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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.
The Dawn of Quantum Physics
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
The Dawn of Quantum Physics
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
The Dawn of Quantum Physics
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)
The Dawn of Quantum Physics
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
[ ... ]
Photoelectric Effect
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
Photoelectric Effect
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
Photoelectric Effect
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
Photoelectric Effect
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
Photon(s) in the Box
Mode density of a 1D cavity, g(!) d!
n! = 2L
resonancecondition
modedensity (1D)
g(!)d! = 2 " 2Lcd!
two polarisations
Photon(s) in the Box
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
Photons by the Number
# of correlations
Single (few) atom fluorescence• atomic beam of low density• antibunching
H.J. Kimble et al. Phys. Rev. Lett. 39, 691 (1977)
Photons by the Number
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)
Cavity-based Single-Photon SourcesA. Kuhn and D. Ljunggren, Contemporary Physics, in print (2010)
• envelope ! interaction time• equidistant comb ! triggered photon emission• antibunching: g(2)(0)< g(2)(%&) ! single photons on demand
Cavity-based Single-Photon SourcesA. Kuhn and D. Ljunggren, Contemporary Physics, in print (2010)
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
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Bragg scattering with electrons (1925, NP 1937)
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More Experiments with Electrons!"#$%&'()*'+,-#.),/#$0'1,(2+-
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Diffraction of Atoms
Double Slit (Carnal & Mlynek, 1991)
Diffraction of Atoms
microfabricatedtransmissionstructures
Diffraction of Atoms
#=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
Diffraction and Interference with Molecules
C60 (Bucky balls) velocity:
SiN diffraction grating:d = 100 nma = 50 nm
Diffraction and Interference with Molecules
C60 (Bucky balls) velocity:
SiN diffraction grating:d = 100 nma = 50 nm
Diffraction and Interference with Molecules
Extension to a Mach-Zehnder like Interferometer:
Diffraction and Interference with Molecules
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-Particle Dualism
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!
Wave-Particle Dualism
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
Uncertainty Principle
Superposition principle
any linear superposition of wave functions is also a possible wave function:
$(x,t)=%1$1(x,t)+%2$2(x,t)
Uncertainty Principle
Heisenberg‘s uncertainty Principle:
a point-like particle cannot be localised both in position and momentum beyond the limit
!x!p " ! / 2
• single-slit diffraction• dispersion of wave packets
Summary
Superposition of plane wavesmomentum distribution defines k rangeinterference leads to localised wave packetsHeisenberg‘s uncertainty principle
!x!p " ! / 2
Quantised Energy Levels
Atomic spectra (from discharge lamps)
hydrogen atoms helium atoms
Quantised Energy Levels
Frank-Hertz Experiment (1914, NP 1925)
electrons accelerated in mercury vapour
Quantised Energy Levels
Frank-Hertz Experiment (1914, NP 1925)
electron current drops every 4.9V
'
inelastic collisionsoccur whenever
Eel = 4.9 eV
Bohr‘s Model
classical electron orbitcircumference = integer multiple of &dB
quantisation of angular momenta
L = mvr = n!
2!r = n"dB
Bohr‘s Model
The Hydrogen Spectrum
Quantisation of Matter - Summary
Interference & Diffraction - De Broglie• Matter-wave diffraction (Bragg)• Wave-particle duality
Uncertainty principle
Quantised energy levels in hydrogen• Bohr‘s model• Frank-Hertz experiment 2!r = n"dB
p = k = h/λ
!x!p " ! / 2
The Mathematical Framework
Schrödinger‘s wave equationtime-independent formulation (TISE)potential barriers:
classically allowedclassically disallowedtunnel effectparticle in a box ! eigenstates and energies
superposition ! time dependence
Schrödinger‘s Wave Equation
i!!
!t" (x,t) = # !
2
2m!2
!x2" (x,t) +V (x)" (x,t)
E = !!! (x,t) = "(x)e# i$ 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
|inside = α|a+ βeiφ|b
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
|inside = α|a+ βeiφ|b
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
Stern-Gerlach Experiment
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
Stern-Gerlach Experiment
Silver ! one valence electronexpectation: random orientation of µ and B
µ = !e!2me
Stern-Gerlach Experiment
Silver ! one valence electronexpectation: random orientation of µ and B
µ = !e!2me
two angles found ! only two µz values
Stern-Gerlach Experiment
Electron Spin quantised !
two angles found ! only two µz values
s = 32!
sz = ±! 2
µz = ±e!2me
Stern-Gerlach Experiment
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
different axes ) incompatible ) mutually exclusive
s =32!
sx = ±! 2 or sy = ±! 2 or sz = ±! 2 or s! = ±! 2
orientationdependent
any
direction!
Electron Spin
modulus
projection
different axes ) incompatible ) mutually exclusive
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
different axes ) incompatible ) mutually exclusive
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
different axes ) incompatible ) mutually exclusive
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
EPR with Entangled Photons
Alain Aspect‘s experiment (1982)
spontaneous photon emission
cascade
EPR with Entangled Photons
Alain Aspect‘s experiment (1982)
• correlation of photon polarisation• depends only on relative orientation
'• polarisation of individual photons undefined• relative polarisation well known
EPR with Entangled Photons
Kwiat, Weinfurter, Zeilinger (1995)
photons fromparametric down
conversion (PDC)
EPR with Entangled Photons
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
EPR with Entangled Photons
Correlated Photon Polarisation
|φ− = (|H1V2 − |V1H2)/√2
EPR with Entangled Photons
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)
Quantum Cryptography
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)
Quantum Cryptography
Secure key distribution . quantum effectsthe BB84 protocol (Bennett and Brassard 1984)
Quantum Cryptography
Secure key distribution . quantum effectsthe BB84 protocol (Bennett and Brassard 1984)
Quantum Cryptography
BB84 & single photons ! Grangier 2002
Quantum Cryptography
BB84 & weak pulses ! Weinfurter (now)
Quantum Cryptography
BB84 & weak pulses ! Weinfurter (now)
Quantum Cryptography
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
!!
!!
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