We describe several experimental studies of Bose-Einstein condensation in a dilute gas of sodium atoms. These include studies of static and dynamic behavior of the condensate, and of its coherence properties.
© Optical Society of America
The recent observation of Bose-Einstein condensation (BEC) in alkali vapors [1–3] was the realization of many long-standing goals: (1) To cool neutral atoms into the ground state of the system, thus exerting ultimate control over the motion and position of atoms limited only by Heisenberg’s uncertainty relation. (2) To generate a coherent sample of atoms all occupying the same quantum state (this was used to realize a rudimentary atom laser, a device which generates coherent matter waves). (3) To produce degenerate quantum gases with properties quite different from the quantum liquids He-3 and He-4. This provides a testing ground for many-body theories of the dilute Bose gas which were developed many decades ago but never tested experimentally. BEC of dilute atomic gases or of excitons is a macroscopic quantum phenomena with similarities to superfluidity, superconductivity and the laser phenomenon.
Bose-Einstein condensation is based on the wave nature of particles, which is at the heart of quantum mechanics. In a simplified picture, atoms in a gas may be regarded as quantum-mechanical wavepackets which have an extent on the order of a thermal de Broglie wavelength (the position uncertainty associated with the thermal momentum distribution). The lower the temperature, the longer is the de Broglie wavelength. When atoms are cooled to the point where the thermal de Broglie wavelength is comparable to the interatomic separation, then the atomic wavepackets “overlap” and the indistinguishability of particles becomes important (Fig. 1). Bosons undergo a phase transition and form a Bose-Einstein condensate, a dense and coherent cloud of atoms all occupying the same quantum mechanical state. The relation between the transition temperature and the peak atomic density n can be simply expressed as = 2.612, where the thermal de Broglie wavelength is defined as λdB = (2πħ 2/mkBT)1/2 and m is the mass of the atom.
The realization of Bose-Einstein condensation requires techniques to cool gases to sub-microkelvin temperatures and atom traps to confine them at high density and keep them away from the hot walls of the vacuum chamber. Over the last 15 years, such techniques have been developed in the atomic physics and low-temperature communities. The MIT experiment uses a multistage process to cool hot sodium vapor down to temperatures where the atoms form a condensate[2,7]. A beam of sodium atoms is emitted from an atomic beam oven at a density of about 1014 atoms per cm3, similar to the eventual density of the condensate. The gas is cooled by nine orders of magnitude from 600K to 1μK by first slowing the atomic beam, then by optical trapping and laser cooling the atoms[8,9], and finally by magnetic trapping and evaporative cooling.
The first experimental demonstrations of BEC [1–3, 11] were followed by several experimental studies and numerous theoretical papers (See Refs. [10, 12–18] for reviews). We refer to our previous review for the historical context, for an account of the developments which led to BEC, and for an overview of the techniques used to realize BEC. In this paper, we summarize some experimental studies of Bose-Einstein condensation and illustrate them with animations of experimental results. These illustrations display another important aspect of Bose-Einstein condensation. Since a Bose condensate is characterized by a macroscopic population of a single quantum state, the imaging of condensates and their dynamical behavior constitutes a dramatic visualization of quantum-mechanical wavefunctions and give wavefunctions a new level of reality. The animations and many figures in this paper have not been published before, whereas all experimental results have been previously reported in more technical papers.
2. Identifying the Bose-Einstein condensate
Bose-Einstein condensation was achieved by evaporatively cooling a gas of magnetically trapped atoms to the transition temperature. In the first observations[1,2], four features were used to identify the formation of a Bose-Einstein condensate :
- The sudden increase in the density of the cloud.
- The sudden appearance of a bimodal cloud consisting of a diffuse normal component and a dense core (the condensate).
- The velocity distribution of the condensate was anisotropic in contrast to the isotropic expansion of the normal (non-condensed) component.
- The good agreement between the predicted and measured transition temperatures.
The first three points are illustrated in Fig. 2. It shows time-of-flight pictures of expanding clouds released from the magnetic trap by suddenly switching off the trap. These images, taken during our second data run which produced Bose-Einstein condensation, were recorded by illuminating the cloud with resonant laser light and imaging the shadow of the cloud onto a CCD camera.
The animation in Fig. 3 shows the suddenness of the formation of the condensate. Each frame required a new loading and cooling cycle. Frames were taken for various final frequencies of the rf sweep which controlled the evaporative cooling process. To a very good approximation, the temperature of the cloud is linearly related to the final rf frequency. Discrete frames were interpolated in such a way that the rf frequency decreases at a constant rate during the length of the animation. Since close to the phase transition the radiofrequency was swept linearly in time, Fig. 3 represents the temporal dynamics of the cooling process during the last fraction of a second (the whole evaporation process took only seven seconds).
Fig. 4 shows the formation of the condensate observed by directly imaging the trapped condensate. One can regard Figures 2 and 3 as showing condensation in energy (or momentum space), whereas Fig. 4 demonstrates that condensation also takes place in configuration space. BEC is always a condensation phenomenon into the lowest energy state. In a homogeneous gas, however, the ground state has the same spatial extension as the excited state - therefore BEC is only a condensation in momentum space. In contrast, in a harmonic oscillator potential the ground state has the smallest extension, and one can observe BEC in configuration space. In his second paper on quantum statistics, Einstein used the notion of the saturated ideal gas to describe the condensation phenomenon. Fig. 4 shows directly the “droplet” formation when the atomic vapor reaches “quantum saturation.”
Spatial images like the one in Fig. 4 are not taken by absorption imaging, but by dispersive imaging[20, 21]. The reason for this is that the trapped condensates are very dense. If the image were to be taken on resonance, the probe light would be fully absorbed, and close to resonance, it would be strongly refracted due to the index of refraction of the trapped cloud. A solution is to go far off resonance (in our case about 350 half linewidths) where absorption is negligible and dispersion is small, and use imaging techniques which image the spatial distribution of the index of refraction. Techniques such as dark-ground imaging and phase-contrast imaging are well known in microscopy and can be applied to the imaging of atoms. A major advantage of dispersive over absorptive imaging is that one can obtain much higher signal levels for the same amount of heating. In dispersive imaging, one collects the photons which have been elastically scattered into the forward direction. The momentum transfer due to this scattering is negligible. Instead, the major heating mechanism in dispersive imaging is due to large angle Rayleigh scattering. The ratio of the rates for elastic scattering and Rayleigh scattering are proportional to the (resonant) optical density, which is about 200 in our case. Therefore, one can image about two orders of magnitude more dispersively scattered photons than absorbed photons for the same amount of heating. Since dispersive imaging is almost non-destructive, it can be used to record “real-time movies” of the dynamics of a condensate by using multiple probe laser pulses in combination with a fast CCD camera.
Fig. 5 shows the expansion of a mixed cloud, demonstrating the different properties of a condensate and the normal component. The frames were recorded by varying the time-of-flight in successive cooling cycles. They show atoms released from the cloverleaf magnetic trap. At early times the cloud is optically dense and absorbs almost all of the incident light (red color). Between 10 and 25 msec one can clearly see the isotropic expansion of the thermal cloud. This reflects the fact that the velocity distribution of a gas is isotropic irrespective of the shape of the container. In contrast, the condensate shows a strong anisotropy in the expansion related to the anisotropy of the magnetic confinement. A condensate of an ideal non-interacting gas occupies the ground state of the system. The width of the velocity distribution of the expanding cloud is given by Heisenberg’s uncertainty relation and is inversely proportional to the spatial extension of the ground state wavefunction. As a result, the aspect ratio of an expanding condensate approaches the inverse aspect ratio of the trapped condensate. For a weakly interacting condensate, the situation is different because the repulsion between the atoms weakens the trapping potential and leads to a larger size of the condensate. After the trap is switched off, the internal energy (mean-field energy) is converted into kinetic energy in an anisotropic way: the accelerating force due to the internal mean field energy is proportional to the gradient of the mean field energy and therefore proportional to the density gradient. This means that for an initially cigar shaped cloud, the radial acceleration is much larger than the axial one, and the aspect ratio of a freely expanding condensate inverts. For very long expansion times, the aspect ratio was predicted to be (π/2) times the inverse of the aspect ratio of the trapped cloud.
3. Collective excitations of a Bose condensate
Collective excitations of liquid helium played a key role in determining its superfluid properties. It is now well understood that the phonon-nature of the low-lying excitations imply superfluidity up to a critical velocity which is given by the speed of sound. The low-lying excitations of a trapped Bose condensate show discrete modes due to the small finite size of the trapped sample. They correspond to standing sound waves. The few lowest-lying excitations were studied in Boulder and at MIT .
Fig. 6 shows our first observation of collective excitations. The cloud is contracting along the axial direction while expanding radially and vice versa and therefore corresponds to a quadrupole mode of a spherical cloud. The oscillations were excited with a time-dependent modulation of the trapping potential. A variable time delay was introduced between the excitation and the release of the cloud. In this way, the free time evolution of the system after the excitation was probed. The cloud was observed by absorption imaging after a sudden switch-off of the magnetic trap and 40 msec of ballistic expansion. The measured frequency of oscillation were in excellent agreement with predictions based on the non-linear Schroedinger equation[27, 28].
In order to understand the observed damping time of 250 msec, studies have recently been extended to finite temperatures[28, 29]. Our studies were done using direct observation of the spatial oscillation by dispersive imaging. Since this method is much less destructive than absorption imaging, “real-time movies” with up to 30 pictures of the same oscillating condensate could be taken. Fig. 7 shows the observation of the axial dipole motion (center of mass motion) which was excited by periodically moving the center of the magnetic trap. The dipole motion is undamped in a harmonic trapping potential. Although the dipole mode by itself doesn’t reveal anything about the nature of the Bose condensate, an accurate measurement of its frequency is important since it is needed to normalize the other collective excitation frequencies in order to compare them with theory. Images like those in Fig. 7 allow a single-shot determination of trapping frequencies with 0.2 % precision.
When the cloud was excited by modulating the axial confinement, the quadrupole-type oscillation could be observed in the spatial domain (Fig. 8). In Fig. 9, thirteen pictures like Fig. 8 but with various delays were combined into an animation. It shows the dynamics of the condensate over a period of one second. One can clearly see the damping of the oscillations of the shape of the condensate, whereas the center-of-mass motion is undamped. Several theoretical schemes have recently been developed to describe the damping of collective excitations as a function of temperature (see references in ).
4. Realization of an atom laser
An atom laser is a device which generates an intense coherent beam of atoms through a stimulated process. It does for atoms what an optical laser does for light; whereas the optical laser emits coherent electromagnetic waves, the atom laser emits coherent matter waves. The condition of high intensity requires many particles per mode or quantum state. A thermal atomic beam has a population per mode of only 10-12 compared to values much greater than 1 for an atom laser. The realization of an atom laser therefore required methods to largely enhance the mode occupation. This was done by cooling to sub-microkelvin temperatures to the onset of Bose-Einstein condensation.
Laser light is created by stimulated emission of photons, a light amplification process. Similarly, an atom laser beam is created by stimulated amplification of matter waves. The conservation of the number of atoms is not in conflict with matter wave amplification: The atom laser takes atoms out of a reservoir and transforms them into a coherent matter wave similar to the manner in which an optical laser converts energy into coherent electromagnetic radiation. An atom laser is possible only for bosonic atoms because the accumulation of atoms in a single quantum state is a result of Bose-Einstein statistics. In a normal gas, atoms scatter among a myriad of possible quantum states. But when the critical temperature for Bose-Einstein condensation is reached, they scatter predominantly into the lowest energy state of the system. This abrupt process is closely analogous to the threshold for operation of an optical laser. The presence of a Bose-Einstein condensate causes stimulated scattering into the ground state. More precisely, the presence of a condensate with N0 atoms enhances the probability that an atom will be scattered into the condensate by a factor of N0+1, in close analogy to the optical laser.
There is some ongoing discussion what defines a laser, even in the case of the optical laser[30, 31]; e.g. it has been suggested that stimulated emission is not necessary to obtain laser radiation. In our discussion, we don’t attempt to distinguish between defining features and desirable features of a laser.
A laser requires a cavity (resonator), an active medium, and an output coupler. Various “cavities” for atoms have been realized, but the most important ones are magnetic traps (which use the force of an inhomogeneous magnetic field on the atomic magnetic dipole moment) and optical dipole traps (which use the force exerted on atoms by focused laser beams).
The purpose of the output coupler is to extract atoms out of the cavity, thus generating a pulsed or continuous beam of coherent atoms. A simple way to accomplish this is to switch off the atom trap and release the atoms. This is analogous to cavity dumping for an optical laser and extracts all the stored atoms into a single pulse. A more controlled way to extract the atoms requires a coupling mechanism between confined quantum states and a propagating mode. Such a “beam splitter” for atoms was realized using the Stern-Gerlach effect (Fig. 10) . A short rf pulse rotated the spin of the trapped atoms by a variable angle, and the inhomogeneous magnetic trapping field separated the atoms into trapped and outcoupled components. By using a series of rf pulses, a sequence of coherent atom pulses could be formed (Fig. 11). The crescent shape of the propagating pulses can be qualitatively explained as the result of the forces of gravity and of the mean field. In the absence of gravity, one would expect a hollow shell or a loop propagating mainly in the radial direction; the center is depleted by the repulsion of the trapped condensate. However, the gravitational acceleration is 5.65 MHz/cm or 27 nK/μm and is comparable to the maximum acceleration by the mean field. Therefore, atoms are not coupled out upward, resulting in a crescent instead of a loop. The two animations (Figs. 11 and 12), observed from the top and from the side, give a compete picture of the three-dimensional propagation of the outcoupled atom pulses.
Interference between two condensates
The spatial coherence of individual outcoupled pulses was proven in an interference experiment. Long range coherence is closely related to the existence of a macroscopic phase. As with any quantum mechanical phase, one cannot measure the absolute phase of a single atom laser pulse, but only the relative phase between two pulses. It is a non-trivial question whether two independent condensates have a well-defined relative phase. The analogous question for superfluids is this: If two superfluids are brought into a weak contact, is there a relative phase which would result in an observable dc Josephson current? There is now general agreement that even if the phase is not initially defined, every realization of the experiment will show a distinct phase. This is often called spontaneous symmetry breaking, and is a consequence of the quantum measurement process. In the case of two independent overlapping condensates one expects to observe a high-contrast interference pattern, but the phase of the pattern should randomly vary between different realizations.
Two independent condensates were produced by evaporatively cooling atoms in a double-well potential. This potential was created by magnetic trap divided in half by optical forces of a focused far-off resonant laser beam. After the trap was switched off, the atom clouds fell down due to gravity, expanded ballistically, and eventually overlapped (Figs. 13 and 14). The interference pattern was observed using a resonant light beam which was absorbed preferentially at the interference maxima. As shown in Fig. 15, the interference pattern consisted of straight lines with a spacing of about 15 μm, a huge length for matter waves; the matter wavelength of atoms at room temperature is only 0.05 nm, less than the size of an atom. The interference experiment provided direct evidence for first-order coherence and long-range correlations, and for the existence of a relative phase of two condensates. Higher order coherences were observed in measurements of the internal energy of the condensate and in measurements of three-body collisions. All these studies were consistent with the standard assumption that a Bose condensate is coherent to first and higher order and can be characterized by a macroscopic wavefunction, but further studies are worthwhile and should lead to a more accurate characterization of the coherence properties of condensates.
The basic phenomena of Bose-Einstein condensation in gases was predicted 70 years ago. The experimental realization required, first, the identification of an atomic system which would stay gaseous at the conditions of BEC and not preempt BEC by forming molecules or clusters, and second, the development of cooling and trapping techniques to reach those conditions. After this had been accomplished, several studies of BEC confirmed theories which had been formulated decades ago but had never been experimentally tested. But BEC has already gone beyond the confirmation of old theories; it has motivated several extensions of the theory. A microscopic picture has been developed of how a macroscopic phase is created . Bose condensates might become a model system for dissipation and coherence in quantum systems. Different statistical ensembles have been discussed. They agree in the thermodynamic limit but not for small Bose condensates. The theory of a weakly-interacting Bose gas has been extended to the situation of a harmonic trapping potential. The collapse of a condensate with attractive interactions has been studied, both theoretically[39, 40] and experimentally. Furthermore, the direct visualization of macroscopic wavefunctions has added a very intuitive component to the understanding of many-body physics.
Finally, we want to mention some of the challenges ahead: to achieve quantum degeneracy for different atoms including fermionic isotopes, to study various mixtures of Bose condensates, to observe vortices, superfluidity, and Josephson tunneling, and to improve the output characteristics of the atom laser. Ultimately, Bose condensates and atom lasers should find use in atom optics and precision experiments.
We are grateful to our collaborators with whom the experiments have been carried out: M.R. Andrews, K.B. Davis, N.J. van Druten, S. Inouye, M.-O. Mewes, H.-J. Miesner, D.M. Stamper-Kurn, and C.G. Townsend. This work was supported by the Office of Naval Research, the National Science Foundation, Joint Services Electronics Program (ARO), and the David and Lucile Packard Foundation.
References and links
2. K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75, 3969 (1995). [CrossRef]
3. C.C. Bradley, C.A. Sackett, and R.G. Hulet, “Bose-Einstein Condensation of Lithium: Observation of Limited Condensate Number”, Phys. Rev. Lett. 78, 985 (1997). [CrossRef]
4. K. Huang, “Imperfect Bose Gas”, in Studies in Statistical Mechanics, vol. II, edited by J. de Boer and G.E. Uhlenbeck (North-Holland, Amsterdam, 1964) p. 3.
5. A. Griffin, D.W. Snoke, and S. Stringari (editors), Bose-Einstein Condensation (Cambridge University Press, Cambridge, 1995).
6. K. Huang, Statistical Mechanics, second edition (Wiley, New York, 1987).
7. M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.M. Kurn, D.S. Durfee, and W. Ketterle, “Bose-Einstein condensation in a tightly confining dc magnetic trap”, Phys. Rev. Lett. 77, 416 (1996). [CrossRef]
8. Background information for the 1997 Nobel prize in physics for laser cooling, http://www.nobel.se/announcement-97/phyback97.html
9. Links to research groups with atom traps, http://www-atoms.physics.wisc.edu/OtherSites.html
10. C.G. Townsend, N.J. van Druten, M.R. Andrews, D.S. Durfee, D.M. Kurn, M.-O. Mewes, and W. Ketterle, “Bose-Einstein condensation of a weakly-interacting gas”, in Ultracold Atoms and Bose-Einstein-Condensation, 1996, K. Burnett, ed., OSA Trends in Optics and Photonics Series, Vol. 7 (Optical Society of America, Washington D.C., 1996) p. 2.
11. Indirect evidence was reported in: C.C. Bradley, C.A. Sackett, J.J. Tollet, and R.G. Hulet, “Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions”, Phys. Rev. Lett. 75, 1687 (1995). [CrossRef]
12. W. Ketterle, M.R. Andrews, K.B. Davis, D.S. Durfee, D.M. Kurn, M.-O. Mewes, and N.J. van Druten, “Bose-Einstein condensation of ultracold atomic gases”, Phys. Scr. T66, 31 (1996). [CrossRef]
13. BEC home page of the Georgia Southern University, http://amo.phy.gasou.edu/bec.html
14. Home page of our group, http://amo.mit.edu/~bec
15. N.J. van Druten, C.G. Townsend, M.R. Andrews, D.S. Durfee, D.M. Kurn, M.-O. Mewes, and W. Ketterle, “Bose-Einstein condensates - a new form of quantum matter”, Czech. J. Phys. 46 (S6), 3077 (1996). [CrossRef]
16. D.S. Jin, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, “Quantitative Studies of Bose-Einstein Condensation in a Dilute Atomic Vapor”, Czech. J. Phys. 46 (S6), 3070 (1996). [CrossRef]
17. C.A. Sackett, C.C. Bradley, M. Welling, and R.G. Hulet, “Bose-Einstein Condensation of Lithium”, Braz. J. Phys. 27, 154 (1997).
18. N.P. Proukakis, K. Burnett, M. Edwards, R.J. Dodd, and C.W. Clark, “Theory of Bose-Einstein condensed trapped atoms”, in Ultracold Atoms and Bose-Einstein-Condensation, 1996, K. Burnett, ed., OSA Trends in Optics and Photonics Series, Vol. 7 (Optical Society of America, Washington D.C., 1996) p. 14.
19. A. Einstein, “Quantentheorie des einatomigen idealen Gases. II”, Sitzungsber. K. Preuss. Akad. Wiss. Phys. Math. Kl, 3 (1925).
21. M.R. Andrews, D.M. Kurn, H.-J. Miesner, D.S. Durfee, C.G. Townsend, S. Inouye, and W. Ketterle, “Propagation of sound in a Bose-Einstein condensate”, Phys. Rev. Lett. 79, 553 (1997). [CrossRef]
22. E. Hecht, Optics, 2nd edition (Addison-Wesley, Reading, 1989).
24. A. Griffin, Excitations in a Bose-condensed liquid (Cambridge University Press, Cambridge, 1993). [CrossRef]
26. M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.M. Kurn, D.S. Durfee, C.G. Townsend, and W. Ketterle, “Collective Excitations of a Bose-Einstein condensate in a Magnetic Trap”, Phys. Rev. Lett. 77, 988 (1996). [CrossRef]
28. D. Stamper-Kurn, H.-J. Miesner, S. Inouye, M.R. Andrews, and W. Ketterle, “Excitations of a Bose-Einstein Condensate at Non-Zero Temperature: A Study of Zeroth, First, and Second Sound”, Phys. Rev. Lett. (1998), submitted.
29. D.S. Jin, M.R. Matthews, J.R. Ensher, C.E. Wieman, and E.A. Cornell, “Temperature-Dependent Damping and Frequency Shifts in Collective Excitations of a Dilute Bose-Einstein Condensate”, Phys. Rev. Lett. 78, 764 (1997). [CrossRef]
30. H.M. Wiseman, “Defining the (atom) laser”, Phys. Rev. A 56, 2068 (1997). [CrossRef]
31. D. Kleppner, Phys. Today, Aug. 1997, p. 11; Jan. 1998, p. 90.
32. M.-O. Mewes, M.R. Andrews, D.M. Kurn, D.S. Durfee, C.G. Townsend, and W. Ketterle, “Output coupler for Bose-Einstein condensed atoms”, Phys. Rev. Lett. 78, 582 (1997). [CrossRef]
33. P.W. Anderson, “Measurement in Quantum Theory and the Problem of Complex Systems”, in The Lesson of Quantum Theory, J.d. Boer, E. Dal, and O. Ulfbeck, ed. (Elsevier, Amsterdam, 1986) p. 23.
36. W. Ketterle and H.-J. Miesner, “Coherence properties of Bose-Einstein condensates and atom lasers”, Phys. Rev. A 56, 3291 (1997). [CrossRef]
37. E.A. Burt, R.W. Ghrist, C.J. Myatt, M.J. Holland, E.A. Cornell, and C.E. Wieman, “Coherence, Correlations, and Collisions: What One Learns About Bose-Einstein Condensates form Their Decay”, Phys. Rev. Lett. 79, 337 (1997). [CrossRef]
38. P. Navez, D. Bitouk, M. Gajda, Z. Idziaszek, and K. Rzazewski, “Fourth Statistical Ensemble for the Bose-Einstein Condensate”, Phys. Rev. Lett. 79, 1789 (1997). [CrossRef]
39. P.A. Ruprecht, M.J. Holland, K. Burnett, and M. Edwards, “Time-dependent solution of the nonlinear Schršdinger equation for Bose-condensed trapped neutral atoms”, Phys. Rev. A 51, 4704 (1995). [CrossRef]