mirrors, while the rst and last pairs of laser beams create p /2 pulses, thus acting as beam splitters. The paths are separated according to the momentum associated with the atoms internal state and when they are recombined the populatio n amplitudes of the two states superpose to give jwi final ˆ aj1i bj3i. The values of a and b can be calculated by tracing the amplitude s through the interferometer. For the case of the p /2 ± p ± p /2 pulse sequence depicted in gure 6, the state amplitudes in the superposition are shown in gure 7 at each stage of the interferometer. Each time an atom changes state after a photon interaction, the phase of the light, U 1, is imprinted on the new state. In the case of emission, the new state receives U 1 and for absorption, it receives U 1. Thus for a Raman pulse which involve s an emission and an absorption from two separate beams, the phase change on the new state is governed by the relative phase of those two beams. Let the relative phases be U 1, U 2 and U 3 for the p /2, p and p /2 pulses respectively. The nal amplitudes in the two states are then found to be a ˆ 1 2 exp i U 2 U 1 Š 1 exp i U 3 2U 2 U 1 Š b ˆ i 2 exp iu 2 1 exp i U 3 2U 2 U 1 Š, which lead to the nal populations of Pop 1 / jaj 2 / cos U 3 2U 2 U 1 Š 3 Pop 1 / jbj 2 / cos U 3 2U 2 U 1 Š 4 By keeping U 1 and U 2 xed and by scanning U 3, familiar cosine interference fringes may be seen if the population of atoms in a particular internal state is detected. This is an example of the way in which the phase shift between the two interferometer arms may be changed while the path lengths are kept equal. We will refer to this type of interferometer as an internal state interferometer since the interference is between the populatio n amplitudes in the internal states. 4. Interferometers as instruments The rst detailed design for an atom interferometer was patented in 1973 by Altshuler and Frantz [17]. There then followed an eighteen year delay before one was achieved experimentally. The main problem was the design and fabrication of the coherent beam splitters and mirrors. For beam splitters relying on light, advancements in the eld of laser technology helped enormously as `oœthe shelf lasers

7 Prospects for atom interferometry 83 Figure 6. An internal state interferometer where the beam splitters separate and recombine the paths according to the internal atomic state. The detection records the population of atoms in a particular internal state. Figure 7. The boxes give the amplitudes of the components in states j1i and j3i after each pulse of light. Every time one state is transferred into another, the relative phase of the laser beams in the Raman pulses (U 1,U 2,U 3 ) is imparted to the new state, along with an additional phase of p / 2.

8 84 R. M. Godun et al. could be purchased at a variety of wavelengths. The rst demonstration of a beam splitter was in 1985 by Moskowitz et al. [18] using lasers to create standing waves which acted as diœraction gratings for atoms. Alternative beam splitters, using material gratings, relied on advancements being made in nanofabricatio n technology to produce gratings with a su ciently small periodicity to act as a beam splitter for the atom waves. Such nanofabricated structures were used in 1991 by Carnal and Mlynek [19] and Keith et al. [12] to create the world s rst atom interferometers. With the experimental advances in the eld of atom optics [20,21] and, in particular, the development of diœerent coherent beam splitting mechanisms, other interferometers rapidly followed [22 ± 27]. It is only ten years since the rst atom interferometer was built and there are many atom interferometers throughout the world with numerous diœerent con gurations. More signi cantly, there are new experiments and proposals being frequently reported. Why is there still such interest? The answer lies in the fact that atom interferometry has gone beyond the `proof of principle stage and is nding many application s as an approach to measurement. With the techniques for atom interferometry developing all the time, there is much potential for improved instruments in the future. As an overview of the range of application s of atom interferometers, a summary is given of some of the physical phenomena which have already been measured The MIT group The rst example of an interferometer that we will examine was used in 1995 to measure directly the electric polarizability of sodium atoms [23] at the Massachusetts Institute of Technology. This experiment is the only one to date which has permitted a physical barrier to be placed between the two arms of an interferometer. The de nition of electric polarizability, a, is given by U ˆ a e2 2, where U is the Stark potential felt by the atoms when an electric eld, e, is applied. The experiment therefore consisted of exposing atoms in one path to an electric eld and nding the potential energy shift by measuring the resulting phase shift of the interference fringes, D Uˆ Us / ± h, where s is the time for which the atoms experience the interaction. The design was based on a Mach ± Zehnder interferometer as pictured in gure 8. The beam splitting mechanism consisted of material gratings which diœracted the atomic matter waves into diœerent orders which gave rise to the separated paths. The de Broglie wavelength of the sodium atoms was only 17 pm and so a nanofabricatio n process was used to produce gratings with a period of just 200 nm. A beam of the sodium atoms was split and recombined with three of these gratings. To observe interference fringes, the third grating was translated along its length. This allowed the separated paths to be recombined with a phase shift which varied according to the grating s position. To allow the electric eld to be applied to only one interferometer arm without aœecting the other, a thin metal foil was inserted between the two arms, directly after the second diœraction grating. At this point in the interferometer, the two atomic beam widths were 40 lm (FWHM) and the beam centres were separated by 55 lm. This was just enough separation to allow a 10 lm thick foil to be inserted. The thin foil was stretched to try and remove wrinkles but it nevertheless cast a 20 ± 30 lm shadow on the Figure 8. The atom interferometer at MIT, using nanofabricated diœraction gratings as the beam splitters. The interaction region consists of a metal foil held symmetrically between two side electrodes, allowing an electric eld to be applied to one arm only.

9 ± Prospects for atom interferometry 85 detector due to remaining imperfections. It was important for the atoms to have as long an interaction time with the electric eld as possible to give a large phase shift and hence increased sensitivity. The interaction region was therefore made 10 cm long. Any longer than this would have led to serious clipping of the atomic beam by the foil. The interferometer was able to measure phase shifts to a precision of 10 mrad when averaged over 1 min of data taking. The reference phase was taken as the position of the interference fringes when no electric eld was applied to the atoms. This phase was found to drift by about 1 rad h 1 so it was measured frequently to try and reduce errors. External vibrations also had to be overcome to allow the interferometer to perform at the required sensitivity. Other sources of error arose from uncertainty in the velocity distribution of the atomic beam, and the geometry of the interaction region. The results of these measurements gave the atomic polarizabilit y with an error of just 0.3%, an order of magnitude improvement on previous direct measurements. It is important to have precise data on atomic polarizabilities as they allow determinations of many atomic properties such as dielectric constants and refractive indices, van der Waals forces between two polarizable systems and Rayleigh scattering cross-sections. This experiment was the rst implementation of an atom interferometer as a measurement tool Atomic clock The primary frequency standard which is created for atomic clocks uses the physics of an internal state atom interferometer. Precise time and time interval data are needed for many application s including telecommunications networks, electricity generation, computer network synchronizatio n and navigation. Calibration laboratorie s are constantly striving for a more precise de nition and realization of the second. The second is currently de ned as the duration of periods of the radiation in the transition F ˆ 3! F ˆ 4 in the ground state of the caesium atom. An atomic clock can therefore be made by creating a frequency standard based on this atomic transition and using it as a reference to give `ticks to determine the speed of a clock. The simplest way to nd the frequency would be to scan a microwave frequency source over the line and detect when the atoms undergo the transition with the highest probability. A much more accurate technique, however, is to use Ramsey s method of separated oscillatory elds [28] which is illustrated in gure 9. Since this is the method used in an atomic clock, we will now look in detail at how the internal states are manipulated. The two states involve d are the magnetically insensitive m ˆ 0 components of the hyper ne levels, F ˆ 3 and F ˆ 4 in the ground state of caesium. The beam splitting mechanism is a pulse of microwaves at frequency, x, close to the transition frequency from F ˆ 3 to F ˆ 4. Atoms in the F ˆ 3 level are exposed to a p /2 pulse of these microwaves, which have a frequency, intensity and pulse length such that the atomic wavefunction is `split into an equal superposition of the F ˆ 3 and F ˆ 4 levels. This is similar to the p /2 pulse described in relation to the Raman pulses in gure 5. The diœerence, however, is that the microwaves transfer the atoms directly from one ground state to the other with a single photon. The microwaves have a frequency of Hz, about four orders of magnitude less than the optical frequencies required in the Raman transition. This means that the momentum transferred to the atoms on absorption of a microwave photon is negligible. The components in the two internal states therefore remain spatially overlapped throughout the clock sequence. After a time T, a second p /2 pulse of microwaves is applied which recombines the components in the two levels. The nal population in each individual state depends on the relative phase, U, between the atomic states and the microwaves at the time of the second pulse. We now consider the origin of this relative phase, U. During the time between pulses, T, a phase diœerence evolves between the F ˆ 3 and F ˆ 4 atomic levels, U a t ˆ x 0 T, due to their energy diœerence, hx 0. The microwaves at frequency x also evolve a phase in the time interval so that U m w ˆ xt. If the microwave frequency is such that U m w ˆ U a t after the time interval, then the second p /2 pulse will add to the rst, producing the overall eœect of a p pulse and transfer all the atoms into the F ˆ 4 level. If, however, U m w and U a t become p radians out of phase during the time interval, then the second p /2 pulse will cancel the eœect of the rst and all the atoms will be returned to F ˆ 3. As the phase diœerence Uˆ U m w U a t varies, the population in each of the internal states will be seen to oscillate. This phase diœerence can be controlled by scanning the microwave frequency, x, since Uˆ (x x 0 )T. A fringe pattern will then be formed, such as that shown in gure 10, if the population of atoms in the F ˆ 4 level is detected. The fringes are essentially the cosine pattern which we saw earlier, but with a modulating envelope. The origin of this extra modulation arises from the fact that as the microwaves are detuned further from resonance, the atoms are less and less likely to make a transition into the F ˆ 4 level and the signal falls oœ to zero. The shape of the envelope is what would be seen if only one of the p /2 pulses of microwaves were applied. The advantage of the separated oscillatory eld method is that it sets up fringes which have a much narrower width than that created by a single pulse. This allows x 0 to be found much more precisely. The longer the time interval between the two pulses, T, the more critically the microwave frequency, x,

10 86 R. M. Godun et al. Figure 9. The manipulation of the internal atomic states with two p /2 microwave pulses separated by a time, T. The rst pulse places the atom in an equal superposition of the two states. The second pulse alters the superposition according to the relative phase, U, between the atomic states and the microwaves. must match the atomic transition frequency, x 0. Hence the width of the fringes decreases as 1/T so that increasing the time between the pulses allows a reference source to be locked to the central frequency more precisely. Experimentally, the regions where the microwaves are applied to the atoms will be xed in space and so the pulse interval, T, will depend on the speed of the atoms between these regions. Cold atoms with slower speeds will therefore give the longest pulse separation times and hence the most precisely de ned reference frequency. An additiona l technique which can be used to increase the pulse separation time is to launch the atoms in a so-called fountain arrangement as shown in gure 11. The atoms pass through a microwave cavity once on their way up, and again on their way down. Times between the pulses can then approach 1 s with a fountain height of 1 m. An extra advantage of passing the atoms through the same microwave cavity twice is that this eliminates many systematic eœects. Figure 12 shows the real set-up of just such a caesium fountain at the National Physical Laboratory, Teddington, UK. The vacuum system in which the atoms are launched is on the right of the picture and the optics and electronics which control the atoms are towards the left. These new experiments are a long way beyond the simple pendulum which acts as the frequency standard in a grandfathe r clock. The accuracy and stability, however, are similarly far removed. Currently, clocks making use of the frequency standard derived from atomic fountains can achieve an accuracy of one part in with a stability of one part in by averaging results over a day [29]. The dominant factors limiting the accuracy arise from uncertainties in the frequency shift induced by cold inter-atomic collisions and the ac Stark shift induced by

11 Prospects for atom interferometry 87 Figure 10. Interference fringes in the population of atoms in the F ˆ 4 ground state of caesium as the frequency of the microwaves is scanned. The middle of the central fringe is precisely the transition frequency, x 0, which is to be found. blackbody radiation. For more information on errors, see [30] The Stanford group Perhaps the most exciting possibilities oœered by atom interferometry are in precision measurement and this has been the centrepiece of this group s research. One of their atom interferometers is based on a fountain of cold caesium atoms, similar to the atomic clock. The key diœerence is the use of optical Raman pulses instead of microwaves to manipulate the superposition of internal states, thus creating a spatial separation between the two interferometer arms. A schematic of the apparatus is shown in gure 13. The instrument has been used to make measurements of the Earth s gravitational acceleration, g, in 1991 and subsequent years [31]. The result has been a value of g with an accuracy equivalent to that attainable from falling corner cube experiments [32], currently among the most sensitive gravimeters in the world. The interferometer is precise enough to show variations in g due to tidal eœects over the period of a day. Due to the diœerent momenta imparted by the Raman pulses, the internal states trace out diœerent trajectories in the fountain, one reaching a greater height than the other, as shown in gure 14. Note that an additional p pulse is therefore needed at the top of the trajectory to impart momentum so as to ensure that the paths are brought back to spatially overlap at the last p /2 pulse. To see the Figure 11. The basic principle of an atomic fountain for a primary frequency standard. The atoms start in one state and are exposed to a p /2 pulse of microwaves, placing them in an equal superposition of both internal states as they are launched upwards. On their way back down, they are exposed once again to a p / 2 pulse of microwaves. The resulting superposition of states depends on the relative phase accumulated between the atoms and microwaves as outlined in gure 9. interference fringes, the relative phase of the Raman beams in the last p /2 pulse, U 3, is scanned. The nal atomic population s in the two states at the end of the interferometer sequence are therefore given by equations (3) and (4), but due to gravity there is an additiona l phase oœset [33], D U ˆ k R gt 2, where k R ˆ k 1 k 2 is the eœective wavevector of the Raman transition (with magnitude k 1 k 2. 2k, g is the acceleration due to gravity and T is the time between each of the pulses in the p /2 ± p ± p /2 sequence. From measurements of this phase oœset, a value for g can be found. The origin of this gravitationa l phase shift can be seen in simple terms by considering the relative phase between the atoms and the light at the time of each pulse. An atom at rest would see x T/2p oscillations of an applied eld in a time T. If the atom moves a distance D z in the same direction as the light, however, it will see k R D z/2p fewer

12 88 R. M. Godun et al. Figure 12. The atomic caesium fountain used to create the primary frequency standard at the National Physical Laboratory, UK. The photograph shows the optics and electronics on the left which control the atoms in the vacuum chamber on the right. [We thank D. Henderson for allowing us to reproduce this photograph from NPL, Teddington, UK.] oscillations in the same time. The atomic phase therefore keeps a count of how many periods of the light eld the atom has moved across. The applicatio n of the rst Raman pulse causes the two components of the superposition to separate from one another whilst travelling upwards. At the time of the second pulse, both components are in new positions in the light eld and consequently experience diœerent phase shifts. Therefore, although the rst two Raman pulses are applied with the same relative phase (U 1 ˆ U 2 ˆ 0), the phases imprinted onto the atoms are diœerent. Between the second and third Raman pulses, there is a further change in relative position and hence another phase shift. In the absence of gravity, these accumulated phases in the rst and second halves of the p /2 ± p ± p /2 sequence cancel each other out. In the presence of gravity, however, the fountain trajectories are asymmetric as shown in gure 14 and there is an overall phase oœset in the nal interference fringes. In eœect, the interferometer measures the diœerence between the fountain trajectories against the very accurate ruler created by the light wave. The most challenging part of this experiment has been to minimize the random and systematic errors. The largest random errors are due to vibrations and rotations of parts of the apparatus. Systematic eœects in the instrument include uncertainty in the rf phase shift, the Coriolis eœect, the wavelength of the caesium D1 transition to which the frequencies of the Raman lasers are referenced, the laser lock oœset, gravity gradients, ac Stark shift and precise pulse timings. Nevertheless, the resulting interferometer is accurate enough to allow phase shifts 10 mrad to be detected, leading to a value of g accurate to one part in The Yale group Atom interferometers based on the analogue of optical Mach ± Zehnder interferometers, as described in section 1, can be used to make very accurate gyroscopes. Sensitive

13 Prospects for atom interferometry 89 Figure 14. A schematic of the fountain trajectories traced out by the diœerent components of the atomic superposition in the Stanford interferometer. Time increases towards the right. Figure 13. The Stanford atom interferometer, using a MOT of cold caesium atoms launched into a fountain trajectory with Raman beams providing the beam splitting mechanism. rotation measurements are required in navigation, geophysical studies and can even be used to make tests of general relativity. The Yale group have used their atom interferometer to measure the Earth s rotation rate from 1996 onwards [34,35]. The interferometer consists of a horizontally propagating atomic caesium beam which is manipulated with transverse Raman pulses, like that shown in gure 6. As mentioned before, rotations introduce phase shifts in the interference fringes proportiona l to the rate of rotation, X, given by equation (2) D U atom ˆ 4p kv X A, where k is the atom s de Broglie wavelength and v its velocity. We see that the phase shift is also proportional to A, the area enclosed by the interferometer. Thus larger area interferometers make more sensitive gyroscopes. The reason for using atomic beams rather than cold atoms in this experiment is therefore to make the enclosed area as large as possible, 22 mm 2 in this case. From accurate measurements of the phase shift, the Earth s rate of rotation has been determined with a short term sensitivity comparable with that of the best active ring laser gyroscopes [36,37]. To achieve such impressive results, there were of course many engineering di culties which had to be overcome. Once again, vibrations would be extremely detrimental to the instrument s sensitivity and so the whole system was designed with vibration isolation in mind. The Raman beam alignment is also very critical and has to be correct to within 10 4 rad to observe interference fringes at all. The Raman beams themselves propagate in tubes to reduce optical phase shifts from air currents. The short term sensitivity could be further improved by increasing the atomic ux as this would give a better signalto-noise ratio. Long term mechanical stability, however, is much more di cult to achieve in a system such as this which is over 2 m long. One possible way to reduce the consequent systematic errors is to create a second

14 90 R. M. Godun et al. gyroscope, with an atomic beam counter-propagatin g relative to the rst which uses the same laser beams for its atom optical elements. The area vectors of the two gyroscopes will then have opposite signs, as will the rotational phase shifts. Subtracting the two phase shift measurements leads to common mode rejection of many systematic errors which do not reverse sign with the direction of the atomic beam and thus enhances the instrument s long term sensitivity. In a second and completely separate device, the Yale group have built a gravity gradiometer [38]. The principle of this experiment involves simultaneously measuring the phase shifts due to gravity in two identical interferometers separated by a height of about 1 m, as shown in gure 15. The diœerence in the value of g yielded by the two interferometers then gives the change in gravity over their separation. The two interferometers are very similar in design to the Stanford interferometer. The atomic sources are caesium MOTs and the beam splitters are vertically propagatin g Raman pulses. The main diœerence is that the Yale group simply allows their atoms to fall, rather than launching them in a fountain trajectory. The Yale group tested their system by measuring the gradient of the Earth s gravitational eld. Two sets of interference fringes were obtained with signal-to-noise ratios su ciently high to see phase shifts between the two fringe patterns of the order of 50 mrad. These measurements yielded a value for the Earth s gravity gradient consistent with the expected value, assuming an inverse square law scaling for g. Figure 15. A schematic of the gravity gradiometer. Two interferometer regions, separated by about a metre, detect phase shifts due to gravity. The diœerence in their phase oœsets gives a measure of the gradient of gravity. The experiment is designed to reduce systematic eœects by having as many elements as possible in common to both interferometers. These include the Raman and detection beams which are both vertical. This means that platform vibration eœects, likely to be a large source of error in a single interferometer, can be eliminated when the two interferometer phase shifts are subtracted from one another. Data can also be taken with the Raman beam direction reversed. This changes the sign of the gravitational gradient whilst leaving the sign of the systematic errors unaœected. Once the data has been suitably analysed to remove these eœects, the main sources of systematic errors that remain are time-varying magnetic elds, ac Stark shifts from the laser beams and platform rotations. Nevertheless, the ability to minimize the major error, that of vibrations, gives a device stable enough to take measurements over periods of several hours. It has also led to this instrument being seen as a viable method for measuring gravity gradients on board a moving platform such as a ship or aircraft. 5. Current limits and possible improvements From the examples we have looked at, it is clear that external factors, such as vibrations, are a major limitation to the achievable sensitivities of existing interferometers. Combinations of passive isolation and active feedback have been used to reduce vibration eœects as much as possible. Passive isolation typically consists of rubber feet under the apparatus to damp out high frequency vibrations. Active feedback works by having an accelerometer, such as a light interferometer, attached to the apparatus. The output which is produced in response to a vibration can be fed back to a control device to provide the reverse motion and compensate the vibration. Such a device responds best to frequencies in the range 0.2 ± 5 Hz, the region which is most detrimental to measurements. It is due to these external factors that both atom and light interferometers currently have the same precision. In other words, they are both limited by the same eœects. If, however, experiments could be performed in an environment where vibrations are naturally much less of a problem, then the intrinsic advantages of atom interferometers would allow them to make much more precise measurements than light interferometers. Such an environment might be a satellite in free-fall around the Earth. It is known that this can be almost vibration-free and there are currently several groups working towards putting atom interferometers in space. Although atom interferometers in a vibration-free environment would start to out-perform light interferometers, the gain would still not be the 11 orders of magnitude mentioned in section 3 as being theoretically achievable. Atom interferometers still have much smaller areas and particle uxes than their counterparts in light and this too is

15 Prospects for atom interferometry 91 preventing them from attaining their ultimate sensitivity. The current limits to the area of atom interferometers are two-fold. First, the atoms cannot simply be made to travel for longer times to create bigger areas. Since gravity is always acting on them, after a certain length of time the atoms would fall out of the regions where the atom optics takes place. Secondly, the beam splitting mechanisms which impart momentum to separate one path from the other do not scale well to higher momenta. For example, applying two Raman pulses instead of one to impart twice as much momentum results in a serious loss of e ciency. Hence there is a limit to the amount of spatial separation which can be achieved between the interferometer arms. Placing the atom interferometer in space would be one way to increase the interferometer s area because the micro-gravity environment would allow interaction times to become very long without the atoms falling out of the experiment. The limit which would then be reached is that the atomic distribution would expand out of the interaction region due to its initial momentum distribution. This expansion could be suppressed by using colder sources, such as BECs, which would allow interaction times of about a minute. Other ways to create larger area interferometers require diœerent techniques for manipulating the atoms, some examples of which will be described in the next section. 6. New techniques Besides improvements to existing atom interferometer schemes, considerable eœort is being invested in the development of completely new techniques. These can be roughly divided into improved sources and novel ways to manipulate the atoms. As mentioned above, Bose ± Einstein condensate sources could provide longer interaction times than those achievable with magneto-optic traps or atomic beams. They could also provide higher particle ux as they are more dense than conventional sources of cold atoms. This could lead to an improvement in the intrinsic sensitivity of interferometers through a better signal-to-nois e ratio. An additiona l advantage of BECs is that some beam splitting mechanisms can have a very uniform e ciency when operating on a narrow momentum distribution. This leads to an increase in the contrast of the interference fringes and hence to better phase shift measurements. A group in Tokyo has already constructed a Mach ± Zehnder interferometer with a BEC and seen almost 100% contrast in the fringes [39]. Aside from very cold sources, more massive sources are also creating interest. This is because the phase shift experienced by a matter interferometer scales linearly with the mass of the interfering particles. A molecular source would therefore give rise to a much higher sensitivity than single atoms. The analogue of a Young s double slit interference experiment has already been demonstrated with a carbon-60 source [40]. Similarly nanostructures and quantum dots are also being investigated. These are manmade structures with an even higher mass. They also possess internal energy states which may allow them to be manipulated with electromagnetic elds in a similar way to atoms. There are several promising new avenues for the further development of atom manipulation. The rst possibility lies in increasing the amount of momentum transferred. This can produce larger spatial separations between the interferometer arms, larger enclosed areas, and hence more sensitive interferometers. A recent method that has been developed in Oxford, called the quantum accelerator mode [41], involves the applicatio n of many pulses of a standing wave of laser light. The key feature is that the standing light wave is translated along its length by a certain amount between each pulse. In this way, atoms with the right initial conditions will receive a momentum transfer of approximately two photon recoils with an e ciency greater than 99% at every pulse. This can be contrasted with the e ciency of a Raman process which at best is 85% for every two photon recoils of momentum transferred. Although this diœerence does not seem overly signi cant, after twenty pulses the fraction of atoms remaining in the Raman case has fallen to 4%, while that in the accelerator mode case is still 82%. The accelerator mode is therefore capable of transferring momenta up to, say, 100 ± hk with high e ciency. Figure 16 shows experimental data where each line is a measurement of the momentum distribution of an atomic ensemble after a given number of light pulses. Each momentum distribution was measured directly by allowing the atoms to fall through a sheet of probe light, the absorption of which revealed the time of ight of the atoms. In gure 16, a fraction of the atomic ensemble which is gaining momentum linearly with pulse number and moving out to high momenta is clearly visible. The atomic source was a MOT which had a momentum distribution such that only about 25% of the atoms satis ed the initial conditions to enter the accelerator mode. However, once atoms are in the accelerator mode, the data show their high probability of remaining there. A technique such as this may therefore be developed into a beam splitter which is capable of transferring large momenta to atoms thus creating a large spatial separation between the interferometer arms. This in turn could create a more sensitive atom interferometer. A diœerent approach to the manipulation of atoms lies in the eld of guided atomic waves and more particularly using miniaturized integrated atom optics. Such systems allow the construction of atom interferometers which are more akin to the optical bre and ring laser interferometers that have proved so successful as robust methods of measuring rotations. These devices are optical interferometers whose enclosed area is greatly enhanced, compared to

16 92 R. M. Godun et al. constant. Atoms which align themselves so as to have a component of magnetic moment parallel to the magnetic eld (l be U > 0) are strong eld seeking and will execute circular orbits around the wire as shown in gure 17 (a). This behaviour is acceptable for wires in free space; however, if the wire is on a surface then it will not be possible to guide the atoms with this con guration since they will come into contact with the surface. A way around this problem is to use weak eld seeking atomic states (l be U < 0) and apply an additional magnetic eld B b ia s which is perpendicular to the wire. This creates a magnetic eld minimum, a distance r s ˆ l0 2p I B bias Figure 16. Experimental data showing the transfer of momentum to atoms using a quantum accelerator mode. The plot shows the variation of the atomic momentum distribution with increasing number of pulses. The fraction of the initial distribution that satis es the conditions to enter the mode is approximately 25%. Once in the mode, the probability to remain there is over 99% per pulse. their outward physical dimensions, by guiding light many times around a looped bre, or a laser cavity [37]. These compact systems are routinely used for inertial navigatio n in aircraft, missiles and even the tips of drills used for oil prospecting. We will now review several of the potential methods for guiding atom waves which would allow the construction of analogous atom optical devices. Experiments at Innsbruck [16] and Harvard [42] have taken the rst steps in this direction by using microfabricated wires on the surface of a chip. Both of these experiments are based on guiding atoms using the magnetic eld gradient produced by current carrying wires. The potential experienced by a neutral particle with magnetic moment, l, in a magnetic eld, B, is V ˆ l B. If the gradient of the magnetic eld is non-zero then the particle will experience a force. In the case of a current carrying wire the magnetic eld is given by B ˆ l0i 2p 1 r b e U, where I is the current in the wire, r is the radial distance from the wire and be U is a unit vector in the azimuthal direction around the wire. Assuming that the magnetic eld direction experienced by the atom changes slowly compared to the Larmor precession of the magnetic moment (the adiabatic approximation), we can treat l be U as a away from the centre of the wire. Thus the weak eld seeking atoms are trapped in a long tube parallel to the wire, see gure 17 (b), which can now be mounted onto a surface. Many other types of wire con guration are possible. For example, at Harvard four wires have been mounted on a chip to create a magnetic potential with a minimum above the surface, with no need for an external bias eld. This shows that `integrated atom optics is truly becoming a reality. The group at Innsbruck have also demonstrated atomic beam splitters using these methods, both in free space and on a surface. The essential idea is simply to divide the current carrying wire into two at a `Y-junction. The ratio of the currents in the arms of the Y determines the beamsplitting ratio. Although the coherence preserving properties of such atom optical elements have yet to be con rmed, there is no reason to suspect that they should not be usable in an atom interferometer with a suitable atomic source. The main criterion for such a source is that only a single mode of the atomic waves should propagate in the waveguide. For the currents and wire sizes which seem to be feasible this will mean using a Bose ± Einstein condensate. It is possible to build atomic sources, such as a magneto-optic trap, right onto the chip containing the atom optics. Eventually it should be feasible to create a BEC on the chip and couple it directly into the single mode guide created by the wire. This type of technology might enable atom optics and interferometry to move into the realm of mass production with a consequent increase in the applications for such devices. Another route towards building better atom interferometers is the use of hollow bres for atom guiding. By using either magnetic or optical forces atoms can be channelled through the hollow core of the bre and be kept away from the walls. The magnetic technique has been demonstrated at the University of Sussex [43] and is accomplished by using special bres which are manufactured to have four wires embedded in their walls as pictured in gure 18. The

17 Prospects for atom interferometry 93 Figure 18. The con guration of wires used in the Sussex experiments to create quadrupolar magnetic elds to guide atoms down the central hole of the bre. Figure 17. The magnetic elds and the resulting atom trajectories in the cases of (a) circular eld lines from a current carrying wire and (b) the same current carrying wire with an oœset bias magnetic eld. In each case the atoms are guided along the path de ned by the wire. [We thank D. Cassettari for allowing us to reproduce this gure from the Innsbruck group.] current carrying wires produce a magnetic potential which traps the weak eld seekers in the bre core. In the Sussex experiments the core had a radius of 261 lm and it was found that atoms which were coupled into the guide remained within a 100 lm radius of the centre of the bre. Even though the atoms were allowed to propagate for tens of centimetres the only signi cant loss appeared to be caused by collisions with the background gas. Optical forces can also be used to guide atoms through a hollow bre. When an atom is placed in an optical eld which is detuned by a frequency D from an atomic transition and has an intensity I, the energy of the atom shifts by an amount which is proportional to I/D. This is the ac Stark shift and can give rise to what are known as dipole forces if a gradient in the intensity exists [20]. For example, if the light is tuned below the atomic resonance (negative D ) the atom will experience a force towards regions of higher intensities (high eld seeking), while it will be attracted towards lower intensities (low eld seeking) when the light is blue-detuned (positive D ). A typical application of the dipole force is the re ection of atoms from a glass surface in which blue-detuned light has been totally internally re ected. In such a situation an evanescent wave will be formed on the outside of the glass. This wave will have an intensity pro le which falls oœ exponentially as the distance from the surface is increased. Since the atoms in this situation are weak eld seeking they will be repelled from the glass, eœectively realizing an evanescent wave mirror. This process can also be used for atom wave guiding in a hollow core bre. If blue-detuned light is coupled into the walls of such a bre, an exponentially decaying evanescent light eld will leak out of the walls and into the bre core. Although a group at JILA in Boulder, Colorado were able to guide atoms through a 6 cm long piece of hollow bre [44] using this idea as the basis of their technique, there still remain a number of signi cant problems to overcome. For example, the attenuation of the light in the bre walls and the existence of laser speckle which can lead to regions in the bre where the intensity of the light is greatly reduced will both need to be addressed. However, it should eventually be possible to develop atom beamsplitters and then atom interferometers which are based on these methods. 7. New experiments With the improvements in atom interferometry that may soon be availabl e there are many new fundamental and

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