Ketterle and C. Wieman in but its roots are more that 75 years older. The idea remained abstract for a long time even for the authors; Einstein himself wrote: The theory is pretty but is there also some truth to it? Solid line shows population of atomic ground state versus temperature in units of critical temperature TC TC depends on atomic density N. Dots are experimental results from [3] reproduced with kind permission of the authors.
Below certain critical temperature, the integral in Eq. This populating of the ground state occurs as phase transition at a critical temperature, as visualized in Fig.
Matter wave picture The de Broglie wavelength of the particles depends on their momentum, and therefore, on the temperature. One speaks then about quantum degeneracy, i. The two solutions are not equivalent as the latter enhances the inter-particle interactions and one cannot speak any more of free particles. It is instructive to compare orders of magnitude typical for the thermal and condensed gas samples.
Note that for treating an atom as boson or fermion, the statistical prop- erties of an atom as a whole need to be taken into account. Kammerlingh Onnes awarded with Nobel Prize in Penrose and L.
Onsager, s describing long-range order in the highly correlated bosonic system. Lee, D. Richardson in , Nobel Letokhov and A. Balykin, V. Letokhov, and W. Chu and W. Dalibard, D. Pritchard, and S. Phillips, S. Chu, and C. Cohen-Tannoudji awarded with Nobel Prize for development of cooling methods.
Quest for BEC in atomic hydrogen The search for BEC in hydrogen was dictated by the desire to study a medium that remains in a gas phase even at lowest temperatures. Nosanov and W. Kleppner and T. Greytak MIT and by I. Silvera and J. Hess develops evaporation cooling technique. How to cool atoms?
Cooling of atomic beam Let us start by showing the possibility of stopping the atomic beam. Consider a beam of atoms and a counter-propagating laser beam at a fre- quency of the atomic resonance Fig. Principle of laser deceleration of collimated atomic beams. The momenta of absorbed laser photons are therefore ac- cumulating to a nonzero value, while the net momentum change in spontaneous emission is zero.
This gives rise to a light-pressure force in a direction of a laser beam. Subsequent relaxation to the ground energetic state by spontaneous emission also changes the atomic momentum. There is asymmetry between these two processes due to the fact that all absorbed photons have equal wave-vectors, whereas the spontaneously emitted photons have arbitrary direction.
Each photon causes only minimal change of atomic velocity so that one needs some 20 photon absorptions to stop an atom. An atom can be stopped in 1 ms, on a distance of 0. Cooling of atomic gas For the case of gas contained in a cell, one has atoms moving in arbitrary directions and thus to cool them along e. Principle of 1D laser cooling of atomic gas. The forces associated with the two beams of Fig. The same process can be analyzed from the energetic point of view where one notes that less energy is absorbed than reemitted, which leads to cooling of atoms.
It has thus a character of a viscous force. How to trap cold atoms? To hold atoms in a prescribed spatial position to trap them one needs a position-dependent force. This is done by tuning atomic levels into, or out of, resonance with laser beams by the Zeeman shift of atomic levels, as shown in Fig. The cooling process is based on a momentum transfer process and dissipation of energy in a cyclic pro- cesses of absorption and spontaneous emission.
This leads to the decrease of average atomic velocity, which tends to zero. At the same time, however, velocity dispersion is not zero and even increases. Principle of a 1D magneto-optical trap. On the other hand, the radiation imprisonment, i. As the magnetic forces are conservative, another cooling mechanism must be invoked.
The required cooling is done by forced evaporation of most energetic atoms. A 3D realization of the magneto-optical trap. The remaining atoms should thermalize to lower temperature. This happens indeed, if the rate of collisions between atoms leading to thermalization is high enough, while the rate of other collisions, which could expel cold atoms from the trap, is so low as to provide enough time for the thermalization process.
The evaporative cooling technique invented for the case of hydrogen has been fruitfully applied to other elements as well. Achieving Bose-Einstein condensation The standard way to achieve Bose-Einstein condensation of atomic gases consists of three steps: 1.
Cooling atoms to the lowest temperatures. Next, a magnetic trap is used and atoms are cooled further by a forced evaporation below nK. By lowering the transition frequency, less energetic atoms in a trap are addressed.
After each evaporation step, the remaining atoms are allowed to thermalize so that they can reach equilibrium distribution with appropriately lower temperature. After about 10 ms of free expansion the cloud is illuminated with resonant light and the shadow of the atomic sample is recorded by a CCD camera. In Fig. Cornell and C. Wiemann group in JILA [4]. The table below presents comparison of parameters for the BEC obser- vation in the liquid He and in alkali atoms.
It clearly indicates that the alkali case is much closer to the ideal free-particle model. Liquid helium vs. We refer also to the Nobel Lectures of the last year laureates [9]. When imaged as illustrated in Fig. Andrews et al. Copyright American Association for the Advancement of Science.
Experiments with BEC 8. Coherent matter-wave optics First experiments with BEC concentrated on the coherent properties of the condensate and continued the development of atom optics, demonstrated already before the advent of BEC. A property of coherent waves, well known in standard and atom optics, is their ability to interfere. Observation is performed by a resonant imaging beam block arrow that allows recording of shadow images. Appearance of the atom-laser beams emitted by various lasers.
Ketterle b Yale laser reprinted with permission from ref. Anderson and M. Thus, the Hamiltonian does not include the term corresponding to the interaction between two direct magnetoexcitons in a single GL.
In equation 5. Substituting equation 5. Below, we estimate this temperature. According 0 to equation 6. In both the cases, the polaritons are considered in a harmonic potential trap. At low temperature, the Hamiltonian of two-dimensional exciton polaritons in a slowly varying external parabolic potential acting on the exciton energy and bringing it into resonance with a cavity photon mode corresponds directly to the case of a weakly interacting Bose gas with an effective mass and effective pairwise interaction in a harmonic potential trap.
The mixing with the photon states leads to a smaller lifetime for high-energy states, that is, an evaporative cooling effect, but does not fundamentally prevent condensation. In our calculations, we have assumed that the system under consideration is in thermal equilibrium. This assumption is valid if the relaxation time is less than the quasiparticle lifetime.
Although the magnetopolariton lifetime is short, thermal equilibrium can be achieved within the regime of a strong pump. Porras et al. We expect a similar characteristic time for magnetopolariton—magnetoexciton scattering in graphene.
However, the consideration of pump and decay in a steady state may lead to results which are different from those presented in this paper. Even though the estimated BEC critical temperatures for polariton condensates are relatively high, the current experiments are still performed at cryogenic temperatures.
The reason for this is that the transition under consideration is determined by the Rabi splitting Littlewood The Rabi splitting is limited by fundamental properties of the material. References Abrikosov, A. Agranovich, V. Moscow, Russia: Nauka. Anderson, M. Science , — A 44, — B 79, B 63, B 70, B 73, Matter 19, B 77, B 78, A , — B 80, Matter 16, R Castro Neto, A. B 58, — B 75, R. Natl Acad. USA , 15 —15 A 24, B 64, Eisenstein, J. Nature , — Low Temp. B 75, B 30, — Amsterdam, The Netherlands: Elsevier.
A 54, — A 56, — The observed clouds To make use of such a novel source, new atom optical were reflected up to three times before they moved out elements that preserve the atomic coherence are being of the field of view due to a slight slope in the orienta- developed. A thorough understanding of the interaction tion of the light sheet. Figure 3 see also Electronic Sup- of coherent matter waves with these elements is elemen- plementary Material shows a time-of-flight series of tary to further experiments.
In comparison with single- BECs bouncing off the mirror. The light sheet shows up atom optics these interactions are much more complex as a sharp lower edge in the fourth frame. Further increasing the drop that need to be developed and characterized in the field height or reducing the laser power results in a partial atom optics.
Mirrors, beamsplitters and optical fibres are transmission through the mirror. As the condensate reapproached the initial altitude, it Nonetheless, similar elements in atom optics have only developed an interesting structure of bright and dark been developed recently and their application to BEC is fringes frames 6—8 in Fig. Electronic Material. These fringes do not occur for tem- peratures above the critical temperature for BEC. They can be explained as an interference pattern.
Such pat- Atomic mirrors terns arise whenever multiple waves are overlapped at one point. The waves can add constructively, forming re- Probably the most important optical element is the gions of large amplitude, or destructively, cancelling mirror. It may have a similar importance in the field of each other out.
Since ment regimes, since increasing the power in the beam they have travelled along different paths during the leads to tighter confinement of the atoms. This The experiments described here studied the transfer phase leads to the observation of dark and bright inter- of BECs into such a waveguide. To avoid heating the ference fringes. Indeed numerical results agree well with atoms, the final stages of RF-evaporation towards BEC the experimental observations and clearly explain the were carried out in a combined potential consisting of self-interference structure observed in BECs.
Most im- both a magnetic trap and the hollow laser beam. By portantly, these interferences prove the persistence of ramping down the magnetic trapping field, the BEC matter—wave coherence for BECs reflected off the dipole was subsequently transferred into the pure waveguide. Figure 4 see also Electronic Supplementary Material In addition to creating an atom mirror with reflectivi- shows examples of these measurements for evolution ty close to unity, partially reflecting mirrors and a phase times of atoms in the waveguide up to ms.
The ex- shifter can be created by reducing the intensity of the pansion of a BEC is in excellent agreement with the the- light sheet. These elements can in the future be applied oretical prediction Castin and Dum of an expan- to develop atom interferometers for BECs.
They may sion dominated by the energy related to the atomic inter- even be used to systematically characterize the coher- actions. Hence the coherence of the cloud during the first few milliseconds after trans- While most of the experimental work so far has concen- fer into the waveguide was investigated with an interfer- trated on BECs in three-dimensional systems, there is ometer scheme.
A An interferometer is a device that superimposes two one-dimensional waveguide for atoms is the equivalent wave phenomena to produce a pattern of constructive of a single-mode optical fibre, a common tool in optics.
Such a pattern can The transfer of a BEC into a one-dimensional system then be analysed to deduce information about the initial may also provide important information about the funda- waves. In our case the BEC itself corresponds to the mental nature of quantum gases Petrov et al. A initial wave. To create an interferometer the BEC has wealth of new phenomena are expected to occur in 1D to be divided into two samples and brought to overlap collisional physics, which may be studied during the ex- at some later point.
Figure 5 see also Electronic Sup- pansion of a dense ensemble in a waveguide. These repel atoms from the light. If atoms can be loaded into light pulses are equivalent to the beamsplitters used in an the centre of such a beam they are tightly confined in the optical interferometer.
They preserve the internal state of radial two directions and free to move in the longitudi- the atom while changing its velocity. The two conden- nal third one. Such laser beams, sometimes also called sate wavepackets created by the first pulse separate dur- doughnut beams, can readily be produced. They are well ing the subsequent free evolution time. The situation is differ- ent in weakly interacting atomic gases. On the other hand, evidence for the super- fluidity of these systems remained elusive.
The observa- tion of these phenomena in dilute Bose systems provides an ideal testing ground for theoretical approaches and in the future, development of these theories might lead to a deeper understanding of superfluidity in more complex systems. The observation of phonon-like collective exci- tations was a first step towards establishing this behav- iour Mewes et al. Only very recently effects more directly associated with superfluidity, such as the direct observation of frictionless flow Raman et al.
Scissors mode excitations A path to probe the superfluidity of a weakly interacting Bose condensate using the so-called scissors mode Fig.
The broad arrows indicate the two light pulses that act as Stringari The method is well known in nuclear beamsplitters. A sample of the observed interference signal is physics but had not previously been used in the case of shown an atomic gas.
The general idea is based on our knowledge of the behaviour of superfluids. Unless singularities such as exit ports of the interferometer as shown in Fig. The displace- field of a superfluid can not have any rotation. In re- ment led to interference fringes in both exit ports due to sponse to a perturbation it must hence display irrotation- the well defined phase within the condensate.
The exis- al flow. Imagine giving a superfluid in a container a tence of such a phase demonstrates the coherence of small shake. It will then display some form of irrotation- BEC after an evolution in the waveguide Bongs et al.
If a known pat- Shaking a normal fluid or gas in a the behaviour of BEC in geometries of lower dimension- container will result in both rotational and irrotational ality. A wealth of new phenomena, such as phase-fluctu- flow. Again, the characteristic frequencies of motion can ating condensates, are predicted to occur in this situation be calculated for known flow patterns. Hence a reaction and are currently being explored both experimentally at two frequencies in the case of a normal gas is expect- Dettmer et al.
The scissors mode is a type of excitation that produc- ly Petrov et al. It is called scissors mode since the sample is tilted from its equilibrium position and then allowed to swing Excitations and superfluidity in BECs back and forth around its centre like the blades of a pair of scissors, as shown in Fig. These phenomena were the transition to BEC. In such a cloud two oscillation fre- discovered in the s but only in was it quencies were observed in response to a scissors mode ex- shown that BEC is closely related to these observations citation.
Then an atomic sample was cooled further, below Bogoliubov A comprehensive overview of the BEC transition temperature. In this case only one os- superfluid behaviour can be found in Tilley and Tilley cillation frequency, corresponding to the irrotational flow The white lines represent a guide to the eye This experiment provided clear evidence for the ex- pected connection between BEC and superfluidity in gaseous condensates.
In addition it provides a means of probing the interaction between the condensate and non- condensed, thermal atoms in its vicinity. Generally was used to modify the confining trap briefly on one side they are waves which propagate through a medium with- of the BEC. If the laser power and interaction time are out changing their shape, i. This process is commonly called optics are examples of such a behaviour.
Once a phase has served in BECs Burger et al. In this will propagate with a speed corresponding to the size of case the small nonlinear interactions between the atoms the phase jump. In addition a density wave is formed that cause the stability of the waves. In the case of attractive propagates in the opposite direction. These small struc- interactions, bright solitons that correspond to density tures only become visible when the atomic sample is re- maxima are formed.
In the more common case of repul- leased from its magnetic container and expands. Figure 7 sive interactions, dark solitons propagate as density min- see also Electronic Supplementary Material shows the ima through the sample. One can imagine that the repul- evolution of a soliton within the condensate. Hence the repulsion prevents the tons. It will be interesting to observe whether solitons density minimum from changing its shape. It will also be exciting to observe the behaviour of fluidity of the system.
Nonetheless their creation and solitons in one-dimensional geometries such as the propagation can only be explained by the equivalence waveguide presented above. However in a real situa- tion such a phase change is not abrupt and the slowly The study of vortices was one of the prime motivations changing phase shift across it leads to a velocity of the for the development of the Gross-Pitaevskii theory soliton, which — as a characteristic pattern for dark soli- Pitaevskii , which has formed the basis for most tons — is smaller than the speed of sound.
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