Light-Matter Interactions
Paul Eastham
February 15, 2012
The model
= Single atom in an electromagnetic cavity
Mirrors
Single
atom
Realised experimentally
Theory:
“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
Ĥ = (∆/2) (|eihe| − |gihg|) + ~ω ↠â +
Dipole coupling energy
Energy difference between atomic levels
~Ω
2
(â|eihg| + ↠|gihe|).
How to get there
Show that ∼ Ĥ = Ĥatom + Ĥfield − Ê.(er̂)
Write electron position operator r̂ in basis
– eigenstates of Ĥatom == atomic orbitals
Approximate to one mode of field and two atomic levels
Neglect non-resonant “wrong-way” terms
(like electron drops down orbital and emits photon)
How to get there
Show that ∼ Ĥ = Ĥatom + Ĥfield − Ê.(er̂) ←−
Write electron position operator r̂ in basis
– eigenstates of Ĥatom == atomic orbitals
Approximate to one mode of field and two atomic levels
Neglect non-resonant “wrong-way” terms
(like electron drops down orbital and emits photon)
Atom-Field Interaction Energy
Hamiltonian for Atom+Field
Field
X
~ωi âi† âi
modes
Ĥ = ĤEM + Ĥatom + Ĥint
~2 2
e2
Atom −
∇ −
2me e 4π0 |re − R0 |
(nucleus fixed @ R0 )
Interaction energy ?
Atom-Field Interaction Energy
Interaction energy: classical field
Ĥatom =
1 2
p̂ + V (r̂).
2m
With vector potential A, p̂ → (p̂ − eA)
and scalar potential φ, Ĥatom → Ĥatom + eφ
Ĥatom−field =
(minimal coupling)
1
(p̂ − eA)2 + eφ + V (r̂).
2m
Atom-Field Interaction Energy
Interaction energy: A.p form
Choose the Coulomb gauge where ∇.A = 0, φ = 0
Ĥatom−field =
p̂2
− e A.p̂ +
2m m
e2 2
2m A
+ V (r )
= Ĥatom + Ĥint
(Can use this form directly – not in this course)
Atom-Field Interaction Energy
Interaction energy: dipole approximation
Interested in interaction with light waves
A = A0 ei(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)
Atom-Field Interaction Energy
Interaction energy: E.r form
Coulomb gauge form:
Ĥatom−field =
1
(p̂ − eA)2 + eφ(= 0) + V (r ).
2m
Now change gauge :
A → A + ∇χ(r, t)
∂χ(r, t)
φ→φ−
,
∂t
so
Ĥatom−field =
1
∂χ
(p̂ − e(A + ∇χ))2 − e
+ V (r ).
2m
∂t
Atom-Field Interaction Energy
Interaction energy: E.r form
1
∂χ
(p̂ − e(A + ∇χ))2 − e
+ V (r ).
2m
∂t
Choose χ(r, t) = −A.r
Ĥatom−field =
so that ∇χ = −A
∂χ
∂A
and
=−
.r = E.r
∂t
∂t
Ĥatom−field =
∂A
∂A
E = −∇φ −
=−
∂t
∂t
– (A, φ-Coulomb gauge)
1
(p̂2 + V (r )) − er̂.E(t).
2m
Atom-Field Interaction Energy
Hamiltonian: E.r form
Quantum form: E(r) → Ê(r0 ) at the position of the atom
So for our cavity quantization + one electron atom:
Ĥ = Ĥatom +
X
~ωn ân† ân
n
s
+
X
n,s
~ωn
sin(kn zat )(ân + ân† )es .(−er̂).
0 V
Atom-Field Interaction Energy
How to get there
Show that ∼ Ĥ = Ĥatom + Ĥfield − Ê.(er̂)
Write electron position operator r̂ in basis
– eigenstates of Ĥatom == atomic orbitals ←−
Approximate to one mode of field and two atomic levels
Neglect non-resonant “wrong-way” terms
(like electron drops down orbital and emits photon)
Atom-Field Interaction Energy
Second quantization: general
Generally have Z indistinguishable electrons
⇒ Atomic eigenstates labelled by occupation of orbitals
(1s2 2s1 etc)
These states – |ii – form a complete set (for Z electrons)
This allows us to formally write atomic operators
– in terms of “transition operators” |iihj|
Atom-Field Interaction Energy
Second quantization: Hamiltonian
1̂ =
X
|iihi|.
i
Formal representation of Ĥ :
Ĥ = 1̂Ĥ 1̂
X
X
=
|iihi|Ĥ
|jihj|
i
=
X
j
|iihi|Ej |jihj|
i,j
=
X
i
Ei |iihi|.
(Ĥ|ji = Ej |ji, hi|ji = δij )
Atom-Field Interaction Energy
Second quantization: One-body operators
Eigenstates of Ĥ – |ii – form a complete set for Z electrons, so
X
|iihi|.
1̂ =
i
Formal representation of D̂ =
X
i
D̂ = 1̂D̂1̂
X
X
=
|iihi|D̂
|jihj|
i
j
X
=
hi|D̂|ji|iihj|
i,j
er̂i :
Atom-Field Interaction Energy
Second quantization: One-body operators
D̂ = 1̂D̂1̂
X
X
=
|iihi|D̂
|jihj|
i
j
X
=
hi|D̂|ji|iihj|
i,j
If we know the orbitals in real-space, can calculate
Z
Dij = hi|D̂|ji = dxdydzψi∗ (r)(er)ψj (r).
Atom-Field Interaction Energy
Dipole matrix elements
Z
Dij = hi|D̂|ji =
dxdydzψi∗ (r)(er)ψj (r).
Dij only non-zero between some states ⇒ selection rules
i.j different parity.
∆l = ±1 (if l good quantum number)
Magnitude |D| ≈ e× 1 Å
Atom-Field Interaction Energy
Atom-field Hamiltonian
So for our cavity problem
Ĥ =
X
~ωn ân† ân
n
+
X
Ei |iihi|
i
+
XX
n,s
ij
En sin(kn zat )(ân + ân† )es .Dij |iihj|.
Atom-Field Interaction Energy
How to get there
Show that ∼ Ĥ = Ĥatom + Ĥfield − Ê.(er̂)
Write electron position operator r̂ in basis
– eigenstates of Ĥatom == atomic orbitals
Approximate to one mode of field and two atomic levels
Neglect non-resonant “wrong-way” terms
(like electron drops down orbital and emits photon)
Atom-Field Interaction Energy
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, gi, |n − 1, ei
∴ can focus on the physics of one mode
+ nearly resonant atomic transition
Atom-Field Interaction Energy
Rotating-Wave approximation
Interaction is
~Ω
[ â|eihg| + ↠|gihe| + â|gihe| + ↠|gihe| ].
2
Energy changes ≈ 0
≈ ±2∆
∴ drop these terms
Jaynes-Cummings Model in Rotating-Wave Approx
Ĥ = (∆/2)(|eihe| − |gihg|) + ~ω ↠â
~Ω
+
(â|eihg| + ↠|gihe|).
2
Atom-Field Interaction Energy
Summary
= One field mode, two atomic states
Energy of photon in field mode
Ĥ = (∆/2) (|eihe| − |gihg|) + ~ω ↠â +
Dipole coupling energy
Energy difference between atomic levels
~Ω
2
(↠|eihg| + â|gihe|).