Download Einstein`s prediction

Einstein’s Legacy
(and its relation to laser
cooling and trapping)
Chinese Academy of Sciences
Beijing
13 December, 2006
“There are at least two reasons for
giving a talk. Either the material is so
new that no one has heard it,
or else the material is so old that
people have forgotten it.”
R.H. Dicke
3rd International Conference on
Quantum Electronics, 1964
Albert
Einstein
1905
Einstein’s Legacy
• Photons are quantized (photo-electric effect),
carry momentum (special relativity)
• Photons are affected by gravitational fields
(General Theory of Relativity)
• Stimulated emission and lasers
• Brownian motion (the weighing of atoms and
confinement in optical molasses)
• Bose-Einstein condensation
The spectrum of blackbody radiation
The spectrum of light emitted from a blackbody
is the same as the light emitted out of a hole in
a metal cavity.
According to classical physics, the energy in each
mode of radiation can assume any value.
Radiation intensity
Classical
calculation
Measured
spectrum
Frequency
In order to derive the frequency spectrum in a
cavity filled with radiation that is consistent with
experiment, Planck assumed that the energy  at
each frequency component  is equal to
 = s , where s is an integer.
“We consider, however – and this is the essential
point of the whole calculation – [the energy] as
made up of an entirely determined number of
finite equal parts, and we make use of the natural
constant h = 6.55 x 10-27 erg sec.”
Max Planck
If the energy is divided into an integer number
of “quanta”, the possibilities are 0, 1, 2, 3 …
S=3
S=3
S=2
The average number in
thermal equilibrium s
is found by summing the
probabilities of finding the
box with s photons.

S=1
S=0
P( s)
s   s P ( s)
s 0
The energy at frequency
 is E = s  .
exp( E / kBT )  exp( s  / kBT )
• The energy spectrum is the sum of the
thermal energy of light at each frequency.
• Planck initially regarded his radical
assumption at a mathematical “trick” to get
the right answer ….
Does light really come in
integral units of energy?
Photoelectric Effect
Direct evidence for the quantization of light!
Emax = h - 
Emax
e-
c
h
•Decreasing the intensity of light  fewer electrons
•Decreasing the frequency of light below a critical
value c  no electrons.
Quantization of light
 quantization of photon momentum
What is the momentum of
each photon?
Maxwell
volume 


 Density of of
light 
electromagnetic
Energy
of
the
light
pulse

energy (ergs/cm3) pulse 





area
of
Radiation
 



 Momentum impulse
light (t) (c)
pressure
=



(dynes/cm

of light
pulse
2)
pulse 





Momentum carried by pulses of light
is used to make a new type of atom
interferometer
B
C
A
B´
Kasevich and Chu, (1991)
Measurement of gravity compared to
Earth tide models
Measured values of gravity compared to solid
Measured acceleration, g (10-8 m/s2)
earth tide model with and without ocean loading effects
A. Peters,
K.-Y. Chung and S. Chu, Nature (1999).
100
0
-100
Data
Solid earth tide only
With ocean loading
-200
Residual acceleration (10-8 m/s2)
20
10
0
-10
-20
12 mn
12 noon
12 mn
12 noon
12 mn
High gravity
Low gravity
• Photons are quantized (photo-electric effect),
carry momentum (special relativity)
• Photons are affected by gravitational fields
(General Theory of Relativity)
• Stimulated emission and lasers
• Brownian motion (the weighing of atoms and
confinement in optical molasses)
• Bose-Einstein condensation
Photons not only carry momentum,
they transport mass
How can this be true if the rest
mass of a photon is zero?
(rest mass of photon)  E  ( pc)  0
2
2
2
L
M
Pcylinder  M v   E / c
x
c.m. does  M x   m
photon L
not move
E L
x  v t  
Mc c
EL
2
 m photon L  E  mphoton c
2
c
a)
b)
Huge
mass
The equivalence of acceleration and
gravity implies that light must bend in a
gravitational field.
Light travels in straight lines.
Einstein used the path taken
light as the definition of a
straight line.
Gravity also affects time!
E  mphoton c
2
U 
Gmphoton
r
U  
Newton’s Law of Gravity assumes
space can be defined by a three
dimensional geometry due to Euclid.
In Euclidian geometry, the sum of the angles
of a triangle add up to 180°.
Without matter, space is described by
Euclid’s geometry.
Einstein’s General Theory of Relativity says
mass distorts the geometry of space and time.
Georg Fredrich Bernard Riemann
In curved space,
the angles can add
up to be more or
less than 180°,
Or less than than
180°.
Carl Frederick Gauss
The vacuum has fields that fluctuate
Evidence for the existence for
quantum fluctuations:
•Emission of light by atoms
•Corrections to the energy levels of atoms
allow predictions accurate to a part in a
trillion.
•The long range attraction of atoms to each
other.
The fundamental problem
•The size of these quantum fluctuations increase
inversely with respect to the region of space we
look at.
x  p  h
•Energy fluctuations make the geometry of
space-time fluctuate. At a small enough size
scale, the assumption of a smooth, continuous
space and time breaks down.
This length scale is ~10-33 cm.
One way out of this
dilemma is to say
that there is a
smallest size of
space and time
that is
allowed…the size
of elementary
“strings”.
Quantum mechanics and the
general theory of relativity
The Fate of the Universe
Accelerating
Average distance
between galaxies
Slowing
Forever
Big Crunch
Time
5th Solvay Conference, Brussels, 1927
The debates of Bohr and Einstein
“Einstein came down to breakfast and expressed his
misgivings about the new quantum theory. Every time
[he] had invented some beautiful experiment from which
one saw that [the theory] did not work … Pauli and
Heisenberg, who were there did not pay much attention,
“Ah well, it will be all right, it will be all right…”
Bohr, on the other hand, reflected on it with care and in
the evening … he cleared up the matter in detail.”
Otto Stern recollection of the 1927 Solvay Meeting
The Bohr-Einstein debate: Round II
After the Solvay Conference, Einstein
writes to Schrödinger a year later:
“The soothing philosophy – or religion?
– of Bohr and Heisenberg is so cleverly
concocted that for the present it
offers the believers a soft pillow from
which they are not easily chased away…
This religion does damn little for me.”
Einstein’s strikes back with the EPR paradox.
Leon Rosenfeld recalls: “This onslaught came down
upon us as a bolt from the blue … as soon as Bohr
heard my report of Einstein’s argument, everything
else was abandoned: we had to clear up the
misunderstanding at once …
In great excitement, Bohr immediately started dictating
to me the outline of such a reply. Very soon, however,
he became hesitant; “No, this won’t do, we must try all
over again…we must make it quite clear.” So it went on
for a while, with growing wonder at the unexpected
subtlety of the argument ...he would turn to me: “What
can they mean? Do you understand it?” There would
follow some inconclusive exegesis…”
Bohm’s version of the
Einstein-Podolsky-Rosen paradox
A system with L = 0 decays
into two particles, each
carrying L = ½.
When Alice measures +½,
Bob must measure -½.
The situation is analogous to
the initial state consisting of a
black and white marble.
Bob is free to rotate his polarizer so that it measures
spin along x or y, while Alice always measures along y.
If Bob measures +x, then
Alice must see –x. Since Alice
measures along y, she has a
50-50 chance of getting -y.
Now suppose Bob chooses to measure y and gets +y.
Then Alice has to measure –y.
The paradox: How can Bob affect Alice’s measurement?
Quantum mechanics describes the
state before any measurement as the
combined (“entangled”) state
|+xBob, -xAlice  + |-xBob, +xAlice  .
The measurement “collapses” the state
to either of the two states.
Einstein: the system was already in one of
the two states before Bob’s measurement. If
Bob decided to rotate his polarizer at the last
minute, he could not affect Alice’s
measurement.
There is an objective reality to Nature
Causality demands that measurements
separated by light years (space-like
separation) can not affect each other.
Quantum Mechanics provides
probabilistic predictions of the
outcome because it is incomplete.
There may be “hidden variables”, and a
complete knowledge of all the
parameters of a system would make
the theory deterministic.
If these “hidden variables” were to
remain permanently hidden, Einstein’s
conjecture is not falsifiable.
In 1964, John Bell derived an inequality
using three angles of polarization that
would differentiate Quantum Mechanics
from any local hidden variable theory.
Bell’s argument:
•List all possible experimental outcomes of
the two measurements. (Bob measures “this”
and Alice measures “that”.) The states “this”
and “that” existed before the measurement.
•There are 8 choices allowed by physics.
These choices can occur with any predetermined probability.
•For appropriate choice of angles, Quantum
Mechanics gives a different prediction,
independent of the probabilities.
Bell’s inequalities were tested in the
1970’2 and 1980’s.
Clauser & Freedman, 1972
Aspect, Dalibard, and Roger, 1982
22.5°
Local hidden
variable prediction
67.5°
Does the verification of Quantum
Mechanics make the quantum
measurement problem and EPR
paradox go away?
No!
Quantum measurement is still a
problem…and we still don’t understand it.
• Photons are quantized (photo-electric effect),
carry momentum (special relativity)
• Photons are affected by gravitational fields
(General Theory of Relativity)
• Stimulated emission and lasers
• Brownian motion (the weighing of atoms and
confinement in optical molasses)
• Bose-Einstein condensation
Einstein used the Planck distribution to
predict the existence of stimulated emission.
E2, N2
A21
B12 u()
B21 u()
E1, N1
In order for the population of atoms to
agree with the thermal distribution
according to Boltzmann and Planck, there
must be another process.
Stimulated emission is the essential
ingredient for a laser
Incoming
radiation
Atoms or
molecules
In order to get more molecules in the excited
state than in the ground state, Charles Townes
(the co-inventor of the maser and the laser) came
up with the following scheme:
• Photons are quantized (photo-electric effect),
carry momentum (special relativity)
• Photons are affected by gravitational fields
(General Theory of Relativity)
• Stimulated emission and lasers
• Brownian motion (the weighing of atoms and
confinement in optical molasses)
• Bose-Einstein condensation
Brownian motion of a small
particle in water.
The constant jiggling of the particle was
not due to the fact that it was alive!
If the erratic motion was due to collisions
with individual molecules of water, the
early estimate of the mass of the water
was to large by many orders of magnitude.
• Einstein showed that Brownian motion was due to
fluctuations in large number of molecules hitting from
opposing sides.
• There is also a viscous drag force acting on the
particle.
dv
m
dt
  v + F (t )
The solution to this differential equation gives
x
2

2 k BT

t  2D t
This simple equation opens up the first quantitative
way to measure the weight of a water molecule!
A measurement of Boltzmann’s constant, the
specific heat of a gas at constant pressure (CP)
and volume (CV) determines Avogadro’s Number,
NA, the number of molecules in 18 grams of water.
CP  Cv  N A kB
“Optical
molasses”
v
F
Force is opposite the motion
F = - v
1 2 1
mv  
2
2
1 2 1
mv  
2
2
• Photons are quantized (photo-electric effect),
carry momentum (special relativity)
• Photons are affected by gravitational fields
(General Theory of Relativity)
• Stimulated emission and lasers
• Brownian motion (the weighing of atoms and
confinement in optical molasses)
• Bose-Einstein condensation
When atoms move slowly, they
become big-fuzz balls
d

Einstein’s prediction: When   d, the atoms
will condense into a single gigantic wave
The reason for Bose-Einstein condensation
comes from fundamental ideas of statistical
mechanics and quantum mechanics!
Consider a small box of gas atoms that is
released into a larger volume.
The Second law of Thermodynamics
• The entropy S of a closed system will remain
constant or increase in time.
The entropy of the atoms in the larger box will
increase when no longer confined to the
smaller box.
S  kB log  number of accessible states
• All atoms are equally likely to be found
anywhere in the larger box. The most likely
configurations have atoms more or less
equally distributed.
• A thermodynamic system tries to minimize its
energy and maximize its entropy.
•This trade-off is described in terms of
minimizing the “Free Energy”, F of the system.
F  E  TS
As the atoms in our box are cooled to lower
temperatures, the entropy S decreases.
WHY?
Quantum mechanics:  = 2 h / p
d
As the atom fuzz balls get bigger, the number of
accessible states decreases, decreasing the entropy S.
In order that the Free Energy, F = E – TS not increase,
a macroscopic number of atoms begin to drop into the
lowest energy state.
Atoms condensed
into the lowest
energy state.
Einstein
Faraday
Newton
Maxwell
How can mere
mortals hope to
make contributions
that compare to
these Giants?
“The knowledge we acquire in science is
additive. At its core is our ability to build on
the knowledge of others. As scientists, we
hope that others take note of what we have
done, and use our work to go in directions
we never imagined. In this way, we continue
to add to our collective scientific legacy….”
Nobel Lecture, 1997