light-matter interactions · light-matter interactions weak (cf. energy differences in uncoupled...

Light-Matter Interactions

Paul Eastham

February 15, 2012

The model

= Single atom in an electromagnetic cavity

MirrorsSingleatom

Realised experimentallyTheory:“Jaynes Cummings Model”⇒ Rabi oscillations

– energy levels sensitive to single atom and photon

– get inside the mechanics of “emission” and “absorption”

Where we’re going

= One field mode, two atomic states

Energy of photon in field mode

H = (∆/2) (|e〉〈e| − |g〉〈g|) + ~ω a†a + ~Ω2 (a|e〉〈g|+ a†|g〉〈e|).

Dipole coupling energy

Energy difference between atomic levels

How to get there

Show that ∼ H = Hatom + Hfield − E .(er)

Write electron position operator r in basis– eigenstates of Hatom == atomic orbitalsApproximate to one mode of field and two atomic levelsNeglect non-resonant “wrong-way” terms(like electron drops down orbital and emits photon)

How to get there

Show that ∼ H = Hatom + Hfield − E .(er)←−Write electron position operator r in basis– eigenstates of Hatom == atomic orbitalsApproximate to one mode of field and two atomic levelsNeglect non-resonant “wrong-way” terms(like electron drops down orbital and emits photon)

Atom-Field Interaction Energy

Hamiltonian for Atom+Field

Field∑

modes

~ωi a†i ai

H = HEM + Hatom + Hint

Atom[− ~2

2me∇2

e −e2

4πε0|re − R0|

](nucleus fixed @ R0)

Interaction energy ?


Interaction energy: classical field

Hatom =1

2mp2 + V (r).

With vector potential A, p→ (p−eA) (minimal coupling)and scalar potential φ, Hatom → Hatom + eφ

Hatom−field =1

2m(p− eA)2 + eφ+ V (r).


Interaction energy: A.p form

Choose the Coulomb gauge where ∇.A = 0, φ = 0

Hatom−field =p2

2m− e

m A.p + e2

2m A2 + V (r)

= Hatom + Hint

(Can use this form directly – not in this course)


Interaction energy: dipole approximation

Interested in interaction with light waves

A = A0ei(k.r−ωt) + c.c.

For an atom the wavefunction extends over about 1Å

For light |k| = 2π/λ ≈ (500nm)−1

∴ A approximately constant in space over the atom,

– A(r, t)→ A(r = 0, t)


Interaction energy: E.r form

Coulomb gauge form:

Hatom−field =1

2m(p− eA)2 + eφ(= 0) + V (r).

Now change gauge :

A→ A +∇χ(r, t)

φ→ φ− ∂χ(r, t)∂t

,

so

Hatom−field =1

2m(p− e(A +∇χ))2 − e

∂χ

∂t+ V (r).


Interaction energy: E.r form

Hatom−field =1

2m(p− e(A +∇χ))2 − e

∂χ

∂t+ V (r).

Choose χ(r, t) = −A.r

so that ∇χ = −A

and∂χ

∂t= −∂A

∂t.r = E.r

[E = −∇φ− ∂A

∂t= −∂A

∂t

]– (A, φ-Coulomb gauge)

Hatom−field =1

2m(p2 + V (r))− er.E(t).


Hamiltonian: E.r form

Quantum form: E(r)→ E(r0) at the position of the atomSo for our cavity quantization + one electron atom:

H = Hatom +∑

n

~ωna†nan

+∑n,s

√~ωn

ε0Vsin(knzat)(an + a†n)es.(−er).


How to get there


Write electron position operator r in basis– eigenstates of Hatom == atomic orbitals←−Approximate to one mode of field and two atomic levelsNeglect non-resonant “wrong-way” terms(like electron drops down orbital and emits photon)


Second quantization: general

Generally have Z indistinguishable electrons

⇒ Atomic eigenstates labelled by occupation of orbitals(1s22s1 etc)

These states – |i〉 – form a complete set (for Z electrons)

This allows us to formally write atomic operators– in terms of “transition operators” |i〉〈j |


Second quantization: Hamiltonian

1 =∑

i

|i〉〈i |.

Formal representation of H :

H = 1H1

=∑

i

|i〉〈i |H∑

j

|j〉〈j |

=∑i,j

|i〉〈i |Ej |j〉〈j | (H|j〉 = Ej |j〉, 〈i |j〉 = δij)

=∑

i

Ei |i〉〈i |.


Second quantization: One-body operators

Eigenstates of H – |i〉 – form a complete set for Z electrons, so

1 =∑

i

|i〉〈i |.

Formal representation of D =∑

i

eri :

D = 1D1

=∑

i

|i〉〈i |D∑

j

|j〉〈j |

=∑i,j

〈i |D|j〉|i〉〈j |


Second quantization: One-body operators

D = 1D1

=∑

i

|i〉〈i |D∑

j

|j〉〈j |

=∑i,j

〈i |D|j〉|i〉〈j |

If we know the orbitals in real-space, can calculate

Dij = 〈i |D|j〉 =

∫dxdydzψ∗i (r)(er)ψj(r).


Dipole matrix elements

Dij = 〈i |D|j〉 =

∫dxdydzψ∗i (r)(er)ψj(r).

Dij only non-zero between some states⇒ selection rulesi.j different parity.∆l = ±1 (if l good quantum number)

Magnitude |D| ≈ e× 1 Å


Atom-field Hamiltonian

So for our cavity problem

H =∑

n

~ωna†nan

+∑

i

Ei |i〉〈i |

+∑n,s

∑ij

En sin(knzat)(an + a†n)es.Dij |i〉〈j |.


How to get there


Write electron position operator r in basis– eigenstates of Hatom == atomic orbitalsApproximate to one mode of field and two atomic levelsNeglect non-resonant “wrong-way” terms(like electron drops down orbital and emits photon)


Why two levels?

Light-matter interactions weak(cf. energy differences in uncoupled problem)

⇒ small effects which can be treated in perturbation theory

except: if ~ω ≈ ∆, degeneracy between |n,g〉, |n − 1,e〉

∴ can focus on the physics of one mode+ nearly resonant atomic transition


Rotating-Wave approximation

Interaction is

~Ω

2[ a|e〉〈g|+ a†|g〉〈e| + a|g〉〈e|+ a†|g〉〈e| ].

Energy changes ≈ 0 ≈ ±2∆

∴ drop these terms

Jaynes-Cummings Model in Rotating-Wave Approx

H = (∆/2)(|e〉〈e| − |g〉〈g|) + ~ωa†a

+~Ω

2(a|e〉〈g|+ a†|g〉〈e|).


Summary

= One field mode, two atomic states

Energy of photon in field mode

H = (∆/2) (|e〉〈e| − |g〉〈g|) + ~ω a†a + ~Ω2 (a†|e〉〈g|+ a|g〉〈e|).

Dipole coupling energy

Energy difference between atomic levels

light-matter interactions · light-matter interactions weak (cf. energy differences in uncoupled...

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