Atomic Physics
“Driving Doppler Down”
Doppler-free methods
Laser cooling, trapping,
quantum computing
COOLING – FUN demos…
Cool (2.1MB and loads of
fun)
Exsetup Movie (469KB)
Cold Cesium (130KB)
This .Zip file includes the .Exe, .Hlp, and .Dll files as well as
other supporting files for Visual Basic. Download the .Zip
and unzip it. Then, run the Setup.exe program. Enjoy!!
Here it is!! (after much sweat and conversion) This .AVI
shows the experimental setup for laser cooling and trapping.
Play it in Cool or on any .AVI movie player
This is a movie shows a cold, dense cloud of cesium atoms
created in the laboratory!!
Trap! (257KB)
This .AVI shows the Magneto-Optic Trap (MOT). Its starts
with optical molasses and then suddenly the magnetic field
gradient is applied and the trap forms. The field is turned off
and the cloud slowly spreads.
Optical Molasses (129KB)
Here is a simulation created using Cool. It demonstrates
what "Cool" people call Optical Molasses. Try it out! (or just
make your own!)
Dark MOT (161KB)
Another simulation which demonstrates a dark magnetooptic trap (MOT). Download this or make your own!
Light MOT (91 KB)
Another simulation which demonstrates a light magnetooptic trap (MOT). Download this or make your own!
Light to Dark MOT (243KB)
This a simulation that starts as a light MOT then changes to
a dark MOT
Dark to Light MOT (252KB)
This simulation starts as a dark MOT then changes
Motion of an atom affects the absorption frequency, and its emission frequency
A reminder:
Question: Estimate the halfwidth at room temperature
(a) for Balmer-alpha, (b) for the cesium resonance line
From kinetic theory, we know the fraction of atoms with velocity between v and v+dv
f(v)dv = {M/(2πkT)}1/2 exp {-Mv2/(2kT)} dv = (1/u√π) exp(-v2/u2) dv
Where u is the most probable velocity u = √(2kT/M)
Since the fractional shift in frequency is δ/ω0 = v/c
Then the line profile is GDoppler(ω) = (c/(uω0√π)exp {-(c2/u2)(ω – ω0)2/ω02}
Example 1: Crossed beams
Doppler broadening depends on the collimation of the 2 beams – generally this is
much larger for the atoms – α typically about 10-3 radians
But also broadening due to the short interaction time – “transit-time broadening”.
Typical widths are of the order of 1 MHz for visible spectra.
“Co-linear” – Saturated Absorption
spectroscopy
A laser beam approaching the atoms with a weak intensity, and a wavelength
off-resonance by a small amount – the atoms with a certain velocity relative to
the laser will be promoted to level 2, “burning a hole” in the Gaussian thermal
distribution of level 1 atoms.
Saturated Absorption spectroscopy
Improvement in “halfwidth” is
typically at least 1000, with
consequent improvement in
precision of measuring the
resonance frequency.
Most experiments
(a) utilize double-beam
geometries – comparing the
difference of the 2 beams –
one crossing, the other notcrossing.
(b) And/or chop the pump beam
to remove time variations.
2-photon spectroscopy
Excitation can be to a virtual level, midway between the initial and final state (of
opposite parity) – e.g. a 1s to 2s transition in hydrogen
Lyman-α has a wavelength of 126 nm; hence each of the 2 photons needs a
wavelength of 252 nm; each of these produced by frequency –doubling a green
line (at 504 nm)
Two vital factors for precision
measurements
1. Measuring frequency directly
2. Cooling the source
Measuring the
wavelength/frequency
A femtosecond laser is so short that its energy distribution can span frequencies
from f to 2f – then direct comparisons of different frequency-doubled components
(from the doubling crystal) allows measurement of the mode number. The
unknown frequency can then be compared with these accurately known absolute
frequencies. – to precisions of parts in 1012 to 1014.
Important corollary: Stepping up frequency ranges from a direct lower frequency
measurement (say, 109 Hz) to frequencies in the 1014 Hz range
Laser cooling of atoms
Photon momentum can be used to
apply a force to each atom:
radiation of intensity I exerts a force
on an area A: Frad = IA/c
In the figure, the atom then radiates
in all directions so that its
momentum long the beam is reduced
– i.e. the beam is cooled.
Example: a sodium beam at T=900K (from the oven shown) v0 = 1000m/s
Acceleration is a = - (ħk/M) (γ/2) = v(recoil) (γ/2)
For sodium τ = 1/ γ = 16 ns, v(recoil) = 3 cm/s
Hence the stopping distance x = 2v02 τ/v(recoil) = 1.1 m
But note - if the velocity changes the resonant wavelength changes – so how do we
fix that?
Use Zeeman splitting to match
cooling velocity!! (the first time)
Matching the magnetic field can bring the atom to rest just at the end of the
solenoid – such an experiment indicates a concentration of atoms which
gradually move away perpendicular to the beam, where no laser cooling has
occurred - W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596-599 (1982).
So now let’s do it in all directions… (three are enough) => TRAP
Alternative longitudinal cooling:
Tuning the laser slightly below the resonance absorption of a stationary sodium atom.
The atom sees the head-on photon as Doppler shifted upward toward its resonant
frequency and it is therefore more strongly absorbed than a photon traveling in the
opposite direction which is Doppler shifted away from the resonance. For room
temperature sodium atom, the incoming photon is Doppler shifted up 0.97 GHz, so to get
the head on photon to match the resonant frequency would require that the laser be tuned
below the resonant peak by that amount. This method of cooling sodium atoms was
proposed by Theodore Hansch and Arthur Schawlow at Stanford University in
1975 and achieved by Chu at AT&T Bell Labs in 1985.
Sodium atoms were cooled from a thermal beam at
500K to about 240 mK. The experimental technique
involved directing laser beams from opposite
directions upon the sample, linearly polarized at 90°
with respect to each other. Six lasers could then
provide a pair of beams along each coordinate axis.
The effectively "viscous" effect of the laser beams in
slowing down the atoms was dubbed "optical
molasses" by Chu.
Three-dimensional cooling
– optical molasses
The effective
molasses force
(off resonance by kv)
Another way: Use laser “chirping”
to match cooling velocity!!
Vary the laser frequency to match the change in velocity: we must vary the frequency
by GHz in milliseconds! – how do we do that?
Electro-optic modulators and rf techniques can do it fast enough – the crystals produce
sideband frequencies which can then be varied rapidly by applying an rf signal.
Velocity distribution of
“chirped” atoms
The slower atoms in the distribution have been brought to zero velocity,
to give a narrow low velocity peak.
Chirping of a laser
The chirp of an optical pulse is usually understood as the time dependence of its
instantaneous frequency. Specifically, an up-chirp (down-chirp) means that the
instantaneous frequency rises (decreases) with time.
Example: consider a pulse with a Gaussian envelope and a quadratic temporal phase:
This is associated with a linear
chirp, i.e., with a linear
variation of the instantaneous
frequency: (Fourier transform)
Magnetic field trap
The “wrongly-connected” Helmholtz coils
give zero magnetic field at the center,
increasing in all directions away from the
center.
Combination – the MOT
Adding the (6) molasses cooling lasers
yields an imbalance of the net forces
always towards the center.
The physical geometry
The Zeeman splitting in the magnetic field.
Dipole cooling
– example: the atomic fountain
An intense laser beam can change the
energy levels of an atom (AC Stark shift).
If the laser frequency is less than the
resonance frequency, this forms a potential
well attracting the atoms into a volume of
high laser intensity – this is called a
“dipole-force” trap which can be loaded
with the “molasses cooled” atoms.
This can be produced neatly on a microscopic scale by producing a standing wave,
and thus a string of small traps.
After the cold atoms have been trapped,
the lasers can be turned off, allowing the
atoms to fall – and then be detected lower
down by a probe laser…
The atomic fountain
– in principle, and in reality
The fountain geometry increases the time between the two interrogations by gently tossing
the atoms up and letting them fall back down under the influence of gravity, all under high
vacuum. Atoms are collected and then launched through a single microwave cavity, which
interrogates the atoms both on the way up and again on the way down. The atoms are then
detected optically to determine the information about the microwave frequency. This cycle
is then repeated. The longer time between interrogations improves the precision of the
measurement,.
Who was Sisyphus?
The gods had condemned Sisyphus to ceaselessly
rolling a rock to the top of a mountain, whence the
stone would fall back of its own weight. They had
thought with some reason that there is no more
dreadful punishment than futile and hopeless labor.
Sisyphus, being near to death, rashly wanted to test his wife's love. He ordered
her to cast his unburied body into the middle of the public square. Sisyphus woke
up in the underworld. And there, annoyed by an obedience so contrary to human
love, he obtained from Pluto permission to return to earth in order to chastise his
wife. But when he had seen again the face of this world, enjoyed water and sun,
warm stones and the sea, he no longer wanted to go back to the infernal darkness.
Recalls, signs of anger, warnings were of no avail. Many years more he lived
facing the curve of the gulf, the sparkling sea, and the smiles of the earth. A
decree of the gods was necessary. Mercury came and seized the impudent man by
the collar and, snatching him from his joys, led him forcibly back to the
underworld, where his rock was ready for him.
A modern Sisyphus – the atom
Manipulating atoms, part 1
The Ioffe-Pritchard trap adds magnetic
fields from coils (much further apart
but with the Helmholtz phase) to pinch
and “Ioffe” coils which hold the atoms
in the center.
By slowly reducing the fields,
evaporative cooling can take place –
i.e. the hotter atoms jump out of the
potential lattice, leaving the cooler
atoms.
Such methods can lead to atom
temperatures less than 10-9K. There is
no theoretical limit – just the number
of atoms trapped.
TIME
August 2009: The NIST ytterbium
clock is based on about 30,000 heavy
metal atoms that are cooled to 15
microkelvins (close to absolute zero)
and trapped in a column of several
hundred pancake-shaped wells
N.B. The NIST Cs-clock is stable to
1 second in 10,000 years!
This photo shows about 1 million ytterbium atoms illuminated by a blue laser in an
experimental atomic clock that holds the atoms in a lattice made of intersecting laser
beams. The photo was taken with a digital camera through the window of a vacuum
chamber. NIST is studying the possible use of ytterbium atoms in next-generation
atomic clocks based on optical frequencies, which could be more stable and accurate
than today's best time standards, which are based on microwave frequencies.
Manipulating atoms, part 2
- to a Bose-Einstein condensate As the temperature gets colder, the interatomic interactions will maintain
their “collision memory”, leading to coherence in any scattering (and a
“coherence time”. The atoms then behave as a single quantum entity. If they
are bosons, this can lead to a Bose-Einstein condensation.
The thermal DeBroglie wavelength can be defined as
λ = h (2πMkT)-1/2
When the inter-atomic spacing reaches (about) this value the bosons tend to
condense…
when N/V ≈ (λ)-1/3 = 2.6 (λ)-1/3
see Bose-Einstein statistics for more exact formulation…
Bose-Einstein condensate and
Quantum Computing
The atoms then fit into the potential well, with all
energy levels filled up to an effective Fermi level.
-> well potential example on left.
-> picture of cooling “atom blob” below.
-> optical density cuts on right.
Quantum computing comes next!
Manipulation of a string or volume of
optical traps…. Quantum mechanics
allows optical entanglement and further
manipulation of phases as well as just
populations in each trap.
The Qubit
The qubit is the quantum analogue of the bit, the classical fundamental unit of
information. It is a mathematical object with specific properties that can be
realized physically in many different ways as an actual physical system. Just as
the classical bit has a state (either 0 or 1), a qubit also has a state.
Note: any linear combination (superposition) is physically possible. In general,
thus, the physical state of a qubit is the superposition ψ = α |0>+ β |1> (where α
and β are complex numbers).
The state of a qubit can be described as a vector in a two-dimensional Hilbert
space, a complex vector space .The states |0> and |1> form an orthonormal basis
of quantum states for this vector space.
Fundamental theorem quantifying the improved speed of quantum computers
(phase information) first formulated in Shor’s algorithm (1994).
Visualizing the Qubit
Any general 2-component state can be written
|Ψ> = cos(θ)|0> + eiφsin(θ)|1>,
where the numbers θ and φ define a point on the unit three-dimensional sphere,
as shown here. This sphere is often called the Bloch sphere, and it provides a
useful means to visualize the state of a single qubit.
Note that the act of measurement yields either the
|1> state or the |0> state,
but the computer can store much more (an infinite?)
amount of information!
This is relevant to the first definition of a
“computer” by Alan Turing in 1936.
The Turing machine (1936) was essentially a table
of look-up values for any calculation.
For a good history of the development of quantum computing see
http://plato.stanford.edu/entries/qt-quantcomp/#2.1
Atomic quantum systems in optical micro-structures by T. M¨uther et al
Abstract (Journal of Physics: Conference Series 19 (2005) 97–101)
We describe an experiment on evaporative cooling in a far-detuned optical dipole trap for 87Rb.
The dipole trap is created by a solid state laser at a wavelength of 1030 nm. To achieve high
initial phase space densities allowing for efficient evaporative cooling, we have optimised the
loading process from a magneto-optical trap into the dipole trap. These investigations aim at
the creation of an ‘all-optical’ BEC based on a simple experimental scheme. As an example,
we present the transport of atoms in a ring-shaped guiding structure, i.e. optical storage ring,
for cold atoms which is produced by a micro-fabricated ring lens.
Absorption images after 10
ms TOF (left), and
measured temperatures
(right, in nK) for different
laser powers.
Schematic of the ring lens
(left) and fluorescence
image of the atoms in the
storage ring (right)
Quantum computing differs from classical computing in that a classical
computer works by processing “bits” that exist in two states, either one or zero.
Quantum computing uses quantum bits, or qubits, which, in addition to being
one or zero can also be in a "superposition," which is both one and zero
simultaneously. This is possible because qubits are quantum units like atoms, ions,
or photons that operate under the rules of quantum mechanics instead of classical
mechanics. The "superposition" state allows a quantum computer to process
significantly more information than a classical computer and in a much shorter time.
The area of quantum computing took off about 14 years ago after Peter Shor
created a quantum algorithm that could factor large integers much more efficiently
than a classical computer. Though researchers are still many years away from
creating a quantum computer capable of running the Shor algorithm, progress has
been made. Kumar’s group, which uses photons as qubits, found that they can
entangle two indistinguishable photons together in an optical fiber very efficiently
by using the fiber’s inherent nonlinear response. They also found that no matter how
far you separate the two photons in standard transmission fibers they remain
entangled and are "mysteriously" connected to each other’s quantum state.
2 graduate students at
Georgia Tech
Science Daily (Dec. 8, 2008) — Physicists have taken a significant step toward
creation of quantum networks by establishing a new record for the length of time
that quantum information can be stored in and retrieved from an ensemble of very
cold atoms. Though the information remains usable for just milliseconds, even that
short lifetime should be enough to allow transmission of data from one quantum
repeater to another on an optical network.
The new record – 7 milliseconds for rubidium atoms stored in a dipole optical
trap – is scheduled to reported December 7 in the online version of the journal
Nature Physics by researchers at the Georgia Institute of Technology. The previous
record for storage time was 32 microseconds, a difference of more than two orders
of magnitude.
Optical entanglement
Entangled photons remain interconnected even when separated by large distances.
Merely observing one instantaneously affects the properties of the other. The
entanglement can be used in quantum communication to pass an encryption key that
is by its nature completely secure, as any attempt to eavesdrop or intercept the key
would be instantly detected!
Example 1 – 2 linearly polarized
photons of perpendicular polarizations
See http://www.davidjarvis.ca/entanglement/
Example of a photon entangler
An ultraviolet laser sends a single photon
through Beta Barium Borate. As the
photon travels through the crystal, there
is a chance it will split into 2 photons,
each of half the energy (twice the
wavelength).
If it splits, the photon will exit from the
Beta Barium Borate as two photons. The
resulting photon pair are entangled!
Result
a Bell-state quantum eraser
The Bell-state quantum eraser has
one more feature: each slit is covered
by a substance that filters the
(circular) polarization of a photon.
Consequently, the left-hand slit will receive photons with a counter-clockwise polarization,
and the right-hand slit will pass photons with a clockwise polarization.
Note: Polarization does not affect interference patterns.
Initially, neither detector shows an interference pattern. Since we control the polarization of
photons passing through the slits and we know the polarization accepted by each slit, we can
deduce which way the photons travelled (counter-clockwise through the left; clockwise
through the right). Thus no interference patterns are detected.
However, if we rotate the polarizing filter in front of detector A so that the polarizations of
the photons that hit the detector are the same (that is, we can no longer distinguish between
clockwise and counter-clockwise polarizations), then the interference pattern appears at both
detectors!
How do the photons arriving at detector B know that the polarizations have been
"erased" at detector A?
Entangled source point-to-point link
(information going to “Bob” and “Alice”)
fiber or free-space transmission
near IR
or
telecom IR
Single-photon or number sensitive
continuous
or
pulsed
phase or polarization qubits
Generation
Propagation
Detection
Quantum Dots - 1
Top: Cross-section scanning tunneling
microscope (STM) image shows indium
arsenide quantum dot regions embedded in
gallium arsenide. Each 'dot' is approximately
30 nanometers long–faint lines are individual
rows of atoms. (Color added for clarity.)
Bottom: Schematic of NIST-JQI experimental
set up. Orienting the resonant laser at a right
angle to the quantum dot light minimizes
scattering (Credit: Top: J.R. Tucker; Bottom:
Solomon/NIST)
Quantum dots are nanoscale regions of a semiconductor material similar to the material in computer
processors but with special properties due to their tiny dimensions. Though they can be composed of
tens of thousands of atoms, quantum dots in many ways behave almost as if they were single atoms.
Unfortunately, almost is not good enough when it comes to the fragile world of quantum
cryptography and next-generation information technologies. When energized, a quantum dot emits
photons, or “particles” of light, just as a solitary atom does. But imperfections in the shape of a
quantum dot cause what should be overlapping energy levels to separate. This ruins the delicate
balance of the ideal state required to emit entangled photons.
Quantum Dots - 2
Two lasers—one shining from above the quantum dot and the other illuminating it from
the side—the researchers were able to manipulate energy states in a quantum dot and
directly measure its emissions.
Andrew Shields at Toshiba and colleagues at the University of
Cambridge, produced entangled photons with an efficiency of
70% -- compared to a previous best figure of 49%. The
improved performance approaches that required for useful
applications, which means that devices emitting entangled
light could one day be as common as lasers and light-emitting
diodes New J. Physics 8, 29 (2006)
The team produced entangled photons from a crystal just 12 nm in diameter made from
indium arsenide embedded within a gallium arsenide and aluminium arsenide cavity.
When excited by a laser pulse, the quantum dot captures two electrons and two holes to
form a "biexciton" state in the dot. One of the electrons recombines with a hole to create a
photon, leaving behind an intermediate "exciton" state in the dot of one electron and one
hole. The other electron-hole pair then combines to create a second photon.
In a Bose Einstein Condensate there is a macroscopic number of atoms
in the ground state
In 1995 teams in Colorado and Massachusetts
achieved BEC in super-cold gas. This feat earned
those scientists the 2001 Nobel Prize in physics.
S. Bose,
1924
Light
A. Einstein,
1925
Atoms
E. Cornell
C. Wieman
W. Ketterle
Using Rb and Na atoms
When atoms are illuminated by laser beams they
feel a force which depends on the laser intensity.
Two counter-propagating beams
Standing wave
V ( x)  Sin 2 (kx)
Perfect Crystals
Mimic electrons in
solids: understand
their physics
Atomic Physics
Quantum
Information
Information is physical!
• Any processing of information is always performed by physical
means
• Bits of information obey laws of classical physics.
Every 18 months microprocessors double in speed:
1946
2000
Faster=Smaller
?
Atoms ~
ENIAC ~ m
Microchip ~ 0.000001 m
0.0000000001 m
Size
Year
Computer technology will reach a point where classical
physics is no longer a suitable model for the laws of
physics. We need quantum mechanics.
weirdness
Bits
• Fundamental building blocks
of classical computers:
• STATE: 0 or 1
• Definitely 0 or 1
Qubits
• Fundamental building blocks
of quantum computers:
• STATE: |0 or |1
• Superposition: a |0 +b |1
n
2n
2 bits
4 states: 00, 01, 10, 11
3 bits
8 states
10 bits
1024 states
30 bits
1 073 741 824 states
500 bits
More than our estimate of the number
of atoms in the universe
• A classical register with n bits can be in one of the 2n
posible states.
• A quantum register can be in a superposition of ALL
2n posible states.
A quantum computer can perform 2n operations at the same time
due to superposition :
However we get only one answer when we measure the result:
F[000]
F[001]
F[010]
.
.
F[111]
Only one
answer
F[a,b,c]
• Classical bit: Deterministic. We can find out if it is in state 0
or 1 and the measurement will not change the state of the bit.
• Qubit: Probabilistic
We get either |0
or |1
probabilities |a|2 and |b|2
|Y =a |0 +b |1
with corresponding
|a|2+|b|2=1
The measurement changes the state of the qubit!
|Y
|0
or |Y
|1
Strategy: Develop quantum algorithms
 Use superposition to calculate 2n values of function simultaneously
and do not read out the result until a useful outout is expected with
reasonably high probability.
Use entanglement: measurement of states can be highly correlated
•“Spooky action at a distance” - A. Einstein
• “ The most fundamental issue in quantum mechanics” –
E. Schrödinger
Quantum entanglement: Is a quantum phenomenon in which the
quantum states of two or more objects have to be described with
reference to each other.
Entanglement
Correlation between observable physical properties
e.g.
|Y
|Y
=|0 0
=( |0A 0B + |1A 1B )/√2
Product states are not
entangled
Use mathematical hard problems: factoring a
large number
870901
172475846743
198043
Shared
privately
with Bob
• Shor's algorithms (1994) allows solving factoring
problems which enables a quantum computer to break
public key cryptosystems.
Classical
172475846743=?x?
Quantum
172475846743= 870901 x198043
 Trapped ions
 Neutral atoms
 Electrons in semiconductors
Many others…..
DiVincenzo criteria
1. Scalable array of well defined qubits.
2. Initialization: ability to prepare one certain state
repeatedly on demand.
3. Universal set of quantum gates: A system in which qubits can be made to
evolve as desired.
4. Long relevant decoherence times.
5. Ability to efficiently read out the result.
a. Internal atomic states
|0
|1
Internal states are well understood: atomic spectroscopy & atomic
clocks.
b. Different vibrational levels
|1
|0
Scalability: the properties of
an optical lattice system do
not change when the size of
the system is increased.
• Internal state preparation: putting atoms in the same internal
state. Very well understood (optical pumping technique is in use
since 1950)
• Motional states preparation: Atoms can be cooled
to motional ground states (>95%)
Only one classical gate (NAND) is needed to
compute any function on bits!
?
1. How many gates do we need to make ?
2. Do we need one, two, three, four qubit
gates etc?
3. How do we make them?
Answer: We need to be able to make arbitrary single
qubit operations and a phase gate
Phase gate:
|0 0
|00
a|0 +b|1
X
c|0 +d|1
|0 1
|01
|1 0
eif |10
|11
|11
1.
Single qubit rotation: Well understood and carried out
since 1940’s by using lasers
|1
Laser
|0
2.
Two qubit gate: None currently implemented but
conditional logic has been demonstrated
Collision
|0102+eif0111+ 1002+1011
Displace
|0102+0111+ 1002+1011
Combine
|(01+11)( 02+12)
initial
|01 02
Experiment implemented in optical lattices
Entangled state
Environment
Classical statistical
mixture
Entangled states are very fragile to decoherence
An important challenge is the design of decoherence resistant
entangled states
Main limitation: Light scattering
Global: Well understood, standard atomic techniques
e.g: Absorption images, fluorescence
Local: Difficult since it is hard to detect one atom
without perturbing the other
Experimentally achieved
very recently at Harvard:
Nature 462 74 (2009).
• All five requirements for quantum computations have
been implemented in different systems. Trapped ions
are leading the way.
• There has been a lot progress, however, there are
great challenges ahead……
Overall, quantum computation is certainly a fascinating
new field.