Kapitza-Dirac Effect: Electron Diffraction from a Standing Light Wave
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Transcript of Kapitza-Dirac Effect: Electron Diffraction from a Standing Light Wave
Kapitza-Dirac Effect:Electron Diffraction from a Standing
Light Wave
Physics 138 SP’05 (Prof. D. Budker)
Victor Acosta
Contents
• History• Introduction• Basic Setup/Results• Theory
– Multi-Slit Analogy– Particle Interaction Picture– QM Treatment
• U. Nebraska 2001 Results• Applications
History• 1804
– Young Proposes Double Slit Experiment• Wave nature of Light
• 1905– Einstein Photoelectric Effect
• Particle nature of Light• 1927
– Davisson and Germer Electron Diffraction (crystalline metal)• Wave nature of matter
• 1930– Kapitza and Dirac propose KDE
• Light Intensity of mercury lamp only allows 10-14 electrons to diffract
• 1960– Invention of Laser
• First Real Attempts at KDE– All 4 were unsuccessful (poor beam quality? Undeveloped
Theory?)• 2001
– KDE seen by U. Nebraska group
Introduction to Kapitza-Dirac Effect (KDE)
Figure 1. Adapted from Kapitza and Dirac's original paper. Electrons diffract from a standing wave of light (laser bouncing off mirror). Figure
from Bataleen group (U. Nebraska).
Analogy)
KDE : Multi-Slit Diffraction
Electron Beam : incident wave
Light Source: grating
Basic Setup/Results
Data for atom diffraction from a grating of ’light’ taken at the University of Innsbruck. Diffraction peak separation = 2 photon recoil momenta. Figure
from Bataleen group (U. Nebraska).
Analogy: Multiple-Slit Diffraction
θ
Assume outgoing waves propagate at θ w.r.t grating axis (z>>d).
d
z
Path Length Difference (PLD) = dSin[θ] Bragg Condition satisfied iff PLD = nλ → dSin[θ] = nλ
Detector
d
Kapitza-Dirac Effect: Particle Interaction Picture
Figure from Bataleen group (U. Nebraska).
Kapitza-Dirac Effect: Particle Interaction PictureStimulated Compton Scattering (bi-level process):
Step 1: Particle absorbs photon from one of the beams
Step 2: counterpropagating light beam stimulates the emission of anoher photon (in the counterpropagating direction)
Condition 1: Energy and Momentum must be conserved (see above figure)
Condition 2: n DB dlight Sin,
Quantum Mechanical Theory
• Need full QM treatment to understand nature of diffraction peaks
• First find H using Classical E+M• Then solve Time-Dependent Schroedinger
Equation
Ponderomotive PotentialPotential describing average electron motion in Laser-field.
Figure from Bataleen group (U. Nebraska).
Ponderomotive PotentialStart with Lorentz Equation:
1 m vt eE v B
The laser is reflected two counterpropagating light waves:
2 E EoCos t k z Cos t k zz 2EoCosk zCos tz
Looking only at oscillatory motion in z direction gives:
3 m vosc
t 2e EoCosk zCos tzThe solution to 3 is just:
4 vosc 2e Eo
m Cosk zSin tzThe Ponderomotive Potential is defined by:
5 Vp 12 mvosc2
Inserting 4 in 5 gives:
6 Vp 2e2Eo2Cosk z2m2 Sin t2 2e2Eo2Cosk z2
m2 2 t t 2 Sin t '2 t ' e2Eo2Cosk z2
m2
We can relate the laser Intensity to the E-field by I oc Eo2 . The Vp becomes:
8 Vp I e2Cosk z2ocm2
Time-Dependent Schroedinger Equation (TDSE) for Particle in a Laser PotentialIn order to really understand Kapitza-Dirac Effect, a full QM treatment is necessary:
Hamiltonian for an electron in a laser beam:
1 H 2
2m 2Vz 2
2m 2
z2 2VoCosk z2Vo I e2
2ocm2 for an electron (see Ponderomotive Potential Slide)Vo 1
4oc I for an atom ( atomic polarizability)
TDSE:
2 H tWe use a general Power Law trial function:
3 z, tncntkn z ETn
tETn Vo 2kn2
2m ncn2 1 kn 2n nok pincident 2no k
Time-Dependent Schroedinger Equation (TDSE) for Particle in a Laser Potential
Making these substitutions elaborates 3 to:
4 z, tncnt2nnok z2 knno22m Vo
tn cntnz, t
TDSE becomes, using Cos2k z 14 2 2k z 2k z:
5 n2 knno22m Vo Vo
2 2k z 2k zcnn ncnt 2 knno22m Vocnn
Simplifying and equating terms with common phase gives:
6 cnt Vo2 Eot2Eonnotcn1 2Eonnotcn1
Eo 2 k no22m
Solutions to 6 are analytic in two regimes:
Time-Dependent Schroedinger Equation (TDSE) for Particle in a Laser PotentialEon2 Vo (Raman-Nath regime or thin slit diffraction)
Here all exponential factors are approximately unity so we are left with:
6 cnt Vo2 cn1 cn1
All modes possible: pparticle 2n k
7 cn nVot JnVot
Mode Population Probability of finding the particle in the nthdiffraction order given by:
8 cn2 JnVot2Eo2 Vo (Bragg Regime or thick slit diffraction)
Interference condition requires that n 2no. Let no 1
2
9 c02 CosVot2210 c12 SinVot22
Legend:
Bragg Regime (Top)
Raman-Nath (Bottom):
n=0 (red)
n=1 (blue)
n=2 (green)
0.5 1 1.5 2Vot0.2
0.4
0.6
0.8
1
Mode Population
5 10 15 20Vot
0.1
0.2
0.3
0.4Mode Population
Note: Even mode peaks occur at roughly the same value for Vot. The rough explanation is that since there is an equal probability that each collision will transfer +k or -k momentum to the incident particle. For example, the 4th n=0 peak corresponds 2 collision with a +k photon and 2 collisions with -k photon. Odd mode peaks also occur at the same Bessel function argument for similar reasons.
Bragg and Raman-Nath Regimes: Position-Momentum Uncertainty Picture
Figure from Bataleen group (U. Nebraska).
Bragg and Raman-Nath Regimes: Position-Momentum Uncertainty Picture
Uncertainty Relation for Photon:
1 x px 2
Photon's momentum in Uncertainty Picture:
2 px p Sin p 2
This makes the Uncertainty Relation:
3 x 4
Interpretation:Laser width w x 1
Thus as w, photons available in larger range of angles wider range of electron that satisfy Bragg condition Many possible Diffraction Orders Raman-Nath Regime
w , only possible incident angle is Bragg only 1 mode Bragg Regime
Bragg and Raman-Nath Regimes: Energy-Time Uncertainty Picture
w t 1E
Large w small E only one possible transition is allowed in order for E-Conservation to hold.
Small w more room for system energy to change (without violating E-conservation) many more modes are possible Raman-Nath Regime
Figure from Bataleen group (U. Nebraska).
U. Nebraska 2001 Results: Raman-Nath Regime
Laser off (Top) and Laser on (bottom)
Plaser= 10 W Ilaser= 271 GW/cm2
Vp= 7.18 meV. Eo = 5.31 µeV Ve=.0367c
U. Nebraska 2001 Results: Bragg Regime
Laser off (Top) and Laser on (bottom)
Plaser= 1.4 W Ilaser= 0.29 GW/cm2
Vp= 7.66 µeV. Eo = 5.31 µeV Ve=.0367c
Applications
• Coherent Electron Beam Splitter• Electron Interferometry
– Greater Sensitivity than Atomic Version• λelectron ~ 10-11 > .1λatom
– Low electron energies possible• Microscopic Stern-Gerlach Magnet?
– Would separate Electron’s by spin• Need light grating that isn’t standing wave