Download Bose-Einstein condensation in dilute atomic gases

Bose-Einstein condensation in dilute atomic gases:
atomic physics meets condensed matter physics
W. Ketterle
Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA
Abstract
Bose-Einstein condensed atomic gases are a new class of quantum fluids. They are produced by cooling a dilute
atomic gas to nanokelvin temperatures using laser and evaporative cooling techniques. The study of these quantum
gases has become an interdisciplinary field of atomic and condensed matter physics. Topics of many-body physics
can now be studied with the methods of atomic physics. Many long-standing predictions of the theory of the weakly
interacting Bose gas have been verified, including thermodynamic properties of the phase transition and dynamic
properties such as shape oscillations and sound propagation. Stimulated light scattering was used to determine the
dynamic structure factor both in the phonon and free-particle regime. Atomic Bose condensates show a variety
of novel phenomena which include multi-component spinor condensates, magnetic domain formation, miscibility
and immiscibility of quantum fluids, and finite-size effects.
Keywords: Bose-Einstein condensation; collective excitations; light scattering; spinor condensates
1. Introduction
To receive the Fritz London Memorial Award
is very special to me—it is the token of a wonderful experience I have had during the last few
years. I was trained as an atomic physicist and,
in the early ’90s, focused on the development of
trapping and cooling techniques trying to reach
lower temperatures of atomic gases. In 1995, while
cooling gaseous clouds to sub-microkelvin temperatures, we realized an intriguing new many-body
system—the gaseous Bose-Einstein condensate.
The years since then have been a rewarding intellectual experience—to learn and study the old and
new physics of a new quantum fluid. But those
Preprint submitted to Physica B
years have also been an enormous personal experience. I made many new friends and got to know
many new colleagues who taught me about lowtemperature physics. I want to thank all of them,
and also acknowledge the past and present collaborators who have shared both the excitement and
the hard work: M.R. Andrews, A.P. Chikkatur,
K.B. Davis, D.S. Durfee, A. Görlitz, S. Gupta,
Z. Hadzibabic, S. Inouye, M.A. Joffe, M. Köhl,
C.E. Kuklewicz, A. Martin, M.-O. Mewes, H.-J.
Miesner, R. Onofrio, T. Pfau, D.E. Pritchard, C.
Raman, D.M. Stamper-Kurn, J. Stenger, C.G.
Townsend, N.J. van Druten, and J. Vogels.
This award is even more special to me since
it carries the name of Fritz London. After Bose12 October 1999
Einstein condensation was predicted in 1925, many
people regarded it as a mathematical oddity. It was
Fritz London who connected it with a real phenomenon when he suggested in 1938 that the recently discovered superfluidity of liquid helium was
related to Bose-Einstein condensation [1,2].
BEC in dilute atomic gases is a new subfield
which is interdisciplinary between atomic and condensed matter physics. This subfield deserves the
name ultralow temperature physics, since the temperatures of the gaseous BEC systems are usually
in the nanokelvin range. Ultracold atomic gases
and nuclear spin systems (for which picokelvin spin
temperatures were obtained in Helsinki [3]) are
the coldest matter studied so far. Alternatively,
one might call the gaseous BEC ultralow density
condensed matter physics. Bose-Einstein condensation in atomic gases is observed at number densities of 1014 cm−3 which are similar to a room
temperature gas at a pressure of 10−2 mbar! The
low density is crucial for the (meta-) stability of
the cold atomic gas against inelastic collisions, in
particular three-body recombination. The system
develops many-body correlations at such low density because the thermal de Broglie wavelength at
nanokelvin temperatures exceeds 1 µm, more than
the average distance between atoms.
Research on gaseous BEC can be divided into
two areas: In the first (which could be labeled “The
atomic condensate as a coherent gas”, or “Atom
lasers”), one would like to have as little interaction as possible—almost like photons in an optical laser. Thus the experiments are preferentially
done at low densities. The Bose-Einstein condensate serves as an intense source of ultracold coherent atoms for experiments in atom optics, in precision studies or for explorations of basic aspects of
quantum mechanics. The second area could be labeled as “BEC as a new quantum fluid” or “BEC
as a many-body system”. The focus here is on the
interactions between the atoms which are most
pronounced at high densities. In the spirit of the
London Award Lecture, I will put the emphasis on
the second area and illustrate with some examples
how novel many-body physics is explored with the
methods of atomic physics.
2. Studies of Bose-Einstein condensation
Since long, macroscopic quantum phenomena
have been studied in strongly interacting systems,
most notably in the superfluids 4 He and 3 He, and
in superconductors. The quest to realize BEC in
a dilute weakly interacting gas was pursued in
at least three different directions: liquid helium,
excitons and atomic gases. Experimental [4] and
theoretical work [5] showed that the onset of superfluidity for liquid helium in Vycor shows features of dilute-gas Bose-Einstein condensation. At
sufficiently low coverage, the helium adsorbed on
the porous sponge-like glass behaved like a dilute
three-dimensional gas. Excitons, which consist of
weakly-bound electron-hole pairs, are composite
bosons. The physics of excitons in semiconductors is very rich and includes the formation of an
electron-hole liquid and biexcitons. As nicely discussed in [6,7], there are systems, most notably
Cu2 O, where excitons form a weakly interacting
gas with a lifetime long enough to equilibrate to a
Bose-Einstein distribution and to show evidence
for Bose-Einstein condensation [8].
Dilute atomic gases are distinguished from the
condensed-matter systems discussed above by
the absence of strong or complex interactions.
Interactions at the density of a liquid or a solid
considerably modify and complicate the nature of
the phase transition. Early suggestions [9,10] that
spin-polarized hydrogen would remain gaseous
down to zero temperature triggered several experimental efforts, most notably by Silvera and
Walraven in Amsterdam [11], and Greytak and
Kleppner at MIT who finally accomplished BEC
in 1998 [12,13]. Major efforts have also been underway to reach quantum degeneracy in a twodimensional gas of spin-polarized hydrogen which
was observed in 1998 by the Turku/Moscow collaboration [14]. Most of the current efforts focus
on alkali atoms. BEC in alkali gases was observed
2
shadow cast by the atom cloud is imaged onto
a CCD camera. In the latter case, dispersively
scattered photons are collected creating an image
of the spatially varying index of refraction. The
BEC phase transition can be directly observed in
the spatial domain [21]. Fig. 1 shows a series of
such spatial images above and below the phase
transition. They show the sudden appearance of
a high-density core of atoms in the center of the
distribution—the Bose-Einstein condensate. Lowering the temperature further, the condensate
number grows and the thermal wings of the distribution become shorter. Finally, the temperature
drops to the point where only the central peak
remains.
Similarly, the BEC phase transition can be observed by imaging the shadow cast by an atom
cloud which expands ballistically after suddenly
switching off the magnetic trap. The signature of
BEC is the sudden appearance of a slow component
with anisotropic expansion [15,16]. This can be regarded as observing BEC in momentum space.
250
m
200 nK
2 K
Lower Temperature
Fig. 1. Phase contrast images of a trapped sodium gas
across the BEC phase transition. As the final radio frequency used in evaporative cooling is lowered, the temperature is reduced (left to right). Images show the onset of
Bose condensation, the growth of the condensate fraction
and contraction of the thermal wings, and finally a pure
condensate with no discernible thermal fraction. The axial
and radial frequencies are about 17 and 230 Hz, respectively.
in 1995, first for rubidium in Boulder [15], and a
few months later for sodium at MIT [16]. At the
same time, evidence for quantum degeneracy in
lithium was obtained by the Rice group [17]. Since
then, more than fifteen other groups have generated alkali Bose condensates of either rubidium or
sodium [18].
Alkali atoms are precooled by laser cooling and
collected in an optical trap. After transfer into
a magnetic trap, further evaporative cooling reduces the temperature to 1 µK or below where
the BEC phase transition is reached [19,20]. Experiments with alkali atoms don’t need cryogenic
cooling. The combination of ultrahigh vacuum
and magnetic trapping provides enough insulation
between nanokelvin atoms and room-temperature
walls. Thermal radiation is not important because
the atoms can only absorb the negligible fraction of the impinging radiation which is resonant
with an atomic transition. A room-temperature
vacuum chamber allows for easy optical access to
the atoms and the implementation of powerful
imaging techniques.
Atom clouds are observed either by absorptive
or dispersive techniques. In the first case, the
3. Sound in gaseous Bose-Einstein
condensates
Collective excitations of liquid helium played a
key role in determining its superfluid properties
[22,23]. The low-lying excitations are phonon-like
and are characterized by the speed of sound. In
a Bose-Einstein condensate, the speed of sound
could be directly determined by creating localized density perturbations using a focused faroff-resonant blue detuned laser beam [24]. Such
light creates a repulsive potential due to the ac
Stark effect. The repulsive optical dipole force expelled atoms from the center of the condensate,
creating two density peaks which propagated symmetrically outward. Fig. 2 shows the propagation
of density perturbations observed by sequential
phase-contrast imaging of a single trapped cloud.
We observed one-dimensional axial propagation
of sound at a constant velocity near the center of
3
t<0
For a more general discussion of sound, we have
to include finite temperatures and a hierarchy of
length scales. The presence of the normal cloud in
addition to the condensate can be described by a
two-fluid model, similar to the normal and superfluid phases of liquid helium. Two fluids should
give rise to two different modes. In the collisionless
regime, the wavelength of the excitation is much
smaller than the mean-free path λ̄ex lmfp . This
regime applies at zero temperature and at low densities of the thermal cloud. In this regime, the thermal cloud behaves ballistically and does not show
collective (sound-like) behavior whereas the condensate shows Bogoliubov sound (sometimes also
called zero sound).
At higher densities of the normal component,
when λ̄ex lmfp , both collective modes are hydrodynamic, and one expects two phonon-like excitations which are the in-phase and out-of-phase
oscillations of two fluids (the normal fraction and
the superfluid). The presence of two hydrodynamic
modes is similar to the case of bulk superfluid 4 He,
where they are known as first and second sound.
Superfluid 4 He has a small coefficient of thermal
expansion. Thus the two eigenmodes decouple into
density modulations (first sound) and temperature
modulations (second sound), with both fluids participating equally in both modes. In contrast, a gas
has a large coefficient of thermal expansion. This
results in the oscillations of each fluid being nearly
uncoupled. The in-phase oscillation, which is analogous to first sound, involves mainly the thermal
cloud. The out-of-phase oscillation, which is analogous to second sound, is confined mainly to the
condensate [26,27].
First and second sound occur only in the socalled hydrodynamic regime, where the mean-free
path is shorter than the wavelength of sound. The
longest possible wavelength is given by the size
of the sample. These low-lying excitations are observed as shape oscillations. The early studies of
sound in a BEC were in the collisionless regime.
In the finite-temperature experiment at MIT, the
hydrodynamic regime was approached for the first
time [28]. The onset of hydrodynamic behavior was
t=0
Density (arb. unit.)
t>0
-200
0
200
Position (µm)
Fig. 2. Observation of sound propagation in a Bose-Einstein
condensate. (top) Basic scheme for exciting wave packets
in a condensate. A condensate is confined in the potential
of a magnetic trap. At time t = 0, a focused, blue-detuned
laser beam is suddenly switched on and, by the optical
dipole force, creates two positive perturbations in density
which propagate at the speed of sound for t > 0. (bottom) Profiles through images taken every 1.3 ms, beginning
1 ms after perturbing the trapping potential by focusing
a far-off resonant laser beam into the center of the cloud.
The perturbation created the “dip” in the center, and two
“blips” traveled outward with the speed of sound. Figures
adapted from ref. [24]
the cloud, where the axial density varies slowly.
The density dependence of the speed of sound cs
was studied using adiabatically expanded or compressed condensates, yielding maximum condensate densities n0 ranging from 1 to 5 ×1014 cm−3 .
The data were in good agreement with the prediction of Bogoliubov theory, cs = (4πh̄2 an/m2 )1/2
where m is the atomic mass, a the scattering
length, and n the radially averaged density [25].
These experiments were done with almost pure
condensate (at temperatures well below the critical temperature). The spatial extent of the propagating density perturbations was about 20 µm.
4
Light scattering from a condensate
indicated by collisional frequency shifts and damping of the thermal cloud signaling the transition towards first sound. An anti-symmetric dipole oscillation was observed where the condensate and the
thermal cloud oscillated out of phase - this mode
has analogies to second sound [29].
The discussion of sound demonstrates that
atomic condensates and liquid helium are in many
respects complementary. Properties which were
difficult to measure in liquid helium were easy in
Bose condensed gases, and vice versa. The early
observation of second sound in superfluid 4 He was
dramatic evidence for the presence of two fluids
and played a major role in developing the theory
of superfluidity. In contrast, in the trapped Bose
gases, a clear signature of second sound has yet to
be observed. However, the coexistence of the normal and the superfluid components was directly
observed by imaging the cloud, either trapped
or in ballistic expansion. Other examples are superfluidity and vortices. Their observations in
liquid helium were clear evidence for the presence
of a macroscopic phase. In the gaseous systems,
the study of vortices and superfluid flow is more
difficult, partially due to the mesoscopic size of
the system. However, a macroscopic phase was
directly observed in “matter wave” interference
experiments [30–32].
(b)
(a)
Condensate
Measure momentum q
and frequency ν
dynamic
structure
factor
S(q,ν)
Laser light
+ excitation
analogous to neutron scattering from 4He
(c)
Optical stimulation
Atoms scatter off a
light grating =
Bragg spectroscopy
dynamic
structure factor
S(q,ν)
(d)
Light scatters off
a matter wave
grating =
Superradiance
Matter wave
stimulation
Fig. 3. Light scattering from a Bose-Einstein condensate.
When a photon is scattered, it transfers momentum to the
condensate and creates an excitation (upper left). Therefore, an analysis of the scattered light allows the determination of the dynamic structure factor, in close analogy
to neutron scattering experiments with superfluid helium
(upper right). The signal is greatly increased by stimulating the light scattering by a second laser beam and detecting the scattered atoms (lower left)—this is the scheme
for Bragg spectroscopy. Light scattering can also be stimulated by adding a coherent atomic field (lower right). This
led to superradiant scattering of light and atoms.
Until recently, the observations have been consistent with the assumption that a Bose condensate
is a cold dilute cloud of atoms that scatters light
as ordinary atoms do. On resonance, the condensate strongly absorbs the light, giving rise to the
well-known “shadow pictures” of expanding condensates where the condensate appears black. For
off-resonant light, the absorption can be made negligibly small, and the condensate acts as a dispersive medium bending the light like a glass sphere.
This regime has been used for non-destructive insitu imaging of Bose-Einstein condensates (see Fig.
1).
Our group has recently looked more closely at
how coherent atoms interact with coherent light.
Light scattering imparts momentum to the condensate and creates an excitation (Fig. 3). Consequently, the coherence and collective nature of excitations in the condensate can strongly affect the
optical properties. As we discuss here, the use of
light scattering to characterize atomic Bose condensates is analogous to the use of neutron scattering in the case of superfluid helium [22,35].
4. Light scattering from a Bose-Einstein
condensates
In the early ’90s, before Bose-Einstein condensation was realized in atomic gases, there were lively
debates about what a condensate would look like.
Some researchers thought it would absorb all the
light and would therefore be “pitch black”, some
predicted it would be “transparent” (due to superradiant line-broadening [33]), others predicted
that it would reflect the light due to polaritons [34]
and be “shiny” like a mirror.
All the observations of Bose condensates have
employed scattering or absorption of laser light.
5
Since the light scattered from a sample containing only 107 atoms is hard to detect when it is
distributed over the full solid angle, we used a second laser beam to stimulate the scattering of light
with a frequency and direction, which was predetermined by the laser beam rather than postdetermined by analyzing scattered light (Fig. 3).
This scheme, which we call Bragg spectroscopy, establishes a high-resolution spectroscopic tool for
Bose-Einstein condensates which is sensitive to the
momentum distribution of the trapped condensate
as well as the effects of interactions [36]. We studied Bragg scattering in two regimes differing by the
amount of momentum transfer.
Bogoliubov theory predicts that for a momentum transfer which is smaller than the speed of
sound (times the atomic mass) phonons are excited, whereas for larger momentum transfer, the
excitations are free-particle like (Fig. 4). In the
regime of large momentum transfer, the impulse
approximation is valid, and the resonance shows a
Doppler broadening due to the zero-point motion
of the condensate, i.e. it can be used to measure
the momentum distribution of the condensates as
pursued for superfluid 4 He [37,35]. More generally,
Bragg spectroscopy can be used to determine the
dynamic structure factor S(q, ν) over a wide range
of frequencies ν and momentum transfers q [38].
Bragg spectroscopy was realized by exposing
the condensate to two off-resonant laser beams
with a frequency difference ν. The intersecting
beams formed a moving interference pattern from
which atoms could scatter when the Bragg condition was fulfilled (i.e. energy and momentum were
conserved). The momentum transfer q is given by
q = 2h̄k sin(ϑ/2), where ϑ is the angle between the
two laser beams with wave vector k. Figures 5 and
6 summarize the results for large and small scattering angles, probing both the phonon and freeparticle regime. In the regime of low momentum
transfer, light scattering was observed to be dramatically reduced (Fig. 6). In this regime, where
atoms cannot absorb momentum “individually”
but only collectively, the suppression arises from
destructive interference of two excitation paths.
2 GHz
Energy
5 1014 Hz
nU0
q2/ 2m
100 kHz
Momentum
Fig. 4. Probing the dispersion relation of a Bose-Einstein
condensate by off-resonant light scattering. The Bogoliubov dispersion relation is phonon-like (linear) for small
momenta. For large momenta, it is particle-like (quadratic)
with a mean-field shift nU0 ≈ 2 kHz indicated in the figure.
The momentum is transferred by scattering of visible light
which is about 2 GHz detuned from the atomic resonance.
The suppression provides dramatic evidence for
the presence of correlated momentum excitations
in the many-body condensate wavefunction. A
similar suppression would occur when a sufficiently
dense condensate scatters light spontaneously—
turning a “pitch-black” condensate transparent!
A condensate which reflected the incident light
was encountered when it was illuminated with a
single intense laser beam [41]. When a condensate
has scattered a photon, an imprint is left in the
form of long-lived excitations. These excitations
form a periodic density modulation which diffracts
light into the same direction as the first scattered
photon. The more photons have been scattered
into a certain direction, the larger is the density
modulation left behind and the larger the increase
of the scattering rate into this direction. This
self-acceleration of scattering can be described as
bosonic stimulation of the scattering by the population of the final (quasi-particle) state (see Fig.
3).
The gain for this process is highest when the
light is scattered along the long axis of the cylindrically shaped condensate and leads to the generation of directed beams of atoms (Fig. 7). This is
accompanied by directed emission of light—a new
form of superradiance where a density modulation
6
Laser beam
14
Condensate
1.0
1.2 mm
Doppler width (kHz)
3
10
Heisenberg limit
0
10
12
Size [mm]
2
1
0
14
5
m/h (kHz)
10
Fig. 6. (a) Static structure factor S(q) and (b) shift of the
line center from the free-particle resonance. S(q) is the ratio
of the line strength at a given chemical potential µ to that
observed for free particles. As the density and µ increase,
the structure factor is reduced, and the Bragg resonance
is shifted upward in frequency. Solid lines are predictions
of a local-density approximation for light scattering by
14 degrees. The dotted line indicates a mean-field shift
of 4µ/7h as measured in the free-particle regime using a
scattering angle of 180 degrees. Figure taken from Ref. [40]
1
8
(b)
3
0
2
6
6
0.0
rms width
~ 2 kHz
-5
0
5
Frequency n-n0 [kHz]
-3
cm )
0.5
Line shift (kHz)
Number of
scattered atoms
Frequency
-10
Peak density (10
2
4
(a)
S(q)
Bragg
scattered
atoms
0
16
Fig. 5. Measuring the momentum distribution of a condensate [36]. A condensate was exposed to two counterpropagating laser beams. Atoms absorbed a photon from
one beam and were stimulated to re-emit it by the other
beam, resulting in the transfer of recoil momentum to the
atoms, as observed in ballistic expansion using absorption
imaging after 20 ms time-of-flight (upper part). The number of Bragg scattered atoms showed a narrow resonance
when the difference frequency between the two laser beams
was varied (upper and middle part). The width of the resonance is caused by Doppler broadening and is therefore
proportional to the condensate’s momentum uncertainty
∆p. It was determined for various sizes ∆x of the condensate. The agreement with the Heisenberg limit ∆p ≈ h̄/∆x
proves that the Doppler width of the resonance is only due
to the zero-point motion of the condensate, or equivalently,
that the coherence length of the condensate is equal to
its physical size. This demonstrates that a condensate is
one “coherent matter wave”! An analogous measurement
in the time domain has been done by W.D. Phillips’ group
in Gaithersburg [39].
Fig. 7. Superradiance and matter-wave amplification in a
Bose-Einstein condensate [41]. The figure shows the velocity distribution of the atoms after a condensate (highest
peak at the back of each image) was illuminated by laser
light. Normal light scattering (rear image) is random, and
the velocity distribution was smeared out in the direction
of the incident light. In contrast, collective “superradiant”
scattering created several highly directional “atom laser”
beams (front image). The images are absorption images of
a cloud which expanded ballistically after the light scattering. The two horizontal axes represent two velocity components and the vertical axis is the column density of the
observed
atoms. The closest spacing between the peaks is
√
2 times the single-photon recoil velocity.
spontaneously develops which makes the condensate “reflect” light like a mirror.
5. Spinor Bose-Einstein condensates
In a magnetic trap, the atomic spin adiabatically
follows the direction of the magnetic field. Thus, although alkali atoms have internal spin, their BoseEinstein condensates are described by a scalar or7
der parameter similar to the spinless superfluid
4
He. One exception is the two-component condensate which was discovered by the Boulder group,
when they trapped atoms in both the upper and
lower hyperfine states of 87 Rb [42]. This observation was surprising because a large rate of inelastic
collisions had been predicted for this system. The
suppression of these spinflip collisions turned out
to come from a fortuitous equality in the scattering lengths in the two hyperfine states.
A general method for creating multi-component
condensates is to employ an optical trap that can
confine condensates with arbitrary orientations of
the spin thus liberating the spin as a new degree of
freedom. Our group used an optical trap to study
condensates with arbitrary population in the three
orientations m = 1, 0, −1 of the ground state of
sodium which has a total spin F = 1 [43,44]. These
condensates have a three-component vectorial order parameter. A variety of new phenomena have
been predicted for such spinor condensates such
as spin textures, spin waves, and the coupling between atomic spin and superfluid flow [45–47]. Such
phenomena cannot occur in condensates with a single component order parameter such as in 4 He and
resemble more the complex features of the superfluid phases of 3 He.
If the components are not coupled (i.e. transformed into each other), they can be regarded as
multi-species condensates (“condensate alloys”).
Both the group in Boulder and our group have
studied the dynamics of the phase-separation of
these components [48,49]. We observed long-lived
metastable structures which could tunnel through
each other and reach the equilibrium configuration
[50]. By selecting two of the three states of the
F = 1 spinor condensates, we could realize twocomponent condensates which were either miscible or immiscible [44]. Multi-component condensates are promising systems for the study of interpenetrating superfluids, a long-standing goal since
the early attempts in 1953 using 4 He-6 He mixtures
[51]. New phenomena arise when the three components are coupled by spinflip collisions, as displayed in Fig. 8.
mF
Evolution time in the trap
0s
1s
2s
4s
7s
0
+1
-1
0s
Trapped
condensate
at t=0
.5 s
2s
3s
5s
Absorption images after
25 ms time-of-flight
Trapped
condensate
in equilibrium
Fig. 8. Observation of coupled spinor condensates. The
components of the condensate are states with different orientations (m = −1, 0, 1) of the total spin F = 1 confined
in an optical trap. These components are coupled by an
anti-ferromagnetic spinflip interaction which drives them
into an equilibrium domain structure. After a variable holding time, the condensate was analyzed by time-of-flight
absorption imaging. During the ballistic expansion, a magnetic-field gradient acted as a Stern-Gerlach filter and separated the components with different spin orientation as
indicated by the arrows. The upper image shows how a
condensate, initially in a pure m = 0 state, developed a
spin domain structure. The same equilibrium state was
reached when the condensate started in an equal mixture
of m = +1 and −1 states [44,49]. In addition to the simple
schematic shown on the right the m = +1 and −1 components showed a bimodal density distribution due to their
miscibility.
6. Evidence for a critical velocity in a BEC
The existence of a macroscopic order parameter implies superfluidity of a gaseous condensate.
Observing frictionless flow is a challenge given the
small size of the system and its metastability. We
have taken a step towards this goal by studying
dissipation when an object was moved through the
fluid [52]. This is in direct analogy with the wellknown argument by Landau [25] and the vibrating
wire experiments in superfluid helium [53]. Instead
of a massive macroscopic object we used a blue
detuned laser beam which repelled atoms from its
focus to create a moving boundary condition.
The beam created a “hole” with a diameter of 13
µm which was scanned back and forth within the
condensate which was cigar-shaped with ThomasFermi diameters of 45 and 150 µm in the radial and
8
7. A new window into the quantum world
1.0
0.8
450
0.6
167 Hz
83 Hz
56 Hz
0.4
400
The direct observations of the condensate’s density distribution can be regarded as a direct visualization of the magnitude of the macroscopic
wavefunction. The time evolution of the squared
wavefunction of a single condensate has even been
recorded non-destructively in real time [21,24]. A
wavefunction is a probabilistic description of a system in the sense that it determines the distribution of measurements if many identical wavefunctions are repeatedly probed. In BEC, one simultaneously realizes millions of identical copies of the
same wavefunction, and thus the wavefunction can
be accurately determined while affecting only a
small fraction of the condensed atoms by the measurement process. On the other hand, we have already observed quantum correlations which go beyond the simple single-particle picture [40].
Questions that have been triggered by BoseEinstein condensation include the comparison of
different statistical ensembles (microcanonical,
canonical etc.) which agree in the thermodynamic
limit, but not for small Bose-Einstein condensates
[57–59]. The creation of a relative phase between
two condensates could be discussed both in the
framework of spontaneous symmetry breaking and
of quantum measurement theory, and has lead
to new insight [60,61]. Another question is under
what conditions is it possible to have an absolute
phase reference for condensates [62–64].
The rapid pace of developments in atomic BEC
during the last few years has taken the community
by surprise. After decades of an elusive search nobody expected that condensates would be so robust
and relatively easy to manipulate. Also, nobody
imagined that such a simple system would pose so
many challenges, not only to experimentalists, but
also to our fundamental understanding of physics.
The list of future challenges is long and includes
the exploration of superfluidity, vortices, and second sound in Bose gases, the study of quantumdegenerate molecules and Fermi gases, the development of practical “high-power” atom lasers, and
Temperature (nK)
Thermal Fraction
500
350
0
2
4
6
Velocity (mm/s)
Fig. 9. Evidence for a critical velocity. Shown is the final
temperature after a laser beam was scanned through the
condensate at variable velocity for 900 ms using different
scan frequencies. The dashed line separates the regimes
of low and high dissipation. The peak sound velocity is
marked by an arrow. The data series for 83 and 167 Hz
showed large shot-to-shot fluctuations at velocities below
2 mm/sec. The solid line is a smoothing spline fit to the 56
Hz data set to guide the eye. Figure taken from ref. [52].
axial directions, respectively. After exposing the
condensate to the scanning laser beam for about
one second, the final temperature was determined.
As a function of the velocity of the scanning beam,
we could distinguish two regimes of heating separated by a critical velocity. For low velocities, no
dissipation was observed, the condensate appeared
immune to the presence of the scanning laser beam.
For higher velocities, the heating increased, until
at a velocity of about 6 mm/s the condensate was
almost completely depleted after the stirring. The
cross-over between these two regimes was quite
pronounced and occurred at a velocity of about 1.6
mm/s which was a factor of four smaller than the
speed of sound at the peak density of the condensate (Fig. 9).
These observations are in qualitative agreement
with numerical calculations based on the nonlinear Schroedinger equation which predict the
onset of vortex nucleation at such subsonic velocities [54–56]). Because of surface effects and the
non-zero temperature, we expect dissipation even
at low velocities and a smooth crossover between
low and high dissipation. More precise measurements of the heating should allow us to study
these finite-size and finite-temperature effects.
9
their application in atom optics and precision measurements.
This work was supported by the ONR, NSF,
ARO, NASA, and the David and Lucile Packard
Foundation. I want to thank A. Görlitz for his help
in preparing this paper.
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Hulet, Phys. Rev. Lett. 75 (1995) 1687.
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