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专题三：玻色－爱因斯坦凝聚
1. 经典统计分布函数
2. 量子统计分布函数
3. 玻色－爱因斯坦凝聚
1. 经典统计分布函数
一绝热系统处在势场中：
x  x  dx
y  y  dy
p x  p x  dp x
z  z  dz
p y  p y  dp y

3
2
dN  N 0 ( 2mkT ) e

EK  EP
KT
p z  p z  dp z
dxdydzdp x dp y dp z
N 0：在E p  0处的单位体积内粒子数
E K : 动能；E p : 势能
对动量积分
1
EK
( p x2  p 2y  p z2 )
2m
x  x  dx ,
y  y  dy ,
z  z  dz :
Ep
dN '  N 0 e
kT
dxdydz
Ep
单位体积内的粒子数 N  N 0 e
kT
玻尔兹曼分布
一个自由度为f的体系，其能量是广义坐标、广义动量的函数
E  E (q1 ,..., q f ; p1 ,..., p f )
在相空间体积元d  dq1 ...dq f dp1 ...dp f 中的粒子数是d
(q1 ,...q f ; p1 ,..., p f )  Ce
 E ( q1 ,..., q f ; p1 ,..., p f ) / kT
称为正则分布
对多自由度体系，有按自由度的能量分配定理：
自由度i，  i ~ p i2 :  i  bp i2
E   i  E ' (q1 ,..., q f ; p1 ,..., p i 1 , p i 1 ,... p f )
E '是除自由度i能量以外的能量，它与p i 无关：
i 
e
  i / kT
e
 i dpi
  i / kT
dpi

b e
 pi2 b
e
 pi2 b
pi2 dpi
dpi
1
, 
kT
分母：

e


分子： 

i 

 pi2 b


 pi2 b
e

1
1
 kT
2
2
dpi 

1
b



dp i  
[

b
1

]

1
b 2 3 / 2
与b无关！
1
如 i  cq , 则 i  kT，与c无关。
2
2
i
一个服从经典统计的系统处于温度T的热平衡中，其能量
的每一独立的平方项都有平均值1/2kT.
考虑由电子和质子组成的等离子体，虽然它们质量不同(相当于b不同),
但平均动能相等。
不同硬度的大量谐振子体系(相当于c不同), 其平均势能也相同。
如粒子动力学只容许分立能量：
对量子谐振子：E  nh

E
 nhe
n 0

e
 nh  / kT
 nh  / kT
n 0
令x  h / kT ,
分母：
1  e  x  e  2 x  ... 
1
1  ex
d   nh / kT
h e  x
分子： －h
( e
)
dx n0
(1  e  x )2
E
h
e
h / kT
1
普朗克能量均分定理
利用它可推导出黑体辐射公式
2. 量子统计分布函数
两个全同粒子，  , 
B (1,2) 
F (1,2) 
:
1
2
1
2
[ (1) ( 2)   ( 2) (1)]
[ (1) ( 2)   ( 2) (1)]
B (1,2) 
2 (1) (1)
两个玻色子处于相同状态的几率是经典几率的2倍
F (1,2)  0  Pauli不相容原理
推广到n个全同粒子（玻色子），n 个玻色子处于相同状态
的几率n！倍于其经典几率。
问题：将一个玻色子放在一个状态上去的几率和状态原有的玻
色子数目有何关系？
经典：在态上放n个粒子的几率
Pnc  ( P1 ) n
量子： PnB  n! Pnc  n! ( P1 ) n
PnB1  ( n  1)! Pnc1
PnB1  (1  n) P1 PnB
再放一个的几率是相应经典几率的(n+1)倍，增强因子（1+n）
费米子：
(1  n) P1 , (n  0,1)
为得量子统计分布，还需精细平衡原理。
在处于平衡的体系中考虑两个能级i与j,上面分别有ni及nj个
粒子。令Ri->j代表从i到j的跃迁几率。
精细平衡：
ni
nj
Ri  j
R ji
ni R i  j  n j R j  i
对所有能级成立。
经典统计，有：
n1
 e ( E1  E2 ) / kT
n2
n1 R2c1 e  E1 / kT
 c   E2 / kT
n2 R12 e
对玻色子，有增强因子1+n:
R1B 2  (1  n2 ) R1c 2
R2B1  (1  n1 ) R2c1
R2B1
n1
1  n1 e  E1 / kT
 B 
n2
1  n2 e  E 2 / kT
R1 2
n1
n2
e  E1 / kT 
e  E2 / kT    e  (T ) ;只与温度有关的常数
1  n1
n2  1
n
1 n


0
e
 E / kT
 e  (T )
n B ( E ) N ( E )dE  N
nB ( E ) 
1
e e E / kT  1
总粒子数
玻色－爱因斯坦分布函数
是和粒子数密度和温度有关的量，
且和系统能级密度有关。
费米子：1－n取代1+n
n  E / kT
1
 ( T )
F
费米－狄拉克分布函数
e
e
n ( E )   E / kT
1 n
e e
1
对自由电子，令e  e  F / kT ,  F : 化学势，Fermi 能量
n (E) 
F
1
e ( E  EF ) / kT  1
1
F
n ( F )  (化学势的分布)
2
Quantum Statistics
3. 玻色－爱因斯坦凝聚
Predicted 1924…
…Created 1995
Q1: What Is Bose-Einstein Condensation?
De Broglie 德布罗意 (1929 Nobel Prize winner) proposed that all matter is
composed of waves. Their wavelengths are given by

 = de Broglie wavelength
h = Planck’s constant 普朗克常数
m = mass
v = velocity
h
mv
Against Our Intuition?!
In most everyday matter, the de Broglie wavelength is much shorter than the
distance separating the atoms. In this case, the wave nature of atoms cannot be
noticed, and they behave as particles.
 The wave nature of atoms become noticeable when the de Broglie wavelength is
roughly the same as the atomic distance.
 This happens when the temperature is low enough, so that they have low
velocities.
 In this case, the wave nature of atoms will be described by quantum physics,
e.g. they can only stay at discrete energy states (energy quantization).

Bose and Einstein
• In 1924 an Indian physicist named Bose studied the quantum behaviour of a
collection of photons.
• Bose sent his work to Einstein, who realized that it was important.
• Einstein generalized the idea to atoms, considering them as quantum particles
with mass.
• Einstein found that when the temperature is high, they behave like ordinary
gases.
• However, when the temperature is very low, they will gather together at the
lowest quantum state. This is called Bose-Einstein condensation.
Fermions (費米子) and Bosons (玻色子)
Not all particles can have BEC. This is related to the spin of the particles.
The spin quantum number of a particle can be an integer or a half-integer.
Single protons, neutrons and electrons have a spin of ½. They are called fermions.
They cannot appear in the same quantum state. BEC cannot take place.
 Some atoms contain an even number of fermions. They have a total spin of whole
number. They are called bosons.
 Bosons show strong “social” behaviour, and can have BEC.
 Example: A 23Na atom has 11 protons, 12 neutrons and 11 electrons.



The Material For BEC

BEC was found in alkali metals e.g. 87Rb (铷), 23Na (钠), 7Li (锂) because:
They are bosons.
Each atom is a small magnetic compass, so that a cooling technique called
magnetic cooling can work.
The atoms have a small repulsion, so that they do not liquefy or solidify down
to a very low temperature.
Cooling Down the Atoms
See the animation:
http://www.colorado.edu/physics/2000/bec/what_is_it.html
 When the temperature is high, the atoms have high energies on average. The
energy levels are almost continuous. It is sufficient to describe the system
by classical physics.
 When the temperature is low, the atoms have low energies on average. It is
necessary to describe the system by quantum physics.
 When the temperature is very low, a large fraction of atoms suddenly crash
into the lowest energy state. This is called Bose-Einstein condensation.

The Strange State of BEC
When all the atoms stay in the condensate, all the atoms are absolutely identical.
There is no possible measurement that can tell them apart.
 Before condensation, the atoms look like fuzzy balls.
 After consdensation, the atoms lie exactly on top of each other (a superatom).

Q2: How Is BEC Made?
Laser beam
Other equipment: laser equipment, computer, electronics
Cost less than US$100,000
Laser Cooling (激光冷卻)
Chu 朱棣文 Cohen-Tannoundji
Phillips
The technique of laser cooling was developed by the winners of the 1997 Nobel
Prize winners.
 In the physical world, the lowest temperatures approach a limit of –273oC. This is
called the absolute zero. Nothing can be as cold as the absolute zero because all
atomic and subatomic motions stop.
 Laser cooling can get to the low temperature of 0.18K (1 K微開= 10-6K).

Ping-pong Balls
Photons are particles. They carry momenta like ping-pong balls.
You can slow the motion of an atom by bouncing laser light off the atoms.
 See the animation http://www.colorado.edu/physics/2000/bec/lascool1.html.


Tuning the Laser
Only laser light with the correct colour (frequency) can be absorbed by the
atoms.
 If the colour is wrong, the atoms cannot absorb the photons.
 See the animation http://www.colorado.edu/physics/2000/bec/lascool2.html

Using the Doppler Effect
Problem: The laser can slow the
approaching atoms, but it can also
blast off the receding ones.
 Solution: Use Doppler shift.
 When the atom is receding from
the laser source, the wavelength is
lengthened and there is a redshift.
 When the atom is approaching the
laser source, the wavelength is
shortened and there is a blueshift.
 See the animation:
http://www.astro.ubc.ca/~scharein/a3
11/Sim.html

Laser Trapping (激光陷阱)
Suppose the laser has the right colour for the photons to be absorbed by an
approaching atom, then the atom will be slowed down.
 However, the laser will not have the right colour for the photons to be
absorbed by the receding atom because of Doppler effect. Hence the atom
will not change in this case.
 When lasers are sent in from all the different directions, the atoms can get
cold very quickly.
 This is called laser trapping, and the trapped atoms form an optical molass
(光學粘膠).
 See the animation: http://www.colorado.edu/physics/2000/bec/lascool4.html

Magnetic Trapping (磁性陷阱)
Problem: Laser cooling can cool the atoms down to 10K, because atoms can
spontaneously emit the absorbed photon. This is still too hot for BEC.
 Solution: Evaporative cooling
 The atoms behave as tiny compasses. They can be pulled by magnetic fields.
 A magnetic field can be designed to push the atoms inwards from both sides,
forming a magnetic trap.
 See the animation: http://www.colorado.edu/physics/2000/bec/mag_trap.html

Evaporative Cooling (挥发冷却)
Principle: Evaporation takes heat. A cup of tea gets cool after steam escapes,
because faster atoms escape from the cup, leaving behind the slower ones.
 Technique: Lower the height of the trap quickly, so that there are still enough
atoms left in the trap to get involved in BEC.
 Try to trap the largest number of atoms in BEC in the animation:
 http://www.colorado.edu/physics/2000/bec/evap_cool.html

Can You Break This Record?
Q3: What Does a Bose-Einstein Condensate Look Like?
There is a drop of condensate at the centre.
 The condensate is surrounded by uncondensed gas atoms.
 The combination looks like a cherry with a pit.
 See the movie when it cools from 400 nK to 50 nK (1 nK= 10-9K). :
http://www.colorado.edu/physics/2000/bec/what_it_looks_like.html

The Nobel Prize in Physics
2001
USA(Eric A.Cornell,1961)
Germany(Wolfgang Kettlerle,1957)
USA(Carl E.Wieman,1951)
“for the achievement of Bose-Einstein
condensation in dilute gases of alkali atoms,
and for early fundamental studies of the
properties of the condensates"
What is an atom laser?
• An atom laser is analogous to an optical laser, but it emits matter
waves instead of electromagnetic waves.
• Its output is a coherent matter wave, a beam of atoms which can be
focused to a pinpoint or can be collimated to travel large distances
without spreading. The beam is coherent, which means, for instance,
that atom laser beams can interfere with each other.
• Compared to an ordinary beam of atoms, the beam of an atom laser is
extremely bright. One can describe laser-like atoms as atoms
"marching in lockstep".
• Although there is no rigorous definition for the atom laser (or, for that
matter, an optical laser), all people agree that brightness and coherence
are the essential features.
Ketterle‘s home page
Atomic trap
ATOMIC TRAP cools by means of
two different mechanisms. First, six
laser beams (red) cool atoms,
initially at room temperature, while
corralling them toward the cen-ter
of an evacuated glass box. Next, the
laser beams are turned off, and the
magnetic coils (copper) are energized. Current flowing through the
coils generates a magnetic field that
further confines most of the atoms
while allowing the energetic ones to
escape. Thus, the average energy of
the remaining atoms decreases,
mak-ing the sample colder and even
more closely confined to the center
of the trap. Ultimately, many of the
atoms attain the lowest possible
energy state allowed by quan-tum
mechanics and become a single
entity known as a Bose-Einstein
condensate.
Sci. Am., March 1998, p. 27.
The parts of an atom laser
• A laser requires a cavity (resonator), an active medium, and an output
coupler.
• In the MIT atom laser, the "resonator" is a magnetic trap in which the
atoms are confined by "magnetic mirrors".
• The active medium is a thermal cloud of ultracold atoms, and the
output coupler is an rf pulse which controls the "reflectivity" of the
magnetic mirrors.
Ketterle‘s homepage
The gain process in an atom laser
• The analogy to spontaneous emission in the optical laser is elastic
scattering of atoms (collisions similar to those between billiard balls).
• In a laser, stimulated emission of photons causes the radiation field to
build up in a single mode.
• In an atom laser, the presence of a Bose-Einstein condensate (atoms
that occupy a "single mode" of the system, the lowest energy state)
causes stimulated scattering by atoms into that mode.
• More precisely, the presence of a condensate with N atoms enhances
the probability that an atom will be scattered into the condensate by
N+1.
Ketterle‘s homepage
The gain process in an atom laser
• In a normal gas, atoms scatter among the many modes of the system.
But when the critical temperature for Bose-Einstein condensation is
reached, they scatter predominantly into the lowest energy state of the
system, a single one of the myriad of possible quantum states. This
abrupt process is closely analogous to the threshold for operating a
laser, when the laser suddenly switches on as the supply of radiating
atoms is increased.
• In an atom laser, the "excitation" of the "active medium" is done by
evaporative cooling - the evaporation process creates a cloud which is
not in thermal equilibrium and relaxes towards colder temperatures.
This results in growth of the condensate. After equilibration, the net
"gain" of the atom laser is zero, i.e., the condensate fraction remains
constant until further cooling is applied.
Ketterle‘s homepage
The gain process in an atom laser
Unlike optical lasers, which sometimes radiate in several modes (i.e. at
several nearby frequencies) the matter wave laser always operates in a
single mode. The formation of the Bose condensate actually involves
"mode competition": the first excited state cannot be macroscopically
populated because the ground state "eats up all the pie".
Ketterle‘s homepage
The output of an atom laser
• The output of an optical laser is a collimated beam of light. For an
atom laser, it is a beam of atoms. Either laser can be continuous or
pulsed - but so far, the atom laser has only been realized in the pulsed
mode.
• Both light and atoms propagate according to a wave equation. Light is
governed by Maxwell's equations, and matter is described by the
Schroedinger equation.
• The diffraction limit in optics corresponds to the Heisenberg
uncertainty limit for atoms. In an ideal case, the atom laser emits a
Heisenberg uncertainty limited beam.
Ketterle‘s homepage
Optical versus atom laser: differences
• Photons can be created, but not atoms. The number of atoms in an atom laser is
not amplified. What is amplified is the number of atoms in the ground state,
while the number of atoms in other states decreases.
• Atoms interact with each other - that creates additional spreading of the output
beam. Unlike light, a matter wave cannot travel far through air.
• Atoms are massive particles. They are therefore accelerated by gravity. A
matter wave beam will fall like a beam of ordinary atoms.
• A Bose condensates occupies the lowest mode (ground state) of the system,
whereas lasers usually operate on very high modes of the laser resonator.
• A Bose condensed system is in thermal equilibrium and characterized by
extremely low temperature. In contrast, the optical laser operates in a nonequilibrium situation which can be characterized by a negative temperature.
There is never any population inversion in evaporative cooling or Bose
condensation.
Ketterle‘s homepage
Oscillations
See the animation:
http://cua.mit.edu/ketterle_group/Animations/Oscillations.htm
Note the “shape” motion and “sloshing” motion.
Interference Pattern (干涉圖像)
When two Bose-Einstein condensates spread out, the interference pattern
reveals their wave nature.
 See the animation:
http://cua.mit.edu/ketterle_group/Animations/TOFsplit.htm

Vortices
When the condensate is rotated, vortices appear. The angular momentum of
each of them has a discrete value.
Q4: What Is Bose-Einstein Condensation Good For?
This is a completely new area. Applications are too early to predict.
 The atom laser can be used in:
 atom optics (studying the optical properties of atoms)
 atom lithography 光刻 (fabricating extremely small circuits)
 precision atomic clocks
 other measurements of fundamental standards
 hologram 全息图
 communications and computation.
 Fundamental understanding of quantum mechanics.
 Model of supernova explosion 超新星爆炸.
 Model of black holes 黑洞.

References
Homepage of the Nobel e-Museum (http://www.nobel.se/).
 BEC Homepage at the University of Colorado
(http://www.colorado.edu/physics/2000/bec/).
 Ketterle Group Homepage (http://www.cua.mit/ketterle_group/).
 The Coolest Gas in the Universe (Scientific American, December 2000, 92-99).
 Atom Lasers (Physics World, August 1999, 31-35).

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