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