Quantum Ideas lecture notes

Quantum IdeasDr. Axel Kuhn, Oxford 2010

lecture noteshttp://www.physics.ox.ac.uk/atomphoton/presentations.html

Trinity term, weeks 1 – 4: Mon, Tue, Wed @ 11am

Quantum Ideas

Some Literature

J. Polkinghorne: Quantum TheoryA Very Short Introduction (OUP 2002)

D. F. Styer: The Strange World of Quantum Mechanics (Cambridge 2000)

T. Hey and P. Walters: The New Quantum Universe (Cambridge 2003)

Basdevant and Dalibard: Quantum Mechanics (Springer 2002)

Quantum Ideas

What is wrong with “classical” physics? ! The dawn of quantum physics

Quantum nature of light ! particle-like behaviour of waves

Quantum nature of matter ! wave-like properties of matter

Does God play dice?! Uncertainty principle, superposition, and entanglement! Schrödinger‘s cat and other paradoxes

Modern applications! Quantum cryptography, teleportation and computing

wave-particledualism

The Dawn of Quantum Physics

Classical Physics ! Deterministic„For an intellect which ... would know all forces that set nature in motion, and all positions of all items of which nature is composed, ... , nothing would be uncertain and the future just like the past would be present before its eyes“

Laplace‘s demon (Pierre-Simon Laplace, 1814)

• fails to describe nature correctly• very intuitive and easy to interpret

Quantum IdeasSyllabus:

The success of classical physics, measurements in classical physics. The nature of light, the ultraviolet catastrophe, the photoelectric effect and the quantisation of radiation. Atomic spectral lines and the discrete energy levels of electrons in atoms, the Frank-Hertz experiment and the Bohr model of an atom.

Magnetic dipoles in homogeneous and inhomogeneous magnetic fields and the Stern-Gerlach experiment showing the quantisation of the magnetic moment. The Uncertainty principle by considering a microscope and the momentum of photons, zero point energy, stability and size of atoms. Measurements in quantum physics, the impossibility of measuring two orthogonal components of magnetic moments. The EPR paradox, entanglement, hidden variables, non-locality and Aspect's experiment, quantum cryptography and the BB84 protocol. Schrödinger's cat and the many-world interpretation of quantum mechanics. Interferometry with atoms and large molecules. Amplitudes, phases and wavefunctions. Interference of atomic beams, discussion of two-slit interference, Bragg diffraction of atoms, quantum eraser experiments. A glimpse of quantum engineering and quantum computing. Schrödinger's equation and boundary conditions. Solution for a particle in an infinite potential well, to obtain discrete energy levels and wavefunctions.


Classical Physics:

• classical mechanics (Newton; F = m a)

• electricity and magnetism (Coulomb, Faraday, Maxwell)

• electromagnetic waves (rf ... light ... x-ray ... gamma)

• thermodynamics (energy conservation, equilibration, statistical mechanics)

• accurate measurement of all observables (position x(t)and momentum p(t) )

Quantum Physics:

• probabilistic - not deterministic (Einstein: „Good does not play dice“)

• probability wave function ψ(x,t) to describe a particle

• superposition and entanglement

• non-local behaviour („Spooky interaction at a distance“ that bothered Einstein)

• uncertainty principle: Δx Δp ≥ ħ/2 and ΔE Δt ≥ ħ/2


Quantum Physics ! Probabilistic• quantisation of energy (discrete energy levels)

• wave-particle duality and interference of particles

• Schrödinger‘s cat and other paradoxes

• uncertainty relation ! limit to accurate measurements

• most successful description of nature• not intuitive and many interpretations


Failures of classical physics... explained by quantum mechanics

photoelectric effect(Albert Einstein, 1905, NP 1921)

black-body radiation(Max Planck, 1900, NP 1918)

atomic spectra and structure(Niels Bohr, 1913, NP 1922)


Failures of classical physics... continued

interference of particles(L. de Broglie, 1924, NP 1929)

Creation of quantum mechanicsuncertainty principle(W. Heisenberg, 1927, NP 1932)

Schrödinger‘s wave equation(E. Schrödinger, 1926, NP 1933)

The Quantum Nature of Light

Photoelectric effectPhoton(s) in the Box: Blackbody radiationMomentum of a photonPhoton-particle collisions (laser cooling)Photons by the number

Newton postulates light “particles”• reflection and refraction

Interference phenomena ! light waves• Young‘s double-slit experiment• Maxwell‘s equations ! electromagnetic wave• diffraction, interference

Quantised radiation ! Photons• Photoelectric effect and black-body radiation• Compton scattering

The Quantum Nature of Light

Some History

Photoelectric Effect

Ekin = h! " Ebind

quantised?

wavelength !frequency "=c/!photon energy h" = hc/! ! Quantisation of Radiation

[ ... ]


Ekin = Ephoton − Ebind

Ephoton = hν = hc/λ

-1,5

-1

-0,5

0

0,5

1

4 5 6 7

1014 Hz)

Ekin h Ebind

Ekin (eV)

Eh= E


Planck‘s constant h = 6.626 #10-34 Js

• different colours (frequencies)• threshold gate voltage = max. kinetic energy

electron binding energy ! quantised interactionno delay ! quantised radiation

Ekin = h! " Ebind


time

Intensity I

time

Current I ! area ! t = Ebind

t

t

no delay!

Photoelectric Effect:Example:

illumination with a mercury lamp, filtering a single spectral line.

Cathode metal with a binding energy (or work function) of Ebind = 2.02 eV

! yellow, 578nm, 5.19E+14 Hz, Ekin = 0.13 eV! green, 546nm, 5.50E+14 Hz, Ekin = 0.27 eV! blue, 436nm, 6.88E+14 Hz, Ekin = 0.81 eV! violet, 405nm, 7.41E+14 Hz, Ekin = 1.02 eVPlanckʻs constant is obtained from the slope of the kinetic energy, Ekin(ν)

Minimum Duration for a 1st Photoelectron:

assuming radiation of λ=598 is continuous and impinges on the cathode with

Interaction Cross section with a single surface atom

Work function Ebind = 1.75 eV

time t needed to accumulate enough energy, with

yields

Hints for quantisation:a) threshold (minimum frequency required): ! • resonance phenomenon! • quantised medium or light

b) linear in the intensity (for ν=const). ! • electron number proportional to photon number

c) photo current insensitive to ν (provided hν > Ebind) ! • no change of the electron current if photon flux constant! • albeit the intensity is increasing: Iphoto ∝ ν

d) no delay ⇒ direct evidence!

! • it lasts seconds until a single atom accumulates enough energy, ! so the radiation cannot be continuous

I = 1pW/cm2

σ = λ2/2

I · σ · t = Ebindt = Ebind/(I · σ) = 164s


Observations:Threshold: h! < Ebind ! Iphoto = 0Linearity: h! > Ebind ! Iphoto " Ilight

Constancy: Iphoto (!)= const. for h! > Ebind No delay! Photocurrent starts immediately!

Energy increase: Ekin = h! " Ebind

Photon Shot Noise

brightillumination

1.5 photons / pixel 20 photons / pixel

1 shot

16 shots

The Story so far

The dawn of quantum mechanics

classical physics insufficient

Photoelectric effect

E = h !

Energy = h " frequency

Single-photon shot noise

Planck‘s constant h = 6.626 #10-34 Js

[ ... ]

So what is a “Photon” ?

smallest amount of light of a given frequency! a photon was there, when my detector clicks (FAPP) ! energy quantum of a field mode (Planck)

cannot be divided! effect of half-silvered mirrors?

wave- and particle like behaviour! (wave amplitude)2 ! probability of photodetections! interference ... only with itself (Dirac)

Photon(s) in the Box

standing wave in aone-dimensional cavity

standing wave ona violin string

nodes at the end points


Mode density of a 1D cavity, g(!) d!

n! = 2L

resonancecondition

modedensity (1D)

g(!)d! = 2 " 2Lcd!

two polarisations


Generalisation to 3D: A black body ...absorbs all incident radiation and radiates in accordance with its temperature, not dependent upon the incident radiation.

The radiated energy ...can be considered to come from standing waves in a three-dimensional hollow cavity.

Spectral energy density:$(") d" = g(") d" ##E (")$= mode density # avg. Energy

Blackbody = Cavity?

- Multiple reflections -> absorption of incident light- Thermal equilibrium -> Walls <-> Cavity modes- Spectral energy density ρ(ν)dν ∝R(ν)dν (Radiance through whole)

- Boundary conditions: Nodes on the walls- Standing waves along x,y,z

Consider a 2D problem and decompose λ into

! λx = λ / cos(α)! λy = λ / sin(α)

with nx λx = 2L etc... ==> nx = (2L/ λ) cos(α) and ny = (2L/ λ) sin(α)

square and add these conditions (generalise into 3D):

(2L/ λ)2 = nx2 + ny2 + nz2

Number of Modes in the cavity with frequencies smaller than ν:

- sphere of radius R = √(nx2 + ny2 + nz2) = 2L/ λ = 2L/c × ν- mode number N(ν) = 4/3 π R3 × 2/8 = ...- same for N(ν+dν) = ...

Mode number in the interval ν...ν+dν

! ΔN = N(ν+dν) - N(ν) = 8π ν2 L3 / c3 dν

Spectral density per unit volume

! ρ(ν)dν = ΔΝ/L3 = 8π ν2 / c3 dν

Blackbody Radiation

Spectral energy density:$(") d" = g(") d" !#E (")$= mode density # avg. Energy

Mode density of a hollow cavity, g(!) d! number of resonant cavity modes within [!... !+d!]

Average energy per cavity mode, #E (!)$

Blackbody Radiation

Mode density of a hollow cavity, g(!) d!

nx2 + ny

2 + nz2 =

4L2

!2

resonancecondition (3D)

number of modesin the “sphere”:

N(!n " !) =43#2L$

%&'

()*

3+28=8#L3

3c3! 3 g(!)d! = 8"!

2

c3d!!

Blackbody Radiation

Average Energy (using classical physics)

• Boltzmann distribution

• average Energy

P(E) = 1kBT

e!E kBT

E = EP(E)dE = kBT!

Wrap-up from Yesterday

Spectral energy density:$(") d" = g(") d" !#E (")$= mode density # avg. Energy

N(!n " !) =43#2L$

%&'

()*

3+28=8#L3

3c3! 3 g(!)d! = 8"!

2

c3d!!

Boltzmann distribution

E = EP(E)dE = kBT!

!(")d" = kBT #8$" 2

c3d"

Rayleigh-Jeans law

ultravioletcatastrophe

Blackbody Radiation - quantised

Average Energy(quantised radiation)

• Boltzmann distribution

• average Energy

P(En )! e"En kBT with En = nh#

E =En P(En )!P(En )!

=h"

eh" kBT #1

Rayleigh-Jeans law !ultraviolet catastrophe

Planck‘s law

!(")d" = kBT #8$" 2

c3d"

!(")d" = h"eh" kBT #1

$8%" 2

c3d"

Blackbody Radiation

Blackbody RadiationAverage energy <E>per frequency mode, using the Boltzmann distribution P(E)

E =EnP(E)

n=0

∞

∑

P(E)n=0

∞

∑=hν ne−

nhνkT

n=0

∞

∑

e−nhν

kT

n=0

∞

∑= hν A

B

with B − Bexp −hν kT( ) = 1and A − Aexp −hν kT( ) = Bexp −hν kT( )∴

AB=

1exp hν

kT( ) −1and E =

hνexp hν

kT( ) −1

Planckʻs lawSpectral energy density (energy per unit volume in the frequency range ν...ν+dν):

ρ(ν)dν = 8πν2

c3hν

exp hνkT( ) −1

=8πhν 3

c31

exp hνkT( ) −1

total energy density:

ρ = ρ(ν)dν = 8π 5k 4

15(hc)3× T 4 = σT 4∫

Blackbody Radiation

shift of the “most probable” wavelength with TWien‘s

displacement law

!maxT = const = 0.2014 hckB

total radiancy

RT = !(")d" = #T 4$

Planck‘s Postulate

Any physical quantity which oscillates in time has a total energy E which satisfies

E = nh! with n = 0,1,2,3,...

" is the frequency andh is Planck‘s constant

E=0

E=h

E=5h

E=4h

E=3h

E=2h

E=0

classical Planckh=6.626 "10-34 Js

Black-Body Radiation - Summary

Blackbody radiationRayleigh-Jeans lawUltraviolet catastrophePlanck‘s law ! Planck‘s postulate

E = nh! with n = 0,1,2,3,...

!(")d" = h"eh" kBT #1

$8%" 2

c3d"

see also“Quantum Physics”

by Eisberg & Resnick

Wave-Particle Dualism

Luis de Broglie, 1924: De-Broglie wavelength

! = h p or p = !k

Any particle of momentum p and energy E=#$

! (x,t) = exp i(kx "#t)[ ]

with %=h/2& and '=2&" • plane wave• not localised

Compton Scattering

elastic photon-electron scattering

Arthur Compton, 1922

Photon-Particle Collisions

Modern Atomic Physics !Atom OpticsLaser cooling and trapping of AtomsBose-Einstein condensation (BEC)

Control and manipulate motional degrees of freedom with light pf = pi ± #k

opticalmolasses

Magneto-Optical Trap

0

1000

2000

3000

4000

5000

6000

7000

0

1

2

0

1

Photo

n C

oun

t R

ate

(H

z)

X P

ositio

n (m

m)

Y Position (mm)

Figure 5: Two-dimensional CCD image of the fluorescence from franciumatoms trapped in a MOT. (From Ref. [23]).

Maxwell-Boltzmann distribution. The remaining atoms thermalize duringwall collisions and form a new Maxwell-Boltzmann distribution. From thisthe MOT can again capture the low velocity atoms. The trapping efficiencydepends on the number of wall contacts that an atom can make before leavingthe system. Since alkali atoms tend to chemisorb in the glass walls, specialcoatings can prevent the loss of an atom [26]. If the wall is coated, the atomphysi-sorbs for a short time, thermalizes and then is free to again cross thecapture region and fall into the trap.

The capture range of the MOT is enhanced with the help of large andintense laser beams. Gibble et al. [27] reported that for their large trapthey captured atoms with initial velocities below about 18 % of the averagethermal velocity at room temperature. However, the fraction of the Maxwell-

22

• 109 alkalis (Rb, Cs, Na)• T below 100 µK

• 3 pairs of laser beams• magnetic field gradient

Summary of the 1st week

Quantisation of radiationEnergy of a single photonEnergies of a frequency mode

Wave-particle dualism (to be continued)

De-Broglie wavelengthMomentum transfer

Applications in modern physicsLaser cooling and trapping of atomsPhotons by the number

E = h!En = nh!

! =

hp

or p = !k

Photons by the Number

•“spatio-temporal” modes carrying single photons

• photo detection% ! mode is empty% ! no 2nd detection

• antibunching% g(2)(0)< g(2)(%&)

g(2) (!" ) =p(t)p(t +!" )

p(t) 2

time


# of correlations

Single (few) atom fluorescence• atomic beam of low density• antibunching

H.J. Kimble et al. Phys. Rev. Lett. 39, 691 (1977)


Laser

Single-ion fluorescence• ion in an RF Paul trap• antibunching

• effect of the laser intensity: weak ( ! (broad minimum strong( ! (narrow minimum

F. Diedrich and H. Walther, Phys. Rev. Lett. 58, 203 (1987)

Cavity-based Single-Photon Sources

==

single atomor trapped ion

A. Kuhn and D. Ljunggren, Contemporary Physics, in print (2010)

two-mirror cavity100,000

round trips

100 µm length'

light travels 20 m

Cavity-based Single-Photon SourcesA. Kuhn and D. Ljunggren, Contemporary Physics, in print (2010)

• A. Kuhn et al., Phys. Rev. Lett. 89, 067901 (2002)• J. McKeever et al., Science 303, 1992 (2004)• M. Keller et al., Nature 431, 1075 (2004)


• envelope ! interaction time• equidistant comb ! triggered photon emission• antibunching: g(2)(0)< g(2)(%&) ! single photons on demand


Quantum Nature of Matter

Light Free Particles Both

Energy

Wavelength

Momentum

Position

Time

IntensityE=(n)h"

E=p2/2m=E=h" E=(n)h"

!=c/" !=h/p wave-particle dualism

p=h/! p=mv p=%k

P(x)=)*) x P(x)=)*)*x*p+%/2

exp(i[kx-'t]) trajectory *E*t+%/2

Interference of Matter Waves

double slit / grating

Bragg diffraction Interferometer

light

electrons

neutrons

atoms

molecules

cats

" " "

" " "

" " "

" " "

" " "

(")

Davidson-Germer Experiment

Bragg scattering

!"#$%&$' ' ()'*)'+,-./'

!

*-01234-53'67'!89:,;:' <8103-2=>?)/6;' @5,A-2:,39'67'+.62,/1'

!

!"#$%&'()*+,-.#/0#/$123#/(0#,4$$

5/&66$7-8&9$:;&44#/(,6$<!"#$#%&=>$

"#$%&'()!*+(!%,-**()&$.!/62'2-7.-;3,650!#1!7-

/&9*!1)#2!*3#!-'4-,($*!56-$(%!#1!-*#2%!/,-66('!

7)-..!56-$(%08!!9#!-!.##'!-55)#:&2-*&#$!*+(!

3&4"$?#,64"$)(@@#/#,;#!;(*3(($!)-<!=!-$'!)-<!

>!&%?$

!%&$>=> /222 "#"$ ?!

3+()(!)!&%!*+(!-*#2&,!%5-,&$.!-$'!!!&%!*+(!

@.)-A&$.B!-$.6(8!!9+C%!2-:&2-6!,#$%*)C,*&D(!

&$*()1()($,(!3&66!#,,C)!-*!

!?>?=?E>

%&$ %%"" 4/

4&

! !!3&*+!*+(!=%*!#)'()!2-:&2C2!-*!

/>%&$ =

&! " 8!

$%&'()*+,-.#/0#/$<'(')$*#%&+,%&$'-.+#/+!"#$#%&=>$

9+(!1&)%*!(D&'($,(!#1!F(!7)#.6&(G%!+<5#*+(%&%!,-2(!&$!ABCD!3+($!%&'()*+,$

&,)$.#/0#/!%,-**()('!(6(,*)#$%!#11!-!H&!,)<%*-68!!9+(!%5-,&$.!)!1#)!H&!3-%!

I$#3$!1)#2!JK)-<!%,-**()&$.!*#!;(!EFEBC$,08!!F-D&'%#$!-$'!L()2()!C%('!

3&*+!-!I&$(*&,!($().<!#1!DG$#H8!!9+(!F(!7)#.6&(!3-D(6($.*+!#1!-$!(6(,*)#$!

3&*+!-!.&D($!I&$(*&,!($().<!&%!

BC;4BC

8;

;4;4BC

8;

;0

8;

0

8

--- 0/>0/0/ >>>>>> '"

#'"""& $

M#)!$#$K)(6-*&D&%*&,!(6(,*)#$%!NO!PP!2(,>!-$'!

BC;4

8;

BC;4BC

8;

-

;4BC

-

-0/>0/> >>>

>((( )('

"**

& F$

M#)!DG$#H!(6(,*)#$%!&$!$EFAIJ$,0!-$'!!

QER8E0EQ>8E/>

=ST8E

>%&$ = +""

54

54

/

&! !-$'!!=!!!SU

#8!

9+(!@'(*(,*#)!-$.6(B!&%!)(6-*('!*#!*+(!@.)-A&$.!-$.6(B!;<!"UE> == +#" !,- 8!!9+(!D-6C(!#1!DE

+!-.)(('!3&*+!*+(!

(:5()&2($*-6!'-*-V!!K+4#$4"&4$4"#$(,4#/@#/#,;#$3&44#/,$+;;L/*$

#'#,$M(4"$+,?9$+,#$#?#;4/+,$(,$4/&,*(4$&4$&$4(0#N!

! ! ! ! ! ! !

)

O6/&P(,6$&,6?#Q.!.

!"#$$%&'(#)%*+#,,-"./$$

!.

R,;()#,4$/&9$A! :;&44#/#)$/&9$A!

R,;()#,4$/&9$C! :;&44#/#)$/&9$C!

!. !.

)*(,!!)*(,!!

!.!.

O)#4#;4+/$&,6?#Q$

.-.

%#4#;4+/$S,6?#$-$

KL0T#/$+@$:;&44#/#)$

DG$#H$

DE+$

!"#$%&$' ' ()'*)'+,-./'

!

*-01234-53'67'!89:,;:' <8103-2=>?)/6;' @5,A-2:,39'67'+.62,/1'

!

!"#$%&'()*+,-.#/0#/$123#/(0#,4$$

5/&66$7-8&9$:;&44#/(,6$<!"#$#%&=>$

"#$%&'()!*+(!%,-**()&$.!/62'2-7.-;3,650!#1!7-

/&9*!1)#2!*3#!-'4-,($*!56-$(%!#1!-*#2%!/,-66('!

7)-..!56-$(%08!!9#!-!.##'!-55)#:&2-*&#$!*+(!

3&4"$?#,64"$)(@@#/#,;#!;(*3(($!)-<!=!-$'!)-<!

>!&%?$

!%&$>=> /222 "#"$ ?!

3+()(!)!&%!*+(!-*#2&,!%5-,&$.!-$'!!!&%!*+(!

@.)-A&$.B!-$.6(8!!9+C%!2-:&2-6!,#$%*)C,*&D(!

&$*()1()($,(!3&66!#,,C)!-*!

!?>?=?E>

%&$ %%"" 4/

4&

! !!3&*+!*+(!=%*!#)'()!2-:&2C2!-*!

/>%&$ =

&! " 8!

$%&'()*+,-.#/0#/$<'(')$*#%&+,%&$'-.+#/+!"#$#%&=>$

9+(!1&)%*!(D&'($,(!#1!F(!7)#.6&(G%!+<5#*+(%&%!,-2(!&$!ABCD!3+($!%&'()*+,$

&,)$.#/0#/!%,-**()('!(6(,*)#$%!#11!-!H&!,)<%*-68!!9+(!%5-,&$.!)!1#)!H&!3-%!

I$#3$!1)#2!JK)-<!%,-**()&$.!*#!;(!EFEBC$,08!!F-D&'%#$!-$'!L()2()!C%('!

3&*+!-!I&$(*&,!($().<!#1!DG$#H8!!9+(!F(!7)#.6&(!3-D(6($.*+!#1!-$!(6(,*)#$!

3&*+!-!.&D($!I&$(*&,!($().<!&%!

BC;4BC

8;

;4;4BC

8;

;0

8;

0

8

--- 0/>0/0/ >>>>>> '"

#'"""& $

M#)!$#$K)(6-*&D&%*&,!(6(,*)#$%!NO!PP!2(,>!-$'!

BC;4

8;

BC;4BC

8;

-

;4BC

-

-0/>0/> >>>

>((( )('

"**

& F$

M#)!DG$#H!(6(,*)#$%!&$!$EFAIJ$,0!-$'!!

QER8E0EQ>8E/>

=ST8E

>%&$ = +""

54

54

/

&! !-$'!!=!!!SU

#8!

9+(!@'(*(,*#)!-$.6(B!&%!)(6-*('!*#!*+(!@.)-A&$.!-$.6(B!;<!"UE> == +#" !,- 8!!9+(!D-6C(!#1!DE

+!-.)(('!3&*+!*+(!

(:5()&2($*-6!'-*-V!!K+4#$4"&4$4"#$(,4#/@#/#,;#$3&44#/,$+;;L/*$

#'#,$M(4"$+,?9$+,#$#?#;4/+,$(,$4/&,*(4$&4$&$4(0#N!

! ! ! ! ! ! !

)

O6/&P(,6$&,6?#Q.!.

!"#$$%&'(#)%*+#,,-"./$$

!.

R,;()#,4$/&9$A! :;&44#/#)$/&9$A!

R,;()#,4$/&9$C! :;&44#/#)$/&9$C!

!. !.

)*(,!!)*(,!!

!.!.

O)#4#;4+/$&,6?#Q$

.-.

%#4#;4+/$S,6?#$-$

KL0T#/$+@$:;&44#/#)$

DG$#H$

DE+$

!"#$%&$' ' ()'*)'+,-./'

!

*-01234-53'67'!89:,;:' <8103-2=>?)/6;' @5,A-2:,39'67'+.62,/1'

!

!"#$%&'()*+,-.#/0#/$123#/(0#,4$$

5/&66$7-8&9$:;&44#/(,6$<!"#$#%&=>$

"#$%&'()!*+(!%,-**()&$.!/62'2-7.-;3,650!#1!7-

/&9*!1)#2!*3#!-'4-,($*!56-$(%!#1!-*#2%!/,-66('!

7)-..!56-$(%08!!9#!-!.##'!-55)#:&2-*&#$!*+(!

3&4"$?#,64"$)(@@#/#,;#!;(*3(($!)-<!=!-$'!)-<!

>!&%?$

!%&$>=> /222 "#"$ ?!

3+()(!)!&%!*+(!-*#2&,!%5-,&$.!-$'!!!&%!*+(!

@.)-A&$.B!-$.6(8!!9+C%!2-:&2-6!,#$%*)C,*&D(!

&$*()1()($,(!3&66!#,,C)!-*!

!?>?=?E>

%&$ %%"" 4/

4&

! !!3&*+!*+(!=%*!#)'()!2-:&2C2!-*!

/>%&$ =

&! " 8!

$%&'()*+,-.#/0#/$<'(')$*#%&+,%&$'-.+#/+!"#$#%&=>$

9+(!1&)%*!(D&'($,(!#1!F(!7)#.6&(G%!+<5#*+(%&%!,-2(!&$!ABCD!3+($!%&'()*+,$

&,)$.#/0#/!%,-**()('!(6(,*)#$%!#11!-!H&!,)<%*-68!!9+(!%5-,&$.!)!1#)!H&!3-%!

I$#3$!1)#2!JK)-<!%,-**()&$.!*#!;(!EFEBC$,08!!F-D&'%#$!-$'!L()2()!C%('!

3&*+!-!I&$(*&,!($().<!#1!DG$#H8!!9+(!F(!7)#.6&(!3-D(6($.*+!#1!-$!(6(,*)#$!

3&*+!-!.&D($!I&$(*&,!($().<!&%!

BC;4BC

8;

;4;4BC

8;

;0

8;

0

8

--- 0/>0/0/ >>>>>> '"

#'"""& $

M#)!$#$K)(6-*&D&%*&,!(6(,*)#$%!NO!PP!2(,>!-$'!

BC;4

8;

BC;4BC

8;

-

;4BC

-

-0/>0/> >>>

>((( )('

"**

& F$

M#)!DG$#H!(6(,*)#$%!&$!$EFAIJ$,0!-$'!!

QER8E0EQ>8E/>

=ST8E

>%&$ = +""

54

54

/

&! !-$'!!=!!!SU

#8!

9+(!@'(*(,*#)!-$.6(B!&%!)(6-*('!*#!*+(!@.)-A&$.!-$.6(B!;<!"UE> == +#" !,- 8!!9+(!D-6C(!#1!DE

+!-.)(('!3&*+!*+(!

(:5()&2($*-6!'-*-V!!K+4#$4"&4$4"#$(,4#/@#/#,;#$3&44#/,$+;;L/*$

#'#,$M(4"$+,?9$+,#$#?#;4/+,$(,$4/&,*(4$&4$&$4(0#N!

! ! ! ! ! ! !

)

O6/&P(,6$&,6?#Q.!.

!"#$$%&'(#)%*+#,,-"./$$

!.

R,;()#,4$/&9$A! :;&44#/#)$/&9$A!

R,;()#,4$/&9$C! :;&44#/#)$/&9$C!

!. !.

)*(,!!)*(,!!

!.!.

O)#4#;4+/$&,6?#Q$

.-.

%#4#;4+/$S,6?#$-$

KL0T#/$+@$:;&44#/#)$

DG$#H$

DE+$

Davidson-Germer Experiment

Bragg scattering with electrons (1925, NP 1937)

!"#$%&$' ' ()'*)'+,-./'

!

*-01234-53'67'!89:,;:' <8103-2=>?)/6;' @5,A-2:,39'67'+.62,/1'

!

!"#$%&'()*+,-.#/0#/$123#/(0#,4$$

5/&66$7-8&9$:;&44#/(,6$<!"#$#%&=>$

"#$%&'()!*+(!%,-**()&$.!/62'2-7.-;3,650!#1!7-

/&9*!1)#2!*3#!-'4-,($*!56-$(%!#1!-*#2%!/,-66('!

7)-..!56-$(%08!!9#!-!.##'!-55)#:&2-*&#$!*+(!

3&4"$?#,64"$)(@@#/#,;#!;(*3(($!)-<!=!-$'!)-<!

>!&%?$

!%&$>=> /222 "#"$ ?!

3+()(!)!&%!*+(!-*#2&,!%5-,&$.!-$'!!!&%!*+(!

@.)-A&$.B!-$.6(8!!9+C%!2-:&2-6!,#$%*)C,*&D(!

&$*()1()($,(!3&66!#,,C)!-*!

!?>?=?E>

%&$ %%"" 4/

4&

! !!3&*+!*+(!=%*!#)'()!2-:&2C2!-*!

/>%&$ =

&! " 8!

$%&'()*+,-.#/0#/$<'(')$*#%&+,%&$'-.+#/+!"#$#%&=>$

9+(!1&)%*!(D&'($,(!#1!F(!7)#.6&(G%!+<5#*+(%&%!,-2(!&$!ABCD!3+($!%&'()*+,$

&,)$.#/0#/!%,-**()('!(6(,*)#$%!#11!-!H&!,)<%*-68!!9+(!%5-,&$.!)!1#)!H&!3-%!

I$#3$!1)#2!JK)-<!%,-**()&$.!*#!;(!EFEBC$,08!!F-D&'%#$!-$'!L()2()!C%('!

3&*+!-!I&$(*&,!($().<!#1!DG$#H8!!9+(!F(!7)#.6&(!3-D(6($.*+!#1!-$!(6(,*)#$!

3&*+!-!.&D($!I&$(*&,!($().<!&%!

BC;4BC

8;

;4;4BC

8;

;0

8;

0

8

--- 0/>0/0/ >>>>>> '"

#'"""& $

M#)!$#$K)(6-*&D&%*&,!(6(,*)#$%!NO!PP!2(,>!-$'!

BC;4

8;

BC;4BC

8;

-

;4BC

-

-0/>0/> >>>

>((( )('

"**

& F$

M#)!DG$#H!(6(,*)#$%!&$!$EFAIJ$,0!-$'!!

QER8E0EQ>8E/>

=ST8E

>%&$ = +""

54

54

/

&! !-$'!!=!!!SU

#8!

9+(!@'(*(,*#)!-$.6(B!&%!)(6-*('!*#!*+(!@.)-A&$.!-$.6(B!;<!"UE> == +#" !,- 8!!9+(!D-6C(!#1!DE

+!-.)(('!3&*+!*+(!

(:5()&2($*-6!'-*-V!!K+4#$4"&4$4"#$(,4#/@#/#,;#$3&44#/,$+;;L/*$

#'#,$M(4"$+,?9$+,#$#?#;4/+,$(,$4/&,*(4$&4$&$4(0#N!

! ! ! ! ! ! !

)

O6/&P(,6$&,6?#Q.!.

!"#$$%&'(#)%*+#,,-"./$$

!.

R,;()#,4$/&9$A! :;&44#/#)$/&9$A!

R,;()#,4$/&9$C! :;&44#/#)$/&9$C!

!. !.

)*(,!!)*(,!!

!.!.

O)#4#;4+/$&,6?#Q$

.-.

%#4#;4+/$S,6?#$-$

KL0T#/$+@$:;&44#/#)$

DG$#H$

DE+$

E=54 eV and !=0.167 nm

!"#$%&$' ' ()'*)'+,-./'

!

*-01234-53'67'!89:,;:' <8103-2=>?)/6;' @5,A-2:,39'67'+.62,/1'

!

!"#$%&'()*+,-.#/0#/$123#/(0#,4$$

5/&66$7-8&9$:;&44#/(,6$<!"#$#%&=>$

"#$%&'()!*+(!%,-**()&$.!/62'2-7.-;3,650!#1!7-

/&9*!1)#2!*3#!-'4-,($*!56-$(%!#1!-*#2%!/,-66('!

7)-..!56-$(%08!!9#!-!.##'!-55)#:&2-*&#$!*+(!

3&4"$?#,64"$)(@@#/#,;#!;(*3(($!)-<!=!-$'!)-<!

>!&%?$

!%&$>=> /222 "#"$ ?!

3+()(!)!&%!*+(!-*#2&,!%5-,&$.!-$'!!!&%!*+(!

@.)-A&$.B!-$.6(8!!9+C%!2-:&2-6!,#$%*)C,*&D(!

&$*()1()($,(!3&66!#,,C)!-*!

!?>?=?E>

%&$ %%"" 4/

4&

! !!3&*+!*+(!=%*!#)'()!2-:&2C2!-*!

/>%&$ =

&! " 8!

$%&'()*+,-.#/0#/$<'(')$*#%&+,%&$'-.+#/+!"#$#%&=>$

9+(!1&)%*!(D&'($,(!#1!F(!7)#.6&(G%!+<5#*+(%&%!,-2(!&$!ABCD!3+($!%&'()*+,$

&,)$.#/0#/!%,-**()('!(6(,*)#$%!#11!-!H&!,)<%*-68!!9+(!%5-,&$.!)!1#)!H&!3-%!

I$#3$!1)#2!JK)-<!%,-**()&$.!*#!;(!EFEBC$,08!!F-D&'%#$!-$'!L()2()!C%('!

3&*+!-!I&$(*&,!($().<!#1!DG$#H8!!9+(!F(!7)#.6&(!3-D(6($.*+!#1!-$!(6(,*)#$!

3&*+!-!.&D($!I&$(*&,!($().<!&%!

BC;4BC

8;

;4;4BC

8;

;0

8;

0

8

--- 0/>0/0/ >>>>>> '"

#'"""& $

M#)!$#$K)(6-*&D&%*&,!(6(,*)#$%!NO!PP!2(,>!-$'!

BC;4

8;

BC;4BC

8;

-

;4BC

-

-0/>0/> >>>

>((( )('

"**

& F$

M#)!DG$#H!(6(,*)#$%!&$!$EFAIJ$,0!-$'!!

QER8E0EQ>8E/>

=ST8E

>%&$ = +""

54

54

/

&! !-$'!!=!!!SU

#8!

9+(!@'(*(,*#)!-$.6(B!&%!)(6-*('!*#!*+(!@.)-A&$.!-$.6(B!;<!"UE> == +#" !,- 8!!9+(!D-6C(!#1!DE

+!-.)(('!3&*+!*+(!

(:5()&2($*-6!'-*-V!!K+4#$4"&4$4"#$(,4#/@#/#,;#$3&44#/,$+;;L/*$

#'#,$M(4"$+,?9$+,#$#?#;4/+,$(,$4/&,*(4$&4$&$4(0#N!

! ! ! ! ! ! !

)

O6/&P(,6$&,6?#Q.!.

!"#$$%&'(#)%*+#,,-"./$$

!.

R,;()#,4$/&9$A! :;&44#/#)$/&9$A!

R,;()#,4$/&9$C! :;&44#/#)$/&9$C!

!. !.

)*(,!!)*(,!!

!.!.

O)#4#;4+/$&,6?#Q$

.-.

%#4#;4+/$S,6?#$-$

KL0T#/$+@$:;&44#/#)$

DG$#H$

DE+$

!"#$%&$' ' ()'*)'+,-./'

!

*-01234-53'67'!89:,;:' <8103-2=>?)/6;' @5,A-2:,39'67'+.62,/1'

!

!"#$%&'()*+,-.#/0#/$123#/(0#,4$$

5/&66$7-8&9$:;&44#/(,6$<!"#$#%&=>$

"#$%&'()!*+(!%,-**()&$.!/62'2-7.-;3,650!#1!7-

/&9*!1)#2!*3#!-'4-,($*!56-$(%!#1!-*#2%!/,-66('!

7)-..!56-$(%08!!9#!-!.##'!-55)#:&2-*&#$!*+(!

3&4"$?#,64"$)(@@#/#,;#!;(*3(($!)-<!=!-$'!)-<!

>!&%?$

!%&$>=> /222 "#"$ ?!

3+()(!)!&%!*+(!-*#2&,!%5-,&$.!-$'!!!&%!*+(!

@.)-A&$.B!-$.6(8!!9+C%!2-:&2-6!,#$%*)C,*&D(!

&$*()1()($,(!3&66!#,,C)!-*!

!?>?=?E>

%&$ %%"" 4/

4&

! !!3&*+!*+(!=%*!#)'()!2-:&2C2!-*!

/>%&$ =

&! " 8!

$%&'()*+,-.#/0#/$<'(')$*#%&+,%&$'-.+#/+!"#$#%&=>$

9+(!1&)%*!(D&'($,(!#1!F(!7)#.6&(G%!+<5#*+(%&%!,-2(!&$!ABCD!3+($!%&'()*+,$

&,)$.#/0#/!%,-**()('!(6(,*)#$%!#11!-!H&!,)<%*-68!!9+(!%5-,&$.!)!1#)!H&!3-%!

I$#3$!1)#2!JK)-<!%,-**()&$.!*#!;(!EFEBC$,08!!F-D&'%#$!-$'!L()2()!C%('!

3&*+!-!I&$(*&,!($().<!#1!DG$#H8!!9+(!F(!7)#.6&(!3-D(6($.*+!#1!-$!(6(,*)#$!

3&*+!-!.&D($!I&$(*&,!($().<!&%!

BC;4BC

8;

;4;4BC

8;

;0

8;

0

8

--- 0/>0/0/ >>>>>> '"

#'"""& $

M#)!$#$K)(6-*&D&%*&,!(6(,*)#$%!NO!PP!2(,>!-$'!

BC;4

8;

BC;4BC

8;

-

;4BC

-

-0/>0/> >>>

>((( )('

"**

& F$

M#)!DG$#H!(6(,*)#$%!&$!$EFAIJ$,0!-$'!!

QER8E0EQ>8E/>

=ST8E

>%&$ = +""

54

54

/

&! !-$'!!=!!!SU

#8!

9+(!@'(*(,*#)!-$.6(B!&%!)(6-*('!*#!*+(!@.)-A&$.!-$.6(B!;<!"UE> == +#" !,- 8!!9+(!D-6C(!#1!DE

+!-.)(('!3&*+!*+(!

(:5()&2($*-6!'-*-V!!K+4#$4"&4$4"#$(,4#/@#/#,;#$3&44#/,$+;;L/*$

#'#,$M(4"$+,?9$+,#$#?#;4/+,$(,$4/&,*(4$&4$&$4(0#N!

! ! ! ! ! ! !

)

O6/&P(,6$&,6?#Q.!.

!"#$$%&'(#)%*+#,,-"./$$

!.

R,;()#,4$/&9$A! :;&44#/#)$/&9$A!

R,;()#,4$/&9$C! :;&44#/#)$/&9$C!

!. !.

)*(,!!)*(,!!

!.!.

O)#4#;4+/$&,6?#Q$

.-.

%#4#;4+/$S,6?#$-$

KL0T#/$+@$:;&44#/#)$

DG$#H$

DE+$

!"#$%&$' ' ()'*)'+,-./'

!

*-01234-53'67'!89:,;:' <8103-2=>?)/6;' @5,A-2:,39'67'+.62,/1'

!

!"#$%&'()*+,-.#/0#/$123#/(0#,4$$

5/&66$7-8&9$:;&44#/(,6$<!"#$#%&=>$

"#$%&'()!*+(!%,-**()&$.!/62'2-7.-;3,650!#1!7-

/&9*!1)#2!*3#!-'4-,($*!56-$(%!#1!-*#2%!/,-66('!

7)-..!56-$(%08!!9#!-!.##'!-55)#:&2-*&#$!*+(!

3&4"$?#,64"$)(@@#/#,;#!;(*3(($!)-<!=!-$'!)-<!

>!&%?$

!%&$>=> /222 "#"$ ?!

3+()(!)!&%!*+(!-*#2&,!%5-,&$.!-$'!!!&%!*+(!

@.)-A&$.B!-$.6(8!!9+C%!2-:&2-6!,#$%*)C,*&D(!

&$*()1()($,(!3&66!#,,C)!-*!

!?>?=?E>

%&$ %%"" 4/

4&

! !!3&*+!*+(!=%*!#)'()!2-:&2C2!-*!

/>%&$ =

&! " 8!

$%&'()*+,-.#/0#/$<'(')$*#%&+,%&$'-.+#/+!"#$#%&=>$

9+(!1&)%*!(D&'($,(!#1!F(!7)#.6&(G%!+<5#*+(%&%!,-2(!&$!ABCD!3+($!%&'()*+,$

&,)$.#/0#/!%,-**()('!(6(,*)#$%!#11!-!H&!,)<%*-68!!9+(!%5-,&$.!)!1#)!H&!3-%!

I$#3$!1)#2!JK)-<!%,-**()&$.!*#!;(!EFEBC$,08!!F-D&'%#$!-$'!L()2()!C%('!

3&*+!-!I&$(*&,!($().<!#1!DG$#H8!!9+(!F(!7)#.6&(!3-D(6($.*+!#1!-$!(6(,*)#$!

3&*+!-!.&D($!I&$(*&,!($().<!&%!

BC;4BC

8;

;4;4BC

8;

;0

8;

0

8

--- 0/>0/0/ >>>>>> '"

#'"""& $

M#)!$#$K)(6-*&D&%*&,!(6(,*)#$%!NO!PP!2(,>!-$'!

BC;4

8;

BC;4BC

8;

-

;4BC

-

-0/>0/> >>>

>((( )('

"**

& F$

M#)!DG$#H!(6(,*)#$%!&$!$EFAIJ$,0!-$'!!

QER8E0EQ>8E/>

=ST8E

>%&$ = +""

54

54

/

&! !-$'!!=!!!SU

#8!

9+(!@'(*(,*#)!-$.6(B!&%!)(6-*('!*#!*+(!@.)-A&$.!-$.6(B!;<!"UE> == +#" !,- 8!!9+(!D-6C(!#1!DE

+!-.)(('!3&*+!*+(!

(:5()&2($*-6!'-*-V!!K+4#$4"&4$4"#$(,4#/@#/#,;#$3&44#/,$+;;L/*$

#'#,$M(4"$+,?9$+,#$#?#;4/+,$(,$4/&,*(4$&4$&$4(0#N!

! ! ! ! ! ! !

)

O6/&P(,6$&,6?#Q.!.

!"#$$%&'(#)%*+#,,-"./$$

!.

R,;()#,4$/&9$A! :;&44#/#)$/&9$A!

R,;()#,4$/&9$C! :;&44#/#)$/&9$C!

!. !.

)*(,!!)*(,!!

!.!.

O)#4#;4+/$&,6?#Q$

.-.

%#4#;4+/$S,6?#$-$

KL0T#/$+@$:;&44#/#)$

DG$#H$

DE+$

e-

d=0.092 nm (Ni crystal)

More Experiments with Electrons!"#$%&'()*'+,-#.),/#$0'1,(2+-

3"#4)55(31,)2+#5(2*#'67'#

89#:-)+7#3#7206#.)('#

;<2'(-1/=#>?@AB

C8"D.2EF0),#4)55(31,)2+#89#

:-)+7#3#1/3(7'6#5)8('

1"#D/'#'0'1,(2+#*)1(2-12&'#)-#:-'6#)+#*:1/#25#

*3,'()30-#3+6#8)2027)130#-1)'+1'#3+6#/3-#3#('-20:,)2+#

25#G">#+*"

Diffraction of Atoms

Double Slit (Carnal & Mlynek, 1991)


microfabricatedtransmissionstructures


#=0.056 nm

#=0.103 nm

Interference pattern

convolution with detector

resolution(2µm)

Diffraction of Atoms from a Light Wave

standing wave

planes of equal intensity

short interaction time:diffraction grating

long interaction time:Bragg scattering

Pritchard, 1988 (atomic beam)Rempe, 1996 (MOT)

344 EUROPHYSICS LETTERS

Fig. 1. – Scheme of the experimental setup used to measure the diffraction of slow atoms from anear-resonant standing-wave light field.

We now discuss this in more detail. Atomic excitation can be neglected when the atom-fielddetuning ∆ is much larger than the peak travelling-wave Rabi frequency R, i.e. ∆ R. Forweak coupling, the stationary states of the atom-field system can be described using the conceptof band theory, effectively treating the electromagnetic field as a “light crystal” [7]-[9]. Thestate of an incoming atom travelling with transverse momentum pz = −nhk (n = 1, 2, ...)is energetically degenerate with the state of a diffracted atom travelling with transversemomentum p

z = −pz = nhk. The atom-field interaction lifts this degeneracy by an amount2hΩn which depends on the diffraction order n. If the interaction is turned on and off slowlyenough for an adiabatic evolution, this two-level model predicts an oscillation between thepopulations of the two momentum states [10]. The probability Πn to find an atom in thescattered state as a function of time t is given by

Πn = sin2(Ωnt). (1)

The Pendellosung frequency Ωn depends on the scattering order n and the two-photon couplingstrength, i.e. the light-shift parameter χ = R2/(4∆). In particular, Ω1 = χ for first-orderBragg scattering, and Ω2 = χ2/ (4ωrec) for second-order diffraction [8].

Figure 1 shows the experimental setup. The measurements were performed with 85Rb atomsfrom a vapour-cell magneto-optical trap [11]. After turning off the magnetic field, atoms loadedinto the trap are further cooled in optical molasses to a temperature of typically 10 mK [12].The cloud of atoms is then released in zero magnetic field by switching off the molasses beams.Next a small bias magnetic field of about 100 mT is applied in the vertical x-direction to define

Diffraction of Atoms from a Light Wave

30 mW must be used to achieve a value of c = 2p ´ 50 kHz for the ac-Stark shift

parameter.

The solid lines in ®gures 6 (a), 6 (b) and 6 (c) are ®ts representing the theoretical

prediction, equation (13), with c being the only ®t parameter. The initial

distribution qin( pz ) is a Gaussian whose parameters are measured independently

(see ®gure 4). The best-®t value c = 2p ´ 48 kHz agrees very well with the

expected value of c = 2p ´ 50 kHz.

We now brie¯y comment on a subtle distinction between diraction experi-

ments employing a tightly focused cw light ®eld on the one hand and a large

diameter pulsed light ®eld on the other hand. Although both experimental

approaches are well described by the one-dimensional theoretical model men-

tioned above, which predicts a change of the transverse kinetic energy of the

Standing wave diraction with a beam of slow atoms 1877

Figure 6. Momentum distribution of atoms after diraction from a standing light wave in

the regime of short and intermediate interaction times. In part (a), (b), and (c) the

solid lines represents ®ts based on the Raman±Nath theory. In part (d ), (e) and ( f ) the

Raman±Nath approximation in clearly not valid.

short interaction timeoptical grating for atoms

! final (p)" !in (p # 2nprec )#$

$

%with prec = !k

photon exchange with both wavesdistribution depends on intensity, detuning and interaction time

Diffraction and Interference with Molecules

C60 (Bucky balls) v = 210 m/s; m = 720 amu = 1.2#10-24 kg!db = 2.5 pm = 0.0025 nmM. Arndt, A. Zeilinger, 1999


C60 (Bucky balls) velocity:

SiN diffraction grating:d = 100 nma = 50 nm


C60 (Bucky balls) velocity:

SiN diffraction grating:d = 100 nma = 50 nm


Extension to a Mach-Zehnder like Interferometer:


also with optical diffraction gratings (as with atoms):

Summary - Diffraction & Interference of Matter

Interference of matter waves

electronsneutronsatomsmolecules !C60, DNA molecules

! =

hp

or p = !k

• Bragg scattering• double slit• diffraction grating• light (standing wave)


Wave !Wavefunctionwell-defined & and momentumcompletely delocalised

Particle !Trajectory well-defined position& (and hence momentum!?) undefined

! (x,t)" exp i kx #$t( )%& '(

p = !kP(x) =! *! = const.

ddt x = v and d

dt v = F / m

x(t) = v(t)dt!


Description of a particle ! Wave function "(x,t)

... describes the state of a particle of mass m at time t.

Probability

... to find it in [x .. x+dx] at t is P(x)dx = |)(x,t)|2 dx

Normalisation

The particle is somewhere ⇒ , P(x)dx = , |)(x,t)|2 dx = 1

Uncertainty Principle

Solutions in free space ! plane waves

delocalised: P(x)dx = |)(x,t)|2 dx = const.not normalised , P(x)dx = , |)(x,t)|2 dx = -

! something seems to be missing ...

Consider single-slit diffraction in transverse direction slit position (x=0) is well knownslit width *x defines uncertainty in positiontransverse momenta ±*p lead to ±1st minima:

!x!p = h


Superposition principle

any linear superposition of wave functions is also a possible wave function:

$(x,t)=%1$1(x,t)+%2$2(x,t)


Heisenberg‘s uncertainty Principle:

a point-like particle cannot be localised both in position and momentum beyond the limit

!x!p " ! / 2

• single-slit diffraction• dispersion of wave packets

Summary

Superposition of plane wavesmomentum distribution defines k rangeinterference leads to localised wave packetsHeisenberg‘s uncertainty principle

!x!p " ! / 2

Quantised Energy Levels

Atomic spectra (from discharge lamps)

hydrogen atoms helium atoms


Frank-Hertz Experiment (1914, NP 1925)

electrons accelerated in mercury vapour


Frank-Hertz Experiment (1914, NP 1925)

electron current drops every 4.9V

'

inelastic collisionsoccur whenever

Eel = 4.9 eV

Bohr‘s Model

classical electron orbitcircumference = integer multiple of &dB

quantisation of angular momenta

L = mvr = n!

2!r = n"dB

Bohr‘s Model

The Hydrogen Spectrum

Quantisation of Matter - Summary

Interference & Diffraction - De Broglie• Matter-wave diffraction (Bragg)• Wave-particle duality

Uncertainty principle

Quantised energy levels in hydrogen• Bohr‘s model• Frank-Hertz experiment 2!r = n"dB

p = k = h/λ

!x!p " ! / 2

The Mathematical Framework

Schrödinger‘s wave equationtime-independent formulation (TISE)potential barriers:

classically allowedclassically disallowedtunnel effectparticle in a box ! eigenstates and energies

superposition ! time dependence

Schrödinger‘s Wave Equation

i!!

!t" (x,t) = # !

2

2m!2

!x2" (x,t) +V (x)" (x,t)

E = !!! (x,t) = "(x)e# i$ t

!!2

2m"2

"x2#(x) +V (x)#(x) = E#(x)

Time-Independent Schrödinger Equation

TISE:

V(x) time independent! separation of variables

Boundary Conditions & Linearity

Continuity:• !(x) continuous for all x• (!(x) continuous (while V(x) finite)

Non-diverging Eigenstates:• "|!(x)|2 dx=1 (bound states normalisable)

Linearity in !: !(x)= !1(x)+!2(x)

!!2

2m"2

"x2#(x) +V (x)#(x) = E#(x)TISE:

State of a Free Particle

!(x) = Aeikx

k = ± 1

!2m(E !V0 )

E >V0

!(x) = Aeikx + Be" ikx

E <V0

! =

1!2m(V0 " E)

!(x) = Ae" x + Be#" x

No force: V(x) = V0 = const.Approach:

!!2

2m"2

"x2#(x) +V (x)#(x) = E#(x)TISE:

Potential Barrier

!1(x) = Aeik1x + Be" ik1x !2 (x) = Ce

ik2 x + De" ik2 x

k1 =

1!2m(E !V1)

k2 =

1!2m(E !V2 )

R = BA

!"#

$%&

2

=k1 ' k2k1 + k2

!

"#$

%&

2

=E 'V1 ' E 'V2E 'V1 + E 'V2

!

"#

$

%&

2

x

V(x)

V1

V2

E

Potential Barrier (E>V1, E>V2)

Potential Barrier - Total Reflection

!1(x) = Aeik1x + Be" ik1x

k1 =

1!2m(E !V1)

x

V(x)

V1

V2E

classically forbidden

!2 (x) = Ce"# 2 x + De+# 2 x

! 2 =

1!2m(V2 " E)

R = B*BA*A

=k1 ! i" 2k1 + i" 2

#

$%&

'()

k1 + i" 2k1 ! i" 2

#

$%&

'(= 1

Potential Barrier (E>V1, E<V2)

Tunnel Effect

!k = 2mE !! ' = 2m(V0 " E)

T =16k2! '2

(k2 +! '2 )2e"2! 'aTransmission

coefficient

x

V(x)

V0

E

classically forbidden

0 a

transmission?

Particle in the Box

0-a a x

V=0

Particle in the Box

0-a a x

V=0

!(x) = Aeikx + Be" ikx

!("a) = 0!(+a) = 0

#$%

kn = n&2a

and En =!2& 2

8ma2n2

!n (x) =1asin kn (x + a)( ) = 1

acos(knx) n oddsin(knx) n even"#$

Approach:

Continuity:

Particle in the Box

Superposition of States

! 1(x,t) =1acos "

2ax#

$%&'(exp )i*1t( ) with *1 =

!" 2

8ma2

! 2 (x,t) =1asin "

ax#

$%&'(exp )i*2t( ) with *2 =

!" 2

2ma2

! (x,t) = 12! 1(x,t) +! 2 (x,t)[ ]

!(x,t) =" *(x,t)" (x,t) = 12a

cos2 (k1x) + sin2 (k2x) + 2cos(k1x)sin(k2x)cos(#$t)( )

different Energies ! time dependent

linear superposition:

probability density oscillates with #$ = $2 % $1

Modern Quantum Technology

Physical reality & binary quantum logicRole of the observer, break-down of the wavefunctionSchrödinger‘s cat paradoxElectron spin (Stern-Gerlach) & Photon polarisation

Joint properties of quantum systemsEPR paradox (Einstein‘s Gedankenexperiment)Entanglement (Alain Aspect)

QIPC: Quantum information processing & communication

Quantum cryptography

Elements of Physical Reality

Classical Observable: x, p, E, l

Wave function .(x,t) ! probability density $(x,t)=.*.

Expectation values (average values)

x = x!(x,t)" dx = # *(x,t) x# (x,t)dx with x = x"p = p!(p,t)" dp = # *(x,t) p# (x,t)dx with p = $i!

%

%x"

E = E!(E,t)" dE = # *(x,t) H# (x,t)dx with H = $!2

2m%2

%x2+V (x)"

Binary Quantum Logic

Orthonormal eigenstatesWavefunctionProbability for either eigenstate

Observation ! break-down (reduction)

φa · φb = δab

Ψ = αφa + βφb

Pa,b =

φ∗a,b · Ψ

2

= |α|2 or |β|2

Ψ −→ Ψcond = φa,b

Schrödinger‘s Cat

Observation: Cat is dead or aliveNo observation: Superposition stateInterference of probability amplitudes

Photons ! Field Modes

Beamsplitter ! One observable, two possible values

!1,in " ei!k!x

vacuum

!A

!B

!1,out = " #!A + $ #!B

PA = ! 2= "A

*"1,out#2

PB = ! 2= "B

*"1,out#2

!"A

!"B

Photons ! Field Modes

Beamsplitter ! One observable, two possible values

!1,in " ei!k!x

vacuum

!A

!B

!2,in = " #!A + $ # ei% #!B

!2,out = "" '+ ## '$ ei%( ) $!C + "# '& #" '$ ei%( ) $!D

Mach-Zehnder Interferometer

-1 0 1 2 3 4

0,5

1

|out = (αα + ββeiφ)|a+ (αβ − βαeiφ)|b

=1

2(1 + eiφ)|a+ 1

2(1− eiφ)|b

Pa = | |2 = cos2(φ/2)

Pb = | |2 = sin2(φ/2)

|in |a

|b

α = β =

α = β

= 1/√2

|inside = α|a+ βeiφ|b

P

φ/π

Breakdown of the Wavefunction|in |a

|b

α = β =

α = β

= 1/√2


measurement inside (eg photon in a)|inside = |a |out = α|a+ β|b

|out = (|a+ |b)/√2

α = β

Pa = Pb = 0.5

-1 0 1 2 3 4

0,5

1

P

φ/π

Orthonormal Wavefunctions|in |a

|b

α = β =

α = β

= 1/√2

φ = 0

φ = π

|+ = (|a+ |b)/√2

|− = (|a − |b)/√2

a|a = b|b = 1a|b = b|a = 0

+|+ = −|− = 1+|− = −|+ = 0


new set of

orthonormal

states

Stern-Gerlach Experiment

Atom beam & B field gradient ! deflection

Electron Spin ! Magnetic Dipole

Orbiting electron ! magnetic dipole

µ = I ! A = " e

2me

! Ln = "e!2me

n

B field ! torque on the dipole

!! =!µ "!B

V = !!µ "!B

L = mvr = n!

Force on Magnetic Dipoles

Dipole in magnetic fields ! torque

B field gradient ! net force

! l =e2me

B

!! =!µ "!B

Larmor precession

V = !!µ "!B

N

S

µ

F

F = −∇V = ∇(µ · B) =

µi∇Bi −→µz

d

dzBz

z


Deflection of dipoles by a B-field gradient

t = L / v

!vz =tmFz =

lvm

µz"B"z

# =!vzv

=l

v2 mµz"B"z

• interaction time• velocity change• deflection angle


Silver ! one valence electronexpectation: random orientation of µ and B

µ = !e!2me


Silver ! one valence electronexpectation: random orientation of µ and B

µ = !e!2me

two angles found ! only two µz values


Electron Spin quantised !

two angles found ! only two µz values

s = 32!

sz = ±! 2

µz = ±e!2me


Silver ! single electron spin ! two angles

Electron Spin

modulus

projection

different axes ) incompatible ) mutually exclusive

s =32!

sx = ±! 2 or sy = ±! 2 or sz = ±! 2 or s! = ±! 2

equal probabilities:25% each

(50% beam blocker)

any

direction!

Electron Spin

modulus

projection


s =32!

sx = ±! 2 or sy = ±! 2 or sz = ±! 2 or s! = ±! 2

orientationdependent

any

direction!

Electron Spin

modulus

projection


s =32!

sx = ±! 2 or sy = ±! 2 or sz = ±! 2 or s! = ±! 2

all going through!sz measured!sz well defined

any

direction!

Electron Spin

modulus

projection


s =32!

sx = ±! 2 or sy = ±! 2 or sz = ±! 2 or s! = ±! 2

incompatibility of sz and sx:! sz defined ! sy determined ! sz lost (equal probabilities)

50%

25% 12.5%

12.5%

any

direction!

Electron Spin

modulus

projection


s =32!

sx = ±! 2 or sy = ±! 2 or sz = ±! 2 or s! = ±! 2

any

direction!

p!+ = "!

+" z+#2= cos2 (! / 2)

p!$ = "!

$" z+#2= sin2 (! / 2)

particle found with sz=+%/2 ! projection into! z+

z

z probabilities interdependent

Electron Spin ! Photon Polarisation

z

z

V 0

H 90

+90

Spin 1/2 particle• two spin values

Polarised Photons• two polarisations (H,V)

p!+ = "!

+" z+#2= cos2 (! / 2) p! = z " e!

2= cos2 (!)

conditional detection probabilities

Quantum Bits

ϕa or ϕb | ↑, | ↓ |H, |V

ϕa ± ϕb√2

| ↑± | ↓√2

|H±| V √2

ϕ+ or ϕ− | →, | ← | + 45o, | − 45o

location spin polarisation

0+ or 1+

mutual exclusive

0x or 1x

bra-ket notation

ϕa −→ |aϕ∗

a −→ a|b|a ≡

φ∗

bφa

SA AB B

? ?

Joint Systems ! Entanglement

Einstein, Podolsky and Rosen Paradox

Source emitting“entangled” particles:

joint properties A or B (e.g. spin up/down or left/right)

particle 1 in A ! particle 2 in A‘ particle 1 in B ! particle 2 in B‘

knowledge of A and B mutually exclusive

Nonlocality ! “spooky interaction at a distance”

EPR with Entangled Photons

Particles with entangled properties to A and B

Polarisation-entangled photon pair


Alain Aspect‘s experiment (1982)

spontaneous photon emission

cascade


Alain Aspect‘s experiment (1982)

• correlation of photon polarisation• depends only on relative orientation

'• polarisation of individual photons undefined• relative polarisation well known


Kwiat, Weinfurter, Zeilinger (1995)

photons fromparametric down

conversion (PDC)


Parametric Down Conversion (PDC)

photon 1 #!1| #!1|#$$ photon 20 #V1| -|H2$ 90

90 #H1| |V2$ 18045 #H1|+#V1| |V2$-|H2$ 135

/#V1|cos(/)+#H1|sin(/)

sin(/)|V2$-cos(/)|H2$

/+90


Correlated Photon Polarisation

|φ− = (|H1V2 − |V1H2)/√2


Correlated Photon Polarisation

• %=& ! strong H-V correlation•*%-&+= 45 ! no correlation (random)

|φ− = (|H1V2 − |V1H2)/√2

EPR with Entangled PhotonsCorrelated Photon Polarisation

|φ− = (|H1V2 − |V1H2)/√2

Entanglement - Summary

Classical:correlation in + basis, not in x basisstate separable into

Entangled:correlation in any basis (+, x, ,)state not separable into

no correlation if 1 and 2 analysed in different bases (45-deg rotated)

|HH or |V V

|φ = (|HH+ |V V )/√2

|φ1 ⊗ |φ2

|φ1 ⊗ |φ2

Entanglement - Summary

Bell-Basis:

maximally entangled states

Bell states not separable into independent quantum states:

|ψ+ = (|H1H2+ |V1V2)/√2

|ψ− = (|H1H2 − |V1V2)/√2

|φ+ = (|H1V2+ |V1H2)/√2

|φ− = (|H1V2 − |V1H2)/√2

|φ1 ⊗ |φ2 = (A|H1+B|V1)⊗ (C|H2+D|V2)= AC|HH+AD|HV +BC|V H+BD|V V = |ψ± [ or |φ±]

Quantum Cryptography

first cryptographic machine: a Skytale

Sparta, 500 B.C.

• transposition (highly insecure)

-• Enigma (20th century)


Key encryption

one-time pad-

100% safe

unless the key is revealed

1001 original message1010 key0011 XOR encoded message

0011 message at receiver1010 same key1001 XOR decoded message

Alice

Bob0011 over a public channel (Internet)


Secure key distribution . quantum effectsthe BB84 protocol (Bennett and Brassard 1984)


Secure key distribution . quantum effectsthe BB84 protocol (Bennett and Brassard 1984)


BB84 & single photons ! Grangier 2002


BB84 & weak pulses ! Weinfurter (now)





quantum key distribution over 144 km

La PalmaTenerife

The End

classical description incomplete

quantum ideas: E = ωp = k

∆p∆x ≥ /2λ = h/p

success! Bohr‘s hydrogen atom Matter waves Black-body radiation Photoelectric effect

formalism− 22m

∆+ V (x)

Ψ(x, t) = i ∂

∂tΨ(x, t)

H|ψ =E|ψ

a|b = δab|ψ = α|a+ β|b

quantum engineering Schrödinger‘s cat, EPR, entanglement,Quantum Cryptography & Computing

!!

!!

Documents

Quantum Ideas lecture notes